654 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953^ 



So far we have two relations (6) and (7) between the four constants 

 Ai , Ati Bi y 82- To determine these constants, we need two more 

 equations. These are supplied by the boundary conditions. We assume 

 that the recombination rate is proportional to the density at the bound- 

 aries (z = =ba). The recombination then has the formal appearance of 

 a current through the surface, the factor of proportionality s playing 

 the part of a "recombination velocity."^' ^ That current through the 

 surface must equal the current arriving at the surface from the interior 

 of the filament, under steady-state conditions. The boundary condi- 

 tions are thus: 



where Si , S2 are the surface recombination velocities of the faces x == 

 +a, —a respectively. The minus sign on the right-hand side of the second 

 of these equations is due to the fact that the recombination current at 

 X = — a is along the —x direction. Defining the recombination param- 

 eters 



and substituting the solutions Wi yW2m the last two equations we obtain 



(8) 



B.|= UA^e*''-""'" + Br) 



52*= UAie-*^'-^""" + Bi) 



Equations (6), (7) and (8) suffice to determine Ai, At, Bi, B,. We 

 easily find that 



*Z) A 



_ 2a (1 - a^*"+"'°'") 

 ' " *D A 



_ 2a ( 1 + ce-*"-"^-"") 

 ' " *D A 



(9) 



