FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1197 



In the absence of time variation, (86) shows that recombination is neces- 

 sarily absent, too, so the results reduce to 



(t^) 



OLCOC) = ^A l^-:^] (100) 



with 3C(x, y, z) any harmonic function and A and B arbitrary constants. 

 This solution for the case grad 3C f^ 0, together with that given by (59b) 

 and (60) (with G = 0) for the case grad 3C = 0, constitute a veritable gold 

 mine of useful solutions because of the arbitrary harmonic function involved. 

 An example involving a particular choice of JC will be examined in Section R. 



Case 3: 



^ ?^ 0, grad (grad h)' = 0. 



In this case (grad hY is a function of t so that (75) can be written in the 

 form 



From this it follows (because div grad ^ = 0) that 



h{x, y, z, t) = a{t)h{x, y, z) + c{l) (101) 



with 



div grad b{x, y, z) = 0. (102) 



The condition grad (grad hY = now requires further that 



grad (grad hf = 0. (103) 



But any h{x, y, z) satisfying both (100) and (101) can, by suitable choice 

 of axes, be written 



h = Sx (S: constant). 



This leaves us with exactly the same totality of solutions as we could have 

 obtained by setting^ = ^(x, /), 3C = 3C(x, /) in the first place. So we replace 

 /r by X in (74) and (75) and obtain: 



dX^!r__!^ (104) 



dx 7 + 'It 



