HOT ELECTRONS IN GERMANIUM AND OHM's LAW 1015 



error is introduced provided the subsequent integrations are always in the 

 direction of increasing values for the variables concerned. 



A. 3 The Allowed Ranges for Py 



An electron with an energy corresponding to room temperature can 

 change its energy by only a small fraction in a one phonon transition. The 

 extremes occur for ^ = tt corresponding to complete reversal of direction. 

 For this case we have 



Si = (pI - p\)/2m = -dftycP^ 

 = -dnyc{P2-{- Pi) 



(A3.1) 



so that 



P2 - Pi = -dfiy 2mc = - 28ny Po. (A3.2) 



Thus the limiting values of P2 differ from Pi by 



±2Po = ± 2mc (A3.3) 



in keeping with the results shown in Fig. 4. [For Vi = Pi/m = 10 cm/ sec, 

 corresponding to Si = 0.025 electron volts, andc = 5.4 X 10^ cm/ sec, it is 

 seen that P2 and Pi differ by 10%.] For this case the range of Py is 



phonon emission, 8na = +1, Pa from to 2 (Pi — Po) (A3. 4) 



phonon absorption, dnp = — 1, P^ from to 2(Pi + Po). (A3. 5) 



A singularity occurs for Pi = Pq. For this case phonon emission becomes 

 impossible and the inner curve of Fig. 4 shrinks to zero; in Fig. Al we show 

 the sequence of shrinkage. The value of 81 for this condition corresponds to 

 thermal energy for a temperature of less than 1°K. Under the conditions 

 for which we shall compare theory and experiment, a negligible number of 

 electrons lie in this range. Accordingly we shall use the above limits in cal- 

 culations and neglect the small errors introduced. 



A A The Matrix Element and the Mean Free Path 

 The matrix element may be written in the form 



\U f = SL {2ny + 1 + 8ny)Pyc/4Vca (A4.1) 



where 8in is the derivative of the edge of the conduction band in respect 

 to dilatation of the crystal and cu is the elastic constant for longitudinal 



^8 See E and H in S, page 528, equation 31. The second expression in equation 31 

 of this reference is in error by omission of a factor hujka = cPy. 



