1018 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



Pi, which should be taken to be l/n where n is the mean free time, is 

 then 



- \lr^ = W^i = (4m^Pi)-i 4P'i = {Pi/m)/t = Vi/t. {M.I) 



This is just the relationship appropriate to the interpretation of / as a mean 

 free path between colUsions/^ It does not follow that ri is the relaxation 

 time for the current, however, unless the average velocity after collision 

 is zero. We shall next show that the average velocity after collision is zero 

 to the same degree of approximation used above by showing that scatter- 

 ing into any solid angle of directions is simply proportional to the solid 

 angle, i.e. the direction of motion after collision is random. 



The conclusion that the scattering is nearly isotropic follows from the 

 approximate proportionality of probability to PydPy. Since P2 is nearly equal 

 to Pi and substantially independent of 0, we may write 



Pa dPa = i d(PaY = ^d {2P\ - 2P\ COS o) 



= - Pld COS e = Pi sin e dd. (A4.8) 



The last term is simply proportional to the soHd angle lying in range dd; 

 hence the end states are distributed with uniform probability over all direc- 

 tions and the scattering is isotropic. 



A5. Approximate Equivalence to Elastic Sphere Model 



In this section we shall show that on the average the energy exchange 

 between the electron and the phonons when the electron is scattered through 

 an angle 6 is very similar in form to that corresponding to elastic collisions 

 between spheres with the phonons represented by a mass much greater 

 than the electrons. If dPa and dPp correspond to scattering through angles 

 between 6 and 6 -\- dd, then the energy loss for phonon emission is cPa and 

 the energy gain for absorption is cP^. The relative probabilities of loss and 

 gain are given by equations (A4.6) and from these it is found that the 

 average energy gain is 



/ V ^ cPp[l - (cP0/2kT)]P0 dPp - cPall + (cPj2kT)]PadP„ 



^^^^ [1 - {cP,/2kT)]P^ dP^ + [1 + {cPj2kT)]P^ dPa , ^ , 



(A5.1) 



^ C[PI dPff - Pi dPg] - (cy2kT){Pl dPff -h Pi dPg) 



[1 - (cP0/2kT)]Pp dPff + [1 + {cPa/2kT)]Pa dPa ' 



Since we are concerned chiefly with cases in which Po <3C Pi , P2 — Pi , 

 and cPfi/kT <<C 1, we may set 



P, = Pp^ I\ = 2Pi sin (0/2) (A5.2) 



'* See E and H in 5, Chapter 11. 



