CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 573 



cal symmetry and it must be defined in terms of the volume elment df, 

 lying at the vector distance r, rather than in terms of the spherical shell 

 of volume, 47rr dr. In reference (44) it is shown that 



T{r) = exp [-4Tr'N/3]c{r) 



(7.18) 



where 6(7) is the density function in the non-equilibrium case, and is 

 determined by the equation 



IcT 



Vc + cV" t// + Vc • Vi/' = 



(7.19) 



after suitable boundary conditions have been appended. The quantity 

 ^, designates the local electrostatic potential, determined by the ions as 

 well as the applied field. These equations are restricted specifically to 

 the semiconductor case in w^hich the negative ion is unable to move. 



The current carried by nearest neighbors in the volume element dr 

 in unit volume of solution is 



Jir) = -exp[-47rrW/3]c(?)/xoV[i/' + (kT/q) tn c(f)] (7.20) 



Using these equations it proves possible in reference 45 to provide a 

 more refined version of (7.15) in which the mobility of nearest neighbors 

 inside r = h need not be considered zero, nor those outside r = 6 be con- 

 sidered perfectly free and possessed of the mobility )Uo . In fact the aver- 

 age mobility of a nearest neighbor separated by a distance r from its 

 immobile partner proves to be 



^ 2(1 - F) \L3r2 ^ 3r ^ _ 



exp (- e/r) + 2F 



.3r 



- 1 



where 



and 



£ = q/KkT 



F = (7^+§ + l)exp(-£/a) 



,2 



,2a2 a 



I'or values of ?■ greater than e (7.21) can be approximated by 

 p. 1 / £" , 4£ ^ 



i"0 



2 V3r2 



+ - + 2 exp (- e/r) 



(7.21) 



(7.22) 



(7.23) 



(7.24) 



and is therefore a function of e/r. Fuoss's b corresponds to r = £/2 or to 

 r/r = 2. Fig. 14 contains a plot of jl/no versus r for T = 400°K, a = 

 cm, 5 = 4.77 X 10"^" statcoulombs, and k = 16. Note that 



-*.5 X 10" 



