1198 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



and 



dx 



ox 



![■ 



+ ii «<* + £?] = »■ ""' 



Any 1l(x, /) satisfying (105) can be substituted into (104) to obtain 3C{x, I) 

 from 



5C(x, /) = 7(0 + / 



^-^- ^^^^"^a^ ^^ (106) 



7 + It 



(/(/): arbitrary function). 



If recombination is absent, (R(*U) disappears from (105). If time variation 



is absent, — disappears from (103) andjXO and/(/) are replaced by arbitrary 

 dt 



constants. In the latter case, the standard change of variables 



WCU) for g 'W('l^)^ f°^ Tx ^^°^^ 



reduces the solution of the second order equation (105) to the solution of a 

 first-order equation followed by a quadrature. If both recombination and 

 time variation are absent, the substitution (107) reduces the solution of (105) 

 to two quadratures. 



A set of equations equivalent to the steady-state f — = j forms of (104) 



and (105) has been the subject of an extensive numerical investigation by 

 W. van Roosbroeck (Reference 1) for the recombination rate functions 

 given in (37) and (38). 



M. Solutions with V = V{h, /), (P = (P{h, /), Grad (9 9^ 0, 

 Div Grad /? = 0, .V = 



For these conditions (21) and (23b) yield 



-(grad/.) =_L(R((P)+-(^__ + _jJ 



I' 

 and 



(107) 



5 e-,Tr-S] *-«•=»■ »««' 



