1024 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



approximation will give the same results as the Maxwellian distribution. 

 We use this procedure in the following section. 



As pointed out in Section A. 5, the problem treated here is closely analo- 

 gous to electronic conduction in gasses. For this case, an exact treatment 

 has been given for high fields such that the motion of the large masses M 

 may be neglected. The drift velocity is found to be"' 



v^ = {iy"(T/4.y"{m/Myi'(qEC/myfyT{il (A6.27) 



Substituting kT/c^ for M and using (A6.7) to eliminate /, we obtain 



Vd = i7^/6yf'T{i)QjLoEcyi' 



(A6.28) 

 = 1.23 (jjioEcyi\ 



This value lies intermediate between the two simple approximations con- 

 sidered above and leads to a critical field of 



Ec = 1.51 c/fio (A6.29) 



at a velocity of 



Vdc = fioEc = 1.51 c. (A6.30) 



Since the exact case gives va = ijlqE for low fields and Vd = noiEEcy^ for 

 large fields, it is evident that either the Maxwellian or single energy dis- 

 tribution will approximate it well (provided suitable choices of juo or aiq and 

 Ee or Ec are made) except for a small error near Ec. 



The distribution in energy for the gas case leads to a probability of 

 finding the electron in a range vtov-\-dv proportional to 



Texp - j mv dv/{kT + SMiqCE/SjnvYU v dv (A6.31) 



For high fields this reduces to 



[exp - {Zm/^M){v''m/qtE)''y dv 



It is seen that this distribution weights the low energies less heavily than 

 does the Maxwellian for the same average energy and thus gives a lower 

 mobility. It weights them more heavily than does the single energy and, 

 therefore, gives a higher mobility than it. 



^Dniyvesteyn Physica 10, 61 (1930). See also S. Chapman and T. G. Cowling, "The 

 Mathematical Theory of Non-Uniform Gases," Cambridge at the University Press, 1939, 

 page 351. 



