Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials

Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials

1.08 Ab Initio Electronic Structure Calculations for Nuclear Materials J.-P Crocombette and F Willaime Commissariat a` l’Energie Atomique, DEN, Service de Recherches de Me´tallurgie Physique, Gif-sur-Yvette, France ß 2012 Elsevier Ltd All rights reserved 1.08.1 1.08.2 1.08.2.1 1.08.2.2 1.08.2.2.1 1.08.2.2.2 1.08.2.3 1.08.2.3.1 1.08.2.3.2 1.08.2.3.3 1.08.3 1.08.3.1 1.08.3.1.1 1.08.3.1.2 1.08.3.2 1.08.3.2.1 1.08.3.2.2 1.08.3.2.3 1.08.3.2.4 1.08.3.2.5 1.08.3.3 1.08.3.3.1 1.08.3.3.2 1.08.3.4 1.08.4 1.08.4.1 1.08.4.1.1 1.08.4.1.2 1.08.4.1.3 1.08.4.2 1.08.4.2.1 1.08.4.2.2 1.08.4.2.3 1.08.4.2.4 1.08.4.2.5 1.08.4.3 1.08.5 1.08.5.1 1.08.5.1.1 1.08.5.1.2 1.08.5.1.3 1.08.5.1.4 1.08.5.1.5 1.08.5.2 Introduction Methodologies and Tools Theoretical Background Codes Basis sets Pseudoization schemes Ab Initio Calculations in Practice Output Cell sizes and corresponding CPU times Choices to make Fields of Application Perfect Crystal Bulk properties Input for thermodynamic 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 Insulators Silicon Carbide Point defects Defect kinetics Defect complexes Impurities Extended defects Uranium Oxide 224 224 224 225 225 226 227 227 227 228 228 229 229 229 229 229 230 230 230 230 230 230 230 231 231 232 232 234 235 236 236 236 237 237 238 238 240 240 240 241 242 242 243 243 223 224 Ab Initio Electronic Structure Calculations for Nuclear Materials 1.08.5.2.1 1.08.5.2.2 1.08.5.2.3 1.08.5.2.4 1.08.6 References Bulk electronic structure Point defects Oxygen clusters Impurities Conclusion Abbreviations bcc CTL DFT DLTS EPR fcc FLAPW FP GGA LDA LSD LVM PAW PL RPV SIA SQS TD-DFT Body-centered cubic Charge transition levels Density functional theory Deep level transient spectroscopy Electron paramagnetic resonance Face-centered cubic Full potential linearized augmented plane waves Fission products Generalized gradient approximation Local density approximation Local spin density approximation Local vibrational modes Projector augmented waves Photo-luminescence Reactor pressure vessel Self-interstitial atom Special quasi-random structures Time dependent density functional theory 1.08.1 Introduction Electronic structure calculations did not start with the so-called ab initio calculations or in recent years The underlying basics date back to the 1930s with an 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 importance of these somewhat historical calculations should not be overlooked However, in the following sections, we focus on ‘ab initio’ calculations, that is, density functional theory (DFT) calculations One must acknowledge that ‘ab initio calculations’ is a rather vague expression that 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 243 244 244 245 245 246 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 1.08.2 Methodologies and Tools 1.08.2.1 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 Schroădinger 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 Ab Initio Electronic Structure Calculations for Nuclear Materials 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 Recently 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 usually involve 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 pseudoization In the community of nuclear materials, most calculations are done for periodic systems, that is, 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 Pseudoization 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 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 pseudoization schemes exist (see Section 1.08.2.2.2) Beyond ground-state properties, other theoretical developments allow the ab initio calculations of additional features Detailing these developments is 225 beyond the scope of this text; let us just mention among others time-dependent 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 1.08.2.2 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 that 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 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 pseudoization scheme and the type of their basis set We will not describe many other numerical or programming differences, even though they can influence the accuracy and speed of the codes The possible choices in terms of basis sets and pseudoization are discussed in the following paragraphs Pseudoization scheme and basis set are intricate as some bases not need pseudoization and some pseudoizations presently exist only 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 to completeness on that matter Furthermore, we 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 1.08.2.2.1 Basis sets For what concerns the basis sets we briefly present plane wave codes, codes with atomic-like localized basis sets, and all-electron codes All-electron codes involve no pseudoization 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 226 Ab Initio Electronic Structure Calculations for Nuclear Materials 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 Wien2k5 code, 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 z 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 pseudoization (see the next section) though nothing forbids the use of more advanced schemes Among this family of codes, SIESTA7,8 is often used in nuclear material studies Finally, 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 1.08.2.2.2 Pseudoization schemes As explained above, pseudoization schemes are especially relevant for plane wave codes All pseudoization 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, that is, 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 pseudoization schemes 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 used 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 pseudoization 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 PAWatomic 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 with norm-conserving and ultrasoft pseudopotentials and PAW formalism, and ABINIT17 with normconserving pseudopotentials and PAW formalism Note that for a specific pseudoization 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 pseudoization process We present in the following a series of practical choices to be made when one wants to perform Ab Initio Electronic Structure Calculations for Nuclear Materials 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, and so on, and so it proves quite difficult to change codes in the 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) 1.08.2.3 Ab Initio Calculations in Practice In this paragraph, we try to give some indication 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 1.08.2.3.1 Output We describe in this section the output of ab initio calculations in general terms The possible applications in the nuclear materials field are 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, which will indicate whether the material is metallic, semiconducting, or insulating (or at least what the code predicts it to be), its possible magnetic structure, and so on Additional calculations are able to provide additional information on the electronic excitation spectra: optical absorption, X-ray spectra, and so on 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 to perform various calculations:  Atomic relaxations to the local minimum for the atomic positions 227  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 socalled frozen phonon approach, or by first-order perturbation theory (if such feature is implemented in the code) In this last case, the third-order derivative of the energy (Raman spectrum, phonon lifetimes) can also be computed  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 (see Chapter 1.09, 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 1.08.2.3.2 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 selfconsistent 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 strongly limits the number of atoms and 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 and 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 228 Ab Initio Electronic Structure Calculations for Nuclear Materials 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 ten complete relaxations, and that each molecular dynamics simulation time step (of about fs) needs a complete ground-state calculation Overall, 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 hundreds of thousands or millions of CPU hours 1.08.2.3.3 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 made 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 the flavor of the exchange and correlation functional that will be used to describe the electronic interactions Most of the time one chooses either an 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 (see Section 1.08.5) The second and more definitive choice is the one of the pseudopotential We not mean here the choice of the pseudoization scheme but the choice of the pseudopotential itself Indeed, calculated energies vary greatly with the chosen pseudopotential, so 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 that need fine sampling of the Brillouin zone, but convergence with respect to 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 that also use FFT) Once again the larger the number of plane waves, the more accurate and 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) 1.08.3 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 the following: metals, particularly iron, tungsten, zirconium, and plutonium; alloys, especially iron alloys (FeCr, FeC to tackle steel, etc.); models of fuel materials, UO2, U–PuO2, and uranium carbides; structural carbides (SiC, TiC, B4C, etc.); waste materials (zircon, pyrochlores, apatites, etc.) In this section, we rapidly expose the types of studies that can be done with ab initio calculations The last two sections on metallic alloys and insulating materials will allow us to go into detail for some specific cases Ab Initio Electronic Structure Calculations for Nuclear Materials 1.08.3.1 Perfect Crystal 1.08.3.1.1 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 semi-conductors, are basically well reproduced, though some problems remain, especially for nonconductors (see Section 1.08.5.1 on SiC) However, difficulties arise when one wishes to tackle the properties of highly correlated materials such as uranium oxide (Section 1.08.5.2) For instance, no ab initio code, whatever the complexity and refinements, is able to correctly predict the fact that plutonium is nonmagnetic In such situations, the nature of the chemical bonding is still poorly understood, so 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, and so on of less difficult materials 1.08.3.1.2 Input for thermodynamic 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 (see Chapter 1.17, Computational Thermodynamics: Application to Nuclear Materials) 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) and the exploration of details of the 229 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 in 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) 1.08.3.2 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 (FP)) (see Chapter 1.02, Fundamental Point Defect Properties in Ceramics and Chapter 1.03, Radiation-Induced Effects on Microstructure) 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 that 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 permits accurate calculation of these energies and also consideration of (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 1.08.3.2.1 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, 230 Ab Initio Electronic Structure Calculations for Nuclear Materials 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, and so on; ab initio gives access to the solution energy of these impurities, which allows one to determine their most favored positions in the crystal: interstitial position, substitution for host atoms, and so on 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 molecular dynamics We are aware of studies in GaN27 and silicon carbides.28,29 The procedure is the same as that 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 interatomic 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 1.08.3.2.3 Point defect assemblies 1.08.3.3.2 Electronic stopping power it is possible to go beyond the 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 1.08.4.2.3) 1.08.3.2.2 Hetero-defects In this class, one can include the calculation of interstitial assemblies as well as the complexes built with impurities and vacancies One then has access to the binding of monoatomic defects to the complexes,24 possibly with the associated kinetic energy barriers 1.08.3.2.4 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 e.g., the works on pure iron25 or FeCu26and Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects) 1.08.3.2.5 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 in the next section on studies on dislocations 1.08.3.3 Ab Initio for Irradiation Irradiation damage, especially cascade modeling, is usually preferentially dealt by larger scale methods such as molecular dynamics with empirical potentials rather than ab initio calculations However, recently ab initio studies that directly tackle irradiation processes have appeared 1.08.3.3.1 Threshold displacement energies First, the increase in computer power has allowed the calculations of threshold displacement energies by 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 time-dependent DFT (TD-DFT) Two kinds 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 here is 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 Ab Initio Electronic Structure Calculations for Nuclear Materials 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, which should be large enough to prevent interaction between the periodic images Following this approach, Pruneda and coworkers33 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 homogeneous electron gas only.34,35 1.08.3.4 Ab Initio and Empirical Potentials Ab initio calculations are often compared to and sometimes confused with empirical potential 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 not explicitly consider electrons Thus, roughly speaking, ab initio calculations deal with electronic structure and give access to good energetics, whereas 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 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 that are really related to the electronic structure itself For instance, the calculations of electronic excitations (e.g., optical or X-ray spectra) are conceptually impossible with empirical potentials In the 231 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 that 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 Conversely, 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 However, this potential assumes a tendency for ferromagnetic order, while ab initio calculation can (in principle) predict what the magnetic order will be Therefore, ab initio is very often used as a way to get 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 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 classical interatomic potentials dealing only with atoms or ions can never fully reproduce all the aspects of bonding in a material 1.08.4 Metals and Alloys The vast majority of DFT calculations on radiation defects in metallic materials have been performed in body-centered cubic (bcc) 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 Ab Initio Electronic Structure Calculations for Nuclear Materials DFT calculations have been intensively used to predict atomistic defect configurations and also transition pathways An overview of these results is presented below, complete with examples in other bcc transition metals, in particular tungsten, as well as hcp-Zr These examples illustrate how DFT data have changed the more or less admitted energy landscape of these defects and also how they are used to improve empirical potentials In the final part of this chapter, a brief overview of typical works on dislocations (in iron) is presented 1.08.4.1 55 Fe 50 45 P-bcc (PW) 40 35 E (mRy) 232 30 P-bcc (LSD) 25 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, see Chapter 4.03, Ferritic Steels and Advanced Ferritic–Martensitic Steels From a basic science point of view, the modeling of these materials starts with that of pure iron, in the ferromagnetic bcc structure Iron presents several difficulties for DFT calculations First, being a three dimensional (3D) metal, it requires rather large basis sets in plane wave 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 exchangecorrelation 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 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 few exceptions, they are indeed quite independent on the form of the GGA functional, the basis set (plane wave or localized), and the pseudopotential or the use of PAWapproaches 1.08.4.1.1 Self-interstitials and selfinterstitial clusters in Fe and other bcc metals The structure and migration mechanism of selfinterstitials in iron is a very good illustrative example of the impact of DFT calculations on radiation defect P-fcc (PW) 20 F-bcc (LSD) 15 10 P-fcc (LSD) F-bcc (PW) 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 s arbitrary units (a.u.) Figure Calculated total energy of paramagnetic (P) bcc and fcc and ferromagnetic (F) bcc iron as a 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) The curves are displaced in energy so that the minima for F bcc coincide Energies are in Ry (1 Ry ¼ 13.6057 eV) and distances in bohr (1 bohr ¼ 0.5292 A˚) Reproduced from Derlet, P M.; Dudarev, S L Prog Mater Sci 2007, 52, 299–318 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 128ỵ1 atoms could then be obtained with a sufficient accuracy In 2001, Domain and Becquart reported that, in agreement with the experiment, the h110i dumbbell was the most stable structure.41 Quite unexpectedly, the h111i dumbbell was predicted to be $0.7 eV higher in energy, at variance with empirical potential results that 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 in the h111i dumbbell in Fe The important consequence of this result in Fe, which has been confirmed repeatedly since 234 Ab Initio Electronic Structure Calculations for Nuclear Materials 0.8 DFT-GGA Mendelev Ackland Energy barrier (eV) 0.6 0.4 0.2 0.0 [011] [110] [111] Crowd [111] Figure Left: Johnson translation–rotation mechanism of the h110i dumbbell; white and black spheres indicate the initial and final positions of the atoms, respectively Reproduced from Fu, C C.; Willaime, F.; Ordejon, P Phys Rev Lett 2004, 92, 175503 Right: Comparison between the DFT-GGA result and two EAM potentials for the energy barriers of the Johnson mechanism and the h110i to h111i transformation Reproduced from Willaime, F.; Fu, C C.; Marinica, M C.; Torre, J D.; Nucl Instrum Meth Phys Res B 2005, 228, 92 Figure New low-energy configurations of SIA clusters in Fe, which revealed discrepancies between DFT and empirical potentials and between various approximations within DFT Reproduced from Terentyev, D A.; Klaver, T P C.; Olsson, P.; Marinica, M C.; Willaime, F.; Domain, C.; Malerba, L Phys Rev Lett 2008, 100, 145503 with less transferable ultrasoft pseudopotentials with VASP and norm-conserving pseudopotentials with SIESTA.46 Such a discrepancy is not common in defect calculations in metals Further investigations are required to understand more precisely its origin, in particular the possible role of magnetism The structures of the most stable SIA clusters in Fe, and more generally of their energy landscape, remain an open question One would ideally need to combine DFT calculations with methods for exploring the energy surface, such as the Dimer47 or ART48 methods Such a combination is possible in principle, and it has indeed been used for defects in semiconductors,49 but due to computer limitations this is not the case yet in Fe The alternative is to develop new empirical potentials in better agreement with DFT energies in particular for these new structures, to perform the Dimer or ART calculations with these potentials, and to validate the main features of the energy landscape thus obtained by DFT calculations To summarize, the energy landscape of interstitial type defects has been revisited in the last decade driven by DFT calculations, in synergy with empirical potential calculations 1.08.4.1.2 Vacancy and vacancy clusters in Fe and other bcc metals DFT has some limitations in predicting accurate vacancy formation energies in transition metals The exceptional agreement with the experiment obtained initially within DFT-LDA50 was later shown to result from a cancellation between two effects First, the Ab Initio Electronic Structure Calculations for Nuclear Materials Ackland et al.94 Mendelev et al.45 DFT-GGA 1.2 Migration energy (eV) structural relaxation, which was neglected by Korhonen et al.50 is now known to significantly reduce the vacancy formation energy, in particular in bcc metals.51 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).52 However, the effect is much weaker for migration energies.52 A new functional, AM05, has been proposed to cope with this limitation.53 Less spectacular effects are expected in vacancytype 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 DFT-GGA, 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 eVand 0.59–0.71 eV.41,43,54 These values are in agreement with experimental estimates at low temperatures in ultrapure iron, namely 2.0 Ỉ 0.2 eV and 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 divacancy is the nextnearest-neighbor configuration, with a binding energy of 0.2–0.3 eV.25,55,56 The migration can occur by two different two-step processes, with an intermediate configuration that is either nearest neighbor or fourth nearest neighbor.56 A quite unexpected result of DFT calculations was the prediction of rather low migration energies for the tri- and quadrivacancies, 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).57 Stronger deviations from empirical potential predictions for divacancies are observed in DFT calculations performed in other bcc metals The most dramatic case is that of tungsten, where the nextnearest-neighbor interaction is strongly repulsive (0.5 eV) and the nearest-neighbor interaction is 235 1.0 0.8 0.6 0.4 0.2 0.0 V V2 V3 V4 V5 Figure Migration energies of vacancy clusters in Fe, as a function of cluster size Reproduced from Fu, C C.; Willaime, F (2004) Unpublished vanishing.58 This result does not explain why voids are formed in tungsten under irradiation 1.08.4.1.3 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,59 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 The magnetic contribution is important in iron Spin fluctuations were shown to be the origin of the strong softening of the C0 elastic constant observed as the temperature increases up to the aÀg transition temperature,60 and it drives, for instance, the temperature dependence of relative abundance of and interstitial loops formed under irradiation.61 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 this effect for point defects quantitatively from DFT calculations This is probably one of the important challenges in the field 236 Ab Initio Electronic Structure Calculations for Nuclear Materials 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.62 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.63 1.08.4.2 Beyond Pure Iron 1.08.4.2.1 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, see Chapter 1.06, 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.64–67 An improved Fe–He pair potential was then obtained by fitting to the DFT results.68 DFT (b) 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.66 The energy landscape of the helium–divacancy complex also revealed unexpected configurations, in particular, for the lowest energy configuration where the helium atom is located halfway between two nearestneighbor vacancies (see Figure 6) More generally, a systematic study of the energetics of all small HenVm clusters in iron was performed,64 giving very useful data for the kinetic modeling of helium–vacancy clustering and dissociation.69 Quite interestingly, interstitial He atoms are found to attract one another, even in the absence of vacancies.24,64 This clustering of helium atoms may then yield the emission of selfinterstitials 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 tungsten58,70 and 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.71 1.08.4.2.2 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 (d) (a) (e) Energy (eV) 1.2 (f) (c) 0.8 0.4 0.0 nn nn Reaction coordinate nn nn Figure Schematic representation of the energy landscape of the HeV2 complex Reproduced from Fu, C C.; Willaime, F Phys Rev B 2005, 72, 064117 Ab Initio Electronic Structure Calculations for Nuclear Materials interaction DFT calculations reproduce the wellknown fact that carbon is located in octahedral sites, and they also confirm the strong attraction between interstitial carbon and a monovacancy, with a binding energy of about 0.5 eV.72–74 This strong attraction is the origin of the confusing discrepancy between the vacancy migration energy in ultrapure iron, $0.6 eV, and the effective vacancy migration energy in iron with carbon or in steels, that is, $1.1 eV, which corresponds to first order to the sum of the vacancy migration energy and the carbon-vacancy binding energy.74 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 eV,72–74 and VC2 complexes are expected to play a very important role The interaction between carbon and selfinterstitials is also attractive but weaker In agreement with experiments,75 DFT calculations confirmed a binding energy of $0.2 eV76 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 This shortcoming of empirical potentials was overcome recently with an improved potential derived taking into account information from the electronic structure.77 The strong interaction of carbon with vacancies also affects the energetics of helium–vacancy clusters, and it is important to take this into account to reproduce, for example, thermal helium desorption experiments performed in iron.78 Similar calculations have been performed with nitrogen.72 1.08.4.2.3 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 237 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 vacancyformation energy in metastable bcc–Cu (which is not known experimentally since bulk Cu is fcc) is 0.9 eV, that is, 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.79 A systematic study of the interaction of monovacancies and self-interstitials with all transition-metal solutes has been reported recently (see Figure 8).80 1.08.4.2.4 From dilute to concentrated alloys: the case of Fe–Cr In the approach described earlier, 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% Nevertheless, first results have been obtained by considering the interaction of defects with one or two Cr atoms in the Fe matrix.81 These data could ideally be used to fit an improved empirical potential, but the Fe–Cr system is rather difficult to model because of the strong interplay between magnetic and chemical interactions This is also clearly one of the challenges in the field Figure Structure of carbon–vacancy and carbon–self-interstitial complexes in iron, predicted from DFT calculations Reproduced from Fu, C C.; Meslin, E.; Barbu, A.; Willaime, F.; Oison, V In Theory, Modeling and Numerical Simulation of Multi-Physics Materials Behavior, 2008; Vol 139, pp 157–164, 168 238 Ab Initio Electronic Structure Calculations for Nuclear Materials Blochl14 Zunger et al.20 and Becquart and Domain55 Djurabekova et al.56 Fu and Willaime57 and Becquart and Domain58 ce nn -0.6 34 di st an -0.3 E bvac-3d (eV) 0.3 Ti V Cr Mn Fe Co Ni Cu Zr Nb Mo Tc Ru Rh Pd Ag Hf Ta W Re Os Ir Pt Au nc e n 23 n d ista -0.3 -0.6 E bvac-4d (eV) 0.3 nc e -0.3 12 nn dis ta E bvac-5d (eV) 0.3 -0.6 Figure DFT-GGA solute-vacancy binding energies in iron for 3D, 4D, and 5D elements for 1–5 nn relative positions Reprinted with permission from Olsson, P.; Klaver, T P C.; Domain, C Phys Rev B 2010, 81, 054102 Copyright (2010) by the American Physical Society 1.08.4.2.5 Point defects in hcp-Zr Point defects in hcp-Zr have also been studied via DFT calculations It was found in particular that the vacancy migration energy is lower by $0.15 eV within the basal plane than out of the basal plane.82 The situation for the self-interstitial is quite complex, since among the known configurations, at least three configurations are found to have almost the same formation energy (within 0.1 eV): the octahedral (O), split dumbbell (S), and basal octahedral (BO) configurations.83,84 1.08.4.3 Dislocations The collective behavior of dislocations can be described thanks to dislocation dynamics codes In order to reinforce the physical foundation, input data such as 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 section on the properties of dislocations in bcc metals and especially iron In these materials, dislocation 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 in the boundary conditions of the calculations For instance, considering h111i screw dislocations, the two cell geometries proposed in the literature – the cluster approach85 and the periodic array of dislocation dipoles86 – have been thoroughly compared.87 The calculations of dislocations are extremely demanding as they can include up to 800 atoms, so studies usually use fast codes such as SIESTA.8 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 calculations87 although the cluster approach with flexible boundary conditions can be considered a reference method when no energies are necessary (i.e., only structures) Ab Initio Electronic Structure Calculations for Nuclear Materials DFT calculations in bcc metals such as Mo, Ta, Fe, and W85,87–91 predict a nondegenerate structure for the core, as illustrated in Figure using differential displacement maps as proposed by Vitek.92 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.93 Thanks to good control of energy, it is also possible to obtain quantitative results on the Peierls potential; namely, the 2D 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), where a high symmetry direction of the Peierls potential is sampled: the line going between two easy core positions along the glide direction, that is, the Peierls barrier These calculations 239 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 Ackland–Mendelev potential for iron,45 which gives the correct nondegenerate core structure unlike most other potentials, has been tested against the obtained DFT results It appears that it compares well with the DFT results for the g-surfaces, but discrepancies exist on the deviation from anisotropic elasticity of both edge and screw components and on the Peierls potential Indeed, the empirical potential results not predict any dilatation elastic field exerted by the core Besides, the Peierls barrier displayed by the Ackland–Mendelev potential yields a camel hump shape, as illustrated in Figure 10(a), and at the halfway position, the core spreads between [110] (a) [111] [112] (b) (c) Figure (a) Differential displacement map of the nondegenerate core structure of a screw dislocation in Fe, as obtained from SIESTA GGA (b) Same as (a) after subtraction of the Volterra anisotropic elastic field and magnified by a factor of 20 (c) Same as (b) for the displacement in the (111) plane (or edge component) and a magnification by a factor of 50 Reproduced from Ventelon, L.; Willaime, F J Comput Aided Mater Des 2007, 14, 85–94 200 100 40 30 20 SIESTA GGA Ackland Ackland–Mendelev Dudarev–Derlet -100 10 0.0 (a) SIESTA GGA SIESTA LDA Ackland–Mendelev Energy (meV/b) Energy barrier (meV/b) 50 0.2 0.4 0.6 Reaction coordinate 0.8 -200 0.0 1.0 (b) 0.2 0.4 0.6 Polarity 0.8 1.0 Figure 10 (a) Peierls barrier in Fe calculated with the Ackland–Mendelev potential45 and with SIESTA using the two exchange-correlation functionals, LDA and GGA Reproduced from Ventelon, L.; Willaime, F J Comput Aided Mater Des 2007, 14, 85–94 (b) Dependence of the dislocation core energy with the modulus of its polarization calculated using SIESTA and the three empirical potentials, namely, the Ackland,94 Ackland–Mendelev,45 and Dudarev–Derlet38 potentials Reproduced from Ventelon, L.; Willaime, F Philos Mag 2010, 90, 1063–1074 240 Ab Initio Electronic Structure Calculations for Nuclear Materials two easy core positions, whereas it exhibits a single hump barrier within DFT and a nearly hard-core structure at halfway position The effect of the exchange-correlation functional within DFT appears to be significant.87 More insight into the stability of the core structure can be gained by looking at the response of the polarization of the core, as represented in Figure 10(b) In the Ackland–Mendelev and DFT cases, these calculations confirm that the stable core is completely unpolarized, and they prove that there is no metastable polarized core.95 Finally, the methodology exists for calculating the structure and formation and migration energies of single kinks, but using it with DFT96 remains challenging because cells with about 1000 atoms are needed, together with a high accuracy 1.08.5 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, whereas 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, that is, 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 research 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 1.08.5.1 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 with 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 microelectronics applications Silicon carbide is a band insulator whose bulk structural properties are well reproduced by usual DFT calculations The 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 eV,97 whereas standard DFT-LDA calculations give 1.30 eV.97 1.08.5.1.1 Point defects The first DFT calculations of point defects in silicon carbide,98 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 divacancies were bound Ab Initio Electronic Structure Calculations for Nuclear Materials Vacancies were thoroughly studied at the turn of the century.97,99–101 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 and coworkers,102 the electron paramagnetic resonance (EPR) spectra of annealed samples of irradiated SiC were measured103 and compared with calculated hyperfine parameters This showed that silicon vacancies are metastable with regard to a carbon vacancy–carbon antisite complex (VC–CSi); a fact that has since been consistently confirmed by the other calculations Interstitials were less studied than vacancies One should however mention a study104 devoted to carbon and silicon in interstitials in silicon carbide Beyond these studies dedicated to one type of defect, very complete and comprehensive work on both vacancies and interstitials was also published One should cite Bernardini et al.105 devoted to the formation energies of defects, while Bockstedte et al.106 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 monointerstitial and that all these structures must be considered in the calculation (see Figure 11) As examples, the results of these various studies on what concerns formation energies and CTL of vacancies are summarized in the following tables One can see a general agreement in 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 Si CTC Front 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 hand, calls for a better description of the exchangecorrelation 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.107 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 SiO2108) or hybrid functionals.109 For what concerns nuclear materials, and especially SiC, GW corrections and excitonic effects will allow further comparisons with experiments (Table 1).110 1.08.5.1.2 Defect kinetics Before the aforementioned work by Bockstedte and coworkers106 almost no work was devoted to migration properties of point defects in SiC We should, however, cite previous preliminary works by the same group,111,112 a work on the mechanisms of formation of antisite pairs,113 and a work on vacancy migration published in 2003.114 The comprehensive study of migration barriers in Bockstedte et al.106 showed, first of all, that vacancies have much higher migration energies than those of interstitials: higher than eV for the former in the neutral state, around eV for the latter (0.5 for IC, 1.4 for ISi) Another C CTSi CHex 241 Csp Csp CspSi Side Figure 11 Possible geometries for a carbon interstitial in cubic SiC Reprinted with permission from Bockstedte, M.; Mattausch, A.; Pankratov, O Phys Rev B 2003, 68, 205201 Copyright (2003) by the American Physical Society 242 Ab Initio Electronic Structure Calculations for Nuclear Materials Table Formation energies for vacancies and their charge transition levels according to various authors VC VSi References ỵ/0 þ/þþ þ/0 0/À À/ÀÀ 98 191 99 97 192 106 105 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 The values are for the 3C-polytype in silicon-rich conditions Values are expressed in eV 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 eV going from the neutral to the 2ỵ charged state, whereas the silicon vacancy finds its migration barrier reduced by eV when its charge goes from neutral to 2À 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 2ỵ and 3ỵ 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 The energy barriers of recombinations of close interstitial vacancy pairs have also been tackled.115–117 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.118 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 eV for C) that are in agreement with currently accepted values (coming from experimental evaluations that are, however, largely dispersed) These calculations prove 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 1.08.5.1.3 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 have been attempted for some of them Crucial to 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 complexes119,120 as well as divacancy complexes121–123 were called upon for the attribution of PL or EPR peaks Various kinds of carbon clusters were studied in detail theoretically.124–127 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 is high: from 3.9 to 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 work107 deals with the stability and dynamics of such silicon clusters (see Figure 12) 1.08.5.1.4 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 Ab Initio Electronic Structure Calculations for Nuclear Materials Ie2 Energy barrier (eV) ITC + ITC Ia2 Ie2 Ia2 Ib2 Ia2 Ic2 Ia2 Id2 ISisp + I 243 Id2 3.0 3.0 2.5 2.5 2.0 2.0 Emigra(I) 1.5 Ebind(I2e) 1.5 Etrans(I2e-a) 1.0 0.5 Emigra(I2a) Ebind(I2d) Etrans(I2c-a) in-plane Erotation(I2a) Eopen(I2a-c) out-plane Erotation (I2a) 1.0 0.5 0.0 0.0 Reaction coordinate Figure 12 Energetic landscape of silicon mono- and di-interstitial in cubic SiC Reproduced from Liao, T (2009) Unpublished the results would not be very different in cubic SiC One can find calculations dealing with boron129,130 as an acceptor and nitrogen131,132 or phosphorus133 as a donor Other impurities were studied: transition metals,134–136 oxygen,137 important for the behavior of the SiO2/SiC interface, hydrogen,138–140 rare gases,141 and palladium.142 A systematic study of substitutional impurities has recently appeared,143 which focuses on the trends of carbon vs silicon substitution according to the position of species in the periodic table 1.08.5.1.5 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 devices144 and the remarkable enhancement of dislocation velocity under illumination in the hexagonal phase.145 Nevertheless, some studies have been done for cubic SiC on the electronic structure of stacking faults146–151 and various types of dislocations.152–155 Obviously, a lot of work remains to be done on the extended defects in b SiC 1.08.5.2 Uranium Oxide 1.08.5.2.1 Bulk electronic structure Due to its technological importance and the complexity of its electronic structure, uranium oxide has become 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 being the work of Arko and coworkers156) 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 LDAỵU correction in which a Hubbard U term acting between f electrons is added ‘by hand’ to the Hamiltonian.157,158 This method allows the opening of an f–f gap.157 However, it suffers from the existence of multiple minima in the calculations, so the search for the real ground state is rather tricky as the calculation is easily trapped in metastable states.159 Hybrid functionals are another type of advanced methods that 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; it has been applied to UO2 has been made by Kudin et al.160 These methods are very promising for solid-state nuclear materials However, the same problem of metastability as in LDAỵU exists for such hybrid functionals,161 and the computational load is much heavier than that in common or LDAỵU calculations Recently, an alternative to LDAỵU has been proposed: the so-called local hybrid functional for correlated electrons162 in which the hybrid functional is applied only to the problematic f electrons An application on UO2 is available.163 244 Ab Initio Electronic Structure Calculations for Nuclear Materials 1.08.5.2.2 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 point defects.164–166 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 More recent studies using the ỵU correction have been published Most of them still focus on neutral defects.159,167–169 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 2.6170 and 6.5 eV.159 This suggests some hidden problem in the calculations, probably related to the possible occurrence of metastable minima in the calculations We are aware of only one study of the charge state of point defects This work,171 done within LDAỵU, predicts the following charge states: À4 charge for uranium vacancy, À2 for oxygen interstitial, and from ỵ2 to for oxygen vacancy depending on the position of the Fermi level However, in this last work, the formation energies of the composite defects (Frenkel and Schottky) built from charged defects are in not as good agreement with experiments as the ones obtained with neutral defects Together with the large spread of values mentioned here, this underestimation shows that the correct reproduction of point defects in UO2 with ab initio seems not yet at hand Beyond the formation energy of isolated defects, some studies focus on their migration Gupta and coworkers172 calculated the migration energy of the oxygen vacancy (1.0 eV) and interstitial (1.1 eV) In this last case, they found that the stable position for a monointerstitial is a dumbbell configuration This point is refuted by others173 who calculated the migration energy of oxygen mono- (0.81 eV) and di-interstials (0.47 eV) and implemented this information in a Kinetic Monte-Carlo model, showing that the di-interstitial configuration, though less abundant than the single interstitial, may play a dominant role in oxygen diffusion in hyperstoichiometric oxide 1.08.5.2.3 Oxygen clusters Another point of interest beyond point defects is the clustering of oxygen interstitials Indeed, oxygen interstitial clustering has been deduced from diffraction experiments174 many years ago However, a debate remains on the exact shape of such clusters Two configurations are contemplated: the so-called Willis clusters174,175 or cubo-octahedral clusters that have been observed by neutron diffraction in U4O9176 and U3O7.177 These clusters are made of 12 oxygen and uranium atoms and amount for 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 cluster (Figure 13).178 The three interstitial–1 vacancy cluster has been found independently by other authors173,180 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.178,179 O O OV OЈ U U OЈЈ OV OV OЈЈ (a) (b) Figure 13 Relaxation process of a Willis cluster of oxygen interstitials in UO2 Reproduced from Geng, H Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M Appl Phys Lett 2008, 93 Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 14 Cubo-octahedral cluster of oxygen interstitials in UO2 Reprinted with permission from Geng, H Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M Phys Rev B 2008, 77, 180101(R).180 Copyright (2008) by the American Physical Society 1.08.5.2.4 Impurities Lattice sites and solution energies of FP are of major importance in fundamental studies of nuclear fuels, see Chapter 2.20, Fission Product Chemistry in Oxide 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 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.181 Indeed, one then has to distinguish between the incorporation energy, defined as the energy to incorporate the FP in a preexisting 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 UO2ỵx A positive (respectively, negative) solution energy then means that the FP is insoluble (respectively, soluble) in UO2ỵx The first DFT study of the incorporation of a FP in UO2 is the one by Petit et al.182 on krypton in the late 1990s It was performed within the LMTO-ASA formalism, which could give only qualitative results 245 Crocombette181 used more modern plane wave formalism to calculate the insertion of some FP (krypton, iodine, cesium, strontium, and helium) but neglected atomic relaxation, which limits the accuracy of the results Freyss et al.183 considered He and Xe All these calculations were performed with standard LDA More recent works always included a ỵU correction While the first papers dealt only with interstitial and monovacancy sites, more recent works may also consider divacancy 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,184 iodine,185 xenon,186 strontium,186 cesium,186–188 molybdenum,189 and zirconium.189 Yun et al.190 dealt with helium and went beyond the solution energies as they also considered migration and clustering energies 1.08.6 Conclusion It is hoped that the examples discussed above have 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 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 material, shows that a lot of information can be obtained, for example, for the solution energies of FP 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 or GW calculations, with the drawback that these advanced methods 246 Ab Initio Electronic Structure Calculations for Nuclear Materials 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 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, and so on Acknowledgments The authors thank Drs Fabien Bruneval, Chu Chun Fu, Guido Roma, and Lisa Ventelon for their valuable input 31 32 33 34 35 36 37 38 39 40 41 References 42 43 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Harrison, W A Electronic Structure and the Properties of Solids; Freeman: San Francisco, 1980 Kohanoff, J Electronic Structure Calculations for Solids and Molecules; Cambridge University Press, 2006 Martin, R M Electronic Structure; Cambridge University Press, 2004 Springborg, M Methods of Electronic-Structure Calculations; Wiley: Chishester, 2000 WIEN2K: http://www.wien2k.at/ Schwartz, K.; Blaha, P.; Madsen, G H K Comp Phys Commun 2001, 147, 71–76 Soler, J M.; Artacho, E.; Gale, J D.; et al J Phys Condens Matter 2002, 14, 2745 SIESTA: http://www.icmab.es/siesta/ Pickett, W E Comput Phys Rep 1989, 9, 115 Gaussian: http://www.gaussian.com/index.htm BigDFT: http://inac.cea.fr/L_Sim/BigDFT/ Troullier, N.; Martins, J L Phys Rev B 1991, 43, 1993 Vanderbilt, D Phys Rev B 1990, 41, 7892 Blochl, P E Phys Rev B 1994, 50, 17953–17979 VASP: cms.mpi.univie.ac.at/vasp/ Quantum Espresso: www.quantum-espresso.org/ ABINIT: www.abinit.org Car, R.; Parrinello, M Phys Rev Lett 1988, 60, 204 Methfessel, M.; Paxton, A T Phys Rev B 1989, 40, 3616 Zunger, A.; Wei, S H.; Ferreira, L G.; Bernard, J E Phys Rev Lett 1990, 65, 353 Van de Walle, C G Nat Mater 2008, 7, 455 Clouet, E.; Barbu, A.; Lae, L.; Martin, G Acta Mater 2005, 53, 2313–2325 Olsson, P.; Abrikosov, I A.; Vitos, L.; Wallenius, J J Nucl Mater 2003, 321, 84–90 Fu, C C.; Willaime, F J Nucl Mater 2007, 367, 244–250 Fu, C.; Dalla Torre, J.; Willaime, F.; Bocquet, J.-L.; Barbu, A Nat Mater 2005, 4, 68–74 Soisson, F.; Fu, C C Phys Rev B 2007, 76, 12 Xiao, H Y.; Gao, F.; Zu, X T.; Weber, W J J Appl Phys 2009, 105, Gao, F.; Xiao, H Y.; Zu, X T.; Posselt, M.; Weber, W J Phys Rev Lett 2009, 103, Lucas, G.; Pizzagalli, L Phys Rev B 2005, 72, 161202(R) Campillo, I.; Pitarke, J M.; Eguiluz, A G Phys Rev B 1998, 58, 10307–10314 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Campillo, I.; Pitarke, J M.; Eguiluz, A G.; Garcia, A Nucl Instrum Methods Phys B 1998, 135, 103–106 Pitarke, J M.; Campillo, I Nucl Instrum Methods Phys B 2000, 164, 147–160 Pruneda, J M.; Sanchez-Portal, D.; Arnau, A.; Juaristi, J I.; Artacho, E Phys Rev Lett 2007, 99, 235501 Barriga-Carrasco, M D Nucl Instrum Meth Phys Res A Accelerators Spectrometers Detectors Associated Equipment 2009, 606, 215–217 Barriga-Carrasco, M D Phys Rev E 2009, 79, 027401 Finnis, M W.; Sinclair, J E Philos Mag A Phys Condens Matter Struct Defects Mechanic Properties 1984, 50, 45–55 Rosato, V.; Guillope, M.; Legrand, B Philos Mag A Phys Condens Matter Struct Defects Mechanic Properties 1989, 59, 321–336 Derlet, P M and Dudarev, S L., Prog Mater Sci 2007, 52, 299–318 Bagno, P.; Jepsen, O.; Gunnarsson, O Phys Rev B 1989, 40, 1997–2000 Kresse, G.; Joubert, D Phys Rev B 1999, 59, 1758–1775 Domain, C.; Becquart, C S Phys Rev B 2002, 65, 024103 Nguyen-Manh, D.; Horsfield, A P.; Dudarev, S L Phys Rev B 2006, 73, 020101 Fu, C C.; Willaime, F.; Ordejon, P Phys Rev Lett 2004, 92, 175503 Willaime, F.; Fu, C C.; Marinica, M C.; Torre, J D Nucl Instrum Meth Phys Res B 2005, 228, 92 Mendelev, M I.; Han, S.; Srolovitz, D J.; Ackland, G J.; Sun, D Y.; Asta, M Philos Mag 2003, 83, 3977–3994 Terentyev, D A.; Klaver, T P C.; Olsson, P.; et al Phys Rev Lett 2008, 100, 145503 Henkelman, G.; Jonsson, H J Chem Phys 1999, 111, 7010–7022 Mousseau, N.; Barkema, G Physical Rev E 1998, 57, 2419–2424 El-Mellouhi, F.; Mousseau, N Appl Phys A Mater Sci Process 2007, 86, 309–312 Korhonen, T.; Puska, M J.; Nieminen, R M Phys Rev B 1995, 51, 9526–9532 Satta, A.; Willaime, F.; de Gironcoli, S Phys Rev B 1999, 60, 7001–7005 Mattsson, T R.; Mattsson, A E Phys Rev B 2002, 66, 214110 Armiento, R.; Mattsson, A E Phys Rev B 2005, 72, 085108 Huang, S Y.; Worthington, D L.; Asta, M.; Ozolins, V.; Ghosh, G.; Liaw, P K Acta Mater 2010, 58, 1982–1993 Becquart, C S.; Domain, C Nucl Instrum Methods Phys B 2003, 202, 44–50 Djurabekova, F.; Malerba, L.; Pasianot, R C.; Olsson, P .; Nordlund, K Philos Mag 2010, 90, 2585–2595 Fu, C C.; Willaime, F unpublished 2004 Becquart, C S.; Domain, C Nucl Instrum Methods Phys B 2007, 255, 23–26 Kormann, F.; Dick, A.; Grabowski, B.; Hallstedt, B.; Hickel, T.; Neugebauer, J Phys Rev B 2008, 78, 033102 Hasegawa, H.; Finnis, M W.; Pettifor, D G J Phys F Metal Phys 1985, 15, 19–34 Dudarev, S L.; Bullough, R.; Derlet, P M Phys Rev Lett 2008, 100, 135503 Lucas, G.; Schaublin, R Nucl Instrum Methods Phys B 2009, 267, 3009–3012 Mattsson, T R.; Sandberg, N.; Armiento, R.; Mattsson, A E Phys Rev B 2009, 80, 224104 Fu, C C.; Willaime, F Phys Rev B 2005, 72, 064117 Willaime, F.; Fu, C C Mater.Res Soc Symp Proc 2007, 981, JJ05–04 Ab Initio Electronic Structure Calculations for Nuclear Materials 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 Seletskaia, T.; Osetsky, Y.; Stoller, R E.; Stocks, G M Phys Rev B 2008, 78, 134103 Zu, X T.; Yang, L.; Gao, F.; et al Phys Rev B 2009, 80, 054104 Juslin, N.; Nordlund, K J Nucl Mater 2008, 382, 143–146 Ortiz, C J.; Caturla, M J.; Fu, C C.; Willaime, F Phys Rev B 2007, 75, 100102 Becquart, C S.; Domain, C J Nucl Mater 2009, 385, 223–227 Heinola, K.; Ahlgren, T J Appl Phys 2010, 107, 113531 Domain, C.; Becquart, C S.; Foct, J Phys Rev B 2004, 69, 144112 Forst, C J.; Slycke, J.; Van Vliet, K J.; Yip, S Phys Rev Lett 2006, 96, 175501 Fu, C C.; Meslin, E.; Barbu, A.; Willaime, F.; Oison, V Theory, Modeling and Numerical Simulation of MultiPhysics Materials Behavior 2008, 139, 157–164 Takaki, S.; Fuss, J.; Kugler, H.; Dedek, U.; Schultz, H Radiat Eff Defects Solids 1983, 79, 87–122 Malerba, L., Ackland, G J., Becquart, C S., et al J Nucl Mater.In Press, Corrected Proof Hepburn, D J.; Ackland, G J Phys Rev B 2008, 78, 165115 Ortiz, C J.; Caturla, M J.; Fu, C C.; Willaime, F Phys Rev B 2009, 80, 134109 Meslin, E.; Fu, C C.; Barbu, A.; Gao, F.; Willaime, F Phys Rev B 2007, 75, 094303 Olsson, P.; Klaver, T P C.; Domain, C Phys Rev B 2010, 81, 054102 Olsson, P.; Domain, C.; Wallenius, J Phys Rev B 2007, 75, 014110 Ve´rite´, G.; Willaime, F.; Fu, C C Solid State Phenomena 2007, 129, 75 Willaime, F J Nucl Mater 2003, 323, 205–212 Domain, C.; Legris, A Philos Mag 2005, 85, 569–575 Woodward, C.; Rao, S I Phys Rev Lett 2002, 88, 216402 Cai, W.; Bulatov, V V.; Chang, J P.; Li, J.; Yip, S Philos Mag 2003, 83, 539–567 Ventelon, L.; Willaime, F J Comput Aided Mater Des 2007, 14, 85–94 Frederiksen, S L.; Jacobsen, K W Philos Mag 2003, 83, 365–375 Segall, D E.; Strachan, A.; Goddard, W A.; IsmailBeigi, S.; Arias, T A Phys Rev B 2003, 68, 014104 Romaner, L.; Ambrosch-Draxl, C.; Pippan, R Phys Rev Lett 2010, 104, 195503 Domain, C.; Monnet, G Phys Rev Lett 2005, 95, 215506 Vitek, V Crystal Lattice Defects 1974, 5, 1–34 Clouet, E.; Ventelon, L.; Willaime, F Phys Rev Lett 2009, 102, 055502 Ackland, G J.; et al Philos Mag A 1997, 75, 713 Ventelon, L.; Willaime, F Philos Mag 2010, 90, 1063–1074 Ventelon, L.; Willaime, F.; Leyronnas, P J Nucl Mater 2010, 403, 216–216 Torpo, L.; et al Appl Phys Lett 1999, 74, 221 Wang, C.; Berholc, J.; Davis, R F Phys Rev B 1988, 38, 12752 Zywietz, A.; Furthmuăller, J.; Bechstedt, F Phys Rev B 1999, 59, 15166 Umeda, T.; et al Phys Rev B 2005, 71, 913202 Umeda, T.; et al Phys Rev B 2004, 70, 235212 Rauls, E.; et al Phys Stat Sol 2000, 217, R1 Lingner, T.; et al Phys Rev B 2001, 64, 245212 Lento, J M.; et al J Phys Condens Matter 2004, 16, 1053 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 247 Bernardini, F.; Mattoni, A.; Colombo, L Eur Phys J B 2004, 38, 437 Bockstedte, M.; Mattausch, A.; Pankratov, O Phys Rev B 2003, 68, 205201 Liao, T.; Roma, G.; Wang, J Y Philos Mag 2009, 89, 2271–2284 Martin-Samos, L.; Roma, G.; Rinke, P.; Limoge, Y Phys Rev Lett 2010, 104, 075502 Heyd, J.; Scuseria, G E.; Ernzerhof, M J Chem Phys 2006, 124, 219906 Bockstedte, M.; Marini, A.; Gali, A.; Pankratov, O.; Rubio, A Mater Sci Forum 2009, 600-603, 285–290 Bockstedte, M.; Pankratov, O Mater Sci Forum 2000, 949, 338–342 Mattausch, A.; Bockstedte, M.; Pankratov, O Mater Sci Forum 2001, 323, 353–356 Rauls, E.; et al Mater Sci Forum 2001, 435, 353–356 Rurali, R.; et al Comp Mater Sci 2003, 27, 36 Rauls, E.; et al Physica B 2001, 645, 308–310 Bockstedte, M.; Mattausch, A.; Pankratov, O Phys Rev B 2004, 69, 235202 Lucas, G.; Pizzagalli, L J Phys.: Condens Matter 2007, 19, 086208 Roma, G.; Crocombette, J P J Nucl Mater 2010, 403, 32–41 Eberlein, T A G.; et al Phys Rev B 2002, 65, 184108 Gao, F Appl Phys Lett 2007, 90, 221915 Torpo, L.; Staab, T E M.; Nieminen, R M Phys Rev B 2002, 65, 085202 Gerstmann, U.; Rauls, E.; Overhof, H Phys Rev B 2004, 70, 201204 Son, N T.; et al Phys Rev Lett 2006, 96, 055501 Gali, A.; et al Phys Rev B 2003, 68, 215201 Mattausch, A.; Bockstedte, M.; Pankratov, O Phys Rev B 2004, 69, 045322 Mattausch, A.; Bockstedte, M.; Pankratov, O Phys Rev B 2004, 70, 235211 Gali, A.; Son, N T.; Janze´n, E Phys Rev B 2006, 73, 033204 Liao, T unpublished 2009 Rurali, R.; et al Phys Rev B 2004, 69, 125203 Bockstedte, M.; Mattausch, A.; Pankratov, O Phys Rev B 2004, 70, 115203 Eberlein, T A G.; Jones, R.; Briddon, P R Phys Rev Lett 2003, 90, 225502 Gerstmann, U.; et al Phys Rev B 2003, 67, 205202 Rauls, E.; et al Phys Rev B 2004, 70, 085202 Overhof, H Mater Sci Forum 1997, 677, 258–263 Barbosa, K O.; Machado, W V M.; Assali, L V C Physica B 2001, 308–310 Yurieva, E I Comp Mater Sci 2006, 36, 194 Gali, A.; et al Phys Rev B 2002, 66, 125208 Aradi, B.; et al Phys Rev B 2001, 63, 245202 Gali; et al Appl Phys Lett 2002, 80, 237 Gali, A.; et al Phys Rev B 2005, 71, 035213 Van Ginhoven, R M.; Chartier, A C.; Meis, C.; Weber, W J.; Corrales, L R J Nucl Mater 2006, 348, 51–59 Roma, G J Appl Phys 2009, 106 Miyata, M.; Higashiguchi, Y.; Hayafuji, Y J Appl Phys 2008, 104, 123702 Skowronski, M.; Ha, S J Appl Phys 2006, 99, 011101 Galeckas, A.; Linnros, J.; Pirouz, P Phys Rev Lett 2006, 96, 025502 Lambrecht, W R L.; Miao, M S Phys Rev B 2006, 155316, 73 Iwata, H.; et al J Phys Condens Matter 2002, 14, 12733 Iwata, H.; et al Phys Rev B 2002, 65, 033203 Lindefelt, U.; et al Phys Rev B 2003, 67, 155204 248 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 Ab Initio Electronic Structure Calculations for Nuclear Materials Iwata; et al J Appl Phys 2003, 93, 1577 Iwata; et al J Appl Phys 2004, 94, 4972 Sitch, P K.; et al Phys Rev B 1995, 52, 4951 Blumenau, A T.; et al Phys Rev B 2003, 68, 174108 Savini, G.; et al Physica B 2007, 62, 401–402 Savini, G.; et al New J Phys 2007, 9, Arko, A J.; Koelling, D D.; Boring, A M.; Ellis, W P.; Cox, L E J Less Common Metals 1986, 122, 95 Dudarev, S L.; Mahn, D N.; Sutton, A P Phil Mag B 1997, 75, 613 Savrasov, S L D G B S.; Temmermen, Z S W.; Sutton, P A Phys Stat Sol (a) 1998, 166, 429 Dorado, B.; Amadon, B.; Freyss, M.; Bertolus, M Phys Rev B 2009, 79, 235125 Kudin, K.; Scuseria, G.; Martin, G Phys Rev Lett 2002, 89, 266402 Kasinathan, D.; Kunes, J.; Koepernik, K.; et al Phys Rev B 2006, 74, 195110 Tran, F.; Blaha, P.; Schwarz, K.; Novak, P Phys Rev B 2006, 74, 155108 Jollet, F.; Jomard, G.; Amadon, B.; Crocombette, J P.; Torumba, D Phys Rev B 2009, 80, 235109 Petit, T.; Lemaignan, C.; Jollet, F.; Bigot, B.; Pasturel, A Phil Mag B 1998, 77, 779 Freyss, M.; Petit, T.; Crocombette, J P J Nucl Mater 2005, 347, 44–51 Crocombette, J P.; Jollet, F.; Petit, T T.N., L.Phys Rev B 2001, 64, 104107 Iwasawa, M.; Chen, Y.; Kaneta, Y.; Ohnuma, T.; Geng, H Y.; Kinoshita, M Materials Transactions 2006, 47, 2651–2657 Gupta, F.; Brillant, G.; Pasturel, A Philos Mag 2007, 87, 2561–2569 Geng, H Y.; Chen, Y.; Kaneta, Y.; Iwasawa, M.; Ohnuma, T.; Kinoshita, M Phys Rev B 2008, 77, 104120 Yu, J G.; Devanathan, R.; Weber, W J J Phys Condens Matter 2009, 21 Nerikar, P.; Watanabe, T.; Tulenko, J S.; Phillpot, S R.; Sinnott, S B J Nucl Mater 2009, 384, 61–69 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 Gupta, F.; Pasturel, A.; Brillant, G Phys Rev B 2010, 81, 2561–2569 Andersson, D A.; Watanabe, T.; Deo, C.; Uberuaga, B P Phys Rev B 2009, 80(R), 060101 Willis, B T M Proc Br Ceram Soc 1964, 1, Willis, B T M J Chem Soc., Faraday Trans 1987, 83, 1073–1081 Bevan, D J M.; Grey, I.; Willis, B T M J Solid State Chem 1986, 61, 1–7 Garrido, F.; Ibberson, R M.; Nowicki, L.; Willis, B T M J Nucl Mater 2003, 322, 87–89 Geng, H Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M Appl Phys Lett 2008, 93 Geng, H Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M Phys Rev B 2008, 77(R), 180101 Andersson, D A.; Lezama, J.; Uberuaga, B P.; Deo, C.; Conradson, S D Phys Rev B 2009, 79, 024110 Crocombette, J P J Nucl Mater 2002, 305, 29–36 Petit, T.; Jomard, G.; Lemaignan, C.; Bigot, B.; Pasturel, A J Nucl Mat 1999, 275, 119 Freyss, M.; Vergnet, N.; Petit, T J Nucl Mater 2006, 352, 144–150 Gryaznov, D.; Heifets, E.; Kotomin, E PCCP 2009, 11, 7241–7247 Dorado, B.; Freyss, M.; Martin, G Eur Phys J B 2009, 69, 203–209 Nerikar, P V.; Liu, X Y.; Uberuaga, B P.; Stanek, C R.; Phillpot, S R.; Sinnott, S B J Phys Condens Matter 2009, 21 Gupta, F.; Pasturel, A.; Brillant, G J Nucl Mater 2009, 385, 368–371 Brillant, G.; Gupta, F.; Pasturel, A J Phys Condens Matter; 2009, 21 Brillant, G.; Pasturel, A Phys Rev B 2008, 77, 184110 Yun, Y.; Eriksson, O.; Oppeneer, P M J Nucl Mater 2009, 385, 72–74 Wimbauer, T.; et al Phys Rev B 1997, 56, 7384 Torpo, L.; et al J Phys Condens Matter 2001, 13, 6203 ...224 Ab Initio Electronic Structure Calculations for Nuclear Materials 1.08. 5.2.1 1.08. 5.2.2 1.08. 5.2.3 1.08. 5.2.4 1.08. 6 References Bulk electronic structure Point defects... wants to perform Ab Initio Electronic Structure Calculations for Nuclear Materials ab initio calculations But the first and certainly most important of these choices is that of the ab initio code... ab initio calculations The last two sections on metallic alloys and insulating materials will allow us to go into detail for some specific cases Ab Initio Electronic Structure Calculations for