Atomic scale simulations of noble gases behaviour in uranium dioxide

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Université Libre de Bruxelles Studiecentrum voor Kernenergie — Faculté des Sciences Appliquées Centre d’étude de l’ Énergie Nucléaire

Service de m´ etrologie nucl´ eaire Institute of Nuclear Materials Science

Atomic scale simulations of noble gases behaviour in uranium dioxide

Th` ese pr´ esent´ ee par Kevin Govers en vue de l’obtention du titre de docteur en Sciences de l’Ing´ enieur.

Directeur de th` ese : Monsieur le professeur Alain Dubus

Co-promoteur : Monsieur le professeur Marc Hou Mentors (SCK

^•

CEN) : Dr. Sergei E. Lemehov Dr. Marc Verwerft

Ann´ ee acad´ emique 2007 – 2008

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Remerciements / Word of thanks

A doctorate thesis is a huge scientific adventure, involving a lot of people, much more than initially expected. I will try to include everyone that contributed, but inevitably a few of them will be forgotten. Usually a doctorate thesis is done inside the academic world or, less often, in a research center. In my case I had the opportunity to have a mix of both worlds, on one hand at my Alma Mater, the Universit´ e Libre de Bruxelles, and on the other hand at the Belgian nuclear research center SCK

^•

CEN, which funded this PhD. I am very grateful to SCK

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CEN for this, but above all for the very rich environment, both in terms of knowledge and expertise, that SCK

^•

CEN could provide me.

I am thankful to all the people working at the Laboratory for High and Medium Activities (LHMA) ; especially I would like to express my gratitude to my mentor at SCK

^•

CEN, Dr. Sergei E. Lemehov, for his guidance, his fruitful scientific advises and views on my progresses. I also would like to thank Dr. Marc Verwerft for his interest in my work, and all his comments. Their deep knowledge of the nuclear fuel, and of physics in general, lead to very interesting (and sometimes animated) discussions.

Je tiens aussi ` a saluer, au sein du groupe “Fuel Materials”, Fr´ ed´ eric (qui sera bientˆ ot le plus flamand des Fran¸cais) et Anne, ainsi qu’Arnaud et Andr´ e, qui ont bri` evement fait un petit d´ etour par mon bureau avant de re-d´ em´ enager. Ook een speciaal bedankje aan Yves (en zijn Xe/Kr ratio), Ben, Marlies en Milan, met wie ik ’s middags ga eten, en Sven voor zijn hulp en adviezen.

Quant ` a l’ULB, je tiens avant tout ` a remercier Robert Beauwens qui me mit en contact, il y a quelques ann´ ees d´ ej` a, avec le SCK

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CEN, ainsi qu’Alain Dubus, qui a acccept´ e d’ˆ etre mon promoteur et m’a fait remarqu´ e que de toute fa¸con, la dynamique mol´ eculaire, c’est simple, une fois qu’on sait que F ~ = m · ~a ;-) Je salue tous les membres du service de m´ etrologie nucl´ eaire ( plus particuli` erement, courage ` a Julien pour la fin de sa th` ese) ainsi que les ´ etudiants qui m’ont subi lors des exercices du cours de PRN-I.

Enfin, un grand merci ` a Marc Hou d’avoir ´ et´ e le co-promoteur de cette th` ese.

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This thesis enabled to make fruitful contacts and cooperations through Net- works of Excellence such as Actinet or the F-Bridge European 7

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Framework Program that started one month ago. I would like to thank all members of our work package in Actinet for the interesting discussions that could be made between

“experimentalists” and “theoreticians”, and particularly its coordinator, Philippe Garcia. I learned that the comparison of modelling and experiment has some in- terest, but their combination allows to go much deeper in our understanding of phenomena.

Another challenging cooperation could be made with CEA, more specifically with Laurent Van Brutzel, Jean-Paul Crocombette and Alain Chartier, in the domain of thermal spikes. I am very thankful to them, on one hand for having recognized dislocations appearing during my simulations, and on the other hand for their work on thermal spikes.

Outre les aspects scientifiques, il y a aussi la vie de tous les jours, la famille, les loisirs. Je tiens ` a remercier ma famille pour son soutien pendant ces quatre ann´ ees (et aussi celles qui ont pr´ ec´ ed´ e) : mes parents, mon fr` ere Benoˆıt, ma sœur Anne- Laure, mes grands-parents (dont mon grand-p` ere qui rappelle sans cesse qu’il m’a appris ` a compter par deux quand j’´ etais petit).

De mani` ere moins conventionnelle, mais tout cavalier le comprendra, j’y inclus aussi mes chevaux : Taco, que j’ai perdue il y a maintenant trois ans, Waldo (Wally pour les intimes) mon “cornichon poilu” actuel, ainsi que tous les chevaux dont j’ai pu m’occuper, pour un jour ou plus, ces derni` eres ann´ ees. Une liste serait trop longue ici !

Pour ne pas sortir du monde ´ equestre, je finirai par remercier mes amis ` a Hof te Bever, pour tous ces bons moments, les soir´ ees pass´ ees ` a monter dans le froid, les promenades, agit´ ees ou non. . . : Alice, qui bientˆ ot pourra nous pr´ eparer un petit spectacle de dressage avec Gringo ; Cathy ; Hugues, notre v´ eto et propri´ etaire du man` ege ; Jessica ; Laurence, dont la ponctualit´ e m’´ epatera toujours ; L´ eo et Vincent ; Sandra, qui n’attend qu’une chose, re-galoper sur la plage cet ´ et´ e ; Vanessa, qui revient de loin, bonne chance avec Shrek ; Christophe, qui a bien du m´ erite de supporter tout cela ; Virginie, qui rep` ere tout ce qui se passe sur internet en mati` ere de chevaux ; ainsi que tous les autres !

I finally would like to mention and thank the Open Source community, for the

various programs, especially in computer simulation techniques, that are freely

available.

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R´ esum´ e

Les performances du combustible nucl´ eaire sont fortement affect´ ees par le com- portement des gaz de fission, et ce particuli` erement lorsqu’un taux d’´ epuisement

´ elev´ e est atteint, puisque d’importantes quantit´ es de gaz sont alors produites et peuvent potentiellement ˆ etre relˆ ach´ ees. Les enjeux, entre autre ´ economiques, li´ es au relˆ achement de gaz de fission ont donn´ e lieu ` a d’importants efforts, tant sur le plan exp´ erimental que th´ eorique, afin d’accroˆıtre notre compr´ ehension des diff´ erentes ´ etapes du processus, et d’am´ eliorer sans cesse les mod` eles. Les exten- sions ` a des taux d’´ epuisements encore plus ´ elev´ es ainsi que l’int´ erˆ et croissant pour de nouveaux types de combustible tels que les matrices inertes, envisag´ ees en vue de la transmutation des actinides mineures, font qu’` a l’heure actuelle, le besoin permanent d’une meilleure mod´ elisation, bas´ ee sur une compr´ ehension et une de- scription physique des diff´ erentes ´ etapes du processus de relˆ achement de gaz de fission, est toujours de mise.

Les simulations par ordinateur ont ainsi ´ et´ e consid´ er´ ee comme un nouvel angle de recherche sur les processus ´ el´ ementaires se produisant ` a l’´ echelle atomique, ` a la fois afin d’obtenir une meilleure compr´ ehension de processus tels que la diffusion atomique ; mais aussi afin d’avoir acc` es ` a certains processus qui ne sont pas ob- servables par des voies exp´ erimentales, tels que le comportement du combustible lors de pointes thermiques.

Dans ce travail, deux techniques, bas´ ees sur l’utilisation de potentiels inter- atomiques empiriques, ont permis d’´ etudier le dioxyde d’uranium, dans un premier temps en l’absence d’impuret´ es. Cette partie ´ etait principalement centr´ ee sur le comportement des d´ efauts ponctuels, mais a aussi concern´ e diff´ erentes propri´ et´ es

´ elastiques, ainsi que le processus de fusion du compos´ e.

Ensuite l’´ etude a ´ et´ e ´ etendue aux comportements de l’h´ elium et du x´ enon. Pour

ce qui a trait ` a l’h´ elium, la diffusion dans diff´ erents domaines de stœchiom´ etrie

a ´ et´ e consid´ er´ ee. Les simulations ont permis de d´ eterminer le coefficient de dif-

fusion ainsi que le m´ ecanisme de migration lui-mˆ eme. Quant au x´ enon, outre les

propri´ et´ es de diffusion, l’intention fut de se diriger vers la mod´ elisation des petites

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bulles intragranulaires, et plus pr´ ecis´ ement vers leur interaction avec les pointes

thermiques, cr´ e´ ees lors du recul des fragments de fission. Une description sim-

plifi´ ee de ce processus a ´ et´ e propos´ ee, qui offre de nouvelles perspectives dans ce

domaine.

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Abstract

Nuclear fuel performance is highly affected by the behaviour of fission gases, par- ticularly at elevated burnups, where large amounts of gas are produced and can potentially be released. The importance of fission gas release was the motivation for large efforts, both experimentally and theoretically, in order to increase our understanding of the different steps of the process, and to continuously improve our models.

Extensions to higher burnups, together with the growing interest in novel types of fuels such as inert matrix fuels envisaged for the transmutation of minor ac- tinides, make that one is still looking for a permanently better modelling, based on a physical understanding and description of all stages of the release mechanism.

Computer simulations are nowadays envisaged in order to provide a better description and understanding of atomic-scale processes such as diffusion, but even in order to gain insight on specific processes that are inaccessible by experimental means, such as the fuel behaviour during thermal spikes.

In the present work simulation techniques based on empirical potentials have been used, focusing in a first stage on pure uranium dioxide. The behaviour of point defects was at the core of this part, but also the estimation of elastic and melting properties.

Then, in a second stage, the study has been extended to the behaviour of he-

lium and xenon. For helium, the diffusion in different domains of stoichiometry

was considered. The simulations enabled to determine the diffusion coefficient and

the migration mechanism, using both molecular dynamics and static calculation

techniques. Xenon behaviour has been investigated with the additional intention

to model the behaviour of small intragranular bubbles, particularly their interac-

tion with thermal spikes accompanying the recoil of fission fragments. For that

purpose, a simplified description of these events has been proposed, which opens

perspectives for further work.

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Notations and symbols used

Table 1: Nomenclature

Name Definition Typical Units

param. value

A Parameter of the potential types (re- pulsive term) :

Lennard-Jones potential – eV ˚ A

¹²

Buckingham potential – eV ˚ A

a

₀

Lattice parameter at 0 K 5.45 ˚ A

a Lattice parameter

α

_s

Fixed sink length 10

¹⁵

m

⁻²

b Probability of destruction of a bubble (and of resolution of a Xe atom present in a bubble considering the model pro- posed in this article)

10

⁻³

s

⁻¹

B

_T

Bulk modulus 200 GPa

B Parameter of the Lennard-Jones poten- tial (dispersion term) :

– eV ˚ A

⁶

C Parameter of the Buckingham potential (dispersion term) :

– eV ˚ A

⁶

C Concentration of defect (in general)

C

_ij

Elastic constants 100 GPa

C

_p

Specific heat at constant pressure J/K

C

v

Specific heat at constant volume J/K

C(r, t) Volumic concentration of Xe atoms in the lattice in r,at time t

10

²⁵

– 10

²⁶

(at.) · m

⁻³

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C

₀

Volumic concentration of gas atoms in the matrix (lattice + intragranular bubbles)

10

²⁵

– 10

²⁶

(at.) · m

⁻³

C

_b

Volumic concentration of bubbles in the matrix

10

²³

– 10

²⁴

(bub.)· m

⁻³

C

_bub^gas

Amount of gas atoms present in in-

tragranular bubbles per matrix volume unit

- (at.) · m

⁻³

C

_lat^gas

Amount of gas atoms present the lat- tice per matrix volume unit

- (at.) · m

⁻³

D Diffusion coefficient (in general) m

²

s

⁻¹

D

_eff

Effective diffusion coefficient 10

⁻¹⁷

– 10

⁻²¹

m

²

s

⁻¹

D

_Xe

In-pile diffusion coefficient of Xe atoms 10

⁻¹⁷

– 10

⁻²¹

m

²

s

⁻¹

D

i

i = 1, 2, 3 Partial contribution to the

diffusion coefficient expression estab- lished by Turnbull. Each term is domi- nant in a particular temperature range.

m

²

s

⁻¹

∆H

_v

Variation of enthalpy associated to a vacancy

1 eV

∆S

_vib,v

Variation of vibrational entropy associ-

ated to a vacancy

– eV/K

∆S

_conf

(n

_V

) Variation of configurational entropy as- sociated to the presence of n

_v

vacancies

– eV/K

E Young’s modulus 400 GPa

E Energy of the system J, eV

E

^∗

Migration energy eV

E

_def

Defect formation energy eV

E

_act

Activation energy eV

epsilon

₀

Static dielectric constant

epsilon

_∞

High frequency dielectric constant f Correlation factor for diffusion

F ˙ Fission rate 10

¹⁸

– 10

¹⁹

(fiss.) · m

⁻³

s

⁻¹

F (R

b

, t) Flux of atoms at the surface of a bubble of radius R

_b

, at time t

- (at.)·s

⁻¹

g Gradient of the potential energy eV/˚ A

g Probability per unit of time for a Xe atom to be trapped by a bubble

- s

⁻¹

G Gibbs free energy J, eV

Γ Total jump frequency s

⁻¹

γ(T ) Gruneisen parameter (as a function of

the temperature)

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NOTATIONS AND SYMBOLS USED

H Hamiltonian of the system J, eV

H

el

Hamiltonian for the electron motion, atom positions being simple parame- ters

J, eV

H Hessian matrix of the system J/m

²

j

_v

Cation (uranium) vacancy jump rate : j

_v

≈ ω

_D

exp(−2.4eV/k

_B

T )

s

⁻¹

K

⁰

Rate of defect production per atom 10

⁴

– 5×10

⁵

(def.)/(fiss.)

~k reciprocal-space vector

k

_B

Boltzmann constant 1.38 × 10

⁻²³

J·K

⁻¹

λ Adimensional parameter describing bubble growth in an infinite medium (see § 4.4.3)

0 – 0.1 -

µ

_{f f}

Length of a fission track 6 – 9 µm

N Number of sites or atoms N

_i

Number of interstitial sites

N

_b

(R

_b

) Number of gas atoms contained in a bubble of radius R

_b

- (at.)/(bub.)

ν Poisson’s ratio

ν Vibrational frequency O

_I

Oxygen interstitial

ω

_D

Debye frequency of the non-disturbed lattice

≈ 10

¹³

s

⁻¹

Ω

_Xe

Volume occupied by a gas atom inside

an intragranular bubble

0.036 nm

³

p

_J

Structural factor representing the frac- tion of sites available for migration P Correction factor proposed bu L¨ os¨ onen

expressing the probability of interac- tion of a fission fragment with a bubble at a distance lower than Z

₀

from the track centerline

0 – 1 -

P

_des

Probability of destruction of a bubble per unit of time, this value can depend or not on the bubble radius according to the chosen model

s

⁻¹

Ψ(r, q) Wavefunction of the system

R

_b

Bubble mean radius 0.5 – . . . nm

R

_b

(t) Radius of the growing bubble at time t - nm

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R

_b,τ

Radius of a bubble at time τ , it is an approximation of R

b

- nm

R

³_b

Average of R

³_b

from the bubble popula- tion

- nm

³

R ˜

_b

Bubble most probable radius (peak value in the bubble distribution)

- nm

R

cv

Radius of the capture volume 6 – 10 nm

ρ Parameter of the Buckingham poten- tial :

– ˚ A

S

^∗

Variation of entropy between the equi- librium and the transition state

eV/K S(~k) Structure factor

S(r, t) Volumic production rate of gas atoms in r, at time t

- (at.)m

⁻³

s

⁻¹

T Temperature - K

T “Instantaneous” temperature during a MD run

τ Bubble mean lifetime. If the destruc- tion process is independant of bubble size, τ = 1/b

- s

U

_I

Uranium interstitial

U

_real^Coulomb

Component of the coulombic energy in

real space

U

_recip^Coulomb

Component of the coulombic energy in

reciprocal space

V Vacancy concentration - (vac.)/(at.)

V

₀

Thermodynamic concentration of va- cancies : exp(−2.4eV/k

_B

T )

(vac.)/(at.)

V Bubbles mean volume - nm

³

V (R

_b

) Volume of a bubble of radius R

_b

: V (R

_b

) =

^4π₃

R

_b³

- nm

³

V (r) Potential energy surface / Interatomic potential – Energy of the electronic configuration when atoms coordinates r are taken as parameters

J, eV

V (r) Pair potential eV

V

_O

Oxygen vacancy

V

_U

Uranium vacancy

W Configurational partition function

x

_v

proportion of interstitials

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NOTATIONS AND SYMBOLS USED

x

_v

proportion of vacancies

Z Number of sites around a defect from which recombination is inevitable

100 Z

₀

Radius around the fission fragment

path in which bubbles are entirely de- stroyed

7 nm

Table 2: Abbreviations used Name Definition

FGR Fission Gas Release MD Molecular Dynamics

NPH Constant Number of particles, Pressure, enthalpy (H) (ensemble) NPT Constant Number of particles, Pressure and Temperature (ensemble) NVE Constant Number of particles, Volume and Energy (ensemble)

NVT Constant Number of particles, Volume and Temperature (ensemble) NV ˜ T pseudo-NVT ensemble

OFP Oxygen Frenkel Pair

PBC Periodic Boundary Conditions RFO Rational Function Optimization SC Static calculations

Sch Schottky defect

UFP Uranium Frenkel Pair

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Introduction

The safe exploitation of a nuclear reactor requires an adequate knowledge of the behaviour of its constituents. The particularities of nuclear materials are the ex- treme conditions at which they are used : temperature and temperature gradients (temperature of about 400

^◦

C for the periphery of fuel pellets and reaching 1000 – 1500

^◦

C at the center) ; pressure of 150 bar in the primary circuit ; irradiation by high energy gammas, neutrons and fission products.

The fission of heavy atoms leads to the production of a variety of elements, among which noble gases : xenon and, to lesser extent, krypton. These gases tend to escape from the fuel. This “fission gas release” process involves atomic scale diffusion, clustering into nanoscale bubbles, growth, destruction and merging of such bubbles. All these processes are influenced by the radiation fields and the dynamic processes related to local destruction and recovery of crystallographic order of the matrix, due to the fission events. Its intrinsically complex nature together with the fact that fission gas release is one of the determining factors in the service life of nuclear fuel, and as such has a direct economic impact, makes that, since many decades, researchers and engineers are continuously improving our understanding of the processes at play in fission gas release. In addition to this, the development of novel types of fuels for the new generation of reactors (“Gen- IV” systems), together with inert matrix fuels considered in view of actinides transmutation, will require an extension of the codes actually devoted to uranium dioxide. Therefore the quest for a better understanding and modelling of all steps of the phenomena will continue.

Considering the computer capabilities that are nowadays available, atomic scale

simulations can help understanding the basic mechanisms of the diffusion process,

in order to improve the actual models and to extrapolate to new fuel types and

reactor conditions, instead of fitting (semi-)empirical models. In addition, atomic

scale simulations can even provide insight in phenomena that are up to now in-

accessible by experimental means, such as the rapid events occurring e.g. in the

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wake of fission events.

A particular aspect of fission gas release, its behaviour inside the grains, involv- ing atomic diffusion, precipitation in small bubbles and the subsequent re-solution of the gas in the matrix will be studied here. Some aspects of fission gas release were already envisaged in my final year thesis where the in-pile Xe diffusion co- efficient was studied by a careful analysis of experimental results coupled to the analytical modelling of the growth and destruction of intragranular bubbles.

The intention of this work was to investigate further, using atomic scale sim- ulations, the assumptions made in the model and the results obtained. It would provide the necessary basis for improvement of the fission gas release models in fuel performance codes ; and, in the future, for an extension to other compounds, for which much less is known experimentally. In view of the large amounts of He that will be produced by α-decay in MOX and Gen-IV fuels, the behaviour of this element in uranium dioxide has also been envisaged.

Since these were, at SCK

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CEN and ULB, the first steps made for atomic scale simulations of the nuclear fuel, it has been preferred to make a detailed description of the basis necessary for the rest of the present study. Therefore the first part will provide some background on the atomic aspects of solid and liquid phases, particularly the defects present in a crystal and subsequently their diffusion, which is the central subject of this research. All this will be described in chapter 1.

The technical part of the simulations (static calculations and molecular dynamics techniques) will then be tackled in chapter 2.

The second part will cover the general properties of uranium dioxide (chap- ter 3), that will serve to validate and select interatomic potentials. The following chapter, chapter 4, discusses the behaviour of fission products in the nuclear fuel, specifically noble gases. It contains the above-mentioned model on bubble growth and destruction, which has been later improved during this work, see chapter 4 and the publications [1, 2]. It showed a very low activation energy for in-pile diffusion compared to out-of pile results (0.8 eV compared to 3.9 eV), which was the original motivation for investigations, with molecular dynamics simulations, of noble gases behaviour at the atomic scale.

After these considerations, the first steps in modelling uranium dioxide at the atomic scale will be presented. It consists of exploring properties of the uranium dioxide system with empirical interatomic potentials. More than twenty potentials have been published in the open literature, each time in order to study one or a few specific domains. The range of validity of these potentials for other types of simulations was rather unknown, since no general comparison of them existed.

For this reason it was decided to assess all existing potentials on the same set of

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INTRODUCTION

properties. At the same time it enables to make a selection for the following steps of the research. This comparison will be found in chapter 5. It has been published in two parts, the first one for the static calculations [3] and the second one for the MD simulations [4]. These results were also presented at different conferences [5–

7]. Apart from comparing the potentials with each other, this stage also showed how care has to be taken for modelling ionic compounds, with the specific difficulty to accurately model charged defects.

Then the last part focused on the atomic-scale behaviour of noble gases in uranium dioxide. Chapter 6 concerns He atomic diffusion, which was investigated in the framework of SCK

^•

CEN ’s participation in the EFP-6 Actinet Network of Excellence. The migration mechanisms operating in different temperature regimes and defect concentrations, as well as the single atom diffusion coefficient, could be identified using both energy minimization and molecular dynamics techniques.

This part has been presented at the last Actinet JRP 05-04 meeting [8] and will be sent for publication in a near future [9].

Considering the importance of fission gas release, underlined in chapter 4, molecular dynamics and static calculations have been envisaged in chapter 7 in order to gain a better view on atomic scale aspects of xenon behaviour. The low diffusion coefficient of Xe in the UO

₂

matrix made it difficult to fully characterize Xe diffusion using MD techniques, because of the much longer calculation times that would be required to have good statistics. For this reason more efforts were devoted to determine the activation energy using static calculation. However, MD simulations could be used for the investigation of Xe behaviour in liquid uranium dioxide, and during the solidification process. It enabled to “observe” the forma- tion of small clusters of a few Xe atoms, a first step toward bubble formation.

The last chapter, chapter 8, contains the first attempts made in order to sim- ulate thermal “spikes” accompanying the recoil of fission fragments through the material. Instead of making simulations of very energetic (in the MeV range) displacement cascades, which are not accessible in view of the computing power available nowadays or in the near future, thermal spikes were described as a very hot cylindrical zone inside a solid matrix. It has the advantage that the energy is already split over a larger number of atoms than in a displacement cascade simulation, which enables to use larger timesteps, and hence to envisage longer simulation times. The initial goal of these simulations was to assess the model of bubble destruction during the recoil of fission fragments presented in chapter 4.

However, these simulations also showed unexpected results with the formation of

dislocations. This part opens the road for deeper investigations, in the future. . .

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Part I

Background

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C h a p t e r 1

A

tomistic aspects of solid and liquid phase properties

1.2.1 Pair potentials . . . . 12

Lennard-Jones potential . . . . 13

Buckingham and “Buckingham-4-ranges” potentials . . . 13

Morse potential . . . . 14

Rydberg potential . . . . 14

1.2.2 Many body potentials . . . . 14

1.2.3 Electrostatic interactions . . . . 15

Ewald summation . . . . 16

Rigid ion / shell-core models . . . . 17

Variable charge models . . . . 17

1.2.4 Fitting of potentials . . . . 18

1.3 Lattice and crystal structures . . . 19

1.3.1 Generalities . . . . 19

1.3.2 Lattice planes and directions . . . . 20

1.3.3 Defects in crystals . . . . 21

Point defects . . . . 21

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Proportion of point defects . . . . 22

1-D, 2-D and 3-D defects . . . . 25

1.4 Diffusion in crystals . . . 25

1.4.1 Diffusion equation . . . . 26

1.4.2 Diffusion mechanism in terms of atomic jumps . . . . 26

1.4.3 Total jump frequency . . . . 29

1.5 Characterization of the solid and liquid phases. . . 30

1.5.1 Atomic description . . . . 31

1.5.2 Structure factor . . . . 31

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1.1. ATOM INTERACTIONS

1.1 Atom interactions

All the matter that surrounds us is constituted of aggregates of atoms. In most physical processes, atoms are not destroyed or created...

Exceptions to this rule are radioactive properties and nuclear reactions... which form the general framework of this thesis.

However, this work will focus on the behaviour of some of the fission products – principally xenon and krypton – and decay products (helium) in the nuclear fuel after their production. Xe and Kr have a major influence on the fuel performance, especially at high burnup, through what is called the fission gas release process (see chapter 4). Helium, which is produced in large amounts by radioactive decay of minor actinides, has a similar behaviour, but is produced in lesser amounts in conventional UO

₂

fuels. It is gaining nowadays interest in view of new types of fuels considered for transmutation of actinides.

The bonding between atoms is due to interactions between the electrons and nuclei present in the system. Formally, the system state is determined by the Schr¨ odinger equation of the system,

HΨ(r, q) = EΨ(r, q) (1.1)

for which the Hamiltonian is (other terms involving e.g. electron coupling have not been incorporated, but their addition does not result in a loss of generality in the following developments.) :

H = X

i

p

²_i

2m

_i

+ X

n

s

²_n

2m

_e

+ 1

2 X

i,j

Z

_i

Z

_j

e

²

|~ r

_i

− ~ r

_j

| + 1 2

X

n,n⁰

e

²

|~ q

_n

− ~ q

_n⁰

| − X

i,n

Z

_i

e

²

|~ r

_i

− ~ q

_n

| (1.2) where the i and j indexes indicate nuclei ; n and n

⁰

electrons ; ~ r

_i

, ~ p

_i

, m

_i

, Z

_i

are position, momentum, mass and atomic number of the nucleus i ; ~ q

_n

and ~ s

_n

the position and momentum of the electron n ; m

e

is the electron mass. For clarity, the set of all ~ r

_n

is described by r, the same for p, q and s.

The Born-Oppenheimer approximation suggests treating separately the elec- trons from the nuclei, since their associated time scales differ by about two orders of magnitude. The total wavefunction can be written under the form

Ψ(r, q) = Θ(r)Φ(q; r) (1.3)

where Θ(r) describes the nuclei only ; Φ(q; r) the electrons when the nuclei

coordinates are simple parameters.

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Eq. (1.1) can now be expressed as a system of two equations :



 



 



H

el

Φ(q; r) = ˜ V (r)Φ(q; r) (a)

"

X

i

p

²_i

2m

_i

+ 1

2 X

i,j

Z

_i

Z

_j

e

²

|~ r

_i

− ~ r

_j

| + ˜ V (r)

#

Θ(r) = EΘ(r) (b) (1.4) where H

_el

includes all terms related to the electrons :

H

_el

= X

n

s

²_n

2m

_e

+ 1

2 X

n,n⁰

e

²

|~ q

_n

− ~ q

_n⁰

| − X

i,n

Z

_i

e

²

|~ r

_i

− ~ q

_n

| (1.5) V ˜ (r) incorporates all electronic effects in view of the nuclei motion, and is called the interatomic potential. Eq. (1.4 b) can be re-written in a more general form, incorporating the term

¹₂

P

i,j ZiZje²

|~ri−~rj|

– that also depends on the nuclei coordinates only – into the interatomic potential :

"

X

i

p

²_i

2m

_i

+ V (r)

#

Θ(r) = EΘ(r) (1.6)

The Born-Oppenheimer approximation is valid whenever the potential energy surfaces V

k

(r) that are solutions of eq. (1.4 a.) are well separated :

V

₀

(r) V

₁

(r) V

₂

(r) · · · (1.7) The solution of the system (1.4) is at the very heart of condensed matter physics. Many solid state physics books (e.g. [10, 11]) are mainly devoted to the solution of the electronic part of the system, assuming that nuclei form a periodic lattice (the crystal), and generally in order to derive properties of metals.

When the interest is in the nuclei motion, it is of common practice to treat eq. (1.6) classically, using Newton’s dynamics instead of quantum mechanics. Some fields of condensed matter physics try to solve numerically (with some approxima- tion) both parts of the system (1.4). These simulations are called first-principles or ab initio simulations. They require large computing power, and are limited to the study of very small systems.

Other fields of solid state physics focus on the nuclei behaviour only, approxi-

mating the interatomic potential in different ways. In this work, we will consider

the approximation by empirical interatomic potentials, with the objective to simu-

late uranium dioxide. Two techniques have been used, molecular dynamics (MD)

and energy minimization algorithms. They will be reviewed in Chapter 2.

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1.2. EMPIRICAL INTERATOMIC POTENTIAL

1.2 Empirical interatomic potential

The basic idea behind empirical interatomic potentials is to approximate the in- teratomic potential V (r) of eq. (1.6) using “simple” analytical functions describing nuclei (atoms) interactions as a function of their positions. Different forms of inter- atomic potentials exist, because of the difficulty, even the impossibility, to model atom interactions in very different environments (e.g. an oxygen atom in an oxide and in an O

₂

molecule) with only one analytical function.

One generally can distinguish between four types of bonding, which differ by the behaviour of the outer shell electrons [12] :

• Metallic bonding, in which delocalization of the electrons occurs, through the whole crystal.

• Covalent bonding, in which electrons, initially localized in one atom state, are shared between two atoms, in a molecular state.

• Ionic bonding, in which one atom loses his outer shell electrons, the other atom filling his outer shell ; electrostatic attraction occurs between the two charged ions.

• Weak chemical bonds, that appear between neutral atoms. In the case of noble gases, the fluctuation of the electrons in the outer shell creates mo- mentary dipoles which interact with each other, creating a weak attraction.

Metals (and metal alloys) can be seen as the “most developed” materials in terms of interatomic potentials, for several reasons. The first one being the diffi- culty to properly model them with simple pair potentials. A second reason is that atom interactions in metals are very short-range, generally limited to the first or second neighbour. It means that the number of interactions of one atom that have to be calculated is limited. Therefore more complex forms of potentials could be used while maintaining a reasonable computing time.

We will not extend much more on metallic systems since this work mainly

concerns ceramic nuclear fuels. The three other types of bonding are of greater

interest for us. The ionic character of ceramic nuclear fuels is obvious, but some

models also suppose a partial covalence of the O–U bonds (more details will be

given in chapter 3). The weak chemical bonds, also called Van der Waals interac-

tions, are of interest not only for the noble gases description, but also has a small

contribution between ions.

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1.2.1 Pair potentials

Without loss of generality, the interatomic potential V (r) can be expanded in Taylor series :

V (r) = E

₀

+ X

j

U

₁

(~ r

_j

) + X

j, k>j

φ

₂

(|~ r

_j

− ~ r

_k

|) + X

j, k>j, l>k

θ (~ r

_j

, ~ r

_k

, ~ r

_l

) + · · · (1.8)

The zeroth-order term, E

₀

is a constant, and will just influence the energy reference of the system. Therefore, it will be omitted from here. The first order term, U

₁

describes applied fields on one atom, which can be an “internal field”

such as in the mean field theory ; or an external one, e.g. an electrical field.

The second order term, φ

₂

describes pairwise interactions. It depends only on the relative distance between two atoms because the potential should be invariant by translation and rotation.

Up to here, the development in eq. (1.8) does not contain any approximation if enough terms are included. However, for most systems, the contribution of each term decreases progressively with its order ; one can therefore choose to stop the development beyond a certain order. In addition to this, a compensation for the approximation made can be to parameterize the remaining terms in the devel- opment (1.8) [13]. One then loses the generality of the potential, because the truncation of a Taylor development still provides the best approximation over the whole coordinate-space, but a re-parameterization can provide a better approxi- mation over a sub-domain.

The simplest approximation is to stop the development of eq. (1.8) at the second order, keeping only the pairwise interactions. This approximation is very poor for metals or semi-conductors, but gives very good results for ionic compounds, since electrostatic forces alone, which are pairwise, contribute to more than 90% of the system energy. A similar observation is made for noble gases, since dispersion forces (Van der Waals interactions, due to temporary dipole), are also pairwise, and decay as 1/r

⁶

. At very short interatomic distances, electron clouds start interacting, giving rise to an additional repulsive interaction. Since this repulsion only involves two atoms, it can again be described by a pair potential.

Therefore, the modeling of uranium dioxide and of noble gases in this compound

could be envisaged in this work using pair potentials. The specific treatment of

the electrostatic interactions will be tackled in a separate paragraph, § 1.2.3, we

will now review the most important types of pair potentials [13, 14]. Most of them

will be used later in this work.

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1.2. EMPIRICAL INTERATOMIC POTENTIAL

Lennard-Jones potential

One of the first empirical ways of describing the sharp repulsion between two atoms is :

V (r) = A

r

ⁿ

(1.9)

where A and n are parameters. The value of n ranges from 5 to 12 according to the radius of the considered ion [12]. The Van der Waals interactions also obey eq. (1.9), with n = 6, but are attractive. A negative value of A is therefore needed. The composition of the repulsive and attractive interactions provides the well-known Lennard-Jones potential :

V (r) = A r

¹²

− B

r

⁶

(1.10)

where A and B are the parameters. A more generalized form also exists where the values of the exponents can also be chosen. Lennard-Jones potentials are very common for describing noble gas interactions.

Buckingham and “Buckingham-4-ranges” potentials

A second function, based on quantum-mechanical calculations, is extensively used for the repulsive part of the potential : the Born-Mayer form

V (r) = A exp

− r ρ

(1.11) Its combination with the Van der Waals interactions provides the Buckingham potential, of the form :

V (r) = A exp

− r ρ

− C

r

⁶

(1.12)

It presents the inconvenience that at very small distances, the 1/r

⁶

term creates a non-physical zone of attraction. Care has therefore to be taken not to enter this zone during MD simulations, e.g. because of a too high kinetic energy. To circumvent this effect at very small distances, a modified form of the Buckingham potential has been proposed, the “Buckingham-4-ranges” potential :

V (r) =



 



 

 A exp

−

_ρ^r

if r ≤ r

₁

,

5th-degree polynomial if r

1

< r ≤ r

min

, 3rd-degree polynomial if r

min

< r ≤ r

2

,

−

_r^C6

if r > r

₂

.

(1.13)

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Morse potential

The preceding potential forms described the repulsion of two electronic clouds, and their attraction at larger distances thanks to dipole-dipole interactions, but cannot model covalent bonds. The Morse potential aims at describing such bonds :

V (r) = D

[1 − exp (β(r − r

^∗

))]

²

− 1 (1.14) In ionic crystals, it is used in addition to a Buckingham potential – and to the electrostatic interactions – to model partially covalent bonds :

V

_ij

(r) = f

₀

(b

_i

+ b

_j

) exp

a

_i

+ a

_j

− r b

_i

+ b

_j

− c

_i

c

_j

r

⁶

+ D

_ij

n

1 − exp β

_ij

(r − r

_ij^∗

)

2

− 1 o

(1.15) The indexes i and j refer to atoms of the type i and j. Note that the different parameters (a

_i

, b

_i

, c

_i

) are intrinsic to atoms of type i, once they are fitted, one also obtains the potential for a bond between different types of atoms. This is given by eq. (1.15). Generally a partial ionicity is assumed for the atom charges with this model.

Rydberg potential

Another type of potential describing covalent bonds is the Rydberg potential : V (r) = A [1 + α (r − ρ)] exp (−α (r − ρ)) (1.16) This form of potential was, up to now, not used for the description of uranium dioxide.

1.2.2 Many body potentials

As mentioned in the previous paragraph, pair potentials are not sufficient to model properly metals and semi-conductors, and also some glasses where special bond orientations need to be predicted. Therefore attempts were made to include some- what the effects of the surrounding atoms – coordination number, orientation of the angles – into the effective potential. Many body potentials for metals are generally based on the following analytical form [15] :

V (r) = X

i,j>i

φ

₂

(r

_ij

) + X

i

U(ρ

_i

) (1.17)

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1.2. EMPIRICAL INTERATOMIC POTENTIAL

where the first term represents a classical pair potential. The potential U stands for the fact that interaction between atoms depends on the number of other neighbour atoms through the local electron density ρ :

ρ

i

= X

j,j6=i

f (r

ij

) (1.18)

where f(r

_ij

) represents the effect of the electron density of an atom j on an atom i situated at a distance r

ij

. The difference between well-known potentials for metallic systems (e.g. Embedded Atom Method, Second moment tight binding models, glue model, ...) lies in the analytical form of f and U [15, 16]. It is to be noted that, even if formally these types of potentials vary with the coordination number, they still require computing only pair distances, and therefore they do not increase too much the simulation time. One of the advantages of such potentials is that once the parameters are fitted, they can be used for e.g. alloys without having to establish the potential between the different elements.

The second general type of many-body potential involves the computation of three-body terms (or even larger-orders). They can be seen as penalties for de- viation from the expected angle, when three-body interactions represent covalent bonds with specific orientations :

U

_ijk

= 1

2 k

₂

(θ − θ

₀

)

²

(1.19)

or an Axilrod-Teller potential that describe, in ionic materials, three-center dispersion contribution [16] :

U

_ijk

= k 1 + 3 cos(θ

ijk

) cos(θ

jki

) cos(θ

kij

)

r

_ij³

r

_jk³

r

³_ik

(1.20) These many-body potentials are presented here for completeness. EAM poten- tials have been used in attempts to develop a variable-charge potential (see § 1.2.3) for uranium dioxide. However, these attempts remained unsuccessful and will not be discussed in more details in this text.

1.2.3 Electrostatic interactions

Electrostatic interactions, despite their simple analytical form : U

_ij^Coulomb

= q

_i

q

_j

4π

₀

r

_ij

(1.21)

are the most time-consuming parts to evaluate for a periodic system because it

forms a conditionally convergent series. This is easily explained since the interac-

tion between two ions decreases slowly, as 1/r, but at the same time the number

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of ions to take into consideration increases as 4πr

²

. The most used method to cir- cumvent this problem is the Ewald summation (see below), but other techniques have also been developed.

Ewald summation The series

U

^Coulomb

= X

i,j>i

q

i

q

j

4π

₀

r

_ij

(1.22)

is slowly convergent and would require very long computing time to proceed.

For periodic systems, the convergence of the series can be accelerated using the Ewald summation. It consists of making a Laplace transform of the series, and separating it into two components, one that converges quite fast in the real space, the other one in the reciprocal space ; and one additional term for the self-energy (for the details, see e.g. [14, 16, 17]) :

U

_real^Coulomb

= 1 4π

0

X

i,j>i

q

_i

q

_j

r

ij

erfc (ηr

_ij

)

U

_recip^Coulomb

= 1

₀

V

X

~k>~0

1 k

²

exp

− k

²

4η

²







X

i

q

_i

cos(~k · ~ r

_i

)

2

+

X

i

q

_i

sin(~k · ~ r

_i

)

2





 U

_self^Coulomb

= η

4π

³²

₀

X

i

q

_i²

(1.23) This separation can conceptually be viewed as adding a Gaussian charge (of equal magnitude as the ion charge), centered on each ion, which screens the in- teraction between ions. The same Gaussian charge has then to be subtracted in order to recover the original distribution, but it will be summed in the reciprocal space. The parameter η controls how the sum is split over the real and reciprocal parts. A tuning of this parameter allows gaining computing time (which is crucial for large system simulations) without any compromise on the precision. Then a cutoff needs to be defined in both real and reciprocal spaces, in order to stop the summation when the desired precision is reached.

Even with this technique, the number of interactions that needs to be con-

sidered makes it the slowest part of a MD run. For that reason, when very

large systems (100 000 to 400 000 atoms) were considered, the reciprocal part

was turned-off, at the expenses of some precision.

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1.2. EMPIRICAL INTERATOMIC POTENTIAL

Rigid ion / shell-core models

The basic model in which atoms are described by a point charge – the rigid ion model – cannot model the polarizability of the ions, which can have a large influ- ence on the estimation of defect properties. Therefore other atom descriptions have been developed in order to take the ion polarizability into account. The most used description is the so called “shell-core model” due to Dick and Overhauser [18], in which an ion is represented as one massive point charge – standing for the nucleus and the inner electron shells – and one massless shell – standing for the valence electron shell. Generally this conceptual view of the ion stops here, since in many parameterizations, cation shells will have a positive charge.

In the shell-core model, all species interact electrostatically except when they belong to the same ion, in that case they interact by a harmonic potential :

V (r) = k · r

²

(1.24)

This was for the coulombic part ; on the other hand the short-range interatomic potential (which presents the same analytical form as for rigid ion potentials) acts between the shells only. It is to be noted that, even if the shell-core model presents these advantages over the rigid ion potential, the rigid ion model is far more effective for MD simulations, since the treatment of the motion of massless shells, whose vibration frequency is higher than that of cores, implies to use much smaller timesteps [13, 16, 19] (∼ 0.1 fs instead of 1 to 3 fs). In addition to this, the number of interactions to compute increases by a factor 4.

Variable charge models

The inconvenience of modeling ionic compounds with empirical interatomic po- tentials is that one attributes initially a charge to all types of atoms. However, depending on its environment, the charge of an atom could change. This is par- ticularly true for defects, where the defect charge should be compensated in the system, in order to keep it electrically neutral. Different possibilities exist for that purpose :

• The simplest way is to assume that a uniform background charge is added to the system. This charge has no influence on the forces acting on each atom and simply adds a correction to the system energy [13, 14, 16, 17].

• A second possibility consists into attributing (manually) a different charge to some atoms of the system, e.g. the closest neighbours to the defect [20, 21].

It can easily be done for static calculations, but in a MD run, it will not be

possible to adapt atom charges according to the motion of a defect through

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the system. In this work we could establish that this method had to be used in order to accurately predict defect relaxation volumes (see § 3.3.2).

• The most elegant way is to develop a potential with continuously variable charges, as a function of the atom environment. The basics of these po- tentials are that charges are calculated in such a way that the electrostatic energy (including the ionization energy) is minimized and at the same time electroneutrality is conserved. Example of such a potential was proposed by Streitz and Mintmire [22] in order to model a metal–oxide interface. We have tried to adapt such a model for the uranium–oxygen system, fitting simultaneously the parameters to properties of different phases (α-uranium, γ-uranium, UO

₂

, UO

₃

). Because of an intrinsic limitation of these vari- able charge potentials that dramatically increases atom charges when they approach too close to each other [23], like in defect configurations, correct defect formation energies could never be achieved. These new techniques, actually under improvement, seem, however, very promising for future de- velopments.

1.2.4 Fitting of potentials

The fitting of any potential is crucial in the sense that it contains most of the system information. It has already been mentioned that it is impossible to obtain a potential valid in all circumstances, therefore the properties on which the potential is fitted are also of great importance. No fitting of potential was made in this work, but a careful assessment of published potentials.

Regardless of the properties on which the potential will be fitted, one needs to evaluate how good the fitting is. The most common way to do it is to define a penalty function, usually a sum of squares (e.g. [13, 24–26]), which is minimized, ideally zero. Different weights can be attributed to each property on which the fitting is based, according to its importance in view of the further use of the potential, according to the confidence in the experimental value, or according to its magnitude. The function to minimize by varying the potential parameters is

F = X

properties

w

_prop

(f

_{f it}

− f

_obs

)

²

(1.25) where f

f it

is the calculated quantity, f

obs

is the reference value and w

prop

is the weight given to the specific property.

Two types of references values can be used :

• Ab initio calculation results, which are sometimes easier to obtain than ex-

perimental ones. The types of properties generally used are energy surfaces

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1.3. LATTICE AND CRYSTAL STRUCTURES

– series of configuration of which the energy has been calculated –, lattice parameter, elastic properties, defect formation energies, etc.

• Experimental data, which provides information on the correctness of the po- tential. The most used properties are the lattice structure, lattice parameter, elastic constants, dielectric constants, and sometimes defect formation ener- gies. For crystals with high symmetry, many sets of parameters could predict the same lattice parameter ; information on the first and second derivatives of the potential are necessary. In these perspectives, the elastic and dielectric constants play a very important role [16].

1.3 Lattice and crystal structures

1.3.1 Generalities

The purpose of this section is not to give detailed information on lattices and crystal structures, which can be found in many textbooks like [10, 11, 27–30], but to recall some properties of lattices and crystal structures and some notations that will be used further in this work.

A lattice consists into the periodic arrangement of points in space. The ensem- ble of points constituting the lattice can be determined from the knowledge of one of them, ~ r

₀

together with the fundamental translation vectors ~a, ~b, ~c :

~

r

⁰

= ~ r

₀

+ l~a + m~b + n~c (1.26) where l, m and n are integers.

The different types of lattice can be divided in 7 crystal systems (Triclinic, Monoclinic, Orthorhombic, Tetragonal, Rhombohedral, Hexagonal, Cubic), and 14 Bravais lattices. These classifications only refer to the regular arrangement of points of the lattice and the translational and rotational symmetries between those points.

The lattice is simply a mathematical abstraction, the crystal structure is ob- tained when the same basis of atoms is placed at each point of the lattice. In simple words, lattice + base = crystal structure. The crystal structures can be clas- sified into crystallographic point groups (or crystal classes) and space groups. The crystallographic point group is a mathematical group that comprises all symme- tries operations that leave at least one point unchanged and keep the appearance of the crystal unchanged. For 3-D systems, there are 32 possible crystal classes.

Atomic scale simulations of noble gases behaviour in uranium dioxide

Service de m´ etrologie nucl´ eaire Institute of Nuclear Materials Science