HOT ELECTRONS IN GERMANIUM AND OHM's LAW 1027 



on the basis of a somewhat different treatment. According to (4.4) 



Vd = ii) {2hv/myi^ = {hp/2myf^ 



(A7.8) 

 = 0.63 X 10 cm/sec. 



a result smaller than (A7.6) by a factor of 2^^^ 



A correct treatment of the optical modes together with acoustical modes 

 would involve solving the Boltzmann equation to find the steady state 

 distribution. This will obviously present problems of considerable com- 

 plexity. In particular scattering will suddenly begin to increase when the 

 electron acquires an energy, Si > hv and it seems unlikely that analytic 

 solutions can be obtained. Even the Maxwellian distribution leads to 

 somewhat complicated integrals. 



A.7b. Estimate of the Matrix Element 



In order to proceed further we must estimate the order of magnitude of 

 the optical scattering matrix element. For this purpose we introduce a 

 deformation potential" coefficient for the optical modes by the equation 



S2n = de>c/d{x/Xo) (A7.9) 



where x is the displacement parallel to the :r-axis of one sublattice in respect 

 to the other and Xq is the a;-component of relative displacement of the sub- 

 lattices for equilibrium conditions. The same reasoning as used in treating 

 mobility by deformation potentials may then be applied and the matrix 

 element evaluated by analogy with the dilatation waves. For the latter the 

 matrix element may be written in the form 



\Ua\' = e,ln{A')/2 (A7.10) 



where A is the dilatation and (A^) is the average (dilatation)^ for the 

 mode before and after transition. Since half the energy in a running wave 

 is potential 



i c^^(A2) V = ihvX [average of (n -f J)] 



(A7.11) 

 = hpi2n+ 1 + 5w)/4. 



This leads to the form introduced in equation (A4.1). By analogy, for the 

 optical modes we should take 



I Uo, \' = e>ln{{x/xoy)/2 (A7.12) 



26 W. Shockley and J. Bardeen Phys. Rev. 77, 407-408 (1950) and J. Bardeen and 

 W. Shockley, Phys. Rev. 80, 72 (1950). 



27 See E and H in S, page 528. 



