Tien Hien Nguyen

Atomic Energy Agency 2 Fuel made of uranium dioxide U O 2 (Uranium Oxide) 3 Fuel made of the mixture U O 2 and P uO 2 , (U, P u)O 2 (Mixed Oxide)

Cette thèse est consacrée à l'étude de la déstabilisation structurale du dioxyde d'uranium monocristallin. L'irradiation avec des ions Xe de 470 keV , Ce de 500 keV et La de 500 keV (correspondant à un parcours projeté Rp ∼ 85 nm et à l'écart type ∆Rp ∼ 40 nm selon le code de calcul SRIM) ont été réalisées pour étudier la déstabilisation du monocristal, induit, d'une part, par la création de défauts dû au processus de ralentissement nucléaire des fragments de fission à la fin de leur parcours (contribution balistique), et, d'autre part, par l'incorporation de produits de fission à forte concentration (contribution chimique). L'énergie cinétique des éléments incorporés a été choisie de sorte que leurs parcours dans le solide soient identiques afin de comparer directement les effets induits par des espèces solubles (La et Ce) et insolubles (Xe) dans le dioxyde d'uranium. La Spectrométrie de Rétrodiffusion Rutherford en canalisation (RBS/C) a été appliquée pour caractériser et quantifier les défauts générés par l'irradiation. Les données ont été analysées par simulation Monte-Carlo en considérant un modèle original comportant de deux-type de défauts (i) des atomes aléatoirement déplacés dans le cristal (RDA) et (ii) des distorsions des rangés atomiques (BC). L'accumulation de la fraction d'atomes déplacés RDA avec l'augmentation de la fluence conduit à une forte augmentation du nombre de défauts observé entre 4 et 7 dpa, indépendamment de la nature des ions. Une seconde augmentation spectaculaire est observée à partir de 300 dpa (correspondant à une concentration excédant 5 % atomique d'ions implantés) pour le cas spécifique des cristaux irradiés avec des ions Xe. Un comportement différentié est clairement observé pour les espèces solubles par rapport à leurs homologues insolubles. Cette différence se traduit d'une part par l'augmentation spectaculaire de RDA lorsque le cristal est implanté à très haute concentration dans le cas d'espèce insolubles, et, d'autre part, par un endommagement plus prononcé entre 7 et 300 dpa. Ce phénomène est notamment la conséquence des différences de taille entre les rayons atomiques de La et de Ce, très inférieure à celles du Xe. A contrario, l'accumulation de défauts de type distorsion des rangées atomiques BC avec l'augmentation de la fluence conduit à une évolution très similaire quelle que soit la nature de l'élément incorporé. Cette augmentation se produit principalement entre 10 et 100 dpa -correspondant à un plateau pour l'évolution des défauts de type RDA -et traduit une importante évolution structurale du dioxyde d'uranium irradié, explorée pour la première fois dans ce travail de thèse. 3.1 Irradiation with noble gas ions . . . . . . . . . . . . . . . .

Frédérico GARRIDO for the continuous support of my Ph.D study and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. Through my PhD, I am very much benefited from his guidance and knowledge. More important, he always has strong faith on students even long before they realize what they can really achieve. His vision on science always motivates me to explore more in the field. In every sense, none of this work would have been completed without him.

I am deeply grateful to Dr. Aurélien DEBELLE and research director, Dr. Lionel Thomé for their insightful suggestions, encouragement discussion and times. Dr.

Aurélien DEBELLE answered many of my questions regarding the X-ray diffraction analysis, taught me how to use the techniques, explained me and helped me a lots in analysing the data. All these are invaluable not only for this thesis but also for my future career.

Dr. Lionel THOM É was always available to answer any of my questions whether they concern the theory, the experimental technique or the results that I have. It is a huge pleasure for me to work with them. Their enthusiasm towards the research and kindness to people are something I really much long for. I would like to address my sincere thank to Dr. Stamatis MYLONAS for his kind helps and interesting discussions with TEM images. He taught me how to polish a sample especially a so fragile material like uranium dioxide, and helped me a lots in recording these images.

Many thanks go to the JANNuS staffs for their kind assistance during my experiments. Special thanks are due to Cyril BACHELET, he has always managed to give me beam time to finish my experiments.

I thank the direction of the CSNSM laboratory for welcoming me in this laboratory, for helping me with the administrations during my time working at the laboratory and for giving me the opportunity to attend several conferences. I thank the high graduate school "MIPEGE" for the financial support for my thesis.

Last but not the least, I would like to thank my family, escpecially my parents for supporting me spiritually throughout my life. Thanks all my friends in Orsay, who have always been with me during my stay in France.

A big thank to whom I forgot to mention!

Abstract

This thesis is dedicated to the investigation of the structural destabilisation of U O 2 single crystal. Irradiations with 470-keV Xe, 500-keV Ce and 500-keV La ions (with corresponding ion range of Rp ∼ 85 nm and range straggling of ∆Rp ∼ 40 nm according to SRIM calculation) have been performed to investigate the destabilisation of U O 2 single crystals induce by (i) the radiation damage effects due to the nuclear stopping process of a fission fragment at the end of their trajectories (ballistic contribution) and by (ii) the incorporation of a fission product at high concentration (chemical contribution). The energies and masses of bombarding ions were deliberately chosen so that they would have very similar projected range in U O 2 in order to compare the effects induced by solubles (La and Ce) versus non soluble Xe species in U O 2 . Rutherford Backscattering Spectrometry in channelling geometry (RBS/C) was applied to study the defects induced. Channelling data were analysed afterwards by Monte-Carlo simulation with McChasy code assuming a two-class model of defects comprising (i) the randomly displaced atoms (RDA) and the bent channels (BC) defects. The accumulation of RDA with increasing ion fluence leads to a steep increase (build-up of defects) observed from 4 to 7 dpa regardless of nature of ions and a dramatic increase observed from 300 dpa (corresponding to ≥ 5 at. % of implanted ions) only for Xe-irradiated crystal. The difference due to the soluble versus insoluble species was clearly observed. Such a difference was observed via the dramatic increase of RDA when the crystal is implanted at very high concentration only for crystal implanted with insoluble species. Moreover, the difference is also observed via the higher fraction of RDA created in the crystal irradiated with insoluble element. This phenomenon is mostly due to the size of implanted species in the matrix. Insoluble Xe atoms have the atomic radius which is larger than twice the atomic radius of U sub-lattice while soluble La and Ce atoms have the atomic radii of similar size as compared to U atom. Xe creates a much stronger stress field in U O 2 crystal in comparison to La or Ce; a higher fraction of RDA is thus created. Conversely, the accumulation of BC with increasing ion fluence leads to very similar evolution versus ion fluence in all crystals implanted with the three elements.

A regular increase of BC versus fluence which reveal the dramatic structural modification is observed. Such a modification is hidden according to the evolution of RDA.

Introduction

Nuclear energy, as for over 50 year, has proved to be one of the most important and the most used non -CO 2 energy source for the human being. During that period, many types of nuclear fuels making use of Uranium fissile isotope ( 235 U ) were studied and used as the main fuel material in the majority of the commercial nuclear reactors all over the world.

At the present time, there are over 400 nuclear reactors in operation all over the world which provide a non-negligible contribution of about 14 % of the world's electricity.

Moreover, the nuclear energy continues to be the first choice of new energy resources in the future especially when the fossil fuels are coming to an end. According to the recent report on application of nuclear source for generating electricity of IAEA, 1 about 60 reactors are under construction and about 20 countries plan for nuclear power programs.

As the world's population is increasing the consumption of energy and electricity increases. In addition, the worldwide development of industry necessitates an increase in production of the electricity. Because of that, scientists all over the world have to search for every available sources of energy so as to meet the demand of not only the energy (in general) but also the need of electricity for the global development. Therefore, energy and particularly electricity always presents as one the biggest challenges to the mankind that force the scientific community to look for all possible sources of energy and electricity for the world's development amongst which nuclear energy is still the most promising candidate.

During the last 50 year, nuclear energy has been intensively used in nuclear reactors for the purpose of generating electricity all over the world. During that time, many types of nuclear reactor technologies have been studied, tested and applied. At the present time, the pressurized water reactor (PWR) -the nuclear reactor of generation II -presents as the most used nuclear reactor type. This technology, which has been widely used with great success, uses U OX 2 and M OX 3 fuels as the heat source for driving the electrical generator's turbine. However, due to the strict regulation on the safety and due to the economical requirements, the new generation of advanced nuclear reactors (generation III and IV) that are safer and more powerful are being developed. Along with the appearances of nuclear reactors generation III and IV, scientist has to come back to the research on the nuclear fuel so as to find out the advanced fuel that can be used in these newly Introduction designed reactors. In particular, the problem arises in the nuclear reactor generation IV where fast neutrons at high flux are used for the two purposes: (i) fast neutrons are used to trigger the fragmentation of fissile isotopes so as to generate heat sources and to produce secondary neutrons in a chain fission reaction and (ii) they are also used in the second application in order to transform fertile into fissile isotopes via the neutron capture reaction followed by the β -emissions, e.g. 238 U (n, 2β -) 239 P u. With the presence of neutron at high flux, the fuel is submitted to an extremely deleterious environment where continuous bombardments by neutrons, fission fragments, alpha particles and heavy recoil nuclei lead to the strong destabilisation of the fuel's structure. In addition, in the new generation of nuclear reactor, the aim is not only to increase the safety of the reactor but also to increase the burn-up 4 so as to increase the production of electricity for economical purposes. The increase of burn-up leads to the increase of number of energetic fission fragments created which consequently gives more contribution to the irradiation damaging sources to the fuel structure. Moreover, due to the long time operation, the fuel is literally poisoned by the created fission products (FP). The incorporation of the FP modifies the chemical composition of genuine fuel, which eventually changes the physical and mechanical properties (thermal conductivity, structural stability, hardness. . . ) of the fuel. Finding new form of nuclear fuel that possesses high stability and resistance to irradiation and possesses a high capability of accommodating large fraction of FP is therefore one crucial challenge to the application of future generation of nuclear reactor.

Besides the success of the nuclear industry, the nuclear security issue still remains as one of the biggest challenges to mankind. Over the time, human has witnessed several disasters due to the failure of a nuclear facility. Those disasters caused thousands of death (direct effect) and hundred thousands of severe diseases years latter (indirect effect).

Along with the effects on people's healths, the release of radioactive materials leads to the abandon of a wide area and destroys the entire environments within. Learning from those disasters, many researches have been being conducted in order to minimize the risk of a nuclear accident in which the research on the nuclear fuel is one important subject.

As stated here before, the stability of the fuel is affected principally by two elements:

(i) the radiation effects and (ii) the incorporation of impurities leading to the modification of the chemical composition. Nuclear fuel figures a quite rich source of radioactive radiations including the high flux neutron particles, the created fission fragments, the alpha particles and heavy recoil nuclei (daughter nuclei produced in a disintegration of Uranium atom and minor actinides). The fuel materials themselves figure as non-negligible source of radiation because they are radioactive element with long half-life of millions of years. In operation, after capturing a neutron (thermal neutron in thermal reactor or fast neutron in fast reactor), fissile isotope of U in U OX fuel or U and P u in M OX fuel fission and give birth to two fission fragments and gamma radiation (one of radioactive radiation) and some neutrons along with. The fission fragments are, in general, unstable nuclei. This means that they are radioactive, they decay (or follow a chain decays) and emit α, β -or energetic γ radiations and finally become stable elements (known as fission products).

Not only fission fragments presenting as sources of radiation in nuclear fuel, minor actinides are also found as part of the radioactive sources. During operation, 238 U isotope captures one neutron and does not pass afterwards a fission. It becomes 239 U which is unstable and β -radioactive with half-life of 23.47 minutes. It emits quickly one β - particle and becomes 239 N p or it can otherwise capture one more neutron to become 240 U which is another β -radioactive isotope with half-life of 14.1 hours. 239 N p and 240 N p (daughter nuclei of 239 U and 240 U ) are themselves β -radioactive nuclei. They emit one β -particle to become P u isotopes. The chain of neutron capturing and β -decays continue and a series of minor actinides (Am, Cm, . . . ) nuclei are produced (see the figure below). Those minor actinides are also long-live sources of α radiation that lead to the storage of nuclear waste to thousands of year before reaching the normal level of radioactive radiation.

Introduction

along with the two fission fragments. Each of them has the energy of ∼ 2 M eV (total of ∼ 4.8 M eV ). This fission reaction also releases about 7 M eV in prompt gamma radiation. In operation, the flux of neutron particles varies (due to the reactor controlling action to meet the demand) around 10 14 cm -2 s -1 which gives a high level of neutron radiation over whole life time of a reactor core. Every atom that make up the reactor vessel displace twice in average due to the bombardment of neutron radiation. All the concerning information around one fission reaction confirm the fact that nuclear fuel is of the highest level of radioactive radiations.

Since nuclear fuel is submitted to such a rich source of radioactive radiations, its structure will therefore be affected. A thorough investigation of its structural stabilisation due to (i) the radiation effect and (ii) the modification of the chemical composition of the genuine fuel is necessary to understand the stability of fuel in this environment so as to help finding out the suitable fuel for the new generation of nuclear reactors. Knowledge of the fundamental process plays an important role and hence in this work, we investigate the fundamental mechanism of the fuel's structural destabilisation due to both the radiation and chemical contribution Investigating the nuclear real fuel taken out from a nuclear reactor core presents as a big challenge to scientist since it requires a lot of special safety conditions as well as special equipment. Experiments on the real fuel are difficult to perform and interpret because of the high radioactivity and because of the mixing of the various contributions coming from the several radiation sources (e.g. neutrons, gamma and fission fragments). Conversely, the use of external beams of mono-energetic ions allows one to perform screening studies in well-defined conditions (choice of ions, energy, fluence, flux, temperature of sample, etc). In this work, the destabilisation of U O 2 single crystals due to the irradiation and due to the incorporation of foreign elements is investigated as a model system to the real fuel. Moreover, by using specified ion, the modification of the chemical composition (leading to the modification of chemical and physical properties of nuclear fuel) due to the incorporation of an impurity will be parallelly investigated. The results obtained from this work are not going to be the complete knowledge but will be useful to explain the structural modification of nuclear fuel submitted to the bombardment of high neutron flux and of internal radiation (energetic fission products). This thesis will therefore be organised in the following 4-chapter structure. In the first chapter, introductions will be given to the materials that can be used in the nuclear industry that have to submit to the high irradiation environments . These material can be used as the fuel, potential matrix for incinerating long-live isotopes and waste storage matrix. Afterwards, the theory of the interaction between a charged particles with material will be discussed. The processes of creation of defect due to this interaction will Tien Hien NGUYEN Université Paris-Sud This chapter gives some descriptions of materials used for nuclear industry in which oxide-based materials such as urania, zirconia (ceramics of fluorite-type structure) and magnesium spinel will be mentioned and analyzed as examples of investigated materials.

To explain the radiation-induced effects, the fundamental (physical) processes occurring at atomic scale when bombarding a solid by a charged particle will be discussed. In the end, the chapter presents a summary of investigation on the behaviour of a solid's structure submitted to the irradiation of an ion beam (process of reorganization of defects and the structural transformation).

Materials for nuclear energy

Materials used for nuclear energy production refer not only to the nuclear fuel but also other materials for the claddings, the reactor vessels, the inert matrix for transmutation of minor actinides that are created during operation and for storage of nuclear wastes and other equipments. Amongst them, the choices for the fuel, cladding, reactor vessel and storage matrix seem to be the most important because those materials have to bear an extremely high irradiation condition during their life time in operation and in long-term storage.

Many classes of materials were investigated and applied into the commercial market depending on the varieties types and on different generations of nuclear reactors. Fuels a large fraction of radioactive elements. However, even though being studied for a long time these material need to be carefully investigated for final application.

Urania

Amongst the possible materials, ceramic materials for instance, the oxide-based ceramics (such as U O 2 , ZrO 2 ) figure the most popular for nuclear energy production.

Uranium dioxide exhibits a fluorite-type crystalline structure in which uranium atoms form a fcc structure with uranium atoms occupy the 8 vertices and 6 centres of the cube [START_REF] Willis | Structures of UO 2 , UO 2+x and U 4 O 9 by neutron diffraction[END_REF]. Urania is used mainly as nuclear fuel for fuel rods in nuclear reactors, specifically as U O 2 (U OX) or as a mixture of U O 2 and P uO 2 (plutonium dioxide). (U, P u)O 2 is called a mixed oxide (M OX fuel) in which P uO 2 exhibits the same fluorite-type structure and the mixture (U, P u)O 2 also exhibits the fluorite-type structure. U O 2 is used as nuclear fuel since it possesses several relevant properties such as a high melting temperature (see table 1.1) and a high resistance to irradiation. Moreover, U O 2 also emerges as a good "trap" of fission products and minor actinides. During operations, a certain amount of fission products and minor actinides are produced. Those elements are mostly highly radioactive and chemically toxic. But thanks to the U O 2 , those elements are confined within the crystalline structure of U O 2 without being released. Though having very interesting aforementioned characteristics, U O 2 still has some disadvantages as being the choice for fuel. The first disadvantage is that its thermal conductivity is low and this leads to the high gradient of temperature of the fuel pin. The temperature of the fuel pellet is always higher at the center than the edge (∆T ∼ 600 • C from the center to the surface of the fuel pellet). This difference in temperature could result in the destruction of the fuel pellet and hence the fuel rod. Another disadvantage of Urania is that it is not stable in a rich-oxygen environment. Urania quickly interacts with oxygen and becomes U 4 O 9 , U 3 O 7

or U 3 O 8 giving the risk of destroying the U O 2 structure. To avoid the contact of U O 2 with oxygen in a reactor, the fuel rod are usually filled with Helium gas at typical pressure of 20 bars. Inside the reactor, uranium dioxide is used in a sintered form in which it has been sintered into a pellet of dimension 0, 82 cm in diameter and 1, 3 cm in height (as in French reactor). Beside that most valuable application, depleted U O 2 (DU O 2 ) can also be used as a material for radiation shielding due to the high atomic number.

Apart from uranium, plutonium (created during operation by the neutron capture reaction of uranium followed by the β -emissions) can be as well used as nuclear fuel.

But the risk of the proliferation requires that the plutonium produced has to be burned (incinerated or consumed) for electricity generating (peaceful) purposes. In order to consume the plutonium, M OX fuel has been used as one of the alternative solutions.

But the used of the M OX fuel has its own disadvantages. It can help using plutonium as one of the fissile isotopes for the fission reaction but an equivalent amount of plutonium Tien Hien NGUYEN Université Paris-Sud can be reproduced again since there is uranium in the fuel. This means that the amount of plutonium stays almost unchanged. To solve the problem, several other materials have been investigated to be used as the host transmutation matrices. Those transmutation matrices are used so that they can keep plutonium inside the matrix to be burned without being recreated. Among the potential candidates used as transmutation matrix, Zirconia and Spinel are two of the most promising.

Zirconia

Zirconium dioxide or Zirconia (ZrO 2 ) is found in natural form of rare baddeleyite mineral but it is mostly produced from zirconium silicates sand (ZrSiO 4 ) by thermal treatment at high temperature followed by chemical treatment. Zirconia has high melting temperature and a low thermal conductivity (see table 1.1). Pure zirconia undergoes two crystallographic transformations between room temperature and its melting point: monoclinic to tetragonal at roughly 1170 • C and tetragonal to cubic at about 2370 • C.

The volume changes associated with these transformations are sufficiently high so that the thermal deviations through the transformations tend to break objects made of pure zirconia and that makes it quite impossible to be used in many applications. The hightemperature tetragonal and cubic forms can be stabilized with a variety of oxide additions (dopant) such as CaO, M gO, T iO 2 , Y 2 O 3 . . . For instance, doping Yttria allows the zirconia to be stabilized in the cubic form in a large domain of composition and temperature.

Once the process completes, a F SZ (Fully Stabilized Zirconia) is obtained. Yttria-dopped zirconia gives the Y SZ -"Yttria Stabilized Zirconia" which exibits an oxygen-deficient fluorite-type structure. The zirconium and yttrium atoms are located in an fcc lattice, i.e., at the corners of the cubic elementary cell as well as at the halves of the 110 directions.

Oxygen atoms lie at the quarters of 111 directions. in the formation of an oxygen vacancy. The concentration of the oxygen vacancy is fixed by the doping fraction. The actual chemical composition of Y SZ is hence

E d (U ) = 40 E d (Zr) = 40 E d (M g) = 30 E d (O) = 20 E d (O) = 40 E d (Al) = 30 E d (O) = 60
Y x Zr 1-x O 2-x/2 .
The relevant phase diagram is shown in figure 12. According to the diagram, cubic zirconium dioxide exists down to room temperatures when the concentration of yttria is higher than 8 mol %. In the nuclear industry, Y SZ is a good candidate for inert matrix for transmutation due to its high radiation resistance and the ability to confine radio-toxic elements. cations moves to octahedral positions. To characterize for a partially inverse structure the inversion parameter i, which represents the fraction of Al 3+ cations in tetrahedral sites, is used. The structure formula for a partially inverse magnesia spinel (M gAl 2 O 4 ) may be then written as:

Spinel

[M g 2+ 1-i Al 3+ i ] tetra [M g 2+ i Al 3+ 2-i ] octa O 4 [15]
The first square bracket represents the mean occupancy of tetrahedral sites while the second one represents the mean occupancy of octahedral sites by M g 2+ cations. For a normal spinel, i = 0; in a spinel where cations arrange randomly, i = 2/3, and for a total inverse spinel, i = 1. The crystalline structure spinel of obeys the F d 3m space group.

M g 2+ cations occupy 8a Wyckoff positions, while Al 3+ cations occupy 16d sites. The O 2- anions are displaced from ideal 48f sites in 111 directions away from the tetrahedral M g 2+ cations and towards the Al 3+ cations [START_REF] Sickafus | Structure of spinel[END_REF][START_REF] Rouse | Space group of the spinel structure: A neutron diffraction study of MgAl 2 O 4[END_REF].

The natural sources of spinel have never been discovered, therefore spinel has to be produced synthetically [START_REF] Schmidtmeier | Magnesium aluminate spinel raw materials for high performance refractories for steel ladles[END_REF]. There exists two main methods to produce spinel: sintering cooling. In the center, the impurities are concentrated due to the lowest melting point.

Thus, homogeneous fused spinel can only be made by very careful process. Magnesia spinel possesses some excellent chemical properties, i.e., high chemical inertness, thermal, dielectrical, mechanical and optical properties. Along with the above properties, it has the high melting temperature, high chemical inertness as well as the good thermal shock resistance that makes it a good usability in radio-technical applications.

Radiation effects on ceramic materials

This section discusses the interactions between a charged particle with materials and the possible defects created when a charged particle travels through the materials.

First of all, the main interactions between ion and a solid (consequently the process of transferring energy from ion to a solid including the nuclei and orbital electrons) will be first mentioned. The notation of "Stopping power" 1 will be then defined as the quantity characterizing for the energy transferred of interaction between ion and materials: the quantity which represents the energy loss of ion per unit of materials' thickness traversed.

Finally, the process of creation of randomly displaced atom defect and vacancy (Frenkel pair), and then the reorganization into extended defects, dislocation loop, grain division, ion track. . . resulting from the passing of an ion will be discussed.

Ions-materials interaction: processes at the atomic scale

Charged particles refer to ions (heavy charged particles), electrons or positrons and other particles. All the mentioned particles cause their own effects while travelling through a solid but this research focuses only on the effects of ion irradiation on materials since ions are of the most deleterious cause to the materials' structure (while the effects of other charged particles are almost negligible). For that reason, the concept of "charged particle" mentioned in this section could be restricted to the notation of "ion" only.

Physically, charged particles interact with materials by two interactions: inelastic (electronic) and elastic (nuclear) interactions. The former one refers to the continuous coulomb interaction of ions with the orbital electrons surrounding the nucleus while the latter one refers to the nuclear collisions with the atomic nuclei. When interacting with materials, ion loses part of its energy by collision and hence slows down inside the materials. Depending upon its energy (or its velocity), the interactions of charged particles with a solid differs. At high energy, the inelastic interaction is dominant while at low energy the elastic interaction is dominant. Due to interactions with a solid, charged particles lose part of the initial energy and slow down and finally stop inside a solid. The energy loss in the inelastic interaction is due to interaction with the electrons via electronic processes 1 Usually used in literature but has the nature of a stopping force.

such as the excitation or the ionization an atom or electronic capture. In an elastic interaction, the energy loss is due to the ballistic collision with the atomic nucleus. The two interactions both result in the loss of energy of charged particles but at different rates with different mechanisms. The energy loss due to electronic collision is continuous through the presence of the electrical fields created by both ion and atoms (mostly the electronic cloud) whilst that due to nuclear collision is not continuous because it happens only when ion gets close enough to the atomic nucleus. For an energetic particle (∼ 100 M eV ), the major part of energy loss is due to the electronic interaction (about 90 %).

The elastic collision

The basic assumptions of elastic collisions are the conservation of kinetic energy and momentum of the system. Suppose that we have two particles, particle 1 having the mass m 1 moves at velocity -→ v towards particle 2 having the mass m 2 at rest (as in the usual situation of irradiating a material, the ions move and the sample is supposed to be a target at rest). 2 After collision, m 1 moves at velocity v 1 , in the direction making an angle θ with the incident direction; while m 2 moves at velocity v 2 , in the direction forming an angle ϕ with the incident direction.

Applying the laws of conservation of kinetic energy and momentum to the system of elastic collision between two bodies, we have:

m 1 - ! v, E m 2 Incident T arget m 1 - ! v 1 , E 1 Scattered m 2 Recoil - ! v 2 , E 2 θ ' BEFORE COLLISION AFTER COLLISION
E sys = E 1 + E 2 = m 1 v 2 2 + 0 = E ′ 1 + E ′ 2 = m 1 v 2 1 2 + m 2 v 2 2 2 (1.1) --→ P sys = -→ P 1 + -→ P 2 = m 1 - → v + - → 0 = -→ P ′ 1 + -→ P ′ 2 = m 1 - → v 1 + m 2 - → v 2 (1.2)
In projected form on the inter-bodies axis direction and on the direction perpendic-

2
The thermal vibration of atoms is assumed to be negligible compared to the velocity of ions Chapter 1

ular to the inter-bodies axis, the conservation of momentum is written:

m 1 v = m 1 v 1 cos θ + m 2 v 2 cos ϕ and m 1 v 1 sin θ = m 2 v 2 sin ϕ
Supposing that T is the energy transferred from m 1 to m 2 , by solving the system of equations (1.1) and (1.2) we finally obtain the expression of energy transferred in a binary elastic collision:

T (ϕ) = 4m 1 m 2 (m 1 + m 2 ) 2 m 1 v 2 2 cos 2 (ϕ)
or

T (ϕ) = 4m 1 m 2 (m 1 + m 2 ) 2 E 1 cos 2 (ϕ)
and the maximum energy transferred is given by:

T max = 4m 1 m 2 (m 1 + m 2 ) 2 E 1 (1.3)
After transferring part of energy to the target, the incident ion slows down and the energy left is expressed as:

E ′ 1 = E 1 -T = K • E 1
in which K is called the kinematic factor of the collision.

K = m 2 2 -m 2 1 sin 2 θ 1/2 + m 1 cos θ m 1 + m 2 2 (1.4)
In a special case in which the interacting angle ϕ = 0, the energy transferred from the incident ion to the target is maximum (T max ). If m 1 < m 2 , the collision with angle ϕ = 0 results in the backscattering of incident ion. If m 1 = m 2 the transfer of kinetic energy is maximum and if m 1 > m 2 the incident ion will only scatter in the area with ϕ ≤ 90 • . Finally, in case that m 1 ≫ m 2 , the energy transferred is very small and is negligible. This is the case of collision between ion and an electron. The energy transferred to electron is so small and becomes negligible so that the collision between ion and atom at low energy is usually considered as a binary elastic collision.

The inelastic collision

The inelastic collision refers to the Coulomb interactions between ion (nucleus and surrounding electrons) with the solid (the cloud of electrons and the nuclei). It is a complex process due to the accumulation of attractive (ion-electron) and repulsive (ionion or electron-electron) forces. Unlike the elastic collision, the internal energy of colliding parters may change as well and so the kinetic energy of a collision system does not conserve in an inelastic collision.

Once entering into a solid, charged particles interact immediately and simultaneously with many electrons. In such single interaction, the electron "feels" the attractive Coulomb force as the particle passes by the electron's vicinity. Depending on the distance of interaction, this attractive Coulomb force may be sufficient to either raise the electron to a high shell level (excitation) or kick it completely off from the atom (ionization). The energy that has been transferred to the electron originates from the particle and therefore the particle decelerates as a result of the interaction. This amount of energy transferred is very small in a single interaction compared to the initial energy. The primary particle must lose its energy in many such interactions during its passage in the solid. At any time, the particle interacts with many electrons and decelerates continuously until the particle is stopped.

Tien Hien NGUYEN Université Paris-Sud

Chapter 1

Evolution of the charge state of moving ion While crossing a solid, the population of the surrounding electrons of an ion varies depending on its velocity. This change in the electronic population results in the modification of the equilibrium charge state of ion and therefore it affects the electronic interactions of ion along its path [START_REF] Penetrante | Evolution of ion-charge-state distributions in an electron-beam ion trap[END_REF].

Practically, the slowing down process of ion can be divided into three velocity domains with respect to the mean quadratic velocity (v e = v 2

Z ef f = Z 1/3 v v Bohr where v Bohr = e 2 4πε 0 = αc = 2.2 × 10 6 m/s (1.5)
At low velocity domain (v ≪ v e ), the process of electron capture becomes more and more probable. The practical charge state of ion decreases as ion slows down in the material. At the end, ion becomes slightly charged and enters into the domain in which nuclear interaction starts.

The stopping power

While traveling through solids, ion gradually transfers energy to orbital electrons and nuclei (constituents of the material) and therefore loses energy along their path. This amount of energy transferred is not constant and depends on collisions to collisions. In addition, it depends instantly on several parameters such as the characteristics of the ion (the nature and the initial energy of ion), the constituents of the material through which it passes. . . In order to characterize for the energy loss of ion in interaction with materials, the stopping power is introduced. The stopping power (usually measured in M eV.µm -1 , keV.nm -1 or eV. Å-1 ) is defined as the energy loss of the charged particle per unit of thickness of material covered. The stopping power depends on the nature and energy of particle and on the composition of materials through which it crosses (see, for example, Table 1.2). It is, therefore, treated as one property of the material (even it describes what happens to the charged particle). Mathematically, the stopping power is expressed as:

- dE dx (E)
where the minus sign expresses the loss of energy of ion. Since ion interacts with materials via two mechanisms, the stopping power is also separated in two terms: electronic and nuclear stopping powers. The electronic stopping power is defined as the amount of energy loss via electronic interaction per unit of distance covered, while the nuclear stopping power is specified as the amount of energy loss per unit of distance covered via elastic nuclear collision between ion and nuclei: The overall stopping power is then the sum of the two contributions:

- dE dx (
- dE dx (E) electronic - dE dx (E) nuclear (1.6)
Practically, the stopping power depends on many parameters. But for a defined ion and material, the stopping power depends only on the energy of ion. Figure 1-5 shows the evolution of stopping power of Cerium ion in U O 2 as a function of its energy. The data are obtained by using SRIM simulation code for both electronic and nuclear stopping power in the range from 10 -2 keV to 10 9 keV . Typically, the diagram of electronic stopping of Cerium can be dived into 4 domain with the energy corresponds to: (i) above 50 GeV (ii) between 2 GeV and 50 GeV (iii) between 0.4 GeV and 2 GeV and (iv) below 0.4 GeV .

For ions having energy higher than 50 GeV , it is governed by the the theory of relativity since its velocity is fairly comparable to the velocity of light. The electronic stopping in this domain is estimated following the Bethe and Bloch theory taking into account the relativistic correction [START_REF] Anderson | Relativistic corrections to stopping powers[END_REF]. For ions with energy in the domain from 2 GeV to 50 GeV , it is totally stripped, i.e. all the orbital electrons surrounding the ion are removed and ion has the charge state of 58+. The energy lost via electronic interaction is described by the theory proposed by Bethe [21] and Bloch [22]:

- dE dx = 4π m e c 2 • N Z 2 β 2 • e 2 4πε 0 2 • ln 2m e c 2 β 2 I • (1 -β 2 ) -β 2 - i c i Z 2 -δ (1.7)
where v, E, Z are respectively the velocity, the energy and the charge of the particle;

c is the speed of light;

β = v/c;
e and m e are the charge and the mass of the electron;

N is the electron density of the target;

I is the mean excitation potential of the target;

ε is the vacuum permittivity;

i c i Z 2 is the shell correction; δ is the Fermi's density correction.
According to this formula, the electronic stopping power of Cerium ion at high energy is inversely proportional to the square of the its velocity. That explains why we observe the increasing electronic stopping while ions slow down in material over this energy range. When ions enter into the domain from 0.4 GeV to 2 GeV , the velocity of ion becomes comparable to the velocity of orbital electrons. In this domain, the Bethe and Bloch formula is no longer valid. The electronic stopping reaches the maximum value (Bragg peak observed around v Bohr Z 2/3 which corresponds to the energy of projectile of 1 GeV ) and tends to fall down after. It is quite complicated to estimate theoretically the electronic stopping in this case because the process of capture of electron becomes more and more important. Ion is no longer considered as fully stripped. To calculate the electronic stopping, the practical approach is to extend the validity of the Bethe and Bloch theory to this domain by introducing the concept of effective charge (Z ef f ) [START_REF] Yarlagadda | Effective-charge theory and the electronic stopping power of solids[END_REF]. It was first introduced by Bohr [START_REF] Bohr | Scattering and stopping of fission fragments[END_REF][START_REF] Bohr | Velocity-range relation for fission fragments[END_REF] and Lamb [START_REF] Lamb | Passage of uranium fission fragments through matter[END_REF]. This very first model supposed that the effective charge of ion is directly proportional to the speed of ion (see formula 1.5).

In 1982, Brandt [START_REF] Brandt | Effective stopping-power charges of swift ions in condensed matter[END_REF] proposed himself a theory with which a large summary of stopping power data for heavy ions was successfully established. Once overcoming this domain, ions enter into the domain of low velocity in which the speed of orbital electrons is higher than the speed of incident ions and ions. After successfully capturing many electrons during its passage, ion return back to its actual ionization state. At low velocity, the electronic stopping of ions is supposed to be proportional to the speed of projectile ions. This theory was proposed by Lindhard and Scharff [START_REF] Lindhard | Energy dissipation by ions in the keV region[END_REF] and independently by Firsov [29,30,31,32] in which the electronic stopping of ions is expressed as:

- dE dx = 8πN e 2 4πε 0 a 0 Z 7/6 1 Z 2 (Z 2/3 1 + Z 2/3 2 ) 3/2 • v v Bohr
where a 0 = 4πε 0 2 /me 2 = 5.29×10 -11 m is the Bohr radius.

By the end of the path inside the solid, ions slows down into the domain of energy below few M eV . In this energy range, the electronic interaction becomes less important and the main interaction is the elastic nuclear collision between ions and nuclei. The nuclear stopping is practically 100 times greater compared to the electronic stopping.

The introduction of the effective charge has helped us to join the domain of high and low speed together with the maximum of the electronic stopping power. A full picture of electronic stopping has been well established.

Radiation induced defects

During passage inside the material, ion transfers part of its energy to atoms and this process could lead to the formation of structural defects. The types and the number of defects created depend on the nature of interactions and on the quantity of energy transferred. The accumulation of effects for a material submitted to a long time irradiation could result in significant modification of the material's properties. These properties include physical dimensions, strength and hardness, thermal and electrical conductivity, corrosion resistance and many others. Knowledge of effects of ion bombardment is therefore necessary in order to understand the behaviour of the material submitted to highly irradiation condition and to predict the life-time of the material.

Defects creation via elastic nuclear interaction

Tien Hien NGUYEN Université Paris-Sud

Chapter 1

As discussed previously, the major interaction between low energy ion and material is elastic nuclear collision. This situation happens practically when ion comes to the end of its trajectory inside the solids. This colliding process results in the transferring of energy from ion to atomic nucleus with the maximum of energy transferred is predicted by formula 1.3. Depending on the amount of energy transferred, defects might be created. If the energy transferred to a lattice atom is higher than the minimum energy needed to permanently displace an atom from its regular lattice site, which is called the "displacement threshold energy" E d , atom will be ejected from its position in the lattice. Likewise, the displacement of an atom leads to the formation of an empty lattice site called a vacancy.

The ensemble of point defect and vacancy is named as the "Frenkel pair" defect. The creation of a Frenkel pair is possible provided that the interstitial point defect is sufficiently far away from the vacancy so that the recombination of interstitial-vacancy does not take place.

The displaced atom (called the primary knock-on atom or PKA in short as it is the first to be attacked by external ion) might carry enough kinetic energy to induce a secondary displacement (internal radiation). The secondary displaced atom in turn could itself create others displacements etc. The process continues and a series of displacements might be induced by a single projectile before it comes to rest. Finally, a highly disordered region around the path of ion is generated as it passes through the material. This process (known as a collision cascade) happens by the end of ion's trajectory via uniquely elastic collision and the set of point defects created is known as a displacement cascade.

According to the model proposed by Kinchin and Peace [START_REF] Kinchin | The displacement of atoms in solids by radiation[END_REF] the number of displacements can be estimated as following:

         N (T ) = 0 if T < E d N (T ) = 1 if E d ≤ T < 2E d N (T ) = T /2E d if T ≥ E d
in which T is the kinetic energy of the primary knock-on atom. Taking into account the contribution of electronic stopping, the number of displacements induced by PKA of energy greater than 2E d is expressed as:

N (T ) = K T -E ε 2E d with K ≈ 0.8
As a consequence of the collision cascades, the total number of displacements is caused not only by the direct ion-atom collision but also via all the secondary PKA-atom collisions. With the assumption made by Kinchin and Peace, the total mean number of Tien Hien NGUYEN Université Paris-Sud displacements along the entire ion range is given by:

N tot = E c1 2E d
which is dependent of the energy given to the projectile ion (E c1 ) and the displacement threshold of the target atom (E d ) and is totally independent of the atomic numbers. Due to the bombardment, an atom might be displaced several times or might not be displaced at all. The average number that one target atom has been displaced over the entire process is called the "displacements per atom (dpa)". The notation of displacements per atom is a more convenient quantity in irradiation experiments rather than the flux (mostly in particles per unit of surface and time) or the fluence (in particles per unit of surface) for the investigation of the elastic collisions effects.

During collision cascades, interstitials and vacancies can be produced so close one to each other that they might easily recombine. The recombination results in the number of defects induced become practically much smaller. Due to the close proximity of the clustering of point defects, many of interstitials and vacancies created during collision cascades annihilate. Only as low as about few percent of the initially created point defects can survive and are capable of successfully producing a radiation induced defect.

Defects creation via inelastic electronic interaction

The energy transferred from ion to material is generally separated into two parts:

(i) the continuous energy loss due to the interaction with the cloud of orbital electrons leading to the excitation and ionization of atoms but not to the direct displacements of atoms and (ii) the discrete energy loss due to the binary elastic ion-atom collision leading to the creation of Frenkel pair or collision cascades (previously discussed).

At high energy (high velocity), the cross section of nuclear interaction between ion an atom is low. The main interaction is the electronic interaction (coulomb interaction) between ion and the cloud of orbital electrons. Through the coulomb interaction, ion continuously transfers energy to the material via process of excitation and ionisation which finally transform to the heat of the target. Depending on the energy transferred, the defects may be created if the energy transferred leads to the high increase in temperature (intense excitation) or no defect is created but just the small increase of temperature observed (weak excitation).

When the energy transferred to the electrons is small but high enough to remove them completely from the shell level, a pair of electron-hole is created. In a metallic materials, this e-hole pair will be annihilated due to the high mobility electrons. The energy will quickly be transformed to the thermal vibration of the target. No localisation of charge or energy deposition is observed and no defect is induced. Inversely, in a non-

Tien Hien NGUYEN Université Paris-Sud Chapter 1
metallic materials, the e-hole has an increased stability.

When the energy transferred to the electrons is much higher (as normally the case of swift heavy ion), ion loses a few hundreds of eV to several target atoms over each interatomic distance. Atoms located in a cylindrical area of about 1 nm along ion trajectory are multi-ionized in typically 10 -16 s after the passage of the ion. Electrons that have been removed spread into the material and lose energy by numerous interactions, an electromagnetic cascade is therefore created. Due to the formation of the electromagnetic cascade, the initial deposited energy dilutes progressively leading to a radial distribution of energy transferred to the target. The radial distribution of the energy density follows a dependence with the inverse of the radius square (1/R 2 ) [START_REF] Toulemonde | Transient thermal process after a high-energy heavy-ion irradiation of amorphous metals and semiconductors[END_REF].

As the cylindrical region center along the path of the ion is highly ionized, all the positive charged atoms within the cylinder form a strongly unstable zone due to the strong Coulomb repulsion between particles of same type of charge. This can cause an explosion and therefore leads to the displacements of atoms (creation of damage due to electronic interaction) within the cylinder forming the latent track. The displacements of atoms require that the energy given to atoms should be higher than the binding energy to the lattice, say > 25 eV [START_REF] Wiss | Radiation damage in UO 2 by swift heavy ions[END_REF]. The mechanism was developed by Fleischer, Price and Walker in 1965 as the well-known "ion explosion model" to explain the formation of amorphous latent track registration in solids [START_REF] Fleischer | Ion explosion spike mechanism for formation of charged-particle tracks in solids[END_REF].

In 1960, Lifchits proposed the theory called the "thermal spike" which was then extended to fast heavy ions by Toulemonde in 1992 [START_REF] Lifshits | On the theory of radiationinduced changes in metals[END_REF][START_REF] Toulemonde | Transient thermal process after a high-energy heavy-ion irradiation of amorphous metals and semiconductors[END_REF]. The theory supposes that the target is considered as two continuous physical systems: the electron gas and the atomic lattice of cylindrical geometry. The theory supposes that under intense electronic excitation, the space and temperature of the electron gas and the atomic lattice is governed by the following coupled system of differential equations:

             C e ∂T e ∂t = 1 r ∂ ∂r rK e (T e ) ∂T e ∂r -g(T e -T l ) + A(r, t) C l ∂T l ∂t = 1 r ∂ ∂r rK l (T l ) ∂T l ∂r + g(T e -T l ) in which
T e , T l are the temperature of the electron gas and the atomic lattice, respectively;

C e , C l , K e K l are respectively the specific heats and the thermal conductivities of electron and lattice atom;

A(r, t) is the energy obtained through the electron cascade after the passage of ion;

g is the electron-lattice coupling constant.

This coupled system of differential equations has no analytical solution but numerical solution shows that ion deposits an extremely high density of energy in a short time that it results in the high temperature of the target within the cylindrical vicinity. As a consequence, the inter-atomic liaisons between atoms in this area are broken. Atoms located within this area become highly disordered that they lead to the fusion of solid in the cylinder. This cylinder is so called the amorphous latent ion track which exists only in a very short time but the trace of ion track can be found under form of point defect or even as a molten to frozen zone in some cases.

Behaviour of ceramics under irradiation

Ceramic materials (e.g. uranium dioxide U O 2 , yttria-stabilized zirconia Y SZ, magnesia spinel M gAl 2 O 4 . . . ) have important applications in the nuclear industry. They are candidates for cladding materials and the promising matrices for immobilization and/or transmutation of nuclear wastes. To these applications, the knowledge of their behaviour upon irradiation is necessary. For that reason, many investigations have been being conducted so as to understand the behaviour of ceramics submitted to the bombardment of ion within them. Both the effects of the two main slowing down regimes (Sn and Se)

were separately investigated: low-energy ions are utilized in order to study the effects of nuclear collision (lattice disorder, dislocation, amorphization . . . ) whilst swift ions are used so as to study the effects of electronic slowing down (formation of defect cluster, latent tracks . . . ).

Various classes of defects can be produced as low-energy ion slows down in ceramics.

Due to the nuclear elastic collision which is prominent at the end of ion path, Frenkel pairs defects are created provided that the energy transferred to lattice target atoms is higher than the displacement threshold. Even though the number of survival Frenkel pairs defects is of few percent only, it is still high enough that they can finally accumulate and lead to the transformation of the structure. Interstitial defects, for instance, gather together forming the defect clusters while vacancy defects, on the other hand, concentrate in a bulk forming the cavity defects. When their concentration continue to increase upon irradiation, interstitial defects can accumulate in the further step leading to the formation of dislocation loops and hence dislocation networks. Those classes of defects are produced in a higher quantity (growth of defects) as the material is further irradiated. As a result, they cause the damage and transformation of the structure resulting in the swelling, polygonalization and also modification of the chemical, electronic, thermal, mechanical and other properties of the material.

Many investigations regarding the radiation effects on ceramic materials for nuclear energy have been conducted. Amongst them, investigations of ceramics single crystals Tien Hien NGUYEN Université Paris-Sud

Chapter 1

as the simplest model play an important role for achieving the very basic knowledge of radiation effects on ceramics structure. Several techniques including the Rutherford Backscattering in Channelling condition (RBS/C), X-Ray diffraction analysis (XRD) and Transmission Electron Microscopy (TEM), were applied for the investigation.

Behaviour of zirconia under irradiation

As discussed previously (see Sec. 1.1.2), pure zirconia exists under only monoclinic structure. When the temperature increases, pure zirconia undergoes two crystallographic transformations and the associated volume change is quite large that make it impossible to be used in applications. However, adding some oxides material can help pure zirconia to fully stabilize in cubic form (fcc form) which is of importance in application. This section is a summary of research carried out on both pure monoclinic and fully stabilized zirconia.

The mechanism of the radiation-induced phase transformation, monoclinic to fcc, in pure ZrO 2 was investigated by Wittels [START_REF] Wittels | Radiation effects in uranium-doped zirconia[END_REF] in 1962. The effects of fast neutrons, fission fragments, and uranium chemical impurities were distinctly studied. The results indicated that: (i) fission fragment fluxes of 10 15 -10 16 f issions/cm 3 produced a monoclinic to fcc phase change in ZrO 2 , and the presence of U in ZrO 2 in concentrations less than 0.38 wt. % did not act as a nucleation or stabilizing atom in the process; (ii) the stability of the phases of ZrO 2 under reactor irradiation depended entirely upon the action of a fission fragment flux of sufficient magnitude; (iii) fast neutrons, solely, could not induce the phase transition at doses up to 6.3 × 10 20 n/cm 2 ; (iv) in twenty-one analyses for U and T h in randomly selected reactor grade zirconium and its alloys, fissionable impurities were found in twenty cases. Natural uranium concentrations varied between 1.8 and 6.3 ppm, and thorium concentrations varied between 0.29 and 50 ppm. Two different models were given to explain the mechanism of fission fragments in producing the phase transition. Both models required the collapse of small regions due to the pressure exerted by interstitials.

In one case the action took place at the beginning of the fission fragment's range via an ionization displacement process, and in the other instance the process occurred near the end of the fission fragment's range where elastic collisions predominate.

Sickafus [START_REF] Sickafus | Radiation damage effects in zirconia[END_REF], in 1999, performed another investigation on the evolution of radiationinduced damage in pure, unstabilized monoclinic zirconia and in Y , Ca and Er F SZ by RBS/C along with XRD and TEM. The crystals were irradiated with 340-400 keV Xe 2+ from 170 to 300 K. The main results obtained by RBS/C showed that the damage accumulation in F SZs irradiated by Xe ion progresses in three stages and no amorphisation was observed (see figure 123456). Unstabilized monoclinic zirconia was observed to transform to a higher symmetry, tetragonal or cubic phase, upon 340-400 keV Xe no loss of P t was observed after annealing at 1373 K. XRD results provided evidence for the formation of a monoclinic phase of ZrO 2 in the damage region after 1373 K annealing. For all annealed samples, P t 200 preferred orientation crystallites were observed by XRD, while the P t 111 peak was almost not visible for 1123 K annealed samples.

Sickafus [START_REF] Sickafus | Radiation damage effects in cubic-stabilized zirconia irradiated with 72 MeV I + ions[END_REF], in 1998, studied the cubic-stabilized zirconia single crystals irradiated with 72-M eV I + ions (to simulate a typical U or P u fission fragment). Irradiations were performed over the fluence range from 1 × 10 14 to 5 × 10 15 cm -2 , at temperatures of 300, 770, and 1170 K. At ambient temperature and at the highest I + fluence (5×10 In 2004, Jagielski and his colleagues [START_REF] Jagielski | Channeling study of the damage induced in zirconia irradiated with high-energy heavy ions[END_REF] investigated the zirconia single crystals irradiated with 450 M eV Xe ions with fluences from 10 13 up to 2 × 10 14 cm -2 . A methodology based on Monte-Carlo simulations of RBS/C data was proposed in order to quantitatively obtain the damage induced. The results showed that the concentration of defects was almost constant at depths above ∼ 500 nm, with a decreasing value in the near-surface region of the crystal. The defect accumulation kinetics exhibited a saturation at a fluence of 5×10 13 cm -2 and could be reproduced by assuming the formation of tracks (diameter ∼ 2 nm) induced by electronic excitation processes.

A. Gentils and co-workers [START_REF] Gentils | Damage production in cubic zirconia irradiated with swift heavy ions[END_REF] applied the MC simulation methodology proposed previously [START_REF] Jagielski | Channeling study of the damage induced in zirconia irradiated with high-energy heavy ions[END_REF] to the investigation of the damage produced in cubic zirconia single crystals irradiated with swift heavy ions. A large variety of swift heavy ions: 20 N e at 70 M eV , 40 Ar at 100 M eV , 84 Kr at 732 M eV , 129 Xe at 450 M eV and 208 P b at 931 M eV were used. Conclusions were that: (i) relatively flat damage profiles were obtained up to depths higher than 2 µm; (ii) The damage fraction increased strongly with the electronic energy loss with a threshold for damage formation at 20 -25 keV /nm and (iii) the creation of damage at high electronic stopping force (dE/dx) e likely resulted from the formation of non-amorphous tracks.

Moll et al. [START_REF] Moll | Radiation effects in yttria-stabilized zirconia: Comparison between nuclear and electronic processes[END_REF] investigated Y SZ single crystals irradiated with heavy ions at energies ranging from a few M eV to a few GeV in order to compare the effects of nuclear collisions and electronic excitations. The damage created by irradiation was measured by RBS/C. Monte-Carlo simulations of RBS/C data showed that both the depth distributions of radiation damage and the damage build-up strongly depend on the type of irradiation.

At low energy (4 M eV Au ions) the damage exhibited a peak around the ion projected range (∼ 500 nm) and it accumulated with a double-step process; at high energy (940 M eV P b ions) the damage profiles were flat up to several micrometers and the damage accumulation was monotonous (one-step). RBS/C and X-ray diffraction results showed that the various irradiations never lead to the amorphization, even at the highest fluences.

In summary, zirconia presents a high resistance to ion irradiation. Amophization was observed only in case of irradiation by 400 keV Cs ion [START_REF] Wang | Amorphization of cubic zirconia by caesium-ion implantation[END_REF] to 1 × 10 [START_REF] Sickafus | Radiation resistance of the oxide spinel: the role of stoichiometry on damage response[END_REF]. Irradiations with both 400 keV and 1.5 M eV Xe ions under cryogenic and room temperature conditions were made, in order to compare the irradiation damage response of these isostructural compounds. Overall, stoichiometric spinel was found to be more resistant to irradiation damage (microstructural alterations) than the non-stoichiometric compound. At room temperature, non-stoichiometric spinel transformed to a "metastable crystalline structure"3 at an ion dose corresponding to 1 to 2 dpa, while this transformation was not observed in stoichiometric spinel to a dose of 10 dpa. Furthermore, under cryogenic conditions, both stoichiometric and non-stoichiometric spinel underwent a metastable phase transformation, followed by a transformation to an aperiodic structure at similar ion doses. ions. The disordered zones overlapped each other after irradiation up to 10 12 cm -2 . Clear lattice fringes are observed in HR images even inside the ion tracks, indicating that the spinel crystals were not amorphized but partially disordered along the ion tracks. High angular resolution electron channelling X-ray spectroscopy (HARECXS) analysis showed that cation disordering progressed successively with ion fluence, and the disordered regions were found to extend over 12.8 ± 0.9 nm in diameter for Al ions and 9.6 ± 0.6 nm for M g ions along the ion tracks. This chemically disordered region was much larger than the strained volume detected by BF and HR images.

Effects of swift heavy ion irradiation on magnesium alunimate spinel was also investigated elsewhere [START_REF] Thomé | Quantitative analysis of radiation-induced disorder in spinel crystals[END_REF][START_REF] Jagielski | Accumulation of radiation damage in spinel crystals: Quantitative analysis and interpretation of RBS/C data[END_REF]. excitation. The elimination rate of the loops was found to be one order higher than for α-alumina. The disappearance of dislocation loops by a dissociation mechanism into isolated interstitials was discussed through analysis of the growth-and-shrink process of the loops. HARECXS analysis on cross section specimens irradiated with 350 M eV Au ions showed the progress of cation disordering along ion tracks to be a function of electronic stopping power, (dE/dx) e . Cations were found to exchange their sites toward a random configuration. Such disordering appeared from (dE/dx) e = 10 keV /nm, and increased in size with increasing (dE/dx) e to reach nearly 10 nm in diameter at 30 keV /nm, under an assumption of a fully disordered configuration.

Behaviour of urania under irradiation

Point defect clusters and dislocation loops produced in bulk uranium dioxide by fission damage were studied by transmission electron microscopy [START_REF] Whapham | Radiation damage in uranium dioxide[END_REF]. In this work, clusters of 25 Å in diameter were first observed at a dose of 4.3 × 10 15 f issions/cm 3 . With increasing dose, these clusters quickly grow into resolvable prismatic dislocation loops. At high doses the loops grow and coalesce to form a network of dislocations, but at a dose of 1.6 × 10 20 f issions/cm 3 and a post-irradiation anneal at 1100 • C, a new set of dislocation loops was observed to form between the existing dislocation network while a second set of dislocation loops is thought to be formed from the collapse of platelets of vacancies.

The radiation damage induced by 10 keV energetic recoils in U O 2 was investigated by molecular dynamic simulations [START_REF] Tian | Dynamical simulations of radiation damage induced by 10 keV energetic recoils in UO 2[END_REF]. The results revealed that, the effects of the PKA direction on the numbers of displaced atoms and Frenkel pair versus time was not observed. Most of the displaced atoms did not result in defects. seconds just after the passage of ion. Afterwards, the diffusion of heat leading to the decrease of temperature is observed until 10 -10 seconds. The maximum temperature is observed to be much higher than the melting point (3073 K) and is located in a narrow zone of 4 nm around the ion's path. This similarity in size between the experimentally observed diameter and the molten zone clearly indicates that the track is formed by a local (and likely instant) melting of the fuel in the close vicinity of ion's trajectory.

T d'un échantillon d'UO irradié avec des ions xénon de 210 M

U O 2 single crystal implanted with 340 M eV Xe ion was studied by Garrido et al.

in 1997 [START_REF] Garrido | Structural modifications in uranium dioxide irradiated with swift heavy ions[END_REF]. RBS/C was applied to study the structural transformation of U O 2 crystal under irradiation. This work reported the good stability of the crystalline structure under irradiation. The observed damaged volume fraction derived form this work is lower compared to the theoretical value. The result demonstrates that strong recombination takes place also under irradiation in the electronic stopping regime, which additionally confirms the stability of the U O 2 structure.

The radiation stability of urania single crystals submitted to intense electronic exci- Relevant key parameters were found to be: the number of displaced lattice atoms created by the slowing-down of energetic ions during the implantation process; the concentration of noble gas atoms in the solid which cause the formation of large stress fields surrounding gas aggregates. The first step was reported to be due to the radiation effect scaled with the number of displacement per atoms while the second step due to the formation of inert gas bubbles at high implanted concentration.

Tien Hien NGUYEN Université Paris-Sud

Chapter 1

In an interaction, ions can knock matrix atom out of their regular lattice location (Primary Knock-on Atoms, PKA). The PKA atoms in turn interact with all surrounding atoms and displace these atoms out of their location if they transfer enough energy to surrounding atom(s) [START_REF] Lewis | Fission product release from nuclear fuel by recoil and knockout[END_REF]. A collision cascade is induced. Van to 7 × 10 14 cm -2 with 390 keV Xe 3+ ions at 873 K. The results reported the presence of nanometer size bubbles above a fluence of 6 × 10 12 Xe/cm 2 and an increase in the bubble density was observed between 6 × 10 12 and 2 × 10 14 Xe/cm 2 . Above 2 × 10 14 Xe/cm 2 , the density saturates at 4 × 10 23 ± 0.5 × 10 23 m -3 .

The fission gas release mechanisms was studied by Marchand et al. [START_REF] Marchand | Xenon migration in UO 2 under irradiation studied by SIMS profilometry[END_REF]. by decoupling thermal and irradiation effects and by assessing the Xenon behavior by profilometry.

The samples were first implanted with 800 -keV 136 Xe then annealed either in the temperature range 1400-1600 • or by irradiated with heavy energy ions (182 -M eV Iodine) at room temperature, 600 • or 1000 • . It is reported that Xenon is mobile during irradiation at 1000 • while, in contrary, thermal treatments do not induce any Xenon migration process:

these results were also reported to correlate to the formation of Xenon bubbles observed by Transmission Electron Microscopy.

In summary, the radiation effects on ceramics materials have been intensively investigated. The investigation was performed to study the effects due to various irradiation conditions such as the nature of ions, energies (effects due to electronic versus nuclear stopping processes), temperature, etc. Many experimental techniques have been applied for the characterisation of irradiated materials including Transmission Electron Microscopy (TEM), X-ray diffraction analysis (XRD), Rutherford Backscattering Spectrometry in Channelling geometry (RBS/C), etc. The investigations show that ceramic materials have high resistance to irradiation. The crystalline to amorphised phrase transformation was reported only for Y SZ crystal irradiated with 400-keV Cs ions at room temperature to extremely high fluence of Φ = 10 17 cm -2 [START_REF] Wang | Amorphization of cubic zirconia by caesium-ion implantation[END_REF]]. Nevertheless, the concentration of Cs at which amorphisation of Y SZ took place is about 8 at. %, far larger than typical value that could be reached in a fuel matrix.

U O 2 crystalline structure submitted to irradiation was also studied but no amorphisation was ever reported. However, the presence of inert noble gases was clearly reported [START_REF] Sabathier | In situ tem study of temperature-induced fission product precipitation in UO 2[END_REF][START_REF] Michel | An in situ TEM study of the evolution of Xe bubble populations in UO 2[END_REF][START_REF] Marchand | Xenon migration in UO 2 under irradiation studied by SIMS profilometry[END_REF][START_REF] Hj | Precipitation of Xe and Cs into bubbles, kinetics of bubble migration and alternative release processes[END_REF]. This phenomenon could lead to a strong structural destabilisation in case of high density noble gas bubbles presenting in the matrix. Nevertheless, a detailed com- low-energy ions. The thesis focused especially on the comparison of the effects induced by typical fission products of close masses and atomic numbers but different solubilities in U O 2 . This thesis also deals with the effects of doping of impurities to very high concentration particularly the doping of an inert noble gas to about 10 at. %.

Tien Hien NGUYEN Université Paris-Sud

Chapter 2

Methodology

Investigation of the radiation effects on a materials can be performed in several way.

In order to study the radiation effects in a real situation, samples can be putted in the reactor core where they will be bombarded by neutrons at high flux, by fission fragments at typical energy of 70 M eV , by α particles and heavy recoil nuclei. However, the eventual radiation effects on the samples is a complex process due to the integration of various types of irradiations with different initial energies. It is quite impossible to understand the effects at different condition such as the effects of low versus high energy particles (nuclear versus electronic mechanisms), the effects due to irradiation by particles of very different masses. . . Moreover, such investigation are very difficult to perform due to the high radioactive level of the samples as they were implanted with highly unstable fission

fragments. An alternative approach that is usually followed to investigate the radiation effects in a material is to irradiate the material by an accelerated ion beam then study by various experimental techniques.

Experimental simulation of the radiation effects in nuclear fuel

Investigation of the nuclear real fuel (spent fuel) is a challenge due to various reasons.

One of the main challenge that could lead to a difficulty in investigating the nuclear fuel lies in the high level of radioactivity of the sample. After a long time operation in the reactor core, the genuine fuel's composition changes due to the incorporation of newly created elements including fission fragments and minor actinides. These elements are highly unstable that disintegrate afterwards via β -and α emission enclosed with energetic gamma rays. The radioactivity of the spent fuel is therefore a parameter that strongly limits the investigation. For instance, such investigation of the spent fuel requires special equipments and regulations that presents as a one of the most important challenge due to the high cost of safety.

Fortunately, those difficulties could be avoided by investigating a material that models the nuclear real fuel submitted to external accelerated irradiation. Moreover, by using an accelerated ion beam, the investigation could be performed with a well defined irra-

diation

Samples preparation

The first step in the investigation is the preparation of the samples. Uranium dioxides (0.3 % 235 U ) single crystal of 100 -orientation has been used as simple model to modelise the nuclear fuel for investigating the effects of ion irradiation. The samples were first cut off from a block with the crystallographic orientation being specified by Laue X-Ray crystallography method. They were then mechanically polished until mirror-liked finish using diamond paste down to 0, 5 µm. Afterwards, the samples underwent the annealing process in the environment containing the mixture of H 2 /Ar gases of 10% H 2 at 1400 • C in order to remove any defects created during cutting and polishing processes and to fix the stoichiometry ratio of U/O to 1/2 1 . The quality of the crystal was then checked by Rutherford Backscattering and Channelling (RBS/C) technique on un-irradiated crystals. Theoretical value of the ratio of minimum axial over random yields for a virgin crystal is χ min ∼ 1 ÷ 2 %.

Experiments

The samples are then irradiated with well-defined ions condition (nature and energy of ion) at room temperature at the JANNuS facility at CSNSM laboratory at the University Paris-Sud at Orsay France using a 190 kV implanter IRMA. The crystals to be irradiated were glued on a special sample holder making 7 • tilting angle with respect to the main crystallographic orientation of the crystals to prevent any channelling of bombarding ions. Irradiations were made with accumulated fluence ranging from 1 × 10 14 cm -2 to 8 × 10 16 cm -2 (which corresponds to a concentration of incorporated impurities

1 U O 2+x + xH 2 H2/Ar environment -------------→ U O 2 + xH 2 O
Tien Hien NGUYEN Université Paris-Sud tively [START_REF] Soullard | High voltage electron microscope observations of UO 2[END_REF]. The calculated depth-distribution of implanted ions is applied to compute the maximum fraction of doping ion (the concentration of implanted impurities); a useful parameter for comparing the effects of doping to high concentration. Details of SRIM calculation and results can be found on Table 2.1 while the transformation from fluence to dpa and fraction of implanted species will be shown in section 3.2.

Although being used as the nuclear fuel for decades, the destabilisation of U O 2 due to the radiation effects and due to the chemical effects is still not fully understood. For instance, the incorporation of fission products up to high concentration could leads to the polygonisation of the high burn-up fuel's structure close to the periphery of the fuel pellet; this phenomenon is referred to as the formation of the High Burn-up structure. The present work investigates the destabilisation of the urania crystalline structure due to the irradiation of a fission product liked elements at the end of their range when the process of defect creation is essentially elastic collision (the so-called ballistic contribution). Moreover, since fission fragments slow down and finally stay in the solid, the effects of their incorporation at high concentration (the so-called chemical contribution, as the incorporation of a fission product eventually modifies the chemical composition of the genuine fuel) are investigated in parallel. For that purpose, the elements of similar masses and atomic numbers have been deliberately chosen in order to investigate both the creation of defects due to nuclear stopping process and due to the incorporation of soluble (La, Ce)

versus insoluble (Xe) species in U O 2 single crystals.

Techniques for characterisation 2.4.1 Rutherford Backscattering Spectrometry

Several experimental techniques can be used for characterizing the defects of the material. Amongst them, Rutherford Backscattering and Channelling (RBS/C) technique represents the outstanding feature which is convenient in material science [START_REF] Garrido | Channeling investigation of the crystalline structure of U 4 O 9-y[END_REF][START_REF] Feldman | Materials Analysis by Ion Channeling: Submicron Crystallography[END_REF][START_REF] Pathak | Study of crystal defects using ion channeling and channeling radiation[END_REF][START_REF] Shao | Methods for the accurate analysis of channeling rutherford backscattering spectrometry[END_REF][START_REF] Swanson | The study of lattice defects by channelling[END_REF][START_REF] Swanson | Channeling studies of defects[END_REF][START_REF] Thomé | Use of channeling for the study of radiation effects in nuclear materials[END_REF]. Such that conveniences could be the ability of well identifying and quantifying the compositions, the capability of profiling the depth distribution of defects and the ability to localize the impurities in a material.

Rutherford Backscattering Spectrometry (RBS)

RBS is a technique which is based on the elastic nuclear interaction between the projectile and target nucleus. Light elements such as Hydrogen or Helium are usually used as projectile because they figure the least destructive elements to the structure. The energy of projectile ranges from few hundreds of kilo electron volts to few mega electron volts which, as a consequence, results in the analyzed depth up to few micrometers.

Typically, in this research, we investigate the effects in the limits of few micrometers from the surface of the material which is totally in consistency to the technique.

Tien Hien NGUYEN Université Paris-Sud

The principle of RBS is demonstrated in figure 2-2. An incident beam of energy E c1 is directed to the material. After interaction with lattice atoms, this ion beam scatters in all directions. A detector is placed at angle θ with respect to the incident beam to detect ions that backscatter in that direction. Knowledge of the energy E ′ c1 (x) of the backscattered ion is a crucial parameter of the technique. It helps us identifying the nature of target atom and the depth at which target atom locates. Besides, the number of detections on specified elements allows one to determine the stoichiometry ratio of all elements constituting the material [START_REF] Chu | Backscattering spectrometry[END_REF].

(m 2 , Z 2 ) (m 1 , Z 1 , E c1 ) T hickness x π-θ E ′ c1 (x) D e t e c t o r

Scattering cross section

The applicability of the technique depends strongly on the scattering probability of the projectile. This probability is characterized by the cross section of the scattering reaction dσ/dΩ in the solid angle dΩ along direction θ. In the theory of Rutherford interaction, this cross section is calculated as following [START_REF] Chu | Backscattering spectrometry[END_REF]:

dσ dΩ = 1 4πε 0 × Z 1 Z 2 e 2 4E 2 × 4 sin 2 θ 1 -(m 2 1 /m 2 2 ) sin 2 θ 1/2 + cos θ 2 1 -(m 2 1 /m 2 2 ) sin 2 θ 1/2
where Z 1 , m 1 and Z 2 , m 2 are respectively the atomic number and mass of projectile and target. ε 0 is the vacuum permittivity. E is the energy of incident particle, e is the elementary charge and θ is the scattering angle.

This formula says that the cross section of the Rutherford scattering reaction is inversely proportional to the square of energy of incident particle. This means that the more energy the particle possesses the smaller the scattering cross section is. It is worth noticing from this formula that, the cross section is proportional to the square of atomic number of target atom. As a result, in a compound the sensitivity of the technique to

Channelling

Channelling is a technique that applies the principle of RBS only on single crystal.

The importance of channelling technique is that the well aligned crystalline lattice site target atoms in the material are not seen as diffraction grating as in the case of X-Ray diffraction technique. They are practically viewed as rows or planes of periodically repeated atoms between which ion channels (figure 23). The channelling results in the much smaller number of detections of backscattered ions as most of ions channel through the crystal with low probability to be backscattered. Only a small fraction (about 1%) of backscattered event is recorded compared to the situation in which ion beam is randomly oriented towards the crystal. When the ion beam is oriented along the crystallographic orientation of a perfect crystal (without any presence of defect), the backscattering events decrease dramatically as most of incident ions channel through the crystal. In this configuration, only a very small fraction of ion scatters on the lattice atoms and the consequential backscattering spectrum presents a very low signal as compared to the random spectrum. The green curves shown on figure 2-4 illustrates such channelling spectrum issued from a nondefective crystal. The ratio of backscattering yield of the axial channelling spectrum over that of the random spectrum integrated in a given energy windows is known as the minimum axial yield (χ min ). The theoretical value of χ min for a perfect crystal is typically between 1 and 2 % depending on crystals.

If the ion beam is oriented along the crystallographic orientation of a crystal containing defects, the backscattering will be improved as compared to the perfect crystal.

As the defects are present in random position within the crystalline structure, they function as both direct and indirect obstruction to the channelling of the incident beam. The obstruction to the incident ion beam at a given depth from the surface of the crystal is proportional to the local concentration of defect. The crystal containing defect will result in the higher channelling spectrum compared to the virgin crystal. The red curve represented on figure 2-4 is a typical example of the channelling spectrum recorded on defective crystal. As it can be seen, the spectrum consists of: (i) a surface peak due to the backscattering of ion upon atoms located at the surface of the crystal; (ii) a damage peak results from direct backscattering upon atoms that have been randomly distributed Tien Hien NGUYEN Université Paris-Sud

Chapter 2

within the lattice structure (the level of the observed damaged peak directly represents the local concentration of disordered atoms at given depth); and (iii) a de-channelling signal owing to the small angle scattering of incident ion upon both lattice atoms and defects. Actually, lattice atoms and defects along a channel contribute to the deviation of the incident ions by a small angle and the integration effect eventually kicks ions out of the channel. At this stage, ions will be scattered in all direction including the direction where the detector is placed. One backscattering signal is hence recorded.

Depth scaling

The depth profiling of RBS/C requires the necessity of relating the energy of the backscattered ion to the depth in the sample at which the backscattering takes place [START_REF] Tesmer | Handbook of Modern Ion Beam Materials Analysis[END_REF]. This relation depends on the energy loss of ion while crossing the sample, on the kinematic factor of the scattering and on the angle formed by the incident beam and by the detector's normal vector. The depth scaling is then estimated by assuming that the stopping powers on the way in as well as on the way out are constant (the so-called surface approximation). Therefore, the energy of detected ion that backscatters at depth x from the surface can be given as following:

E(x) = KE • -xS.
Where, E(x) is the detected energy of ion that backscattered at depth x, E • is the initial energy of the incident beam and S is the energy loss factor. In this formula, S is calculated as following (with the help of knowledge of the kinematic factor, scattering angles and the incoming and outgoing stopping powers assuming that the stopping in random and aligned directions are the same):

S = K dE dx in 1 cos θ 1 + dE dx out 1 cos θ 2
Thus the depth at which the backscattering takes place will be:

x = KE • -E(x) S
Finally, a distribution of defects versus depth can be well constructed.

X-Ray diffraction Principle

The X-Ray radiation is generated when bombarding an anode by an electron beam produced by a Tungsten filament under vacuum. It can be generated also when high speed electron beam or high energy charged particles decelerate suddenly. The generation of Tien Hien NGUYEN Université Paris-Sud X-Ray is made via the emission of a photon of energy on the order of magnitude keV when an electron at the outer shell, e.g. L shell, falls down to the K shell from which another electron has been kick off by the bombarding beam. This configuration leads to the generation of an X-ray called K α radiation. X-rays are actually considered as an electromagnetic wave that have a wavelength in the range of 0.01 to 10 nm, corresponding to frequencies in the range 3 × 10 16 Hz to 3 × 10 19 Hz and energies in the range 100 eV to 100 keV . X-rays have a much shorter wavelength than visible light, which makes it possible to probe structures much smaller than what can be seen using a normal optical microscope. However, not all X-Ray can be used for diffraction experiment. As the typical lattice spacings (d hkl between atomic planes of a given orientation) are between 2 and 3 Å, the wavelength of the X-Ray has to be smaller than twice the lattice spacing but should not be two small otherwise θ (the angle formed by the incident ray and the crystal's surface) is too small to be observed. Therefore, the wavelength of X-ray is usually between 0.5 and 2.5 Å for the use in material characterisation.

Bragg condition

X-ray crystallography can be used to study the crystalline structure and to determine the positions of atoms in crystals. When directed on a periodically arranged solid (a crystalline structured solid), in given direction, constructive interferences give rise to scattering of X-rays in a coherent way. For this diffraction phenomenon to occur, the inter-planar spacing d hkl and diffracted angle have to satisfy the Bragg condition, i.e.

2d hkl sin θ B = n • λ
where θ B is the Bragg angle; λ is the wavelength and n is an integer that represents the diffracting order. Once the Bragg condition is satisfied, it can allow one to determine (i) the atomic planar family that is the responsible to the diffraction and (ii) the inter-planar distance between two adjacent atomic planes. Since the wavelength and the inter-planar distance are fixed, constructive interference will be observed, therefore, only at certain angle.

In order to observe the diffraction from a given lattice plane, the Bragg's law must be satisfied. This condition is often obtained by varying either the wavelength, λ, or the Bragg angle, θ. For a monochromatic X-ray beam, the only possible way is to vary the θ angle to meet the Bragg condition.

Laue condition

Bragg's law can also be rewritten under the following alternative form:

sin θ B = n • λ 2d hkl Tien Hien NGUYEN Université Paris-Sud
where the angles of diffraction are inversely proportional to the spacing of the reflecting planes in the crystal lattice. Therefore, it is usually helpful to establish a new coordinate system whose axes have the dimension of inverse length: The Reciprocal Space. The reciprocal lattice is therefore the inverse of the real lattice whose basic lattice vectors a * , b * and c * are defined from the translation vectors in the crystal lattice:

a * = b × c a • b • c ; b * = c × a a • b • c and c * = a × b a • b • c
In the reciprocal space, the translation vectors a * , b * and c * are perpendicular to the planes indexing ( 100), ( 010) and (001), respectively. And if In the reciprocal space, the Bragg's condition can be rewritten differently as the Laue condition that can be derived as following:

H hkl = h a * + k b * + l c * ,
Let us consider an X-ray, with the wavevector K 0 , is directed on a crystal and the diffracted X-ray wave has the wavevector K. Assuming that the wavelength stays unchanged (which is true in the case of elastic scattering of a photon by an electron), the magnitudes of the wavevectors K 0 and K are identical and equal to the inverse of wavelength:

| K| = | K 0 | = 1 λ
and the question raised is to what condition the relationship between wavevectors will the maximum constructive interference of the diffracted X-ray occur? Since the crystal is strictly periodic, it is sufficient to consider two atoms related by translation through a lattice vector r hkl = h a + k b + l c as illustrated on figure 2345. The waves diffracted from atoms A and C will result in maximum constructive interference if the difference between the optical paths is an integer multiple of the wavelength λ. As shown on figure 2-5, the optical paths difference is given by:

∆ = AD -BC = r hkl • K | K| -r hkl • K 0 | K 0 | = λ r hkl • ( K -K 0 )
Thus, the required condition must be satisfied is that the optical path equals a multiple of the wavelength, that is:

r hkl .∆ K = n
where ∆ K = K -K 0 is called the scattering vector and n is an integer. This condition shows that diffraction takes place when the scattering vector ∆ K coincides with a reciprocal space vector H whose magnitude is equal to multiple of the distance between two nodes of a reciprocal space row, that is to say:

∆ K = K -K 0 = H Ewald's sphere
Laue condition can be figured following the construction of the Ewald's sphere. This construction, in turn, allows determination of the direction that results in the constructive interference of the diffracted beam. Ewald's sphere is defined by the center C and the radius R which is equal to the magnitude of the wavevector K 0 . on the surface of the Ewald's sphere. As a convention, the extremity of the incident wavevector is considered as the origin of the reciprocal space, the condition for a constructive interference of diffracted beam is satisfied when the extremity of the diffracted wavevector coincides with a node in the reciprocal lattice space.

Symmetric geometry

The basic measurement geometry the most frequently used of x-ray diffraction instrument is the θ -2θ geometry including symmetric geometry. In the symmetric geometry, the angles of both the incoming and the exiting beam equal θ with respect to the specimen surface. The diffraction pattern is collected by varying the incidence angle of the incoming X-ray beam (θ) and the scattering angle (2θ) while measuring the scattered intensity I as a function of 2θ. Two angles vary during a θ -2θ scan and various types of diffractometers are in use. For one set of instruments, the X-ray source remains fixed while the sample is rotated around θ and the detector moves by 2θ. For other systems the sample is fixed while both the x-ray source and the detector rotate by θ simultaneously, but clockwise and anticlockwise, respectively. The diffraction on crystallographic planes that satisfy the Bragg condition yields a diffracted peak. This peak locates exactly at the angle 2θ and the shape of the peak can be a Gaussian or Lorentzian or more practically a combination of them, known as a Pseudo-Voigt. The enlargement of the peak is often observed and consists of two contributions: (i) the practical resolution of the goniometer and (ii) the intrinsic contribution of the investigated sample. The former one is unavoidable but identical in all measurements while the latter one is usually due to the presence of micro strains and/or finite size of crystallites.

Asymmetric geometry

In this geometry, the incident angle does not equal the scattering angle. This geometry is often applied to a scan known as the rocking-curve in which the angle between incident and diffracted beams is maintain to be equal to 2θ B while the incident angle varies around 2θ B /2. In the real space, this geometry corresponds to the rocking of the sample around the direction of the vector

K B that is determined when ω = θ B . K B is
the scattering vector at the maximum of the peak in a θ -2θ scan on the same-oriented plans. The scan in omega is similar to the situation where the module of the diffracted vector maintained constant.

In the framework of this thesis, the X-ray diffraction technique has also been applied

to study the microstructural evolution and the elastic deformation of the U O 2 crystals submitted to irradiation with 500-keV Ce ions. 5 samples of 100 orientation and 5 samples of 111 orientation have been irradiated at JANNuS Orsay to the fluence of 1 × 10 14 , 3 × 10 14 , 5 × 10 14 , 7 × 10 14 and 9 × 10 14 cm -2 . In this experiments, the implanted samples were characterised at the Institut des Sciences Chimiques de Rennes in the University of Rennes 1. A Bruker D8 diffractometer equipped with a standard Cu tube was used. An intense and monochromatic beam of wavelength λ = 0.15406 nm was obtained by using a multilayer mirror behind the tube followed by a four-crystal monochromator (4×Ge220) in asymmetric configuration; the resulting primary-beam divergence was ∼ 18 arcsec (0.005 • ). A 1D solid detector (LynxEye) was used to record X-ray scattered intensity; the point detector mode was used for recording scattering signals.

Transmission Electron Microscopy

The transmission Electron Microscopy (TEM) is a well established tool for studying crystallographic defects present in crystalline materials. Basically, a transmission electron microscope consists of an electron gun and an assembly of electromagnetic lenses mounted in a vertical column. The column is evacuated to a pressure better than 10 -5 T orr. The thin sample (< 0.1 µm thick) to be investigated is mounted on a goniometer capable of tilting the sample along two orthogonal axes so that an appropriate set of crystallographic planes can be brought into a diffracting condition. Since the wave-length of high energy electrons (e.g. wavelength of 100 keV electron is ≃ 3.7 × 10 -3 nm) is comparable to the magnitude of the lattice strain around a defect, a shadow is formed in the image when lattice planes affected by the defects are in diffracting condition. The resulting contrast in the image is called diffraction contrast. Owing to the small wavelength of electrons, TEMs are capable of imaging at a significantly higher resolution than light microscopes.

This allows for the examination of very fine details in materials, down to a single column of atoms. In a typical modern microscope the final magnification of displayed images can be varied between 100 to several 10 6 times.

The electron microscope can be operated in two principal modes; imaging or diffrac- Regarding the sample preparation, unfortunately a destructive method, the two modes of observation require two different methods of preparation. The plan view geometry requires mechanical back-thinning, using a tripod, until the edge of the sample reaches or is close to electron transparency; if required, the last step can be completed either by chemical etching or ion thinning at low incidence angle (in this case 8 • ). The sample preparation of cross section specimen is more complicated involving slicing of the sample in small pieces and gluing them together, surface against surface; then from the resulting block of material small square slices are cut so that the interface of interest lies in the middle of the piece. The steps then are the same as for a plan view sample with the additional difficulty of preferably thinning a very specific area, the area of interest close to the interfaces.

In the frame work of this thesis, some crystals were irradiated ex situ at selected ion fluences for investigation by Transmission Electron Microscopy. In order to obtain the images shown here a JEOL 3010 microscope was used. The microscope was equipped with a LaB 6 filament and can be operated at an accelerating voltage of 300 kV . A schematic drawing of the microscope is shown in figure 234567. The choice of the microscope was driven Tien Hien NGUYEN Université Paris-Sud by the necessity to use 300 keV electrons in order to obtain electron transmission through the U O 2 specimen; the microscope is designed for high resolution imaging, with a point resolution of 0.19 nm. us to have access to a quantitative measurement of defects versus depth assuming the presence of randomly displaced atoms but it becomes very difficult to apply for a material containing various classes of defects especially the problem arises in a crystal containing extended defects. In this situation, an ordinary method is no longer suitable and Monte-Carlo simulations performed on a computer seems to play the role [START_REF] Turos | Monte carlo simulations of ion channeling in crystals containing extended defects[END_REF][START_REF] Smulders | Computer simulation of channeling in single crystals[END_REF][START_REF] Jagielski | Monte carlo simulations of channeling spectra recorded for samples containing complex defects[END_REF][START_REF] Doolittle | Algorithms for the rapid simulation of rutherford backscattering spectra[END_REF]. In this thesis we used the Monte-Carlo simulation code called McChasy developed at the Andrzej Soltan Institute for Nuclear Studies in Warsaw, Poland as a tool to analyse the channelling data.

Computer tool

Principle of Monte-Carlo simulation of channelling phenomenon

Monte-Carlo computation is a technique of calculating integrals by repeated sampling probability to obtain numerical results. In case of an ion-crystal interaction, the sampling is made at each atomic plane by calculating the probability of close encounter.

In each calculation, the instantaneous distance between ion and surrounding atoms is taken into account and when it becomes small enough the nuclear interaction takes place.

Due to the atomic thermal vibrations, this probability has a Gaussian-shaped distribution around a lattice location. In Cartesian coordinates xyz (assuming that the z axis coincides with the channelling axis), the probability P j i of close encounter with an atom located on plane j is given by [START_REF] Barrett | Monte carlo channeling calculations[END_REF]:

P i j = cos ψ j i 2πN u 2 A exp - (x i j -x A j ) 2 + (y i j -y A j ) 2 2u A 2
where i is used to number the acts of sampling and (x j i , y j i ) is the coordinates of ion at the plane j during sampling i while (x j A , y j A ) is the position of target atom A at the plane j. u A denotes the thermal vibration of atom A and ψ j i is the angle between projectile velocity vector and the channelling axis; N is the sampling number.

An integration that gives a statistical estimation of number of backscattering events corresponding to the depth z (in the z, z + ∆z depth interval) is given by:

K A (z) = σ A N tot Ω j N i P i
in which, σ A is the Rutherford cross section, N tot is the total number of projectiles and Ω is the open solid angle of detection. The summation is made for all acts of sampling and all atomic planes in the indicated depth interval. The "depth-spectrum" K A (z), i.e. the series of K A values calculated for subsequent depth intervals, can be transformed to the energy spectrum by taking into account the energy loss of projectiles on the entrance and exit paths. Such spectra, obtained for each element in the crystal, are finally added together to the final backscattering spectrum. In case the sampling is made uniformly at each atomic plane in the crystal, the random spectrum is produced. Since the accuracy of the Monte-Carlo calculation is proportional to the total number of projectile (∼ N -1 tot ) it can be improved by increasing N tot .

In order to simulate the channelling of ion, the most basic feature is the incorporation of the lattice structure. Since the channelling of ion takes place along a channel or between atomic planes, the lattice crystalline structure is usually viewed as composed of rows. This situation becomes realistic as long as the ion beam is oriented close to the considered direction (typically a few degree off). It is also advantageous that in this approach the next important interaction will generally be the next atom along the row. Taking into account this approach, many crystalline structures were incorporated in Monte-Carlo computational program. Another alternative approach to incorporate the lattice structure is made by providing the basic locations of atoms in a unit cell and all translation operators related to the crystal. By doing this, the program performs a search procedure to find the next lattice atom (or atoms) with which the projectile will interact.

Others features that are also important to the simulation are the experimental divergence of the ion beam around its main direction and the potential function which governs the ion-atom Coulomb interaction. The ion beam's divergence was generally implemented by using a distribution of the probability of finding projectile ion from the beam's centre while the Molière's approximation to the Thomas-Fermi screening function was most widely used as screening function since it provides a universal function and is reasonably accurate [START_REF] Barrett | Methods of channeling simulation[END_REF].

Thermal vibration of atoms plays also an important role in computational simulation. Typically, atoms are supposed to locate around its lattice location where the exact position is predicted by a Gaussian distribution having the width calculated from the Debye theory. beam divergence), (iv) the energy resolution (the factors that define the energy interval in one acquisition channel). In the second step, the crystal is usually divided in thin layer of few nanometres to few hundreds nanometres. In each layer, a given fraction of defects is defined as input for simulation. Simulation can be then performed and the output is a channelling spectrum that contains information on the backscattering events at each acquisition channel. The final step is to normalize the obtained spectra for comparison with experimental data.

McChasy

In a simulation, a backscattering energy spectra for He channelling in crystalline materials and/or an angular scan are usually obtained as the output. As the most important application of the RBS/C technique, this program also allows the calculation of the damage fraction and the distribution of defects versus depth in the crystal. In this situation, the program can take into account various classes of defects such as point defects, defect clusters, dislocations and even the polygonisation of the crystal.

McChasy uses the ZLB potential [START_REF] Ziegler | The stopping and range of ions in matter[END_REF] in order to takes into account the electron screening on the ion-atom interaction [START_REF] Nowicki | Modern analysis of ion channeling data by Monte-Carlo simulations[END_REF]. An important feature is the option to take into account the substitutions of atoms in the structure. Such option becomes important when the crystal is heavily irradiated by ions to very high concentration. In this situation, the incorporation of foreign ions modifies to large extent the composition of the structure and thus influences the eventual results. Typically, one simulation that gives an acceptable statistic (N tot ≃ 40000) takes 120 minutes.

Tien Hien NGUYEN Université Paris-Sud

Chapter 3

Irradiation effects on the urania structure

This chapter focuses on the experimental results obtained in this thesis. In the first part, channelling spectra recorded on the crystal implanted with the noble gas Xe is presented as an illustrative example of the used methodology. Then, the analysis of channelling spectra recorded on a crystal implanted with low-energy ion (applying the two-beam approximation approach) is explained and applied to the decomposition of channelling spectra. Afterwards, the distributions of defects versus depth and the evolution of the kinetics of damage build-up are extracted so as to understand the kinetics of the defects induced. Finally, similar experiments regarding the irradiations by lanthanide ions are presented in the second part. displaced from their regular positions), whilst the channelled component is supposed to interact only with atoms that are located within the channels. Backscattered ions of the channelled beam gives contribution to the damage peak and directly reflect the fraction of randomly displaced atoms in the matrix. The channelling spectrum is therefore composed of three components: the surface peak, the dechannelling signal and the damage peak.

Irradiation with noble gas ions

The surface peak represents the backscattering of He ions on atoms located at the surface of the crystal. For the sake of simplicity both the surface peak and the damage peak are fitted assuming a Gaussian-liked distribution. In addition, the dechannelling background is modelled by a Fermi-Dirac-shaped function [START_REF] Cerofolini | A fast technique for the quantitative analysis of channeling RBS spectra[END_REF].

An example of the decomposition of the channelling spectra recorded for a crystal implanted at the fluence of Φ = 1×10 15 cm -2 is depicted on figure 3-2. The decomposition of channelling spectra recorded for crystal implanted to high concentration is slightly different due to the extra contribution coming from implanted species and is explained later on. In general, in order to analyse a channelling spectrum using the TBA approach, one has to separate the damage peak from the channelling spectra. As a consequence, we obtain (i) a damage peak that reflects the local concentration of defects at given depth and

(ii) a dechannelling signal that reflects the progressive increasing signal with increasing depth. The decisive point, which figures as the clue of this decomposition method, is the choice of shape of the dechannelling signal). In this work, the S-shaped background signal has been modelled assuming a Fermi-Dirac-like function that can be written by the following simple analytical expression:

F dech (ch i ) = k - a 1 + exp -ch i -b c
where ch i corresponds to the channel number i used for recording the signal during the acquisition of a backscattering spectrum. The decomposition is made in channel (thus in energy) and then converted to the depth z. The parameters k, a, b, c are the characteristic parameters that define the height, the width and the position . . . of the dechannelling signal.

For low implantation fluence, the procedure used to decompose a channelling spectrum is to fit the channelling spectrum by the sum of the three individual components:

(i) the S-shaped signal which represents the dechannelled fraction of analysing beam, (ii) the surface peak which represent the fraction of analysing beam backscattered from atoms located at the surface of the crystal and (iii) the damage peak which represents the fraction of analysing beam backscattered from defective atoms that are randomly displaced within the matrix. When the best fit to the channelling spectrum is achieved, the Fermi-Dirac-liked dechannelling contribution is subtracted to get the fraction of defects Tien Hien NGUYEN Université Paris-Sud F d , which is calculated according to the following formula:

F d = F d (ch i ) = Y aligned (ch i ) -Y surf ace.peak (ch i ) -F dech (ch i ) Y random (ch i ) -F dech (ch i )
in which: Y aligned (ch i ) and Y random (ch i ) are the normalized backscattering yields recorded in axial channelling condition (when the analysing beam is aligned along the crystal's orientation) and in random conditions respectively. Y surf ace.peak (ch i ) and F dech (ch i ) are the data obtain by the fit.

Finally, to extract the distribution of defect versus depth a transformation from energy into the depth is performed applying the tabulation of the stopping power measured for 4 He in U O 2 by using the data provided by the RUMP code.

For channelling spectrum recorded at high implanted fluence, the decomposition has to be modified since there is an extra contribution to the channelling spectrum due to the presence of impurities in the matrix (see figure 3-3). This phenomenon is observed via the appearance of a peak located at around 2680 keV on both aligned and random spectra (see figure 3-1.d). As it can be seen on this figure, the backscattering signal due to impurities overlaps on the damage peak. It is then necessary to isolate the backscattering signal on impurities so as to obtain the damage peak that reflects the pure contribution of the U sub-lattice. In this work, we performed that decomposition by supposing that the contribution of impurities to the axial channelling spectrum is equal to that contribution to the random spectrum. This approach assumes that all Xe atoms are located in random position similar to the case where impurities locate randomly within the matrix and give Tien Hien NGUYEN Université Paris-Sud contribution to the channelling spectra, as if spectrum was recorded in random condition.

Finally, the eventual damage fraction at a given depth extracted from channelling spectra recorded for high implanted fluence is calculated taking into account the subtraction of contribution of impurities. The fraction of defects F d is hence calculated using the following modified formula:

F d = F d (ch i ) = Y aligned (ch i ) -Y surf ace.peak (ch i ) -F dech (ch i ) -Y impurity (ch i ) Y random (ch i ) -F dech (ch i ) -Y impurity (ch i )
where Y impurity (ch i ) is the normalized backscattering yield extracted from the random spectrum for impurities. (c) high fluence ranges. Depth distribution were extracted from analysis of channelling spectra using the two-beam approximation method. Fluences are expressed in unit of cm -2 . Filled area represents the distribution of implanted Xe calculated by the SRIM calculation code (in arbitrary unit).

Chapter 3

32 %) when the maximum depth of the created damage exceeds the maximum depth of Xe atoms calculated by SRIM code. At high fluence (over 1 × 10 16 cm -2 ), a marked behaviour of damage induced is observed. First of all, a maximum depth of about 300 nm was observed on the damage profiles for the highest investigated fluence. The distribution of defects induced at high ion fluence can be separated into two regions: the surface region and the bulk one. On the bulk region (the region which is located beyond the distribution of the implanted ion, larger than Rp + ∆Rp, i.e. 150 nm), the fraction of defects created remains almost constant. On the contrary, a dramatic increase of the defects is observed from the projected range of Xe ions towards the free surface of the crystal (0 ≤ z ≤ Rp).

At the highest ion fluence used in this investigation (8×10 16 cm -2 ), the fraction of defects almost reaches the random level, i.e., 100 % of atoms are randomly displaced from their regular positions.

dpa = D T RIM × 10 8 Φ N V U O 2 and C = (N /V ) ion (N /V ) U O 2 × 100%
where D T RIM is the maximum collision events calculated by the SRIM code expressed in atoms/cm 3 /atoms/cm 2 ; (N/V ) U O 2 is the theoretical density of U O 2 ; (N/V ) ion is the maximum density of ion calculated by SRIM expressed in atoms/cm 3 and Φ is the actual ion fluence (in expressed in cm -2 ). at. %. Moreover this build-up of damage is created for the same number of dpa and is therefore independent of the nature of the bombarding ion. Even though the mass and the energy chosen are different (for the sake of having the similar implanted range as discussed in previous session), one can conclude that this build-up step that appears at the same position regardless of the nature of ions originates from the same radiation-induced effects, due to the collision cascades generated by nuclear interaction.

Beside the same build-up step observed, the evolutions of the damage accumulation reveal important differences due to the effects of implantation with Xe compared to the effects of implantation with lanthanide ions. The first difference lies in the saturation plateaus. Distinct levels of saturation are clearly evidenced. The plateaus of saturation are almost the same for La and Ce but Xe ions results in a higher plateau of saturation.

A significant difference of 10 % of absolute damage fractions induced is obtained from the data analysis (corresponding to a 30 % relative difference in the damage fraction).

Secondly, the major difference is characterised by the lack of the second build-up step at very high concentration. A huge fraction of damage was created by the incorporation of Xe ions when its concentration exceeds 4 at. %, but such a dramatic increase is not evidenced for irradiation with La and Ce ions.

Tien Hien NGUYEN Université Paris-Sud

Microstructural evolution under irradiation

The X-ray diffraction technique has been applied to study the microstructural evolution and the elastic deformation of the U O 2 crystals due to irradiation with 500-keV Ce ions at low fluence to investigate the first step of the radiation damage, previously observed by RBS/C. In this investigation, the θ -2θ scans around peak index (400) for the 100 -oriented crystals were recorded while the same data were recorded around peak index (222) for the 111 -oriented crystals. Bragg reflection for 100 -oriented crystals. The intensity is plotted in logarithmic scale in arbitrary unit as a function of 2θ as well as a function of the normalized deviation from the reciprocal lattice vector (ε N = -q N /H 400 ). This quantity directly reflects the out-ofplane elastic strain of the crystal (i.e. the relative variation of the interplanar distance between (400) planes in the direction perpendicular to the crystal's surface). All recorded spectra present an intense peak located at 2θ = 68.567 • which is the consequence of the diffraction by the bulk part of the crystal, i.e. the underlying un-damaged substrate. This peak was recorded because X-rays allow probing the crystals structure up to the depth of about 5 µm from the surface while the irradiated part is present at the close surface up to the maximum depth of about 300 nm, as observed by RBS/C measurements (see figure 3-4, 3-7 and 3-9). This peak can be named, therefore, as the un-damaged peak.

Tien Hien NGUYEN Université Paris-Sud

Chapter 3

The spectra presented on this figure are very similar for fluence from Φ = 1 × 10 14 to 7 × 10 14 cm -2 . That is to say, these spectra display three similar important features:

(i) the signal is composed of fringe pattern; (ii) the signal arising from the damaged part is located on the lower 2θ angles with respect to the un-damaged peak. According to the Bragg condition, this feature demonstrates that the damage layer is characterised by a larger lattice parameter and thus it exhibits a positive normal strain; (iii) The position of the last peak observed at the lowest 2θ angle (which is indicated by the small arrows on figure 3-11) shifts towards the lower angle when the ion fluence increases. This shift reveals an increase of the maximum normal strain ε max N up to the fluence of Φ = 7 × 10 14 cm -2 . For the crystal implanted at 9 × 10 14 cm -2 , the fringe pattern vanishes. The θ -2θ scan recorded at this fluence presents an intense peak due to the substrate along with a very wide peak due to the damaged part. This wide peak is attributed to the diffuse scattering arising from a highly defective crystal [START_REF] Dederichs | The theory of diffuse X-ray scattering and its application to the study of point defects and their clusters[END_REF][START_REF] Debelle | XRD contribution to the study of Cs-implanted cubic zirconia[END_REF]. It is worth noticing that at this fluence, this peak moves towards the un-damaged peak, a phenomenon that reveals the strain relief in the damaged layer. Such phenomenon has been well reported for U O 2 single crystal irradiated with 20 -keV He ion [START_REF] Debelle | Strain and stress build-up in He-implanted UO 2 single crystals; an x-ray diffraction study[END_REF] and also for other crystalline structures as in cubic zirconia [START_REF] Moll | Multistep damage evolution process in cubic zirconia irradiated with mev ions[END_REF] and pyrochlores [START_REF] Sattonnay | Mechanical stresses induced in ceramic oxides by ion irradiation[END_REF]. cm -2 , the corresponding spectra clearly show the fringe patterns on the lower 2θ angles.

While increasing ion fluence, the last peak located on the lower 2θ angle shifts towards smaller values of 2θ; this feature is related to the increase of the maximum normal strain due to irradiation. At the fluence of Φ = 9 × 10 14 cm -2 , the fringe pattern disappears.

The spectrum reveals the consequence of diffuse scattering by the defective part which is observed via the wide peak located on lower 2θ angles side as the perfect atomic arrangements is severely perturbed. In addition, this wide peak is also observed to move towards the higher 2θ angle, indicating a partial relief of the normal strain.

(ε max N = ∆d d hkl = -q N H N = -q N H cos ψ )
where ∆d represents the variation of the inter-planar spacing induced by irradiation; -q N H N represents the normalized deviation expressed in the reciprocal space.As clearly shown on this figure, the maximum normal strain experimentally measured depends on the orientation of the investigated crystals. The values observed for 111 -oriented crystals are higher than those for 100 -oriented crystals.

Regarding the evolution with irradiation, the maximum normal strain in both orientations increases with increasing ion fluence up to 7 × 10 14 cm -2 , corresponding to 5.9 dpa.

Maximum values of 0.65 % and 1.02 % are observed when the irradiation reaches the fluence of 7 × 10 14 cm -2 for both 100 -and 111 -oriented families, respectively. When Tien Hien NGUYEN Université Paris-Sud Chapter 3

the fluence exceeds 7 × 10 14 cm -2 the maximum normal strain in both families drops down. This phenomenon reveals a partial relief of the strain in the crystal [START_REF] Debelle | XRD contribution to the study of Cs-implanted cubic zirconia[END_REF]. At this stage, there exists a transition from elastic to plastic deformation in the crystal. It is likely that at the first step the irradiation creates point defects leading to the increase of lattice parameter of the irradiated part. When the relaxation of the crystal takes place, the deposited energy leads to the creation of extended defect clusters in the matrix. These phenomena, i.e. the relief of strain field and the formation of defects clusters, are observed in connection to the steep increase of RDA observed by the RBS/C measurement at 4 to 7 dpa (see figure [3][START_REF] Kurepin | Thermodynamic properties of hyperstoichiometric urania in the UO 2 -UO 2.25 solid solution range[END_REF][START_REF] Sheindlin | Experimental determination of the thermal conductivity of liquid UO 2 near the melting point[END_REF][START_REF] Banerjee | Specific heats of thoria urania solid solutions[END_REF][START_REF] Leclercq | Thermal conductivity of zirconia-based ceramics for thermal barrier coating[END_REF][START_REF] Degueldre | Specific heat capacity and Debye temperature of zirconia and its solid solution[END_REF][START_REF] Azom | Zirconia -ZrO 2 , Zirconium Dioxide[END_REF][START_REF] Slack | Thermal conductivity of MgO, Al 2 O 3 , MgAl 2 O 4 , and Fe 3 O 4 crystals from 3 to 300 K[END_REF]. The steep increase of RDA is concomitant to the relief of the strain field under continuous irradiation.

The θ -2θ scans recorded at each fluence step can allow one to determine the maximum out-of-plane strain. Based on the maximum normal strain, the corresponding strain/stress state of the damage layer can be derived. The following analysis of the strain/stress state, which is based on a two-step model [START_REF] Rao | The measurement of elastic stresses and energy in cubic single-crystal films by x-ray diffraction[END_REF][START_REF] Debelle | XRD investigation of the strain/stress state of ionirradiated crystals[END_REF], holds for single crystal of cubic crystalline structure submited to irradiation with low energy ion.

The analysis of the strain/stress state can be made assuming that: (i) only the irradiated layer responds to the mechanical solicitation and this response is merely elastic;

(ii) the in-plane strain does not exist, there is only an out-of-plane strain and (iii) the swelling of the damaged layer is free in the out-of-plane direction due to the free surface.

The two-step evolution of the strain/stress state occurs as the following. In the first step, the thin layer that is submitted to the mechanical solicitation (i.e ion irradiation) can be considered as a free thin film inside which the radiation induced defects lead to the free swelling or shrinking of the surrounding matrix and thus to the local volume modification:

∆V V def = 2ε def // + ε def N
where ε def // and ε def N denote the in-plane and out-of-plane strains due to defects, respectively. Supposing that the lattice modification is isotropic,

ε def // is equal to ε def N and therefore ∆V V def = 3ε def N
In the second step, since the damage layer can be considered as the thin film deposited on a substrate (the thick un-damage layer), it is constrained by the substrate and hence there exists a stress imposed by the substrate to the damaged layer in the direction parallel to the crystal's surface (i.e. the in-plane direction). As a result, the substrate exerts a fixation stress (σ SR // , SR denotes the reaction of the substrate) which finally cancels the two in-plane strain components appearing in the damage layer as a consequence of the radiation damage (ε SR // = -ε def // ). The applied fixation stress consequently leads to an additional normal strain acting on the damaged layer and the deformation takes places only in the out-of-plane direction, i.e., the deformation evolves along the free surface of the crystal where there is no stress imposed. This additional strain due to the reaction of the substrate is proportional to the initial elastic strain induced by irradiation:

ε SR // = α SR hkl ε def N
where α SR hkl , which depends on the orientation and the elastic constants of the material, reflects the reaction of the substrate acting on the damage layer. This factor has been presented in the work of S.I. Rao and C.R. Houska [START_REF] Rao | The measurement of elastic stresses and energy in cubic single-crystal films by x-ray diffraction[END_REF]:

α SR hkl = 2C 12 -C an Ω hkl C 11 + C an Ω hkl
In this expression, Ω hkl is a geometric factor which is equal to 0 for 100 orientation and to

ε total N = ε def N 1 + α SR hkl
In summary, the irradiated layer is characterized by a zero in-plane strain and a non-zero out-of-plain strain which is normal to the crystal's surface.

Considering the stress exerting on the damage layer, this layer exhibits only the in-plane fixation stress since the third component is zero due to the free surface. As mentioned above, the in-plane strain of the damage layer is totally cancelled by the substrate; the fixation stress can be, therefore, calculated according to the work of Rao and Houska developed for cubic single crystal [START_REF] Rao | The measurement of elastic stresses and energy in cubic single-crystal films by x-ray diffraction[END_REF]:

σ f ix // = - C 11 + (1 -α SR hkl )C 12 -C an 1 + α SR hkl Ω 1 1 + α SR hkl ε total N
where Ω 1 equals 0 for 100 orientation and -2/3 for 111 orientation. ions exhibited stress level of -2.8 GP a [START_REF] Debelle | Strain and stress build-up in He-implanted UO 2 single crystals; an x-ray diffraction study[END_REF]. The difference is believed to be due to the solubility and concentration of implanted species. Indeed, the maximum relaxation stress is measured in the crystal implanted with 20-keV He ions at almost 5 at. % of implanted He, while the stress measured in this work is at less than 0.1 at. % of implanted Ce.

In addition, He -an inert noble gas -is well known to form bubbles leading to the high strain in the damage layer (ε def N = 1.03%), thus higher substrate reaction stress.

Referring to the evolution of damage build-up measured by RBS/C technique (see sections 3.1 and 3.2), both evolution of maximum elastic normal strain and maximum inplane stress are observed since the very first stage of damage evolution when irradiation creates a the small fraction of defects. At this stage, the main type of induced defect is essentially point defects (interstitial defects). As irradiation reaches over 5.9 dpa (the same dpa at which the steep increase of radiation damage starts), a transformation of point defects to extended defects takes place. It is at this point we observed the relaxation of the crystalline structure, i.e. the transition from elastic to plastic deformation.

Comparing the two orientations, it is apparent that the elastic strain exhibited by the damaged layer shows no difference (see figure 3 a larger strain to be developed at the same irradiation condition. Similarly, the in-plane axial stress experienced by the damaged layers differs for the two orientations. Maximum values derived from the strain/stress analysis are -1.8 and -1.3 GPa for 100 and 111 orientations, respectively. The results obtained in this investigation is therefore totally in agreement to the recently reported work [START_REF] Debelle | Influence of elastic properties on the strain induced by ion irradiation in crystalline materials[END_REF] where the elastic strain arising in 100 and 111 orientations is almost perfectly equal in irradiated cubic zirconia.

The same strain level exhibited by both crystallographic orientations can be explained by following argumentations. Firstly, the irradiation induced defects are isotropically distributed within the crystal. This argument is in consistency with the fact that the process of defects creation occurring during a collision cascades is random. Secondly, the damage fraction produced at given fluence seems to be equal regardless of the orientation.

Indeed, the figure 3 orientations. This phenomenon is due to the same amount of energy deposited during a collision cascade. The eventual total normal strain experimentally measured is different because the reaction of the thick substrate layer is different. This reaction is dependent to the orientation of the crystal. The results discussed here is in good consistency with that reported for cubic zirconia crystals irradiated by 300 keV Cs + ions [START_REF] Debelle | Influence of elastic properties on the strain induced by ion irradiation in crystalline materials[END_REF]).

TEM observation of the radiation damage

The Transmission Electron Microscopy technique has been applied to investigate, at the atomic scale, the formation of defects in irradiated U O 2 crystals due to irradiation with 500-keV Ce ions at low fluence according to the first step of the radiation damage previously observed by RBS/C. In this investigation, high resolution images were recorded for both plan-view and cross-section geometries on the same crystal irradiated at Φ = 5 × 10 14 cm -2 .

The plan view images obtained, as displayed in figure 3-15, show the existence of dislocation loops in the irradiated sample. The dislocations appear as small ellipses and in some cases, as on the left-hand side in the image shown, they appear to be linked.

The wide almost straight contrasts that extend all over the image are due to thickness variations; as indeed sample preparation induced a thickness gradient in the thinned area, away from the edge. The exact depth at which these dislocations are located is difficult to determine due to the sample preparation.

The spot shown in figure 3-16 corresponds to a projectile impact. The plan view high resolution image was taken using a 100 type zone axis and the corresponding diffraction pattern is shown in the inserted image. The selected area diffraction (SAD) was obtained using a selection aperture of ∼ 150 nm in diameter projected on the sample. The contrast Tien Hien NGUYEN Université Paris-Sud difference around the impact is an indication that the angle of incidence of the projectile was not exactly normal to the surface (as predicted when the sample was fixed on a 7 • sample holder to avoid channelling of bombarding ion). Worth mentioning is the absence of any other defects in the impact region meaning that a complete recovery exists, at least at the level that the image was taken. In order to observe the depth and the character of the damage caused by the ion irradiation, a series of images with increasing depth from the surface were obtained from a cross-section specimen, and are shown in figure 3 the cross-section tries to emulate the path of the projectiles, as shown in figure 3-17a.

From these images, it is apparent that the most damaged part, that is defects and wide dislocation loops, is distributed over the first 120 nm down from the surface, while, as expected, the maximum damage is between 90 and 130 nm. These values are in excellent Finally, part of the images were processed using FFT filtering, in order to obtain some more accurate characteristics regarding the defects.

Description of the model of defects

The radiation-induced defects are very complex and consist of various types. Their influences on the dechannelling of incident probing ions are partially, on one hand, obstructing the incoming ions and deflecting their trajectories, on the other hand. In the case of irradiation with low-energy ions, the created defects are located close to the surface of the crystal and their presence in the crystal are viewed on the axial spectrum via the appearance of a damage peak and via the dechannelling signal which is at higher level than the virgin axial spectrum. These two components are obviously the results of the presence of radiation defects with their own characteristics for each type (and hence distinct defect scattering factors). Such a complex configuration may be nevertheless simulated by a simplified model of defects, where the two dominant types are taken into account:

Tien Hien NGUYEN Université Paris-Sud 4.1. Description of the model of defects (i) pure interstitial atoms, small enough in order to virtually produce no lattice distortion and referred to as randomly displaced atoms; (ii) distorted channels characterised by a local curvature, hereafter referred to as bent channels, resulting in some extra transverse energy given to channelled ions. It is worth to stress that those two types of defects correspond in fact to extreme conditions of pure obstruction-type channelling and pure distortion-type channelling, respectively (i.e. defect scattering factor equals 1 and 0, respectively). Every complex mixture of defects involving different types and numbers can therefore be in principle reproduced by a distribution of defects using both types.

Let us to point out that the higher dechannelling signal observed at low energy, i.e. at depth larger than the damage profile, does not mean that the defects (or more precisely, the distortion-type defects) are created at the corresponding depth as compared to the virgin spectrum. In fact, the increasing dechannelling signal is due to the continuous interactions of probing ions with matrix ions with all types of defect including the randomly displaced atoms (RDA) and bent channels (BC). These interactions lead to the small-angle deviations of the channelled ions and finally kick off them out of the channel at a given depth. In other words, each contribution to the dechannelling signal results from the integration of small interaction over the whole thickness covered including damage area where major deviation of ions is caused by the presence of defects. For the sake of simplicity, a two-class defect model is proposed. This model assumes that both the RDA and BC are created on the same defected zone in the near-surface region, in agreement with the creation of radiation-induced defects by low-energy ions.

The RDA is responsible for the damage peak, which reflects the local concentration of interstitial defects with a small contribution to the dechannelling yield, while the BC is supposed to have the major influence on the dechannelling signal. The concentration of BC is assumed to be constant over the whole damaged layer, whilst a given arbitrary Tien Hien NGUYEN Université Paris-Sud Chapter 4

distribution of RDA is allowed over the same region (see figure 4-1). Such an approach allows one to limit the number of free parameters in such a way that the obtained results can be directly compared at various ion fluence and different type of projectiles. This simplified assumption can be considered as a thorough primary analysis in which the BC has been averaged. The limitation of such an approach will be discussed in section 4.2.3. Ben channels defect refers to atomic channels that are modified under irradiation.

It refers actually to the distortion of atomic arrangement in the investigated direction.

In this work the distorted atomic row is modelled by an Arctangent function, which is parameterized by the length L of the curved region and by the angle η with respect to the direction of the undistorted channels (figure 4-2). The volume fraction of bent channels is a variable and it varies in the range of 0 to 1. In addition, the fraction of randomly displaced atoms can simply be fixed at every depth from 0 to 1 (i.e. up to the hypothetical case of 100 % displaced atoms corresponding to the creation of an amorphous structure). U O 2 crystals [START_REF] Pryor | Thermal vibration amplitudes in fluorite lattices[END_REF][START_REF] Garrido | 4 He channelling studies of U 4 O 9[END_REF] and data derived from neutron investigations [START_REF] Dolling | The crystal dynamics of uranium dioxide[END_REF]. Typical values vary from u 2 U = 6.0 pm to 7.5 pm and u 2 O = 9.0 pm to 12.0 pm.

Simulation of axial channelling spectra by Monte-

The role played by randomly displaced atoms (RDA) to channelling spectra

As a first simplified approach, a defect model comprises only a distribution of RDA can be considered as an extreme configuration in which the fraction of BC equals zero, i.e., irradiation leads solely to the creation of RDA. In this configuration, the depthdistribution of RDA is the sole input data (the fraction of RDA on both sub-lattices has to be specified at every depth) and the axial spectrum calculated by Monte Carlo simulation is compared to the experimental spectrum. A typical example of analysis carried out for crystal implanted with Xe ion at a chosen fluence, namely Φ = 2 × 10 15 cm -2 , is shown on figure 4-3.

The data was analysed at the first step by the TBA approach, by which the damage peak was separated from the dechannelling signal and hence a depth-distribution of defects was calculated (see section 3.1). The distribution of defects derived from the TBA calculation has been used as an input parameter for simulation, assuming that all defects are considered as RDA, in order to test the ability to reproduce the channelling spectrum.

Hence, an identical profile of RDA compared to the depth-distribution of defects extracted from the TBA approach is used for simulation (see figure 4 identical profile as the depth-distribution of defects issued from the TBA method cannot reproduce the experimental channelling data. A discrepancy between experimental and simulation is clearly witnessed.

Let us discuss first the dechannelling signal at a depth larger compared to the damage peak (i.e. in the undamaged zone at large depth). It is observed that the presence of RDA clearly leads to an increasing dechannelling yield as compared to the virgin spectrum, but the dechannelling yield is by far too low compared to the experimental spectrum.

Considering now the damage peak, the Monte-Carlo simulation fails to reproduce the height of the damage peak; such a feature is partially due to the underestimation of the dechannelling signal discussed here before. A much lower damage peak is obtained in comparison to experimental spectrum. In addition, the simulated damage peak presents a narrower width. It is seemingly that the fraction of RDA is neither high nor wide enough Tien Hien NGUYEN Université Paris-Sud to reproduce the damage peak. Therefore, figure 4-3 clearly demonstrates that the use of the RDA with a depth-distribution directly extracted from the TBA does not allow fitting neither the damage peak nor the dechannelling signal.

An improved profile of RDA can then be applied so as to try to fit the channelling spectrum. In fact, another profile of RDA is defined as input for simulation (a table of the fraction of randomly displaced atoms on both U and O sub-lattices at every depth is defined) and the input table is modified until a good agreement between simulation and experimental data is finally reached. As it can be seen on figure 4-3.d, the damage peak is nicely reproduced by simulation: a good agreement is found both the height and the width and the peak. As shown on figure 4-3.c, the profile of RDA shows a higher fraction and a wider distribution of randomly displaced atoms versus depth as compared to the depth-distribution of defect extracted from TBA approach. However, even if these additional defects slightly improve the dechannelling yield, a much lower dechannelling signal is still evidenced. The underestimation of the dechannelling yield provides strong evidence that in order to fully fit the spectrum either (i) a broader defects distribution is necessary or (ii) other classes of defect need to be taken into account to compensate the difference between experimental and calculated dechannelling yields.

The first hypothesis is tested in this section. The profile of RDA is extended to larger depth (at depths larger compared to the damage peak) to increase the level of the dechannelling yield. As seen from the figure 4

The role played by the bent channel (BC) class of defects

In the previous section we show that the use of a distribution of Randomly Displaced Atoms does not provide a satisfactory fit to the channelling data. As an alternative approach, channelling data was analysed assuming that the contribution of RDA is fully cancelled and that bent channels are the sole type of defects created under irradiation Tien Hien NGUYEN Université Paris-Sud Chapter 4

with low-energy ions. This configuration shares strong similarities with the case where only dislocations are located in the defective crystal. ). The red solid line denotes the simulated spectrum and the corresponding distribution of BC assuming both the depth distribution and the concentration derived from the TBA. Long-dashed green line denotes the simulated spectrum and the corresponding sharp distribution of BC (up to 95 % of BC). Short-dashed blue line represents the calculated spectrum assuming a constant distribution of BC from the surface up to a given depth. Dotted line help showing that the sharp distribution of BC is centred at the ion's implanted range.

As it can be seen from the figure 4-4.b, the use of a profile of BC with the depthdistribution extracted from the TBA method does not provide a satisfactory agreement between experimental and calculated axial spectra. The first tested distribution of BC appears to be at a clear variance with the experimental data. As observed on the figure 4-4.b, the simulated spectrum shows an increasing dechannelling yield with decreasing energy of the backscattered ions. The dechannelling yield has a similar trend compared with the experimental dechannelling yield but an underestimation in the yield is clear seen. Moreover, the backscattering signal on the defected zone close to the surface does not show any damage peak as compared to the experimental spectrum. Only a steep increasing of the signal on the damage layer is observed with increasing depth, a feature Tien Hien NGUYEN Université Paris-Sud which was anticipated since the presence of bent channels does not involved the presence of direct-scattering centres. Thus, the use of a distribution of BC with the profile issued from TBA approach fails to reproduce the channelling spectrum especially the damage peak.

In order to check if the presence of the damage peaks could be accounted for by the sole contribution of BC, a sharp profile of BC is employed. As viewed on figure 4-4.b, a new distribution of BC is used as input data up to the same depth at which a significant fraction of defects extracted by the TBA method is still observed, i.e. ∼ 160 nm. The maximum fraction of 95 % of BC is fixed at around 80 nm, i.e. at a depth corresponding to the maximum fraction of 470-keV Xe ions implanted in U O 2 . The resulting spectrum obtained from simulation reveals once again that even though a very sharp distribution of BC is used, no damage peak can be evidenced. The simulated spectrum shows the same increasing spectrum with decreasing energy of the analysing ions previously observed.

Conversely, the dechannelling yield shows a higher level (due to the higher fraction of BC) but a similar trend (slope) of the dechannelling yield is witnessed compared to the experimental data. As a conclusion, the dechannelling yield can be easily reproduced with a suitable distribution of BC.

Since the use of BC essentially affects the level of the dechannelling yield, a constant distribution of BC should also be tested. As seen on figure 4-4.b, a constant distribution of BC from 0 up to 175 nm with a fraction of 30 % is used as input. The result obtained is quite interesting since the spectrum obtained is very similar to the one calculated assuming a sharp distribution of BC. On the defected zone, no damage peak is observed:

the signal increases rapidly with increasing depth. Beyond this zone, the dechannelling signal reveals a similar behaviour compared to the dechannelling yield obtained using the sharp profile of BC. The result discussed here lead to a very important conclusion: since the calculated dechannelling yield reveals to be eventually insensitive of the exact details of the depth distribution of BC, a constant profile of BC is then assumed for the sake of simplicity in the following Monte-Carlo simulations to account for the radiation damage created by ion irradiation.

To summarise, simulation with the sole contribution of BC can allow reproduction of the dechannelling yield but it fails to reproduce the damage peak. BC merely leads to a gradual increasing dechannelling yield irrespective of the exact details of the depth distribution of BC defects. In fact, the BC does not act as an obstruction centre to the probing ions and hence do not lead to a significant increase of direct backscattering events.

They influence the pathway of probing ions through a series of small-angle scattering. As a result, the backscattering of ions takes place at a certain depth transversed. The deeper the depth transversed by ions, the higher the probability to be dechannelled and hence Tien Hien NGUYEN Université Paris-Sud Chapter 4

the higher the signal is recorded. A complete analysis of the channelling data recorded on low-energy ion-irradiated crystals requires a model involving the contribution of a more complex type of defects, since the sole contribution of BC does not allow a satisfactory description of the radiation damage.

Monte

Carlo simulations with the complete model of defects: RDA-and BC-type

In the section 4.2.2 and 4.2.3, Monte Carlo simulations performed with the unique contribution coming from either the RDA or the BC type of defects fail to reproduce the experimental channelling spectra. The RDA-type of defects can reproduce the damage peak but it clearly underestimates the dechannelling yield at depths larger than the implantation profile. Conversely, the use of the BC-type of defects allows a nice fitting of the dechannelling contribution but the presence of the damage peak cannot be obtained.

Therefore, the BC-type of defects should be used in combination with another type of defects that allows the reproduction of the damage peak. The RDA-type of defect figures, of course, as an obvious candidate that fulfils those requirements. In fact, the RDA-type of defects allows one to fit the damage peak whilst BC can be used as a complementary factor to improve the fitting of the dechannelling signal.

To fit a channelling spectrum with the combination of RDA-and BC-types of defects, the same procedure is followed for every channelling spectra recorded at a given implanted fluence. In the first step, a profile of RDA is employed as the sole class of defect to fit the damage peak. As a consequence (see section 4.2.2), the simulated spectrum shows an underestimated dechannelling yield compared to the experimental spectrum. In the next step, the BC-type of defects is applied to improve the fit. The constant fraction of BC over the entire thickness of damage zone is progressively increased from 0 by small steps until the best fit is finally achieved. It is worth stressing that the presence of BC implies the need to modify the profile of RDA at every step to maintain the quality of the fit to the damage peak. In fact both BC-type and RDA-type of defects contribute to the dechannelling signal, whilst only the RDA-class of defects has a significant contribution to the direct scattering yield. As it was demonstrated in the section 4. parameters that are the length and the angle of bent channel (see section 4.1). Therefore, the choice of these two parameters directly affects the contribution of BC to the channelling spectrum. However, these two parameters result in the subsequent contribution of BC to the simulated channelling spectra in opposite ways. The angle of the distorted part has a straight influence on the dechannelling yield as it refers to the imperfection of the atomic channel. This imperfection results in an extra transverse force acting on the channelling ions. This force is perpendicular to the trajectory of channelling ions that progressively deviates and eventually dechannels the probing ions. Therefore, the higher the angle of distortion is, the higher probability to be dechannelled the ions have and the higher the dechannelling yield is obtained. A clear example that represents the influence of the angle of distorted part is shown on figure 4-5.a. Conversely, the length of the distorted part has inverse influence on the dechannelling yield. The higher length of the distorted part will reduce the intensity of interaction (intensity of transverse force over unit of thickness covered by analysing ion) that leads to the deviation of the channelled ions. The higher the length of distortion is, the smaller the intensity of transverse force becomes. Hence, the smaller angle the ion is deviated (scattered) and therefore the lower dechannelling yield is consequently obtained. The direct influence of the length of distorted part to the simulated channelling spectra is demonstrated on figure 4-5.b. The two parameters affect the contribution of BC to the channelling spectra in two opposite ways. It is then important to specify the suitable intervals for the two parameters such that (i) the combination of these two parameters reasonably reflects the true created defects and (ii) the combination of BC + RDA allows the reproduction the channelling spectra. These parameters also need to be in reasonable Tien Hien NGUYEN Université Paris-Sud Chapter 4

agreement with the defects that are observed by other alternative investigating technique, e.g. the HRTEM observation on dislocation. The following discussion will focus on the use of the complete model of defects (RDA + BC) to analyse channelling data in which the angle and the length of distortion are fixed to be L = 5 nm and η = 1.5 • for all three crystals. The choice of these values will be discussed in section 4.3.4.

Damage evolution

As a result of analysis, the distributions of RDA and BC versus depth have been extracted from the channelling spectra. This section focuses on the analysis of channelling data by Monte-Carlo simulations and on the evolutions of both distributions of RDA and BC defects versus ion fluence. Afterwards, this section discusses also the evolutions of the fraction of the defects created versus the accumulated ion fluence. Results obtained for irradiation with all the three implanted elements will be shown and commented. Moreover, a comparison between the effects due to the irradiation by ions of different chemical nature is discussed.

Analysis of channelling spectra

Channelling data corresponding to the 470-keV Xe-implanted crystal are analysed by Monte-Carlo simulation using the complete model of defects. All the spectra have been fitted with the combination of RDA and BC. Experimental channelling spectra are plotted in symbols for fluences of Φ = 3 × 10 14 cm -2 (red circle), 7 × 10 14 cm -2 (green triangle up), 8 × 10 14 cm -2 (yellow triangle down), 1 × 10 15 cm -2 (violet square), 2.5 × 10 15 cm -2 (pink diamond), 5 × 10 15 cm -2 (cyan hexagon), 1 × 10 16 cm -2 (gray star), 2 × 10 16 cm -2 (orange cross) and 3 × 10 16 cm -2 (red crossed-circle). Fits were performed assuming L = 5 nm and η = 1.5 • .

The evolution of the shape of the random spectra versus implanted ion fluence deserves a discussion. In fact, as it was already mentioned, the MC simulations were performed assuming the presence of the implanted species located at the octahedral positions in the lattice structure with a content corresponding to the ion fluence. As it can be seen from the figure 4-6.b, the exact shape of experimental random spectra is fully accounted for the presence of extra-incorporated atoms (see for instance the shape of the random spectrum recorded when the crystal is irradiated at the highest fluence of 8 × 10 16 cm -2 ). The incorporation of Xe ions to high concentration leads to a random spectrum with both (i) a lower backscattering yield at the surface due to the modification of the stopping power on the implanted layer compared to the un-irradiated crystal and

(ii) the appearance of a peak located in between 2600 and 2700 keV originated from the Tien Hien NGUYEN Université Paris-Sud backscattering of probing ions on Xe atoms located in the matrix (the front of backscattering signal due to Xe atoms found at the surface of the crystal locates at 2737 keV ).

All those modifications of the random spectra are reproduced by the incorporation of Xe into U O 2 crystalline structure with a profile corresponding to implantation. cm -2 (red circle), 7×10 14 cm -2 (green triangle up), 8×10 14 cm -2 (yellow triangle down), 1×10 15 cm -2 (violet square), 2.5 × 10 15 cm -2 (pink diamond), 5 × 10 15 cm -2 (cyan hexagon), 1 × 10 16 cm -2 (gray star), 3 × 10 16 cm -2 (red crossed-circle) and 4 × 10 16 cm -2 (green crossed-square). Fits were performed assuming L = 5 nm and η = 1.5 • .

A similar analysis has been performed regarding channelling data recorded for crystal implanted with lanthanide atoms, i.e. 500 -keV La and 500 -keV Ce ions. Analyses have been performed employing the same procedure as for crystal implanted with Xe.

The similar evolution of the radiation damage was considered for La and Ce implanted in a good agreement between experimental and simulated spectra is revealed over a large energy range of the probing ions, as it was the case for Xe-implanted crystals. A discrepancy was also observed at the depths exceeding the damaged zone for some fluence below Φ = 1 × 10 16 cm -2 . This discrepancy is systematic as it presents in all three analysis.

U O 2 .
However, this discrepancy does not limit the usefulness of the analysis with the complete model of defects. As a main conclusion of this section, all the results confirm that axial channelling spectra can be accurately reproduced by MC simulation with the assumption of defects comprising a profile of RDA and a constant concentration of BC-type of defects.

It is remarkable that the simulation of the axial spectra can be performed with the same value of the BC parameters (L, η) over an extensive range of fluence. Such a feature allows a comparison of the parameters over the entire fluence range.

Evolution of the randomly displaced atoms defects

As already mentioned in the description (see section 3. The difference is well understood since the model of defects applied into the simulation assumes two types of defects while the TBA approach gives a distribution of defects that contains all types of defects.

(iii) In the high fluence range, the distribution of RDA presents both similarities and differences for irradiation with noble gas Xe ion compared to La and Ce ions (figure 4-11.a). The figure shows that in all cases, the creation of radiation damage seems to reach a depth limit. To be precise, the creation of damage does not exceed a depth which is estimated to be 300 nm. Beyond this depth, no radiation damage is created. For the crystal irradiated with La and Ce ions, the distributions of RDA calculated are similar.

As revealed on the figure 4-11.a, the creations of RDA both saturate at the same level, i.e 20 %. Distribution shows no dramatic increase of RDA even when the crystals are irradiated to very high fluence.

(iv) As when the crystal is implanted with Xe ions at very high fluence (corresponding to fraction of implanted Xe in the matrix of over 5 at. %), a striking behaviour is Tien Hien NGUYEN Université Paris-Sud over which the BC is created remains almost constant at high ion fluence range. In all the three cases, the BC is created up to the thickness of about 260 to 300 nm. The almost constant BC fraction observed demonstrates that the creation of BC seems to saturate over the high ion fluence range. In fact, this phenomenon is not so surprising since the channelling of probing ion is known to be more sensitive to the low concentration than to the high concentration of defects. The presence of a higher concentration of defects might not necessarily results in the higher fraction of dechannelled ions as the crystal is already heavily damaged. For that reason, this phenomenon can be partially accounted for by the limited sensitivity of the analysing ion beam to the variation of damage at high concentration. As it can be seen, the bottom of the dip integrated from 40 to 130 nm, in comparison to the experimental data, was nicely fitted both the level and the width (i.e. the shape of the dips) supporting the idea that the model of defects used in this investigation together with the choice of L and η have totally accounted for the damage peak observed on the channelling spectra; a great achievement of the approach that was used to enlighten a new method to analysing channelling data.

Regarding the undamaged part of the crystal from 415 to 615 nm (corresponding to the energy window from 2370 to 2540 keV ), the integration on this part of the channelling spectra have reproduced satisfactorily the level and the shape of the experimental dip. It is worth pointing out that the calculated width of the angular scan is fairly well simulated.

However, the agreement is not as good as those obtained for the integration over the

BC type defects

The channelling of ion (in case of well oriented along a crystallographic orientation of a crystal) depends strongly on the quality of the crystal. Every imperfection of the crystal results immediately in the dechannelling of ion thus increases the level of axial backscattering spectra recorded. In case of a crystal containing defects, the channelling of ion is affected by two way: the presence of defects leads to (i) direct obstruction or (ii) small angle scattering of channelled ion. However, the sensitivities of the direct obstruction and of the small angle scattering depend on the nature and concentration of defects in the crystal.

RDA refers to obstruction-type defects that directly results mostly in the direct backscattering of channelled ion. The backscattering of ion is so sensitive to the presence of RDA within the lattice that every small variation in the concentration of RDA will results immediately in the corresponding modification of the backscattering events. On the contrary, the presence of BC in the lattice, which refers to the distortion-type defects, affects the backscattering of ion in a different way. The backscattering of analysing beam is much more sensitive to the presence of BC at low concentration than at high concentration. As a consequence, a small variation at low concentration of BC leads to a more important modification of the backscattering event as compared to a larger variation at higher concentration.

Following these remarks, the uncertainties in the fraction of RDA and BC can be then specified. At first, the procedure used to fit a channelling spectrum described in section 4.2.4 is followed so as to obtain the best reproduction to the channelling spectra by simulation. Then in a second step, the uncertainties for RDA and BC can be specified as below:

• Uncertainties for RDA: To specify an uncertainty for RDA, the fraction of BC that Tien Hien NGUYEN Université Paris-Sud Chapter 4

gives the best reproduction of axial channelling spectra was kept constant while the fraction of RDA was varied by small steps. In this case, a variation of 1 % of RDA can still give acceptable agreement between simulation and experimental channelling spectra. Any variation of larger than 1 % breaks this agreement since the backscattering of channelling ion is very sensitive to the concentration of RDA whether this concentration is low or high. Therefore, the errors bar for RDA was taken to be 1 % at all fluences.

• Uncertainties for BC: Conversely, in order to specify the error bar for BC, the fraction of RDA was kept constant while the constantly distributed fraction of BC was varied step by step. In this situation, the amount of BC variation that still gives an acceptable agreement between simulation and experimental channelling spectra depends on the concentration of BC. As the channelling of analysing ion is more sensitive to the low concentration of BC (at corresponding low ion fluence) than the higher concentration (at corresponding higher fluence), the uncertainty for BC varies from 0.1 % for low fluence to 5 % for high fluence, respectively. The exact value obtained varies, therefore, from fluence to fluence. For the sake of clarity, these values is displayed on the figure that shows the evolution of BC versus ion fluence (see figure 45678910111213141516).

Kinetics of damage accumulation

The kinetics of the created defects is established by following the evolutions of both RDA and BC defects are plotted. The evolutions of RDA show similar features that are observed on the evolutions of radiation damage extracted from channelling data by the TBA approach (see figure [3][START_REF] Kurepin | Thermodynamic properties of hyperstoichiometric urania in the UO 2 -UO 2.25 solid solution range[END_REF][START_REF] Sheindlin | Experimental determination of the thermal conductivity of liquid UO 2 near the melting point[END_REF][START_REF] Banerjee | Specific heats of thoria urania solid solutions[END_REF][START_REF] Leclercq | Thermal conductivity of zirconia-based ceramics for thermal barrier coating[END_REF][START_REF] Degueldre | Specific heat capacity and Debye temperature of zirconia and its solid solution[END_REF][START_REF] Azom | Zirconia -ZrO 2 , Zirconium Dioxide[END_REF][START_REF] Slack | Thermal conductivity of MgO, Al 2 O 3 , MgAl 2 O 4 , and Fe 3 O 4 crystals from 3 to 300 K[END_REF]. That is to say, the evolutions of RDA reveal once again that: (i) a sharp increase step appears around 4 to 7 dpa (corresponding to the concentration of implanted ions of less than 0.1 at. %). This step is observed to appear at the same radiation condition regardless of the nature of ions. It is, therefore, a clear consequence due to the radiation effect (creation of defect due to nuclear stopping) and is totally independent of the nature of ions. (ii) After the steep increase, the creation of RDA saturates over a wide range of irradiation that is observed via the plateau of saturation from 10 to 300 dpa. In addition, Tien Hien NGUYEN Université Paris-Sud of each individual step and can therefore be described by the following equation:

f d = m i=1 f sat d,i -f sat d,i-1 G{1 -exp [-σ i (Φ -Φ i-1 )]}
where f d is the amount of accumulated damage; f sat d,i is the level of damage at saturation at the step number i; σ i is the cross section for the damage formation at the step i; Φ is the irradiated fluence; Φ i-1 is the value of accumulated fluence at which the step number i starts; G is a function that transforms negative values into 0 and leaves the positives unchanged; m represents the total number of individual steps. As compared to the 3-step damage accumulation process in Xe-implanted crystal, the evolutions of RDA in La-and Ce-irradiated crystals are fitted by the 2-step damage accumulation. As it can be seen on figure 4-15, the first damage accumulation step is observed on the same ion fluence range as compared with Xe-irradiated crystal. This step saturates at the fraction of RDA of 5 % for both La-and Ce-implanted crystals.

The second step begins essentially at the same irradiation condition ( The evolution of the fraction of bent channels was also examined. The evolutions of BC exhibit a similar evolution for the three elements (see figure [START_REF] Kurepin | Thermodynamic properties of hyperstoichiometric urania in the UO 2 -UO 2.25 solid solution range[END_REF][START_REF] Sheindlin | Experimental determination of the thermal conductivity of liquid UO 2 near the melting point[END_REF][START_REF] Banerjee | Specific heats of thoria urania solid solutions[END_REF][START_REF] Leclercq | Thermal conductivity of zirconia-based ceramics for thermal barrier coating[END_REF][START_REF] Degueldre | Specific heat capacity and Debye temperature of zirconia and its solid solution[END_REF][START_REF] Azom | Zirconia -ZrO 2 , Zirconium Dioxide[END_REF][START_REF] Slack | Thermal conductivity of MgO, Al 2 O 3 , MgAl 2 O 4 , and Fe 3 O 4 crystals from 3 to 300 K[END_REF][START_REF] Devanathan | Structure and mechanical properties of irradiated magnesium aluminate spinel[END_REF][START_REF] Klemme | Low-temperature heat capacities of MgAl 2 O 4 and spinels of the MgCr 2 O 4 -MgAl 2 O 4 solid solution[END_REF][START_REF] St | Thermophysical properties of sapphire, AlN and MgAl 2 O 4 down to 70 K[END_REF][START_REF] Scott | Phase relationships in the zirconia-yttria system[END_REF][START_REF] Murphy | Nonstoichiometry in MgAl 2 O 4 spinel[END_REF][START_REF] Sickafus | Structure of spinel[END_REF]. At low fluence (before 7 dpa), the fraction of created BC is very small. Significant increase of BC is observed almost at the same irradiation condition for all the three case. BC starts to increase at about 7 dpa, i.e at higher ion fluence compared to the steep increase of RDA.

Note that the fraction of BC at each fluence depends on the exact choice of length (L) and angle (η) of distortion (see section 4.3.4) but the evolution of BC, which reflects the effect of irradiation of the U O 2 structure, is independent of the choice of these parameters. That is to say, the three evolutions, in which the fraction of BC follows the similar evolution for all the three crystals, clearly demonstrate that irradiation has led to similar structural modifications.

At high implanted fluence, the creation of BC saturates. This saturation is observed in all three implanted crystals where the saturation level is observed to be higher in Tien Hien NGUYEN Université Paris-Sud La-then in Ce-and Xe-implanted crystals. In fact, this difference is likely not as significant as it seems due to the following reason. Over this fluence range, the crystals were so heavily irradiated that the structural modification appears to be constant via the observation of BC. The fraction of the created defects reaches so high level that the indirect backscattering of channelled ions is almost insensible to a large variation of defects.

The comparison between RDA and BC shows that, the results reveal some following important features. First of all, the evolution of BC is shifted to larger fluence as compared to the steep increase of RDA observed at 4 to 7 dpa. This shift makes that the clear and significant increase of BC defect is observed when the saturation of RDA starts. The two phenomena, i.e. the saturation of RDA and the increase of BC, take place almost at the

General discussion

In the channelling analysis, the separation of the direct and indirect backscattering contribution (by the TBA method) gives the depth-distribution of the radiation damage in a crystal. However, the radiation damage extracted from channelling spectra is generally a mixture of many types of defects and it is quite impossible to distinguish and separate these classes of defects. Therefore the defects created are usually supposed to be of single type, RDA only. Since the separation of individual type of defects is impossible, the results may lead to a misunderstanding and/or to the incomplete knowledge regarding the radiation effects. For example, the evolution of radiation damage extracted form channelling spectra by the TBA method (see figure 3-10) clearly shows a steep increasing step around 4 to 7 dpa. After this step, the evolution of radiation damage saturates over a very wide range of irradiation (from 10 to 300 dpa for Xe irradiation and even larger for La and Ce irradiation). It is seemingly that there is no modification of the crystalline structure over this range. There might be some consequences due to irradiation on this range of ion fluence but it is in fact impossible to explore these consequences via the evolution of radiation damage extracted by the TBA approach.

Computer simulation offers the nice opportunity to allow such a separation of individual type of defects with an appropriate assumption, i.e with a relevant model of defects to be tested. Therefore, it might be possible to discover some modifications of the crystalline structure under irradiation which was hidden in the TBA analysis. for crystal submitted to elastic deformation, i.e, the distance between atomic planes was elongated but atoms still keep their strictly periodic arrangement. In case of crystal is plastically deformed, the use of an X-ray beam on the damaged layer never results in the constructive interference since there is no strict periodic arrangement of atoms. In this situation, the measurement of elastic strain is no longer possible.

In addition, as BC, which is characterised by the length (L) and the angle (η) of the distorted atomic arrangement, can be considered as extended defects in the crystal such as the formation of dislocations in the matrix. Indeed, the formation of a dislocations in the matrix consequently results in the bending of atomic rows around the dislocation edge.

Since the presence of dislocation leads to the modification of not only one but several atomic arrangement surrounding (as observed on the FFT processed TEM image, see show that crystals were elastically deformed.

It is also at the low fluence range, the accumulation of point defects with increasing ion fluence lead to the transformation from point defects to extended defects. This phenomenon was observed via (i) the steep increase of RDA from 4 to 7 dpa and (ii) the relaxation of crystalline structure (observed via the strain/stress relief). The steep increase of RDA was observed to be independent to the nature of implanted ions.

(ii) In the medium fluence range, the accumulation of radiation effects results in significant destabilisation of the crystalline structure. Furthermore, the differences due to the irradiation of a soluble versus insoluble elements was seen. In general, the radiation damage effects over this fluence range reveal two important features.

As it can be observed via the distribution of defects versus depth (see figure 4-10), the creation of defects occurs at a larger depth as compared to the implantation range of ions. As increasing ion fluence, the profiles of defects literally shift towards greater depth and the damage zone broadens. This shift is observed on both two sides of the profiles of RDA, i.e. both on the shallower and the deeper part. More importantly, it extends to the depth that is much deeper with respect to the ions implantation profiles. This phenomenon is seemingly due to the diffusion of defects into the deeper layer. Such a process can be explained in the following two-stage process. In the early stage (i. defects into deeper depths. This atomistic mechanism was also reported for Y SZ single crystal irradiated by 400-keV Xe ions where the damage layer was observed to extend to 160 nm well beyond the maximum irradiated depth of 110 nm below the crystal surface [START_REF] Yu | In situ observation of defect growth beyond the irradiated region in yttria-stabilized zirconia induced by 400 keV xenon ion-beam at -90 and 30℃[END_REF].

Over the medium fluence range the RDA saturates while the BC increase regularly.

This feature was totally hidden according to the TBA approach since this approach assumes the presence of only randomly distributed atoms. This feature reveals, therefore, an important structural modification under irradiation. However, this phenomenon is not visible via the RDA as the fraction of RDA remains almost constant. More precisely, the creation of RDA reaches its limitation and therefore no further structural modification is observed. Conversely, the increase of the BC defects with increasing ion fluence is clearly seen. Irradiation agglomeration does not lead to the higher fraction of point defects but to the higher fraction of extended defects; a feature that reveals the structural modification due to irradiation. Such that structural modification can be mostly explained by the formation of dislocations at higher density that they finally aggregate to form a networks of tangled dislocations (extended defects). Worth mentioning is that the two phenomena for La and +3 or +4 for Ce. However, due to the charge compensation, Ce has probably the charge state of +3 in U O 2 . The radius of inert Xe atom is r Xe = 218 pm while the radii of La and Ce at charge state +3 are r La = 117 pm and r Ce = 115 pm, respectively [START_REF] Emsley | The Elements. Oxford chemistry guides[END_REF]. Compared to the radius of U in the crystal (r U = 97 pm), the radius of Xe is greater than twice the radius of U while the radii of La or Ce in compound is totally comparable to that of U . It is clear that Xe creates a higher stress field than La or Ce. It can affect the position of all surrounding atoms and pushes them into the location where they appear as an obstruction to the analysing beam. As a consequence, the fraction of defects created is higher as compared to La or Ce ions.

A second mechanism may explain the observed difference in the saturation plateaus:

since Xe is present as an insoluble species in U O 2 matrix, Xe ions tend to gather together forming nano-size bubbles. Such a presence of Xe bubbles in U O 2 matrix was clearly reported [START_REF] Sonoda | Transmission electron microscopy observation on irradiation-induced microstructural evolution in high burn-up UO 2 disk fuel[END_REF][START_REF] Sabathier | In situ tem study of temperature-induced fission product precipitation in UO 2[END_REF][START_REF] Michel | An in situ TEM study of the evolution of Xe bubble populations in UO 2[END_REF][START_REF] Marchand | Xenon migration in UO 2 under irradiation studied by SIMS profilometry[END_REF][START_REF] Matzke | Fission gases in nuclear fuels[END_REF][START_REF] Baker | The fission gas bubble distribution in uranium dioxide from high temperature irradiated SGHWR fuel pins[END_REF][START_REF] Nogita | High resolution TEM observation and density estimation of Xe bubbles in high burnup UO 2 fuels[END_REF] while La and Ce are totally soluble in U O 2 .

They can form solid solution with U O 2 up to high concentration (up to 80 %) [START_REF] Kleykamp | The chemical state of the fission products in oxide fuels[END_REF][START_REF] Fujino | Thermodynamics of fluorite type solid solutions containing plutonium, lanthanide elements or alkaline earth metals in uranium dioxide host lattices[END_REF][START_REF] Grimes | The stability of fission products in uranium dioxide[END_REF][START_REF] Diehl | Das system UO 2 -UO 3 -LaO 1.5[END_REF][START_REF] Gmelin | Gmelin handbuch der anorganischen chemie: U, Uran. Ternére und polynére oxide des urans. systemnummer 55[END_REF][START_REF] Brillant | Fission products stability in uranium dioxide[END_REF][START_REF] Imoto | Chemical state of fission products in irradiated UO 2[END_REF], and stay most probably in substitutional or empty octahedral positions matrix [START_REF] Grimes | Site preference and binding of iodine and caesium in uranium dioxide[END_REF][START_REF] Turos | Lattice location of fission products in UO 2 single crystals[END_REF]. Indeed At the highest implanted fluence (Φ = 8 × 10 16 cm -2 ) the maximum fraction of RDA created reaches 80 % which is much higher compared to the saturation plateau observe on the medium fluence range but it is lower compared to the maximum damage fraction extracted from the channelling spectrum by the TBA approach, i.e. at almost 100 %.

This fact indicates that the defects created in U O 2 crystal irradiated at very high fluence are actually a mixture of complex defects and that U O 2 crystal implanted with Xe to very high concentration still remains the crystalline structure. It is similar to the situation where the crystalline structure becomes structure with small domains that slightly missoriented one with respect to each other by small angle; it was never amorphised as the case of swift heavy ion irradiation in U O 2 .

Over this fluence range, the creation of BC seems to reaches the limitation as observed via the constant fractions (see figure 45678910111213141516). Such a phenomenon was observed in the three cases. It is seemingly that the crystals were so heavily irradiated that the structural modification appears to be constant. It is also possible that since the fraction of the created defects reaches so high level that their variation leads to negligible influences on the small angle scattering of channelling ions. As a consequence, the fraction of BC, which reflects essentially the indirect backscattering of channelled ions, remains constant.

To summarise, channelling data recorded for U O 2 crystals implanted with low-energy ions have been analysed in much more details with an advanced method. Such a method Tien Hien NGUYEN Université Paris-Sud is based on the application of Monte-Carlo simulation using a two-class defect model comprising the randomly displaced atoms and the bent channels. In the framework of this investigation, the new advanced approach has been successfully applied to the analysis of a large number of channelling data recorded for U O 2 crystals implanted with different ions. Along with the similar results (similar to that obtained by the TBA approach), the new method has helped exploring important information regarding the radiation effects which was totally hidden in the analysis by TBA method. Despite some limitations of the model of defects, the new method developed in this work is very promising and can be applied to the analysis of channelling data recorded for low-energy ion irradiated crystal.

Channelling data analysis by Monte-Carlo simulation can be further improved in several ways, both experimentally or theoretically.

From the experimental point of view, the channelling measurement using an analysing beam at various energies can be performed on the defective crystals to test the energydependence of the spectra recorded in the channelling axis at selected fluence. In such an experiment, the backscattering of the analysing beam on different types of defects will be a crucial evidence to determine the nature of defects. Similarly, angular scans performed on selected fluence with an analysing beam at different energies will gives a more decisive conclusion on the exact nature of defects. As a matter of fact, defects cross-sections exhibit a different behaviour depending on the exact nature of defect; for instance, the dechannelling cross section for point defects decreases with the analysing beam energy, while constrained regions leads to an increase of the cross section. Though these experiments were performed in the past for simple systems, involving one defect type only, their extensions to complex irradiated systems would require the need of a large number of beam-time day and a full analysis by Monte-Carlo simulation. An "Energy-dependence" experiment of channelling technique could be therefore a crucial key in order to analyse channelling data by Monte-Carlo simulation at a better level.

On a theoretical point of view, a better channelling data analysis by Monte-Carlo simulation can be improved by testing a more complex model of defects. For instance, a model comprising the polygonisation along with the RDA and BC could be worth tested.

In this situation, a more realistic model of defect could allow a better reproduction of the channelling data, thus better interpretation. On the contrary, a more realistic defective crystalline structure can be assumed to take into account a true defective structure. For example, a new structure in which the true form of a dislocation can be incorporated can be tested. This possibility allows to take into account both the presence of the dislocation edge and of the distortion of the surrounding atomic arrangements. In another possibility, a more complex structure which assumes the incorporation of impurities under the form of precipitates or bubbles would be worth tested, especially for crystals implanted at Tien Hien NGUYEN Université Paris-Sud

Conclusions and perspectives

This PhD thesis has been focused on studying the behaviour of uranium dioxide single crystals submitted to low-energy ion irradiation. For this purpose, a literature review was given on the behaviour of ceramics materials in general and uranium dioxide in particular. From the bibliographical review, it is found that uranium dioxide is radiation tolerant; amorphisation was never reported. However, a full knowledge of the mechanisms of radiation damage remains unclear in particular the knowledge of the behaviour of uranium dioxide due to the irradiations with soluble versus insoluble elements. To our knowledge, the problem of doping in U O 2 by a soluble versus insoluble elements to high concentration was never explored before. As in a real situation, the fuel is literally poisoned by a mixture of all possible fission products, such an investigation of the behaviour due to individual species is necessary for a full understanding. For these reasons, a series of in situ experiments was performed to investigate the radiation damage effects in uranium dioxide single crystal submitted to external irradiation of well-defined condition.

Low-energy irradiations were used for studying the behaviour of uranium dioxide crystal due to (i) radiation effect and (ii) effects due to the doping to high concentration of a sol- • At very low ion fluence (Φ ≤ 7 × 10 14 cm -2 corresponding to a dose less than 5.9 dpa), irradiation leads to the elastic deformation of the crystal. The main type Tien Hien NGUYEN Université Paris-Sud of defect induced is point defects and their presence at low dimension results in the strain/stress build-up in the crystal. The damage layer exhibits a tensile strain along the direction normal to the surface of the crystal. The crystal's relaxation takes place when the irradiation exceeds 5.9 dpa. A partial strain/stress relief is observed. At this stage, there is a transformation from point defects to extended defects in the crystal.

• After the relaxation the radiation damage increases rapidly. This steep increase of is observed at the fluence range between 4 to 7 dpa regardless of the nature of ions.

This fluence range corresponds to the very small fraction of implanted ions of less than 0.1 at. % which is too small to play any important role due to the nature of ions. Therefore, the steep increase of defects clearly demonstrates that the radiation damage is due to elastic nuclear collision in the crystal, leading to collision cascades.

• The saturation of RDA is observed over a wide range (from 10 to 300 dpa) of irradiation for all three crystals. It is at this stage a clear difference due to irradiations by soluble versus insoluble ions was revealed. Insoluble species results in higher fraction of defects (higher saturation plateau) as compared to soluble species. The difference is believed to be due to the solubilities of implanted species. Soluble La and Ce species form solid solution with U O 2 and locate most probably at substitutional or octahedral positions. Because U O 2 can accommodate a large fraction of these elements in its structure (up to 80 %) and because their radii in solid solution form (i.e. at corresponding charge state of +3 for both species) are totally comparable to the radius of U +4 , the consequential stress field arising from their presence in the crystal is much smaller as compared to the presence of insoluble Xe in the matrix.

Since the radius of insoluble Xe is almost twice the radius of U +4 and that Xe tends to accumulate together forming nanometre-size bubbles in the matrix, they create a higher stress field and thus higher fraction of defects in the U O 2 single crystal.

• Significant structural modification is observed in correlation with the saturation

  be made clear and then a bibliography on investigations of the radiation effects on these material will be summarised. The second chapter discusses in details the experiments and experimental techniques applied to characterise the irradiated crystalline structure. The third chapter presents the results of the experiments concerning the investigations of the structural modification of uranium dioxide single crystal under irradiation. Chapter 4 discusses the measurement of the radiation effects on U O 2 single crystal by the channelling technique where the data are analysed in more details with the help of Monte-Carlo simulation. The discussion to the whole picture about the investigation of the destabilisation of the U O 2 single crystal submitted to irradiation made in this work will be rised. The thesis ends with the conclusion to this work and future orientation for further investigation.

  of oxide-based materials such as U OX and M OX are basically used in PWR (the most popular nuclear reactor -Generation II). Other carbide or nitride based materials are being investigated for the application as advanced fuels in the future reactors (Generation III and IV). Stainless steel and zirconium based alloys are both investigated and used as fuel cladding but zirconium based alloy (zircaloy) is mostly used in the present generation of nuclear reactor since they are transparent to neutrons. As for transmutation matrices and for long time storage containers, other ceramics such as Yttria Stabilized Zirconia (Y SZ) and magnesia spinel (M gAl 2 O 4 ) are currently investigated as potential candidate since they possess a high resistance to radiation and a high capability of accommodating Tien Hien NGUYEN Université Paris-Sud Chapter 1

Figure 1 - 1 :

 11 Figure 1-1: Uranium dioxide crystal with uranium atoms located at fcc positions (black) and oxygen atoms located at tetrahedral positions (red).

  and fusion. Synthesis is made either in a vertical furnace for sintered spinel or in an electric arc furnace for fusion. Each of the two method has its own advantages. Sintering is a continuous process with controlled feed-rate and homogeneous temperature distribution in the furnace and this leads to a very homogeneous product with crystal sizes of 30 -80 µm. On the other hand, spinel synthesized by fusion is often produced in a batch. For obtaining a large ingot, an extended cooling time is required which results in the inhomogeneities in micro-structure. At the outside, spinel crystals are smaller due to the faster Chapter 1

Figure 1 - 3 :

 13 Figure 1-3: Schematic drawing of the two-bodies elastic collision.

Figure 1 - 1 Figure 1 - 4 :

 1114 Figure 1-4 shows the ratio of maximum energy transferred over the initial energy of the projectile as a function of the ratio of the masses of colliding partners. The maximum of energy transferred becomes higher if the mass of the projectile gets closer to the mass of target atom. When the mass of the projectile equals the mass of the target the maximum

e 1 / 2 )

 12 of orbital electron: the high, intermediate and low velocity domains. When ion cross the material at high velocity (v ≫ v e ), ion is fully stripped. All the surrounding electrons are removed and the ion exists only as a bare nucleus. The charge of ion at high velocity (fully stripped) is equal to the atomic number (Z ef f = Z). In the intermediate domain, the speed of ion is comparable with the speed of electron (v ∼ v e ). In this domain, the process of continuous capture and emission of electron occurs and ion is partially stripped. The effective charge of ion (Z ef f ) is introduced as being proportional to the velocity of ion (for v ≤ v Bohr Z 2/3 )

Figure 1 - 5 :

 15 Figure 1-5: Stopping power of Cerium ions in U O 2 calculated according to the SRIM code.

  2+ ion irradiation to Xe fluences in excess of 5 × 10 14 cm -2 (equivalent to ∼ 2 dpa). No amorphisation was Tien Hien NGUYEN Université Paris-Sud observed up to Xe fluence equivalent to 680 dpa.

Figure 1 - 6 : 1 Figure 1 - 7 :

 16117 Figure 1-6: Damage accumulation recorded in F SZs [39].

Thomé and colleagues studied

  the damage production in yttria-stabilized cubic zirconia single crystals irradiated with medium-energy (from 30 to 450 keV ) heavy ions (from He to Cs)[START_REF] Thomé | Radiation damage in ion-irradiated yttria-stabilized cubic zirconia single crystals[END_REF]. The disorder created in the two sub-lattices (Zr 4+ and O 2-) of the crystals and the lattice sites of heavy ions were determined by in situ Rutherford backscattering and channelling experiments using a 3 -M eV 4 He ion beam. Damage was created at a depth close to the ion projected range at low fluences and growths towards greater depths with increasing fluences once the saturation was reached. The kinetics of the damage accumulation process revealed three stages, which (excepted for He) essentially depended on the number of displacements per atom (dpa) induced by irra-Tien Hien NGUYEN Université Paris-Sud Chapter 1 diating ions (ballistic contribution). Channelling results show that the lattice location of the heaviest atoms (Xe, Cs and I) varied with the nature of implanted species (chemical contribution). The position of Xe and I ions is almost random over the whole fluence range investigated, meanwhile a large fraction of Cs ions is located in substitutional lattice sites at low ion fluences. At high fluences, Cs loses its substitutionality and this could be due to the precipitation of implanted species or to the formation of compounds (such as Cs 2 ZrO 3 ), both processes being driven by a radiation-enhanced diffusion mechanism.

  at dose rates up to 100 µA/cm 2 , to a total dose of 3.0 × 10 16 cm -2 was studied by Kishimoto[START_REF] Kishimoto | Metal nanocrystal formation in magnesium aluminate spinel and silicon dioxide with high-flux Cuions[END_REF]. Nanocrystal morphology and optical absorption (∼ 0.5 -5 eV ) varied depending on flux. At high fluxes, M gAl 2 O 4 showed spontaneous precipitation of nanoparticles but, in contrast, neither long-range rearrangement of implants nor particle coarsening, up to high fluxes.Structural changes in magnesium aluminate spinel (M gO • nAl 2 O 3 ) single crystals, which were irradiated with 900 keV electrons or 1 M eV N e + ions at 873 K, were examined by electron channelling-enhanced X-ray microanalysis[START_REF] Soeda | Cation disordering in magnesium aluminate spinel crystals induced by electron or ion irradiation[END_REF]. Un-irradiated M gO • Al 2 O 3 has a tendency to form the normal spinel configuration, where M g 2+ ions and Al 3+ ions occupy mainly the tetrahedral and the octahedral sites, respectively. Electron irradiation induces simple cation disordering between the tetrahedral sites and the octahedral sites in M gO • Al 2 O 3 . In addition to cation disordering, slight evacuation of cations from the tetrahedral sites to the octahedral sites occurs in a peak-damaged area in M gO • Al 2 O 3 irradiated with N e + ions. In contrast, cation disordering is suppressed in M gO • 2.4Al 2 O 3 irradiated with electrons. The structural vacancies, present in the non-stoichiometric compound, appear to be effective in promoting irradiation damage recovery through interstitial-vacancy recombination. The damage accumulation of M gO • n(Al 2 O 3 ) single crystal implanted with 60 keV Cu -at a flux up to 100 µA/cm 2 was studied by RBS/C [58]. No complete amorphization was observed over the flux range examined, although the crystalline lattice was significantly damaged. The damaged region on the Al sub-lattice extended to a deeper region with increasing flux. Fluence and flux dependence of Al damage peak was greatly influenced by the stoichiometry n of spinel. The Al damage peak of M gO • 1.0(Al 2 O 3 ) was smaller for higher flux and fluence. Explanation was that the recombination of radiationinduced interstitials with structural vacancies effectively reduces damage of the Al sublattice. In the case of M gO • 2.4(Al 2 O 3 ), the Al damage peak, however, increased with increasing fluence and flux. The RBS/C results indicated that the recombination mechanism was more effective in the stoichiometric spinel than the non-stoichiometric one. Tien Hien NGUYEN Université Paris-Sud The effects of swift heavy ion irradiation on nearly stoichiometric magnesium aluminate spinel (M gO • 1.1Al 2 O 3 ) was analysed by transmission electron microscopy. Irradiation was made with 200 M eV Xe 14+ ions (S e = 25 keV /nm) up to 5.0 × 10 11 and 2.0 × 10 12 cm -2 at ambient temperature [59, 60]. Bright-field (BF) images revealed that 200 M eV Xe 14+ ions produced heavily disordered columns about 4 -5 nm in diameter along the tracks, tiny pits were formed at the entrance of ion tracks. Electronic excitation from one particle of 200 M eV Xe 14+ caused significant disordering in region over 10 nm in diameter. The disordering of Al 3+ ions took place over a larger area than that of M g 2+

  Structural defects in the surface region of M gAl 2 O 4 crystals irradiated with 450 M eV Xe ions were analyzed by using 4 He RBS/C. Monte-Carlo simulations applied for the interpretation of channelling spectra permitted to determine the concentration of radiation-induced defects. The depth distribution of defects was obtained. The results showed that the defect distributions were depth-dependent, likely due to annihilation of defects at the surface of the crystals. The cross-section for the formation of defects by swift Xe ions, and consequently the diameter of an ion track, were estimated using single-impact-model calculations. The chemical and structural modifications in M gAl 2 O 4 single crystals submitted to Cs ion implantation and thermal treatments were investigated by RBS/C experiments [63]. The changes of composition, the damage created and the behaviour of implanted ions were studied as a function of the Cs fluence and annealing temperature. Heat treatments showed that the lattice disorder was clearly maximum just after implantation and decreased with increasing annealing temperature. Heat treatments above 800 • C lead to huge chemical and structural modifications. A drastic change of the sample stoichiometry (loss of M g atoms) was exhibited, without noticeable Cs diffusion up to 1500 • C (below concentrations of 3-4 at. %). The formation of radiation induced defects via electronic excitation in M gAl 2 O 4 and their stability were studied by TEM [64]. It was observed that small interstitial-type dislocation loops dissociated into isolated interstitials under electron-induced electronic Tien Hien NGUYEN Université Paris-Sud Chapter 1

14 to 2 ×

 142 77 % and 96.3 % of the displaced uranium and oxygen atoms located at crystalline positions, creating replacements. The numbers of oxygen Frenkel pairs was about 2.5 times higher than the uranium Frenkel pairs. 89.5 % of the total interstitial clusters were composed of the cluster involves 2 -3 point defects. The fractions of vacancies and interstitials in clusters are 65.6 % and 44.5 %, respectively. Nogita et Une investigated by High-resolution TEM the small intragranular bubbles in the outer region of high burnup U O 2 pellets. The results reported the presence of 4-8 nm size Xe -Kr bubbles [67]. Nogita et al. [68] studied the irradiation effects in U O 2 polycrystalline due to 100 M eV Iodine ions (stopping power at surface is 23 keV /nm) over a fluence range of 1×10 10 15 cm -2 . The surface of the specimens was analyzed by scanning electron microscopy (SEM) and X-ray diffractometry (XRD), and then the depth profiles of incident iodine ions and defect clusters were measured by secondary ion mass spectrometry (SIMS) and transmission electron microscopy (TEM), respectively. Lattice parameter change, which is associated with point defect accumulation, increased with ion fluence. Defect clusters of dislocations and dislocation loops were recognized, and their depth profiles were in good agreement with the calculated damage profile. These profiles of iodine ions and dislocation loops in both U O 2 and (U, Gd)O 2 were discussed in terms of inelastic and elastic collisions. Sonoda et al. [69] investigated the 210 -M eV Xe implanted U O 2 to fluence of 5 × 10 11 ion/cm 2 . By the cross-sectional observation of U O 2 under at 300 • C, elliptical changes of fabricated pores that exist till ∼ 6 µm depth and the formation of dislocations have been observed in the ion fluence over 5 × 10 11 cm -2 (figure 1-8). The drastic changes of surface morphology and inner structure in U O 2 indicate that the overlapping of ion tracks will cause the point defects, enhance the diffusion of point defects and dislocations, and form the sub-grains at relatively low temperature.

Figure 1 - 8 : 1 17 cm - 2 .

 181172 Figure 1-8: Typical images of ion tracks in U O 2 under irradiation with 210 M eV Xe 14+ ions to fluence of 5 × 10 11 cm -2 at room temperature [69].

Figure 1 -

 1 Figure 1-9 show that the temperature increases rapidly between 10 -15 and 10 -13

Figure 1 - 9 :

 19 Figure 1-9: Calculated evolution of the U O 2 radial temperature distribution along a track in U O 2 as a function of time. The calculations are for 11.4 M eV /u U ions in the first nm of the target, i.e. dE/dx = 60 keV /nm with λ = 6 nm and a melting temperature of T m = 3150 K [35].

Figure 1 - 10 :

 110 Figure 1-10: Molecular dynamic simulation of displacement cascade induced by 80 -keV U ion irradiated in U O 2 at two different times: (a) at 2.15 ps after injection and (b) at 20 ps corresponding to the very end of the cascade [76].

  parison of radiation effects due to insoluble versus soluble species (for instance of typical fission products) was never done. Such a comparison will help understand the complete behaviour of U O 2 structure submitted to irradiation of different conditions such as the effects of the solubility. This question gives rise to a research topic that has been carried out in the framework of this thesis. The following chapters of this thesis will present the investigation of the radiation effects in U O 2 single crystals submitted to irradiation of Tien Hien NGUYEN Université Paris-SudChapter 1

  conditions that is very useful for understanding the fundamental mechanisms of Tien Hien NGUYEN Université Paris-Sud Chapter 2 radiation effects such as the choice of nature and energy of irradiation, the temperature at which the investigation if performed, the accumulated implantation fluence, irradiation flux. . . Such option is the investigation of the natural or depleted U O 2 that does not contain any radioactive elements mentioned above. In this work, the experimental simulation has been done in order to investigate the structural destabilisation of the nuclear fuel (i) due the effects of the irradiation with low-energy ions and (ii) due to the doping of the foreign elements, especially at high incorporated concentration. The samples that simulate the nuclear fuel (depleted U O 2 (0.3% 235 U ) single crystals) have been irradiated with external accelerated ion to simulate the radiation effects and to simulate the effects due to the doping of the insoluble versus soluble fission products. Irradiated samples are characterized by various investigating techniques including the Rutherford Backscattering Spectrometry in Channelling geometry, the X-Ray diffraction and the Transmission Electron Microscopy.

Figure 2 - 1 :

 21 Figure 2-1: JANNuS nuclear facility, located at Orsay and managed by the "Centre de Sciences Nucléaire et de Sciences de la Matière" (CSNSM), belongs to CNRS-IN2P3-Université Paris-Sud. It couples a 2 M V Tandem accelerator (ARAMIS) and a 190 kV ion implanter (IRMA) to a 200 kV Transmission Electron Microscope (TEM FEI Tecnai G 2 20).

Figure 2 - 2 :

 22 Figure 2-2: RBS geometry

Figure 2 - 3 :

 23 Figure 2-3: Sketch of channelling phenomenon

Figure 2 - 4 :

 24 Figure 2-4: Typical channelling spectrum recorded on a crystal containing defects in a region of the crystal (red). Blue line represents the random spectrum and the green line represents the channelling spectrum recorded for perfect crystal.

  which denotes a translation vector in the reciprocal lattice, then the ensemble of indices h, k and l represents one node in the reciprocal lattice and H hkl is normal to the plane (hkl) in the real lattice and the magnitude of H hkl is the inverse of the inter-planar spacing d hkl between two adjacent (hkl) planes.

Figure 2 - 5 :

 25 Figure 2-5: Derivation of the Laue diffraction condition

Figure 2 - 6 :

 26 Figure 2-6: Construction of the Ewald's Sphere

  tion. The electrons transmitted through the sample are brought to focus in the back focal plane of the objective lens, forming a diffraction pattern. In the image mode, the image formed by the objective is further magnified by the intermediate and projector lenses while in the diffraction mode, the intermediate lens strength is reduced so that the back focal plane of the objective is focused on the viewing screen, forming a diffracting pattern from the illuminated area of the specimen.A Transmission Electron Microscope is ideal for a number of different fields of application amongst which the application in material science is widely seen. In the material science, the TEM is usually applied to study the structure of the material including crystalline material with the presence of defects. For example, TEM with in situ ion irradiation is unique among experimental techniques in that it allows the direct observation of the internal microstructure of materials on the nanoscale while they are being subjected Tien Hien NGUYEN Université Paris-SudChapter 2to irradiation. Invaluable insights into the underlying atomistic processes at work can be gained through direct observation of radiation induced and/or radiation enhanced effects such as: phase changes and segregation; mechanical and structural changes; compositional changes; grain growth and shrinkage; precipitation and dissolution; defect/bubble formation, growth, motion, coalescence, removal and destruction; ionization; diffusion; and collision cascades. . . There are two geometries used for analysis of the irradiated samples, namely in plan-view or as a cross-section. The plan view can offer images of the surface or at a certain depth that is not always straightforward to determine accurately. The cross section offers the possibility to observe the effects of irradiation in depth, knowing at the same time the exact location regarding the observable depth.HRTEM is an imaging mode that allows the imaging of the crystallographic structure of a material at an atomic scale. At such small scales, individual atoms and crystalline defects can be viewed. As opposed to conventional microscopy, HRTEM does not use electron wave absorption by the sample for image formation. Instead, contrast arises from the interference in the image plane of the electron wave with itself. When the electron beam penetrates into the sample, the sample no longer changes the amplitude of the incoming electron wave function but does modify its phase, resulting in a phasecontrast imaging. In practice, this technique requires very high quality samples with ultra thin areas. It is important to note that the recorded image with this technique is not a direct representation of the sample's crystallographic structure. Image processing is usually needed afterwards for final interpretation.

Figure 2 - 7 :

 27 Figure 2-7: Schematic diagram of a High Resolution Transmission Electron Microscope.

  Ion channelling technique, for long time, is frequently used for measuring the structural defects located on the surface layer of a crystalline material over a typical depth of few µm. Experimental results obtained by this technique are usually composed of backscattering spectra measured only major axis and planes and angular scans. Interpretation of such data is by no way straightforward and the difficulty is present while extracting quantitative information from the spectra. A simplified channelling data analysis based on the two-beam approximation can be performed and the results obtained are usually considered to be sufficiently accurate. Moreover, such that approximation allows Tien Hien NGUYEN Université Paris-SudChapter 2

  McChasy (abbreviation for Monte-Carlo CHAnnelling SYmulation) is a computer computational program designed for calculating the ion-crystal interaction in channelling condition for a variety of crystalline structure including the fluorite-type structure exhibited by U O 2 . Analysis by McChasy program is made with the following steps. In a very first step, it is mandatory to declare several parameters concerning (i) the crystal to be investigated (structure, chemical composition, thermal vibration of each component), (ii) the geometry of such RBS/C measurement (solid angle of detector, incident and backscattering angle, etc), (iii) the analysing beam (nature of analysing ion, energy, Tien Hien NGUYEN Université Paris-Sud Chapter 2

Figure 3 - 3 92 2

 332 Figure 3-1 displays a typical example of channelling spectra recorded on a U O 2 single crystal recorded in aligned and random directions. Crystals were implanted with 470 -keV Xe ions at both low fluence (Φ ≤ 5 × 10 15 cm -2 ) and high fluence (Φ ≥ 5 × 10 15 cm -2 ) ranges and alternatively characterised by RBS/C after each implantation step. The figure also displays the channelling spectrum recorded on the same sample before irradiation. Note that the random spectrum recorded consists of a front at 2887 keV (the signal due to the backscattering of He ions on U sub-lattice at the surface of crystal), a plateau below 2887 keV (the signal due to backscattering on U atoms at increasing depth) and a characteristic peak located at about 1045 keV (see figure 3-1.a). The appearance of this peak is the consequence of the elastic resonant scattering at 3038 keV of 4 He ions with 16 O atoms, 16 O( 4 He, 4 He) 16 O, which enhances the backscattering on O sublattice target. Actually, the Rutherford elastic backscattering of 4 He ions on O sub-lattice results in the backscattered signal with a front (that should appear at 1130 keV ) and a continuous increasing signal as decreasing energy of backscattered ions. However, since the Rutherford backscattering cross section is proportional to the square of atomic number of target atoms, the backscattering signal coming back from O sub-lattice (Z 2 = 8 2 = 64) is so small compared to the backscattering signal coming back from U sub-lattice (Z 2 =

Figure 3 - 1 :

 31 Figure 3-1: Typical (a and b) and high-energy part (c and d) of RBS/C spectra recorded in random (full circle) and 100 -aligned directions (empty circle) on un-implanted and implanted U O 2 crystal with 470 -keV Xe ions. Fluences are measured in unit of cm -2 .

Figure 3 - 2 :

 32 Figure 3-2: Typical example of a decomposition of channelling spectrum applying the TBA approach (a) and the corresponding distribution of defects versus depth (b).The channelling spectrum is fitted with three components: the dechannelling signal (red long-dashed), the Gaussian-shaped surface peak (pink short-dashed) and the Gaussian-shaped damage peak (blue dash-dotted).

Chapter 3 Figure 3 - 3 :

 333 Figure 3-3: Decomposition of a channelling spectrum recorded at high implanted fluence taking into account the contribution of implanted impurities (indicated on figure a) and the corresponding distribution of defects versus depth (b).

Figure 3 -Figure 3 - 4 :

 334 Figure 3-4 displays the distributions of damage fraction versus depth for various implantation fluences of Xe ions at low-fluence (figure 3-4.a), medium-fluence (figure 3-4.b) and high-fluence (figure 3-4.c) ranges. The distributions were extracted from the decomposition of signals according to the three components previously described. The distribution of Xe ions implanted to the matrix sub-lattice calculated by the SRIM code is also displayed so as to compare with the experimentally measured damage profiles. At low fluence (ranging from 0 to 1 × 10 15 cm -2 ) the radiation damage is created essentially around the implanted range of Xe in U O 2 , i.e. around Rp = 85 nm. It is also apparent that the fraction of damage increases progressively with increasing ion fluence. At the medium fluence range (from 1×10 15 to 1×10 16 cm -2 ), two important features are observed on the distribution of damage versus depth. Firstly, the radiation damage literally grows thicker. As the ion fluence increases, the damage created spreads into the deeper layer of the crystal up to 250 nm in depth, far beyond the calculated implantation profile.Secondly, the concentration of damage gradually extends deeper and saturates (at about

Figure 3 - 5 :

 35 Figure 3-5: Evolution of the fraction of randomly displaced atoms calculated at the maximum of the profile damage accumulation in the U O 2 single crystal as a function of ion fluence.

3. 2 3 Figure 3 - 6 :

 2336 Figure 3-6: Channelling spectra (a and b) and high energy part of channelling spectra (c and d) recorded for U O 2 crystals irradiated with La ions. Fluences are measured in unit of cm -2 .

Figure 3 -Figure 3 - 7 :

 337 Figure 3-10 shows a fast rising of damage induced is observed in all three cases at low fluence. In other words, all the evolutions present a build-up step that is induced around 4 to 7 dpa corresponding to a minor fraction of incorporated ions of about 0.1

Figure 3 - 8 :

 38 Figure 3-8: Channelling spectra (a and b) and high energy part of channelling spectra (c and d) recorded for U O 2 crystals irradiated with Ce ions. Fluences are measured in unit of cm -2 .

Figure 3 - 9 :Chapter 3 Figure 3 - 10 :

 393310 Figure 3-9: Distributions of the defects versus the depth extracted (using the same TBA method discussed here before) for U O 2 crystals implanted with Ce ions. Fluences are expressed in unit of cm -2 . The distribution of Ce (filled area) calculated by SRIM program is plotted in arbitrary unit.

Figure 3 -

 3 Figure 3-11: θ -2θ experimental scan recorded around the (400) Bragg reflection for 100 -oriented U O 2 crystals implanted to indicated fluences. The arrows indicate the last peak arising from the maximum strain value exhibited by the damaged layer.

Figure 3 -

 3 Figure 3-11 represents the θ -2θ experimental scans recorded around the (400)

Figure 3 -

 3 Figure 3-12: θ -2θ experimental scan around the (222) atomic planes for 111 -oriented U O 2 crystals implanted to indicated fluences. The arrows indicate the last peak arising from the maximum strain value exhibited by the damaged layer.

Figure 3 -

 3 Figure 3-12 displays the θ -2θ experimental scans around the (222) Bragg reflection for 111 -oriented crystals. Similar features are observed as compared to the same scans obtained on 100 -oriented crystals. The peak located at 2θ = 58.395 • corresponds to

Figure 3 - 13 :

 313 Figure 3-13: Evolution of the maximum total strain as a function of implantion fluence for 100 -and 111 -oriented crystals.

Figure 3 -

 3 Figure 3-13 depicts the evolution of the maximum normal strain ε max N arising in the damage part as a function of the ion fluence for both 100 and 111 orientations. The value of maximum normal strain is obtained directly from the position of the last peak located on the lowest angle on the θ -2θ scans with respect to the un-damaged peak:

4/ 3

 3 for 111 orientation; C an = C 44 -(C 11 -C 12 ) /2 with C ij being the elastic stiffnesses of the investigated material. The values used in this investigation were experimentally obtained [111]: C 11 = 396 GP a, C 12 = 121 GP a and C 44 = 64 GP a. Thus, the total normal strain experimentally measured via the (θ -2θ) scan can be expressed as the following [110, 109, 112]:

Figure 3 - 3 Figure 3 - 14 :

 33314 Figure 3-14 depicts the evolution of the maximum elastic normal strain induced by irradiation and the maximum in-plane fixation stress due to the substrate reaction arising in both the 100 -and 111 -oriented crystals as a function of the implantation dose (expressed in dpa). The evolutions of the damage fraction measured by RBS/C are also represented. As it can be seen, the maximum elastic normal strain ε def N due to defects

  -14.a) while the total normal strains experimentally measured depend on the orientations. Such value measured for 111 orientations is clearly higher than 100 orientations. However, the measured total normal strain is higher for 111 than 100 orientations because of the elastic anisotropy of U O 2 . This quantity is negative for U O 2 (C an = C 44 -(C 11 -C 12 )/2 = -73.5 GP a) meaning that the (111) planes are more compliant than the (100) planes. Therefore, they allow Tien Hien NGUYEN Université Paris-Sud Chapter 3

  -14.c displays the kinetics of maximum damage fraction measured by RBS/C versus implantation fluence where the damage fraction is in both 111 and 100

Figure 3 - 15 :

 315 Figure 3-15: Plan view image of a U O 2 single crystal irradiated with 500 -keV Ce ions, at a fluence of 5 × 10 14 cm -2 . Dislocations loops are marked with arrows. The white arrows show two linked dislocations.

Figure 3 - 16 :

 316 Figure 3-16: Plan view image of a U O 2 single crystal irradiated with 500 -keV Ce ions, at a fluence of 5 × 10 14 cm -2 ; Projectile impact, close to the surface of the specimen, and associated SAD pattern. The zone axis is of 100 type, as attested by the four-fold symmetry of the diffraction pattern.

  -17b. The path of the images along Tien Hien NGUYEN Université Paris-Sud Chapter 3 (a) General area where the high resolution sequence was taken. (b) Composite high resolution image ( 011 type zone axis) over a depth of 270 nm.

Figure 3 - 17 :

 317 Figure 3-17: Cross-section image of a U O 2 single crystal irradiated with 500 -keV Ce ions, at a fluence of 5 × 10 14 cm -2 . Figure (b) is displayed as rotating 180 • from figure (a) for easy following.

Figure 3 - 18 :

 318 Figure 3-18: Cross-section image of a U O 2 single crystal irradiated with 500 -keV Ce ions, at a fluence of 5 × 10 14 cm -2 . Enlarged view (a) on the top of the composite image (upper region marked as (1) with a rectangle in figure 3-17b) and (b) in the middle of the composite image (lower region marked as (2) with a rectangle in figure 3-17b). The black arrows in figure (b) indicate selected dislocation loops lying in the (111) planes.

Figure 3 -

 3 Figure 3-18a is an enlarged view of the top-most area in the sample (see rectangle (1) in figure 3-17b) in high resolution mode, as it was obtained. The (200) planes parallel to the sample surface, as well as some close packed (111) planes are indicated, and local disorientations are clearly visible on the (111) planes. Moreover, the vertical broader lines are reminiscent of irradiation tracks. This picture shows that whatever damage initially introduced by the ion irradiation, the sample area close to the surface is in almost fully recovered, as the broad dark contrast lines are a residual stress signature of the trajectory traces, whereas the atomic planes only show local distortions but no global or large-sized defects. This feature is totally consistent with the distribution of defects versus depth as measured by RBS/C (see figure 3-4.a) where significant fraction of defects is observed starting from ∼ 30 nm below the surface. The specimen region shown enlarged in figure 3-18b (see lower rectangle in figure 3-17b) is located roughly ∼ 120 nm away from the sample surface. As in the previous image, vertical contrasts due to irradiation tracks are still present; and the (111) type planes are clearly visible. In this region the dislocation loops, some of them being indicated by the

Figure 3 - 19 :

 319 Figure 3-19: Processed image, filtered FFT and the corresponding power spectrum of U O 2 single crystal irradiated with 500 -keV Ce ions, at a fluence of 5 × 10 14 cm -2 and associated SAD pattern. The white arrows indicate selected dislocation edges.

Figure 3 -

 3 [START_REF] Penetrante | Evolution of ion-charge-state distributions in an electron-beam ion trap[END_REF] shows an example of an FFT filtered image and the corresponding power spectrum. A number of edge-type dislocations are indicated by the arrows. A highly disordered zone is clearly observed where dislocations are present at high density. Such an image gives the density of dislocations to be approximately of few 10 12 cm -2 . A thorough analysis of one dislocation (indicated by the two black lines on this figure) shows that the length of the distorted part of the two closest atomic rows amounts to ∼ 4.8 nm and the angle measured between the atomic row connected to the dislocation edge and the closest one (between the two indicated black lines) is about ∼ 4 • .To summarize, irradiation with Xe, La and Ce ions of low energy have clearly led to the formation of radiation damage in U O 2 crystals. The results measured by RBS/C and analysed by the TBA approach reveal that the radiation effects lead to a three stages for irradiation with La and Ce ions and four stages for irradiation with Xe ion of damage evolutions. At the first stage, irradiation lead to the elastic deformation of the crystalline structure. This stage is characterised by the creation of point defects of low dimension that leads to the expansion of the damaged layer toward the free surface. At the second stage, the strain/stress build-up triggers the transformation from point defects to defects cluster. This stage is characterised by the steep increase of damage observed at 4 to 7 dpa which is due to the creation of damage via ballistic collision. This stage is also observed to be independent to the nature of implanted ions. At the third stage, the fraction of created damage saturates over a long range of irradiation (from 10 to 300 dpa for irradiation with Xe ion and from 10 dpa to the highest investigated fluence for irradiation with La and Ce ions). It is also at this stage, the difference behaviour of U O 2 crystalline structure submitted to irradiation with inert noble gas (Xe) versus soluble species (La and Ce) is observed. Irradiation with inert noble gas leads to the higher damage fraction than soluble species. This clear difference between insoluble and soluble ions was observed over a large range of irradiation but such that difference due to irradiation with soluble species (La vs. Ce) is not evident. In fact, the U O 2 crystalline structure seems to response very similarly to irradiation with La and Ce ions. The fourth stage of damage evolution is observed only for implantation with noble gas Xe ions up to very high concentration.When the incorporation of Xe ions exceeds 5 at. % the dramatic increase of damage fraction takes place. This dramatic increase localises at the close surface of the crystal on the region that corresponds to the implantation range of Xe in U O 2 . At this stage, due to the incorporation up to high concentration, the incorporated Xe ions tend to gather together in large quantity forming the bubbles that eventually result in the frustration of the crystalline structure. Consequently, the damage fraction increases dramatically as it can be seen on the evolution of damage measured by RBS/C. The channelling data were analysed in the framework of the TBA approach. This approximation implicitly assumes that all created defects can be described as randomly displaced atoms. TEM results are at clear variance with such an interpretation, since the presence of extended defects, such as dislocations, were revealed. Nevertheless, channelling data are indeed intrinsically sensible to both randomly displaced (short-range order) atoms and distortions of the lattice (medium-range order). The two components can be extracted by an advanced analysis of the channelling data provided by the use of Monte-Carlo simulation codes. In the next chapter, we discuss the development of a new model -Tien Hien NGUYEN Université Paris-Sud comprising two classes of defects -that was developed to shed some light on the evolution of the populations of defects versus ion fluence and their influence on the destabilisation of the crystalline structure under irradiation. Tien Hien NGUYEN Université Paris-Sud Chapter 4 Towards an advanced description of the evolution of the urania structure under irradiation: a two-defect class model The analysis of the structural reorganisation induced in uranium dioxide single crystals bombarded with low-energy ions used to mimic the fate of fission fragments has led to the several important conclusions. Two important steps in the process of damage formation are established: (i) at low fluence the presence of the first step scales with the number of displacements per atom and is therefore related to the radiation damage generated by atomic collision cascades; (ii) at large fluence, corresponding to a very large concentration of incorporated impurities, a huge second step is observed, specifically for crystals implanted with Xe ions, whilst, conversely, the structure of lanthanide-implanted crystals seems to be essentially insensible to the further creation of damage and incorporation of impurities up to a very large number of dpa. Those two steps are separated by almost constant plateaus, for which the structure for the crystal seems to be in a quasi-steady state characterised by a constant fraction of randomly displaced atoms. The coupling of the ion channelling method with Rutherford Backscattering Spectrometry has been extensively used in the past to quantitatively investigate the depth distribution of displaced atoms from their regular atomic positions. As a first approximation, analysis of the data was performed in the framework of the so-called two-beam approximation (TBA, see the previous chapter). This model supposes the existence of two simultaneously processes: (i) direct backscattering of ions on atoms randomly located within the channels and (ii) scattering of ions that continuously interact with matrix atoms and atoms located within the matrix and thereby leave the channel at a certain depth traversed in the crystal. The latter process occurs as a consequence of a continuous series of small-angle collisions and eventually forms the dechannelling contribution to a spectrum recorded across the main crystallographic direction. The channelling spectrum comprises therefore the signals coming from the direct backscattering superimposed over Tien Hien NGUYEN Université Paris-Sud Chapter 4 the dechannelling contribution (signal coming from forwards scattering). Let us stress that the quantitative investigation of the damage depth distribution implicitly assumes the knowledge of the nature of radiation-induced defects. The analysis performed in the previous section assumes that only randomly displaced atoms are generated during irradiation. An advanced knowledge of the mechanisms of the matrix destabilisation under irradiation requires a complete analysis of the evolution of the various created defects versus fluence. Although the channelling technique is intrinsically limited when applied to the case of a highly defective material, including the presence of various types of defects, defect in crystals can nevertheless be classified according to their influence with respect to a channelled ion. As a matter of fact, they are responsible for (i) obstruction-type dechannelling (typically an atom off its regular atomic position, a stacking fault) and (ii) distortion-type dechannelling (e.g. lattice distortion, dislocations). Many defects contribute of course to both obstruction and distortions of the lattice leading to a composite dechannelling. Although the complete evaluation of the influence of various types of defects is almost impossible, the use of Monte-Carlo simulation codes allows us to propose a simplified model of defects involving both the presence of (i) randomly displaced atoms and (ii) bent channels to account for the dechannelling contributions coming from obstruction and distortion, respectively. In the present work, such a model is used to shed some light on the mechanisms of the radiation damage production in fluorite-structured uranium dioxide. Main objectives include the determination of the damage evolution for the various types of defects leading to obstruction and distortion, as well as the understanding of the observed differences between noble-gas-irradiated and lanthanide-irradiated crystals, in relation with the nature of incorporated species.

Figure 4 - 1 :

 41 Figure 4-1: A simplified model of defects: a profile of RDA defects (blue) and a constant fraction of BC defects (red) are incorporated on the damaged layer.

Figure 4 - 2 :

 42 Figure 4-2: A schematic representation of bent channel defect characterised by the length (L) and the angle (η) of the distortion with respect to the channel direction.

Carlo 4 . 2 . 1

 421 Definition of the structural parameters Monte Carlo simulations to the channelling data recorded on U O 2 crystals were performed by means of the McChasy simulation code. Positions of both U and O atoms are derived from the crystalline structure of urania (space group F m 3m): U atoms are located at 4(a) positions (0,0,0), and O atoms are located at 8(c) positions 1 cell parameter is a = 547.0 pm. The incorporated element implanted in the lattice (Xe or lanthanide ions) is assumed to be located at octahedral positions of the fluorite-type structure with an occupancy corresponding to the fraction of the foreign specie, which depends on the implantation profile and thus on depth z. The influence of the implanted element is almost negligible at low fluence, i.e. below Φ = 1×10 16 cm -2 (the concentration at maximum is about 1.4 % according to SRIM calculation), but it has an increasing Tien Hien NGUYEN Université Paris-Sud significant contribution at larger concentration (see chapter 3). The distribution profile of implanted species (characterised by Rp and ∆Rp) is calculated by the careful analysis of the recorded experimental random RBS spectra for crystals implanted at very large fluence, whose shape in the surface region of the U signal allows the determination of the range and range straggling (see the specific shape of the random spectra on figure 3-1.d, 3-6.d and 3-8.d). Such parameters are in good agreement with values calculated by means of the SRIM simulation code (Rp = 83 nm and ∆Rp = 39 nm). These parameters are used to calculate the distribution profile at low impurity concentration, i.e. below Φ = 1 × 10 16 cm -2 , since RBS is not sensible enough to give an accurate estimation in these experimental conditions. Root-mean-square amplitudes of U and O atoms are measured on pristine crystals before implantation with ions. Axial channelling spectra and angular scans are recorded and they are compared to Monte Carlo simulations performed assuming a defect-free U O 2 single crystal with various values of thermal vibrations. Both values of thermal vibrations measured on U and O sub-lattice vary slightly with the crystalline quality of a given specimen and they are all in good agreement with values previously recorded on virgin

4 Figure 4 - 3 :

 443 Figure 4-3: Fits to channelling spectra (b and d) assuming that RDA distributions are the sole type of defects (a and c). The red solid lines denotes the distribution of RDA (in figure (a), indicated in the legend as TBA) (see chapter 3) and the corresponding spectrum obtained from simulation (in figure (b)) with the profile of defects derived from the TBA. Long-dashed green lines denote distribution of RDA (in figure (a) and (c), indicated in the legend as RDA) that allows reproducing the damage peak and the corresponding spectrum (figure (b) and (d)). Short-dashed blue lines represent the distribution of RDA (indicated in the legend as extended RDA) that allows fitting the entire spectrum over the whole energy range of backscattered ions.

  -3.b, the new profile of RDA allows an excellent fit to the channelling spectrum up to very larger depth (up to typically over 2 µm). However, a much broader depth-distribution of RDA is necessary to account for the observed calculated axial spectrum. A non-zero fraction of RDA is needed up to a depth far larger than the implanted profile of Xe ions (represented by filled area on figure 4-3.c). As observed on the figure 4-3.a, a significant fraction of RDA is created up to 1 µm meanwhile the maximum depth of Xe-implanted in U O 2 calculated by SRIM is only 200 nm (see the filled area). Such a result is physically unacceptable. The discussed results reveal that channelling data analysed by simulation with the sole contribution of RDA fails. The resulting spectrum shows either an underestimated dechannelling signal or an unphysical distribution of RDA versus depth. Other classes of defect definitely need to be taken into consideration.

Figure 4 - 4 .

 44 a displays the channelling spectrum recorded for the Xe-implanted crystal at a fluence Φ = 2×10 15 cm -2 (the same fluence that has been analysed by RDA) and the fitted spectra for comparison. At first, the experimental spectrum was fitted assuming a distribution of BC identical to the profile of radiation damage extracted by the TBA method. As a second possibility, a very sharp distribution of BC is constructed based on the actual implanted profile of Xe ions in the damage area (a maximum BC fraction of 95 % located at the depth of 80 nm, i.e. around the Xe implanted range) is employed for simulation. Finally, a constant distribution of BC is also tested for the sake of simplicity. All simulated spectra and the corresponding profiles of BC are shown on figure 4-4.a and figure 4-4.b, respectively.

Figure 4 - 4 :

 44  Fit to channelling spectrum assuming the sole presence of bent channels as radiationinduced defects (L = 5 nm and η = 1.5 • ). The red solid line denotes the simulated spectrum and the corresponding distribution of BC assuming both the depth distribution and the concentration derived from the TBA. Long-dashed green line denotes the simulated spectrum and the corresponding sharp distribution of BC (up to 95 % of BC). Short-dashed blue line represents the calculated spectrum assuming a constant distribution of BC from the surface up to a given depth. Dotted line help showing that the sharp distribution of BC is centred at the ion's implanted range.

2 . 3 ,

 23 the use of BC improves not only the dechannelling yield but also gives an extra contribution to the backscattering yield on the damage region, i.e. the damage peak. Experimental channelling spectra are fitted by MC simulation until the combination of RDA and BC produces the best agreement between experimental and simulations. Both depth distributions of RDA and BC are then extracted and they provide quantitative measurements of the defects distribution measured by the ion channelling technique. As indicated in the description of the model of defects, BC is characterised by two Tien Hien NGUYEN Université Paris-Sud

Figure 4 - 5 :

 45 Figure 4-5: Examples showing the influence of (a) the length L and (b) the angle η of distorted channels to the simulated channelling spectra. The distribution of RDA and the constant distribution of BC are fixed according to the best fit which is obtained for L = 5 nm and η = 1.5 • .

Figure 4 -

 4 6 displays the best fits to experimental channelling spectra obtained from simulation for Xe implanted crystal at various ion fluences. It is worth noticing that experimental RBS/C spectra are remarkably fitted and that a good agreement between experimental data and simulation is obtained over a large energy window. Only at large depth, i.e. at depths largely exceeding the damaged zone z > 1 µm, a discrepancy is observed for some fluences. Such feature is likely due to the multiple scattering processes (that were not taken into account in the computational code) and to the inaccuracy of the stopping cross sections at large depth. They do not limit the present Monte Carlo investigations. The agreement between experimental axial spectra and the defect model involving both RDA and BC is really excellent in the energy window from 2000 to 2887 keV (corresponding to a depth from 0 to 1 µm), as it can be seen from figure4-6.a and figure 4-6.b. Figure 4-6 also reveals that the damage peaks, observed on the channelling spectra at the indicated ion fluence, have been nicely reproduced by the combination of RDA and BC. Both the increase and the broadening into the larger depth of the damage peak as increasing ion fluence have been totally accounted for by simulation. Even at very high implanted fluence (larger than Φ = 1 × 10 16 cm -2 ) the dramatic increasing of the signal observed on the damage peak (close to the surface of the crystal) has been fitted satisfactorily. The complete model of defects (RDA + BC) has, therefore, succeeded in reproducing the channelling spectra Tien Hien NGUYEN Université Paris-Sud recorded in crystal implanted with low-energy ions.

Figure 4 - 6 : 4 Figure 4 - 7 :

 46447 Figure 4-6: Best MC simulations to RBS/C spectra recorded for Xe-implanted crystal in random condition (dashed line) and in axial aligned condition for virgin (dotted) and U O 2 implanted crystal (solid line). Experimental channelling spectra are plotted in symbols and the fluences are expressed in unit of cm -2 . (c) and (d) are zooms in the surface region of figure (a) and (b). Fits were performed assuming L = 5 nm and η = 1.5 • .

Figure 4 - 8 :

 48 Figure 4-8: Best MC simulations to RBS/C spectra recorded for Ce-implanted U O 2 crystal at low (a) and high (b) fluences. The zooms on high energy part are shown on figure (c) and(d) respectively. Dashed lines and dotted lines denote the best fits to random and axial virgin spectra. Solid lines denote best MC simulations to axial aligned spectra recorded for irradiated crystal. Experimental channelling spectra are plotted in symbols for fluences of Φ = 2 × 10 14 cm -2 (red circle), 7×10 14 cm -2 (green triangle up), 8×10 14 cm -2 (yellow triangle down), 1×10 15 cm -2 (violet square), 2.5 × 10 15 cm -2 (pink diamond), 5 × 10 15 cm -2 (cyan hexagon), 1 × 10 16 cm -2 (gray star), 3 × 10 16 cm -2 (red crossed-circle) and 4 × 10 16 cm -2 (green crossed-square). Fits were performed assuming L = 5 nm and η = 1.5 • .

Figure 4 -

 4 7 and figure 4-8 display best Monte-Carlo simulations to the experimental backscattering data obtained along the main crystallographic direction, recorded on both La-and Ce-implanted single crystals, respectively. As it can be seen from the figures, Tien Hien NGUYEN Université Paris-Sud Chapter 4

Figure 4 - 9 :Figure 4 - 11 : 4 (

 494114 Figure 4-9: Distributions of RDA (a) and BC (b) versus depth measured for U O 2 crystal implanted with Xe, La and Ce at low fluence (Φ ≤ 1 × 10 15 cm -2 ) extracted from MC simulations. BC is parameterized by L = 5 nm and η = 1.5 • . Fluences are measured in the unit of cm -2 . Xe, La and Ce implantation profiles calculated by SRIM (represented by filled area) are plotted in arbitrary unit.

Figure 4 - 12 :

 412 Figure 4-12: Distributions of RDA versus depth measured for Xe-implanted U O 2 crystal at very high fluence (corresponding to over 5 at. % of implanted Xe in the matrix) derived from simulations. BC is parameterized by L = 5 nm and η = 1.5 • . Fluences are measured in the unit of cm -2 . Xe implantation profile calculated by SRIM (represented by filled area) is plotted in arbitrary unit.

4. 3 . 4 4 Figure 4 - 13 :

 344413 Figure 4-13: Experimental angular scans (symbols) recorded across the main 100 axis for U O 2 single crystal implanted with La ions at Φ = 2 × 10 16 cm -2 . The yields are normalized to random yield. The angular scans are integrated from 40 to 130 nm (red circle) and from 415 to 615 nm (blue star). Solid lines are the corresponding Monte-Carlo simulations of angular scans. All the dips are reproduced by the two-class model of defects where BC was characterised by the length and angle of L = 5 nm and η = 1.5 • , respectively.

4 . 3 . 5

 435 damage peak region. Angular scans recorded at lower fluence show a similar discrepancy but the difference is less. The limitation when fitting the angular scans might be necessary the use of other classes of defects along with the RDA and BC. These differences clearly point out the limitation of such an approach, where the two-class of defect model is supposed to reproduce all types of defects, from the low fluence range up to the highest one. In summary, the two-class model of defects have been developed to analyse chan-Tien Hien NGUYEN Université Paris-Sud nelling data recorded on low-energy ion irradiated crystals. Such a model has been successfully applied with Monte-Carlo simulation to reproduce a large number of channelling spectra and angular scans recorded at various ion fluence for all crystals implanted with the three different ions. In this work, the length and angle of distortion were estimated based on analysis of HRTEM image (see figure 3-19) to be approximately L = 5 nm and η = 1.5 • . The values used in this work have allowed a great success in analysing channelling data recorded for Xe-, La-and Ce-implanted crystals. Nevertheless, slight discrepancies still exist during analysis which would suggest a more complete model where other classes of defects (such as the polygonisation at high fluence) should be taken into consideration. Estimation of uncertainties in the fraction of RDA and

Figure 4 -

 4 14 displays the evolution of the maximum fraction of RDA (i.e. recorded at the absolute maximum of the distribution at each ion fluence) and for the three studying cases as the function of the number of dpa and as the function of concentration of implanted species. All the evolutions of RDA induced in crystals implanted with Xe, La and Ce are plotted on the same figure in order to compare the radiation effects of ions of different nature and to compare the effects due to the incorporation of soluble versus insoluble species.

Figure 4 - 14 :

 414 Figure 4-14: Evolutions of RDA extracted from channelling data by Monte-Carlo simulation for U O 2 crystals implanted with Xe (red circle), La (yellow square) and Ce (blue star) ions, respectively.

Figure 4 - 15 :

 415  Fits to the evolution of RDA extracted from channelling spectra by MSDA model for U O 2 crystals irradiated with Xe (red circle), La (blue star) and Ce (yellow square). Solid, dashed and dash-dotted lines represent respectively the fits to the evolution of RDA.

Figure 4 -

 4 Figure 4-15 depicts the fits to the evolutions of RDA performed in the framework of the MSDA model for all the three cases. The corresponding parameters that give the best fits are shown on table 4.1. For the crystal implanted with Xe ions, the evolution of the RDA in is fitted nicely by the MSDA model with three damage accumulation steps;the same feature that is observed on the evolution of radiation damage extracted by the TBA method. The first accumulation step is observed at low irradiated fluence. As observed on table 4.1, this accumulation step saturates at 6 % of RDA. The second damage accumulation step starts when the irradiation reaches 4.9 dpa. This step is characterized by the cross section of σ 1 = (16 ± 11) nm 2 and by a fraction of RDA observed at the saturation of 25 %. The evolution of RDA passes to the final damage accumulation step when irradiation reaches 300 dpa (Φ = 4 × 10 16 cm -2 ). This step saturates at a high

Figure 4 - 16 :

 416 Figure 4-16: Evolutions of BC extracted from channelling data by Monte-Carlo simulation for U O 2 crystals implanted with Xe (red circle), La (yellow square) and Ce (blue star). Fits were performed assuming L = 5 nm and η = 1.5 • .

Figure 4 - 17 :

 417 Figure 4-17: A comparison of the evolution of RDA and BC versus dpa and concentration. For the sake of clarity, the error bars are not shown on this figure.

4 Frenkel

 4 As an attempt to analyse the channelling data with a new model of defect (in accordance with the obstruction and the distortion phenomena to the analysing beam), the channelling Tien Hien NGUYEN Université Paris-Sud Chapter 4 data recorded for low-energy implanted U O 2 crystals have been analysed by Monte-Carlo simulations with the McChasy code using the two-class model of defects: (i) the randomly displaced atoms and (ii) the bent channels. The application of MC simulation into the channelling data analysis with the RDA and BC allows the user to take into account the presence of various classes of defects such as point defects, defect clusters, interstitials, stacking-faults, dislocations, dislocation loops, the presence of bubbles or precipitates and even the polygonisation. These defects give both contributions to the obstruction and distortion of the analysing beam which is eventually seen via the damage peak and the dechannelling signal. Such an obstructrion-type defects are most probably point defects, defect clusters, interstitials, stacking-faults, the presence of bubbles or precipitates and polygonisation while the distortion-type defects are dislocations, dislocation loops. As indicated in beginning of section 4.1, the presence of defects of any type in the matrix leads to the backscattering of analysing ion beam via direct backscattering and small angle scattering. Eventually, channelling spectra recorded is the consequence of the backscattering of probing ion that probes the defects as two main type: obstructiontype and distortion-type. Monte-Carlo simulation applied in this work enlightens a new method to analyse hence to extract more than one class of defects from channelling data. As an example, the RDA and BC have been extracted form channelling spectra recorded for crystals implanted with insoluble Xe and soluble La and Ce ions. Let us start the discussion first with the model of defects itself. As used to analyse channelling data, the model of defects need to be consistent with the main features observed on channelling spectra. That is to say, since the damage peak and the dechannelling signal present as the two most important contributions of the channelling spectra (they reflect respectively the direct/indirect backscattering of analysing ion upon defected and matrix atoms), the choice of the model of defect containing the obstruction-type and the distortion-type defects is a good start that can allow the reproduction of both damage peak and dechannelling background. As there as many classes of defects are created during the irradiation process, it is possible (in principle) to assume as many classes of defect as we wanted in the simulation code but the eventual influence seen on channelling spectra is restricted to the obstruction and distortion influences. In addition, the use of many classes of defect is very complicated to interpret. A Two-class model of defects is therefore relevant and sufficient to enlighten a new approach. The use of a model of defects containing the RDA and BC has clearly proved that it is totally possible to analyse channelling data by Monte-Carlo simulation to extract, afterwards, information regarding the radiation damage in the crystal including a profile of RDA (which reflects the local concentration of RDA at given depth) and a constant fraction of BC (that represents the modification of the crystalline structure). Channelling spectra have been fitted satisfactorily up to the large depth using this model of defects (see figure 4-6, figure 4-7 and figure 4-8). Despite some discrepancy observed just after the damage peak for few fluence (see figure 4-6.c, figure 4-7.c and figure 4-8.c) where the underestimation of channelling spectra is observed around 2700 keV ), the model of defect used in this work still succeeds in fitting most of the characteristic features of the channelling spectra and angular scans. This discrepancy is probably a limitation of the model of defects applied in this investigation; other classes of defects might be needed. Itis also possible that since the BC is supposed to be constant over the whole damage region, the backscattering event drops down unexpectedly after the damage peak as the fraction of defect (more precisely the BC) goes suddenly down to 0. Likewise, the disagreement between simulation and experimental spectra observed at large depth (corresponding to the energy below 1000 keV ) is due to the multiple scattering processes that were not taken into account in the computation program. However, these discrepancies do not limit any success of the channelling data analysis by Monte-Carlo simulation.In the framework of this investigation, channelling data have been successfully analysed by Monte-Carlo simulations from which the distributions of RDA and BC versus depth and the consequential the kinetics of damage versus fluence are well extracted from channelling data. The results obtained can be summarised as follows.(i) At low fluence range, a similar-to Gaussian-shaped distribution of RDA is created at the region that is similar to the ion implantation range. Regarding the creation of BC, the fraction of BC is created over the same area with respect to the RDA (as directly assumed by the model) and it is present with a low level. The two features demonstrate that the defects created are clearly due to the radiation effects via elastic nuclear interaction since the fraction of incorporated projectile is very small and hence their incorporation has a negligible effect on the process of damage formation in the crystal. Moreover, the similar depth observed for the distribution of defects and distribution of implanted ions (which is confirmed by TEM analysis, see section 3.4) indicates that the defects are created by the irradiation process is a clear fact. It is likely that irradiation at low fluence mainly creates point defects or more precisely Frenkel pair defect (pair of interstitial and vacancy) leading to the destabilization of the crystalline structure. Such a destabilisation has been observed via the buildup of strain/stress arising in the damaged layer (see section 3.3). The presence of RDA in interstitial position together with the presence of vacancies affects the regular arrangement of atoms in a row (modification of channels) that is eventually seen as the RDA and BC. It is worth stressing that the fraction of defect created at a given depth is essentially proportional to the concentration of implanted ions at that depth (similar position of the depth-distributions). It is also important to remind that due to the efficient reorganization of the crystalline structure (the annihilation of Tien Hien NGUYEN Université Paris-Sud Chapter pair defect), we observed the residual effects as after annihilation atoms come back to regular lattice location and the modification is only affected by the creation RDA and vacancies. Over this fluence range, irradiation is believed to created at first a small fraction of point defects. Afterwards, the accumulation of point defects transforms into the form of extended defects and/or defect clusters. The creation of point defects leads to the elastic deformation of the crystal and to the weak modification of the crystalline structure. This phenomenon is indirectly viewed via the presence of elastic strain arising in the damaged layer and via the very low level of BC (see figures 3-14.a and 4-9.b). Indeed, XRD measurement performed on Ce-implanted crystal over this fluence range reveals the presence of tensile elastic strain in the damaged layer. Such measurement is only possible

figure 3 -

 3 figure[3][START_REF] Kurepin | Thermodynamic properties of hyperstoichiometric urania in the UO 2 -UO 2.25 solid solution range[END_REF][START_REF] Sheindlin | Experimental determination of the thermal conductivity of liquid UO 2 near the melting point[END_REF][START_REF] Banerjee | Specific heats of thoria urania solid solutions[END_REF][START_REF] Leclercq | Thermal conductivity of zirconia-based ceramics for thermal barrier coating[END_REF][START_REF] Degueldre | Specific heat capacity and Debye temperature of zirconia and its solid solution[END_REF][START_REF] Azom | Zirconia -ZrO 2 , Zirconium Dioxide[END_REF][START_REF] Slack | Thermal conductivity of MgO, Al 2 O 3 , MgAl 2 O 4 , and Fe 3 O 4 crystals from 3 to 300 K[END_REF][START_REF] Devanathan | Structure and mechanical properties of irradiated magnesium aluminate spinel[END_REF][START_REF] Klemme | Low-temperature heat capacities of MgAl 2 O 4 and spinels of the MgCr 2 O 4 -MgAl 2 O 4 solid solution[END_REF][START_REF] St | Thermophysical properties of sapphire, AlN and MgAl 2 O 4 down to 70 K[END_REF][START_REF] Scott | Phase relationships in the zirconia-yttria system[END_REF][START_REF] Murphy | Nonstoichiometry in MgAl 2 O 4 spinel[END_REF][START_REF] Sickafus | Structure of spinel[END_REF][START_REF] Rouse | Space group of the spinel structure: A neutron diffraction study of MgAl 2 O 4[END_REF][START_REF] Schmidtmeier | Magnesium aluminate spinel raw materials for high performance refractories for steel ladles[END_REF][START_REF] Penetrante | Evolution of ion-charge-state distributions in an electron-beam ion trap[END_REF], where the presence of dislocation clearly leads to the modification of several atomic arrangements in the vicinity), it leads to the modification of atomic channels that eventually results in higher dechannelling signal of the channelling spectra. Such phenomenon is, to a large extent, similar to the presence of BC as the second class of defects or, conversely, the fraction of BC reflects the concentration of dislocations and/or network of tangled dislocations induced by irradiation. It reflects, therefore, a modification of the crystalline structure submitted to irradiation. This modification could be the gradual modification of the defective crystalline structure, which finishes by the formation of low angle boundary domain. A further step due to the radiation effects is possibly the sub-grains division process. Over this fluence range, the presence of BC at low concentration (see figure3-9.b) reveals either that the modification of the crystalline structure is so weak that it simply leads to the elongation of distance between periodic atoms or that the BC measured over this fluence range is just a secondary effects of point defects that results in the small angle scattering of channelled ions. Both two assumptions were confirmed by the presence of elastic tensile strain measured by XRD analysis which

  figure 4-10.a). This phenomenon seems to be due to the radiation-enhanced diffusion in the crystal. A high concentration of non-equilibrium point defects can readily annihilate within the irradiated region (saturation of defects) or diffuses out of the irradiated region and annihilates at sinks or form stable extended defect structures (expansion of damage zone). Only the inward fluxes of point defects were expected to contribute to the growth of defect structures beyond the irradiated region. Two factors could cause the inward flux including (a) the concentration of the generated point defects and (b) the strain/stress field induced by implanted ions and stabilized defects. Analysis by XRD technique presented in section 3.3 has confirmed the presence of the strain/stress in the damaged layer. This strain /stress field may provide a significant driving force for inward diffusion of point

  (i.e. the saturation of RDA and the regular increase of BC) are observed to occur at the same irradiation condition (∼ 7 dpa). It is a strong evidence confirming that when the ion fluence increases, the inward accumulation of point defects transforms into the extended defects which is finally observed via the increase of BC.The extension of the damage region and the increase of BC (observed besides the saturation of RDA) can be considered as the second response of the U O 2 crystalline structure submitted to irradiation over the medium fluence range. In fact, at low fluence we observe the strain/stress build-up arising in the damage layer as a consequence of the creation of point defect. It is also at low fluence range we observed the transformation from point defects to extended defects (most probably defects clusters). After this first step, irradiation seems to not affect the U O 2 crystalline structure but evolution of BC defect shows that U O 2 crystalline structure undergoes dramatic transformation that could weaken its stability. Such that transformation from RDA to BC can be considered as the second response of U O 2 crystalline structure under irradiation.As indicated in section 4.3.6, the fraction of RDA observed at saturation plateaus (i.e. over the medium fluence range) due to insoluble versus soluble species is clearly different. Insoluble Xe ion results in higher level whilst soluble La and Ce ions both result in the same lower saturation plateaus. The difference of plateaus of saturation can be mostly explained by the size and the solubility of implanted species. Being an inert noble gas, the size of atomic Xe atoms is much higher as compared to La and Ce atoms in the matrix. Inert Xe does not form any compound with U O 2 while La and Ce (that Tien Hien NGUYEN Université Paris-Sud 4.4. General discussion 123 are totally soluble in U O 2 ) form compound with U O 2 at the possible charge state of +3

  , Monte-Carlo simulations with McChasy program performed along the main 100 type axis assuming that implanted species are located in substitutional and octahedral positions in U O 2 matrix reveal no difference. Therefore, the position of implanted species does not seem to explain the difference of the fraction of RDA observed on the satuation plateaus. As Xe ions form bubbles in U O 2 matrix, they create a much stronger stress field in U O 2 crystalline structure. On the contrary, La and Ce stay most probably as individual impurities in substitutional or octahedral position (since there are 4 octahedral positions are available in a U O 2 basic cell structure), they create a much weaker stress field in U O 2 . The fact that U O 2 can accommodate La and Ce in substitutional and empty octahedral positions to a large concentration somehow eliminates the formation of precipitates and hence diminishes to the large extent the stress field arising in the crystal. Such a situation is not the case in Xe-implanted crystal where Xe does not form any compound with U O 2 and thus sticks together in the form of nano-size bubbles. The presence of nano-size bubbles affects the process of damage creation twofold. They limit on one hand the annihilation of point defect and vacancy during a collision cascade and create on the other hand a very strong stress field that leads to the creation of supplement defects in the crystal. Taking into account the size and the formation of Xe bubbles in the crystal as compared to La and Ce ions, the difference in saturation plateaus can be explained by the solubility of implanted species. In practice, it would have been nice to be able to locate Xe, La and Ce in U O 2 to verify this assumption but it is impossible to perform Tien Hien NGUYEN Université Paris-Sud Chapter 4 by channelling technique, since U atoms are too heavy compared to Xe, La and Ce. (iii) In the high fluence range, irradiation with Xe leads to the dramatically creation of RDA as Xe ions are incorporated at high concentration (over 5 at. %).It is seemingly that starting from the fluence of Φ = 4 × 10 16 cm -2 , the incorporation of Xe at high concentration leads likely to the formation of Xe bubbles of larger size. The impact of Xe bubbles becomes so strong that they lead to the frustration of the U O 2 crystalline structure, known as the polygonisation of U O 2 single crystal, which is the origin of the dramatically increase of RDA at the surface. It is worth recalling that the dramatically increase of RDA is solely observed on this part (from the surface to 120 nm correspond to Rp + ∆Rp of Xe implantation profile) where the large fraction of Xe is present. While on the deeper part, which is located beyond the maximum implantation range of Xe, the implanted Xe ion is always present at low concentration. Therefore, the radiation damage on the deeper part (deeper than the depth z > Rp + ∆Rp) remains essentially unchanged; no dramatically increase of RDA is observed. It is undoubtedly that this frustration occurs on the area of the highest concentration of Xe and evolves towards the free surface not to the constrained part (which is attached to the thick underlying substrate). The dramatic damage creation of RDA observed at the close surface is obviously the consequence of the incorporation of a non soluble atom of larger size in the crystal at very high concentration.

  uble versus insoluble elements at normal temperature condition. Due to their importance in the nuclear fuel -since their abundance as fission products is high -Xe, La and Ce with similar masses were chosen. The experiments were designed to apply the ion implantation technique to induce damage in the crystalline structure of U O 2 . The creation of defects was monitored principally with the ion channelling technique. Channelling data recorded were first analysed by the Two-Beam approximation which can gives a reasonably accurate information regarding the damage creation. Monte-Carlo simulation with McChasy computer program that allows the simulation of the channelling phenomenon of ion in a crystalline structured material was applied afterwards for a thorough analysis of the experimental data.With the Monte-Carlo simulation, channelling data were analysed in much more details with a model of defects comprising the Randomly Displaced Atom (RDA) and the Bent Channels (BC). The RDA and BC are supposed to model, in a first approach, many classes of actual defects including point defects, extended defects such as defects clusters or dislocations and even the polygonisation of the solid. The results obtained can be summarised as below:

4 - 5 4 - 6 4 - 7 4 - 8 4 - 9

 4546474849 figure as an interesting reseach topic.First of all, a characterisation of the defect by ion channelling technique at various energies for the crystals irradiated at selected fluences should be performed. Angular scans will be very useful for a more detailed investigation. According to the classified ion fluence ranges, the angular scan can be performed (i) at low fluence range for instance at Φ = 5 × 10 14 cm -2 ) where the elastic deformation of the crystal is observed and at and Φ = 7 × 10 14 cm -2 ) where the steep increase of RDA is observed, (ii) at medium fluence range where the saturation of RDA and the regular increase of BC are visible (for instance at Φ = 2 × 10 15 and Φ = 5 × 10 15 cm -2 ) and (iii) at high fluence range, say Φ = 2 × 10 16 and Φ = 5 × 10 16 cm -2 , where the dramatical increase of RDA in the Xe-implanted crystal takes place. Such that angular scans by ion channelling technique at various energies (at indicated fluences) will allow a more detailed access to the radiation damage, particularly to the true nature of RDA and BC assuming a complete simulation of the data by Monte-Carlo.Secondly, a better description of radiation-induced defects created by ion irradiation should be performed. The two-type defect model developed in this thesis is only a first step towards this description. The description of a realistic model for dislocation, involving dislocation core and distorted row is currently being developed at Warsaw. In a similar spirit, the description of a heterogeneous system, precipitates or gas bubbles embedded into a crystalline structure, would be a good model system to obtain a better description of the evolution of the irradiated crystal at high fluence.Last but not least, in a real situation the nuclear fuel is bombarded by fission fragments at operating temperature. Therefore, the behaviour of the fuel is different due

Table 1 . 1 :

 11 Some selected physical properties.

Table 1 . 2 :

 12 Stopping power in U O 2 calculated by SRIM.

	100	25.5	21.2	21.2
	10	6.3	5.7	5.7
	1	4.2	4.4	4.3
	0.5	4.3	4.5	4.4

Ion's energy (M eV ) Xe (keV /nm) Ce (keV /nm) La (keV /nm)

  Electronic stopping power of Cerium in U O 2 Nuclear stopping power of Cerium in U O 2 The stopping power of Cerium in U O 2

												Chapter 1
		40
		35
	Stopping power (keV /nm)	10 15 20 25 30
		5
		10 -2 10 -1 0	10 0	10 1	10 2	10 3	10 4	10 5	10 6	10 7	10 8	10 9
				Energy of incident ion (keV )
	Tien Hien NGUYEN								Université Paris-Sud

  Cs-ion implantation to 1 × 10 17 cm -2 at room temperature. Amorphization of Y SZ was caused by the large size incompatibility and low mobility of Cs ions in the Y SZ structure, which reflects a relatively low solubility of Cs in Y SZ. Nevertheless, the Cs concentration at which amorphization of Y SZ occurred is about 8 at. %, well above the

	typical value that could be reached in an inert fuel matrix.
	Afanasyev-Charkin [44] investigated the effects of 340 keV Xe 2+ irradiations on
	Y SZ with fluences ranging from 1×10 11 to 1.5×10 17 cm -2 . RBS/C data and TEM images
	revealed no amorphization of Y SZ after irradiation even to the highest investigated ion
	fluence. An analytical model was developed, using ion range and damage distributions
	calculated based on Monte Carlo simulations for ion-solid interactions, to assess sputtering
	effects at high fluence. Analysed results and experimental data revealed that at high
	fluences, the implanted ion and damage distribution profiles were significantly modified
	by sputtering.
	Yasuda investigated the defect clusters induced in F SZ irradiated with ions and/or
	electrons [45]. Micro-structure evolution of Y SZ, ZrO 2 -13 mol % Y 2 O 3 , was investigated
	through transmission electron microscopy. Anomalous formation of large defect clusters
	was found under electron irradiation subsequent to ion irradiation, such as 300 keV O + ,
	100 keV He + and 4 keV Ar + ions. Such defect clusters were not formed solely with
	ion irradiation. The defect clusters were transformed into dislocation network when they
	reached a critical diameter of about 1.0 -1.5 µm, and processes of nucleation, growth
	and transformation were repeated under electron irradiation. The defect clusters were
	assumed to be oxygen platelets induced through selective displacements of oxygen ions in
	Y SZ with electron irradiation.
	In 2005, Hojo and his co-workers studied the effects of irradiation on Y SZ and the
	influence of temperature [46]. In situ electron microscope was used to observe the defects
	induced in Y SZ single crystal irradiated by helium and xenon ion. Damage evolution
	was observed during irradiation with 35 -keV He + with a flux of 5 × 10 13 cm -2 .s -1 at
	15 I + /cm 2 ), 1073, 1273 and 1473 K and during the irradiation with 60 -keV Xe 2+ with a flux of
	RBS/C measurements revealed a high degree of lattice disorder. Specifically, the dechan-5×10 12 cm -2 .s -1 at 1073 and 1473 K. The formation of Xe bubbles was clearly observed.
	nelling parameter χ min varied from 80 % to greater than 90 % over the depth probed Dislocation loops were formed by Xe ion irradiation at both temperatures but were not
	by RBS/C (∼ 1 µm). Nano-indentation measurements on the same sample indicated a formed by He. The results had revealed the clear dependence of loop formation on the
	decrease in elastic modulus, E, and hardness, H (both by about 9 %). These results ion species.
	suggest that an alteration in structure beyond simple defect accumulation occurs under
	these irradiation conditions. However, transmission electron microscopy observations and
	in particular micro-diffraction measurements failed to reveal any structural transforma-
	tions in the irradiated material.
	Wang et al. [43], in 2000, observed the amorphization of Y SZ submitted to the
	400-keV Tien Hien NGUYEN	Université Paris-Sud

  Kr + ions. The Al + and F e + pre-irradiated specimens contained a high density (10 21 -10 23 m -3 ) of interstitial dislocation loops with diameters between 5 and 30 nm prior to the swift heavy ion irradiation. Near-continuous latent ion tracks were observed in all of the specimens irradiated with swift heavy ions. The swift heavy ions also appeared to effinm for the 430 M eV Kr and 614 M eV Xe ion irradiations, respectively. Spinel of M gAl 2 O 4 irradiated with Cu -

	Chapter 1
	ciently destroy pre-existing dislocation loops with diameters 5 nm, whereas larger loops
	remained intact following the swift heavy ion irradiation. The swift heavy ions caused
	structural disordering of the octahedral cautions, but did not appear to produce amor-
	phous cores in the ion tracks. The disordered ion track diameters were ∼ 2.0 and ∼ 2.6
	The micro-structure of polycrystalline stoichiometric M gAl 2 O 4 was examined by
	cross-sectional electron microscopy following 430 M eV Kr + or 614 M eV Xe + ion irra-
	diation near room temperature up to a fluence of 1.1 × 10 16 ions/m 2 [55]. In addition,
	the micro-structure was examined for two specimens which had been pre-irradiated with
	either 2 M eV Al + ions or 3.6 M eV F e + ions and subsequently irradiated with 430 M eV

Table 2 . 1 :

 21 Summary of physical properties and relevant characteristics of implantations performed in U O 2 single crystals. Range and range straggling are calculated by SRIM program assuming the displacement thresholds for uranium and oxygen sub-lattices are E d (U ) = 40 eV and E d (0) = 20 eV , respectively [81]. The stopping power is calculated at indicated initial energy of ions The maximum dpa and concentration are calculated for highest expected fluence of Φ = 8 × 10 16 cm -2

  light element is much smaller compared to heavy element. The signal recorded on light element which overlay on the signal due to heavy element is quite low and unable to be separated from background signal. Light element is seemingly undetectable in compound with heavy element.A series of in-situ experiments were conducted in which a 100 -oriented U O 2 crystals were alternately irradiated and characterized by RBS technique in channelling geometry (see next section) using a 3, 085M eV He 2+ ion beam. Helium was chosen because it is a light element which turns out to be less deleterious as compared to heavier element and because the backscattering of light element is more efficient than heavy element. The energy was chosen to detect uranium cations and to benefit the resonance of interaction He + 16 O → 20 N e * → 16 O + He for enhancing the detection on oxygen sub-lattice[START_REF] Blanpain | The use of the 3.05 MeV oxygen resonance for 4 He backscattering near-surface analysis of oxygen-containing high Z compounds[END_REF][START_REF] Thomé | On the use of the 16 O( 4 He, 4 He) 16 O resonance for the evaluation of radiation damage in oxides[END_REF].

		Chapter 2
	Tien Hien NGUYEN	Université Paris-Sud

  5 dpa) and saturates at similar level (19.9 % and 19.6 % respectively). It is evident that irradiations by soluble La and Ce ions lead to similar consequence while irradiation by soluble Xe ions leads to distinct effects compared to La and Ce ions: irradiations by soluble La and Ce ions do not lead to the third damage accumulation step.

		Fitted	Xe-implanted La-implanted Ce-implanted
		parameters	crystal	crystal	crystal
		f sat d,0 (%)	6 ± 27	5 ± 17	5 ± 2
	Step 0	Φ 0 (dpa)	0	0	0
		σ 0 (nm 2 )	2 ± 16	2 ± 11	4 ± 7
		f sat d,1 (%)	25 ± 1	19.9 ± 0.6	19.6 ± 0.6
	Step 1	Φ 1 (dpa)	4.9 ± 0.3	5.0 ± 0.1	5.0 ± 0.1
		σ 1 (nm 2 )	16 ± 11	17 ± 5	10 ± 3
		f sat d,2 (%)	88 ± 6	-	-
	Step 2	Φ 2 (dpa)	299 ± 1	-	-
		σ 2 (nm 2 )	8 ± 2	-	-

Table 4 . 1 :

 41 Fits to the evolution of RDA by the MSDA model.

This is a disordered structure in which the O 2-anion sub-lattice maintains a "pseudo"-cubic closepacked arrangement (as in spinel), while the cations occupy randomly the octahedral lattice interstices and all tetrahedral interstices are empty[START_REF] Ishimaru | Atomistic structures of metastable and amorphous phases in ion-irradiated magnesium aluminate spinel[END_REF].Tien Hien NGUYENUniversité Paris-Sud

Acknowledgments

First of all, I would like to express the most profound gratitude to my advisor Prof.

List of Figures