CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 585 



the spheres can be superposed so that an assembly of donors A'' in num- 

 ber is contained in the volume 1/A^, at the density A'^ . The problem of 

 relaxation is then the problem of diffusion of these donors to the sink of 

 radius R, at the center of the volume. The bounding shell of the sphere 

 may be considered impermeable, thus enforcing the condition that each 

 donor shall be trapped by its nearest neighbor. Since the diffusion prob- 

 lem has spherical symmetry the radius, r, originating at the center of 

 the sink at the origin may be chosen as the position coordinate. At r = 

 R, the density, p, of diffusant may be considered zero. The radius, L, of 

 the volume, 1/A^, is so large compared to R, that in the initial stages of 

 diffusion L may be regarded as infinite. 



In spherical diffusion to a sink from an infinite field, a true steady 

 state is possible, and this steady state is quickly arrived at when the 

 radius, R, of the sink is small. Under this condition concentration is 

 described by 



p = A -- (10.4) 



r 



where A and B are constants. Furthermore at early times n is still N, 

 the initial concentration at r = L ^ oo , so that 



p(oo) = AT' (10.5) 



In addition we know that 



p(R) = (10.6) 



These boundary conditions suffice to determine A and B in (10.4), and 

 yield 



P = N' 



1-^ 

 r 



(10.7) 



Now the rate of capture (—(dn/dt) in (10.2)) is obviously measured 

 by the flux of ions into the spherical shell of area, 4tR', which marks the 

 boundary of the sink. This flux is given according to Fick's law by 



4.^R'D, (^-^) = - ^ (10.8) 



where Do is the diffusivity of the donor. Substituting (10.7) into (10.8) 

 yields 



r2 r- - C?n 



4:tN'RDo = - 4^ (10.9) 



dt 



During the initial stages of trapping the right side of (10.2) may be 



