ON THE THEORY OF POINT DEFECT RECOMBINATION
IN CRYSTALS
I. B. Azarov, M. S. Veshchunov*
Nuclear Safety Institute (IBRAE), Russian Academy of Sciences 115191, Moscow, Russian Federation
Received March 17, 2014
A new approach to the diffusion-limited reaction kinetics for particles migrating by random walks on discrete lattice sites and reacting when two particles occupy the same site is extended to a more general case of a large reaction radius and applied to the problem of the recombination rate of point defects in cubic lattices. Numerical calculations correctly reproduce the analytic expressions in the limit cases considered in previous work and in the general case represent a step-wise dependence of the reaction rate on the recombination radius.
DOI: 10.7868/S0044451014090132
1. INTRODUCTION
The kinetics of an irreversible diffusion-controlled bimolecular reaction ,4 + B —¥ C (where C does not affect the reaction) is described by the rate equation
nA(r,t) = hB( r,t) = -KABnA(rJ)nB(r,t), (1)
where ha and nB are the respective concentrations (numbers of particles per unit volume) of reacting ,4 and B particles, which diffuse freely, and Kab is the reaction constant [1]. This equation is also applicable to the reaction of point defects, vacancies, and intersti-tials (V +1 —¥ 0) and annihilation in crystals produced by means of high-energy particles or electrons [2].
In the continuum approach, the reactant particles are represented as points or spheres undergoing spatially continuous Brownian motion, with chemical reactions ,4 + B —¥ C occurring instantly when the particles pass within a specified reaction radius RaB between their centers. A method for calculating the reaction rate of reaction partners migrating by three-dimensional diffusion was developed in Refs. [3,4] by generalizing the Smoluchowski theory for coagulation of colloids [5,6]. In this method, which stipulates that the local reaction rate is equal to the diffusive current of particles, the radius of the activated complex (or the "reaction radius") corresponds to the "influence-sphere radius" in the Smoluchowski theory (equal to
E-mail: vms'flibrae.ac.ru
the sum of the radii of two colliding Brownian particles, R = Ra + RB)•
This traditional ("diffusion") approach was critically analyzed in our paper [7]. In particular, it was shown that the approach is applicable only to the special case of small ,4-particle trapping in large U-centres with a large trapping radius, 7 -C Rab -C rb (where 7a « « nj^3 and rB « are the mean inter-particle
distances), and becomes invalid for calculating the reaction rate in the case RaB -C 7a,7b, which is most important for the reaction kinetics and, in particular, corresponds to comparable-size (or point-wise, owing to RaB -C 7a,7b) particles ,4 and B migrating by random walks. In order to resolve this inadequacy of the traditional approach, a new approach to the diffusion-limited reaction rate theory, based on a similar consideration of Brownian coagulation proposed in our papers [8 10], was developed in Ref. [7]. In the new ("kinetic") approach, point-wise particles tend to a homogeneous (in random) spatial distribution owing to their migration and mixing on the scale of the mean inter-particle distance / « 7, with the characteristic diffusion mixing time Td that is generally small in comparison with the characteristic reaction time rc, i.e., Td -C rc.
Indeed, for instance in the simplest case Da « ss Db = D and tia ss tib = n, reactions between ,4 and B particles induce local heterogeneities in the spatial distribution of their probability densities on the length scale of the mean inter-particle distance 7 « n-1/3 u. However, such kind of heterogeneities quickly disappear owing to rapid diffusion relaxation on the length scale
of the moan inter-particle distance r with the characteristic time Td « r2/6D, which is generally much shorter than the characteristic time rc « (Kab>>°f the particle concentration variation, Td -C rc (or equally, in terms of the mean free path A of a particle between its two subsequent collisions, A « (6Dtc)x/2 r, which has a clear physical sense and is valid under the basic "dilution" condition of the theory, n1/3i? « R/r <1).
In the opposite cases, Da Db or ha h>b, the mixing of slow particles (e.g., B) might be incomplete (if « r2/6DB > tc). However, owing to the stochastic character of movement and collisions of particles ,4, the "surviving" particles B are still randomly distributed in space, whereas rapidly moving particles ,4 heal up local heterogeneities in the ,4-particlo distribution induced by reactions ("rarefied zones" in the vicinity of two-particle reactions) and thus uphold efficient mixing of the reaction system. This ensures the applicability of the kinetic approach to this case with a somewhat reduced, but reasonable accuracy.
This implies that a random distribution of particles is attained during a time step Td -C St -C rc, chosen for calculation of the reaction rate, which can therefore be searched in the kinetic approach as the collision frequency of two particles (.4 and B) randomly located in unit volume. That value can be equally calculated as the rate of volume sweeping S(Vab)/St by the effective particle of the radius Ra + Rb migrating with the diffusivity Da + Db ■
The new approach (based on the "diffusion mixing" condition) was also generalized in Rof. [7] to the reaction kinetics of particles migrating by random walks on discrete lattice sites (with the lattice spacing a), and reacting when two particles occupy the same site, i.e., Rab < u. Similarly to the continuum limit, it was shown that the original multi-particle problem can be readily reduced, owing to rapid diffusion mixing of particles between their mutual collisions, to the calculation of the collision probability between two particles randomly located in unit volume, which in turn can be related to the mean number of distinct sites visited by a fr-stop random walk of the effective particle (a discrete analogue of the swept volume).
The volume swept by a Brownian particle is known as the Wiener sausage [11]. In particular, this quantity equals the probability that a diffusing Brownian point-like particle is absorbed by a single trap of radius Rab in time t (see, e.g., [12]). For this reason, the rate of volume sweeping coincides with the condensation rate constant for small particles sinking in a large trap, Ta -C Rab -C rs\ for comparable-size particles (or Rab -C ^i.^b). it eventually determines the
Smoluchowski constant in Eq. (1), as is justified in the new kinetic approach [7,8].
The discrete analogue of the Wiener sausage was related to the survival probability for a Brownian particle in the prosonso of random immobile traps in the Rosonstock approximation [13] or other related problems, e.g., the so-called "target annihilation by scavengers" [14]. In the latter problem, a single particle .4 (target) and N,\ = hbN particles B (scavengers) of a finite concentration ns are randomly located on Ar —¥ oc sites of a 3-dimonsional regular lattice. Particle ,4 is immobile, whereas particles B perform independent, homogeneous discrete-time random walks on the lattice sites (including sites occupied by other particles); particle ,4 annihilates as soon as a particle B reaches it.
In fact, the kinetic approach (based on the diffusion mixing condition) allows extending the solution of the target annihilation problem to consideration of many-body effects in the diffusion-limited reaction kinetics. Indeed, since particles B moves independently from each other, the probability of the target annihilation between time t and t+St reduces to the probability of a two-particle (.4 B) collision, wab(t)St, times Atb-In the case of mobile particles ,4 with a finite concentration nyi(0), the problem also reduces to the analyzis of two-particle collisions, if rapid diffusion mixing of particles occurs between their mutual collisions. Actually, after each annihilation event (at a moment t) when a certain lattice position (where the collision occurred) becomes definitely unoccupied, the random (oquiprob-ablo) spatial distribution of particles over lattice sites is rapidly restored during the mixing time Td -C St, and a configuration similar to the initial configuration (i.e., random location of particles ,4 and B on lattice sites), but with the new (diminished) particle concentrations ila,b (t + St) = ila,b (t) — 'h'a'H'buk\b{t)St, can be considered in the subsequent time step, if St -C rc. In the case St f « 16R2iB/irDAB, which is generally valid because rc f [7], a steady-state value of it-ab(t) « wab(oo) = wab is attained in the time step St, and thus the reaction rate equation takes the form of Eq. (1) with the rate constant Kab = u-ab» which does not depend on time explicitly (as opposed to condensation of small particles in a large trap, considered in the diffusion approach).
It is important to note that in contrast to the target annihilation problem where sites can be occupied by several particles, two (or more) point-like defects (of the same type) cannot occupy the same site. However, under the basic assumption of the reaction rate theory, nA,ns -C 1, the ,4 ,4 and B B "collisions" (i.e.,
occupation of one site by two identical particles) can be generally neglected in calculating the ,4 B reaction rate. Indeed, the incorporation of these events during the time step St -C rc, which is used in the derivation of the rate equation and calculation of the reaction constant Kab in the kinetic approach, requires considering two simultaneous or successive collisions (.4 ,4 and .4 B, or B B and B ,4) in unit volume during St, with the respective probabilities waab^- oc ti\tibStand w abb St oc n%n ¿St, which can be neglected, owing to nA,ns -C 1, in comparison with the probability WAsSt oc iia'!isSt of a single pair-wise ,4 B collision during St in unit volume. Therefore, the influence of the for
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