586 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



written as ^lA'', i.e., 



hN^-'^ (10.10) 



Equating the left sides of (10.9) and (10.10) gives 



A;i = 4:7rNRDo 

 or 



^ = r = A A^r. (10-11) 



It now remains to choose a value for the capture radius, R. A reason- 

 able guess may be made as follows: Around each acceptor there is a 

 coulomb potential well of depth 



V = -(IIkt . (10.12) 



Since the average thermal energy is kT, it seems reasonable to regard an 

 ion as trapped when it falls to a depth kT in this well. Thus, inserting kT 

 on the left of (10.12) and R for r on the right leads to 



R = qlKkT (10.13) 



and upon substitution in (10.11) we obtain 



KkT 



(10.14) 



4xgWZ)o 



This result, obtained by crude reasoning, is actually quite close to the 

 more rigorous value derived below. Furthermore, the above derivation 

 is useful in providing insight into the physical meaning of the relaxation 

 time. 



The chief difficulty with the preceding lies in the arbitrary choice of 

 72, and is a direct consequence of the long range nature of coulomb forces. 

 Another difficulty arises because the distribution of donors about ac- 

 ceptors is eventually specified by (7.7) so that at r = i^ = q/KkT 



Be 



Since this slope has a negative value the trap exhibits some aspects of a 

 source rather than a sink which could only produce a positive concen- 

 tration gradient. This last objection will not be serious when h is very 

 small since, then the final value of c{r) beyond r = q/KkT = R will be 

 effectively zero, as would be required for a perfect sink. 



