1002 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



(3.8) 



(3.9) 



SO that the change in magnitude of momentum is 



P2 - Pi = 2mc. (3.10) 



For an electron with energy kT and "thermal velocity" 



Vt = {2kT/myi\ (3.11) 



corresponding to 10' cm/sec at 300°i<r, the fractional change in momentum is 



(P2 - Pi)/Pi = 2c/vt = 2 X 5.4 X lOyiO' = 0.11. (3.12) 



Thus the phonon absorption surface Hes only 11% outside the constant 

 energy sphere. Figure 4 is drawn for the case of Pi = 6 mc, or Vi = Vt for 

 32°K, for purposes of exaggerating the differences in the surfaces. (A fur- 

 ther discussion of Fig, 4 is given in the appendices.) 



For transitions with optical phonons, in the range of interest, hvy is 

 nearly independent of Py . Furthermore, the optical phonons have hvy = 

 k 520°K so that they are nearly unexcited at room temperature and have 

 w-y = so that only bUy = +1 is allowed. For this case transitions can 

 occur only if 81 is greater than hvoY> and the end surface is a sphere with 



S2 = Si — hvox> . (3.13) 



We shall neglect the role of the optical phonons until after a comparison 

 between acoustical phonon processes and experiment has been made. We 

 shall then show that they play an essential role in explaining Ryder's data. 



3d. Energy Exchange and The Equivalent Sphere Problem 



We shall here give in brief some results derived in the Appendices which 

 permit us to show the equivalence of the problem of acoustical phonon scat- 

 tering to a problem in gas discharges. This has two advantages: it enables 

 us to take over the solution to the statistical problem from gas discharge 

 theory, the second problem mentioned at the end of Section 3a, and to 

 concentrate on the problem of the mechanism. In addition the equivalence 

 makes it much easier to visualize the mechanism of energy losses. 



According to the theory of phonon scattering, an electron is equally hkely 

 to be scattered from its initial direction of motion to any other. This implies 

 that after an interaction the electron is equally likely to end in any unit 



