Dynamic particle aggregation induced by elastic interactions

3 DYNAMIC PARTICLE AGGREGATION INDUCED BY ELASTIC INTERACTIONS 

Up to now we have been discussing in this Chapter many-particle effects in bimolecular reactions between non-interacting particles. However, it is well known that point defects in solids interact with each other even if they are not charged with respect to the crystalline lattice, as it was discussed in Section 3.1. It should be reminded here that this elastic interaction arises due to overlap of displacement fields of the two close defects and falls off with a distance r between them as U r) — —Xr for two symmetric (isotropic) defects in an isotropic crystal or as U r) = — A[o 4]r , if the crystal is weakly anisotropic [50, 51] ([04] is an angular dependent cubic harmonic with ( = 4). In the latter case, due to the presence of the cubic harmonic 0 4 an interaction is attractive in some directions but turns out to be repulsive in other directions. Finally, if one or both defects are anisotropic, the angular dependence of U r) cannot be presented in an analytic form [52]. The role of the elastic interaction within pairs of the complementary radiation the Frenkel defects in metals (vacancy-interstitial atom) was studied in [53-55] it was shown to have considerable impact on the kinetics of their recombination, A -f- B — 0. 

Mobile H centres in alkali halides are known to aggregate in a form of complex hole centres [64] this process is stimulated by elastic attraction. It was estimated [65, 66] that for such similar defect attraction the elastic constant A is larger for a factor of 5 than that for dissimilar defects - F, H centres. Therefore, elastic interaction has to play a considerable role in the colloid formation in alkali halides observed at high temperatures [67]. In this Section following [68] we study effects of the elastic interaction in the kinetics of concentration decay whereas in Chapter 7 the concentration accumulation kinetics under permanent particle source will be discussed in detail. 

In this Section we consider the following problem. Defects B are mobile (r B 0) and interact with each other elastically as I7bb( ) = — Ar - we call hereafter this interaction dynamical. Their counter-partners A involved into the bimolecular recombination, A + B 0, could be both immobile. Da = 0, and mobile. Da = Db- Obviously to calculate the kinetics of this reaction, we have to go beyond the framework of the traditional approach. Section 4.1, which neglects the interaction of similar particles. 

In the dimensionless units r — r/ro, f = Dt/rQ, K t ) = K t)/ 4TrDro), n t ) = 4nrln t) and using relative diffusion coefficients S)a = 2 , B = 2(1 — /c) where k = Da./D (primes are omitted below) the joint correlation functions describing spatial correlations of dissimilar, Y r,t), and similar, X r,t), particles obey the following non-linear kinetic equations 

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