A Pineau

B Chimier

S X Hu

G Duchateau

Improved modeling of the solid-to-plasma transition of polystyrene ablator for laser direct-drive inertial confinement fusion hydrocodes

The target performance of laser direct-drive inertial confinement fusion (ICF) can be limited by the development of hydrodynamic instabilities resulting from the non-homegeneous laser absorption at the target surface, i.e. the laser imprint on the ablator. To understand and describe the formation of these instabilities, the early ablator evolution during the laser irradiation should be considered. In this work, an improved modeling of the solid-to-plasma transition of a polystyrene ablator for laser direct-drive ICF is proposed. This model is devoted to be implemented in hydrocodes dedicated to ICF which generally assume an initial plasma state. The present approach consists of the two-temperature model coupled to the electron, ion and neutral dynamics including the chemical fragmentation of polystyrene. The solid-to-plasma transition is shown to significantly influence the temporal evolution of both free electron density and temperatures, which can lead to different shock formation and propagation compared with an initial plasma state. The influence of the solid-to-plasma transition on the shock dynamics is evidenced by considering the scaling law of the pressure with respect to the laser intensity. The ablator transition is shown to modify the scaling law exponent compared with an initial plasma state.

I. INTRODUCTION

Laser direct-drive inertial confinement fusion consists in focusing high-energetic lasers onto a target in order to compress it and create nuclear fusion reactions [1][2][3][4]. Typical targets are usually made of a cryogenic deuterium-tritium (DT) shell filled with a gaseous DT mixture. The cryogenic DT shell is covered with a plastic layer, namely the ablator, which is often polystyrene [5]. By absorbing the laser energy, the ablator initially in a solid state is converted into a plasma and expands leading to the compression of the target through the so-called rocket effect. Intensity modulations coming from laser speckles or beam-to-beam interactions can lead to a non-homogeneous laser absorption and density modulations at the target surface, which can be seeds of Rayleigh-Taylor and Richtmyer-Meshkov instabilities, i.e. the laser-imprinting process [6]. The latter reduces drastically ICF target performances [7][8][9]. To understand and control the formation of such instabilities, the ablator evolution during the laser irradiation should be taken into account, especially the solid-toplasma transition.

The solid-to-plasma transition is generally neglected in ICF hydrocodes by assuming a negligible transition time compared with the total laser pulse duration. Simulations start with an ablator in the plasma state with an electron density higher than the critical density of the corresponding laser. However, a recent study has shown that the timescale of this transition is of the order of 100 ps for a conventional laser pulse irradiation used in ICF [5], i.e. of the order of magnitude of the picket duration [10]. Moreover, the influence of the solid-to-plasma transition on the hydrodynamic procceses, resulting from the laser absorption, has been highlighted by measuring the laser induced shock velocities in polystyrene [11]. Significant differences have been observed between experimental results and hydrodynamic simulations where the solid-to-plasma transition is not considered. Indeed, in the plasma state, the relative variations of the pressure scales linearly with the laser intensity following a linear absorption process, whereas a non linear behavior has been shown to be necessary to account for experimental observations [11]. This non linear behavior has been attributed to the primary step of plasma formation through two-photon absorption of polystyrene.

A first microscopic modeling of the solid-to-plasma transition has been proposed for a polystyrene ablator [5]. The ablator heating eventually leading to phase transition and plasma formation is evaluated owing to a two-temperature model describing the coupled evolution of both electron and lattice temperatures, and including the free electron density evolution induced by the laser irradiation. The free electron density is estimated by describing processes of photoionization, electron impact ionization, and electron recombination. Although this first microscopic modeling laid the foundations of the description of the solid-to-plasma transition, it must be improved for its introduction in a hydrocode due to several reasons. (i) The densities of carbon and hydrogen ions are not predicted whereas they are required to describe properly the plasma formation. The chemical fragmentation of initial solid polystyrene leading to atomic species has thus to be described (ii) The description of electron collisions is rather crude despite it provides correct orders of magnitude. To predict accurate electron and lattice heating (leading to phase transitions), a more accurate description of collisional processes is required. (iii) In the perspective of a model properly describing the material decomposition into a plasma where various transient chemical species are involved (electrons, polymer, molecules, atoms), adapted equations of state (EOS) have to be implemented.

The first goal of the present work is to introduce an improved microscopic modeling of the solid-to-plasma transition where above mentioned three assumptions of the first modeling attempt are fixed. The chemical fragmentation and improved collision frequencies are introduced following developments proposed in [12]. Both the energy transfer and the laser induced electron dynamics including photoionization, electron impact ionization, and electron recombination, are adapted to the state of matter. These theoretical developments are presented in Section II. Section III presents numerical simulations performed by using laser intensity profiles pertaining to ICF conditions. The solid-to-plasma transition is investigated where in particular the evolution of the chemical composition of the ablator is highlighted. Comparisons with typical initial conditions of ICF hydrocodes are also performed. The second goal of the present work is to address the evolution of the pressure as a function of the laser intensity following observations in [11]. A non linear behavior is evidenced in section V confirming the role of the solid-to-plasma transition on the subsequent plasma hydrodynamics. Finally, conclusions and outlooks are drawn in Sec. VI. Owing to the fact it includes all main physical processes driving the solidto-plasma transition, and provides densities and temperatures of formed plasma, the present model is devoted to be introduced in current hydrocode to improve their accuracy and correctly describe the onset of hydrodynamics which is expected to influence the subsequent implosion of ICF targets.

II. MODELING OF THE SOLID-TO-PLASMA TRANSITION

The phenomenology of the laser induced phase transition of a polystyrene ablator, i.e. chemical fragmentation, and its laser interaction are first summarized following a previous work [12] in order to introduce the forthcoming theoretical developments. Let define T il the temperature of the so called ion-lattice system [5] which includes various chemical species as polymer, molecules, or atoms with different ionization degree [12]. If T il < 58 meV, the ablator is constituted of solid polystyrene which is a polymer. In that case, there is a band structure constraining the electron dynamics, and the electrons undergo collisions with phonons. If T il > 58 meV, the ablator consists of several species, namely styrene, benzene, acetylene molecules, or carbon and hydrogen atoms, with various ionization degrees, depending on temperatures. In that case, the electron dynamics consists of successive ionizations of these molecules and atoms. Within this phase, electrons undergo collisions with ions and neutral species. As a final product of the laser induced and thermal molecule dissociations, the ablator consists of a mixture of ionized carbon and hydrogen atomic species (i.e. at the end of the solid-to-plasma transition). Such a scenario for the chemical fragmentation of polystyrene implies that laser induced electron dynamics, electron collision frequencies, and energy transfer between various species populations, have to be adapted to a given chemical composition which depends on ion-lattice temperature.

For typical laser pickets used in ICF experiments, i.e. Gaussian profile with a maximum laser intensity about 10 14 W.cm -2 and a duration of 100 ps, the solid-to-plasma takes place over a timescale of about 100 ps [5] which is smaller than the hydrodynamic timescale. Hydrodynamic processes can thus be neglected a priori (based on the forthcoming results, an estimation of the hydrodynamic expansion will confirm this assumption a posteriori). In addition, thermal diffusion is also neglected. The smallest spatial scale where absorption is not uniform corresponds to the intensity profile of one laser speckle which characteristic size is of the order of 1 µm. Due to the variations of absorption and energy deposition on this spatial scale, the resulting temperature gradient may lead to thermal diffusion in general. By using the expression of thermal conductivity provided in [12], calculations show that the characteristic diffusion time becomes short enough (∼ 1 ps) to induce heat diffusion only when the solidto-plasma transition is finished, i.e. when a fully plasma state is formed. Heat diffusion thus can be neglected during the solid-to-plasma transition.

The ablator is a system out of equilibrium (temperatures of electrons and ion-lattice are different in general) during the solid-to-plasma transition since electrons are first excited by the laser, and the ion-lattice system is then heated subsequenly through electron collisions which transfer their energy. The evolution of the electron and ion-lattice temperatures (T e and T il , respectively) is modeled with the two-temperature model which reads:

C e ∂T e ∂t = -μe ∂n e ∂t - ∂U e ∂t collisions + ∂U e ∂t laser (1)
C il ∂T il ∂t = -∑ s μs ∂n s ∂t + ∂U e ∂t collisions (2)
where ions and neutral species are assumed to have the same temperature. C e and C il correspond to the electron and ionlattice heat capacities, respectively. The first term in the right hand side of both equations correspond to the variation of energy due to the evolution of the number of particules ( μe and μs are effective chemical potentials and the sum is over the different chemical species constituting the ion-lattice system, see Appendix A). The second terms account for the energy transfer from the electrons to the ion-lattice system due to electron collisions. The last term in Eq. 1 corresponds to the laser heating. Except C e and (∂ t U e ) laser which exhibit generic expression, all the quantities appearing in Eqs. ( 1) and (2) depend on the chemical composition of the ablator and have to be defined according to the ion-lattice temperature.

A. Laser heating

The laser energy absorbed by electrons (∂ t U e ) laser is given by the Drude model [5]:

∂U e ∂t laser = e 2 n e ν c m e (ω 2 L + ν 2 c ) E 2 L (3)
where ω L and E L are the laser frequency and laser electric field, respectively. The electron collision frequency ν c is given by a model dedicated to the solid-to-plasma transition of the polystyrene ablator [12]. If the lattice-ion temperature is smaller than 58 meV, ν c is evaluated from collisions between electrons and both acoustic and optical phonons of solid polystyrene. This collision frequency is evaluated with the Fermi golden rule. If the lattice-ion temperature is higher than 58 meV, electron-ion collisions take place, which are described by the Lee & More model [13] using the average collision time. A Coulomb logarithm different from the Lee & More model is used [14] since collisions may take place in a strongly coupled regime for electron temperatures of a few eV.

We have checked by evaluating the electron and ion coupling parameter that such an estimation for the Coulomb logarithm is suitable for the solid-to-plasma transition. Electron-neutral collisions are described by using classical cross sections.

B. Equations of state

The electron heat capacity C e and the effective chemical potential μe are evaluated for both T il < 58 meV and T il > 58 meV from the general free electron gas equation of state which includes both degenerated limit and ideal gas behavior for temperatures larger than the Fermi temperature (see Appendix B). The ion-lattice heat capacity C il and the ionlattice effective chemical potential μs depend on the considered species and thus depend on the ion-lattice temperature.

For T il < 58 meV, C il is the heat capacity of solid polystyrene. In that case, C il = C m ρ/M with M = 104 g.mol -1 and ρ = 1.1 g.cm -3 the styrene molecular mass and polystyrene density, respectively. The molar heat capactiy C m of polystyrene, which depends on the lattice temperature, is given by a fit of experimental data [15]. μs is associated to the variation of the polystyrene energy due to the variation of the phonon density. Because the internal energy of polystyrene does not depend on the phonon density [START_REF] Ashcroft | Physique des Solides[END_REF], it comes directly μs = 0.

For T il > 58 meV, C il corresponds to the heat capacity of both ions and neutrals (molecular and atomic species). Because a same temperature is assumed, C il = C i + C n with C i and C n the heat capacity of ions and neutrals, respectively. C i is deduced from the Cowan model [START_REF] Atzeni | The Physics of Inertial Fusion[END_REF] which is a standard equation of state used in ICF. For neutrals, an ideal gas is assumed, i.e. C n = 3k B n n /2 with n n the total density of neutrals.

The effective chemical potential of ions and neutrals, μs , is given from the same EOS as the one used for the heat capacities, see Appendix A.

Note that relatively simple EOS are used in this model, in particular for neutrals. Forthcoming results of Sec. III shows that neutrals exist only during a short period of time. Thus, a more suitable EOS for neutrals is not expected to change the results and conclusions of this work. For electrons and ions, the thermal energy was compared to the Fermi energy and the interaction potential. It turned out that screening and quantum effects can be neglected during the solid-to-plasma transition. In addition, the presently used EOS have been compared (results not shown) with ab initio calculations [START_REF] Hu | [END_REF] providing the evolution of the pressure with respect to the temperature at equilibrium (T e = T i ). A good agreement is obtained, definitively confirming the EOS used in this model is suitable for the description of the solid-to-plasma transition.

C. Electron dynamics

Photoionization, electron impact ionization, and electron recombination are general processes taking place for all matter composition: polymer, molecules, or atoms. However the formalism used to describe these processes depends on the transient chemical composition, i.e. on the ion-lattice temperature T il . When the ablator species are molecules and atoms ionized or not, the dissociation of molecules has further to be considered.

Conditions where T il < 58 meV

The ablator is solid polystyrene which is a polymer, i.e. a dielectric material. The band structure consists of two bands, the valence band (VB) and the conduction band (CB), separated by a bandgap of 4.05 eV [19]. Before any interaction, most of the electrons are in the VB while a small number is in the CB. Because a polystyrene target with a temperature different from zero is considered, the CB is initially populated with an electron density of the order of 10 10 cm -3 [20]. Due to photoionization, electrons in the VB are promoted into the CB. Promoted electrons in the CB may reach higher energy levels by successive one-photon absorption. When the energy level becomes higher than the bandgap energy, those electrons may collide valence electrons resulting in two electrons in the bottom of the CB. The latter process is the so called electron impact ionization in dielectric materials [21]. In solid polystyrene, three levels of the CB have thus to be considered [5]. Accounting for these two ionization processes and electron recombination, the laser induced electron dynamics can be described as follows [5]:

∂n 0 ∂t = αW PI + 2αν II n 2 -W 1 n 0 -ν r (n 0 -n cb0 ) (4)
∂n 1 ∂t = W 1 n 0 -W 1 n 1 -ν r n 1 ( 5
)
∂n 2 ∂t = W 1 n 1 -αν II n 2 -ν r n 2 (6)
where n 0 , n 1 and n 2 correspond to the electron densities in the three levels considered in the CB and n cb0 the initial electron density of the CB. ν II = 10 14 s -1 and ν r = 10 12 s -1 correspond to the electron impact ionization and recombination frequencies, respectively. The photoionization rate W PI is evaluated with the Keldysh theory accounting for both tunnel ionization and multiphoton absorption [22]. The onephoton absorption rate is set to

W 1 = 10 -7 E 2 L , in units of s -1
, where E L is the laser electric field [5]. Finally, α = 1 -(n 0 + n 1 + n 2 )/n vb0 , with n vb0 the initial electron density in the VB, accounts for the depletion of the VB.

Conditions where T il > 58 meV

For this range of temperatures, the electron dynamics corresponds to the successive ionizations and dissociations of the molecular and atomic species. Within the ICF framework, the chemical species involved in the chemical fragmentation of the ablator are styrene, benzene, acetylene, carbon and hydrogen [12]. Both neutral and ionized forms have to be considered for each species in order to model the plasma formation. The following mixture is thus considered:

S, S + , S 2+ , A, A + , A 2+ , B, B + , B 2+ , H, H + , C, C + , C 2+ , C 3+ , C 4+ , C 5+ , C 6+ ,
where S, A, B, H and C correspond to styrene, acetylene, benzene, hydrogen and carbon, respectively. The density evolution of those species is related to the electron dynamics including photoionization, electron impact ionization, and electron recombination. Dissociations of styrene, benzene and acetylene molecules must also be described to model the chemical fragmentation of the ablator. The most relevant dissociations reactions are [12]:

S -→ A + B (7a) S + -→ A + B + (7b) S 2+ -→ 1 2 A + 1 2 A + + 1 2 B + + 1 2 B 2+ (7c) A -→ 2C + 2H (7d) A + -→ C +C + + 2H (7e) A 2+ -→ 2C + + 2H (7f) B -→ 6C + 6H (7g) B + -→ 5C +C + + 6H (7h) B 2+ -→ 4C + 2C + + 6H (7i)
which take place when the ion-lattice temperature is about several eV. The dynamics of each population is described as follows:

∂n X j ∂t = ∑ s Q D,s + Q IR (X j-1 ↔ X j ) -Q IR (X j ↔ X j+1 ) (8)
with X ∈ {S, A, B, H,C} denoting the chemical species and j ≥ 0 the ionization state. The first term corresponds to the dissociation rate where the sum is over all the reactions driving the evolution of the population X. For example, in the case of B + , three dissociations rates have to be considered accounting for the reactions (7b), (7c) and (7h). The dissociation rate reads:

Q D = β n D k arr exp - E a k B T il (9)
with β the algebraic stoichiometric coefficient (negative for reactants, positive for products), n D the density of the molecule that dissociates, k arr the Arrhenius law prefactor, and E a the activation energy. We set k arr = 10 13 s -1 corresponding to the characteristic time of dissociation of a molecule [START_REF] Chergui | Femtochemistry, Ultrafast Chemical and Physical Processes on Molecular Systems[END_REF]. For the reactions given by Eqs. (7a)-(7c), one has E a = 2.4 eV and for the reactions given by Eqs. (7d)-( 7i), E a = 6 eV [12]. The second and third term in Eq. ( 8) correspond to the transitions between the j -1-th and j-th ionization state, and the j-th and the j + 1-th ionization state, respectively. It is assumed that these transitions can be due to photoionization (PI), electron impact ionization (EII) and electron recombination (ER) leading to :

Q IR (X j-1 ↔ X j ) = W PI (X j-1 → X j )n X j-1 +W EII (X j-1 → X j )n X j-1 -W ER (X j → X j-1 )n X j (10)
Photoionization of carbon and hydrogen atomic species is described within the Keldysh formalism for atoms [22]. In order to get a tractable model, this Keldysh expression derived for atoms is also used to calculate the photoionization rate of styrene, benzene and acetyelene, assuming the ionization energy is the main parameter driving the electron dynamics. Electron impact ionization is described with the Gryziński's theory [START_REF] Gryziński | [END_REF] as a compromise between simplicity and accuracy. Concerning electron recombination, the approach of Zel'dovich [START_REF] Zel | Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena[END_REF] is adapted by substituting the Coulomb potential by the Debye-Hückel potential to account for screening effects which are significant at solid density: (11) where λ D = k B T e /4πn e e 2 is the screening Debye length and W is the principal branch of the W -Lambert function.

W ER (X j → X j-1 ) = π 2 n 2 e j 3 8k B T e πm e λ D W 2e 2 3λ D k B T e 5
The evaluation of the density of each population with Eq. ( 8) allows to define the densities of electrons n e , ions n i and neutrals n n :

n e = ∑ X, j≥1 Z j n X j ; n i = ∑ X, j≥1 n X j ; n n = ∑ X n X (12)
where Z j is the charge of the ion n X j , and the total density is given by n T = n i + n n .

D. Energy transfer

When T il < 58 meV, the ablator is solid polystyrene and the energy transfer is mainly due to electron-phonon collisions. When T il > 58 meV, the ablator is constituted of molecules and atoms. In that case the energy transfer is mainly supported by electron-ion collisions.

For T il < 58 meV, as a first approximation, it is possible to assume that two optical phonons with energy hω L0,1 = 33 meV and hω L0,2 = 88 meV, and one acoustic phonon exist in polystyrene [12]. Thus, the energy transfer from electrons to phonons is given by :

∂U e ∂t collisions = ∂U e ∂t AC + ∂U e ∂t LO,1 + ∂U e ∂t LO,2 (13
) accounting for the different contributions of each phonon. The electron-acoustic phonon and electron-optical phonon energy transfer are respectively defined by :

∂U e ∂t AC = ν et,ac n e k B (T e -T il ) (14
)

∂U e ∂t LO = ν et,op hω L0 n e [g(T e ) -g(T il )] (15)
where g(T ) = [exp (hω LO /k B T ) -1] -1 corresponds to the Bose-Einstein distribution and ν et,ac and ν et,op are the energy transfer frequencies evaluated in Appendix C. For T il > 58 meV, the ablator is constituted of molecules and atoms. Because they are mostly ionized due to the laser, the energy transfer is mainly due to electron-ion interactions which is described by the energy transfer derived by Landau [START_REF] Lifshitz | [END_REF] :

∂U e ∂t collisions = 4n e n i Z 2 e 4 √ 2πm e ln Λ m e f f (k B T e ) 3/2 k B (T e -T il ) (16)
where m e f f = ∑ X, j m X j n X j / ∑ j n X j is the effective mass of ions. It accounts for the variation of the mass of ions due to the dissociation of the molecular species. ln Λ corresponds to the Coulomb logarithm for which an approximation valid for dense plasmas is used [14]. Note that the energy transfer between electrons and neutral species is neglected. This is supported by the fact that neutrals are only present during a short period of time as shown in section III.

III. RESULTS ON THE DYNAMICS OF THE SOLID-TO-PLASMA TRANSITION

To perform numerical calculations with the model presented previously, a 351 nm-laser is considered with a temporal Gaussian shape I(t) = I 0 exp (-4 ln 2(tt max ) 2 /τ 2 ) where I 0 , t max and τ are chosen such that they correspond to typical laser picket parameters used in ICF experiments [27]. This leads to I 0 = 10 14 W.cm -2 , t max = 200 ps and τ = 100 ps. The initial temperature for electrons and the ion-lattice system is set to 300 K. Simulations begin when the laser intensity is high enough to quickly produce an electron density in the CB of the order of the initial electron density (10 10 cm -3 ), i.e. when the laser beam modifies significantly the material. Calculations are stopped when the electron density remains constant, ensuring the solid-to-plasma transition is ended since no more electrons can be generated. It then turns out that simulations are performed between -75 ps and 100 ps. The laser is already intense enough for t < 0 ps to create a significant free electron density, and the electron density remains constant after t > 100 ps. L /4πe 2 and the laser frequency ω L are given for purpose of comparison. A Gaussian profile is considered for the laser intensity with a maximum of 10 14 .Wcm -2 reached at 200 ps and a duration of 100 ps.

Figure 1 shows the temporal evolution of the electron density, electron and ion-lattice temperatures, and the electron collision frequency between -75 ps and 100 ps.

Regarding the electron density, a monotonic behavior is observed when t < 50 ps. The ablator is solid polystyrene, this variation of the electron density is driven by the photoionization process. The electron impact ionization has a weak influence because the laser is not intense enough to promote a high number of electrons to the third level and initialize the electron breakdown. For t > 50 ps, the ablator is constituted of molecules and atoms and the electron generation is driven by electron impact ionization. The ionization energies of the species constituting the ablator are about 8 eV for the lowest one, simultaneous absorption of three photons is required to photoionize which makes this rate low. Since the electron temperature is higher than 10 eV for t > 50 ps, the impact ionization process becomes dominant. The first increase of the electron density going from n e ≈ 6 × 10 19 cm -3 to n e ≈ 10 22 cm -3 corresponds to the ionization of styrene molecules since the ion-lattice system remains cold and only styrene molecules are present. The second increase where the electron density goes from n e ≈ 10 22 cm -3 to n e ≈ 2.5 × 10 23 cm -3 correspond to the simultaneous ionizations of benzene, acetylene, hydrogen and carbon atomic species. The ionlattice temperature increases from 0.1 eV to 10 eV and all the dissociation reactions given by Eq. (7a)-(7i) take place. Finally, the last two increases of the electron density taking place between 70 ps and 75 ps are associated to the fifth and sixth ionization of atomic carbon. The ion-lattice temperature is about a few hundreds of eV and the ablator is mostly composed of hydrogen and carbon atomic species.

When n e ≈ 3.5 × 10 23 cm -3 , all the species are fully ionized, no more electrons can be generated and the electron density remains constant. Thus, the initial solid ablator has transformed into a fully ionized plasma, and the solid-to-plasma transition can be considered as finished. Note that as long as t < 54 ps, the electron density is smaller than the critical density. For these early times where the intensity is lower than a few 10 11 W.cm -2 , the laser beam can propagate throughout the ablator and be absorbed in the fuel area [28]. This is the shinethrough issue which may reduce the target efficiency [5,29].

Concerning the temperatures and the electron collision frequency, two behaviors associated to t < 50 ps and t > 50 ps can be noticed, as for the electron density evolution. For t < 50 ps, electrons remains relatively cold, the electron temperature does not exceed 0.1 eV. Although the electron collision is important, ν c ≈ 2 × 10 15 s -1 , leading to an efficient laser heating, the energy transfer from electrons to the lattice is also important and the whole laser energy acquired by the electrons is directly transferred to the lattice. This results in a slow increase of the electron temperature. However, even if the internal energy of the lattice increases due to the electronphonon energy transfer, it does not lead to a significant heating of the lattice due to the large value of the polystyrene heat capacity.

For t ≈ 50 ps, the electron temperature increases rapidly due to the drop of the energy transfer. At this moment where T il ≈ 58 meV, the chemical constitution of the ablator changes going from solid polystyrene to styrene which is a molecular specie. The nature of the energy transfer is modified because collisions are no longer between electrons and phonons but between electrons and ions. This drop of the energy transfer leads to a heating of the ions much less efficient and the ion-lattice temperature decreases because, on the other hand, the internal energy of styrene is modified due to ionization. Note that a signature of the phase transition is the drop of the electron collision frequency at t ≈ 50 ps.

For t > 50 ps, the electron temperature and the ion-lattice temperature increase due to the laser heating and electronion energy transfer, respectively. Regarding the behavior of the electron collision frequency, it first increases because the electron density increases due to ionizaition while the electron temperature remains relatively constant around 10 eV. Then, the electron collision frequency decreases because the electron density does not increase much going from 2.5 × 10 23 cm -3 to 3.5 × 10 23 cm -3 while the electron temperature increase significantly.

In order to have a closer look at the evolution of the chemical composition of the ablator in the plasma state, Fig 2 shows the temporal evolution of the density of styrene, acetylene, benzene molecules, and the density of hydrogen and carbon atomic species during the solid-to-plasma transition. They are shown between t ≈ 50 ps (moment when polystyrene dissociates into styrene molecules) and t ≈ 60 ps (densities of benzene and acetylene molecules are small enough so that only carbon and hydrogen atomic species remain). Such a temporal window ensures to capture the entire chemical fragmentation of the ablator.

Because the ion-lattice temperature increases during the solid-to-plasma transition, the ablator is successively constituted of polystyrene, styrene molecules, benzene and acetylene molecules, and finally carbon and hydrogen atomic species. This chemical fragmentation can be observed with the increase of the total density on a timescale of ∼ 10 ps which is not negligible compared with the whole solid-toplasma transition timescale of ∼ 100 ps [5]. Thus the chemical fragmentation of the ablator cannot be neglected in the description of the solid-to-plasma transition. However, benzene and acetylene are only present during a few picoseconds. This is due to the fact that the ion-lattice temperature increases very quickly from 0.1 eV to several tens of eV. Thus, as a possible increase in computational efficiency for purposes of implementation in a hydrocode, it could be assumed that styrene dissociates directly into carbon and hydrogen atomic species.

Regarding the temporal evolution of neutral and ionized species, neutrals exist only during a very short period of time compared to the solid-to-plasma transition timescale. Just after the polystyrene dissociation, the ablator is mainly composed of neutrals molecules. Because the electron temperature is small at this moment, there is no impact ionization. The latter process starts when electrons are hot enough, then neutral and ion densities decreases and increases, respectively. After a fast decrease between t = 52 ps and t = 54 ps because of the ionization of styrene molecules, the neutral density increases between t = 54 ps and t = 58 ps. This is due to the chemical fragmentation of the ablator which leads to the formation of new neutral species. Regarding chemical reactions given by Eqs. (7a)-(7i), it can be observed that neutral species are created during the dissociation of ions. Finally, the neutral density decreases again for t>58 ps because the chemical fragmentation is ended. Neutrals species do no longer appear and already present neutrals disappear because they are ionized by electron impact. Even though neutrals are present during a short period of time, considering them enable not to have to use some tweaks to properly describe the solid state transition to a mixture of ionic species. This approach ensures the continuity of the quantities such as the electron collision frequency.

FIG. 2. (a)

Temporal evolution of the density of styrene molecules n S , acetylene molecules n A , benzene molecules n B , hydrogen atoms n H , carbon atoms n C , and the total density n T . The evolution of n A and n B as well as n H and n C are plotted together because they appear and disappear for the same ion-lattice temperature (See Section II C 2). For each specie, the density corresponds to the density of neutrals plus the density of ions. (b) Temporal evolution of the total ion density n i , total neutral density n n and the total density n T = n i + n n . As a reminder, the total ion and neutral densities n i and n n are defined by Eq. ( 12). The same laser intensity profile as the one used for Fig. 1 is considered.

The solid-to-plasma transition timescale as predicted by this improved model is now addressed. Figure 3 shows the temporal evolution of the electron density and both electron and ion-lattice temperatures for three maximum laser intensities I 0 commonly used in ICF experiments [27]. The solid-toplasma transition timescale is defined as the duration necessary to fulfill two criteria: n e ≥ n c and T il ≥ 0.1 eV. The second criterion ensures the periodicity of the lattice is broken (plasmas in general are disordered matter). Figure 3 shows that the temporal shape of n e , T e and T il during the solid-toplasma transition does not depend on the maximum laser intensity, it mainly consists of a temporal shift. However, the laser intensity has a slight influence on the solid-to-plasma I. More intense is the laser, more energy is absorbed by the ablator, and faster the plasma state is reached. The presently obtained timescale is consistent with previsous estimations providing ∼ 100 ps [5]. Indeed the ablator remains a relatively long time in the solid state before undergoing any phase transition, the electron dynamics is mainly driven by photoionization of polystyrene. Thus, electron impact ionization taking place when the ablator is constituted of molecules and atomic species has only a weak influence on the solid-to-plasma transition timescale.

The model introduced in this work is now compared with initial conditions corresponding to a fully ionized carbonhydrogen plasma. These are standard initial conditions used in ICF hydrocodes where in particular the electron density is over critical. The present model is also compared with the one introduced by Duchateau et al. [5]. For the three models, the initial temperature of electrons and ions is 300 K and a Gaussian laser pulse with the same parameters as previously is used. Figure 4 provides the temporal evolution of the electron density, the electron temperature and the ion-lattice temperature as predicted by the three approaches.

Regarding the electron density, the model presented in this work and the one developed by Duchateau et al. give similar results as long as t < 50 ps, i.e. when solid polystyrene still stands, because the same description of the electron dynamics (photoionization with band structure) is used in both models. For t > 50 ps where chemical fragmentation takes place, the disagreement is due to the fact that Duchateau et al. still assume the same model for electron dynamics as previously whereas impact ionization of molecules and atoms becomes significant. Since the present improved model accounts for full ionization, the electron density of the initial plasma model (∼ 3.5 × 10 23 cm -3 ) is retrieved.

Considering electron and ion-lattice temperatures at the end of the solid-to-plasma transition, similar results are obtained from our improved model and from the case where the ablator is assumed to be initially a fully ionized plasma. Because the laser absorption by the ablator leading to the electron generation and the chemical fragmentation of the ablator takes place for laser intensities much smaller than those associated to the heating of the electrons and the ion-lattice system when the ablator is totally ionized, the total laser energy absorbed by the ablator is about the same whether the solid-to-plasma transition is considered or not. However the initially fully ionized plasma leads to an ablator heated earlier since the electron density is much higher at the beginning of the interaction, which makes more efficient the laser heating of electrons and the electron-ion energy transfer. We thus expect that the ablator expansion takes place later when the solid-to-plasma transition is considered. Compared with the first attempt of model of solid-to-plasma transition, the interest of the present model is clearly shown by the differences in electron and ion temperatures. This discrepancy is due to improved model of collision frequencies and chemical fragmentation affecting the electron dynamics.

Compared with an initial plasma state, previous results have shown that accounting for the solid-to-plasma transition is expected to modify electron density and transient temperatures which drive both the laser dynamics and shock wave formation. Such conclusions were drawn assuming no hydrodynamics. This assumption can now be evaluated a posteriori to confirm previous predictions. For this purpose, the ablator expansion due to laser heating, δ x, is estimated by integrating the ion sound speed c s over the time δ x = t t 0 c s (τ) dτ where the plasma neutrality is assumed. t 0 = -75 ps corresponds to FIG. 4. Electron density (a), electron temperature (b) and ionlattice temperature (c) during the solid-to-plasma evaluated for different models. The red curve corresponds to the model developed by Duchateau et al [5]. The blue curve corresponds to the model presented in this work. The green curve corresponds to initial conditions that can be encountered in current hydrocodes, i.e. the ablator is assumed to be initially a fully ionized carbon-hydrogen plasma. The same laser intensity profile as the one used in Sec. III is considered.

the initial time and the sound speed is [START_REF] Spitzer | Physics of Fully Ionized Gases[END_REF] :

c s = γ e Zk B T e + γ i k B T il m i ( 17
)
where γ e = 1 and γ i = 3 are the electron and ion heat capacity ratios, respectively [START_REF] Kruer | The Physics Of Laser Plasma Interactions[END_REF]. The evolution of the ablator expansion during the solid-to-plasma transition is shown in Fig. 5.

Two main behaviors can be observed as for the electron density, electron temperature and ion-lattice (see Fig. 1). If t < 50 ps, the ion-lattice system remains cold and the ablator expansion is negligible (less then 0.1µm). If t > 50 ps, the ion-lattice system is heated very quickly and reaches high temperatures in excess of several hundreds of eV. The ablator expansion, which may be of several µm, then has to be considered. Polystyrene ablators typically used in ICF experiments have a thickness of several tens of µm [START_REF] Lindl | [END_REF][33][34]. For t ∼ 70 ps where the solid-to-plasma transition is mainly finished, the expansion is of the order of 1 µm, i.e. less than 10% of the ablator thickness. We thus confirm that hydrodynamics can be neglected during the solid-to-plasma transition. However the solid-to-plasma transition may have an influence on the subsequent hydrodynamics as suggested in recent experiments in order to explain differences on the shock velocities modulations between experimental and numerical results [11]. This discrepancy may be explained by the evolution of the matter pressure with the laser intensity which scales differently depending on the state of matter, thus forming different gradients responsible for shock formation and propagation.

V. SCALING LAW OF MATTER PRESSURE WITH RESPECT TO LASER INTENSITY

In above mentioned experiments [11], shock velocities modulations induced in a polystyrene foil by a laser beam including speckles (spatial intensity modulations) were measured with a 2D VISAR to study the laser imprint. By using a state-of-the-art hydrocode assuming an initial plasma state, a good agreement of numerical simulations with experimental observations was obtained by increasing the intensity modulations by a factor of two. The later modification was supported by invoking the role of the initial solid state. By considering a fully ionized plasma, the relative pressure exerted by a beam profile, δ P/P, is equal to the relative intensity variations δ I/I. An ideal gas EOS is assumed with a constant density and a temperature resulting from linear laser absorption. Considering the initial solid state, absorption becomes non linear (ionization through two-photon absorption) and free electron density depends on laser intensity. Then δ P/P = αδ I/I, or P ∝ I α , where the coefficient α accounts for the plasma formation. Compared with an initial fully ionized plasma state, the solid-to-plasma transition is thus expected to modify the scaling law of the pression as a function of the laser intensity. This section is devoted to the investigation of the modification of the previous scaling law exponent due to the solid-toplasma transition.

First, the scaling law exponent is evaluated for a fully ionized plasma. The obtained value will be a reference to check the predictions of the present solid-to-plasma model. The total pressure p is given by considering electron and ion contributions, p = p e + p i , with p e and p i the electron and ion pressures, respectively. Assuming electrons and ions have the same temperature (long enough time) and ideal gas EOS, the total pressure reads:

p = 1 + 1 Z n e k B T e (18)
with Z = n e /n i the averaged ionization degree of the plasma. Thus, the scaling law of the pressure as a function of the laser intensity is due to the evolutions of both electron density n e and electron temperature T e . For a fully ionized plasma where n e is constant, only the electron temperature scaling needs to be evaluated. This is done by Eq. 1 which, after simplification, reads:

∂T e ∂t = 16πe 2 ν c 3ck B m e ω 2 L I L (t) (19)
with c the light velocity. ν c ω L is assumed for a hot enough plasma. Thus, the scaling law of the temperature is obtained owing to the scaling law of the electron collision frequency. The assumption of a hot plasma enables to write

ν c ∝ T -3/2 e ln Λ.
Neglecting the electron temperature dependency of the Coulomb logarithm, the previous equation can be solved leading to:

T 5/2 e -T 5/2 0 ∝ I 0 F(t) (20)
with T 0 the initial electron temperature and F(t) = t -∞ f (t) dt where the laser intensity profile is I L (t) = I 0 f (t) where f (t) is the temporal profile. With T 0 T e , the scaling law of the pressure as a function of the laser intensity for a fully ionized plasma reads:

p ∝ I 2/5 0 (21)
In order to study the modification of this scaling law due to the solid-to-plasma transition, simulations were performed by using the model presented in Sec. II and a Gaussian shape for the temporal laser intensity profile with the same parameters as those used in Sec. III. Simulations were performed for different maximum laser intensities I 0 ranging from 10 12 W.cm -2 to 10 15 W.cm -2 . Figure 6 shows the evolution of the total pressure and the electron density as a function of time and of the laser intensity.

It can be observed on Fig. 6(b) that the pressure evolution exhibits four behaviors. The first three behaviors are related to the evolution of the electron density while the last one is related to the evolution of the electron temperature. When I L < 2.0 × 10 11 W.cm -2 , the ablator is solid polystyrene and the electron density scales as n e ∝ I 2 L as shown by Fig. 6(c). This scaling law is also valid for the pressure, because the electron temperature remains relatively constant for the solid polystyrene state (see Fig 1(b)). In that case the scaling of the pressure is given by the scaling of the electron density. Thus, when I L < 2.0 × 10 11 W.cm -2 , the pressure scales as p ∝ I 2 L . For laser intensities between 2.0 × 10 11 W.cm -2 and 5.0 × 10 11 W.cm -2 , the pressure exhibits a fast increase which is due to the increase in the electron density induced by dissociations and ionizations of all molecular and atomic species present at this moment. A rough fit of the pressure leads to the following scaling law: p e ∝ I 11 L . For laser intensities between 6.0 × 10 11 W.cm -2 and 2.0 × 10 12 W.cm -2 , the evolution of the pressure is due to the increase of the electron density going from 2.5 × 10 23 cm -3 to 3.5 × 10 23 cm -3 . This evolution is due to the last two ionizations of carbon taking place for I L ≈ 10 12 W.cm -2 . A numerical fit leads to p ∝ I 1.2 L . Finally, for I L > 3.0 × 10 12 W.cm -2 , the electron density remains constant and the evolution of the pressure is due to variations of electron temperature. In that case, a fitting of the pressure leads to p ∝ I 0.5 L . This scaling law is similar to the one given by Eq. 21. The exponent difference (0.5 vs 0.4) is due to the dependence of the collision frequency on the electron temperature. Eq. 21 has been obtained assuming a constant Coulomb logarithm, neglecting the electron-ion energy transfer, and assuming ν c ω L to evaluate the energy absorption, the last two assumptions leading to Eq. 19.

These results show that the solid-to-plasma transition strongly modifies the scaling law of the pressure as a function of the laser intensity due to the free electron generation. By fitting the curves over a relevant intensity range, the average scaling law exponent is of the order of 3. This work thus confirms a posteriori the approach discussed in [11] to account for experimental observations. Note that a similar experimental study has been carried out recently with different parameters [35] where the photoionization of the solid target is due to one-photon absorption. As in [11], state-of-the-art numerical simulations with an initial plasma state cannot account accurately of experimental observations. Since the primary ionization process of the solid plastic is due to a linear absorption process, the authors of [35] concluded that the introduction of the solid-to-plasma transition is not enough to explain discrepancies. Here we have shown that the scaling law exponent value is mainly due to variations of the free electron density, regardless the exact ionization process (of course the precise value of the exponent depends on the ionization processes). The solid-to-plasma transition may thus also mainly explain experimental and theoretical discrepancies observed in [35].

VI. CONCLUSION AND OUTLOOKS

An improved description of the solid-to-plasma transition has been proposed in this work enabling to investigate in particular the temporal evolution of the electron density, electron temperature and ion-lattice temperature during the early interaction between the ablator and typical laser pickets used in ICF experiments. They exhibit two distinct behaviors associated to the chemical composition of the ablator. As long as the ion-lattice temperature is smaller than 58 meV, the ablator is in the solid polystyrene state. The electron density is driven by the laser-induced photoionization and the electron and ion-lattice temperatures are driven by collisions between electrons and phonons. For higher temperatures, polystyrene dissociates, transforming into various molecular and atomic species. The electron density is then driven by electron impact ionization while the heating of the electrons and of the ionlattice system is driven by electron-ion collisions. This model leads to a non negligible solid-to-plasma transition timescale of about 100 ps, which is about the order of magnitude of laser pickets duration. The present model can estimate the density of carbon and hydrogen ions at the end of the solidto-plasma transition, as well as electron and ion temperatures, which makes it suitable for its implementation in an ICF hydrocode.

The chemical evolution of the ablator going from polystyrene to a mixture of carbon and hydrogen atomic species has been investigated. It takes place over a period of time of about 10 ps, which is not negligible compared with the solid-to-plasma transition timescale. However, some intermediate chemical species are present only during a few ps and should be neglected for application purposes. By extrapolating the scenario of polystyrene fragmentation, a general scenario of fragmentation of plastics can be deduced. Indeed, in the ICF framework, and more generally within the laserplasma interaction context, polystyrene is only one among a large number of plastics that can be used, in particular of the case of complex target coating or the use of foams [36][37][38]. We thus propose a simpler two-step scenario for the plasma formation: (i) The plastic (polymer) dissociates into monomers around the ceiling temperature; (ii) monomers dissociate into atomic species when the ion-lattice temperature is in between 3 and 8 eV which corresponds to the typical range of chemical binding energies.

Finally, by investigating the evolution of the matter pressure as a function of the laser intensity, we have shown that the strong modification of the scaling law exponent is due to the solid-to-plasma transition where the free electron density varies. The solid-to-plasma transition is thus expected to have a significant influence on the shock formation and propagation for the laser imprint issue, and on the shinethrough issue as already demonstrated with a simplified model [28]. To definitely confirm these expectations, the model presented in this work will be implemented in a state-of-the-art multidimensional radiative hydrodynamic code. By doing so, it will be possible in addition to determine whether the early-time evolution of the ablator has a significant influence on the target implosion and performance such as the neutral yield within direct-drive ICF conditions. By using the free energy F defined by F = U -T S, which its total differential is dF = -SdT -PdV + µdN, the entropy can be defined as S = -(∂F/∂T ) V,N . Because F is a function of T , V and N, the entropy is a function of these same variables and its total differential is:

dS = ∂S ∂T V,N dT + ∂S ∂V T,N dV + ∂S ∂N T,V dN (A3)
By substituting this expression for the differential of the entropy in Eq. (A1), this allows to write the total derivative of the energy as:

dU = C v dT -pdV + μdN (A4)
with C v = T (∂S/∂T ) V,N the heat capacity of a system with a variable number of particles as defined by Landau [39]. The partial derivatives over V and N have been gathered to defined an effective pressure p and an effective chemical potential μ as follows:

p = - ∂U ∂V T,N = P + T ∂S ∂V T,N (A5) μ
= ∂U ∂N T,V = µ + T ∂S ∂N T,V (A6)
Finally, the variations of internal energy is given by:

∂U ∂t = C v ∂T ∂t - p ∂V ∂t + μ ∂N ∂t (A7)
Appendix B: Thermodynamic quantities in the free electron gas approximation

The two electron thermodynamic quantities of interest in the two-temperature model are the electron heat capacity C e and the effective chemical potential µ e f f . As the ablator does not significantly expand during its solid-to-plasma transition, the electron density can be very large. Thus, it is assumed electrons can be degenerated and the free electron gas approximation is used to recover both degenerate and non-degenerate regime. Thus, the electron energy reads:

U e = √ 2m 3/2 e π 2 h3 +∞ 0 ε 3/2 e (ε-µ e )/k B T e + 1 dε (B1)
which can be written in terms of the complete Fermi-Dirac integral:

U e = √ 2m 3/2 e π 2 h3 (k B T e ) 5/2 F 3/2 µ e k B T e (B2)
According to Landau & Lifshitz [39], the heat capacity of a system with a variable number of particles is defined by C v = T (∂S/∂T ) V,N . By using Eq. (A1) and the fact that T , V and N are independent variables, this allows to define C v in terms of U e :

C v = ∂U e ∂T V,N (B3)
Thus, by using Eq. (B2), this leads to:

C v = √ 2m 3/2 e π 2 h3 5 2 k 5/2 B T 3/2 e F 3/2 µ e k B T e + 3 2 (k B T e ) 5/2 F 1/2 µ k B T e ∂ ∂T e µ e k B T e (B4)
where the property of the Fermi-Dirac integral dF j (x)/dx = jF j-1 (x) has been used. Finally, as the electron density is defined by:

n e = √ 2m 3/2 e π 2 h3 (k B T e ) 3/2 F 1/2 µ e k B T e (B5)
the heat capacity can be written as:

C v = 3 2 k B n e 5F 3/2 (µ e /k B T e ) 3F 1/2 (µ e /k B T e ) + T e ∂ ∂T e µ e k B T e (B6)
Because n e and T e are independent variables, the derivative of n e with respect to T e is equal to 0. Thus, by calculating explicitly this derivative from Eq. (B5), we obtain:

∂ ∂T e µ e k B T e = - 3 T e F 1/2 (µ e /k B T e ) F -1/2 (µ e /k B T e ) (B7)
and the heat capacity of electrons is finally given by:

C v = 3 2 k B n e 5F 3/2 (µ e /k B T e ) 3F 1/2 (µ e /k B T e ) -3 F 1/2 (µ e /k B T e ) F -1/2 (µ e /k B T e ) (B8)
In the limit of the ideal gas, µ e /k B T e -1. Thus the Fermi-Dirac integral can be simplified to F j (x) ≈ Γ( j + 1)e x with Γ the Gamma function, which leads to C v = 3k B n e /2. In the limit µ e /k B T e 1, the Fermi-Dirac integrals can be simplified owing to the Sommerfeld expansion leading to the heat capacity of the free electron gas C v ≈ π 2 k 2 B T e n e /2ε F [START_REF] Ashcroft | Physique des Solides[END_REF]. Regarding the effective chemical potential μe , it comes directly: Following the theory developed by Allen [40], the energy transfer resulting from a collision between an electron and an acoustic or optical phonon is defined by:

μe = 3 2 √ 2m 3/2 e π 2 h3 (k B T e ) 3/
∂E e ∂t = 4π h ∑ k,k' hω q |M k,k' | 2 S(k, k')δ (ε k -ε k' + hω q ) (C1)
The term S(k, k') corresponds to the so-called thermal factor, ε k and ε k' are electron energies in states k and k', respectively, and hω q is the phonon energy absorbed or emitted by the electron moving from the state k to the state k', and satisfying the momentum conservation k' = k + q. As the matrix element |M k,k' | 2 given by [START_REF] Ziman | Electrons and Phonons[END_REF]:

|M k,k' | 2 = - h 2ρV ω q
(n q + 1)q 2 U 2 (ω q , q)δ (k -k' + q) (C2) with n q = [exp (hω q /k B T il ) -1] -1 the Bose-Einstein distribution and U (ω q , q) the deformation potential. By writing the energy conservation as:

δ (ε k -ε k' + hω q ) = m e
h2 kq δ cos θ -m e ω q hkq -q 2k (C3) with θ the angle between k and q, where it was used the momentum conservation k' = k + q and assumed a free electron gas, i.e. ε k = h2 k 2 /m e . This allows to write Eq. (C1) as follows:

∂E e ∂t = -2π (8π 3 ) 2 m e q ρ hk (n q + 1)U 2 (ω q , q)S(k, k')δ cos θ -m e ω q hkq -q 2k dk' dk (C4)

Note that ∑ k = (V /8π 3 ) d k has been used assuming close enough electron states and the electron energy E e has been replaced by the electron energy per unit volume E e = E e /V . The thermal factor S(k, k') is defined by

S(k, k') = [ f (ε k ) -f (ε k' )] n q - f (ε k' ) [1 -f (ε k )] with f (ε k ) = [exp ((ε k -µ e )
/k B T e ) + 1] -1 the Fermi-Dirac distribution but can be written as:

S(k, k') = [ f (ε k ) -f (ε k + hω q )]
[n q (T il )n q (T e )] (C5) so that it clearly appears S(k, k') = 0 and ∂ t E e = 0 when T e = T il . Owing to momentum conservation, integration over k' can be replaced by integration over q which gives:

dk' = dq = 2π +∞ 0 1 -1
q 2 dqd cos θ (C6) and Eq. (C4) can be simplified after integration over cos θ in terms of the Heaviside step function Θ as follows:

∂E e ∂t = 1 16π 4 +∞ 0 m e q 3 ρ hk (n q + 1)U 2 (ω q , q) [ f (ε k )f (ε k + hω q )] [n q (T e )n q (T il )]

×Θ 1 -m e ω q hkq -q 2k Θ 1 + m e ω q hkq + q 2k dq dk (C7)

Acoustic phonon:

For acoustic phonons, the relation dispersion leads to ω q = c s q with c s the speed of sound and the deformation potential is a constant [START_REF] Ziman | Electrons and Phonons[END_REF] U (ω q , q) = U . Thus, Eq. (C7) becomes:

∂E e ∂t = 1 16π 4 m e U 2
ρ h 1 k 2k+2m e c s /h 0 q 3 [n(hc s q, T il ) + 1] [ f (ε k )f (ε k + hc s q)] [n(hc s q, T e )n(hc s q, T il )] dq dk (C8)

As it appears that the integral over k depends only on its modulus, one has dk = 4π +∞ 0 k 2 dk and the previous equation can be simplified into:

∂E e ∂t = 1 4π 3 m e U 2 ρ h k BZ 0 k 2k+2m e c s /h 0 q 3 [n(hc s q, T il ) + 1] [ f (ε k ) -f (ε k + hc s q)]
[n(hc s q, T e )n(hc s q, T il )] dq dk (C9)

where the integration is restricted to the first Brillouin zone as during the solid-to-plasma transition, the electron energy does not exceed E(k BZ ). In the limit of small momentum transfer, one has hc s q k B T il and the Bose-Einstein distribution simplifies as n(T il ) + 1 ≈ k B T il /hc s q and n(T e )n(T il ) ≈ k B (T e -T il )/hc s q. Thus, Eq. (C9) becomes: The function I corresponds to the integral over q in Eq. (C10) and can be calculated analytically, leading to:

∂E e ∂t = 1
I(κ) = κ 2 2 - k B T il hc s κ log 1 + exp ε κ + hc s κ -µ e k B T il + k 2 B T 2 il h2 c 2 s F 1 ε κ + hc s κ -µ e k B T il -F 1 ε κ -µ e k B T il (C13)
with ε κ = h2 κ 2 /2m e . F 1 is the Fermi-Dirac integral of order 1.

Optical phonon

For optical phonons, the dispersion relation leads to ω q = ω LO and the deformation potential is given by the Fröhlich formalism [START_REF] Ziman | Electrons and Phonons[END_REF][START_REF] Fröhlich | [END_REF][43]:

U 2 (ω q , q) = 4πe 2 ρ ω 2 LO 1 ε ∞ r - 1 ε 0 r 1 q 4 (C14)
where ε 0 r and ε ∞ r are the dielectric constants at low and high frequencies, respectively. Plugging them into Eq. (C7) gives after integration over q:

E(k BZ ) hω LO [ f (ε k ) -f (ε k + hω LO )] ln 1 + 1 -hω LO /ε k 1 -1 -hω LO /ε k dε k (C16)
after having transformed integration over k to integration over ε k . Finally, as for acoustic phonons, it is possible to exhibit an electron-optical phonon energy transfer rate defined by:

ν et = 1 n e m 2 e e 2 ω LO π 2 h4 1 ε ∞ r - 1 ε 0 r (n(T il ) + 1) E(k BZ ) hω LO [ f (ε k ) -f (ε k + hω LO )] ln 1 + 1 -hω LO /ε k 1 -1 -hω LO /ε k dε k (C17)
giving the following electron-optical phonon energy transfer:

FIG. 1 .

 1 FIG. 1. Evolution of (a) the electron density n e and the laser intensity I L , (b) electron temperature T e and ion-lattice temperature T il and (c) the electron collision frequency ν c as a function of time. The critical density n c = m e ω 2L /4πe 2 and the laser frequency ω L are given for purpose of comparison. A Gaussian profile is considered for the laser intensity with a maximum of 10 14 .Wcm -2 reached at 200 ps and a duration of 100 ps.

FIG. 3 .

 3 FIG. 3. (a) Evolution of the electron density as a function of time. (b) Evolution of the electron temperature T e (thick lines) and the ionlattice temperature T il (thin lines) as a function of time. Simulations are performed with a Gaussian laser shape with a duration of 100 ps for the three following maximum laser intensities I 1 = 10 13 W.cm -2 , I 2 = 10 14 W.cm -2 and I 3 = 10 15 W.cm -2 reached at 200 ps. For the temperatures, solid and dashed lines corresponds to electron and ionlattice temperatures respectively. The critical density n c is shown for purpose of comparison.

FIG. 5 .

 5 FIG. 5. Ablator expansion during the solid-to-plasma transition. The same laser intensity profile as the one used in Sec. III is considered.

FIG. 6 .

 6 FIG. 6. (a) Total pressure as a function of time and of the laser intensity, (b) total pressure as a function of the laser intensity for three different instants, (c) electron density as a function of the laser intensity for three different instants.

4π 3 m e k 2 B U 2 T il ρc 2 sq

 3222 h3 (T e -T il ) [ f (ε k )f (ε k + hc s q)] dq dk (C10)Finally, the energy transfer can be written in such a way that the electron-acoustic phonon energy transfer rate ν et appears:∂E e ∂t = ν et n e k B (T e -T il ) (C11)where ν et is defined by:

m 2 e e 2 ω 2 LO

 2 T il ) + 1][n(T e )n(T il )] 1 k [ f (ε k )f (ε k + hω LO )] ln k + k 2 -2m e ω LO /h kk 2 -2m e ω LO /h dk (C15)As for acoustic phonons, the integration over k depends only on its modulus leading, for the electron-optical phonon energy transfer, to:∂E e ∂t = T il ) + 1] [n(T e )n(T il )]

  ∂E e ∂t = ν et hω L0 n e [n(T e )n(T il )] (C18)

  (x)/dx = jF j-1 (x). By calculating the derivative of n e with respect to itself, one obtains:

				inserting Eq. (B2) into Eq. (A6) and using the property
				dF j ∂n e ∂n e	=	√ 2m 3/2 e π 2 h3	k B T e 2	F -1/2	µ e k B T e	∂µ e ∂n e	(B10)
				Obviously, this derivative is also equal to unity and it is pos-
				sible to have an expression of ∂µ e /∂n e . Inserting this expres-
				sion into Eq. (B9), the effective chemical potential is finally
				given by:
						μe = 3k B T e	F 1/2 (µ e /k B T e ) F -1/2 (µ e /k B T e )	(B11)
				Appendix C: Electron-phonon energy transfer
	2 F 1/2	µ e k B T e	∂µ e ∂n e	(B9)

ACKNOWLEDGMENTS

The authors acknowledge Vladimir Tikhonchuk for his fruitful comments about this work. CEA is also acknowledged for supporting these studies. SXH acknowledges the support by the US Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority.

Appendix A: Two-temperature model with a non-constant number of particles

Let us consider a thermodynamic system with a dependence on the number N of particles. Thus, the total differential of the energy U of the system reads [39]:

where T , S, P, V and µ are the temperature, entropy, pressure, volume and chemical potential, respectively. This equation leads to the definition of P and µ:


Alejandro J Giangreco-Maidana

On the cyclicity of the rational points group of abelian varieties over finite fields

Keywords: group of rational points, cyclic, abelian variety, finite field 2000 MSC: Primary 11G10, 14G15, 14K15

We propose a simple criterion to know if an abelian variety A defined over a finite field F q is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End Fq (A). We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic F q -isogeny classes among certain families of them, when q tends to infinity. Some of these bounds require an additional hypothesis. In the case of surfaces, we prove that this hypothesis is achieved and, over all F q -isogeny classes with endomorphism algebra being a field and where q is an even power of a prime, we prove that the one with maximal number of rational points is cyclic and ordinary.

Introduction

The group structure of elliptic curves (EC) defined over finite fields has a theoretical interest, but also an interest in applications to cryptography and error-correcting codes. More precisely, those elliptic curves with cyclic groups of rational points are of special interest for various problems (see for example [START_REF] Koblitz | Elliptic curve cryptosystems[END_REF], [START_REF] Morain | Building cyclic elliptic curves modulo large primes[END_REF] and [START_REF] Kaliski | One-way permutations on elliptic curves[END_REF]). In many algorithms that use elliptic curves over finite fields one chooses it at random, thus, knowing such statistics can be useful.

The structure of all possible groups for EC over a finite field was discovered (independently) in [START_REF] Schoof | Nonsingular plane cubic curves over finite fields[END_REF], [START_REF] Rück | A note on elliptic curves over finite fields[END_REF], [START_REF] Tsfasman | Group of points of an elliptic curve over a finite field[END_REF] and [START_REF] Voloch | A note on elliptic curves over finite fields[END_REF]. However, this result does not provide immediately the statistics for cyclic EC over a given field or its extensions, which was explored in [START_REF] Vlȃdut | Cyclicity statistics for elliptic curves over finite fields[END_REF] and [START_REF]On the cyclicity of elliptic curves over finite field extensions[END_REF]. See also [START_REF] Banks | On group structures realized by elliptic curves over arbitrary finite fields[END_REF], where the realizability of possible groups for EC was studied from a statistical point of view.

There are many other closely related topics which were considered in the literature. In [START_REF] Shparlinski | Orders of points on elliptic curves[END_REF] and [START_REF] Murty | Group Structure Of Elliptic Curves Over Finite Fields And Applications[END_REF] the order of points on the groups of rational points was studied, in particular lower bounds for the exponent (the largest order), and some applications as well. Also, the exponent under base field extensions was examined in [START_REF] Luca | On the exponent of the group of points on elliptic curves in extension fields[END_REF]. This question is closely related to the question about cyclicity, since cyclicity is equivalent to the fact that the exponent equals the cardinality of the group of rational points.

For both theoretical and practical interests, it is very natural to extend this investigation to abelian varieties of higher dimensions. Many facts are still unknown in this case. Nevertheless, in a recent paper [START_REF] Rybakov | The groups of points on abelian varieties over finite fields[END_REF], Rybakov gives a very explicit description of all possible groups of rational points of an abelian variety in a Email address: ajgiangreco@gmail.com (Alejandro J. Giangreco-Maidana) given isogeny class. His result is formulated in terms of the Newton polygon of the variety and the Hodge polygon of the group. In particular, from this description it follows that varieties with cyclic group of rational points always exist in an isogeny class. In [START_REF] David | Abelian surfaces over finite fields with prescribed groups[END_REF] it is shown that for abelian surfaces, "very split" groups of rational points occur with density zero. This result agree with the general framework of the Cohen-Lenstra heuristics, which establish that groups with a higher number of automorphisms appear less frequently, since the very split groups have many more automorphisms than the cyclic group of the same size.

Another related and widely explored topic is maximal curves, i.e., those with a maximal number of rational points, such curves are very important for applications. Goppa showed that the applications of EC to coding theory are interesting only if the curve has many rational points ( [START_REF] Goppa | Algebraico-geometric codes[END_REF]). Theoretical and practical bounds were explored and many algorithms to construct it were proposed. Relating "maximal" abelian varieties with its structures of rational points does not seem to be a simple task, but with many promising applications.

In cryptography, the Discrete Logarithm Problem is extensively used in such systems and, under some conditions, cyclic groups are suitable for these purposes ( [START_REF] Galbraith | Algebraic curves and cryptography[END_REF]). In many applications, security is based on working with big cyclic subgroups of the group of rational points.

Another interesting related topic is counting isomorphism classes of abelian varieties defined over a fixed finite field. In the 1-dimensional case, this corresponds to compute class numbers of some orders in quadratic number fields, and in the general case, this was treated for example (very recently) in [START_REF] Xue | Counting abelian varieties over finite fields[END_REF]. Isomorphism classes can be grouped in isogeny classes. In [START_REF] Dipippo | Real polynomials with all roots on the unit circle and abelian varieties over finite fields[END_REF], DiPippo and Howe developed some techniques and gave an asymptotic formula (with explicit error terms) for the number of isogeny classes of n-dimensional abelian varieties over the finite field with q elements, for fixed n and q tending to infinity.

In this paper we give some statistical results concerning cyclic isogeny classes, namely, in Theorem 2.2 we present a simple criterion to know if an isogeny class is cyclic, i.e., all its varieties have a cyclic group of rational points, this criterion is based on the characteristic polynomial of the isogeny class. Further, in Theorems 2.3 and 2.6 we give asymptotic lower bounds on the fraction of cyclic isogeny classes when some coefficients on the characteristic polynomial corresponding to these isogeny classes are fixed. Theorem 2.6 needs an extra condition and, we show that for "good" coefficients, this condition holds true for abelian surfaces. Finally, we prove in Theorem 4.3 that under some restrictions, isogeny classes of abelian surfaces defined over finite fields with maximal number of rational points are cyclic and ordinary. Also, we give some examples of families of cyclic isogeny classes of abelian surfaces.

The rest of this paper is organized as follows: in Section 2 we briefly recall some general facts about abelian varieties over finite fields and we state our results more precisely; in Section 3 we prove Theorems 2.2, 2.3 and 2.6 and, Section 4 is devoted to abelian surfaces.

Preliminaries and Statement of the Results

For the general theory of abelian varieties see for example [START_REF] Mumford | Abelian varieties[END_REF], and for precise results over finite fields, see [START_REF] Waterhouse | Abelian varieties over finite fields[END_REF].

Let q = p r be a power of a prime, and let k = F q be a finite field with q elements. Let A be an abelian variety of dimension g over k. For an extension field K of k, we denote by End K (A) the ring of K-endomorphism of A and by End 0 K (A) = (End K (A)) ⊗ Q its endomorphism algebra, the latter being an invariant of its isogeny class A, we can denote it by End 0 K (A). For an integer n, denote by n the ratio of n to the product of different prime divisors of n; and by A[n] the group of n-torsion points of A over F q , i.e., the kernel of the multiplication endomorphism [n]. Then

A[n] ∼ = (Z/nZ) 2g , p n A[p] ∼ = (Z/pZ) i , 0 ≤ i ≤ g (1)
For a fixed prime ( = p), the Tate module T (A) is defined by lim ← -A[ n ]. The Frobenius endomorphism F of A acts on T (A) by a semisimple linear operator, and its characteristic polynomial f A (t) is called Weil polynomial of A. Tate proved in [START_REF] Tate | Endomorphisms of abelian varieties over finite fields[END_REF] that the isogeny class A of abelian variety is determined by its characteristic polynomial f A (t) and, if A is simple, f A (t) = h A (t) e for some irreducible polynomial h A and the center of End 0 k (A) is isomorphic to the number field

Q(F ) ∼ = Q[t]/(h A (t)
). The cardinality of the group of rational points A(k) of A equals f A (1), and thus, it is an invariant of the isogeny class. For a prime number , v denotes the usual -adic valuation. We denote by P the set of prime integers and by P(n) the set of prime divisors of n.

For an elliptic curve E, we have that E(F q ) is non-cyclic if and only if there exist a prime = p such that E[ ] ⊆ E(F q ). This is not the case for higher dimensional varieties, however, a slightly different criterion holds:

Lemma 2.1. If A is simple, then A(F q ) is not cyclic ⇐⇒ ∃ϕ ∈ End Fq (A), [f A (1)/ ] = ϕ • (1 -F ) for some prime divisor |f A (1).
Note that in this case we do not require = p. Note also that if such ϕ exists, it has to be defined over F q and it belongs to the center of its endomorphism ring. This lemma is an easy consequence of the theory of abelian varieties over finite fields and we will prove it here within the proof of Theorem 2.2. Lemma 2.1 implies that abelian varieties with few endomorphism are more likely to be cyclic. The group structure of an abelian variety is not necessarily determined by its endomorphism ring, but from Lemma 2.1 it follows that the property of being cyclic or not depends on its endomorphism ring, and thus, it is an invariant of it.

We say that A is cyclic if the group A(F q ) is cyclic. Counting cyclic varieties in a given isogeny class is the same as counting some fractional ideals in the center of End 0 Fq (A). In this paper we will focus on cyclic isogeny classes, i.e., those that contain only cyclic varieties. We have the following criterion: Theorem 2.2. Let A be a g-dimensional F q -isogeny class of abelian varieties corresponding to the Weil

polynomial f A (t). Then A is cyclic if and only if f A (1) is coprime with f A (1).
This can be proved by using Rybakov's Theorem ( [START_REF] Rybakov | The groups of points on abelian varieties over finite fields[END_REF]) when End 0 Fq (A) is commutative, and we shall prove it using the Lemma 2.1 in the general case. Note that we do not require A to be simple.

Every Weil polynomial has the form

f A (t) = t 2g + a 1 t 2g-1 + • • • + a g t g + a g-1 qt g-1 + . . . a 1 q g-1 t + q g ∈ Z[t], (2)
and for simplicity, we write it as f q,(a1,...,ag) (t). Denote by S g (q) the set of g-dimensional isogeny classes A of abelian varieties defined over the finite field F q , and by S g c (q) the subset of cyclic isogeny classes. The size of S g (q) is c(g)(1 -1/p)q g(g+1)/4 , when q tends to infinity and c(g) is a constant depending only on the dimension g (see [START_REF] Dipippo | Real polynomials with all roots on the unit circle and abelian varieties over finite fields[END_REF] for details). We write (a 1 , . . . , a g ) ∈ S g (q) if the polynomial f q,(a1,...,ag) (t) defines such an isogeny class.

Our first approach is to fix a vector a = (a 1 , . . . , a g-1 ) ∈ Z g-1 of g -1 integers, and define two sets of possible values of a g :

I a (q) := {z ∈ Z|(a 1 , . . . , a g-1 , z) ∈ S g (q)}, (3)
I a,c (q) := {z ∈ Z|(a 1 , . . . , a g-1 , z) ∈ S g c (q)}. ( 4
)
It is obvious that I a (q) is finite and non empty from a certain value of q = q(a). Fixing a prime p, we are interested on the proportion

r p,a (n) := n i=1 #I a,c (p i ) n i=1 #I a (p i ) , (5)
when n tends to infinity. Finally, we define the polynomial h(X, a) := gf X,(a,z) (1) -f X,(a,z) (1) (in fact, it is independent of z), and we have Theorem 2.3. Given a ∈ Z g-1 and a prime p, then

lim inf r p,a (n) ≥ 1 - p p -1     1 - ∈P(h(p,a)) 1 - 1 2   1 - 1 p g/2 + 1 p g/2   . (6)
Remark 2.4. Note that the limit of r p,a does not necessarily exist. Also, it is easy to see that if p 0, this bound is approximately

∈P(h(p,a)) 1 - 1 2 ,
which is greater than 1/ζ(2) = 0.6 . . . , where ζ is the Riemann zeta function.

Let us follow now a different approach by fixing a vector b = (b 1 , . . . , b g ) ∈ Z g of g integers, we want to study when f q,b (t) defines a cyclic isogeny class over F q when q varies over the power of primes. We consider this problem in two ways: (i)q varies over primes and (ii) we fix a prime p and q varies over the powers of p.

For our next result, we will need an extra hypothesis, which we make explicit as follows:

Hyp(b): there exist integers η(b), t(b) and s(b) with no common factor, such that the only possible divisors of f q,b (1) and f q,b (1) are divisors of η and tq + s.

Remark 2.5. The key idea is that these values, in particular η, are independent of q. Such η can always be obtained by successive Euclidean division, since we can see f and f as polynomials in Q[q]; but it is not sure to have a linear polynomial in q, with the condition of no common factors. Observe that η can be chosen as a product of different prime numbers. For the 2-dimensional case, we will see in Section 4 that this hypothesis holds for a wide range of values of b, and we will explicit them.

Then we are interested on the following proportions:

x b (n) = #{ ≤ n prime|b ∈ S g c ( )} #{ ≤ n prime|b ∈ S g ( )} , (7)
y p,b (n) = #{i ≤ n|b ∈ S g c (p i )} #{i ≤ n|b ∈ S g (p i )} , (8)
when n tends to infinity. Note that the above fractions are defined from a big enough n. Then

1. lim inf x b1 (n) ≥ L 1 and, 2. lim inf y p,b2 (n) ≥ L 2 ,
where

L i = ∈P(η(bi))\P(t(bi)s(bi)) -2 -1 , i = 1, 2.
Remark 2.7. As we can see, this bound depends on the choice of (η, t, s). We can verify that if (η 1 , t 1 , s 1 ) and (η 2 , t 2 , s 2 ) verify Hyp(b) for some fixed b, then ((η 1 , η 2 ), t 1 , s 1 ) also does. Also, if (η, t 1 , s 1 ) and (η, t 2 , s 2 ) verify Hyp(b), and if there exist a prime ∈ P(t 2 s 2 ) \ P(t 1 s 1 ) dividing η, then (η/ , t 1 , s 1 ) also verifies Hyp(b). This allows us to optimize the bound in that way. However, in Section 4 we will see with an example that Theorem 2.6 does not provide necessarily the best bound.

Example 2.8. As example, we discuss the case of elliptic curves. Here we have f q,a (1) = (1 + a) + q and f q,a (1) = 2 + a, thus, Hyp(a) holds for all values of a if we take

   η(a) = 2 + a, t(a) = 1, s(a) = 1 + a.
Also, since η(a) and s(a) are always coprimes, the bound for Theorem 2.6 takes the form

L = ∈P(2+a) -2 -1 .
Concerning Theorem 2.3, we have that h(p, a) = p -1, where a can be considered empty. This provides a complete answer to the fraction of cyclic isogeny classes defined over some finite extension of F p , and for p 0, the bound takes the form

∈P(p-1)
1 -1 2 .

To finish this example, let us say that the question

For which values of q, all isomorphism classes of EC defined over F q are cyclic?

is answered in Theorem 4.1 in [START_REF] Vlȃdut | Cyclicity statistics for elliptic curves over finite fields[END_REF], and in view of Theorem 2.2, this is equivalent to

For which values of q, ( f q,a (1), f q,a (1)) = 1 ∀a ∈ [-2 √ q, 2 √ q]?

General case: proof of theorems

In this section we prove Theorems 2.2, 2.3 and 2.6.

Proof of Theorem 2.2:

First we consider the case when A is simple. For an abelian variety A ∈ A, the center of its endomorphism algebra End Fq (A) ⊗ Q is isomorphic to the number field K = Q(π), where π is a root of f A and represents the Frobenius endomorphism F . For every variety A in A, End Fq (A) ∩ K is an order in K (thus contained in O K ), and O K is the endomorphism ring for some B ∈ A by Theorem 3.13 in [START_REF] Waterhouse | Abelian varieties over finite fields[END_REF]. For any x ∈ K, we have that x ∈ O K holds if and only if its (unitary) minimal polynomial has integer coefficients, that is, is an algebraic integer.

Denote by N the value f A (1), then Lemma 2.1

A(F q ) is not cyclic ⇐⇒ ∃ϕ ∈ End Fq (A), [N/ ] = ϕ • (1 -F ) for some prime divisor |N
follows from the fact that A(F q ) is not cyclic if and only if

A(F q ) = ker(1 -F ) ⊂ ker([N/ ]) = A[N/ ] for some |N , since 1 -F is separable. Since A is simple, if such ϕ exist, it must be in K: if ψ ∈ End Fq (A),
multiplying the previous equation in both sides and subtracting them, we have 0 = (ϕψ -ψϕ) • (1 -F ), thus, ϕ commutes with ψ. Similarly, we can show that this endomorphism is defined over F q . We consider the minimal polynomial p (t) of N (1-π) . All non-leading coefficients of p are of the form

N i-1 i
times an integer, except for the coefficient of t 2g-1 , which equals -f (1) , and that the constant term equals N 2g-1 2g . If A is cyclic, then all elements of { N

(1-π) , |N } are algebraic non integers: if not, for some , N (1-π) ∈ O K and for some abelian variety B with End Fq (B) ∩ K = O K we will have that B(F q ) is not cyclic. Therefore p have some non integer coefficient, either f (1), either v (N ) = 1. In consequence, gcd( N , f (1)) = 1. If gcd( N , f (1)) = 1, then for any |N , either ( , f (1)) = 1, or v (N ) = 1. In both cases, p have some non integer coefficient, and thus, N

(1-π) / ∈ O K . This implies that, for all A ∈ A, there is no ϕ ∈ End(A) with N = ϕ • (1 -F ): if such ϕ exist, it has to be equal to N

(1-π) , which is a contradiction. Now we consider the general case. Then A is isogenous to a product A i of simple isogeny classes, and we have that f A (t) = f i (t), where f i (t) is the characteristic polynomial of A i . Suppose A is not cyclic, we have two possibilities: (i) one of its components is not cyclic or (ii) ∃ prime such that |(f 1 (1), f 2 (1)) (WLOG). In (i), by the first part of the theorem, (

f 1 (1), f 1 (1)) > 1 (WLOG), writing f A (t) = f 1 (t) i>1 f i (t) + f 1 (t)( i>1 f i (t)) we have that ( f A (1), f A (1)) > 1. In (ii), writing f A (t) = f 1 (t)f 2 (t) i>2 f i (t) + f 1 (t)f 2 (t) i>2 f i (t) + f 1 (t)f 2 (t)( i>2 f i (t)) we have that ( f A (1), f A (1)) > 1. Sup- pose now that |( f A (1), f A (1)
), thus, 2 |f A (1) and |f A (1). Again, we have two possibilities: (i) |f 1 (1) and | i>1 f i (1) or, (ii) 2 |f 1 (1) and i>1 f i (1). In (i), A is not cyclic. In (ii), writing f A (t) = f 1 (t) i>1 f i (t) + f 1 (t)( i>1 f i (t)) we see that |f 1 (1), thus, A is not cyclic by the first part of the theorem. Theorem 2.2 is thus proved.

Note that Lemma 3.3.1 in [START_REF] Dipippo | Real polynomials with all roots on the unit circle and abelian varieties over finite fields[END_REF] implies that for fixed a ∈ Z g-1 , f q,(a,z) (t) always defines an ordinary isogeny class from a q big enough when p z, and they are almost all the isogeny classes. This will be important for the proofs of both theorems that follow, and it will be explained at beginning of the next proof.

Proof of Theorem 2.3:

From now on, a and p are fixed. For simplicity, we denote by N i and S i the integers #I a (p i ) and the sets of prime divisors of h(p i , a), with S = S 1 , respectively. From a certain value of i, I a (p i ) is a finite non-empty set of integers, thus we can take its maximum max(I a (p i )) and minimum min(I a (p i )) values, and we set M i = max(I a (p i )) -min(I a (p i )). Also, we write f i,z (t) instead of f p i ,(a,z) (t).

From Weil's "Riemann Hypothesis", if z ∈ I a (p i ), then |z| ≤ 2p ic , where we set c = g/2. For a big enough i, among the integers z between -2p ic and 2p ic , those such that (z, p) = 1 define an ordinary isogeny class (this follows from the inequality in Lemma 3.3.1 in [START_REF] Dipippo | Real polynomials with all roots on the unit circle and abelian varieties over finite fields[END_REF]) and the proportion of such z defining non-ordinary isogeny classes is less than 1/p i/2 (this follows from Proposition 3.1.3 in [START_REF] Dipippo | Real polynomials with all roots on the unit circle and abelian varieties over finite fields[END_REF]). Then we deduce that

N i /M i → p -1 p and M i /4p ic → 1 when i → ∞.
In order to take advantage of Theorem 2.2, the following elementary lemma allows us to identify the primes that can simultaneously divide the arithmetic progressions f i,z (1) and f i,z (1) (as sequences on z): Lemma 3.1. Let x n and y n be two arithmetic progressions of difference d x and d y respectively, and let be a prime such that d x ≡ 0 (mod ). Then |(x i , y i ) for some i if and only if d x y j ≡ d y x j (mod ) for some j.

Proof. First, observe that if the congruence d x y j ≡ d y x j (mod ) is verified for some j, then it verifies for all i:

d x y i ≡ d x (y i -y j + y j ) (mod ) ≡ d x y j + d x d y (i -j) ≡ d y x j + d x d y (i -j) ≡ d y (x j + d x (i -j)) ≡ d y x i
Suppose now that d x y j ≡ d y x j (mod ), then we have to solve the system

x 0 + d x i ≡ 0 (mod ) y 0 + d y i ≡ 0 (mod )
We get that i ≡ -x 0 d -1

x , and thus,

y 0 + d y (-x 0 d -1 x ) ≡ y 0 -(d y x 0 )d -1 x ≡ y 0 -d x y 0 d -1 x ≡ 0 (mod ).
We apply the lemma to the sequences f i,z (1) and f i,z (1), hence the modular condition becomes g(f i,z (1)z) ≡ f i,z (1) -gz (mod ) and thus, from the definition of h(p i , a) only the primes in S i can simultaneously divide f i,z (1) and f i,z [START_REF] Banks | On group structures realized by elliptic curves over arbitrary finite fields[END_REF].

For an integer n and a finite set of integers Z, let denote

σ(n, Z) = i≥1 (-1) i+1 R⊂Z,|R|=i n x∈R x 2 , ξ(Z) = i≥1 (-1) i+1 R⊂Z,|R|=i 1 x∈R x 2 = 1 - x∈Z 1 - 1 x 2 .
Observe that ξ can be defined for infinite subsets of P since ξ(P) converges and, if Z ⊆ Z , ξ(Z) ≤ ξ(Z ). We will study the sequence 1 -r p,a , i.e., we will count non cyclic isogeny classes using the lemma above. Note that if 2 |f i,z (1), then 2 f i,z+ [START_REF] Banks | On group structures realized by elliptic curves over arbitrary finite fields[END_REF], thus after applying the inclusion-exclusion principle, we can see that the number #(I a (p i ) \ I a,c (p i )) of non cyclic isogeny classes for p i is at most σ(M i , S i ).

For every finite set T of prime numbers that does not divide h(p, a) (T ⊂ P \ S 1 finite), we consider the polynomial

g(X) = m(g)X + 1, m(g) = ∈T -1 ,
thus, h(p i , a) ≡ h(p, a) ≡ 0 (mod ) for all ∈ T and for all i ∈ g(N), hence S i ⊂ P \ T for all i ∈ g(N).

We get

1 -r p,a (n) ≤ σ(M i , S i ) N i ≤ i / ∈g(N) M i + i∈g(N) σ(M i , S i ) N i ≈ p p -1 i / ∈g(N) M i + i∈g(N) σ(M i , S i ) M i ≈ p p -1 i / ∈g(N) M i + i∈g(N) M i ξ(S i ) M i ≤ p p -1 i / ∈g(N) M i + i∈g(N) M i ξ(P \ T ) M i ≈ p p -1 i / ∈g(N) (p c ) i + ξ(P \ T ) i∈g(N) (p c ) i (p c ) i
where all the sums are taken over 1 ≤ i ≤ n and ≈ means that both sequences have the same superior limit. The last sequence converges to

p p -1 1 + (ξ(P \ T ) -1) p cm(g) (p c -1) p c(m(g)+1) -1 .
As we have the previous reasoning for all finite set T disjoint to S, thus,

lim sup(1 -r p,a (n)) ≤ p p -1 1 + (ξ(S) -1)(1 - 1 p c ) = p p -1 ξ(S) 1 - 1 p c + 1 p c . Finally lim inf r p,a ≥ 1 - p p -1 ξ(S) 1 - 1 p c + 1 p c .
That finish the proof of Theorem 2.3.

Proof of Theorem 2.6:

We write q for varying (i) over the primes or (ii) over powers p i when i varies over positive integers. As a consequence of Dirichlet's Theorem or Theorem's hypothesis, the values of q are (asymptotically) evenly distributed among the invertible congruence classes modulo η, when q varies according to (i) or (ii), respectively.

Set j(q) = tq + s, then, by Hyp, the condition (j, η) = 1 would imply that b i ∈ S g c (q), from a certain value of q when f q,bi (t) defines an isogeny class.

Observe that (j, η) = 1 if and only if j is invertible in Z/ηZ; then we conclude by using the following lemma.

Lemma 3.2. Let t, s and n be integers with no common factor, then

#{x ∈ (Z/nZ) * |tx + s ∈ (Z/nZ) * } #(Z/nZ) * = ∈P(n)\P(ts) -2 -1 .
Proof. Take an ∈ P(n) and set m = v (n) , we have that #(Z/mZ) * = m -m/ . If ts we have that #{x ∈ (Z/mZ) * |tx + s ∈ (Z/mZ) * } = m -2m/ . If |ts, for all x ∈ (Z/mZ) * we have that tx + s ∈ (Z/mZ) * . We conclude by using the isomorphism Z/nZ ∼ = |n Z/ v (n) Z.

The case of Abelian Surfaces

In this section we consider the case of abelian surfaces, where the characteristic polynomial takes the form

f A (t) = t 4 + at 3 + bt 2 + aqt + q 2 ,
thus, we have b = (a, b).

Theorems applied to Surfaces.

In order to apply Theorem 2.6 to surfaces, we need to verify Hyp(b); and this is a consequence of the proposition below and the conditions on b that follow: Proposition 4.1. Let a, b and q be integers, then the common divisors of

N = 1 + a(q + 1) + b + q 2 and J = 4 + a(q + 3) + 2b are divisors of η(a, b) := (2a + b + 2)(a 2 -4a + 4b -16).
Proof. Let be a prime divisor of N and J, then it divides (q + 3)N -(q + 1)J = (q -1)(q 2 + 4q -1 + b) and 2N -J = (q -1)(2q + 2 + a).

If divides q -1, then N = q 2 -1 + a(q -1) + 2a + b + 2 ≡ 2a + b + 2( mod ). We set γ = q 2 + 4q -1 + b and

θ = 2q + 2 + a.
So, if doesn't divide q -1, then it divides γ and θ, and we have 4γ = θ(2q + 6 -a) + (a 2 -4a + 4b -16).

Finally, the possible common divisors are the divisors of (2a + b + 2)(a 2 -4a + 4b -16).

We have η(a, b) as defined in the proposition and clearly f (1) = aq + 4 + 3a + 2b, thus, we can take

   η(a, b) = (2a + b + 2)(a 2 -4a + 4b -16), t(a, b) = a, s(a, b) = 4 + 3a + 2b.
To finish, and in order to have the condition of no common divisor (with the values previously chosen), the integers in b = (a, b) must verify: (a, b + 2) = 1, and a ≡ 1 (mod 2).

A priori these values are not optimal in the sense of Remark 2.7, and as mentioned previously, Theorem 2.6 does not necessarily provide the optimal bound. For example, observe that Hyp(0, 0) holds with η = 2, t = 1, s = 1, in which case L 1 = L 2 = 0, which are clearly very bad bounds. But by applying directly Theorem 2.2 we can have better bounds, first we have:

f q,b (1) = 1 + q 2 = (q 2 -1) + 2 and f q,b (1) = 4, thus, f q,b (1)
is equivalent to 1 or 2 modulo 4, depending if p equals 2 or not, respectively, thus L 1 = L 2 = 1. This is because Theorem 2.6 uses only a weak result:

(f (1), f (1)) = 1 ⇒ cyclicity,
and not the stronger equivalence from Theorem 2.2:

( f (1), f (1)) = 1 ⇔ cyclicity.
Concerning Theorem 2.3 applied to abelian surfaces, this provides the fraction of cyclic isogeny classes with a fixed trace of Frobenius a and a fixed base field characteristic p. We have that h(p, a) = (p -1)[2(p + 1) + a]. Thus, for a = -4 and a Fermat prime p, the lower bound for Theorem 2.3 is 0.75 and we cannot improve it. Even for primes of form 2 + 1 (with 0 a prime number) we have a good bound, around 0.75.

Maximal Surfaces

Now, let us discuss about the other topic mentioned in the Introduction, varieties with many rational points. Rück's theorem [START_REF]Abelian surfaces and jacobian varieties over finite fields[END_REF] describes polynomials that occurs as Weil polynomials for abelian surfaces with endomorphism algebra being a field: Theorem 4.2 (Rück). The set of f A (t) for all abelian varieties A over F q of dimension 2 whose algebra End(A) ⊗ Q is a field is equal to the set of polynomials f (t) = t 4 + at 3 + bt 2 + aqt + q 2 where the integers a and b satisfy the conditions

1. |a| < 4 √ q, 2|a| √ q -2q < b < a 2 /4 + 2q,
2. ∆ = a 2 -4b + 8q is not a square in Z, and

3. either (a) v p (a) = 0, v p (b) ≥ r/2 and (b + 2q) 2 -4qa 2 is not a square in Z p , (b) v p (b) = 0, or (c) v p (a) ≥ r/2, v p (b) ≥ r, and f (t) has no root in Z p .
Concerning this type of isogeny classes, we have the following result Theorem 4.3. Let q = p r with r even. Consider the F q -isogeny classes A of abelian surfaces with End 0 Fq (A) being a field. Then the isogeny class with maximal number of rational points among such classes is ordinary and cyclic, and corresponds to the Weil polynomial f (t) = t 4 + at 3 + bt 2 + aqt + q 2 , where a = 4 √ q -3 and b = 6q -6 √ q + 1.

Proof. For a fixed q, the number of rational points increases with a and b. From the conditions of Rück's Theorem we have that the biggest possible values are a = 4 √ q -3 and b = 6q -6 √ q + 1 with ∆ = 5, and the ordinariness is clear.

Then we have

f (1) = -1 -2 √ q + 3q + 4( √ q) 3 + q 2 and f (1) = -3 + 9q + 4( √ q) 3
Writing f (1) and f (1) as functions N and j of x = √ q, respectively, we see that

N (x) = x 4 + 4x 3 + 3x 2 -2x -1, j(x) = 4x 3 + 9x 2 -3,
and thus,

(14x 3 + 49x 2 + 28x -14)j(x) -(56x 2 + 98x -7)N (x) = 35 = 5 • 7
Finally, we can see case by case, that for ∈ {5, 7} we cannot have that |j(x) and 2 |N (x) simultaneously:

• for = 5,

x (mod 5) 0 1 2 3 4 j (mod 5) 2 0 0 1 2

x (mod 25) 1 2 6 7 11 12 16 17 21 22 N (mod 25) 5 5 5 5 5 5 5 5 5 5

• for = 7,

x (mod 7) 0 1 2 3 4 5 6 j (mod 7) 4 5 3 1 2 2 4

Surfaces having more number of rational points N than those considered in Theorem 4.3 are only (in decreasing order of N ):

E 2 max , E max × E max-1 , E max × E max-2 , E 2 max-1 and E max-1 × E max-2 ,
where E max , E max-1 and E max-2 are the elliptic curves defined by its Frobenius traces 2 √ q, 2 √ q -1 and 2 √ q -2, respectively. We can easily check, by using Theorem 2.2, that E max is always not cyclic, E max-2

is always cyclic, and E max-1 is cyclic or not, depending on q ≡ 0, 1 (mod 3) or q ≡ 2 (mod 3), respectively (since 3 is the only possible prime divisor of f (1) and f (1)); thus, the cyclicity of E max-1 × E max-2 depends only on the cyclicity of E max-1 , since the cardinalities of E max-1 and E max-2 are relatively prime. On the other hand, note that for q = 23 3 the ordinary class with maximal number of rational points, among the classes considered in Theorem 4.3, is given by f (t) = t 4 + 438t 3 + 72293t 2 + 438(23 3 )t + 23 6 and is not cyclic since 7|f (1) and 49|f (1), so Theorem 4.3 holds only for even powers.

Families of cyclic isogeny classes

We finish this section by giving two sequences of cyclic isogeny classes of abelian surfaces in Proposition 4.4 and 4.5. Basically, we get the cyclicity of these families by choosing appropriate values of q and b. Proposition 4.4. Given an integer b and a power of a prime q = p r > 4, such that:

1. p b 2. 8 √ q -4q ≤ b ≤ 4,
3. N 0 = b -3 -2q + q 2 is coprime with N 0 = 2(b -4)

Let be s = ∈P(N 0 )

( -1). Then for i ∈ Z ≥0 f i (t) = t 4 -4t 3 + b i t 2 -4q i t + q 2 i , where q i = p r+is and b i = b + 2q i , defines a ordinary cyclic isogeny class A i of abelian surfaces over F q i . Proof. Note that f i (t) = f qi,(-4,bi) (t). Every f i defines an ordinary isogeny class of abelian surfaces since p b i , and f i is a Weil polynomial since

8 √ qp is/2 -4qp is ≤ 8 √ q -4q ≤ b ⇒ 8 √ q i -2q i ≤ b + 2q i = b i b i = b + 2q i ≤ 4 + 2q i = (-4) 2 4 + 2q i .
Observe that f qi,(-4,bi) (1) = 4 -4(q i + 3) + 2(b + 2q i ) = N 0 = f q,(-4,b) [START_REF] Banks | On group structures realized by elliptic curves over arbitrary finite fields[END_REF]. Also, we have that f i+1 (1) -f i (1) = (p s -1)[p 2(r+is) (p s + 1) + 2p r+is ].

Thus, (f i (1), N 0 ) = 1 implies (f i+1 (1), N 0 ) = 1, then if A i is cyclic, A i+1 is cyclic as well. We conclude by using the theorem's hypothesis.

It is not hard to find values (q, b) verifying the last hypothesis. Moreover, for a fixed prime p, there are infinite many pairs (r, b) such that (p r , b) verifies it. then for every i ≥ 0, f i (t) = t 4 + bt 2 + q 2 i defines an ordinary cyclic isogeny class A i of abelian surfaces over F q i , where q i = p r+is and s = ∈P(b+2)

( -1).

Proof. It is easy to check that f i (t) is a Weil polynomial and it defines an ordinary isogeny class since p > b. Put c = ∈P(b+2)

,

We have f i (1) = p 2(r+is) -1 + (b + 2), i ∈ Z ≥0 .

Observe that f i (t) = f qi,(0,b) (t) and f i (1) = 2(b + 2). For A i to be cyclic it is enough to have (f i (1), 2(b + 2)) = 1, and since f i (1) ≡ b + 2 ≡ 0 (mod 4), it is enough to have (f i (1), b + 2) = (f i (1), c) = 1. The last statement is true since by hypothesis (f 0 (1), c) = 1 and since c divides the difference f i+1 (1) -f i (1) = p 2(r+is) (p s + 1)(p s -1) (using Fermat's little theorem).

For a fixed p, big enough, take a prime such that 2 < < p and p 2 -1, then set b = -2. This gives b and p satisfying the proposition hypothesis.

Proposition 4 . 5 .

 45 Given an integer b and an odd prime p > b + 2 such that b ≡ 2 (mod 4) and ∃r ≥ 1, (p 2r -1, b + 2) = 1,

Acknowledgement

I would like to thank my advisor Serge Vlȃdut ¸for very fruitful discussions and for his very useful comments and suggestions. I would like to thank also the anonymous reviewers and the editor for their helpful remarks.