T N+1

-∆t N G N,

Alfven-Lawson limit * I A -γ b β b m e c 3 /e -17γ b β b kA

and present a consistency error O(∆t n ). The worst case is the Spitzer regime for Tantalum where C V,i ≈ 10 7 erg.cm -3 .s -1 .K -1 and G ≈ 10 20 erg.cm -3 .s -1 .K -1 that gives ∆t n 100 fs, which is much less restrivtive than the CFL criterion for solving the M1 equations (7.50) or (7.52). The heat equation for the background electron temperature is computed according to the 2nd order explicit scheme

Dedicated to my grandmother Louise Halimi Laloum who died the 1st of february 2015 at the age of 95 years old still without knowing how to write nor read. Dedicated also to my friend Jonathan Sandler (30 years old), his two sons Aryeh (3 years old) and Gabriel (6 years old) and their little friend Myriam Monsonego (8 years old) murdered at the entrance of their school Ozar Hatorah in Toulouse (France) that horrible March 19, 2012, just because they were Jews.

"It can be said that anti-Semitism is one particular case of intolerance; that for centuries it had a prevailingly religious character; that in the Third Reich it was exacerbated by the nationalistic and military predisposition of the German people and by the "differentness" of the Jewish people; that it was easily disseminated in all of Germany-and in a good part of Europe-thanks to the efficiency of the fascist and Nazi propaganda, which needed a scapegoat on which to load all guilts and resentments; that the phenomenon was heightened to paroxysm by Hitler, a maniacal dictator.

But these commonly accepted explanations do not satisfy me. They are reductive-not commensurate with, nor proportionate to, the facts that need explaining. In rereading the chronicles of Nazism, from its murky beginnings to its convulsed end, I cannot avoid the impression of a general atmosphere of uncontrolled madness. Thus I prefer the humility with which some of the most serious historians confess to not understanding the furious anti-Semitism of Hitler and of Germany back of him. [...] For this reason, it is the duty of everyone to meditate on what happened. Everybody must know, or remember, that Hitler and Mussolini, when they spoke in public, were believed, applauded, admired, adored like gods. They were "charismatic leaders"; they possessed a secret power of seduction that did not proceed from the credibility or the soundness of the things they said, but from the suggestive way in which they said them. And we must remember that their faithful followers, among them the diligent executors of inhuman orders, were not born torturers, were not (with a few exceptions) monsters: they were ordinary men. Monsters exist, but they are too few in number to be truly dangerous; more dangerous are the common men, the functionaries ready to believe and to act without asking questions.

Since it is difficult to distinguish true prophets from false, it is well to regard all prophets with suspicion. Yet it is clear that this formula is too simple to suffice in every case. A new fascism, with its trail of intolerance, abuse, and servitude, can be born outside our country and imported into it, walking on tiptoe and calling itself by other names; or it can loose itself from within with such violence that it routs all defenses. At that point, wise counsel no longer serves, and, and one must find the strength to resist. But then, too, the memory of what happened in the heart of Europe, not very long ago, can serve as support and warning."

Primo Levi

Publications related to this thesis Some results from this PhD research work have given or are going to give rise to the following publications:

1. M Touati, J-L Feugeas, Ph Nicolaï, J J Santos, L Gremillet and V T Tikhonchuk. A reduced model for relativistic electron beam transport in solids and dense plasmas. New J. Phys 16, 073014 (2014) 2. S Gus'kov, X Ribeyre, M Touati, J-L Feugeas, Ph Nicola¨i and V T Tikhonchuk. Ablation Pressure Driven by an Energetic Electron Beam in a Dense Plasma. Phys. Rev. Lett. 109, 255004 (2012) 3. J J Santos, B Vauzour, M Touati, L Gremillet, J-L Feugeas, T Ceccotti, R Bouillaud, F Deneuville, V Floquet, C Fourment, M Hadj-Bachir, S Hulin, A Morace, Ph Nicolaï, P d'Oliveira, F Reau, A Samaké, O Tcherbakov, V T Tikhonchuk, M Veltcheva and D Batani. Ultra-high pressure hydrodynamics driven by fast electrons in solid targets. To be submitted 4. X Vaisseau, J J Santos, M Touati, J-L Feugeas, Ph Nicolaï, V Tikhonchuk et al., Laser-generated fast electron beam collimation in laser-shocked vitreous carbon foils. To be submitted

The model has also been used by colleagues for studies that have given or are going to rise to the following publications:

1. Ph Nicolaï, J-L Feugeas, M Touati, X Ribeyre, S Gus'kov, V T Tikhonchuk. Deleterious effects of nonthermal electrons in shock ignition concept. Phys. Rev. E 89, 033107 (2014) 2. L Volpe, J-L Feugeas, Ph Nicolai, J J Santos, M Touati, J Breil, D Batani, and V Tikhonchuk.

CONTENTS

List of symbols used

Linear algebra notations

Let us note a a scalar, A, B and C three vectors in a 3-dimensional space (x, y, z), D 2 , E 2 and F 2 second order tensors (or 3 × 3-dimensional matrices) and G 3 a third order tensor in the same 3-dimensional space.

Scalar product

A.B = i=x,y,z

A i B i .

Linear transformations

B = D 2 .A implies ∀i ∈ {x, y, z}, B i = j=x,y,z D 2,ij A j , C = A.D 2 implies ∀j ∈ {x, y, z}, C j = i=x,y,z A i D 2,ij , D 2 = E 2 .F 2 implies ∀(i, j) ∈ {x, y, z} 2 , D 2,ij = k=x,y,z E 2,ik F 2,kj , D 2 = G 3 .A implies ∀(i, j) ∈ {x, y, z} 2 , D 2,ij = k=x,y,z G 3,ijk A k and E 2 = A.G 3 implies ∀(j, k) ∈ {x, y, z} 2 , E 2,jk = i=x,y,z A i G 3,ijk .

Page xiii CONTENTS

Transposition A = B T is the transposition of B and

D 2 = E T 2
is the transposition of E 2 .

Vectorial product

A × B =     A y B z -A z B y A z B x -A x B z A x B y -A y B x     .
Tensorial products

T 2 = A ⊗ B implies ∀(i, j) ∈ {x, y, z} 2 , T 2,ij = A i B j and
T 3 = A ⊗ B ⊗ C implies ∀(i, j, k) ∈ {x, y, z} 3 , T 3,ijk = A i B j C j .
Twice contracted product

a = D 2 : E 2 = i=x,y,z j=x,y,z D 2,ij E 2,ji .

Differential operators notations

Let us note x = (x, y, z) T a vector expressed in 3-dimensional Cartesian coordinates in the basis (0, e x , e y , e z ), r = (z, r, θ) T a vector expressed in 3-dimensional cylindrical coordinates in the basis (0, e z , e r , e θ ) and p = (p, θ, ϕ) T a vector expressed in 3-dimensional spherical coordinates in the basis (0, Ω = p/p, e θ , e ϕ ). Also, we note f = f (x), g = g(r) and h = h(p) scalar functions of

x, r and p, respectively, A = A x (x)e x + A y (x)e y + A z (x)e z , B = B z (r)e z + B r (r)e r + B θ (r)e θ and C = C p (p)Ω + C θ (p)e θ + C ϕ (p)e ϕ vectorial functions of x, r and p, respectively and D = [D ij (x)], (i, j) ∈ {x, y, z} 2 , E = [E ij (r)], (i, j) ∈ {z, r, θ} 2 and F = [F ij (p)], (i, j) ∈ {p, θ, ϕ} 2 2nd order tensorial functions of x, r and p, respectively. Also, we note v = (v x , v y , v z ) T , (v z , v r , v θ ) T or (v p , v θ , v ϕ ) T a velocity in the choosen system of coordinates.

Gradient

p        . Curl ∂ ∂x × A =        ∂A z ∂y - ∂A y ∂z ∂A x ∂z - ∂A z ∂x ∂A y ∂x - ∂A x ∂y        , ∂ ∂r × B =        1 r ∂ ∂r (rB ϕ ) - 1 r ∂B r ∂ϕ 1 r ∂B z ∂ϕ - ∂B ϕ ∂z ∂B r ∂z - ∂B z ∂r        and ∂ ∂p × C =        1 p sin θ ∂ ∂θ (sin θC ϕ ) - 1 p sin θ ∂C θ ∂ϕ 1 p sin θ ∂C r ∂ϕ - 1 p ∂ ∂p (pC ϕ ) 1 p ∂ ∂p (pC θ ) - 1 p ∂C p ∂θ       
.

CONTENTS

Laplacian of a scalar function

∂ 2 f ∂x 2 = ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 + ∂ 2 f ∂z 2 , ∂ 2 g ∂r 2 = ∂ 2 g ∂z 2 + 1 r ∂ ∂r r ∂g ∂r + 1 r 2 ∂ 2 g ∂θ 2 and ∂ 2 h ∂p 2 = 1 p 2 ∂ ∂p p 2 ∂h ∂p + 1 p 2 sin θ ∂ ∂θ sin θ ∂h ∂θ + 1 p 2 sin 2 θ ∂ 2 h ∂ϕ 2 .

Laplacian of a vectorial function

∂ 2 A ∂x 2 =        ∂ 2 A x ∂x 2 ∂ 2 A y ∂x 2 ∂ 2 A z ∂x 2        , ∂ 2 B ∂r 2 =        ∂ 2 B z ∂r 2 ∂ 2 B r ∂r 2 - 2 r 2 ∂B θ ∂θ - B r r 2 ∂ 2 B θ ∂r 2 + 2 r 2 ∂B r ∂θ - B θ r 2       
and

∂ 2 C ∂p 2 =         ∂ 2 C p ∂p 2 - 2 p 2 ∂C θ ∂θ - 2 p 2 sin θ ∂C ϕ ∂ϕ -2 C p + cot θC θ p 2 ∂ 2 C θ ∂p 2 + 2 p 2 ∂C p ∂θ - 2 cos θ p 2 sin 2 θ ∂C ϕ ∂ϕ - C θ p 2 sin 2 θ ∂ 2 C ϕ ∂ 2 p 2 + 2 p 2 sin θ ∂C p ∂ϕ + 2 cos θ p 2 sin 2 θ ∂C θ ∂ϕ - C ϕ p 2 sin 2 θ         .

Components of advection terms

In Plasma Hydrodynamic Theory, the relation

∂ ∂x . (nv ⊗ v) = n ∂ ∂x .v v + v. ∂ ∂x (v)
describes the convection term. Therefore, we will note : (Here r is the vector position expressed in whatever the coordinates system)

v. ∂ ∂x (A) =        v x ∂A x ∂x + v y ∂A x ∂y + v z ∂A x ∂z v x ∂A y ∂x + v y ∂A y ∂y + v z ∂A y ∂z v x ∂A z ∂x + v y ∂A z ∂y + v z ∂A z ∂z        , v. ∂ ∂r (B) =       v z ∂B z ∂z + v r ∂B z ∂r + v θ r ∂B z ∂θ v z ∂B r ∂z + v r ∂B r ∂r + v θ r ∂B r ∂θ - v θ B θ r v z ∂B θ ∂z + v r ∂B θ ∂r + v θ r ∂B θ ∂θ + v θ B r r       and v. ∂ ∂p (C) =        v p ∂C p ∂p + v θ p ∂C p ∂θ + v ϕ p sin θ ∂C p ∂ϕ - v θ C θ + v ϕ C ϕ p v p ∂C θ ∂p + v θ p ∂C θ ∂θ + v ϕ p sin θ ∂C θ ∂ϕ + v θ C p -v ϕ cot

Symbols of fundamental plasma parameters

The electron temperature T e is expressed in eV, the electron and ion densities n e and n i in cm -3 .

Physical quantity

Relativistic electron beam transport

The electron temperature T e of the plasma, where the beam propagates through, is expressed in eV, the plasma ion density n i is in cm -3 , the beam density n b is in [10 21 cm -3 ], the beam radius r b is in [10 µm], the electron velocity v and the beam electrons mean velocity v b of a collimated and monoenergetic electron beam ( * ) are in cm.s -1 and the thermal Maxwell-Juttner (M-J) beam temperature T b is in keV.

CONTENTS

Physical quantity Symbol Formula order of magnitude (cgs) All experiments and observations carried out until today show that there is a universal quantity which is conserved during all evolutions and all transformations of matter and fields : the energy. According to Noether's theorem, conservation of energy is a direct mathematical consequence of the translational symmetry of the quantity conjugate to the energy, namely the time. In other words, it is nothing else than the intuition we have that physics laws do not change with time. In 1845, this quantity has been introduced by James Prescott Joule in order to understand the link between mechanical work and generation of heat. In order to pay homage to him, one calls "Joule" and one notes "J" the SI energy unit, based on the amount transferred to an object by the mechanical work of moving it 1 m against a force of 1 N. Since the development of the thermodynamics theory in the XIXth century, the energy conservation principle has become an indispensable element of understanding of any physical process. All developments of modern physics are related to studying different ways of converting energy from one form into another. To give an order of magnitude, the nuclear reactions of fission of 0.01 g of Uranium provide approximatively 1 kWh (1 kWh = 3.6 MJ) of heat energy in a nuclear power plant. The same amount of heat energy can be obtained by burning approximatively 100 g of oil, coal or gas (1 million of tonnes of oil or oil equivalent produces about 4.4 TWh of electricity on a modern power station), by condensing 1.6 kg of water vapor or by capturing solar radiations energy on a surface of 1 m 2 during one hour (The sun light intensity on Earth is about 0.1 W.cm -2 during a sunny day). 1 kWh represents also the gravitational potential energy of 3 tonnes of water falling from 100 m of altitude in a hydroelectric power plant or the kinetic energy of 20000 m 3 of air moving with a velocity of 60 km.h -1 in a wind power plant or the energy needed by a human being of 65 kg to climb a mountain peak of 3000 m. In addition to this important concept of energy conservation, the thermodynamic theory has also led to the Industrial Revolution : little by little, a manual labor has been replaced by machines, a horse-drawn carriage has been replaced by steam powered or thermal engines transportation vehicles and a lot of new manufacturing have been developed. In the XXth, thanks to this concept, a technological and scientific knowledge has exponentially grown up and has led to incredibly improved quality of human life, allowing the world population to considerably increase (see the World Population estimates of the United Nations). It is striking to notice how the Gross National Product of a today country is strongly correlated with its energy "consumption". The fossil fuels such as oil, gas and coal are used today mainly for energy "production" and represent ≈ 90% in the world energy "consumption" as shown in Figure 1. The nuclear and hydroelectrical energy represent each one only ≈ 6% of the world energy "consumptions". Even if other renewable energies from wind, biomass, waste, solar and geothermal power plants are more and more used thanks to governments subsidies, their contribution is still negligible. One must say that the expressions such as energy "consumption" or "production" are misused.

Indeed, by definition, the energy is conserved during all transformations/evolutions. The sense that we attribute to the words "production" and "consumption" of energy is that one converts one amount of energy which already exists in a certain form into the same quantity of energy in another form.

In this way, there is no "clean energy", the energy "production"/"consumption" is nothing else than the pure modification of our environment. Thus, like all continous functions, one can show that each energy resource extraction from Earth will attain its maximum at a certain time and then will decrease until the resource will totally disappear from Earth. The BP Statistical Review of world energy 2014

estimates that the oil and natural gaz reserves-to-"production" ratios are available for about 50 years, while the coal reserves-to-"production" ratio will be available for about 100 years. These evaluations are probably underestimated because of financial reasons but, on the other hand, these estimates are based on the 2013 data of the World population, while it is expected to grow up to more that 11 billions of people in ≈ 2100 (compared to ≈ 7.1 billions in 2013) according to extrapolations carried out by the United Nations in 2012. In addition, this strong population growth is expected to happen in developing countries where the resource demands will be the higher than average. One understands easily, according to Figure 1, which highlights our huge consumption of coal, gaz and oil, that the cost of energy consumption will be continuously increasing until each of them disappears if one does not find an alternative. Even if it is the more efficient way of producing energy and the less harmful for the environment, the governments policy is expected to limit the use of nuclear power plants due to the long life time of the radioactive waste and the growing public opposition -which plays an important role during elections-due to disasters such as Fukushima (2011), Chernobyl (1986) and Three Mile Island (1979). However, the reality is quite opposite. One can easily demonstrate that nuclear power 0.1. THERMONUCLEAR PLASMA FUSION AS THE SOLUTION FOR THE XXITH CENTURY ENERGY PRODUCTION plants are responsible for less human deaths than fossil fuel burning plants if one takes into account the induced skin and lung cancers and the death of mine workers.

In addition, each way of "producing" or "consuming" energy produces actually entropy due to the irreversible modifications it operates in the Nature. The entropy is also a physical quantity which has been defined during the development of thermodynamics in the XIXth century. It is a measure of the number of specific ways in which a thermodynamic system may be arranged. It can therefore be understood as the measure of the system disorder. According to the second law of thermodynamics, the entropy of an isolated system never decreases and such systems spontaneously evolve towards thermodynamic equilibrium, that is to say, the configuration with a maximum entropy. The Earth is not an isolated system due to its continuous irradiation by Cosmic Rays and its self-emission of thermal infrared radiations. Besides, one can show according to the second principle of Thermodynamics taking into account these processes, that the World "Entropy consumption/production" is responsible for the rise of the Earth temperature (≈ +1 o since the beginning of the XXth century). Indeed, since 1965, the carbon dioxide emission, known to be one of the major factor responsible for the Earth greenhouse effect, has increased from ≈ 3500 kg/year/capita to ≈ 5000 kg/year/capita according to the BP Statistical Review of world energy 2014 workbook and the United nations world population estimates.

Consequently, even if the hydro power plants are the most efficient way of "producing" energy among all ecologically friendly ones, it will also be affected by the climate changes due to the droughts induced by this increase of temperature. For example, Europe is expected to lose 20 -30% of precipitations until 2100 leading to a smaller hydroelectricity production efficiency (The majority of European hydro power plants does not work well yet at summer). Finally, other "renewable energies" such as wind power plants or photovoltaic panels are unsufficient to supply the World demand especially in the industrial zones. As a conclusion, there is an important and challenging Energy issue for the Humanity in the XXIth century and will surely lead to important conflicts between Nations if one does not find an alternative (or reduce our consumption which seems to be impossible).

Thermonuclear Fusion

Four fundamental interactions have been discoverded by physicists : the Gravitational, the Electromagnetic, the Strong Nuclear and the Weak Nuclear interaction. Effective only at a distance of a few fm, the Strong Nuclear force is 137 times stronger than the Electromagnetic one, 10 25 times stronger than the Weak Force interaction and 10 38 times stronger than the Gravitational interaction. This ensures the stability of ordinary matter, in confining the elementary particles quarks into hadrons such as proton and neutron both called nucleons, the largest components of the mass of ordinary matter.

Most of the mass-energy of a common proton m p c 2 or neutron m n c 2 is in the form of the Strong Force Field Energy; the individual quarks provide only about 1% of the mass-energy of a proton. Also, experiments show that the mass of an atomic nucleus M is always smaller than the sum of its nucleons ). Lightweight atoms are stable when the number of their neutrons N approximatively equals the number of protons Z which defines the nucleus electrical charge. The atoms heavier than iron (A > 54) are stable if N ≈ 1.5Z. Again according to the Aston curve, two kinds of nuclear reactions are exoenergetic : the nuclear fission of one heavy nucleus, which is a process already used in fission nuclear power plants, but also the nuclear fusion of two lightweight nuclei. In the 1950's, the idea of controlling thermonuclear combustion of lightweight atoms for Energy "production" was born shortly after the development of a theoretical model of the fusion reactions of lightweight elements to explain the conversion of nuclear binding energy into heat in stars [Bethe, 1939]. Since then, "bringing the star power on Earth" has been the dream of many physicists and seems to be a promising solution to solve the World Energy issue of the XXIth century. The problem is that while the Gravitational and Electromagnetic forces act over potentially infinite distance, the another two Nuclear forces act over minuscule subatomic distances and are more difficult to access. According to the Electromagnetic theory, an energy of about 1 MeV is needed to counteract the Coulomb barrier between two nuclei and make them getting close enough to fuse. Actually, an energy of about 10 keV is sufficient thanks to the quantum tunneling effect [Gamow, 1928]. The fusion of two nuclei has been achieved many times thanks to particle accelerators and a lot of stable nuclei have been discovered thanks to them.

Recently, the yet-unnamed element 117 have been created by physicists at the GSI Helmholtz Center for Heavy Ion Research, an accelerator laboratory located in Darmstadt, Germany. But, concerning the energy "production", the quantity of accelerated isotopes is too small to generate more energy than the energy needed to accelerate them. Also, even if the cold fusion or muon-catalyzed fusion is a well known process since the 1980's [START_REF] Jones | Experimental Investigation of Muon-Catalyzed d -t Fusion[END_REF], it does not allow to get high gains. Indeed, 0.1. THERMONUCLEAR PLASMA FUSION AS THE SOLUTION FOR THE XXITH CENTURY ENERGY PRODUCTION a net energy production is impossible because of a high energy required to create muons, their short 2.2 µs life time and a high probability that a muon will bind to a new alpha (He) particle.

The only way to "produce" energy thanks to fusion reactions consists in creating a thermonuclear plasma in order to achieve high reaction rates and consequently high gains like in the Stellar Temperature-Density conditions (see the right panel of Figure 2). In addition to this constraint, there are other requirements : the nuclear reaction must be exothermic, it must employ the lightest possible nuclei to limit the Coulomb barrier, which is proportional to their atomic number Z, it must have a large cross section (probability), implying consequently only 2 nuclei, it must conserve the proton and neutron numbers in order to limit the weak nuclear interaction and finally it must produce at least one neutron in addition to the heavier nucleus in order to heat the blanket coolants and to produce electricity. 80 fusion reactions satisfy these criteria but the most probable one is the fusion reaction of the two Hydrogen isotopes :

2 1 D + 3 1 T → 2 2 He (3.5 MeV) + 1 0 n (14.1 MeV) (1)
due to the existence of an intermediate resonant nuclear state in this reaction. In Figure 3, the reaction rate of such a T(d,n) 4 He fusion nuclear reaction σv is plotted. Here, σ is the fusion reaction cross section i.e. the effective area of a targeted Deuterium (or Tritium) nuclei seen by a projectile Tritium (Deuterium, respectively), v is the relative velocity of the projectile in a collision and the angle brackets mean that the rate is averaged over a Maxwell-Boltzmann distribution function (thermodynamic equilibrium). One can see that the reaction probability is maximum for a thermonuclear DT plasma at a temperature T ≈ 100 keV, which is not yet achievable with the today technology.

However, a plasma temperature of 10 keV can be achieved and would be sufficient. Three other fusion nuclear reactions can also occur at this temperature but with smaller probabilities : 3 He(d,p) 4 He, D(d,p)T and D(d,n) 3 He. Their reaction rates are also shown in Figure 3.

From a practical point of view, concerning the energy "production", there is an almost infinite quantity of Deuterium 2 1 D on the Earth with 33 g in each tons of sea water; That is why one calls the DT fusion the blue energy. The Tritium 3 1 T can be produced in a fusion reaction between the escaping neutrons 1 0 n (14.1 MeV) and lithium nuclei 6 3 Li or 7 3 Li which are abundant on Earth. So, contrary to the fission nuclear power plants, which are using limited resources of Uranium, Plutonium and Thorium, there is no resources problems. Also, the DT fusion presents no risk of a runaway chain reactions and no long life time radioactive waste. The fusion reaction products are stable and only activation of construction materials by fast neutrons is expected. As a conclusion, an eventual nuclear fusion power plant would have all the pros of the nuclear fission ones without its cons that is to say, without polluting the environment, without eventual nuclear catastrophes and not facing the problem of limited terrestrial resources. for the Ignition (right panel) both estimated from [START_REF] Bosch | Improved formulas for fusion cross-sections and thermal reactivities[END_REF] Even more than 99% of visible matter in the universe can be found in the plasma state and it has been studied for many decades. However, it is still difficult to create and maintain a thermonuclear plasma today. In order to evaluate the conditions needed to create and control such an equimolar DT plasma, one has to take into account on the one hand the plasma power losses P loss and on the other hand the total power gain P gain . By definition, a plasma is made of a large number of charged particles and it is characterized by a collective behavior of particles due to the long distance electromagnetic forces. Therefore, the total power losses consist in the thermal, mechanical and radiation losses (every accelerated charged particles lose energy by emitting light). The total power gain consists in the external energy brought by the "driver", that is to say, the external power needed to create and maintain the plasma P ext and the fraction P α = F α (3.5/17.6)P fus released by the fusion reactions P fus in a form of alpha particles ( 2 2 He), as only alpha-particles are depositing their energy. The neutrons are electrically neutral and are consequently leaving from the plasma without collisions. Their energy is recovered in the blanket and used to produce the Tritium and heat. By considering a stationary energy balance P loss = P gain = P α + P ext and assuming that all α particles deposit their energy inside the plasma (F α = 1), one may define the energy gain G = P fus /P ext . Then, by evaluating the plasma life time or confinement time by the ratio of its internal energy divided by the total power losses τ c = 3n e k B T /P loss where n e is the plasma electron density and T the plasma temperature, one finds a criterion to achieve the ignition of such a little star on Earth defined by G → ∞ of an equimolar DT plasma [Lawson, 1957] n e τ c > 12 3.5 MeV

Lawson Criterion

k B T σ DT v
≈ 10 15 cm -3 .s at T ≈ 10 keV.

(

) 2
Due to chaotic motion of charged particles at a temperature of T ≈ 10 keV, the plasma tends

INERTIAL CONFINEMENT FUSION (ICF)

naturally to expand and it is difficult to maintain it during the needed time τ c with a sufficiently high density n e , because these two parameters T and τ c are linked according to the Lawson criterion. In the star cores, the plasma confinement is accomplished naturally thanks to the Gravitational attraction.

For example, the mass of the Sun (about 10 30 kg) is sufficiently high to attract and compress matter to densities up to 10 32 cm -3 during its whole life of about 10 billion years, which corresponds to the time it needs to consume all its fusion fuel. The fact that plasma particles are electrically charged has naturally led to the idea of using strong magnetic fields in order to confine the thermonuclear plasma.

In the 1950's, the Soviet physicists Igor Tamm and Andrei Sakharov proposed a device called tokamak in the shape of a torus allowing to confine a thermonuclear plasma thanks to a toroidal magnetic field produced by magnetic coils that surround the torus. In addition, a poloidal magnetic field which is created by a toroidal electric current that flows inside the plasma allows to heat it. The international project ITER is currently building the world's largest experimental tokamak nuclear fusion reactor and aims to make the long-awaited transition from experimental studies of plasma physics to full-scale electricity-producing fusion power plants. In 2003, the ITER prototype Tore Supra has obtained the world record by confining a thermonuclear plasma of n e ≈ 10 15 cm -3 more than 6 minutes and 30 seconds during which time, energy on the order of 300 kWh was injected and extracted.

Inertial Confinement Fusion (ICF)

Inertial confinement fusion (ICF) is an alternative way to control fusion reactions. It is based on scaling down a thermonuclear bomb explosion to a small size, applicable for a power production. In this approach, achieving the energy gain through fusion reactions relies firstly on a fast compression to a high density (up to 1000 g.cm -3 in the fuel) of a mm-scale capsule filled with a mixture of Deuterium and Tritium by the use of an ablative rocket effect. Then, a conversion of the implosion kinetic energy into the internal energy results in heating of the central zone called "hotspot" up to temperatures T > 5 keV, allowing to initiate the fusion reactions of the DT fuel in agreement with the Lawson criterion. Thus, instead of magnetic fields, here, the plasma is confined by its own inertia. Besides, this process lasts only a few ns so that this approach presents significant technological difficulties due to a high repetition rate of 10 Hz needed to continuously produce electricity. However, by reaching very high densities during a short confinement time, the ICF approach would be much more efficient in terms of gain, than magnetic confinement fusion, which aims to fuse the DT fuel at low densities but long confinement times.

Conventional ICF Schemes

Since the invention of lasers [Maiman, 1960], it came naturally the idea to use many laser pulses to strongly compress DT fuel capsules. In the 1970's, scientists began experimenting with powerful laser beams in France [START_REF] Colin | Laser Produced Plasmas from Solid Deuterium Targets[END_REF], in the United States of America (USA) [START_REF] Nuckolls | Laser induced thermonuclear fusion[END_REF] and in the Union of Soviet Socialist Republics (USSR) [START_REF] Basov | Experiments on the observation of neutron emission at a focus of high-power laser radiation on a lithium deuteride surface[END_REF]]. In the "direct drive" approach to ICF, powerful beams of laser light are focused on a small spherical pellet containing micrograms of Deuterium and Tritium (see Figure 4). A rapid heating caused by the laser driver makes the outer layer of the target explode. According to the momentum conservation law, the remaining portion of the target is driven inwards in a rocket-like implosion, causing compression of the fuel inside the capsule and the formation of a shock wave. The latter heats the fuel in the very center and results in ignition of fusion reactions which are propagating the fusion burn wave and release of more nuclear energy than was initially deposited. In the "indirect drive" method, the lasers heat the inner walls of a gold cavity called a hohlraum containing the pellet, creating a hot plasma which radiates a uniform "bath" of soft X-rays (see Figure 4). The X-rays rapidly heat the outer surface of the fuel pellet, causing a high-speed ablation, or "blowoff," of the surface material and the fuel capsule implosion as if it had been hit with the lasers directly. In both approaches (direct drive and indirect drive), symmetrically compressing the capsule with radiations creates a central "hot spot" where fusion processes set in, the plasma is self-heated and the fusion burn propagates outward through the cooler, outer regions of the capsule much more rapidly than the capsule can expand. The Direct and Indirect drive approaches [START_REF] Campbell | The National Ignition Facility -applications for inertial fusion energy and high-energy-density science[END_REF] Direct and indirect drive schemes have their advantages and drawbacks. The former has a higher laser-target coupling efficiency but is less uniform in laser irradiation due to the use of discrete laser beams. Beam smoothing techniques have a key role in the direct drive. The indirect drive by soft X rays, which are generated at the inner surface of a hohlraum, may produce a better uniformity.

INERTIAL CONFINEMENT FUSION (ICF)

This allows to reduce the growth of perturbations due to Rayleigh-Taylor instabilities. The soft X ray drive has much higher ablation rates and is less sensitive to hot electron preheat. A potential disadvantage of indirect drive is the larger scale of the plasma crossed by the laser beam from the inlet hole to the hohlraum wall. Parametric instabilities (i.e. the unstable decomposition of the incoming laser radiation into two daughter waves) in hohlraums are responsible for a significant energy loss and production of energetic electrons. One of the most important advantages of the indirect drive approach is a radiation drive concept which allows to use another drivers such as Z-pinch of heavy ions.

The National Ignition Facility (NIF) at the Lawrence Livermore National Laboratory (LLNL) is operational since March 2009 and a variety of experiments have already been completed. The NIF is a Nd : Glass laser facility, which is now routinely operating at 1.6 MJ of ultraviolet (3ω) light on target with a very high reliability. It recently reached its design goal of 1.8 MJ laser energy and 500 TW power of 3ω light on target, and has performed experiments with 1.9 MJ at peak powers of 410 TW [Moses, 2009]. In addition to this impressive success, the National Ignition Campaign (NIC) on the NIF has allowed to achieve the world record indirect-drive neutron yield of 6.1 10 15 neutrons representing a gain G of few % during one indirect-implosion [START_REF] Hurricane | Fuel gain exceeding unity in an inertially confined fusion implosion[END_REF]. Even if the primary break-even goal (G > 1) has not been achieved yet, the NIC allowed to obtain new experimental results and code developments, generating a large body of knowledge and corrections of the previous models. In France, the Laser MegaJoule (LMJ) is under construction near Bordeaux at the Cesta center of the "Commissariat à l'Energie Atomique et aux Energies Alternatives" (CEA).

The project implies a construction of 176 laser beam lines (instead of 240 at the beginning) delivering more than 1 MJ to a DT target using, as the NIF, the indirect drive method. The first laser shots with 8 beam lines were successfully delivered in the end of 2014. The laser lines of LMJ will be assembled in quads of four beams. Each quad will deliver more than 30 kJ of energy within a few ns. Direct drive ICF has been studied with smaller pellets at sub-ignition scale for many years, in particular, at the University of Rochester (USA), on the Omega laser (60 beams delivering a total of 30 kJ on target) and at the University of Osaka (Japan), on the Gekko laser (12 beams delivering 15 kJ on target).

Problems facing the ICF Conventional Schemes

The conventional schemes rely on the ignition of an isobaric hotspot where the DT fuel must reach a temperature of T ≈ 7 keV and an areal density of ρR ≈ 0.25 g.cm -2 during a confinement time of τ c ≈ 40 ps. In order to achieve this extreme conditions, many ns laser pulses representing a total energy of E L ≈ 1 MJ are needed to uniformly irradiate the solid shell. From irradiation by laser pulses (direct approach) or x-rays (indirect approach), the outer shell layers are ablated and the resulting laser-generated plasma expands. The time evolution of the laser pulses is chosen according to the Nuckolls-Kidder law such that the ablation pressure launches a spherically converging shock wave and the shock ignition scheme c) [Atzeni, 2009] followed by a continuous succession of spherically converging compressional waves, which arrive at the same time on the internal surface of the shell. At that moment, a shock wave is transmitted in the DT gas, while a rarefaction wave is reflected in the shell. When the latter arrives at the ablation surface, the shell undergoes a strong acceleration and the capsule implodes. Due to its spherical symmetry, the DT fuel in the shell is compressed to the desired density. A conversion of the imploded target's kinetic energy into internal energy results in the creation of an isobaric hotspot, where a self-sustained reaction of Deuterium and Tritium fusion is initiated. The fusion reactions generate a spherically diverging thermonuclear combustion wave followed by a detonation, which burns the denser part of the shell.

The confinement time τ c corresponds to the hotspot lifetime before its hydrodynamic expansion.

Thus, the simultaneous compression and heating processes of the fuel impose several constraints on the target and driver designs which make it difficult to obtain significant energy gains. These constraints are multifactorial. Firstly, parametric instabilities may reduce the conversion efficiency of laser energy deposited in the target and create the pressure inhomogeneities. Secondly, the generation of fast electrons due to laser-plasma interaction processes results in the target preheat, which leads to the increase of the target entropy and limits the shell compression. Finally, the hydrodynamic instabilities may mix the hot and cold fuel and can break the shell during its implosion.

Fast Ignition and Shock Ignition Alternative Schemes

Since the discovery of the Chirped Pulse Amplification (CPA) by [START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF], short pulse laser technology has grown steadily. In the 1990's, the threshold intensity value of 10 18 W.cm -2 has been attained, allowing to reach the relativistic laser-matter interaction regime where high currents of relativistic electrons can be generated. In order to relax the constraints on the driver and the target . [START_REF] Norreys | Fast electron energy transport in solid density and compressed plasma[END_REF] imposed by the simultaneous compressing and heating processes, it was thus proposed to separate the target compression and fuel heating phases. In this scheme, the shell is imploded at a lower velocity in a more stable regime. The ignition is achieved by using such a relativistic laser pulse generating a fast electron beam, which creates a hot spot in the dense part of the fuel just after the end of the compression phase at the capsule stagnation [START_REF] Tabak | Ignition and high gain with ultrapowerful lasers@f[END_REF]. In this fast ignition scenario, the required density is much smaller (300 g.cm -3 instead of 1000 g.cm -3 ) due to the fact that the hot spot is not created anymore in the center of the target but in the lateral denser region due to the heating by the fast electrons, as shown in Figure 5. Thus, the constraints on the shock wave convergence and on the implosion symmetry are reduced. Furthermore, the compression phase needs much less energy (200 -300 kJ) than in the conventional scheme and the compression driver cost is consequently lower. This is definitely an advantage for a future fusion power plant. The shell implosion velocity is also much smaller than in the conventional scenario. This allows a greater tolerance concerning the hydrodynamic instabilities and a lower risk of breaking the shell. Moreover, a laser-produced relativistic electron beam may provide a more efficient heating of the dense material and one may expect much higher gains than in the conventional scenario. Finally, the fast ignition has the advantage of creating a much denser isochoric hot spot due to the fact that the heating time by the fast electrons is much less than the hot spot hydrodynamic expansion time.

However, the fast ignition opens new problems to resolve. Due to the fact that the Ultra-High Intense (UHI) laser pulse cannot propagate beyond the critical density n c of the plasma, there is a problem to transport the electron beam to the dense fuel. Moreover, it has been demonstrated that the generated relativistic electron beam is divergent and cannot deliver the energy into a sufficiently small hot spot. Two approaches have been proposed to compensate the beam divergence. First, one may use a first ultra intense laser pulse to create a channel through the process of hole boring letting a second UHI laser pulse to generate the ignitor electron beam as close as possible to the dense area.

However, an efficient hole boring was not demonstrated so far. Another approach relies on a cone inserted in the target as shown in Figure 6. It allows to reduce the distance to ≈ 100 µm between the electron beam generation zone and the dense part of the fuel. The parameters required to reach ignition have been estimated thanks to hydrodynamic simulations assuming the fast electron beam energy being deposited in a spherical zone of a compressed fuel [Atzeni, 1999] E ign = 18 ρ 300 g.cm -2 -1.85 kJ, τ ign = 21 ρ 300 g.cm -2 -0.85 ps and r ign = 20 ρ 300 g.cm -2 -0.97 µm.

(3) It was found that, assuming a fuel density ρ = 300 g.cm -3 , one needs to deposit the energy of E ign ≈ 18 kJ during the time τ ign ≈ 21 ps in a sphere of the radius r ign ≈ 20 µm. This corresponds to a hot spot areal density of ρR = 0.6 g.cm -3 . According to numerical simulations, the optimal ignitor energy and pulse duration scale with the density as E ign ∝ ρ -1.85 and τ ign ∝ ρ -0.85 , while optimal beam radius of the resulting accelerated electrons scales with the density as r ign ∝ ρ -0.97 .

The electrons having a stopping length of 40 µm can be generated by a laser pulse with an intensity of [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF] I ign = 6.8 10 19 η L→e ρ 300 g.cm -2 0.95

W.cm -2 (4)
where η L→e is the laser-to-electron beam conversion efficiency.

Experiments conducted on the Gekko-XII ns laser system coupled with a PW laser beam at the Institute of Laser Engineering of the Osaka University in Japan has already demonstrated a significant increase in the number of neutrons released by the fusion reactions compared to the direct scheme scenario [START_REF] Kodama | Fast heating of ultrahigh-density plasma as a step towards laser fusion ignition[END_REF]] [Kodama R. et al., 2002]. In another experiment conducted in 2010 on the Gekko XII laser coupled with the new LFEX PW laser (that can deliver an energy up to 10 kJ in a 0.5 -20 ps pulse), the neutron enhancement was confirmed. However, a relatively modest neutron yield of 3.5 10 7 has been obtained with a short pulse laser energy of 300 J on the target, which is smaller than the yield obtained in 2002 [START_REF] Shiraga | Fast ignition integrated experiments with Gekko and LFEX lasers[END_REF]. Some sub-ignition scale Fast-ignition experiments were also performed on the OMEGA/OMEGA EP laser at the University of Rochester in the USA. With optimal timing, the OMEGA EP pulse produced up to 1.4 10 7 additional neutrons, which is a factor of ≈ 4 more than without short-pulse heating [START_REF] Theobald | Initial cone-in-shell fast-ignition experiments on OMEGAa)[END_REF]. One quad of NIF beams is undergoing conversion to high-intensity picosecond-duration pulses to provide an Advanced Radiographic Capability (ARC). These beams will deliver up to 10 kJ in a 5 ps pulse that can be used as a sub-scale ignitor pulse to study fast electron core heating in integrated fast ignition experiments. In France, the PETAl project consists in coupling the LMJ facility to a PW laser with an energy of 3.5 kJ and a pulse duration of 0.5-5 ps.

Another method to separate the assembly and ignition phases of the DT fuel is Shock Ignition.

It consists in igniting a central hot spot (see Figure 5) heated by the ignitor shock generated by an 2) is the ignitor pulse (left) and the corresponding logarithmic pressure gradients in the target versus space and time (right) [START_REF] Ribeyre | Shock ignition: an alternative scheme for HiPER[END_REF] ablation pressure of 300 Mbar at the end of compression phase. It increases its strength as it converges in the imploding shell and collides with the return shock [Zhou andBetti, 2005] [Betti et al., 2007].

Massive cryogenic shells at a low implosion velocity and a low adiabat can be used in this shock ignition scheme leading to fuel assemblies with large areal densities. The igniting shock creates the hot-spot pressure greater than the surrounding dense fuel pressure. Such a nonisobaric assembly features a lower energy threshold for ignition than the conventional isobaric one. The ignitor shock can be launched by a spike in the laser power which is compatible with the performance of NIF and LMJ. The thermonuclear gain can be significantly larger than in the conventional isobaric ignition for an equal driver energy. Compared with fast ignition, shock ignition presents the advantages that it does not require any complex cone-in-a-shell targets or high power lasers. Also, the physics at work in this scheme is standard laser driven hydrodynamics, a relatively well-known and proven discipline.

Yet the latter observation must be mitigated considering the failure of the NIC. Besides, as it involves low velocity implosions, this scheme is relatively robust with regards to hydrodynamic instabilities during the shell acceleration and it mitigates the Rayleigh-Taylor instability at the stagnation time.

The required power for a 300 Mbar shock delivery corresponds to a 120-200 TW final spike, the actual value depending on the ablator material and focal spot dimensions, independently of the irradiation pattern. This power is an order of magnitude lower than the power required by fast ignition and it is achievable with the NIF-LMJ technology. Nevertheless, the coupling of this final pulse to the target presents several unsolved issues such as parametric instabilities, hydrodynamic instabilities or the role of the fast electrons generation in the ignitor shock creation. An ignitor shock generated by an ablation pressure close to the required 300 Mbar has been achieved recently in Omega experiments with ≈ 500 µm -diameter solid plastic ball targets and a laser spike of 4 10 15 W.cm -2 laser pulse [START_REF] Theobald | Demonstration of 200-Mbar Ablation Pressure for Shock Ignition[END_REF]. In the non relativistic regime of laser-plasma interaction where I L λ 2 10 18 W.cm -2 .µm 2 (with I L being the laser pulse intensity and λ its wavelength), the material is ionized by the laser electric field during the first ps of the Laser-Solid-Interaction (LSI) [Keldysh, 1965]. This quasi-instantaneous multi-photon absorption process takes place for laser intensity above I L ≈ 10 11 W.cm -2 . Due to their low inertia, the electrons rapidly gain a kinetic energy from the laser in collisions with atoms and ions. The laser energy transfer to the ions is done indirectly through the energy exchange in collisions with electrons. While the heated plasma consequently expands, the laser pulse cannot penetrate the plasma where the electron density n e is above the value of n c ≈ 10 21 cm -3 /λ[µm] 2 called the critical density. Indeed, in this denser part of the plasma, the Langmuir frequency ω pe = 4πn e e 2 /m e , at which the electrons oscillate, is greater than the laser frequency ω = 2πc/λ so that the laser field is screened. Thus, a part of the laser energy is absorded in the sub-critical region and another part is reflected. The denser part where n e ≥ n c is heated by the electron thermal conduction lauched from the absorption zone, that is to say, by the collisions of hotter electrons with cooler ones. The electron heat conduction transports the absorbed energy to the ablation front where it is transformed into the energy of vapors. The reaction of ejected vapors creates the ablation pressure which is responsible for the rocket effect and the launching of shock/compression waves depending on the laser intensity temporal profile.

Concerning the Shock Ignition scheme, at the moment of the laser peak, the absorption zone called corona has a larger scale and a higher temperature than in the conventional ICF designs. This changes considerably the conditions of exciting the parametric instabilities. As the laser intensity is higher than in the conventional schemes, more absorbed energy is transported by hot electrons which can reach kinetic energies up to several 10th keV as pointed out by [START_REF] Klimo | Particlein-cell simulations of laser-plasma interaction for the shock ignition scenario[END_REF] according to kinetic simulations. Thus, one of the main issues of Shock Ignition concerns the transport of hot electrons.

At the moment of the ignition spike's arrival, the shell is already compressed, its radius is reduced by a factor of 2-3 and its areal density is increased by a factor of ten or twenty approaching a level of about 10 mg.cm -2 . This value is comparable to the range of a 100 keV electron. For this reason, the hot electrons generated in the corona with lower energies than 100 keV may not present a danger for the fuel compression contrary to the conventional scenario [START_REF] Ribeyre | Shock ignition: an alternative scheme for HiPER[END_REF]. Depending on their characteristics (kinetic energy spectrum and number) as well on the target hydrodynamic properties (density gradient and density value) at the moment of the spike arrival, the hot electrons may play an important role in the creation of the ablation pressure ≈ 300 Mbar required for launching the ignitor shock. Also, the electron transport may affect the implosion symmetry because of a large distance between the zone of electron generation in the underdense corona and the ablation surface. The hot electrons may smooth out the small scale inhomogeneities improving consequently the shell stability and suppressing the fuel mix with the hot-spot material at the internal surface of the shell. All these observations about the role of the fast electrons in the shock ignition scheme need further experimental as well as numerical studies.

Fast Electron Beam Divergence in the context of Fast Ignition

Figure 9: Schematic of the transport of laser-driven fast electrons in a dense plasma [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF] Interaction of Ultra-High Intensity (UHI) laser pulses, I L λ 2 10 18 W.cm -2 .µm 2 , with solid targets leads to a forward acceleration of electrons up to several MeV [START_REF] Beg | A study of picosecond lasersolid interactions up to 1019 W cm-2[END_REF]. Such a relativistic electron beam of a density n b and a current density j b , propagating through a plasma or a solid having an electron density n e n b , generates an electric field, which tends to eject the plasma electrons out of the beam volume in order to equilibrate the total charge. The charge neutralisation proceeds on a timescale of the order of the plasma electron-ion collision time [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] or the Coulomb explosion time [START_REF] Cox | Reverse Current Induced by Injection of a Relativistic Electron Beam into a Pinched Plasma[END_REF], depending on the plasma temperature-density conditions. Over the same timescale, the electric field accelerates the plasma electrons thus creating a return current j e ≈ -j b , and decelerates the beam electrons in order to cancel the total current density in agreement with Lenz's law [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF]. In case of irradiation of an insulator by a UHI laser pulse, the laser-generated electron beam is electrostatically neutralized [START_REF] Debayle | Target ionization by a high current relativistic monoenergetic electron beam[END_REF]. Electrons of the material are firstly ionized at the beam front by the charge-space electrostatic field generated by the fast escaping electrons. Then, they are accelerated by this electrostatic field thus creating a current density j e ≈ -j b . This "return current", ionizing the material through collisions with the bound electrons of the material, leads to the electric neutralization of the fast electron beam. In both cases, metal/plasma or insulator, the return current allows for the propagation of electron currents in excess of the Alfven-Lawson limit which defines the maximum relativistic electron beam current in vacuum [Alfvén, 1939]. However, due to the imperfect current neutralization, a magnetic field is induced that can deflect the beam electrons. The plasma electron temperature-density crossed gradients, plasma resistivity gradients and the beam current density curl are the main sources of this residual magnetic field. Later, this magnetic field begins to diffuse leading to a separation of the beam and the return current [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF]. Besides, the resulting system of two counterpropagating high currents is very unstable and may lead to the electron beam resistive filamention [START_REF]Multidimensional electron beam plasma instabilities in the relativistic regime[END_REF]. The collisions of the relativistic electrons with plasma electrons and ions also contribute to the scattering and the slowing down of the beam.

The experimental studies of fast electron generation by a UHI laser pulse have shown a significant beam divergence angle [START_REF] Green | Effect of Laser Intensity on Fast Electron Beam Divergence in Solid-Density Plasmas[END_REF]. Concerning the Fast Ignition scheme, this strong divergence strongly limits the estimates (3) of energy deposition in the hot spot [START_REF] Bellei | Fast ignition: Dependence of the ignition energy on source and target parameters for particle-in-cell-modelled energy and angular distributions of the fast electrons[END_REF]. Several methods have been proposed to collimate the electron beam. It has been noticed that the plasma resistivity gradients naturally induced in the heated material by the electron beam can collimate the beam [START_REF] Bell | Resistive Collimation of Electron Beams in Laser Produced Plasmas[END_REF]. This self-collimation, however, is not sufficient for the beam guiding because of the radial dependence of the divergence angle [START_REF] Debayle | Divergence of laser driven relativistic electron beams[END_REF]. It was also proposed to guide a relativistic electron beam in a magnetic channel created by a relativistically intense, picosecond laser prepulse followed by the main pulse [START_REF] Robinson | Artificial Collimation of Fast-Electron Beams with Two Laser Pulses[END_REF]] [Scott et al., 2012]. The prepulse serves to create such a collimating magnetic field structure due to the laser-produced electron beam propagation. This magnetic channel then helps guide the fast-electron beam generated by the second pulse. Other methods consist in using targets containing a high-resistivity-core-low-resistivity-cladding structure or a low-density-core-high-density-cladding structure. These field structures can be generated during the beam transport, hence enabling the beam to self-collimate [START_REF] Cai | Magnetic collimation of fast electrons in specially engineered targets irradiated by ultraintense laser pulses[END_REF]. In the high-resistivity-core-low-resistivity-cladding targets, the magnetic field at the interfaces is generated by the resistivity gradients and the fast electron current, while in low-density-core-high-density-cladding targets, the magnetic field is generated by a rapid change of the flow velocity of the background elec-

FAST ELECTRON GENERATION IN THE CONTEXT OF ICF

trons in the transverse direction (perpendicular to the flow velocity) caused by the density jump. A similar idea, based on a more complex array of resistivity gradients consisting of alternating layers of different Z materials have also been proposed [START_REF] Robinson | Focusing of Relativistic Electrons in Dense Plasma Using a Resistivity Gradient-Generated Magnetic Switchyard[END_REF]] [Schmitz et al., 2012a]. Adapted to the Fast Ignition cone-in-shell target, this "elliptical mirror" may improve the coupling efficiency into the hot spot by a factor of 3-4 [Robinson and Schmitz, 2013]. Also, it has been shown that a target composed of resistive filaments with a decreasing background density, adiabatically converts the beam transverse energy into longitudinal energy [START_REF] Debayle | Reduction of the fast electron angular dispersion by means of varying resistivity structured targets[END_REF]. Another possibility is to apply an external magnetic field parallel to the beam direction in the fuel [START_REF] Strozzi | Fast ignition transport studies: Realistic electron source, integrated particle in cell and hydrodynamic modeling, imposed magnetic fields[END_REF]. With a field strength exceeding 2 kT, relativistic electrons are trapped by the magnetic field lines and the lateral transport of these electrons is strongly suppressed. In the paper by [START_REF] Daido | Generation of a strong magnetic field by an intense CO 2 laser pulse[END_REF], it has been demonstated that kilotesla magnetic fields can be generated by using a capacitor-coil target.

More recently, a magnetic field of 1.5 kT was generated [START_REF] Fujioka | Kilotesla Magnetic Field due to a Capacitor Coil Target Driven by High Power Laser[END_REF]. Extended double cones have also been proposed to confine the fast electrons escaping from the cone by electrostatic and magnetic fields formed in the vacuum gap region of several micrometres width between the two walls [START_REF] Johzaki | Pre plasma effects on core heating and enhancing heating efficiency by extended double cone for FIREX[END_REF]. It has been shown through numerical simulations of electron transport that an extended double cone may enhance the core heating rate by more than a factor four compared to single cones, under otherwise similar conditions. However, despite these progresses in the understanding of relativistic electron beam transport, there is still a need of further numerical and experimental studies.

Existing Simulation Methods for Fast Electron Transport

Modeling

The equation which takes into account both the collisional and collective processes of fast electron transport in the context of shock and fast ignition is the Vlasov-Fokker-Planck (V-F-P) equation for the beam distribution function f b [START_REF] Landau | Physical kinetics[END_REF]. Due to the complexity of fast electron transport imposed by the coupling of the VFP equation with the Maxwell equations and the temperature-dependence of the transport coefficients, numerical tools are needed for preparation and interpretation of experiments. This system of equations has been extensively studied for 30 years and several numerical methods have been developed [START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]. Several families of codes can be identified. The first family consists in solving the V-F-P equation by a Particle-In-Cell (PIC) method [START_REF] Birdsall | Clouds in clouds, clouds in cells physics for many body plasma simulation[END_REF] and by interpolating the resulting macroparticle positions and velocities to compute the electromagnetic fields. Historically, this method was used to solve physical problems where collisional processes can be neglected. All electrons (both plasma and beam electrons) are sampled by macroparticles which consequently leads to accurate but time-consuming computations.

Moreover, in order to limit the non-conservative force associated with the particle-grid mapping which leads to self-heating and numerical instabilities, the space resolution has to be comparable to the Debye screening length. This poses a strong constraint in the case of dense and/or cold plasmas. Collisional processes are usually treated by Monte-Carlo methods [START_REF] Takizuka | A binary collision model for plasma simulation with a particle code[END_REF]. The second family of codes employs the same method but restricted to the beam electrons, introducing a low-energy cut off [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF]. The plasma electrons are taken into account via hydrodynamic equations of conservation or simplified ones. This approach belongs to the family of hybrid PIC codes. Other authors solve the full V-F-P equation [START_REF] Yokota | Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets[END_REF]] [Duclous et al., 2009] or use a decomposition of the distribution function in the momentum space with their corresponding hybrid versions. It has been shown that a spherical harmonic decomposition of the distribution function enables modeling of arbitrary local anisotropy for large enough expansion orders [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]. Besides their accuracy and the rapid progress in high performance computing resources, all these codes are time consuming because of the Courant-Friedrichs-Lewy condition that restricts the time step computation (a fraction of a plasma period) combined with the high resolution needed (a fraction of the Debye length), the large number of the distribution function variables (x, y, z, v x , v y , v z , t) and the large spatial (of the order of mm) and temporal (tens of ps) scales needed to study fast electrons transport in the context of the fast or shock ignition of fusion pellets.

Objectives of the thesis and Plan of the Manuscript

The goal of this PhD thesis consists in developing a new reduced 3D-3V hybrid relativistic Vlasov-Fokker-Planck model, which must be as accurate and time efficient as possible for the study of fast electron transport in solids and dense plasmas in the context of ICF. Firstly, the model will be applied to interprete experiments of laser-generated fast electron transport in solids or dense plasmas.

Secondly, the model will allow us to study the collimation methods for laser-generated electron beams in a fusion pellet. Thirdly, the model will be coupled with a hydrodynamic code for studying of the role of fast electrons in the shock ignition scheme.

In the first part of this manuscript, the state of the art of this problematic is presented:

Chapter 1 reviews the physics of laser-plasma interaction and the main electron acceleration mechanisms relevant to the Shock and Fast Ignition. We also estimate the laser-to-electron coupling efficiency as well as the spatial, energetic and angular properties of laser-generated fast electron beam.

Chapter 2 is dedicated to the electromagnetic neutralization of laser-generated relativistic electron beam. Self-consistent electromagnetic fields of a relativistic electron beam propagating through vacuum are derived in order to highlight the need of its electromagnetic neutralization allowing it to overpass the Alfven-Lawson limit. The electric and magnetic neutralization of fast electron beams are presented separately, showing the main differences between metal/plasma or insulator targets.

Chapter 3 focuses on the collective effects of electromagnetically neutralized fast electron beam transport. The quasi-static approximation in the Maxwell equations is introduced as well as the instability theory of two counterpropagating currents and the temperature dependance of fast electron beam transport. These collective effects play an important role in the Fast Ignition scheme since the proposed collimation techniques are based on the self-generated electromagnetic field.

Chapter 4 deals with the collisional effects of fast electron beam transport. The slowing down and scattering of fast electrons from collisions with free electrons, bound electrons and screened free electrons is detailed as well as their angular scattering from collisions with background ions and electrons. These effects play an important role in the Fast Ignition scheme since they are responsible for the fuel heating and the pellet ignition. Finally, we present the relativistic Vlasov-Fokker-Planck (VFP) equation based on the Belyaev-Budker small-angle collision tensor. This equation describes both the collective and collisional processes ruling fast electron transport.

Chapter 5 presents the numerical methods used to solve the relativistic V-F-P equation. Often, these codes are based on the "hybrid assumption", that is to say, they solve the V-F-P equation only for the high-energy component of the electron population. The dynamics of background particles is computed according to hydrodynamic equations or simplified ones. A comparison between the PIC method, the full "Vlasov-Fokker-Planck" method and other methods, based on the decomposition of the distribution function, is presented. Most of V-F-P (full or expanded) models do not use the Belyaev-Budker collision tensor but a Landau-like relativistic collision operator. Moreover, except for the relativistic V-F-P code of [START_REF] Yokota | Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets[END_REF], the dependence of Coulomb logarithms on the relativistic mass γm e is often neglected.

In the second part of the manuscript, a new fast electron model is presented including. Its numerical implementation and its validation are also presented:

In Chapter 6, the Landau-like relativistic collision operator mentioned above is derived from the Belyaev-Budker collision operator in the context of relativistic electron beam transport. It allows us to derive an expression for the relativistic Coulomb logarithm, starting from the fast electron stopping power term, in the V-F-P equation. It thus relates naturally the fast electron angular scattering rate due to the collisions with free, bound and screened free background electrons with the corresponding stopping powers. According to Chapter 5, the best compromise between the accuracy and the numerical cost can be obtained with hybrid and expanded relativistic V-F-P methods. As a consequence, the model developed in this PhD thesis consists in solving the two first angular moments of the V-F-P equation in order to make computations as fast as possible.

Besides, truncating the distribution function expansion at the first angular harmonic may lead to non-physical results in a case of strong anisotropy. Consequently, a special closure relation based on the Minerbo approach of a maximum angular entropy [Minerbo, 1977] [Minerbo, 1978], Chapter 7 is dedicated to the numerical tools developed to solve the equations of the model.

developed
A key point is the use of a numerical model developed previously for the radiative transfer equations which ensure that the 0th order angular moment stays positive and that the 1st order angular moment stays smaller than the 0th order one for all electron energies, times and space locations.

Chapter 8 presents a simple academic test case of fast electron transport in a warm and dense

Fast Electron Generation

"Philosophy is written in that great book which ever lies before our eyes -I mean the universebut we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it;

without which one wanders in vain through a dark labyrinth."

Galileo Galilei

LASER-SOLID INTERACTION AT HIGH INTENSITIES

In the non relativistic regime of laser-plasma Interaction (where I L λ 2 10 18 W.cm -2 .µm 2 with I L being the laser pulse intensity and λ its wavelength), the laser pulses may have temporal durations from several tens of fs with an energy of several mJ to several ns with an energy of several kJ.

For long pulse durations like those used in the conventional ICF schemes, the material is ionized by the laser electric field during the first ps of the Laser-Solid-Interaction (LSI) and heat the plasma.

While the heated plasma consequently expands, the laser pulse cannot penetrate the plasma where the electron density n e is above the critical density n c ≈ 10 21 cm -3 /λ[µm] 2 . Concerning the Shock Igntion scheme, at the moment of the laser spike's arrival, the sub-critical zone has a larger scale and a higher temperature than in the conventional scheme. This changes considerably the conditions of laser-plasma interaction and it is expected that part of the laser energy is converted into fast electrons. For short laser pulse durations, the plasma has no time to expand so that the laser pulse interacts with a steep gradient density. Since the discovery of the Chirped Pulse Amplification (CPA)

by [START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF], short-pulse laser technology has grown steadily. In the 1990's, the threshold value of a laser intensities with I L λ 2 10 18 W.cm -2 .µm 2 has been attained allowing to reach the relativistic laser-matter interaction regime. However, the intense short laser pulse is preceded by a lower intensity prepulse of several ps duration. Therefore, the plasma may have time to expand. The relativistic laser plasma interaction leads to the generation of very energetic electrons. This chapter is dedicated to the desciption of laser-plasma interaction in general and a special attention is devoted to the different mechanisms responsible for fast electron generation in the context of conventional ICF, shock ignition and fast ignition.

1.1 Laser-Solid Interaction at High Intensities

Collisional versus "Collisionless" Absorption mechanisms

The laser pulse propagation is described by the equation for the electric field E obtained from the Maxwell equations

∂ 2 ∂ 2 r (E) - ∂ ∂r ∂ ∂r .E - 1 c 2 ∂ 2 E ∂t 2 = 4π c 2 ∂j e ∂t (1.1)
where j e = -n e ev e is the plasma electron current density. The plasma response to electomagnetic fields is described by the plasma electron hydrodynamic equations (see Appendix B, section B.1.2).

By neglecting the electron viscosity, the electron pressure, the magnetization effects and the thermal force, these equations read

∂n e ∂t + ∂ ∂r .
(n e v e ) = 0 (a) for the electric field at normal incidence

∂v e ∂t + v e . ∂ ∂r (v e ) = - e m e E + v e c × B -ν ei v e (b)
d 2 E z dx 2 + ω 2 c 2 (x, ω) E z (x) = 0. (1.3)
Here, E is the time Fourier transform of the electric field and (x,

ω) = 1 -ω p (x) 2 / [ω (ω + iν ei (x))]
is the coronal permitivity. According to the WKB approximation (from the names of its founders G.

Wentzel, H. A. Kramers and L. Brillouin), one can derive the laser energy absorption coefficient in the corona [Dawson andOberman, 1962] [Mora, 1982]

η abs = 1 -exp -2 ∞ xc ν ei (x) c n e (x) n c 1 - n e (x) n c -1/2 dx = 1 -exp - 8 3
Lν c c .

(1.4)

where ν c = ν ei (x c ). The calculation of Integral (1.4) shows that, typically in the conventional ICF schemes, 50 % of the laser absorption takes place in the immediate vicinity of the critical density where . Thus, by equating the absorbed energy flux η abs I L to the energy flux necessary to maintain this self-similar isothermal expansion from the critical density to the vacuum 4n c T e c s , one finds that the temperature scales with the laser intensity I L and the laser wavelength λ as T e ∝ η abs I L λ 2 2/3 . By injecting this expression in (1.4), one obtains [Mora, 1982]

η abs = 1 -exp - I 0 η abs I L 2/3
; α = 1 + (π/2) 1/4 /2 ± 1 -(π/2) 1/4 /2 and β = (1/2) 1/3 + (π/2) 1/6 /2 ± (1/2) 1/3 -(π/2) 1/6 /2)
; the signs ± come from the incertitudes on frontiers between the different regimes, evaluated according to the conditions of validity of each mechanism. (Right panel) Plot of the corresponding absorption coefficients as a function of the laser pulse intensity I L evaluated for Copper with λ = 1 µm at t = 10 fs and t = 1 ps after the beginning of the LSI according to Equations (1.9), (1.10),(1.11) and (1.12) (right)

For laser pulse duration ∆t FWHM shorter than used in conventional ICF schemes such as L/c s ∆t FWHM , the laser-ionized plasma has no time to expand hydrodynamically. Let us as-sume so that the laser pulse at normal incidence interacts directly with a solid steep density gradient (L → 0). In this case, the resolution of {(1.1), (1.2)} shows that the laser pulse penetrates the solid over a skin-depth L s ≈ c/ω pe while a stationary wave consisting in the incident laser pulse and the reflected parts is formed, standing in vacuum. For high electron temperatures and high laser frequencies, the previous model developed for large density gradients can be extended by taking the limit L → 0 and then by replacing L by the penetration depth L s in (1.4 Two "collisionless" mechanisms have been firstly identified : the Sheath Inverse Bremsstrahlung (SIB) [START_REF] Catto | Sheath inverse bremsstrahlung in laser produced plasmas[END_REF] and the Anormal Skin Effect (ASE) [Weibel, 1967a]. It has been demonstrated later that they are in fact two limits of the same "collisionless" absorption mechanism [START_REF] Yang | Absorption of laser light in overdense plasmas by sheath inverse bremsstrahlung[END_REF]. Usually discussed separately, the absorption coefficients for these four mechanisms can be derived within a common theory by linearizing Equation (1.1) coupled with the classical kinetic equation expressed within the BGK approximation (for the name of its founders P. L. Bhatnagar, E.P.

Gross and M. Krook, see Appendix A, section A.3.1)

∂f e ∂t + ∂ ∂r . (vf e ) - ∂ ∂v . e m e E + v c × B f e = -ν (f e -f M ) . (1.6)
where ν is the effective electron collision frequency. The results obtained by [START_REF] Rozmus | A model of ultrashort laser pulse absorption in solid targets[END_REF],

according to this method, are summarized in the following table where the mean effective collision frequency ν has been evaluated by ν ei for simplicity. By assuming a square temporal laser pulse shape and that the absorption coefficient η abs depends on the electron temperature, one can find a self-similar solution of the heat equation (1.2 c) with the form T e (x, t) = T 0 τ 2/9 F (ζ) [START_REF] Zel'dovich | Physics of Sock Waves and High Temperature Phenomena[END_REF], where ζ = ξ/τ 7/9 , ξ = x/v 0 t 0 , τ = t/t 0 ,

Conditions of validity

L s η abs CA L s v T he ν ei v T he ω c ω pe 2ν ei ω pe NSE L s v T h,e ω v T h,e ν ei c ω pe 2ν ei ω ω ω pe 8ν ei ω ASE L s v T h,e ω v T h,e ν ei 2 π 1/6 v T h,e ω 1/3 c ω pe 2/3 8 3 √ 3 2 π 1/6 v T h,e c 1/3 ω ω pe 2/3 SIB v T h,e ω L s v T h,e ν ei c ω pe 8 2 π v T h,e c
T 0 = m e v 2 0 /k B , t 0 = (2κ 0 /3n e )(m e /k b ) 5/2 v 3 0 , v 0 = (2η abs I L /3n e m e ) 1/3 and d dζ F 5/2 dF dζ - 2F 9 + 7ζ 9 dF dζ = 0. (1.7)
This equation can be solved numerically, giving an expression for the temperature at the LSI interface [Gibbon, 2005] (1.12)

T I = T e (x = 0, t) = 250 n i 10 23 cm -3 -2/9 η abs I L 10 15 W.cm
Even if the temperature dependence of the electron collision frequency

ν ei = ν 0 T -3/2 e
and the thermal electron velocity v th,e = k B T e /m e has not been taken into account in Equation (1.8), these results are in good agreement with the more rigorous calculations of NSE and ASE absorption coeeficients found by [START_REF] Rozmus | Skin effect and interaction of short laser pulses with dense plasmas[END_REF]. It must be emphasized that collisions play an important role in the ASE and SBI mechanisms. The term "collisionless" comes from the fact that the electron collision frequency ν ei does not appear explicitly in the absorption coefficient η abs , but the electrons escaping the skin layer are thermalized in the bulk target.

Resonant Absorption

Let us consider now a laser pulse interacting with a large scale density gradient at an incidence angle θ with respect to the target normal direction e x (see Figure 8) such that the wave vector has the form k = |k| (cos θe x + sin θe y ). If the laser pulse is s-polarized i.e. if E = E z (x) exp (iky sin θ)e z where the double hat means that we have also performed the y-Fourier transform of the electric field, one gets Equation (1.3). According to the WKB approximation, the absorption coefficient can be estimated. It reads

d 2 E z dx 2 + ω 2 c 2 (x, ω) -sin 2 θ E z (x) = 0 (1.13) instead of
η abs = 1 -exp - 8 3
Lν c cos 3 θ c . (1.14)
Thus, the turning point in the oblique incidence case, i.e., the position where the laser pulse is reflected is no longer at the critical surface x c but at the position where the density n e = n c cos 2 θ. In the case where the laser pulse is p-polarized with the laser electric field in the plane (x, y), it is more convenient to work with the magnetic field. One gets

d 2 B z dx 2 - 1 (x, ω) d dx d B z dx + ω 2 c 2 (x, ω) -sin 2 θ B z (x) = 0 (1.15)
while the electric field can be found from the Maxwell-Ampere law : E x = -c sin θ B z / and Equation (1.15), it has been shown that the plasma resonance may cause a significant laser energy absorption [START_REF] Freidberg | Resonant Absorption of Laser Light by Plasma Targets[END_REF]. In addition to the collisional absorption mechanism taken into account in Equation (1.15), it describes another absorption mechanism which does not depend on the collision frequency ν c but only on the incidence angle θ, the laser frequency ω and the density gradient length L [Ginzburg, 1961]. The underlying physical process is the conversion of the incident laser wave into the resonant plasma wave near the critical density. [Ginzburg, 1961] provides the estimate of the absorption coefficient

E y = -i(c 2 /ω )d B z /dx. Starting from
η abs ≈ 2.65 ωL c 2/3 sin 2 θ exp - 4 3 ωL sin 3 θ c . (1.16)
It means that, due to the presence of the longitudinal component of the electric field, electron plasma waves can be excited near the critical density. Indeed, at the turning point where the real part of the dielectric function vanishes, the magnetic field takes a finite value B z0 . It follows that E x = -c sin θ B z0 / becomes very large, which means that the incident electromagnetic wave resonantly excites an electron plasma wave. A rigorous account for the collisional and/or thermal effects is needed to describe the plasma wave structure. Indeed, the excited electron plasma oscillations are acceleration. The energy of fast electrons scales with the intensity as [START_REF] Forslund | Theory of Hot-Electron Spectra at High Laser Intensity[END_REF] T fast e-≈ 14 T e 1 keV

I L 10 15 W.cm -2 λ 1 µm 2 1/3
keV.

(1.17)

The resonant absorption can also occur in the case of a steep gradient density. In this case, the results obtained in this section can be generalized by taking the limit of L → 0.

Ponderomotive Force and Parametric Instabilities

The 1st order response of the plasma electrons in the sub-critical zone in the cold plasma approximation

(1.1) is given by

m e ∂ ∂t δv (1) e = -eE (1.18)
where E = δE (1) is the laser electric field assumed here to be a 1st order term in the perturbation expansion. The 1st order response consists in the plasma electron oscillations along the laser electric field direction at the laser frequency ω, which are screening the laser field over a distance of the order of the skin depth L s behind the critical surface. The 2nd order response of plasma electrons is given by m e ∂ ∂t δv (2) e + δv (1) e .

∂ ∂r δv

(1) e = -e δv (1.20)

This non-linear force corresponds to the second order plasma response at the frequencies 0 and of 2ω.

The ponderomotive force is the average of this quantity over a laser period. From a particle point of view, it can be seen as a pressure force exerted by the laser photons. For a monochromatic plane wave linearly polarized propagating in the x-direction with E = E 0 (r, t) sin (ωt + kx)e y where E 0 (r, t) is the slowly varying laser electric field envelope, the laser intensity has the form I L (r, t) = cE 0 (r, t) 2 /8π. In this case, the mean ponderomotive force over a laser period can be written

F pond = - ∂ ∂r 1 2 m e δv (1) e 2 = - ∂ ∂r e 2 E 2 2m e ω 2 = - 2πe 2 ω 2 m e c ∂ ∂r (I L ) .
(1.21)

In the local plane, the laser intensity profile can be taken to be I L (r, t) = I max exp -4 ln 2(r/∆r FWHM ) 2 f (t) where r = y 2 + z 2 and ∆r FWHM is the spatial Full Width Half Maximum (FWHM) of the focal spot. Consequently, the ponderomotive force tends to eject plasma electrons from the focal spot center where the laser intensity is highest, leading to the ion density modification on the hydrodynamic time scale. Moreover, in the rising part of the laser pulse, the ponderomotive force is directed along the laser propagation axis and tends to push the plasma inward the target [START_REF] Lee | Theoretical derivation of laser induced plasma profiles[END_REF].

The three-wave parametric instabilities consist in a decomposition of the laser wave at ω into two daughter waves ω 1 and ω 2 where ω 1 > 0 and

ω 2 = ω -ω 1 (Stokes configuration) or ω 1 < 0 and ω 2 = ω -ω 1 (anti-
≈ ω IAW ω 2 ω n e ≈ n c Stimulated Brillouin Scattering DEW ω 1 ≈ ω IAW ω 2 ω n e ∈]0, n c ] Two-Plasmon-Decay EPW ω 1 ≈ ω/2 EPW ω 2 ≈ ω/2 n e = n c /4 Stimulated Raman Scattering DEW ω 1 ≈ ω/2 EPW ω 2 ≈ ω/2 n e ∈]0, n c /4]
Experiments on indirect or direct drive ICF have shown that several tens of per cent of laser energy can be reflected due to stimulated Raman and Brillouin scatterings while part of the scattered light is absorbed in the corona thus modifying the absorption and accelerating fast electrons [START_REF] Ebrahim | Hot Electron Generation by the Two Plasmon Decay Instability in the Laser Plasma Interaction at 10.6 µm[END_REF]] [Lindl, 1995]. It is expected in the Shock Ignition scheme that the stimulated Raman scattering plays an important role in the generation of energetic electrons [START_REF] Klimo | Particlein-cell simulations of laser-plasma interaction for the shock ignition scenario[END_REF] [ [START_REF] Klimo | Twodimensional simulations of laser-plasma interaction and hot electron generation in the context of shock-ignition research[END_REF]. This simple analysis shows that a strong EPW can trap and accelerate electrons to energies up to

K ≈ (γ ϕ -1) m e c 2 where γ ϕ = 1/ 1 -(v ϕ /c) 2
written with the relativistic formalism here. In order to give an order of magnitude, let us assume that k ≈ 0.3λ -1 Debye (value at which the Landau damping is optimal) and that the electron temperature scales as T e ∝ η abs I L λ 2 2/3 , according to analysis presented in section 1.1.1.. Then, one finds that the energy of accelerated electrons scales with the

LASER SOLID INTERACTION AT ULTRA-HIGH INTENSITIES

absorption coefficient η abs , the laser intensity I L and the laser wavelength λ as

T fast e-≈ (γ ϕ -1) m e c 2 ≈ 12k B T e ≈ 33 A Z η abs 10% I L 10 15 W.cm -2 λ 1 µm 2 2/3 keV. (1.23)
Concerning the Shock Ignition scheme (I L ≈ 10 15 W.cm -2 ), fully kinetic large-scale two-dimensional simulations have been conducted recently [START_REF] Klimo | Particlein-cell simulations of laser-plasma interaction for the shock ignition scenario[END_REF]] [Klimo et al., 2014]. They show that a significant amount of the laser spike's energy is reflected and absorbed close to n c /4 due to stimulated Raman scattering and that electrons are accelerated to energies up to several tens of keV in agreement with the previous estimate.

1.2 Laser solid interaction at Ultra-High Intensities

Single Electron Motion in an Ultra-High Intense Laser Pulse

Since the invention of CPA by [START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF], the laser pulse durations have come down from 1 ps to less than 5 fs, leading to an increase of laser pulse intensities by more than six orders of magnitude allowing to overpass the relativistic threshold of I L λ 2 10 18 W.cm -2 .µm 2 . In order to introduce this regime, we first consider the motion of a single electron in such a Ultra-High-Intense (UHI) laser pulse traveling in the positive x-direction. The wave vector potential is

A = 0, δA 0 f (t) cos φ, √ 1 -δ 2 A 0 f (t)
dp dt = -e E + v c × B = e ∂A ∂t + v. ∂ ∂r (A) -e ∂ ∂r ⊗ A.v (
                   t = a 2 0 4 T 2π 2δ 2 -1 2 sin 2φ + 1 + a 2 0 4 T 2π φ x = a 2 0 4 λ 2π 2δ 2 -1 2 sin 2φ + a 2 0 4 λ 2π φ y = a 0 λ 2π δ sin φ z = -a 0 λ 2π √ 1 -δ 2 cos φ (1.28)
Here, T = λ/c and a 0 = eA 0 f /m e c = 2e 2 I L λ 2 /πm 2 e c 5 where I L is the laser intensity. Equation (1.28) shows that an electron interacting with a laser pulse starts to drift with an average momentum p = (a 2 0 /4)m e c e x , corresponding to a drift velocity

v D = p γ m e = c a 2 0 /4 1 + a 2 0 /4 e x .
(1.29) Indeed, by performing the Lorentz transformation from the laboratory frame to the average rest frame moving at the velocity v D :

t 0 = γ D t -v D x/c 2 , x 0 = γ D (x -v D t), y 0 = y and z 0 = z where γ D = 1/ 1 -(v D /c) 2 = (1 + a 2 0 /4)/ 1 + a 2 0 /2, one gets                        t 0 = a 2 0 /4 1 + a 2 0 /2 T 0 2π 2δ 2 -1 2 sin 2φ 0 + T 0 2π φ 0 x 0 = a 2 0 /4 1 + a 2 0 /2 λ 0 2π 2δ 2 -1 2 sin 2φ 0 y 0 = a 0 1 + a 2 0 /2 λ 0 2π δ sin φ 0 z 0 = - a 0 1 + a 2 0 /2 λ 0 2π √ 1 -δ 2 cos φ 0 (1.30)
where the subscripts 0 denote the values in the average rest frame and for which the phase invariance φ 0 = φ as well as the longitudinal Doppler-Fizeau relationship drift is negligible (a 2 0 /4 a 0 ) and we find the transverse oscillations at the frequency ω considered in sections 1.1. In the opposite ultra-relativistic case where I L λ 2 10 18 W.cm -2 .µm 2 (a 2 0 /4 a 0 ), this longitudinal drift, accompanied with oscillations at 2ω is the main process. In the general case, the angle between the electron momentum and the laser pulse propagation direction is

T 0 /T = λ 0 /λ = (1 + v D /c)/(1 -v D /c) = 1 + a 2 0 /
θ = arctan |p ⊥ | p x = arctan 2 γ -1 . (1.31)
UHI laser experiments are usually conducted with short laser pulses interacting with a solid target with a steep density gradient. The analysis concerning the single electron response shows that electrons interacting with a UHI propagating laser plane wave stop moving immediately after the laser pulse ended. The electron motion is adiabatic and it cannot acquire energy directly from the laser pulse.

Moreover, contrary to the non-relativistic case, the colisional friction force is negligible here due to a large amplitude of the laser electromagnetic fields and high temperatures in the skin-depth imposed by the laser absorption. Indeed, the surface temperature T I at the solid-vacuum interface scales with the laser intensity as I 4/9 L according to Equation (1.8). It can reach several keV for laser intensities

I L 10 18 W.cm -2 .
Consequently, in the relativistic regime, collisions cannot explain the break of this adiabaticity. The UHI Laser Solid Interaction has been therefore intensively studied for more than twenty years and a large number of collisionless processes can be found in the litterature to explain the experimentally observed accelerated electron.

Cold Plasma Approximation

In a fluid framework, the mechanisms of UHI laser electron acceleration can be described by replacing (n e v e ) = 0 (a)

∂p e ∂t + v e . ∂ ∂r (p e ) = -e E + v e m e c × B (b) (γ e -1)m e c 2 = m 2 e c 4 + p e 2 c 2 -m e c 2 (c) (1.32)
where p e = γ e m e v e is the mean electron momentum, γ e is the mean Lorentz factor and v e the mean velocity. It must be noted here that, in difference from the cold plasma approximation in the classical regime, the relativistic cold plasma approximation assumes that the electron distribution (

n e v e ) = 0 (a) ∂ ∂t (p e -eA) = F + ∂ ∂r (eΦ) (b) (γ e -1)m e c 2 = m 2 e c 4 + p e 2 c 2 -m e c 2 (c) (1.34)
where

F = - ∂ ∂r (γ e -1) m e c 2 (1.35)
is the non averaged Relativistic Ponderomotive Force which was extended from the classical theory [START_REF] Stroscio | Relativistic derivation of laser induced plasma profiles[END_REF]. This force leads to exotic effects compared to the non-relativistic regime and will be described more accurately in the next paragraph.

Let us consider a UHI elliptically polarized plane wave traveling in an underdense plasma ω pe ω (ω pe = 4πZn i e 2 /m e is the classical plasma frequency) in the positive x-direction represented by its

vector potential A = 0, δA 0 f (t) cos φ, √ 1 -δ 2 A 0 f (t) sin φ T .
In contrast with the single electron case, here, φ = 2πc(t -x/v φ )/λ where v φ = ω/k is the wave phase velocity. The system of equations (1.34) is coupled with the Maxwell equations (1.1) expressed for the vector potential A and the electrostatic potential Φ in the Coulomb gauge ((∂/∂r).A = 0)

∂ 2 ∂r 2 (Φ) = 4πe (n e -Zn i ) (a) ∂ 2 ∂t 2 (A) -c 2 ∂ 2 ∂r 2 (A) = 4πj e . (b)
(1.36)

Assuming plasma electrons are initially at rest and f = 1, the system of Equations {1.34, 1.36} reads [START_REF] Akhiezer | Theory of wave motion of an electron plasma[END_REF]]

n e = γ e m e v φ γ e m e v φ -p e,x
Zn i (a)

d 2 p e,⊥ dφ 2 = - 1 γ e ω pe ω 2 γ e m e v φ γ e m e v φ -p e,x v φ 2 v φ 2 + c 2 p e,⊥ (b) d 2 dφ 2 γ e m e c 2 -p e,x v φ = 1 γ e ω pe ω 2 γ e m e v φ γ e m e v φ -p e,x v φ p e,x . (c)
(1.37)

According to [START_REF] Kaw | Relativistic Nonlinear Propagation of Laser Beams in Cold Overdense Plasmas[END_REF], Equations (1.37 b) and (1.37 c) in the limit p e,x γ e m e v φ as well as v φ 2 /(v φ 2 + c 2 ) ≈ 1 and γ e m e c 2 p e,x v φ take the non-relativistic form with ω pe / √ γ e instead of ω pe . This can be interpreted as a deeper penetration of electromagnetic waves in the target with an increased skin-depth L s ≈ c √ γ e /ω pe instead of L s ≈ c/ω pe , due to the higher electron inertia γ e m e .

This relativistic effect is called the Self-Induced Transparency. While this phenomenon has been observed in kinetic simulations [Lefebvre andBonnaud, 1995] [Guérin et al., 1996], its experimental validation is still controversial [Gibbon, 2005]. Equations (1.37) provide the adequate theoretical formalism to extend some mechanisms to the relativistic regime as the Relativistic Parametric Instabilities [START_REF] Guérin | Modulational and Raman instabilities in the relativistic regime[END_REF] and the Relativistic Linear Resonant Absorption [START_REF] Yu | Model for fast electrons in ultrashort pulse laser interaction with solid targets[END_REF]].

These two mechanisms can be responsible for laser energy absorption by driving strong EPWs and accelerating fast electrons. Also, this formalism can be used to predict the Direct Laser Acceleration of fast electrons by Channeling the laser pulse [START_REF] Pukhov | Particle acceleration in relativistic laser channels[END_REF]] [Li et al., 2011]. However, these mechanisms take place in an underdense plasma (n e < n c ) and they cannot explain the laser energy absorption reported in UHI laser-solid interaction experiments where the absorption may reach up to 70 % [START_REF] Sauerbrey | Reflectivity of laser produced plasmas generated by a high intensity ultrashort pulse*[END_REF]. Also, they cannot explain the fast electron energy spectrum reaching up to 10 MeV as measured experimentally by [START_REF] Beg | A study of picosecond lasersolid interactions up to 1019 W cm-2[END_REF]. The following sections present the main absorption mechanisms operating in laser-solid interaction experiments in the relativistic regime.

Relativistic Ponderomotive Force and j × B heating

The ponderomotive force may push the plasma electrons inward the target at the center of the focal spot. Then, the plasma ions are accelerated due to the induced electrostatic fields. In the relativistic regime, the ponderomotive force (1.35) push the surface of a modestly overdense plasma over a distance of several laser wavelengths deep in a moderately overdense plasma on the sub-ps timescale [START_REF] Wilks | Absorption of ultra intense laser pulses[END_REF]. This so-called Hole boring process has been demonstrated experimentally at a laser intensity of 10 18 W.cm -2 by [START_REF] Kalashnikov | Dynamics of Laser Plasma Interaction at 10 18 W/cm 2[END_REF] by measuring the Doppler red-shifted reflected light. This effect was at the basis of the fast ignition scheme [START_REF] Tabak | Ignition and high gain with ultrapowerful lasers@f[END_REF] (1.41)

The ponderomotive force is usually defined by the average part of this force and it reads in both cases

F pond = F = - e 2 4γ e m e ∂ ∂x A 0 2 e x .
(1.42) [Marburger andTooper, 1975] [Siminos et al., 2012]. Neglecting the laser energy absorption, one finds

A 0 (x) =                      -2 m e c e 2 1 - ω 2 ω pe 2 cosh ω x -x m c ω pe 2 ω 2 -1 ω pe 2 ω 2 -1 -cosh 2 ω x -x m c ω pe 2 ω 2 -1 if x x d 2A L cos 2π x -x d λ + ϕ = 1 4 A 0 (x) 2 + dA 0 dx x=c/2ω x=x d cos 2π x -x d λ + ϕ if x < x d (1.43)
and

n e (x) =      Zn i 1 + ω 2 ω 2 pe e 2 2γ e m e 2 ω 2 1 γ e 2 dA 0 dx 2 + A 0 d 2 A 0 dx 2 if x x d 0 if x < x d (1.44)
where

γ e (x) = 1 + eA m e c 2 = 1 + eA 0 m e c 2 ,
(1.45)

A L is the initial normally incident laser pulse amplitude, ϕ is a constant phase delay, x m is the position where the vector potential is maximum [START_REF] Bauer | Vacuum heating versus skin layer absorption of intense femtosecond laser pulses[END_REF]]. However, in the limit ω p,e ω, an approximate solution reads

A 0 (x m ) = 2(m e c/e) 2(ω pe 2 /ω 2 )[(ω pe 2 /ω 2 ) -1] and x d = -(1/2πen i )A 0 (x d )(dA 0 /dx)| x=x d /γ(x d )
A 0 (x) =        2 ω ω p,e A L exp - x -x d L s if x x d 2A L cos 2π x -x d λ + ϕ if x < x d (1.46)
where L s = c/ω (ω pe 2 /ω 2 ) -1 ≈ c/ω pe . The oscillating component of the ponderomotive force (1.41)

in the linearly polarized case can lead to laser energy absorption [START_REF] Kruer | J×B heating by very intense laser light[END_REF]. This mechanism is called j×B heating (and not ponderomotive heating!) due to the fact that it is induced by the oscillating component of (1.41), coming from the Lorentz force (v e /c) × B term of (1.32). One can explain this mechanism as follows :

1. During a quarter of the laser cycle, between t = -T /8 and t = T /8, the cosine of (1.41) is positive and electrons from the plasma surface are ejected in vacuum by the Lorentz force.

2. A strong electrostatic field Φ is generated due to the induced charge separation according to Equation (1.36 a).

3. During the following quarter of the laser cycle, the cosine becomes negative and the electrons are consequently recalled by this ponderomotive force component but this time with an amplification due to the force (∂/∂x)(eΦ).

4. Thus, the electrons are reinjected in the overdense plasma where the local electromagnetic fields vanish behind the skin depth L s .

Consequently, the adiabaticity of electron motion is broken at the moment when they are crossing the plasma skin layer and they acquire the kinetic energy gained from the electric force (∂/∂x)(eΦ). Thus, accelerated electrons escape from the laser solid interaction zone thanks to their residual energy. This acceleration mechanism has been confirmed by kinetic simulations [START_REF] Wilks | Absorption of ultra intense laser pulses[END_REF]. The authors find a good agreement between their numerical results and the estimate of the mean kinetic energy often called "temperature" of the accelerated electrons

T e ≈   1 + ep e,⊥ m e c 2 -1   m e c 2 = 1 + a 0 2 2 -1 m e c 2 (1.47)
according to Equation (1.38 a) where . means that the value has been averaged over a laser cycle and a 0 = 2e 2 I L λ 2 /πm e 2 c 5 . This expression is commonly called the ponderomotive scaling due to the fact that it corresponds to the ponderomotive potential of a single electron [START_REF] Bauer | Relativistic Ponderomotive Force, Uphill Acceleration, and Transition to Chaos[END_REF].

Besides, even if numerical simulations confirm that this process makes a significant contribution to the laser energy absorption, this temperature scaling differs from experimental results [START_REF] Ping | Absorption of Short Laser Pulses on Solid Targets in the Ultrarelativistic Regime[END_REF]. fields in this configuration was demonstrated by [Brunel, 1987] thanks to kinetic simulations and a simplified analytical model. Thus, it would have been possible to place this subsection in the previous section. However, due to the fact that this mechanism plays an important role in UHI laser-plasma interaction, we have prefered to place it here. This mechanism can be understood in a one-dimensional approach, neglecting of the Lorentz force (v e /c)×B in Equation (1.32). Thus, the system of Equations {(1.34), (1.36)} reduces to Consequently, the electron motion adiabaticity is broken and the electron gains a kinetic energy from the electrostatic field. Moreover, the accelerated electrons escape from the interaction zone thanks to this residual energy. Except for the time periodicity and the spatial configuration of the process, the Brunel heating mechanism is similar to the j × B one. By using the capacitor approximation for the step 2, approximating v e,x ≈ 2eA 0 /m e for the step 3 and assuming that all electrons are lost in the solid for the step 4, one can estimate the laser energy absorption [Gibbon, 2005] .49) where f = 1 + 1 -(4a 0 sin 3 θ/π cos θ) where the fast electrons "temperature" has been assumed

Brunel Effect and Vacuum heating

A =      2A L cos (ωt + ϕ) cos (kx cos θ) sin θe x if x 0 ∝ exp - x -x d L s if x > 0 (a) ∂ 2 Φ ∂x 2 = 4πe (n e -Zn i ) (b) ∂n e ∂t + ∂ ∂x (n e v e,x ) = 0 (c) dp e,x dt = e ∂A ∂t + e ∂Φ ∂x (
η abs ≈ 1 πa 0 f 1 + (f a 0 sin θ) 2 -1 sin θ cos θ . ( 1
T e ≈ 1 + v e,x c 2 -1 m e c 2 ≈ 1 + 4a 0 2 -1 m e c 2 .
(1.50)

The maximum laser energy absorption is obtained with θ = 75 o according to Equation (1.49). Actually, it has been shown that not all electrons pushed out into vacuum return to the target each laser period [Brunel, 1988] [Gibbon and[START_REF] Gibbon | [END_REF]. Due to the presence in vacuum of non-stationary electromagnetic fields and a low density plasma, the electron orbits are more complex than those predicted in the capacitor approximation. According to kinetic simulation results, the maximum of laser absorption is obtained for θ ≈ 45 o and the temperature scales with the laser intensity as T e ∝ (I L λ 2 ) α with α between 1/3 and 1/2. For these reasons, and notably because of recalling aspects of disorder, the term Vacuum heating is used to make the distinction. It is considered as one of the main absorption mechanisms.

Anharmonic Resonant Absorption

In the case of a linearly polarized incident laser pulse A L cos [ωt -k (x cos θ + y sin θ)] (-sin θe x + cos θe y )

totally reflected by a steep density at normal or at oblique incidence, the Relativistic Harmonic Resonance Absorption mentioned in the introduction of this section may be mitigated due to the absence of any rarefaction wave. Moreover, the critical-surface oscillations driven by the laser ponderomotive force may lead to the broadening and the splitting of the harmonics [START_REF] Ding | Bulk resonance absorption induced by relativistic effects in laser-plasma interaction[END_REF] as well as an Figure 1.9: Test electron trajectory position x(t) and momentum p x (t)/m e c (black lines), total electric field E x (white line) at position x(t) plotted on the electric field (x, t) map from a kinetic simulation done with a constant laser pulse a 0 = 0.3 with an oblique incidence θ = 45 o focused on a steept gradient with fixed ions Zn i /n c = 81 [START_REF] Mulser | Collisionless Laser-Energy Conversion by Anharmonic Resonance[END_REF] .

efficient laser energy absorption mechanism [START_REF] Mulser | Collisionless Laser-Energy Conversion by Anharmonic Resonance[END_REF]. Noticing that a break of adiabaticity under steady state conditions is obtained when j e E ∼ sin (ωt + ϕ) cos (ωt) = (1/2) sin ϕ = 0 according to the Poynting theorem, the authors conclude that only an anharmonic resonance in the self-generated plasma potential may provide conditions for an efficient laser energy absorption as observed in UHI laser-solid interaction experiments. Indeed, by considering a constant density Zn i , an electron layer of thickness d, oscillating with the amplitude ξ and by approximating the non-averaged ponderomotive force (1.41) by F ≈ F 0 f (t) sin (2ωt)e x , one can derive from Equation

(1.36 a)

E x (ξ) = -e ∂Φ ∂x =        4πZn i e 1 - |ξ| 2d ξ if ξ d 4πZn i e ξd 2|ξ| if ξ > d (1.51)
which along with Equation (1.38 b) gives the following equation of motion for the electron layer (in the non-relativistic approximation)

d 2 ξ dt 2 = F 0 f (t) sin (2ωt) -        ω 2 pe 1 - |ξ| 2d ξ if ξ d ω 2 pe d 2|ξ| ξ if ξ > d (1.52)
The anharmonic resonance heating can thus be understood as follows.

1. According to Equation (1.52), the electron layer oscillates at the frequency ω pe inside the plasma where there is no driver (F 0 = 0).

2. With increasing oscillation amplitudes, the electron layer escapes from the plasma and experiences the non-averaged ponderomotive force F 0 f (t) sin (2ωt).

3. According to Equation (1.52), the increasing oscillation amplitude ξ 0 d leads to a reduction of the oscillation frequency enabling a resonance condition at 2ω pe .

4. Finally, due to the strong driver F 0 f (t) sin (2ωt), the oscillation amplitudes may rise to ξ 0 → ∞ with a pulsation ≈ ω 2 pe d/2ξ 0 → 0 This means that between step 1 and step 3, there is a moment when the electron layer oscillation frequency is 2ω (0 2ω ω pe ). At this moment, the electron layer oscillates in phase with the driver and enters the anharmonic resonance regime. This mechanism has been explained by considering the electron trajectories obtained in kinetic simulations as illustrated in Figure 1.9. According to the authors, this Anharmonic Resonance Heating in the self-generated plasma waves represents one of the leading electron acceleration mechanisms in UHI laser-solid interaction experiments and may explain energies of accelerated electrons many times the ponderomotive scaling observed in experiments [START_REF] Cerchez | Absorption of Ultrashort Laser Pulses in Strongly Overdense Targets[END_REF].

Stochastic heating

While the motion of an electron in a single laser wave is deterministic, its motion in two counterpropagating electromagnetic waves can become chaotic if the frequencies of these two waves are different [Mendonça, 1983] [Forslund et al., 1985]. By "chaotic", one means that small differences in the initial conditions produce a large divergence in the electron trajectory with time. Thus, in other words, two counter-propagating electromagnetic waves with slightly different frequencies can break the adiabaticity of the electron motion. This happens in the case of large-scale density gradients and high laser intensities or a sufficiently long pulse 1 -10 ps like in the Fast Ignition scenario. Let us consider a linearly polarized laser pulse A 1 = A 1,0 cos φ 1 e y with φ 1 = ω 1 t -k 1 x normally incident on a target with a large scale density gradient and a counterpropagating one

A 2 = A 2,0 cos φ 2 (cos αe y -sin αe x ) with φ 2 = ω 2 t -k 2,
x -k 2,⊥ y + α 0 which may originate from the reflection of the incident laser pulse or a Raman-backscattered wave as suggested by [START_REF] Sheng | Stochastic Heating and Acceleration of Electrons in Colliding Laser Fields in Plasma[END_REF]. The Hamiltonian of a single electron in such a configuration can be expressed in the extended space by [Jackson, 1975]

H = γ 2 m e c 2 + (P + eA 1 + eA 2 ) 2 c 2 -m e c 2
(1.53)

where P = p -eA 1 -eA 2 . Indeed, in this case, H = 0 according to the momentum quadrivector invariance along directions of translational invariance only. According to [Rax, 1992], by assuming A 2 A 1 , α = -π/2 and by changing of variables from (x, y, z, ct, p x , p y , p z , γm e c) to the angleaction variables (θ, ϕ, φ, E, P , P ⊥ ), it can be shown that H can be written as

H(x, y, z, ct, p x , p y , p z , γm e c) = H (0) (θ, ϕ, φ, E, P , P ⊥ ) + δH (1) (θ, ϕ, φ, E, P , P ⊥ ) (1.54)
where V N E, P , P ⊥ cos

H (0) = γ 2 m e c 2 + (P + eA 1 ) 2 c 2 -m e c 2 = E 2 -P ⊥ 2 -P 2 -M 2 , (1.55) with M = 1 + (a 2 1 /2) and a 1 = eA 1,0 /m e c
k 2, c ω 1 ϕ + k 2,⊥ c ω 1 θ + ω 2 ω 1 φ + N (ϕ + φ) (1.56) 1.2. LASER SOLID INTERACTION AT ULTRA-HIGH INTENSITIES
is the 1st order perturbation induced by the counterpropagating wave where a 2 = eA 2,0 /m e c,

V N = m∈Z n∈Z 2 h=-2 δ N h+m+2n U h J m a 1 m e c ω 1 P ⊥ ω 2 -k 2, c P -(E/c) 2 + a 1 m e c ω 1 k 2,⊥ c P -(E/c) × J n k 2, c -ω 2 2ω 1 (eA 1,0 ) 2 P -(E/c) 2 ,
and

U h = 2 P ⊥ m e c cos α - P m e c sin α δ 0 h + a 1 cos α + P ⊥ sin α P -(E/c) δ 1 |h| + a 1 2 4 m e c sin α P -(E/c) δ 2 |h|
with J m and J n the Bessel functions. It is interesting to notice that according to the change of variable, E = γ m e c 2 , P ⊥ = p y and P = p x where . which means that the values are averaged over the laser pulse cycle 2π/ω 1 . Indeed, in the case where A 2 = 0, the perturbation disappear and H (0) = 0 can be integrated and one finds that the unperturbed electron motion consists in the figure-of-eight found in section 1.2.1

E P , P ⊥ = M 2 + P 2 + P ⊥ 2 .
(1.57) According to the author, such a perturbation scheme fails to converge because of the occurence of small resonant denominators when the cosine of the phase remains stationary which means that the system cannot be integrated whenever it exists one N ∈ Z such that

k 2, c P m e c + k 2,⊥ c P ⊥ m e c -ω 2 E m e c 2 + N ω 1 P m e c - E m e c 2 = 0. (1.58)
It means that the electron motion is not anymore deterministic and becomes chaotic. The degree of chaoticity for each resonance N can be measured using the Chirikov criterion [START_REF] Bourdier | Stochastic heating in ultra high intensity laser plasma interaction[END_REF] but in the general case, kinetic simulations of the UHI laser-solid interaction are needed to find the stochastically accelerated electrons properties.

Efficient electron heating in two counter propagating electromagnetic waves was demonstrated in 1D relativistic kinetic simulations by [START_REF] Yu | Electron Acceleration by a Short Relativistic Laser Pulse at the Front of Solid Targets[END_REF]. A linearly polarized, Gaussian laser pulse with a peak amplitude a 1 = 7.5, a wavelength λ 1 = 1 µm, and a width L 1 = 15 µm was normally incident and reflected from a foil of density n e = 10n c . The electron heating was observed in a Deuterium preplasma of a width 30 µm and a density n e = 10 -3 n c . In Figure 1.10, one can clearly see the effect of the preplasma on the accelerated electrons momenta. Although the electron heating is probably due to stochastic effects, [START_REF] Yu | Electron Acceleration by a Short Relativistic Laser Pulse at the Front of Solid Targets[END_REF] do not mention it in their paper. Instead, they explain the electron heating up to several ponderomotive energies as follows. Firstly, the incident laser pulse carries the accelerated electrons toward the target. As the preplasma behind the pulse maximum becomes positively charged, a relativistic electron return current is created. These counterpropagating electrons with negative momenta cover the entire preplasma and even extend beyond it into the vacuum area.

In addition, the target electrons localized near the density jump are accelerated due to the j × B

Heating. Highly accelerated electrons can be seen at the far right of the frame, showing that the forward accelerated electrons escaping from the incident laser pulse have passed through the foil target. As the reflected laser pulse propagates backwards in the preplasma, it accelerates a small number of electrons to energies several times those of the forward ones, as shown on the far left of For the authors, this strong backward acceleration is therefore attributed to the relativistic electrons return-current, which is missing in the case without preplasma.

As a conclusion, in a presence of a preplasma, laser-accelerated electrons may attain energies exceeding several times the ponderomotive energy, thus forming a hot tail in the energy spectrum, and have a significant angular divergence [Kemp et al., 2009]. This Stochastic heating dominates in the underdense plasma in the case of large-scale density gradients according to kinetic simulations and experiments. It largely prevails over the Direct Laser Acceleration as well as the Linear Resonant Absorption mechanisms [START_REF] Kemp | Laser plasma interactions for fast ignition[END_REF].

1.3 Particle-in-Cell method for Laser-Plasma Interaction simulations

Numerical simulations of kinetic processes in laser-plasma interactions are usually conducted with Particle-In-Cell codes. In Part 3 of these manuscript, kinetic simulations will be used for characterization of the laser-generated fast electron transport. This section is dedicated to a brief description of the Particle-In-Cell modeling of Laser-Plasma Interaction.

Phase-Space Discretization for the Vlasov Equation

In order to simulate the interaction between an incident laser pulse and a plasma, one has to solve the Maxwell equations for the laser pulse propagation coupled with the Vlasov equation (see Appendix A, section A.1.1) for the kinetic plasma response to the electromagnetic fields. We use the relativistic formalism here and assume that the ions remain immobile. Let us note f e = f e (r, p, t) the distribution function of plasma electrons. We do not consider the collisions here; they will be discussed in the next

dr l dt = p l γ l m e dp l dt = -e E + v l c × B .
(1.60)

From this observation, it comes naturally the idea to discretize the phase-space (r, p) into macroparticles {r l , p l }, l ∈ [1..N p ] which are solutions of (1.60) depending only on their initial position r l (t = 0) and momentum p l (t = 0) in order to approach the solution of (2.1) [START_REF] Birdsall | Clouds in clouds, clouds in cells physics for many body plasma simulation[END_REF].

Of course, the number of macropaticles N p is much less than the number N of electrons in the system. Indeed, N = n c ∆x∆y∆z ≈ 10 9 /λ[µm] 2 at the critical density with numerical cells ∆x = ∆y = ∆z = 1 µm; The best actual computer technology has allowed a maximum of N p ≈ 10 10 particles and calculation of 10 10 time steps, requiring 10 4 hours of CPU. Besides, in the most cases, N p N is sufficient to describe accurately the relevant long-range physical processes. The split explicit leapfrog scheme commonly called the Boris scheme is usually used to solve the electromagnetic macroparticle pusher (1.60) (see [START_REF] Birdsall | Plasma Physics via Computer Simulation. The Adam Hilger Series on Plasma Physics[END_REF] for other numerical methods). This numerical scheme reads :

1 o ) p - l -p n-1/2 l ∆t/2 = eE n 2 o ) p * l -p - l = T p -× B n |B n | where T = tan -e|B n | 2γ -m e with γ -= 1 + p - l m e c 2 3 o ) p + l -p * l = 2T 1 + T 2 p * × B n |B n | 4 o ) p n+1/2 l -p + l ∆t/2 = eE n 5 o ) v n+1/2 l = p n+1/2 l γ n+1/2 l m e where γ n+1/2 l = 1 + p n+1/2 l m e c 2 5 o ) r n+1 l = r n l + v n+1/2 l ∆t n (1.61)
where n is the discrete time (t n = n∆t n ). The complications come from the temporal and spatial variations of the electromagnetic fields in (1.60). These are the solutions of the Maxwell equations

E = - ∂A ∂t - ∂Φ ∂r B = c ∂ ∂r × A ∂ 2 ∂r 2 (Φ) = 4πρ ∂ 2 A ∂t 2 -c 2 ∂ 2 ∂r 2 (A) = 4πj e (1.62)
which depend on the electrical charge density ρ and the current density j e . These source terms must consequently be interpolated at each time t n from knowing the N p macroparticle positions and velocities. For example, the electrical charge is estimated as

ρ n,i,j,k = Np k=1 W r i,j,k -r n l (1.63)
where r i,j,k = [x i , y j , z k ] T with x i = x min + (i -1)∆x, y j = y min + (j -1)∆y and z k = z min + (k -1)∆z.

The function

W r i,j,k -r n l = x i +∆x/2 x i -∆x/2 dx y j +∆y/2 y j -∆y/2 dy z k +∆z/2 z k -∆z/2 dzΠ(x i -x n l , y j -y n l , z k -z n l ) (1.64)
is called the macroparticle weight. It depends on the interpolation function Π which can be a Dirac distribution δ 3 [r] (NGP method for Nearest-Grid-Point [START_REF] Birdsall | Clouds in clouds, clouds in cells physics for many body plasma simulation[END_REF]), a linear interpolating function (CIC method for Clouds in Clouds [START_REF] Birdsall | Clouds in clouds, clouds in cells physics for many body plasma simulation[END_REF]) and so on. Currently, third order (or more) interpolations are used in high performance PIC codes. The higher the interpolation order is, the higher the accuracy and the computational cost are. The electromagnetic fields are computed according to the well known finite difference schemes of the propagation equation for the vector W r i,j,kr n l E n (r i,j,k ), B n (r i,j,k ) (1.65)

where N x , N y and N z are the grid number respectively in the x, y and z directions. The interpolation function must be the same as for the particle weighting in order to limit the self-force induced by the macroparticles motion in the spatial mesh grid.

Binary Collision Modelling

Coulomb binary collisions are due to the mutual electromagnetic fields created by two interacting particles. The PIC method cannot take into account the binary collisions effects because the forces acting on the particles in a PIC scheme correspond to macroscopic fields, and because the collisional spatial scales are not resolved. As a result, inter-particle forces inside grid cells are underestimated.

The binary collisions are described by introducing a Coulomb collision operator inside each cell. The usual method consists of a Monte-Carlo scheme simulating the interaction between each particle pair.

Most of Coulomb collision operators used in today PIC codes are based on the binary collision model introduced by [START_REF] Takizuka | A binary collision model for plasma simulation with a particle code[END_REF]. The main idea is based on the fact that, the plasma being

PARTICLE-IN-CELL METHOD FOR LASER-PLASMA INTERACTION SIMULATIONS

essentially collisionless, the binary Coulomb collisions occur only between particles separated by a distance of the order of the Debye length λ Debye . Since a typical size of the PIC cell is close to λ Debye , the interaction between particles in neighboring cells can be neglected. According to this method, all particles are firstly grouped to the cells (i, j, k) where they are located. Then, these particles are paired in a random way, so that one particle has at least one partner. Thus, selected pairs are collided according to the Monte-Carlo method based on the collision probability of collision proportional to the differential collision cross section. For example, the probability of a small angle and non-relativistic collision between an electron (1) and an electron or ion ( 2) is proportional to the Rutherford differential cross section (see Appendix A, section A.2.1). During the time step ∆t n , the scattering angle θ in the 2D collision plane follows the Gaussian stochastic process [START_REF] Takizuka | A binary collision model for plasma simulation with a particle code[END_REF]]

p(θ) = θ θ 2 n exp - θ 2 2 θ 2 n with θ 2 n = 4π n α Z α e 4 (1/2)µ 2 V 3 ln Λ eα ∆t n (1.66)
where Z α is the charge of the particle (2), n α is the density of particle ( 2), µ = m e m 2 /(m e + m 2 ) is the reduced mass in the center of mass frame and V = v 1 -v 2 is the relative velocity. Inverting the relation (1.66) and randomly choosing a value p R between 0 and 1 (uniform stochastic process) for the probability p, one obtains the scattering angle in the center of mass frame θ = -2 θ 2 ∆tn ln p R .

(1.67)

Correspondingly, the azimuthal angle ϕ is chosen randomly between 0 and 2π according to the uniform statistic process. If plasma is nearly uniform, one can introduce a cumulative binary collision operator, as proposed by [Nanbu, 1997], which allows to increase the time step of the collision procedure.

Computational Constraints

The main time step restriction of PIC codes comes from the propagation equation for the vector potential A. The time step must fulfill the Courant-Friedrichs-Lewy condition

∆t n < 1 c 1 ∆x 2 + 1 ∆y 2 + 1 ∆z 2 -1/2
.

(1.68)

In addition, if the cumulative binary collision operator approximation cannot be introduced, the time step must be shorter than the characteristic collision time, ν ei ∆t n < 1. Moreover, the value of the laser frequency ω 0 limits the time step in the Boris scheme to ω 0 ∆t n < 2. Concerning simulations of laser plasma interaction, the critical density zone where, typically ω 0 = ω pe needs to be considered with special attention because of strong laser-driven electron plasma waves. In that case, the spatial grid must satisfy the condition ∆x ≈ ∆y ≈ ∆z ≈ c/ω pe . In order to limit the non-conservative force associated with the particle-grid mapping, which leads to self-heating and numerical instabilities, the space resolution has to be comparable to the plasma Debye length ∆x ≈ ∆y ≈ ∆z < v T h,e /ω pe < c/ω pe . This imposes a strong constraint in the case of dense and/or cold plasmas! PIC codes represent one the most time consuming numerical tools used in Physics. To give an order of magnitude of a Laser Solid Interaction simulation in the context of fast ignition, a 100 µm size plasma needs 6250 spatial cells in 1D, 3.9 10 7 in 2D and 3.4 10 11 in 3D (c/ω pe = 0.016 µm) while the oscillation time is about 1/ω pe = 0.05 fs. Correspondingly, about 4 10 5 time steps are needed for a 20 ps simulation [START_REF] Kemp | Laser plasma interactions for fast ignition[END_REF]. Still according to [START_REF] Kemp | Laser plasma interactions for fast ignition[END_REF], a number of macroparticles of 10 7 in 1D, 5 10 9 in 2D and 5 10 11 in 3D needs respectively 10 2 , 5.6 10 4 and 5.6 10 6 time steps.

PIC codes are key tools to study the laser-plasma interaction in general, and the acceleration of fast electrons in particular in the context of the physics of fast and shock ignition.

1.4 Brief Summary of Laser Solid Interation and Laser-

Generated Relativisitic Electron Beam Properties

High-Intensity Laser-Plasma Interaction (HILPI) has been studied for many years, motivated inter alia by the Inertial Confinement Fusion (ICF) concept. In the conventional ICF schemes, ns laser pulses interact with a sub-critical, long-scale length plasma (corona) and the dominant laser energy absorption mechanisms are inverse bremsstrahlung heating and the resonant absorption. Besides, parametric instabilities such as the acoustic decay, the stimulated Brillouin scattering, the two-plasmon-decay and the stimulated Raman scattering may be an origin of strong electrostatic fields. The Landau damping of such electrostatic plasma waves may be responsible for fast electron acceleration up to 100s of keV. For shorter laser pulses of several 10s or 100s of fs in the HILPI regime, the plasma has no time to expand so the laser pulse interacts with a solid steep gradient density. In this case, various absorption mechanisms may operate depending on the plasma temperature and density conditions :

the collisional absorption, the normal skin effect, the sheath inverse bremsstrahlung and the anormal skin effect. In the context of Shock Ignition, the laser spike has a duration of several 100s of ps and interacts with a hot and large-scale expanding corona. As a consequence, this changes considerably the conditions of excitation of parametric instabilities and a large amount of fast electrons is expected. This regime is still under investigation through extensive Particle-In-Cell simulations (briefly introduced in section 1.3) [START_REF] Klimo | Twodimensional simulations of laser-plasma interaction and hot electron generation in the context of shock-ignition research[END_REF]. Ultra-High Intensity Laser-Plasma interaction (UHILPI) has been studied since the 1990s, after the development of the chirped pulse amplification technique. It is an intense field of research with several unresolved questions. The details of absorption processes are still not well understood and new explanations for the break of adiabaticity in the laser-induced electron motion have been recently proposed [START_REF] May | Mechanism of generating fast electrons by an intense laser at a steep overdense interface[END_REF]] [Mulser et al., 2012] [Sanz et al., 2012]. The complexity of the laser-solid interaction due to strong nonlinearities and various competing processes summarize the obtained results as follows. While the particle acceleration is always due to the electric field work, what differentiates the main mechanisms at play is the source of the electric field. In the case of an obliquely incident laser pulse onto a steep gradient target, the standing wave structure consisting of the incident and reflected waves combined with the electrostatic field is the source of the vacuum heating. In the case of a normally incident laser pulse onto a steep gradient plasma, it is the longitudinal component of the Lorentz force, along with the electrostatic field, which is responsible for the j × B heating. In both cases, an electrostatic field induced by the plasma surface oscillation is responsible for the anharmonic resonant absorption of the laser energy. Actually, due to hole boring of the target, the vacuum heating operates also in the case of a normally incident UHI laser pulse. The electron acceleration depends also on the laser pulse contrast (I L,max /I L,min ). A strong laser prepulse may ionize the surface, causing the plasma to expand. Then, the main laser pulse interacts with a large-scale expanding plasma. In this case, the Raman backward scattered laser light or the reflected laser pulse coupled with the incident laser pulse may lead to stochastic heating of the electrons. Due to a relatively long laser pulse duration (≈ 10 ps) in the Fast Ignition scheme, the fast electron acceleration mechanisms may change with time and extensive particle-in-cell simulations of the LPI are needed. The state of the art of UHILPI in the context of Fast Ignition can be found in [START_REF] Kemp | Laser plasma interactions for fast ignition[END_REF].

The physical processes described in sections 1.2.3, 1.2.4, 1.2.5 and 1.2.6 allow us to make some conclusions concerning the properties of laser-generated relativistic electron beam Properties.

Firstly, it can be shown, by using the electron momentum conservation in the plane perpendicular to the target normal that, in the case of a laser plane wave, obliquely incident on a steep gradient density profile, the angle between the propagation direction of a forward accelerated electron and the target normal reads [START_REF] Sheng | Angular Distributions of Fast Electrons, Ions, and Bremsstrahlung x/ γ Rays in Intense Laser Interaction with Solid Targets[END_REF]]

θ 0 = arctan 2 (γ -1) (1 + δΦ) -δΦ 2 (γ -1 -δΦ) 2 sin -2 θ + tan -2 θ -1/2
(1.69)

where δΦ = [eΦ(z, t) -eΦ 0 ]/m e c 2 is the variation of the electrostatic potential and θ is the laser incidence angle.

Concerning the time dependence of the forward accelerated electrons, one can assume that, at the target surface z = 0, the temporal envelope of the electron distribution follows the laser Gaussian shape of duration ∆t FWHM:

f z (z -v 0 t) = 1 2π (v 0 ∆t) 2 8 ln 2 exp -4 ln 2 z -v 0 t v 0 ∆t 2 .
(1.70)

Here, the internal temporal structure of the electron bunches with the modulation at ω or 2ω has been omitted since it is usually not resolved in fast electron transport hybrid simulations (see Chapter 5).

v 0 is the mean velocity associated with the energy flux density of the forward accelerated electrons in the mean propagation z-direction such that the fast electron beam intensity

I b = n b ε v 0 = η abs I L
where η abs is the laser-to-electrons conversion efficiency, I L the laser pulse intensity, n b the fast electron beam density and ε their mean kinetic energy. As explained in Chapter 2, the fast electron beam generates a counterpropagating return current in order to cancel the total electrical current. As explained in Chapter 3, this system of two counterpropagating currents may be unstable, leading to the generation of small-scale magnetic fields with amplitudes comparable to the laser magnetic field. Such self-generated magnetic fields can strongly deflect the electrons [START_REF] Adam | Dispersion and Transport of Energetic Particles due to the Interaction of Intense Laser Pulses with Overdense Plasmas[END_REF]] [Pérez et al., 2013] thus producing a divergent beam. Another source of the electron beam divergence is the curvature of the electron acceleration region due to hole boring effects [Schmitz et al., 2012b]. The dependence of the electron beam divergence on the intensity can be described by the following empirical scaling law

θ 1/2 ≈ 15 o + 30 o log 10 I L 10 18 W.cm -2
(1.71)

where θ 1/2 is the cone apex angle. This scaling was deduced from experimental data by [START_REF] Green | Effect of Laser Intensity on Fast Electron Beam Divergence in Solid-Density Plasmas[END_REF]. Since the more energetic electrons are less deviated by these strong stochastic magnetic fields, the divergence angle θ 1/2 depends also on the electron energy. Moreover, due to the laser transverse ponderomotive force and propagation effects in the preplasma, the divergence angle of the accelerated electrons increases with the radial distance [START_REF] Debayle | Divergence of laser driven relativistic electron beams[END_REF]. The resulting fast electron angular distribution can be approximated with the following form where α 0 = 2 or 4, θ 0 (r, ε, t) is the mean angle of electron emission which increases with the radial distance r from the beam axis and depends on the electron kinetic energy ε and the time t according to (1.69). ∆θ(ε, t) is the dispersion angle which decreases with increasing electron kinetic energy. θ e is the half apex angle of the cone oriented in the direction defined by θ 0 .

f θ (θ e ) = 1
Experiments and simulations show that the laser-produced electron beam has approximately an axisymetric Gaussian radial distribution of the form

f r (r) = 1 2π ∆r 2 8 ln 2 exp -4 ln 2 r ∆r 2 (1.73)
in the case of a normally incident laser pulse. here, r = x 2 + y 2 is the distance from the beam axis and the beam radius FWHM ∆r is two or three times the laser pulse radial FWHM. In the case of oblique incidence, in the (z, x) plane, the spatial distribution of the accelerated electrons can be 1.4. BRIEF SUMMARY OF LASER SOLID INTERATION AND LASER-GENERATED RELATIVISITIC ELECTRON BEAM PROPERTIES written as

f x (x) = 1 2π ∆x 2 8 ln 2 exp -4 ln 2 x -x 0 ∆x 2 (1.74)
where x 0 and ∆x may depend on time. The distribution in the perpendicular direction f y can be written similarly by replacing x by y and x 0 by y 0 in the previous Equation. Concerning the energy spectrum of the forward accelerated electrons, several interpolations have been proposed. There is a simple 1D exponential distribution :

f ε (ε) = 1 k B T b exp - ε k B T b (1.75)
where T b can be estimated according to (1.47), (1.50) or with other scalings. Other proposed interpolations such as a combination of two exponential functions

f ε (ε) = α 1 k B T b exp ε α 2 k B T b + α 3 k B T b exp ε α 4 k B T b , (1.76) a power law f ε (ε) = α 5 ε k B T b α 6 (1.77) or f ε (ε) = α 7 k B T b exp ε α 8 k B T b + α 9 ε exp ε α 1 0k B T b , (1.78)
where {α i } and T b are parameters found from the fitting of experimental or simulation results. The total number of fast electrons can be written

N 0 = η abs E L ε (1.79)
where ε is the mean electron kinetic energy and η abs means the laser energy conversion into the forward accelerated fast electrons kinetic energy. By collecting available experimental and numerical results, two interpolations were proposed for the time integrated laser energy absorption coefficient [Davies, 2009] η abs = I L λ 2 3.37 10 20 W.cm -2 .µm (1.80)

Note that the instantaneous laser energy absorption is expected to depend on time. The distribution function of the forward accelerated electron beam at the Laser-Solid Interface z = 0 can be written as follows

Ψ b (x, y, z = 0, ε, θ, ϕ, t) = N 0 f x (x)f y (y)f z (-v 0 t)f ε (ε)f θ (θ e (θ, ϕ)) . (1.81)
It is the number of fast electrons emitted from the Laser Plasma Interaction zone per unit of volume As described in the previous chapter, the interaction of (ultra) high-intensity short-pulse lasers with solid targets of density n e generates large numbers of energetic electrons of density n b n e .

d 3 r = dxdydz,
These fast electrons can only penetrate into the solid if the solid can supply an equivalent charge that electrically neutralize the fast electron beam. This electrical neutralization is done transversally to the beam propagation direction in the case of solid conductors and longitudinally in the case of solid insulators. While this electrostatically induced "return current" j e tends to compensate exactly the fast electron beam current j b for insulators, the magnetic neutralization of the fast electron beam occurs longitudinally for conductors. In the latter case, this is the magnetic neutralization of the beam that induces the generation of the return current j e which tends to exactly compensate the fast electron beam j b in agreement with the Lenz law. It allows the beam to propagate through the solid, overpassing the Alfvén-Lawson limit. This chapter is dedicated to describing these electromagnetic neutralization processes. Assuming a monoenergetic, collimated and homogeneous semi-infinite electron beam for simplicity, the self-consistent electromagnetic fields are derived in order to highlight the need of these electromagnetic neutralization processes allowing the beam to propagate. Assuming the beam to be rigid, its electric neutralization is derived in both cases highlighting the difference between the insulator and conductor cases. Then, the magnetic neutralization of the fast electron beams is derived in the case of laser-irradiated solid conductors. one has the relations for each event (r, t) and for each beam electron of momentum p and velocity v.

             t = γ b t - v b c 2 z r = r θ = θ z = γ b (z -v b t) and              γ = γ b γ 1 - vv b c 2 p r = p r p θ = p θ p z = γ b γm e (v z -v b )
where 2 is the beam Lorentz factor, c is the velocity of light and m e is the electron mass. In order to estimate the equilibrium (if it exists!) between the beam and the electric field it generates in the beam rest frame, one has to solve the Vlasov equation

γ b = 1/ 1 -(v b /c)
∂f b ∂t + ∂ ∂r . v f b - ∂ ∂p . e E + v c × B f b = 0 (2.1)
coupled with the Maxwell equations

∂ 2 Φ ∂r 2 = 4πen b (2.2) and ∂ 2 A ∂r 2 - 1 c 2 ∂ 2 A ∂t 2 = - 4π c 2 j b + 1 c 2 ∂ ∂t ∂Φ ∂r (2.3)
where f b (r , p , t ) = f b (r , p , t ) is the electron beam distribution function in the beam rest frame (see Appendix A, section A.1.1). Also, it has been noted Φ and A the electrostatic and vector potential

       H = γ m e c 2 -eΦ P z = p z -eA z p θ = γ m e x v y -y v x .
(2.4)

Any function of these constants of motion is a solution of the Vlasov equation (2.1). Under our assumptions of a semi-infinite, homogeneous, axisymmetric, monoenergetic and collimated electron beam, the distribution function reads consequently

f b (r , p , t ) = n b0 Π r b -r Π -z 1 2πm e δ H -m e c 2 δ p z -eA z (2.5)
where it has been noted Π the Heaviside distribution and δ the Dirac distribution. Starting from this distribution function, one can obtain the beam density in the beam rest frame by integrating it over the whole momentum space. It reads

n b (r , z , t ) = ∞ -∞ p r dp r ∞ -∞ dp z 2π 0 dθf b (r , p , t ) = n b0 Π r b -r Π -z 1 m e ∞ -∞ p r dp r ∞ -∞ dp z δ H -m e c 2 δ p z -eA z = n b0 Π r b -r Π -z 1 m e ∞ -∞ p r dp r δ   m e c 2 ( 1 + p r m e c 2 + eA z m e c 2 -1) -eΦ   = n b0 Π r b -r Π -z m e c 2 ∞ 1 γ dγ δ γ -1 m e c 2 -eΦ = n b0 Π r b -r Π -z 1 + eΦ m e c 2 (2.6)
Injecting this expression in (2.2) using E = -(∂Φ /∂r ) due to the fact A = 0 (because j b = 0 and Φ does not depend on the time t in the beam rest frame), one gets the following equation for the electrostatic potential

∂ 2 Φ ∂r 2 + 1 r ∂Φ ∂r + ∂ 2 Φ ∂z 2 - 1 λ b 2 Π r b -r Π -z Φ = 4πen b0 Π r b -r Π -z (
E(r → 0, z ≤ v b t, t) = E(r → 0, z, t ≥ z/v b ) = E (r → 0, z > 0, t ) = E 0 exp γ b (z -v b t) λ b e z
(2.9)

where E 0 = -4πen b0 λ b . It means physically that each beam slice of thickness ≈ λ b generates a longitudinal electric field which is screened by the electromagnetic counter-reaction of its closest neighboring slices of same thickness so that the slices are screened one by one by each other except for the last slices close to the beam-front-vacuum interface at z = 0. Thus, one can roughly estimate the longitudinal self-electric field generated at the beam front on the z -axis by approximating the beam as a disk of density of effective charge per surface unit σ b = -2n b0 eλ b located at z = 0. According to the Gauss theorem, one gets in this case

E (r → 0, z > 0, t ) ≈ 2πσ b   1 - z r b 2 + z 2   e z (2.10)
and consequently

E(r → 0, z > v b t, t) = E(r → 0, z, t < z/v b ) ≈ E 0   1 - γ b (z -v b t) r b 2 + γ b 2 (z -v b t) 2   e z .
(2.11)

The factor 2 in the expression of the density of charge per surface unit σ b has been chosen in order to respect the continuity of the electric field at z = 0 (z = v b t). Also, far away from the beam front in the limit z → -∞, one can neglect the longitudinal spatial variations of the electrostatic potential compared to the radial ones. The resulting equation has already been obtained and solved by [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] while considering an infinite electron beam. This solution which respects the continuity conditions at r = r b and the boundary condition

Φ (r = 0, z → -∞, t ) = 0 is Φ (r , z → -∞, t ) =        - m e c 2 e 1 -I 0 r λ b if r ≤ r b - m e c 2 e 1 -I 0 r b λ b + m e c 2 e r b λ b I 1 r b λ b ln r r b if r > r b (2.12)
where it has been noted I ν the modified Bessel functions of the first kind. By applying the Lorentz transformation, one gets [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF]] and consequently .14) In the case where r b λ b , which implies r b λ b , one recovers the well known electric field generated by a homogeneous, infinite and cylindrical rigid electron beam which can be obtained according to the Gauss law :

Φ(r, z → -∞, t) = Φ(r, z, t → ∞) = γ b Φ (r , z → -∞, t ) -v b A z (r , z → -∞, t ) =        - γ b m e c 2 e 1 -I 0 r λ b if r ≤ r b - γ b m e c 2 e 1 -I 0 r b λ b + γ b m e c 2 e r b λ b I 1 r λ b ln r r b if r > r b (2.13)
E(r, z → -∞, t) = E(r, z, t → ∞) =        γ b E 0 I 1 r λ b e r if r ≤ r b γ b E 0 r b r I 1 r b λ b e r if r > r b . ( 2
E ∞ =      -4πen b0 r 2 e r if r ≤ r b -4πen b0 r 2 b 2r e r if r > r b
(2.15) (I 1 (x) ∼ x/2 when x → 0). The term "rigid" means that it is assumed that the beam electrons are not affected by the electromagnetic fields they generate as well as by external electromagnetic fields.

Figure 2.2: Artist's view taken from [START_REF] Macchi | Ion acceleration by superintense laser plasma interaction[END_REF] showing a laser-generated electron beam trying to get out a solid target and the resulting electron cloud and accelerated ions.

We are more interested here in the opposite case where r b λ b since in relativistic laser-solid interaction experiments, the laser-generated electron beam has a radius r b ≈ 10 µm while n b0 is necessarily less than the critical electron density n c ≈ 10 21 cm -3 for a laser pulse of a 1 µm wavelength (cf. Chapter 1). Thus, λ b is typically less than λ c ≈ 0.2 µm. Moreover, in such experiments, the Lorentz factor γ b is typically less than 10 so that one can consider r b λ b . In this case, the solution (2.13) is not physical because the linear density of electric energy generated by the beam is greater than the linear density of the beam energy itself (I 1 (x) ∼ exp (x)/ √ 2πx when x → ∞). This is, of course, physically impossible so that an electron beam cannot propagate in vacuum without being electrically neutralized by a background media or by an accompanying positively charged beam. In the case where there is no electric neutralization, the transport is necessarily inhibited. This is exactly what happens when a laser-generated electron beam tries to get out in vacuum from the solid target rear side (see Figure 2.2). When the beam electrons are escaping in vacuum, they create a spatial charge separation which prevents the electrons with energies smaller than the target potential from escaping the target. While the major part of beam electrons reflux inside the target, those with energies higher than the surface potential can escape, thus creating a net positive charge at the surface which is responsible for the generation of strong Electromagnetic pulses [START_REF] Dubois | Target charging in short pulse laser plasma experiments[END_REF].

Also, this strong electrostatic potential is responsible for the acceleration of light ions originated from impurities, i.e., thin layers of water or hydrocarbons which are ordinarily present on solid surfaces under standard experimental conditions. Such a positively charged ion beam accompanied by electrons is commonly detected in laser-solid interaction experiments. A large number of theoretical and experimental studies of accelerated ions have been published recently because of their interesting properties such as ultrashort duration, high brilliance and low emittance comparable with ion beams generated by a classical accelerator [START_REF] Macchi | Ion acceleration by superintense laser plasma interaction[END_REF]. Let us consider the propagation in a conducting media of the semi-infinite electron beam studied in the previous subsection. Such a conducting media can be a plasma or a metal. Due to their huge inertia compared to the electrons and the time scale considered here, the ions (lattice) can be considered as immobile. For simplicity, one assumes that the plasma (metal) is infinite, homogeneous and one notes n i the fixed ion density (or the density of atoms for metals), n e the density, j e the current density and f e the distribution function of the conducting electrons of the media where the electron beam is propagating through. For metals, these electrons are those of the conduction band ; n e0 = Z c n i0 with for example Z c = 3 for Aluminum, Z c = 1 for Copper, ... For plasmas, they are the ionized electrons: equilibrium of the beam electrons with the self-generated electric field is physically impossible. One assumes so for simplicity that the electron beam is rigid in order to determine the electric response to the beam propagation in the conducting media. Also, one neglects the collisions of the beam electrons with the atoms, electrons or ions of the media which will be discussed in detail later on.

Electric

n e0 = Z * n i0 at t =
Thus, instead of solving Equation (2.1) for the beam electrons, one has to determine the conducting electrons dynamics. This can be done by solving the Vlasov-Fokker-Planck equation in the BGK approximation for the conducting electrons (cf. Appendix A, section A.3.1)

∂f e ∂t + ∂ ∂r . v f e - ∂ ∂p . e E + v c × B f e = -ν f e -f M - δn e n e0 f M (2.16)
coupled to the Maxwell equations

∂ ∂r .E = -4πe n b + n e -Z * n i (2.17)
(for metals, Z * must be replaced by Z c ),

∂ ∂r × E = - 1 c ∂B ∂t , (2.18)
and

∂ ∂r × B = 4π c j b + j e + j i + 1 c ∂E ∂t (2.19)
where the term

δn e = R 3 f e -f M
f M (r , p , t ) = γ 2 b f M (r, p, t) = γ 2 b n e0 (2πm e k B T e ) 3/2 exp - p x 2 + p y 2 2m e k B T e -γ 2 b p z + γ b m e v b 2 2m e k B T e
where k B T e m e c 2 is the conducting electron temperature in the laboratory frame. For simplicity, one assumes that ν does not depend on the velocity of particles and that it is equal to the conducting electron-ion collision frequency in the beam rest frame ν ei . Also, in order to ensure the Lorentzinvariance of Equation (2.16), one has ν ei = ν ei /γ b where ν ei is the non-relativistic electron-ion collision frequency in the laboratory frame (see Appendix A, section A. where the ξ 0 are the values at t = 0 without the perturbation induced by the presence of the beam.

One has consequently

f e0 = f M , n e0 = R 3 f M d 3 p = γ b n e0
in agreement with the Lorentz transformation of the charge/current density quadrivector,

p e0 = 1 n e0 R 3 p f M d 3 p = -γ b m e v b which implies v e0 = -v b and j e0 = γ b n e0 ev b .
The unperturbed plasma is initially quasineutral so that Z * n i0 = n e0 , j i0 = -j e0 and consequently where δj e = -n e0 eδv e + δn e ev b . In the 1960's, advances in the production of high-current beams of relativistic electrons using a long coaxial capacitor [START_REF] Graybill | OBSERVATIONS OF MAGNET-ICALLY SELF FOCUSING ELECTRON STREAMS[END_REF], [START_REF] Roberts | The pinch effect in pulsed streams at relativistic energies[END_REF] stimulated experimental [START_REF] Andrews | ON THE PROPAGATION OF HIGH CURRENT BEAMS OF RELATIVISTIC ELECTRONS IN GASES[END_REF] and theoretical [START_REF] Cox | Reverse Current Induced by Injection of a Relativistic Electron Beam into a Pinched Plasma[END_REF] research. In this context, [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] and [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF] under some assumptions we are going to explain. The authors used the Laplace-Fourier method (Laplace for t and Fourier for r ) in order to replace all spatial and temporal derivatives by simple multiplications. Besides, interested in the behaviour of such beam-plasma system after the initial transients, i.e., far away from the beam front in the laboratory frame (z → -∞), they looked for solutions in the limit t → ∞ using the final-value theorem of Laplace transform theory:

E 0 = B 0 = 0.
lim t →∞ δξ(k, t ) = lim s→0 s δξ(k, s)
where δξ is the Fourier transform of δξ and δξ its Laplace-Fourier transform. Thus, due to the fact that the Fourier transform of the beam density is

n b = 2πr b J 1 (k ⊥ ) k ⊥ 0 -∞ dz 0 exp -ik z z 0 ,
where J ν is the Bessel function of the first kind, they found that all these quantities δξ in the limit t → ∞ can be written

δξ(r , t → ∞) = 1 (2π) 3 ∞ 0 k ⊥ dk ⊥ ∞ -∞ dk z lim s→0 s δξ(k, s) = 1 2π ∞ 0 k ⊥ dk ⊥ 0 -∞ dz 0 ∞ -∞ dk z P ξ (k ⊥ , k z )J 1 (k ⊥ r b )J ν ξ (k ⊥ r ) D 0 (k ⊥ , k z ) exp ik z z -z 0
(2.28)

where P ξ (k ⊥ , k z ) are polynomials and ν ξ = 1 or 0 depending on ξ while

D 0 (k ⊥ , k z ) = k z 2 + k ⊥ 2 k z + i ν ei v b + k z λ e 2 k z k z + i ν ei v b - c γ b v b λ e 2 .
(2.29)

Here, λ e = c/ω p = λ e / √ γ b is the plasma skin depth in the beam rest frame and ω p = √ γ b ω p is the Langmuir plasma frequency. In order to perform such k z -integrations in the integrals (2.28) using the residue theorem, one has to determine the zeros k ν for which D 0 (k ⊥ , k ν ) = 0.

The second bracket term of (2.29) is easy to factorize and one finds the poles

k 1 = -i ν ei 2v b + c γ b v b λ e 2 - ν ei 2v b 2 = -i ν ei 2v b + c γ b v b λ e + o ν ei ω p
(2.30) and

k 2 = -i ν ei 2v b - c γ b v b λ e 2 - ν ei 2v b 2 = -i ν ei 2v b - c γ b v b λ e + o ν ei ω p . (2.31)
The first one is more complicated and needs some approximations. [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] have

k z 2 + k ⊥ 2 k z + i ν ei v b + k z λ e 2 = k z k z 2 + k ⊥ 2 + 1 λ e 2 + o ν ei ω p
by considering the weakly-collisional plasma case ω p /ν ei 1. Thus, they found the poles

k 3 ≈ i 1 λ e 1 + λ e k ⊥ 2
(2.32) and

k 4 ≈ -i 1 λ e 1 + λ e k ⊥ 2 . (2.33)
Actually, this is in the laboratory frame where the collisionless plasma condition must be verified so that it is the condition ω p /ν ei 1 which must be considered and not ω p /ν ei 1. Besides, in the original paper of [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF], the authors consider that ω p = ω p and not ω p = ω p / √ γ b . However, this does not change the poles obtained by [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] because ω p /ν ei 1 still implies λ e ν ei /v b 1. k 5 is then found by identification :

k z 2 + k ⊥ 2 k z + i ν ei v b + k z λ e 2 = (k z -k 3 ) (k z -k 4 ) (k z -k 5 )
which gives

k 5 ≈      -i ν ei v b (λ e k ⊥ ) 2 if λ e k ⊥ 1 -i ν ei v b if λ e k ⊥ 1
Page 91

ELECTRIC NEUTRALIZATION OF A MONOENERGETIC, COLLIMATED AND HOMOGENEOUS RIGID RELATIVISTIC ELECTRON BEAM PROPAGATING IN SOLIDS OR DENSE PLASMAS

Thus, [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF]] approximate [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF], one can neglect the pole k 5 . Indeed, concerning laser-generated electron beams, the typical length scale we are interested in is in ranges from the collisional relaxation length

k 5 ≈ -i ν ei v b (λ e k ⊥ ) 2 1 + (λ e k ⊥ ) 2 (2.
l b = v b /ν ei = l b /γ b to the beam length L b ≈ v b τ L = L b /γ b
where τ L is the laser pulse duration. In current experiments, τ L is typically less than 1 -10 ps and is consequently very small compared to the characteristic magnetic diffusion time

τ d = r 2 b λ 2 e ν ei = τ d γ b 3 (2.35)
found by [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF]. Thus, for z -z 0 > 0, the k z -integration can be performed according to the residue theorem along the contour defined by the real axis including the pole k z = k 3 completed by a half-circle in the upper half k z plane such as represented in the left panel of gives [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF]]

                                                       δE r (r , z < 0, t → ∞) = 4πn b er b γ 2 b F 1 (r ) v 2 b c 2 -g c (z ) δE θ (r , z < 0, t → ∞) = 0 δE z (r , z < 0, t → ∞) = -4πn b er b γ b v b ω p r b G 2 (z ) -g s (z )F 2 (r ) δB r (r , z < 0, t → ∞) = 0 δB θ (r , z < 0, t → ∞) = -4πn b er b γ 2 b v b c F 1 (r ) 1 -g c (z ) δB z (r , z < 0, t → ∞) = 0 δn e (r , z < 0, t → ∞) = -n b G 1 (z ) + γ 2 b -1 ω p r b c F 2 (r ) 1 - c v b g c (z ) δv er (r , z < 0, t → ∞) = - n b n e0 γ b ω p r b g s (z )F 1 (r ) δv ez (r , z < 0, t → ∞) = - n b n e0 v b G 1 (z ) - ω p r b c F 2 (r ) 1 - c v b g c (z ) (2.36)
where

g c (z ) = cos ω p z γ b v b exp ν ei z 2v b , g s (z ) = sin ω p z γ b v b exp ν ei z 2v b , G 1 (z ) = 1 -g c (z ) if r ≤ r b 0 if r > r b , G 2 (z ) = g s (z ) if r ≤ r b 0 if r > r b , F 1 (r ) =        I 1 r λ e K 1 r b λ e if r ≤ r b I 1 r b λ e K 1 r λ e if r > r b
and

F 2 (r ) =        I 0 r λ e K 1 r b λ e if r ≤ r b -I 1 r b λ e K 0 r λ e if r > r b .
Here, I ν and K ν are the modified Bessel functions respectively of the first and second kind (There is a mistake in [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF], Eq. ( 110) : it is "+I 1 (r b /λ e )K 1 (r /λ e )" and not "-I 1 (r b /λ e )K 1 (r /λ e )" in the lower line of the second bracket). The discontinuities at r = r b of the functions G 1 and G 2 are due to the discontinious beam current profile and the cold plasma assumption. According to the Lorentz transformations, one can now deduce the electric response to the beam propagation of the conducting electrons far away from the beam front (|z

-v b t| λ e /γ 3/2 b ) in the laboratory frame                      δn e (r, z < v b t, t τ d ) = -n b      1 -cos ω p √ γ b z -v b t v b exp ν ei (z -v b t) 2v b if r ≤ r b 0 if r > r b δv er (r, z < v b t, t τ d ) = -c n b n e0 r b λ e sin ω p √ γ b z -v b t v b exp ν ei (z -v b t) 2v b F 1 (r) δE r (r, z < v b t, t τ d ) = -4πn b er b cos ω p √ γ b z -v b t v b exp ν ei (z -v b t) 2v b F 1 (r)
.

(2.37) Thus, the electric field generated by a relativistic electron beam far away from the beam front (2.15) expels radially a small fraction (n b /n e0 1) of the conducting electrons out of the beam volume.

This electron current generates a radial electric field which counteracts the electric field generated by the beam so that the full radial electric field E r = δE r vanishes. This electric neutralization of the beam occurs within a time scale of few ν -1 ei with oscillations at the plasma frequency ω p = ω p √ γ b evaluated in the beam rest frame. This frequency can become significant for very large values of γ b .

In the opposite case of a collisional plasma where ν ei ω pe , k 1 becomes purely imaginary :

k 1 = -i 1 γ 2 b τ e v b + o ω p ν ei 3 (2.38)
where :

τ e = ν ei /ω p 2 = τ e /
                     δn e (r, z < v b t, t τ d ) = -n b      1 -exp γ b z -v b t τ e if r ≤ r b 0 if r > r b δv er (r, z < v b t, t τ d ) = c n b n e0 r b λ e ω p ν ei √ γ b exp γ b z -v b t τ e F 1 (r) δE r (r, z < v b t, t τ d ) = -4πn b er b exp γ b z -v b t τ e F 1 (r)
.

(2.39) Thus, the electron beam is electrically neutralized in a time scale of τ e /γ b . In agreement with [START_REF] Cox | Reverse Current Induced by Injection of a Relativistic Electron Beam into a Pinched Plasma[END_REF], the oscillations at the plasma frequency ω p have disappeared and the magnitude of the radial velocity is lower because of a greater influence of the electron-ion collisions.

Electric Neutralization of a Monoenergetic, Collimated and Homogeneous Rigid Relativistic Electron Beam Propagating in a Dielectric

The propagation of a semi-infinite electron beam in a non-conducting media is more complicated.

The experiments show that the laser-generated electron beam can propagate deeply in the target while there is no free electrons in dielectrics (Z * = 0) to electrically neutralize the beam. This paradox is solved by noticing that the atoms in a dielectric can be ionized in collisions with the beam electrons (or with the newborn electrons released by ionization), by the self-consistent electric field at the beam front (2.11) or by the electrostatic field induced by the space-charge separation.

Let us consider that the electron beam is generated in the laser-solid interaction zone at z 0 over a small thickness λ s (cf. Chapter 1) and is propagating in a semi-infinite dielectric (z > 0). [Tikhonchuk, 2002] has shown that the ionization of dielectric atoms by the self-consistent electric field at the beam front (2.11) is much more important than their collisional ionization by the impact of beam electrons. Indeed, according to our estimate (2.11) and assuming the typical values of lasergenerated electron beam parameters n b ≈ n c and γ b ≈ 1 -10, the typical value of the self-consistent electric field at the beam front is E 0 = -4πen b0 λ b ≈ -10 12 V.m -1 which represents ≈ 10% of the atomic electric field E Bohr,n = -Z 3 e/n 2 r 2 Bohr ≈ -5(Z 3 /n 2 )10 11 V.m -1 in the Bohr hydrogenoid approximation (for example for Carbon, Z = 6 and n = 2). Thus, the field is sufficiently high to induce a tunelling ionization with the characteristic rate of ≈ 1 fs -1 according to [Keldysh, 1965] while the collisional ionization probability does not exceed ≈ 100 ns -1 according to [Tikhonchuk, 2002].

Besides, still according to the author, the self-consistent electric field is not sufficient to fully explain a deep penetration of laser-generated electron beam in dielectrics as observed in experiments due to the screening of the self-consistent electric field by newborn electrons. The space-charge-separation electric field must so be taken into account. [START_REF] Debayle | Target ionization by a high current relativistic monoenergetic electron beam[END_REF] have developed a quasi-stationary 1D model of the laser-generated electron beam transport through a dielectric material that we are going to present here. In this model, the atoms and new born ions are assumed to be at rest due to their huge inertia compared to the electrons and the small time scale considered here. The new born free electrons dynamics is resolved according to the hydrodynamic equation for the conduction electron density (see Appendix B, section B.1.2)

∂n e ∂t + ∂ ∂z (n e v ez ) = ∂n i ∂t = - ∂n n ∂t = ν E (n n -n i ) + ν en (n n -n i ) -ν rec n i (2.40)
where ν E = ν E (E z ) is the electric field ionization rate, ν en = ν en (n e , T e ) is the collisional ionization rate depending on the conducting electrons density n e and their temperature T e and ν rec is the three-body recombination rate, the hydrodynamic equation for the conduction electron momentum 2.41) where the left hand side takes into account the temporal part of the electron inertia and the hydrodynamic equation for the conduction electron internal energy

∂j e,z ∂t = e 2 m e n e E z -ν ei j e,z (
3 2 ∂ ∂t (n e k B T e ) = j e,z E z -2I p ∂n i ∂t (2.42)
where the second term in the right hand side, depending on the mean ionization potential I p , accounts for the energy losses due to the ionization and finally the Saha's equation .43) based on the detailed equilibrium between the collisional ionization process and the three body recombination. In a 1D model, the magnetic field is neglected while the electric field is estimated according to the Maxwell-Gauss equation

n i n n -n i = ν en ν rec , ( 2
∂E z ∂z = -4πe (n e + n b -n i ) -4π I p E z ∂n i ∂z (2.44)
where the last term in the right hand side accounts for the dielectric polarization P z induced by the field ionization process such that (∂P z /∂z) = (I p /E z )(∂n i /∂z) [START_REF] Debayle | Target ionization by a high current relativistic monoenergetic electron beam[END_REF]. Also, for simplicity, only the first ionization of atoms is taken into account, leading to an ionization state Z * = 1. One can notice that direct collisional ionization of atoms by the beam electrons is neglected here as the pressure of the new born electrons, because their thermal velocity is small compared to their mean velocity. The beam is assumed to be rigid since the ionization losses are relatively small for present day laser-generated electron beam currents. The 1D approximation consists in considering only the beam transport close to the z-axis rejecting the beam radius r b to the infinity. Consequently, the self-consistent electric field at the beam front cancels. One can roughly estimate the space-charge-

MAGNETIC NEUTRALIZATION OF A MONOENERGETIC, COLLIMATED AND HOMOGENEOUS RIGID RELATIVISTIC ELECTRON BEAM PROPAGATING IN A CONDUCTOR

separation electric field on the time scale t ν -1 E by solving (2.44) assuming

       n e z, t ν -1 E = (n e0 -n b ) (Π[z + λ s ] -Π[z]) n i z, t ν -1 E = n i0 (Π[z + λ s ] -Π[z]) n b (z, t) = n b0 (Π[z] -Π[z -v b t])
where n e0 = n i0 is the free electron/ion density of the laser-dielectric interaction zone. This condition accounts for the lack of the forward accelerated electrons in this zone as well as their propagation inside the target. The resulting space-charge separation electric field reads

E z 0 < z < v b t, t ν -1 E ≈ 4πe(2n b0 ) (v b t -z) . ( 2
∂E z ∂t = -4π (j b + j ez ) - ∂P z ∂t . (2.46)
This Ampère-like equation (2.46) shows that the charge-space-separation electric field induces electrostatically a "return current" in the dielectric which tends to cancel the total current. This return current generates an Ohmic electric field according to Equation (2.41). The new born electrons are strongly heated by Joule effect according to Equation (2.42) and participate in the collisional ionization of the dielectric according to Equation (2.40). Finally, the beam current is electrically neutralized in agreement with the the charge conservation equation except close to the beam front where the "return current" induces a lack of free electrons. These time-dependent processes are leading to an additional beam energy loss compared to laser-metal interaction as shown experimentally by [START_REF] Pisani | Experimental evidence of electric inhibition in fast electron penetration and of electric-field-limited fast electron transport in dense matter[END_REF].

Magnetic Neutralization of a Monoenergetic, Collimated and Homogeneous Rigid Relativistic Electron Beam Propagating in a Conductor

Let us consider here a monoenergetic cylindrical and semi-infinite electron beam propagating in a medium assuming that it is already electrically neutralized. In a plasma or a conducting metal, the beam is neutralized electrically by the radial expulsion of background electrons out of the beam volume while in a dielectric material, this is the electrostatically induced "return current" which longtudinally neutralizes the beam. However, we show in this subsection that, even electrically neutralized, a high current of electron beam cannot propagate without its magnetic neutralization. We start from Equations (2.1), (2.3) and the Maxwell-Gauss equation

∂ 2 Φ ∂r 2 = 0 (2.47)
which accounts for the electric beam neutralization. By performing an analysis similar to that of Section 2.1, one can show that, far away from the beam front i.e. for |z| λ b , the beam remains homogeneous n b = n b0 Π(r b -r), the electrostatic potential Φ = 0 and the vector potential

A z r , z → -∞, t =        A 0 1 -I 0 r λ b if r ≤ r b A 0 1 -I 0 r b λ b -A 0 r b λ b I 1 r b λ b ln r λ b if r > r b (2.48)
where A 0 = -γ b β b m e c/e. Consequently, according to the Lorentz transformation of the fields, the magnetic field in the laboratory frame reads

B (r, z → -∞, t) =        A 0 λ b I 1 r λ b e θ if r ≤ r b A 0 λ b I 1 r b λ b r b r e θ if r > r b .
(2.49)

In the case where r b λ b , one retrieves the well known magnetic field generated by a homogeneous, infinite and cylindrical rigid electron beam, which can be obtained according to the Ampère law :

B ∞ =      4π c j b r 2 e θ if r ≤ r b 4π c j b r 2 b 2r e θ if r > r b (2.50) (I 1 (x) ∼ x/2 when x → 0).
In order to explain observations concerning cosmic rays, [Alfvén, 1939] has studied the trajectories of beam electrons in this self-consistent magnetic fields (2.50). Indeed, thanks to the electrical neutralization and to the fact that n b is constant inside the beam, the beam electron kinetic energy is constant. By integrating the beam electrons equation of motion, [Alfvén, 1939] showed that if the beam current I b = j b πr 2 b is sufficently small, their motion is approximatively sinusoïdal as illustrated by the trajectory a in According to [Lawson, 1959], the threshold value I A can be defined as the current for which the beam electron Larmor radius R L in the maximum self-magnetic field, is equal to the half of the beam radius. axis of a uniform and electrically neutralized relativistic electron beam. Solid (dashed) curves represent particle trajectories with net motion forward (inward) taken from [Alfvén, 1939] One obtains according to this criterion the well-known Alfvén-Lawson limit (2.49) is not physical because we obtain that the linear density of magnetic energy generated by the beam is greater than the linear density of the beam energy (I 1 (x) ∼ exp (x)/ √ 2πx when x → ∞). This is of course physically impossible. In order to break this paradox, we will show here that, in addition to the electrical neutralization, an electron beam cannot propagate without being also magnetically neutralized.

I A = -γ b β b m e c 3 e ≈ -17γ b β b kA. ( 2
Let us consider so a rigid relativistic electron beam propagating through a conducting medium (plasma or metal) and let us calculate the magnetic response of the medium to the beam propagation.

The system of equations to be solved {(2.16), (2.17), (2.18), (2.19)} was already introduced in the previous section 2.2.1. According to [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF], assuming n b n e , a weakly collisional (ν ei ω p ) and cold plasma (neglecting the pressure effects) and considering the time scales small compared to the diffusion time τ d , one obtains the solutions (2.36). Then performing the Lorentz-transformations, one finds

                                                                       δE z (r, z < v b t, t τ d ) = -4πn b er b sin ω p √ γ b z -v b t v b exp ν ei (z -v b t) 2v b ×        v b c λ e r b -I 0 r λ e K 1 r b λ e if r ≤ r b I 1 r b λ e K 0 r λ e if r > r b δj e,z (r, z < v b t, t τ d ) = -j b        1 - r b λ e I 0 r λ e K 1 r b λ e if r ≤ r b r b λ e I 1 r b λ e K 0 r λ e if r > r b + j b √ γ b cos ω p √ γ b z -v b t v b exp ν ei (z -v b t) 2v b ×        1 - c v b r b λ b I 0 r λ e K 1 r b λ e if r ≤ r b c v b r b λ b I 1 r b λ e K 0 r λ e if r > r b δB θ (r, z < v b t, t τ d ) = 4π c j b r b        I 1 r λ e K 1 r b λ e if r ≤ r b I 1 r b λ e K 1 r λ e if r > r b . (2.52)
That solution shows that, within the time scale of the beam electrical neutralization (few ν -1 ei ), a temporal variation of the magnetic field generated by the beam induces a longitudinal electric field. This electric field accelerates a return current of plasma electrons δj e,z which tends to cancel the total net current (j b + j e,z = 0) over the time scale ν -1 ei except in the narrow zone at the beam edge r b ± λ e . This is in agreement with the Lenz law which stipulates that the effects of the magnetic field generated by the beam counteract its cause, that is the total net current here. The longitudinal electric field δE z vanishes over the time scale ν -1 ei . Besides, as the two counterpropagating currents do not cancel each other exactly, the magnetic field generated by the return current does not completely cancel the magnetic field generated by the beam. There is a residual magnetic field δB θ localized at the beam edge r b ± λ e . The difference between expressions (2.52) and the original results of [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] comes from the fact that here, the relationship ω p = √ γ b ω p has been taken into account (and not ω p = ω p like [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF]). Also, as pointed out by [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF], the magnetically induced return current and consequenly the residual magnetic field diffuse over the time scale τ d estimated by Equation (2.35) (see also Chapter 3).

According to [START_REF] Cox | Reverse Current Induced by Injection of a Relativistic Electron Beam into a Pinched Plasma[END_REF], in the case of a collisional plasma (ν ei ω pe ), the time scale of the magnetic neutralization is defined by the Coulomb explosion time τ e /γ b . In this case, the resolution of the system of Equations {(2.16), (2.17), (2.18), (2.19)} using the same methodology as

MAGNETIC NEUTRALIZATION OF A MONOENERGETIC, COLLIMATED AND HOMOGENEOUS RIGID RELATIVISTIC ELECTRON BEAM PROPAGATING IN A CONDUCTOR

[ [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF] gives

                                                                       δE z (r, z < v b t, t τ d ) = -4πn b er b exp γ b (z -v b t) τ e ×        v b c λ e r b -I 0 r λ e K 1 r b λ e if r ≤ r b I 1 r b λ e K 0 r λ e if r > r b . δj e,z (r, z < v b t, t τ d ) = -j b        1 - r b λ e I 0 r λ e K 1 r b λ e if r ≤ r b r b λ e I 1 r b λ e K 0 r λ e if r > r b + j b ω p ν ei exp γ b (z -v b t) τ e ×        1 - c v b r b λ e I 0 r λ e K 1 r b λ e if r ≤ r b c v b r b λ e I 1 r b λ e K 0 r λ e if r > r b δB θ (r, z < v b t, t τ d ) = 4π c j b r b        I 1 r λ e K 1 r b λ e if r ≤ r b I 1 r b λ e K 1 r λ e if r > r b . (2.53)
In contrast to the collisionless case treated by [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF], the oscillatory component of the return current and the longitudinal electric field have disappeared here. Also, the magnitude of the time-dependent component of the return current is smaller than in the collisionless case because of greater influence of collisions.

Magnetic neutralization proceeds differently through an insulator. In this case, the "return current" is already generated due to the induced charge-space-electric field. Thus, one cannot separate the electric and magnetic neutralization of the beam but the resulting beam-"return current" system is actually very similar : there is also a residual magnetic field at the beam edge. Thus, in both cases (insulator or conductor), at the end of the beam electromagnetic neutralization, the beam-plasma system consists in a relativistic electron beam, a quasi-opposite counterpropagating return current and immobile ions. Such a system can be extremely unstable. The state of the art of the instablity theory is presented in the next Chapter 3.

Chapter 3

Collective Effects of Relativistic Electron Beam Transport in Solids and

Dense Plasmas

"In relativity, movement is continuous, causally determinate and well defined, while in quantum mechanics it is discontinuous, not causally determinate and not well defined."

David Bohm 101

QUASI-STATIC APPROXIMATION

When propagating through a material, laser-generated fast electron beams are electromagnetically neutralized over a time scale of the background electron-ion collision time ν -1 ei or the background Coulomb explosion time τ e /γ b depending on the temperature and density conditions. In solids and dense plasmas, these time scales are typically of the order of a few fs. The electric field generated by the electron beam expels radially a small part of the background electrons ∼ n b n e out of the beam volume in case of metals or plasmas. In case of insulators, the longitudinal electric field generated by the fast electrons at the beam front initiates the field ionization of the material and accelerates the newborn free electrons. The field ionization is followed by the collisional ionization by these accelerated free electrons in the zone where the space-charge electrostatic field is screened. This electrostatically induced "return current" j e tends to exactly compensate the beam current density j e ≈ -j b . In both cases (insulators or conductors), the resulting electric field cancels at the end of the neutralization process. Concerning metals or plasmas, the temporal variation of the magnetic field generated by the beam induces a longitudinal electric field that accelerates a return current of background free electrons. This magnetically induced return current j e tends also to exactly compensate the beam current density j e ≈ -j b in agreement with the Lenz law. However, in both cases (insulator or conductor), the magnetic neutralization is not perfect. A fine surface around the beam edge of the order of the background skin depth λ e remains non neutralized and a residual magnetic field remains locally. However, this residual magnetic field must be mitigated in case of smoother radial gradients of the beam where [(1/n b )(∂n b /∂r)] -1 λ e . This chapter presents the collective effects, taking place at time scales larger than the beam neutralization time. Indeed, in this PhD studies, we are interested in a time scale ranging from a few fs to a few hundreds of ps since the studied laser-pulse durations τ L are typically of about 10 fs -100 ps.

3.1 Quasi-static Approximation

Background Electrons Dynamics after the Beam Electromagnetic Neutralization

In dense background media such as solids or dense plasmas, the background electrons can be assumed sufficiently collisional and close to equilibrium (Maxwell-Boltzmann or Fermi-Dirac distribution functions) so that they may be modelled by a non-relativistic fluid approach. Such a fluid model for the background requires that departures from collisional equilibrium are small. For example, fluid models break down when electric fields are greater than m e ν ei v Te /e because they may be responsible for the acceleration of runaway electrons that are not taken into account by the hydrodynamic theory. The non-relativistic assumption comes from the fact that the beam density n b is small compared to the background conducting electrons density n e . Thus, the background response to the beam propagation can be considered as a small perturbation and the induced background electron velocities are small compared to v b ≤ c. Also, over the time scale considered here, ions/atoms can be considered immobile because of their huge inertia compared to the electron mass. Thus, the background response to the beam propagation can be described by the hydrodynamic equations

∂n e ∂t + ∂ ∂r .
(n e v e ) = 0 (3.1) and

m e n e ∂ ∂t + v e . ∂ ∂r (v e ) = -n e e E + v e c × B - ∂ ∂r . (P e I -τ e ) + R ei (3.2)
for the background free electrons, only (see Appendix B, section B.1.2). The relativistic electron beam and the background electrons interact via the macroscopic electromagnetic fields E and B.

Direct collisions of background electrons with beam electrons do not appear in Equations ( 3.1) and (3.2). This is due to the fact that, since n b n e , they are negligible compared to collisions with background ions/atoms or background electrons. Here, we focus on the electromagnetic fields; the collisional effects of relativistic electron beam transport are considered in the next Chapter 4. In the time scale considered here, the beam has already been neutralized electrically so that the charge neutrality equation

n e + n b = i Z i n i . (3.3)
replaces the Maxwell-Gauss Equation. The ion charge state Z i depends on the local temperature and properties of the material. n b being small compared to n e , the latter equation is valid on length scales much longer than the Debye length like in the usual hydrodynamic approach. Also, over time scales larger than the Langmuir wave time scale ω -1 p , one can neglect the electron inertia (left hand side term in Equation (3.2)). In this case, oscillations of the laser-generated electron beam at the laser frequency ω L or 2ω L cannot be resolved properly and are not taken into account. Also, the fluid viscosity is neglected and the Braginskii or the Lee-More transport coefficients R ei = en e η.j-k B β.

∂T

E = η.j e - 1 n e e ∂ ∂r (n e k B T e ) (3.4)
commonly called the Ohm's law. Assuming that the cyclotron frequency ω c = |eB|/m e c is small compared to ν ei , the magnetization of the background electrons is neglected and the resistivity tensor is usually taken to be isotropic η = ηI. Also, the thermal force is neglected since, in general, it is small compared to the friction force. The electromagnetic fields are defined by the Maxwell-Faraday

equation ∂B ∂t = -c ∂ ∂r × E (3.5)
and the Maxwell-Ampère equation

∂E ∂t = c ∂ ∂r × B -4π (j e + j b ) = 0 (3.6)
with the initial conditions of zero field divergences (∂/∂r).B = 0 and (∂/∂r).E = 0. The quasi-static approximation consists in neglecting the displacement current (∂E/∂t) in the Maxwell-Ampere equation (3.6). This is fully justified as the background electron inertia is neglected, as one considers time scales larger than ω -1 p and space scales larger than the plasma skin depth λ e . Sometimes, due to the fact that j e ≈ -j b on spatial scales larger than λ e , the Maxwell-Ampère Equation (3.6) is not resolved and j e is directly replaced by -j b in (3.4). The system of Equations (3.6), (3.4) and (3.5) describes the self-generated electromagnetic field

E = -ηj b + ηc 4π ∂ ∂r × B - 1 n e e ∂ ∂r (n e k B T e ) (3.7) and 1 c ∂B ∂t + ∂ ∂r × ηc 4π ∂ ∂r × B = η ∂ ∂r × j b + ∂η ∂r × j b - k B n e e ∂n e ∂r ×
∂T e ∂r .

(3.8)

The second term in the left hand side of Equation (3.8) describes the magnetic field diffusion. One can understand easily now the characteristic time scale (2.35) proposed by [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF]. Indeed, considering space scale of the order of the beam radius r b , the diffusion operator gives the time scale

τ d = 4πr 2 b ηc 2 = r 2 b λ 2 e ν ei
because η = 4πν ei /ω 2 p , by definition. There are three source terms for magnetic field generation in Equation (3.8), depending on the beam current density, the electrical resisitivity and the background electron temperature and density gradients, while the self-generated electric field E is mainly given by -ηj b as already explained. These self-generated electromagnetic fields play an important role in the relativisitic electron beam transport. The magnetic field due to the curl of the beam current density tends to pinch the relativistic electron beam, the magnetic field due to the resistivity gradients tends to move the relativistic electrons from low electrical resistivity zones to higher ones, while the resistive electric field slows down the relativistic electrons [START_REF] Davies | Shortpulse high-intensity laser-generated fast electron transport into thick solid targets[END_REF]. The magnetic field generated by the temperature-density crossed gradients in (3.8) may modify the beam transport on a time scale of a few picoseconds [START_REF] Nicolai | Effect of the plasma-generated magnetic field on relativistic electron transport[END_REF] but on shorter time scales, it can be neglected. Using the same assumptions and methodology as [START_REF] Hammer | Propagation of High Current Relativistic Electron Beams[END_REF], [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF] derived the electron background response to the propagation of a semi-infinite, monoenergetic, cylindrical, collimated and rigid relativistic homogenous electron beam on a time scale large compared to ν -1 ei or τ e /γ b . Following the notations introduced in Chapter 2, [START_REF] Lee | Return Current Induced by a Relativistic Beam Propagating in a Magnetized Plasma[END_REF]] considered only the TRANSPORT IN SOLIDS AND DENSE PLASMAS pole k 5 . Their solution reads in the limit |z -

v b t| λ e n e r, z < v b t, t ν -1 ei or τ e = -n b 1 if r ≤ r b 0 if r > r b (3.9)
for the background electron density and

j e,z r, z < v b t, t ν -1 ei or τ e = -j b r b ∞ 0 dk ⊥ J 0 (k ⊥ r)J 1 (k ⊥ r b ) 1 + (k ⊥ λ e ) 2 exp ν ei (z -v b t) v b (k ⊥ λ e ) 2 1 + (k ⊥ λ e ) 2 ≈ -j b 1 -exp γ 2 b v b τ d 4 (z -v b t)
when r λ e .

(3.10)

for the diffused background return current. This last expression shows that the return current starts from the perfect neutralization j e,z = -j b and then decreases in the time scale of τ d and in a spatial scale of τ d v b .

Electric and Magnetic Fields Radial Profiles

Since laser-generated electron beam lengths are small compared to the diffusion length

v b τ L v b τ d ,
let us consider here time scales shorter than the diffusion time (2.35). Due to the fact that there are many orders of magnitude between ν -1 ei or τ e /γ b and τ d , one has to solve accurately Equations (3.7) and (3.8). In order to estimate the radial profiles of the electric and magnetic field induced by a laser-generated electron beam, [Fill, 2001] assumed a homogenous conducting background with a constant resistivity η 0 , an axisymetric rigid electron beam of the form

j b =      j b0 sin 2 π t -z/v b 2τ L exp - r 2 2r 2 b if z ≤ v b t < z + 2v b τ L 0 else
and the scaling .11) where δr -1 b ∼ ∂/∂r.

τ d τ L r b τ L c 2 , τ d τ L r b δr b 2 , ( 3
Starting from (3.8), he thus obtained for the magnetic field

B θ (r, z, t) = η 0 c t 0 ∂j b ∂r dt = j b0 2πr c τ L τ d exp - r 2 2r 2 b at t = τ L .
(3.12) Indeed, under the assumption (3.11), the diffusion term can be neglected, while the homogenous background assumption cancels all the source terms except the one due to the curl of the beam [Fill, 2001] current density. The background current density is deduced by using the Maxwell-Ampere law (3.6) knowing the magnetic field (3.12). It reads j e,r = j n,r and j e,z = -j b + j n,z (3.13) where the net total current density Since the 1950's with the discovery of the longitudinal electrostatic "two-stream instability" by [START_REF] Bohm | Theory of Plasma Oscillations. B. Excitation and Damping of Oscillations[END_REF], it is known that such a system of two counterpropagating electron beam is unstable. Later, in order to determine the physical mechanism responsible for the purely transverse instability growth found by [Weibel, 1959] and associated with an anisotropic two-temperature Maxwellian plasma , [Fried, 1959] found a second class of instabilities by modelling the Weibel-unstable distribution function by two cold counterpropagating electron beams as illustrated in Figure 3.2. This purely transverse instability is called the "filamentation instability". However, these two designations are often used interchangeably in the litterature (Weibel or filamentation) even if this equivalence holds only for symmetric beams. The longitudinal and transverse instabilities are two limits of a more general instability called the "oblique instability" [START_REF] Bludman | Statistical Mechanics of Relativistic Streams[END_REF]. Let us consider here the propagation in the z-direction of an axisymmetric relativistic electron beam in a dense plasma on the time scale t ≈ τ L such that ν -1 ei or τ e /γ b τ L τ d . In this case, while plasma ions can still be assumed immobile, the electrical and the magnetical neutralization of the beam has already occured and one has

j n = c 4π ∂ ∂r × B (3.14) TRANSPORT IN SOLIDS AND DENSE PLASMAS and consequently                        j n,r = - c 4π ∂B θ ∂z ≈ - 1 4π ∂B θ ∂t ≈ - η 0 c 4π ∂j b ∂r ≈ j b0 r cτ d exp - r 2 2r 2 b at t = τ L j n,z = c 4π 1 r ∂ ∂r (rB θ ) = η 0 c 2 4π t 0 1 r ∂j b ∂r + ∂ 2 j b ∂r 2 dt = j b0 τ L τ d 1 - r 2 2r 2 b exp - r 2 2r 2 b at t = τ L (3.15) assuming v b ≈ c.
n (0) e + n (0) b -Z * n i = 0 and j (0) e + j (0) b = 0 with n (0) b n (0) e .
(3.16)

The supscript " (0) " denotes here the initial equilibrium in charge and in current between the electron beam denoted by the subscript " b " and the background conducting electrons denoted by the subscript " e ". Contrary to the previous Chapter 2, the beam is not considered here as a perturbation but as a part of the equilibrium (0) . One notes n j (r, t)f j (p, t), j ∈ {e, b} the distribution functions in the laboratory frame of both populations. We neglect the collisional effects so that n j f j are solutions of the Vlasov equation with the electromagnetic fields given by the Maxwell equations (see Appendix A, section A.1.1). Also, we neglect the boundary conditions assuming that the perturbation space scale in the z-direction is much smaller than the characteristic size of the system. In order to address relativistic thermal spreads k B T j , we model the initial distribution functions by drifting Maxwell-Juttner distribution functions [Jüttner, 1911] 3.17) where γ = 1 + p 2 /(m e c) 2 , v j0 = β j c the initial drift velocity in the z-directions of species j, γ j = 1/ 1 -β 2 j the corresponding Lorentz factors and K 2 the modified Bessel function of the second kind. The standard method for studying instabilities consists in working in the Fourier's space

f (0) j (p) = k B T j /m e c 2 4πγ j (m e c) 3 K 2 m e c 2 γ j k B T j exp - m e c 2 k B T j γ -β j p z m e c (
∀ξ ∈ {n e , n b , j e , j b , E, B} , ξ(k, ω) = R 3 d 3 r ∞ -∞ dt ξ(r, t) exp (ik.r -iωt)
and consider a small perturbation δ ξ ξ (0) of the initial equilibrium ξ (0) such that ξ = ξ (0) + δ ξ in order to determine eventual temporally exponentially increasing terms exp (δt). Here, δ = Im{ω(k)} and depends on the excitation mode k. Applying this method to our considered equilibrium (0) , one can look for unstable modes, characterized by their linear growth rate δ, by determining a solution of the dispersion relation in the form ω = ω r + iδ where ω r = Re{ω} for which δ > 0. According to [START_REF]Multidimensional electron beam plasma instabilities in the relativistic regime[END_REF], without specifying any favored direction for k, the linearized system of equations consisting in the two Vlasov equations for the two distribution functions n j f j coupled with the Maxwell TRANSPORT IN SOLIDS AND DENSE PLASMAS equations for the electromagnetic fields (E, B) gives the dispersion relation

       xx ω 2 -k 2 c 2 = 0 (a)
or

ω 2 zz -k 2 y c 2 yy ω 2 -k 2 z c 2 -yz ω 2 -k y k z c 2 2 = 0 (b) (3.18)
where Whereas the first factor may yield unstable modes, the remaining dispersion equation zz = 0 defines modes with an electric field aligned with the flow. These are the two-stream modes, which are therefore purely longitudinal. If we now consider wave vectors normal to the flow, with k z = 0, we recover the dispersion equation for the filamentation instability

∀(α, β) ∈ {x, y, z} 2 , αβ (k, ω) = δ αβ + i ω 2 j ω 2 R 3 d 3 p p α γ ∂f (0) j ∂p β + i ω 2 j ω 2 R 3 d 3 p p α p β γ 2 k. ∂f (0) j ∂p m e ω - k.p γ is the dielectric tensor, ω j = 4πn j e 2 /
yy zz ω 2 - k 2 y c 2 ω 2 = yz . (3.20)
Thus, contrary to a common assumption, the filamentation instability is generally not purely transverse. It is purely transverse only in the case where yz = 0 and consequently zz - Finding analytical expressions for the different growth rates δ is difficult due to the presence of the Lorentz factor γ in the integrals defining the dielectric tensor components. However, these triple integrals can be reduced to much more tractable one-dimensional quadratures using a change of variables mentioned in [START_REF] Wright | Relativistic distribution functions and applications to electron beams[END_REF]. It allows to find scaling laws for the maximum of each instability in the case of high γ b and high T b . It reads still according to [START_REF]Multidimensional electron beam plasma instabilities in the relativistic regime[END_REF]]

k 2 y c 2 ω 2 = 0, ( 3
δ max F ∝ n b n e 3/2 γ b -1/2 T b -3/2 , δ max O ∝ n b n e γ b -1/3 T b -1 and δ max T S ∝ n b n e γ b T b -1 , ( 3

.22) TRANSPORT IN SOLIDS AND DENSE PLASMAS

respectively for the filamentation instability, the oblique instability and the two-stream instability. In the case where the fast electron beam temperature is sufficiently small

k B T b m e c 2 < 3 2 10/3 n b n e 2/3 γ 1/3 b 1 + γ -2 b 2/3 1 + γ -1 b 2 , (3.23)
the background electrons and the beam electrons can be considered as cold fluids according to [Bret et al., 2010a]. In this particular case, starting from the hydrodynamic equations with the cold approximation for both populations, coupled with the Maxwell-Gauss equation, one obtains the system of equations

                 ik z δ E z = -4πe (δ n b + δ n e ) (ω -k z v e0 ) δ n e = k z n (0) e δ v e,z (ω -k z v b0 ) δ n b = k z n (0) b δ v b,z im e (ω -k z v e0 ) δ v e,z = eδ E z iγ 3 b m e (ω -k z v b0 ) δ v b,z = eδ E z (3.24)
for the two-stream instability, assuming that the problem is one-dimensional (k x = k y = 0 and thus, neglecting the magnetic field). The combination of the previous equations provides the dispersion relation

1 - ω 2 e (ω -k z v e0 ) 2 - ω 2 b γ 3 b (ω -k z v b0 ) 2 = 0. (3.25)
The imaginary part of unstable mode δ = Im{ω} > 0 presents a maximum at the wave number k z = ω e /v b0 and it is cut off at higher wave numbers. This maximum can be approximated by

δ max T S (T b → 0) ≈ √ 3ω e 2γ b n (0) b 2n (0) e 1/3 . (3.26)
For the filamentation instability, assuming k z = k y = 0, the hydrodynamic equations in the cold approximation for both populations, coupled with the Maxwell equations gives

                                           ik x δ E x = -4πe (δ n b + δ n e ) ωδ B y = - k x c δ E z ik x δ B y = - 4π c e n (0) e δ v e,z -v e0 δ n e + n (0) b δ v b,z + v b0 δ n b -i ω c δ E z ωδ n e = k x n (0) e δ v e,x ωδ n b = k x n (0) b δ v b,x im e ωδ v e,z = eδ E z im e ωδ v e,x = e δ E x - v e0 c δ B y iγ 3 b m e ωδ v b,z = eδ E z iγ b m e ωδ v b,x = e δ E x + v b0 c δ B y (3.27) In the limit n (0) b n (0)
e , the determinant of this system provides the following dispersion relation

ω 2 -ω 2 e ω 4 -k 2 x c 2 + ω 2 e ω 2 -ω 2 e 1 + γ -1 b v 2 e0 k 2 x -ω 4 e 1 -γ -1 b 2 v 2 e0 k 2 x = 0. (3.28)
According to the equation, the unstable solution δ = Im{ω} > 0 saturates at high wave numbers

ω max = ω e β b n (0) b n (0) e √ 3γ b -1 γ b . (3.29)
In the limit ω ω e , the unstable solution can be written

δ max F (T b → 0) ≈ ω max k x v e0 k 2 x c 2 + ω 2 e .
(3.30)

Non-linear Evolution and Saturation Effects

The instability enters a nonlinear phase when the perturbations δ ξ become of the same order of magnitude as ξ (0) . In this case, the linearization of the equations is no longer valid and some saturation effects occur. The non-linear behaviour of these instabilities can be studied with PIC codes. However, some key aspects of saturation effects can be understood from physical considerations. The twostream instability may give rise to a periodic chain of holes in the electron phase space. This coherent structuring implies that a single mode eventually dominates the unstable spectrum. Oppositely, a broad perturbation spectrum, like for laser-generated electron beam, may cause a quasilinear relaxation of the beam. Scattering in the velocity space of primary unstable waves outside the beam-resonant region limits their growth and the related beam energy loss. The instability saturation comes from the growing waves that are trapping electrons which oscillate and form a vortex in the phase-space.

A simplified description of such electron traping by Landau dumping in strong electron plasma waves has already been presented in Chapter 1, section 2.1.4.

The nonlinear development of the filamentation instability can be understood as follows. Electrons of both beams interact through their microscopic currents. Electrons moving in opposite directions repel each other. Thus, the initial charge and current neutral equilibrium is unstable. The magnetic fields grow due to the rearrangement of beam electrons into spatially separated current filaments, until it becomes sufficiently strong to confine the particles within a filament. The laser-generated beam electrons are therefore strongly compressed while the electrons of the denser return current are expelled. The tenuous beam electrons are channeled into filaments, which are immersed in an almost uniform background return current. Such currents filaments can be remarkably stable according to PIC simulations [START_REF]Multidimensional electron beam plasma instabilities in the relativistic regime[END_REF]. This magnetic trapping was identified as the main mechanism responsible for quenching the initial filamentation growth.

The main collisionless instability concerning laser-generated relativistic electron beam transport

Resistive Filamentation Instability

Collisions of the return current electrons are expected to influence the development of the instabilities in high density, low temperature regions due to the scaling of the background electron-ion collision

frequency ν ei ∝ n i /T 3/2 e
(see Appendix A, section A.3.2). As a consequence, the previous results concerning the different instabilities inherent to the transport of laser-generated relativistic electron beam transport through heated solids or dense plasmas may be affected by these collisions. An analysis of the collisional two-stream instability for Maxwell-Juttner electron distribution functions and using the electron-ion Landau operator (see Appendix A, section A.2.2) has been recently carried out, as illustrated in the left panel of Figure 3.5, according to [START_REF] Robinson | Theory of fast electron transport for fast ignition[END_REF]. In the presence of collisions, the peak growth rate drops from δ T S = 5.3 × 10 -3 to 1.1 × 10 -3 , while the dominant wave number remains approximately the same. If strong enough, collisions may completely stabilize the two-stream mode. The oblique modes are affected by collisions in a similar fashion, exhibiting complete stabilization in the strongly collisional limit [START_REF] Hao | Collisional effects on the oblique instability in relativistic beam-plasma interactions[END_REF]. As first demonstrated by [Molvig, 1975], an opposite phenomenon occurs for the filamentation modes. For a dilute and energetic enough relativistic electron beam, collisions keep it unstable regardless of the transverse temperature. The right panel of Figure 3.5, which is extracted from [START_REF] Fiore | Relativistic effects on the collisionless-collisional transition of the filamentation instability in fast ignition[END_REF] by [START_REF] Robinson | Theory of fast electron transport for fast ignition[END_REF], illustrates this tendency by comparing the wave number dependence of the T b⊥ = 0.5 keV (red). Both plots are taken from [START_REF] Robinson | Theory of fast electron transport for fast ignition[END_REF] collisionless and collisional filamentation growth rates for waterbag distributions (squared distribution functions in the momentum space fitting roughly Maxwell-Juttner distributions but allowing to separate the longitudinal temperature T j,z and the transverse one T j,⊥ ). A BGK collision model (see Appendix A, section A.3.1) is employed with ν = 0.5ω p . While the instability is weakened and confined to smaller wave numbers as the beam transverse temperature is increasing, it is also enhanced in the presence of collisions, especially in the large temperature limit. PIC simulations also confirmed the predicted robustness of the collisional filamentation and the generation of filamentary structures [START_REF] Karmakar | Three-dimensional filamentary structures of a relativistic electron beam in fast ignition plasmas[END_REF]. In this collisional regime, the filamentation instability is called the resistive filamentation instability. Let us derive a simple dispersion relation for this instability. We still assume n e n b and we still neglect collisions of beam electrons with plasma particles so that f b still satisfies the Vlasov equation. Also, we still consider the quasi-neutrality n e -Z * n i = 0 and we neglect the displacement current in the Maxwell-Ampere equation so that the Maxwell equations reduce to

∂ ∂r × B = 4π c j e -e R 3 p γm e f b d 3 p (3.31) and ∂ ∂r × E = - 1 c ∂B ∂t . (3.32)
The background ions are still assumed immobile and the background electron dynamic is described by the Ohm's law (3.4)

E = ηj e (3.33)
for which we retain only the main term. Also, we consider a constant resistivity η = η 0 in space and time. Finally, we still consider that the initial beam distribution function is a drifted Maxwell-Juttner distribution function. The set of equations for the self-generated electromagnetic fields gives

∂B ∂t + ∂ ∂r × η 0 c 2 4π ∂ ∂r × B = η 0 c ∂ ∂r × j b , (3.34) ∂E ∂t + ∂ ∂r × η 0 c 2 4π ∂ ∂r × E = -η 0 ∂j b ∂t (3.35) and consequently            δE z = i 4πω c 2 k x 2 -i 4πω η 0 c 2 δj b,z (a) δE x = -η 0 δj b,x (b)
.

(3.36)

The linearized Vlasov equation reads

δf b = ien (0) b ω     δE + p. δE 1 - k x p x γm e ω k x γm e ω e x     . ∂f b (0) ∂p (3.37)
and gives consequently

δj b,z = - in (0) b e 2 m e ω R 3 p z γ     δE x ∂f b (0) ∂p x + δE z ∂f b (0) ∂p z + k x γm e ω p x δE x + p z δE z 1 - k x p x γm e ω ∂f b (0) ∂p x     d 3 p = -     in (0) b e 2 m e ω R 3 p z γ 1 1 - k x p x γm e ω ∂f b (0) ∂p x d 3 p     δE x -     in (0) b e 2 m e ω R 3 p z γ     ∂f b (0) ∂p z + k x p z γm e ω 1 - k x p x γm e ω ∂f b (0) ∂p x     d 3 p     δE z (3.38)
By neglecting δE x compared to δE z in (3.38), due to the fact that in general the longitudinal beam current density j bz is greater than the transverse component j bx , and injecting (3.38) in (3.36 a), one finally finds the dispersion relation for the resistive filamentation instability 1 -

ω 2 b k x 2 c 2 1 -i 4πω k x 2 η 0 c 2     R 3 p z γ ∂f b (0) ∂p z d 3 p + R 3 p z γ k x p z γm e ω 1 - k x p x γm e ω ∂f b (0) ∂p x d 3 p     = 0. (3.39)
By performing the integrals for a low-temperature Maxwell-Juttner distribution function, the dispersion relation can finally be written with the form [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF]]

-i 4πω η 0 ω 2 b + k x c ω b 2 + 1 γ 3 b + 1 γ b β b β T h,b 2 F 1 β T h,b ω k x c = 0 (3.40)
where F can be used (F (ξ) ∼ ξ -2 when ξ 1), which leads to [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF] 4π

F (ξ) = 1 √ π 1 (u -ξ) 2 exp -u 2 du
η 0 ω b δ ω b 3 + k x c ω b 2 + 1 γ 3 b δ ω b 2 - 1 γ b β b k x c ω b 2 = 0 (3.41)
In the high-k The key ingredient for fast electron transport models are the transport coefficients and the ionization states of the material through which the fast electrons propagate. An ideal (fully ionized, weakly coupled, non degenerate) plasma model is insufficient. At high density, it is important to account for the Fermi-Dirac electron statistics. The exclusion principle causes the electrons to have random momentum even at zero temperature. These quantum effects become significant when the electrons temperature is below the Fermi energy

x limit k x c/ω b 4πβ b /η 0 ω b √ γ b ,
k B T e < E F = 2 2m e 3π 2 n e 2/3 . (3.42)

Ionization state

The ionization state Z * of the material defines the ratio of free and bound electrons in the material.

Based on the Thomas-Fermi model, [More, 1985] has proposed a useful formula for the ionization state as a function of the density ρ = n i m i and electron temperature T e of the material. It reads

Z * = Z * T F T e Z 4/3 , n i m u Z = Z X Z 1 + X Z + 1 + 2X Z (3.43)
where Z is the atomic number of the matrial, m u is the atomic mass unit and

T 0 = T e [eV ]/Z 4/3 , R = n i m u /Z, T f = T 0 1 + T 0 , A Z = 0.003323 T 0.9718 0 + 9.26148 10 -5 T 3.10165 0 , B Z = -exp -1.763 + 1.43175 T f + 0.31546 T 7 f , C Z = -0.366667 T f + 0.983333, Q 1 = A Z R B Z , Q = R C Z + Q C Z 1 1/C Z and X Z = 14.3139 Q 0.6624 .
This formula is accurate for Z 10 and it badly estimates the ionization state at a low temperature.

Indeed, the Thomas-Fermi model is inadequate for capturing the metal-insulator transition since it neglects any atomic structure effects on the ionization equilibrium and thus gives a too high ionization level at low temperatures. Therefore, [Desjarlais, 2001] proposed a weighted blend of Thomas-Fermi and a single ionization Saha model with a pressure ionization correction. This model provides smooth transitions between the Thomas-Fermi model and the non-ideal Saha limit. The Saha contribution is determined from

f e = 1 2 K 2 + 4K -K where K = 2 g 1 g 0 1 n i 2πm e k B T e 2 3/2 exp    - I k B T e   1 - 1.5e 2 I(3/4πn i ) 1/3 3/2      .
The statistical weights g 0 and g 1 correspond to the ground state of the neutral atom and singly ionized ion respectively, I is the first ionization energy, n i is the total neutral plus ion number density. For both Cu and Al, g 0 = 2 and g 1 = 1, but in general the level degeneracy g = 2J + 1 is used. The second term in parentheses in the exponential gives a semi-empirical pressure ionization correction.

The ionization state is finally given by (3.44) according to [Desjarlais, 2001].

Z * = f 2/Z * T F 2 e Z * T F + 1 -f 2/Z * T F 2 e Z * T F f e ,

Electrical resistivity

The electrical resistivity is a key parameter for relativistic electron beam transport. According to the Drude model, it can be written as

η = m e ν e n e e 2 (3.45)
where ν e is the background electron relaxation rate. The Lee-More model [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] is usually used to estimate the relaxation rate. It takes into account the electron degeneracy by using the Fermi-Dirac distribution (see Appendix B, section B.2.4 ) that covers the domain of temperatures from k B T ≈ E F to the hot Spitzer regime [START_REF] Cohen | The Electrical Conductivity of an Ionized Gas[END_REF]Härm, 1953]. For lower temperatures in the solid and liquid phase, a different electron collision time is used. It is evaluated

according to τ = 1 ν e = max {τ ec , τ melt , τ min } (3.46)
where the electron collision frequency τ -1 ec = τ -1 ei + τ -1 en accounts for the collisions on ions τ -1 ei with the Lee-More degeneracy corrections and on atoms τ -1 en improved by [Desjarlais, 2001] while τ melt and τ min stem from a Bloch-Gruneisen melting model [Ziman, 1961] and a minimum time based on inter-atom

spacing τ min = (3/4πn i ) 1/3 / (k B T e + E F )/m e .
The melt model gives τ melt = 50(T melt /T )τ min with the material dependant constant 50 decreasing somewhat for T > T melt . In the left panel of Figure 3.7, the electrical resistivity of Beryllium with ρ = 1.84 g.cm -3 taken from [START_REF] Robinson | Theory of fast electron transport for fast ignition[END_REF] is plotted versus the temperature and is also compared with numerical results allowing to determine the free parameters of the Lee-More model. Also, the hot Spitzer resistivity with and without the electron-electron collisions correction factor (see Appendix B, section B.2.2) is plotted with the cold melting model. For Hydrogen plasmas, as suggested by [START_REF] Lambert | On the transport coefficients of hydrogen in the inertial confinement fusion regime a)[END_REF], comparing it with ab initio molecular dynamic computations, the electrical resistivity can be evaluated as η

-2 = η Hubbard -2 + η -2
Spitzer where η Spitzer is the hot temperature Spitzer plasma transport coefficient and η Hubbard is the low temperature transport coefficient found by [Hubbard, 1966]. Comparisons between such expression for Hydrogen plasmas with different densities are plotted in the right panel of Figure

3.7.

The main disadvantage of such models (Lee-More and Hubbard-Spitzer) is that they assume T i = T e = T and they do not account for a different ion/lattice temperature T i than the conducting electron temperature T e . A two-temperature model for the electrical resistivity of Copper and Aluminum has been proposed by [START_REF] Eidmann | Hydrodynamic simulation of subpicosecond laser interaction with solid-density matter[END_REF] and later improved by [START_REF] Chimier | Heating model for metals irradiated by a subpicosecond laser pulse[END_REF]. In this model, the collision frequency of the electrons ν e is evaluated by taking the weighted average of the electron relaxation rate in different temperature regimes

ν -2 e = (ν e-ph + ν e-e ) -2 + ν -2 c + ν -2
Spitzer .

(3.47)

In the low temperature regime, the mean free path is evaluated by v e / (ν e-ph + ν e-e ) where ν e-ph is the electron-phonon collision rate, ν e-e is the electron-electron collision rate and v e = (2E F + T e )/m e is the electron velocity. In the high temperature regime, the mean free path is evaluated by v e /ν Spitzer . In the intermediate range of temperatures, the mean free path is written as v e /ν c where resistivity, a least square fit from the ion-ion coupling parameter Γ table of [Hubbard, 1966] is used to avoid discontinuities. For dielectric materials (insulators), the situation is much more complicated and one usually needs extensive quantum molecular dynamic computations. For example, the left panel of Figure 3.9 shows huge differences between the electrical resistivity given by the Lee-More

ν c = v e /(3/4πn i ) 1/3

Heat Equations

The electrical resistivity of background electrons may change significantly due to the heating induced by the beam energy deposition, the Ohmic heating by the return current and the resistive filamentation instability. This heating needs to be accounted for in fast electron transport models. Under the assumption T e = T i = T , the energy conservation equation reads (see Appendix B, section B.3.1)

C V ∂ ∂t + u. ∂ ∂r (T ) + ∂ ∂r .q = -P ∂ ∂r .u + τ ∂ ∂r ⊗ u + j.E + W (3.48)
where

C V = 3 2 k B (Z * + 1) n i
is the plasma thermal capacity (electrons + ions). According to the assumptions presented in the section 3.1.1 of this chapter, the magnetization and the ion conduction are neglected so that the thermal flux reduces to q = q e = -κ e ∂T e ∂r (3.49) where κ e is the background electron thermal conduction evaluated within the same models as presented in the previous subsection for the electrical resistivity η. These two transport coefficients are indeed related by the Lorenz relation

κ e σT e = γ L (3.50)
where γ L is the Lorenz factor (see Appendix B, section B.2.4). The background fluid is assumed to be incompressible (∂/∂r).u = 0, the viscosity is neglected and the ions are assumed immobile. Thus, the fluid current density reduces to the background electron return current j = j e and the electric field is given by the system of equations {(3.7), (3.8)} presented in section 3.1.1. The direct collisional heating of the background electrons by the beam electrons is taken into account via the heating source term

W = - R 3 ε ∂f b ∂t coll d 3 p (3.51)
which is nothing else than the energy lost by the beam electrons due to collisions with the background electrons (cf. Chapter 4).

In the two-temperature model, which is notably the case for relativistic electron transport in solids, the background electrons are firstly heated due to the beam energy deposition. Then, they transfer their energy to ions. Therefore, Equation (3.48) must be replaced by the two energy conservation equations

C V,e ∂ ∂t + u e . ∂ ∂r (T e ) + ∂ ∂r .q e = W + Q ei . (3.52)
and

C V,i ∂ ∂t + u i . ∂ ∂r (T i ) = Q ie (3.53)
where

Q ie = -Q ei + ηj e 2 = 2 m e m i C V,e ν ei (T e -T i ) . (3.54)
For plasmas, ν ei may be expressed by the [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] formula accounting for the electrons

degeneracy at T e ≈ E f (see Appendix B, section B.2.4) while C V,e = 3 2 k B Z * n i and C V,i = 3 2 k B n i (3.55)
are the plasma electron and ion thermal capacities. However, these last expressions need to be improved for low temperature materials in the solid/liquid phase.

Plasma Heating by an Electron Beam

A simple model have been proposed by [START_REF] Bell | Resistive Collimation of Electron Beams in Laser Produced Plasmas[END_REF]] and [Davies, 2003] in order to account for the effects of the background electron heating. These are the self-collimation of the beam and the beam hollowing, respectively. The model assumes that the return current equals the beam current as explained in section 3.1.1

j b + j e = 0. (3.56)
The background electron dynamic is modelled according to the Ohm's law retaining only the main term

E = ηj e = -ηj b (3.57)
with a resistivity of the form

η = η 0 T T 0 α (3.58)
where T 0 is a characteristic temperature. The background electron heating is taken into account according to Equation (3.52), neglecting the ion heating and the thermal conduction. In addition, [START_REF] Bell | Resistive Collimation of Electron Beams in Laser Produced Plasmas[END_REF]] assumes the plasma is fully ionized and thus estimates the electron thermal capacity according to C V,e = (3/2)k B Zn i . Contrary to [START_REF] Bell | Resistive Collimation of Electron Beams in Laser Produced Plasmas[END_REF]], [Davies, 2003] just assumes a constant thermal capacity C V,e . The heat equation (3.52) thus reduces to

C V,e ∂T e ∂t = ηj 2 b . (3.59)
This equation can be generalized to the case of a metallic target with the solid Sommerfeld thermal capacity C V,e = C V,e0 (T e /T 0 ), replacing in the expressions below the parameter α by α + 1 and C V,e by C V,e0 . The magnetic field is given by the Maxwell-Faraday equation

∂B ∂t = -c ∂ ∂r × E = ηc ∂ ∂r × j b + c ∂η ∂r × j b . (3.60)
The beam is assumed to be rigid, collimated and axisymmetric of the form

j b = j b0 (r) Π t - z v b -Π(t) e z with j b0 = -j 0 exp - r 2 r 2 b (3.61)
where j 0 > 0 and Π is the Heaviside function. It is emitted from z = 0 and moves with the constant velocity v b = v b e z . The Gaussian radial shape corresponds to the shape of the laser pulse. Therefore, the magnetic field is azimuthal B = B θ e θ and the electric field is longitudinal E = E z e z . The beam transport strongly depends on the parameter α. In the case where α > 1 which corresponds to cold solid/liquid temperatures (0 < α < 2 for k B T e < E F according to the resistivity models presented in section 3.3.1), one finds diverging solutions for the temperature (T e → ∞) at very short times

t -(z/v b ) = C V,e T 0 /(α -1)η 0 j 2 b0 .
However, this unphysical behaviour must be mitigated because the Page 123

TEMPERATURE DEPENDENCE OF FAST ELECTRONS TRANSPORT

temperature growth for α is actually limited by the Fermi temperature k B T e ≈ E F since that, at this temperature, the parameter α becomes negative and then reach the value α = -3/2 of the hot temperature plasma regime. Thus, according to [Davies, 2003], (3.59) gives

T =          T 0 exp η 0 j 2 b0 τ C V,e T 0 if α = 1 T 0 1 + (1 -α) η 0 j 2 b0 τ C V,e T 0 1/(1-α) if α < 1 (3.62)
where it has been noted τ = t -(z/v b ). Knowing the temperature, one can express the electric field according to Ohm's law

E z =          -η 0 j b0 exp η 0 j 2 b0 τ C V,e T 0 if α = 1 -η 0 j b0 1 + (1 -α) η 0 j 2 b0 τ C V,e T 0 α/(1-α) if α < 1 (3.63)
And, the magnetic field follows from the Maxwell-Faraday equation by making the change of variables

t → τ B θ =        -c dj b0 dr C V,e T 0 j 2 b0 1 + 2η 0 j 2 b0 τ C V,e T 0 -1 T T 0 if α = 1 -c dj b0 dr C V,e T 0 j 2 b0 1 + 1 + α 1 -α T T 0 - 2 1 -α η η 0 if α < 1 . (3.64)
Self-collimation of the beam In the limit of weak heating τ /τ 0 1 where

τ 0 = C V,e T 0 η 0 j 2 b0 , (3.65)
one obtains from (3.63) and (3.64) to first order in τ /τ 0

E z = -η 0 j b0 1 + α τ τ 0 (3.66)
and

B θ = -η 0 c dj b0 dr τ (3.67)
for α ≤ 1. This results is similar to the one obtained by [Fill, 2001], presented in section 3.1.2. The magnetic field is negative and it reaches a minimum value at r = r b / √ 2, as illustrated in Figure 3.10.

This magnetic field may be responsible for the beam pinching. [START_REF] Bell | Resistive Collimation of Electron Beams in Laser Produced Plasmas[END_REF]] estimated the angle θ that a fast electron will be deflected while propagating over the distance

r b / tan θ 1/2 by θ = |eB θ |r b γm e c tan θ 1/2 (3.68)
where the unperturbed beam doubles its radius. here, θ 1/2 is the divergence half-angle of the beam (which has not been taken into account in the previous derivations). The collimation condition can be estimated as θ = θ 1/2 . For example, [START_REF] Bell | Resistive Collimation of Electron Beams in Laser Produced Plasmas[END_REF] Consequently, considering the experimental scaling (1.71) for the divergence half-angle θ 1/2 , we deduce that θ 1/2 is in general too high for allowing the self-collimation of the beam.

Beam hollowing

According to [Davies, 2003], in the opposite limit of a strong heating τ τ 0 , one obtains

E z =        -η 0 j b0 exp η 0 j 2 b0 τ C V,e T 0 if α = 1 -η 1/(1-α) 0 j (1+α)/(1-α) b0 (1 -α)τ C V,e T 0 α/(1-α) if α < 1 (3.70)
and This chapter review the theory of collisions of laser-generated relativistic electrons with particles of the material they are propagating through, such as free electrons, bound electrons, screened free electrons or ions. Due to the fact that the beam electrons travel at a relativistic velocity much greater than background electron or ion velocities, the principal effects of collisions are the beam electrons energy losses and their angular scattering. As the beam electron density is much less than the background electron density n b n e , a single fast particle model provides an adequate desciption of these drag and scattering processes. After describing the differential cross section for collisions of a relativistic electron with a background particule at rest, the following subsections are devoted to the slowing down and angular scattering theories of a relativistic electron. The last subsection presents the relativistic Vlasov-Fokker-Planck equation for the laser-generated electron beam that describes these effects.

B θ ≈ -(1 + α) ηc dj b0 dr τ (3.71)

Binary Collisions of a Relativistic Electron with

Background Particles

Electron-Electron Binary Collisions

Let us consider firstly a collision of a relativistic electron with a momentum p with a target electron α ∈ {free e -, bound e -or free screened e -} with a momentum p α such that

|p α | |p|. (4.1)
Thus, the target electron can be assumed to be initially at rest and one has the following relations between the center of mass frame and the laboratory frame :

                       ε * = γ * m e c 2 = γ + 1 2 m e c 2 (a) p * = γ + 1 2 p (b) sin θ * = γ + 1 2 sin 2θ 1 + γ + 1 2 -1 sin 2 θ (c) (4.2)
where * means that the values are taken in the center of mass frame, γ is the Lorentz factor of the relativistic electron projectile and θ is the scattering angle. Consequently, one can obtain a relation between the normalized energy loss in one binary collision w = ∆γ γ -1 and the scattering angle is the plasma screening length as explained for the non-relativistic case in Appendix A, section A.2.2. Besides, b e,min is directly choosen to be the De Broglie wavelength here. Indeed, in the case of a relativistic electron projectile, the De Broglie wavelength is always much larger than the Landau length. In the relativistic approach considered here, it is more convenient to deal with w instead of b. The equivalent boundaries are therefore w min < w < w max . w max = 1/2 must be taken due to the indiscernability of the electrons. That means that it is the more energetic electron outgoing from the collision which is considered as the primary electron (projectile). w min = w c is a cut-off used to distinguish the binary collisions from the collective interaction. It is assumed that w min = w c is much less than the beam electron energy and much greater than the energy of any electron in the material. One may estimate it by (b e,min /b e,max ) 2 . with an impact parameter b > b e,max for collisions with a small momentum transfer from the relativistic electron to a bound or screened free plasma electron target, which are affected by a collective contribution of the surrounding electrons. The equivalent boundary is w < w c and

θ * w = 1 -cos θ * 2 , ( 4

BINARY COLLISIONS OF A RELATIVISTIC ELECTRON WITH BACKGROUND PARTICLES

this collective part will be treated in the next subsection devoted to the fast electron slowing down.

with an impact parameter b < b e,min for collisions of beam electrons with a large momentum transfer. Due to the fact that the probability of such a collision is small compared to the two previous ones, we neglect this contribution.

For the binary part, one can use the Möller scattering formula [Möller, 1932]. It consists in an approximate solution to the Dirac equation to order αv/c (the first Born approximation) where α is the fine structure constant. Indeed, [Möller, 1932] considered two interacting electrons, described by the two component Dirac wave-functions; He ignored the two components referring to negative energy states. He expressed the charge and current densities corresponding to the transition of the electron 1 from the initial state to the final state thanks to the obtained Dirac matrix tensor. In accordance with a procedure proposed by Klein in 1927, he then expressed the corresponding retarded potentials he obtained thanks to the Maxwell equations with the deduced charge and current densities of the particle 1 as sources. Finally, he solved the Dirac equation for the particle 2 with the presence of these fields and he identified the transition probability for the corresponding two-body system. Although the method was controversial because of its unsymmetrical approach and the lack of electromagnetic fields quantization, the result was readily confirmed in experiments [START_REF] Champion | The Scattering of Fast β-Particles by Electrons[END_REF] and a few years later by more rigorous quantum electrodynamic treatments [Bhabha, 1936], as well explained in [START_REF] Beretstetskii | of A Course of Theoretical Physics[END_REF]. The Moller differential cross section reads

dσ dΩ * e,f = e 4 p * 2 c 2 + ε * 2 2 4p * 4 ε * 2 4 sin 4 θ * - 3 sin 2 θ * + p * 2 c 2 p * 2 c 2 + ε * 2 2 1 + 4 sin 2 θ * . (4.9)
The first term in the square brackets correspond to the relativistic generalization of the Rutherford differential cross section formula (see Appendix A, section A.2.1) while the other terms account for the quantum spin and exchange effects. Usually, as already mentioned, one may use the equivalent and simpler form The quantum relativistic differential cross section of such a binary collision has been obtained by [Mott, 1932]. It reads

dσ dw e,f = 2πe 4 (γ -1) β 2 m 2 e c 4 1 w 2 + 1 (1 -w) 2 + γ -1 γ 2 - 2γ -1 γ 2 w (1 -w) . ( 4
dσ dΩ i = (Z * e 2 ) 2 ε 2 4p 4 sin 4 θ 2 1 - v 2 c 2 sin 2 θ 2 . (4.13)
The first term in the parenthesis is the relativistic generalization of the Rutherford differential cross section and the second term is the quantum spin effect correction. In the first order in the small parameter m e /m i , the electron conserves the energy in a collision with a target ion. It can be compared with a collision of a tennis ball with a rigid wall. The tennis ball does not lose its energy but it is strongly deflected. Instead of working with the maximum impact parameter b α,max when integrating the differential cross section over all impact parameters b, some authors prefer to add a screening factor in the expression of the differential cross sections (4.9) and (4.13) and integrate from b = b α,min to b → ∞ (see for example [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF]).

Slowing Down of a Relativistic Electron in Solids

and Dense Plasmas

Stopping Powers of relativistic Electrons

The stopping power of an electron is the electron energy loss dε per unit path length ds due to collisions with the particles of the medium. It is defined as

dε ds = (γ -1) m e c 2 n i 1/2 0 w dσ dw dw. (4.14)
Collisions with ions do not contribute to the slowing down of electrons due to their large mass compared to the electron mass. One may separate in this integral the contributions of the binary collisions with free electrons (w > w c ) from those of the bound electrons and/or with the screened free electrons (w < w c ).

Concerning the binary part, one can use the Moller differential cross section (4.10). The integration

= - 2πn i Ze 4 m e v 2 ln 1 4w c + 1 - 2γ -1 γ 2 ln 2 + 1 8 γ -1 γ 2 (4.15)
In case of plasmas, the contribution of collisions with background free electrons can be obtained by replacing Z by Z * in the previous Equation ( 4.15) :

dε ds e,f = - 2πn i Z * e 4 m e v 2 ln 1 4w c + 1 - 2γ -1 γ 2 ln 2 + 1 8 γ -1 γ 2 . (4.16)
The first term in the square brackets corresponds to the well known Coulomb logarithm from the non-relativistic theory (see Appendix A, section A.2.3). That is why the term w c may thus be estimated by (b e,min /b e,max ) 2 . This corresponds to the non relativistic stopping power Coulomb logarithm, in the limit γ → 1 [Jackson, 1975]. The other terms account for the quantum relativistic effects.

For a combination of historical and mathematical reasons, the energy loss due to the collective response of the material is artificially divided into two parts

The "Bethe part" for collisions with bound electrons (acounting for the mean excitation energy per atom I ex ) for a impact parameters b smaller than the interatomic/interionic distance. This corresponds to the beam electrons interacting with one atom/ion.

The "density effect" or "plasmon part" for impact parameters b greater than the interatomic/interionic distance for which the relativistic electron projectile interacts with many electrons at the same time.

The contribution of collisions with bound electrons in a cold solid has been determined by [Bethe, 1932], considering the energy transferred to the excitation of an atom by the electric field

I ex /m e c 2 2 -β 2 .
(4.17)

A striking results is that the cut-off w c at which the two models are patched together cancel out for cold solids when adding the two integral contributions (4.15) and (4.17) [START_REF] Rohrlich | Positron-Electron Differences in Energy Loss and Multiple Scattering[END_REF] dε The complexities of dealing with coupled, quantized oscillations of multiple bound electrons are hidden in the mean excitation potential I ex . In very general terms, it can be written as the logarithm mean of all possible transitions of bound electrons between the energy levels E i and E j , weighted by the

transition probability f ij ln I ex = i,j f ij ln (E i -E j ). (4.20)
In the simple case of a single, undamped, harmonic oscillator at frequency ω, one has I ex = ω. This is a good approximation for a plasma, giving the mean excitation I ex = ω p . An approximate model for the mean excitation potential of bound electrons in an ion was proposed by [More, 1985]. In this simplified theoretical model known as the local plasma approximation, ln

I ex = R 3 f e (r) ln ( ω p )d 3 r, (4.21)
where f e is the bound electron probability density function. In order to obtain the electron distribution around an ion, [More, 1985] used the Thomas-Fermi model and found that the result could be described by

I ex (Z * ) = I ex (0) exp 1.29(Z * /Z) 0.72-1.18(Z * /Z) 1 -(Z * /Z) (4.22)
where I ex (0) = ZE 0 with E 0 = 10 eV empirically chosen to fit the quantum calculus made by [McGuire, 1982] for Aluminum, Krypton and Gold.

The second part of the collective contribution to the stopping power is the density effect correction δ. It has been firstly derived by [Fermi, 1940], using a purely classical calculation representing the plasma electrons response to the electron projectile perturbation by a single harmonic oscillator. It gives a reduction in the energy loss due to the electric field generated by the fast electron, shielded by

SLOWING DOWN OF A RELATIVISTIC ELECTRON IN SOLIDS AND DENSE PLASMAS

the collective response of electrons in the material. It is called the density effect correction because it increases with the electron density. According to [Fermi, 1940], it can be obtained analytically in the limit of ultrarelativistic electron projectile and reads in this case

δ = 2 ln γ ω p I ex -β 2 . (4.23)
For a plasma, where I ex = ω p , this expression is valid for all cases of interest. For solids and consequently bound electrons, where typically I ex ω p , the expression obtained by [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF]]

dε ds p = 2πn i Z * e 4 m e v 2 δ = - 4πn i Z * e 4 m e v 2 ln c γ 2 -1/γ ω p b e,max (4.24)
may be used. This formula was originally derived for a non-relativistic electron projectile. However, it is valid in the limit γ → ∞ and for intermediate γ as well.

A relativistic electron can also lose energy by emitting bremsstrahlung radiations. It is a small correction to the total collisional energy losses in the range of beam electron energies considered in this manuscript. However, the radiative stopping power

dε ds b = - 4π(Z -Z * )(Z -Z * + 1)n i e 4 m e c 2 /γ α π ln (2γ) - 1 3 , (4.25)
obtained by [START_REF] Heitler | Stopping of Fast Particles with Emission of Radiation and the Birth of Positive Electrons[END_REF] is taken into account. This formula is valid in the case where 1 γ 1/αZ 1/3 according to [START_REF] Bethe | On the Stopping of Fast Particles and on the Creation of Positive Electrons[END_REF] and it is sufficient for our needs. Indeed, according to [START_REF] Berger | Tables of energy losses and ranges of electrons and positrons[END_REF] there is an uncertainty at low energies due to the lack of shell corrections which are required when the velocity of the incident electron is comparable to the velocities of the atomic electrons, especially those in the inner shells. As a consequence, the formula (4.27) is valid for laser-generated relativistic electrons of kinetic energies greater than ≈ 1 -10 keV.

As illustrated in Figure 4.1, it is worth noticing that, even if the contributions of the bound electrons and the free electrons depend strongly on the temperature via their densities depending on the ionization state Z * , the total stopping power weakly depends on the temperature (logarithmically).

The contribution of the bound electrons at a low temperature is essentially balanced by the contribution of the free electrons at a higher temperature. It can be understood qualitatively because, in both cases, the amount of electrons encountered by the electron projectile is the same. The total stopping power (4.27) is plotted in Figure 4.2 versus the electron kinetic energy (γ -1)m e c 2 for materials that will be studied in this manuscript. One can see a change of slope around ε = m e c 2 ≈ 511 keV. In the non-relativistic side, the stopping power decreases with the non-relativistic kinetic energy

(1/2)m e v 2 .
This is a consequence of the Rutherford cross section which decreases with the transferred momentum. In the relativistic domain, an opposite behavior appears; the stopping power increases with the electron kinetic energy. This is due to the fact that the velocity has reached its maximum value of c and does not change anymore while the electron inertia γm e in the relativistic Coulomb logarithm is increasing so that the kinetic energy exchanged during each collision increases with the kinetic energy.

Moreover, for more energetic electrons, the radiation loss becomes predominant compared to the collisional ones. It is well known that the more energetic the electron is, the more the electron radiates. In report 37 [Brice, 1985]. Discrepancies between Formula (4.27) and ESTAR appear for relativistic electrons with kinetic energies greater than 10 MeV mainly due to the simple expression for the radiation stopping power (4.25). In the ESTAR database, the total collision stopping power is also calculated from the theory by [Bethe, 1932] but with a more accurate density-effect correction evaluated accord- ing to [Sternheimer, 1952] [Sternheimer et al., 1982] and the mean excitation potential I ex adopted in the ICRU Report 37 [Brice, 1985]. The uncertainties of the calculated collision stopping powers for electrons are estimated [Brice, 1985] to be 1 % to 2 % above 100 keV, 2 % to 3 % (in low-Z materials) and 5 % to 10 % (in high-Z materials) in the range between 10 keV and 100 keV. The radiative stopping powers are evaluated in ESTAR with a combination of theoretical bremsstrahlung cross sections described by [START_REF] Seltzer | Bremsstrahlung spectra from electron interactions with screened atomic nuclei and orbital electrons[END_REF]. Analytical formulas (using a high-energy approximation) are used above 50 MeV, and accurate numerical results of [START_REF] Pratt | Bremsstrahlung energy spectra from electrons of kinetic energy 1 keV ≤ {T1} ≤ 2000 keV incident on neutral atoms 2 ≤ Z ≤ 92[END_REF] For an equimolar DT plasma (Z = 1), still according to [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF], one gets by expanding the logarithm of (ln Λ rel ee ) * around ρ = 300 g.cm -3 and by assuming f * = 0

f (γ) = ln γ + 1 (γ -1) γ - ln 2 + (1/8) γ + (1/2) ln 2 + (9/16) γ 2 . ( 4
R [µm] ≈ 23.7 ε 0 [MeV] 2 0.34 + 0.66ε 0 [MeV] 300g.cm -3 ρ 1.066 . (4.33)
However, this formula assumes that the beam electron trajectory is a straight line and consequently overestimates the effective penetration length (along the initial electron velocity) of a relativistic electron into a plasma owing to its angular scattering by colliding the background ions and electrons.

Angular Scattering of a Relativistic Electron in

Solids and Dense Plasmas

In materials with Z 1, the fast electron scattering is dominated by elastic collisions with ions and impact parameters b much greater than the De Broglie wavelength. For low Z materials such as Hydrogen plasmas, angular scattering on target electrons also plays an important role. We may describe the angular scattering of a relativistic electron colliding with target electrons, by following the method used for the estimate of the total stopping power of relativistic electrons, as presented in the

1 sin 4 θ + (γ + 1) 2 4 cos 4 θ + (γ 2 -1)/γ 2 (γ -1) sin 2 θ + 2 2 - (2γ -1)(γ + 1) 2γ 2 sin 2 θ cos 2 θ . (4.35)
According to [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF], although one can retain all terms, it suffices to consider the leading term at small θ corresponding to the relativistic generalization of the Rutherford differential cross section (see Appendix A, section A.2.1)

dσ dΩ e = 4e 4 cos θ (pv) 2 sin 4 θ . (4.36)
Indeed, the scattering at small angle θ gives the dominant contribution to the mean scattering angle θ 2 , as described in the next sections 4.3.1 and 4.3.2. This approximation does not lead to significant errors in the resulting range of laser-generated electron's energies.

Multiple Scattering Theory by Lewis

We start here from the simpler multiple scattering theory by [Lewis, 1950] and we restrict this subsection to fully ionized plasmas as done by [START_REF] Solodov | Stopping power and range of energetic electrons in dense plasmas of fastignition fusion targets[END_REF] and [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF] avoiding collisions with θ < θ c . In the Lewis theory, the mean scattering angle can be evaluated according to

cos θ = exp (-k 1 s) (4.37)
where s is the relativistic electron path length and k -1 1 is the relativistic electron mean free path.

Expanding both sides of equation ( 4.37) for small values of the arguments (θ 1 and k 1 s 1)

considering only small angle collisions, one gets

θ 2 = 2k 1 s. (4.38)
The mean free path of the relativistic electron -which is the key parameter of such a model-has two contributions : its angular scattering due to binary collisions with electrons and its angular scattering due to binary collisions on ions

k 1 = k 1,e + k 1,i (4.39)
where

k 1,α = 2πn α θα,max θ α,min (1 -cos θ) dσ dΩ α sin θdθ. (4.40)
According to the quantum theory of diffraction, θ α,min (which corresponds to the cut-off θ c ) can be estimated by

θ α,min = 4π b α,max b α,min , (4.41)
where b α,min and b α,max are the limiting impact parameters defined by (4.7), (4.8), (4.11) and (4.12).

For scattering on electrons, θ e,max is the deflection angle corresponding to the maximum energy loss w max = 1/2 and reads consequently θ e,max = arcsin 2 γ + 3 (4.42) according to (4.2). For the scattering on target ions, due to their greater inertia, the maximum angular deflection is

θ i,max = π. (4.43)
According to [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF], one gets consequently for the scattering on target electrons

k 1,e = 4π Zn i e 4 (pv) 2 ln b e,max b e,min - 1 + ln [2(γ + 3)] 2 , (4.44)
retaining only the leading terms. Following the same procedure, one gets for target ions

k 1,i = 4π Z 2 n i e 4 (pv) 2 ln b i,max b i,min - 1 + β 2 2 . (4.45)
One can see from (4.44) and ( 4.45) that the contribution of binary collisions with ions is Z times greater than the contribution of binary collisions with electrons.

Multiple Scattering Theory by Moliere

The multiple scattering theory by [Moliere, 1948], later improved by [Bethe, 1953] and [START_REF] Nardi | Energy deposition by relativistic electrons in high-temperature targets[END_REF], provides a more accurate description for the angular distribution function of a relativistic electron beam. It consists in solving the transport equation for the angular

ANGULAR SCATTERING OF A RELATIVISTIC ELECTRON IN SOLIDS AND DENSE PLASMAS

distribution function f (Ω, s) of the electron beam

∂f ∂s = α=e, i n α f (Ω -Ω , s) -f (Ω, s) dσ dΩ α dΩ (4.46)
where Ω -Ω is the direction of electron velocity before the last scattering event, s is the electron path length and the differential cross sections are given by (4.34) and (4.36). For simplicity, we will only consider here the binary collisions of beam electrons with target ions. The equation ( 4.46) is solved by expanding the distribution function on the spherical harmonic functions Y lm (θ, ϕ) depending on the unassociated Legendre polynomials P l (cos θ). In the small angle approximation, the Legendre polynomials can be approximated by the 0th order Bessel function of the first kind P l (θ) ≈ J 0 (lθ), and the sum over l can be replaced by a continuous integral. Thus, one obtains

f (θ, s) = ∞ 0 lJ 0 (lθ) exp 2πn i s θ i,max θ i,min dσ dΩ i J 0 (lθ ) -1 θ dθ dl (4.47)
where θ i,min is given by (4.41) and θ i,max by (4.43). By expanding the exponential and performing the integrals, one obtains

f (θ, s) = ∞ n=0 f (n) (θ, s) = ∞ n=0 1 n!B n R uJ 0 θu χ c √ B exp - u 2 4 u 2 4 ln u 2 4 n du (4.48)
where

B = W -1, - θ i,min χ c 2 with W (-1, x) the Lambert W-function (the inverse function of x = y exp y on the intervals y ∈ [-∞, -1], x ∈ [-1, 0[) and χ c 2 = 4πn i s Zr e γβ 2 2 .
In the limit of small angle scattering θ, the 0th order function

f (0) (θ, s) = 2 exp - θ 2 χ c 2 B . (4.49)
is sufficient to describe the angular distribution of the relativistic electrons. Indeed, as all collisions are independant, one would have deduced, according to the central limit theorem, that the deflection probability f (θ, s) can be described by such a normal law. Thus, the quadratic mean angle can be deduced. It reads

θ 2 = χ c 2 ln χ c θ i,min . ( 4

Belyaev-Budker Collision Integral

A more general framework for description of the relativistic electron beam transport is the relativistic kinetic equation.

The collision integral of this equation has been obtained by [Belyaev and Budker, 1956] generalizing the Landau collision integral (see Appendix A, section A.2.3) to the relativistic regime. Here, we present the phenomenological approach from [START_REF] Landau | Physical kinetics[END_REF], used to derive the equation. The general form of the electron kinetic

equation reads ∂f b ∂t + ∂ ∂r (vf b ) - ∂ ∂p e E + v c × B f b = ∂f b ∂t coll . (4.51)
Conservation of the number of particles implies that the collision integral has the form

∂f b ∂t coll = - ∂ ∂p .F coll (4.52)
where F coll is the particle flux in momentum space due to binary collisions. Let us consider a small area near a point p in momentum space, perpendicular to the p µ -axis where µ ∈ {x, y, z}. The flux component F coll,µ is a difference between the number of electrons crossing this area from left to right per time unit and those crossing it from right to left due to binary collisions. If a particle α ∈ {i, e} receives in a collision a µ-component of momentum equal to ∆p µ > 0, it will cross the small area from left to right, thus increasing the value of its momentum from p µ -∆p µ to p µ . Hence, the total number of particles crossing the area from left to right is α=i, e ∆pµ>0

d 3 ∆p R 3 d 3 p α pµ pµ-∆pµ P (p, p α , ∆p)f b (r, p, t)f α (r, p α , t)dp µ where P (p, p α , ∆p)f b (r, p, t)f α (r, p α , t)d 3 p α d 3 ∆p
is the number of collisions occurring per time unit between an electron with momentum p and a particle α with momentum p α in the range d 3 p α . After the collision, the particles acquire the momenta p+∆p and p α -∆p, respectively. Similarly, the number of particles crossing that area from right to left may be written as

α=i, e ∆pµ>0

d 3 ∆p R 3 d 3 p α pµ pµ-∆pµ P (p, p α , -∆p)f b (r, p + ∆p, t)f α (r, p α -∆p, t)dp µ .
From now, we omit the variables r and t for brievity in the equations since the collisions take place locally in space and time. By virtue of the principle of detailed balance, the probability density of such a momentum exchanged is necessarily a symmetrical function with regard to the interchange of the initial and final particles states :

P (p, p α , ∆p) = P (p, p α , -∆p).
Thus, after adding these two contributions and replacing the integration over dp µ by a multiplication by ∆p µ , one obtains the Boltzmann result (see Appendix A, section A.1.4)

F coll,µ = α=i, e ∆pµ>0 d 3 ∆p R 3 d 3 p α P (p, p α , ∆p) [f b (p)f α (p α ) -f b (p + ∆p)f α (p α -∆p)] ∆p µ . (4.53)
One can express the probability density P in terms of the differential collision-cross-section as

P d 3 ∆p = v r, α d 2 σ α (4.54)
where v r,α = c γ2 α -1/γ α is the relative velocity of one particle in the rest frame of the other during their collision, γα = γγ α /γ 2 * and γ * = 1/ (1 -v.v α /c 2 ). This expression tends to |v -v α | in the non-relativistic limit. Under the assumption of a small momentum transfer ∆p compared to p and p α (also called the small angle assumption, see Appendix A, section A.2.2), one can expand the difference in the square brackets of (4.53) to give

F coll,µ = α=i, e ν=x, y, z R 3 d 3 p α U α µν f b (p) ∂f α ∂p α, ν -f α (p α ) ∂f b ∂p ν (4.55)
where

U α µν = 1 2 ∆p µ ∆p ν v r,α d 2 σ α . (4.56)
For a small angle deviation, the exchanged momentum ∆p is perpendicular to the velocity v (v α ) in the electron projectile rest frame (in the α particle rest frame, respectively). The tensor U α µν is therefore transverse to these vectors. According to [Belyaev and Budker, 1956], the only one possible particle-symmetric tensor leading to a Lorentz-invariant collision integral and satisfying

U α .v = U α .v α = 0 (4.57) where v = γ 2 * γ α v + γ α -1 v 2 α v.v α -γ α .v α and v α = γ 2 * γ v α + γ -1 v 2 v.v α -γ .v
are the relative velocities of one particle expressed in the rest frame of the another (and having the same norm v r,α ) is

U α µν = 1 2 U α 0 δ µν - 1 (γ 2 α -1) p µ p ν (m e c) 2 - p α,µ p α,ν (m α c) 2 + γα p µ p α,ν m e m α c 2 + p α,µ p ν m e m α c 2 . (4.58)
The scattering potential coming from the diagonal terms reads

U α 0 = 1 2 ∆p 2 v r,α d 2 σ α . (4.59)
However, in the litterature, one can find the Belyaev-Budker scattering potential expressed with the relativistic generalization of the Rutherford differential cross-section (A.49), mentioned in the introduction for both ions and electrons :

U α 0 = 4π q 2 α e 2 ln Λ eα v r,α
where ln Λ eα = ln (b α,max /b α,min ) and q α = e for α = e and q α = Ze for α = i (see for example [START_REF] Braams | Differential form of the collision integral for a relativistic plasma[END_REF]). Obviously, the Belyaev-Budker collision tensor tends to the Landau collision tensor (see Appendix A, section A.2.2) in the non-relativistic limit. 2). The principal difference between the PIC method and another particle method, like a molecular dynamics model, is that the interactions between particles are mediated by the electromagnetic fields calculated on a finite-difference grid, while the particles are allowed to occupy an arbitrary location in the phase-space. This approach circumvents the necessity to compute a huge number of binary interactions between individual particles while retaining N -body particle dynamics. However, by representing the whole electron population by N p macroelectrons

q i , i ∈ [1, N p ],
weighted by the number of electrons each macroelectron represents, the PIC method does not account for the binary collisions. The collisions must be taken into account separately using a Monte-Carlo collision operator. Indeed, if N p = N , the self-consistent electromagnetic fields computed from the Maxwell equations would account naturally for the binary collisions. But, the fact that N p N because of computational restrictions and that binary collision space scales, much smaller than the Debye length, are usually not resolved, only collective effects are taken into account.

Therefore, a Monte-Carlo method for modelling collisions is crucial. As already explained in Chapter 1, [START_REF] Takizuka | A binary collision model for plasma simulation with a particle code[END_REF] developed a robust algorithm for dealing with binary collisions. The key computational step is to pair particles within a cell randomly and perform a rotation in the centerof-mass frame. This method has been extended for relativistic particles by [START_REF] Sentoku | Numerical methods for particle simulations at extreme densities and temperatures: Weighted particles, relativistic collisions and reduced currents[END_REF] [ [START_REF] Pérez | Improved modeling of relativistic collisions and collisional ionization in particle-in-cell codes[END_REF] and their model can be generalized to arbitrary particle weights as proposed by [Nanbu, 1997]. However, the momentum and energy are not always conserved in each individual collision, and it has been shown that this method does not relax to the Maxwell-Juttner equilibrium distribution function [START_REF] Peano | Statistical kinetic treatment of relativistic binary collisions[END_REF]. In addition, the non-conservative force associated with the particle-grid mapping leads to self-heating and in some cases to a numerical instability, thus modifying the plasma properties. Nevertheless, if the resolution is high enough, such that the numerical space step ∆x ≤ λ Debye , the effects associated with the aliasing terms are unimportant. The PIC codes which resolve λ Debye are the best tools for studying Laser-Plasma Interaction, turbulences in tokamaks, plasma-based accelerators, relativistic shocks, ion propulsion, and many other problems.

However, concerning the problem of fast electron transport in solids or dense plasmas, the Debye length is very small compared to the characteristic scale of the problem, and the simulations become too expensive. Moreover, by focusing on small-scale collisional effects, one loses precision in resolving large scale collective effects such as introduced in Chapter 3. It has been early proposed to use so implicit PIC methods without resolving λ Debye by relating the fields at the new time to the particle positions and momenta at the new time [START_REF] Cohen | Implicit time integration for plasma simulation[END_REF] or by using only the moments of the distribution function needed in the Maxwell equations (ρ and j) at the new time [Mason, 1981].

More recently, [START_REF] Welch | Implementation of an non-iterative implicit electromagnetic field solver for dense plasma simulation[END_REF] have implemented an implicit electromagnetic field solver in the commercialized 3-Dimensional PIC code LSP.

Hybrid Particle-in-Cell methods

Even if PIC codes are perfect tools for studying hot and relatively low density plasmas, they do not incorporate all the Physics needed for describing the fast electron transport in solids and dense plasmas such as the background electrons degeneracy, the presence of bound electrons and in general atomic or molecular structure of materials and their equation of states. Therefore, it has been proposed more recently by [START_REF] Davies | Shortpulse high-intensity laser-generated fast electron transport into thick solid targets[END_REF]] [Davies, 2002] and [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF]

dp k dt = q k E + v k c × B -ν d p k + R (5.1)
where ν d is choosen such that the total stopping power of the electrons contained by the macroelectron

k reads dε ds (γ k ) = 1 v k dγ k dt m e c 2 = 1 v k p k γ k (m e c) 2 dp k dt coll = -ν d p k ⇒ ν d = - 1 p dε ds (5.2)
and the Langevin term R is treated by a Monte-Carlo method constructed in such a way that the mean angular scattering value θ 2 is defined by the multiple scattering theory of Moliere or Lewis.

This equation is implemented in the PaRIS [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF] and ZUMA [START_REF] Strozzi | Fast ignition transport studies: Realistic electron source, integrated particle in cell and hydrodynamic modeling, imposed magnetic fields[END_REF] hybrid PIC codes. For example, in the case of a fully ionized plasma, the multiple scattering theory of Lewis (4.38) reads

θ 2 s = 2k 1 (5.3)
where k 1 is given (4.39), according to [START_REF] Solodov | Stopping power and range of energetic electrons in dense plasmas of fastignition fusion targets[END_REF] or [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF].

There are some cases where the "hybrid assumption" is not fully justified. For example, it is not possible to distinguish between the fast and slow electrons in thin laser-irradiated targets where fast electrons are crossing the laser-plasma interaction zone many times. In addition, even if the "hybrid assumption" is justified, as in the case of thick targets, one needs to extract from the full PIC simulation the properties of the fast electrons in order to initialize the hybrid simulation. More recently, an improved model, named the two-region PIC, was proposed by [START_REF] Cohen | Simulation of laser-plasma interactions and fast-electron transport in inhomogeneous plasma[END_REF]. In this model, the simulation box is separated into a low plasma density region of the laser-plasma interaction and a high density region where the laser-generated fast electrons are propagating through. The position of the boundary between these two regions is taken at the density 100n c . In the low density region, the plasma is described by a full PIC algorithm with collisions taken into account. In the high density region, the Maxwell's equations are reduced to Ohm's law for the electric fields and the Ampère's law for the magnetic field. This reduced field solver is similar to the one used in the hybrid PIC models, whereas the background plasma comprises macroparticles as in a traditional PIC model. However, the difficulty of this model arises from the continuity of electromagnetic fields near the boundary between the two regions which can be violated due to the noise of the full Maxwell solver. This noise is usually several orders of magnitude higher than the one of the reduced field solver, which may mask the real value of the field given to the reduced field solver after a long period of simulation. This issue may be partially solved by increasing the spatial resolution and using a large number of macroparticles per cell.

5.2 Vlasov-Fokker-Planck methods

Full Vlasov-Fokker-Planck methods

Another approach consists in solving the Vlasov-Fokker-Planck equation (4.51) with finite-difference schemes for the spatial advection, advection in the momentum space due to the electromagnetic fields and the collisional friction and diffusion. This is a fundamentally different approach for the description of a system of particles compared to the PIC method. It considers a continuous distribution function and solves the kinetic equation on a Eulerian grid. These methods are very expensive in terms of numerical cost due to a huge number of variables that have to be resolved (3 in space and 3 in momentum). Therefore, hybrid versions have also been proposed for the same reasons as for PIC codes. Besides, due to the robustness and the long history of PIC codes (see for example [START_REF] Birdsall | Plasma Physics via Computer Simulation. The Adam Hilger Series on Plasma Physics[END_REF]), the finite-difference techniques have only been developed recently after the pioneering work by [START_REF] Bell | Elecron Energy Transport in Steep Temperature Gradients in Laser-Produced Plasmas[END_REF]. Besides, I did not find in the literature codes that solve the full Vlasov-Belyaev-Budker equation ( 4.51), using, for example, the Rosenbluth potentials proposed by [START_REF] Braams | Differential form of the collision integral for a relativistic plasma[END_REF]. The published codes solve the Vlasov equation with the Landau collisional operator, which is only valid for relativistic electrons colliding with non-relativistic background particles. Indeed, in the case of a "hybrid model", as it will be demonstrated in the next Chapter 6, the Belyaev-Budker collision tensor can be simplified into a Landau-like collision operator. Considering the spherical coordinates (p, θ, ϕ) for the momentum space (see Figure 5.2) and assuming a fully ionized plasma, the kinetic equation reads, following the notations used by [START_REF] Yokota | Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets[END_REF]

∂f b ∂t + ∂ ∂x [v sin θ cos ϕf b ] + ∂ ∂y [v sin θ sin ϕf b ] + ∂ ∂z [v cos θf b ] + 1 p 2 ∂ ∂p p 2 [F x sin θ cos ϕ + F y sin θ sin ϕ + F z cos θ] f b + 1 p sin θ ∂ ∂θ {sin θ [F x cos θ cos ϕ + F y cos θ sin ϕ -F z sin θ] f b } + 1 p sin θ ∂ ∂ϕ {[-F x sin ϕ + F y cos ϕ] f b } = m 2 e p 2 ∂ ∂p γ 2 Y ee n e m e + Y ei n i m i f b + 1 2 (Y ee n e + Y ei n i ) γm e p 3 1 sin θ ∂ ∂θ sin θ ∂f b ∂θ + 1 sin 2 θ ∂ 2 f b ∂ϕ 2 .
(5.4)

where

F x = -e[E x -(v/c) cos θB y + (v/c) sin θ sin ϕB z ], F y = -e[E y + (v/c) cos θB x -(v/c) sin θ cos ϕB z ], F z = -e[E z + (v/c) sin θ cos ϕB y -(v/c) sin θ sin ϕB x ]
are the components of the force depending on the self-generated electromagnetic fields (E, B) while Y ee = 4πe 4 ln Λ rel ee and Y ei = 4πZ 2 e 4 ln Λ rel ei are the coefficients coming from the stopping power/angular scattering formulas of the relativistic electrons respectively by colliding electrons and ions. Besides, except for the code developed by [START_REF] Yokota | Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets[END_REF], the Vlasov-Landau codes developed for fast electron transport do not take into account the γ-dependence in the relativistic Coulomb logarithms ln Λ rel ee and ln Λ rel ei in Y ee and Y ei [Sherlock, 2009] [Duclous et al., 2009]. While this approximation is fully justified for non-relativistic electrons, this γ-dependence defines the energy loss comportment of relativistic electrons, as explained in Chapter 4, section 4.2.1. Thus, by fixing numerical values for ln Λ rel ee and ln Λ rel ei , one underestimates the relativistic electron energy loss and the material heating. Even in the original paper by [START_REF] Yokota | Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets[END_REF], the factor [(Y ee n e /m e ) + (Y i n i /m i )] of the collisional friction term appears outside the p-derivative which means that the factors ln Λ rel ee and ln Λ rel ei are supposed to be independant of p. An excellent review dedicated to Vlasov-Fokker-Planck numerical modeling is given by [START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]. The computational cost of solving (5.4) can be prohibitive. Indeed, even if one considers mod-

Distribution Function Expansion methods

Spherical Harmonic Expansion method

VLASOV-FOKKER-PLANCK METHODS

estly 10 2 cells for each dimension, one needs to memorize 10 6×2 = 10 12 values of the distribution function f b at each time step which represents 8 To in double-precision. Depending on the Courant-Friedrichs-Lewy condition which defines the time step ∆t and the simulation time t f = N f ∆t, Vlasov-Fokker-Planck simulations can become extremely expensive in terms of computer memory and/or time computation. In practice, the full Vlasov-Fokker-Planck equation is resolved in 1D-1V, 1D-2V, 1D-3V or at maximum 2D-3V. Another approach consists in expanding the distribution function f b in spherical harmonics in the momentum-space, as initially proposed by [START_REF] Bell | Fast electron transport in laser-produced plasmas and the KALOS code for solution of the Vlasov-Fokker-Planck equation[END_REF].

This expansion reads

f b (r, p, t) = N l l=0 l m=-l f m l (r, |p|, t) P |m| l (cos θ) exp (imϕ) (5.5)
where f -m l equals the complex conjugate of f m l and P |m| l (cos θ) are the associated Legendre polynomials (including the Condon-Shortley phase (-1) m )

P m l (x) = (-1) m 1 -x 2 m/2 d m P l dx with P l (x) = 1 2 l l! d l dx l x 2 -1
the unassociated Legendre polynomials. For example, the first associated Legendre polynomials read

P 0 0 (x) = 1, P 0 1 (x) = x, P 1 1 (x) = -1 -x 2 1/2 , P 0 2 (x) = 1 2 3x 2 -1 , P 1 2 (x) = -3x 1 -x 2 1/2 and P 2 2 (x) = 3 1 -x 2 .
The spherical harmonic expansion (5.5) is exact for N l → ∞ but, in practice, N l is chosen, depending on the degree of anisotropy of a given physical problem. The larger the indices (l, m) are, the more directional/anisotropic the harmonics are. Recently, [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF] developed a relativistic Fokker-Planck code, called OSHUN, following this approach initially proposed for non-relativistic electrons (KALOS code by [START_REF] Bell | Fast electron transport in laser-produced plasmas and the KALOS code for solution of the Vlasov-Fokker-Planck equation[END_REF]). [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF] shown that N l ≥ 13 (≈ 100 harmonics f m l ) are sufficient to study the relativistic collisionless two-stream and filamentation instabilities presented in Chapter 3. However, it is expected that collisional effects reduce the needed degree of anisotropy N l . In addition to reducing the number of variables from 6 (x, y, z, p x , p y , p z ) to only 4 (x, y, z, |p|), the advantage of the spherical harmonic decomposition is that the Y m l are the TRANSPORT eigenfunction of the Laplace-Beltrami operator

∆ θ,ϕ (f b ) = 1 sin θ ∂ ∂θ sin θ ∂f b ∂θ + 1 sin 2 θ ∂ 2 f b ∂ϕ 2
of the Vlasov-Fokker-Planck diffusion term in (5.4). Indeed, by noting

Y m l (θ, ϕ) = 2l + 1 4π (l -m)! (l + m)! P m l (cos θ) exp (imϕ), -l ≤ m ≤ l (5.6)
the spherical harmonics, one has

∆ θ,ϕ (Y m l ) = -l(l + 1)Y m l .
(5.7)

This method gives rise to a set of coupled differential equations for the coefficients f m l , obtained starting from the Vlasov-Landau equation (5.4). They read

∂f m l ∂t -A m x,l -A m y,l -A m z,l -E m x,l -E m y,l -E m z,l -B m l = ∂ ∂p 1 v 2 Y ee n e m e + Y ei n i m i f m l - l(l + 1) 2 (Y ee n e + Y ei n i ) γm e p 3 f m l .
(5.8)

The advection terms in space are given for m = 0 by

A m z,l = -v ∂ ∂z l -m 2l -1 f m l-1 + l + m + 1 2l + 3 f m l+1 , A m x,l + A m y,l = v 2 ∂ ∂x -i ∂ ∂y f m-1 l-1 2l -1 - ∂ ∂x + i ∂ ∂y (l -m) (l -m -1) f m+1 l-1 2l -1 - ∂ ∂x -i ∂ ∂y f m-1 l+1 2l + 3 + ∂ ∂x + i ∂ ∂y (l + m + 1) (l + m + 2) f m+1 l+1 2l + 3
and for m = 0 by

A 0 x,l + A 0 y,l = Re -v ∂ ∂x + i ∂ ∂y - l (l -1) 2l -1 f 1 l-1 + (l + 1) (l + 2) 2l + 3 f 1 l+1 .
The advection terms in the momentum space due to the electric field are given by

E m z,l = eE z l -m 2l -1 p l-1 ∂ ∂p p -(l-1) f m l-1 + l + m + 1 2l + 3 p -(l+2) ∂ ∂p p l+2 f m l+1
and, for m = 0 by

E m x,l + E m y,l = e 2 E x -iE y 2l -1 p l-1 ∂ ∂p p -(l-1) f m-1 l-1 - E x + iE y 2l -1 (l -m) (l -m -1) p l-1 ∂ ∂p p -(l-1) f m+1 l-1 - E x -iE y 2l + 3 p -(l+2) ∂ ∂p p l+2 f m-1 l+1 + E x + iE y 2l + 3 (l + m + 1) (l + m + 2) p -(l+2) ∂ ∂p p l+2 f m+1 l+1
and for m = 0 by

E 0 x,l + E 0 y,l = Re e (E x + iE y ) - l (l -1) 2l -1 p l-1 ∂ ∂p p -(l-1) f 1 l-1 + (l + 1) (l + 2) 2l + 3 p -(l+2) ∂ ∂p p l+2 f 1 l+1 .
Finally, the rotational term in the momentum space due to the magnetic field is given for m = 0 by

B m l = -i eB z γm e c mf m l + e 2γm e c (l -m) (l + m + 1) (B y -iB x ) f m+1 l -(B y + iB x ) f m-1 l
and for m = 0 by

B 0 l = l (l + 1) e γm e c
Re (B y -iB x ) f 1 l .

The electron-electron collision operator in OSHUN [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]] [Tzoufras et al., 2013] is based on the non-relativistic Rosenbluth potentials [START_REF] Rosenbluth | Fokker-Planck Equation for an Inverse-Square Force[END_REF]]

∂f b ∂t coll,ee = - 4πe 4 ln Λ ee m 2 e ∂ ∂v . f b ∂h[f b ] ∂v - 1 2 ∂ ∂v ⊗ ∂ ∂v : f b ∂ ∂v ⊗ ∂g[f b ] ∂v (5.9) where ∂ 4 g ∂v 4 [f b ] = ∂ 2 h ∂v 2 [f b ] = -8πf b
while the electron-ion collision operator is based on the non-relativistic Lorentzian approximation (see

Appendix A, section A.3.2) ∂f b ∂t coll,ei = - 4πn i Z 2 e 4 ln Λ ei m 2 e v 3 f b = - ν ei 2 f b .
(5.10)

In order to account for this electron-ion collision operator, one just has to replace Y ei by 0 in the collisional friction term of (5.8) (first term in the right hand side) and replace ln Λ rel ei by the classical Coulomb logarithm ln Λ ei (see Appendix A, section A.2.3) in Y ei of the angular diffusion term (second term in the right hand side). The electron-electron collision operator (5.9) is more difficult to solve numerically. In OSHUN, in order to break the interdependance between the amplitudes f m l and allow for rapid numerical calculations, it is linearized assuming that f b is weakly anisotropic TRANSPORT

f b = f 0 0 + δf b .
According to [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF], it gives

∂f 0 0 ∂t coll,ee = 4πe 4 ln Λ ee m 2 e 4π f 0 0 2 + 4πe 4 ln Λ ee m 2 e 1 2 ∂ ∂v ⊗ ∂g[f 0 0 ] ∂v : ∂ ∂v ⊗ ∂f 0 0 ∂v = (4π) 2 e 4 ln Λ ee 3m 2 e 1 v 2
∂ ∂v

1 v ∂W [f 0 0 ] ∂v (5.11)
for the isotropic part where

W [f 0 0 ] = f 0 0 v 0 f 0 0 v 4 dv + v 3 f 0 0 ∞ v f 0 0 vdv -3 ∞ v f 0 0 vdv v 0 f 0 0 v 2 dv
according to [START_REF] Bobylev | On the numerical solution of Landau's kinetic equation[END_REF] and

∂δf b ∂t coll,ee = 8π 4πe 4 ln Λ ee m 2 e f 0 0 δf b + 1 2 ∂ ∂v ⊗ ∂g[f 0 0 ] ∂v : ∂ ∂v ⊗ ∂δf b ∂v + 1 2 ∂ ∂v ⊗ ∂f 0 0 ∂v : ∂ ∂v ⊗ ∂g[δf b ] ∂v (5.12)
for the anisotropic perturbation. In order to account for this electron-electron collision operator, one just has to replace Y ee by 0 in (5.8) and add (5.11) in the right hand side of Equation ( 5.8) with m = l = 0 and (∂f m l /∂t) coll,ee ) coming from the decomposition in spherical harmonics of (5.12) (see [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]) in the right hand side of the other equations (5.8) with l = 0. These collision operators are out of the scope of laser-generated relativistic electron beam transport presented in this manuscript. The details of numerical methods used to discretize these collision operators can be found in [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]. The authors point out that for certain problems it would be extremely expensive to apply an explicit scheme for the collisions of the anisotropic part of the distribution function. Concerning the Vlasov part, i.e. the left hand side of equation (5.8), all advection terms in the phase-space are differenciated by using the central difference scheme of the second order. While periodic and reflecting boundaries in space have been implemented, the behavior of the harmonics at p = 0 has been chosen such that f m l≥1 (p) = f m l (p 1 )(p/p 1 ) l and that the isotropic part of the distribution has an extremum at p = 0 in order to get the p-derivatives at p = p 0 where p k = p 0 + k∆p with p 0 = ∆p/2 and ∆p the numerical momentum step. An iteration loop involving a list of successive operations on each harmonic is performed such that each operator in the (l, m)-equation (5.8) depends only on f m l . These operations are found by starting from (5.8) to find the effect that each f m l has on its neighboring amplitudes in (l, m) space. Thus, (5.8) can finally be written with the form

∂f m l ∂t = F (f m l , t) (5.13)
and be therefore numerically solved by using the Runge-Kutta methods up to the 4th order.

VLASOV-FOKKER-PLANCK METHODS

Cartesian Tensor Scalar Product Expansion method

Another expansion of the distribution function has been proposed by [Johnston, 1960] and [Shkarofsky, 1963] for plasmas. This tensor scalar product expansion reads

f b (r, p, t) = 1 4π f 0 (r, |p|, t) + 3 4π f 1 (r, |p|, t).Ω + N l l≥2 C l 4π f l (r, |p|, t) l Ω (l) (5.14)
where l) is the (l -1) tensor products of Ω with itself

C l = (2l + 1)! 2l! , Ω = p/p = [sin θ cos ϕ, sin θ sin ϕ, cos θ] T (see Figure 5.2), Ω (
Ω (l) = Ω ⊗ ... ⊗ Ω = (Ω i 1 Ω i 2 ...Ω i l ) (i 1 , i 2 , ..., i l )∈{x, y, z} l
and l is the l times contracted product which means that

A B = i 1 =x, y, z i 2 =x, y, z ... i l =x, y, z A i1, i 2 , ..., i l B i1, i 2 , ..., i l
where A and B are lth order tensors. Thus, f 0 is a scalar (1 term f 0 ), f 1 is a vector (3 terms f 1,x , f 1,y and f 1,z ), f 2 is a second order tensor (9 terms f 2,xx , f 2,xy , f 2,xz , f 2,yx , f 2,yy , f 2,yz , f 2,zx , f 2,zy and f 2,zz ), f 3 is a third order tensor (27 terms f 3,i 1 i 2 i 3 , (i 1 , i 2 , i 3 ) ∈ {x, y, z} 3 ) and so on... The expansion (5.14) is exact in the limit N l → ∞ and it is equivalent to the spherical harmonics expansion (5.5) cut at the same order N l . Indeed, by writing the spherical harmonic decomposition (5.5) with the form [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]]

f b (r, p, t) = N l l=0 l m=0 1 0 f lms (r, |p|, t)Y lms (θ, ϕ) (5.15)
where

Y lms (θ, ϕ) = 2l + 1 4π (l -m)! (l + m)! P m l (cos θ)    δ 0s cos (mφ) + δ 1s sin (mφ) √ 2 if m > 0 δ 0s if m = 0
, one has [Johnston, 1960]

N l l=1 C l 4π f l (r, |p|, t) l Ω (l) = N l l=1 l m=0 1 0 f lms (r, |p|, t)Y lms (θ, ϕ).
(5.16)

For example, we have

1 4π f 0 = f 000 √ 4π , and 3 4π f 1 .Ω = 3 4π (f 1,x sin θ cos ϕ + f 1,y sin θ sin ϕ + f 1,z cos θ) = 3 4π f 100 cos θ - f 110 2 sin θ cos ϕ - f 111 2 sin θ sin ϕ .
According to these two first examples, one can clearly see, that the dependences in cos a l θ sin b l θ cos c l ϕ sin d l ϕ are strictly the same where a l , b l , c l , and d l are constants depending on l.

However, it is not true for the second order term

C 2 4π f 2 : Ω ⊗ Ω = 2 m=0 1 0 f 2ms (r, |p|, t)Y lms (θ, ϕ)
because 1, Ω and Ω ⊗ Ω are not orthogonal (with the definition of the scalar product l ) contrary to the spherical harmonics. Actually, this is 1, Ω and Ω ⊗ Ω -(1/3)I that are orthogonal. Indeed, according to (5.15), the relation between the Y lms and the Y m l are straightforward and reads

Y lm0 Y lm1 = 1 √ 2 1 (-1) m -i i(-1) m . Y |m| l Y -|m| l
while the relation between the f lms and the f m l are

f m l = 2l + 1 4π (l -m)! (l + m)! f lm0 -if lm1 √ 2 .
Thus, one gets the relations for the three first components

f 0 = 4πf 0 0 , (5.17
)

f 1 = 4π 3     2Re f 1 1 -2Im f 1 1 f 0 1     , (5.18)
and

f 2 - 1 3 f 0 I = 4π 15     12Re f 2 2 -f 0 2 -12Im f 2 2 6Re f 1 2 -12Im f 2 2 12Re f 2 2 -f 0 2 -6Im f 1 2 6Re f 1 2 -6Im f 1 2 2f 0 2     (5.19)
where I is the 2nd order unit tensor.

In the case where the Cartesian tensor scalar product expansion is cut at the first order N l = 1, each f l corresponds to the lth order angular moment of the distribution function f b :

f l (r, |p|, t) = p 2 S 2 f (r, p, t)Ω ( ) d 2 Ω (5.20)
where it has been noted S 2 the unity sphere in the momentum space, defined by {(θ, ϕ), θ ∈ [0, π], ϕ ∈ [0, 2π[} (see Figure 5.2). Thus, while it is strictly equivalent to the spherical harmonic expansion, the Cartesian tensor scalar product expansion cut at the 1st order has the advantage to directly relate the physical quantities and the expansion terms f l . In our case of relativistic electron beam transport, f 0 yields directly to the beam number density

n b = R 3 f b d 3 p = ∞ 0 f 0 dp (5.21)
and the beam kinetic energy density

K b = n b (γ b -1) m e c 2 = R 3 (γ -1) m e c 2 f b d 3 p = ∞ 0 (γ -1) m e c 2 f 0 dp (5.22)
while f 1 yields directly to the beam current density

j b = -n b ev b = e R 3 vf b d 3 p = -e ∞ 0 vf 1 dp (5.23)
where v b is the mean beam electrons velocity, the mean beam electrons momentum

p b = 1 n b R 3 pf b d 3 p = 1 n b ∞ 0 pf 1 dp (5.24)
and the beam kinetic energy flux

F b = R 3 (γ -1) m e c 2 vf b d 3 p = ∞ 0 (γ -1
) m e c 2 vf 1 dp.

(5.25)

In this cartesian tensor scalar product approach cut at the 1st order, the lth order equation that have to be solved in order to get the expansion term f l can be obtained by integrating over the unity sphere S 2 the Vlasov-Fokker-Planck equation (5.4) multiplied by Ω (l) . According to (5.20), these two first equations of the hierarchy read

∂f 0 ∂t + ∂ ∂r . (vf 1 ) - ∂ ∂p (ef 1 .E) = ∂ ∂p 1 v 2 Y ee n e m e + Y ei n i m i f 0 (5.26) and ∂f 1 ∂t + ∂ ∂r . (vf 2 ) - ∂ ∂p (ef 2 .E) + e p (f 0 I -f 2 ) .E + e γm e c f 1 × B = ∂ ∂p 1 v 2 Y ee n e m e + Y ei n i m i f 1 -(Y ee n e + Y ei n i ) γm e p 3 f 1 .
(5.27)

In addition to the increasing complexity of the obtained equations with increasing l, one can notice that each lth order equation for f l makes appearing the (l + 1)th order component f l+1 . Consequently, the (N l + 1)th order component appearing in the N l th order equation has to be approximated to close TRANSPORT the set of equations. This is done by imposing the PN closure relation

f N l +1 = S 2 f b Ω (N l +1) d 2 Ω ≈ S 2 N l l=0 C l 4π f l l Ω (l) Ω (N l +1) d 2 Ω.
In practice, assuming a small perturbation of the isotropy, the Cartesian models are often limited to the 1st order approximation (P1) in order to make fast computations. This was for example usually done in studies of the non local thermal flux carried out by suprathermal electrons [START_REF] Matte | Electron Heat Flow with Inverse Bremsstrahlung and Ion Motion[END_REF] [ [START_REF] Schurtz | A nonlocal electron conduction model for multidimensional radiation hydrodynamics codes[END_REF]] [Nicolaï et al., 2006] [Schurtz et al., 2007]. In this case, the second order angular moment expression reduces to the isotropic part of (5.19)

f 2 = 1 3 f 0 I. (5.28)
Injecting it in the 1st order equation, one gets the so-called diffusive approximation characterized by the two equations and Expanded meaning using the spherical harmonic or Cartesian tensor expansion. The first three aspects of the codes, "Efficiency", "Time/space constraint" and "Noise/finesse" are rather qualitative.

∂f 0 ∂t + ∂ ∂r . (vf 1 ) - ∂ ∂p (ef 1 .E) = ∂ ∂p 1 v 2 Y ee n e m e + Y ei n i m i f 0 and ∂f 1 ∂t + v 3 ∂f 0 ∂r - ∂ ∂p e 3 f 0 E + e p 2 3 f 0 .E + e γm e c f 1 × B = ∂ ∂p 1 v 2 Y ee n e m e + Y ei n i m i f 1 -(Y ee n e + Y ei n i ) γm e p 3 f 1 .

Conclusion

Efficiency is how the information is stored, it relates to how fast the code is expected to be, and is the most subjective rating. PIC methods store information in a minimal way, and are therefore both listed as "High". Full Vlasov-Fokker-Planck would typically contain the most redundancy of information storage, as the full phase-space needs to be stored. Hybrid and Expanded Vlasov-Fokker-Planck are listed as "Reasonable" as they store momentum phase-space information in a more efficient way than Full Vlasov-Fokker-Planck. Time/space constraint is also somewhat subjective, and is based on running a "typical" code on a "typical" cluster of ≈ 100 processors in terms of what are the maximum length and timescales that could be "reasonably simulated" for a hot solid target interaction, based on the timescales in published results. Noise/finesse entries are more quantitative, in that Vlasov-Fokker-Planck codes are naturally smoothed, whereas PIC methods are subject to noise due to the finite number of discrete particles, with the Hybrid PIC therefore naturally being less subject to noise. It is worth pointing out that while PIC codes tend to be noisy, which in some cases may provide an unreasonably large seed perturbation for instabilities to grow from, Vlasov-Fokker-Planck codes may be unrealistically smoothed [START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]. It is possible that numerical diffusion in Vlasov-Fokker-Planck codes damps real physical instabilities. The last three aspects describe the relative treatment of phenomena in different regimes of ln Λ ei and the relative evaluations are therefore related to the physical equations involved. The most important difference between PIC codes and Vlasov type codes is the issue of graininess, with PIC codes exhibiting artificially high levels of it and Vlasov codes completely lacking it. The presence of granularity in PIC codes is not fundamentally an issue of inadequate statistics, because no matter how high the number of simulation particles is, graininess will never completely disappear from a PIC code (as it is also true for an actual physical system) [START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]. This allows PIC codes to model instabilities from noise and to incorporate physics associated with complex particle trajectories. For inertial confinement fusion plasmas however, the lack of statistical smoothness can obscure and modify the physics [START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]. Furthermore, numerical effects associated with finite size simulation particles can severely compromise the reliability of the results. Vlasov and Vlasov-Fokker-Planck codes produce results without noise, which allow for clear physical pictures to emerge since Vlasov-Fokker-Planck codes represent the plasma using distribution functions, lack of statistics is never an issue [START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]. However, if the distribution function is not described with sufficient details, the physics cannot be modelled accurately, and as a result Vlasov-Fokker-Planck codes are also prone to numerical artifacts.

As a conclusion, both hybrid PIC codes and hybrid Vlasov-Fokker-Planck codes are the best and complementary numerical tools for solving the Vlasov-Belyaev-Budker equation applied to laser-TRANSPORT generated relativistic electron beam transport. Moreover, expanded Vlasov-Fokker-Planck hybrid method are able to limit the numerical cost imposed by the kinetic description of the relativistic electron beam transport. Besides, cutting the expansion of the distribution function N l may lead to unphysical results since the resulting distribution function may become negative if N l is not taken sufficiently large for resolving strong anisotropy [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]. Starting from the V-F-P equation for a Lorentzian gaz of non relativistic electrons, the authors proposed therefore to close the set of equation by using a special closure relation based on the principle of the Minerbo maximum angular entropy approximation [Minerbo, 1977] [Minerbo, 1978] that the laws of human reasoning coincide with the laws governing the sequences of the impressions we receive from the world about us; that, therefore, pure reasoning can enable man to gain an insight into the mechanism of the latter. In this connection, it is of paramount importance that the outside world is something independent from man, something absolute, and the quest for the laws which apply to this absolute appeared to me as the most sublime scientific pursuit in life. "

Max Planck

KINETIC DESCRIPTION OF RELATIVISTIC ELECTRON BEAM TRANSPORT IN SOLIDS AND DENSE PLASMAS

In this chapter, the Landau-like relativistic collision tensor, mentioned in the previous Chapter 5

and allowing to obtain the Vlasov-Fokker-Planck (V-F-P) equation (5.4), is derived from the Belyaev-Budker collision tensor and applied to relativistic electron beam transport in solids and dense plasmas.

Relativistic Electron Beam Transport

We note f b (r, p, t) the distribution function of the relativistic electron beam in the laboratory frame.

It is the solution of the relativistic Vlasov-Belyaev-Budker equation

∂f b ∂t + ∂ ∂r . p γm e f b - ∂ ∂p . e E + p γm e c × B f b = α C α [f b ] (6.1)
where

C α [f b ] = - ∂ ∂p . R 3 U α . f b ∂f α ∂p α - ∂f b ∂p f α d 3 p α (6.2)
with U α the Belyaev-Budker collision operator (see Chapter 4, section 4.4). The collision operator C α [f b ] comes from a 2nd order expansion of the Boltzmann collision integral in the small angle scattering limit:

∆p = ∆p p p + ∆p ⊥ b b with p ∆p ⊥ ∆p , (6.3)
where b is the impact parameter vector and ∆p is the momentum transfer in a collision of a relativistic beam electron with momentum p with a particle α having a momentum p α (cf. Annexe A, section A.2.2). Here, α particles can be ions, bound electrons, free electrons or screened free electrons of the medium where the relativistic electron beam is propagating through. We neglect the binary collisions of the beam electrons with themselves since n b n e . (6.3) means that each binary collision occurs in the 2D plane (p, p α ) and leads to an exchange of momentum mostly in the perpendicular direction to p. Indeed, even if large angle collisions lead to a large change in momentum after each collision, the probability that they occur is small compared to the probability of small angle collisions. The Belyaev-Budker collision tensor reads

U α = 1 2 U α,0 I - 1 (γ 2 α -1) p m e c ⊗ p m e c - p α m α c ⊗ p α m α c + γα p m e c ⊗ p α m α c + p α m α c ⊗ p m e c .
(6.4)

It is often expressed with the scattering potential U α,0 = 4πZ 2 α e 4 ln Λ eα /v r,α expanded within the classical Rutherford cross section (dσ α /dΩ) Ruth [Belyaev andBudker, 1956] [Landau andLifshitz, 1981] [ [START_REF] Braams | Differential form of the collision integral for a relativistic plasma[END_REF]. But, in a more general case i.e. without integrating it within a given cross section, the scattering potential reads with n α the density of the α particles. The integration limits w α,min and w α,max in (6.8) depend on the nature of collisions. For collisions on free electrons, w free,max = 1/2 due to the indiscernibility of electrons and a cut-off w c is used to distinguish between binary collisions and collisions where collective effects take place. That is to say, for collisions on bound and screened free electrons. These latter collisions can be understood respectively in terms of energy transfer to the excitation of bound electrons by the beam electron electric field and in terms of energy transfer to plasma waves in quanta of ω pe (plasmons). For collisions on plasma ions, w i,min and w i,max correspond to the commonly used impact parameters. All details are given in Chapter 4, sections 4.2.1 and 4.2.2 and are summarized in Figure 6.1.

U α,0 = 1 2 ∆p 2 v r,α dσ α . (6.5) v r,α = c γ2 α -1/γ α is
Injecting the expression (6.7) in (6.5) and noticing that v r,α ∼ v under the assumptions (6.3) and (6.6), one finds the following relation between the scattering potential U α,0 and the electron stopping

power (dε/ds) α U α,0 = - m α v n α dε ds α . (6.9)
Moreover, according to assumptions (6.3) and (6.6), one has γ α ∼ 1, γα ∼ γ and the Belyaev-Budker tensor (6.4) can thus be simplified to

U α = U α,0 2 I - p ⊗ p p 2 (6.10)
and its divergence to where Ω = p/p. Instead of f b , we make use of the distribution function Ψ depending on the kinetic energy ε and the propagation direction Ω Ψ (r, ε, Ω, t) = p 2 /v f b (r, p, t), (6.12)

∂ ∂p α .U α = - U α,0 m α v p p . ( 6
where p 2 /v comes from the Jacobian of the change of variables from p to (ε, Ω). By defining S(r, ε, t) =α dε ds α (6.13) the total stopping power (> 0) and

ν(r, ε, t) = - α m α v p 2 dε ds α (6.14)
the total angular isotropization rate, one may integrate the collision integral (6.2) by parts and express the Vlasov-Belyaev-Budker equation in the laboratory frame (6.1) as

∂Ψ ∂t + ∂ ∂r .(vΩ Ψ) - ∂ ∂ε [v (eE.Ω + S) Ψ] - ∂ ∂Ω . (I -Ω ⊗ Ω) . e p E + vΩ c × B Ψ + ∂ ∂Ω ν 2 Ψ = 0 (6.15) where ∂ ∂Ω . [(I -Ω ⊗ Ω) .A] = 1 sin θ ∂ ∂θ (sin θA θ ) + 1 sin θ ∂A ϕ ∂ϕ and ∂ ∂Ω . (I -Ω ⊗ Ω) . ∂f ∂Ω = 1 sin θ ∂ ∂θ sin θ ∂f ∂θ + 1 sin 2 θ ∂ 2 f ∂ϕ 2
is the Laplace-Beltrami operator. It is the same equation as Equation (5.4), but written for Ψ(r, ε, Ω, t) instead of f b (r, p, θ, ϕ, t).

Collisional Effects of Relativistic Electron Transport in Solids and Dense Plasmas

The analysis presented above in the section 6.1.1 shows that the V-F-P equation (6.15) usually derived for free electrons can be extended to a more general case by replacing the Coulomb scattering potential U α,0 by the realistic stopping power according to Equation (6.9). We use the general expression for the stopping powers in solids and dense plasmas

dε ds α = -4π n α Z α 2 e 4
m α v 2 ln Λ rel eα . (6.16)

Z α equals 1 for electrons and equals the nuclear charge Z for ions. As detailed in Chapter 4 section 4.2.1, the Coulomb logarithm ln Λ rel eα is calculated using the M øller cross section [Möller, 1932] for col- lisions with free plasma electrons. The cut-off used to distinguish between the binary part (collision with plasma free electrons) and the collective one (collisions with plasma bound and screened electrons) is evaluated by w c = λ De Broglie 2 / max {λ Debye 2 , (3/4πn i ) 2/3 }. The Mott cross section [Mott, 1932] is used for collisions with ions. An extension of the Bethe formula [Bethe, 1932] is used with a mean excitation potential I ex provided by [More, 1985] to take into account collisions with plasma bound electrons. The Fermi density effect correction [Fermi, 1940] is taken into account according to the Pines and Bohm cross section [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF] for collisions with screened free electrons (plasmons). These expressions for stopping powers are derived in the first Born approximation for low and intermediate Z plasmas and for electrons with kinetic energies greater than approximatively 10 keV.

i ln Λ clas ei - 2γ 2 -1 2γ 2 free e - Z * n i ln Λ clas ee -ln 2 + 1 2 - 2γ -1 2γ 2 ln 2 + 1 16 γ -1 γ 2 bound e -(Z -Z * ) n i ln (γ -1) m e c 2 I ex γ + 1 2 + 1 2γ 2 - 2γ -1 2γ 2 ln 2 + 1 16 γ -1 γ 2 plasmons Z * n i ln   c γ 2 -1/γ ω p,
The expressions for the densities n α and the Coulomb logarithms ln Λ rel eα of (6.16) are summarized in Figure 6.1. Even if Bremsstrahlung losses of the relativistic electrons can be neglected in the considered range of energies (10 keV -10 MeV), a radiative stopping power from [START_REF] Heitler | Stopping of Fast Particles with Emission of Radiation and the Birth of Positive Electrons[END_REF]

dε ds rad = -4π (Z -Z * )(Z -Z * + 1)n i e 4 m e c 2 /γ α π ln (2γ) - 1 3
is added into S. Due to a very small mass ratio m e /m i 1, the contribution of the stopping power on ions (dε/ds) i is negligible compared to those on electrons.

Concerning the angular diffusion of the beam electrons, it is worth noting that the isotropization rate

ν = α ν α with ν α = 4π n α Z α 2 e 4
γ 2 m e 2 v 3 ln Λ rel eα (6.17) deduced from (6.14) and (6.16) does not depend on the α particles mass. It is plotted in the left panel of Figure 6.2 for Aluminum (ρ = 2.7 g.cm -3 ) versus the electron kinetic energy with the separate and T = 100 eV) versus its kinetic energy according to formula (6.17) (black) and the separate contributions due to free electrons (red), bound electrons (blue), screened free electrons/plasmons (green) and ions (magenta).

ν α of collisions with background free electrons, bound electrons, screened free electrons/plasmons and ions at ambiant temperarure (T e = T i = 300 K) and T e = T i = 100 eV. It shows that, for intermediate and high Z plasmas, the electron beam scattering on ions is dominant compared to their scattering on electrons by a factor Z. One can also notice that, as for the total stopping power (see Chapter 4, section 4.2.1), the isotropization rate ν weakly depends on the background temperature (logarithmically). The isotropization rate (6.17 One deduces consequently that the total slowing down rate ν d = (1/p)(dε/ds) introduced in Equation

(5.2) is approximatevely γ times greater than the total isotropization rate ν of the beam electrons :

ν d = γ (ν -ν i ) + 1 p dε ds rad (6.19)
in the limit m e /m i 1.

Background Electrons Dynamics in the "Hybrid" Assumption

As detailed in Chapter 3, section 3.1.1, we consider that the beam is already electromagnetically neutralized. That is to say, (6.24)

n e = Z * n i -n b ≈ Z * n i (n b n e ),
The role of each term in these equations is discussed in Chapter 3, section 3.1.1.

As detailed in Chapter 3, section 3.3.2, the energy deposition by the electron beam produces a heating and a hydrodynamic motion of the plasma. On the picosecond time scale, the dominant effect is the electron heating while the ion motion is much less important. So, in our model, the ions are assumed to be immobile. Both collisional and collective effects contribute to the plasma heating.

The electron temperature T e of the plasma is calculated in our model according to Equation (3.52)

C V,e ∂T e ∂t - ∂ ∂r . κ e ∂T e ∂r = W e -G (T e -T i ) (6.25)
where the the hydrodynamic velocity divergence, the electron viscosity effects and the thermal force have been neglected. Also, the thermal electron conductivity tensor has been assumed to be scalar κ e = κ e I (no magnetization effects) and it has been noted C V,e the background electron heat capacity.

The heating source term (3.51)

W e = dε
C V,i ∂T i ∂t = W i = G (T e -T i ) . (6.28)
Here, C V,i is the background ion thermal capacity. The thermal ion conductivity has been neglected since it is negligible in the considered time scale. Also, the energy loss of the relativistic electron beam from collisions with the ions has been neglected compared to G(T e -T i ).

The energy conservation equation of the full system consiting of the beam electrons, the background electrons, the background ions, the bremsstralung photons and the electromagnetic fields can be obtained starting from the Poynting theorem. It reads

W i + (W e -W i ) + W brem + W em + ∂ ∂r .Π = -W b = dε S 2 d 2 Ω (vSΨ) -j b .E. (6.29)
Here, due to the fact that it is usually small, the pressure force has been neglected compared to the friction force in the Ohm's law (6.21) to write ηj e 2 = j e .E. The part of the beam power density converted into the electromagnetic power density A standard method of resolution of the V-F-P equation (6.15) consists in using a spherical harmonic decomposition (see Chapter 5, section 5.2.2). This approach takes advantage of the fact that the spherical harmonics constitute a full set of orthogonal functions on the unity sphere S 2 and they are the eigen functions of the Laplace-Beltrami operator [START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]. Another approach consists in using a Cartesian tensor scalar product expansion (5.14)

W em = ∂ ∂t E 2 + B 2 8π (6.
Ψ = ∞ l=0 C 4π Ψ Ω ( ) (6.32)
where C = (2 + 1)!/2 !, Ω ( ) equals the (l -1) tensor products of Ω with itself Ω ⊗ ... ⊗ Ω and is the times contracted product. The N th order Lagrange polynomial approximation (Cartesian tensor scalar product expansion with max = N ) is strictly equivalent to the spherical harmonic expansion approximation with max = N [Johnston, 1960]. This method gives rise to a set of differential equations where each equation, describing the th order component Ψ , involves the ( + 1)th order component Ψ +1 . Consequently, the (N + 1)th order component has to be approximated to close the set of equations. This is done by imposing the PN closure relation

Ψ N +1 (r, ε, t) = S 2 ΨΩ (N +1) d 2 Ω ≈ S 2 N l=0 C 4π Ψ Ω ( ) Ω (N +1) d 2 Ω.
In practice, the models are limited to the 1st order approximation (P1) in order to make fast computations. In this approach, the decomposition components Ψ correspond to the th order angular moments of the distribution function Ψ (6.33) and the first two equations can be obtained by integrating (6.15) multiplied by Ω ( ) over the unity sphere S 2 . They read

Ψ (r, ε, t) = S 2 ΨΩ ( ) d 2 Ω,
∂Ψ 0 ∂t + ∂ ∂r . (v Ψ 1 ) - ∂ ∂ε [v (eΨ 1 .E + SΨ 0 )] = 0 (6.34)
and

∂Ψ 1 ∂t + ∂ ∂r . (v Ψ 2 ) - ∂ ∂ε [v (eΨ 2 .E + SΨ 1 )] = - e p (Ψ 0 -Ψ 2 ) .E - e γm e c Ψ 1 × B -ν Ψ 1 . (6.35)
The second order angular moment Ψ 2 is evaluated by using the P1 approximation Ψ ≈ Ψ P1 = Ψ 0 /4π + 3Ψ 1 .Ω/4π of (6.32)

Ψ 2 = ΨΩ ⊗ Ω d 2 Ω ≈ Ψ P1 Ω ⊗ Ωd 2 Ω = 1 3 Ψ 0 I (6.36)
which is the second order angular moment of an isotropic angular distribution function. Consequently, the P1 approximation (6.36) is limited to weakly anisotropic distributions and it does not allow to evaluate the anisotropic part of Ψ 2 . Moreover, Ψ P1 may become negative if the anisotropic part 3Ψ 1 .Ω/4π < 0 is greater than the isotropic part Ψ 0 /4π [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]. Another approach consists in solving the set of equations (6.34) and (6.35) with a more general closure relation. A general expression for a 2nd order tensor in a system with one preferential direction given by the vector

M1 closure

Ψ 1 /|Ψ 1 | reads Ψ 2 = 1 3 Ψ 0 I + µΨ 0 Ψ 1 ⊗ Ψ 1 |Ψ 1 | 2 - 1 3 I (6.37)
where µ = µ (Ψ 0 , Ψ 1 ) is a closure parameter depending on the two first angular moments [Eddington, 1926]. Indeed, let us define the eigenvalues χ 1 , χ 2 and χ 3 of the tensor Ψ 2 /Ψ 0 , associated with the eigenvectors u 1 , u 2 and u 3 , respectively. According to the definitions (6.33) of the angular moments of the distribution function Ψ, we deduce that

Tr Ψ 2 Ψ 0 = χ 1 + χ 2 + χ 3 = 1 Ψ 0 S 2 Tr (Ω ⊗ Ω) Ψd 2 Ω = 1. (6.38)
Due to the fact that we truncate the Cartesian tensor scalar product expansion at the 1st order, there is only one preferential direction n = Ψ 1 /|Ψ 1 | for beam electrons at the space location r and having the kinetic energy ε at time t. As a consequence, the vector (Ψ 2 /Ψ 0 ).n must be invariant under a rotation around the n-axis and so n must be an eigenvector of Ψ 2 /Ψ 0 [Levermore, C.D., 1984]. Let us choose u 1 = n. By symetry, the plane perpendicular to u 1 must also be a sub-eigenspace of Ψ 2 /Ψ 0 so that the eigenspace (u 1 , u 2 , u 3 ) is an orthonormal basis of R 3 . It implies that the corresponding eigenvalues read

χ 2 = χ 3 = 1 -χ 1 2 , (6.39)
according to the trace identity relation (6.38). Let us note {α ij , (i, j) ∈ {1, 2, 3} 2 } the components of the tensor Ψ 2 /Ψ 0 in this orthonormal basis (u 1 , u 2 , u 3 ) :

Ψ 2 Ψ 0 = 3 i=1 3 j=1 α ij u i ⊗ u j . (6.40)
By definition of eigenvectors, we have

∀k ∈ {1, 2, 3}, Ψ 2 Ψ 0 .u k = χ k u k . (6.41)
Injecting (6.40) in (6.41), we obtain ∀k ∈ {1, 2, 3},

3 i=1 3 j=1 α ij u i ⊗ u j .u k = χ k u k ⇒ 3 i=1 3 j=1 α ij (u i .u k ) u j = χ k u k ⇒ 3 j=1 α kj u j = χ k u k (6.42)
and we deduce the components of Ψ 2 /Ψ 0 :

∀(k, j) ∈ {1, 2, 3} 2 , α kj = δ kj χ k . (6.43)
Accounting for (6.43) and the axisymmetry relation (6.39) in (6.40), we finally obtain the closure relation (6.37) with

µ = 3χ 1 -1 2 . (6.44)
In radiation transfer theory, χ 1 is called the Eddington factor [Minerbo, 1977] [ Levermore, 1979] [ Pomeraning, 1981] [Dubroca andFeugeas, 1999]. Finally, according to the definition (6.33) of Ψ 2 , the tensor Ψ 2 /Ψ 0 is symetric and positive-definite since it is the second order angular moment of a positive unit density of probability on the unit sphere S 2 . Thus,

∀k ∈ {1, 2, 3}, 0 ≤ χ k ≤ 1. (6.45)
The closure relation (6.37) is exact for both totally isotropic angular distributions (µ = 0 or χ 1 = 1/3) and totally anisotropic angular distributions (µ = 1 or χ 1 = 1).

According to the method derived by [Minerbo, 1977] [Minerbo, 1978] in the context of radiation transport and extended to electron transport by [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF], the underlying parameter µ can be estimated by maximizing the local angular entropy of beam electrons with a given kinetic energy under the constraints of the definition of the angular moments Ψ 0 and Ψ 1 . Let us demonstrate so the closure relation (6.37) following this approach. The maximum local angular entropy principle is discussed in detail in the next section 6.2.3. The local angular entropy per unit energy of the beam is defined by

H ε [Ψ] = - S 2 Ψ (ln Ψ -1) d 2 Ω. (6.46)
The distribution function Ψ M1 , maximizing the local angular entropy H ε [Ψ], is obtained by the method of Lagrange multipliers. This is a strategy for finding the local maxima and minima of a function subject to equality constraints. For instance, we want to maximize the local angular entropy (6.46) subject to the definition (6.33) of the two first angular moments Ψ 0 and Ψ 1 . We introduce the 4dimensional vector α = (α 0 , α 1 ) T where α 0 and α 1 , called Lagrange multipliers, are living in the same space than the angular moments Ψ 0 and Ψ 1 , respectively. To incorporate these constraints, we introduce the functional of this maximization problem, called Lagrangian, defined by

L ε [Ψ, α] = H ε [Ψ] -α 0 Ψ 0 - S 2 Ψd 2 Ω -α 1 . Ψ 1 - S 2 ΨΩd 2 Ω . (6.47)
The critical points of the Lagrangian (6.47) occur at saddle points. Note that

∂L ε ∂α [Ψ = Ψ M1 , α] = 0 (6.48)
implies the constraints of the definition (6.33) of the two first angular moments with Ψ = Ψ M1 , = 0 and = 1. The maximizing distribution function Ψ M1 is defined by the Equation

∂L ε ∂Ψ [Ψ = Ψ M1 , α] = 0 (6.49)
where ∂/∂Ψ is the functional derivative. It is defined by

S 2 ∂L ε ∂Ψ [Ψ, α]δΨd 2 Ω = lim →0 L ε [Ψ + δΨ, α] -L ε [Ψ, α] (6.50)
where δΦ is an arbitrary function of Ω and δΨ is the variation of Ψ. According to the definition (6.46) of the beam local angular entropy and the Lagrangian (6.47) of this maximization problem, we find

L ε [Ψ + δΨ, α] -L ε [Ψ, α] = S 2 δΨ (-ln Ψ + α 0 + α 1 .Ω) d 2 Ω + O( 2 ) (6.51) which gives S 2 ∂L ε ∂Ψ [Ψ, α]δΨd 2 Ω = S 2 δΨ (-ln Ψ + α 0 + α 1 .Ω) d 2 Ω. (6.52)
Therefore, according to the definition (6.49) of the maximizing distribution function Ψ M1 , we deduce that, whatever the function δΨ,

S 2 δΨ (-ln Ψ M1 + α 0 + α 1 .Ω) d 2 Ω = 0. (6.53)
The only possible solution of the previous equation is

Ψ M1 = exp (α 0 + α 1 .Ω) . (6.54)
Then, the Lagrange multipliers α 0 and α 1 have to be evaluated in terms of physical quantities by using the constraints given by Equation (6.48). One has

Ψ 0 = S 2 Ψ M1 d 2 Ω = exp (α 0 ) 4π sinh |α 1 | |α 1 | (6.55)
and

Ψ 1 = S 2 Ψ M1 Ω d 2 Ω = exp (α 0 ) 4π sinh |α 1 | |α 1 | coth |α 1 | - 1 |α 1 | α 1 |α 1 | . (6.56)
The detailed calculation of these two integrals can be found in [START_REF] Wright | Relativistic distribution functions and applications to electron beams[END_REF]. From (6.55) and (6.56), one deduces an expression for the anisotropy vector

Ω ε = Ψ 1 Ψ 0 = coth |α 1 | - 1 |α 1 | α 1 |α 1 | . (6.57)
It is defined as the mean propagation direction of the electrons having the energy ε at the position r at the time t. Due to the triangular inequality applied to (6.33) with = 1, one has

0 ≤ |Ω ε | ≤ 1. (6.58)
Although the bijective relation (6.57) cannot be inverted analytically, one can fit the real values of α 1 by

α 1 ≈ 3Ω ε 1 - |Ω ε | 2 2 1 + |Ω ε | 2 . ( 6
Ψ M1 → Ψ 0 δ [Ω -Ω ε ]
where δ is the Dirac distribution. By substituting (6.60) in the definition (6.33)

of Ψ 2 , one obtains the closure relation

Ψ 2 ≈ S 2 Ψ M1 Ω ⊗ Ω d 2 Ω = 1 3 Ψ 0 I + µΨ 0 Ψ 1 ⊗ Ψ 1 |Ψ 1 | 2 - 1 3 I (6.61)
with the closure parameter

µ = 1 -3 coth |α 1 | |α 1 | - 1 |α 1 | 2 ≈ |Ω ε | 2 2 1 + |Ω ε | 2 . (6.62)
Here, we refer again to [START_REF] Wright | Relativistic distribution functions and applications to electron beams[END_REF] for the detailed calculation of this integral. This M1 closure relation (6.61) provides an interpolation between the local beam-like case where all electrons at the position r with the energy ε move in the same direction Ω ε = e z and the local isotropic case where all electrons at the position r with the energy ε move in all directions with the same probability

(Ω ε = 0). It preserves consequently the advantage of the P1 model in fast computing while angular distributions are described with a much better precision.

Properties of the M1 closure

angular moments Ψ 0 and Ψ 1 provides the best possible closure relation for electrons, which are charged particles evolving under the action of the Lorentz force. We refer here to Minerbo who justified this closure concerning photons by saying : "In communication theory, it is shown that the information content is the negative of the entropy of the distribution. Thus, by using the maximum entropy criterion, one avoids introducing information that is not available. This approach is conceptually superior to the use of an ad hoc model for the intensity" (i.e. Ψ 1 ). Concerning the relativistic electrons considered here, even if their angular scattering tends to isotropize their angular distribution and increase their angular entropy, self-generated electromagnetic fields may not follow the same trend. One can deduce the local angular entropy dissipation rate (∂H ε /∂t) col starting from the V-F-P equation (6.15) and by integrating it over the unity sphere in the momentum space S 2 . Here, we define the local angular entropy as

H ε [Ψ](r, ε, t) = - S 2 Ψ ln Ψ -1 + ln v p 2 d 2 Ω. (6.63)
It is different from (6.46) by a constant term and therefore does not modify the results obtained in the previous section 6.

dH ε dt = ∂H ε ∂t + ∂ ∂r . S 2 vΩφ d 2 Ω - ∂ ∂ε S 2 v (eE.Ω + S) φ d 2 Ω = ∂H ε ∂t col (6.65) with ∂H ε ∂t col = ν 2 S 2 1 φ ∂φ ∂θ 2 + ∂φ ∂ϕ 2 d 2 Ω -Ψ 0 v p 2 ∂ ∂ε p 2 S .
(6.66)

M1 Closure and Collisional Effects of Fast Electron Transport

The local angular entropy dissipation rate (∂H ε /∂t) col , which is the angular entropy time evolution of a M1 mesoscopic particle following its trajectory in the (r, ε)-space, contains two terms according to the previous equation (6.53). The term depending on the isotropization rate ν is positive and consequently increases the angular entropy H ε with time. But, the sign of (∂H ε /∂t) col depending also on the total stopping power S is not defined. Let us estimate so the contribution of each term assuming that It means that, above these values, the energy exchange between energy groups of beam electrons, due to their slowing down, does not allow to consider each group as a closed system and thus does not allow to justify the maximum angular entropy criterion. This is due to the fact that the slowing down rate is γ times greater than the angular diffusion rate (see Equation (6.19)). This limitation is compatible with the characteristics of laser-generated electrons. The number of electrons with kinetic energies above these material-dependant threshold values represent only a small part of the whole beam electron population (see Chapter 1, section 1.2.7). Moreover, being more energetic, these electrons propagate faster in the target compared to others so that their local angular distribution is quasi-anisotropic and therefore, is well described by the M1 model. Thus, one can conclude that the collisional effects of laser-generated electron fully justify the M1 closure for their transport in solids and dense plasmas. Indeed, by orienting the Cartesian coordinates such that

S 2 1 φ ∂φ ∂θ 2 + ∂φ ∂ϕ 2 d 2 Ω ∼ Ψ 0
α 1 = 8 ln 2 ∆ϕ 2     sin ϕ 0 0 cos ϕ 0     ⇔ Ω ε = coth 8 ln 2 ∆ϕ 2 - ∆ϕ 2 8 ln 2     sin ϕ 0 0 cos ϕ 0     , (6.68)
we obtain the angular distribution function averaged over the polar angle θ π 0

Ψ M1 sin θdθ = Ψ 0 |α 1 | 4 sinh |α 1 | ({I 1 [|α 1 | cos (ϕ -ϕ 0 )] + L -1 [|α 1 | cos (ϕ -ϕ 0 )]} ≈ Ψ 0 2π ∆ϕ 2 8 ln 2 exp -4 ln 2 (ϕ -ϕ 0 ) 2 ∆ϕ 2 (6.69)
which is nothing else than the 0th order statistical normal law obtained in the multiple scattering theory by Moliere (see Chapter 4, section 4.3.2). They are strictly similar in the anisotropic limit and diverge slightly in the isotropic limit, as illustrated in the right panel of Figure 6.4.

The integration has been carried out in the same way as for (6.55), (6.56) and (6.61) that is to say, by expanding the exponential in power series, by performing the integration of the different sinus power according to the formula 3.621 from [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF] and by applying the doubling formula for Gamma functions 8.335.1 from [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF]. Then, the sum was split in odd and even terms which were compared with the definitions 9.6.10 and 12.2.1 from [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] of the modified Bessel function of the first kind I ν and the modified Struve function L ν .

However, even if the collisional effects can justify the M1 closure in the problem of fast electron transport, the third term in the left hand side of (6.65) shows that the self-generated electric field affects the local angular entropy time evolution (i.e. the angular entropy obtained without following the trajectory of a M1 mesoparticle in the (r, ε)-space). This is not the case for the Boltzmann entropy, since by integrating over all kinetic energies (6.65), this term vanishes to give

dH dt = ∂H ∂t + ∂ ∂r . ∞ 0 S 2 vΩφ d 2 Ωdε = ∞ 0 ∂H ε ∂t col dε. (6.70)
It must be emphasized here that, similarly to the maximum local angular entropy criterion, the Boltzmann H-theorem does not apply here. Indeed, contrary to the Boltzmann H-theorem demonstration presented in Appendix A, section A.1.5, the right hand side term of (6.70) is not necessarily positive. It would have been positive if the whole electron population (beam electrons + background electrons) were taken into account in Ψ, which is not the case in our "hybrid" assumption. Energy exchanges between the beam and background electrons do not allow to consider the beam electrons as a closed system.

M1 Closure and Collective Effects of Fast Electron Transport

In order to illustrate the M1 closure in a collisionless case, let us evaluate the dispersion relation (3.39) for the resistive filamentation instability with the M1 model. As in Chapter 3, section 3.2.

Ψ (0) = Ψ M-J = n (0) b γ 2 βk B T b /m e c 2 4πγ b (m e c 2 )K 2 m e c 2 γ b k B T b exp - m e c 2 k B T b γ -β b p z m e c . (6.71)
Being a particular case of distribution functions of the kind (6.60), one can deduce easily the angular moments of the distribution function at the equilibrium. They read

Ψ (0) 0 = 4π sinh |α 1 | |α 1 | n (0) b γ 2 βk B T b /m e c 2 4πγ b (m e c 2 )K 2 m e c 2 γ b k B T b exp - γm e c 2 k B T b , (6.72) Ψ (0) 1 = Ψ (0) 0 coth |α 1 | - 1 |α 1 | α 1 |α 1 | (6.73)
and

Ψ (0) 2 = 1 3 Ψ (0) 0 I + µ (0) Ψ (0) 0   Ψ (0) 1 ⊗ Ψ (0) 1 |Ψ (0) 1 | 2 - 1 3 I   (6.74)
where

µ (0) = 1 -3 coth |α 1 | |α 1 | - 1 |α 1 | 2 and α 1 = m e c 2 k B T b β b p m e c e z .
The linearized M1 equations (6.34), (6.35) and (6.61) read

∂δΨ 0 ∂t + ∂ ∂r . (v δΨ 1 ) -e ∂ ∂ε vΨ (0) 1
.δE = 0, (6.75)

∂δΨ 1 ∂t + ∂ ∂r . (v δΨ 2 ) -e ∂ ∂ε vΨ (0) 2 .δE = - e p Ψ (0) 0 -Ψ (0) 2 .δE - e γm e c Ψ (0) 1 × δB (6.76)
and

δΨ 2 = ∂Ψ 2 ∂Ψ 0 Ψ (0) 0 , Ψ (0) 1 δΨ 0 + ∂Ψ 2 ∂Ψ 1 Ψ (0) 0 , Ψ (0) 1
.δΨ 1 . (6.77)

M1 MODEL FOR RELATIVISTIC ELECTRON BEAM TRANSPORT

Let us note the tensors

T 0 = ∂Ψ 2 ∂Ψ 0 Ψ (0) 0 , Ψ (0) 1 and ∀i ∈ {x, y, z}, T 1,i = ∂Ψ 2 ∂Ψ 1,i Ψ (0) 0 , Ψ (0) 1 such that δΨ 2 = T 0 δΨ 0 + T 1,x δΨ 1,x + T 1,y δΨ 1,y + T 1,z δΨ 1,z . (6.78)
Their general expressions are

∀(i, j) ∈ {x, y, z} 2 , (T 0 ) ij = 1 3 δ ij - Ψ (0) 1 Ψ (0) 0 2   1 + 2 Ψ (0) 1 Ψ (0) 0 2     Ψ (0) 1,i Ψ (0) 1,j Ψ (0) 1 2 - 1 3 δ ij   + µ (0)   Ψ (0) 1,i Ψ (0) 1,j Ψ (0) 1 2 - 1 3 δ ij   (6.79)
and

∀(i, j) ∈ {x, y, z} 2 , (T 1,i ) xj = Ψ (0) 1,i Ψ (0) 0   1 + 2 Ψ (0) 1 Ψ (0) 0 2     Ψ (0) 1,x Ψ (0) 1,j Ψ (0) 1 2 - 1 3 δ xj   + µ (0) Ψ (0) 0 Ψ (0) 1 4 Ψ (0) 1 2 δ ix Ψ (0) 1,j + δ ij Ψ (0) 1,x -2Ψ (0) 1,x Ψ (0) 1,i Ψ (0)
1,j .

(6.80)

Performing the Fourier transform of Equations (6.75), (6.76) and (6.78), neglecting δ E x and using the Maxwell-Faraday equation for the magnetic field (δ B = -(k x c/ω)δ E z e y ), we obtain the matrix relation

A.δ X = -ieY (0) δ E z (6.81)
where

δ X =        δ Ψ 0 δ Ψ 1,x δ Ψ 1,y δ Ψ 1,z        , Y (0) =        (d/dε) vΨ 1,z (0) -k x Ψ 1,z (0) /γm e ω 0 0        and A =        -ω k x v 0 0 k x v(T 0 ) xx k x v(T 1,x ) xx -ω k x v(T 1,y ) xx k x v(T 1,z ) xx k x v(T 0 ) xy k x v(T 1,x ) xy k x v(T 1,y ) xy -ω k x v(T 1,z ) xy k x v(T 0 ) xz k x v(T 1,x ) xz k x v(T 1,y ) xz k x v(T 1,z ) xz -ω        .
Let us consider here the limit T b → 0 for simplicity. In this case, the Lagrange multiplier |α 1 | → ∞ and the equilibrium distribution function (6.71) becomes purely anisotropic and monoenergetic, giving

A =        -ω k x v 0 0 k x v -ω 0 -k x v 0 0 -ω 0 0 k x v 0 -ω        . (6.83)
Then, the solution of equation (6.81) reads

δ X = -ieA -1 .Y (0) δ E z = i e ω 3        (d/dε) vΨ 0 (0) k 2 x c 2 β 2 + ω 2 -k 2 x c 2 β(Ψ (0) 0 /γm e c) k x ω v(d/dε) vΨ 0 (0) -c(Ψ 0 (0) /γm e c) 0 k 2 x v v(d/dε) vΨ 0 (0) -c(Ψ 0 (0) /γm e c)        δ E z . (6.84)
This provides us with the expression for the beam current density perturbation

δ j b = -e ∞ 0 δ Ψ 1,z vdεe z = i k x c ω 2 ω 2 b β 2 b 4πωγ b 1 + 3 γ 2 b δ E z e z (6.85)
where

ω b = 4πn (
+ k x c ω b 2 + 1 γ 3 b δ ω b 2 - 1 γ b β b k x c ω b 2 = 0.
A term ∝ 1/γ 3 b has disappeared in the second term, while another term ∝ 3/γ 2 b has appeared in the third term. This is due to the fact that the Maxwell equations depend on the hydrodynamic moments n b and j b of the beam distribution function Ψ (0) and not directly on its angular moments Ψ 0 and Ψ 1 . Indeed, contrary to the reference case of the Vlasov equation (3.37) where the currents in (3.36) are calculated with p x , p y and p z -integrals, in the M1 approximation there appear only εintegrals since the integration over angles Ω = p/p has already been performed before the coupling of the equations. Therefore, the integration of Y

Conclusion

Since the development of thermodynamics, physicists have discovered two main principles that govern physical processes. The first one is the least action principle. A particle evolves from a state A, at the space location r A and time t A , to a state B at the space location r B and time t B , by following the trajectory from r A to r B that minimizes its action. The action of the particle being its kinetic energy, minus its potential energy, integrated over time between t A and t B , it means that particles will always move following the trajectory that minimizes its kinetic energy compared to its potential energy, imposed by the surrounding force fields (there is a similar law concerning photons called the Fermat's principle). This "energy minimization principle" is at the origin of many theories in Physics from optical geometry and Newton's mechanics to quantum electrodynamics [START_REF] Feynman | The Feynman Lectures on Physics[END_REF].

However, another tendency arises when studying a system consisting of a lot of particles. In principle, one may think that all the physical properties of such a system can be determined by its microscopic state, i.e. by the description of the positions and momenta of all particles. Actually, because the number of particles is so large, the details of the motion of individual particles is mostly irrelevant to the behavior of the system as a whole. Indeed, according to thermodynamics and statistical physics, the macroscopic state of a system is defined by a distribution on the microstates that are accessible to the system in the course of its thermal fluctuations. In order to describe this discrete set (continuous set) of microscopic states of a system, J.W. Gibbs introduced the statistical entropy

H = -k B i p i ln p i H = -k B f ln f , respectively (6.88)
where E i is the energy of the microscopic state i, and p i is the probability that it occurs during the system's fluctuations (f is the particles distribution function, respectively). The difference between this last expression, the Boltzmann entropy (6.64) and the local angular entropy (6.46) or (6.63) simply comes from an additive constant. The above expression of the statistical entropy is also used in communication theory and it is called the Shannon Entropy. It is the opposite of the information content of signals, consisting in a succcession of binary numbers. L. Boltzmann showed that the statistical entropy of an isolated system of particles can only rise with time dH dt ≥ 0 (6.89) (see Appendix A, section A.1.5) and that the thermodynamic equilibrium is obtained when all the accessible microscopic states of the system are equally likely. Thus. the thermodynamic equilibrium is the configuration corresponding to the maximum of a system's entropy for a given set of accessible microscopic states. In other words, the thermodynamic equilibrium is the macroscopic configuration in which the lack of information is maximal. This is the second main principle mentioned previously.

The difficulties happen due to the fact that both principles ("Energy minimization" and "Entropy maximization") are mutually exclusive. For example, according to the least action principle, the particle can move from r B to r A by the same trajectory as he moved from r A to r B . Indeed, the time is reversible in the equations describing the particle dynamics (Newton's fundamental principle of mechanics, Einstein's fundamental principle of mechanics or Schrödinger's fundamental principle of quantum mechanics). It is not the case for a system of many particles. The increase of entropy can proceed in an irreversible way, since the second law of Thermodynamics (6.89) is not time-reversible.

Concerning fast electron transport in solids or dense plasmas, these two principles are in competition. Collective effects are described by the particles dynamics in the self-generated electromagnetic fields and therefore follows the "Energy Minimization Principle". Oppositely, the collisional effects tend to homogenize the particles momentum distribution and are therefore better described by the "Entropy Maximization Principle". In this section, it has been shown that collisional effects ensure that the local angular entropy dH ε dt ≥ 0 (6.90)

for fast electrons with kinetic energies less than a threshold value ε th , depending on the atomic number Z of the material. Concerning laser-generated fast electron beams, this threshold ε th is sufficiently large compared to their typical kinetic energies. Moreover, laser-generated fast electrons with energies greater than ε th propagate deeper in the laser-irradiated target and have consequently a local angular distribution close to the anisotropic limit for which the M1 closure is exact. In order to check the M1 closure in a collisionless case, where there is more arguments in favor of "minimizing the action" instead of "maximizing the angular entropy", we have derived the dispersion relation of the resistive filamentation instability, obtained with the M1 model, in the limit T b → 0, and we have compared it to the reference Vlasov case presented in Chapter 3, section 3.2.3. An advantage of the M1 closure in this derivation is that the angular moments of the Maxwell-Juttner distribution function of the beam Ψ (0) are easy to obtain since the Maxwell-Jutnner distribution is a particular case of the more general distribution function Ψ M1 . That can be explained by the following mathematical relation (6.91) This relation shows that the M1 approximation overestimates the Boltzmann entropy of the beam and that the maximum angular entropy distribution function (6.61) can be obtained by multiplying the maximum entropy Maxwell-Juttner distribution function (6.71) with a Dirac distribution in energy

H[Ψ M-J ] = max Ψ {H[Ψ]} = max Ψ ∞ 0 H ε [Ψ]dε ≤ ∞ 0 max Ψ {H ε [Ψ]}dε ≤ ∞ 0 H ε [Ψ M1 ]dε.
Ψ M1 (r, ε 0 , Ω, t) = δ[ε -ε 0 ]Ψ M J (r, ε, Ω, t), (6.92)
as explained by [START_REF] Wright | Relativistic distribution functions and applications to electron beams[END_REF]. Besides, this study has shown that the M1 model overestimates the instability growth rate for γ b 1. However, the resistive filamention instability is reproduced with an error of less than 15 % for γ b ≥ 3. This result is surprising since there is no reason to maximize the angular entropy in such a collisionless case. It can be explained by the argument given by Minerbo, saying that "by using the maximum entropy criterion, one avoids introducing information that is not available." Nevertheless, it will be shown in Chapter 8 that the closure relation (6.61) allows for a sufficiently accurate and fast computation of the V-F-P equation (6.15) with an arbitrary degree of anisotropy. Moreover, contrary to P1, it continuously relates the anisotropic and isotropic regimes while satisfying the physical constraints Ψ 0 ≥ 0 and 0 ≤ |Ω ε | ≤ 1 thanks to the exponential form of the underlying distribution function (6.60). Several numerical experiments in the non-relativistic regime, carried out for the comparison of the M1 model with the full V-F-P code [START_REF] Duclous | High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications[END_REF], have shown good agreements with a much reduced computation time [START_REF] Mallet | An Entropic Scheme for an Angular Moment Model for the Classical Fokker-Planck-Landau Equation of Electrons[END_REF].

Physics of the Background Material

The response of the background particles to the beam propagation is described by the diffusion equation of the magnetic field (6.24), the Ohm's law (6.23) and the heat equations (6.25) and (6.28). These equations depend on the electrical resistivity η, thermal electron conductivity κ e , thermal capacities C V,e and C V,i and the electron-ion coupling factor G. All these parameters depend on the density and temperatures of the target material. In the case of laser-irradiated solid targets, the solid is initially cold and these different parameters are given by the solid state physics theory. One talks about the lattice thermal capacity and electron-lattice coupling factor instead of the ion thermal capacity and electron-ion coupling factor, in a plasma. However, the collisional and collective losses of the fast electron beam rapidly induce the ionization of the material and the heating of the background free electrons, which in turn heat the lattice. The material enters consequently the liquid and then the plasma state. While the plasma parameters are known, there is no theory predicting the material parameters in the transient domain between these states. In the case of insulators or Warm Dense Matter (WDM), quantum molecular dynamics computations allow to evaluate these parameters. This section is dedicated to the description of these parameters in metals.

PHYSICS OF THE BACKGROUND MATERIAL

The ionization state Z * is an important parameter, which allows us to determine the density of free electrons n e,f = Z * n i and bound electrons n e,b = (Z -Z * )n i in the solid. As explained in Chapter 3, section 3.3.1, Z * can be estimated by the More Formula (3.43) based on the Thomas-Fermi model. However, the Thomas-Fermi model is inadequate for capturing the metal-insulator transition since it neglects any atomic structure effects on the ionization equilibrium. Contrary to the treatment proposed by [Desjarlais, 2001], which consists in imposing a smooth transition from Thomas-Fermi to non-ideal Saha at low temperature, we impose phenomenologically the adequate value of the ionization state Z * = Z c at a low temperature, where Z c is the number of electrons per atom in the conduction band (s-band), as

Z * = {[(1 -f e ) Z c ] a 1 + [f Z * T F ] a 1 } 1/a 1 (6.93)
where Z * T F is the ionization state given by (3.43) and

f = K 1/Z * T F 2 with K = 1 2 1 + tanh k B T e -a 2 E F a 3 E F .
Here, E F is the Fermi energy given by Equation (3.42) with n e = Z c n i , a 1 and a 2 are parameters that allow for adjusting a smooth transition from Z c to Z * T F while a 2 determines the temperature at which the transition occurs. For example, Z c = 3 , a 1 = a 2 = 1 and a 3 = 0.05 for Aluminum, where γ exp is the Sommerfeld parameter. From a theoretical point of view, the thermal capacity of degenerate electrons at low tempearatures is defined as

Z c = 1, a 1 = 2,
C cold V,e = dU e dT e = ∞ 0 (ε -E F ) ∂f F D ∂T e g(ε)dε (6.96)
where U e is the total electron thermal energy, ε is the electron kinetic energy,

f F D (ε, µ, T e ) = 1 1 + exp ε -µ k B T e (6.
C cold V,e = γ th T e with γ th = π 2 2 n e k B E F . (6.99)
However, γ th is usually of the expected magnitude, but often does not agree very closely with the measured value γ exp . Moreover, computational analysis based on first-principles electronic structure calculations of the electron DOS have recently shown large deviations from the commonly used linear approximations (6.95) [START_REF] Lin | Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium[END_REF]] [Bévillon et al., 2014]. An interpolation formula presents a good compromise, allowing to describe the electron thermal capacity in the whole range of considered temperatures :

C V,e = (γT e ) -2 + 3 2 Z * n i k B -2 -1/2 (6.100)
where γ = 912 erg.cm -3 .K -2 for Aluminum, 968 erg.cm -3 .K -2 for Copper and 5428.8 erg.cm -3 .K -2

for Tantalum. The temperature dependence of the electron heat capacity is illustrated in the left panel of Figure 6.7.

At the plasma state, the ion thermal capacity is given by the ideal gas expression (see Annexe B, section B.1.1)

C hot V,i = 3 2 n i k B . (6.101)
In the cold solid phase, the lattice thermal capacity is due to phonons. According to experiments, it can be written

C cold V,i = A exp T i 3 . (6.102)
According to the Debye theory of phonons, the expression for this constant below the Debye temperature T D reads

A th = 12π 4 n i k B 5T D 3 . (6.103)
However, above T e = 300 K, the Einstein model is sufficient to describe the lattice heat capacity. It is calculated by using the Bose-Einstein distribution function for the phonons

f BE (ω, T i ) = 1 exp ω k B T i -1 (6.104)
where ω is the phonon frequency and by assuming the Einstein phonon DOS g(ω) = δ (ω -ω E ). Then, the lattice heat capacity reads

C cold V,i = dU i dT i = ∞ 0 ω ∂f BE ∂T i g(ω)dω = k B n i ω E k B T i 2 exp ω E k B T i exp ω E k B T i -1 2 (6.105)
where ω E is the Einstein temperature. For example, ω E = 284 K for Aluminum, 278 K for Copper and 193 K for Tantalum. Contrary to the Debye temperature, the Einstein temperature is difficult to find in the literature for some metals. In this case, one can use the empirical relation ω E /k B T D ≈ (π/6) 1/3 . However, the Einstein heat capacity tends to k

B n i instead of (3/2)k B n i when T i ω E .
Consequently, a good compromise, allowing to describe the ion thermal capacity in the whole range of considered temperatures, can be obtained by estimating it as As explained in Chapter 3, section 3.3.1, the Lee-More model [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] assumes equal electron and ion temperatures. Another model that takes into account T i = T e was proposed by [START_REF] Chimier | Heating model for metals irradiated by a subpicosecond laser pulse[END_REF]. Here, we propose to merge these two models. In our approach, the electrical resistivity reads η = m e ν e n e e 2

C V,i = 3 2 k B n i ω E k B T i 2 exp ω E k B T i exp ω E k B T i -1 2 . ( 6

1

A α µ k B T e (6.107) and the thermal electron conductivity reads

κ e = γ L T e η = n e k 2 b T e m e ν e A β µ k B T e with γ L = A β µ k B T e A α µ k B T e k B e 2 .
(6.108)

They are plotted in Figure 6.8 for Aluminum, Copper and Tantalum. Here, the free electron density is given by n e = Z * n i where Z * is computed according to (6.93) so that at low temperatures n e = Z c n i is the density of s-band electrons. The Lorenz factor γ L is computed according to the functions A α and A β provided by [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] (see Appendix B, section B. The electron collision frequency is computed according to the harmonic mean

ν e = (ν e-ph + ν e-e ) -2 + ν -2 c + ν -2 hot -1/2 . (6.111)
At hot plasma temperatures, ν e is therefore given by

ν hot = ν ei γ E for η and ν hot = ν ei δ T
for κ e (6.112)

where γ E and δ T are the electron-electron collision correction factors following the notation from [START_REF] Spitzer | Transport Phenomena in a Completely Ionized Gas[END_REF]. They can be fit as a function of Z * according to the table given by [START_REF] Spitzer | Transport Phenomena in a Completely Ionized Gas[END_REF] (see the fits given in Annexe B, section B.2.2). For ν ei , we use the Lee-More electron-ion collision frequency

ν ei = 2 √ 2π(Z * ) 2 n i e 4 ln Λ ei 3 √ m e (k B T e ) 3/2 1 + exp - µ k B T e F 1/2 µ k B T e (6.113)
where the Coulomb logarithm ln Λ ei is given in Appendix A, section A. (6.114)

According to [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] and [START_REF] Chimier | Heating model for metals irradiated by a subpicosecond laser pulse[END_REF], we impose that the mean free path proposed by [START_REF] Chimier | Heating model for metals irradiated by a subpicosecond laser pulse[END_REF], the electron-electron collision frequency is

ν e-e = A ν k B T F T 2 e (6.115)
and the electron-phonon collision frequency is

ν e-ph = k s 2e 2 k B 2 2k B T F m e T i (6.116)
where k s and A ν are parameters depending on the material. At the ambient temperature, the electron-phonon collision frequency ν e-ph is the main contribution in the electron momentum dumping. Therefore, while the parameter k s can be obtained by imposing the measured value of the electrical resistivity m e ν e-ph /n e e 2 at the ambient temperature, the parameter A ν of the electron-electron collision frequency ν e-e is usually unknown. Moreover, it has been recently shown experimentally [START_REF] Fourment | Experimental determination of temperature-dependent electron-electron collision frequency in isochorically heated warm dense gold[END_REF]] that collisions of s-band with d-band electrons make an important contribution to the electron collision frequency ν e for transition and noble metals while the expression (6.115) corresponds to the low one-temperature asymptotics accounting for the collisions between s-band electrons only (Fermi liquid model).

[ [START_REF] Inogamov | Thermal conductivity of metals with hot electrons[END_REF] and [START_REF] Petrov | Thermal conductivity and the electron-ion heat transfer coefficient in condensed media with a strongly excited electron subsystem[END_REF] have developed a semi-analytical model by using the electron kinetic equation, the matrix element for the scattering probability and a screened

Coulomb potential describing the interaction between s-band electrons and s or d-band electrons. In their approach, the electron-electron collision frequency is given by ν e-e = ν s-s + ν s-d . (6.117)

According to [START_REF] Inogamov | Thermal conductivity of metals with hot electrons[END_REF], the s-s electrons collision frequency reads where f F D the Fermi-Dirac electron distribution function (6.97), ε 1 is the energy of electrons 1 and ν s-s (ε 1 ) is the collision frequency for an electron 1 colliding electrons 2 having the energies ε 2 given by

ν s-s = ∞ 0 - ∂f F D ∂ε (ε 1 )ν s-s (ε 1 )dε 1 ∞ 0 - ∂f F D ∂ε (ε 1 )dε 1 (6.118)
ν ss (k 1 ) = 2m 2 e (2π) 3 5 k 1 dq dk 2 dα dβ k 2 U (q) 2 S(k 1 , k 2 , α, β)δ[α -β]. (6.119) U (q) = 4πe 2 q 2 + k screen 2
is the screened Coulomb repulsion of electrons with the screening reciprocical length, estimated by the Lindhard screening length k screen = 4πe 2 /(∂µ/∂n e ).

S(k

1 , k 2 , α, β) = f F D (ε 2 ) [1 -f F D (ε 1 -α)] [1 -f F D (ε 2 + β)] + f F D (ε 1 -α)f F D (ε 2 + β) [1 -f F D (ε 2 )]
is the statistical factor describing the admissible energy states for the electrons 1 and 2 before and after their binary collision, α is the energy loss of the electron 1 and β is the energy gain of the electron 2 during the collision. The Dirac distribution δ[α -β] describes the energy conservation in the collision. First, the authors consider the one-band metal for all s band electrons with the simple parabolic dispersion law ε(k) = 2 k 2 /2m e . The (α, β)-integration is straighforward and gives

s(α) = dα dβ S(k 1 , k 2 , α, β)δ[α -β] = k B T e 1 + exp ε 1 -µ k B T e 1 + exp ε 2 -µ k B T e exp ε 2 -µ k B T e 1 -exp ε 1 + ε 2 -2µ k B T e × ln     exp ε 1 -µ k B T e + exp α k B T e 1 + exp α + ε 2 -µ k B T e     .
(6.120)

Then, the authors split the domain of integration as illustrated in the left panel of Figure 6.12 to finally obtain

ν s-s (k 1 ) = 2m 2 e (2π) 3 5 k 1 (I a1 + I a2 + I b + I c + I d ) (6.121)
where

I a1 = k 1 0 dq k 1 +q k 1 -q dk 2 k 2 U (q) 2 s(α ) -s(β ) , I a2 = ∞ k 1 dq k 1 +q -k 1 +q dk 2 k 2 U (q) 2 s(α ) -s(β ) , I b = ∞ 0 dq ∞ k 1 +q dk 2 k 2 U (q) 2 s(α ) -s(α ) , I c = 0 I d = k 1 0 dq k 1 -q 0 dk 2 k 2 U (q) 2 s(β ) -s(β )
with β = ( 2 /2m e ) q 2 -2k 2 q , β = ( 2 /2m e ) q 2 + 2k 2 q , α = -( 2 /2m e ) q 2 + 2k 1 q and α = -( 2 /2m e ) q 2 -2k 1 q . The dependence of ν ss (k 1 ) on the density is implicitly taken into account via the chemical potential µ according to the relation

n e = Z * n i = ∞ 0 f F D (ε)g(ε)dε (6.122)
where g(ε) is the free electron gas DOS (6.98). In the limit T e → 0 in (6.121), which implies I a1 +

I a2 + I b + I d → I a1 + I a2
and by approximating the s-band electron collision frequency by its value in the vicinity of the Fermi surface k F = 2m e E F / 2 , i.e. by approximating ν s-s ≈ ν ss (k F ), the authors finally estimate

A ν = π 8 E F m e 3 e 4 6 k F 4 (2k F /k screen ) 4 1 + (2k F /k screen ) 2 + 2k F k screen 3 + arctan 2k F k screen . (6.123)
According to [START_REF] Petrov | Thermal conductivity and the electron-ion heat transfer coefficient in condensed media with a strongly excited electron subsystem[END_REF], the s-d electron collision frequency is estimated with the same method but, with two terms in the electron DOS as

g(ε) = g s (ε) + g d (ε) (6.124)
where, for s-band electrons, the authors use

ε = ε s + 2 k 2 2m s (6.125) and g s (ε) =      1 2π 2 2m s 2 3/2 √ ε -ε s if ε ≥ ε s 0 else (6.126)
while for the d-band electrons, they use

ε = ε 1 + 2 k 2
2m s (6.127) and electrons with the periodic lattice potential, the interaction of electrons with phonons or the interaction of electrons with themselves. [START_REF] Petrov | Thermal conductivity and the electron-ion heat transfer coefficient in condensed media with a strongly excited electron subsystem[END_REF] model these effective masses according to

g d (ε ) =      1 2π 2 2m d 2 3/2 ε -ε 1 if ε 1 ≤ ε ≤ ε 2 0 else . ( 6
m s = (3π 2 n s ) 2/3 2 2 (E F -ε s ) (6.129)
and The upper limit ε 2 in (6.128) complicates the calculation of the multiple integrals I µ in Equation (6.121). The electron-electron collision frequency (6.117) is therefore computed numerically for any

m s = (3π 2 n d ) 2/3 2 2 E F -ε 1 (6.

PHYSICS OF THE BACKGROUND MATERIAL

electron temperature T e T F . they can be found for some metals at http://laser.itp.ac.ru. For other metals, one can directly compute the electron collision frequency at a low temperature according to the algorithm proposed in the Appendix of the original paper by [START_REF] Inogamov | Thermal conductivity of metals with hot electrons[END_REF]. It is very useful for determining the parameter A ν . However, we have noticed that the electron-electron collision frequency (6.117) can be fit with a sufficient accuracy according to

ν e-e = ν -2 1 + ν -2 2 -1/2 (6.131) with ν 1 = A ν k B T 2 e T F and ν 2 = B ν k B T e T F (6.132)
where A ν and B ν are parameters to be determined, depending on the material as illustrated in Figure 6.9 for Aluminum, in Last but not least, the electron-ion/lattice energy exchange is described by the parameter G in Equations (6.25) and (6.28). Its value varies considerably in the range from the ambient temperature T e ≈ 300 K to the hot plasma T e ≈ 10 keV. Concerning the high temperature regime, we use the plasma expression [START_REF] Brysk | Thermal conduction in laser fusion[END_REF] (see Appendix B, section B.2.4)

G hot = 2 m e m i C V,e ν ei (6.133)
where ν ei is given by (6.113) allowing to account for electrons degeneracy close to the Fermi temperature. In the limit of low temperatures T e , T i < T F , but T e , T i T D , the rate of energy transfer per unit volume from the electrons to the lattice in a metal can be written according to [START_REF] Kaganov | Relaxation between electrons and the crystalline lattice[END_REF]]

G cold (T e -T i ) = π 2 6 n i m e c s 2 1 τ e (T e , T e ) - 1 τ e (T i , T i ) . ( 6
G cold = G RT 1 + T e + T i ℵT F (6.136)
where G RT = π 2 B i n i m e c s 2 /6 is the room temperature electron-lattice coupling factor and ℵ = B i /A e T F is a parameter chosen from comparison with experimental or theoretical data such as those proposed by [START_REF] Lin | Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium[END_REF] or [START_REF] Petrov | Thermal conductivity and the electron-ion heat transfer coefficient in condensed media with a strongly excited electron subsystem[END_REF]. The cold and hot regime are related according to

G = G -2 cold + G -2 hot -1/2 . (6.137)
The value of ℵ calculated with A ν and k s given in the previous subsection, does not allow for patching the two regimes around the Fermi temperature. A better interpolation can be obtained with ℵ = 0.2.

The room temperature value can be found in the literature : G RT = 1 10 18 erg.s -1 .cm -3 .K -1 for Aluminum, G RT = 3 10 18 erg.s -1 .cm -3 .K -1 for Copper and G RT = 1.5 10 18 erg.s -1 .cm -3 .K -1 for Tantalum. The electron-ion coupling factors are plotted in Figure 6.13.

Conclusion

We The analysis, presented in Chapter 5, of existing numerical methods for solving the obtained Vlasov-Fokker-Planck (V-F-P) equation ( 6.15)

∂Ψ ∂t + ∂ ∂r .(vΩ Ψ) - ∂ ∂ε [v (eE.Ω + S) Ψ] - ∂ ∂Ω . (I -Ω ⊗ Ω) . e p E + vΩ c × B Ψ + ∂ ∂Ω ν 2 Ψ = 0
oriented our choice towards a hybrid and expanded "Vlasov-Fokker-Planck" method. In order to make numerical computations as fast as possible, we limit the angular order of expansion to the 1st order.

We derived the equations (6.34) and (6.35) by integrating over the unity sphere in the momentum space the V-F-P equation (6.15) multiplied by 1 and Ω, respectively :

∂Ψ 0 ∂t + ∂ ∂r . (v Ψ 1 ) - ∂ ∂ε [v (eΨ 1 .E + SΨ 0 )] = 0 and ∂Ψ 1 ∂t + ∂ ∂r . (v Ψ 2 ) - ∂ ∂ε [v (eΨ 2 .E + SΨ 1 )] = - e p (Ψ 0 -Ψ 2 ) .E - e γm e c Ψ 1 × B -ν Ψ 1
where the Ψ are the angular moments of the distribution function Ψ, S is the total stopping power of the beam electrons introduced in Chapter 4 and ν is their total angular isotropization rate. Contrary to an equivalent spherical harmonic decomposition, the beam density and current are directly related to the angular moments :

n b = ∞ ε min Ψ 0 dε and j b = -e ∞ ε min vΨ 1 dε.
Here, the parameter ε min comes from the "hybrid" assumption, which consists in separating the beam electrons population ε > ε min and the background electrons population ε < ε min . Contrary to the widely used P1 approximation, also usually called the "diffusion approximation", which consists in the closure relation Ψ 2 = (1/3)Ψ 0 I, our M1 model accounts for an arbitrary degree of anisotropy by using the closure

Ψ 2 = 1 3 Ψ 0 I + µΨ 0 Ψ 1 ⊗ Ψ 1 |Ψ 1 | 2 - 1 3 I . µ ≈ |Ω ε | 2 2 1 + |Ω ε | 2
is estimated according to the Minerbo maximum angular entropy criterion depending on the anisotropy

vector Ω ε = Ψ 1 /Ψ 0 . Such a closure is exact for local (in space and kinetic energy) angular distribution functions eiher fully isotropic or fully anisotropic, while the parameter µ allows us to relate these limits. Obviously, the first order expansion reduces the information concerning the beam electrons local angular distribution function. However, the maximum angular entropy criterion analysis shows that it allows for a sufficient accuracy for laser-generated fast electron beam transport. Indeed, we showed that collisional effects of laser-generated electron beam transport in solids or dense plasma fully justified such a closure. Besides, comparison of the full kinetic and the M1 approach for the analysis of the collisionless resitive filamentation instability for which there is no reason to maximize the local angular entropy, shows that our model describes the instability growth rate with an error of few 10s of % in the particular case of a monoenergetic electron beam.

This chapter terminates with a discussion of the self-consistent hydrodynamic response of the laser-irradiated target material to the beam propagation as well as the self-generated electromagnetic fields. We consider the time scale greater than the electromagnetic neutralization time of the beam ν -1 ei or τ e /γ e , presented in Chapter 2. Similarly to other hybrid models, our model assumes that the beam is not modified during its electromagnetic neutralization. This is a strong assumption in case of propagation through insulators since the ionization processes occur in this time scale, implying additional energy losses of the electron beam that are omitted here. In our hybrid model, the selfgenerated magnetic field verifies the diffusion equation (6.24)

1 c ∂B ∂t + ∂ ∂r × ηc 4π ∂ ∂r × B = η ∂ ∂r × j b + ∂η ∂r × j b - k B n e e ∂n e ∂r ×
∂T e ∂r with the source terms introduced in Chapter 3 and the self-generated electric field is given by the Ohm's law (6.23)

E = -ηj b + ηc 4π ∂ ∂r × B - 1 n e e ∂ ∂r
(n e k B T e ) also discussed with details in Chapter 3. Thus, we have neglected the magnetization effects, the background electrons viscosity, the collisional friction of the background electrons due to collisions with beam electrons, the displacement current in the Maxwell-Ampère equation (quasi-static approximation), the background electrons inertia and we have considered the ideal gas expression for the equation of state P e = P e (n e , T e ) of background electrons. These assumptions are justified in the case of laser-generated electron beam transport in dense targets as it was explained in Chapter 3.

CONCLUSION

However, the effect of electron inertia and the quasi-static approximation may be responsible for an additional time-dependent heating of the background electrons, as it was demonstrated recently by [START_REF] Sherlock | In-depth Plasma-Wave Heating of Dense Plasma Irradiated by Short Laser Pulses[END_REF], and eventual angular deviations of the fast electrons due to strong local magnetic fields. Nevertheless, the self-generated electromagnetic fields described by Equations (6.23) and (6.24) account for the main processes of laser-generated relativistic electron beam transport in solids or dense plasmas. They depend on the electrical resistivity η of the material and its spatial gradients, which depend on the electron and ion temperatures of the material. Thus, the electron temperature evolution needs to be described self-consistently according to the electron heat equation (6.25)

C V,e ∂T e ∂t - ∂ ∂r . κ e ∂T e ∂r = W e -G (T e -T i )
where

W e = ∞ ε min vS col Ψ 0 dε + ηj e 2 .
The background ion heating is described self-consistently according to the heat equation (6.28)

C V,i ∂T i ∂t = G (T e -T i ) .
In our model, we neglect the ion motion and the ion thermal conductivity, considering times scales smaller than 10th of ps. These heat equations are in agreement with the assumptions made with respect to the self-consistent electromagnetic fields (no viscosities, no magnetization effects, no inertiae and no collisional friction by colliding the beam electrons in η) except for the temporal derivatives of the temperatures that have not been neglected here. We also proposed new expressions for the heat capacities C V,e and C V,i , the electrical resistivity η, the electron thermal conductivity κ e and the electron-ion coupling factor G, allowing to describe metals from the solid state at the room temperature ≈ 300 K through the liquid and Warm and Dense Matter (WDM) states to the hot plasma state with temperatures ≈ 10 keV. The collisions of background free electrons with d-band bound electrons are taken into account according to recent studies showing the importance of this relaxation process. and the code is compiled with ifort which can be freely obtained at https://software.intel.com/. For example, on a Linux terminal, the instruction "ifort -prec-div -prec-sqrt -openmp -openmp-report2 -r8 -o exec acuracy.f90 physical constants.f90 data.f90 special relativity.f90 coulomb logarithms.f90

Fermi integrales.f90 collisional terms.f90 resistivity.f90 M1.f90 diagnostics.f90 initialization.f90 conjugated gradients.f90 MHD.f90 temperatures.f90 boundaries conditions.f90 scheme.f90 " allows for the compilation of the code and the instruction "time OMP NUM THREADS =8 ./exec" allows for its execution on 8 CPU. In the code, the space variables are normalized by L = 1 µm, the time is normalized by τ = 1 fs and the masses are normalized by M = keV/c 2 , except for the distribution function which is normalized by its maximum value. The chosen units correspond to the typical values in laser-generated fast electron transport. For example, the velocity of light is ≈ 0.3 µm/fs and the fast electrons may have energies from a few keV to several MeV. In the following, the variables are discretized as follows.

∀l

∈ [1, N ε ], ε l [keV] = ε min + ∆ε 2 + (l -1)∆ε where N ε = E L ε ∆ε (7.1)
where the index l, is used for the kinetic energy variable ε of the fast electrons, ε min may vary from a few keV to 20 keV and L ε may vary from a few MeV to 10th of MeV depending on the simulation.

Concerning the discretization of space, we use The first section of this chapter is dedicated to the numerical schemes used for the resolution of the M1 equations. The second section deals with the numerical schemes used to compute self-consistently the electromagnetic fields. The third section is dedicated to the numerical schemes used to solve self-consistently the heat equations.

∀i ∈ [1, N x ], x i [µm] = - L x 2 + ∆x 2 + (i -1)∆x where N x = E L x ∆x + 1, ( 7

M1 equations

According to Chapter 6, section 6.2.1 and 6.2.2, the M1 equations that have to solved are

∂Ψ 0 ∂t + ∂ ∂x [v(ε)Ψ 1,x ] + ∂ ∂y [v(ε)Ψ 1,y ] + ∂ ∂z [v(ε)Ψ 1,z ] - ∂ ∂ε S(x, y, z, ε, t)v(ε)Ψ 0 +eE x (x, y, z, t)v(ε)Ψ 1,x + eE y (x, y, z, t)v(ε)Ψ 1,y + eE z (x, y, z, t)v(ε)Ψ 1,z , = 0 (7.6) ∂Ψ 1,x ∂t + ∂ ∂x [v(ε)Ψ 2,xx ] + ∂ ∂y [v(ε)Ψ 2,xy ] + ∂ ∂z [v(ε)Ψ 2,xz ] - ∂ ∂ε S(x, y, z, ε, t)v(ε)Ψ 1,x +eE x (x, y, z, t)v(ε)Ψ 2,xx + eE y (x, y, z, t)v(ε)Ψ 2,xy + eE z (x, y, z, t)v(ε)Ψ 2,xz = - e p(ε) Ψ 0 -Ψ 2,xx E x (x, y, z, t) -Ψ 2,xy E y (x, y, z, t) -Ψ 2,xz E z (x, y, z, t) - e γ(ε)m e c Ψ 1,y B z (x, y, z, t) -Ψ 1,z B y (x, y, z, t) -ν(x, y, z, ε, t)Ψ 1,x (7.7) ∂Ψ 1,y ∂t + ∂ ∂x [v(ε)Ψ 2,yx ] + ∂ ∂y [v(ε)Ψ 2,yy ] + ∂ ∂z [v(ε)Ψ 2,yz ] - ∂ ∂ε S(x, y, z, ε, t)v(ε)Ψ 1,y +eE x (x, y, z, t)v(ε)Ψ 2,yx + eE y (x, y, z, t)v(ε)Ψ 2,yy + eE z (x, y, z, t)v(ε)Ψ 2,yz = - e p(ε) -Ψ 2,yx E x (x, y, z, t) + Ψ 0 -Ψ 2,yy E y (x, y, z, t) -Ψ 2,yz E z (x, y, z, t) - e γ(ε)m e c Ψ 1,z B x (x, y, z, t) -Ψ 1,x B z (x, y, z, t) -ν(x, y, z, ε, t)Ψ 1,y (7.8)
and

∂Ψ 1,z ∂t + ∂ ∂x [v(ε)Ψ 2,zx ] + ∂ ∂y [v(ε)Ψ 2,zy ] + ∂ ∂z [v(ε)Ψ 2,zz ] - ∂ ∂ε S(x, y, z, ε, t)v(ε)Ψ 1,z +eE x (x, y, z, t)v(ε)Ψ 2,zx + eE y (x, y, z, t)v(ε)Ψ 2,zy + eE z (x, y, z, t)v(ε)Ψ 2,zz = - e p(ε) -Ψ 2,zx E x (x, y, z, t) -Ψ 2,zy E y (x, y, z, t) + Ψ 0 -Ψ 2,zz E z (x, y, z, t) - e γ(ε)m e c Ψ 1,x B y (x, y, z, t) -Ψ 1,y B x (x, y, z, t) -ν(x, y, z, ε, t)Ψ 1,z (7.9)
where Ψ 0 = Ψ 0 (x, y, z, ε, t) is the 0th order angular moment and Ψ 1 = Ψ 1 (x, y, z, ε, t) is the 1st order angular moment of the distribution function while Ψ 2 = Ψ 2 (x, y, z, ε, t) is the 2nd order angular moment given by the M1 closure :

∀(i, j) ∈ {x, y, z} 2 , Ψ 2,ij = 1 3 δ ij Ψ 0 + µΨ 0 Ψ 1,i Ψ 1,j Ψ 1,x 2 + Ψ 1,y 2 + Ψ 1,z 2 - 1 3 δ ij (7.10) with µ(x, y, z, ε, t) = 1 2Ψ 0 2 Ψ 1,x 2 + Ψ 1,y 2 + Ψ 1,z 2 1 + 1 Ψ 0 2 Ψ 1,x 2 + Ψ 1,y 2 + Ψ 1,z 2 .
7.1.1 Second Order Explicit HLL Scheme for the Fast Electron Advection in Space and Fast Electron Slowing Down due to the Self-Generated Electric field

First order HLL scheme

Let us consider firstly the 1D-3V case, in order to point out the major features of the numerical scheme that is used to solve the fast electrons advection in space and in kinetic energy space (due to their collective energy losses). In this 1D case, the equations that have to be solved are (7.6) and (7.7) with Ψ 1,y = Ψ 1,z = 0. The 3D-3V case will be further generalized in the section. Therefore, we are here interested in the numerical resolution of two 1D coupled equations

∂Ψ 0 ∂t + ∂ ∂ξ (u ξ Ψ 1,x ) = 0 ∂Ψ 1,x ∂t + ∂ ∂ξ (u ξ Ψ 2,xx ) = 0 (7.11) with Ψ 2,xx = 1 + 2µ 3 Ψ 0 and µ = 1 2 Ψ 1,x Ψ 0 2 1 + Ψ 1,x Ψ 0 2
describing the fast electrons advection in the x-direction in space (ξ = x and u x = v(ε)) or the fast electrons advection in the ε-direction in the kinetic energy space due to their collective energy losses (ξ = ε and u ε = -eE x (x, t)v(ε)), appearing in (7.6) and (7.7).

This hyperbolic system (7.11) can be solved numerically with respect to the realizability domain

A = {(Ψ 0 , Ψ 1,x ) /Ψ 0 ≥ 0 and |Ψ 1,x | ≤ Ψ 0 } (7.12)
by using the HLL scheme (from the name of its founders A. Harten, P. Lax and B. Van Leer) developed for the radiative transfer equations [Harten et al., 1983] [Dubroca andFeugeas, 1999] [ [START_REF] Berthon | A Free Streaming Contact Preserving Scheme for the M1 Model[END_REF]. According to the finite volume method, we define the mean values in each cell :

Ψ n,i,l 0 = 1 ∆x ∆ε x i +∆x/2 x i -∆x/2 ε l +∆ε/2 ε l -∆ε/2 Ψ 0 (x, ε, t n ) dx dε, (7.13) Ψ n,i,l 1,x = 1 ∆x ∆ε x i +∆x/2 x i -∆x/2 ε l +∆ε/2 ε l -∆ε/2 Ψ 1,x (x, ε, t n ) dx dε (7.14) and Ψ n,i,l 2,xx = 1 ∆x ∆ε x i +∆x/2 x i -∆x/2 ε l +∆ε/2 ε l -∆ε/2 Ψ 2,xx (x, ε, t n ) dx dε.
(7.15)

Let us note Φ the vector Φ = (Ψ 0 , Ψ 1,x ) T , i ξ the index i (if ξ = x) or l (if ξ = ε) depending on ξ and F = F (Φ) the vector F = (Ψ 1,x , Ψ 2,xx ) T depending on Φ. According to the HLL scheme, one can update at each time step the new value Φ n+1,i ξ at time t n+1 in the cell i ξ , knowing the previous values Φ n,i ξ at time t n according to

Φ n+1,i ξ = Φ n,i ξ - ∆t n ∆ξ F n,i ξ +1/2 HLL -F n,ix-1/2 HLL (7.16)
where the HLL fluxes are given by

F n,i ξ +1/2 HLL = u n,i ξ ξ (u n,i ξ +1 ξ /u n,i ξ ξ )F n,i ξ +1 + F n,i ξ 2 -u n,i ξ ξ (u n,i ξ +1 ξ /u n,i ξ ξ )Φ n,i ξ +1 -Φ n,i ξ 2 (7.17) and F n,i ξ -1/2 HLL = u n,i ξ ξ F n,i ξ + (u n,i ξ -1 ξ /u n,i ξ ξ )F n,i ξ -1 2 -u n,i ξ ξ Φ n,i ξ -(u n,i ξ -1 ξ /u n,i ξ ξ )Φ n,i ξ -1 2 . (7.18)
In the case where ξ = x for which u ξ = v(ε) does not depend on ξ, the ratios (u

n,i ξ ±1 ξ /u n,i ξ ξ
) appearing in the the HLL fluxes (7.17) and (7.18) are equal to 1. In the case where ξ = ε for which u ξ = -eE x (x, t)v(ε) depends on ξ, the ratios reduce to v l+1 /v l and v l-1 /v l where ∀l ∈ [1, N ε ], v l = v(ε l ). This has been deduced from the case where u ξ does not depend on ξ. Indeed, let us define Φ = vΦ and F = vF . In this case, the realizability domain is still the same :

A = Ψ 0 , Ψ 1,x / Ψ 0 ≥ 0 and Ψ 1,x ≤ Ψ 0 (7.19)
while the equation (7.11) for ξ = ε can be written

∂ Φ ∂t - ∂ ∂p eE x (x, t) F = 0 (7.20)
as (dε/dp) = v. Therefore, the velocity u ε depending on ε in (7.11) has disappeared and, since u p = -eE x does not depend on p, we can compute this equation with the same way as (7.11) for which u ξ does not depend on ξ. We obtain consequently (7.16) with the ratios (u

n,i ξ ±1 ξ /u n,i ξ ξ
) which are equal to 1. By coming back to Φ and F instead of Φ and F , we finally obtain (7.11) with the ratios (u

n,i ξ ±1 ξ /u n,i ξ ξ ).
The 1st order HLL scheme is stable and the numerical solutions are in the realizability domain (7.12) if the Courant-Friedrich-Lewy (CFL) condition

∆t n < ∆ξ max i ξ ∈[1, N ξ ] u n,i ξ ξ (7.21)
is verified. However, according to numerical experiments that I have performed during this thesis, the consistency error of this 1st order explicit HLL scheme

n,i ξ HLL, 1D = Φ n+1,i ξ -Φ n,i ξ ∆t n + F n,i ξ +1/2 HLL -F n,i ξ -1/2 HLL ∆ξ - ∂Φ ∂t n,i ξ + ∂ ∂ξ (u ξ F ) n,i ξ = ∆t n 2 ∂ 2 Φ ∂t 2 n,i ξ - ∆ξ 2 ∂ 2 ∂ξ 2 (|u ξ | Φ) n,i ξ + O(∆t 2 + ∆ξ 2 ) (7.22)
is not sufficiently small for values ∆ξ about a fraction of microns (if ξ = x) or about a keV (if ξ = ε) typical of laser-generated fast electron transport studies. We used therefore the 2nd order explicit HLL scheme.

2nd order HLL scheme

The second order explicit HLL scheme can be obtained by replacing the HLL fluxes (7.17) and (7.18) by

F n,i ξ +1/2 HLL = u n,i ξ ξ (u n,i ξ +1 ξ /u n,i ξ ξ )F n,i ξ +1,-+ F n,i ξ ,+ 2 -|u n,i ξ ξ | (u n,i ξ +1 ξ /u n,i ξ ξ )Φ n,i ξ +1,--Φ n,i ξ ,+ 2 (7.23) and F n,i ξ -1/2 HLL = u n,i ξ ξ F n,i ξ ,-+ (u n,i ξ -1 ξ /u n,i ξ ξ )F n,i ξ -1,+ 2 -|u n,i ξ ξ | Φ n,i ξ ,--(u n,i ξ -1 ξ /u n,i ξ ξ )Φ n,i ξ -1,+ 2 , (7.24)
respectively [Dubroca, 2012]. Here, the sign + indicates

∀i ξ ∈ [1, N ξ ], Φ n,i ξ ,+ = Φ n,i ξ + P n,i ξ ∆ξ 2 and F n,i ξ ,+ = F (Φ n,i ξ ,+ ) (7.25) while the sign -indicates ∀i ξ ∈ [1, N ξ ], Φ n,i ξ ,-= Φ n,i ξ -P n,i ξ ∆ξ 2 and F n,i ξ ,-= F (Φ n,i ξ ,-
). (7.26)

P n,i ξ is the vector defined by

P n,i ξ = min 0, max Φ n,i ξ +1 -Φ n,i ξ ∆ξ , Φ n,i ξ -Φ n,i ξ -1 ∆ξ + max 0, min Φ n,i ξ +1 -Φ n,i ξ ∆ξ , Φ n,i ξ -Φ n,i ξ -1 ∆ξ .
(7.27) However, these 2nd order HLL fluxes (7.23) and (7.24) may lead to numerical solutions outside the realizability domain (7.12); while Ψ 0 ≥ 0 is respected, it is not necessarily the case for the condition

|Ψ 1,x | ≤ Ψ 0 .
Consequently, we introduce a value Θ at each time step and in each cell i ξ defined by Φ n,i ξ ,+ = Φ n,i ξ + P n,ξ Θ ∆ξ 2 and Φ n,i ξ ,-= Φ n,i ξ -P n,ξ Θ ∆ξ 2 (7.28) such that |Ψ 1,x | ≤ Ψ 0 [Dubroca, 2012]. Thus, by injecting (7.28) in the conditions (Ψ

n,i ξ ,+ 1,x ) 2 ≤ (Ψ n,i ξ ,+ 0 ) 2 or (Ψ n,i ξ ,- 1,x ) 2 ≤ (Ψ n,i ξ ,- 0 ) 2
, we obtain the following condition

(Ψ n,i ξ 0 ) 2 -(Ψ n,i ξ 1,x ) 2 ± 2Θ P n,i ξ 0 Ψ n,i ξ 0 -P n,i ξ 1,x Ψ n,i ξ 1,x ∆ξ 2 + Θ 2 (P n,i ξ 0 ) 2 -(P n,i ξ 1,x ) 2 ∆ξ 2 4 ≥ 0. (7.29) Consequently, if Ψ n,i ξ 0 = Ψ n,i ξ
1,x or if the discriminant of the trinomial left hand side is negative, we can impose Θ = 1 without problems. But, if we find solution Θ ∈ [0, 1] cancelling the trinomial left hand side, allowing to obtain (Ψ

n,i ξ ,± 0 ) 2 ≥ (Ψ n,i ξ ,± 1,x
) 2 , we impose Θ = min Θ . In other exotic cases, we come back to the 1st order HLL scheme by imposing Θ = 0. The 2nd order HLL scheme allows for a much smaller consistency error O(∆t + ∆ξ 2 ) compared to the 1st order consistency error (7.22) O(∆t + ∆ξ) while the CFL condition (7.21) remains the same.

Generalization to the 3D-3V case

We note ∀ζ ∈ {x, y, z}, E

ζ (x i , y j , z k , t n ) = E n,i,j,k ζ Ψ n,i,j,k,l 0 = 1 ∆x ∆y ∆z ∆ε x i +∆x/2 x i -∆x/2 y j +∆y/2 y j -∆y/2 z k +∆z/2 z k -∆z/2 ε l +∆ε/2 ε l -∆ε/2 Ψ 0 (x, y, z, ε, t n ) dx dy dz dε (7.30) and Ψ n,i,j,k,l 1,ζ = 1 ∆x ∆y ∆z ∆ε x i +∆x/2 x i -∆x/2 y j +∆y/2 y j -∆y/2 z k +∆z/2 z k -∆z/2 ε l +∆ε/2 ε l -∆ε/2 Ψ 1,ζ (x, y, z, ε, t n ) dx dy dz dε.
(7.31) Also, we define the vector Φ = (Ψ 0 , Ψ 1,x , Ψ 1,y , Ψ 1,z ) T and ∀ζ ∈ {x, y, z} and the vectors

F ζ = F ζ (Φ) = (Ψ 1,ζ , Ψ 2,ζx , Ψ 2,
ζy , Ψ 2,ζz ) T depending on Φ. This subsection is dedicated to the generalization of the numerical resolution of the two 1D coupled equations (7.11) to the numerical resolution of the four "4D" (3D-1ε) coupled equations

∂Φ ∂t + ξ=x, y, z, ε   ζ=x, y, z ∂ ∂ξ (u ξζ F ζ )   = 0 (7.32)
where the u ξζ are recapitulated in Figure 7.3. The 2nd order explicit HLL scheme applied to these

ξ ζ x y z x v(ε) 0 0 y 0 v(ε) 0 z 0 0 v(ε) ε
-eE x (x, y, z, t) -eE y (x, y, z, t) -eE z (x, y, z, t) four "4D" coupled equations reads

Φ n+1,i,j,k,l = Φ n,i,j,k,l -(∆t n /∆x) F n,i+1/2,j,k,l x,HLL -F n,i-1/2,j,k,l x,HLL -(∆t n /∆y) F n,i,j+1/2,k,l y,HLL -F n,i,j-1/2,k,l y,HLL -(∆t n /∆z) F n,i,j,k+1/2,l z,HLL -F n,i,j,k-1/2,l z,HLL -(∆t n /∆ε) F n,i,j,k,l+1/2 ε,HLL -F n,i,j,k,l-1/2 ε,HLL (7.33)
where the HLL fluxes are given by

F n,i+1/2,j,k,l x,HLL = v l F i+1,- x + F i,+ x 2 n,j,k,l -|v l | Φ i+1,--Φ i,+ 2 n,j,k,l , F n,i-1/2,j,k,l x,HLL = v l F i,- x + F i-1,+ x 2 n,j,k,l -|v l | Φ i,--Φ i-1,+ 2 n,j,k,l , F n,i,j+1/2,k,l y,HLL = v l F j+1,- y + F j,+ y 2 n,i,k,l -|v l | Φ j+1,--Φ j,+ 2 n,i,k,l , F n,i,j-1/2,j,k,l y,HLL = v l F j,- y + F j-1,+ y 2 n,i,k,l -|v l | Φ j,--Φ j-1,+ 2 n,i,k,l , F n,i,j,k+1/2,l z,HLL = v l F k+1,- z + F k,+ z 2 n,i,j,l -|v l | Φ k+1,--Φ k,+ 2 n,i,j,l , F n,i,j,k-1/2,l z,HLL = v l F k,- z + F k-1,+ z 2 n,i,j,l -|v l | Φ k,--Φ k-1,+ 2 n,i,j,l , F n,i,j,k,l+1/2 ε,HLL = -eE n,i,j,k x v l (v l+1 /v l )F l+1,- x + F l,+ x 2 n,i,j,k -|eE n,i,j,k x v l | (v l+1 /v l )Φ l+1,--Φ l,+ 2 n,i,j,k + -eE n,i,j,k y v l (v l+1 /v l )F l+1,- y + F l,+ y 2 n,i,j,k -|eE n,i,j,k y v l | (v l+1 /v l )Φ l+1,--Φ l,+ 2 n,i,j,k + -eE n,i,j,k z v l (v l+1 /v l ) F l+1,- z + F l,+ z 2 n,i,j,k -|eE n,i,j,k z v l | (v l+1 /v l )Φ l+1,--Φ l,+ 2 n,i,j,k and F n,i,j,k,l-1/2 ε,HLL = -eE n,i,j,k x v l F l,- x + (v l-1 /v l )F l-1,+ x 2 n,i,j,k -|eE n,i,j,k x v l | Φ l,--(v l-1 /v l )Φ l-1,+ 2 n,i,j,k + -eE n,i,j,k y v l F l,- y + (v l-1 /v l )F l-1,+ y 2 n,i,j,k -|eE n,i,j,k y v l | Φ l,--(v l-1 /v l )Φ l-1,+ 2 n,i,j,k + -eE n,i,j,k z v l F l,- z + (v l-1 /v l )F l-1,+ z 2 n,i,j,k -|eE n,i,j,k z v l | Φ l,--(v l-1 /v l )Φ l-1,+ 2 n,i,j,k
.

Here, for all indexes i ξ ∈ {i, j, k, l} corresponding respectively to the random variable ξ ∈ {x, y, z, ε}, Φ n, i ξ ,+ = Φ n,i ξ + Θ P n,i ξ ∆ξ 2 and Φ n,i ξ ,-= Φ n,i ξ -Θ P n,i ξ ∆ξ 2 (7.34)

where

P n,i ξ = min 0, max Φ n,i ξ +1 -Φ n,i ξ ∆ξ , Φ n,i ξ -Φ n,i ξ -1 ∆ξ + max 0, min Φ n,i ξ +1 -Φ n,i ξ ∆ξ , Φ n,i ξ -Φ n,i ξ -1 ∆ξ and if ∃Θ ∈ [0, 1] / (Ψ n,i ξ 0 ) 2 -(Ψ n,i ξ 1,x ) 2 -(Ψ n,i ξ 1,y ) 2 -(Ψ n,i ξ 1,z ) 2 ± 2Θ P n,i ξ 0 Ψ n,i ξ 0 -P n,i ξ 1,x Ψ n,i ξ 1,x -P n,i ξ 1,y Ψ n,i ξ 1,y -P n,i ξ 1,z Ψ n,i ξ 1,z ∆ξ 2 + Θ 2 (P n,iξ 0 ) 2 -(P n,i ξ 1,x ) 2 -(P n,i ξ 1,y ) 2 -(P n,i ξ 1,z ) 2 ∆ξ 2 4 = 0 then Θ = min Θ else Θ = 1.
This "4D" explicit 2nd order HLL scheme is stable and the numerical solutions are in the realizability domain

A = {(Ψ 0 , Ψ 1 ) /Ψ 0 ≥ 0 and |Ψ 1 | ≤ Ψ 0 } (7.35) if the CFL condition ∆t n < 1 v max 1 ∆x + 1 ∆y + 1 ∆z + ev max ∆ε (E n x,max ) 2 + (E n y,max ) 2 + (E n z,max ) 2 . (7.36)
is verified. Also, we can estimate the consistency error in the worst case (i.e. in the 1st order case) :

n,i,j,k,l HLL = ∆t n 2 ∂ 2 Φ ∂t 2 n,i,j,k,l - ∆ε 2 eE n,i,k,k x + eE n,i,k,k y + eE n,i,k,k z ∂ 2 ∂ε 2 (vΦ) n,i,k,k,l - ∆x 2 ∂ 2 ∂x 2 (vΦ) n,i,k,k,l - ∆y 2 ∂ 2 ∂y 2 (vΦ) n,i,k,k,l - ∆z 2 ∂ 2 ∂z 2 (vΦ) n,i,k,k,l
+ O ∆t 2 + ∆x 2 + ∆y 2 + ∆z 2 + ∆ε 2 (7.37)

Downwind Explicit Scheme for the Fast Electron Collisional Slowing Down

Concerning the slowing down due to collisional effects, we have to solve the equation

∂Φ ∂t - ∂ ∂ε S(x, y, z, ε, t)v(ε)Φ = 0. (7.38)
Since the total stopping power S is always positive, we can use the simple downwind scheme Φ n+1,i,j,k,l = Φ n,i,j,k,l + ∆t n ∆ε S n,i,j,k,l+1 v l+1 Φ n,i,j,k,l+1 -S n,i,j,k,l v l Φ n,i,j,k,l (7.39)

where it has been noted S n,i,j,k,l = S(x i , y j , z k , ε l , t n ) the opposite of the total discretized stopping power of the fast electrons. This 1st order downwind scheme is stable if the CFL condition

∆t n < ∆ε 2S max v max (7.40)
is verified and the consistency error is

n,i,j,k,l S = ∆t n 2 ∂ 2 Φ ∂t 2 n,i,j,k,l - ∆ε 2 ∂ 2 ∂ε 2 (SvΦ) n,i,j,k,l + O ∆t 2 + ∆ε 2 .
(7.41)

Explicit Scheme for the Fast Electron Angular Deviations

The fast electron angular deviations are due to the self-generated electric and magnetic fields and the angular scattering. Let us note γ l = γ(ε l ) the discretized Lorentz factor, p l = p(ε l ) the discretized momenta and ∀ζ ∈ {x, y, z}, B ζ (x i , y j , z k , t n ) = B n,i,j,k ζ the components of the self-generated magnetic field. This subsection is dedicated to the numerical resolution of the equation

∂Φ ∂t = Γ E + Γ B + Γ ν . (7.42) where Γ n,i,j,k,l E = - e p l        0 Ψ n,i,j,k,l 0 -Ψ n,i,j,k,l 2,xx E n,i,j,k x -Ψ n,i,j,k,l 2,xy E n,i,j,k y -Ψ n,i,j,k,l 2,xz E n,i,j,k z -Ψ n,i,j,k,l 2,yx E n,i,j,k x + Ψ n,i,j,k,l 0 -Ψ n,i,j,k,l 2,yy E n,i,j,k y -Ψ n,i,j,k,l 2,yz E n,i,j,k z -Ψ n,i,j,k,l 2,zx E n,i,j,k x -Ψ n,i,j,k,l 2,zy E n,i,j,k y -Ψ n,i,j,k,l 0 -Ψ n,i,j,k,l 2,zz E n,i,j,k z        (7.43)
is the angular deviation term due to the self-generated electric field,

Γ n,i,j,k,l B = - e γ l m e c        0 Ψ n,i,j,k,l 1,y B n,i,j,k z -Ψ n,i,j,k,l 1,z B n,i,j,k y Ψ n,i,j,k,l 1,z B n,i,j,k x -Ψ n,i,j,k,l 1,x B n,i,j,k z Ψ n,i,j,k,l 1,x B n,i,j,k y -Ψ n,i,j,k,l 1,y B n,i,j,k x        (7.44)
is the angular deviation term due to the self-generated magnetic field and

Γ n,i,j,k,l ν = -ν n,i,j,k,l        0 Ψ n,i,j,k,l 1,x Ψ n,i,j,k,l 1,y Ψ n,i,j,k,l 1,z        (7.45)
is the angular deviation term due to the fast electron collisional angular scattering. We use the explicit numerical scheme Φ n+1,i,j,k,l = Φ n,i,j,k,l + ∆t n Γ n,i,j,k,l E + Γ n,i,j,k,l B + Γ n,i,j,k,l ν .

(7.46)

It is stable if the approximated CFL condition ∆t n < 1 ω c,max + ν max + e p min (E n x,max ) 2 + (E n y,max ) 2 + (E n z,max ) 2 (7.47) is verified where ω c,max = e (B n x,max ) 2 + (B n y,max ) 2 + (B n z,max ) 2
/m e c is the most restrictive fast electron cyclotron frequency and ν max is the most restrictive fast electron isotropization rate ν. The consistency error of this scheme is directly given by

n,i,j,k,l A = ∆t n 2 ∂ 2 Φ ∂t 2 n,i,j,k,l + O ∆t 2 .
(7.48)

Summary of the Full Explicit Scheme

As a conclusion, we use the numerical scheme

Φ n+1,i,j,k,l = Φ n,i,j,k,l -(∆t n /∆x) F n,i+1/2,j,k,l x,HLL -F n,i-1/2,j,k,l x,HLL -(∆t n /∆y) F n,i+1/2,j,k,l y,HLL -F n,i-1/2,j,k,l y,HLL -(∆t n /∆z) F n,i,j,k+1/2,l z,HLL -F n,i,j,k-1/2,l z,HLL -(∆t n /∆ε) (-S n,i,j,k,l+1 )v l+1 Φ n,i,j,k,l+1 -(-S n,i,j,k,l )v l Φ n,i,j,k,l -(∆t n /∆ε) F n,i,j,k,l+1/2 ε,HLL -F n,i,j,k,l-1/2 ε,HLL + ∆t n Γ n,i,j,k,l E + Γ n,i,j,k,l B + Γ n,i,j,k,l ν . (7.49)
It is stable with the respect of the approximate CFL condition

1 ∆t n > v max 1 ∆x + 1 ∆y + 1 ∆z + 2S n max v max ∆ε + ev max E n x,max 2 + E n y,max 2 + E n z,max 2 ∆ε + ν n max + ω n c,max + e E n x,max 2 + E n y,max 2 + E n z,max 2
p min .

(7.50)

In practice, the CFL condition is mainly constrained by the resolution of the spatial derivatives and leads to time step ∆t n of a fraction of fs. The resolution of the kinetic energy derivative due to collective energy losses constrains the CFL condition only near the peak of the laser pulse which corresponds to the peak of the self-generated electric field. The consistency error can be estimated by

n,i,j,k,l = ∆t n 2 ∂ 2 Φ ∂t 2 n,i,j,k,l - ∆ε 2 eE n,i,k,k x + eE n,i,k,k y + eE n,i,k,k z ∂ 2 ∂ε 2 (vΦ) n,i,k,k,l - ∆x 2 ∂ 2 ∂x 2 (vΦ) n,i,k,k,l - ∆y 2 ∂ 2 ∂y 2 (vΦ) n,i,k,k,l - ∆z 2 ∂ 2 ∂z 2 (vΦ) n,i,k,k,l - ∆ε 2 ∂ 2 ∂ε 2 (SvΦ) n,i,j,k,l + O ∆t 2 + ∆x 2 + ∆y 2 + ∆z 2 + ∆ε 2 .
(7.51) However, in practice, the consistency error of the simulation results is measured by computing the percentage of error in the time-and-space-integrated energy conservation equation (6.29).

First order Implicit Scheme for the Collisional Terms in the case of Very Dense Plasmas

In the case of laser-generated fast electron transport in a very dense plasma like in the Fast Ignition or the Shock Ignition Scheme for Inertial Confinement Fusion, the total stopping power S and the angular isotropization rate ν may severely restrict the CFL criterion (7.21). Therefore, I also introduce implicit numerical schemes for these two terms in order to relax the CFL condition to the less restrictive one :

1 ∆t n > v max 1 ∆x + 1 ∆y + 1 ∆z + ev max E n x,max 2 + E n y,max 2 + E n z,max 2 ∆ε + ω n c,max + e E n x,max 2 + E n y,max 2 + E n z,max 2
p min .

(7.52)

In this case, we firstly compute all the other terms :

Φ * ,i,j,k,l = Φ n,i,j,k,l -(∆t n /∆x) F n,i+1/2,j,k,l x,HLL -F n,i-1/2,j,k,l x,HLL -(∆t n /∆y) F n,i+1/2,j,k,l y,HLL -F n,i-1/2,j,k,l y,HLL -(∆t n /∆z) F n,i,j,k+1/2,l z,HLL -F n,i,j,k-1/2,l z,HLL -(∆t n /∆ε) F n,i,j,k,l+1/2 ε,HLL -F n,i,j,k,l-1/2 ε,HLL + ∆t n Γ n,i,j,k,l E + Γ n,i,j,k,l B . (7.53)
Then, the collisional effects are taken into account implicitely :

Ψ n+1,i,j,k,l 0 = Φ * ,i,j,k,l 0 + S n,i,j,k,l+1 v l+1 ∆t n ∆ε Φ * ,i,j,k,l+1 0 1 + S n,i,j,k,l v l ∆t n ∆ε (7.54)
and ∀ζ ∈ {x, y, z},

Ψ n+1,i,j,k,l 1,ζ = Φ * ,i,j,k,l 1,ζ + S n,i,j,k,l+1 v l+1 ∆t n ∆ε Φ * ,i,j,k,l+1 1,ζ 1 + S n,i,j,k,l v l ∆t n ∆ε + ν n,i,j,k,l ∆t n . (7.55)
In order to improve the consistency error of the numerical scheme for the collisional slowing down of the fast electrons, I have also tested the centered implicit scheme. However, it led to solutions outside the realizability domain (7.35). That is why, we have finally chosen the simpler implicit and explicit downwind schemes.

Fast Electron Injection and Escaping Boundary Conditions

The fast electron injection in the simulation box is performed by defining the angular moments Φ ini of the distribution function Ψ M1 (x i , y j , z 1 , ε l , θ, ϕ, t n ) in the first cells in the z-direction z = z 1 at each time step t n (see Figure 7.1). The distribution function may depend on parameters introduced in Chapter 1, section 1.4 that are deduced from a Particle-In-Cell simulation of the laser-target interaction. It may happen that fast electrons are still propagating in the simulation box while the laser pulse is off. In this case, we define the laser pulse duration t source such that for times t n ≥ t source , we stop injecting fast electrons by imposing Φ ini = 0. This section is dedicated to the boundary conditions at the simulation box boundaries x = ±L x /2, y = ±L y /2, z = 0 and z = L z as well as the kinetic energy space boundaries ε = ε min and ε = L ε .

In a first attempt, we decided to let the fast electrons escape the target at the target spatial boundaries (see Chapter 9 for the fast electrons refluxing). One may therefore naively think that it would be sufficient to impose for example Φ n,i,j,Nz+1,l = Φ n,i,j,Nz+2,l = 0, which means that there are no fast electrons in vacuum at the target rear side, so that all fast electrons with a positive momentum p l Ψ n,i,j,Nz,l 1,z will escape from the target boundary k = N z . Actually, it is not so easy. Indeed, what really matters is the fast electron fluxes at z = L z which are computed according to the HLL scheme. Thus, concerning this example, this is the HLL flux F n,i,j,Nz,l z,HLL that must be taken equal to F n,i,j,Nz,l z and not directly F n,i,j,Nz,l z . Consequently, for this example, we have to impose the following boundary condition at the target rear side :

If Ψ n,i,j,Nz,l 1,z
> 0 then Φ n,i,j,Nz+2,l = Φ n,i,j,Nz+1,l = Φ n,i,j,Nz,l else Φ n,i,Nz+2,l = Φ n,i,Nz+1,l = 0.

(7.56)

If the fast electrons have a positive momentum p l Ψ n,i,j,Nz,l 1,z in the z-direction at the target rear side k = N z , it means that they are going to escape from the target so that we have to impose Φ n,i,j,Nz+2,l = Φ n,i,j,Nz+1,l = Φ n,i,j,Nz,l such that the HLL flux in the z-direction at the rear side reads

F n,i,j,Nz+1/2,l z,HLL = v l F Nz+1,- z + F Nz,+ z 2 n,i,j,l -|v l | Φ Nz+1,--Φ Nz,+ 2
n,i,j,l = F n,i,j,Nz,l z .

In the opposite case where the fast electrons have a negative momentum p l Ψ n,i,j,Nz,l 1,z at the target rear side k = N z , we impose no fast electrons in vacuum k = N z + 1 and k = N z + 2 so that we do not inject fast electrons according to the HLL flux expression. By generalizing to other target boundaries, we impose the following boundary conditions : If t < t source then Φ n,i,j,0,l = Φ n,i,j,1,l = Φ n,i,j,1,l ini else if Ψ n,i,j,2,l 1,z < 0 and t > t source then Φ n,i,j,0,l = Φ n,i,j,1,l = Φ n,i,j,2,l else Φ n,i,j,0,l = Φ n,i,j,1,l = 0,

If Ψ n,i,1,k,l 1,y < 0 then Φ n,i,-1,k,l = Φ n,i,0,k,l = Φ n,i,1,k,l else Φ n,i,-1,k,l = Φ n,i,0,k,l = 0, If Ψ n,i,Ny,k,l 1,y > 0 then Φ n,i,Ny+1,k,l = Φ n,i,Ny+2,j,k,l = Φ n,i,Ny,k,l else Φ n,i,Ny+1,k,l = Φ n,i,Ny+2,k,l = 0, If Ψ n,1,j,k,l 1,x < 0 then Φ n,-1,j,k,l = Φ n,0,j,k,l = Φ n,1,j,k,l else Φ n,-1,j,k,l = Φ n,0,j,k,l = 0 and If Ψ n,Nx,j,k,l 1,x
> 0 then Φ n,Nx+1,j,k,l = Φ n,Nx+2,j,k,l = Φ n,Nx,j,k,l else Φ n,Nx+1,j,k,l = Φ n,Nx+2,j,k,l = 0.

(7.57)

Concerning the target corners, the priority is given to the z-axis fluxes, assuming that L x and L y have been chosen sufficiently large.

The boundary conditions concerning the HLL fluxes in the kinetic energy space are simpler since in our laser-generated fast electron transport model, fast electrons only lose their kinetic energy.

Consequently, we impose Φ n,i,j,k,-1 = Φ n,i,j,k,0 = Φ n,i,j,k,1 (7.58) in order to let the fast electrons lose all their energy according to the HLL fluxes at the low energy boundary ε = ε min . Since n b n e , these electrons with energies less than ε min are not injected in the background electrons population and are just removed from the system. In the opposite boundary, we impose Φ n,i,j,k,Nε+1 = Φ n,i,j,k,Nε+2 = 0 (7.59) in order to avoid injection of fast electrons at the high energy boundary ε = L ε .

Self-Generated Electromagnetic Fields

According to the previous section, we know exactly the fast electron distribution function Ψ n M1 at each time step n in the whole simulation box. Therefore, we can compute the laser-generated fast electron beam density according to

n n,i,j,k b = n b (x i , y j , z k , t n ) = Nε l=1
Ψ n,i,j,k,l 0 ∆ε (7.60) and the beam current density according to

j n,i,j,k b = j b (x i , y j , z k , t n ) = -e Nε l=1
Ψ n,i,j,k,l 1 v l ∆ε. (7.61) Also, as illustrated in Figure 7.2, the background electron and ion temperatures at time t n , T n e and T n i , are known in each cell, allowing to compute the electrical resistivity η n,i,j,k and the background electron density n n,i,j,k e = (Z * ) n,i,j,k n n,i,j,k,l i in each cell at time t n . This section is dedicated to the resolution of the self-generated electromagnetic fields at each time step t n . The equations that have to be solved self-consistently in 3 dimensions for the self-generated magnetic fields are while the self-generated electric fields can be deduced knowing the self-generated magnetic field according to

∂B x ∂t - ∂ ∂y η(x, y, z, t)c 2 4π ∂B x ∂y - ∂ ∂z η(x, y, z, t)c 2 4π ∂B x ∂z = - ∂ ∂y η(x, y, z, t)c 2 4π ∂B y ∂x - ∂ ∂z η(x, y, z, t)c 2 4π ∂B z ∂x + η(x,
E x = -η(x, y, z, t)j b,x + η(x, y, z, t)c 4π ∂B z ∂y - ∂B y ∂z - k B n e (x, y, z, t)e ∂ ∂x
(n e T e ) , (7.65)

E y = -η(x, y, z, t)j b,y + η(x, y, z, t)c 4π ∂B x ∂z - ∂B z ∂x - k B n e (x, y, z, t)e ∂ ∂y
(n e T e ) (7.66)

and

E z = -η(x, y, z, t)j b,z + η(x, y, z, t)c 4π ∂B y ∂x - ∂B x ∂y - k B n e (x, y, z, t)e ∂ ∂z
(n e T e ) .

(7.67)

The first line of the B-field equations (7.62), (7.63) and ( 7 with the quasi-static approximation. Finally, the last term comes from the pressure force. This latter is usually very small compared to other terms.

Second Order Implicit Scheme describing the Self-Generated

Magnetic Fields Diffusion and Second Order Explicit Schemes describing the Self-Generated Magnetic Fields Sources

Here, we describe the numerical methods used to solve the magnetic fields equations (7.62), (7.63) and (7.64). For simplicity, we present the resolution of the y-component of the B-field (7.63) in each slice y = y j 0 , indexed by j 0 . This numerical scheme can be generalized to the x-component of the B-field (7.62) by permuting x → z, y → x and z → y (j 0 → i 0 , i → k and k → j) and to the z-component of the B-field (7.64) by permuting x → y, y → z and z → x (j 0 → k 0 , i → j and k → i). Thus, let us note B n,i,j 0 ,k y,cour = η n,i,j 0 ,k c∆t n j n,i,j 0 ,k+1 b,x

-j n,i,j 0 ,k-1 b,x 2∆z - j n,i+1,j 0 ,k b,z -j n,i-1,j 0 ,k b,z 2∆x (7.68)
the discretized B y -field generated at the time interval between t n and t n + ∆t n due to the curl of the beam current density, B n,i,j 0 ,k y,res = j n,i,j 0 ,k b,x c∆t n η n,i,j 0 ,k+1 -η n,i,j 0 ,k-1 2∆z -j n,i,j 0 ,k b,z c∆t n η n,i+1,j 0 ,k -η n,i-1,j 0 ,k 2∆x (7.69) the discretized B y -field generated at the time interval between t n and t n + ∆t n due to the resistivity gradients,

B n,i,j 0 ,k y,cross = k B c en e ∆t n n n,i+1,j 0 ,k e -n n,i-1,j 0 ,k e 2∆x T n,i,j 0 ,k+1 e -T n,i,j 0 ,k-1 e 2∆z - n n,i,j 0 ,k+1 e -n n,i,j 0 ,k-1 e 2∆z
T n,i+1,j 0 ,k e -T n,i-1,j 0 ,k e 2∆x (7.70) the discretized B y -field generated at the time interval between t n and t n + ∆t n due to the background electrons temperature-density crossed gradients and

B n,i,j 0 ,k y,3D = -∆t n 1 2∆z η n,i,j 0 ,k+1 c 2 4π B n,i,j 0 +1,k+1 z -B n,i,j 0 -1,k+1 z 2∆y - η n,i,j 0 ,k-1 c 2 4π B n,i,j 0 +1,k-1 z -B n,i,j 0 -1,k-1 z 2∆y + 1 2∆x η n,i+1,j 0 ,k c 2 4π B n,i+1,j 0 +1,k x -B n,i+1,j 0 -1,k x 2∆y - η n,i-1,j 0 ,k c 2 4π B n,i-1,j 0 +1,k x -B n,i-1,j 0 -1,k x 2∆y (7.71)
the discretized B y -field generated between t n and t n + ∆t n due to 3D effects, all expressed fully explicitely with second order schemes. The diffusion of the y-component of the B-field B y in the plane (x, z) is computed semi-implicitely according to the numerical scheme :

B n+1,i,j 0 ,k y -B n,i,j 0 ,k y ∆t n - η n,i+1/2,j 0 ,k c 2 4π∆x B n+1,i+1,j 0 ,k y -B n+1,i,j 0 ,k y ∆x + η n,i-1/2,j 0 ,k c 2 4π∆x B n+1,i,j 0 ,k y -B n+1,i-1,j 0 ,k y ∆x - η n,i,j 0 ,k+1/2 c 2 4π∆z B n+1,i,j 0 ,k+1 y -B n+1,i,j 0 ,k y ∆z + η n,i,j 0 ,k-1/2 c 2 4π∆z B n+1,i,j 0 ,k y -B n+1,i,j 0 ,k-1 y ∆z = 1 ∆t n
B n,i,j 0 ,k y,cour + B n,i,j 0 ,k y,res + B n,i,j 0 ,k y,cross + B n,i,j 0 ,k y,3D

(7.72)

where it has been noted

∀i ξ ∈ {i, k} corresponding to ξ ∈ {x, z}, η i ξ +1/2 = 2η i ξ +1 η i ξ η i ξ + η i ξ +1 and η i ξ -1/2 = 2η i ξ η i ξ -1 η i ξ + η i ξ -1 . (7.73)
Indeed, a fully explicit scheme would lead to a CFL condition

∆t n < 4π 2η max c 2 1 1 ∆x 2 + 1 ∆z 2
, that strongly constrains the time step ∆t n in case of highly resistive material and/or "highly" resolved spatial grids. For example, with η max = 10 -5 Ω.m and ∆x = ∆z = 0.1 µm, an explicit scheme would lead to time step ∆t n < 0.3 fs that can be smaller than the time step imposed by the resolution of the M1 equations. Let us define the vector

x n =           u n,Nz . . . u n,k . . . u n,1           where ∀k ∈ [1, N z ], u n,k =           B n,Nx,j 0 ,k y . . . B n,i,j 0 ,k y . . . B n,1,j 0 ,k y           , (7.74)
the vector

s n =           v n,Nz . . . v n,k . . . v n,1           where ∀k ∈ [1, N z ], v n,k =           B n,Nx,j 0 ,k y,cour + B n,Nx,j 0 ,k y,res + B n,Nx,j 0 ,k y,cross + B n,Nx,j 0 ,k y,3D
. . .

B n,i,j 0 ,k y,cour + B n,i,j 0 ,k y,res + B n,i,j 0 ,k y,cross + B n,i,j 0 ,k y,3D

. . .

B n,1,j 0 ,k y,cour + B n,1,j 0 ,k y,res (7.75) and the matrix (7.76) where

+ B n,1,j 0 ,k y,cross + B n,1,j 0 ,k y,3D          
A n =                            D Nz E Nz (0) C Nz-1 D Nz-1 E Nz-1 (0) (0) C Nz-2 D Nz-2 E Nz-2 (0) (0) . . . . . . . . . . . . . . . (0) C k D k E k (0) . . . . . . . . . . . . . . . (0) (0) C 3 D 3 E 3 (0) (0) C 2 D 2 E 2 (0) C 1 D 1                           
D k =                            d Nx,k 1 d Nx,k 4 0 d Nx-1,k 2 d Nx-1,k 1 d Nx-1,k 4 0 0 d Nx-2,k 2 d Nx-2,k 1 d Nx-2,k 4 0 0 . . . . . . . . . . . . . . . 0 d i,k 2 d i,k 1 d i,k 4 0 . . . . . . . . . . . . . . . 0 0 d 3,k 2 d 3,k 1 d 3,k 4 0 0 d 2,k 2 d 2,k 1 d 2,k 4 0 d 1,k 2 d 1,k 1                            , (7.77) C k =              d Nx,k 3 . . . ( 0
)
d i,k 3 (0) . . . d 1,k 3              and E k =              d Nx,k 5 . . . ( 0
)
d i,k 5 (0) . . . d 1,k 5              (7.78) with d i,k 1 = 1 + η n,i+1/2,j 0 ,k + η n,i-1/2,j 0 ,k c 2 ∆t n 4π∆x 2 + η n,i,j 0 ,k+1/2 + η n,i,j 0 ,k-1/2 c 2 ∆t n 4π∆z 2 , d i,k 2 = - η n,i+1/2,j 0 ,k c 2 ∆t n 4π∆z 2 , d i,k 3 = - η n,i,j 0 ,k+1/2 c 2 ∆t n 4π∆x 2 , d i,k 4 = - η n,i-1/2,j 0 ,k c 2 ∆t n 4π∆z 2 and d i,k 5 = - η n,i,j 0 ,k-1/2 c 2 ∆t n 4π∆x 2 .
(7.79)

In this case, the N x N z coupled equations (7.72) can be written with the linear form

A n .x n+1 = y n with y n = x n + s n (7.80)
where A n is a symetric positively definite matrix of dimension N x N z × N x N z . It would be computationnaly expensive to inverse numerically this linear equation ( 7.80) at each slice y = y j0 and at other slices x = x i0 for (7.62) and z = z k0 for (7.64). Consequently, by noticing that (7.80) can be written with the form df dx x=x n+1 = 0 where f (x) = 1 2 x T .A n .x -x T .y n , (7.81)

x n+1 can approached with the wanted error by using the following conjugated gradients algorithm :

m := 0 x (m) n+1 := y n r m := y n -A.x (m) n+1 p m := r m While |r m | > |y n | do α m := r T m .r m p T m .A n .p m x (m+1) n+1 := x (m) n+1 + α m p m r m+1 := r m -α m A n .p m β m := r T m+1 .r m+1 r T m .r m p m+1 := r m+1 + β m p m m := m + 1 end do.
Also, a Jacobi preconditionner is used so that it is A n .x n+1 = y n that is actually solved instead of A n .x n+1 = y n where A n = P -1 n .A and y n = P -1 n .y n with

P n =                             d Nx,Nz 1 . . . d 1,Nz 1 (0)
. . .

d i,k 1 . . . (0) d Nx,1 1 . . . d 1,1 1                            
in order to work with a better conditioned matrix. Usually, tens of m-iterations are needed to obtain

x n+1 with an error of = 10 -14 . The consistency error in the discretization of y n is O(∆t n + ∆x 2 + ∆y 2 + ∆z 2 ). An option in the Fortran file data.f90 allows for deciding to compute or not the magnetic field diffusion. In the case where the magnetic diffusion is not computed, the magnetic field B n+1 y is directly given by (7.80) with the N x N z -dimensional unity matrix I instead of A n i.e. x n+1 = y n . In the 3D-3V case where the fast electrons are injected parallel to the z-axis (without an angle of incidence), the z-component of the B-field B z is small compared to the other transverse components B x and B y .

In this particular case, the conjugated gradients algorithm does not work since the residues r m are very small. We may deduce the z-component of the B-field B n z by knowing the transverse components (B n

x , B n y ) and by imposing the Maxwell-Thomson equation (∂/∂r).B = 0. However, for simplicity, this method has not been implemented in the code and an option in the Fortran file data.f90 allows for deciding or not the computation of the diffusion of this z-component B n z of the magnetic field.

Deduction of the Self-Generated Electric Field from the Magnetic Fields

Instead of solving similar equations than (7.62), (7.63) and (7.64) for the self-generated electric field, we can compute directly the electric field at each time step t n by knowing the self generated magnetic field at the same time step t n according to

E n,i,j,k x = -η n,i,j,k j n,i,j,k b,x + η n,i,j,k c 4π B n,i,j+1,k z -B n,i,j-1,k z 2∆y - B n,i,j,k+1 y -B n,i,j,k-1 y 2∆z - k B
en n,i,j,k e n n,i+1,j,k e T n,i+1,j,k e -T n,i-1,j,k e n n,i-1,j,k e 2∆x , (7.82)

E n,i,j,k y = -η n,i,j,k j n,i,j,k b,y + η n,i,j,k c 4π B n,i,j,k+1 x -B n,i,j,k-1 x 2∆z - B n,i+1,j,k z -B n,i-1,j,k z 2∆x - k B en n,i,j,k e n n,i,j+1,k e T n,i,j+1,k e -T n,i,j-1,k e n n,i,j-1,k e 2∆y (7.83) and E n,i,j,k z = -η n,i,j,k j n,i,j,k b,z + η n,i,j,k c 4π B n,i+1,j,k y -B n,i-1,j,k y 2∆x - B n,i,j+1,k x -B n,i,j-1,k x 2∆y - k B
en n,i,j,k e n n,i,j,k+1 e T n,i,j,k+1 e -T n,i,j,k-1 e n n,i,j,k-1 e 2∆z .

(7.84)

The second term in the right hand side of each equation (7.82), (7.83) and (7.84) represents the discretized components of the total net current (7.85) multiplied by η n,i,j,k .

j t = j b + j e = c 4π ∂ ∂r × B,

Boundary Conditions

We suppose that the lateral dimensions of the simulation box L x and L y have been chosen sufficiently large to impose vanishing magnetic fields at the transverse boundaries :

B n,-1,j,k = B n,0,j,k = B n,Nx+1,j,k = B n,Nx+2,j,k = 0 (7.86) and

B n,i,-1,k = B n,i,0,k = B n,i,Ny+1,k = B n,i,Ny+2,k = 0. (7.87)
Concerning the target rear side (z = L z ), we impose the same boundary condition assuming the target is sufficiently deep (see Chapter 9 for the fast electrons refluxing) :

B n,i,j,Nz+1 = B n,i,j,Nz+2 = 0. (7.88)

At the laser-irradiated side of the target, we assume that the magnetic field is the same as in the first cells z = z 1 (k = 1) where it is usually maximal :

B n,i,j,-1 = B n,i,j,0 = B n,i,j,1 . (7.89)
Concerning the self-generated electric fields, the same boundary conditions are used i.e. :

E n,-1,j,k = E n,0,j,k = E n,Nx+1,j,k = E n,Nx+2,j,k = 0, (7.90)

E n,i,-1,k = E n,i,0,k = E n,i,Ny+1,k = E n,i,Ny+2,k = 0. (7.91)
E n,i,j,Nz+1 = E n,i,j,Nz+2 = 0 (7.92) and

E n,i,j,-1 = E n,i,j,0 = E n,i,j,1 . (7.93)
Finally, the priority is given to the boundary conditions in the z-direction at the simulation box corners.

Second order Explicit Schemes for the Heat Equations

According to the two previous sections, we know exactly the distribution function Ψ n M1 as well as the electric field E n at the time step t n in the whole simulation box. These data are needed for the

SECOND ORDER EXPLICIT SCHEMES FOR THE HEAT EQUATIONS

numerical resolution of the background electron and ion Heat equations The energy deposition due to the direct collisional energy losses of fast electrons and the Ohmic heating by the return current is deduced from the MHD and M1 packages as illustrated in Figure 7.2 according to

C V,e (x,
W n,i,j,k e = Nε l=1 S n,i,j,k,l col v l Ψ n,i,j,k,l 0 ∆ε + η n,i,j,k (j n,i,j,k e,x
)

2 + (j n,i,j,k e,y

) 2 + (j n,i,j,k e,z ) 2 (7.96)
where it has been noted S n,i,j,k,l col = S col (x i , y j , z k , ε l , t n ) the discretized stopping power of fast electrons due to collisions with background electrons (free, bound and screened free) and j n e = j n t -j n b the return current deduced from the beam current density j n b (7.61) and the total current density j n t depending on the curl of the magnetic field B n . According to the previous section 7.2.2, its discretized values are

j n,i,j,k t,x = c 4π B n,i,j+1,k z -B n,i,j-1,k z 2∆y - B n,i,j,k+1 y -B n,i,j,k-1 y 2∆z , (7.97) j n,i,j,k t,y = c 4π B n,i,j,k+1 x -B n,i,j,k+1 x 2∆z - B n,i+1,j,k z -B n,i-1,j,k z 2∆x (7.98) and j n,i,j,k t,z = c 4π B n,i+1,j,k y -B n,i-1,j,k y 2∆x - B n,i,j+1,k x -B n,i,j-1,k x 2∆y . (7.99)
Also, the discretized values of the electron-ion/lattice coupling factor G n,i,j,k and the thermal capacities

C n,i,j,k V,e
and C n,i,j,k V,i are calculated from the known temperatures T n e and T n i at the time step t n . In what follows, we explain how the electron temperature T n+1 e and the ion temperature T n+1 i at t n+1 are computed.

Let us begin with the equation for the ion temperature. It is computed according to

T n+1,i,j,k i = T n,i,j,k i + ∆t n G n,i,j,k C n,i,j,k V,i T n,i,j,k e -T n,i,j,k i . (7.100)
This numerical scheme is stable with the respect of the approximated CFL condition

∆t n < C V,i G max (7.101) T n,i,j,k e -T n,i,j,k-1 e ∆z (7.102)
where it has been noted ∀i ξ ∈ {i, j, k} corresponding to{x, y, z}, κ

i ξ +1/2 e = 2κ i ξ +1 e κ i ξ e κ i ξ e + κ i ξ +1 e and κ i ξ -1/2 e = 2κ i ξ e κ i ξ -1 e κ i ξ e + κ i ξ -1 e . (7.103)
This scheme is stable with the respect of the approximated CFL condition

∆t n < 1 2 κ e C V,e max 1 ∆x 2 + 1 ∆y 2 + 1 ∆z 2 + G C V,e max (7.104)
and presents a consistency error O(∆t + ∆x 2 + ∆y 2 + ∆z 2 ). In practice, as for the ion temperature equation, this CFL condition is much less restrictive than the CFL for the M1 equations (7.50) or (7.52). In the worst case, in the hot Spitzer regime for Tantalum with a coupling factor G ≈ 10 20 erg.cm -3 .s -1 .K -1 , a thermal capacity C V,e ≈ 10 9 erg.cm -3 .s -1 .K -1 and a thermal conduction κ e ≈ 10 13 erg.cm -1 .s -1 .K -1 , we obtain that ∆t n must be less than several fs, which is larger than a fraction of fs imposed by the CFL condition for the M1 equations. Concerning the boundary conditions, we impose the same temperatures at the target boundaries with a priority for the z-axis at the simulation box corners :

T n,-1,j,k α = T n,0,j,k α = T n,1,j,k α and T n,Nx+1,j,k α = T n,Nx+2,j,k α = T n,Nx,j,k α , (7.105) T n,i,-1,k α = T n,i,0,k α = T n,i,1,k α and T n,i,Ny+1,k α = T n,i,Ny+2,k α = T n,i,Ny,k α (7.106)
and

T n,i,j,-1 α = T n,i,j,0 α = T n,i,j,1 α and T n,i,j,Nz+1 α = T n,i,j,Nz+2 α = T n,i,j,Nz α (7.107)
for both electrons temperature (α = e) and ions temperature (α = i). Sometimes, for short laser pulses with a duration less than 100 fs, the fast electron transport simulation takes only several hundreds

SUMMARY

of fs, and one can neglect the electron thermal conductivity. Therefore, an option in the Fortran file data.f90 allows for deciding to compute or not the thermal electron diffusion. In the case where the thermal electron diffusion is not computed, the background electron temperature T n+1 e is directly

given by (7.102) with κ e = 0 in each cell. Also, sometimes, we only know the transport coefficients η and κ depending on T e = T i = T and not T i = T e . In this case, T n+1 is directly given by (7.102)

with G = 0 while the background ion heat equation is not computed. An option in the Fortran file data.f90 also allows for deciding to compute or not the temperatures T e and T i or only T = T e = T i .

Summary

We have implemented with Fortran 90 the reduced model for fast electron transport in solids or dense plasmas presented in Chapter 6. The numerical resolution of the M1 equations is performed according to the 2nd order explicit HLL scheme for the advection of the fast electrons in space and in the kinetic energy space due to the collective effects. It allows to respect the physical constraints Ψ 0 ≥ 0 and |Ψ 1 | ≤ Ψ 0 that define the realizability domain. The angular deviations of the fast electrons due to the self-generated electromagnetic field are also computed explicitely. Concerning the collisional effects, they can be computed explicitely or implicitely depending on the density of the target material. In the implicit case, the advection in the kinetic energy space due to the collisional slowing down is computed according to the implicit downwind scheme. The implicit centered scheme does not allow for respecting the realizability domain. Therefore, in order to make comparisons between implicit and fully explicit computations, we also use the explicit downwind scheme.

The self-generated electromagnetic fields as well as the background temperature(s) are computed self-consistently according to second order explicit schemes except for the magnetic field diffusion, which is computed implicitly according to the conjugated gradients algorithm. are computed and written separately. It is also performed for the other components B x and B z , etc...

In the case where the CFL condition strongly constrains the time step, it is possible to chose the intervals ∆t diag between the two consecutive time steps at which the simulation results are saved in the text files in order to limit the memory needed to stock the information.

The consistency error of the full numerical model is estimated by measuring the percentage error in the discretized energy conservation equation ( 6.29) integrated in space and time at t = t Nt :

U inj = U brem + U E + U B + U col + U res + U b + U out (7.
eE n,i,j,k x Ψ n,i,j,k,l 1,x +eE n,i,j,k y Ψ n,i,j,k,l 1,y +eE n,i,j,k z Ψ n,i,j,k,l 1,z v l ∆t n ∆x∆y∆z∆ε (7.116)
is the total kinetic energy lost by the fast electrons due to the electric field induced by the magnetic neutralization of the beam. As already mentioned, except for intermediate Z materials like Tantalum, we include the bremsstrahlung losses directly in W e,col and U col even if it is a rough approximation since the bremsstrahlung photons do not deposit their energy locally but propagate in the material depending on its opacity. Also, since j t = (c/4π)(∂/∂r) × B ≈ 0 and -(k B /n e e)(∂/∂r)(n e T e ) is small compared to ηj b , we can consider that the total kinetic energy lost by the fast electrons U res , due to their slowing down by the electric field induced by the magnetic neutralization of the beam, is also This chapter is dedicated to the validation of the model. In a first time, a 2D-3V academic case of a monoenergetic and collimated fast electron beam propagating in a warm and dense hydrogen plasma is presented. It allows us to demonstrate the major features of the M1 approximation and to derive analytical expressions for the various quantities computed by the code. Therefore, it serves to check the simulation results and to validate the numerical methods described in Chapter 7. Secondly, a more realistic simulation of a laser-generated fast electron beam transport in a thin

Aluminum target is presented. The laser-generated fast electron beam distribution function is obtained from a 2D-2V Particle-In-Cell (PIC) simulation of the laser plasma simulation, conducted by [Gremillet, 2012] with the PIC code CALDER [START_REF] Lefebvre | Electron and photon production from relativistic laser-plasma interactions[END_REF]. The resulting 2D-3V M1 simulation of the laser-generated fast electron transport is compared with a 3D-3V fast electron transport simulation conducted by [Gremillet, 2012] with the hybrid PIC code PaRIS [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF] [ [START_REF] Martinolli | Fast electron transport and heating of solid targets in high intensity laser interactions measured by Kα fluorescence[END_REF]. It shows that the M1 approximation is sufficiently accurate to reproduce the hybrid PIC simulation results.

2D-3V Academic Case

Introduction

As a first illustration of the M1 model, we consider the simple case of a quasi-monoenergetic and monodirectional (Ω ε (z = 0) = e z ) relativistic electron beam injected at z = 0 in a 2D box (100 µm × 100 µm) of a Hydrogen plasma with a density ρ = 50 g.cm -3 and an initial temperature T 0 = 1 eV. A

Gaussian distribution centered at ε 0 = (γ 0 -1)m e c 2 = 1 MeV with a 50 keV standard deviation is used for the beam energy spectrum. A Gaussian temporal shape centered at t 0 = 1750 fs with a standard deviation of σ t = 500 fs and a Gaussian spatial shape with a standard deviation of σ x = 10 µm have also been used. The electron beam has a total energy of U = 10 J. As already mentioned in Chapter 6, we neglect the electron-ion energy exchange, assume

Z * = 1, T i = T e = T and C V = (3/2)k B n i
in the heat equation. Even if the background electron thermal conduction can be neglected in this simulation due to the small considered time scale (≈ ps), it is computed according to the Hubbard-Spitzer model (for the electrical resistivity η and the thermal electron conduction κ e ; see Chapter 3, section 3.3.1) in order to check the numerical method for solving the thermal diffusion. The spatial resolution has been chosen ∆x = ∆z = 1 µm while the energy resolution has been chosen ∆ε = 5

keV with ε min = 20 keV and the maximum electron energy L ε = 1.2 MeV, so that the computation time needed is about 5 hours on 20 CPU with the full explicit numerical scheme. In this academic case, due to the high Hydrogen density, which implies strongly collisional beam transport, and a low plasma electrical resistivity because of plasma electrons degeneracy, for the value of the initial beam current density

j b (x, z, t) = j b0 exp - x 2 2σ 2 x - [z -v 0 (t -t 0 )] 2 2(v 0 σ t ) 2 (8.1)
with j b0 = U e/(2π) 3/2 ε 0 σ 2 x σ t = -1.27 10 12 A.cm -2 and v 0 = c(1 -1/γ 0 2 ) 1/2 , the collisional effects are predominant compared to the collective ones. In Figure 8.1, various contributions to the total energy integrated in space and time, as defined in Chapter 7, section 7.4, are plotted versus time.

It shows that the numerical simulation have been sufficiently converged with an error in the energy conservation of ≈ 0.3%.

Plasma Heating and Self-generated Electromagnetic Fields

By assuming a Dirac distribution in energy centered at ε 0 for Ψ 0 and by neglecting the Ohmic heating by the return current as well as the electron thermal conduction, we can evaluate from the heat equation (6.25) the plasma temperature distribution due to collisional losses of the beam close to z = 0 (to ensure the rigid beam approximation): where

T (x, z, t) ≈ T 0 + T 1 exp - x 2 2σ 2 x F (z, t) (8.2)
F (z, t) = 1 -erf[(t 0 + z/v 0 -t)/σ t √ 2]
, erf is the error function and T 1 = S(ε 0 )U/4π 2 C V ε 0 σ 2 x ≈ 17.6 eV. A comparison of the simulation profile at z = 0.5 µm and t = 3.5 ps with the estimate (8.2) shows a good agreement, as illustrated in Figure 8.2 b). That confirms that neglecting the temperature diffusion as well as the indirect electron beam energy deposition via Ohmic heating W e,res is a good approximation. Indeed, the diffusion time of the temperature is about C V σ 2

x /κ e ≈ 100 -1000 ns, which is large compared to the few ps time interval considered here. Along the z-axis, the temperature rises from z = 0 to z ≈ 30 µm reaching a maximum value of T = 21 eV and then it decreases to the initial value T 0 due to the beam's energy losses discussed in the next section.

By neglecting the resistive diffusion of the magnetic field (ηc 2 (t -z/v 0 -t 0 )/2πσ x 2 1 at the considered times of a few ps) and by approximating the temperature dependence of the resistivity in the self-generated magnetic diffusion equation (6.24) as η ≈ η 0 (T /T 0 ) α where η 0 = 9.10 -9 Ω.m and α = 0.25 according to the Hubbard theory in this regime (see Chapter 3, section 3.3.1, Figure 3.8), the estimate (8.2) of the temperature allows us to evaluate the self-generated magnetic field close to z = 0:

B y = B y,j + B y,η (8.3)
where is the contribution due to the beam current density and

B y,j ≈ B 0 T 0 T 1 (x/σ x ) α + 1 T T 0 α+1 -1 (8.4)
B y,η ≈ B 0 T 1 T 0 α(x/σ x ) exp - x 2 σ 2 x 1 + T 1 T 0 exp - x 2 2σ 2 x 1-α F (z, t) (8.5)
is the contribution due to the resitivity gradients. Here, B 0 = j b0 (ηc/σ x )σ t π/2 ≈ -7.95 T and F (z, t) of (8.2) has been approximated by the Heaviside function H(t -z/v 0 -t 0 ) to get these results.

These analytic estimates are plotted and compared with the simulation results in Figure 8.3 b). It confirms that the resistivity gradient makes a significant contribution to the self-generated magnetic field even for Hydrogen temperatures below 20 eV. The temperature-density crossed gradients do not contribute to the magnetic field generation because the plasma electron density is constant.

By neglecting the plasma pressure gradients and the self-generated magnetic field in (6.23), we can also evaluate the self-generated electric field

E z ≈ -ηj b0 exp - x 2 2σ 2 x - [z -v 0 (t -t 0 )] 2 2(v 0 σ t ) 2 . (8.6)
The maximum value of the slowing down electric force -eE.Ω ε is eηj b0 ≈ 0.1 keV.µm -1 . It is very small compared to S (see Chapter 4, section 4.2.1, Figure 4.1). This confirms again that the resistive heating W e,res in the heat equation is negligible. According to the estimates (8.4) and (8.5), the maximum of the beam cyclotron frequency ω c = eB y /γm e c is about eB 0 /m e c ≈ 1 ps -1 at x = σ x , z = 0 and t = 3.5 ps. Consequently, the inequality ω c ν is verified in this particular case (see Chapter 6, section 6.1.2, Figure 6.2). Thus, the effects of the self-generated electromagnetic fields are negligible and the evolution of the electron beam is essentially collisional. where ln Λ e is the sum of the Coulomb logarithms ln Λ rel eα of beam electrons scattering on the bound, free and screened plasma electrons (6.16), r e is the classical electron radius and the stopping power on plasma ions has been neglected since m e /m i 1. For the case of a plasma with degenerate electrons (T = 1 -20 eV T F = 351 eV for ρ = 50 g.cm -3 ), the drag number ln Λ e can be evaluated in the The mean cosine cos θ can be evaluated in the M1 model (6.34), (6.35) and (6.61) by noticing that cos θ = Ω ε .e z . Then, neglecting the self-generated electromagnetic field E and B, we finds

Kinetic evolution of the electron beam

d ds cos θ = -k 1 cos θ - 1 Ψ 0 ∂ ∂r . (Π ε .e z ) (8.10)
where

Π ε = 1 -µ 3 Ψ 0 I + Ψ 0 µ -|Ω ε | 2 Ψ 1 ⊗ Ψ 1 |Ψ 1 | 2 and d ds = 1 v ∂ ∂t + vΩ ε . ∂ ∂r -Sv ∂ ∂ε .
It has also been noted

k 1 = ν v = 4π n e r 2 e γ 2 β 4 ln Λ e + ln Λ rel ei (8.11)
the inverse of the beam electrons mean free path where ln Λ rel ei = ln 2(3/4πn i ) 1/3 /( /m e c) -1 + ln ( γ 2 -1) + 1/γ 2 is the Coulomb logarithm from the stopping power of the beam electrons on ions.

Assuming that |Ω ε | ≈ 1 which implies Π ε ≈ 0, we may neglect the second term in the right hand side of (8.10) and obtain, in agreement with the multiple scattering theory of Lewis (see Chapter 4,

section 4.3.1), that cos θ (ε) ≈ exp - s 0 k 1 (s)ds = exp - ε ε 0 k 1 (ε) dε ds -1
dε .

(8.12)

As it was suggested in [START_REF] Solodov | Stopping power and range of energetic electrons in dense plasmas of fastignition fusion targets[END_REF] and [START_REF] Robiche | Fast electron energy deposition in a magnetized plasma: Kinetic theory and particle-in-cell simulation[END_REF], the ratio ln Λ rel ei / ln Λ e can be considered as a constant (the ratio ln Λ rel ei / ln Λ rel e attains its minimum when γ = 1 with the value 0.50 and it is maximum when γ ≈ 3 with the value 0.64) and we obtain cos θ ≈ (γ -1)/(γ + 1) (γ 0 -1)/(γ 0 + 1)

ln Λ e + ln Λ rel ei 2 ln Λ e . (8.13)

Following the arguments by [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF] (see Chapter 4, section 4.2.2), we neglect the γdependance of f (γ) in (dε/ds) -1 of (8.9) and we note ln Λ * e = ln (m e c 2 / ω pe ) + (9/16) -(ln 2/2) ≈ 7.98 for ρ = 50 g.cm -3 . Moreover, by considering ln Λ e + ln Λ rel ei /2 ln Λ e ≈ 1 in cos θ of (8.7), the mean electron propagation distance can be estimated as Actually, ln Λ e + ln Λ rel ei /2 ln Λ e is minimum when γ = 1 with the value 0.75 and it is maximum when γ ≈ 3 with the value 0.82. Then, the penetration depth of the beam electrons with an initial kinetic energy ε 0 and an initial velocity v 0 = v 0 e z at z = 0 can be written as

z ≈ 1 4πn e r 2 e ln Λ * e γ 0 + 1 γ 0 -1 γ 2 0 -1 γ 0 - γ 2 -1 γ -2 ln γ 0 γ . ( 8
L p = z (ε → 0) = ξR (8.15)
where

R = 0 ε 0 dε ds -1 dε = 1 4πn e r 2 e ln Λ * e (ε 0 /m e c 2 ) 2 1 + ε 0 /m e c 2 (8.16)
is the range of the beam electrons with an initial kinetic energy ε 0 (see Chapter 4, section 4.2.2)

and ξ = γ 0 + 1 γ 0 -1 2 1 β 2 0 β 2 0 -2 ln γ 0 γ 0 (8.17)
is the correction due to angular scattering. It is equal to 2/3 when γ 0 → 1 and it increases to 1 when γ 0 → ∞. These values are in agreement with the approximation of the penetration depth ( 30)

in [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF], which assumes a factor ξ ≈ 0.8 to recover the Monte Carlo simulation. For ρ = 50 g.cm -3 and ε 0 = 1 MeV, we find R ≈ 54 µm and ξ ≈ 0.7, which corresponds to L p ≈ 38 µm.

This is in agreement with our simulation results shown in

Summary

We have derived an analytical estimate of the temperature of a Hydrogen plasma heated through the direct collisional losses of a monoenergetic and collimated fast electron beam. It is in good agreement with the numerical result close to the fast electron injection zone. That fact validates the numerical scheme used to solve the Heat Equations presented in Chapter 7, section 7.3. Moreover, the analytical estimates of the self-generated magnetic fields generated due to the curl of the beam current density and the resistivity gradients, assuming that the Hubbard electrical resistivity follows roughly η = η 0 (T /T 0 ) α with η 0 = 9.10 -9 Ω.m and α = 0.25 in the considered range of temperatures. These estimates are in agreement with the magnetic fields calculated numerically close to the fast electron injection zone. The agreement betwen analytical estimates and numerical solutions concerning the self-generated electromagnetic fields validates the numerical schemes presented in Chapter 7, section 7.2. In addition, by neglecting the collective effects in this academic case, we obtained estimates of the mean propagation angle cos θ and the fast electron penetration depth L p due to collisional effects.

An analytical expression of the correction factor

ξ = γ 0 + 1 γ 0 -1 2 1 β 2 0 β 2 0 -2 ln γ 0 γ 0
due to the fast electrons angular scattering have been derived in the expression of the penetration depth L p in agreement with the empirical value given by [START_REF] Atzeni | Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[END_REF]. The estimates for cos θ and L p are also in good agreement with the M1 simulation results, thus validating the numerical schemes used to solve the M1 equations presented in Chapter 7, section 7.1. where ω 0 is the laser frequency, according to the Particle-In-Cell (PIC) simulation of the laser plasma simulation conducted by [Gremillet, 2012]. The z-axis in all the following section (like in the Left Panel) corresponds to the x-axis of the PIC simulation picture (Right panel) and the x-axis correspond to the y-axis, respectively In this section, the M1 model is compared to a hybrid PIC simulation of a relativistic electron beam propagation in a thin solid target, motivated by an experimental campaign [START_REF] Santos | Supra-thermal electron beam stopping power and guiding in dense plasmas[END_REF].

The target is composed of three successive layers of 1 µm of Aluminum, 3 µm of Copper and 1 µm of Aluminum as illustrated in the left panel of Figure 8.7. A linearly polarized laser pulse with a wavelength λ = 800 nm, a total energy E L = 0.7 J and a 26 fs Full Width at Half Maximum (FWHM) time duration is focused with a peak intensity of I L = 3.10 19 W.cm -2 at a 45 • incident angle. Plasma mirrors have been used in this experiment to avoid prepulse/preplasma so that the main electron acceleration mechanism during the laser-target interaction is the j × B heating (see Chapter 1, section 1.2) and the accelerated electrons propagate mainly in the laser pulse propagation direction. The laser-generated fast electron beam's initial properties are obtained from a 2D-2V fully PIC simulation of the laser-plasma interaction [Gremillet, 2012] using the code CALDER [START_REF] Lefebvre | Electron and photon production from relativistic laser-plasma interactions[END_REF] beam F ε (ε, t), its angular F ϕ (ε, ϕ, t) and spatial F x (x, t) distribution as well as the instantaneous conversion efficiency η L→e (t) from the laser to the beam have been interpolated by [Gremillet, 2012].

Here, we present how they are adapted to initializing the angular moments Ψ 0 and Ψ 1 at z = 0.

Firstly, the cut-off ε min = 20 keV is used to distinguish between the beam electrons and the bulk electrons. Secondly, the initial beam distribution is parameterized according to the PIC simulation: (8.21) which is equivalent to

F ε (ε, t) =      0 if ε > ε max exp c 1 (γ(ε) + c 2 ) c 3 -c 4 γ(ε) -c 5 else , ( 8
α 1 (Ω ε (x, z = 0, ε, t)) = 8 ln 2 ∆ϕ(ε, t) 2     sin ϕ 0 (ε, t) 0 cos ϕ 0 (ε, t)     ,
Ω ε (x, z = 0, ε, t) = coth 8 ln 2 ∆ϕ(ε, t) 2 - ∆ϕ(ε, t) 2 8 ln 2     sin ϕ 0 (ε, t) 0 cos ϕ 0 (ε, t)     (8.22)
in such a way that, in agreement with the angular notations introduced in Chapter 7, Figure 7.1,

π 0 Ψ M1 (x, z = 0, ε, θ, ϕ, t) sin θdθ = Ψ 0 (x, z = 0, ε, t) |α 1 (ε, t)| 4 sinh |α 1 (ε, t)| I 1 |α 1 (ε, t)| cos (ϕ -ϕ 0 (ε, t)) + L -1 |α 1 (ε, t)| cos (ϕ -ϕ 0 (ε, t)) ≈ Ψ 0 (x, z = 0, ε, t) 2π ∆ϕ(ε, t) 2 8 ln 2 exp -4 ln 2 (ϕ -ϕ 0 (ε, t)) 2 ∆ϕ 2 .
(8.23) as already discussed in Chapter 6, section 6.2.3. The first angular moment has been initialized as follows:

Ψ 0 (x, z = 0, ε, t) = N 0 (t)f x (x, t)f z (z = 0, t)f ε (ε, t) (8.24)
where

N 0 (t) = ν L→e (t) E L k B T b0 (t) 2π ∆y 2 8 ln 2 (8.25)
is the number of electrons per unit length in the third y-dimension (not taken into account in this simulation) with ∆y = 8.6 µm,

f x (x, t) = F x (x, t) 2π ∆x(t) 2 8 ln 2 (8.26)
is the normalized transverse spatial distribution function,

f z (z = 0, t) = 1 V b (t) 2π ∆t 2 8 ln 2 exp -4 ln 2 t -t c ∆t 2 (8.27)
is the normalized longitudinal spatial distribution at z = 0 and

f ε (ε, t) = F ε (ε, t) ∞ ε min F ε (ε, t) dε (8.28)
is the normalized energy spectrum of the fast electrons. Here, t c = 40 fs,

k B T b0 (t) = ε (t) = ∞ ε min εf ε (ε, t) dε (8.29)
is the initial "beam temperature" and

V b (t) = εv (t) ε (t) .e z = ∞ ε min εv(ε) k B T b0 (t) |Ω ε (x, z = 0, ε, t)| cos (ϕ 0 (ε, t))f ε (ε, t) dε (8.30)
is the velocity at which the beam kinetic energy is injected in the simulation box at z = 0 in the z-direction. Finally, the first order angular moment is initialized according to (8.21) : The More formula for the ionization state Z * has been used without the corrections introduced in Chapter 6, section 6.3.1. Both background electron and ion heat equations have been computed

Ψ 1 (x, z = 0, ε, t) = Ψ 0 (x, z = 0, ε, t)Ω ε (x, z = 0, ε, t). ( 8
with the plasma expression for the thermal capacities and the electron-ion coupling factor. The spatial resolution has been chosen ∆x = ∆z = 0.25 µm while the energy resolution has been chosen ∆ε = 10 keV, in the range from ε min = 20 keV to 3 MeV so that the computation time needed is about 4 hours and 40 minutes on 20 CPU. Absorbing conditions at the target boundaries have been used so that the refluxing of the beam electrons at both the rear and irradiated sides of the target was suppressed.

As illustrated in Figure 8.9, the percentage of error in the energy conservation equation is about 0.5 %. The total injected energy at z = 0 is ≈ 70 mJ, which represents a conversion efficiency from the laser to the electron beam of ≈ 10 %. The electromagnetic energy is negligible compared to the beam energy by a factor ≈ 1000. The heating of the target due to the return current (≈ 10 mJ)

exceeds by roughly two times the direct collisional heating by the beam electrons. Thus, contrary to the previous academic case, collective effects are here predominant. Indeed, while the maximum initial beam density is close to the critical density n b,max ≈ 10 21 cm -3 , the maximum value of the initial beam current density j b,max is above 10 12 A.cm -2 and the Aluminum and Copper electrical resistivity is much higher than that of Hydrogen (see Figure 3.8). Indeed, Copper is less resistive but denser than Aluminum; if direct collisional losses were dominant, the Copper temperature would be higher than in Aluminum, which is not the case here. Kα photons has been computed by using the empirical expression for K-shell ionization cross section by electron impact [Hombourger, 1998] and the K-shell fluorescence yield probability provided by [Bambynek, 1984]. It is plotted in Figure 8.12 d) at t = 500 fs. There is an important discrepancy between the simulation result and the experimental data concerning the Kα spot size. This can be explained by the refluxing of beam electrons from the target boundaries, which enhances their lateral expansion and consequently increases the Kα emission spot size. Note, however, that, at this stage, collisions between the background free electrons and the d-band electrons in Copper are not taken into account. These two effects will be considered in Part III.

Comparison with the Hybrid PIC Simulation

Conclusion

From an experimental point of view, many methods may be used to diagnose the properties of electrons produced by intense laser pulses, including vacuum electron spectrometry [START_REF] Wei | Observations of the filamentation of high-intensity laser-produced electron beams[END_REF] [ [START_REF] Yabuuchi | Transport study of intense-laser-produced fast electrons in solid targets with a preplasma created by a long pulse laser[END_REF], nuclear activation [START_REF] Hatchett | Electron, photon, and ion beams from the relativistic interaction of Petawatt laser pulses with solid targets[END_REF]] [Ledingham et al., 2000], optical emission from foams [START_REF] Jung | Study of Electron-Beam Propagation through Preionized Dense Foam Plasmas[END_REF], optical probing [START_REF] Norreys | Observation of annular electron beam transport in multi-TeraWatt laser-solid interactions[END_REF]] [Ping et al., 2012], xrays bremstrahlung spectrometry [START_REF] Chen | Bremsstrahlung and Kα fluorescence measurements for inferring conversion efficiencies into fast ignition relevant hot electrons[END_REF]] [Westover et al., 2011], Incoherent Transition Radiation (ITR) or Coherent Transition Radiation (CTR) spectrometry and imagery [START_REF] Santos | Fast Electron Transport in Ultraintense Laser Pulse Interaction with Solid Targets by Rear-Side Self-Radiation Diagnostics[END_REF] [ [START_REF] Storm | High-Current, Relativistic Electron-Beam Transport in Metals and the Role of Magnetic Collimation[END_REF] and Kα fluorescence measurements [START_REF] Stephens | Kα fluorescence measurement of relativistic electron transport in the context of fast ignition[END_REF]] [Baton et al., 2008].

Each technique has advantages and disadvantages for measuring aspects of the electron beam. The vacuum electron spectroscopy measures the properties of electrons escaping the target, which may differ from those of the electrons in the bulk of the material [START_REF] Yabuuchi | Transport study of intense-laser-produced fast electrons in solid targets with a preplasma created by a long pulse laser[END_REF]. The nuclear activation is likewise sensitive to very energetic (> 10 MeV) x-rays, which are produced by highenergy electrons outside the spectral region with the best coupling to the dense core in Fast Ignition [START_REF] Kodama | Nuclear fusion: Fast heating scalable to laser fusion ignition[END_REF]. The x-ray bremsstrahlung spectrometers, that measure the light emitted by the fast electrons thanks to a set of compact filter-stack based x-ray detectors, are sensitive to x-rays in the 10-700 keV range corresponding to energetic electrons (> 1 MeV) [START_REF] Westover | Fast electron temperature and conversion efficiency measurements in laserirradiated foil targets using a bremsstrahlung x-ray detector[END_REF]. The CTR and ITR techniques are operating in the visible domain by detecting the emission produced by fast electrons crossing the rear target. [START_REF] Santos | Fast Electron Transport in Ultraintense Laser Pulse Interaction with Solid Targets by Rear-Side Self-Radiation Diagnostics[END_REF] showed that by imaging this transition radiation, the spatial distribution of electrons emerging from the target can be accurately measured.

When the emerging hot electron flux is modulated at the laser frequency and/or its harmonics, the emitted CTR can be much brighter than the ITR that results from un-bunched electrons. The coherent addition of the transition radiation from periodically bunched hot electrons also yields information on the acceleration mechanism, as the spectrum of the CTR is intimately related to the period of the bunches [START_REF] Baton | Evidence of Ultrashort Electron Bunches in Laser-Plasma Interactions at Relativistic Intensities[END_REF]] [Zheng et al., 2003] [Schroeder et al., 2004] [Bellei et al., 2012].

The most commonly used diagnostic is the imaging of X-rays that are produced in the cold target material as the electrons propagate through the target. The Kα radiation is due to the atomic electron transitions to K-shell holes produced in a collision of a hot electron with an atom of the target material; see Figure 9.1. When an outer shell electron fills the vacancy, the energy is released in a form of Auger electron or an emission of a photon. The cross section for this K-shell electron impact ionization peaks at the electron energy roughly two to three times the ionization energy and decays slowly at higher energies. Since most materials used in experiments have K-shell transitions in the tens of keV range, this X-ray imaging technique is, in principle, sensitive to electrons with energies as low as tens of keV. However, with an electron energy distribution extending to much higher energies, the signal of the lowest energy electrons will be diminished and masked. Imaging this K-shell X-ray emission with a spherically bent diffraction crystal has become so a powerful diagnostic of hot electron beams. However, it has been shown that a Kα image is not solely determined by the initial population of forward directed hot electrons, but also depends upon "delayed" hot electrons, and in fact continues to evolve long after the end of the laser interaction [START_REF] Ovchinnikov | How well do time-integrated Kα images represent hot electron spatial distributions?[END_REF].

Firstly, there is a population of hot electrons created in the laser-plasma interaction that acquire a velocity direction opposite that of the laser and subsequently reflux off the front surface of the target, deflect when they encounter magnetic fields and then spread far from the laser focal spot [START_REF] Pérez | Deflection of MeV Electrons by Self-Generated Magnetic Fields in Intense Laser Solid Interactions[END_REF]. These delayed fast electrons create significant features in the Kα time-integrated images. Secondly, the electrons refluxing from the sides and the rear of the target also contribute to the final Kα image [START_REF] Ovchinnikov | How well do time-integrated Kα images represent hot electron spatial distributions?[END_REF]. Indeed, shortly after the beginning of the laser-plasma interaction, the hot electrons leave the target, making it positively charged. This gives rise to strong electric fields that cause most of the escaping electrons return to the target. In some experiments the refluxing is minimized by increasing the target dimensions and/or by using materials with short stopping distances.

More recently, a novel technique of shadowgraphy coupled to phase contrast imaging has been proposed. The Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory in Stanford (United States of America) is one of few available sources of high peak brightness photons suitable for studies of transient behaviour of hot electrons. They can provide a sub-ps temporal resolution and a spatial resolution of the order of several 10 µm (inaccessible for conventional sources).

The basic principle of this novel technique relies on the shift of the K-edge of the target atoms after the hot electrons have created a K-shell vacancy. This shift is sufficiently large (about 400 eV for an electron in the K-shell of Copper according to Atomic Physic numerical computations) to be used as a femtosecond X-ray switch. This technique will allow to observe traces of hot electron transport, depending on whether electrons have created K-shell hole(s) or not in the atoms, thus leading to a transparent or opaque material for the LCLS photons. The vacancies created by a collisional ionization of cold atoms are also those responsible for the emission of Kα photons. Tuned to a photon energy just above the K-edge, these photons, primarily absorbed before the hot electrons propagate, will be 9.1. COMPUTATIONAL METHODS FOR ESTIMATING THE EMISSION OF Kα PHOTONS transmitted after their propagation. However, this technique, directly related to the Kα emission, can be biased because of refluxing of hot electrons at boundaries. Cross-correlation of this Kα emission with direct observations of the hot electrons inside the sample may remove this ambiguity.

This chapter is dedicated to the implementation of the commonly used Kα photons diagnostic in the M1 model for the fast electron transport in solids or dense plasmas, presented in Part II.

Indeed, thanks to the fast computations allowed by the M1 model, sufficiently long times (several tens of ps) and large spatial dimensions (several hundreds of µm) can be computed with a relatively small computational cost. We will study the recirculation of the fast electrons due to their reflection at the target borders, the K-shell hole dynamics and 3-dimensional effects on photon emission. It will be also possible to apply these developments to the novel technique of shadowgraphy based on the LCLS X-ray source. In a first time, the model for the computation of the emission of Kα photons is presented. Secondly, the model is applied to the interpretation of experiments conducted by [START_REF] Santos | Supra-thermal electron beam stopping power and guiding in dense plasmas[END_REF] on the UHI100 laser facility of the CEA (Saclay) introduced in the previous section Chapter 8, section 2. The source of fast electrons was calculated by [Gremillet, 2012] where p b is the mean momentum of beam electrons. In a first attempt, electron refluxing can be described as specular reflections off the target boundaries.

However, one issue of the M1 model is that, by working with the angular averaged values, it (vΨ 1 ) -∂ ∂ε (vSΨ 0 + evΨ 1 .E) = 0, (9.1)

∂Ψ 1 ∂t + ∂ ∂r . (vΨ 2 ) - ∂ ∂ε (vSΨ 1 + evΨ 2 .E) = - e p (Ψ 0 I -Ψ 2 ) .E - e γm e c Ψ 1 × B -νΨ 1 (9.2)
and

∂Ψ 2 ∂t + ∂ ∂r . (vΨ 3 ) - ∂ ∂ε (vSΨ 2 + evΨ 3 .E) = - e p (Ψ 1 ⊗ E + E ⊗ Ψ 1 -2Ψ 3 .E) - 2e γm e c [Ψ 2 .T(B) -T(B).Ψ 2 ] -3ν Ψ 2 - 1 3 Ψ 0 I (9.3)
where it has been noted T(B) the tensor

T(B x , B y , B z ) =     0 B z -B y -B z 0 B x B y -B x 0     such that ∀V ∈ R 3 , T(B).V = V × B.
The third order angular moment Ψ 3 (third order tensor → 27 terms) must be computed according to the Minerbo maximum entropy criterion in order to close the system. However, this option is outside the scope of this thesis. PHOTONS Two Sets of M1 Equations as a Solution for the M1 Limitation Instead, we choose another possibility that consists in solving two sets of M1 equations for each counterpropagating electron beam in order to account for the fast electrons refluxing : one set of M1 equations (see Chapter 6, section 6.2), for the angular moments Ψ

(1) 0 and Ψ

(1)

1 , is computed for the fast electrons population propagating in the laser pulse propagation direction and another one, for the angular moments Ψ

(2)

0 and Ψ (2)
1 , is computed for the refluxed electrons population that propagate in the opposite direction. While the absorbing boundary conditions, detailed in Chapter 7, section 7.1.6, are not modified, we initialize the electron beam (2) propagating in the backward direction upon assuming specular reflections off the target rear side of the forward-propagating beam (1) . It is expected that we do not lose much of physics since λ D,b is small compared to the considered space scales ∆z and since only a small part of fast electrons is escaping from the target (few percents).

From a numerical point of view, it reads, according to the HLL scheme for the fast electrons spatial advection (see Chapter 7 for the notations) : 2),n,i,j,Nz+2 = φ (2),n,i,j,Nz+1 = 0 (9.4) and If Ψ

If Ψ (1),n,i,j,Nz,l 1,z > 0 then              Ψ (2),n,i,j,Nz+2,l 0 = Ψ (2),n,i,j,Nz+1,l 0 = Ψ (1),n,i,j,Nz,l 0 Ψ (2),n,i,j,Nz+2,l 1,x = Ψ (2),n,i,j,Nz+1,l 1,x = Ψ (1),n,i,j,Nz,l 1,x Ψ (2),n,i,j,Nz+2,l 1,y = Ψ (2),n,i,j,Nz+1,l 1,y = Ψ (1),n,i,j,Nz,l 1,y Ψ (2),n,i,j,Nz+2,l 1,z = Ψ (2),n,i,j,Nz+1,l 1,z = -Ψ (1),n,i,j,Nz,l 1,z else φ (
(2),n,i,j,2,l 1,z < 0 then φ (2),n,i,j,0,l = φ (2),n,i,j,1,l = φ (2),n,i,j,2,l else φ (2),n,i,j,0,l = φ (2),n,i,j,1,l = 0.

(9.5)

Concerning the other boundary conditions at x = ±L x /2, y = ±L y /2, ε = ε min and ε = L ε , we use the boundary conditions explained in Chapter 7, section 7.1.6, assuming L x and L z are taken sufficiently large (several hundreds of µm) to avoid fast electrons refluxing at the target lateral boundaries as it is usually the case in experiments.

Also, the use of large transverse dimensions L x and L y allows us to neglect the electromagnetic fields at the transverse simulation box boundaries ±L x /2 and ±L y /2, as explained in Chapter 7, section 7.2.3. However, contrary to Chapter 7, section 7.2.3, we cannot impose the same boundary condition at the target rear side z = L z . Instead of Equations (7.88) and (7.92), we impose the same boundary conditions as at the irradiated side of the target (7.89) and (7.93). It reads respectively :

B n,i,j,Nz+2 = B n,i,j,Nz+1 = B n,i,j,Nz (9.6) and E n,i,j,Nz+2 = E n,i,j,Nz+1 = E n,i,j,Nz . (9.7)

In the case of a normally incident laser pulse on a solid target, the electrons coming back to the laser-plasma interaction zone may be strongly deviated due to the presence of local magnetic fields or they may enter in the underdense preplasma. In all cases, we assume that these electrons are taken into account by the Particle-In-Cell (PIC) simulation of the laser plasma interaction Φ ini and we consequently let all fast electrons (1) and (2) escape from the target in the laser-irradiated side.

However, in the case of a high contrast, short pulse obliquely incident on the target, the laser-generated fast electrons (1) propagate mainly in the laser pulse propagation direction. The refluxed fast electrons (2) , coming from the target rear side arrive at an abrupt solid-vacuum interface at the laser irradiated side of the target away from the laser plasma interaction zone. Therefore, in this case, we also account for the refluxing of the fast electrons (2) at the front side. Thus, after initializing the laser-generated fast electron (1) as If t < t source then φ (1),n,i,j,0,l = φ (1),n,i,j,1,l = φ n,i,j,1,l ini else if Ψ

(1),n,i,j,2,l 1,z < 0 and t > t source then φ (1),n,i,j,0,l = φ (1),n,i,j,1,l = φ (1),n,i,j,2,l else φ (1),n,i,j,0,l = φ (1),n,i,j,1,l = 0 (9.8) by knowing the fast electron distribution φ ini according to the PIC simulation of the laser plasma interaction, we add the fast electrons component due to the refluxing of the fast electrons population propagating backward (2) as follows.

If Ψ

(2),n,i,j,2,l 1,z < 0 then

            

Ψ

(1),n,i,j,0,l 0 = Ψ

(1),n,i,j,1,l 0 = Ψ

(1),n,i,j,1,l 0

+ Ψ

(2),n,i,j,2,l 0

Ψ

(1),n,i,j,0,l 1,x = Ψ

(1),n,i,j,1,l 1,x = Ψ

(1),n,i,j,1,l 1,x + Ψ

(2),n,i,j,2,l 1,x

Ψ

(1),n,i,j,0,l 1,y = Ψ

(1),n,i,j,1,l 1,y = Ψ

(1),n,i,j,1,l 1,y + Ψ

(2),n,i,j,2,l 1,y

Ψ

(1),n,i,j,0,l 1,z = Ψ

(1),n,i,j,1,l 1,z = Ψ

(1),n,i,j,1,l 1,x -Ψ

(2),n,i,j,2,l 1,z . (9.9)

An option in the Fortran 90 file data.f90 allows for enabling or not the refluxing of fast electrons at

COMPUTATIONAL METHODS FOR ESTIMATING THE EMISSION OF Kα PHOTONS

the target rear side and another one for enabling or not the refluxing of fast electrons at the target irradiated side. This is illustrated in Figure 9.3. Obviously, solving two sets of M1 equations doubles the computational cost. Also, the beam density reads now

n b = Lε ε min Ψ (1) 0 + Ψ (2) 0 dε → n n,i,j,k b = Nε l=1 Ψ
(1),n,i,j,k,l 0 + Ψ

(2),n,i,j,k,l 0 ∆ε (9.10) while the beam current density needed to compute the self-generated electromagnetic fields reads now

j b = -e Lε ε min v Ψ (1) 1 + Ψ (2) 1 dε → j n,i,j,k b = -e Nε l=1 v l Ψ (1),n,i,j,k,l 1 + Ψ (2),n,i,j,k,l 1 ∆ε. (9.11)
In the following, we note Ψ 0 = Ψ

(1)

0 + Ψ (2)
0 and Ψ 1 = Ψ

(1)

1 + Ψ (2)
1 .

Emission of Kα Photons in Hybrid Models due to Lasergenerated Fast Electron Beam

K-shell emission is produced when a fast electron knocks out a K-shell electron from an atom in the solid target is then replaced by an electron from an outer shell -a transition that leads to the emission of a photon with a characteristic energy in the x-ray band (see Figure 9.1). If the outer electron comes from the L-shell, the emission is called Kα; if it comes from the M-shell, it is called Kβ. Depending on the orbital moment of the L-shell electron, one can distinguish between a more energetic Kα 1 photon and a less energetic Kα 2 photon. According to the Dipole Transition Criterion, Kα emission is more probable than Kβ emission while Kα 1 emission is more probable than Kα 2 emission. Therefore, the majority of K-shell diagnostics used in laser solid experiments rely on the detection of Kα 1 photons. A fundamental parameter is the collisional K-shell electron ionization cross section. According to [START_REF] Davies | Copper K-shell emission cross sections for laser-solid experiments[END_REF] for Copper tracer layer, the empirical formula provided by [Hombourger, 1998]

σ K (ε) = 2π r 2 Bohr G r (ε) D(ε) E 0 E K C(ε) (9.12)
is the most acurate expression found by the authors in the literature for the K-shell electron ionization cross section induced by a collision with a laser-generated fast electron with a kinetic energy ε. Here, r Bohr = 5.2918.10 -9 cm is the Bohr radius, E 0 = 13.61 eV is the fundamental Hydrogen state energy, E K is the ionization energy of the K-shell electron depending on the atomic number Z of the material, where U = ε/E K is the normalized kinetic energy of the fast electron and

D(ε) = 3.125 - 4.172 U + 1.877 U 2 ln U U , (9.13)
C(ε) = 2.0305 - 0.3160 U + 0.1545 U 2 (9.14) TRANSPORT IN SOLID TARGETS Material Z E K [eV] hν Kα 1 [eV] hν Kα 2 [eV] I Kα 2 /I
G r (ε) = (1 + 2J) (U + J) 2 (1 + U ) (U + 2J) (1 + J) 2 3/2 (U + 2J) (1 + J) 2 [J 2 (1 + 2J) + U (U + 2J) (1 + J 2 )] 3/2 (9.15)
is the Grysinski coefficient where J = m e c 2 /E K is the normalized electron mass energy. Actually, there is a mistake in the original paper by [Hombourger, 1998]. It is 2J and not only 2 as colored in red in (9.15) [Gryziński, 1965a[START_REF] Gryziński ; Gryziński | Two-Particle Collisions. II. Coulomb Collisions in the Laboratory System of Coordinates[END_REF].

According to [START_REF] Davies | Copper K-shell emission cross sections for laser-solid experiments[END_REF], it is usually assumed that the fraction of atoms with empty K-shells, as a result of collisions with fast electrons, is negligible. Under this assumption, by knowing the K-shell fluorescence yield ω K depending on the atomic number Z of the material, one may directly deduce the number of Kα and Kβ photons emited per unit of time, volume and steradian according to

dn α1 dt Emitted = 1 4π F Kα 1 F Kα ω K 2n i τ b with F Kα 1 = 1 1 + I Kα 2 I Kα 1 and F Kα = 1 1 + I Kβ I Kα , dn α2 dt Emitted = 1 4π F Kα 2 F Kα ω K 2n i τ b with F Kα 2 = 1 1 + I Kα 1 I Kα 2 = 1 -F Kα 1 and dn β dt Emitted = 1 4π F Kβ ω K 2n i τ b with F Kβ = 1 1 + I Kα I Kβ = 1 -F Kα (9.16)
where

τ b = ∞ ε min Ψ 0 σ K v dε -1 . (9.17)
Here, n i is the ion density, I Kα 1 , I Kα according to [START_REF] Thomson | X-RAY DATA BOOKLET[END_REF]. The factor 1/4π comes from the fact that the photons emission is assumed to be isotropic. F Kα i F Kα ω K and F Kβ ω K are the probabilities that a photon Kα i or K β , respectively, is emitted when a K-shell hole is recombined by a L-shell electron or a M-shell electron, respectively. 2n i is the density of holes induced by the ionization of K-shell electrons (there are two electrons in the K-shell of atoms used in experiments). τ b is the characteristic ionization

COMPUTATIONAL METHODS FOR ESTIMATING THE EMISSION OF Kα PHOTONS

time or the inverse of the ionization frequency. It accounts for the fast electron energy distribution Ψ 0 (r, ε, t) at the space location r and time t and the ionization probability σ K (ε)v(ε) that a fast electron with a kinetic energy ε ionize one K-shell electron. According to [START_REF] Kahoul | K-shell fluorescence yields for elements with 6≤Z≤99[END_REF], the K-shell fluorescence yield can be estimated as ω K = 0.985 (Z/30.896) 3.847 1 + (Z/30.896) 3.847 . (9.18) This formula fits the experimental results compiled by [START_REF] Bambynek | X-Ray Fluorescence Yields, Auger, and Coster-Kronig Transition Probabilities[END_REF] and [START_REF] Krause | Natural widths of atomic K and L levels, Kα X ray lines and several KLL Auger lines[END_REF] with an error less than 5 % except Aluminum with a deviation of 14 % compared to [START_REF] Krause | Natural widths of atomic K and L levels, Kα X ray lines and several KLL Auger lines[END_REF]. The time integrated number of Kα 1 photons emitted per unit of volume and steradian

n α 1 (r, t) = t 0 dn α 1 dt Emitted dt (9.19)
is computed self-consistently with the fast electron beam transport hybrid model, as illustrated in

K-shell Hole Density Dynamics

As explained in the previous subsection, the method of computing the emisssion of Kα photons (9.19) is based on five assumptions :

1. The hole lifetime τ K due to K-shell fluorescence, Auger and Coster-Kronig effects is small compared to the ionization time τ b .

2. The numerical time step of the fast electron transport hybrid model ∆t n is large compared to the hole lifetime τ K so that the density of holes n H (r, t n+1 ) attains its maximum value

2n i (r, t n )τ K /τ b (r, t n ).
3. The target is fully transparent for the Kα photons.

4. The K-shell photoionization by the X-rays emitted by the fast electrons or other laser-plasma processes is negligible compared to the collisional ionization by the fast electrons.

5. The K-shell fluorescence yield ω K and the collisional ionization cross section σ K do not depend on the target temperature.

A more detailed analysis of Kα emission was proposed by [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF]. It allows us to demonstrate that, while the assumption 1 is fully justified, one must be careful with the assumption 2, which is depending on the tracer material.

Let us note n H1 the density of atoms with one hole in the K-shell.

According to [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF], its temporal evolution can be described by the following kinetic equation

∂n H1 ∂t = R K1 - n H1 τ K where R K1 = 2 n i -n H1 τ b (9.20)
is the ionization rate depending on the density of available K-shell holes 2(n i -n H1 ). τ K = /Γ K is the hole life time deduced from the K-shell natural level width Γ K ( Heisenberg incertitude principle).

The factor 2 in R K1 comes from the assumption that the K-shell is initially full, as it is the case in the laser-solid experiments. Let us note n H2 the density of atoms in the solid target with two holes in the K-shell. According to [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF], its temporal evolution can be described by the following kinetic equation

∂n H2 ∂t = R K2 - n H2 τ K where R K2 = n H1 -n H2 τ b (9.21)
is the ionization rate of the second K-shell electron and (n H1 -n H2 ) is the density of available atoms with only one K-shell hole. Then, the total number of atoms with one or two holes in the K-shell n H = n H1 + n H2 evolves according to the following kinetic equation

∂n H ∂t + 1 τ b + 1 τ K n H = 2n i τ b . (9.22)

COMPUTATIONAL METHODS FOR ESTIMATING THE EMISSION OF Kα PHOTONS

According to [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF], the K-shell natural level width can be estimated by (9.23) which agrees with the results compiled by [Krause, 1979] with an error less than ≈ 10 % for Z ≥ 10. The hole lifetime τ K = /Γ K is plotted in the right panel of Figure 9.5. As proposed by [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF], one can solve analytically Equation (9.22) to determine in each numerical spatial cell (i, j, k) the hole density n n+1,i,j,k H at t n + ∆t n as a function of the ionization time

Γ K = exp -0.0002725Z 2 + 0.09932Z -2.160 eV,
τ n,i,j,k b = 1 Nε l=1 Ψ n,i,j,k,l 0 σ K (ε l )v l ∆ε (9.24)
and the hole density n n,i,j,k H at time t n . It reads [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF]]

n n+1,i,j,k H = n n,i,j,k H + n n,i,j,k H - 2n n,i,j,k i Γ n,i,j,k τ n,i,j,k b exp -Γ n,i,j,k ∆t n -1 = 1 -Γ n,i,j,k ∆t n n n,i,j,k H + 2n n,i,j,k i ∆t n τ n,i,j,k b + O Γ n,i,j,k ∆t n 2 if Γ n,i,j,k ∆t n 1 = 2n n,i,j,k i Γ n,i,j,k τ n,i,j,k b if Γ n,i,j,k ∆t n 1 (9.25)
where it has been noted Γ n,i,j,k = (1/τ K ) + (1/τ n,i,j,k b

). and E K = 67416 eV). We deduce consequently that the assumption 1 is fully justified and that we can consider Γ n,i,j,k = 1/τ K . However, depending on the material and on the numerical time step ∆t n , the assumption 2 is not necessarily verified as the hole lifetime can be comparable with the numerical time step ∆t n ≈ 0.5 fs (see the CFL condition (7.50) of Chapter 7, section 7.1.4 or (7.52) of Chapter 7, section 7.1.5). Besides, for intermediate Z materials like Tantalum ∆t n τ K = 0.01731 fs, we obtain n n+1,i,j,k

H → 2n n,i,j,k i τ K /τ n,i,j,k b by making Γ → 1/τ K 1/∆t n in (9.25).
According to [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF], the Kα 1 emission rate reads per unit of time, volume and steradian reads

dn α 1 dt Emitted = 1 4π F Kα 1 F Kα ω K n H τ K . (9.27)
Therefore, in the particular case where Γ n,i,j,k ∆t n = ∆t n /τ K + o(τ K /τ b ) 1 as for Tantalum, (9.25) and (9.27) lead to the standard expression (9.19) of the time integrated density of Kα photons emitted per steradian. It is not the case for Copper and Aluminum. Consequently, we decide to take into

COMPUTATIONAL METHODS FOR ESTIMATING THE EMISSION OF Kα PHOTONS

account the hole density dynamics in our model. It will be also useful for studies of the temporal dynamics of the hole density in the context of the X-ray shadowgraphy.

The analytical solution of (9.22) for t ∈ [t n , t n + ∆t n ] is

n i,j,k H (t) = n i,j,k,n H - 2n n,i,j,k i Γ n,i,j,k τ n,i,j,k b exp -Γ n,i,j,k (t -t n ) + 2n n,i,j,k i Γ n,i,j,k τ n,i,j,k b (9.28)
by assuming that τ n,i,j,k b and n n,i,j,k i are constants during the numerical time step. Correspondingly, we compute the Kα 1 , Kα 2 and Kβ photons number emited per unit of time, volume and steradian according to

dn α1 dt n,i,j,k Emitted = 1 4π F Kα 1 F Kα ω K n i,j,k H n τ K , dn α2 dt n,i,j,k Emitted = 1 4π F Kα 2 F Kα ω K n i,j,k H n τ K and dn β dt n,i,j,k Emitted = 1 4π F Kβ ω K n i,j,k H n τ K (9.29)
where

n i,j,k H n = 1 ∆t n tn+∆tn tn n i,j,k H (t) dt = n n,i,j,k H -n n+1,i,j,k H + 2n n,i,j,k i (∆t n /τ n,i,j,k b ) Γ n,i,j,k ∆t n (9.30)
according to the analytical solution (9.28), with n n+1,i,j,k H given by Equation (9.25). The time integrated number of photons emitted from the cell (i, j, k) per unit of volume and steradian at time t Nt is then given by n Nt,i,j,k by [Gremillet, 2012]. We may assume that these simulation results are close to the experimental reality thanks to the high contrast of the laser pulses, avoiding prepulse/preplasma effects on the laser solid interaction that are usually not well characterized. In addition, the laser pulses have been focused with a 45 o angle of incidence. Thus, since the hiher-energy fast electrons propagate mainly in the propagation direction of the laser pulse, the recirculation through the laser plasma interaction zone and their deflections by strong local magnetic fields are minimized here. Therefore, we only take into account refluxing process due to the strong electric fields generated by the fast electrons at the target's rear and irradiated sides. The fast electron propagation is modeled with the M1 code accounting for two, forward and backward propagating groups, and assuming specular reflections of the fast electrons at both target surfaces. The time integrated Cu Kα 1 photon signal, emitted from the Copper tracer layer have been measured experimentally with a cylindrical quartz crystal Von Hamos spectrometer and another spherical quartz crystal completed by two cooled X-ray CCD camera [START_REF] Santos | Supra-thermal electron beam stopping power and guiding in dense plasmas[END_REF]] so that we can compare the time integrated Kα 1 photons signals obtained numerically and those obtained experimentally for all targets.

The M1 simulations presented in this Chapter are performed with the thermal capacities C V,e and C V,i presented in Chapter 6, section 6.3.1, the transport coefficients η and κ e presented in section 6.3.2 and the electron-latice/ion coupling factor G presented in section 6. Except for the 2D-3V M1 simulation presented in 9.2.1, which uses the same numerical conditions as the simulation presented in Chapter 8, section 8.2, all others are performed with the same conditions. Firstly, in agreement with the experiments (≈ mm), the lateral size of the targets are taken sufficiently large (L x = L y = 500 µm) to avoid refluxing at the lateral target boundaries. We choose ∆x = ∆y = ∆z = 1 µm for the spatial mesh size and ∆ε = 30 keV for the kinetic energy cells. This is a compromise between the numerical cost of the simulations and their accuracy, imposed by the available dynamic random access memory of the computer for the 3D-3V simulation. Indeed, with these conditions, we have to allocate at each time step the variables Φ, F x , F y , F z , Γ E , Γ B and Γ ν (see Chapter 7, section 7.1 for the notations) while each one represents a 2 × 4 × 500 × 500 × 5 × 100dimensional table (2 for the two electron populations (1) and (2) , 4 for Ψ 0 , Ψ 1,x , Ψ 1,y and Ψ 1,z , 500 2 × 5

for the spatial cells and 100 for the kinetic energy cells), that needs approximately 8 Go of available dynamic memory in double-precision floating-point format.

The fast electrons are initialized as in Chapter 8, section 8.2, except for the 3D-3V simulation where the third spatial dimension is introduced as

Ψ ini,0 (x, y, z = 0, ε, t) = N 0 (t)f x (x, t)f y (y, t)f z (z = 0, t)f ε (ε, t) (9.32)
where for the thermal capacities C hot V,e and C hot V,i , the Lee-More plasma electron-ion coupling factor G hot (see Chapter 6, section 6.3 for the notations) and the standard formula (9.19) for the Kα emission.

f y (y, t) = 1 2π ∆x(t) 2 8 ln 2 exp -4 ln 2 y ∆x(t) 2 (9.33) and N 0 (t) = ν L→e (t) E L k B T b0 (t) (9.34) while Ψ ini,1 (x, y, z = 0, ε, t) = Ψ ini,0 (x, y, z = 0, ε, t)Ω ε (x, z = 0, ε, t) (9.
The total energy is conserved within an accuracy about 1.7 %, compared to the value of 0.5 % obtained in the reference simulation. As illustrated in Figures 8.9 and 9.7, the energy injected in the target U inc ≈ 70 mJ, the energy escaping from the target rear side U out ≈ 50 mJ, the maximum instantaneous beam energy in the target U b,max ≈ 4 mJ and the total energy loss of the fast electron beam U col + U res ≈ 20 mJ are close to the reference results. However, the maximum of electric energy in the target U E,max ≈ 5 10 -3 mJ is approximatively twice less than the value obtained in the reference simulation, the maximum magnetic energy U B ≈ 20 10 -3 mJ is less than the reference U B,max ≈ 30 10 -3 mJ and the contributions of the collisional and collective losses are different :

U res ≈ 10 mJ and U col ≈ 8 mJ while it has been obtained U res ≈ 13 mJ and U col ≈ 6 mJ in the reference case. Therefore, we conclude that the greater energy conservation error is due to the greater collisional losses of fast electrons, computed with the downwind scheme (1st order). The differences in the contributions of collective and collisional fast electron energy losses indicate a significant effect of solid state physics effects introduced in Chapter 6, section 6.3. As illustrated in Figure 9.10 (to be compared with Figure 8.11), the fast electrons reach the target rear side at approximatively 100 fs in both simulations and the beam propagation do not differ too much. However, Figure 9.11 (to be compared with Figure 8.12) shows differences in the target heating profile (a), that affects the electrical resistivity (b) and consequently the selfgenerated magnetic field (c). The maximum value of the self-generated magnetic field is less than 90 T while it exceeds 100 T in the reference case. Also, we observe a broader transverse profile (in the x-direction) of the heated Copper area compared to both Aluminum layers while we obtained exactly the opposite in the reference case. These discrepancies may be explained as follows. The Lee-More electron-ion coupling factor G hot overestimates the electron-lattice coupling factor G cold at solid/liquid/Warm Dense Matter (WDM) temperatures by one or two order of magnitudes (see Figure 6.13). Consequently, in the reference simulation, the ratios G hot /C V,e and G hot /C V,i were so huge that the target ion and electron temperature were instantaneously equilibrated (T e ≈ T i ), as x max is defined as the position where T e (x max , z) is the maximum electron temperature at a given depth z (red). Refluxing is neglected here.

As illustrated in the right panel of In this subsection, we consider the effects of refluxing on the fast electron transport. We present here the simulation results concerning the thinnest target Al(1 µm)Cu(3 µm)Al(1 µm) and the thickest target Al(1 µm)Cu(3 µm)Al(15 µm). Let us remind here that the fast electron refluxing at the target-vacuum interfaces is modelled assuming their specular reflections. Figure 9.12, illustrates the time evolution of the injected fast electrons kinetic energy U inc , the instantaneous fast electron's energy in the target U b , the collisional and collective losses of the fast electrons U col and U res and the self-generated electromagnetic energies U E and U B for both targets.

The energy conservation errors in these two simulations are approximatively 0.8 and 1.9 %. A larger energy conservation error obtained for the thinner target is due to a greater collisional energy loss by the fast electrons, which is computed according to the downwind scheme (1st order consistency error). We can see that, contrary to the case without refluxing, the simulation must be run up to 1.5 ps in order to allow the fast electrons to lose all their energy and to ensure that they do not contribute anymore to the emission of Kα photons. Actually, a simulation time of 1.5 ps is not sufficient for the thicker target (U b ≈ 5 mJ at this time). However, as indicated by the increasing fast electron energy escaping from the target transverse boundaries x = ±L x /2, the fast electrons still in the target at times t > 1.5 ps are located at the target transverse boundaries and do not contribute to the Kα emission zone we are interested in. As soon as the fast electrons have completed their first recirculation in the target (at ≈ 200 fs for the thinner target and ≈ 250 fs for the thicker target), the fast electron's energy losses are mostly due to collisions. Indeed, the recirculation of fast electrons TRANSPORT IN SOLID TARGETS recirculation induces the presence of two counterpropagating beams of similar current densities in the z-direction. Thus, the background electron current and the self-generated electric field in the z-direction decrease rapidly (j

(1) b,z + j (2) b,z + j e,z ≈ 0 implies j e,z ≈ 0 ⇒ E z = ηj e,z ≈ 0 if j (1) b,z ≈ -j (2)
b,z ). This effect can be seen in Figure 9.13 for the Al(1 µm)Cu(3 µm)Al(15 µm) target and in Figure 9.14 for the Al(1 µm)Cu(3 µm)Al(1 µm) target. Indeed, we clearly see that the beam current density

|j b | = |j (1) b + j (2)
b | ≈ 0 at x ≈ 10 µm and t ≈ 240 fs or t ≈ 490 fs, while there is a lot of fast electrons in these zones according to the corresponding beam density maps.

Figure 9.12 also shows that, for the thin target, the collective energy losses of the fast electrons are comparable with those obtained without refluxing (U res ≈ 15 mJ iSnstead of ≈ 10 mJ). The fact that we observe the same amount of collective energy losses for the thick target (U res ≈ 15 mJ) indicates that the decrease in the beam current density with depth is such that collisional losses of the fast electrons prevail for z 4 µm. That is why we also obtain similar temperatures at the Aluminum rear side layer when comparing the simulations without refluxing in the previous subsection.

In Figure 9.12, we can see that the maximum of magnetic energy U B in the target is greater for the thin target compared to the thick target by a factor ≈ 2. This is a consequence of resistivity gradients at the Al/Cu and Cu/Al interfaces (see B y at t ≈ 1.5 ps in Figure 9.13 and 9.14). They are greater in the thin target compared to the thick target since the transverse component of the beam current density j b,x is greater. this results from the shorter propagation distance of the fast electrons, which have thus a larger density (see Figure 9.13 and 9.14). It must be emphasized here that we have to be careful concerning these self-generated magnetic fields induced by resistivity gradients at material interfaces: like all "hybrid" model, our model is based on the quasi-static approximation and the neglect of the target electron inertia, assuming time scales greater than the beam neutralization time. Consequently, it is valid on space scales much larger than the target skin depth λ e = c/ω p . Therefore, the increase of these self-generated magnetic fields with decreasing spatial cell dimensions ∆z must be mitigated in the case where ∆z is chosen smaller than λ e . Indeed, the temporal growth of this contribution to the self-generated magnetic field reads

∂B y,res ∂t ≈ j b,x c ∂η ∂z ⇒ B n+1,i,k y, res -B n,i,k y, res ∆t n = j n,i,k b,x c η i,k+1 -η i,k-1 2∆z . (9.36)
Thus, if we fix η i,k+1 = η Cu and η i,k-1 = η Al or η i,k+1 = η Al and η i,k-1 = η Cu , the B-field strongly depends on the value of ∆z for a given beam current density j b,x . Even if these huge magnetic fields are initially located over a few ∆z, they subsequently diffuse inside the target, leading to unphysical magnetic energies. It is not the case for all the simulations presented in this thesis (∆z = 0.25 µm is the smaller spatial cell dimension that is used). However, additional simulations were conducted in the course of the thesis for the understanding of experiments using targets made of a Copper layer followed by a vitreous Carbon layer, for which the ratio η C /η Cu was huge (see Figure 3.9 c). As a consequence, the obtained magnetic energy was found greater than the injected fast electron's kinetic x max is defined as the position where T e (x max , z) is the maximum electron temperature at a given depth z.

We can see that in both thin and thick targets, far away from the first recirculation zone (from

x ≈ -10 to x ≈ 20 µm), the Copper tracer layer is hotter than the Aluminum layers (see T e in Figures 9. 13 and 9.14). This is due to the greater density of background electrons (bound, free and screened free) in Copper, leading to greater collisional energy losses (see Chapter 4, section 4.2, Figure 4.2) compared to Aluminum. Besides, the Copper layer is hotter in the thin target due to the same reason as given for the explanation of the resistivity gradients contribution to the self-generated magnetic fields: the fast electrons recirculate more times through the Copper layer in the thin target and thus mainly deposit their energy through collisions. Also, in the M1 simulations, the Ohmic heating of the Copper layer by the return current is greater than the collisional heating of the Aluminum rear side layer during the first 100 fs due to solid state effects (section 9.2.1). Indeed, as illustrated in Figure 9.15, this is not the case for the simulation results obtained with the full collisional PIC code CALDER [START_REF] Lefebvre | Electron and photon production from relativistic laser-plasma interactions[END_REF]] [Nuter et al., 2011] [Pérez et al., 2012] and the hybrid PIC code PARIS [START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF] conducted by [Gremillet, 2014]. The difference between the PIC methods (hybrid or not) and the M1 method concerning the spatial cell dimensions may be striking:

∆z = 1 µm for M1 while ∆z ≤ λ D,b = k B T b /4πγ b n b e 2 ≈ 7.43 10 -3 µm(T b [keV]/γ b n b [10 21 cm -3 ]) 1/2
for the PIC codes. This is due to the conceptual difference between a PIC method and a "Vlasov-Fokker-Planck" method, like M1. For the latter, the whole space 500 µm(×500 µm) × 5( or 19) µm) is discretized in order to describe the distribution function, while for both PIC methods, this is the trajectories of the discretized macro particles following their position that are computed, only. Thus, the PIC method allows for a less expensive numerical cost concerning the space discretization by following

APPLICATION TO THE SACLAY UHI100 EXPERIMENT

the particles trajectories. The counterpart of this advantage is the interpolation and extrapolation of the electromagnetic fields that imposes ∆z ≤ λ D,b and contrains severely the computational cost.

However, while we obtain a significant difference of heating in the Copper layer due to the solid state corrections mentioned above, we obtain the same temperature at the rear side Aluminum layer for the thick target, where the fast electron collisions dominate, is approximately the same (see Figure 9.15).

Besides, the full collisional PIC simulation predicts the temperature of the first Aluminum layer twice greater than the one predicted in both hybrid simulations. Let us try to explain such a discrepancy by the fact that the fast electron current is temporally modulated. For this, we solve the Maxwell equations coupled to the background electron hydrodynamic equations, assuming for simplicity 1. that the problem is one-dimensional, Here, due to the ultra high contrast of the laser pulse, we may expect that the j × B heating is the dominant acceleration mechanism so that we may consider ω 0 = 2ω L . We note The solutions n e0 , v e0 , j e0 , T e0 provided by hybrid models correspond to the quasi-static approximation: they neglect the target electron inertia, assuming times greater than the beam electromagnetic neutralization time. They read consequently We assume constant thermal capacity C V,e and electron relaxation time ν in space and time. In addition, n i = 6 10 22 cm -3 and n e0 = Z * n i ≥ 3n i in Aluminum so that the plasma frequency ω p ≥ 23 rad.fs -1 . The laser frequency is ω L = 2πc/λ = 2.3 rad.fs -1 (λ = 800 nm). Consequently, we can consider ω p ω 0 . Also, according to Figure 6.9, the background electron relaxation rate ν can be consider smaller than 10 fs -1 but larger than 1 fs -1 from T e ≈ 10 -1 eV (T i = T e ) to T e ≈ 1 keV.

j b0 = -n
                           ∂E 0 ∂z = -4πe (n e0 + n b0 -Z * n i ) = 0 with n b0 n e0 = Z * n i ∂E 0 ∂t = -4π (j e0 + j b0 ) = 0 implying j e0 = -j b0 and v e0 = - n b0 n e0 v b ∂n e0 ∂t + ∂ ∂z (n e0 v e0 ) = 0 E 0 = η 0 j e0 = - m e νv
∂ ∂t + v e0 ∂ ∂z δn e = -
Thus, we consider the scaling

ω p ν ω 0 . (9.50)
We assume in addition 7. the non-linear terms are negligible, 8. quasi neutrality δn e = -δn b and 9. (∂δv e /∂z) = 0.

Assumptions 8. and 9. can be justified by noting that the charge neutralization mainly takes place in the transverse direction leading rapidly to δn e = -δn b with spatial variations of δv e mainly in the x-direction. Assumption 7 serves to linearize the equations and to find analytical estimates.

Under these last assumptions, by working in the frame where the target electron envelope is at rest :

δ ξ(ζ, t) = δξ(z, t) with ζ = z -v e0 t
and by coupling (9.45) and (9.46), we obtain the coupled equations

                             ∂δ E ∂t = -4π δ j e + δ j b a)
∂δ n e ∂t = 0 b)

∂δ j e ∂t = ω p 2 4π δ E + ω p 2 4π E 0 δ n e n e0 -νδ j e c) , (9.51)
that finally gives ∂ 2 δ j e ∂t 2 + ν ∂δ j e ∂t + ω p 2 δ j e = -ω p 2 δ j b , (9.52) describing the error δ j e done by the hybrid methods. Considering the limit (9.50), assuming δ j e (ζ, t = 0) = 0 and ∂ t δ j e (ζ, t = 0) = 0, we find

δ j e = - j b0 2 1 -cos (ω p t) exp - ν 2 t . ( 9
δ T e = - j b0 E 0 C V,e t - sin (ω p t) ω p exp - ν 2 t . (9.55)
Finally, averaging over one electron bunch period τ 0 = 2π/ω 0 , we deduce an estimate of the additive temperature component in the first Aluminum layer, obtained by the full PIC simulation and neglected by both hybrid models :

δ T e 1 bunch = 1 τ 0 τ 0 0 δ T e (ζ, t)dt = - j b0 E 0 C V,e τ 0 = T e0 1 bunch . (9.56)
Thus, according to our simple model (9.51), each fast electron bunch generates a weak field of background electron plasma wave according to (9.51 a) and (9.51 c), which are damped due to collisions with the background ions and electrons, according to (9.51 c). This damping results in an additional target electron heating mechanism according to (9.54) that leads to a target electron temperature T e = T e0 + δT e , which is approximatively two times greater than the one predicted by the hybrid models T e0 according to (9.56). This is quite in good agreement with the simulation results, illustrated in the left panel of Figure 9.16. This additional electron heating mechanism was recently pointed out by [START_REF] Sherlock | In-depth Plasma-Wave Heating of Dense Plasma Irradiated by Short Laser Pulses[END_REF]. They proposed therefore to replace the quasi-static Ohm's law E = -ηj b by the dynamic equation

∂ 2 E ∂t 2 + ν ∂E ∂t + ω p 2 E = - 4π c ∂j b ∂t + νj b (9.57)
for the electric field that takes into account the excitation of plasma waves and their collisional damping due to collisions.

Comparison of Time Integrated Kα Emission of with the Experimental Data

Figure 9.16: 2D maps of the number of emitted Kα 1 photons at t = t f = 1.5 ps from the 2D-3V M1 simulations with refluxing and an example of CCD images obtained experimentally [Vauzour, 2014].

The 2D maps of the time integrated number of Kα 1 photons emitted per steradian at t = 1.5 ps from the 2D-3V M1 simulations with refluxing for all targets are plotted in Figure 9.16. We can see that the emission of Kα 1 photons from the Copper tracer layer increases with decreasing target thickness. Indeed, the maximum value of the Kα 1 emission reaches ≈ 30 10 18 cm -3 .sr -1 for the thinner target while this number decreases to ≈ 11 10 18 cm -3 .sr -1 for the Al(1 µm)Cu(3 µm)Al( 6µm) target and ≈ 7 10 18 cm -3 .sr -1 for the Al(1 µm)Cu(3 µm)Al(10 µm) target falling down to 4 10 1 8 cm -3 .sr -1 for the thickest target. Moreover, the location of maximum emission shifts to larger x with increasing target thickness. This is a clear signature of the fast electron's mean propagation direction that makes an angle of ≈ 25 o with the target normal.

Let us compare these 2D-3V M1 simulation results with the experimental Kα 1 signals received by the CCD camera and obtained thanks to the crystal spectrometers. An example of CCD image is shown in Figure 9.16. In order to decrease the effects of noise-to-signal ratios estimated around 10 % according to [Santos, 2014], the experimental signals have been summed over several shots under the same conditions [Vauzour, 2014]. Therefore, we cannot directly compare the absolute values of the signals (since they depend on the number of shots conducted for each target). Rather, we renormalize the experimental signals according to our 2D-3V M1 simulation results:

n Kα,exp → n Kα,exp n Kα z max n max Kα,exp . (9.58)
Here, n Kα z max is the maximum value of the number of emitted Kα 1 photons from the 2D-3V M1 simulations, averaged over the Copper layer thickness

n Kα z (x) = 1 L Cu z 2 z 1 n Kα (x, z, t f )dz. (9.59)
Here, L Cu = 3 µm, t f = 1.5 ps, z 1 = 1 µm and z 2 = 4 µm or z 1 = 3 µm and z 2 = 6 µm or z 1 = 6 µm and z 2 = 9 µm depending on the targets. The resulting experimental signals averaged over 5 pixels (8.6 µm × 7.825 µm) [Vauzour, 2014] are plotted in Figure 9.17. The panels a) and b) present the horizontal (x-axis) and vertical axis (y-axis) of the two-dimensional CCD image as shown in Figure 9.16. In order to make comparable plots, we add a uniform noise of 10 % of the maximum value to the calculated Ka profiles: andd).

n Kα z (x) → n Kα z (x) + b[x] where b[x] = 0.1 n Kα z max 1 x∈[-∞,∞] [x]. ( 9
According to the inset in Figure 17 d), we can see that the Kα 1 emission spot size increases with the tracer depth. It is usually interpreted in experiments as a signature of a strong angular divergence of laser-generated electron beams. However, as already noticed by [START_REF] Ovchinnikov | How well do time-integrated Kα images represent hot electron spatial distributions?[END_REF], it is not so evident. The wings of the emission profiles are contamined by the recirculation of fast electrons inside the target. For example, the normalized emission profiles plotted in the insert of Figure 9.17 c)

would mean that the Kα 1 photon emission spot size increases with increasing target depth. It is false according to the non-normalized values as illustrated in Figure 9.17 c) : due to the fast electron refluxing, the emission spot size increases with decreasing target thickness. According to 9.17 b), the Kα 1 photon signals obtained experimentally are symmetric with respect to the y-axis in the plane x = 0. As illustrated in Figure 9.17 a), it is not the case with respect to the x-axis in the plane y = 0 due to the incidence angle of the laser pulse in the plane y = 0.

Except for the thinner target, Figures 9.17

Three-Dimensional Effects

As illustrated in the inset of the right panel of Figure 9.18, the spot size on x-axis of the Kα 1 photon emission is almost twice less in 3D than in 2D. We explain this as follows. In the 2D simulations, we impose specular reflections of the fast electrons in the plane y = 0 since the third dimension is not taken into account. Therefore, the fast electrons are confined in this plane and the emission in the x-direction is enlarged. It is not the case in the 3D simulation where we allow the fast electrons to propagate in the y-direction. It is important to notice that this 3D effect also lead to a smaller maximum value of Kα 1 photon emission of 1.4 10 18 cm -3 .sr -1 even smaller than in the 2D case without refluxing (see n Kα in Figure 9.11). However, the effective surface of Kα 1 photon emission S α is greater in 3D as we may expect. Indeed, we obtain a total number of Kα 1 photons emitted per [Vauzour, 2014], renormalized according to the simulation (dashed curves) and corresponding horizontal and vertical profiles of the Kα 1 photon emission obtained from the 3D simulation (full curves). The horizontal profile is compared to the corresponding profile from the 2D simulation in the insert.

steradian

N Kα =                    2π ∆y 2 8 ln 2 L x 2 - L x 2 dx z 2 z 1
dz n Kα (x, z, t f ) ≈ 7.8 10 8 sr -1 for the 2D-3V simulation

L y 2 - L y 2 dy L x 2 - L x 2 dx z 2 z 1
dz n Kα (x, y, z, t f ) ≈ 1.5 10 9 sr -1 for the 3D-3V simulation , (9.63)

an averaged emittance of Kα 1 photons per steradian

E Kα =                    L x 2 - L x 2 dx L x z 2 z 1
dz n Kα (x, z, t f ) ≈ 4.3 10 13 cm -2 .sr -1 for the 2D-3V simulation

L y 2 - L y 2 dy L y L x 2 - L x 2 dx L x z 2 z 1
dz n Kα (x, y, z, t f ) ≈ 3.8 10 12 cm -2 .sr -1 for the 3D-3V simulation and can be neglected while the 3D effects contribution has comparable values as the contribution of resistivity gradients. The major contribution to the x-component is mainly due the to the curls of the beam current, in particular due to the term ηc(∂j b,z /∂y). Similarly, B y is mainly due to -ηc(∂j b,z /∂x).

However, since the fast electron beam propagate also in the x-direction, gradients in the x-direction are smoother so that B x is greater than B y . In this subsection, we discuss the photoionization effects on the emission of CuKα 1 photons in order to check the assumptions 3 and 4 of our model in the subsection 9.1.3. According to [START_REF] Verner | Subshell Photoionization Cross Sections and Ionization Energies of Atoms and Ions from He to Zn[END_REF] and [START_REF] Verner | Analytical fits for partial photoionization cross sections[END_REF], the partial photoionization cross section for a photon with an energy hν, colliding with a (nl) atomic shell-electron, reads σ (nl)

P (hν) =        σ 0 hν E 0 -1 2 + y w 2 hν E 0 -Q 1 + hν y a hν E 0 -P if hν ≥ E th 0 else (9.66)
where Q = 5.5 + l -0.5P . This is an interpolation of a series of numerical calculations using the Hartree-Dirac-Slater method. [START_REF] Verner | Subshell Photoionization Cross Sections and Ionization Energies of Atoms and Ions from He to Zn[END_REF] provide the tables for the photoionization threshold = 1 for Copper. We find a good agreement with the total photoionization cross section computed according to the simpler semi-classical Kramers formula σ (nl)

P (hν) =      64 3 √ 3 π 4 m e e 10 h 3 c Z 4 n 5 1 (hν) 3 if hν ≥ E th 0 else (9.68)
However, the Hydrogen-like approximation for the estimates of E th is too rough. Knowing the photoionization cross section, one may deduce the attenuation length 1/n i σ P of a photon hν propagating in Aluminum (n i = 6.0 10 22 cm -3 ) or Copper (n i = 8.5 10 22 cm -3 ). The transmission of a photon beam propagating through Aluminum or Copper of a thickness L is plotted in the right panel of Figure 9.23 according to [START_REF] Henke | X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92[END_REF] (http://henke.lbl.gov/optical constants/).

The photoionization threshold for Cu K-shell electrons is E th = 8.972 keV. According to formula

(1.8), the temperature T I of electrons in the conditions of the UHI100 experiment is around 7 keV.

Photons with energies greater than the Cu K-shell photoionization threshold E th are thus emitted from the laser plasma interaction zone. In addition, according to the right panel of Figure 9.23, the first Aluminum layer is fully transparent for photons with energy greater than 10 keV. Therefore, the bremsstrahlung photons emitted from the laser plasma interaction zone may ionize some Cu Kshell electrons. However, their density is too small to play a significant role in the Cu K-shell hole dynamic (9.22) compared to the collisional ionization of K-shell electrons by the laser-generated fast electrons. Indeed, the radiative stopping power (dε/ds) brem ≈ 10 -8 keV/µm for electrons with a kinetic energy of ε ≈ 10 keV in Aluminum according to [START_REF] Heitler | Stopping of Fast Particles with Emission of Radiation and the Birth of Positive Electrons[END_REF]; see This is 8 orders of magnitude smaller than the collisional stopping power. Therefore, we can neglect the photoionization of K-shell electrons by X-ray bremsstrahlung photons compared to the collisional ionization by the laser-generated fast electrons. Also, X-ray photons may be emitted due to the presence of impurities in the laser plasma interaction zone such as hydrocarbons; see Chapter 2, section 2.1. However, for the same reason as for bremsstrahlung photons, the photoionization of CuK-shell electrons by these transition line photons can also be neglected compared to the collisional ionization by the laser-generated fast electrons. As a conclusion, the assumption 4 is fully justified.

According to the right panel of of Aluminum. This effect is omitted in our model. Moreover, the ionization of CuL-shell electrons by CuKα 1 photons may be responsible for a decrease of the emission of Kα 1 photons. However, an error around 10 % is not sufficient to explain a strong discrepancy between our simulations and the experimental data concerning the size of the Ka emission zone..

Summary and Conclusion

The refluxing of fast electrons in the target may strongly affect the emission of Kα photons. This effect must be taken into account when comparing numerical simulations of the Kα emission with experimental data. In our model, this effect is accounted for by imposing specular reflection of fast electrons at the target-vacuum interfaces and adding a second population in the M1 equations. The first one describes the laser-generated electron population (1) propagating in the laser pulse propagation direction while the second one describes the counterpropagating fast electron poulation (2) . The model of calculation of the emission of Kα photons is revised. It is demonstrated that the numerical time step of the fast electron transport calculation ∆t n may be comparable to the K-hole lifetime τ K in the case of Aluminum and Copper targets. Therefore, we have implemented the self-consistent model

proposed by [START_REF] Thomas | Hybrid Vlasov-Fokker-Planck-Maxwell simulations of fast electron transport and the time dependance of K -shell excitation in a mid-Z metallic target[END_REF] describing the K-shell holes dynamic.

Simulations of fast electron transport in solid targets are compared with the experiments conducted on the UHI100 laser facility, introduced in Chapter 8, section 8.2. By comparing the simulations with different models for the target parameters such as the electron and ion thermal capacities, their temperature equilibration parameter and the transport coefficients, we demonstrated that solid state physics effects must be taken into account. In particular, the electron-ion temperature equilibration time and collisions of d-band with s-band electrons may affect the magnetic field distribution in the Copper target.

By comparing our simulations with refluxing with corresponding hybrid PIC and/or full PIC simulations conducted by [Gremillet, 2014], we confirmed the results obtained by [START_REF] Sherlock | In-depth Plasma-Wave Heating of Dense Plasma Irradiated by Short Laser Pulses[END_REF] concerning the error made in the quasi-static hybrid models neglecting the target electron inertia in the Ohm's law and the displacement current. However, this effect is restricted to the first thin Aluminum layer. Here, the laser-generated bunches of electrons are injected into the target at the laser frequency or twice the laser frequency. each bunch generates an electric field that excites a weak of the 2D simulations. Even if r 2D is greater than r 3D , l is so small compared to r 3D that finally

πr 2 3D > r 2D l.
Figure 9.24: Sketch of the toy model by [Macchi, 2012].

The neglect of secondary electrons in our model may explain the discrepancy between the experimental data and our simulations. However, I think that the main critical assumption in our model concerns the specular reflection of fast electrons at the target-vacuum interfaces. The local magnetic fields generated at the target-vacuum interfaces may deviate strongly the refluxed fast electrons, thus enhancing the off-axis Kα 1 signal. Such magnetic fields have already been observed in experiments [START_REF] Sarri | Dynamics of Self-Generated, Large Amplitude Magnetic Fields Following High-Intensity Laser Matter Interaction[END_REF] and in PIC simulations [Pukhov, 2001] with the laser parameters in the range of the UHI100 experiments. Similar observations have also been reported in the context of the resonant absorption [Sakagami et al., 1979] [Kolodner andYablonovitch, 1979] (see Chapter 1, section 1.1.2). [Macchi, 2012] proposed a simple model called "toy model of the fountain effect", allowing to estimate the value of the magnetic field generated by the fast electrons escaping from the target and reaccelerated back. The term "fountain" comes from the analogy with a fountain of water (electrons) where the water (electrons) fall to the floor (to the target) due to gravity (electrostatic field). The "toy model of fountain" is based on many simplifying assumptions such as non-relativistic electrons, small beam divergence, uniform electric field and many others. However, according to [Macchi, 2012], it provides an order of magnitude of the magnetic field generated at the target-vacuum interfaces. For example, this simple model provides an estimate of ≈ 10 kT in agreement with the experiment by [START_REF] Sarri | Dynamics of Self-Generated, Large Amplitude Magnetic Fields Following High-Intensity Laser Matter Interaction[END_REF] and a corresponding PIC simulation. According to [Macchi, 2012], the magnetic field generated at the target-vacuum interface may be estimated as

B ≈ 8k B T b eEr 0 θ d 2I b r 0 c . (9.69)
The notations are illustrated in Figure 9.24. In order to apply this formula to our case, let us firstly estimate the maximum value of the electrostatic field at the target edge. Assuming that fast electrons escaping from the target form a Boltzmann distribution with a temperature k B T b in the electrostatic potential, we may estimated the maximum electrostatic field as In agreement with the PIC simulation performed by [Pukhov, 2001], who obtains magnetic fields ≈ 1 kT, the main part of fast electrons with low kinetic energies are trapped by the magnetic field while only fast electrons with a momentum greater than If in addition, we neglect the fluid viscosity, the equations in one dimension read: When the perturbation is strong enough (δρ ∼ ρ 0 ), we cannot anymore linearize the equations assuming δρ is a perturbation of the equilibrium ρ 0 . A powerful theoretical tool when studying such non-linear equations is the search for self-similar solutions. It consists in looking for solutions of the equations in the form ρ(x, t) = ρ 0 F (x a t b ) where the scalar parameters a and b may be found according to dimensional reasoning depending on the invariant quantities of the problem, while the function F may be found by injecting the assumed form F (x a t b ) in the equations; see for example the self-similar rarefaction waves in a plasma found by [START_REF] Gurevich | Self-similar Motion of Rarefied Plasma[END_REF]] [Manheimer et al., 1982] [ [START_REF] Fabbro | Planar laser driven ablation: Effect of inhibited electron thermal conduction[END_REF] and illustrated in Figure 10.1 a). When studying the nonlinear equations of conservation (10.1), (10.2), (10.3) and (10.4), neglecting the thermal energy flux q and the source term W e , B. Riemann discovered in the second half of the XIXth century thanks to a self-similar solution that the development of singularities may be possible, in the form of shock waves, as illustrated in Figure 10.1 f ). Actually, if we zoom in on this hydrodynamic density discontinuity to the kinetic scale, i.e. at the spatial scale of the order of the particle mean free path, this discontinuity disappears.

E ≈ k B T b √ 2 λ D,
∂ρ ∂t + ∂ ∂x (ρu) = 0, (10.1) ρ ∂ ∂t + u ∂ ∂x (u) = - ∂P ∂x , (10.2)
C V ∂ ∂t + u ∂ ∂x (T ) = - ∂q ∂x + W e (10.
A few years later, starting from the same conservation equations (10.1), (10.2), (10.3) and (10.4) on the jump

                 ρ 0 D = ρ 1 (D -U ) P 1 -P 0 = ρ 0 DU ρ 0 D ε 1 -ε 0 + U 2 2 = P 1 U h 1 -h 0 + U 2 2 = DU (10.7)
in such a configuration f ), W. J. M. Rankine and H. Hugoniot found the relations between the specific internal energies ε i = C V T i /ρ i , the pressures P i , the specific enthalpies h i = ε i + P i V i and the specific volumes V i = 1/ρ i of the shocked (i = 1) and unperturbed gas (i = 0). These so-called Rankine-

Hugoniot relations read      ε 1 -ε 0 = 1 2 (P 1 + P 0 ) (V 0 -V 1 ) h 1 -h 0 = 1 2 (P 1 -P 0 ) (V 0 + V 1 )
.

(10.8)

Here, U and D are the velocities of the piston and the shock front, respectively. The velocities of the shocked and unperturbed gas have been taken equal to u 0 = 0 and u 1 = U . Let us apply Equation (10.8) to the case of an ideal gas filling the tube with the specific internal energy and enthalpy read

ε = C V ρ T = 1 γ -1 P ρ and h = C P ρ T = γ γ -1 P ρ
(10.9)

where γ = C P /C V = 1 + (2/d) is the adiabatic index depending on the number d of degrees of freedom of particles. In this particular case, if we push the piston such that U c s , a strong shock is generated that propagates in the tube with the velocity

D = γ + 1 2
P 1 ρ 0 (10.10) (P 1 P 0 ) and compresses the gas downstream up to the density .11) according to (10.7), (10.8) and (10.9). Therefore, maximum density ratio for a monoatomic ideal gas with γ = 5/3 is equal to 4. In reality, at high temperatures and pressures, the specific heats and the specific heat ratio are no longer constant because of molecular dissociation and of ionization. However, the density ratio (10.11) remains finite and does not increase without limits : generally, it does not exceed 11-13 according to [START_REF] Zel'dovich | Physics of Sock Waves and High Temperature Phenomena[END_REF].

ρ 1 ρ 0 = γ + 1 γ -1 . ( 10
When a large amount of energy is quasi-instantaneously deposited in a very small volume, we talk about blast waves instead of shock waves. Even if it presents a leading front discontinuity, as in shock waves, a blast wave is followed by a blast wind of negative pressure gradients. For example, the outcome from supernova-explosions may be described as a blast wave. Indeed, when a star has consumed all its thermonuclear fuel (see the section 0.1.2 of the Introduction), the star internal pressure decreases and does not counterbalance anymore the star gravitational force. As a result, the equilibrium is broken, the star implodes and its plasma density increases. Depending on its mass according to [Chandrasekhar, 1931], it may reach the degeneracy pressure and a sudden reignition of nuclear fusions in the degenerate core may lead to its explosion. The subsequent expansion phase, called supernova and illustrated in Figure 10.2, may be described by the adiabatic self-similar blast wave obtained by [Sedov, 1946] and confirmed by [Taylor G., 1950], according to numerical simulations. It allows us to illustrate the "dimensional reasoning" mentioned above concerning the self-similar solutions of the non-linear and coupled equations of conservation (10.1), (10.2), (10.3) and (10.4) that must be expressed here in 3 dimensions, assuming a spherical symetry and neglecting the thermal energy flux q as well as eventual source terms W e . Indeed, let us try to find out the expression of the radius r [cm] of such an expanding spherical supernova according to the dimensional reasoning.

We expect that r depends on the energy released quasi-instantenously, let us say at t = 0, by the star explosion E 0 [erg]. It may thus be considered as the self-similar invariant quantity. We expect also that r depends on the surrounding medium density ρ 0 [g/cm -3 ] and the age of the supernova t [s].

Therefore, let us look for an expression of the radius with the form

r [cm] = ξ 0 (E 0 [erg]) a ρ 0 [g/cm 3 ] b (t [s]) c (10.12)
where ξ 0 , a, b and c are dimensionless constants to be determined. Since 1 erg = 1 g.cm 2 /s 2 , we deduce that we must have necessarily

       (a + b) kg = 0 kg (2a -3b) cm = 1 cm (-2a + c) s = 0 s (10.13)
for dimensional reasons. It thus gives the self-similar variable found by [Sedov, 1946] /5 (10.14) and the blast wave properties can be found by looking for the function F (r), solution of the nonlinear and coupled equations (10.1), (10.2), (10.3) and (10.4) expressed in 3 dimensions, assuming a spherical symetry and neglecting q as well as W e . Applying this procedure for an ideal gas (10.9), [Taylor G., 1950] found that ξ 0 ≈ 1.11 for γ = 5/3 (d = 3). Concerning our thought experiment of a piston compressing a gas filling a semi-infinite tube, a blast wave can be generated by applying a strong hit of hammer to the piston. The resulting blast wave is illustrated in Figure 10.1 e). It has this particular triangular shape due to the fact that the piston is not continuously pushed like in the case of shock waves. The X-ray photographies of the supernovae G299 and Tycho, illustrated in Figure 10.2, are obtained thanks to the strong emission of the hot expanding plasma behind the front.

r = ξ 0 E 0 ρ 0 1/5 t 2
The first direct observation of a laser-driven shock wave was reported by [van Kessel and Sigel, 1974]. A planar solid hydrogen target was irradiated with a 10 J, 5 ns, Nd laser (1.06 µm wavelength) and the propagation of the laser-driven shock wave was measured using a high-speed photography. The estimated pressure in this pioneering experiment was 2 Mbar.

Twenty years after the first published experiment, the Nova laser at the Livermore laboratories in the United States of America (USA) created a pressure of 750 ± 200 Mbar [START_REF] Cauble | X-ray driven flyer foil experiments near 1.0 Gbar[END_REF].

This was achieved in a collision of two gold foils, where the flyer (Au foil) was accelerated by a high-intensity x-ray flux created by the laser-plasma interaction. As explained above, a shock or a blast wave is created in a medium that suffers a sudden impact or in a medium where a large amount of energy is released in a short period of time. As already explained in the section 0. Energetic electrons are commonly considered to be a dangerous effect for ICF; see the section 0.2.2 of the Introduction. Having a long mean free path, they penetrate through the solid shell and deposit their energy in the ablator and Deuterium-Tritium (DT) fuel. This process significantly increases the target entropy H, thus degrading its implosion. The phenomenon of target preheat was the major reason for several milestone events [Lindl, 1998] : cessation of the ICF program based on the CO2 laser in the 1980s, switching to the third harmonic in the Nd:glass ICF lasers, and limiting the "useful" laser intensities to a few PW/cm 2 . All these limitations significantly reduce the ICF operational domain. However, matching the mean free path of fast electrons with the target size may suppress the negative effect of preheat and open the possibility of using the energetic particles for creation of a high ablation pressure [Volosevich andRozanov, 1981] [Gus'kov, 1983] [ Evans, 1983] [Evans, 1986]. Fast ignition is an example of the application of energetic electrons in ICF. Here, a beam of relativistic electrons is supposed to create a small hot spot in the compressed fuel [START_REF] Tabak | Ignition and high gain with ultrapowerful lasers@f[END_REF]; see the section 0.2.3. This scheme, nevertheless, faces serious difficulties related to tight focusing of an intense electron beam; see the section 0.3.2. Energetic electrons may play an important role in the creation of a high ablation pressure, which is interesting for ignition of fusion reactions in the laboratory [START_REF] Betti | Shock Ignition of Thermonuclear Fuel with High Areal Density[END_REF] and in astrophysics concerning thermonuclear supernovae [START_REF] Gamezo | Deflagrations and Detonations in Thermonuclear Supernovae[END_REF] or deflagration (subsonic)-to-detonation (supersonic) transition in premixed combustion wave front [START_REF] Bychkov | Physical Mechanism of Ultrafast Flame Acceleration[END_REF]. In the shock ignition scheme in ICF, the fuel is ignited by a strong shock launched by an intense laser spike at the end of the implosion process; see the section 0.2.3. The laser spike intensity is in the range of 10 PW/cm 2 , certainly well above the threshold of parametric instabilities, and a significant part of laser energy is expected to be deposited in the nonthermal, energetic electrons [START_REF] Klimo | Laser plasma interaction studies in the context of shock ignition-Transition from collisional to collisionless absorption[END_REF]; see the section 0.3.1. It was suggested by [START_REF] Betti | Shock Ignition of Thermonuclear Fuel with High Areal Density[END_REF] and [START_REF] Ribeyre | Shock ignition: an alternative scheme for HiPER[END_REF] that their deleterious effect on target implosion can be mitigated by the fact that, at the moment of spike arrival, the target has already passed halfway through the implosion phase, and its areal density is increased significantly, by a factor of 10-20 at least. If the target areal density would be larger than the range of fast electrons, the latter will be stopped in the imploding shell and may play a positive role by contributing to the ablation pressure.

In this Chapter, we study for the first time the generation of a shock wave by an ablation pressure driven by an energetic electron beam in a dense plasma. For this, we first present the theoretical predictions found by [START_REF] Gus'kov | Ablation Pressure Driven by an Energetic Electron Beam in a Dense Plasma[END_REF]. Then, we describe the coupling of the reduced model for the fast electron transport in solids and dense plasmas presented in Chapter 6 with the radiation hydrodynamic code CHIC, briefly described in Appendix B, section B. 3.2 and B.3.3.

Finally, we compare the predictions of the theory with one-dimensional simulations of an electron beam energy deposition in a DT plasma with a step-like density profile. The parameters of these academic simulations correspond to typical values expected at the time of spike arrival in the shock ignition scheme. These simulations show a positive effect of energetic electrons in the shock ignition scheme, allowing to achieve the ablation pressures above 500 Mbar. Furthermore, in this section we consider another example of a blast wave generated by an ultrashort femtosecond laser pulse. The experimental campaign conducted by [START_REF] Santos | Supra-thermal electron beam stopping power and guiding in dense plasmas[END_REF] was already described in sections 8.2 and 9.2. We demonstrate here by using analytical estimates and numerical radiation hydrodynamic simulations that a strong blast wave can be generated in a thin target due to strong temperature gradients induced by fast electron heating. This interpretation is confirmed by the observation of the shock breakout at the target rear side with a Streaked Optical Pyrometry. There is no general model of pressure formation by an energetic particle beam. The model by [Gus'kov, 1983] applies for nanosecond electron beams where the heat conductivity of the thermal electrons plays an important role. In this case, the laser energy is deposited at the critical density and it is transported to the ablation zone by the thermal electrons. This is a stationary ablation process where the shock wave launched into the solid material is connected to the isothermal rarefaction wave [START_REF] Gurevich | Self-similar Motion of Rarefied Plasma[END_REF]] [Manheimer et al., 1982] [Fabbro et al., 1985], as it is schematically shown in the Figure 8 of the Introduction and in Figure 10.1 a). Formation of the ablation pressure by an energetic ion beam was considered by [Evans, 1983] [Evans, 1986]. A recent paper [START_REF] Bell | Electron transport and shock ignition[END_REF] considers the regime of transition from the thermal electron diffusion to a nonlocal energy transport. However, the fast electron plasma heating is limited to a very short time scale, before the hydrodynamic separation takes place. The fast electrons, similarly to energetic x rays, propagate deeper in the target behind the ablation front created by the thermal electron conduction and produce a second ablation front. However, in contrast to x rays and thermal electrons, the range of fast electrons depends only weakly on the plasma temperature (only logarithmically; see Chapter 4, section 4.2). For this simple reason, the standard stationary isothermal model of plasma expansion does not apply to fast electrons. Fast electron-driven ablation is intrinsically a nonstationary process similar to the ion driven ablation [Evans, 1983] [Evans, 1986]. It can be described by a model of heating and expansion of a finite mass defined by the fast electron range. Consequently, the shock is launched by the ablation pressure created by fast electrons during a finite time interval, the loading time, and after that, the shock transforms in a blast wave and its amplitude decreases slowly with time. Figure 10.3 presents schematically the density and temperature profiles in this regime after the loading time. Thus, the fixed energy deposition range of fast particles implies an optimal time for fast electron beam injection : shorter beams will drive a smaller amplitude shock, while longer beams will be detached from the solid target and deposit their energy in the expanding plasma, thus decreasing the coupling efficiency. The fast electron ablation theoretical model GRT is presented here for a simple case of monoenergetic electrons and a plane geometry.

There are two classes of self-similar solutions of equations of ideal hydrodynamics that describe a rarefaction wave. One of them is the well-known stationary isothermal rarefaction wave, where the temperature is constant and the ablated mass increases linearly with time [START_REF] Gurevich | Self-similar Motion of Rarefied Plasma[END_REF] [ [START_REF] Manheimer | Steady state planar ablative flow[END_REF]] [Fabbro et al., 1985]. Another one describes an isothermal expansion of a given mass plasma with a temperature increasing with time [Imshennik, 1960] [Drake, 2011]. The former applies readily to the energy deposition of laser beams and thermal x rays [START_REF] Fabbro | Planar laser driven ablation: Effect of inhibited electron thermal conduction[END_REF] [ Mora, 2003]. There, the plasma temperature is adjusted in a way that it accommodates the photon stopping length to the plasma density profile. This model, however, does not apply to fast electrons because their stopping power depends only on the fast electron energy ε 0 = (γ 0 -1)m e c 2 . According to section 8.1.3, a monoenergetic and collimated electron beam deposits its energy over the distance is the range of the beam electrons and

ξ = γ 0 + 1 γ 0 -1 2 1 β 2 0 β 2 0 -2 ln γ 0 γ 0 (10.17)
is the correction factor due to their angular scattering. For a fully ionized DT plasma with a density of ρ 0 = 10 g/cm 2 and a temperature of 1 eV, the range R of a beam of collimated electrons is 0.878 µm for the electron energy ε 0 = 30 keV and 6.650 µm for ε 0 = 100 keV while the correction factor reaches the limiting value of 2/3 so that L p = 0.5 µm for ε 0 = 30 keV and L p = 4.4 µm for ε 0 = 100 keV. (10.18)

Collisions on screened free electrons (plasmons) provide the main contribution to the stopping power and the angular scattering correction factor in such a degenerate plasma; see section 8. In the first stage, the plasma is heated by the incident beam of fast electrons and starts expanding.

The plasma energy increases linearly with time. It is redistributed between the areal density of internal energy,

W int = 3 2 ρ 0 c 2 s dx (10.19)
and the areal density of kinetic energy

W kin = 1 2 ρ 0 u 2 dx (10.20)
where c s is the velocity of sound (10.22). The energy conservation of the process reads

W int + W kin = I b t, (10.21)
assuming that the energy flux I b does dot depend on time. In addition, it is assumed that the DT follows the relations (10.9) with d = 3, typical of a monoatomic ideal gas, so that the sound velocity (10.5) can be written

c s = (Z + 1) k B T m i . (10.22)
As a consequence, the repartition between the internal and kinetic energies in the heated layer, where

W kin /W int = ζ(t)
P h = 1 6π 1/3 I b D 0 (10.25)
is the maximum pressure and

D 0 = I b ρ 0 1/3 (10.26)
is the characteristic hydrodynamic velocity found according to the dimensional reasoning. Note that the maximum pressure depends only on the beam intensity and on the target density, while the loading time increases strongly with the electron energy in agreement with [Evans, 1983] [Evans, 1986].

For electron beam intensities in the range of few PW/cm 2 , the heating proceeds so fast that the electron thermal conduction does not play a significant role, and the heated mass undergoes expansion without transferring the internal energy to the adjacent cold plasma. However, the pressure in the heated layer exerts a mechanical work and launches a shock wave in the cold plasma. Therefore, the second stage consists in the expansion of a heated layer of plasma continuously heated by an electron beam. It can be described by the isothermal rarefaction wave of a constant mass [Imshennik, 1960] [ Drake, 2011]. It corresponds to the solution of hydrodynamics equations (10.1), (10.2), (10.

               ρ = 3 2πI b ρ 0 L p t 3/2 exp - 9ρ 0 L p (L p -x) 2 8I b t 3 a) T = m i I b 3 (Z + 1) ρ 0 k B L p t b) u = 3 2 x -L p t b)
, (10.28)

Here, the velocity increases linearly with the coordinate for x < x h , the temperature increases linearly with time, and the density profile has a Gaussian-like shape with the characteristic scale length at the cold plasma interface at x = L p accoding to Equations (10.9), (10.23), (10.28 a and b). The pressure decreases as a square root of time at the second stage.

λ ρ (t) = 1 3 L p (2D 0 t)
As an example, we consider an electron beam with an energy ε 0 = 30 keV and intensity I b = 1 PW/cm 2 incident on a DT plasma with a density ρ 0 = 10 g/cm 3 . Then, the maximum pressure P h rises to the value of 380 Mbar according to Equations (10.24) and (10.25). The loading time t h ≈ 6 ps (D 0 = 10 7 cm/s and L p = 0.5 µm) is relatively short according to (10.23). However, for longer times, the pressure decreases rather slowly according to (10.31). Knowing the pressure, it is straightforward to evaluate the shock wave velocity in the strong pressure limit. According to Equation (10.10), it reads D sh (t) = 4 3 in dense plasmas, described in Chapter 6, with the radiation MagnetoHydrodynamic CHIC code [START_REF] Breil | Multi-material {ALE} computation in inertial confinement fusion code {CHIC}[END_REF], described in Appendix B, sections B. 3.2 and B.3.3. This code is currently used to simulate laser-plasma interaction experiments. It includes thermal coupling of electrons and ions, classical or nonlocal electron heat conduction, and a detailed radiation transport with the tabulated ionization and the opacity data. Equations (6.34) and (6.35) completed with the M1 closure (6.61) of the reduced model for fast electron transport have already been implemented in CHIC by [Regan, 2010] [Regan et al., 2011] [Regan, 2011] for fast ion or electron beam transport in dense plasmas, without electromagnetic fields. An option allows for projecting or not the different quantities from the hydrodynamic Lagrangian grid of the code CHIC to a regular Eulerian grid for the fast particle transport, and vice-versa, at each hydrodynamic time step ∆t n . In the case where there are no projections, the resolution of the fast particle transport has been extended to irregular Eulerian grids, allowing to propagate the fast particles in the Lagrangian CHIC grid [Feugeas, 2011]. The selfgenerated magnetic fields have been implemented by [START_REF] Nicolai | Effect of the plasma-generated magnetic field on relativistic electron transport[END_REF] by adding the source terms η(∂/∂r)×j b and (∂η/∂r)×j b of Equation ( 6.24) into the magnetic field equation (B.81) [Nicolai, 2011].

P m ρ 0 =          D sh (t h ) t t h 1/2 if t < t h D sh (t h ) t h t 1/4 if t ≥ t h
The numerical scheme used to solve the resulting M1 equations (10.35) and

∂ ∂ε S Ψ 0 - ∂ ∂r . Ψ 1 = ∂ Ψ 0 v∂t , (10.34)
∂ ∂ε S Ψ 1 - ∂ ∂r . Ψ 2 = ∂ Ψ 1 v∂t + k Ψ 1 + e pc Ψ 1 × B
Ψ 2 = 1 3 Ψ 0 I + µ Ψ 0 Ψ 1 ⊗ Ψ 1 | Ψ 1 | 2 - 1 3 I (10.36)
has been implemented by [Regan, 2010] in 1 or 2D Cartesian or axisymetric and cylindrical geometry.

Here, ∀i ∈ {0, 1, 2}, Ψ i = vΨ i and k = ν/v is the inverse of the beam electrons mean free path (8.11).

This fully implicit numerical scheme reads

S n+1,l+1 Φ n+1,l+1 -S n+1,l Φ n+1,l ∆ε l - ∂ ∂r . F n+1,l+1 = Φ n+1,l -Φ n,l 0 v l ∆t n + Γ n+1,l ν + Γ n+1,l B (10.37)
where the kinetic energy derivative is computed according to the 1st order downwind scheme and the spatial derivative according to the 2nd order HLL scheme described in section 7.1.1. It considers the kinetic energy derivative as a "time derivative" and it allows to deduce the angular moments Φ n+1,l at kinetic energy ε l from the knowledge of the angular moments Φ n+1,l+1 at kinetic energy ε l . Here,

Φ n,l = v l Φ n,l , F n,l = (F n,l x , F n,l y , F n,l z ), Γ n+1,l ν = k        0 Ψ n+1,l 1,x Ψ n+1,l 1,y Ψ n+1,l 1,z        and Γ n+1,l B = e p l c        0 Ψ n+1,l 1,y B n z -Ψ n+1,l 1,z B n y Ψ n+1,l 1,z B n x -Ψ n+1,l 1,x B n z Ψ n+1,l 1,x B n y -Ψ n+1,l 1,z B n x        , 10
∂ ∂ε S tot Ψ 0 - ∂ ∂r . Ψ 1 = ∂ Ψ 0 v∂t + (k res + k d ) Ψ 0 , (10.43) ∂ ∂ε S tot Ψ 1 + Π ε .E - ∂ ∂r . Ψ 2 = ∂ Ψ 1 v∂t + (k + k d ) Ψ 1 + e pv Ψ 0 E + e pc Ψ 1 × B (10.44)
and

Ψ 2 = 1 3 Ψ 0 I + µ Ψ 0   Ψ 1 ⊗ Ψ 1 | Ψ 1 | 2 - 1 3 .I   (10.45)
where it has been noted ∀i ∈ {0, 1, 2}, (10.46) the total stopping power due to both collective and collisional effects and

Ψ i = p Ψ i = pvΨ i , ν d = S/
Π ε = Ψ 2 -Ψ 1 ⊗ Ψ 1 = pvΠ ε = 1 -µ 3 Ψ 0 I + Ψ 0 µ -|Ω ε | 2 Ψ 1 ⊗ Ψ 1 | Ψ 1 | 2 (10.47)
the "angular pressure tensor" that has already been introduced in (8.10). These notations allow to make appearing the term Ψ 2 .E only in the kinetic energy derivative of the 1st order equation (10.44).

Then, by noticing that Π ε .E = 0 in both anisotropic ( Π ε = 0) and isotropic (E = 0 since j b = 0 for monoenergetic electron beam) limits, we assume (10.48) in all cases. As a consequence, we can use the same numerical scheme as (10.37), by replacing S by S tot in (10.37) and by adding the new source terms expressed at t n+1 for ε l of Equations ( 10.43) and

Π ε .E = 0
(10.44) in the the right hand side of the discretized Equation (10.37). The terms depending on the self-generated electric field in Equations ( 10.43) and (10.44) are discretized semi-implicitely i.e. they are expressed with the electric field E n at time t n . It may happen that the resistive stopping power (10.40) changes sign and transforms into a "resistive pushing power" (S res < 0) for some electron energy groups ε l . However, it does not lead to numerical issues with the downwind scheme in the following simulations since we consider cases where S res S so that the total stopping power (10.46) in the CFL condition (10.38) is always positive. Finally, we use the absorbing boundary conditions, described in section 7.1.6. conducted by [START_REF] Ribeyre | Shock ignition: an alternative scheme for HiPER[END_REF] with the HiPER baseline target designed by [Atzeni et al., 2009a].

Electron driven shock waves

The

Ψ 1,x (x = 0, ε, t) = Ψ 0 (x = 0, ε, t) = N f ε (ε)F z (t) (10.49)
where

F z (t) = Π(t) -Π(t -t f ) (10.50)
(Π is the Heaviside distribution function), (10.51) with δε = 1 keV and

f ε (ε) =    1 2δε if |ε -ε 0 | ≤ δε 0 else
N = I b ε 0 v 0 . (10.52)
The kinetic energy boundaries are ε min = 1 keV and L ε = ε 0 + 5 keV.

The left panel in Figure 10.6 shows the power density W e deposited by the electron beam, the plasma pressure and the density profiles for case 1. The mean free path of fast electrons is initially of the order of L p ≈ 1 µm, that is, twice the theoretical estimate (10.18). This difference is due to the assumption (ln λ e +ln λ rel ei )/2 ln λ e ≈ 1, which allows us to find an analytical estimate for the correction factor (8.17) due to angular scattering of fast electrons. However, the characteristic hydrodynamic velocity D 0 ≈ 100µm/ns and the corresponding loading time t h ≈ 11ps are not so far from the predicted values. The maximum pressure P h rises to 400 Mbar in agreement with the theoretical prediction and the density increases by a factor of 2.7 after the loading time. The assumption of a homogeneous energy deposition, W e ≈ I b /L p , seems to be reasonable even if the beam energy flux evolves in time according to Figure 10.6 a). Then, as time goes on, the pressure drops down to

ROLE OF LASER-GENERATED FAST ELECTRONS IN THE SHOCK IGNITION SCHEME

P sh ≈ 120 Mbar, and the shock takes a triangular shape characteristic for a blast wave. It propagates with a velocity D sh ≈ 60 µm/ns. Consequently, the shock wave power I sh = P sh D sh is about 70 TW/cm 2 . The driver efficiency is thus about I sh /I b ≈ 7 %. The fast electron energy deposition does not follow the shock front, but instead moves out and spreads over the expanding plasma. The front edge of the energy deposition zone coincides with the rear edge of the density compression. Thus, the deposited energy becomes decoupled from the shock, so the shock pressure drops down with time. A comparison of the runs with and without electron thermal conductivity shows that its role is negligible at the loading stage as the plasma temperature in the shock is rather low, just a few eV. Later in time, after ≈ 600 ps, the thermal wave from the hot corona catches up to the shock and broadens the pressure profile.

The electron beams of higher energy and intensity may create even much stronger blast waves.

Conclusion

The limits of the model are threefold. First, the target thickness should be larger than the electron beam stopping length. In practice, having in mind the electron energies of several tens of keV and the stopping lengths L p of a few microns, the target density needs to be in the range of 10 g/cm 3 or more.

Second, the planar model is limited by the two or three-dimensional effects. Thus, the thickness of the expanding plasma layer, ≈ (D 0 t 3/2 / L p , according to Equation (10.29), should be smaller than the characteristic distance in the second dimension r (the shell radius for a spherical target or the electron beam radius for a planar target). This condition limits the time to t < t h (r/L p ) 2/3 . Considering the shell radius of 200 µm, this condition allows the time intervals of a few hundred ps in the examples discussed above. Third, a strong plane shock may become unstable with respect to front modulations if the target is accelerated. However, this effect needs a global description of the target dynamics.

The theory of the fast electron driven shock wave in dense solids has been confirmed by numerical simulations. It shows a possibility to achieve extremely high shock pressures in high density solid materials with the coupling efficiency up to 10 %. The case presented in the left panel of Figure 10.6 corresponds to the fast electron current of 30 GA/cm 2 , which can be generated with high power laser pulses. The numerical simulations of laser plasma interaction [START_REF] Klimo | Laser plasma interaction studies in the context of shock ignition-Transition from collisional to collisionless absorption[END_REF] predict the ELECTRON ENERGY DEPOSITION efficiency of laser absorption more than 70 % with 90 % conversion in fast electrons for the laser intensities exceeding 10 PW/cm 2 at the wavelength 0.351 µm. The laser accelerated electrons with energies less than 100 keV could be efficient drivers of strong shocks for ignition of ICF targets and for other high energy density applications. Such drivers could invest about 10 % of energy in the shock wave in solids with a pressure amplitude at the level of several hundred Mbar or more. However, the electron energy distribution, angular aperture and the target density profile may affect the shock amplitude or the preheat of the imploding shell. The simulations similar to those presented in this section have been conducted with more realistic electron energy spectrum and more realistic density profiles [START_REF] Nicolaï | Deleterious effects of nonthermal electrons in shock ignition concept[END_REF]. It has been shown that the fast electron beam may preheat the DT shell and jeopardizes its compression.

Since this work, another model developed by [START_REF] Piriz | Ablation driven by hot electrons generated during the ignitor laser pulse in shock ignition[END_REF] for the desciption of ablation driven by hot electrons generated during the ignitor laser pulse in shock ignition have been proposed. However, contrary to the model presented here, it assumes the process to be quasi-stationary, which does not agree with our numerical simulations. The possibility to launch shocks of severalhundred Mbar in spherical targets has been recently demonstrated on the OMEGA laser facility [START_REF] Theobald | Demonstration of 200-Mbar Ablation Pressure for Shock Ignition[END_REF]] [Nora et al., 2015]. The ablation pressure has been inferred from the time of shock propagation to the target center by using radiation-hydrodynamic simulations. Peak ablation pressures exceeding 300 Mbar are inferred at absorbed laser intensities of ≈ 3 10 15 W/cm 2 . It has been demonstrated that the shock strength is significantly enhanced by the coupling of suprathermal electrons with a total converted energy of up to 8 % of the incident laser energy. At the end of the laser pulse, the shock pressure is estimated to exceed ≈ 1 Gbar because of convergence effects.

10.2 Blast Wave generation in solid targets by the quasiisochoric heating by laser-generated Electron Beam

Analytical Estimates

Here, we consider another example of a blast wave driven by an intense short laser pulse. The experimental campaign conducted by [START_REF] Santos | Supra-thermal electron beam stopping power and guiding in dense plasmas[END_REF], was described in sections 8.2 and 9.2. PIC simulations of laser-solid interaction with a laser pulse intensity of I L = 7.5 10 17 W/cm 2 and a duration of 500 fs, [START_REF] Sentoku | Isochoric heating in heterogeneous solid targets with ultrashort laser pulses[END_REF] showed that the quasi-isochoric heating of the target by the laser-generated fast electrons may excite shock waves that compress the plasma beyond solid density and to keV temperatures. According to their simulations, shocks with pressures up to gigabar can be launched inside the target with ultrashort laser pulses.

As illustrated in

Therefore, in order to explain this increase of visible light emission in the experiments conducted by [START_REF] Santos | Supra-thermal electron beam stopping power and guiding in dense plasmas[END_REF], we developed a model of a blast wave generation by strong temperature gradients induced by a fast target heating. For this, we solve Equations (10.1), (10.2), (10.3) and

(10.4) with a nonlinear heat flux q = -κ 0 T n ∂T ∂x (10.53)

induced by an instantaneous energy deposition W e ∝ δ(t).

(10.54) Indeed, according to section 9.2.2, the target heating by the laser-generated fast electrons lasts ≈ 1 -2 ps; see Figure 9.12. On the hydrodynamic time scale, this heating may thus be considered as instantaneous and isochoric due to the large ion inertia. Actually, some ions are accelerated on the ps time scale, but they do not affect the target dynamics. The main part of ions may be considered as immobile. Since in our conditions the electron and radiation heat fluxes are comparable (10.55) according to [START_REF] Ditmire | Supersonic Ionization Wave Driven by Radiation Transport in a Short-Pulse Laser-Produced Plasma[END_REF] Let us assume that at the time t = 0 after the end of the laser pulse, the electron heating resulted in an exponential temperature profile T (x, t = 0) = T m exp (-x/d 0 ), (10.56)

κ rad κ SH ≈ 5 10 18 (k B T e [eV]) 5/2 n e [cm -3 ] > 1
with T m = 400 eV and d 0 = 5 µm; see Figure 9.15. Contrary to the explanation given by [START_REF] Sentoku | Isochoric heating in heterogeneous solid targets with ultrashort laser pulses[END_REF], the blast wave generation is not related to the Copper tracer layer in our case. Therefore, we assume a simple target made of Aluminum, only (ρ = 2.69 g/cm 3 and n i = 6 10 22 cm -3 ). We model therefore the thermal capacity as

C V = 3 2 n e k B = C st with n e = Z * n i (10.57)
and the velocity of sound as

c s = Z * k B T m i . ( 10.58)
The ionization state may be estimated by its averaged value :

Z * = 1 L x Lx 0 Z * m exp - x d 0 dx ≈ 5 (10.59)
with L x being the target thickness and Z * m = 10 the ionization state in the first micron of Aluminum. Due to the high temperature T m , the plasma expands rapidly in vacuum at the irradiated side of the target with the velocity c s (T m ) = Z * m k B T m /m i e x corresponding to few hundreds of µm/ns, in agreement with the self-similar rarefaction wave expanding in vacuum [START_REF] Gurevich | Self-similar Motion of Rarefied Plasma[END_REF]. Due to the plasma expansion, the temperature at x = 0 decreases rapidly to low temperatures in agreement with the 1D academic simulation so that we can consider the boundary condition ∀t > 0, T (x = 0, t) = 0.

(10.60)

This boundary condition corresponds to the dipole-type solution found by [Barenblatt and Zel'dovich, 1957] when studying the self-similar process of gas filtration in a porous medium. Here, the invariant quantities are the heat diffusivity (10.64)

a = κ 0 C V [cm 2 .s -1 .K -n ] ( 10
Here, ξ 1 and M are dimensionless constants while a ≈ 4.7 10 -14 cm 2 .s -1 .K -5/2 for the electron thermal conductivity (C V = 6.2 10 7 erg.K -1 .cm -3 and ln Λ ei = 3) and a ≈ 8.5 10 -29 cm 2 .s -1 .K -5 for the radiation thermal conductivity assuming

κ rad = 16σ SB λ R T 3 3 (10.65)
and the Rosseland mean free path for Hydrogen-like plasmas λ R ≈ 8.7 10 6 T 2 Z * 2 n i (10.66)

according to [Eliezer, 2002]. By looking for the function F such that

T (x, t) = τ (t).F x x f (t) (10.67)
is solution of (10.3), we find according to [Barenblatt and Zel'dovich, 1957]

F (ξ) = ξ 1/(n+1) 1 -ξ (n+2)/(n+1) , (10.68) M = n 2(n + 2) 1/n ≈ 0.599 if n = 5/2 0.814 if n = 5
(10.69) and compression waves that propagate towards the target rear side as

ξ 0 = (n + 2) 1/2 2 n (n + 1) B 1 + 1 n , n + 1 n + 2 + 1 -n 1 2(n + 1) ≈ 1.918 if n = 5/
c s (t) ≈ Z * k B T (t) m i (10.71)
where

T (t) = τ (t) n n + 2 (10.72)
is the mean temperature in the dipole-type profile illustrated in Figure 10.8. Then, by equating the compression wave velocity at the steepening time with the thermal wave front velocity

c s (t s ) ≈ dx f dt (t s ) = x f (t s ) 2(n + 1)t s , (10.73) we obtain t s ≈ ξ 1 2 (n + 2) 4M n(n + 1) 2 . m i Z * k B P (n+1)/2n (aP n ) 1/n ≈         

16

T m 400 eV Knowing the steepening time, we can also estimate the steepening depth as the thermal wave front position at the steepening time t s . It reads

x s ≈ x f (t s ) ≈         

7

T m 400 eV This expression explains why a strong increase in visible light emission has been observed experimentally for the thickest target and not for the thinnest target. This is due to the fact that the thin target does not have a the sufficient thickness to allow for the blast wave generation. Interestingly, we obtain the same steepening time of 16 ps in both regimes of thermal conduction for the set of parameters corresponding to the experiment. However, the compression waves steepening depth is slightly larger in the case of radiation thermal transport. Since the heat transport regime changes with depth, we estimate between 7 and 10 µm the value of the steepening depth.

We have demonstrated analytically that the target temperature gradients, due to the isochoric and instantaneous target heating by the laser-generated fast electrons, may be responsible for the generation of a blast wave in the thicker target. The blast wave breakout at the target rear side is in turn responsible for a rapid increase of the temperature at the target rear side. It thus increases the visible light emission from the target rear side that has been observed in the experiment. Let us estimate now the time t bo needed for the blast wave to reach the target rear side. By estimating the energy released in the target, responsible for the blast wave generation as We thus obtain the same order of magnitude of a few hundreds of ps as observed in the experiment for the thicker target. The target rear side temperature of ≈ 10 eV that drives a rarefaction wave may explain the discrepancy between the analytical estimate and the experiment as we are going to explain in the next section. For example, if we conduct the same estimate with L x = 19 µm instead of L x -x s , we obtain t b0 = 514 ps.

E 0 = ∞ 0 2πrdr Lx 0 dxC V T (x, t = 0) exp -4

Radiation Hydrodynamic Simulations

We have performed hydrodynamic simulations with the hydrodynamic code CHIC, which described the radiative transfer in the diffusive approximation and the visible light emission at λ = 405 nm from the target rear side. Opacities were computed according to the approximate method proposed by [START_REF] Tsakiris | An approximate method for calculating Planck and Rosseland mean opacities in hot, dense plasmas[END_REF]. The blast wave was generated with or without accounting for the radiation thermal conduction, confirming the results obtained in section 10.2.1. In order to account for the 3-dimensional effects of the blast wave propagation, the simulations are conducted in the 2D- and pressure produce a rapid plasma expansion in vacuum at the target boundaries. At the laserirradiated side of the target, the temperature is much higher so that the expansion is stronger and a rapid decrease of the temperature corresponds well to our theoretical model (10.60). The gradients of the temperature and the pressure drive both a thermal wave and a compression wave inside the target.

The thermal wave front is caught up by the compression wave for the thicker target, producing a blast wave. It is not the case for the thinner target, which simply explodes. A shock is formed at a depth of x s ≈ 10 µm (in agreement with our theoretical predictions) at a time between 20 and 60 ps for the thicker target. It is not evident to distinguish between the expansion of the Copper tracer layer in the rear side Aluminum layer and the shock front at these times. The pressure at the shock front is ≈ 50

Mbar at 60 ps as illustrated in the left panel of Figure 10.10 and decreases with time, as illustrated in Figure 10.12. The downstream plasma temperature behind it is more or less homogeneous and on the level of T ≈ 50 eV. We can see that the plasma expansion at the target rear side drives a rarefaction wave that decreases strongly the blast wave strength on-axis. As a result, we deduce that the increase of visible light observed experimentally is mainly due to the blast wave front breakout far away from the fast electron propagation axis. It may also explain why using L x = 19 µm in (10.77) instead of L x -x s allows for a better agreement with the experimental observations. The analysis of existing numerical methods for solving the Vlasov-Fokker-Planck equation (V-F-P) oriented our choice towards a hybrid and expanded "Vlasov-Fokker-Planck" method. In order to make numerical computations as fast as possible, we limited ourselves to the first two angular moments.

The "hybrid" assumption consists in separating the beam electron population and the background electron population. Contrary to the widely used P1 approximation, also usually called the "diffusion approximation", our M1 model accounts for an arbitrary degree of anisotropy. The closure relation is deduced from the Minerbo maximum angular entropy criterion depending on the anisotropy vector.

Such a closure is exact for fully isotropic local (in space and kinetic energy) angular distribution function and for fully anisotropic local angular distribution function. While the first order expansion reduces the information concerning the local angular distribution function, it provides a sufficient accuracy for the laser-generated fast electron beam transport. Such a closure is fully justified in the collisional limit. Besides, a comparison of the full kinetic and the M1 approach for the analysis of the resitive filamentation instability shows that our model describes the instability growth rate with an error of a few 10s of % in the case of collisionless fast electrons.

Our model assumes time scales greater than the electromagnetic neutralization time of the beam and that the beam is not modified during its electromagnetic neutralization. This is a strong assumption in the case of propagation through insulators since the ionization processes occur in this time scale, implying additional energy losses of the electron beam that are omitted here. In our hybrid model, the self-generated magnetic field verifies the diffusion equation with source terms depending on the resistivity gradients, curls of the beam current and temperature-density crossed gradients while the self-generated electric field is given by the quasi-static Ohm's law. Thus, we have neglected the magnetization effects, the background electron viscosity, the collisional friction of the background electrons due to collisions with beam electrons, the displacement current in the Maxwell-Ampere equation (quasi-static approximation), the background electron inertia and we have considered the ideal gas expression for the equation of state of background electrons. These assumptions are justified in the case of laser-generated electron beam transport in dense targets. The self-generated electromagnetic fields depend on the electrical resistivity of the material and its spatial gradients, which depend on the electron and ion temperatures of the material. Thus, the electron temperature evolution needs to be described self-consistently according to the electron heat equation. Also, the background ion heating is described self-consistently according to a simplified heat equation. In our model, we neglect the ion motion and the ion thermal conductivity, considering times scales smaller than 10-100s of ps. We also proposed new expressions for the heat capacities, the electrical resistivity, the electron thermal conductivity and the electron-ion coupling factor, allowing to describe metals from the solid state at the room temperature ≈ 300 K through the liquid and Warm and Dense Matter (WDM) states to the hot plasma state with temperatures ≈ 10 keV. The collisions of background free electrons with d-band bound electrons are taken into account according to recent studies showing the importance of this relaxation process.

The numerical schemes used for the solution of the M1 equations have been described. The M1 equations are computed with second order explicit schemes except for the fast electron collisional slowing down, which is computed according to the 1st order downwind scheme. The advection terms are computed according to the HLL schemes allowing to ensure a positive number of electrons and a norm of the mean propagation vector less than 1. Implicit schemes have also been implemented in order relax the CFL condition in case of fast electron transport in a very dense plasma. The self-generated electromagnetic fields are computed according to 2nd order schemes except the selfgenerated magnetic diffusion that is resolved semi-implicitely thanks to 2nd order discretizations and a conjugated gradients algorithm. Finally, both heat equations are computed according to explicit numerical schemes. The numerical schemes used to solve the equations of the model have been validated thanks to a 2D-3V academic case of a monoenergetic and collimated fast electron beam propagating in a warm and dense hydrogen plasma. Also, it allowed to demonstrate the major features of the M1 approximation and to derive analytical expressions for the various quantities computed by the code such as the fast electrons penetration-depth-to-the-range ratio due to their angular scattering.

A realistic simulation of a laser-generated fast electron beam transport in a thin Aluminum target Firstly, we demonstrated that the numerical time step of the fast electron transport calculation may be comparable to the K-hole lifetime in the case of Aluminum and Copper targets. Therefore, we have implemented the K-shell hole dynamics in our model. Secondly, we demonstrated that the solid state corrections must be taken into account. In particular, the electron-ion temperature equilibration time and collisions of d-band with s-band electrons may affect the magnetic field distribution in the Copper target. Thirdly, we confirmed with a simple analytical model, the result obtained by [START_REF] Sherlock | In-depth Plasma-Wave Heating of Dense Plasma Irradiated by Short Laser Pulses[END_REF] concerning the effect of the target electron inertia in the Ohm's law and the displacement current. Each laser-generated bunch of electrons, injected in the target at the laser frequency or twice the laser frequency excites a weak field of background electron plasma waves.

Collisional damping of these plasma oscillations results in an additional target electron heating , which is not accounted for in hybrid models. Fourthly, we confirmed a strong contribution of refluxing electrons in the Kα photon emission. The thinner the target is, the stronger the refluxing electrons contribute to the Kα photon emission. Fifthly, we demonstrated significant differences between 2D and 3D simulations, especially in what concerns the self-generated magnetic fields, the size of the Kα emission zone and the local density of emitted photons. In particular, the size of the Kα emission zone is underestimated in our calculations as compared to the experimental data. We analyzed the assumptions of our model concerning the Kα emission that may explain this discrepancy. The photon re-absorption may introduce an error of only ≈ 10 % in our computations. Another candidate is the contribution of the secondary electrons. However, it seems to me that the main critical assumption in our model is concerned with the specular reflection of fast electrons at the target-vacuum interfaces.

A simple estimate of the magnetic field generated by the "fountain" effect indicates that these local fields may significantly deviate the refluxed fast electrons, thus enhancing the Kα 1 signal spotsize.

We coupled our reduced model for fast electron transport with a radiation hydrodynamic code and investigated the generation of strong shock by energetic electron beam. One example concerns the fast electron driven shock wave in dense plasmasin the shock ignition conditions. It confirms the theoretical estimates by [START_REF] Gus'kov | Ablation Pressure Driven by an Energetic Electron Beam in a Dense Plasma[END_REF] and shows a possibility to achieve Gbar shock pressures in high density solid materials with the coupling efficiency up to 10 %. In contrast, we disapprove another model developed by [START_REF] Piriz | Ablation driven by hot electrons generated during the ignitor laser pulse in shock ignition[END_REF] as the quasi-stationary assumption of that paper does not agree with our numerical simulations. Another example concerns the excitation of a blast wave with a femtosecond laser pulse. We have demonstrated analytically and numerically that a blast wave is generated due to the temperature gradients induced by the laser-generated fast electron energy deposition in the target tens of picoseconds after the end of the laser pulse. The analytical estimates and hydrodynamic simulations showed a good agreement with the experimental data.

The hybrid model of fast electron transport developed in this thesis has been already used in several other studies. More realistic simulations of the ablation pressure driven by an electron beam in the context of shock ignition by [START_REF] Nicolaï | Deleterious effects of nonthermal electrons in shock ignition concept[END_REF] show that an exponential electron energy spectrum and a more realistic target density profiles may reduce the ablation pressure and preheat the target thus reducing the shock strength. The model have been applied by [START_REF] Volpe | Controlling the fast electron divergence in a solid target with multiple laser pulses[END_REF] for studies of the multiple pulse scheme proposed by [START_REF] Robinson | Artificial Collimation of Fast-Electron Beams with Two Laser Pulses[END_REF]] [Scott et al., 2012] to control the divergence of laser-generated electron beam in the context of fast ignition. It is shown that a sequence of three ps laser pulses allows to improve the collimation of the beam at least by a factor 2 compared to the double pulse scheme. Another series of simulations has been performed

for the understanding of a laser-generated transport experiment reported by [Vaisseau, 2014]. In this experiment, a collimation of the fast electron beam in a planar Carbon target has been demonstrated and explained with hybrid numerical simulations. Finally, we conducted estimates of the emission of Kα photons in future PETAL laser pulse experiments for radiography applications. Simulations with the M1 code have been compared with 3D Monte Carlo simulations conducted by [Boutoux, 2014] with the codes GEANT4 [Agostinelli et al., 2003] and PENELOPE [START_REF] Sempau | An algorithm for Monte Carlo simulation of coupled electron-photon transport[END_REF].

The perspectives of this work are manyfold. Firstly, the wake field losses of modulated electron beams may be implemented in the hybrid code. This will allow to resolve the laser-generated electron bunches and to implement the CTR diagnostic in the code. The secondary electrons and the magnetic field generated at the target edges by the escaping electrons can also be taken into account.

However, these physical effects require resolving very small spatial scales of the order of the fast electron Larmor radius and, therefore, represent an important investment in terms of vectorization and parallelization of codes. Also, it would be interesting to compute self-consistently the fast electron and Kα photon transport in the target in order to account for opacity effects that are usually neglected. Concerning the application of fast electron transport to the generation of shocks in ICF, more realistic simulations must be conducted combining both laser-and-electron-generated shocks in a convergent geometry. Concerning fast ignition, the interaction of the petawatt laser pulse with the cone is still an intense field of research [START_REF] Kemp | Laser plasma interactions for fast ignition[END_REF]. The M1 model may be useful to perform a parametric study of the target density and temperature conditions and the properties of the ignitor electron beam. Different ways to collimate the fast electron beam can be also considered.

For example, simulations at the ignition scale of the fast ignition of fusion pellets with an engineered cone-in-shell [START_REF] Robinson | Elliptical magnetic mirror generated via resistivity gradients for fast ignition inertial confinement fusion[END_REF] or with an external magnetic field [START_REF] Fujioka | Kilotesla Magnetic Field due to a Capacitor Coil Target Driven by High Power Laser[END_REF] can be performed with plasma magnetization and three-dimensional effects taken into account. where ρ is the plasma electric charge density and j the electric current density. In a quantum approach, which is out of scope in this thesis, the electromagnetic fields are considered as an ensemble of photons with quantized energy hν described by the Klein-Gordon equation for bosons.

Part IV

Résumé en Français

The retroaction of these electromagnetic fields on the charged particles must be also resolved. For with given initial conditions p α,n (t = 0) = p α,n,0 and r α,n (t = 0) = r α,n,0 where r α,n and p α,n are the position and the momentum at time t of the particle denoted by (α, n). Here, α = e for electrons, α = i for ions and n ∈ [1..N α ]. Thus, the particle charges read q e = -e while q i = Z * e. Also, it has been noted γ α,n = 1 + (p α,n /m α c) 2 the Lorentz factor of the particle (α, n) and v α,n its velocity.

A.1.4 Vlasov-Boltzmann Equation

In order to find an expression for the right hand side of Equation (A.12), we consider the BBGKY hierarchy at the 1st order. Thus, one has to simplify the second order equation (A.14) to express the right hand side of the first order equation of the hierarchy (A.12). Let us introduce the normalized distribution functions f e,1 = f e,1 /N e and f e,2 = f e,2 /N e (N e -1). If electrons 1 and 2 were independent, one has f e,2 (r 1 , p 1 , r 2 , p 2 , t) = f e,1 (r 1 , p 1 , t) f e,1 (r Again, by assuming that f 2 evolves in space mainly due to changes in f 1 , rather than to changes in the pair correlations g 2 , we neglect the space derivative of g 2 in the first term of (A.18). One expects that f e,2 (r 1 , p 1 , r e,2 , p 2 , t) is varying slowly in the binary collision center of mass coordinates . f e,1 (r 1 , p 1 , t) f e,1 (r -f e,1 (r 1 , p 1 -∆p ee , t) f e,1 (r 1 , p 2 + ∆p ee , t) , A.23) is the electron-electron Boltzmann collision integral and

(
C B ei [f e,1 , f i,1 ] (r 1 , p 1 , t) = - R 3 d 3 p 2 R 3 d 2 σ ei |v 1 -v 2 |
. f e,1 (r 1 , p 1 , t) f i,1 (r 1 , p 2 , t)

-f e,1 (r 1 , p 1 -∆p ei , t) f i,1 (r 1 , p 2 + ∆p ei , t) The case A(p α,1 ) = 1 is obvious because the double integral of the second term cancels directly the double integral of the first term of (A.24) or (A.23). The demonstration for A(p α,1 ) = p α,1 and A(p α,1 ) = p α,1 2 /2m α is also evident. These three functions A(p α,1 ) are called the Boltzmann integral invariants. These conservation properties come from the fact that each binary collision conserves the number (1), momentum (p α,1 ) and kinetic energy p α,1 2 /2m α of the particles. For the same mathematical reasons, one can also show that It was demonstrated that the Vlasov-Boltzmann equation has only these three integral invariants [START_REF] Decoster | Modeling of collisions[END_REF]. Thus, any other functions which is Boltzmann integral invariant, is a linear combination of these three functions.

H-Theorem

The Boltzmann collision operator property is that for any A(p e,1 ) for which the subscript " 1 " and " 2 " have been omitted for brevity. Thus, if f e (p e )f α (p α ) > f e (p e -∆p eα )f α (p α + ∆p eα ), the logarithm is positive and in the opposite case where it is negative, the logarithm is also negative. As a conclusion, as Boltzmann has noticed, the function H so called the plasma electrons Entropy is such that dH dt ≥ 0. (A.32)

This property is called the H-theorem and it is at the origin of the 2nd principle of Thermodynamics.

Maxwell-Boltzmann distribution function

According to the H-theorem (A.32), the entropy H of any closed system will asymptotically reach its as explained in [Chandrasekhar, 1943], the Landau collision operator (??) can also be derived assuming that the distribution function f e (p e , t) is a stochastic process which does not depend at t = t 0 on its whole history t < t 0 but only on the infinitezimal time interval t 0 -∆t with ∆t t 0 which however, is sufficiently large compared to the time between two consecutive electron-α collisions (definition of a Markovian stochastic process). In the textbook by [Balescu, 1963], a similar approach is presented to derive (A.57) from the Boltzmann collision operator while [START_REF] Landau | Physical kinetics[END_REF] derived it by using physical considerations on the electron fluxes in the momentum space. Equation (A.57) can also be derived from the Lenard-Balescu equation, as done in the textbook by [START_REF] Decoster | Modeling of collisions[END_REF], where g is an analytical fit of numerical Thomas-Fermi calculations provided by [More, 1985].

assuming

A.2.4 Properties of the Landau equation

One can check that the Landau equation preserves the properties of the Boltzmann equation. The functions 1, p α,1 and p α,1 2 /2m α are still integral invariant i.e. for α = e or i, to reach the maximum-entropy equilibrium distribution function f M [START_REF] Bhatnagar | A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems[END_REF]. This is called the BGK approximation from the names of its founders. It describes the relaxation of the distribution function to the Maxwell-Boltzmann distribution in agreement with the H-theorem. The parameters in the function f M (A.40) are defined according to the conservation of the number of particles, the total momentum and energy.

R 3 C L αα [f α,1 , f α,1 ] (r, p α,1 , t) 1 d 3 p α,1 = 0 R 3 C L αα [f α,1 , f α,1 ] (r, p α,1 , t) p α,1 d 3 p α,1 = 0 R 3 C L α [f α,1 , f α,1 ] (

A.3.2 Lorentzian Plasmas

The Lorentz assumption concerns only the electron-ion collision term (??). Because of the large mass ratio m i /m e 1, the friction term (A.61) is much smaller than the diffusion term (A.62). By assuming that the distribution function of particles α is given by f α (r, p α , t) = n α (r, t)δ[p α ], where δ is the Dirac distribution, in order to account for only small α particle velocities, the diffusion term (A. According to Appendix A, section A.2., plasma electrons can be described on the kinetic scale according to the Vlasov-Fokker-Planck-Landau equation for the electron distribution function

f e = f e (
2∆t = 4πe 2 q α 2 ln Λ eα R 3 |v -v α | 2 I -(v -v α ) ⊗ (v -v α ) 2|v -v α | 3 f α (
q α R 3 f α m α (v α -u α ) 2 2 (v α -u α ) d 3 p α
Under these definitions, one can relate the internal kinetic energy U α with the mean particles kinetic energy K α , the mean particle velocity u α and the temperature T α as follows

U α (r, t) = n α K α -n α m α u 2 α 2 = C V,α T α (B.7)
where it has been noted C V,α = (3/2)n α k B the α particles thermal capacity. Also, the kinetic pressure tensor P α is related with the scalar kinetic pressure P α , the temperature T α and the viscosity tensor The thermal energy flux q α can be related with the kinetic energy flux Q α , the mean particles velocity u α , the scalar pressure P α and the viscosity tensor τ α as follows forms a set of 10 equations with 10 hydrodynamic unknowns n e , u e , T e , n i , u i , T i assuming that the hydrodynamic fluxes P e , q e , P i , q i , Q ei and R ei are known. Considering the time scales much less than ∼ 100 ps, we neglect the ion conductivity q i and the ion velocity u i in the ion conservation equations. Also, the viscosity tensors τ i and τ e and the mechanical work -P e (∂/∂r).u e of (B.26) are neglected. In what follows, we evaluate the transport coefficients allowing to express the friction force R ei , the electron-ion energy exchange Q ei and the electron thermal flux q e in order to close this set of 10 equations.

q α = Q α - m α u α 2 2 n α u α - 5

B.2.1 Lorentz Approximation

The Concerning the electron-ion energy exchange Q ei , it can be obtained directly by using the Maxwell-Boltzmann distribution function (B.5) and the full Landau electron-ion collision integral, i.e. without neglecting the term ∝ m e /m i (see for example [START_REF] Landau | Physical kinetics[END_REF] p.173). It reads (B.40) [START_REF] Spitzer | Transport Phenomena in a Completely Ionized Gas[END_REF] have shown that the electron-electron collision contribution to the electrical and thermal conductivities can be expressed as follows where γ E and δ T are correction factors, which can be fitted by functions of the ionization state Z * from the tabulations provided by [START_REF] Spitzer | Transport Phenomena in a Completely Ionized Gas[END_REF] The expression proposed by [START_REF] Brysk | Thermal conduction in laser fusion[END_REF] for δ T is less accurate than the expression proposed here because it does not reproduce the Spitzer results (κ Sp /σ Sp T e ) → (κ L /σ L T e ) = 4(k B /e) 2 when Z * → ∞ [Ribeyre, 2014]. The parameters δ T and δ E account for the electron-electron collisions while accounts for the reduction of the heat flux due the electrostatic field induced by temperature gradients in a steady state.

Q ie (r, t) = R 3 m i (v -u i ) 2 2 C L ie [f i , f e ] (
One may ask himself why electron-electron collisions modify the electric conductivity since the hydrodynamic moments of the electron-electron collision terms vanish, giving R ee = 0 and Q ee = 0, according to the properties of the Landau collision integral. This is actually a purely kinetic effect due to the "rearrangement" of the electron momentum spectrum, due to electron-electron collisions, such that the electron-ion collisions (∝ 1/v 3 ) are enhanced leading to larger slowing downs of the electrons by colliding the ions. Indeed, in the limit Z * → ∞ where electron-electron collisions are negligible compared to electron-ion collisions, one can notice that σ Sp → σ L , κ Sp → κ L and that the Lorenz factor (κ Sp /σ Sp T e ) → (κ L /σ L T e ) = 4(k B /e) 2 . For other specific values of Z * , one gets according to [START_REF] Spitzer | Transport Phenomena in a Completely Ionized Gas[END_REF] :

κ

B.2.3 Transport Coefficients in an External Magnetic Field

The transport in the external homogeneous magnetic field was considered by [Braginskii, 1965] The notations and ⊥ define the components for the generalized forces (current and temperature gradient) parallel and perpendicular to the magnetic field direction. The first term R u is due to the friction of electrons on the ions as already presented in the previous section while R T is due to the temperature gradient effect. Here, j is the plasma current density evaluated in the quasineutral approximation : j = -en e u e + Z * en i u i = ρ c u -en e (u e -u i ) = -en e (u e -u i ) .

Analogously, the electron heat flux consists of two parts q e = q u + q T (B. The components of the tensors are

κ ⊥ = n e k B 2

B.2.4 Electron Degeneracy Corrections

The transport coefficients in the case of degenerated electrons were considered by [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF].

They considered a linearized kinetic equation in the BGK approximation (see Appendix A section By noting with a supscript i the index of the fluid streamline denoted by i, V m = 1/ρ m and ∀ξ ∈ {ρ m , V m , u, P e , P i , P, T e , T i , ε e , ε i , Q ei , W e,ext , Q ie } , ξ i (t) = ξ( r i , t) where d r i dt = u i (t) = u r i , t knowing the initial condition r i (t = 0), these equations read where E = U m +(u 2 /2) (There is a mistake in [START_REF] Breil | Multi-material {ALE} computation in inertial confinement fusion code {CHIC}[END_REF] [START_REF] Maire | A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems[END_REF] and a high order Lagrangian finite volume scheme based on the acoustic generalized Riemann problem solved using a least squares procedure followed by a slope limitation procedure [Maire, 2009]

ρ i m d V i m dt - ∂
                   ρ i,n m V * m -V n m ∆t * i - ∂ ∂r .
u 2. Then, the energy transfer between ions and electrons and the electron heating is solved on the interval between t * and t n+1 using a Newton algorithm while the non linear conduction is solved implicitly with the discretization of the thermal diffusion operator [START_REF] Breil | A cell-centered diffusion scheme on twodimensional unstructured meshes[END_REF]] where it has been noted ∀α ∈ {e, i} , C V,α,m = dε e /dT α the specific thermal capacity.

       ρ i m C i V,

B.3.3 CHIC MHD Package

In order to study the effects of the magnetic fields on the thermal conduction according to the Braginskii expression for the thermal conduction coefficient κ e , a MHD package has been implemented in CHIC by [START_REF] Schurtz | Revisiting Nonlocal Electron-Energy Transport in Inertial Fusion Conditions[END_REF] and [START_REF] Breil | Modelling of the magnetic field effects in hydrodynamic codes using a second order tensorial diffusion scheme[END_REF]. This has been motivated by an experimental campaign on the LIL facility of CEA. This section is devoted to the description of this package. By neglecting the viscosity tensor (τ = 0), the plasma electrons inertia ( n e m e (∂u e /∂t)+n e m e (u e .∂/∂r)(u e )

) and by assuming that the plasma is quasineutral (ρ c = 0), the hydrodynamic conservation equation (B.80)

The first line of this equation accounts for the magnetic diffusion, the second line accounts for its advection by the plasma ions (u i ) and the plasma heat flux via the Nernst effect (u Nernst ) and the third line accounts for the magnetic field generation due to electron temperature-density crossed gradients. Focused on the heat-conduction phenomena, the electrical resistivity has been assumed isotropic, such that η = ηI with η -2 = η -2 Hubbard + η -2 Spitzer . Also, due to the huge ion mass, the electron contribution to the plasma fluid velocity is neglected u = u i . Thus, Equation (B.80) the results of this computation.

2. Secondly, the diffusion of the magnetic field is solved in the same way as the thermal diffusion [START_REF] Breil | A cell-centered diffusion scheme on twodimensional unstructured meshes[END_REF] .

  Controlling the fast electron divergence in a solid target with multiple laser pulses. Phys Rev E 90, 063108 (2014) 3. D Del Sorbo, J-L Feugeas, Ph Nicolaï, M Olazabal-Loumé, B Dubroca , S Guisset, M Touati and Vladimir T. Tikhonchuk. Reduced Entropic Model for Studies of Multidimensional Nonlocal Transport in High-Energy-Density Plasmas. Submitted to Phys. of Plasma 4. A Colaïtis, G Duchateau, X Ribeyre, Y Maheut, G Boutoux, L Antonelli, Ph Nicolaï, D Batani and V Tikhonchuk. A multiscale model for laser-plasma interaction. Submitted to Phys. Rev.LetterFinally, some results have been partially presented in the following conferences 1. Fast ignition workshop, Napa Valley, California,United-States, 2012 (oral)

  in the text that (∂/∂x) = (∂ x , ∂ y , ∂ z ) T , (∂/∂r) = (∂ z , ∂ r , (1/r)∂ θ ) T or (∂/∂p) = (∂ p , (1/p)∂ θ , (1/p sin θ)∂ ϕ ) T areused as vectors in expressions such as presented previously in the paragrah Linear algebra notations (see for example curl ) Infinitezimal volume d 3 x = dx dy dz, d 3 r = rdz dr dθ and d 3 p = p 2 sin θdp dθdϕ.

  momentum p b γ b m e v b Beam * electrons kinetic energy ε b (γ b -1)m e c 2 Beam (M-J) thermal velocity β T,b 2k B T b /γ b m e c 2 0.06T b 1/2 Beam * neutralization time (ω p /ν ei 15 T e 3/2 /n i Z * 2 ln Λ ei Beam * neutralization time (ω p /ν ei 1)) τ e /γ b Chapter 2.2 1.2 10 -15 Z * ln Λ ei /γ b T e /Z * ln Λ ei Beam ( * and M-J) plasma frequency ω b 4πn b e 2 /γ b m e 1.78 10 15 (n b /γ b ) 1/2 Beam (M-J) Debye screening length λ D k B T b /4πγ b n b e 2 7.43 10 -7 (T b /γ b n b ) 1/2 Beam ( * and M-J) electrons inertia length λ b c/ω b 1.68 10 -5 γ b /n bIntroduction"Physics is like sex: sure, it may give some practical results, but that's not why we do it."

Figure 1 :

 1 Figure 1: World consumption per capita (left) estimated from the BP Statistical Review of world energy 2014 workbook and World Electricity Consumption per capita (right) estimated from data published by the World Bank and the International Energy Agency. The world population is estimated from the United Nations Revision of the World Population

Figure 2 :

 2 Figure 2: The Aston curve of the strong nuclear binding energy per nucleon B(Z, N )/A (left) and a NASA density-temperature map of existing plasmas

Figure 3 :

 3 Figure 3: Reaction rate σv of the different fusion nuclear reactions (left panel) and Lawson criterionfor the Ignition (right panel) both estimated from[START_REF] Bosch | Improved formulas for fusion cross-sections and thermal reactivities[END_REF]

Figure 4 :

 4 Figure 4: Classical ICF schemes :The Direct and Indirect drive approaches[START_REF] Campbell | The National Ignition Facility -applications for inertial fusion energy and high-energy-density science[END_REF]

Figure 5 :

 5 Figure 5: Fuel configuration at ignition for the conventional scheme a), the fast ignition scheme b)and the shock ignition scheme c)[Atzeni, 2009]

Figure 6 :

 6 Figure 6: Schematic of cone-guided fast igntion (a) and photograph of cone-attached shell target used for integrated experiments on OMEGA (b).[START_REF] Norreys | Fast electron energy transport in solid density and compressed plasma[END_REF]

Figure 7 :

 7 Figure 7: Laser pulses shape for schock ignition : (1) corresponds to the Nuckolls-Kidder law for the compression phase and (2) is the ignitor pulse (left) and the corresponding logarithmic pressure gradients in the target versus space and time (right)[START_REF] Ribeyre | Shock ignition: an alternative scheme for HiPER[END_REF]

0. 3

 3 Fast Electron Generation in the context of ICF 0.3.1 Role of Fast Electrons in the Shock Ignition Scheme

Figure 8 :

 8 Figure 8: Schematic view of the laser-shell interaction

  for the radiative transfer theory, is adapted here to the fast electron transport. It allows us to close the set of equations by evaluating the 2nd order angular moment of the distribution function needed in the 1 st order angular moment equation. Contrary to the largely used approximation of the distribution function with one Legendre polynomial often called P1, this M1 model allows to describe the distribution function with an arbitrary local anisotropy. It is shown that the model is exact in the limits of fully isotropic and fully anisotropic local angular distribution functions. As the laser-generated relativistic electron beams may have a wide energy spectrum and an arbitrary angular distribution functions, the equation of the local angular entropy is derived and the limitations of the model model are discussed. Developments of new plasma transport coefficients necessary to model the self-generated electromagnetic fields are also proposed.

  Hydrogen plasma, allowing us to present the major features of the model. Analytical expressions are also derived to check the numerical schemes. Then, a comparison with a hybrid PIC simulation is presented to validate the model. It deals with a realistic laser-produced fast electron beam deduced from the PIC simulation of an experiment conducted on the UHI100 laser facility of the CEA (Saclay).The third part of the manuscript is dedicated to applications of the model in the context of ICF: Chapter 9 is dedicated to studies of the Kα emission induced in a plasma or in a solid by the fast electron transport. The theory of Kα emission is presented and the simulation results are compared to experimental data. It is shown that 3-dimentional effects as well as the photoionization process assuming the specular reflexion of fast electrons at the solid target edges are not sufficient to recover the results obtained experimentally. Chapter 10 presents two applications of the model concerning the generation of shock waves by the fast electron energy deposition. The first one deals with the same experimental campaign considered in two previous chapters. It is shown that, in agreement with the target temperature evaluated with the M1 model, a UHI laser pulse with the energy less than 1 J can heat a solid target and generate temperature gradients that drive a blast wave of ≈ 50 Mbar. A theoretical model of such a blast wave generation and transport is proposed and compared with the experiment and hydrodynamic simulations. The second application concerns with the role 0.4. OBJECTIVES OF THE THESIS AND PLAN OF THE MANUSCRIPT of the fast electrons in the shock ignition scheme. The M1 model is coupled to the radiation hydrodynamic code CHIC using a simplified numerical scheme. The simulation results are in good agreement with theoretical predictions. It is shown that the hot electrons accelerated by the ignitor laser spike can deposit a sufficient energy in the dense shell and generate a 300 Mbar shock required to reach the ignition. The original results obtained in this thesis are summarized in the Conclusion. Perspectives of this present work are also proposed. Appendix A is dedicated to the classical plasma kinetic theory while Appendix B presents the classical hydrodynamic theory. It introduces the non relativistic kinetic approach as well as the Spitzer, Braginskii and Lee-More transport coefficients. The radiation hydrodynamic monofluid and two-temperatures CHIC code is also presented.

e

  ion collision frequency with ν 0 = (1/3) 2/πZn e e 4 ln Λ ei /( √ m e (k B ) 3/2 ), Z the mean ionization state of the material and κ e = κ 0 T Ze 4 ln Λ ei is the Spitzer-Härm transport coefficient (see Appendix B, section B.2.2). In all what follows, we neglect the advection term of the temperature in the left hand side of Equation (1.2 c).

Figure 1 . 1 :

 11 Figure 1.1: Inverse bremsstrahlung absorption coefficient η abs as a function of ν c L/c according to Equation (1.4) (left) and I L /I 0 according to Equation (1.5) (right)

1. 1 .

 1 LASER-SOLID INTERACTION AT HIGH INTENSITIES the density is between 0.88 n c and n c in the case of a low absorption (Lν c c). In the opposite case of an almost total absorption of the laser energy (Lν c c), more energy is absorbed in the lower density zone. According to the self-similar isothermal expansion, L scales with c S t ∝ T 1/2 e where c S = Zk B T e /m i is the sound velocity while ν c ∝ n c T -3/2 e

  I 0 = 4.8 10 11 W.cm -2 (1.06 µm/λ) 5 (2Z/A) 5/4 [Z (ln Λ c /6) (t/100 ps)] 3/2 . This laser energy absorption mechanism called Inverse Bremsstrahlung Absorption (IBA) is the main absorption mechanism for laser intensities below 10 15 W.cm -2 used in the conventional ICF schemes. The implicit Equation (1.5) can be solved numerically and the resulting IBA absorption coeeficient is plotted in the right panel of Figure 1.1. η abs decreases as the intensity increases and is larger for shorter wavelengths.

Figure 1 . 2 :

 12 Figure 1.2: (Left panel) Parameters domain in the (ν ei , ω) plane where occurs each absorption mechanism during the LSI with a laser pulse and a steep density gradient; α = 1 + (π/2) 1/4 /2 ± 1 -(π/2) 1/4 /2 and β = (1/2) 1/3 + (π/2) 1/6 /2 ± (1/2) 1/3 -(π/2) 1/6 /2); the signs ± come from the incertitudes on frontiers between the different regimes, evaluated according to the conditions of validity of each mechanism. (Right panel) Plot of the corresponding absorption coefficients as a function of the laser pulse intensity I L evaluated for Copper with λ = 1 µm at t = 10 fs and t = 1 ps after the beginning of the LSI according to Equations (1.9), (1.10),(1.11) and (1.12) (right)

3 ω pe ω 2

 32 The domains in the plane (ω pe , ν ei ), where each absorption mechanism occurs, are shown in the left panel of Figure 1.2. These absorption mechanisms lead to the target isochoric heating over the skin-depth L s . In order to evaluate the absorption coefficient, one has first to relate the electron 1.1. LASER-SOLID INTERACTION AT HIGH INTENSITIES temperature and the laser intensity according to Equation (1.2 c). At the LSI interface, the energy flux conservation -κ e (∂T e )/∂x) = η abs I L imposes the boundary condition and ensures a unique solution.

  n i is the initial atomic density. This expression is implicit since the absorption coefficient depends on the temperature. However, by injecting this expression in the absorption coefficients, one can deduce the explicit expressions of the absorption coefficient for the CA, the NSE, the ASE and the SBI mechanisms assuming for simplicity ln Λ ei = 4. It reads respectively η abs = 2.64 10 -2 Z

Figure 1 . 3 :

 13 Figure 1.3: LSI resonant absorption scheme inspired by [Gibbon, 2005] (left) and resonant absorption coefficient of a p-polarized laser pulse η abs as a function of the incidence angle θ (right)

  noticing that the curl of Equation (1.18) provides -(m e /e) (∂/∂t) (∂/∂r) × δv ∂r) × E = -(1/c)(∂B/∂t) and consequently B = (m e c/e)(∂/∂r) × δv

  Stokes configuration). The Stokes decomposition is unstable and leads to an energy transfer from the laser pulse into the two daughter waves. According to the hydrodynamic equations (1.2), the laser electromagnetic waves scatters in the expanding corona (see Figure8), off the density perturbations due to the excitation of Electron Plasma Waves (EPWs) or Ion Acoustic Waves (IAWs) at the frequency ω 1 . An electron current is generating the electromagnetic fields at the frequency ω 2 .It will be resonant if it corresponds to the electromagnetic dispersion relation. Then, as illustrated in Figure 1.4, this electromagnetic field perturbation coupled with the laser pulse electromagnetic field drives a ponderomotive force which resonantly amplifies the density perturbation if ω 1 = ω -ω 2 Page 53 1.1. LASER-SOLID INTERACTION AT HIGH INTENSITIES corresponds to the dispersion relation of the longitudinal plasma waves (EPWs or IAWs). These nonlinear processes are important because they can scatter the laser pulse and prevent it from reaching the absorption zone. In addition, part of the laser energy is transferred to EPWs, which can then transfer part of their energy into fast electrons. Also, the stimulated Brillouin scattering as well as the stimulated Raman scattering can lead to an energy transfer into the Daughter Electromagnetic Wave (DEW) propagating in the backward direction, which can destroy the laser optics.

Figure 1 . 4 :

 14 Figure 1.4: Parametric instabilities scheme

1. 1 . 4 Figure 1 . 5 :

 1415 Figure 1.5: Solutions of (1.22) plotted in the phase-space (V = dX/dt as a function of X) for K = 3eE 0 /k (1) and (6), K = eE 0 /m e k (2) and (5), K = eE 0 /4m e k (3), K = -0.92eE 0 /m e k (4).

  sin φ T where φ = 2π (ct -x) /λ is the wave phase, λ is the laser wavelength, f (t) is a slowly varying laser pulse temporal envelope and δ is a polarization parameter such that δ ∈ {-1, 0, 1} corresponds to a linearly polarized wave and δ = ±1/ √ 2 corresponds to a circular wave. Using the relations E = -∂A/∂t and B = c(∂/∂r) × A as well as v × [(∂/∂r) × A] = [(∂/∂r) ⊗ A].v -[v.(∂/∂r)] (A), the motion of an electron in the presence of this electromagnetic field is given by the Lorentz equation

  p = γm e v is the electron momentum, γ = 1 + (p/m e c) 2 its Lorentz factor and v its velocity. The perpendicular component of Equation (1.24) reads dp ⊥ a function of the phase φ, only. Thus, on one hand, γ = 1 + (p x /m e c) while in the other hand, γ = 1 + (p x /m c ) 2 + (p ⊥ /m e c) 2 ), by definition. These two relations give consequently dφ/dt = (∂φ/∂t)+v x (∂φ/∂x) = ω/γ and by changing the variable p = γm e (dr/dt) = γm e (dφ/dt)(dr/dφ) = (2πm e c/λ)(dr/dφ), the electron trajectory in the laboratory frame can be integrated starting from (1.26) and (1.27) assuming it is initially at the origin and neglecting the slowly varying terms. It reads

  Figure 1.6: Plot of x vs t (top left), y vs t (top right), y vs x (down left) and y 0 vs x 0 (down right) for different values of the laser intensity with f (t) = 1 and δ = 1 corresponding to a 0 = 0.27, 0.85 and 2.70 according to Equations (1.28) and (1.30).

  is monoenergetic and anisotropic. Indeed, due to the nonlinear relation between the single electron velocity and momentum p = γm e v, the relation p e = γ e m e v e is valid only if all electrons have the same velocity v e and consequently the same mass γm e = γ e m e . This explains why Equation (1.2 c) for the electron temperature is replaced by Equation (1.32 c) with zero electron temperature. By working with the laser vector potential A and the electrostatic potential Φ and by noticing that 2γ e (m e c) 2 (∂/∂r)(γ e -1) = (m e c) 2 (∂/∂r)(γ 2 e ) = (∂/∂r)(p e .p e ) = 2[p e .(∂/∂r)](p e ) + 2p e × [∂/∂r) × Page 59 1.2. LASER SOLID INTERACTION AT ULTRA-HIGH INTENSITIES p e ] ⇒ [v e .(∂/∂r)](p e ) = (∂/∂r)(γ e -1)m e c 2 -v e × [(∂/∂r) × p e ], one can rewrite (1.32 b) as ∂ ∂t (p e -eA) = -∂ ∂r (γ e -1) m e c 2 + ∂ ∂r (eΦ) + v e × ∂ ∂r × (p e -eA) .Thus, by taking the curl of this equation, one gets

Figure 1 . 7 :

 17 Figure 1.7: Schematic view of a UHI LSI in a steep gradient density

  is the target depth where the electrons have been displaced by the laser pulse ponderomotive force. These three parameters ϕ, x m and x d can be found by connecting the electron density n e and the field A 0 at the depletion edge x = x d (see Figure 1.7). In the region x < x d , the field behaves as in a vacuum. Assuming that the standing wave results from the superposition of two progressive waves of amplitude A L (the laser energy absorption is neglected), one obtains the lower line of Equation (1.43). The upper line of Equation (1.43) comes from the resolution of the system of equations {(1.34), (1.36)} in the region x > x d . In the limit of a very dense solid n e n c , this expression reduces to A 0 ∝ exp [-(x -x m )/L s ] with L s = c/ω pe being the skin-depth. The linear polarization case A = A L cos (ωt)e ⊥ is more complicated because one cannot find the analytical expression for γ e as (1.45) obtained for the circular polarization case 1.2. LASER SOLID INTERACTION AT ULTRA-HIGH INTENSITIES

Figure 1 . 8 :

 18 Figure 1.8: Schematic view of a UHI p-polarized laser pulse incident on a steep gradient density

  ky sin θ is a constant phase parameter. The Brunel heating can be explained as follows 1. In the first half of the laser cycle, between t = -(T /2) -ϕ/ω and t = -ϕ/ω, the cosine of (1.48 a) is negative. Thus, a sheath of electrons at the plasma interface (x = 0 -) is experiencing the longitudinal electric field (1.48 a) and is pulled out in vacuum up to a distance ≈ ∆x according to Equations (1.48 c) and (1.48 d). 1.2. LASER SOLID INTERACTION AT ULTRA-HIGH INTENSITIES 2. According to Equation (1.48 b), an electrostatic field is created in this zone [-∆x, 0]. 3. During the following half of the laser cycle, the cosine of (1.48 a) is positive and the electrons in this zone are consequently recalled back but this times with an amplification due to the electric force (∂/∂x)(eΦ), according to Equations (1.48 c) and (1.48 d).4. The electrons are reinjected in the overdense plasma where the local electromagnetic fields vanish beyond the skin depth L s according toEquation (1.48 a)

Figure 1 . 10 :

 110 Figure 1.10: Distribution of longitudinal electron momenta normalized by m e c in space at t = 110(2πc/λ) [Yu et al., 2000] obtained from a 1D relativistic kinetic simulation of a linearly polarized, Gaussian laser pulse with a 1 = 7.5, λ 1 = 1 µm, and width L 1 = 15 µm normally incident on a foil of density n e = 10n c (dashed lines) with (c) or without (d) the presence of a Deuterium preplasma of width 30 µm and density n e = 10 -3 n c . The time averaged laser electric field normalized by m e cλ 1 /2πc is also plotted in (c) (solid line).

Following

  standard perturbation techniques, one can solve the Hamilton equations by plugging the unperturbed motion (θ = (P ⊥ /m e c)ω 1 τ , ϕ = (P /m e c)ω 1 τ and φ = -(E/m e c 2 )ω 1 τ where τ is the proper time) in the argument of the perturbating cosines of the right hand side of Equation (1.56).

Figure 1 .

 1 Figure 1.10 (c). For the authors, this strong backward acceleration is therefore attributed to the

  potential and the Poisson equation for the scalar potential providing the electromagnetic fields. At the next time step, the fields are reinterpolated at the positions of the macroparticles to compute their trajectory at the following next time step. (E n (r l ), B n (r l )) =

(

  see section 1.2) require the use of numerical Particle-In-Cell kinetic simulations. One can briefly 1.4. BRIEF SUMMARY OF LASER SOLID INTERATION AND LASER-GENERATED RELATIVISITIC ELECTRON BEAM PROPERTIES

2. 1

 1 Electric Field Generated by a Monoenergetic, Collimated and Homogeneous Semi-infinite Relativistic Electron Beam Propagating in Vacuum

Figure 2 . 1 :

 21 Figure 2.1: Schematic of a semi-infinite -∞, O , axisymmetric, uniform, monoenergetic and collimated electron beam of radius r b moving at the velocity v b = v b e z compared to the laboratory frame (O, x, y, z, t) and its associated beam rest frame (O , x , y , z , t ) such that O ≡ O at t = 0.

(

  B = c(∂/∂r ) × A and E = -(∂Φ /∂r ) -(∂A /∂t ) ) with the Coulomb gauge ( (∂/∂r ).A = 0 ), n b = n b /γ b the beam charge density and j b = 0 the beam current density. According to the least action principle, the constants of motion for an electron in the beam volume are the Hamiltonian H , the canonical axial momentum P z and the angular momentum p θ which are given by

  2.7) where λ b = c/ω b = √ γ b λ b is the beam skin depth in the beam rest frame and ω b = 4πn b0 e 2 /m e = ω b / √ γ b its natural plasma frequency. Even if one can find solutions of Equation (2.7) in each separated subspace (z > 0), r > r b , z ≤ 0 and r ≤ r b , z ≤ 0 , it is impossible to find an analytical solution valid in the whole space by connecting continuously these electrostatic potentials and their spatial derivatives (the self-electric field) found in these three subspaces. Besides, one can estimate the longitudinal electric field by neglecting the radial spatial variation of the potential compared to the longitudinal one close to the z -axis and the vacuum-beam interface at z = 0. Equation (2the longitudinal self-electric field E z = -(∂Φ /∂z ) vanishes inside the beam front over approximately the beam skin depth λ b . According to the Lorentz transformation,

d 3 p

 3 is added to ensure the conservation of the number of conducting electrons. Assuming the latter are not relativistic in the laboratory frame, one has γ = γ M = γ b . Consequently, d 3 r d 3 p = (d 3 r/γ b )(d 3 p/γ b ) and one has

Figure 2 . 3 :

 23 Figure 2.3: Integration contour used for the k z -integration of (2.28) when z -z 0 > 0 (left) and when z -z 0 < 0 (right )

Figure 2 . 3 . 2 and k 4

 2324 Forz -z 0 < 0, it can be performed along the contour defined by the real axis including the pole k z = k 1 , k completed by a half-circle in the lower half k z plane such as represented in the right panel of

Figure 2 . 3 .

 23 Figure 2.3. Finally, the z 0 and k ⊥ -integrations can be performed exactly in the limit |z | λ e and

  .45) Contrary to the self-consistent electric field (2.11), this electric field (2.45) is positive. Its maximal value is close to 4πen b0 v b /ν E which has the same order of magnitude as E 0 and can consequently lead to the tunnel ionization of the neutral atoms. Moreover, by coupling the Maxwell-Gauss equation (2.44) with the charge conservation equation -e(∂/∂t)(n e + n b -n i ) = (∂/∂z)(j e,z + j b ), one gets

Figure 2 . 4 .

 24 As the current I b increases, the trajectory passes through the beam axis at a greater angle (trajectory b) until I b = I A for which the particle passes through the axis perpendicular to it (trajectory c). If I b is increased still further, the net particle motion is backward, as shown by orbit e and the extreme case of orbit f. It means that the electron beam propagation is stopped due to the action of the self-generated magnetic field on the beam electrons.

Figure 2 . 4 :

 24 Figure 2.4: Trajectories of beam electrons starting in the z-direction at various distance from theaxis of a uniform and electrically neutralized relativistic electron beam. Solid (dashed) curves represent particle trajectories with net motion forward (inward) taken from[Alfvén, 1939]

  .51) Concerning laser-generated electron beams, the typical current is greater than I A by many orders of magnitude while experiments show clearly the signature of the beam propagation deep inside the irradiated target. Moreover, tfor laser-generated electron beam, we have typically r b λ b , which is the opposite to the condition for which the solution (2.50) was obtained. In the case where r b λ b ,

Figure 3 . 1 :

 31 Figure 3.1: Radial profile of beam current density j b , return current density j e,z (j c,z with the Fills notations), net current density j n,z (j n ), azimuthal magnetic field B θ (B ϕ ), and radial current density j e,r = j n,r (j c,r ) at peak of a pulse t = τ L . The beam current is assumed to have a Gaussian radial profile. The various quantities are normalized with respect to the beam current density amplitude j b0 . In addition, the following scaling factors are used: τ L /τ d for j n,z , 2πr b τ L /τ d c for B θ . The radial current has the same spatial profile as the B-field. Its scaling factor is r b /τ d c. The return current is drawn for a ratio τ d /τ L = 100[Fill, 2001]

  Finally, by using Ohm's law(3.7) and by neglecting the pressure term, the electric field can be deduced knowing the return current j e . The profiles of the current and the magnetic field are shown in Figure3.1. The small factor τ L /τ d in (3.15) explains why the magnetic curl in the Maxwell-Ampere equation(3.6) is usually neglected and why the return current density is usually considered as the exact opposite to the beam current density j e = -j b . However, such an "initial situation" j e,z = -j b is extremely unstable. One has to solve self-consistently Equation (3.7) for the background electrons and the relativistic kinetic equation for the fast electron beam to model correctly the laser-generated electron beam transport.

Figure 3 . 2 :

 32 Figure 3.2: Typical distribution functions subject to the Weibel and the Filamentation instabilities [Bret et al., 2010b].

  m e is the electron plasma frequency of species j and δ αβ is the Kronecker symbol. This dispersion equation has two main branches. The first one, defined by Equation (3.18 a), pertains to modes with an electric field along the x-axis. Such modes are therefore purely transverse for any k such that k.e x = 0. The second branch (3.18 b) defines modes with an electric field lying within the (y, z) plane, which contains longitudinal and transverse components. When considering wave-vectors k in the flow direction such that k y = 0, the off-diagonal term yz vanishes and (3.18 b) reduces to yy ω 2 -

  .21) which corresponds to two perfectly symmetric counterpropagating electron beams which is not our case since n b /n e 1 (and therefore v b v e ). The domain of preponderance of each instability class has been numerically computed in the (n b , n e , T b ) parameters space by [Bret et al., 2010b] for a fixed plasma hot temperature k B T e = 5 keV. The surfaces that delimit regions governed by different instability classes are displayed in Figure 3.3 and coloured according to the local maximum growth rate in the k-space. The two-stream instability prevails for non relativistic beam drift energies (γ b -1 1) as well as in the weakly relativistic systems with hot enough beams. Filamentation modes govern systems where the beam and plasma densities are similar. Oblique modes are dominant for our case of laser-generated electron beam propagation in a dense plasma. Also, oblique modes dominate for hot enough relativistic beams. These results are illustrated by the lower panels of Figure 3.3 showing the plasma density observed in 2D PIC simulations each ruled by a distinct intability class.

Figure 3 . 3 :

 33 Figure 3.3: (top) Hierarchy of the two-stream, oblique and filamentation modes in the (n b , n e , T b )parameter space for Maxwell-Juttner distribution functions. (bottom) Plasma density profiles at the end of the linear phase as predicted by 2D PIC simulations, each ruled by a specific instability class. Here, the plasma temperature is T e = 5 keV and the beam flows in the y-axis[START_REF] Robinson | Theory of fast electron transport for fast ignition[END_REF]

Figure 3 . 4 :

 34 Figure 3.4: 3D PIC simulation of a Maxwell-Juttner beam-plasma system with n b /n e = 0.1, γ b = 3, T b = 50 keV and T e = 5 keV. Isosurfaces of the beam (upper plots) and plasma (lower plots) density profiles at successive times. The beam flows rightward [Bret et al., 2010b]

Figure 3 . 5 :

 35 Figure 3.5: (Left panel) Growth rate of the two-stream instability (Maxwell-Juttner initial distribution function) normalized by ω e versus k z normalized by the plasma skin depth λ e = c/ω e with (red, black) and without (blue) electron-ion collisions for n e = 10 23 cm -3 , n b /n e = 0.01, T b = 100 keV, γ b = 3, T e = 1 keV, Z * = Z = 10 and a Coulomb logarithm (see Appendis A, section A.2.3) ln Λ ei = 2; (Right panel) Filamentation growth rate (Maxwell-Juttner initial distribution function) normalized by ω e versus k z normalized by the plasma skin depth λ e = c/ω e in collisionless (dashed lines) and collisional (solid lines) configurations for γ b = 5, T e = 10 keV, n b /n e = 0.1, ν ei = 0.5ω p and increasing transverse temperature T b⊥ = 0.5 keV (blue), T b⊥ = 0.5 keV (magenta) andT b⊥ = 0.5 keV (red). Both plots are taken from[START_REF] Robinson | Theory of fast electron transport for fast ignition[END_REF]

  is the derivative of the Fried-Comte function F and β T h,b = 2k B T b /γ b m e c 2 is the thermal beam velocity normalized by c. The growth rate δ of the resistive filamentation instability can be deduced by looking for the solution of the dispersion relation (3.40) in the form ω = iδ. It is plotted in the left panel of Figure 3.6. The growth rate δ increases when increasing the plasma resistivity η 0 . For

Figure 3 .

 3 Figure 3.6: (Left panel) Growth rate of the resistive filamentation instability Ω normalized by the beam plasma frequency as a function of the wave vector in the transverse direction k normalized by the beam skin depth c/ω b for β T h,b /β b = 0.001 (solid), 0.01 (dotted) and 0.1 (circles). The beam density is n b = 10 20 cm -3 , the beam Lorentz factor is γ b = 2, and the target resistivity is η 0 = 10 -6 Ω.m; (Right panel) 3D Isosurface associated to j b = 4.8 10 11 A.cm -2 of a Gaussian-shaped beam of monoenergetic 0.5 MeV electron beam penetrating a 0.5-eV silica solid at normal incidence. The beam current density is plotted at time t = 405 fs from a 3D PaRIS hybrid PIC simulation. Both plots are taken from[START_REF] Gremillet | Filamented transport of laser-generated relativistic electrons penetrating a solid target[END_REF].

  when the magnetic diffusion is faster that the pinching of the perturbation, the instability growth rate is saturated at δ = ω b β b / √ γ b . 3.3 Temperature Dependence of Fast Electrons Transport 3.3.1 Temperature Dependence of the Background Electrical Resistivity and Ionization State

Figure 3 .

 3 Figure 3.7: (Left panel) Electrical resistivity of Beryllium with ρ = 1.84 g.cm -3 taken from [Robinson et al., 2014] versus the temperature (cyan solid curve) compared with numerical results (black solid curve). The hot Spitzer resistivity (red) with (solid line) and without (dashed line) the electron-electron collisions correction factor (see Appendix B, section B.2.2) is also plotted with the cold melting model (dashed cyan line). (Right panel) Electrical conductivity (σ = 1/η) taken from [Lambert et al., 2011] obtained numerically for Hydrogen plasmas with different densities as a function of temperature compared with Hubbard-Spitzer, Lee-More, and Ichimaru models.

Figure 3 . 8 :

 38 Figure 3.8: Electrical resistivity η for Aluminum (black) and Copper (red) as a function of the electron temperature plotted in two particular cases : T i = 300K (dashed curves) and T i = T e (solid curves) and for Hydrogen (ρ = 50 g.cm -3 ) with T i = T e (solid blue curve).

  model and such a computation, solving the the Kubo-Greenwood equation[START_REF] Maclellan | Annular Fast Electron Transport in Silicon Arising from Low-Temperature Resistivity[END_REF].For a same material,[START_REF] Mckenna | Effect of Lattice Structure on Energetic Electron Transport in Solids Irradiated by Ultraintense Laser Pulses[END_REF] found strong differences between the electrical resistivity of amorphous and diamond Carbon, as illustrated in the right panel of Figure3.9.

Figure 3 .

 3 Figure 3.9: (Left panel) Theoretical calculations of the resistivity of silicon as a function of temperature: The black line represents ab initio quantum molecular dynamic calculations coupled with the Kubo-Greenwood equation, the red line the Lee-More model, and the blue line the Spitzer model [MacLellan et al., 2013]. (Right panel) Structure of (a) diamond and (b) vitreous carbon. (c) Electrical conductivity as a function of temperature for both carbon allotropes computed with a quantum molecular dynamic code [McKenna et al., 2011].

Figure 3 .

 3 Figure 3.10: Magnetic field B θ normalized by j 0 τ 0 /c versus the radius r normalized by r b at differenttimes τ /τ 0 taken from[Davies, 2003]

  considered the Spitzer resistivity (α = -3/2) and a fast electron beam with a mean kinetic energy k B T b and a current I b , that is related to the laser intensity I L as I b = |e|η L→e I L where η L→e is the laser-to-electrons conversion efficiency. Approximating dj b0 /dr ≈ -j 0 /r b where j 0 = I b /k B T b , they conclude that the beam collimation occurs if θ 1/2 tan θ 1/2 = 3|e|n e k B T b γm e cβη L→e I L

Figure 3 .

 3 Figure 3.11: Magnetic field B θ normalized by j 0 τ 0 /c versus the radius r normalized by r b at differenttimes τ /τ 0 taken from[Davies, 2003]

  .[START_REF]Laser Plasma Institute (ILP) conference[END_REF] starting from the momentum 4-vector invariance and using the momentum and energy conservation relations. It implies that α /dΩ * ). By using the fundamental principle of Einstein's mechanics for the effective particle motion in the Coulomb central force, one can relate the collision impact parameter b and the normalized energy loss w by the following formula w = 2e 4 m e v 2 1 (γ -1)m e c 2 1 b 2 (4.6) As a consequence, similarly to the non relativistic case (see Appendix A, section A.2.1 and A.2.3), one has to distinguish between collisions with an impact parameter b e,min < b < b e,max for collisions with a small momentum transfer from the relativistic electron to a free plasma electron target where b e,min = 2p * (4.7) is the De Broglie wavelength of electrons in the center of mass frame and b e,max = max λ Debye ,

  .10)4.1.2 Electron-Ion Binary CollisionsLet us consider now a collision of a relativistic electron with a momentum p with a target ion with a charge Z * |e| and a momentum p i statisfying (4.1) with α = i. One can assume that the ion is initially at rest. Here, one consider only the "the binary part" where b i,min < b < b i,max . In this case, the laboratory frame coincides with the center of mass frame, since m e /m i 1. One has consequently b i,min = 2p(4.11) which is much larger than b e,min for ultra relativistic electron projectiles while b i,max = b e,max . (4.12)

  of a charged electron moving at constant velocity. It reads dε ds e,Bethe = (γ -1) m e c 2 n i wc 0 w dσ dw e,b dw = -2πn i Ze 4 m e v 2 ln 2w c γ 2 -1 (γ -1)

Figure 4 . 1 :

 41 Figure 4.1: Total stopping power of a relativistic electron in Aluminum (ρ = 2.7 g.cm -3 ) versus its kinetic energy according to formula (4.27) (black) and the different contributions due to free electrons (red), bound electrons (blue), screened free electrons/plasmons (green) and bremsstrahlung losses (magenta) at ambiant temperarure T e = T i = 300 K (solid lines) and T e = T i = 100 eV (dashed lines).

Figure 4 . 2 ,

 42 Figure 4.2, the formula (4.27) is also compared with the total stopping power provided by the online database ESTAR (http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html) based on the ICRU

Figure 4 . 2 :

 42 Figure 4.2: Total stopping power of a relativistic electron versus its kinetic energy according to formula (4.27) (solid line) compared to the ESTAR database (dashed lines) plotted for different materials : dense Hydrogen plasma (Z=1) with ρ = 10 g.cm -3 (magenta), amorpheous solid Carbon (Z=6) with ρ = 2.1 g.cm -3 (green), solid Aluminum (Z=13) with ρ = 2.7 g.cm -3 (blue), solid Copper (Z=29) with ρ = 8.96 g.cm -3 (red) and solid Tantalum (Z=73) with ρ = 16.69 g.cm -3 (black).

  .30) The (undeflected) range of an electron with the initial kinetic energy ε 0 = (γ 0 -1)m e c 2 is given by R = m e c 2 it by replacing f (γ) by a constant value f * in the integral. It reads, by noting (ln Λ rel ee ) * = ln (m e c 2 / ω p ) -(ln 2/2) + (1/16) + f * , according to [Atzeni et al., 2009b] R = (γ 0 -1) 2 γ 0 (m e c 2 ) 2 4πZn i e 4 (ln Λ rel ee ) * . (4.32)

  .50) THROUGH SOLIDS AND DENSE PLASMAS

5. 1

 1 Particle-In-Cell methods 5.1.1 Full Particle-in-Cell methods

Figure 5 . 1 :

 51 Figure 5.1: Schematic view of the discretization in the 6D phase-space of the full distribution function f = f b + f e (a) in a serie of discrete charge clouds with eventually different weights q i (b) according to the PIC method [Gibbon, 2005].

  to use the PIC method only for the laser-generated fast electron component f b of the full electrons distribution function by using a cut-off in the kinetic energy ε min = 1 -20 keV separating the fast electrons > ε min and the background electrons < ε min . In this "hybrid" method, the background electron dynamics are resolved according to hydrodynamic equations or simplified ones, such as introduced in Chapter 3, allowing to account for the response of background electrons via their transport coefficients η and κ e , their thermal capacities C V,e , the electron-ion equilibration Q ie which may vary from the solid state to the plasma state as well as eventual equation of states P e = P e (ρ, T e ) different from the ideal gas. This natural separation of the electrons into two interlinked populations assumes that the background plasma responds instantaneously to the fast electron beam transport in order to ensure the electromagnetic beam neutralization, introduced in Chapter 2, without modifying the beam. Thus, splitting of the populations is only valid when n b n e and when the fast electrons' mean kinetic energy k B T b is much greater than the mean thermal energy of the background electrons k B T e . The Ohmic approximation (approximating the hydrodynamic momentum conservation equation by Ohm's law) is usually acompanied by the neglect of the displacement current in the Maxwell-Ampère equation. Therefore, it does not allow for modeling of Langmuir plasma waves and/or perturbations of the quasineutrality. Binary collisions of beam electrons with background particles are taken into account in the electromagnetic particle pusher by adding a continuous slowing down term -ν d p and a Langevin term R representing a random rotation of p due to the angular scattering. The equation of motion of a macroelectron k ∈ |1, N p ] can therefore be written

Figure 5 . 2 :

 52 Figure 5.2: Spherical coordinates corresponding to Equation (5.4). Ω = p/p = [sin θ cos ϕ, sin θ sin ϕ, cos θ] T , e θ = [cos θ cos ϕ, cos θ sin ϕ, -sin θ] T and e ϕ = [-sin ϕ, cos ϕ, 0] T .

Figure 5 . 3 :

 53 Figure 5.3: Iso-surfaces of the first 10 spherical harmonics Y m l , 0 ≤ m ≤ l ≤ 3 multiplied by exp -p 2[START_REF] Tzoufras | A Vlasov-Fokker-Planck code for high energy density physics[END_REF]

Figure 5 . 4 :

 54 Figure 5.4: Qualitative summary of the benefits and drawbacks of different approaches[START_REF] Thomas | A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[END_REF]

"

  , from radiative transfer theory. It allows to close the set of equations by evaluating the 2nd order angular moment f 2 of the distribution function needed in the 1st order angular moment equation. Contrary to the largely used approximation of the distribution function with the two first Legendre polynomial P1, this M1 model allows to describe the distribution function with an arbitrary local anisotropy. My original decision to devote myself to science was a direct result of the discovery which has never ceased to fill me with enthusiasm since my early youth-the comprehension of the far from obvious fact

  It allows us to introduce the relativistic Coulomb logarithms, coming from the relativistic electron stopping powers, in the V-F-P equation. Moreover, it allows us to naturally relate the fast electrons angular scattering mean free path by colliding free, bound and screened free background electrons and background ions with the corresponding stopping powers. It has been shown in Chapter 5 that the best compromise between the accuracy and the numerical cost of a fast electron transport model can be obtained with a hybrid and expanded relativistic V-F-P method. Such a new reduced kinetic model, developed in this PhD, is presented. It consists in computing the two first angular moments of the distribution function, according to Equations (5.27) and(5.28) in order to make computations as fast as possible. However, in order to preserve the accuracy of calculations in case of strong anisotropy, a special closure relation based on the Minerbo maximum angular entropy approximation[Minerbo, 1977] [Minerbo, 1978], from radiative transfer theory, has been adapted. It allows to close the set of equations by evaluating the anisotropic part of the 2nd order angular moment f 2 of the distribution function needed in the 1st order angular moment equation (5.28). Contrary to the largely used approximation of the distribution function expanded on the two first Legendre polynomials, often called P1, this M1 model allows to describe the distribution function with an arbitrary local anisotropy. It is shown that the model is exact for fully isotropic and fully anisotropic local angular distribution functions. Furthermore, it relates both of these extrema in the expression of f 2 . The equation of the local angular entropy of fast electron beam is derived and the limitations of the model are discussed. Developments of new plasma transport coefficients necessary to model the self-generated electromagnetic fields are also proposed. 6.1 Kinetic Description of Relativistic Electron Beam Transport in Solids and Dense Plasmas 6.1.1 The Relativistic Vlasov-Belyaev-Budker Equation Applied to

  the relative velocity of one particle in the rest frame of the another, γα = γγ α /γ 2 * and γ * = 1/ (1 -v.v α /c 2 ) (cf. Chapter 4, section 4.4). In addition to the small angle scattering assumption (6.3), we make the assumption that the target particles α remain non relativistic after each binary collision with a relativistic beam electron. That is to say, we neglect high energy secondary electrons|p α | |∆p| m α c. (6.6)Under the assumption (6.3), the 2D binary collision problem consists in solving 6 unknown variables (the momenta and energy of each particle after the collision) while having 7 equations (1 energy conservation equation, 4 momenta conservation equations and the 2 Einstein relationships between energies and momenta). Consequently, there is 1 relation between 2 chosen free parameters which are for example the scattering angle θ and the relative energy loss of the relativistic electron w = ∆γ/ (γ -1) in the laboratory frame. So, one is free to work with dσ α /dw instead of dσ α /dΩ. Moreover, under the assumption (6.6), the energy conservation equation for one collision, ∆γ α m α c 2 + ∆γm e c 2 = 0, provides ∆p ⊥ 2 = -2m α ∆γm e c 2 . (6.7)This naturally leads to a relation between the stopping power of relativistic electrons due to collisions with the α particles and the scattering potential U α,0 . Knowing the differential cross section dσ α /dw, one can define the loss of electron kinetic energy ε = (γ -1)m e c 2 per unit path length ds as follows (cf. Chapter 4, section 4

  .11) Instead of momentum, it is more convenient to work with the kinetic energy ε of the relativistic electrons. Besides, the structure of the collision tensor motivates to use spherical coordinates (Ω, e θ , e φ )

3 Figure 6 . 1 :

 361 Figure 6.1: Expressions for the Coulomb Logarithms ln Λ rel eα and the densities n α of the stopping powers (6.16). Z * is the ionization state and ln Λ clas eα = max {2, ln (b max /b min,α )} is the "classical" Coulomb logarithm where b max = max {λ Debye , (3/4πn i ) 1/3 } is the upper impact parameter and b min,α = /(2m e c γ 2 -1) if α = ions (b min,α = /(2m e c (γ -1)/2) if α = free electrons) is the lower impact parameter.

Figure 6 . 2 :

 62 Figure 6.2: (Left panel) Total isotropization rate ν in Aluminum (ρ = 2.7 g.cm -3 ) versus its kinetic energy according to formula (6.17) (black) and the different contributions due to free electrons (red), bound electrons (blue), screened free electrons/plasmons (green) and ions (magenta) at ambient temperarure T e = T i = 300 K (solid lines) and T e = T i = 100 eV (dashed lines). (Right panel) Total isotropization rate ν in Hydrogen (ρ = 10 g.cm -3and T = 100 eV) versus its kinetic energy according to formula (6.17) (black) and the separate contributions due to free electrons (red), bound electrons (blue), screened free electrons/plasmons (green) and ions (magenta).

  ) is also plotted for Hydrogen (ρ = 10 g.cm -3 and T = 100 eV) versus the electron kinetic energy in the right panel of Figure6.2 with the separate contributions of collisions with background free electrons, bound electrons, screened free electrons/plasmons and ions. It shows that for the Hydrogen plasmas, the scattering on both plasma ions and electrons provides comparable contributions. One also sees that, in the case of beam electrons with kinetic energies less than ≈ 100 keV, propagating in a degenerate Hydrogen plasma, collisions on screened free electrons/plasmons provide the main contribution to the beam electrons angular scattering. The linearization of the Belyaev-Budker collision tensor has allowed us to determine a relation between the stopping power on α particles (dε/ds) α and the corresponding isotropization rate ν α = v/λ α,lpm where λ α,lpm = 1/k 1,α is the angular scattering mean free path. This relation readsν α = m α γm e

  and we neglect the displacement current in the Maxwell-Ampere equation ∂ ∂r × B = 4π c (j e + j b ) , (6.20) considering times greater than the beam electromagnetic neutralization time (see Chapter 2). The plasma dynamics is taken into account by the generalized Ohm equation (3.4) E = ηj e -1 n e e ∂ ∂r (n e k B T e ) . (6.21)Considering a space scale larger than the plasma skin depth λ e = c/ω pe , which is typically less than a fraction of microns, the electron inertia has been neglected in (6.21). It has been assumed an isotropic resistivity tensor η = ηI (no magnetization effects) and the ideal gas expression for the electron pressure P e = n e k B T e . As also discussed in Chapter 3, section 3.1.1, the electron viscosity, the thermal force, the magnetic force and the friction force due to collisions with beam electrons have been neglected compared to the friction force by colliding with background particles. In order to account for the induced electric field, one has to add the Maxwell-Faraday equation (3

S 2 d 2 Ω

 22 (vS col Ψ) + ηj e 2 (6.26) is evaluated by calculating the direct energy loss of the beam electrons in collisions with the background electrons according to (6.15) and the friction of background electrons on background particles -R ei .v e = ηj 2 e . Thus, S col reads S col = -dε ds free e - is the electron-ion equilibration coupling factor. The ion temperature is evaluated fromEquation (3.53)

2 d 2

 22 30) is negligible compared to W e as well as the divergence of the Poynting vector Π = E × B/4πc. The term W brem = -dε S bremsstrahlung power losses of beam electrons and W b is the full power lost by the fast electron beam. For low Z material, we have shown in Chapter 4, section 4.2.1 that bremsstrahlung losses of beam electrons are negligible compared to their collisional losses. As the transport of bremsstralung photons is not taken into account in our model, we make an implicit assumption that photons deposit directly their energy in the material and therefore include W brem in W e . However, for intermediate Z materials and/or depending on the material opacity, this assumption could not be sufficient and W brem is considered separately.6.2 M1 Model for Relativistic Electron Beam Transport6.2.1 Spherical Harmonic and Cartesian Tensor Scalar Product Expansions

Figure 6 . 3 :

 63 Figure 6.3: Closure parameter µ as a function of the anisotropic parameter |Ω ε | = |Ψ 1 |/Ψ 0 (solid blue curve) plotted within the approximation (6.59). The dots are the exact values of µ for some values of |Ω ε |.

  .59) It is compared with the real value of α 1 for some values of |Ω ε | in Figure 6.3, where the dependence of the anisotropy vector Ω ε on the closure parameter µ is plotted. Consequently, one deduces an explicit expression for the approximate beam distribution function Ψ M1 = Ψ 0 |α 1 | 4π sinh |α 1 | exp (α 1 .Ω). (6.60) In the isotropic case where |α 1 | 1 (|Ω ε | 1), the M1 model reduces to the one-polynomial approximation P1. But, in the opposite case of a strong anisotropy |α 1 | → ∞ (|Ω ε | → 1), the function

2 . 2 . 3 f

 223 This definition allows us to relate the local angular entropy with the Boltzmann entropy H[f b ] (see Appendix A, section A.1.5) by the simple relation H[f b ](r, t) = -R b (ln f b -1) d 3 p = φ = Ψ[1 -ln (v/p 2 )] -Ψ ln Ψ, the equation for the time derivative of the local angular entropy reads

Figure 6 . 4 :

 64 Figure 6.4: (Left panel) Ratio ξ versus the kinetic energy of electrons according to Equation (6.67) for dense Hydrogen (ρ = 10 g.cm -3 , Z = 1), Aluminum (ρ = 2.7 g.cm -3 , Z = 13), Copper (ρ = 8.96 g.cm -3 , Z = 29) and Tantalum (ρ = 16.65 g.cm -3 , Z = 73) at T = 100 eV. (Right panel) Comparison between the M1 approximation (8.23) (full curves) and the statistic normal law (dashed curves) for ∆ϕ = 180 o (green), ∆ϕ = 90 o (blue), 45 o (black) and 20 o (red).

Figure 6 . 5 :

 65 Figure 6.5: Comparison between the resistive filamentation growth rate in the M1 approximation (full curves) and the resistive filamentation growth rate (3.41) (dashed curves) for η 0 = 10 -6 Ω.m, T b → 0, n (0) b = 10 21 cm -3 (ω b = 1.78 fs -1 ) and different values of γ b = 1.5 (green), 3 (blue), 7 (black) and 14 (red).

  0) b e 2 /m e is the electron beam plasma frequency. Injecting (6.85) in (3.36 a), we obtain the dispersion relation for the unstable mode ω = iδ

  over ε leads to a different result. The instability growth rates δ > 0, solutions of (6.86) and(3.41), are plotted in Figure 6.5 for η 0 = 10 -6 Ω.m, n (0) b = 10 21 cm -3 (ω b = 1.78 fs -1 ) and different values of γ b . The M1 approximation largely overestimates the filamentation resistivity growth rate in the short wavelength region k x c/ω b 1 for low mean electron energies γ b 1. Indeed, according to (6.86), the growth rate obtained in the larger electron energies, above γ b ≈ 3 ( ε ≈ 1 MeV), the M1 model provides the solution with an error less than ≈ 15%. For γ b 7 ( ε ≈ 3 MeV), the M1 model makes an error less than ≈ 3%. This example partially confirms the argument given by Minerbo, and quoted in the introduction of this section, concerning the justification of the M1 closure (6.61) when neglecting collisions of beam electrons with background particles.

6. 3 .

 3 Figure 6.6: Ionization state given by (6.93) (red) compared to the More formula (3.43) (blue) for Copper (Left panel) and Tantalum (Right panel).

a 2 =

 2 35 and a 3 = 1.925 for Copper and Z c = 2, a 1 = 2, a 2 = 20 and a 3 = 1.25 for Tantalum provide good estimates for the charge state, as illustrated in Figure 6.6. By taking Z * = Z c at the solid temperature instead of Z * ≈ 0 as [Desjarlais, 2001], we empirically account for the collisions between free electrons and neutral atoms in the calculation of transport coefficients η and κ e by taking into account collisions between free electrons and phonons (see the next section 6.3.2).

Figure 6

 6 Figure 6.7: (Left panel) Electron thermal capacity (6.100) for Tantalum (solid black curve) and comparison between the electron thermal capacity (6.100) (full curve) and the results by [Lin et al., 2008] (dashed curves) for Aluminum (blue) and Copper (red). (Right panel) Ion thermal capacity (6.105) for Tantalum (black), Copper (red) and Aluminum (blue)

  97) is the Fermi-Dirac distribution function depending on the chemical potential µ and the Fermi energy E F (3.42), expressed with the electron density n e = Z * n i and g(ε) is the electron Density of States(DOS). By assuming that g(ε) ≈ g(E F ) and by approximating the DOS g by the free electron gas expression

  .106) It is plotted in the right panel of Figure 6.7 for Aluminum, Copper and Tantalum.

Figure 6 . 8 :

 68 Figure 6.8: Electrical resistivity (6.107) (Left panel) and thermal electron conductivity (6.108) (Right panel) for Aluminum (blue), Copper (red) and Tantalum (black) at T i = 300 K (dashed curves) and at T i = T e (full curves).

2 . 4 )Figure 6 . 9 :

 2469 Figure 6.9: Contributions of the different electron collision frequencies in the electrical resistivity of Aluminum (6.107) with (6.111) (Z s = 3, Z d = 0). The electron frequency contributions are indicated in the Figure.

2 . 3 ,

 23 allowing to account for the electron degeneracy close to the Fermi temperature (see Appendix B, section B.2.4). At a low temperature, ν e is given by a sum of the electron-phonon collision frequency and the electron-electron collision frequency ν cold e = ν e-ph + ν e-e .

v

  e /ν e of the electrons cannot exceed the mean interionic distance v e /ν c in the intermediate range of temperatures where ν c = v e /(3/4πn i ) 1/3 , v e = 3k B (T F + T e )/m e and T F = E F /k B . In the model

Figure 6 . 10 :

 610 Figure 6.10: Contributions of the different electron collision frequencies in the electrical resistivity of Copper (6.107) with (6.111) (Z s = 1, Z d = 10). The electron frequency contributions are indicated in the Figure.

Figure 6 . 11 :

 611 Figure 6.11: Contributions of the different electron collision frequencies in the electrical resistivity of Tantalum (6.107) with (6.111) (Z s = 2, Z d = 3). The electron frequency contributions are indicated in the Figure.

  .128) Here, ε s , ε 1 , ε 2 are parameters determined by fitting the best as possible the real electron DOS of the metal while m s and m d are the effective mass of s-band and d-band electrons, respectively. Such effective mass approximation is common in Solid States Physics in order to model the interaction of

  130) where n s and n d are the densities of s-band electrons (n s = Z * n i ) and d-band electrons. An example is given for Tungsten in the right panel of Figure 6.12.

Figure 6 .

 6 Figure 6.12: (Left panel) Regions in plane qk 2 over which integrals I a1 , I a2 , I b and I d of (6.119) are taken. The heavy segment indicates the image of the Fermi surface at T e → 0 and k 1 = k F (Fermi wave number) [Inogamov and Petrov, 2010]. (Right panel) Example of fit (6.124) for Tungsten (solid black curve) compared to ab initio calculation done by [Lin et al., 2008] (solid red curve); the s-band electrons contribution g s in (6.124) (dashed green curve) is plotted with m s = 0.9m e (E F -ε s = -9.2 eV) and the d-band electrons contribution g d in (6.124) (dashed magenta curve) is plotted with m d = 1.7m e (E F -ε 1 = -6.6 eV and ε 2 = E F + 5.5 eV); the Fermi energy value is E F = 17.969 eV (dashed blue curve).

  Figure 6.13: Electron-ion coupling factor (6.137) for Aluminum (blue), Copper (red) and Tantalum (black) at T i = 300 K (dashed curves) and at T i = T e (full curves).

2 1

 2 Figure 6.9, Figure 6.10 and Figure 6.11), one can write τ e (T e , T i ) = 1 A e T 2 e + B i T i

  have linearized the Belyaev-Budker collision tensor by applying it to the study of laser-generated fast electron beam transport in solids or dense plasmas, assuming a small momentum transfer in Page 201 6.4. CONCLUSION a collision. The production of secondary electrons is neglected, assuming the residual energy of background electrons after a collision with a beam electron is smaller than the exchanged momentum ∆p of a consecutive collision with another beam electron. These assumptions allowed us to obtain a more simple Landau-like collision tensor. Moreover, it allows us to relate the angular scattering collision frequency by colliding background particles (free electrons, bound electrons, screened free electrons/plasmons or ion nuclei) with the corresponding stopping powers according to an Einsteinlike relation, similar to the one obtained for Brownian motion of particles. This allows us to obtain more accurate expressions compared to the angular scattering theories presented in Chapter 4, by retaining all terms in the Moller relativistic Coulomb logarithm instead of the relativistic generalization of Rutherford term, only. However, our model is limited to low density beams n b n e since the collisions of beam electrons with themselves and the production of secondary electrons are neglected.

Figure 7 . 1 :

 71 Figure 7.1: Spatial simulation box (Left panel) and Momentum Simulation box (Right panel)

. 2 )

 2 indexed by i for the position in the x-direction, ∀j ∈ [1, N y ], x j [µm] =for the position in the y-direction and ∀k ∈ [1, N z ], z k [µm] = ∆z 2 + (k -1)∆z where N z = E L z ∆z , (7.4) indexed by k for the position in the z-direction according to Figure 7.1. Finally, we discretize the

Figure 7 . 2 :

 72 Figure 7.2: Synoptic Diagram describing the code.

Figure 7 . 3 :

 73 Figure 7.3: Expression of u ξ,ζ in (7.32) depending on ξ and ζ.

  .64) describes the diffusion of the B-field component B ξ , ξ ∈ {x, y, z} in the plane perpendicular to the ξ-axis. The second line describes the coupling between the different B-field components, which cancels in two dimensions. The third line describes the B-field source term due to the curl of the beam density current while the fourth line describes the B-field source due to resistivity gradients. Finally, the fifth line describes the B-field source term due to the background electron temperature-density crossed gradients. The first term in the left hand side of the E-field equations (7.65), (7.66) and (7.67) provides the main contribution due to the beam density component j b in the return current expression j e = j t -j b . The second term comes from the total net current j t = (c/4π)(∂/∂r) × B according to the Maxwell-Ampere equation

  fast electrons' kinetic energy injected in the simulation box from t = 0 to t = t Nt (it also accounts for eventual escaping electrons at the irradiated side of the target z = 0), the total fast electrons' kinetic energy escaped from the simulation box from t = 0 to t = t Nt at z = L z (first line), x = -L y /2 (second line), y = L y /2 (third line), x = -L x /2 (fourth line) andx = L x /2 (fifth line),is the instantaneous beam kinetic energy at t = t Nt , electric energy in the simulation box at t = t Nt , magnetic energy in the simulation box at t = Nt , kinetic energy lost (gain) by the fast electrons (background electrons) by colliding the background electrons (by being collided by fast electrons), kinetic energy lost (gained) by the fast electrons (bremsstrahlung photons) by radiating bremsstrahlung photons (emitted by the fast electrons) and finally

Figure 8 . 1 :

 81 Figure 8.1: Instantaneous beam energy U b × 10 (solid black), integrated beam energy balanced between injected and escaping electrons at z = 0 U inc (solid red), escaped energy U out from the simulation box (dashed red), total collisional energy loss U col (solid magenta), total "collective" U res energy loss (solid blue), instantaneous electric energy U E × 10 8 (solid green) and instantaneous magnetic energy U B × 10 6 (solid cyan) from the M1 simulation according to the formula given in Chapter 7, section 7.4.1.

Figure 8 . 2 :

 82 Figure 8.2: Hydrogen temperature T [eV] at t = 3.5 ps from the simulation (a) and a comparison of the temperature T profiles at z = 0.5 µm from the simulation (solid black) and the estimate (8.2) (dashed red) at t = 3.5 ps (b).

Figure 8 . 3 :

 83 Figure 8.3: Magnetic field B y [T] at 3.5 ps from the simulation (a) and comparison between the magnetic field B y = B j,y + B η,y profile at z = 0.5 µm and t = 3.5 ps (solid black) where B y,j is the contribution due to the beam current density gradients (solid red) and B j,η the contribution due to the resistivity gradient (solid blue) from the simulation and their respective estimates (8.3) (dashed black), (8.4) (dashed red) and (8.5) (dashed blue) (b).

Figure 8 . 4 :

 84 Figure 8.4: Electron beam density n b [cm -3 ] at t = 1750 fs from the simulation. The electron trajectory (oscillating red curve) is arbitrary and is presented here to illustrate the definitions of the mean cosine of the angle between the z-axis and the position of the beam electron cos θ and the path length following the electron trajectory in the (r, ε) space s.

Figure 8 . 5 :

 85 Figure 8.5: Comparison between the simulation results at t = t 0 : arccos Ω max ε,z versus z where Ω max ε,z= Ω ε,z (x = 0, z, ε m , t 0 ) = max ε {Ω ε,z (x = 0, z, ε, t 0 )} (black curve), arccos Ω mean ε,z

  Figure 8.2 a) and Figure 8.4. The estimates (8.13) and (8.14) predict the mean position z and the mean diffusion angle arccos cos θ of ≈ 10 µm and ≈ 20 o respectively for p z /m e c = 2.5, ≈ 20 µm and ≈ 30 o for p z /m e c = 2 and ≈ 30 µm and ≈ 45 o for p z /m e c = 1.7. This is in agreement with the numerical results obtained for the electron beam distribution function in the M1 approximation (6.60) as illustrated in Figure 8.6, panels b), c)and d), respectively. Close to z = L p , the analytic estimates z and cos θ differ from the numerical results as shown in Figure8.5. This is due to the singularity at the penetration depth L p in this particular case of monoenergetic electron beam in (8.13) and (8.14) and to the fact that the last term in the right hand side of (8.10) cannot be neglected anymore when the local angular distribution is close to be isotropic.

Figure 8 . 6 :

 86 Figure 8.6: Electron beam distribution function f b [cm -3 .m e c -3 ] from the simulation on the z-axis at t = 1750 fs, p y = 0 and at different depth z = 1 µm (a), z = 10 µm (b), z = 20 µm (c) and z = 30 µm (d). The dashed red curves represent the analytical estimates of arccos cos θ (ε) evaluated at the kinetic energies ε corresponding to z (ε) = 10 µm (b), 20 µm (c) and 30 µm (d).

Figure 8

 8 Figure 8.7: (Left panel) Scheme of the thin solid target irradiated by the ultra-high intense laser pulse. (Right panel) 2D spatial map of the mean kinetic energy of the laser-generated fast electrons normalized by n c m e c 2 , at t = 168ω -1 0

  as illustrated in the right panel of Figure 8.7. The energy distribution of the laser-generated electron

Figure 8 . 8 :

 88 Figure 8.8: Propagation direction angle ϕ 0 (ε, t) (full curves) from PIC simulation and anisotropic parameter |Ω ε (ε, t)| (dashed curves) deduced from ∆ϕ(ε, t) for the different considered energy ranges 20-50 keV (black), 50-100 keV (blue), 100-500 keV (cyan), 500-1000 keV (magenta) and 1000-5000 keV (red).

Figure 8 . 9 :

 89 Figure 8.9: Instantaneous beam energy U b × 10 (solid black), integrated beam energy balanced between injected and escaping electrons at z = 0 U inc (solid red), escaped energy U out at z = 5 µm (dashed red), total collisional energy loss U col (solid magenta), total "collective" U res energy loss (solid blue), instantaneous electric energy U E × 10 3 (solid green) and instantaneous magnetic energy U B × 10 3 (solid cyan) from the M1 simulation according to the formula given in Chapter 7, section 7.4.1.

Figure 8 . 10 :

 810 Figure 8.10: Distribution function f b (p x , p z ) from the M1 simulation at t = 77 fs and different target depth : z = 0.125 µm (Up Left), z = 1.125 µm (Up Right), z = 2.125 µm (Down Left) and z = 3.125 µm (Down Right). Each plot corresponds to the transverse position x where the beam density n b is maximum at the given depth z.

Figure 8 .

 8 Figure 8.10 shows the evolution of the fast electron distribution function f b with the target depth. It is plotted in the plane (p x , p y ) since the M1 distribution function is locally axisymmetric around the mean propagation direction Ω ε . At z = 0.125 µm, we can distinguish between the different initial angular distributions, depending on the energy range (ε[keV] ∈ [20, 50], [50, 100], [100, 500], [500, 1000] or [1000, 5000]), as explained above. In the first cells of Copper located at z = 1.125 µm, the lower energy electrons are already almost fully isotropized while the more energetic ones retain their initial anisotropy. The angular spread rises with increasing depth z while the mean energy steadily decreases.

Figure 8 .

 8 Figure 8.12: Plasma electron temperature T e [eV] in a logarithmic scale (a), plasma electrical resistivity η [Ω.m] in a logarithmic scale (b), self-generated magnetic field B y [T] (c) and time integrated density of Kα photons emitted per steradian n Kα [cm -3 .sr -1 ] (d) from the M1 simulation at t = 500 fs.

Figure 8 .

 8 Figure 8.13: Slices at z = 0.375 µm of the resistivity (black curve), the electron temperature (red curve) and the ion temperature (blue curve) at t = 26.6 fs (Left panel) and t = 499.3 fs (Right panel).

Figure 8 . 14 :

 814 Figure 8.14: Comparison at t = 500 fs between the mean electron temperature < T e > over |x max -x| < 5 µm profile obtained with M1 (blue) and the one obtained with the hybrid PIC code Paris (green)[Gremillet, 2012]. x max is defined as the position where T e (x max , z) is the maximum electron temperature at a given depth z (red).

Figure 9 . 1 :

 91 Figure 9.1: Scheme illustrating the collisional ionization of a K-shell electron of a cold atom by a laser-generated fast electron (Left panel) and the recombination of the electron-generated hole by a L-shell electron of the cold atom, which results in the emission of a Kα photon (Right panel).

  with a fully Particle-In-Cell (PIC) simulation. The electron transport is described with the M1 model by taking into account the reflexing effect. The emission of Kα photons emitted by the Copper tracer layer located at different depths is calculated and compared with the experimental results.9.1 Computational Methods for Estimating the Emission of Kα Photons 9.1.1 Fast Electron Recirculation and M1 Model for Fast Electron Transport Fast Electron Recirculation in the M1 Model The Kα images obtained in the laser plasma interaction experiments are contaminated by the electrons refluxing off the target sides. Indeed, the propagation of fast electrons in vacuum is limited by the Alfven-Lawson limit I max ≈ -17γ b β b kA and the space charge electric field. Most of the beam electrons are thus strongly decelerated and then accelerated in the opposite direction by the space charge electric field at the target boundary in the beam Debye sheath λ D,b = k B T b /4πΓ b n b e 2 ≈ 7.43 10 -3 µm T b [keV]/Γ b n b [10 21 cm -3 ], which is less than one micron. Here, k B T b is the beam electrons temperature (and not their mean kinetic energy!) and Γ b = Γ(p b ) is the bulk Lorentz factor

Figure 9 . 2 :

 92 Figure 9.2: Schematic illustrating the M1 limitation in the case of two counterpropagating anisotropic and monoenergetic electron beam having the same energy.

Figure 9 . 3 :

 93 Figure 9.3: Schematic of the method used to account for the fast electrons refluxing at the laserirradiated side and/or the rear side of the target by solving two sets of M1 equations for two counterpropagating fast electron beams.

Figure 9 . 4 :

 94 Figure 9.4: Values of a K-shell electron ionization energy E K , energy of a Kα 1 , Kα 2 and Kβ photon hν Kα 1 , hν Kα 2 and hν Kβ , and ratios I Kα 2 /I Kα 1 and I Kβ /I Kα of the K-lines intensities depending on the atomic number Z of the material.

2 ,

 2 I Kα and I K β are the intensities of a single Kα or Kβ photon signal, respectively. Their values, depending on the atomic number Z, are given in Figure 9.4

Figure 8 .

 8 Figure 8.12 d) of Chapter 8, section 8.2. It can be directly compared with the experimental time-integrated Kα 1 signal.

Figure 9 . 5 :

 95 Figure 9.5: (Left panel) Collisional K-shell electron ionization cross section σ K on the electron kinetic energy ε for Aluminum (blue curve), Copper (red curve) and Tantalum (black curve) according to Equation (9.12); (Right panel) Dependence of the hole lifetime τ K = /Γ K on the atomic number Z according to Equation (9.23).

Figure 9 . 6 :

 96 Figure 9.6: Scheme of the different targets irradiated by the UHI100 laser pulses in the experimental campaign conducted by [SANTOS et al., 2013]; courtesy of J. Santos.

  3.3. We start with a 2D-3V M1 simulation of the Al(1 µm)Cu(3 µm)Al(1 µm) target without refluxing but with these solid state physics corrections. It can be compared with the reference M1 simulation presented in Chapter 8, section 8.2, computed with the Eidmann-Chimier model for η and κ e (see Chapter 3, section 3.3.1) and the hot plasma expressions for G, C V,e and C V,i . Moreover, we will confirm that the assumption 2 made in the reference simulation is not verified for Copper and Aluminum. Secondly, we present the M1 simulation results with refluxing for the Al(1 µm)Cu(3 µm)Al(1 µm) and Al(1 µm)Cu(3 µm)Al(15 µm) targets. It allows us to highlight the dependence of refluxing effects upon the target thickness. Thirdly, we present the 2D-3V M1 simulation results with refluxing for all targets, focusing on the emission of Kα 1 photons. The simulations reproduce qualitatively the experimental data. Finally, three-dimensional effects are analyzed by considering the the 3D-3V M1 simulation results with refluxing for the Al(1 µm)Cu(3 µm)Al(1 µm) target.

  photons and Kα photons interchangeably and we omit the emission of Kα 2 photons. The latter can be deduced from the Kα 1 emission and the contribution F Kα 2 to the K-shell fluorescence.

Figure 9 . 7 :

 97 Figure 9.7: Instantaneous beam energy U b × 10 (solid black), integrated beam energy balanced between injected and escaping electrons at z = 0 U inc (solid red), escaped energy U out at z = 5 µm (dashed red), total collisional energy loss U col (solid magenta), total "collective" U res energy loss (solid blue), instantaneous electric energy U E × 10 3 (solid green) and instantaneous magnetic energy U B × 10 3 (solid cyan).

Figure 9 . 8 :

 98 Figure 9.8: Slices at z = 0.375 µm of the resistivity (black curve), the electron temperature (red curve) and the ion temperature (blue curve) at t = 26.6 fs (Left panel) and t = 499.3 fs (Right panel).

  Figure 6.8, the electrical resistivity is higher in Aluminum than in Copper in the solid state regime with T i T e but it is much lower in the liquid/Warm Dense Matter (WDM) state. Since the Ohmic heating by the return current and the self-generated magnetic field generation strongly depend on the beam current and the electrical resisitivity, the temperature decreases at both the Al/Cu and Cu/Al interfaces. By contrast, the reference simulation shows an increase in electron temperature at the Cu/Al interface (see Figure 9.9). This is due to the Eidmann-Chimer model predicting a greater

Figure 9 . 9 :

 99 Figure 9.9: Comparison at t = 500 fs between the mean electron temperature < T e > over |x max -x| < 5 µm profile obtained with M1 (blue) and the one obtained with the hybrid PIC code Paris (green).x max is defined as the position where T e (x max , z) is the maximum electron temperature at a given depth z (red). Refluxing is neglected here.

  Figure 9.8 (to be compared with the right panel of Figure 8.13), we observe the decrease of the electrical resistivity in the hot plasma regime, typical of the beam hollowing effect. This different Ohmic heating by the return current in the first target layers is therefore due to the introduction of the parameter B ν that saturates the target electron-electron collision rate (6.113) in the liquid/WDM states. Thus, collisions of s-band electrons on d-band electrons may play a significant role in Copper. Due to the short time scale considered here, the target electron thermal conduction κ e does not play any role. Also, C V,i does not impact the results since it weakly varies from the solid state to the plasma state according to the Einstein model. Both simulations give approximately the same 2D distribution of the time-integrated number of emitted Kα photons, as illutrated in Figure 9.11 d) (to be compared with Figure 8.12 d)). Both densities of time integrated Al and Cu Kα 1 photons are noted n Kα even if they are computed from different formulae depending on the atomic number of the material. It confirms the analytical estimates obtained for Copper and Aluminum, showing that the hole density dynamics plays an important rolein the Kα emission in cases where ∆t n is comparable to τ K . Indeed, we obtain a maximum number of emitted of Kα photons about 3 10 18 cm -3 .sr -1 from the Copper layer, which is four times less than the value of 12 10 18 cm -3 .sr -1 , obtained in the reference case. In what follows, we will only discuss the Kα 1 and Kβ emissions from the Copper layer.

Figure 9 . 10 :

 910 Figure 9.10: Electron beam density n b [cm -3 ] from the M1 simulation at t = 25 fs (a), t = 50 fs (b), t = 75 fs (c), and t = 100 fs (d). Refluxing is neglected here.

Figure 9 . 11 :

 911 Figure 9.11: Plasma electron temperature T e [eV] in a logarithmic scale (a), plasma electrical resistivity η [Ω.m] in a logarithmic scale (b), self-generated magnetic field B y [T] (c) and the number of emitted Kα photons n Kα [cm -3 .sr -1 ] (d) from the M1 simulation at t = 500 fs. Refluxing is neglected here.

  Figure 9.12: (Left panel) Al(1 µm)Cu(3 µm)Al(1 µm) target; (Right panel) Al(1 µm)Cu(3 µm)Al(15 µm) target; Instantaneous beam energy U b (solid black), integrated beam energy injected at z = 0 U inc (solid red), total collisional energy loss U col (solid magenta), total "collective" energy loss U res (solid blue), instantaneous electric energy U E × 10 4 (solid green) and instantaneous magnetic energy U B × 10 3 (solid cyan).

Figure 9 .

 9 Figure 9.14: 2D maps of the beam density n b [cm -3 ] at t = 127 fs, t = 172 fs, t = 240 fs and t = 492 fs, corresponding 2D maps of the beam current density |j b | and target electron temperature T e , self-generated magnetic field B y and the number of Kα and Kβ photons n Kα and n Kβ , emitted from the Copper tracer layer at t = 1487 fs.

Figure 9 .

 9 Figure 9.15: (Left panel) Longitudinal profiles of background electron temperature T e (averaged over |x max -x| < 5 µm) obtained with the M1 code (blue points) with ∆z = 1 µm, the hybrid PIC code PaRIS [Gremillet, 2014] (green curve) and the full collisional PIC code CALDER [Gremillet, 2014] (red curve) for the Al(1 µm)Cu(3 µm)Al(1 µm) target. (Right panel) Longitudinal profiles of T e obtained with the M1 code (blue points) with ∆z = 1 µm and with the hybrid PIC code PaRIS [Gremillet, 2014] (green curve) for the Al(1 µm)Cu(3 µm)Al(15 µm) target. x max is defined as the position where T e (x max , z) is the maximum electron temperature at a given depth z.

  2. a negligible pressure force of the plasma electrons, 3. a negligible electron thermal conductivity, 4. negligible collisionnal effects of fast electrons transport, 5. a negligible electron-ion energy exchange and 6. a rigid electron beam with a beam density n b (z, t) = n b0 + δn b where δn b = -n b0 sin 2 ω 0 2 t such that n b (z, t) = n b0 cos 2 ω 0 2 t . (9.37) n b0 = n b0 (z -v b t) represents the temporal/spatial envelope of the laser-generated beam density and δn b accounts for the fast electron bunches injected in the target at the frequency ω 0 = ω L or ω 0 = 2ω L depending on the acceleration mechanisms (see Chapter 8, section 8.2, Figure 8.7).

  b0 ev b the rigid beam current density envelope with v b the beam velocity, assumed to be constant, and δj b = -δn b ev b the bunches component. While assumptions 1 and 6 are made for simplicity in order to obtain analytical estimates, assumptions 2, 3, 4 and 5 are fully justified over simulation times of t ≈ 100 fs. The equations that have to be solved are the Maxwell-Gauss equation ∂E ∂z = -4πe (n e + n b -Z * n i ) , (9.38) the Maxwell-Ampère equation ∂E ∂t = -4π (j e + j b )

  analytically the errors ∀ξ ∈ {n e , v e , j e , T e }, δξ = ξ -ξ 0 (9.44) done by hybrid codes. Assuming ξ 0 vary slowly compared to δξ, we obtain, by injecting (10.56) and (10.60) in the previous equations : ∂ ∂z δE = -4πe (δn e + δn b ) , (9.45) ∂ ∂t δE = -4π (δj e + δj b ) ,(9.46)

  .53) TRANSPORT IN SOLID TARGETS Solving then the heat equation (9.49) for the error in temperature, we findC V,e∂δ T e ∂t = 2E 0 δ j e + Ej e0 9, assuming δ T e (ζ, t = 0) = 0, the ordering (9.50) and n b0 n e0 for neglecting of the second term in the right hand side of Equation (9.54), the solution reads

2 - L x 2 n

 22 .60) Also, we account for the average over 5 pixels (8.62 µm in the x-direction) of Kα 1 signals by convolving (9.60) with a Gaussian function that has a Full Width at Half Maximum (FWHM) of ∆x 0 = 10 µm n Kα z (x) = L x Kα z (x 0 ) offset implied by the artificial noise b is accounted for by substracting 0.1 n Kα z max from (9.61). The processed numerical "signals" are plotted in Figure 9.17 c) for targets of variable thickness and in Figure 9.17 d) for targets of variable target depth.

Figure 9 . 17 :

 917 Figure 9.17: Experimental horizontal a) and vertical b) profiles of Kα 1 photon signals averaged over 5 pixels (8.62 µm × 7.825 µm) [Vauzour, 2014] for targets of variable thickness (renormalized according to the corresponding 2D-3V M1 simulations). Horizontal profiles of the Kα 1 emission obtained from the simulations by averaging it over the whole Copper layer thickness (dashed curves), adding an uniform noise and convolving it with a Gaussian function of 10 µm FWHM (full curves) for targets of variable thickness c) and variable tracer depth d). Corresponding normalized profiles are plotted in the inserts of Figures c) and d).

n

  a) and c) show that the 2D-3V M1 simulations reproduce well the experimental results. However, these 2D-3V calculations must be revised with regard to three-dimensional effects. The right panel of Figure9.18 compares the 3D-3V simulation result and the experimental data. There is a strong discrepancy concerning the Kα photon emission spot size, even worse than for the 2D-3V simulation. The comparison is conducted under the same conditions as in the 2D-3V simulations. Firstly, the experimental signal is renormalized to the 3D-3V M1 simulation according to (9.58). The maximum value of the Kα 1 emission is averaged over the Copper layer thickness n Kα z (x, y) Kα (x, y, z, t f )dz (9.62)with L Cu = 3 µm, t f = 1.5 ps, z 1 = 1 µm and z 2 = 4 µm. A uniform noise of 10 % of the maximum value is added in x and y directions according to (9.60) and the result was convolved with two Gaussian functions of ∆x 0 = ∆y 0 = 10 µm FWHM according to (9.61). The offset of 0.1 n Kα z max is then substracted from the final result.

Figure 9 .

 9 Figure 9.18: Left panel : Instantaneous beam energy U b (solid black), integrated beam energy bal-anced between injected at z = 0 U inc (solid red), total collisional energy loss U col (solid magenta), total "collective" energy loss U res (solid blue), instantaneous electric energy U E × 10 4 (solid green) and instantaneous magnetic energy U B × 10 2 (solid cyan) from the 3D-3V simulation for the thinner target. Right panel : Experimental horizontal and vertical profiles of Kα 1 photon signals averaged over 5 pixels (8.62 µm × 7.825 µm) for the thinner target[Vauzour, 2014], renormalized according to the simulation (dashed curves) and corresponding horizontal and vertical profiles of the Kα 1 photon emission obtained from the 3D simulation (full curves). The horizontal profile is compared to the corresponding profile from the 2D simulation in the insert.

  Figure 9.19: 3D-3V run : Slices of the beam density (Up Left) and current density at t = 235.1 fs (Up Right), target electron temperature (Middle Left), target electrical resistivity (Middle Right), time integrated density of Kα 1 photons emitted per steradian (Down Left) and magnetic fields at t = 1062.1 fs (Down Right); Slices are indicated in the Figure.

Figure 9 .

 9 Figure 9.21: 3D-3V run : Slices at z = 0.5 µm (Left) and z = 2.5 µm (Right) of the x-component (Up), y-component (Middle) and z-component (Down) of the self-generated magnetic field B.

Figure 9 .

 9 Figure 9.22: Slices of the contributions to B x due to curls of the beam current density, resistivity gradients, temperature-density crossed gradients and 3D effects at x = 0 (Left), z = 0.5 µm (Middle) and z = 2.5 µm (Right).

9. 2 . 5

 25 Figure 9.23: (Left panel) Photoionization cross section σ P in Copper (red curves) and in Aluminum (blue curves) dependence on the photon energy hν according to [Verner et al., 1993] (full curves) and the semiclassical formula (9.68) (dashed curves). Kα 1 photon energy from Copper is indicated by a vertical dashed line. (Right panel) Transparency of an Aluminum or Copper layer with a thickness L for a photon having the energy hν according to [Henke et al., 1993] (http://henke.lbl.gov/optical constants/).

E

  th and the parameters σ 0 , E 0 , y w , y a and P , depending on the quantum numbers n and l, that fit their numerical results for atomic numbers Z, Z * ≤ 30 and photon energies hν ≤ 50 keV. n = 1 for K-shell electrons, n = 2 for L-shell electrons, n = 3 for M-shell electrons and n = 4 for N-shell electrons while l = 0 for s-band electrons, l = 1 for p-band electrons, l = 2 for d-band electrons and l = 3 for f-band electrons. The total photoionization cross section σ p (hν) = the left panel of Figure9.23 for Aluminum and Copper in the worst case, that is to say : for cold solid atoms (Z * = Z c ). Therefore, n max = 3, N

Figure 4

 4

Figure 9 . 2 . 4 ,

 924 Figure 9.23, only ≈ 90 % of Kα 1 photons are transmitted through ≈ 3 µm of Copper or ≈ 15 µm

  field of background electron plasma waves. These plasma oscillations are damped due to collisions with target electrons and ions, resulting in an additive target electron heating component that is taken into account by full PIC simulations but neglected by hybrid models. The hybrid quasi-static model agrees well with the full PIC simulation in the following Copper tracer layer. This may be explained by the fast electron collisions in Copper (Z = 29) which degrades the coherence of the fast electron bunches and the decrease of the Ohmic heating by the return current with the target depth. The profiles of Kα 1 photon emission from the Copper tracer layers obtained in to 2D and 3D simulations are compared to the experimental signals. Except for the thinnest target, the 2D simulation qualitatively reproduce the experiments. However, we found significant differences between 2D and 3D simulations, especially in what concerns the self-generated magnetic fields, the size of the Kα emission zone and the absolute value of the local number of photons emitted per unit of volume. The simulation predicts a Kα 1 emission spot size 2-3 times smaller than the experimentally measured. We analyzed the assumptions of our model concerning the Kα emission. The photon re-absorption may introduce an error of ≈ 10 % in our computations but it cannot explain the discrepancy concerning the Kα 1 spot size. We also checked the effective surface of Kα emission. Even if the 3D simulation predicts a smaller spot size in the x-direction, it predicts a larger effective surface of emission of about ≈ 200 µm × 200 µm, as compared to ≈ 40 µm × 40 µm obtained in the 2D case. This apparent paradox comes from simple geometrical reasons. Let us note r 3D the mean radius of the Kα spot size obtained in 3D, r 2D the radius obtained in 2D (x-axis) and l = 2π∆y 2 /8 ln 2 the thickness of the slice y = 0

  b e where λ D,b = k B T b 4πγ b0 n b0 e 2 is the relativistic Debye screening length. (9.70) The electron divergence angle θ d can be roughly estimated by the angle ≈ 25 o of the fast electron propagation direction during their first passage in the target and the "fountain" beam radius r 0 by L z arctan θ d ≈ 2.1 µm. According to our simulation, ε (z = L z ) ≈ 70 keV (→ γ b0 ≈ 1.13), n b ≈ 10 20 cm -3 and I b ≈ -2 MA at the target rear side at t ≈ 75 fs. Then, assuming roughly k B T b ≈ ε (z = L z ), we obtain λ D,b ≈ 0.18 µm, E ≈ 7.6 10 11 V/m and B ≈ 30kT. (9.71) We deduce that fast electrons has a Larmor radius of ρ L = γvm e c eB ≈ 0.056γβ µm.(9.72)

p > 3 ."

 3 Figure 10.1: (Left panel) Schematic view of a rarefaction wave arising from the motion of a piston and corresponding distributions of the density (a) and gas velocity (b). (Middle panel) Schematic view of the equilibrium bewteen the gaz and the ambiant pressure and corresponding distribution of the density (c). (Right panel) Schematic view of a compression wave generated by an accelerated piston : (d) is the distribution of the gas velocity of a compression wave arising from the slow motion of the piston, (e) is the distribution of the gas velocity of a blast wave arising from an instantaneous and rapid motion of the piston and (f) is the distribution of the gas velocity of a shock wave arising from a continuous and rapid motion of the piston.

4 )

 4 The notations are explained in Appendix B, section B.3.1. We have omitted the subscript " m " for brievity. Compression waves generated when pushing the piston, as illustrated Figure10.1 c), can be described by assuming a perturbation of the equilibrium ρ = ρ 0 , u = 0, T = T 0 , H = H 0 . By noting δρ and u the 1st order perturbation terms of this equilibrium andc s = ∂P ∂ρ H (10.5)the velocity of sound in the gas, the 1st order linearized equations (10.1) and (10.2compression waves propagating at the velocity of sound in the gas filling the tube, as illustrated in Figure10.1 d).

Figure 10 . 2 :

 102 Figure 10.2: Supernovae G299 (Left panel) and SN 1572, also called Tycho's Supernova because of Tycho Brahe's extensive work [Tycho, 1573] (Right panel). Both are expected to be supernovae of Type Ia i.e. from a thermonuclear explosion of a white dwarf star in a tight orbit with a companion star. Pictures are taken from the NASA's telescope Chandra X-ray Observatory, called Chandra to pay homage to the astrophysicist and mathematician S. Chandrasekhar. Chandra orbits above Earth's atmosphere at an altitude of 139,000 km. The Smithsonian's Astrophysical Observatory in Cambridge, MA (USA), hosts the Chandra X-ray Center which operates the satellite, processes the data, and distributes it to scientists around the world for analysis.

2 . 1

 21 of the Introduction concerning the conventional schemes of Inertial Confinement Fusion (ICF), a high-power laser pulse creates a very hot plasma at the target surface. This plasma exerts a high pressure on the surrounding material, acting like the piston of our though experiment, that leads to the formation of an intense shock wave, moving into the interior of the target. The momentum of the out-flowing plasma balances the momentum imparted to the compressed medium behind the shock front. The thermal pressure together with the momentum of the ablated material drives the shock wave.

10. 1 .Figure 10 . 3 :

 1103 Figure 10.3: Schematic of the density and temperature profiles in the hot electron beam ablation non-stationary regime after the loading time.

  2. Since the fast electron range depends only logarithmically on the plasma temperature, the electron beam 10.1. ROLE OF LASER-GENERATED FAST ELECTRONS IN THE SHOCK IGNITION SCHEME will deposit its energy in the same mass even when this mass expands. This reasoning enables to introduce the following two-stage model of plasma expansion driven by a monoenergetic electron beam transporting the energy flux I b = n b ε 0 v 0 , where v 0 is the initial electron velocity and n b the beam density.

  3) and (10.4) with the energy deposition rate defined by the flux of fast electrons W e ≈ I b L p (10.27) in the right hand side of Equation (10.3) and neglecting the thermal energy flux q. Assuming the boundary conditions of zero flow velocity at x = L p and zero density at x → ∞, this self-similar solution reads :

2

 2 Figure 10.4: Schematic view of the coupling between the reduced model for fast electron transport and the radiation hydrodynamic CHIC code.

  p the collisional slowing down frequency (5.2), k d = ν d /v the corresponding mean free path, S tot = S res + S

Figure 10 . 5 :

 105 Figure 10.5: (Left panel) DT plasma density profile at the moment of spike arrival obtained from a CHIC simulation [Ribeyre et al., 2009]; Courtesy of X. Ribeyre. (Right panel) Initialization of the 1D academic simulations assuming an idealized target density profile at the time of ignitor spike arrival.

  initial dimensions of the capsule are 1044 µm of external radius and 211 µm of shell thickness. It is supposed to be imploded at a constant in-flight adiabat α ≈ 1. At the moment of the spike launch the shell is already compressed by a factor of 40. Consequently, we model here the shell as a DT plasma layer of a steplike profile with a maximum density of ρ 0 = 10 g/cm 3 , a temperature of T 0 = 10 eV and a thickness of 100 µm. This idealized target density profile is illustrated in the right panel of Figure 10.5. The plasma thickness is much larger than the fast electron range, thus allowing us to observe the creation and propagation of the shock wave for a sufficiently long time of the order of t f = 1 ns. The energy flux of monoenergetic and collimated electron beams is maintained constant in time during the simulation. Two representative cases with I b = 1 PW/cm 2 and ε 0 = 30 keV (case 1) and I b = 10 PW/cm 2 and ε 0 = 100 keV (case 2) have been tested. In both cases, because of a high plasma density, the resistive losses (10.40) are small and the electron energy deposition is due to the collisional effects of in a dense plasma. The fast electrons propagate in the Lagrangian CHIC grid, as explained in section 10.1.2. The initialized angular moments of the beam distribution function thus read

Figure 10 . 6 :

 106 Figure 10.6: Distributions of the deposited power density W e (a), the plasma density ρ (b) and DT plasma pressure P (c) from the simulations of a shock drive by a monoenergetic and collimated beam of electrons with a kinetic energy and an energy flux of 30 keV, 1 PW/cm 2 (Left panel) and 100 keV, 10 PW/cm 2 (Right panel). The numbers near the curves indicate the time in ps. The dashed line in the right panel b shows the self-similar solution (10.28 a).

For the case 2

 2 Figure 10.6 b with a dashed line. It agrees rather well with the numerical solution shown with a red line corresponding to the time of 100 ps. The density profile in the shock in the later time, t > 200 ps, takes a two-humps shape. The second hump is driven by the thermal wave catching up the shock at the time of 1 ns. The driving efficiency of the beam remains at the same level of 7 % as in case 1.

  Figure 10.7: Schematic view of the laser-irradiated target and the Streaked Optical Pyrometry (SOP) diagnostic a) and a SOP image of SOP result at λ = 532 nm for the Al(1 µm)Cu(3 µm)Al(15 µm) target b); Courtesy of J. Santos.

  .61) and P [K.cm 2 ] = ∞ 0 zT (z, t = 0) dz = T m d 0 2 ≈ 1.16 K.cm 2 (10.62) with T m = 400 eV and d 0 = 5 µm instead of a [cm 2 .s -1 .K -n ] and T 0 [K] for the self-similar solution 10.2. BLAST WAVE GENERATION IN SOLID TARGETS BY THE QUASI-ISOCHORIC HEATING BY LASER-GENERATED ELECTRON BEAMwith a constant temperature T 0 at x = 0. Consequently, we can construct the two self-similar variablesx f (t) [cm] = ξ 1 (aP n t)

2 1.680 if n = 5 .

 25 (10.70) The dipole-type solution is illustrated in Figure10.8.Knowing the temperature profile, let us now estimate the target depth and the time at which the blast wave is generated. A criterion called hydrodynamic separation consists in estimating the time needed for compression waves to "overshoot" the thermal wave front. According to Equation (10.6) that describes the propagation of compression waves, we estimate the propagation velocity of

Figure 10

 10 Figure 10.8: (Left panel) Dipole-type solution profile for the heat equation at 10 ps. (Right panel) Plots of the thermal wave front and compression wave velocities versus time.

  WAVE GENERATION IN SOLID TARGETS BY THE QUASI-ISOCHORIC HEATING BY LASER-GENERATED ELECTRON BEAM

Figure 10 . 9 :

 109 Figure10.9: Comparison between the temperature profile obtained with the hybrid PIC simulation conducted by[Gremillet, 2012] and the initialization of the Radiation Hydrodynamic simulation for the thinner and thicker targets with the Copper traer layer at z = 1 µm.

Figure 10 . 10 :

 1010 Figure 10.10: Distribution of the plasma temperature and pressure at t = 60 (Left panel) and 600 ps (Right panel).

Figure 10

 10 Figure 10.11: Distributions of the target density ρ (Left), temperature T (Middle) and pressure P (Right) from the CHIC simulation at t = 39, 239, 499 and 979 ps for the 5-microns thick target.

Figure 10

 10 Figure 10.13: (Left panel) Comparison between the experimental signals and the simulation results for both targets. (Right panel) Streak camera-like images from the CHIC simulation for the thicker target.

  has been conducted and compared to a hybrid PIC simulation. It shows that the M1 approximation is sufficiently accurate to reproduce the hybrid PIC simulation results. We have applied the model to the study of the emission of Kα photons. The refluxing of fast electrons is accounted for by imposing the specular reflection of fast electrons at the target-vacuum interfaces and adding a second population in the M1 equations. The first one describes the laser-generated electron population propagating in the laser direction, while the second one describes the counterpropagating fast electrons. The model of calculation of the emisssion of Kα photons is revised. The conclusion of this study are manyfold.

  photons émis localement par unité de volume. Néanmoins, la taille de la tâche d'émission Kα obtenue reste beaucoup plus petite que celle déduite des données expérimentales même dans la simulation tridimensionnelle beaucoup plus réaliste. Nous avons donc analysé toutes les hypothèses de notre modèle concernant l'émission de photons Kα afin de déterminer qu'est ce qui pourrait expliquer cette différence. L'opacité de la cible peut introduire une erreur de seulement ≈ 10 % dans nos calculs. Une autre possibilité serait que les électrons secondaires, négligés dans notre modèle, influent sur l'émission Kα. Aussi, la forte augmentation de la température pourrait éventuellement jouer sur la physique atomique de l'émission de photons Kα. Cependant, il me semble que l'hypothèse la plus critique dans

  a very diluted classical gas consisting of N i ions and N e = Z * n i electrons, one can solve direcly the dynamic equations for allN = N i + N e particles dp α,n dt = q α E ext (r α,n , t) + E (r α,n , t) + v α,n c × [B ext (r α,n , t) + B (r α,n , t)] dr α,n dt = v α,n = p α,n γ α,n m α (A.2)

=

  Finally, (E ext , B ext ) are eventual external electromagnetic fields. The plasma generated electromagnetic fields (E, B) can be deduced by solving self-consistently the Maxwell's equations (A.1) with the source termsj (r, t) = -e Ne n=1 v e,n (t)δ 3 [r -r e,n (t)] + Z * e N i n=1 v i,n (t)δ 3 [r -r i,n (t)] ρ (r, t) = -e Ne n=1 δ 3 [r -r e,n (t)] + Z * e N i n=1 δ 3 [r -r i,n (t)] . (A.3)For example, in the static approch i.e. by neglecting the time delay terms in the Maxwell's equations (A.1), the electromagnetic fields readE (r, t) = -∂Φ ∂r (r, t) B (r, t) = c ∂ ∂r × A (r, t) (A.4)This equation depends on the 2-body distribution functionf e,2 (r 1 , p 1 , r 2 , p 2 , t)= N e (N e -1) Ne n=3 d 3 r n d 3 p n f e,Ne (r 1 , p 1 , r 2 , p 2 , ..., r n , p n , ..., r Ne , p Ne , t). (A.13) The equation for this function can be obtained by integrating the Liouville Equation (A.11) over the 6(N e -2) phase coordinates. It reads∂f e-d 3 r 3 d 3 p 3 e 2 r 1 -r 3 |r 1 -r 3 | 3 . ∂ ∂p 1 + e 2 r 2 -r 3 |r 2 -r 3 | 3 . continue that procedure until s = N e -1 with ∀s ∈ [3, N e -1], f e,s (r 1 , p 1 , ..., r s , p s , t) = N e ! (N e -s)! Ne n=s+1 d 3 r n d 3 p n f e,Ne (r 1 , p 1 , r 2 , p 2 , ..., r n , p n , ..., r Ne , p Ne , t). (A.15)This chain of equations is called the BBGKY hierarchy (from the name of its founders N.N. Bogoliubov, M. Born, H.S. Green, J.G. Kirkwood and J. Yvon). It formally simplifies the N e -body problem by approaching the solution. In practice, it allows to evaluate the right hand side of Equation (A.12) that accounts for collisions by evaluating the two-body distribution function f e,2 from Equation (A.14).

R

  12 = (r 1 +r 2 )/2 while it exhibits fast variations over the relative coordinates r 12 = r 2 -r 1 . Therefore, (∂ f e,2 /∂r 12 ) (∂ f e,2 /∂R 12 ) and (∂ f e,2 /∂r 12 ) = (∂ f e,2 /∂r 2 ) = -(∂ f e,2 /∂r 1). Thus, by integrating (A.18) over the positions r 2 and momenta p 2 of the electron 2, we make appearing the collision integral of Equation (A.17) 2 d 3 r 12 (v 2 -v 1 ) . ∂ ∂r 12

3 f 3 f 3 C 3 d 3 p α, 2 R 3 d 2 σ 3 d 2 σ

 3333323232 -ion Boltzmann collision integral. Similarly to the Vlasov equation (A.8), the electromagnetic fields (E, B) verify the Maxwell equations (A.1) with the plasma charge and current densitiesρ (r, t) = -e R e,1 (r, p 1 , t) d 3 p 1 + Z * en i j (r, t) = -eR e,1 (r, p 1 , t) v 1 d 3 p e,1 + 0. can easily notice that for α = e or i,R 3 C B αα [f α,1 , f α,1 ] (r, p α,1 , t) 1 d 3 p α,1 = 0 R 3 C B αα [f α,1 , f α,1 ] (r, p α,1 , t) p α,1 d 3 p α,1 = 0 R B αα [f α,1 , f α,1 ] (r, p α,1, t) change of variables p α,1 -∆p αα → p α,1 and p α,2 + ∆p αα → p α,2 in the second double integrald 3 p α,1 R αα |v α,2 -v α,1 | f α,1 (p α,1 -∆p αα ) f α,1 (p α,2 + ∆p αα ) A(p α,αα |v α,2 -v α,1 | f α,1 (p α,1 ) f α,1 (p α,2 ) A(p α,1 + ∆p αα )where A(p α,1 ) = 1, p α,1 or p α,1 2 /2m α .

R 3 C 3 C 3 C 3 C 3 C

 33333 B ei [f e,1 , f i,1 ] (r, p e,1 , t) 1 d 3 p e,1 = R B ie [f i,1 , f e,1 ] (r, p i,1 , t) 1 d 3 p i,1 = 0 R B ei [f e,1 , f i,1 ] (r, p e,1 , t) p e,1 d 3 p e,1 + R B ie [f i,1 , f e,1 ] (r, p i,1 , t) p i,1 d 3 p i,1 = 0 R B ei [f e,1, f i,1 ] (r, p e,1 , t)

R 3 A 3 ( 3 v 3 d 3

 33333 (p e,1 )C B eα [f e,1 , f α,1 ] d 3 p e,1 p e,1 ) + A(p e,2 ) -A(p e,1 -∆p eα ) -A(p e,2 -∆p eα )] C B eα [f e,1 , f α,1 ] d 3 p e,1 . (A.28) This equality can be demonstrated similarly to the Boltzmann integrals by performing the change of variables p e,1 -∆p eα → p e,1 and p α,2 + ∆p eα → p α,2 in the integrals. Then, starting from the Boltzmann equation (A.22), one can also show that the function defined as H (r e,1 , t) = -R f e,1 ln f e,1 -f e,1 ) d 3 p ee,1 (f e,1 ln f e,1 -f e,1 ) d 3 p e,1 = -α=e,i R 3 ln f e,1 C B eα [f e,1 , f α,1 ] d 3 p e,1 (A.30) since for all µ ∈ [t, r e,1 , p e,1 ] ∂ ∂µ (f e,1 ln f e,1 -f e,1 ) = ∂f ep e d 2 σ |v α -v e | ln f e (p e )f α (p α ) f e (p e -∆p eα )f α (p α + ∆p eα ) × f e (p e )f α (p α ) -f e (p e -∆p eα )f α (p α + ∆p eα ) (A.31)

3 f e d 3 3 f 3 f e v e d 3

 33333 maximum value. Let us find the distribution function f M which maximizes the entropy H under the constraints of the definition of the electron density n e (r, t) = R following the standard procedure of maximization problems, one introduces the Lagrange multipliers α 0 , α 1 and α 2 associated with these three constraints, respectively. Then, in order to find f M , one has to solvedL df e [f M ] = 0 (A.36)where L is the Lagrangian of this maximization problem :L[f e ] = H[f e ] + α 0 n e -R e d 3 p e + α 1 . n e u e -R (A.36) is an exponential function f M = exp -α 0 + α 1 .v e + α2 m in the definition of the electron density (A.33), the mean electron flux (A.34) and the mean electron kinetic energy (A.35), one finds respectively n e = exp -α 0 r e , p e , t) = n e (r e , t) (2πm e k B T e ) 3/2 exp -m e (v e -u e ) 2 2k B T e (A.40) called the Maxwell-Boltzmann distribution function. It is the stationary solution of the Boltzmann equation (A.22). Thus, one deduces from (A.47) differential Coulomb collision cross-section. Coming back to the laboratory frame, one finds the exchange momentum in the collision ∆p= p 1,i -p 1,f = p 2,f -p 2,i = ∆P = m 1 m 2 m 1 + m 2 |v 1,i -v 2,i | sin θ * b b -(1 -cos θ * ) v 1,i -v 2,i |v 1,i -v 2,i | . (A.50)A.2.2 Small-angle Collisions

Figure A. 3 : 2 v 2 I 3 U 3 d 3 p

 322333 Figure A.3: Schematic view of the diffusion of the fictitious particle of charge e and mass µ by the Coulomb diffusion center q α .

  the permittivity (k, ω) = 1 + (1/3k 2 λ D 2 ) that takes into account electron plasma waves with thermal corrections. The Landau collision integral can be extended to degenerate electrons by replacing the Debye length λ D by a more general expression evaluated in the Debye-Huckel theory electron temperature T e has been replaced by T e 2 temperature and Z * the ionization state which can be evaluated using the formula Z

A. 2 . 3

 23 Coulomb Logarithm ln Λ eα The integral defined in the previous section as the Coulomb logarithm ln Λ eα = bmax b min db b diverges at both the lower and upper boundaries of integration if b min = 0 and b max = ∞. These limits need to be defined separately. As the Landau collision operator (A.57) is valid only for small angle collisions θ * 1 or b λ L according to (A.47), the low boundary has to be defined by b min = λ L . Moreover, the Landau collision operator must be cut at the screening length b max = λ D . This also explains why assuming kλ D 1 (k ∼ b -1 ), the Lenard-Balescu collision operator also allows to describe this Debye screening effect. Thus, the Coulomb logarithm can be evaluated as ln Λ eα = This classical expression is valid only if the relative collision velocity is sufficiently small so that the Landau length is larger than the effective De Broglie length λ DB = 2µv rel (A.68) (see Figure A.3). In this case, the lower limit must be chosen by b min = λ DB . This quantum limit must be chosen for the electron energies larger than 10 eV. Concerning collisions of electrons on ions, in the case of dense plasmas, the Debye-Huckel treatment of screening breaks down due to strong ion-ion correlation effects. If the Debye length λ D becomes less than interparticle distance must be evaluated by r i [Lee and More, 1984]. The Coulomb logarithm depends on the colliding particles velocities v e and v α . In a plasma at the thermal equilibrium, the particles follow a Maxwell-Boltzmann distribution function and the effective lengths in the Coulomb logarithm can be replaced by their average values 3k B T e /m e . Finally, one can write the Coulomb logarithm as ( the minimum value of 2 accounts for the non-ideal plasma effects) ln Λ eα = max 2= max {r i , λ D } and b min = max λ L , λ DB with

  (A.81) expressed in the spherical coordinates in the electron momentum space (p e , θ e , ϕ e ).

  r, p α , t) d 3 p α .(B.4) According to the H-theorem (cf Appendix A, section A.1.5), there is a time t ∼ ν -1 needed for the electrons and ions to reach the maximum entropy equilibrium where their distribution functions are close to the Maxwell-Boltzmann distribution functionf α (r, p α , t) = n α (r, t) (2πm α k B T α ) 3/2 exp -m α (v α -u α ) 2 2k B T α . (B.5)The kinetic time scale ν -1 can be estimated by the averaged electron-ion collision frequency (A.83) distribution function f e being assumed to be locally a Maxwell-Boltzmann distribution in the considered time scales t ν -1 ei , we only need to find the electron density n e , the mean electron velocity u e and the electron temperature T e in order to fully characterize the electron distribution function f e . This approximation is called the Local Thermodynamic Equilibrium.This Appendix is dedicated to the Classical Plasma Hydrodynamic Theory (without Relativistic effects) based on this Local Thermodynamic Equilibrium, quoted throughout this thesis. It also presents the radiation hydrodynamic code CHIC that has been used in this work. For simplicity, we neglect external electromagnetic fields (except in the derivation of the Spitzer, Lee-More or Braginskii transport coefficients) and assume that the plasma consists of two species -electrons and ions.

B. 1 3 f 3 f 3 f 3 f

 13333 Two fluids Hydrodynamic Equations B.1.1 Definitions Knowing the distribution function f α = f α (r, p α , t) for the particles α, one defines the α d 3 p α Mean particles fluxn α u α (r, t) R α v α d 3 p α Mean particles kinetic energy K α (r, t) 3 f α (v α -u α ) 2 d 3 p α Mean particles momentum flux tensor Π α (r, t) m α R α v α ⊗ v α d 3 p α Kinetic pressure tensor P α (r, t) m α R α (v α -u α ) ⊗ (v α -u α ) d 3 p α Scalar kinetic pressure P α (r, t) m α 3 R 3 f α (v α -u α ) 2 d 3 p α

τ

  α = Π α -P α I -n α m α u α ⊗ u α (B.8)as followsP α = P α Iτ α (B.9) with P α (r, t) = 1 3 Tr [P α ] = n α k B T e .(B.10)

2 P 3 m

 23 α u α -τ α .u α . (B.11) B.1.2 Local Thermodynamic Equilibrium By integrating (B.1) over the momentum space d 3 p, one gets the hydrodynamic particles conservathe Landau collision integral property of the section A.3.4 of Appendix A R 3 C L eα [f e , f α ] (r, p, t) d 3 p = 0. By integrating (B.1) multiplied by the particles momentum m e v over the momentum space d 3 p v, one gets the hydrodynamic particle momentum conservation equation m e ∂ ∂t (n e u e ) + ∂ ∂r . (n e u e ⊗ u e ) = -n e e E + u e c × B -∂ ∂r . (P e Iτ e ) + R ei . (B.13) Indeed, according to the Landau collision integral property (see Appendix A, section A.2.4), one has R 3 vC L ee [f e , f e ] (r, p, t) d 3 p = 0. Concerning electron-ion collisions, we define the friction force R ei (r, t) = R e vC L ei [f e , f i ] (r, p, t) d 3 p. (B.14) By developing (B.13) and simplifying it using (B.12), one can also deduce the hydrodynamic e Iτ e ) + R ei . (B.15) By integrating (B.1) multiplied by the electron kinetic energy m e v 2 /2 over the momentum space d 3 p (m e v 2 /2), one gets the hydrodynamic electron energy conservation equation

3 vδf e d 3 2 Z∂ 2 3 m e v e 2 2

 332232 hydrodynamic fluxes are found from the kinetic equation by evaluating the deviation of the electron distribution function δf e from the local equilibrium Maxwellian function f M . Here, we account only for the electron-ion collisions in the electron kinetic equation and also neglect the magnetic component of the Lorentz force. Then, the electron kinetic equation takes the following from (see electric field E is assumed to be homogeneous, not varying with time and with a value|E|E run ∼ m e ν ei v T,e /e sufficiently small such that the electric field E can be seen as a perturbation of the maximum entropy equilibrium f e (0) = f M with u e (0) = 0 (see Appendix A, section A.2.4).By noting sof e (r, p, t) = f M (r, p, t) + δf e (r, p) , (B.29) one can linearize the equation assuming the electric field is a first order term E = δE to get expanding δf e on the Legendre polynomial basis and by noticing that the left hand side of this equation does not depend on the azimuthal angle ϕ, one deduces δf e and consequently δj e = j e = -e R electron electrical conductivity and expressed with the averaged electron-ion collision frequency (see Appendix A, section A.3.2). Recalling that the electric current is proportional to the electron mean velocity, one can find from (B.31) the expression for the friction force in the electron momentum hydrodynamic equation : R ei = en e η L j e (* e 2 ln Λ ei (k B T e ) 3/2 (B.34) called the plasma electrical resistivivity. It does not depend on the plasma density (in this approximation) but only on the plasma temperature. To obtain an expression for the electron heat flux, one has to solve the linearized equation v. ∂f M ∂r -δf e ∂ϕ 2 . (B.35) One can express the electron current density δj e and the electron heat flux δq e as a function of the moments of δf e which depend on (∂T e /∂r) and E. One gets the expression of the electric field E by considering δj e = 0 in order to avoid charge accumulations. Injecting the obtained expression of the electric field in the expression of δq e , one finally obtains q e = δq e = R v e δf e d 3 p e = -κ L called the Lorenz constant.

  r, p, t) d 3 p = 2 m e m i C V,e ν ei (T e -T i ) = -Q ei (r, t) -R ei (u e -u i ) . (B.39) B.2.2 Electron-electron Collision Contribution to the Hydrodynamic Fluxes By taking into account the electron-electron collision term, the linearized equation (B.35) becomes v. ∂f M ∂r -eE. ∂f M ∂p = C L ee [f M , δf e ] + C L ee [δf e , f M ] +

σ

  Sp = γ E σ L and κ Sp = δ

  . The magnetic field may strongly affect the electron fluxes if the electron cyclotron frequency ω ce = |eB|/m e c is of the same order of magnitude than the electron-ion collision frequency ν ei . The electron kinetic equation in the external magnetic field B (B.40) reads v.∂fM ∂r -eE. ∂f M ∂p -e v c × B ∂δf e ∂p = C L ee [f M , δf e ] + C L ee [δf e , f M ] + ν ei (vThe magnetic field introduces an anisotropy of the fluxes that are not parallel to the generalized forces any more. Consequently, transport coefficients become tensors :R ei = R u + R T (B.43)with R u = en e η.j = en e η j + en e η ⊥ j ⊥ + en e η ∧

  44)withq u = k B T e n e e β.j = β k B T e n e e j + β ⊥ k B T e n e e j ⊥ + β ∧ k B T e n e e

  ei n e e 2 1 -α 1 x 2 + α 0 ∆ η ∧ = 1 σ ∧ = m e ν ei n e e 2 x α 1 x 2 + α 0 ∆ β = n e β 0 δ 0 β ⊥ = n e β 1 x 2 + β 0 ∆ and β ∧ = n e x β 1 x 2 + β 0 ∆where ∆ = x 4 + δ 1 x 2 + δ 0 and x = ω ce /ν ei while the transport coefficients η and κ in the direction are defined by equations (B.41). The parameters in these expressions depend only on the ionization state Z * :

C(

  Figure B.1: CHIC main packages[START_REF] Breil | Multi-material {ALE} computation in inertial confinement fusion code {CHIC}[END_REF]

  concerning the second term of the left hand side of this equation [?]). These equations are solved in 2D Cartesian (x, y) or in 2D cylindrical (z, r) assuming an axial symetry (azimuthal invariance assumption). Numerical Schemes : The numerical computation of Equations (B.69), (B.70), (B.71) and (B.72) is split in two steps 1. On the interval between the time step t n and t * = t n + ∆t * , the system of Equations (B.69), (B.70) and (B.73) is solved without taking into account the thermal conduction according to a cell-centered discretization of each fluid mesoparticle i [

  assuming that the electron evolution at this time step is isentropic in such a way that the entropy is deposited into the ion internal energy, one has to solve deduceε i i, * = E i, * -ε i, * e -[ u i, * 2 /2].

  Debye screening length λ D or λ Debye v T,e /ω p 7.43 10 2 (T e /n e ) 1/2

		Symbol	Formula	order of magnitude (cgs)
	Plasma ionization state	Z -3/2
	Coulomb explosion time 1.2 10 -15 Z Mean ions distance τ e ν ei /ω p 2 r i (3/4πn i ) 1/3 0.6n i -1/3
	Electron Larmor radius	r L or R L	v T,e /ω c	2.88T e	1/2 B -1
	Electron inertia length	λ e	c/ω p	5.31 10 5 n e	-1/2

* Z * = n e /n i 0 ≤ Z * ≤ Z Electron-ion Coulomb logarithm ln Λ ei Annexe A.2.3 > 2 Electron Thermal velocity v T,e or v T h,e k B T e /m e 4.19 10 7 √ T e Ion sound velocity c s Z * k B T e /m i 4.19 10 7 √ T e Electron plasma frequency ω p , ω e or ω p,e 4πn e e 2 /m e 5.64 10 4 √ n e Electron gyrofrenquency ω c or ω ce eB/m e c 1.76 10 7 B Electron-ion collision rate ν ei , ν ei or ν Annexe A.3.2 3.9 10 -6 n i Z * 2 ln Λ ei T e * ln Λ ei T e -3/2 Electron De Broglie wave length λ DB or λ De Broglie /2m e v T,e 1.38 10 -8 T e -1/2 Electron-electron Landau length λ L or λ Landau e 2 /k B T e 1.44 10 -7 T e -1

  ). This laser energy absorption mechanism is called Collisional Absorption (CA). For lower laser frequencies and colder and/or denser solids, the electron collision frequency becomes larger than the laser frequency. In this case, well-known in metal optics, this is the Normal Skin Effect (NSE) which is responsible for the laser absorption in the skin-depth. This is what happens for example when a light is reflected by a

mirror. These two collisional absorption mechanisms are valid only if the electron mean free path v T h,e /ν ei and the mean distance travelled by electrons during one laser period v T he /ω are smaller than the skin-depth L s where v T h,e = k B T e /m e is the electron thermal velocity. With increasing laser intensities and electron temperatures, "collisionless" absorption mechanisms become dominant.

  e c)2 . Therefore, the distribution function f e (r l , p l , t) is constant at the electron trajectories defined by the equations of motion

	1.3. PARTICLE-IN-CELL METHOD FOR LASER-PLASMA INTERACTION
	SIMULATIONS
	where γ = 1 + (p/m
	subsection. The Vlasov equation reads
	∂f e ∂t	+	∂ ∂r	.	p γm e	f e -	∂ ∂p	. e E +	v c	× B f e = 0	(1.59)

  per unit of kinetic energy dε and per unit of steradian d 2 Ω = sin θdθdϕ at time t. It

	2.1. ELECTRIC FIELD GENERATED BY A MONOENERGETIC, COLLIMATED AND
	HOMOGENEOUS SEMI-INFINITE RELATIVISTIC ELECTRON BEAM PROPAGATING
	IN VACUUM
	Chapter 2
	Electromagnetic Neutralization of a
	Laser-generated Relativistic Electron
	Beam
	"The effects are always opposed to the causes that gave rise to them."
	Heinrich Lenz

  0 where Z * is the plasma ionization state which depends on the plasma density and the plasma electron temperature T e . As mentioned in the previous subsection, the electrical

	2.2. ELECTRIC NEUTRALIZATION OF A MONOENERGETIC, COLLIMATED AND
	HOMOGENEOUS RIGID RELATIVISTIC ELECTRON BEAM PROPAGATING IN
	SOLIDS OR DENSE PLASMAS

  , v e , j e , E , B }, ξ = ξ 0 + δξ

	and
	∂ ∂t	+ v e .	∂ ∂r	p e = -e E +	v e c	× B -ν p e -p e0 .	(2.21)
	Here, the pressure tensor term has been neglected in (2.21) assuming the thermal velocity of the plasma
	electrons can be neglected in comparison with the beam velocity i.e. k B T e /m e /γ b ≤ k B T e /m e
	v b . Concerning laser-generated relativistic electron beams, the beam density n b is necessarily less than
	the critical density n c ≤ n e0 and one has γ b ≈ 1 -10. Thus, one can linearize the non-linear set of
	equations {(2.17), (2.19), (2.18), (2.20), (2.21)} with respect to the small parameter n b /n e = n b /γ 2 b n e .
	One notes
			∀ξ ∈ {n e
								3.2). By integrating (2.16) and (2.16)
	multiplied by p over the whole momentum space, one gets
				∂n e ∂t	+	∂ ∂r	. n e v e = 0	(2.20)

  γ b 2 is the Coulomb explosion time evaluated in the beam rest frame and k 2 coincides now with k 5 (in the limit k ⊥ λ e 1) while k 3 , k 4 and k 5 are unchanged. By repeating the

	2.2. ELECTRIC NEUTRALIZATION OF A MONOENERGETIC, COLLIMATED AND
	HOMOGENEOUS RIGID RELATIVISTIC ELECTRON BEAM PROPAGATING IN
	SOLIDS OR DENSE PLASMAS
	procedure described above, one obtains in this case for |z -v b t|	λ e /γ	3/2 b

  below 2 MeV. Cross sections in the intermediate energy region from 2 MeV to 50 MeV are obtained by interpolation, a procedure whose accuracy was confirmed by more detailed calculations for a few cases. The uncertainties of the radiative stopping powers are estimated to be 2 % above 50 MeV, 2 % to 5 % between 50 MeV and 2

	number Z can be estimated as	dε ds	= -4π	Zn i e 4 m e v 2 ln Λ rel ee			(4.28)
	with the relativistic electron-electron Coulomb logarithm, usually called drag number,
	ln Λ rel ee	= ln = ln	γ + 1 m e c 2 ω p -(γ -1)m e c 2 I ex ln 2 2 + 16 + f (γ) -ln 2 2 1	+	1 16	-	ln 2 + (1/8) γ	+	(1/2) ln 2 + (9/16) γ 2	-	δ 2	(4.29)
	with
	MeV, and 5 % below 2 MeV.
	4.2.2 Range of a Relativistic Electron Propagating in a Dense
		Plasma

By neglecting the bremsstrahlung losses (4.25) and using the relation

(4.23)

for the density effect, the total stopping power of a relativistic electron propagating in a fully ionized plasma with the atomic THROUGH SOLIDS AND DENSE PLASMAS

  4.3. ANGULAR SCATTERING OF A RELATIVISTIC ELECTRON IN SOLIDS AND DENSE PLASMASprevious section 4.2.1. Collisions on target electrons with the scattering angle above θ c (respectively below w c ) need a statistical treatment of the electric field due to random thermal fluctuations and will not be described in this section. Here, we will only describe the angular scattering of a relativistic electron by colliding with target electrons with a scattering angle below θ c (respectively above w c ), i.e.,

	we will only consider the electrons angular scattering due to binary collisions. The binary collisions
	with target ions are described by the Mott scattering formula (4.13) which can be written with the
	simpler form
			dσ dΩ i	=	2 4(pv) 2 sin 4 θ/2 (Z * e 2 )	1 -β 2 sin 2 θ/2 .	(4.34)
	Concerning binary collisions of a relativistic electron with free electrons of the material, one can use
	the Möller scattering formula (4.9) expressed in the laboratory frame according to the relation (4.2
	c). It reads
	dσ dΩ e	=	4e 4 cos θ (pv) 2

  The numerical computation of the Vlasov-Fokker-Planck equation(4.51) is crucial for understanding the physics of laser-generated relativistic electron beam transport. The relativistic kinetic equation takes into account both collective and collisional effects and it is coupled with Maxwell's equations.The numerical computation of this system of equations is extremely challenging because of the large number of variables of the distribution function and the extremely small time and spatial numerical steps compared to the several ps and hundreds of µm needed for fast electron transport studies in the context of inertial confinement fusion. Compared with a hydrodynamic fluid model, a kinetic model is computationally expensive, as such a model not only contains spatial information but also momentum coordinates, and is therefore of high dimensionality and rich in information. In this chapter, we review the existing numerical tools. We start from the Particle-in-Cell (PIC) methods and then pro-

	5.1. PARTICLE-IN-CELL METHODS
	Chapter 5
	Existing Simulation Methods for Fast
	Electron Transport
	John Von Neumann
	145

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." ceed to describe the Eulerian methods commonly called "Vlasov-Fokker-Planck methods". Actually, there is no Vlasov-Fokker-Planck codes which solve the Vlasov-Belyaev-Budker equation (

4

.51) but, as we will demonstrate in Chapter 6, applied to laser-generated relativistic electron beam transport, the Belyaev-Budker collision tensor can be simplified into a Landau-like collision tensor. Finally, we describe the less expensive "Vlasov-Fokker-Planck methods" based on expansions of the distribution function.

  3, we neglect the pressure force in (6.21), consider k y = k z = 0 and assume |E z | |E x |. While the system of equations {(6.20), (6.22), (6.21)} provides the same relations as (3.36), for the electric field, the linearization of the M1 equations (6.34), (6.35) and (6.61) is different. The distribution function at the equilibrium is the Maxwell-Juttner distribution function (3.17) expressed as a function of (ε, Ω) :

  The numerical model is constrained by the CFL condition of the resolution of the M1 equations. The package diagnostics (see Figure7.2) allows to write the computational results at each time step (Ψ M1 , (E, B), T e , T i , ...) in 86 text files ".dat" with one text file per quantity. For example, the contribution of each source term to the B y -field is written in a separate file. That is the contribution of the the curl of the beam

	the contribution due to the temperature-density crossed gradients
	∂B y,cross ∂t	-	∂ ∂z		ηc 2 4π	∂B y,cross ∂z	-	∂ ∂x	ηc 2 4π	∂B y,cross ∂x	=	k B c n e e	∂n e ∂x	∂T e ∂z	-	∂n e ∂z	∂T e ∂x
	and the contribution due to 3D effects
	∂B y,3D ∂t	-	∂ ∂z		ηc 2 4π	∂B y,3D ∂z		-	∂ ∂x	ηc 2 4π	∂B y,3D ∂x	= -	∂ ∂z	ηc 2 4π	∂B z ∂y	-	∂ ∂x	ηc 2 4π	∂B x ∂y
	current density :
	∂B y,cour ∂t	-	∂ ∂z	ηc 2 4π	∂B y,cour ∂z	-	∂ ∂x	ηc 2 4π	∂B y,cour ∂x	= ηc	∂j b,x ∂z	-	∂j b,z ∂x
	the contribution due to the resistivity gradients :
		∂B y,res ∂t	-	∂ ∂z	ηc 2 4π	∂B y,res ∂z	-	∂ ∂x	ηc 2 4π	∂B y,res ∂x	= j b,x c	∂η ∂z	-j b,z c	∂η ∂x

  In order to relate these fits with the angular moments Ψ 0 and Ψ 1 of the M1 model, we assume

				.18)
	F ϕ (ε, ϕ, t) = exp -4 ln 2	ϕ -ϕ 0 ∆ϕ	2	(8.19)
	and
	F

x (x, t) = exp -4 ln 2 x -x 0 ∆x 2 (8.20)

where c 1 (t), c 2 (t), c 3 (t), c 4 (t), c 5 (t), ε max (t), x 0 (t), ∆x(t), ν L→e (t) have a polynomial time dependence and ∆ϕ(ε, t) and ϕ 0 (ε, t) have a polynomial time dependence depending on the energy range (ε[keV] ∈ [20, 50], [50, 100], [100, 500], [500, 1000] or [1000, 5000]) as illustrated in Figure 8.8.

  Kα 1 hν Kβ [eV] I Kβ /I Kα

	Al	13 1559.6	1486.70	1486.27	0.5	1557.45	0.02
	Cu	29	8979	8047.78	8027.83	0.51	8905.29	0.12
	Ta	73	67416	57532.	56277	0.57	65223	0.22

  + E 0 δj e -E 0 n e0 eδv e .(9.49)

	and
			C V,e	∂ ∂t	+ v e0	∂ ∂z	δT e = -C V,e δv e	∂ ∂z	δT e + E 0	δv e v e0	δj e
									n e0	∂ ∂z	δv e -	∂ ∂z	(δn e δv e ) where δv e = -	δj e + δn e ev e0 n e0 n e0 e 1 + δn e	,	(9.47)
	∂ ∂t	+ v e0	∂ ∂z	δj e = -	∂ ∂z	(δv e δj e ) -j e0	∂ ∂z	δv e +	ω p 4π 2	δn e n e0	δE +	ω p 4π 2	δE +	ω p 4π 2	E 0	δn e n e0	-νδj e	(9.48)

  , increases with time. The duration of this stage t h , called the loading time, is defined by the time of propagation of the rarefaction wave across the heated layer, t h ≈ L p /c s . The coefficient ζ can be evaluated by requesting a continuity of the plasma density and pressure at the time t h with the self-similar solution. During the expansion phase, the absorbed energy is equally divided between the kinetic and internal energy. Thus, ζ(t h ) = 1 at this stage, and the loading time

	and the pressure read consequently
	t h = 2	W kin I b	=	9 2π	1/3 L p D 0	(10.23)
	and
		P m = P h	t t h		(10.24)

  . As this self-similar solution corresponds to an infinitely thin initial heated layer, it formally diverges at t = 0, but it has a physical sense for times longer than the loading time t h . We can check according to Equation (10.28 a) that the maximum of density ρ m is obtained at t = t h and x = L p that gives the expression of the loading time (10.23). Also, we can check that, by

	10.1. ROLE OF LASER-GENERATED FAST ELECTRONS IN THE SHOCK IGNITION
	SCHEME
	and pressure
	P m = P h	t h t	(10.31)
			3/2	(10.29)
	increasing with time as t 3/2 injecting the sound velocity (10.22) expressed with the temperature (10.28 b) in the areal density of
	kinetic energy (10.19) at the loading time t h , we find (10.23), as previously explained. The effect of
	expanding plasma on the shock wave formation depends on the values of plasma density
	ρ m = ρ 0	t h t	3/2	(10.30)

  .1. ROLE OF LASER-GENERATED FAST ELECTRONS IN THE SHOCK IGNITION SCHEME following the notations introduced in section 7.1. The CFL condition reads

		min r S n+1,l+1 (r)
	∆ε l <	1 ∆x	+	1 ∆y	+	∆z 1	.	(10.38)
	Here, we introduce the self-generated electric field in the electron transport equations. For this,
	we consider only the main term of Equation (6.23) i.e. we assume the simpler Ohm's law
		E = -ηj b .			(10.39)
	In order to solve Equations (6.34) and (6.35) with the self-generated electric field according to the
	numerical scheme (10.37), we introduce the resistive stopping power
		S res = eE.Ω ε ,			(10.40)
	the resistive slowing down frequency
		ν res =	S res p			(10.41)
	and the inverse of the fast electrons resistive mean free path
		k res =	ν res v	.			(10.42)
	Then Equations (6.34), (6.35) and (6.61) multiplied by p read

  -3 ) at temperatures greater than k B T e ≈ 100 eV, according to the Ditmire criterion (10.55).

, we do not specify the values of n and κ 0 in the thermal energy flux (10.53). Here, κ SH is the Spitzer-Harm thermal electron conductivity (see Appendix B, section B.2.1 and B.2.2). Indeed, thermal waves are driven by radiation transport for solids (n e ≈ 10 23

cm

  Toutes les expériences et les observations menées jusqu'à aujourd'hui montrent qu'il existe une quantité universelle qui est conservée au cours de toutes évolutions et/ou transformations de la matière et des champs : l' Énergie. En d'autres termes, cela signifie que les lois de la Physique ne changent pas au cours du temps. Tous les développements de la physique moderne sont liés à ce principe de conservation de l'énergie et consiste en l'étude des différentes manières de convertir l'énergie existante sous une forme donnée en une autre forme d'énergie. Par exemple, les réactions nucléaires de fission de 0.01 g d'uranium fournissent approximativement 1 kWh (1 kWh = 3.6 MJ) d'énergie thermique dans une centrale nucléaire. La même quantité d'énergie thermique peut être obtenue par combustion d'approximativement 100 g de pétrole, de charbon ou de gaz (1 million de tonnes de pétrole ou équivalent pétrole produit environ 4,4 TWh d'électricité dans une centrale électrique moderne), par condensation de 1,6 kg de vapeur d'eau ou en capturant le rayonnement solaire sur un panneau d'une surface de 1 m 2 pendant une heure (l'intensité de la lumière du soleil sur la Terre est d'environ 0,1 Le nucléaire et l'énergie hydroélectrique ne représentent que seulement 6 % de la consommation mondiale en énergie. Même si d'autres énergies renouvelables tels que l'éolien, la combustion de biomasse ou de déchets, l'énergie solaire ou encore les centrales géothermiques sont de plus en plus utilisés grâce aux subventions gouvernementales, leurs contributions restent toutefois négligeables. Comme toute fonction continue positive partant de zéro et arrivant à zéro, on peut montrer que l'extraction de chaque ressource naturelle terrestre atteindra tôt ou tard un maximum à un moment donné de notre Histoire, puis diminuera jusqu'à ce que la ressource ait totalement disparu de la surface de la Terre. On estime ainsi que nous disposons de 50 ans de réserves en gaz et en pétrole ainsi que d'environ 100 ans de réserves en charbon avant d'avoir consommé Cependant, en plus de ces contraintes de températures et de densités, la réaction de fusion nucléaire choisie doit être exothermique. Pour cela, il faut donc que la réaction consomme des noyaux atomiques les plus légers possibles afin d'avoir à vaincre une barrière Coulombienne moins énergétique. Aussi, la section efficace de la réaction de fusion (ou probabilité de réaction de fusion) doit être la plus grande possible. Cela implique par conséquent une réaction nucléaire consommant seulement deux noyaux, conservant le nombre de protons et de neutrons afin de limiter la force nucléaire faible et, enfin, produisant au moins un neutron en plus du noyau plus lourd produit afin de chauffer le liquide caloporteur circulant dans la couverture de la chambre de la centrale, permettant ainsi de chauffer de l'eau pour la production d'électricité à l'aide d'une turbine. 80 réactions de fusion nucléaire satisfont à ces critères. Cependant, la réaction la plus probable compte tenu de la technologie actuelle est la réaction de fusion des deux isotopes de l'atome d'Hydrogène confiné pendant le temps nécessaire τ c avec une densité suffisamment élevée n e . En effet, d'après le critère de J. Lawson, la température T ainsi que le temps de confinement τ Par exemple, la masse du Soleil d'environ 10 30 kg est suffisamment élevée pour attirer et comprimer le plasma stellaire à des densités pouvant aller jusqu'à n e ≈ 10 32 /cm 3 et des températures de l'ordre de T ≈ 10 keV pendant toute sa durée de vie d'à peu près τ c ≈ 10 milliards d'années. Du fait que, dans un plasma, les particules sont chargées électriquement, les physiciens soviétiques I. Tamm et A. Sakharov ont proposé dans les années 1950 l'idée d'utiliser de puissants champs magnétiques afin de confiner le plasma thermonucléaire. Ce dispositif expérimental appelé tokamak se présente sous la forme d'un tore. Il permet de confiner le plasma thermonucléaire grâce à un champ magnétique toroïdal produit par des bobines magnétiques supraconductrices entourant le tore alors qu'un autre champ magnétique poloïdal est créé par un courant électrique à l'intérieur du tore permettant ainsi le chauffage du plasma. Le projet internationale ITER planifie la construction d'un tel dispositif sur le site du CEA (Commissariat à l' Énergie Atomique et aux Énergies Alternatives) à Cadarache, en France, d'ici quelques années. mais sur des temps de confinement τ c beaucoup plus longs. Les schémas FCI conventionnels impliquent l'allumage d'un point chaud central de manière isobarique où le combustible de DT atteint une température de T ≈ 7 keV et une densité surfacique ρ.R d' environ 0,25 g/cm 2 , où R est le rayon de la coquille solide, pendant un temps de confinement τ c d'environ 40 ps. Afin d'atteindre ces conditions extrêmes, de nombreuses impulsions laser nanosecondes représentant une énergie totale de E L ≈ 1 MJ peuvent être utilisées afin d'irradier uniformément la coquille solide renfermant le combustible de DT. L'irradiation de la coquille peut se faire directement avec les impulsions laser (Attaque directe) ou par des rayons X produits par interaction laser-matière (Attaque indirecte). au même instant au niveau de la surface interne de la coquille, une onde de choc est transmise dans le DT gazeux, tandis qu'une onde de raréfaction est réfléchit dans la coquille. Lorsque cette dernière arrive à la surface d'ablation, la coquille est mise en vol et subit une forte accélération centripète. Du fait de sa symétrie sphérique, la capsule implose et le combustible de DT est alors comprimé à la densité souhaitée. La conversion de l'énergie cinétique de l'implosion en énergie interne à la fin de la phase de compression entraîne la création d'un point chaud central isobare à la température voulue. Des réactions de fusion auto-entretenues entre les atomes de D et de T sont alors initiées, générant une onde de combustion thermonucléaire sphérique divergente suivie d'une détonation, qui brûle la partie dense du combustible dans la coquille. Le temps de confinement tau c correspond à la durée de vie du point chaud centrale juste avant son explosion hydrodynamique. Les processus de compression et de chauffage de la cible imposent de nombreuses contraintes sur la rugosité et la symétrie sphérique des cibles ainsi que sur l'uniformité de l'irradiation par les impulsions laser pour l'attaque directe ou du rayonnement X pour l'attaque indirecte. Toutefois, l'allumage rapide présente toujours de nombreux problèmes à résoudre. En raison du fait que l'impulsion laser ultra-intense, utilisée pour générer le faisceau d'électrons, ne peut pas pénétrer les zones de la couronne ayant une densité supérieure à la densité critique du plasma, il est difficile de focaliser le faisceau d'électrons rapides d'allumage sur une toute petite zone de la partie dense du combustible. De plus, il a été démontré expérimentalement et numériquement que les faisceaux d'électrons relativistes accélérés par laser présentent nécessairement un angle de divergence important. Depuis, de nombreuses méthodes ont alors été proposées pour tenter de collimater le faisceaux d'électrons d'allumage. Cependant, de nombreuses expériences et simulations numériques sont toujours nécessaires afin de pouvoir les confirmer et les valider. Plus récemment en 2006, une autre méthode séparant les phases de compression et d'allumage du combustible de DT a été proposé par R. Betti. Il s'agit du schéma d'allumage par choc qui consiste à allumer un point chaud central à la fin de la phase de compression en générant un fort choc à Cesta près de Bordeaux, en France. Aussi, le gain thermonucléaire obtenu par allumage par choc peut être significativement plus grand que dans le cas isobare classique pour une énergie laser donnée. Enfin, vu qu'il implique des implosions de cible à faible vitesse de même que pour l'allumage rapide, le schéma d'allumage par choc est aussi plus robuste en ce qui concerne les instabilités hydrodynamiques au cours de l'accélération de la coquille. La puissance laser nécessaire pour générer une pression d'ablation de 300 Mbar dépend du matériau de la coquille et de ses dimensions. Néanmoins, elle peut être estimée à 120-200 TW. Cela représente une puissance laser un ordre de grandeur inférieure à la puissance requise par l'impulsion laser d'allumage rapide. des électrons donnés, ainsi que pour des distributions angulaires locales totalement anisotropes. Un paramètre dépendant de la direction moyenne locale de propagation des électrons du faisceau permet de relier ces deux régimes extrêmes. De manière évidente, le fait d'arrêter l'expansion de la fonction de distribution du faisceau d'électrons à l'ordre 1 entraîne nécessairement une perte d'informations concernant la distribution angulaire locale des électrons du faisceau. Par conséquent, nous avons dérivé l'équation d'évolution de l'entropie angulaire du faisceau afin d'étudier le critère que nous avons choisi. Notre analyse montre que le critère de G. N. Minerbo est justifié pour le transport de faisceaux d'électrons rapides accélérés par laser en raison de la diffusion angulaire des électrons du faisceau et de la propagation anisotrope des électrons les plus énergetiques. En outre, l'étude menée sur l'instabilité faisceau-plasma de filamentation résistive, pour laquelle il n'y a aucune raison de maximiser localement l'entropie angulaire, montre que notre modèle décrit le taux de croissance de l'instabilité avec une erreur de l'ordre de 10particulier d'un faisceau d'électrons monoénergétique. inertie des électrons du milieu et nous avons considéré l'expression de la pression d'un gaz parfait d'électrons. Ces hypothèses sont justifiées dans le cas du transport de faisceaux d'électrons accélérés par laser dans des cibles denses ainsi que dans le cas où la fréquence cyclotron des électrons du milieu est négligeable devant leur fréquence de collisions. Les champs électromagnétiques autogénérés dépendent donc de la résistivité électrique du matériau dans lequel se propage le faisceau ainsi que de ses gradients spatiaux. Ces derniers dépendent fortement de la température des électrons et du réseau cristallin/ions du matériau. Par conséquent, l'évolution des températures électronique et du réseau/ions du matériau sont aussi décrites de manière auto-cohérente suivant les équations bien température dans les équations de la chaleur qui, ici, ne sont pas négligées. Nous avons également proposé de nouvelles expressions pour les capacités thermiques, la résistivité électrique, la conductivité thermique électronique ainsi que le facteur de couplage des températures électronique et ionique, dans le cas particulier des métaux. En effet, sous l'action de son chauffage par le faisceau d' électrons accélérés par l'impulsion laser, un métal initialement à l'état solide à température ambiante (≈ 300 K) devient rapidement liquide avant de devenir un plasma avec des températures pouvant aller jusqu'à plusieurs keV. Les collisions des électrons du matériau avec les vibrations (phonons) et les électrons libres du réseau cristallin ainsi qu'avec les électrons liés aux atomes sont pris en compte, en accord avec des études récentes montrant l'importance de ces processus de relaxation. Les schémas numériques utilisés pour la résolution des équations M1 sont décrits en détail. Elles sont résolues avec des schémas explicites du second ordre, sauf pour le terme d'advection en énergie dû au ralentissement collisionnel des électrons rapides qui est résolu par le schéma d'advection décentré numériques utilisés pour résoudre les équations du modèle ont été validées grâce à leur application à un cas d'école de transport de faisceau d'électrons rapides mono énergétique et collimaté dans un plasma dense et tiède d'Hydrogène pour lequel nous avons pu déterminer des solutions analytiques afin de les comparer aux différentes quantités physiques calculées par le code. Par exemple, l'expression analytique de la distance de pénétration d'un électron relativiste dans un plasma dense et tiède d'Hydrogène que nous avons trouvé reproduit bien les résultats de la simulation. par le laser dans la cible irradiée. Dans notre modèle, la recirculation des électrons rapides dans la cible est prise en compte en imposant la réflexion spéculaire des électrons rapides au niveau des interfaces cible-vide ainsi qu'une seconde population d'électrons rapides décrite par un second système d'équations M1. La première population décrit les électrons rapides se propageant dans la direction de propagation de l'impulsion laser et la seconde population décrit les électrons rapides se propageant en sens contraire, réaccélérés dans la cible par les forts champs électriques de charge d'espace générés aux interfaces cible-vide lorsque la première population tente de s'échapper de la cible. Le modèle de calcul de l'émission de photons Kα est ensuite révisé. Les conclusions de cette étude sont multiples. Tout d'abord, nous avons montré que, dans le cas de cibles faites d'Aluminium ou de Cuivre, le pas de temps numérique imposé par la résolution des équations du transport d'électrons rapides peut être comparable à la durée de vie des absences électroniques ou "trous" du niveau d'énergie K des électrons atomiques, responsables de l'émission Kα. Par conséquent, nous avons pris en compte dans notre modèle la dynamique des trous en couche K des atomes du solide irradié par le laser. Deuxièmement, nous avons démontré que les propriétés du solide à basses températures doivent être prises en compte. En particulier, le facteur de couplage entre la température électronique

	tous les avantages des centrales à fission nucléaire sans ses inconvénients, c'est-à-dire, sans polluer l'environnement, sans provoquer d'éventuelles catastrophes nucléaires ou encore sans problèmes liés à la limitation des ressources terrestres. Seule la radioactivation éventuelle de matériaux environnant peut poser problème. Du fait du mouvement chaotique des particules chargées portées à de grandes températures T ≈ 10 keV (1 eV = 11600 K), un plasma thermonucléaire tend naturellement à s'expandre et il est difficile pulsion laser relativiste sur la cible. L'impulsion laser relativiste génère alors un faisceau d'électrons rapides qui va déposer son énergie en profondeur dans la partie plus dense du combustible juste à la fin de la phase de compression, créant ainsi un point chaud latéral isochore. La densité centrale requise est donc beaucoup plus faible (300 g/cm 3 au lieu de 1000 g/cm 3 ) que dans les schémas clas-siques d'allumage et les contraintes sur la convergence de l'onde de choc, la vitesse et la symétrie de l'implosion sont réduits. Notamment, la vitesse d'implosion étant plus faible, les contraintes imposées Ainsi, contrairement à l'allumage rapide, l'allumage par choc présente l'avantage qu'il ne nécessite pas d'installation laser de ultra forte puissance. Cependant, le couplage de l'impulsion laser d'allumage Dans un second temps, une simulation plus réaliste de transport d'électrons rapides accélérés par laser dans une cible mince d'aluminium a été comparée à une simulation PIC (Particle-In-Cell) hybride. de le maintenir Cependant, l'allumage des réactions de fusion est amorcé dans un second temps en focalisant une im-sur la cible et les impulsions laser vis-à-vis des instabilités hydrodynamiques se retrouvent amoindries avec la cible en implosion présente plusieurs problèmes toujours non résolus tels que les instabilités Elle montre que le modèle M1 pour le transport de faisceaux d'électrons relativistes est suffisamment

W/cm 2 lors d'une journée ensoleillée). 1 kWh représente aussi l'énergie potentielle gravitationnelle de 3 tonnes d'eau chutant de plus de 100 m d'altitude dans une centrale hydroélectrique, l'énergie cinétique de 20000 m 3 d'air se déplaçant à une vitesse de 60 km/h poussant une pâle d'éolienne ou encore l'énergie nécessaire pour un être humain de 65 kg pour grimper jusqu'à un sommet de montagne situé à une altitude de 3000 m.

Depuis le XIXe siècle, la croissance exponentielle des connaissances technologiques et scientifiques, rendue possible grâce à ce concept de conservation d'énergie, a conduit à une incroyable amélioration de la qualité de vie de l'Homme sur Terre ainsi qu'une augmentation fulgurante de la population mondiale. A titre d'exemple, il est frappant de constater la forte corrélation qu'il existe entre le produit national brut d'un pays et la consommation en énergie de ses habitants à l'heure actuelle.

Cependant, la combustion de ressources fossiles tels que le pétrole, le gaz et le charbon qui est toujours majoritairement utilisée aujourd'hui afin de produire de l'énergie. Ils représentent à eux seuls environ 90 % des ressources en énergie consommées dans le monde. tout ce qu'il en reste sur Terre. Ces évaluations sont probablement sous-estimées pour des raisons financières. Cependant, elles sont fondées sur des données datant de 2013 concernant le nombre d'être humains sur Terre alors que l'on s'attend à être plus de 11 milliards en 2100 (par rapport à environ 7,1 milliards en 2013). En outre, cette forte croissance démographique devrait se produire dans les pays où les demandes en ressources naturelles seront les plus élevés. On comprend donc facilement que, si nous la production d'électricité. D + T → He (3, 5 MeV) + n (14, 1 MeV) en raison de l'existence d'un état intermédiaire résonant lors de la réaction. D'un point de vue pratique concernant la production d'énergie électrique, il existe sur Terre une quantité quasi-infinie de Deutérium (D) avec une concentration de 33 g par tonne d'eau de mer; C'est pour cette raison que la production d'énergie par fusion d'un plasma de DT s'appelle l'énergie bleue. Le Tritium (T) peut être produit directement dans la centrale à l'aide d'une seconde réaction de fusion entre les neutrons (n), s'échappant du plasma thermonucléaire, et des noyaux de Lithium (Li), eux aussi très abondants sur Terre, préalablement placés dans la couverture de la chambre de la centrale. Ainsi, contrairement aux centrales nucléaires de fission préexistantes, qui utilisent des ressources limitées comme l'Uranium, le Plutonium ou le Thorium, l'énergie bleue ne fait face à aucun problème de limitation de ressources. En outre, la fusion d'un plasma thermonucléaire de DT ne présente aucun risque d' emballement des réactions en chaîne et ne produit que des déchets radioactifs à courte durée de vie (moins de 10 ans). En conclusion, une éventuelle centrale à fusion thermonucléaire aurait donc c d'un plasma thermonucléaire sont reliés par la densité n e du plasma si on veut pouvoir extirper du plasma plus d'énergie de fusion que d'énergie investie pour le créer et le maintenir confiné. Dans les Étoiles, le confinement du plasma est accompli naturellement par l'attraction gravitationnelle de l' Étoile sur elle-même.

La Fusion par Confinement Inertiel (FCI) est un autre moyen de produire et contrôler de l'énergie de fusion thermonucléaire. Suivant cette approche, les conditions de densités extrêmes (jusqu'à 1000 g/cm 3 ) sont obtenues grâce à la compression rapide d'une capsule sphérique solide de dimension millimétrique remplie d'un mélange de D et de T gazeux et cryogénisé. La conversion de l'énergie cinétique de l'implosion en énergie interne à la fin de la phase de compression entraîne le chauffage de la zone centrale, communément appelé "point chaud", jusqu'à une température T > 5 keV. Les réactions de fusion du combustible de DT sont ainsi initiés en accord avec le critère de J. Lawson.

Contrairement à la Fusion par

Confinement Magnétique (FCM), le plasma thermonucléaire est ici confiné par l'inertie de sa propre masse et non grâce à des champs magnétiques extérieurs. En outre, l'implosion des cibles ne dure que quelques nanosecondes. Par conséquent, cette approche entraîne des difficultés technologiques supplémentaires dues au taux de répétition du processus à 10 Hz, imposé par la production continue d'électricité. Cependant, en atteignant des densités n e si élevées pendant un temps de confinement τ c si bref, l'approche FCI est beaucoup plus efficace en termes de gain de production par rapport à la FCM qui, elle, vise à fusionner les isotopes de D et T à de faibles densités n e Dans les deux cas, les couches externes du solide irradié sont ablatés par la lumière. Cette ablation de matière entraîne ensuite l'implosion de la cible par effet fusée, i.e. par conservation de la quantité de mouvement. L'évolution temporelle des impulsions laser de Nuckolls-Kidder est choisie de telle sorte que la pression d'ablation de la coquille génère une onde de choc suivie d'une succession continue d'ondes de compression sphériques convergentes dans la cible. Au moment précis où les ondes de compression arrivent Ces contraintes sont multifactorielles. Tout d'abord, les instabilités paramétriques liées à l'interaction des impulsions lasers avec le plasma en expansion, appelé "couronne", peuvent réduire l'efficacité de conversion de l'énergie laser en pression d'ablation de la coquille ainsi qu'entraîner des inhomogénéités de la surface d'ablation. D'autres part, la génération d'électrons rapides inhérent à l'interaction laser-plasma dans la couronne entraîne un préchauffage de la cible. Cela conduit à l'augmentation de l'entropie de cette dernière et limite par conséquent la bonne compression du combustible. Enfin, du fait d'une vitesse d'implosion élevée et de la non-uniformité de la surface d'ablation, les instabilités hydrodynamiques peuvent briser la coquille pendant son implosion et entraîner le mélange du combustible chaud avec le combustible froid. Afin de relaxer ces contraintes liées à la compression et au chauffage simultanés du combustible, M. Tabak a proposé en 1994 de séparer la phase de compression de la phase de chauffage. Dans ce schéma qualifié d'allumage rapide, la coquille est implosée et le combustible est densifié de manière analogue aux schémas conventionnels de FCI. et le risque de briser la coquille lors de l'implosion est réduit. Aussi, un faisceau d'électrons relativistes accéléré par laser peut fournir un chauffage beaucoup plus efficace de la matière dense et des gains beaucoup plus élevés peuvent être obtenus comparé aux scénarios classiques. En outre, la phase de compression a besoin de beaucoup moins d'énergie (200 -300 kJ) comparée aux schémas conventionnels et le coût d'une éventuelle centrale à fusion thermonucléaire par allumage rapide serait donc plus faible. l'aide d'une pression d'ablation supérieure à 300 Mbar. En convergeant ensuite vers le centre de la coquille en implosion, la force du choc va en augmentant. Lorsque ce dernier entre en collision avec sa propre réflexion au centre de la coquille, le point chaud central est boosté, libérant ainsi l'énergie de fusion thermonucléaire. Pour ce schéma d'allumage, des coquilles cryogéniques de grande masses ainsi qu'une faible vitesse d'implosion et un faible adiabat (quantité physique mesurant l'entropie de la cible) peuvent aussi être utilisés. Par conséquent, le combustible de DT atteint de grandes densités surfaciques et permet d'atteindre l'allumage des réactions de fusion avec une énergie inférieure à celle de l'allumage central isobarique conventionnel. Le choc d' allumage peut être lancé par une puissance laser compatible avec les lasers existants pour l'étude de la FCI comme le NIF (National Ignition Facility) du LLNL (Lawrence Livermore National Laboratory) aux Etats-Unis d'Amérique ou le LMJ (Laser MegaJoule) du CEAparamétriques dans la couronne, les instabilités hydrodynamiques de la coquille et le rôle des électrons rapides accélérés dans la couronne sur l'ablation de la coquille dans ce régime particulier d'interaction laser-plasma. Cette thèse s'inscrit directement dans ce contexte. Elle a en effet consisté à développer un nouveau énergie cinétique Dans notre modèle, nous considérons des échelles de temps grandes devant le temps caractéristique de la neutralisation électromagnétique du faisceau d'électrons. Par conséquent, de même que les autres modèles hybrides, notre modèle suppose que le faisceau ne subit pas de modifications importantes durant sa neutralisation en charge et en courant électriques. Ceci est une hypothèse grossière dans le cas particulier du transport de faisceau d'électrons accélérés par laser dans des matériaux isolants du fait des processus d'ionisation qui se produisent à cette échelle de temps et qui impliquent une perte en énergie supplémentaire pour le faisceau. Dans notre modèle hybride, le champ magnétique auto-généré par le faisceau vérifie l'équation de diffusion couramment utilisée avec les différents termes sources dus aux gradients de résistivité électrique du milieu, au rotationnel du courant électrique du faisceau ainsi qu'aux gradients croisés de température et de densité des électrons du milieu tandis que le champ électrique auto-généré par le faisceau est donné par la loi d'Ohm quasi-statique classique. Ainsi, nous avons négligé l'aimantation du milieu, la viscosité des électrons du milieu, leur friction sur les électrons du faisceau, le courant de déplacement dans l'équation de Maxwell-Ampère (approximation quasi-statique), l'connues de la chaleur. Dans notre modèle, nous négligeons le mouvement des ions et la conductivité thermique ionique, considérant des échelles de temps plus petites que la dizaine ou la centaine de picosecondes. Par ailleurs, les équations de la chaleur résolues sont en accord avec les hypothèses faites à l'égard des champs électromagnétiques auto-générés (pas de viscosités, aucune aimantation, pas d'inertie et pas de friction sur les électrons du faisceau), sauf en ce qui concerne les dérivées temporelles de la d'ordre 1. Concernant les termes d'advection spatiale et d'advection en énergie dûs au champ électrique auto-généré, nous avons utilisés les schémas HLL (A. Harten, P. Lax et B. Van Leer) permettant d'assurer un nombre positif d'électrons et une norme du vecteur d'anisotropie (vecteur associé à la direction moyenne locale de propagation des électrons) inférieure à l'unité. Des schémas implicites ont également été développés pour les effets collisionnels afin de diminuer la contrainte sur le pas de temps numérique imposée par la condition CFL (R. Courant, K. Friedrich et H. Lewy) dans le cas d'un transport de faisceau d'électrons rapides dans des plasmas très denses. Les équations d'évolution des champs électromagnétiques auto-générés sont résolues à l'aide de schémas explicites du second ordre sauf en ce qui concerne le terme de diffusion magnétique qui est résolu semi-implicitement grâce à une discrétisation de l'opérateur de diffusion au second ordre ainsi qu'à l'inversion de la matrice de diffusion obtenue par la méthode des gradients conjugués. Enfin, les deux équations de la chaleur sont discrétisées à l'aide de schémas numériques explicites du second ordre. L'ensemble de ces schémas précis pour reproduire les résultats de la simulation PIC hybride. Une fois le modèle théorique et numérique validé, nous avons pu appliqué le modèle à l'étude de l'émission de photons Kα induit par le transport d'électrons relativistes dans des cibles solides ou des plasmas denses. Lors des expériences d'interaction laser-matière, ce processus est en effet souvent utilisé pour diagnostiquer le passage des électrons accélérés

  Obviously, g 2 is particle-symmetric, i.e. it has the same values by permuting the two electrons. By injecting(A.16) in the first order equation (A.12), we obtain , t)= d 3 r 2 d 3 p 2 e 2 r 1 -r 2 |r 1 -r 2 | 3 f e,1 (r 2 , p 2 , t)is the Coulomb electrostatic force averaged over the momenta and positions of the electron 2 and∂ f e,1 ∂t d 3 r 2 d 3 p 2 e 2 r 1 -r 2 |r 1 -r 2 | 3 . ∂g 2 ∂p 1is the collision integral that has to be estimated according to the second orderEquation (A.14). In a first attempt, we neglect the right hand side of Equation (A.14) accounting for 3-body correlations.Also, assuming that f 2 evolves in time mainly due to changes in f 1 , rather than to changes in the pair correlations g 2 , we neglect the time derivative of f 2 in (A.14) to get + e 2 r 1 -r 2 |r 1 -r 2 | 3 .

	∂ f e,1 ∂t	+ v 1 .	∂ f e,1 ∂r 1	+ F 1 .	∂ f e,1 ∂p 1	=	∂ f e,1 ∂t	col	(A.17)
	where
	F 1 = e (r 1 v 1 . ∂Φ ∂r 1 ∂ ∂r 1 + v 2 . ∂ ∂r 2	f e,2 ∂ ∂p 1	-	∂ ∂p 2	f e,2 = 0.	(A.18)

2 , p 2 , t). The binary collisions between electrons 1 and 2 can be accounted for by the correlation function

g 2 f e,2 (r 1 , p 1 , r 2 , p 2 , t) = f e,1 (r 1 , p 1 , t) f e,1 (r 2 , p 2 , t) + g 2 (r 1 , p 1 , r 2 , p 2 , t). (A.16) col = -1 N e

  2 , p 2 , t)(A.19)By renormalizing the distribution functions and performing the integration of this last equation, introducing the differential cross section d 2 σ and the exchanged momentum ∆p in such a electronelectron binary collision, we finally obtain the Boltzmann equation -d 3 p 2 d 2 σ |v 2 -v 1 | . f e,1 (r 1 , p 1 , t) f e,1 (r 1 , p 2 , t) -f e,1 (r 1 , p 1 -∆p, t) f e,1 (r 1 , p 2 + ∆p, t) (A.21)By generalizing this result to the electron-ion collisions and by taking into account all terms in the one-body force F 1 as in the Vlasov equation (A.8), we obtain the Vlasov-Boltzmann equationB ee [f e,1 , f e,1 ] (r 1 , p 1 , t) + C B ei [f e,1 , f i,1 ] (r 1 , p 1 , t) ee [f e,1 , f e,1 ] (r 1 , p 1 , t) = -2 σ ee |v 2 -v 1 |. f e,1 (r 1 , p 1 , t) f e,1 (r 1 , p 2 , t)

	where the Boltzmann collision integral reads
	∂f e,1 ∂t col = ∂f e,1 ∂t + ∂ ∂r 1 = ∂f e,1 ∂t col	(v 1 f e,1 ) -= C (A.22) ∂ ∂p 1 e E (r 1 , t) + v 1 × B (r 1 , t) f e,1 c
	where
	C B		d 3 p 2
		R 3	R 3
	∂f e,1 ∂t	+ v 1 .	∂f e,1 ∂r 1	+ F 1 .	∂f e,1 ∂p 1	=	∂f e,1 ∂t col	(A.20)

d

  r, p α,1 , t) Secondly, one can show that if f e,1 is the solution of the Landau equationE ext + E p (r e,1 , t) + v e,1 c × (B ext + B p (r e,1 , t)) f e,1 = ∂f e,1 ∂t coll = C L ee [f e,1 , f e,1 ] (r e,1 , p e,1 , t) + C L ei [f e,1 , f i,1 ] (r e,1 , p e,1 , t) , .3 Simplified Forms of the Collision Integral A.3.1 BGK Approximation A simplified form of the collision term was proposed by P. L. Bhatnagar, E. P. Gross and M. Krook. (f e,1 -f M ) (A.77) where ν is a velocity-dependent effective collision frequency which characterize the time needed by f e,1

	It reads
						∂f e,1 ∂t col	= -ν
									(A.71)
									2 2m α p α,1	d 3 p α,1 = 0
	and
	R 3
	R 3	C L ei [f e,1 , f i,1 ] (r, p e,1 , t)	p e,1 2m e 2	d 3 p e,1 +	R 3	C L ie [f i,1 , f e,1 ] (r, p i,1 , t)	2 2m i p i,1	d 3 p i,1 = 0. (A.72)
	∂f e,1 ∂t	+	∂ ∂r e,1	(v e,1 f e,1 ) +	∂ ∂p e,1	e (A.73)
	thus, the entropy defined as
					H (r e,1 , t) = -	(f e,1 ln f e,1 -f e,1 ) d 3 p e,1	(A.74)
							R 3
	verifies the H-theorem
							dH dt	≥ 0.	(A.75)

C L ei [f e,1 , f i,1 ] (r, p e,1 , t) 1 d 3 p e,1 = R 3 C L ie [f i,1 , f e,1 ] (r, p i,1 , t) 1 d 3 p i,1 = 0 R 3 C L ei [f e,1 , f i,1 ] (r, p e,1 , t) p e,1 d 3 p e,1 + R 3 C L ie [f i,1 , f e,1 ] (r, p i,1 , t) p i,1 d 3 p i,1 = 0

Finally, it can be shown that the Maxwell-Boltzmann distribution, maximizing the entropy H, f M (r e , p e , t) = n e (r e , t)

(2πm e k B T e )

3/2 exp -m e (v e -u e ) 2 2k B T e (A.76) is still the stationary solution of the Landau equation (A.73).

A

  eα (v e ) |p e | 2 Ip e ⊗ p e 2 (A.78) where ν eα (v e ) = 4πn α e 2 q α 2 ln Λ eα m e 2 |v e | 3 . (A.79) One can directly notice that the ratio of the two isotropization rates 10 assuming ln Λ ei and ln Λ ee have approximatively the same order of magnitude. Thus, the omission of the electron-electron diffusion term compared to the electron-ion one is fully justified for a plasma with Z * 10 and the Lorentz approximation leads to ∂f e ∂t col = ν ei (v e )|v e | 3 ∂ ∂p e . |p e | 2 Ip e ⊗ p e 2|v e | 3

										62)
	reads :
	∆p ⊗ ∆p α 2∆t = ν ν ei ν ee = Z * ln Λ ei ln Λ ee	1.			(A.80)
	for material with Z .	∂f e ∂p e
	=	ν ei (v e ) 2	1 sin θ e	∂ ∂θ e	sin θ e	∂f e ∂θ e	+	1 sin 2 θ e	∂ 2 f e ∂ϕ e 2

*

  r, p, t). It reads L ee [f e , f e ] (r, p, t) + C L ei [f e , f i ] (r, p, t)where for α = i or e,C L eα [f e , f α ] (r, p, t) = -

	∂f e ∂t	+	∂ ∂r	. (vf e ) -	∂ ∂p	. e E (r, t) +	v c	× B (r, t) f e =	∂f e ∂t col	(B.1)
	with
			∂f e ∂t coll	= C ∂ ∂p	.	∆p α ∆t	f e -	∆p ⊗ ∆p α 2∆t	.	∂f e ∂t	(B.2)
	with	∆p α ∆t	= -4π	e 2 q α	2 ln Λ eα m e	R 3

vv α |v -v α | 3 f α (r, p α , t) d 3 p α (B.3)

and ∆p ⊗ ∆p α

  Indeed, according to the Landau collision integral property (see Appendix A, section A.2.4), one hasR 3 m e v 2 2 C L ee [f e , f e ] (r, p, t) d 3 p = 0.Concerning electron-ion collisions, we define the energy exchange rateW ei (r, t) = ei [f e , f i ] (r, p, t) d 3 p. (B.17)By substracting the work of the friction force, one gets the relationW ei (r, t) = Q ei + R ei .u e (B.18)whereQ ei (r, t) = R 3 m e (v -u e ) 2 2 C L ei [f e , f i ] (r, p, t) d 3 pis the thermal electron-ion equilibration power. By developing (B.16) and using the hydrodynamic electron momentum conservation equation (B.13), one finds the electron hydrodynamic electron energy By multiplying (B.15) by m e u e , one can find the hydrodynamic electron kinetic energy conservation = -n e eE.u e -u e . ∂ ∂r .(P e Iτ e ) + R ei .u e .By performing the same calculi for the ions starting from the Vlasov-Fokker-Planck-Landau equation for the ion distribution function f i = f i (r, p, t), one gets respectivelyIτ i ) + R ie . Iτ i ) .u i ] -∂ ∂r .q i + n i Z * eu i .E + W ie ie = -R ei , W ie = -W ei and thus Q ie = -Q ei + R ei . (u e -u i ) .

	∂ ∂t equation in the form . [(P equation 1 2 n e m e u e 2 + n e ∂ 3 2 ∂t = -∂ ∂r n e n e k B T e + n R 3 ∂ ∂r . 3 2 n e k B T e + 1 2 m e v 2 2 C L + u e . ∂ ∂r 1 2 m e u e 2 + 3 2 k B T e ∂ ∂t + u e . ∂ ∂r 1 2 m e u e 2 Substracting (B.19) by (B.20), one gets the hydrodynamic electron internal energy equation C V,e ∂ ∂t + u e . ∂ ∂r (T e ) + ∂ ∂r .q e = -P e ∂ ∂r .u e + τ e ∂ ∂r ⊗ u e + Q ei . ∂n i ∂t + ∂ ∂r . (n i u i ) = 0, m i ∂ ∂t (n i u i ) + ∂ ∂r which consequently provides the non conservative form m i n i ∂ ∂t + u i . ∂ ∂r (u i ) = n i Z * e E + u i c × B -∂ ∂r . (P i (B.24) (B.20) (B.21) (B.22) Also, one obtains ∂ ∂t 1 2 n i m i u i 2 + 3 2 n i k B T i + ∂ ∂r . 3 2 n i k B T i + 1 2 n i m i u i 2 .u i = -∂ ∂r . [(P i (B.25) which provides the non conservative form C V,i ∂ ∂t + u i . ∂ ∂r (T i ) + ∂ ∂r .q i = -P i ∂ ∂r .u i + τ i ∂ ∂r ⊗ u i + Q ie (B.26) where obviously R (B.27) . (n B.2. PLASMA TRANSPORT COEFFICIENTS B.2 Plasma Transport Coefficients

e m e u e 2 .u e = -∂ ∂r . [(P e Iτ e ) .u e ] -∂ ∂r .q e -n e eu e .E + W ei (B.16) e Iτ e ) .u e ] -∂ ∂r .q e -n e eE.u e + R ei .u e + Q ei (B.19

) i u i ⊗ u i ) = n i Z * e E + u i c × B -∂ ∂r . (P i Iτ i ) + R ie (B.23)

The system consisting of the hydrodynamic equations {(B.12), (B.15, B.21), (B.22), (B.24), (B.26)}

  + 0.2 ln Z * + 3.44 .

	and	δ T 0.4	≈	Z	Z *

. They read

γ E ≈ Z * + 0.9833 Z * + 2.4101 *

  Also, for numerical reasons which will be explained further,Equation (B.62) is simplified according to (B.58) and expressed in the Lagrangian formalism to give

											∂r	.u ( r i , t) = 0,	(B.69)
				ρ i m	d u i dt	+	∂ ∂r	(P e + P i ) ( r i , t) = 0,	(B.70)
	ρ i m	d ε i e dt	+ P i e	d V i m dt			-	∂ ∂r	. κ e	∂T e ∂r	( r i , t) = -Q i ie + W i e,ext	(B.71)
	and
		ρ i m		d ε i i dt	+ P i i	d V i m dt	-	∂ ∂r	. κ i	∂T i ∂r	( r i , t) = Q i ie .	(B.72)
	Indeed, as mentioned, the time derivative of ξ following a fluid particle reads
				d ξ i dt	=	∂ξ ∂t	r i , t + u r i , t .	∂ξ ∂r	r i , t .
		ρ i m	d E i dt	+		∂ ∂r	. (P u) ( r i , t) = -	∂ ∂r	. (q) ( r i , t)	(B.73)

  of electron momentum (B.24) is written with the Braginskii collisional terms as which accounts for the Nernst effect. Coupled to the Maxwell equations in the quasi-static approximation, the generalized Braginskii Ohm's law gives

	E -	j n e ec	× B +	u i c	× B +	k B en e	β.	∂T e ∂r	-η.j +	1 n e e	∂P e ∂r	= 0.	(B.77)
	This equation is called the generalized Ohm's law. The term
						E Hall =	j n e ec	× B	(B.78)
	accounts for the Hall effect and will be neglected since |j|	n e ec while in the 2D Cartesian or 2D
	axisymetric configuration, one can write
				k B en e	β.	∂T e ∂r	=	u Nernst c	× B	(B.79)
	where
			u Nernst,⊥ =	k B c n e e|B|	β ⊥	∂T e ∂r ⊥
	and
			u Nernst,∧ =		k B c n e e|B|	β ∧	B |B|	×	∂T e ∂r
			∂B ∂t	+		∂ ∂r	×		ηc 2 4π	.	∂ ∂r	× B
				-= -∂ ∂r k B c × [(u i + u Nernst ) × B] n e e ∂n e ∂r × ∂T e ∂r .

  can be written in the Lagrangian form (in the 2D Cartesian or 2D axisymetric geometry!) B.3. RADIATION HYDRODYNAMIC CODE CHIC Numerical scheme : The numerical computation of Equation (B.81) is presented here in the 2D Nernst,z(rB θ )] ( r i , t) + ∂ ∂r [u Nernst,r (rB θ )] ( r i , t)The 2D Cartesian case can be obtained in the same way by removing all terms ∝ 1/r in the right hand side of this equation, by replacing z by x, r by y and rB θ by B y . The compution is split in two steps 1. Firstly, the magnetic field generation due to density-temperature crossed gradients and the Nernst advection are solved explicitely using a finite volume method. Let us note Nernst,z (rB θ )] Nernst,r (rB θ )]

	axisymetric case :
	d dt	r i B θ	i	-	∂ ∂z	ηc 2 4π	∂ ∂z	(rB θ ) ( r i , t) -	∂ ∂r	ηc 2 4π	∂ ∂r	(rB θ ) ( r i , t)
					+ [u = ∂ ∂z 1 r u Nernst,r (rB θ ) ( r i , t)	-	1 r	ηc 2 4π	∂ ∂r	(rB θ ) ( r i , t)
					-	k B c n e e	∂n e ∂r	×	∂T e ∂r		( r i , t)	.
	S n B = r i B θ	i n	-∆t	k B c n e e	∂n e ∂r	×	∂T e ∂r	i,n	-∆t	∂ ∂z	[u i,n	-∆t	∂ ∂r	[u i,n
						d B i dt	+			∂ ∂r	×	ηc 2 4π	.	∂ ∂r	× B ( r i , t)
									-× (u = -∂ ∂r k B c n e e ∂n e ∂r	×	∂T e ∂r	( r i , t).	(B.81)

Nernst × B) ( r i , t)

  to deduce r i B θ

						i n+1	from
	r i B θ	i n+1	-∆t	∂ ∂z	ηc 2 4π	∂ ∂z	(rB θ )	n+1	-∆t	∂ ∂r	ηc 2 4π	∂r ∂	(rB θ )	n+1
		= S n B + ∆t	1 r	u Nernst,r (rB θ )	i,n	-∆t	1 r	ηc 2 4π	∂ ∂r	(rB θ )

i,n

Figure 8.11: Electron beam density n b [cm -3 ] from the M1 simulation at t = 25 fs (a), t = 50 fs (b), t = 75 fs (c), and t = 100 fs (d),

Figure 9.13: 2D maps of the beam density n b [cm -3 ] at t = 127 fs, t = 172 fs, t = 240 fs and t = 492 fs, corresponding 2D maps of the beam current density |j b | and target electron temperature T e , self-generated magnetic field B y and the number of Kα and Kβ photons n Kα and n Kβ , emitted from the Copper tracer layer at t = 1487 fs.

as we expected. The left panel of Figure9.18 illustrates the different energy balance for the 3D simulation. By comparing it with the left panel of Figure9.12 for the corresponding 2D simulation, one can clearly see strong 3D effects. While the maximum electric energy in the target weakly varies, the maximum magnetic energy U B ≈ 40 10 -2 mJ (compared to U B ≈ 60 10 -3 mJ) is greater than in the 2D simulation since the self-generated magnetic field diffuse in the third dimension. Also, since we let the fast electrons propagate in the third dimension, the longitudinal beam current density j b,z is lower in 3D during the first passage of the fast electrons through the target. Consequently, the Ohmic heating by the return current U res ≈ 8 mJ is smaller than in 2D, where U res ≈ 15 mJ. It also explains why the number of Kα photons emitted in the 3D simulation is greater than in the 2D simulation. Indeed, fast electrons lose less energy due to their slowing down by the self-generated electric field compared to the 2D simulation. Therefore, the fast electrons are more energetic and ionize more K-shell electrons. This explains also why the 3D simulation shows greater collisional losses.

de densité et de température optimales de la cible au moment de l'allumage ainsi que les propriétés optimales du faisceau d'électrons lui-même. Par ailleurs, différentes méthodes ont déjà été proposées afin de collimater le faisceau d'électrons rapides d'allumage. Par exemple, des simulations à l'échelle de l'allumage rapide d'une cible de FCI avec un cône préalablement inséré dans la coquille et présentant des gradients de résistivités du type de ceux proposés par A.Robinson (2013) ou encore avec un champ magnétique externe comme proposé par S.Fujioka (2013) peuvent être réalisées en prenant en compte l'aimantation de la cible ainsi que les effets tridimensionnels.

Chapter 7

Numerical Implementation of the Model "I have tried to read philosophers of all ages and have found many illuminating ideas but no steady progress toward deeper knowledge and understanding. Science, however, gives me the feeling of steady progress: I am convinced that theoretical physics is actual philosophy. It has revolutionized fundamental concepts, e.g., about space and time (relativity), about causality (quantum theory), and about substance and matter (atomistics), and it has taught us new methods of thinking (complementarity) which are applicable far beyond physics."

Max Born

Chapter 8

Validation of the Model

"There are two possible outcomes: if the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery." We conclude that there is a strong dependence of refluxing effects on the fast electron transport with the target thickness and that the refluxing effects on the emission of Kα photons decrease with the target thickness.

In the Copenhagen quantum approach, electrons trajectories does not exist anymore and one has to solve the Dirac equation for fermions, verified by the N e -body wave function. Also, the electron dynamics cannot be resolved without considering the electron antiparticles namely the positrons.

However, in the De Broglie-Bohm quantum approach, also usually called the pilot wave approach, developped in the non relativistic case, one can solve Newton-like particle trajectory equations with a quantum force depending on the N e -boby wavefunction, which is a solution of the Schrodinger equation [Bohm, 1952a] [Bohm, 1952b]. It can be done exactly with the same technique as one uses to solve the Lorentz-equation depending on the electromagnetic field (respectively the wave function) which are the solution of the Maxwell's equations (respectively the Schrodinger equation). In this approach, which is mathematically strictly equivalent to the Copenhagen approach, particles trajectories remains true, r and p not being hidden variables [Bell et al., 1964]. This quantum approach is however out of scope in this thesis and we will only consider the Classical and not Quantum Physics.

The resolution of the N -body problem could be the more accurate method for plasma studies.

However, it is impossible to solve it analytically and it can be done numerically only for a small number of particles N = N e + N i . In the usual plasma physics cases of a very large number of N e electrons and N i ions, one may adopt a statistical approach. In this case, the electron and ion populations are described by the N α -body distribution functions

where r n is the random variable for the position in space of the particle (α, n) and p n is the random variable for its momentum.

This Appendix is dedicated to the Classical Plasma Kinetic Theory (without Relativistic effects), allowing to estimate the distribution functions f α,Nα . For simplicity, we neglect external electromagnetic fields and assume that the ions are immobile.

A.1 BBGKY Hierarchy

A.1.1 Vlasov Equation

In a first attempt, let us neglect collisions between electrons and between electrons and ions. In this particular case, one can assume that all electrons are independent. It follows from the Theory of Probability that

and the problem is reduced to study of the 1-body distribution function

The 1-body electron distribution function f e,1 (r, p, t) is the probable number of electrons per unit of the phase-space infinitezimal volume d 6 V 1 = d 3 rd 3 p located between (r, p) and (r + d 3 r, p + d 3 p)

with a given initial condition f e,1 (t = 0). Indeed, since all electrons evolve in the same way according to (A.6), it is sufficient to characterize their properties in the 6-dimensional phase space (r, p). By assuming that the total number N e of electrons is conserved in this 6-dimensional infinitezimal phase space (no chemichal reactions, no quantum electrodynamic effects, no radiation losses, . Here, v = p/m e is the the velocity random variable of the electrons and

is the Lorentz force acting on the electrons deduced from the self-consitent Maxwell's equation (A.1)

with the source terms ρ = -e

where n i is the ion density. It can be shown that the Vlasov equation (A.8) remains true in the relativistic regime [Weibel, 1967b]. The only difference is that, one has to account for the relativistic relation between the velocity and momentum v = p/γm e in (A.8) and (A.9) where γ = 1 + (p/m e c) 2

is the Lorentz factor.

A.1.2 Liouville Theorem

The N e -body electron distribution function f e,Ne = f e,Ne (r 1 , p 1 , ..., r n , p n , , ..., r Ne , p Ne , t) is the density of probabilty of all electrons in the the phase space infinitezimal volume d 6Ne V located between (r 1 , p 1 , ..., r n , p n , , ..., r Ne , p Ne ) and (r 1 + d One can also consider that the 6N e -dimensional infinitezimal phase space volume is incompressible.

Then, the conservation of N e electrons between t and t + dt in the infinitezimal phase-space volume

A.1.3 BBGKY Hierarchy

In this section, we assume the static approximation (A.4) and neglect the magnetic force in order to simplify the presentation. Equation (A.11) can be reduced to the to 6-dimensional phase space by integrating it over 6(N e -1) phase coordinates i.e. over r 2 , p 2 , ..., r Ne , p Ne :

∂f e,2 ∂p 1 .

(A.12)

A.2 Vlasov-Fokker-Plank-Landau Equation A.2.1 Differential Rutherford Cross Section Let us consider the binary Coulomb collision between a non-relativistic charged particle (1) of mass m 1 and a charge q 1 moving with the velocity v 1,i before t = 0 with the momentum p 1,i and a non-relativistic charged particle (2) of mass m 2 and a charge q 2 moving with the velocity v 2,i before t = 0 with the momentum p 2,i in the laboratory frame. These particles collide at the time t = 0 at the origine O. Let us note v 1 or 2,f and p 1 or 2,f their velocities and momenta after the collision time t = 0. The system consisting of these two particles {(1) + ( 2)} interacting with each other at t = 0 is equivalent to a system of one effective particle of mass µ = m 1 m 2 /(m 1 + m 2 )

with a velocity v rel = v 1 -v 2 , a momentum P = µv rel and a charge q 1 interacting with a Coulomb potential q 2 r/|r| 3 located at the origine O com ≡ O in the center-of-mass frame moving at the velocity

. It has been noted r = r 1 -r 2 . This equivalence follows from the conservation of the energy and momentum of the system, which means that the center of mass motion is not affected by the collision:

where it has been noted the values in the center of mass frame within the supscript * (see Figure

A .2). According to the Newton's law expressed in the center of mass frame, the momentum variation of the fictitious particle reads

According to the law of conservation of angular momentum for a central force, one has necessarily

where θ * is the angle between the momenta of the fictitious particle before (P i ) and after (P f ) its diffusion while

is the Landau length i.e. the classical minimal distance between the fictitious particle and the diffusion center. According to the law of energy conservation for the fictitious particle (A.41),

It leads consequently to

Finally, due to the fact that u = cos (θ * /2)b/b -sin (θ * /2)P i /P i , one deduces from (A.43) and (A.45) the variation of the momentum of the fictitious scattered particle where the averaged electron-ion collision frequency is given by

(B.48)

The expression of the electron-ion coupling power Q ei (B.39) depending on ν ei is therefore also modified by the electron degeneracy according to (B.48). Here,

are the Fermi-Dirac integrals (There is a mistake in the original paper [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] concerning the numerator of A α , it is F 2 and not F 3 ). Thus, according to [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] the Lorenz factor reads

In the non-degenerate limit i.e. when µ/k B T e → -∞, one has A α → 32/3π and A β → 128/3π which reproduces the transport coefficients in the non-degenerate case considered by [START_REF] Spitzer | Transport Phenomena in a Completely Ionized Gas[END_REF] without the electron-electron correction factors. At the complete degeneracy limit i.e. when µ/k B T e → ∞, the conductivities of totally degenerated plasmas are recovered. Indeed, one has A α → 1, A β → π 2 /3 and ν ei → 3π 3 /2m e Z * e 4 ln Λ ei (second mistake in [START_REF] Lee | An electron conductivity model for dense plasmas[END_REF] where it is written 4 and not 2 in the denominator) which agree with the results given by [Hubbard, 1966] [START_REF] Franz | Ueber die Wärme-Leitungsfähigkeit der Metalle[END_REF]] is also recovered. The only difference with the results found by [Hubbard, 1966] is the Coulomb logarithm which is defined in [Hubbard, 1966] as ln Λ ei = 1/G Γ (κ F )

where Γ = Z * 2 e 4 /k B r i T i is the ion-ion coupling parameter and κ F = r i (2m e k B T F ) 1/2 / the Fermi wavenumber. [Hubbard, 1966] provides a tabulation of G Γ depending on Γ and κ F .

B.3 Radiation Hydrodynamic