The PF equations introduced in Section 1.15.2, that

is, eqns [2] and [3], are phenomenological, and one particular consequence is that they lack an absolute length scale. All scales observed in PF simulations are expressed in units of the interfacial width we of the appropriate field variable. As discussed in the previous section, for the case of one scalar conserved order parameter, this width we and the excess inter­facial free energy a are directly related to the gradi­ent energy coefficient к and the energy barrier between the two stable compositions Dfmax (see eqns [15] and [16]).

Beyond the difficulty of parameterizing к and Afmax to accurately reflect the properties of a given alloy system, the phenomenological nature of these coefficients creates additional problems. In particular, as the number of mesh points used in a simulation increases, the interfacial width, expressed in units of mesh point spacing, remains constant if no other parameter is changed. Increasing the number of mesh points thus increases the physical volume that is simulated but does not increase the spatial resolution of the simulations. If the intent is to increase the spatial resolution, one would have to increase к so that the equilibrium interface is spread over more mesh points. Equilibrium interfacial widths in alloy systems typically range from a few nanometers at high temperatures to a few angstroms at low tem­peratures. In the latter case, if the interface is spread over several mesh points, it implies that the volume assigned to each mesh point may not even contain one atom. This raises fundamental questions about the physical meaning of the continuous field vari­ables, and practical questions about the merits of PF modeling over atomistic simulations.

 

Another important problem related to the lack of absolute length scale in conventional PF modeling concerns the treatment of fluctuations. Fluctuations arise owing to the discrete nature of the microscopic (atomistic) models underlying PFMs. Furthermore, fluctuations are necessary for a microstructure to escape a metastable state and evolve toward its global equilibrium state, such as during nucleation. Fluctua­tions, or numerical noise, will also determine the initial kinetic path of a system prepared in an unstable state. The standard approach for adding fluctuations to the PF kinetic equations is to transform them into Langevin equations, and then to use the fluctuation- dissipation theorem to determine the structure and amplitude of these fluctuations. For instance, in the case of one conserved order parameter, the Cahn-Hilliard diffusion equation, that is, eqn [2], is transformed into the Langevin equation:

 

dC(r, t) dt

 

V

 

x(rF)

 

[19]

 

where X(r, t) is a thermal noise term. The structure of the noise term can be derived using fluctuation — dissipation105,106:

 

(x(r;t )) = 0 (X(r, t)X(r’/)) = -2кв TMV2d(r — r’)8(t — t’) [20;

 

where the brackets () indicate statistical averaging over an ensemble of equivalent systems. However, eqn [20] does not include a dependence of the noise amplitude with the cell size, which is not physical. Even if this dependence is added a posteriori, it is observed practically that this noise amplitude gives rise to unphysical evolution, as reported by Dobretsov eta/.107 While these authors have proposed an empirical solution to this problem by filtering out the short-length-scale noise in the calculation of the chemical potentials, a physically sound treatment of fluctuations requires a derivation of the PF equations starting from a discrete description. Recently, Bronchart et a/.100 have clearly demon­strated how to rigorously derive the PF equations from a microscopic model through a series of con­trolled approximations. We outline here the main steps of this derivation. The interested reader is referred to Bronchart et a/.100 for the full derivation. These authors consider the case of a binary alloy system in which atoms migrate by exchanging their position with atoms that are first nearest neighbors on a simple cubic lattice. A microscopic configuration is defined by the ensemble of occupation variables, or

spin values, for all lattice sites, C = {s}, where S = ±1 when the site i is occupied by an A or a B atom, respectively. The evolution of the probability distribution of the microscopic states is given by the following microscopic Master Equation (ME):

@PC: = -£ W (C! CiJ )P(C)

i; J

+ W(CiJ! C)P(CiJ) [21]

where the * symbol in the summation indicates that it is restricted to microscopic states that are connected to C through one exchange of the i and J nearest neighbor atoms, resulting in the configuration CiJ. The next step is to coarse-grain the atomic lattice into cells, each cell containing Nd lattice sites. It is then assumed that local equilibrium within the cells is achieved much faster than evolution across cells. The composition of the cell n, cn, is given by the average occupation of its lattice site by B atoms, and thus cn = 0, 1/Nj,, Nj/Nj. A mesoscopic config­uration is fully defined on this coarse-grained system by C = {cn}. A chemical potential can be defined within each cell and, if this chemical potential varies smoothly from cell to cell, the microscopic ME, eqn [21], can be coarse-grained into a mesoscopic ME: 2 *

lmn(C)exp

n, m

+ gain term

where a is the lattice parameter and d the number of lattice planes per cell (i. e., Nd = (d/a)3), в is the attempt frequency of atom exchanges, lmn(C) is a mobility function that is directly related to the microscopic jump frequency, b = (B T) 1, and mn(C) is the chemical potential in cell n. The * symbol over the summation sign indicates that the summation over m is only performed over cells that are adjacent to the cell n; the first term on the right — hand side of eqn [22] represents a loss term, and there is a similar gain term, which is not detailed.

The mesoscopic ME eqn [22] can be expanded to the second order using 1/Nd as the small parameter for the expansion. The resulting Fokker-Planck equa­tion is then transformed into a Langevin equation for the evolution of the composition in each cell n:

dc a2 в (n) ~ ~ ~

~@t = d2 к T^ ^l”m(C)[mm(C) — mm(C)] + Cn(t) [23]

B m

where the noise term Zn (t) is a Gaussian noise with first and second moments given by

<Cn(0> = 0

2 a2 (n)

<Cn(t)Cn(t’)> = — J2 lnf (C )8(t — t0)

d p 2 a2 ~

<Cn(t)Zm(t0)> = —- ^2 lnm(C )d(t — О [24]

Nd d2

While the structure of eqns [23] and [24] is quite similar to that of the phenomenological eqns [19] and [20], there are several key differences in these two descriptions. First, thermodynamic quantities such as the homogeneous free-energy density and the gradient energy coefficient are now cell-size dependent. These quantities can be evaluated sep­arately using standard Monte Carlo techniques.1 Second, the mobility coefficients, and thus the correlations in the Langevin noise, are func­tions of the local concentration, as well as of the cell size.

Bronchart eta/.100 applied their model to the study of nucleation and growth in a cubic A1-cBc system for various cell sizes, d = 6a, d = 8a, and d = 10a. The supersaturation is chosen to be small so that the critical nucleus size is large enough to be resolved by these cell sizes. As seen in Figure 4, for a given supersaturation, the evolution of the volume fraction of precipitates is independent of the cell size and in very good agreement with fully atomistic kinetic Monte Carlo (KMC) simulations (not shown in Figure 4).

The above results are important because they show that it is possible to derive and use PF equations that retain an absolute length scale defined at the atomistic level. The point will be shown to be very important for alloys under irra­diation. On the other hand, the work by Bronchart et a/.10 clearly highlights the difficulty in using quantitative PF modeling when the physical length scales of the alloy under study are small, as for instance in the case of precipitation with large supersaturation, which results in a small critical nucleus size, or in the case of precipitation growth and coarsening at relatively moderate temperature, which results in a small interfacial width. In these cases, one would have to reduce the cell size down to a few atoms, thus degrading the validity of the microscopically based PF equations since they are derived by relying on an expansion with respect to the parameter 1/Nd.

Many experimental studies ofRIS were carried out in the 1970s in model binary or ternary alloys, as well as in more complex and technological alloys (especially in stainless steels). It became apparent quite early on that RIS was a pervasive phenomenon, occurring in many alloys and with any kind of irradiating particle (ions, neutrons, or electrons). Extensive reviews can be found in Russell,1 Holland et al.,2 Nolfi,3 Ardell,4 and Was5: here, we present only the general conclu­sions that can be drawn from these studies.

1.18.2.3.1 Segregating elements

From the previous discussion, it is clear that it is difficult to predict the segregating element in a given alloy because of the competition between sev­eral mechanisms and the lack ofprecise diffusion data (especially concerning interstitial defects). As will be shown in Section 1.18.3, only the knowledge of the phenomenological coefficients Lj provides a reliable prediction of RIS. Nevertheless, on the basis of the body of RIS experimental studies, several general rules have been proposed. In dilute binary AB alloys, thermal self-diffusion coefficients D^ and impurity diffusion coefficients are generally well known, at least at high temperatures. Tracer diffusion or intrinsic diffusion coefficients in some concentrated alloys are also available.34 RIS experiments do not reveal a sys­tematic depletion of the fast-diffusing and enrichment of the slow-diffusing elements near the point defect sinks4,29: this suggests that the IK effect by vacancy diffusion is usually not the dominant mechanism. On the other hand, it seems that a clear correlation exists between RIS and the size effect33; undersized atoms usually segregate at point defect sinks, oversized atoms usually do not. This suggests that interstitial diffusion could control the RIS, at least for atoms with a significant size effect. There are some excep­tions: in Ni-Ge and Al-Ge alloys, the segregation of oversized solute atoms has been observed. Neverthe­less, as pointed out by Rehn and Okamoto,33 no case of depletion of undersized solute atoms in dilute alloys has ever been reported. According to Ardell,4 this holds true even today.

Irradiation of materials with energetic particles drives them from equilibrium, and in alloys, this becomes manifest in a number of ways. One of them concerns nonequilibrium segregation. The creation of large supersaturations of point defects leads to persistent defect fluxes to sinks. In many cases, these point defect fluxes couple with solutes, resulting in either the enrichment or depletion of solutes at these sinks. This effect was first discovered by using in situ electron
irradiations in a high voltage electron microscope,27 and it has been systematically investigated subse­quently using ion irradiations,28 as the surface sink provides a convenient location to measure composi­tion changes. Unlike neutron irradiation, moreover, the damage created by ions is generally inhomoge­neous, reaching a peak level at some depth in the sample. As a consequence, point defect fluxes ema­nate from these regions. An example of this effect is shown in Figure 15 where a Ni—12.7at.% Si alloy was irradiated with protons. As the alloy is supersat­urated with Si prior to irradiation, Ni3Si precipitates

form in the sample. At the location of peak damage, the concentration of interstitials is the highest, and hence these defects flow outward from this region. These interstitials form interstitial-solute complexes with Si, resulting in a Si flux out of this area as well, depleting the region of Si. As a consequence, a region depleted of Ni3Si precipitates is observed at the peak damage depth. Note too that the surface sink for interstitials leads to enrichment of Si, resulting in a surface layer of Ni3Si. The region just below the surface accordingly becomes depleted of Si, leaving a zone depleted of Ni3Si precipitates.

While irradiation induced segregation can lead to nonequilibrium segregation and precipitation in sin­gle phase alloys, irradiation can also lead to dissolu­tion of precipitates in nominally two-phase alloys. An interesting example of this behavior concerns Ni-12 at.% Al alloys irradiated with 300 keV Ni ions.29 These alloys were first annealed at high tem­peratures to develop a two-phase structure of Ni3Al (g0) and Ni-10.5 at.% Al (g). The initial precipitate size, depending on the annealing time was 2.5 or 4.6 nm. As shown in Figure 16, the precipitates dis­order during irradiation at room temperature, owing to atomic mixing in cascades. The rate of disordering depends on the size of the precipitates, being slowest for homogeneous Ni3Al sample and fastest in the alloy with the smallest precipitates. The authors

suggest that the reason for this dependence on pre­cipitate size is that atomic mixing reduces the concentration of Al in the precipitates, which thereby accelerates the disordering. When the same irradiation is performed at higher temperatures, and radiation-enhanced diffusion takes place, the system does not completely disorder, but rather remains partially ordered, owing to a competition between disordering in the displacement cascades and reor­dering by radiation-enhanced diffusion. Noteworthy, however, is the size of the precipitate, as shown in Figure 16(c), where it is observed that the precipi­tates initially shrink in size, but then reach a steady state radius. Therefore, unlike in thermal aging, pre­cipitates in irradiated alloys can reach a stable steady state size that is a function of irradiation intensity and temperature. Similar behavior has been observed in two-phase immiscible alloys in which case a steady state size of precipitates is formed.3 This so-called ‘patterning’ phenomenon has been explained on the basis of a competition between disordering by atomic mixing in energetic collision events and reordering during thermally activated diffusion. For patterning, however, it is required that the atomic relocation distances during collisional mixing are significantly larger than the nearest neighbor distance. An inter­esting consequence of this requirement in regard to the present discussion of using ion irradiation to simulate neutron damage is that electron and proton irradiations, which do not produce energetic cascades or long relocation distances, should not induce com­positional patterning, but heavy ions and fast neutron irradiation, which do produce cascades, will cause patterning. Further details can be found in Enrique31 and Enrique et al31

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 p 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

The two-band model projects the magnetism onto each atom. It does not properly describe magnetic interactions, so it cannot distinguish between ferro, para, and antiferromagnetism. In order to do so, we need to include an interionic exchange term.

Pauli repulsion arises from electron eigenstates being orthogonal. While its nature on a single atom is complex, its interatomic effects can be modeled as a pairwise effect of repulsion between electrons of similar spins. The secondary effect of magnetization is that there are more electrons in one band than in another, and more same-spin electron pairs to repel one another, and so the repulsion between those bands is enhanced.

Conceptually, this can be captured in two pairwise effects, the standard nonmagnetic screened-Coulomb repulsion of the ions plus the core-core repulsion, and an additional Si dependent term arising from Pauli repulsion between like-spin electrons.

V (r, j) = Vo(rt,) + (S" Sj" + S# S#) Vm(rj) [21]

Note that an antiferromagnetic state with Si + Sj = 0 would have a lower repulsive energy. In the tight — binding picture, this would be compensated by a much reduced hopping integral, and hence, lower W. If we insist on Si > 0, then we suppress these solutions and can model ferromagnetic or diamagnetic iron. Also, as with DFT-GGA/LDA, the spin is Ising-like.

At the time of writing, no good parameterization of this type of potential exists. The difficulty is that determining the spins Sj is a nonlocal process: the optimal value ofthe spin on site i depends on the spin at site j. The only practical way to proceed appears to be to treat the spins as dynamical variables, in which case it is probably better to treat them as noncollinear Heisenberg moments.

All models use periodic boundary conditions in the direction of the dislocation line so that the dislocation is effectively infinite in length. If the model contains one obstacle, the length, L, of the model in the periodic direction represents the center-to-center obstacle spacing along an infinite row of obstacles. It is the treatment of the boundaries in the other two directions that distinguishes one method from another. A versatile atomic-scale model should allow for the following.10

1. Reproduction of the correct atomic configuration of the dislocation core and its movement under the action of stress.

2. Application of external effects such as applied stress or strain, and calculation of the resultant response such as strain (elastic and plastic) or stress and crystal energy.

3. Possibility of moving the dislocation over a long distance under applied stress or strain without hindrance from the model boundaries.

4. Simulation of either zero or non-zero temperatures.

5. Possibility of simulating a realistic dislocation density and spacing between obstacles.

6. Sufficiently fast computing speed to allow simula­tion of crystallites in the sizes range where size effects are insignificant.

A comprehensive review ofmodels developed so far is to be found in Bacon et at.? and so here we merely present a short summary of the pros and cons of some models used most commonly. Historically, the earliest models consisted of a small region of mobile atoms surrounded in the directions perpendicular to the dislocation direction by a shell of atoms fixed in the positions obtained by either isotropic or anisotropic elasticity for displacements around the dislocation of interest.11 This model was used successfully to inves­tigate dislocation core structure and, being simple and computationally efficient, can use a mobile region large enough to simulate interaction between static dislocations and defects and small defect clusters. Its main deficiencies are its inability to model dislocation motion beyond a few atomic spacings because of the rigid boundaries (condition 3) and its restriction to temperature T = 0 K (condition 4).

The desirability of allowing for elastic response of the boundary atoms due to atomic relaxation in the inner region, for example, when a dislocation moves, has led to the development ofseveral quasicontinuum models. The elastic response can be accounted for by using either a surrounding FE mesh or an elastic Green’s function to calculate the response of bound­ary atoms to forces generated by the inner region. Such models are accurate but computationally
inefficient and have not found wide application so far.4 Furthermore, their use for simulation of T> 0K (condition 4) is still under development.12 Nevertheless, quasicontinuum models, especially those based on Green’s function solutions, can be employed in applications where calculation of forces on atoms is computationally expensive and a signifi­cant reduction in the number of mobile atoms is desirable.13

The models now most widely applied to simulate dislocation behavior in metals are based on the peri­odic array of dislocations (PAD) scheme first intro­duced for simulating edge dislocations.14,15 In this, periodic boundary conditions are applied in the direction of dislocation glide as well as along the dislocation line, that is, the glide plane is periodic. This means that the dislocation is one of a periodic, 2D array of identical dislocations. The success of PAD models is because of their simplicity and good computational efficiency when applied with modern empirical IAPs, for example, embedded atom model (EAM) type. They can be used to simulate screw, edge, and mixed dislocations.4,10,16 With a PAD model containing ^106-107 mobile atoms, essentially all con­ditions 1-6 can be satisfied. Their ability to simulate interactions with strong obstacles of size up to at least ~10 nm makes PAD models efficient for investi­gating dislocation-obstacle interactions relevant to a radiation damage environment. Practically all impor­tant radiation-induced obstacles can be simulated on modern computers using parallelized codes and most can even be treated by sequential codes.

Details of model construction for different dislo­cations can be found elsewhere.4,10,16 Here we just present an example of system setup for screw or edge dislocations in bcc and fcc metals interacting with dislocations loops and SFTs, as presented in Figure 1.

There are two types of DL in an fcc metal: glissile perfect loops with bL = 1/2(110) and sessile Frank loops with bL = 1/3(111). There are two types of glissile loop with Burgers vectors 1/2(111) and (100) in a body-centered cubic (bcc) metal.

Visualizing interaction mechanisms is a strong feature of atomic scale modeling. The main idea is to extract atoms involved in an interaction and visua­lize them to understand the mechanism. Usually these atoms are characterized by high energy, local stresses, and lattice deformation. The techniques used are based on analysis of nearest neighbors,17 central symmetry parameter,18 energy,19 stress,20 dis­placements,10 and Voronoi polyhedra.21 A relatively simple and fast technique, for example, was suggested for an fcc lattice.16 It is based on comparison of position of atoms in the first coordination of an atom with that of a perfect fcc lattice. If all 12 neigh­bors of the analyzed atom are close to that position, it is assigned to be fcc. If only nine neighbors corre­spond to perfect fcc coordination, the atom is taken to be on a stacking fault. Other numbers of neighbors can be attributed to different dislocations. Modifica­tions of this method have been successfully applied in hexagonal close-packed (hcp)22 and bcc23 crystals. Another improvement of this method for MD simu­lation at T> 0 K was introduced24 in which the above analysis was applied periodically (every 10-50 time — steps depending on strain rate e) and over a certain time period (100-1000 steps). A probability of an atom to be in different environment was estimated and the final state was assigned to the maximum over the analyzed period. Such a probability analysis can be applied to other characteristics such as energy or stress excess over the perfect state and it provides a clear picture when the majority of thermal fluctua­tions are omitted.

It has been suggested that deviations of the SIA cluster diffusion from pure 1D mode may signifi­cantly alter their interaction rate with stable sinks.2 These deviations could have different reasons, such as thermally activated changes of the Burgers vector of glissile SIA clusters, as observed in MD simulation studies for clusters of two and three SIAs. The reac­tion rate in the case has been calculated previ — ously25,27; here we present the main result only.

If tch is the mean time delay before Burgers vector change and I = J2D1D’tch is the corresponding MFP, then the reaction rate can be approximated by the following function25:

8l21/2 C

/2 Tch

which gives the correct value in the limiting case of pure 1D diffusion, when tch! 1, and a correct description of increasing reaction rate with decreasing tch. The analysis is valid for values of I larger than the mean void and dislocation capture radii, and overesti­mates reaction rates in the limiting case of 3D diffu­sion, see paragraph 6 in Barashev et a/.25 for details. Similar functional form of the reaction rate is obtained by employing an embedding procedure,27 which gives a correct description over the entire range of / in the case when voids are the dominant sinks in the system.

Once dislocation sources are unlocked from their decoration atmospheres causing a yield drop, they additionally interact with the surrounding random field of defect clusters (e. g., vacancy loops or SFTs). We investigate here the interaction between emitted F-R dislocations and vacancy-type defect clusters as a possible mechanism of radiation softening imme­diately beyond the yield point. Numerical computer simulations are performed for the penetration of undissociated slip dislocation loops emitted from active F-R sources (i. e., unlocked from defect dec­orations) against a random field of SFTs or sessile Frank loops of the type: 1(111){111}. At the present level of analysis, there is no distinction between SFTs and vacancy loops as they are modeled as point obsta­cles to dislocation motion that can be destroyed once an assumed critical force on them is reached. The long-range elastic field of these small obstacles is ignored. The random distribution of SFTs is gener­ated as follows: (1) The volumetric density of SFTs is used to determine the average 3D position of each generated SFT; (2) A Gaussian distribution function is used (with the standard deviation being 0.1-0.3 of the average spacing) to assign a final position for each generated SFT around the mean value. (3) The inter­section points of SFTs with glide planes are com­puted by finding all SFTs that intersect the glide plane. For simplicity, we perform this procedure assuming that SFTs are spherical and uniform in size.

Initially, one slip dislocation loop is introduced between two fixed ends and a search is performed for all neighboring SFTs on the glide plane. Subsequent nodal displacements (governed by the local velocity) are adjusted such that a released segment interacts with only one SFT at any given time. The interaction scheme is a dynamic modification to Friedel statis — tics,36 where the asymptotic maximum plane resis­tance is found by assuming steady-state propagation

of quasistraight dislocation lines. While Friedel cal­culates the area swept as the average area per particle on the glide plane, we adjust the segment line shape dynamically over several time steps after it is released from an SFT When a segment is within 5a from the center of any SFT, it is divided into two segments with an additional common node at the point of SFT intersection with the glide plane. The angle between the tangents to the two dislocation arms at the com­mon node is then computed, and force balance is performed. When the angle between the two tangents reaches a critical value of Fc, the node is released, and the two open segments are merged into one. If the force balance indicates that the segment is near equilibrium, no further incremental displace­ments of the node are added, and the segment of the loop is temporarily stationary. However, if a net force acts on that segment of the loop, it is advanced and the angle recomputed. It is possible that the angle between tangents will reach the critical value even though the segment is out of equilibrium. Sun eta/.37 have shown that the elastic interaction energy between a glissile dislocation and an SFT is not suffi­cient to transform the SFT into a glissile prismatic vacancy loop. They proposed an alternate mechanism for the destruction of SFTs by passage of jogged and/ or decorated dislocations close to the SFT. The energy released from recombination of a small frac­tion of vacancies in the SFT was estimated to result in its local rearrangement.

In the present calculations, we assume that vacan­cies in the SFT are absorbed in the dislocation core of the small contacting dislocation segment, forcing it to climb and form atomic jogs. With this mechanism, the entire SFT is removed from the simulation space, and jogged dislocations continue to glide on separate planes, thus dragging atomic-size jogs with them. Successive removal of SFTs from nearby glide planes can easily lead to channel formation and flow locali­zation in the channel, because the passage of consec­utive dislocation loops emitted from the F—R source is facilitated with each dislocation loop emission. The matrix density of SFTs in irradiated copper at low temperature (0.22-0.27 Tm) is taken from experimen­tal data (Singh et a/.38). Figure 5 shows the results of computer simulations for propagation of plastic slip emanating from a single Frank-Read source in cop­per irradiated and tested at 100°C (Singh et a/.38). The density of SFTs is 4.5 x 1023 m~3 and the aver­age size is 2.5 nm. In this simulation, the crystal size is set at ^1.62 pm, while the initial F-R source length is 1600a (^576 nm). A uniaxial applied tensile stress

along [100] (ffn) is incrementally increased, and the dislocation line configuration is updated until equilibrium is reached at the applied stress. Once full equilibrium of the dislocation line is realized, the stress is increased again, and the computational cycle repeated. At a critical stress level (flow stress), the equilibrium dislocation shape is no longer sustainable, and the dislocation line propagates until it is stopped at the crystal boundary, which we assume to be impene­trable. All SFTs interacting with the dislocation line are destroyed, and plastic flow on the glide plane is only limited by dislocation-dislocation interaction through the pileup mechanism. During the initial stages of deformation, small curved dislocation seg­ments unzip, forming longer segments, which are stuck until they are unzipped again by increasing the applied stress. It is noted that, particularly at higher stress levels, the F-R source dislocation elongates along the direction of the Burgers vector, as a result of the higher stiffness of screw dislocation segments as compared to edge components. The F-R source con­figuration is determined by (1) the character of its

initial segment (e. g., screw or edge), (2) the distribution of SFTs intersecting the glide plane, and (3) any other dislocation-dislocation interactions. This aspect is illu­strated in Figure 6, where two interacting F-R sources in copper are shown for a displacement dose of 0.01 dpa, an SFT density of 2.5 x 1023 m~3, and an average size of 2.5 nm. All other conditions are the same as in Figure 5. The two F-R sources are sepa­rated by 20a (~7.2 nm).

Interdiffusion experiments have been performed in austenitic and ferritic steels.34 The determination of the intrinsic diffusion coefficients requires the mea­surement of the interdiffusion coefficient and of the Kirkendall speed for each composition.68 In general, an interdiffusion experiment provides the Kirkendall speed for one composition only, leading to a pair of intrinsic diffusion coefficients in a binary alloy. Therefore, few values of intrinsic diffusion coeffi­cients have been recorded at high temperatures and on a limited range of the alloy composition. More­over, experiments such as those by Anthony happened to be feasible in some Al, Cu, and Ag dilute alloys. As a result, a complete characterization of the L-coefficients of a specific concentrated alloy (even limited to the vacancy mechanism) has, to our knowledge, never been achieved. In the case of the interstitial diffusion mechanism, the tracer diffusion measurements under irradiation were not very con­vincing and did not lead to interstitial diffusion data. The interstitial data, which could be used in RIS models,12 were the effective migration energies deduced from resistivity recovery experiments.

1.18.3.2 Determination of the Fluxes from Atomic Models

First-principles methods are now able to provide us with accurate values of jump frequencies in alloys,
not only for the vacancy, but also for the interstitial in the split configuration (dumbbell). Therefore, an appropriate solution to estimate the L-coefficients is to start from an atomic jump frequency model for which the parameters are fitted to first-principles calculations.

1.18.3.4.1 Jump frequencies

In the framework of thermally activated rate theory, the exchange frequency between a vacancy V and a neighboring atom A is given by:

G ( D EAVg

kB T

if the activation energy (or migration barrier) AeAV is significantly greater than thermal fluctuations kB T (a similar expression holds for interstitial jumps). aeAV? is the increase in the system energy when the A atom goes from its initial site on the crystal lattice to the saddle point between its initial and final posi­tions. One of the key points in the kinetic studies is the description of these jump frequencies and of their dependence on the local atomic configuration, a description that encompasses all the information on the thermodynamic and kinetic properties of the system.

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 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.