## monte carlo models

The MD simulation of displacement cascades enables the quantification of the irradiation damage in a small volume (typically few tens of nm) and few tens of ps. The evolution in time and space of this damage necessitates the diffusion of atoms and point defects on the crystal lattice which can be modeled by Monte Carlo methods or Statistical Physics. Both tools, since they include a cohesive and kinetic model based on inter-atomic potentials, are able to treat nucleation and growth of precipitation or segregation, in multi-elements materials.

The Monte Carlo (MC) method is a stochastic, statistical mechanics computational tool used very widely in materials science. In particular, the Metropolis MC (MMC) method samples the possible microstates of a system of atoms interacting according to a known cohesive model. It therefore has a similar range of applications to MD, of which it can be seen as an alternative, or as a prolongation. Like MD, its physical reliability lies more in the cohesive model used than in the approximations made in the method. The MMC technique is used in this project mainly for two purposes: (i) to calculate thermodynamic averages in a system of atoms at finite temperatures; (ii) to simulate the annealing of a system of atoms in order to find configurations corresponding to possible energy minima. The main disadvantage of the MMC algorithm, especially compared to MD, is that the convergence towards equilibrium does not follow the real physical mechanisms that lead to a transformation from an arbitrary state to the final one.

Atomic Kinetic Monte Carlo (AKMC): Atomistic kinetic Monte Carlo (AKMC) models share many features with MMC models. they include all atoms of all chemical species of interest, as well as defects (hence the adjective “atomistic”), generally located at known positions on a lattice. Secondly, they allow the system to evolve by extracting random numbers, whereby a decision is taken about what to do next in probabilistic terms (hence the “Monte Carlo” nature). Finally, they can be used to make a system of atoms evolve towards a state of equilibrium. In AKMC simulations, the atomic configuration evolves by thermally activated point defect jumps, i, characterised by specific frequencies.

In this expression, the exponential part is a Boltzmann-type probability which takes into account the effect of temperature, given the jump activation energy (or “migration barrier”). This migration barrier depends on the type of point-defect that jumps, on the atom with which it exchanges position and on the local chemical environment. the attempt frequency is often replaced by a constant value, 0.