CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 569 



ions to interact with but themselves. Thus the potential energy of inter- 

 action, for near nearest neighbors will be 



- - (7.6) 



KV 



For small values of r, therefore, c(r) can be derived from Boltzmann's 

 law and is given by 



c(r) = hexp[q-/KkTr] (7.7) 



where his a constant. Guided by the requirement that c(r) should equal 

 A^ at infinite distance from the central negative ion, h was set equal to 

 N giving, finally, 



c{r) = N exp [g'/KkTr] (7.8) 



The assumption that a theory could be developed depending only on 

 near nearest neighbors proved reasonable, but the choice of /t = A'' in 

 (7.8) leads to certain logical diflEiculties. Thus the average volume domi- 

 nated by a given negative ion is evidently 1/A^. If (7.8) is summed over 

 this volume the result, representing the number of positive ions in 1/A'', 

 should be unity since there are equal numbers of positive and negative 

 ions. Unfortunately, the i-esult exceeds unity by very large amounts ex- 

 cept for very small values of iV, i.e., for veiy dilute solutions. We shall 

 return to this point later. 



If (7.8) is inserted into (7.4) the resulting g{r) has the form typified 

 by Fig. 13. First, there is an exponential maximum occurring at r = a, 

 followed bj^ a long low minimum, and this by another maximum which 

 like the one in Fig. 12 occurs, not far from r = (3/47rA'')*'^, if N is not 

 too large. For small values of N the minimum occurs at 



r = h = q/2KkT (7.9) 



The function g{r) is actually normalized in (7.4) so that the area under 

 the curve is unity. The second maximum corresponds to the most proba- 

 ble position for a nearest neighbor in a random assembly, i.e., to the maxi- 

 mum in Fig. 12. Essentially the first maximum has been grafted onto 

 Fig. 12 by the interaction at close range which makes it probable that 

 short range neighbors will exist. At high values of N the region under the 

 first maximum becomes so great that enough area is drained (by the con- 

 dition of normalization) from the second maximum to make it disappear 

 entirely. At this point the minimum is replaced by a point of inflection. 

 More will be said concerning this phenomenon later. 



Fuoss chooses to define all sets of nearest neighbors inside the mini- 



