Clustering theory of atomic defects

Clustering of atomic defects leads to changes in the microstructure of mate rials, and hence induces drastic variations in their properties. In many te chnical fields, the role of defect clustering is very significant, and is s ometimes limiting to further progress. We present here a comprehensive revi ew of the theory of atomic defect clustering under non-equilibrium conditio ns, particularly encountered during irradiation of materials with energetic particles, as well as during material processing by energetic sources. The se conditions are met in a wide range of technical applications, ranging fr om nuclear and fusion energy to microelectronics and surface engineering. W e first present a general stochastic framework for the evolution of atomic clusters, and show how this can be described within the context of death-an d-birth processes. This leads to the well-known master equation for microsc opic atomic clusters. In the limiting case of a Poissonian process for the transition probabilities between cluster sizes, the master equation tends, in the macroscopic limit, to the mean-field approximation embodied by the t heory of rate processes. When atomic clusters grow or shrink by the absorpt ion of single atomic defects, a continuum Fokker-Planck approximation can b e derived. Within this approximation, the evolution of interstitial loops, voids, bubbles, and general clusters of complex phases is presented, and in some cases, good agreement with experiments is obtained. It is shown that because of coalescence reactions, the evolution of surface atomic clusters during atom deposition processes is best described by kinetic moment equati ons, directly derived from rate equations. It is shown that breaking the sy mmetry of space or time leads to drastic variations in the size and space d istributions of defect clusters. Examples are given for pulsed irradiation conditions, where it is shown that non-linear rate processes enhance cluste r formation during on-time, and could lead to their dissolution during the off-time at high temperature. On the other hand, fluctuations are shown to result in instabilities and spatial self-organization of defect clusters. D escription of pattern formation during irradiation, such as void and inters titial loop lattices, is very well described by a Ginzburg-Landau type equa tion, reminiscent of phase transitions under thermodynamic equilibrium cond itions.