CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 571 



ate 12 at infinite dilution, and the value so obtained employed at higher 

 concentrations. 



Besides the inconsistency of the choice, h = N, the form (7.4) contains 

 another objectionable feature. This is revealed by a more rigorous treat- 

 ment devised recently by Reiss, and has to do with the factor, 



exp [— 47r / x^c(r) dx], 



Ja 



in (7.4). It can be shown that this factor is inconsistent with the suppo- 

 sition that the nearest neighbor to a given negative ion interacts only 

 with that ion and no other. Fortunately, in Fuoss's scheme g(r) given by 

 (7.4) needs to be used only at infinite dilution, and then only for such 

 values of r as lie inside h. Under this condition and in this range the ex- 

 ponential factor in question can be replaced by unity from which it de- 

 viates only slightly. Thus the form of g(r) used eventually is 



g(r) = Awf^N exp [q^KkTr-] (7.11) 



U is computed as follows. At infinite dilution P tends toward zero so 

 that (7.10) becomes 



^ = fiA^ (7.12) 



But P/N is the fraction of ions paired which by definition is the fraction 

 of nearest neighbors lying inside r ^ h. From the definition of g{r), 

 P/N is evidently given by 



^ = I g(r) dr = 4.tN [ r' exp [q^KkTr] dr (7.13) 



iV Ja "a 



which upon substitution in (7.12) yields 



j^ = 47r f r' exp [q'/KkTr] dr (7.14) 



The evaluation of in this way permits one to base the entire theory on 

 the distribution of near nearest neighbors, so that all the assumptions 

 \\hich demand this procedure are validated. 



Using the computed 12 in (7.10) P can be evaluated, and also N — P 

 which as the concentration of free ions of one species measures the ther- 

 modynamic activity of that species. In this manner it is possible to calcu- 

 late the equilibrium effects of coulomb interaction insofar as solution 

 properties are concerned. To treat transport phenomena such as ionic 

 mobility in an applied electric field Fuoss assumes that paired ions repre- 



