1006 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



The steady state condition then leads to 



vi = {qlElmf'iMlmf^ (3.31) 



and to 



n = {qeE/m)"\m/My'' 



(3 32) 

 = iV^cqCE/rnvTY'^ ^ (cuoEY'^ 



The treatment^ based on accurate statistics for the equivalent sphere model 

 leads to 



Vd = L23{cfioEy'\ (3.33) 



The transition between the high field behavior and low field behavior 

 should occur in the neighborhood of a critical field Ec at which both Umiting 

 forms give the same Vd : 



Vd = noEc = 1.23{cuLoEcy'\ (3.34) 



leading to 



Ec= 1.51 c/fjLo (3.35) 



and to a drift velocity, which shall be referred to as the critical velocity, of 



Vdc = 1.51c (3.36) 



if Ohm's law held to a field as high as Ec . The drift velocity can be ex- 

 pressed in terms of Ec by the equation 



Vd = noiEEcY'' (3.37) 



for values of E much greater than Ec. 



It is interesting to note that this initial field is just that which would give 

 electrons a drift velocity corresponding to the thermal motion of M-masses. 

 This seems a natural critical field. For it the effect of random motion of M 

 would be suppressed by the systematic drift velocity so that the transfer 

 of thermal energy to the electrons would be much reduced. This value of 

 Ec corresponds to much smaller initial fields than are sometimes proposed. 

 For example, one frequently encounters proposals that Ohm's law should 

 hold up to the condition that Vd = Vt . This would correspond to 10 times 

 higher field at 300°K than that obtained. Another criterion is that the 

 energy gained in one mean free path, qtE, should be equal to kT. This is 

 substantially equivalent and corresponds to Vd = Vt/2. 



' Druyvesleyn Physica 10, 61, 1930. This paper is reviewed by S. Chapman and T. G 

 Cowling in "The Mathematical Theory of Non-Uniform Gases," Cambridge at the Uni- 

 versity Press, 1939, page 347. (The factor is 0.897 (187r/8)i^* = 1.23.) 



