1004 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



for the large mass. The second term in (3.19) is just the average of (3.15) 

 over all directions of motion after collision and represents the energy loss 

 that would arise if M were initially stationary. The first term represents an 

 energy transfer from M to m due to the thermal motion of the large mass. 



Furthermore, if the light and heavy masses are perfectly elastic spheres, 

 the scattering of m will be isotropic, just as is the case for the phonons. This 

 shows that there is an almost perfect correspondence between the two 

 mechanisms of scattering so that we are justified in using previously derived 

 results for the sphere case and applying them to the phonon case. 



To complete the equivalence we should introduce a density of large spheres 

 so as to get the correct mean free path. There is nothing unique about this 

 procedure, as there is about the mass M and temperature T, and we may 

 make a large selection of choices for number of M-spheres per unit volume 

 and radii of interaction so as to obtain the desired mean free path. Once 

 any choice is made, of course, it will give the same mean free path inde- 

 pendent of electron energy and may be held constant independent of the 

 electric field. 



3e. Acoustical Phonons and Electric Fields 



We shall next give a very approximate treatment of mobility in low and 

 high electric fields. The emphasis will be upon the interplay of the physical 

 forces, the mathematical details being left to the Appendices or to references. 



In the Ohm's law range, the field E is so small that the electrons have 

 the temperature of the lattice. They have a velocity of motion of approxi- 

 mately 



Vt = (2kT/my'' (3.21) 



and a mean free time between collisions of 



r = ^/vt . (3.22) 



The electric field accelerates the electron at a rate 



a = qE/m (3.23) 



and imparts a velocity ar in one mean free time. Since the collisions are 

 spherical, the effect of the field is wiped out after each collision. The drift 

 velocity is thus approximately 



Vd = ar = {qt/mvT)E. (3.24) 



An exact treatment which averages over the Maxwellian velocity distribu- 

 tion gives a value smaller by 25% and leads to 



/io = ^qt/Sr^'^rnvT . (3.25) 



