562 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



the system the other begins to take effect. This symmetry, of course, is 

 necessary for explaining the symmetrical locus of the points around Curve 

 B in Fig. 9. 



The scheme (5.2) can be treated quantitatively by applying the mass 

 action principle, but now the symbol D^ can not be used for the solu- 

 bility of lithium since the totality of dissolved lithium is distributed 

 between LiE~ and Li^, and the symbol only applies to the latter. We 

 therefore denote the total concentration of lithium by No , and the con- 

 centration of LiB" by C. Then 



No = D^ + C (5.3) 



The same argument applies to boron, so that its total concentration will 

 be designated by 



Na = A- + C (5.4) 



The problem then reduces to specifying No as a function of Na • To 

 accomplish this, to (3.1) and (3.2) is added the mass action expression 

 going with (5.1) 



""^ = ^e-w-) = ^ (5,5) 



D+A-n 



where 7 and jS are constants. It has been assumed that the vacancy con- 

 centration follows a temperature law of the form 7* exp[ — ^*/T] where ^ 

 7* and ;S* like 7 and ^ are constants. This permits the equilibrium con- 

 stant when multiplied by the vacancy concentration to assume the form 

 7 exp[ — /S/r] shown in (5.5). In place of (2.8) a new conservation condi- 

 tion, 



D^-\-p = C+A- + n (5.6) 



is introduced. The combination (3.1), (3.2), (5.3), (5.4), (5.5) and (5.6) 

 can be solved so that No , the lithium solubility appears as a function of 

 the total boron concentration A^^ . Thus 



^" - 1 + Vl + (2n,/No^y + y {1 + Vl + (2n,/iV.o) j + ^^^""^' 



^_ TNANpyjl + Vl + {2n~/N7)'] 

 2 -I- wiNo'fil + Vl + (27ii/No'y\ 



In this equation No like Do in (3.4) is the solubility of lithium in un- 

 doped silicon, i.e., in silicon from which boron is absent. 



All the parameters in (5.7) are independently measurable save x 





