DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1049 



We assume: 



N(v) = A ch (qv + B) (20) 



and substitute in (11), (12) and (7). In view of the sharp maximum in 

 the integrands of these expressions, it is permissible to set the limits 

 which should correspond to the edges of the gap or of the state distribu- 

 tion equal to ± <» . The integrals are conveniently evaluated by the con- 

 tour method (see Appendix 1) and yield the following results: 



( -^ j = Attq cosec TQ ch [B — q{Y — /n X)] 

 /aS,\ ^ _ATrq cosec vq ch [B - q{Y - (n X)] X 



where 



'y = }' - (n X - (n X 

 (S, = B - qtnx 



(21) 



(22) 



(23) 



{VrnVrp)"-' (24) 



= I (X -f X~^)(o-„(Tp)^'^ni£ 2x A sh qy ch (B cosec irq cosech 'y 



Comparing (21) with (15), we see that the delta-function approxima- 

 tion is in error to the extent that it replaces irq cosec irq by 1. With the 

 value of q found experimentally, this is not too bad; we can now, how- 

 ever, by fitting the right-hand side of (21) to the experimental facts, 

 (13) and (14), obtain exact solutions for A''(j'): 



22.6 ohm-cm n-type 



iY(j;) = 3.6 chiQMu + 0.8) (for u < 4) 



8.1 ohm-cm p-type (25) 



N(v) = 8.3 chiOMp + 0.5) (for p < 4) 



The question arises as to whether this solution for the distribution is 

 unique. We have already pointed out that the mathematical methods 

 fail if the distribution is discontinuous. It seems that (25) represents the 

 only solution that is slowl3^-^'arying, in the sense used in the previous 

 section; its correctness could presumably be checked by carrying out 

 experiments at different temperatures. For v > 4:, the abo\-e expressions 



