CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 567 



VII. THEORIES OF ION PAIRING 



Fuoss begins by considering a solution of dielectric constant k, con- 

 taining equal concentrations, A'', of ions of opposite sign. When equilib- 

 rium has been achieved each negative ion will have another ion (most 

 probably positive) as a nearest neighbor, a distance r away from it. 

 Fuoss discounts the possibility that the nearest neighbor will be another 

 negative ion, and proceeds to calculate what fraction of such nearest 

 neighbors lies in spherical shells of volumes, ixr^ dr, having the negative 

 ions at their origins. If this fraction is denoted by g{r) dr, it may be evalu- 

 ated as follows. 



In order for the nearest neighbor to be located in the volume, Airr" dr, 

 two events must take place simultaneously. First the volume, 47rr /3, 

 enclosed by the spherical shell must be devoid of ions, or else the ion in 

 the shell would 7iot be the nearest neighbor. Since g(x)dx is the proba- 

 bility that a nearest neighbor lies in the shell, ^TX^dx, the probability 

 that a nearest neighbor does not lie in this shell is 1 — g(x)dx. From this 

 it is easily seen that the chance that the volume 47rrV3 is empty is 



E(r) = I - f gix) dx (7.1) 



where a is the distance separating the centers of the two ions of opposite 

 sign when they have approached each other as closely as possible. 



The second event which must take place is the occupation of the shell 

 ;: 47rr^ dr by any positive ion. The chance of this event depends on the time 

 average concentration of positive ions at r. This concentration is bound 

 to exceed the normal concentration A^ by an amount depending on r, 

 because of the attractive effect of the negative ion at the origin. It may 

 be designated by c{r). The probability in question is then 



47rr'c(r) dr (7.2) 



The chance g{r) dr that the nearest neighbor lies in the shell A-wr dr is 

 therefore the product of (7.1) by (7.2), i.e., the product of the proba- 

 bilities of the two events required to occur simultaneously. This leads 

 to the relation 



g{r) = (l - £ gix) dA ^^^Mr) (7.3) 



an integral equation whose solution is 



g(r) = exp — 47r / x^cix) dx 4Trr^c(r) (7.4) 



Ja 



