420 BELL SYSTEM TECHNICAL JOURNAL 



However, in two limiting cases good approximations have been ob- 

 tained. 



In the first case AI is so small a quantity that 



ilf-i exp. lm{u- + y- + w-)/2kT^ » 1. 



It then follows that 



A£ = nh^myG{2irkT)^ (14) 



and that equation (13) is the same as equation (7) for the classical 

 treatment. 



In the second case M is a large quantity and the 1 in the denom- 

 inator of equation (13) cannot be neglected. Sommerfeld ^* has shown 

 that in this case 



M = exp. {K/kT), (15) 



where 



^ A2(3w/47rG)' 



-fv = ^ 



2m 



r {2jmkTYI Zn Y"' ] ,,.. 



The second term in the brackets is usually a very small numerical 

 quantity and can nearly always be neglected. 



If we assume that n, the number of electrons/cm.^, is equal to the 

 number of atoms/cm.' in a metal or a small factor times this number, 

 we can compute M for case 1 by equation (14) or for case 2 by equation 

 (15). In either case M turns out to be a large quantity. Hence for 

 metals the second case is applicable while the first case is not. Hence 



n{u, V, w)dtidvdw 



X r^ — r^r-, — ^r-, -s E^n dmlvdw, (17) 



exp.j^ rr J + ^ 



smce 



ikf-i = exp. (- K/kT). 



Integrating this from — co to -|- <» with respect to v and w Nordheim ^^ 

 has shown that the number of electrons per cm.-"* having velocities in 

 the range (w, du), i.e., between ii and u + du, is 



, s , lirGm-kT , 

 n{u)du = Y~3 '" 



1 + exp. 



kT 



du. (18) 



The number of electrons striking a surface normal to the u direction 

 per cm.- per sec. and having velocities in the range («, du) is given 1)>' 



