1020 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



since M{v'^m)/'^ — kT/2. In addition, however, there is an effect due to 

 relative velocity: when Vm is directed so as to increase the closing velocity, 

 the probability of collision is increased. Due to this effect, colUsions with 

 higher relative velocity are favored and as a result the energy transferred 

 due to the Vm effect is just doubled" leading to a total contribution of 



(energy transfer due to Vm) = 4 mkT/M. (A5.10) 



This is just to the first term of (A5.5) when averaged over all values of 6. 



Thus the average of the gain in energy term in (A5.5) is just that corre- 

 sponding to interactions with heavy masses M in thermal agitation. The 

 difference is that in the sphere model the average energy gain term is inde- 

 pendent of ^, whereas in the phonon case it varies as P'y/P\ and approaches 

 zero for forward scattering so that the dependence upon angle is different. 

 The energy loss term, however, is correctly represented by the sphere model. 



The average value of (58) averaged over all values of d is denoted by 

 (58)/»i. Since Py averaged over B is 2Pi, we obtain 



{bZ)p, = 4cPo - cP\/kT 



(A5.12) 

 = ^mc (1 - PI/4 mkT). 



From this expression it is seen that an electron with energy Pl/2m = 2kT 

 keeps the same energy on the average after M coUision. We shall use ex- 

 pression (A5.12) in Section A.6. 



For high electric fields, the electron energies are higher than thermal and 

 the loss terms predominate. Furthermore, the scattering in both cases is 

 nearly isotropic if Af » w and the colliding particles are spheres. Hence, 

 the analysis of kinetic theory of ionized gases can be applied to a high 

 degree of approximation to estimate electron behaviors. 



It should be stressed that several approximations are involved in this 

 treatment. In particular it is assumed that (I) cPy < kT and that (II) 

 Pi » Po. If this is true, then 



vi/c = Pi/Po»l (A5.13) 



so that the approximation used in considering the heavy spheres to be 

 moving slowly holds and the mass of the heavy spheres is much greater 

 than the electron mass: 



M/m = {kT/c')/i2kT/v\) = vl/2c\ (A5.14) 



If conditions (I) and (II) are not satisfied, the approximations leading to 

 (A5.5) will require revision. 



*" If vin is the velocity towards centers, then the probability of collision is weighted by 

 [1 4- (vu/vin)] and the term linear in vm in the energy, which is (i«i)(4»i«»ji), con- 

 tributes 2{v]it)m to the average transfer. 



