614 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



where [Li"'"Ga~] represents an ion pair, whose concentration we denote 

 by P. Na and No will be the total densities of acceptor and donor re- 

 spectively and A" and D'^ the densities of acceptor and donor ions in the 

 unpaired state. 



As in the main text, n and p will represent the concentrations of holes 

 and electrons. The following relations are then to be expected on the 

 basis of definition, mass action, and charge balance. 



Na = A- + P (A2) 



AT^ = £)+ + P (A3) 



D^n = K* (A4) 



np = n/ (A5) 



^ = n (A6) 



A+D- 



D'^ + p = A~ -j- n (A7) 



Equations (A4), (A5), and (A7) are just reproductions of (3.1), (3.2), 

 (2.8), while (A6) is the same as (9.4). The problem is to express the solu- 

 bility of lithium, Nd , as a function of Na ■ Manipulation of the pre- 

 ceding set of equations gives this result as 



N. = (^^ - ^")(1 + "^") (AS) 



with A~ given by the solution of 

 Na - A~ A- 



QA- 



A- 



2 



(A9) 



+n2 



/i+d 



i + ^/i + i|-;^' 



+ (Do*) 



where Do^ is defined by (3.3). Equation (A9) generally needs to be solved 

 numerically for A~. • 



To see what these relations predict in a special case consider the 

 solubility of lithium in gallium-doped germanium at 300°K. At this 

 temperature the values of Ui and Do^ and 12 are 



rii = 2.8 X 10^' cm~^ 



2)/ = 7 X 10'' cm"' (AlO) 



fi = l.Gl X 10^'cm"l 



