482 BELL SYSTEM TECHNICAL JOURNAL 



surface is smaller than diflfusion to the surface so that x is not large. The values 

 of di, over which the sum is to be taken, may be estimated as follows: in 

 each interval of di of the form iit to (n + (^))7r, di tan Oi varies from to oo, 

 giving one solution to di tan di = x- For x small, the solutions are approxi- 

 mately 



^0 - sin do - tan ^o - Vx 



^1 - 7r + x/t^; — sin di = tan di = x/t 



dn - nw + x/nr; (-1)" sin dn - tan dn - x/nrr 

 From this we see that the terms in the sum are as follows: 



aoo-4x /x -i = «'oo 

 anO-2(x/«7r)V(«7r)^ = ocn(P-x/nTr 



.4/4 4 8 



From this it is evident that unless x is large, the series converges very 

 rapidly. (This conclusion is not altered when the increase in anm with ^„^m is 

 considered.) Thus the dominant term in the admittance is 



Aw^qupo (1 + iwTQoy'^/\^DTm 

 where 



1/roo = 2 ^1^ {dl) + 1/r 



This expression is valid only for sw/D small so that dl = sw/D. The term 

 s/iw/l) represents the rate of decay due to holes recombining on the surface, 

 s having the dimensions of velocity. For co » 1/too , the admittance becomes 

 4-d'^qfjLpo{io)''Dy^-, the same value as given in equation (4.12) for large co and 

 an area -iw^. 



The conclusion from this appendix is that for x small, the effect of surface 

 recombination is simply to modify the effective value of t and otherwise leave 

 the theory of Section 4 unaltered. 



For very large values of x, it is necessary to consider higher terms in the 

 sum and several values of r will be unportant. Under these conditions the 



