The ‘irradiation’ rate, that is, the rate of impinging particles in the case of neutron and ion irradiation, is usually transformed into a production rate (number per unit time and volume) of randomly distributed displacement cascades of different energies (5, 10, 20, … keV) as well as residual Frenkel pairs (FPs). New cascade debris are then injected randomly into the simulation box at the corresponding rate. The cascade debris can be obtained by MD simulations for different recoil energies T, or introduced on the basis of the number of FP expected from displace­ment damage theory. In the case of KMC simulation of electron irradiation, FPs are introduced randomly in the simulation box according to a certain dose rate, assuming most of the time that each electron is responsible for the formation of only one FP. This assumption is valid for electrons with energies close to 1 MeV (much lower energy electrons may not produce any FP, whereas higher energy ones may produce small displacement cascades with the forma­tion of several vacancies and SIAs).

The dose is updated by adding the incremental dose associated with the scattering event of recoil energy T, using the Norgett-Robinson-Torrens expression8 for the number of displaced atoms. In this model, the accumulated displacement per atom (dpa) is given by:

0 8 T

Displacement per subcascade = — [3]

where T is the damage energy, that is, the fraction of the energy of the particle transmitted to the PKA as kinetic energy and ED is the displacement threshold energy (e. g., 40 eV for Fe and reactor pressure vessel (RPV) steels51).

1.14.4.1 Transmutation Rate

The rate of producing transmutations can also be included in KMC models, as deduced from the reac­tion rate density determined from the product of the neutron cross-section and neutron flux. Like the irra­diation rate, the volumetric production rate is used to introduce an appropriate number of transmutants, such as helium that is produced by (n, a) reactions in the fusion neutron environment, where the species are introduced at random locations within the material.

1.14.4.2 Diffusion Rate

Usually the rates of diffusion can be obtained from the knowledge of the migration barriers which have to be known for all the diffusing ‘objects’; that is, for the point defects in AKMC, OKMC, and EKMC or the clusters in OKMC or EKMC. For isolated point

defects, the migration barriers can be from experi­mental data, that is, from diffusion coefficients, or theoretically, using either ab initio calculations as described in Caturla et a/.49 and Becquart and Domain50 or MD simulations as described in Soneda and Diaz de la Rubia.22 Since the migration energy depends on the local environment of the jumping species, it is generally not possible to calculate all of the possible activation barriers using ab initio or even MD simulations. Simpler schemes such as broken bond models, as described in Soisson et a/.,52 Le Bouar and Soisson,53 and Schmauder and Binkele,54 are then used. Another kind of simpler model is based on the calculation of the system configurational energies before and after the defect jump. In this model, the activation energy is obtained from the final Ef and the initial E as follows:

DE

Ea0 + ~2~ where Ea0 is the energy of the moving species at the saddle point. The modification of the jump activation energy by DE represents an attempt to model the effect of the local environment on the jump frequen­cies. Indeed, detailed molecular statics calculations suggest that this represents an upper-bound influence of the effect,55 and although this is a very simplified model, the advantage is that this assumption maintains the detailed balance of jumps to neighboring positions.

The system configurational energies E and Ef, as well as the energy of the moving species at the saddle point Ea0 can be determined using interatomic poten­tials as described in Becquart et a/.,26 Bonny et a/.,44 Wirth and Odette,55 and Djurabekova et a/.56 when they exist. However, at present, this situation is only available for simple binary or ternary alloys. This approach allows one to implicitly take into account relaxation effects as the energy at the saddle point which is used in the KMC and is obtained after relax­ation of all the atoms. The challenge in that case is the total number of barriers to be calculated, which is determined by the number of nearest neighbor sites included in the definition ofthe local atomic environ­ment. Without considering symmetries, this number is sN, where s is the number of species in the system. In spite of using the fast techniques that were devel­oped to find saddle points on the f/y such as the dimer method,57 the nudged elastic band (NEB) method,58 or eigen-vector following methods,59 this number quickly becomes unmanageable. Ideally, the alterna­tive should be to find patterns in the dependence of the energy barriers on the configuration. This is the
approach chosen by Djurabekova and coworkers,56 using artificial intelligence systems. For more complex alloys, for which no interatomic potentials exist, Ei and Ef can be estimated using neighbor pair interactions60- 63 A recent example of the fitting procedure of a neigh­bor pair interactions model can be found in Ngayam Happy et a/63 A discussion of the two approaches applied to the Fe-Cu system has been published by Vincent et a/64 Also note that in the last 10 years, methods in which the possible transitions are found in some systematic way from the atomic forces rather than by simply assuming the transition mechanism a priori (e. g., activation-relaxation technique (ART) or dimer methods)65-68 have been devised. The accuracy of the simulations is thus improved as fewer assumptions are made within the model. However, interatomic poten­tials or a corresponding method to obtain the forces acting between atoms for all possible configurations is necessary and this limits the range of materials that can be modeled with these clever schemes.

The attempt frequency (nX in eqn [1]) can be calculated on the basis of the Vineyard theory69 or can be adjusted so as to reproduce model experiments.