HOT ELECTRONS IN GERMAISTIUM AND OHM's LAW 1017 



For the cases with which we shall be concerned, the values of ity may- 

 be approximated by classical equipartition. This may be seen from the 

 fact that largest energy phonons correspond approximately to an energy of 



cPy = 2cPi = (Acm/Pi)Pl/2m 



(A4.4) 

 = WvOkT. 



Their energies will, therefore, be considerably less than kT. For large in- 

 creases in electron energy in high fields, however, this approximation may 

 not be adequate. At room temperature cPy/kT = 0.2 and the critical 

 range will correspond approximately to an increase of about 10 fold of 

 electron temperature above the temperature of the crystal. Under these 

 conditions cPy deduced from equation (A4.4) will be about 2kT for the 

 most energetic phonons; for this condition, however, Py lies at the edge of 

 the Brillouin zone and dispersive effects must be considered. In this treat- 

 ment we shall not investigate further these limits and shall in general 

 assume that cPy < kT. 



We shall next derive an expression for the mean free path and verify 

 that the scattering is isotropic. These results can be derived more simply 

 and directly from the matrix element by neglecting (Pq/Pi) and (cPa/kT) 

 from the outset. For the treatment of energy losses that follows, we cannot 

 make these approximations. We shall, however, make them in the re- 

 mainder of this section thus establishing that our more general formulation 

 reduces correctly to the more convenient and simpler formulation usually 

 used. 



For the condition under which equilibrium apphes we may approximate 

 ity as follows: 



ny = l/[(exp cPy/kT) -1] = (kT/cPy) - i (A4.5) 



Then, for the phonon emission or a case, the contribution to Wu d&2 be- 

 comes 



Wi2d&2 = (4w^Pi)-i[l + {cPa/2kT)]Pa dPa (A4.6a) 



and for the phonon absorption case, it becomes 



Wi2d&2 = (4w^Pi)-'[l - {cPfi/2kT)\PfidPfi. (A4.6b) 



We shall use these expressions later for the calculation of energy exchange, 

 in which case the terms in cPj2kT, which favor phonon emission, play an 

 important role. In order to check the expression for mean free path we neg- 

 lect these terms, however, and also approximate the integral of Pa dPa by 

 2P\ rather than 2 (Pi — Po)'. The total probability of transition from state 



