Ab initio Electronic Structure Calculations for Nuclear Materials

Ab initio Electronic Structure Calculations for Nuclear Materials$ JP Crocombette and F Willaime, CEA, DEN, Service de Recherches de Métallurgie Physique, Gif-sur-Yvette, France r 2016 Elsevier Inc. All rights reserved.

1 2 2.1 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 3 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.3 3.3.1 3.3.2 3.4 4 4.1 4.1.1 4.1.2 4.1.3 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.3.1 4.3.2 5 5.1 5.1.1 5.1.2 5.1.2.1 5.1.3 5.1.4 5.1.5 5.2 5.2.1 5.2.2 5.2.3

Introduction Methodologies and Tools Theoretical Background Codes Basis sets Pseudization schemes Ab initio Calculations in Practice Outputs Cell sizes and corresponding CPU times Choices to make Fields of Application Perfect Crystal Bulk properties Inputs for thermodynamical models Defects Self-defects Hetero-defects Point defect assemblies. Kinetic models Extended defects Ab initio For Irradiation Threshold displacement energies Electronic stopping power Ab initio and Empirical Potentials Metals and Alloys Pure Iron and Other bcc Metals Self-interstitials and self-interstitial clusters in Fe and other bcc metals Vacancy and vacancy clusters in Fe and other bcc metals Finite temperature effects on defect energetics Beyond Pure Iron Helium-vacancy clusters in iron and other bcc metals From pure iron to steels: The role of carbon Interaction of point defects with alloying elements or impurities in iron From dilute to concentrated alloys: the case of Fe−Cr Point defects in hcp-Zr Dislocations Dislocations in bcc metals Dislocations in hcp zirconium Insulators Silicon Carbide Point defects Defect kinetics Recombinations Defect complexes Impurities Extended defects Uranium oxide Bulk electronic structure Point defects Oxygen clusters

2 2 2 3 3 3 4 4 4 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 9 11 11 12 12 12 13 13 14 15 15 16 17 17 17 18 18 19 19 19 20 20 20 20

☆

Change History: May 2015. J.P. Crocombette and F. Willaime. Part ‘4.1.1 Self-interstitials and self-interstitial clusters in Fe and other BCC metals’ has been rewritten. Figure 4 has been changed. Part ‘4.2.4 From dilute to concentrated alloys: the case of Fe–Cr’ has been extended. Part ‘4.3 Dislocations’ has been updated. Figure 10 has been changed. Part ‘5.2.2 Point defects’ has been completed. References have been updated and renumbered.

Reference Module in Materials Science and Materials Engineering

doi:10.1016/B978-0-12-803581-8.00658-5

1

2

Ab initio Electronic Structure Calculations for Nuclear Materials

5.2.4 6 References

1

Impurities Conclusion

20 21 22

Introduction

Electronic structure calculations did not start with the so-called ‘ab initio’ calculations. Underlying basics date back to the 1930s with the understanding of the quantum nature of bonding in solids, the Hartree and Fock approximations and the Bloch theorem. A lot was understood of the electronic structure and bonding in nuclear materials using semiempirical electronic structure calculations, for example, tight binding calculations.1 The interest of these, somewhat historical, calculations should not be overlooked. However, in the following, we shall focus on ‘ab initio’ calculations, i.e., Density Functional Theory (DFT) calculations. One must acknowledge that ‘ab initio calculations’ is a rather vague expression which may have different meanings depending on the community. In the present Chapter we use it, as most people in the materials science community do, as a synonym for DFT calculations. The popularity of these methods stems from the fact that, as we shall see, they provide quantitative results on many properties of solids without any adjustable parameters, though conceptual and technical difficulties subsist that should be kept in mind. The presentation is divided as follows. Methodologies and tools are briefly presented in the first section. The next two sections focus on some examples of ab initio results on metals and alloys on one hand and insulating materials on the other.

2 2.1

Methodologies and Tools Theoretical Background

In the following a very basic summary of the DFT is given. The reader is referred to specialized textbooks2–4 for further reading and mathematical details. Electronic structure calculations aim primarily at finding the ground state of an assembly of interacting nuclei and electrons, the former being treated classically and the latter needing a quantum treatment. The theoretical foundations of DFT were set in the 1960s by the works of Hohenberg and Kohn. They proved that the determination of the ground state wave function of the electrons in a system (a function of 3N variables if the system contains N electrons) can be replaced by the determination of the ground state electronic density (a function of only three variables). Kohn and Sham then introduced a trick in which the density is expressed as the sum of squared single particle wave functions, these single particles being fictitious noninteracting electrons. In the process, an assembly of interacting electrons has been replaced by an assembly of fictitious noninteracting particles, thus greatly easing the calculations. The electronic interactions are gathered in a one electron term called ‘the exchange and correlation potential,’ which derives from an exchange and correlation functional of the total electronic density. One finally obtains a set of one electron Schrödinger equations whose terms depend on the electronic density, thus introducing a self consistency loop. No exact formulation exists for this exchange and correlation functional, so one has to resort to approximations. The simplest one is the Local Density Approximation (LDA). In this approximation the density of exchange and correlation energy at a given point depends only on the value of the electronic density at this point. Different expressions exist for this dependence so there are various LDA functionals. Another class of functionals pertains to the Generalized Gradient Approximation (GGA), which introduces in the exchange and correlation energy an additional term depending on the local gradient of the electronic density. These two classes of functionals can be referred to as the standard ones. Most of the ab initio calculations in materials science are performed with such functionals. Recent effort has been put into the development of a new kind of functional, the so-called hybrid functionals, which include some part of exact exchange in their expression. Such functionals which have been used for years in chemistry, have begun to be used in the nuclear materials context, though they involve usually much more time consuming calculations. One of their interests is that they give a better description of the properties of insulating materials. We finish this very brief theoretical introduction by mentioning the concepts of k point sampling and pseudization. In the community of nuclear materials, most calculations are done for periodic systems, i.e., one considers a cell periodically repeated in space. Bloch theorem then ensures that the electronic wave functions should be determined only in the irreducible Brillouin zone which is in practice sampled with a limited number of so-called k points. A fine sampling is especially important for metallic systems. Most ab initio calculations use pseudopotentials. Pseudization is based on the assumption that it is possible to separate the electronic levels in valence orbitals and core orbitals. Core electrons are supposed to be tightly bound to their nucleus with their states unaffected by the chemical environment. In contrast, valence electrons fully participate in the bonding. One then first considers in the calculation that only the valence electrons are modified while the core electrons are frozen. Second, the true

Ab initio Electronic Structure Calculations for Nuclear Materials

3

interaction between the valence electrons and the ion made of the nucleus and core electrons is replaced by a softer pseudopotential of interaction which greatly decreases the calculation burden. Various pseudization schemes exist (see Section 2.2.2). Beyond ground state properties, other theoretical developments allow the ab initio calculations of additional features. Detailing these developments is beyond the scope of this text, let us just mention among others: Time Dependent Density Functional Theory (TD-DFT) for electron dynamics, GW calculations for the calculation of electronic excitation spectra, Density Functional Perturbation Theory for phonon calculations and other second derivatives of the energy.

2.2

Codes

Ab initio calculations rely on the use of dedicated codes. Such codes are rather large (a few hundred thousand lines) and their development is a heavy task which usually involves several developers. An easy, though oversimplified way to categorize codes is to classify them in terms of speed on one hand and accuracy on the other hand. The optimum speed for the desired accuracy is of course one of the goals of the code developers (together with the addition of new features). Codes can primarily be distinguished by their pseudization scheme and the type of their basis set. Many other numerical or programming differences exist that we will not describe, even though they can influence the accuracy and speed of the codes. The possible choices in terms of basis sets and pseudization are exposed in the following paragraphs. Pseudization scheme and basis set are intricate as some bases do not need pseudization and some pseudizations presently only exist for specific basis sets. These methodological choices intrinsically lead to accurate but heavy, or conversely fast but approximate, calculations. We also mention some codes, though we have no claim for completeness on that matter. Furthermore, we do not comment on the accuracy and speed of the codes themselves as the developing teams are making continuous efforts to improve their codes, which make such comments inappropriate and rapidly outdated.

2.2.1

Basis sets

For what concerns the basis sets we briefly present plane wave codes, codes with atomic-like localized basis sets and allelectron codes. All-electron codes involve no pseudization scheme as all electrons are treated explicitly, though not always on the same footing. In these codes a spatial distinction between spheres close to the nuclei and interstitial regions is introduced. Wave functions are expressed in a rather complex basis set made of different functions for the spheres and the interstitial regions. In the spheres spherical harmonics associated with some kind of radial functions (usually Bessel functions) are used while in the interstitial regions wave functions are decomposed in plane waves. All electron codes are very computationally demanding but provide very accurate results. As an example one can mention the Wien2k5code which implements the FLAPW (Full potential Linearized Augmented Plane Wave) formalism.6 At the other end of the spectrum are the codes using localized basis sets. The wave functions are then expressed as combinations of atomic-like orbitals. This choice of basis allows the calculations to be quite fast since the basis set size is quite small (typically 10–20 functions per atom). The exact determination of the correct basis set, however, is a rather complicated task. Indeed, for each occupied valence orbital one should choose the number of associated radial ζ basis functions with possibly an empty polarization orbital. The shape of each of these basis functions should be determined for each atomic type present in the calculations. Such codes usually involve a norm conserving scheme for pseudization (see the next section) though nothing forbids the use of more advanced schemes. Among this family of codes, SIESTA7,8 is often used is nuclear material studies. Last but not least, many important codes use plane waves as their basis set.9 This choice is based on the ease of performing fast Fourier transform between direct and reciprocal space, which allows rather fast calculations. However, dealing with plane waves means using pseudopotentials of some kind as plane waves are inappropriate for describing the fast oscillation of the wave functions close to the nuclei. Thanks to pseudopotentials, the number of plane waves is typically reduced to 100 per atom. Finally, we should mention that other basis sets exist, for instance gaussians as in the eponymous chemistry code10 and wavelets in the BigDft project,11 but their use is at present rather limited in the nuclear materials community.

2.2.2

Pseudization schemes

As explained above, pseudization schemes are especially relevant for plane wave codes. All pseudization schemes are obtained by calculations on isolated atoms or ions. The real potential experienced by the valence electrons is replaced by a pseudopotential coming from mathematical manipulations. A good pseudopotential should have two apparently contradictory qualities. First it should be soft, meaning that the wave function oscillations should be smoothened as much as possible. For a plane wave basis set this means that the number of plane waves needed to represent the wave functions is kept minimal. Second, it should be transferable, which means that it should correctly represent the real interactions of valence electrons with the core in any kind of chemical environment, i.e., in any kind of bonding (metallic, covalent, ionic), with all possible ionic charges or covalent configurations conceivable for the element under consideration. The generation of pseudopotentials is a rather complicated task but nowadays libraries of pseudopotentials exist and pseudopotentials are freely available for almost any element, though not with all the pseudization schemes.

4

Ab initio Electronic Structure Calculations for Nuclear Materials

One can basically distinguish Norm Conserving pseudopotentials, ultrasoft pseudopotentials and PAW formalism. Norm conserving pseudopotentials were the first ones designed for ab initio calculations.12 They involve the replacement of the real valence wave function by a smooth wave function of equal norm, hence their name. Such pseudopotentials are rather easy to generate and several libraries exist with all elements of the periodic table. They are reasonably accurate although they are still rather hard and so they are less and less in use in plane wave codes but are still used with atomic-like basis sets. Ultrasoft pseudopotentials13 remove the constraint of norm equality between the real and pseudowave functions. They are thus much softer though less easy to generate than norm conserving ones. The Projector Augmented Wave14 formalism is a complex pseudization scheme close in spirit to the ultrasoft scheme but it allows the reconstruction of the real electronic density and the real wave functions with all their oscillations, and for this reason this method can be considered an all-electron method. When correctly generated, PAW atomic data are very soft and quite transferable. Libraries of ultrasoft pseudopotentials or PAW atomic data exist, but they are generally either incomplete or not freely available. Plane wave codes in use in the nuclear materials community include VASP15 with ultrasoft pseudopotentials and PAW formalism, Quantum-Espresso16 which is equipped with norm conserving and ultrasoft pseudopotentials and PAW formalism, and ABINIT17 with norm conserving pseudopotentials and PAW formalism. Note that for a specific pseudization scheme many different pseudopotentials can exist for a given element. Even if they were built using the same valence orbitals, pseudopotentials can differ by many numerical choices (e.g., the various matching radii) that enter the pseudization process. We present in the following a series of practical choices to be made when one wants to perform ab initio calculations. But the first and certainly most important of these choices is that of the ab initio code itself as different codes have different speeds, accuracies, numerical methods, features, input files, etc., and so it proves quite difficult to change codes in a middle of a study. Furthermore, one observes that most people are reluctant to change their usual code as the investment required to fully master the use of a code is far from negligible (not to mention the one to master what is in the code).

2.3

Ab initio Calculations in Practice

In this paragraph we try to give some flavor of what can be done with an ab initio code and how it is done in practice. The calculation starts with the positioning of atoms of given types in a calculation cell of a certain shape. That would be all if the calculations were truly ab initio. Unfortunately, a few more pieces of information should be passed to the code; the most important ones are described in the final section. The first section introduces the basic outputs of the code and the second one deals with the possible cell sizes and the associated CPU times.

2.3.1

Outputs

We describe in this section the outputs of ab initio calculations in general terms. Possible applications in the nuclear materials field will be given below. The basic output of a standard ab initio calculation is the complete description of the electronic ground state for the considered atomic configuration. From this, one can extract electronic as well as energetic information. On the electronic side, one has access to the electronic density of states that will indicate whether the material is metallic, semiconducting or insulating (or at least what the code predicts it to be), its possible magnetic structure, etc. Additional calculations are able to provide additional information on the electronic excitation spectra: optical absorption, X-ray spectra, etc. On the energetic side, the main output is the total energy of the system for the given atomic configuration. Most codes are also able to calculate the forces acting on the ions as well as the stress tensor acting on the cell. Knowing these forces and stress it is possible to chain ground state calculations in order to perform various calculations: - Atomic relaxations to the local minimum for the atomic positions. - From the relaxed positions (where forces are zero), one can calculate second derivatives of the energy to deduce, among other things, the phonon spectrum. This can be done either directly, by the so-called frozen phonon approach, or by first order perturbation theory (if such feature is implemented in the code). In this last case, third order derivative of the energy (Raman spectrum, phonon lifetimes) can be computed too. - Starting from two relaxed configurations close in space, one can calculate the energetic path in space joining these two configurations, thus allowing the calculation of saddle points. - The integration of the forces in a Molecular Dynamics scheme leads to so-called ab initio molecular dynamics. Car-Parrinello molecular dynamics18 calculations, which pertain to this class of calculations, introduce fictitious dynamics on the electrons to solve the minimization problem on the electrons simultaneously with the real ion dynamics.

2.3.2

Cell sizes and corresponding CPU times

The calculation time of ab initio calculations varies–to first order - as the cube of the number of atoms, or equivalently of electrons, (the famous N3 dependence) in the cell. If a fine k point sampling is needed this dependence is reduced to N2 as the number of k points decreases in inverse proportion with the size of the cell. On the other hand, the number of self-consistent cycles needed to reach convergence tends to increase with N. Anyway, the variation of calculation time with the size of the cell is huge and thus

Ab initio Electronic Structure Calculations for Nuclear Materials

5

strongly limits the number of atoms but also the cell size that can be considered. On one hand calculations on the unit cell of simple crystalline materials (with a small number of atoms per unit cell) are fast and can easily be performed on a common laptop. On the other hand when larger simulation cells are needed, the calculations quickly become more demanding. The present upper limit in the number of atoms that can be considered is of the order of a few hundreds. The exact limit of course depends on the code but also on the number of electrons per atoms and other technicalities (number of basis functions, k points, available computer power, etc.) so it is not possible to state it precisely. Considering such large cells leads anyway to very heavy calculations in which the use of parallel versions of the codes is almost mandatory. Various parallelization schemes are possible: on k points, fast Fourier transform, bands, spins; the parallelization schemes actually available depend on the code. The situation gets even worse when one notes that a relaxation roughly involves at least ten ground state calculations, a saddle point calculation needs about 10 complete relaxations and that each molecular dynamics simulation timestep (of about 1 fs) needs a complete ground state calculation. All in all, one can understand that the CPU time needed to complete an ab initio study (which most of the time involves various starting geometry) may amount up to hundred of thousands or millions of CPU hours.

2.3.3

Choices to make

Whatever the system considered and the code used, one needs to provide more inputs than just the atomic positions and types. Most codes suggest some values for these inputs. However their tuning may still be necessary as default values may very well be suited for some supposedly standard situations and irrelevant for others. Blind use of ab initio codes may thus lead to disappointing errors. Indeed not all these choices are trivial, so mistakes can be hard to notice for the beginner. Choices are usually made out of experience, after considering some test cases needing small calculation time. One can distinguish between choices that should be done only once at the beginning of a study and calculation parameters that can be tuned calculation by calculation. The main unchangeable choices are: the exchange and correlation functional and the pseudopotentials or PAW atomic data for the various atomic types in the calculation. First, one has to choose which flavor of the exchange and correlation functional will be used to describe the electronic interactions. Most of the time one chooses either a LDA or a GGA functional. Trends are known about the behavior of these functionals: LDA calculations tend to overestimate the bonding and underestimate the bond length in bulk materials, the opposite for GGA. However things can become tricky when one deals with defects as energy differences (between defect-containing and defect-free cells) are involved. For insulating materials or materials with correlated electrons the choice of the exchange and correlation functional is even more difficult. The second and more definitive choice is the one of the pseudopotential. We do not mean here the choice of the pseudization scheme but the choice of the pseudopotential itself. Indeed, calculated energies vary greatly with the chosen pseudopotential, so that energy differences that are thermodynamically or kinetically relevant are meaningless if the various calculations are performed with different pseudopotentials. The determination of the shape of the atomic basis set in the case of localized bases is also of importance and it is close in spirit to the choice of the pseudopotential except that much less basis sets than pseudopotentials are available. More technical inputs include: - the k point sampling. The larger the number of k points to sample the Brillouin zone, the more accurate the results but the heavier the calculations will be. This is especially true for metallic systems which need fine sampling of the Brillouin zone, but convergence with respect the number of k points can be accelerated by the introduction of a smearing of the occupations of electronic levels close to the Fermi energy. The shape and width of this smearing function is then an additional parameter.19 - The number of plane waves (obviously for plane wave codes but also for some other codes which also use FFT). Once again the larger the number of plane waves, the more accurate but the heavier the calculation. - the convergence criteria. The two major convergence criteria are the one for the self-consistent loop of the calculation of the ground state electronic wave functions and the one to signal the convergence of a relaxation calculation (with some threshold depending on the forces acting on the atoms).

3

Fields of Application

Ab initio calculations can be applied to almost any solid once the limitations in cell sizes and number of atoms are taken into account. Among the materials of nuclear interest that have been studied one can cite: metals-particularly iron, tungsten, zirconium and plutonium; alloys, especially iron alloys (FeCr, FeC to tackle steel, etc.); models of fuel materials, UO2, U-PuO2, uranium carbides; structural carbides (SiC, TiC, B4C, etc.); waste materials (zircon, pyrochlores, apatites, etc.). In this section we will rapidly expose the types of study that can be done with ab initio calculations. The last two sections on metallic alloys and insulating materials will allow us to go into details for some specific cases.

6

Ab initio Electronic Structure Calculations for Nuclear Materials

3.1

Perfect Crystal

3.1.1

Bulk properties

Dealing with perfect crystals, ab initio calculations provide information about the crystallographic and electronic structure of the perfect material. The properties of usual materials, such as standard metals, band insulators or semiconductors are basically well reproduced, though some problems remain, especially for nonconductors. But difficulties arise when one wishes to tackle the properties of highly correlated materials such as uranium oxide. For instance, no ab initio code, whatever the complexity and refinements, is able to correctly predict the fact that plutonium is non-magnetic. In such situations the nature of the chemical bonding is still poorly understood so that the correct physical ingredients are probably not present in today's codes. These especially difficult cases should not mask the very impressive precision of the results obtained for the crystal structure, cohesive energy, atomic vibrations, etc. of less difficult materials.

3.1.2

Inputs for thermodynamical models

The information on bulk materials can be gathered in thermodynamical models. Most ab initio calculations are performed at zero temperature. Even with this restriction they can be used for thermodynamical studies. First, ab initio calculations enable one to consider phases that are not accessible to experiments. It is thus possible to compare the relative stability of various (real or fictitious) structures for a given composition and pressure. Considering alloys it is possible to calculate the cohesive energy of various crystallographic arrangements. Solid solutions can also be modeled by so-called Special Quasi-random Structures (SQS).20 Beyond a simple comparison of the energies of the various structures, when a common underlying crystalline network exists for all the considered phases, the information about the cohesive energies can be used to parameterize rigid lattice inter-atomic interaction models (i.e., pair, triplet, etc. interactions) that can be used to perform computational thermodynamics. For instance these interactions can then be used in mean field or Monte Carlo simulations to predict phase stabilities at nonzero temperature.21 As examples of this kind of studies one can cite the determination of solubility limits (e.g., Zr and Sc in aluminum22), the exploration of details of the phase diagrams (e.g., the inversion of stability in the iron rich side of the Fe–Cr diagram23). Directly considering nonzero temperature in ab initio simulation is also possible, though more difficult. First, one can calculate for a given composition and structure the electronic and vibrational entropy (through the phonon spectrum), which leads to the variation of heat capacity with temperature. Nontrivial thermodynamic integrations can then be used to calculate the relative stability of various structures at nonzero temperature. Second, one can perform ab initio molecular dynamics simulations to model finite temperature properties (e.g., thermal expansion).

3.2

Defects

Point defects are of course very important in a nuclear complex as they are created either by irradiation or by accommodation of impurities (e.g., fission products). More generally they have a tremendous role in the kinetic properties of the materials. It is therefore not surprising if countless ab initio studies exist on point defects in nuclear materials. Most of them are based on a supercell approach in which the unit cell of the perfect crystal is periodically repeated up to the largest possible simulation box. A point defect is then introduced and the structure is allowed to relax. By difference with the defect-free structure one can calculate the formation energy of the defect which drives its equilibrium concentration. Some care must be taken in writing this difference as the number and types of atoms should be preserved in the process. Point defects are also the perfect object for the saddle point calculations that give the energy that drives their kinetic properties. Ab initio allows to calculate accurately these energies and also to consider (for insulating materials) the various possible charge states of the defects. They have shown that the properties of defects can vary greatly with their charge states. Many different kinds of defects can be considered. A list of possible defects follows with the characteristic associated thermodynamical and kinetic energies.

3.2.1

Self-defects

Vacancies and interstitials, with the associated formation energy driving their concentration and migration energy driving their displacement in the solid; the sum of these two energies is the activation energy for diffusion at equilibrium. For such simple defects it is possible to go beyond the 0 K energies and to access the free energies of formation and migrations by calculating the vibrational spectra in the presence of the defect in the stable position and at the saddle point (see Section 4.2.3).

3.2.2

Hetero-defects

In the nuclear context such defects can be fission products in a fuel material, actinide atoms in a waste material, helium gases in structural materials etc. ab initio gives access the solution energy of these impurities, which allows one to determine their most favored positions in the crystal: interstitial position, substitution for host atoms, etc. The kinetic energies of migration of interstitial impurities are accessible as well as the kinetic barrier for the extraction of an impurity from a vacancy site.

Ab initio Electronic Structure Calculations for Nuclear Materials 3.2.3

7

Point defect assemblies.

In this class one can include the calculation of interstitial assemblies as well as the complexes built with impurities and vacancies. One has then access to the binding of monoatomic defects to the complexes24 possibly with the associated kinetic energy barriers.

3.2.4

Kinetic models

As for perfect crystals the information obtained by ab initio calculations can be gathered and integrated in larger scale modeling especially kinetic models. Many kinetic Monte Carlo models were thus parameterized with ab initio calculations (see for instance the works on pure iron25 or FeCu26).

3.2.5

Extended defects

Even if the cell sizes accessible by ab initio calculations are small, it is possible to deal with some extended defects. Calculations then often need some tricks to accommodate the extended defect in the small cells. Some examples are given below on studies on dislocations.

3.3

Ab initio For Irradiation

Irradiation damage, especially cascade modeling, is usually preferentially dealt by larger scale methods such as empirical potentials molecular dynamics, rather than ab initio calculations. However, recent ab initio studies appeared that directly tackle irradiation processes.

3.3.1

Threshold displacement energies

First, the increase in computer power has allowed the calculations of Threshold Displacement Energies by ab initio molecular dynamics. We are aware of studies in GaN27 and silicon carbides.28,29 The procedure is the same as with empirical potentials: one initiates a series of cascades of low but increasing energy and follows the displacement of the accelerated atom. The threshold energy is reached as soon as the atom does not return to its initial position at the end of the cascade. Such calculations are very promising as empirical potentials are usually imprecise for the orders of energies and inter-atomic distances at stake in threshold energies. However, they should be done with care as most pseudopotentials and basis sets are designed to work for moderate interatomic distances and bringing two atoms too close to each other may lead to spurious results unless the pseudopotentials are specifically designed.

3.3.2

Electronic stopping power

Second, recent studies have been published in the ab initio calculations of the electronic stopping power for high velocity atoms or ions. The framework best suited to address this issue is TD-DFT. Two flavors of TD-DFT have been applied to stopping power studies so far. The first approach relies on the linear response of the system to the charged particle. The key quantity is here the density– density response function that measures how the electronic density of the solid reacts to a change in the external charge density. This observable is usually represented in reciprocal space and frequency, so it can be confronted directly with Energy Loss measurements. The density–density response function describes the possible excitations of the solid that channel an energy transfer from the irradiating particle to the solid. Most noticeably the (imaginary part of the) function vanishes for an energy lower than the band gap and shows a peak around the plasma frequency. Integrating this function over momentum and energy transfers one obtains the electronic stopping power. Campillo, Pitarke, Eguiluz and Garcia have implemented this approach and applied to some simple solids, such as aluminum or silicon.30–32 They showed that there is little difference between the usual approximations of TD-DFT: the Random Phase Approximation which means basically no exchange-correlation included, or Adiabatic LDA which means that the exchange-correlation is local in space and instantaneous in time. The influence of the band structure of the solid accounts for noticeable deviations from the homogeneous electron gas model. The second approach is more straightforward conceptually, but more cumbersome technically. It proposes to simply monitor the slowing down of the charged irradiated particle in a large box in real space and real time. The response of the solid is hence not limited to the linear response: all orders are automatically included. However, the drawback is the size of the simulation box that should be large enough to prevent interaction between the periodic images. Following this approach Pruneda and co-workers33 calculated the stopping power in a large band gap insulator, lithium fluoride, for small velocities of the impinging particle. In the small velocity regime, the nonlinear terms in the response are shown to be important. Unfortunately, whatever the implementation of TD-DFT in use, the calculations always rely on very crude approximations for the exchange-correlation effects. The true exchange-correlation kernel (the second derivative of the exchange-correlation energy with respect to the density) is in principle nonlocal (it is indeed long-ranged) and has memory. The use of novel approximations of the kernel was recently introduced by Barriga–Carrasco but for the homogeneous electron gas only.34,35

8

Ab initio Electronic Structure Calculations for Nuclear Materials

3.4

Ab initio and Empirical Potentials

Ab initio calculations are often compared to and sometimes confused with empirical potentials calculations. We will now try to clarify the differences between these two approaches and highlight their point of contacts. The main difference is of course that ab initio calculations deal with atomic and electronic degrees of freedom. Empirical potentials depend only on the relative positions of the considered atoms and ions. They do not consider explicitly electrons. Thus, roughly speaking, ab initio calculations deal with electronic structure and give access to good energetics, while empirical potentials are not concerned with electrons and give approximate energetics but allow much larger scale calculations (in space and time). Going into some details, we have hopefully shown that ab initio gives access to very diverse phenomena. Some can be modeled with empirical potentials, at least partly; others are completely outside the scope of such potentials. In the latter category one will find the phenomena which are really related to the electronic structure itself. For instance, the calculations of electronic excitations (e.g., optical or X-ray spectra) is conceptually impossible with empirical potentials. In the same way, for insulating materials, the calculation of the relative stability of various charge states of a given defect is impossible with empirical potentials. Other phenomena which are intrinsically electronic in nature can be very crudely accounted for in empirical potentials. The electronic stopping power of an accelerated particle is an example. As indicated above it can be calculated ab initio. Coversely, from the empirical potential perspective one can add an ad hoc slowing term to the dynamics of fast moving particles in solids whose intensity has to be established by fitting experimental (or ab initio) data. In a related way, some forms of empirical potentials rely on electronic information, for instance the Finnis–Sinclair36 or Rosato et al.37 forms. In the same spirit, a recent empirical potential has been designed to reproduce the local ferromagnetic order of iron.38 But this potential assumes a tendency for ferromagnetic order while ab initio calculation can (in principle) predict what the magnetic order will be. Then, ab initio is very often used as a way to get very accurate energies for a given atomic arrangement. This is the case for the formation and migration energies of defects, the vibration spectra and so on. These phenomena are conceptually within reach of empirical potentials (except the ones that reincorporate electronic degrees of freedom such as charged defects). Ab initio is then just a way to get proper and quantitative energetics. Their results are then often used as reference for fitting empirical potentials. However, the fit of a correct empirical remains a tremendous task especially with the complex forms of potentials nowadays and when one wants to correctly predict subtle, out of equilibrium, properties. Finally, one should always keep in mind that cohesion in solids is quantum in nature so that classical inter-atomic potentials dealing only with atoms or ions can never fully reproduce all the aspects of bonding in a material.

4

Metals and Alloys

The vast majority of DFT calculations on radiation defects in metallic materials have been performed in body-centered cubic iron based materials, for obvious application reasons of ferritic steels, but also because of the more severe shortcoming of predictions based only on empirical potentials. A number of accurate estimates of energies of formation and migration of self-interstitial and vacancy defects, as well as small defect clusters and solute-vacancy or solute-interstitial complexes have been obtained. DFT calculations have been intensively used to predict atomistic defect configurations, but also transition pathways. An overview of these results are presented below, complete with examples in other bcc transition metals, in particular tungsten, as well as hcp-Zr. These examples illustrated how DFT data have changed the more or less admitted energy landscape of these defects, but also how they are used to improve empirical potentials. In the final part a very brief overview of typical works on dislocations (in iron) is presented.

4.1

Pure Iron and Other bcc Metals

Ferritic steels are an important class of nuclear materials, which include reactor pressure vessel (RPV) steels and high chromium steels for elevated temperature structural and cladding materials in fast reactors and fusion reactors. From a basic science point of view, the modeling of these materials starts with that of pure iron, in the ferromagnetic body-centered cubic (bcc) structure. Iron concentrates several difficulties for DFT calculations. First, being a 3D metal, it requires rather large basis sets in planewave calculations. Second, the calculations need to be spin polarized, to account for magnetism, and this at least doubles the calculation time. But most of all, it is a case where the choice of the exchange correlation functional has a dramatic effect on bulk properties. The standard LDA incorrectly predicts the paramagnetic face centered cubic (fcc) structure to be more stable than the ferromagnetic bcc structure. The correct ground state is recovered using gradient corrected functionals,39 as illustrated in Figure 1. Finally, it was pointed out that pseudopotentials tend to overestimate the magnetic energy in iron,40 and therefore some pseudopotentials suffer from a lack of transferability for some properties. In practice, however, in the large set of the results obtained over the last decade for defect calculations in iron a quite remarkable agreement is obtained between the various computational approaches. With a very few exceptions, they are indeed quite independent on the form of the GGA functional, the basis set (planewave or localized), and the pseudopotential or the use of PAW approaches.

Ab initio Electronic Structure Calculations for Nuclear Materials

9

Figure 1 Calculated total energy of paramagnetic (P) bcc and fcc and ferromagnetic (F) bcc iron as function of Wigner–Seitz radius (s). The dotted curve corresponds to the Local Spin Density (LSD) approximation and the solid curve corresponds to the GGA functional proposed by Perdew and Wang in 1986 (PW) (after Ref. 39). The curves are displaced in energy so that the minima for F bcc coincide. Energies are in Ry (1 Ry¼ 13.6057 eV) and distances in bohr (1 bohr¼0.5292 Å ).

4.1.1

Self-interstitials and self-interstitial clusters in Fe and other bcc metals

The structure and migration mechanism of self-interstitials in iron is a very good illustrative example of the recent impact of DFT calculations on radiation defect studies. Progress in methods, codes and computer performance made this archetype of radiation defects accessible to DFT calculations in the early 2000s, since total energy differences between simulation cells of 128 þ 1 atoms could then be obtained with a sufficient accuracy. In 2001 Domain and Becquart indeed reported that, in agreement with experiment, the 〈110〉 dumbbell was the most stable structure.41 Quite unexpectedly, the 〈111〉 dumbbell was predicted to be B0.7 eV higher in energy, at variance with empirical potential results which predicted a much smaller energy difference. DFT calculations performed in other BCC metals revealed that this is a peculiarity of Fe,42 as illustrated in Figure 2, and magnetism was proposed to be the origin of the energy increase of the 〈111〉 dumbbell in Fe. The important consequence of this result in Fe, which has been confirmed repeatedly since then, is that it excludes the SIA migration to occur by long 1D glides of the 〈111〉 dumbbell followed by on-site rotations of the 〈110〉 dumbbell, as predicted previously from empirical potential MD simulations. Moreover DFT investigation of the migration mechanism yielded a quantitative agreement with experiment for the energy of the Johnson translation-rotation mechanism (see Figure 3), namely B0.3 eV.43 These DFT calculations were followed by a very successful example of synergy between DFT and empirical potentials. The DFT values of interstitial formation energies in various configurations and inter-atomic forces in a liquid model have indeed been included in the database for a fit of EAM type potentials by Mendelev et al.45 This approach has resulted in a new generation of improved empirical potentials, albeit still with some limitations. When considering SIA clusters made of parallel dumbbells, the Mendelev potential agrees with DFT for predicting a crossover as a function of cluster size from the 〈110〉 to the 〈111〉 orientation between four and six SIA clusters.44 A new class of interstitial clusters in BCC metals has recently been observed in electronic structure calculations.46 A three dimensional periodic structure appears for self-interstitial clusters in body-centered-cubic metals, as opposed to the conventional two dimensional loop morphology. The underlying crystal structure corresponds to the C15 Laves phase. The new three dimensional structures generalize previous observations.47 From the methodological point of view this study46 is an example of how to go beyond the standard techniques in the exploration and characterization of complex energy landscape as those of point defects using an Eigenvector Following method coupled to lattice dynamics free energy calculations,48,49 it was possible to reach regions of phase space inaccessible by standard molecular techniques. Finally using DFT calculations on typical configurations, it was shown that in a-iron these C15 aggregates are highly stable and immobile and that they exhibit large antiferromagnetic

10

Ab initio Electronic Structure Calculations for Nuclear Materials

Figure 2 Formation energies of several basic SIA configurations calculated for bcc transition metals of group 5B (left) and group 6B (right), taken from.42 Reproduced from Nguyen Manh, D., Horsfiels, A.P., Dudarev, S.L., Phys. Rev. B 2006 73, 020101. Data for BCC Fe are taken from Ref. 43.

Figure 3 Left (after Ref. 43): Johnson translation-rotation mechanism of the 〈110〉 dumbbell; white and black spheres indicate the initial and final positions of the atoms respectively. Right (after Ref. 44): Comparison between the DFT-GGA result and two EAM potentials for the energy barriers of the Johnson mechanism and the 〈110〉 to 〈111〉 transformation.

moments. The impact of the exchange-correlation functional as well as the type of pseudopotential used in the relative stability between the C15 cluster and traditional loops has been further investigated.50 It was found that using more cumbersome but more exact calculations in the PAW formalism with semicore states results in an improved the stability of C15 cluster in alpha. These clusters form directly in displacement cascades and they can grow by capturing self-interstitials. Even if the density and kinetic of such C15 clusters is still under debate,51 these new morphologies of self-interstitial clusters are an important element to consider to tackle the microstructural evolution of iron base materials under irradiation, Figure 4. To summarize, the energy landscape of interstitial type defects has been profoundly revisited in the last decade driven by DFT calculations, in synergy with empirical potential calculations.

Ab initio Electronic Structure Calculations for Nuclear Materials

11

Figure 4 Structure of small C15 interstitial clusters in a BCC lattice. (A) Representation by vacancies (blue cubes) and interstitials (orange spheres) of the tetra-interstitial cluster. (B) Same as (A) in a skeleton representation, i.e., without the vacancies and the cubic lattice. (C) unit cell of the Mg Cu2 C15 Laves structure, with the Mg atoms in green and the Cu atoms in orange. (D) The structures traditional planar tetra-interstitial cluster, the atoms in interstitial position are represented by pink spheres and vacancies by dark blue cubes.

4.1.2

Vacancy and vacancy clusters in Fe and other bcc metals

DFT has some limitations for predicting accurate vacancy formation energies in transition metals. The exceptional agreement with experiment obtained initially within DFT-LDA,52 was later shown to result from a cancellation between two effects. First, the structural relaxation, which was neglected in Ref. 52, is now known to significantly reduce the vacancy formation energy, in particular in BCC metals.53 Second, due to limitations of exchange-correlation functionals at surfaces, DFT-LDA tends to underestimate the vacancy formation energy. This discrepancy is even larger within DFT-GGA, and it increases with the number of valence electrons. It is therefore rather small for early transition metals (Ti, Zr, Hf,), but it is estimated to be as large as 0.2 eV in LDA and 0.5 eV in GGA-PW1 for late transition metals (Ni, Pd, Pt).54 However, the effect is much weaker for migration energies.54 A new functional, AM05, has been proposed to cope with this limitation.55 Less spectacular effects are expected in vacancy type defects than in interstitial type defects when going from empirical potentials to DFT calculations. The discussion on vacancy type defects in Fe will be restricted to the results obtained within DFTGGA, due to the superiority of this functional for bulk properties. For pure Fe, DFT-GGA vacancy formation and migration energies are in the range of 1.93–2.23 eV and 0.59–0.71 eV.41,43,56 These values are in quite good agreement with experimental estimates at low temperatures in ultra-pure iron, namely 2.070.2 eV and 0. 55 eV respectively. These values can be reproduced by empirical potentials when included in the fit, but one discrepancy remains with DFT concerning the shape of the migration barrier. It is indeed clearly a single hump in DFT25 and usually a double hump with empirical potentials. Concerning vacancy clusters, the structures predicted by empirical potentials, namely compact structures, were confirmed by DFT calculations but there are discrepancies in the migration energies. In both cases, the most stable di-vacancy is the next-nearestneighbor configuration, with a binding energy of 0.2–0.3 eV.25,57,58 The migration can occur by two different two-step processes, with an intermediate configuration that is either nearest-neighbor or fourth nearest-neighbor.58 A quite unexpected result of DFT calculations was the prediction of rather low migration energies for the tri- and quadri-vacancies, namely 0.35 and 0.48 eV.25 Depending on the potential, this phenomenon is either not reproduced or only partly reproduced (see Figure 5).59 Stronger deviations from empirical potential predictions for di-vacancies are observed in DFT calculations performed in other bcc metals. The most dramatic case is that of tungsten, where the next-nearest-neighbor interaction is strongly repulsive (0.5 eV), and the nearest-neighbor one is vanishing.60 This result does not explain why voids are formed in tungsten under irradiation.

4.1.3

Finite temperature effects on defect energetics

The properties of radiation defects at high temperature may change due to three possible contributions to the free energy: electronic, magnetic and vibrational. These three effects can be well modeled in bulk bcc iron,61 but they are more challenging for defects. The electronic contribution, which exists only in metals, arises due to changes in the density of states close to the Fermi level. The electronic entropy difference between, for example, two configurations is, to first order, proportional to the temperature, T, and the change in density of states at the Fermi level. This electronic effect is straightforward to take into account in DFT calculations. It was shown in tungsten to decrease the activation free energy for self diffusion by up to 0.4 eV close to the melting temperature. Thus, although this effect is relatively small in general, it cannot be neglected at high temperature.

12

Ab initio Electronic Structure Calculations for Nuclear Materials

Figure 5 Migration energies of vacancy clusters in Fe, as function of cluster size.59

The magnetic contribution is important in iron. Spin fluctuations were shown to be the origin of the strong softening of the C' elastic constant observed as the temperature increases up to the a  g transition temperature,62 and it drives, for instance, the temperature dependence of relative abundance of 〈100〉 and 〈111〉 interstitial loops formed under irradiation.63 It is also known to have a small effect on vacancy properties, but to the authors knowledge there is presently no tractable method to predict quantitatively this effect for point defects from DFT calculations. This is probably one of the important challenges in the field. Finally, vibrational entropy effects can in principle be obtained either in the quasi-harmonic approximation from phonon frequency calculations, or directly from first principles molecular dynamics. There are very few examples of such calculations in the literature. The vibrational modes of vacancies and self-interstitials in iron have been investigated by DFT calculations, and their formation entropies have been estimated.64 As illustrated recently in Mo, it is also possible to calculate the temperature dependence of the vacancy formation enthalpy, from DFT Molecular Dynamics simulations, including anharmonic effects, as well as the defect jump frequency, going beyond the transition state approximation.65

4.2 4.2.1

Beyond Pure Iron Helium-vacancy clusters in iron and other bcc metals

Irradiation of metals by neutrons produces, besides point defects, rare gases by transmutation reactions. Helium is a major concern since it has a very low solubility in metals. The Effects of Helium in Irradiated Structural Alloys. It is deeply trapped by vacancies and helium-vacancy clustering can ultimately lead to bubble formation and void swelling. At variance with empirical potential predictions, DFT calculations showed that interstitial helium is unambiguously located on tetrahedral sites, not only in iron, but also in all other bcc metals.66–69 An improved Fe-He pair potential was then obtained by fitting to the DFT results.70 DFT calculations account for the very fast migration of interstitial helium, as well as for its deep trapping to vacancies, although not as deep as predicted by empirical potentials. Note that an unexpected effect was observed in Vanadium, where the helium atom in a vacancy is found to be off-centered.68 The energy landscape of the helium di-vacancy complex also revealed unexpected configurations, in particular for the lowest energy configuration where the helium atom is located halfway between two nearest-neighbor vacancies (see Figure 6). More generally, a systematic study of the energetics of all small HenVm clusters in iron was performed,66 giving very useful data for the kinetic modeling of helium-vacancy clustering and dissociation.71 Quite interestingly, interstitial He atoms are found to attract each other, even in the absence of vacancies.66,24 This clustering of helium atoms may then yield the emission of self-interstitials. Finally, the interaction of helium with self-interstitials is, as expected, much weaker but also attractive.24 Similar studies have been performed on small helium-vacancy clusters in tungsten,60,72 but also on the behavior of hydrogen in iron and tungsten. It should be noted that at low temperature, quantum effects must be taken into account in the migration properties in particular for hydrogen.73

4.2.2

From pure iron to steels: The role of carbon

In steels, the presence of carbon, even though its concentration is very low, considerably affects defect properties because of the strong carbon-defect interaction. DFT calculations reproduce the well known fact that carbon is located in octahedral sites, and they also confirm the strong attraction between interstitial carbon and a mono-vacancy, with a binding energy of about 0.5 eV.74–76 This strong attraction is the origin of the confusing discrepancy between the vacancy migration energy in ultra-pure iron, B0.6 eV, and the effective vacancy migration energy in iron with carbon or in steels, i.e., B1.1 eV, which corresponds to first order to the sum of the vacancy migration energy and the carbon-vacancy binding energy.76 More interestingly, DFT calculations predict that the complex formed by a vacancy and two carbon atoms, VC2, is extremely stable, due to the formation of a strong covalent bond between the carbon atoms. The VC–C binding energy is indeed close to 1 eV,74–76 and VC2 complexes are expected to play a very important role.

Ab initio Electronic Structure Calculations for Nuclear Materials

13

The interaction between carbon and self-interstitials is also attractive, but weaker. In agreement with experiments,77 DFT calculations confirmed a binding energy of B0.2 eV78 and predict, at variance with initial empirical potential results, that the nearest-neighbor configurations are repulsive and that the most attractive configuration is that shown in Figure 7. This shortcoming of empirical potentials was overcome recently with an improved potential derived taking into account information from the electronic structure.79 The strong interaction of carbon with vacancies also affects the energetics of helium-vacancy clusters, and it is important to take it into account to reproduce, for example, thermal helium desorption experiments performed in iron.80 Similar calculations have been performed with nitrogen.74

4.2.3

Interaction of point defects with alloying elements or impurities in iron

The diffusion of point defects produced by irradiation may induce fluxes of solutes, for example, toward or away from defect sinks, depending on the defect-solute interactions. DFT is again a very powerful tool to predict such interactions, which can then be used in kinetic models. This approach is also useful in the absence of irradiation and a very interesting example has been obtained in the simulation of the first stages of the coherent precipitation of copper in bcc-Fe. DFT calculations predicted that the vacancy formation energy in metastable bcc-Cu (which is not known experimentally since bulk Cu is fcc) is 0.9 eV, i.e., much smaller than that in bcc iron, namely 2.1 eV. This leads to strong trapping of vacancies by the Cu precipitates. As a result, precipitates containing up to several tens of copper atoms are quite surprisingly predicted to be much more mobile than individual copper atoms in the iron matrix.26 Another very illustrative example is given by the study of atomic transport via interstitials in dilute Fe–P alloys. DFT results indeed predict that Fe–P mixed dumbbells are highly mobile but that they can be deeply trapped by a substitutional P atom.81 A systematic study of of the interaction of mono-vacancies and self-interstitials with all transition-metal solutes has even been reported recently (see Figure 8).82

4.2.4

From dilute to concentrated alloys: the case of Fe–Cr

In the approach described above, which considers low concentrations of solutes and defects, the number of independent configurations is rather small, and they can be easily taken into account in kinetics model. The situation is much more complex when considering Fe–Cr with Cr concentration in the range 10–20%.

Figure 6 Schematic representation of the energy landscape of the HeV2 complex.66

Figure 7 structure of carbon-vacancy and carbon-self-interstitial complexes in iron, predicted from DFT calculations (from Ref. 76).

14

Ab initio Electronic Structure Calculations for Nuclear Materials

Figure 8 DFT-GGA solute-vacancy binding energies in iron for 3d, 4d, and 5d elements for 1-nn to 5-nn relative positions.82

FeCr alloys are composed of elements of similar atomic volumes in a bcc lattice. At variance with most transition-metal alloys, several properties in FeCr alloys are strongly dictated by the magnetism. Since a few decades ago, magnetic properties of FeCr multilayers have been intensively studies by electronic structure methods and experimentally, due to the interests for magnetic devices and the development of giant magneto-resistance. More recently, due to the potential application of these alloys as structural materials for advanced fission and future fusion reactors, many investigations are performed, for instance based on DFT. These studies showed a fundamental link between magnetism and various thermodynamic, segregation, defect and kinetic properties of bulk FeCr alloys.83–89 For instance, FeCr mixing energies show a nonstandard behavior, exhibiting negative values on the dilute-Cr side and positive above 10–15 atomic%-Cr.85,84,90 This indicates a transition from chemical ordering to the phase separation (and precipitation) tendency. The emergence of nano-clusters and precipitates during the so-called a-a' phase decomposition is observed experimentally if the chemical content of the alloy is inside the miscibility gap of the binary FeCr phase diagram. The phase decomposition may also be induced or accelerated by irradiation.91 As mentioned in various studies,84,85,90 the change of mixing tendency is closely linked to magnetic properties of FeCr solid solution, that is, Cr atoms in an extremely dilute alloy are stabilized by the induced magnetic moments, while at higher Cr concentration in a random alloy, magnetic frustration emerges as the number of neighboring Cr atoms increases. Based on these evidences, it is clear that quantitative predictions of energetic and diffusion properties in FeCr alloys can only rely on methods with explicit evaluation of electronic structures and magnetism (DFT, tight-binding, …). Moreover, magnetic frustrations often emerge in FeCr systems, especially in the vicinity of defects. They occur if it is not possible to satisfy simultaneously the ferromagnetic coupling between Fe atoms, and the antiferromagnetic tendency between neighboring Fe–Cr and Cr–Cr atoms. As a result, complex magnetic configurations appear, including noncollinear magnetic arrangements. It is shown to be the case in the vicinity of interfaces between the precipitates and the matrix.92–94 It is worth noting that the magnetism of the precipitates may affect the interfacial energies,93 diffusion of vacancies and solutes92,95–97 as well as the Curie temperature of the alloys.94,98 For an accurate description of magnetic structures, theoretical approaches beyond the standard ones may be required. Recent, noncollinear DFT studies and DFT-based magnetic cluster expansion simulations have addressed some of these features at low and high temperatures, respectively.93,94

4.2.5

Point defects in hcp-Zr

Point defects in hcp-Zr have also been studied via DFT calculations, with a special emphasis on their potential diffusion anisotropy. This anisotropy, which arises from the reduced symmetry of the hcp lattice, is often assumed to explain the self-organization

Ab initio Electronic Structure Calculations for Nuclear Materials

15

of the microstructure observed in irradiated zirconium,99,100 as well as the breakaway growth visible for high irradiation doses.100,101 It was found that the vacancy migration energy is lower by B0.15 eV within the basal plane than out of the basal plane.102,103 An attractive interaction between vacancies is observed when they are in first nearest-neighbor position. The interaction becomes repulsive and cancels out for larger separation distance.103 A similar interaction has been obtained in Ti,104,105 another hcp transition metal belonging to the same column of the periodic table as Zr. This attractive interaction is responsible for vacancy clustering, with cavity-like compact clusters being the most stable for small clusters containing at most seven vacancies.103 This looks at first sight in contradiction with experiments as cavities are seldom observed in irradiated or quenched zirconium. But the modeling of larger vacancy clusters through continuous laws parameterized on ab initio data (surface and stacking fault energies) show that the vacancy loops lying in the prismatic planes, i.e., 〈a〉 loops, are the most stable defects at large sizes,103 thus reconciling atomistic simulations with experiments. Finally, the ab initio study106 of the interaction between vacancy clusters and hydrogen reveals a trapping of H by vacancy loops, with a stronger trapping, thus a stronger stabilization, obtained for vacancy loops lying in the basal planes, i.e., the 〈c〉 loops responsible for the breakaway growth. The situation for the self-interstitial is quite complex, since among the known configurations, at least three configurations are found to have the almost the same formation energy (within 0.1 eV): the octahedral (O), basal split dumbbell (BS), and basal octahedral (BO) configurations.107,108 Ab initio calculations with large enough supercells containing more than 300 lattice sites,100,102 and including an energy correction to remove the spurious elastic interaction of the point defect with its periodic images,109 have concluded that the BO configuration is the most stable. These calculations100,102 also evidenced the existence of some new metastable configurations with lower symmetries than the usually assumed configurations for a self-interstitial atom in a hcp lattice. The diffusion coefficient, and its anisotropy, could be estimated from the ab initio migration energies.110 The diffusion is faster in the basal plane than along the 〈c〉 axis, but the anisotropy is less prominent than for the vacancy. This therefore casts doubt on the generally accepted assumption that the diffusion anisotropy is responsible for the self-organization of the microstructure under irradiation and for the apparition of vacancy loops lying in the basal planes at high irradiation doses. But another explanation has been recently put forward.111 The elastic dipoles of the self-interstitial configurations, i.e., the 2nd rank tensor characterizing the interaction of the point defect with an external applied strain, have been extracted from ab initio calculations. These dipoles evidence a stronger interaction with the components of the external stress field lying in the basal plane, which could result in the aforementioned characteristic of the irradiation microstructure.

4.3

Dislocations

The collective behavior of dislocations can be described thanks to dislocation dynamics codes. In order to reinforce the physical foundation, input data like mobility laws can be obtained from atomistic calculations of individual dislocations. These defects can now be investigated using more accurate ab initio electronics structure methods. We exemplify these studies by focusing in the following on the properties of dislocations in body-centered cubic (bcc) metals (especially iron) and hcp zirconium. In these materials dislocation's properties are known to be closely related to their core structure. When dealing with dislocations special care should be taken in the positioning of the dislocations and on the boundary conditions of the calculations. For instance, considering 〈111〉 screw dislocations in bcc lattice, the two cell geometries proposed in the literature – the cluster approach112 and the periodic array of dislocation dipoles113 – have been thoroughly compared.114 The construction of simulation cells appropriate for such extended defects should be optimized for cell sizes accessible to DFT calculations, and the cell-size dependence of the energetics evidenced in both the cluster approach and the dipole approach for various cell and dipole vectors should be rationalized. The quadrupolar arrangement of dislocation dipoles is most widely used for such calculations114 although the cluster approach with flexible boundary conditions can be considered a reference method when no energies are necessary (i.e., only structures).

4.3.1

Dislocations in bcc metals

DFT calculations in bcc transition metals (V, Nb, Ta, Cr, Mo, W, and Fe),112,114–122 predict a nondegenerate structure for the easy core, as illustrated in Figure 9 for Fe using differential displacement maps as proposed by Vitek.123 The edge component reveals the existence of a significant core dilatation effect in addition to the Volterra field, which can be successfully accounted for by an anisotropic elasticity model.124 Thanks to the good control of energy it is also possible to obtain quantitative results on the Peierls potential, namely the two dimensional energy landscape seen by a straight screw dislocation as it moves perpendicular to the Burgers vector. This is exemplified in the following Figure 10(a) for Fe, where a high symmetry direction of the Peierls potential is sampled: the line going from an easy to a hard-core position along the glide direction, i.e., the Peierls barrier. These calculations were performed by simultaneously displacing the two dislocations constituting the dislocation dipole in the same direction, and by using a constrained relaxation method. In the same work, the behavior of the Mendelev,45 Gordon,125 and Marinica126 potentials for iron, which gives the correct nondegenerate core structure unlike most other potentials, have been tested against the obtained DFT results. It appears that the Peierls barrier displayed by Mendelev and Gordon potentials yield a camel hump shape, as illustrated in

16

Ab initio Electronic Structure Calculations for Nuclear Materials

Figure 9 (a) Differential displacement map of the nondegenerate core structure in Fe, as obtained from SIESTA GGA. (b) Displacement map after subtraction of the Volterra anisotropic elastic field: along [111] (or screw component) and magnified by a factor of 20. (c) Displacement map after subtraction of the Volterra anisotropic elastic field: in the (111) plane (or edge component) and magnified by a factor of 50.114

Figure 10 (a) Peierls barriers in Fe from DFT, and from Mendelev,45 Gordon,125 and Marinica126 potentials. Reproduced from Ventelon, L., Willaime, F., Clouet, E., Rodney, D., 2013. Acta Mater. 61, 3973–3985. (b) Peierls potentials in Fe from DFT, from the analytical fit of Ref. 127, and from Mendelev45 and Marinica potentials.126 Reproduced from Dezerald, L., Ventelon, L., Clouet, E., Denoual, C., Rodney, D., Willaime, F., 2014. Phys. Rev. B 89, 024104–13.

Figure 10(a), and at the halfway position, the core spreads between two easy core positions, which corresponds to the split core configuration, whereas it exhibits a single hump barrier within DFT.119 Marinica potential does not exhibit such an intermediate metastable configuration. It was also shown from DFT calculations that the Peierls barriers in other bcc metals (V, Nb, Ta, Mo, W) are also single-humped.120 More insight into the 2D Peierls potential can be gained by looking at the pathway joining the hard-core to the split core configurations, which can be used along with the Peierls barrier to obtain the 2D Peierls potential from a numerical interpolation, as represented in Figure 10(b) for Fe. In this figure, the 2D Peierls potential obtained with DFT is compared to the simple analytical potential proposed by Edagawa et al.127 and often used in the literature, and to Mendelev45 and Marinica126 potentials. The 2D Peierls potential proposed by Edagawa and those fitted on EAM data are qualitatively similar with a global maximum at the hardcore position and strongly differ from the DFT calculations, which show that the split core is the maximum energy configuration and that the hard-core is a monkey saddle. DFT calculations in other bcc metals show that the former feature is common to all investigated bcc metals while the latter feature is specific to Fe.120 Finally, a line tension model based solely on the line tension and Peierls barrier values allows to predict kink properties from DFT calculations performed in cells containing only a few hundred atoms.128 This methodology is used to determine the kink-pair formation enthalpy in bcc transition metals from DFT.129

4.3.2

Dislocations in hcp zirconium

DFT calculations in hcp zirconium predicts that a screw dislocation of Burgers vector 1/3 〈1-210〉 spontaneously dissociate in two partial dislocations in the prismatic plane,130 because of the existence of a stable stacking fault in this plane for a fault vector 1/6 〈1-210〉. Ab initio calculations predict that the Peierls stress, i.e., the stress which needs to be applied on the dislocation to make it glide without any thermal activation, is lower than 21 MPa. This is in good agreement with experiments showing that screw dislocations can easily glide in their prismatic habit plane in pure zirconium. Another configuration, where the screw dislocation dissociates in a basal plane, may also exist in hcp metals. But according to ab initio calculations, this configuration is unstable in zirconium.130 This contrasts with what is usually obtained with empirical potentials, like EAM potentials, where this basal dissociation is obtained as a metastable configuration, if not the most stable one.

Ab initio Electronic Structure Calculations for Nuclear Materials

17

This competition between a prismatic and a basal dissociation of 1/3 〈1-210〉 dislocations in hcp transition metals is controlled by the electronic filling of the valence d band,131 leading to an angular dependence of the atomic bonding that cannot be caught by simple central-force empirical potentials, thus justifying the need for ab initio approached. At high enough temperatures, not only prismatic glide is observed, but secondary glide systems are also activated: screw dislocations can escape their prismatic habit plane to glide in either pyramidal or basal planes. Ab initio calculations combined with the nudged elastic band method (NEB) have shown that both events share the same thermally activated process with an unusual conservative motion of the prismatic stacking fault perpendicularly to itself.132 The minimum energy path goes through an intermediate metastable configuration, which corresponds to a dissociation of the dislocation in a first order pyramidal plane. Such a dissociation can be rationalized by the existence of a stable stacking fault in this pyramidal plane,132,133 the atomic structure of this fault being equivalent to a two-layer pyramidal twin.

5

Insulators

From the atomistic and electronic structure point of view it is legitimate to distinguish between electrically conducting materials on one hand and insulating or semiconducting materials on the other. Indeed, insulating materials exhibit specific behaviors, especially for the point defects. Due to the existence of a gap in the electronic density of states, the point defects may be charged. There is recent evidence that the properties of the point defects, especially their kinetic properties, such as the migration energy, depend a lot on their charge state. The charge of a given point defect depends on the position of the Fermi level within the band gap: a low lying Fermi level (close to the valence band) favors positively charged defects while a Fermi level close to the conduction band favors negatively charged defects. The positions of the Fermi level corresponding to transitions between charge states are called Charge Transition Levels (CTL). The correct determination of these CTL allows the correct prediction of the charge states of the defects, as piloted by Fermi level position, i.e., the doping conditions for the material. Standard DFT methods fail to reproduce accurately these CTL and the research of more accurate methods is presently a very active field in the electronic structure community with major implications for microelectronic research as well as for nuclear materials, especially in view of the aforementioned variation of point defect kinetic properties with their charge state. All these charge aspects of point defects are completely out of range for empirical potentials. In the last two sections we exemplify the researches on insulating materials by summarizing the available results for two important insulating nuclear materials: silicon carbide and uranium dioxide. Silicon carbide is an important candidate material for fusion and fission applications. Even if it arguably a less crucial material than UO2, we start with this material as its electronic structure is simpler. UO2 is obviously the basic model material for the nuclear fuel of usual reactors.

5.1

Silicon Carbide

This brief survey exemplifies the kind of calculations that can be performed on common insulating materials (as opposed to correlated ones such as UO2) in a nuclear context. Specificities of insulating materials when compared to metallic systems will clearly appear, especially for what concerns the possible charge states of the defects and the difficulties standard DFT calculations have in satisfactorily reproducing the quantities that govern them. SiC exists in many different structures. Nuclear applications are interested with the so-called b structure (3C–SiC), a zinc blende crystal cubic form. We shall therefore focus on this structure, although many additional calculations have been performed on other structures of the hexagonal type which are more of interest for micro-electronics applications. Silicon carbide is a band insulator whose bulk structural properties are well reproduced by usual DFT calculations. Electronic structure of the bulk material is also well reproduced except for the usual underestimation of the band gap by DFT calculations. Indeed the measured gap is 2.39 eV134 while standard DFT-LDA calculations give 1.30 eV.134

5.1.1

Point defects

The first DFT calculations of point defects in silicon carbide,135 dating back to 1988, were burdened by strong limitations in computing time. For this reason they were performed with relatively small supercells (16 and 32 atoms), largely insufficient basis sets (plane waves with energy up to 28 Ry) and further approximations, namely for the relaxation of atomic positions. Moreover they were limited to high symmetry configurations. The results were only qualitative; however, it was already clear that vacancies and antisites could be relatively abundant, at equilibrium, with respect to interstitial defects. The authors dared to approach some defect complexes and could predict that antisite pairs and di-vacancies were bound. Vacancies were thoroughly studied at the turn of the century.134,136–138 The most prominent result may be the metastability of the silicon vacancy. Indeed, following a suggestion coming from a self-consistent DFT based tight-binding calculation by Rauls et al.,139 the electron paramagnetic resonance (EPR) spectra of annealed samples of irradiated SiC was measured140 and compared to calculated hyperfine parameters. This showed that silicon vacancies are metastable with regards to a carbon vacancy–carbon antisite complex (VC–CSi); a fact which has since been consistently confirmed by the other calculations.

18

Ab initio Electronic Structure Calculations for Nuclear Materials

Interstitials were less studied than vacancies. One should however mention a paper141 devoted to carbon and silicon in interstitials in silicon carbide. Beyond these papers dedicated to one type of defect, very complete and comprehensive work on both vacancies and interstitials were also published. One should cite142 devoted to the formation energies of defects, while143 goes further as it also covers migration energetics of basic intrinsic defects (vacancies, interstitials, antisites). It is worth noting that in such covalent compounds there are many possible atomic structures for defects as simple as a mono interstitial and that all these structure must be considered in the calculation (see Figure 11). As examples, the results of these various papers for what concerns formation energies and CTL of vacancies are summarized in the following tables. One can see a general agreement on the formation energies of the neutral defects, especially in the recent references. The small differences are related to k-point sampling or cell size in the calculations. Larger discrepancies appear between the various predicted CTL. They relate to the inaccuracy of standard DFT calculations in treating empty or defect states. A simple example relates directly to the underestimation of the band gap : the silicon interstitial (in the ISi TC configuration) in the neutral state shows up as metallic in standard calculations, the defect states lying inside the conduction band. This fact, on one side, calls for a better description of the exchange correlation potential for these configurations; on the other it makes the convergence with k points and cell size very slow, as has recently been pointed out.144 This drawback of standard DFT-LDA/GGA supercell calculations is common to other defects in SiC. Even when calculated defect states fall within the band gap, their position inside it can be grossly miscalculated with standard DFT calculations. The errors produced by standard DFT calculations for the CTL are well known nowadays. The determination of an accurate method to calculate these CTL is an active field of research with works on advanced methods such as GW (e.g., the results on SiO2145) or hybrid functionals.146 For what concerns nuclear materials, and especially SiC, GW corrections and excitonic effects will allow further comparisons with experiments.147 Vacancies

5.1.2

Defect kinetics

Before the aforementioned work by Bockstedte and co-workers143 almost no work was devoted to migration properties of point defects in SiC. We should, however, cite previous preliminary works by the same group,148,149 a work on the mechanisms of formation of antisite pairs,150 as well as a work on vacancy migration published in 2003.151 The comprehensive study of migration barriers in143 showed, first of all, that vacancies have much higher migration energies than interstitials: higher than 3 eV for the formers in the neutral state, around 1 eV for the latter (0.5 for IC; 1.4 for ISi). Another remarkable finding is the strong variation of the migration energy with the charge state; indeed the migration energy for the carbon vacancy is raised by almost 2 eV going from the neutral to the 2 þ charged state, while the silicon vacancy finds its migration barrier reduced by 1 eV when its charge goes from neutral to 2  . Interstitials are reported to have their lowest migration barriers in the neutral state, except for the ISi TC configuration, which is expected to have an almost zero energy barrier of migration in the 2 þ and 3 þ charge states. Such large changes in the migration energies of defects with their charge should induce tremendous variations in their kinetic behavior under different charge states. 5.1.2.1 Recombinations The energy barriers of recombinations of close interstitial vacancy pairs has also been tackled.152–154 It appears that the energetic landscape for the recombination of Frenkel pairs is extremely complex. One should distinguish the regular recombination of an homo interstitial-vacancy pairs from those of hetero interstitial-vacancy pairs, which leads to the formation of an antisite. Recent works tend to suggest that the latter may, in certain conditions, have a lower energy than the recombination of a regular Frenkel pair. A kinetic bias for the formation of antisites, preliminary to decomposition, may thus be active in SiC under irradiation.155

Figure 11 Possible geometries for a carbon interstitial in cubic SiC from.143

Ab initio Electronic Structure Calculations for Nuclear Materials

19

Calculations of threshold displacement energies from first principles molecular dynamics29 have also been reported. Their results show that this quantity is strongly anisotropic and they found average values (38 eV for Si and 19 for C) that are in agreement with currently accepted values (coming from experimental evaluations that are, however, largely dispersed). These calculations evidence that available CPU power is now large enough to calculate TDE from ab initio molecular dynamics. This is good news as empirical potentials are basically not reliable in the prediction of TDE.

5.1.3

Defect complexes

Several defect complexes have been studied by first principles calculations in silicon carbide. The identification of EPR signals, Deep Level Transient Spectroscopy (DLTS) or photoluminescence (PL) experiments based on calculated properties has been attempted for some of them. Crucial for these identifications is the reliability of the predictions of charge transition levels (for the position of DLTS peaks) and of annealing temperatures, through more or less complicated mechanisms. One of the first, and simplest, defect complex identified through comparison of theory and experiment was the VC–CSi coming from the annealing of silicon vacancies in 6H–SiC, as previously mentioned. More complex antisite defects or antisite complexes156,157 as well as di-vacancy complexes158–160 were called upon for the attribution of PL or EPR peaks. Various kinds of carbon clusters were studied in detail theoretically.161–164 The cited works deal with the stability, electrical properties and Local Vibrational Modes (LVM) of several structures. It was shown that the aggregation of carbon interstitials with carbon antisites can lead to various bound configurations. In particular two, three or even four carbon atoms can substitute one silicon atom forming very stable structures. The binding energy of these structures are high: from 3.9 to 5 eV, according to the charge state, for the (C2)Si, and further energy is gained when adding further carbon atoms. Silicon clusters did not raise as much interest as carbon ones, however a recent work144 deals with the stability and dynamics of such silicon clusters (see Figure 12).

5.1.4

Impurities

The interest in SiC as a large band gap semiconductor for electronic applications has promoted works on typical dopants. Most of the calculations focus on hexagonal SiC but one can reasonably assume that the results would not be very different in cubic SiC. One can find calculations dealing with boron166,167 as an acceptor and nitrogen168,169 or phosphorus170 as donors. Other impurities were studied: transition metals171–173 oxygen,174 important for the behavior of the SiO2/SiC interface, hydrogen,175–177 rare gases178 and palladium.179 A systematic study of substitutional impurities has recently appeared,180 which focused on the trends of carbon vs. silicon substitution according to the position of species in the periodic table.

5.1.5

Extended defects

Another major subject, which has attracted much interest for the hexagonal types of SiC is related to the electronic properties of extended defects, surfaces/interfaces, stacking faults and dislocations. The reason why extended defects have been mainly studied in the hexagonal types of silicon carbide lies in the fact that electronic properties of dislocations and stacking faults are particularly important for understanding the degradation of hexagonal SiC devices181 and the remarkable enhancement of dislocation velocity under illumination in the hexagonal phase.182 Nevertheless, some works have been done for cubic SiC on the electronic structure of stacking faults183–188 and various types of dislocations.189–192 Obviously, a lot of work remains to be done for the extended defects in b SiC.

Figure 12 energetic landscape of silicon mono and di-interstitial in cubic SiC from.165

20 5.2 5.2.1

Ab initio Electronic Structure Calculations for Nuclear Materials Uranium oxide Bulk electronic structure

Due to its technological importance and the complexity of its electronic structure, uranium oxide has become is one of the test cases for beyond LDA methods. Indeed, UO2 comes out as a metal when its electronic structure is calculated with LDA or GGA. This result has been found by many authors using many different codes or numerical schemes (the primary calculation being193). The physical difficulty lies in the fact that UO2 is a Mott insulator. f electrons are indeed localized on uranium atoms and are not spread over the material as usual valence electrons are. The first correction that has been applied is the LDA þ U correction in which a Hubbard U term acting between f electrons is added ‘by hand’ to the Hamiltonian.194,195 This method allows the opening of a f–f gap.194 However it suffers from the existence of multiple minima in the calculations so that the search for the real ground state is rather tricky as the calculation is easily trapped in metastable states.196 Two ways of circumventing this multiple minima problem have been proposed: the so-called occupation matrix control197 and the U-ramping method,198 the former being the more accurate. Hybrid functionals are another type of advanced methods which are very often used nowadays in the quantum chemistry community. Their principle is to mix a part of Hartree-Fock exact exchange with a DFT calculation, an application to UO2 has been made by.199 These methods are very promising for solid state nuclear materials. However, the same problem of metastability as in LDA þ U exists for such hybrid functionals200 and the computational load is much heavier than in common or LDA þ U calculations. Recently an alternative to LDA þ U has been proposed : the so-called Local Hybrid functional for Correlated Electrons201 in which the hybrid functional is applied only to the problematic f electrons. An application on UO2 is available.202

5.2.2

Point defects

While UO2 comes out as a metal with LDA or GGA DFT calculations, its structural properties are quite well reproduced by these standard methods. Based on this observation, some studies using this standard framework have been published on the point defects.203–205 The values obtained for the formation energies for the composite defects (Oxygen and uranium Frenkel pairs and Schottky defect) compare well with experimental estimates. However, as UO2 is predicted to be a metal with such methods, it is impossible to consider the charge state of the defects. The þ U correction has become the standard way to account for the insulating nature of the material. The importance of taking into account the possible charge of defects has long been neglected. So that initial studies using this correction still focused on neutral defects.196,206–208 The discrepancies between the results obtained in these various studies are larger than the spread usually observed in ab initio calculations. for instance the formation energy of the oxygen Frenkel pair is found anywhere between 2.6 eV209 and 6.5 eV.196 This suggests some hidden problem in the calculations probably related to the possible occurrence of metastable minima in the calculations. Since the studies of Nerikar et al.210 and Crocombette et al.211 most calculations take into account the possible charge of defects (e.g., 212 and 213), with at least the consideration of neutral of formally charged defects.214 These recent studies also deal more properly with multiple minima problem highlighted in the previous section. Beyond the formation energy of isolated defects, some studies focus on their migration.215–217 Despite all the recent progress and the multiplication of ab initio work on UO2, some spread remains in the results as well as a large uncertainty in terms of the possible of taking into account the spin-orbit coupling or the Van Der Waals forces. It is not clear whether a satisfactory reproduction of point defects in UO2 with ab initio is yet at hand.

5.2.3

Oxygen clusters

Another point of interest beyond point defects is the clustering of oxygen interstitials. Indeed oxygen interstitial clustering has been deduced from diffraction experiments218 many years ago. However, a debate remains on the exact shape of such clusters. Two configurations are contemplated: the so-called Willis clusters218,219 or cubo-octahedral clusters that have been observed by neutron diffraction in U4O9220 and U3O7.221 These clusters are made of 12 oxygen and eight uranium atoms and amount for four oxygen interstitials. An additional oxygen interstitial may reside in the center of the cluster, forming a so-called filled cube-octahedral cluster (with five interstitials). Recent calculations have proved that Willis clusters are in fact unstable and transform upon relaxation into assemblies of three or four interstitials surrounding a central vacancy clusters (Figure 13).222 The three interstitial- one vacancy cluster has been found independently by other authors215,224 who refer to it as split-di-interstitials. These clusters prove in fact to have a formation energy higher than the cube-octahedral cluster (Figure 14), especially the filled one.222,223 Clusters calculations have recently developed enough to allow tackling the subtleties of the structure of UO2 þ x and ordered over stoichiometric structures (e.g., U4O9).225–228

5.2.4

Impurities

Lattice sites and solution energies of fission products (FP) are of major importance in fundamental studies of nuclear fuels. They pilot the dependence of the behavior of FP on fuel stoichiometry and temperature, as well as their possible release from the fuel in the context of a direct storage of spent fuel. As experimental studies in this field are very difficult, ab initio results are of great value.

Ab initio Electronic Structure Calculations for Nuclear Materials

21

Figure 13 relaxation process of a Willis cluster of oxygen interstitials in UO2 from 222.

Figure 14 Cuboctahedral cluster of oxygen interstitials in UO2 from.233

In such studies one considers the insertion of a Fission atom in interstitial or vacant sites of UO2. A difficulty arises for the latter case.229 Indeed one then has to distinguish between the incorporation energy, defined as the energy to incorporate the FP in a pre-existing vacancy site, and the solution energy, which is the one relevant for full thermodynamical equilibrium, in which the amount of available vacant site is taken into account. One then adds to the incorporation energy the so-called apparent formation energy, which is defined as the logarithm of the vacancy concentration multiplied by the temperature. Such apparent formation energies depend on the stoichiometry of UO2 þ x. A positive (resp. negative) solution energy then means that the FP is insoluble (resp. soluble) in UO2 þ x. The first DFT study of the incorporation of a fission product in UO2 is the one by Petit et al.230 on krypton in the late 1990s. It was performed within the LMTO-ASA formalism, which could give only qualitative results.229 used more modern plane wave formalism to calculate the insertion of some fission products (krypton, iodine, cesium, strontium, and helium) but neglected atomic relaxation, which limits the accuracy of the results. Freyss et al.231 considered He and Xe. All these calculations were performed with standard LDA. More recent works works always included a þ U correction. While the first papers dealt only with interstitial and mono-vacancy sites, more recent works may also consider di-vacancy or tri-vacancy sites that often appear to be the most stable sites for FPs. Many FPs have been recently considered. At the time of writing one could find in the literature, beyond the works already mentioned, calculations on helium,232 iodine,233,234 xenon,235 strontium,235 cesium,235–237 molybdenum,238 and zirconium.238 The present tendency is to go beyond the solution energies and to tackle the migration properties, for example, Yun et al.239 dealing with helium, Wang et al. dealing with various actinides240 and Andersson et al.214,241studying in detail the mechanisms of Xe diffusion.

6

Conclusion

The examples discussed above have hopefully shown the tremendous interest of ab initio calculations for nuclear materials. Indeed they allow the qualitative and most of the time quantitative calculations of the basic energetic and kinetic properties that have a major influence on the behavior of the materials at the atomic scale.

Ab initio Electronic Structure Calculations for Nuclear Materials

22

Table 1

Formation energies and charge transition levels of vacancies in silicon carbide VC

135 242 136 134 243 143 142

VSi

0

þ /0

þ/þ þ

0

þ /0

0/ 

/ 

5.6 – 4.01 – 3.74 3.78 3.84

1.7 – 1.41 – 1.18 – –

1.9 – 1.72 – 1.22 1.29 –

7.6 – 8.74 7.7 8.37 8.34 8.78

– 0.54 0.43 0.50 – 0.18 0.41

– 1.06 1.11 0.56 0.57 0.61 0.88

– 1.96 1.94 1.22 1.60 1.76 1.40

For metallic materials the common theoretical framework works quite well. One can thus nowadays tackle objects of increasing complexity, for example, assemblies of defects or dislocations. The main limit for these materials is the severe restriction in possible cell sizes. Silicon carbide exemplifies the successes of ab initio methods in modeling the properties of a band insulator of interest for the nuclear industry. However, some difficulties remain, especially for what concerns the correct prediction of CTL in these materials. In actinide materials, the case of uranium oxide, by far the most studied of the actinide compounds of interest as a nuclear materials, shows that a lot of information can be obtained, for example, for the solution energies of Fission products or the structure of oxygen interstitial clusters. However, this information remains only qualitative, due to the very complex electronic structure of such actinide compounds with localized f electrons. The solution for these difficulties with insulating materials should come from the current developments of hybrid functional, GW or DFT þ DMFT calculations with the drawback that these advanced methods are at least one order of magnitude heavier than the standard ones. Ab initio methods have thus brought a lot of information for nuclear materials and will certainly continue to do so. Conversely, nuclear materials are a very challenging field for the use of these ab initio methods in many aspects: physical principles, numerical schemes, practical implementation, etc. We would like to thank Drs. Guido Roma, Lisa Ventelon, Fabien Bruneval, Emmanuel Clouet, Chuchun Fu, and Mihai-Cosmin Marinica for their valuable input, Table 1.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Harrison, W.A., 1980. Electronic Structure and the Properties of Solids. San Francisco: Freeman. Kohanoff, J., 2006. Electronic Structure Calculations for Solids and Molecules. Cambridge: Cambridge University press. Martin, R.M., 2004. Electronic Structure. Cambridge: Cambridge University Press. Springborg, M., 2000. Methods of Electronic-Structure Calculations. Chishester: Wiley. WIEN2K. Available at: http://www.wien2k.at/ (accessed 16.9.15). Schwartz, K., Blaha, P., Madsen, G.H.K., 2001. Comp. Phys. Commun. 147, 71–76. Soler, J.M., Artacho, E., Gale, J.D., et al., 2002. J. Phys.: Condens. Matter 14, 2745. SIESTA. Available at: http://www.icmab.es/siesta/ (accessed 16.9.15). Pickett, W.E., 1989. Comput. Phys. Rep. 9, 115. Gaussian. Available at: http://www.gaussian.com/index.htm (accessed 16.9.15). BigDFT. Available at: http://inac.cea.fr/L_Sim/BigDFT/ (accessed 16.9.15). Troullier, N., Martins, J.L., 1991. Phys. Rev. B 43, 1993. Vanderbilt, D., 1990. Phys. Rev. B 41, 7892. Blochl, P.E., 1994. Phys. Rev. B 50, 17953–17979. VASP. Available at: cms.mpi.univie.ac.at/vasp/ (accessed 16.9.15). Quantum Espresso. Available at: www.quantum-espresso.org/ (accessed 16.9.15). ABINIT. Available at: http://www.abinit.org (accessed 16.9.15). Car, R., Parrinello, M., 1988. Phys. Rev. Lett. 60, 204. Methfessel, M., Paxton, A.T., 1989. Phys. Rev. B 40, 3616. Zunger, A., Wei, S.H., Ferreira, L.G., Bernard, J.E., 1990. Phys. Rev. Lett. 65, 353. Van de Walle, C.G., 2008. Nat. Mater. 7, 455. Clouet, E., Barbu, A., Lae, L., Martin, G., 2005. Acta Mater. 53, 2313–2325. Olsson, P., Abrikosov, I.A., Vitos, L., Wallenius, J., 2003. J. Nucl. Mater. 321, 84–90. Fu, C.C., Willaime, F., 2007. J. Nucl. Mater. 367, 244–250. Fu, C., Dalla Torre, J., Willaime, F., Bocquet, J.-L., Barbu, A., 2005. Nat. Mater. 4, 68–74. Soisson, F., Fu, C.C., 2007. Phys. Rev. B 76, 12. Xiao, H.Y., Gao, F., Zu, X.T., Weber, W.J., 2009. J. Appl. Phys. 105, 5. Gao, F., Xiao, H.Y., Zu, X.T., Posselt, M., Weber, W.J., 2009. Phys. Rev. Lett. 103, 4. Lucas, G., Pizzagalli, L., 2005. Phys. Rev. B 72, 161202. (R). Campillo, I., Pitarke, J.M., Eguiluz, A.G., 1998. Phys. Rev. B 58, 10307–10314.

Ab initio Electronic Structure Calculations for Nuclear Materials 31. Campillo, I., Pitarke, J.M., Eguiluz, A.G., Garcia, A., 1998. Nucl. Instrum. Methods Phys. Res. Sect. B-Beam Interact. Mater. Atoms 135, 103–106. 32. Pitarke, J.M., Campillo, I., 2000. Nucl. Instrum. Methods Phys. Res. Sect. B-Beam Interact. Mater. Atoms 164, 147–160. 33. Pruneda, J.M., Sanchez-Portal, D., Arnau, A., Juaristi, J.I., Artacho, E., 2007. Phys. Rev. Lett. 99, 235501. 34. Barriga-Carrasco, M.D., 2009. Nucl. Instrum. Methods Phys. Res. Sect. A-Accel. Spectrom. Detect. Assoc. Equip. 606, 215–217. 35. Barriga-Carrasco, M.D., 2009. Phys. Rev. E 79, 027401. 36. Finnis, M.W., Sinclair, J.E., 1984. Philos. Mag. A-Phys. Condens. Matter Struct. Defects Mech. Prop. 50, 45–55. 37. Rosato, V., Guillope, M., Legrand, B., 1989. Philos. Mag. A-Phys. Condens. Matter Struct. Defects Mech. Prop. 59, 321–336. 38. Derlet, P.M., Dudarev, S.L., 2007. Prog. Mater. Sci. 52, 299–318. 39. Bagno, P., Jepsen, O., Gunnarsson, O., 1989. Phys. Rev. B 40, 1997–2000. 40. Kresse, G., Joubert, D., 1999. Phys. Rev. B 59, 1758–1775. 41. Domain, C., Becquart, C.S., 2002. Phys. Rev. B 65, 024103. 42. Nguyen-Manh, D., Horsfield, A.P., Dudarev, S.L., 2006. Phys. Rev. B 73, 020101. 43. Fu, C.C., Willaime, F., Ordejon, P., 2004. Phys. Rev. Lett. 92, 175503. 44. Willaime, F., Fu, C.C., Marinica, M.C., J, D.T., 2005. Nucl. Instrum. Meth. Phys. Res. B 228, 92. 45. Mendelev, M.I., Han, S., Srolovitz, D.J., et al., 2003. Philosophical Magazine 83, 3977–3994. 46. Marinica, M.C., Willaime, F., Crocombette, J.P., 2012. Phys. Rev. Lett 108. 47. Bacon, D., Gao, F., Osetsky, Y., 2000. J. Nucl. Mater. 276, 1. 48. Barkema, G.T., Mousseau, N., 1996. Phys. Rev. Lett 77, 4358–4361. 49. Marinica, M.C., Willaime, F., Mousseau, N., 2011. Phys. Rev. B 83, 094119. 50. Dézerald, L., Marinica, M.C., Ventelon, L., Rodney, D., Willaime, F., 2014. J. Nucl. Mater. 449, 219−224. 51. Zhang, Y.F., Bai, X.M., Tonks, M.R., Biner, S.B., 2015. Scr. Mater. 98, 5–8. 52. Korhonen, T., Puska, M.J., Nieminen, R.M., 1995. Phys. Rev. B 51, 9526–9532. 53 Satta, A., Willaime, F., de Gironcoli, S., 1999. Phys. Rev. B 60, 7001–7005. 54. Mattsson, T.R., Mattsson, A.E., 2002. Phys. Rev. B 66, 214110. 55. Armiento, R., Mattsson, A.E., 2005. Phys. Rev. B 72, 085108. 56. Huang, S.Y., Worthington, D.L., Asta, M., et al., 2010. Acta Mater. 58, 1982–1993. 57. Becquart, C.S., Domain, C., 2003. Nucl. Instrum. Methods Phys. Res. Sect. B-Beam Interact. Mater. Atoms 202, 44–50. 58. Djurabekova, F., Malerba, L., Pasianot, R.C., Olsson, P., Nordlund, K., 2010. Philos. Mag. 90, 2585–2595. 59. Fu, C.C., Willaime, F.,2004, Unpublished. 60. Becquart, C.S., Domain, C., 2007. Nucl. Instrum. Methods Phys. Res. Sect. B-Beam Interact. Mater. Atoms 255, 23–26. 61. Kormann, F., Dick, A., Grabowski, B., et al., 2008. Phys. Rev. B 78, 033102. 62. Hasegawa, H., Finnis, M.W., Pettifor, D.G., 1985. J. Phys. F-Met. Phys. 15, 19–34. 63. Dudarev, S.L., Bullough, R., Derlet, P.M., 2008. Phys. Rev. Lett 100, 135503. 64. Lucas, G., Schaublin, R., 2009. Nucl. Instrum. Methods Phys. Res. Sect. B-Beam Interact. Mater. Atoms 267, 3009–3012. 65. Mattsson, T.R., Sandberg, N., Armiento, R., Mattsson, A.E., 2009. Phys. Rev. B 80, 224104. 66. Fu, C.C., Willaime, F., 2005. Phys. Rev. B 72, 064117. 67. Willaime, F., Fu, C.C., 2007. Mater.Res. Soc. Symp. Proc. 981, JJ05–04. 68. Seletskaia, T., Osetsky, Y., Stoller, R.E., Stocks, G.M., 2008. Phys. Rev. B 78, 134103. 69. Zu, X.T., Yang, L., Gao, F., et al., 2009. Phys. Rev. B 80, 054104. 70. Juslin, N., Nordlund, K., 2008. J. Nucl. Mater. 382, 143–146. 71. Ortiz, C.J., Caturla, M.J., Fu, C.C., Willaime, F., 2007. Phys. Rev. B 75, 100102. 72. Becquart, C.S., Domain, C., 2009. J. Nucl. Mater. 385, 223–227. 73. Heinola, K., Ahlgren, T., 2010. J. Appl. Phys. 107, 113531. 74. Domain, C., Becquart, C.S., Foct, J., 2004. Phys. Rev. B 69, 144112. 75. Forst, C.J., Slycke, J., Van Vliet, K.J., Yip, S., 2006. Phys. Rev. Lett 96, 175501. 76. Fu, C.C., Meslin, E., Barbu, A., Willaime, F., Oison, V., 2008. Theor. Model. Numer. Simul. Multi-Phys. Mater. Behav. 139, 157–164. 168. 77. Takaki, S., Fuss, J., Kugler, H., Dedek, U., Schultz, H., 1983. Radiat. Eff. Defects Solids 79, 87–122. 78. Malerba, L., Ackland, G.J., Becquart, C.S., et al., J. Nucl. Mater. In Press, Corrected Proof. 79. Hepburn, D.J., Ackland, G.J., 2008. Phys. Rev. B 78, 165115. 80. Ortiz, C.J., Caturla, M.J., Fu, C.C., Willaime, F., 2009. Phys. Rev. B 80, 134109. 81. Meslin, E., Fu, C.C., Barbu, A., Gao, F., Willaime, F., 2007. Phys. Rev. B 75, 094303. 82. Olsson, P., Klaver, T.P.C., Domain, C., 2010. Phys. Rev. B 81, 054102. 83. Ackland, G.J., 2006. Phys. Rev. Lett 97, 015502. 84. Klaver, T.P.C., Drautz, R., Finnis, M.W., 2006. Phys. Rev. B 74, 094435. 85. Nguyen-Manh, D., Lavrentiev, M.Y., Dudarev, S.L., 2007. J. Comput-Aided Mater. Des. 14, 159–169. 86. Senninger, O., Martinez, E., Soisson, F., Nastar, M., Brechet, Y., 2014. Acta Mater. 73, 97–106. 87. Ruban, A.V., Korzhavyi, P.A., Johansson, B., 2008. Phys. Rev. B 77, 5. 88. Ropo, M., Kokko, K., Airiskallio, E., et al., 2011. J. Phys.-Condes. Matter 23, 8. 89. Korzhavyi, P.A., Ruban, A.V., Odqvist, J., Nilsson, J.O., Johansson, B., 2009. Phys. Rev. B 79, 16. 90. Levesque, M., Martinez, E., Fu, C.C., Nastar, M., Soisson, F., 2011. Phys. Rev. B 84, 10. 91. Bachhav, M., Yao, L., Odette, G.R., Marquis, E.A., 2014. J. Nucl. Mater. 453, 334–339. 92. Soulairol, R., Fu, C.C., Barreteau, C., 2011. Phys. Rev. B 83, 11. 93. Lavrentiev, M.Y., Soulairol, R., Fu, C.-C., Nguyen-Manh, D., Dudarev, S.L., 2011. Phys. Rev. B 84, 144203. 94. Fu, C.-C., Lavrentiev, M.Y., Soulairol, R., Dudarev, S.L., Nguyen-Manh, D., 2015. Phys. Rev. B 91, 094430. 95. Olsson, P.r., Domain, C., Wallenius, J., 2007. Phys. Rev. B 75, 014110. 96. Martinez, E., Senninger, O., Fu, C.-C., Soisson, F., 2012. Phys. Rev. B 86, 224109. 97. Wen, H., Ma, P.-W., Woo, C.H., 440 (2013) 428−434. 98 Yamamoto, H., 1964. Jpn. J. Appl. Phys. 3, 745. 99. Woo, C.H., 1988. J. Nucl. Mater. 159, 237–256. 100. Samolyuk, G.D., Golubov, S.I., Osetsky, Y.N., Stoller, R.E., 2013. Philos. Mag. Lett 93, 93–100. 101. Christien, F., Barbu, A., 2009. J. Nucl. Mater. 393, 153–161. 102. Vérité, G., Willaime, F., Fu, C.C., 2007. Solid State Phenom. 129, 75. 103. Varvenne, C., Mackain, O., Clouet, E., 2014. Acta Mater. 78, 65–77.

23

24

Ab initio Electronic Structure Calculations for Nuclear Materials

104. Raji, A.T., Scandolo, S., Mazzarello, R., Nsengiyumva, S., Harting, M., Britton, D.T., 2009. Philos. Mag. 89, 1629–1645. 105. Connetable, D., Huez, J., Andrieu, E., Mijoule, C., 2011. J. Phys.-Condes. Matter 23, 14. 106. Varvenne, C., Mackain, O., Clouet, E., 2014. Acta Mater. 78, 65–77. 107. Willaime, F., 2003. J. Nucl. Mater. 323, 205–212. 108 Domain, C., Legris, A., 2005. Philos. Mag. 85, 569–575. 109. Varvenne, C., Bruneval, F., Marinica, M.C., Clouet, E., 2013. Phys. Rev. B 88, 7. 110. Samolyuk, G.D., Barashev, A.V., Golubov, S.I., Osetsky, Y.N., Stoller, R.E., 2014. Acta Mater. 78, 173–180. 111. Rouchette, H., Thuinet, L., Legris, A., Ambard, A., Domain, C., 2014. Phys. Rev. B 90, 12. 112. Woodward, C., Rao, S.I., 2002. Phys. Rev. Lett 88, 216402. 113. Cai, W., Bulatov, V.V., Chang, J.P., Li, J., Yip, S., 2003. Philos. Mag. 83, 539–567. 114. Ventelon, L., Willaime, F., 2007. J. Comput-Aided Mater. Des. 14, 85–94. 115. Frederiksen, S.L., Jacobsen, K.W., 2003. Philos. Mag. 83, 365–375. 116. Segall, D.E., Strachan, A., Goddard, W.A., Ismail-Beigi, S., Arias, T.A., 2003. Phys. Rev. B 68, 014104. 117. Romaner, L., Ambrosch-Draxl, C., Pippan, R., 2010. Phys. Rev. Lett 104, 195503. 118. Domain, C., Monnet, G., 2005. Phys. Rev. Lett 95, 215506. 119. Ventelon, L., Willaime, F., Clouet, E., Rodney, D., 2013. Acta Mater. 61, 3973–3985. 120. Dezerald, L., Ventelon, L., Clouet, E., et al., 2014. Phys. Rev. B 89, 13. 121. Itakura, M., Kaburaki, H., Yamaguchi, M., 2012. Acta Mater. 60, 3698–3710. 122. Weinberger, C.R., Tucker, G.J., Foiles, S.M., 2014. Phys. Rev. B 87. 123. Vitek, V., 1974. Cryst. Lattice Defects 5, 1–34. 124. Clouet, E., Ventelon, L., Willaime, F., 2009. Phys. Rev. Lett. 102, 055502. 125. Gordon, P.A., Neeraj, T., Mendelev, M.I., 2011. Philos. Mag. 91, 3931–3945. 126. Proville, L., Rodney, D., Marinica, M.-C., 2012. Nat. Mater. 11, 845–849. 127. Edagawa, K., Suzuki, T., Takeuchi, S., 1997. Phys. Rev. B 55, 6180–6187. 128. Proville, L., Ventelon, L., Rodney, D., 2013. Phys. Rev. B 87, 8. 129. Dezerald, L., Proville, L., Ventelon, L., Willaime, F., Rodney, D., 2015. Phys. Rev. B 91, 7. 130. Clouet, E., 2012. Phys. Rev. B 86, 11. 131. Legrand, B., 1984. Philos. Mag. B-Phys. Condens. Matter Stat. Mech. Electron. Opt. Magn. Prop. 49, 171–184. 132. Chaari, N., Clouet, E., Rodney, D., 2014. Phys. Rev. Lett 112, 5. 133. Chaari, N., Clouet, E., Rodney, D., 2014. Metall. Mater. Trans. A-Phys. Metall. Mater. Sci. 45A, 5898–5905. 134. Torpo, L., et al., 1999. Appl. Phys. Lett. 74, 221. 135. Wang, C., Berholc, J., Davis, R.F., 1988. Phys. Rev. B 38, 12752. 136. Zywietz, A., Furthmüller, J., Bechstedt, F., 1999. Phys. Rev. B 59, 15166. 137. Umeda, T., et al., 2005. Phys. Rev. B 71, 913202. 138. Umeda, T., et al., 2004. Phys. Rev. B 70, 235212. 139. Rauls, E., et al., 2000. Phys. Stat. Sol. 217, R1. 140. Lingner, T., et al., 2001. Phys. Rev. B 64, 245212. 141. Lento, J.M., et al., 2004. J. Phys. Condens. Matter 16, 1053. 142. Bernardini, F., Mattoni, A., Colombo, L., 2004. Eur. Phys. J. B 38, 437. 143. Bockstedte, M., Heid, M., Pankratov, O., 2003. Phys. Rev. B 67, 193102. 144. Liao, T., Roma, G., Wang, J.Y., 2009. Philos. Mag. 89, 2271–2284. 145. Martin-Samos, L., Roma, G., Rinke, P., Limoge, Y., 2010. Phys. Rev. Lett 104, 075502. 146. Heyd, J., Scuseria, G.E., Ernzerhof, M., 2006. J. Chem. Phys. 124, 219906. 147. Bockstedte, M., Marini, A., Gali, A., Pankratov, O., Rubio, A., 2009. Mater. Sci. Forum 600−603, 285–290. 148. Bockstedte, M., Pankratov., O., 2000. Mater. Sci. Forum 338−342, 949. 149. Mattausch, A., Bockstedte, M., Pankratov, O., 2001. Mater. Sci. Forum 353−356, 323. 150. Rauls, E., et al., 2001. Mater. Sci. Forum 353−356, 435. 151. Rurali, R., et al., 2003. Comp. Mater. Sci. 27, 36. 152. Rauls, E., et al., 2001. Physica B 308−310, 645. 153. Bockstedte, M., Mattausch, A., Pankratov, O., 2004. Phys. Rev. B 69, 235202. 154. Lucas, G., Pizzagalli, L., 2007. J. Phys.-Condes. Matter 19, 086208. 155. Roma, G., Crocombette, J.P., 2010. J. Nucl. Mater. 403, 32–41. 156. Eberlein, T.A.G., et al., 2002. Phys. Rev. B 65, 184108. 157. Gao, F., 2007. Appl. Phys. Lett. 90, 221915. 158. Torpo, L., Staab, T.E.M., Nieminen, R.M., 2002. Phys. Rev. B 65, 085202. 159. Gerstmann, U., Rauls, E., Overhof, H., 2004. Phys. Rev. B 70, 201204. 160. Son, N.T., et al., 2006. Phys. Rev. Lett. 96, 055501. 161. Gali, A., et al., 2003. Phys. Rev. B 68, 215201. 162. Mattausch, A., Bockstedte, M., Pankratov, O., 2004. Phys. Rev. B 69, 045322. 163. Mattausch, A., Bockstedte, M., Pankratov, O., 2004. Phys. Rev. B 70, 235211. 164. Gali, A., Son, N.T., Janzén, E., 2006. Phys. Rev. B 73, 033204. 165. Liao, T., 2009. Unpublished. 166. Rurali, R., et al., 2004. Phys. Rev. B 69, 125203. 167 Bockstedte, M., Mattausch, A., Pankratov, O., 2004. Phys. Rev. B 70, 115203. 168. Eberlein, T.A.G., Jones, R., Briddon, P.R., 2003. Phys. Rev. Lett. 90, 225502. 169. Gerstmann, U., et al., 2003. Phys. Rev. B 67, 205202. 170. Rauls, E., et al., 2004. Phys. Rev. B 70, 085202. 171. Overhof, H., 1997. Mater. Sci. Forum 258−263, 677. 172. Barbosa, K.O., Machado, W.V.M., Assali, L.V.C., 2001. Phys. B 308−310, 308–310. 173. Yurieva, E.I., 2006. Comp. Mater. Sci 36, 194. 174. Gali, A., et al., 2002. Phys. Rev. B 66, 125208. 175. Aradi, B., et al., 2001. Phys. Rev. B 63, 245202. 176. Gali, et al., 2002. Appl. Phys. Lett. 80, 237.

Ab initio Electronic Structure Calculations for Nuclear Materials 177. Gali, A., et al., 2005. Phys. Rev. B 71, 035213. 178. Van Ginhoven, R.M., Chartier, A.C., Meis, C., Weber, W.J., Corrales, L.R., 2006. J. Nucl. Mater. 348, 51–59. 179 Roma, G., 2009. J. Appl. Phys. 106, 123504. 180. Miyata, M., Higashiguchi, Y., Hayafuji, Y., 2008. J. Appl. Phys. 104, 123702. 181. Skowronski, M., Ha, S., 2006. J. Appl. Phys. 99, 011101. 182. Galeckas, A., Linnros, J., Pirouz, P., 2006. Phys. Rev. Lett. 96, 025502. 183. Lambrecht, W.R.L., Miao, M.S., 2006. Phys. Rev. B 73, 155316. 184. Iwata, H., et al., 2002. J. Phys. Condens. Matter 14, 12733. 185. Iwata, H., et al., 2002. Phys. Rev. B 65, 033203. 186. Lindefelt, U., et al., 2003. Phys. Rev. B 67, 155204. 187. Iwata, et al., 2003. J. Appl. Phys. 93, 1577. 188. Iwata, et al., 2004. J. Appl. Phys. 94, 4972. 189. Sitch, P.K., et al., 1995. Phys. Rev. B 52, 4951. 190. Blumenau, A.T., et al., 2003. Phys. Rev. B 68, 174108. 191. Savini, G., et al., 2007. Phys. B 401−402, 62. 192. Savini, G., et al., 2007. New J. Phys. 9, 6. 193. Arko, A.J., Koelling, D.D., Boring, A.M., Ellis, W.P., Cox, L.E., 1986. J. Less Common Met. 122, 95. 194. Dudarev, S.L., Mahn, D.N., Sutton, A.P., 1997. Phil. Mag. B 75, 613. 195. Savrasov, S.L.D.G.B.S., Temmermen, Z.S.W., Sutton, P.A., 1998. Phys. Stat. Sol (A) 166, 429. 196. Dorado, B., Amadon, B., Freyss, M., Bertolus, M., 2009. Phys. Rev. B 79, 235125. 197. Dorado, B., Freyss, M., Amadon, B., et al., 2013. J. Phys.-Condes. Matter 25, 333201. 198 Meredig, B., Thompson, A., Hansen, H.A., Wolverton, C., van de Walle, A., 2010. Phys. Rev. B 82, 195128. 199. Kudin, K., Scuseria, G., Martin, G., 2002. Phys. Rev. Lett 89, 266402. 200. Kasinathan, D., Kunes, J., Koepernik, K., et al., 2006. Phys. Rev. B 74, 195110. 201. Tran, F., Blaha, P., Schwarz, K., Novak, P., 2006. Physical Rev. B 74, 155108. 202. Jollet, F., Jomard, G., Amadon, B., Crocombette, J.P., Torumba, D., 2009. Phys. Rev. B 80, 235109. 203. Petit, T., Lemaignan, C., Jollet, F., Bigot, B., Pasturel, A., 1998. Phil. Mag. B 77, 779. 204. Freyss, M., Petit, T., Crocombette, J.P., 2005. J. Nucl. Mater. 347, 44–51. 205. Crocombette, J.P., Jollet, F., Thien Nga, L., Petit, T., 2001. Phys. Rev. B 64, 104107. 206. Iwasawa, M., Chen, Y., Kaneta, Y., et al., 2006. Materials Transactions 47, 2651–2657. 207. Gupta, F., Brillant, G., Pasturel, A., 2007. Philos. Mag. 87, 2561–2569. 208. Geng, H.Y., Chen, Y., Kaneta, Y., et al., 2008. Phys. Rev. B 77, 104120. 209. Yu, J.G., Devanathan, R., Weber, W.J., 2009. J. Phys.-Condes. Matter 21, 435401. 210. Nerikar, P., Watanabe, T., Tulenko, J.S., Phillpot, S.R., Sinnott, S.B., 2009. J. Nucl. Mater. 384, 61–69. 211. Crocombette, J.-P., Torumba, D., Chartier, A., 2011. Phys. Rev. B 83, 184107. 212. Vathonne, E., Wiktor, J., Freyss, M., Jomard, G., Bertolus, M., 2014. J. Phys.-Condes. Matter 26, 325501. 213. Crocombette, J.P., 2012. Phys. Rev. B 85, 144101. 214. Andersson, D.A., Uberuaga, B.P., Nerikar, P.V., Unal, C., Stanek, C.R., 2011. Phys. Rev. B 84, 054105. 215. Andersson, D.A., Watanabe, T., Deo, C., Uberuaga, B.P., 2009. Phys. Rev. B 80, 060101. (R). 216. Dorado, B., Andersson, D.A., Stanek, C.R., et al., 2012. Phys. Rev. B 86, 035110. 217. Dorado, B., Garcia, P., Carlot, G., et al., 2011. Phys. Rev. B 83, 035126. 218. Willis, B.T.M., 1964. Proc. Br. Ceram. Soc. 1, 9. 219. Willis, B.T.M., 1987. J. Chem. Soc., Faraday Trans. 2 83, 1073–1081. 220. Bevan, D.J.M., Grey, I., Willis, B.T.M., 1986. J. Solid State Chem. 61, 1–7. 221. Garrido, F., Ibberson, R.M., Nowicki, L., Willis, B.T.M., 2003. J. Nucl. Mater. 322, 87–89. 222. Geng, H.Y., Chen, Y., Kaneta, Y., Kinoshita, M., 2008. Appl. Phys. Lett. 93. 223. Geng, H.Y., Chen, Y., Kaneta, Y., Kinoshita, M., 2008. Phys. Rev. B 77, 180101. (R). 224. Andersson, D.A., Lezama, J., Uberuaga, B.P., Deo, C., Conradson, S.D., 2009. Phys. Rev. B 79, 024110. 225. Wang, J., Ewing, R.C., Becker, U., 2013. Sci. Rep 4. 226. Andersson, D.A., Baldinozzi, G., Desgranges, L., Conradson, D.R., Conradson, S.D., 2013. Inorg. Chem. 52, 2769–2778. 227. Brincat, N.A., Molinari, M., Parker, S.C., Allen, G.C., Storr, M.T., 2015. J. Nucl. Mater. 456, 329–333. 228. Geng, H.Y., Song, H.X., Jin, K., Xiang, S.K., Wu, Q., 2011. Phys. Rev. B 84. 229. Crocombette, J.P., 2002. J. Nucl. Mater. 305, 29–36. 230. Petit, T., Jomard, G., Lemaignan, C., Bigot, B., Pasturel, A., 1999. J. Nucl. Mater. 275, 119. 231. Freyss, M., Vergnet, N., Petit, T., 2006. J. Nucl. Mater. 352, 144–150. 232. Gryaznov, D., Heifets, E., Kotomin, E., 2009. Phys. Chem. Chem. Phys. 11, 7241–7247. 233. Dorado, B., Freyss, M., Martin, G., 2009. Eur. Phys. J. B 69, 203–209. 234. Crocombette, J.P., 2012. J. Nucl. Mater. 429, 70–77. 235. Nerikar, P.V., Liu, X.Y., Uberuaga, B.P., et al., 2009. J. Phys.-Condes. Matter 21, 435602. 236. Gupta, F., Pasturel, A., Brillant, G., 2009. J. Nucl. Mater. 385, 368–371. 237. Brillant, G., Gupta, F., Pasturel, A., 2009. J. Phys.-Condes. Matter 21, 9. 238. Brillant, G., Pasturel, A., 2008. Phys. Rev. B 77, 184110. 239. Yun, Y., Eriksson, O., Oppeneer, P.M., 2009. J. Nucl. Mater. 385, 72–74. 240. Wang, J., Becker, U., 2013. J. Nucl. Mater. 433, 424–430. 241. Andersson, D.A., Garcia, P., Liu, X.Y., et al., 2014. J. Nucl. Mater. 451, 225–242. 242. Wimbauer, T., et al., 1997. Phys. Rev. B 56, 7384. 243. Torpo, L., et al., 2001. J. Phys. Condens. Matter 13, 6203.

25