HOT ELECTRONS IN GERMANIUM AND OHM's LAW lOOvS 



area of the surfaces of Fig. 4. The probabihty of being scattered per unit time 

 is simply 



1/ri = v^ll (3.14) 



where Vx is the speed and / the mean free path ; according to the theory I is 

 a function of the temperature T of the phonon system and is independent 

 of V\ . The time r\ is the mean free time or average time between coUisions. 

 Figure 4 shows the average energy after collision. The Figure represents 

 a case in which the average energy is somewhat smaller than the initial 

 energy. This will be the case for a high energy electron, that is one with an 

 energy greater than kT for the acoustical modes. The average loss in energy 

 for a high energy electron is found to be 



(5£) = -c'PyikT (3.15) 



where Py is the momentum change in the coUision. This formula is analogous 

 to the formula for energy loss if a light mass m strikes a heavy stationary 

 mass M and transfers a momentum P2 — Pi = Py to it. The energy transfer 

 is given by (3.15) if 



M = kTl& (3.16) 



since then the kinetic energy of the large mass is simply 



P\I1M = c'P^/lkT. (3.17) 



The value of the mass which satisfies equation (3.16) for room temperature 

 may be calculated from the previously quoted values of Vt and c: 



M = kT/c" = mvfllc = 170m, (3.18) 



a value which may certainly be considered large compared to m. 



Equation (3.15) is not the complete expression for average energy change 

 for a coUision with momentum change Py and another term representing 

 energy gain also occurs. If the complete expression is averaged over all final 

 directions of motion, it is found that the average change of energy, which is 

 obviously the average energy change per coUision, is 



(58) = 4mc\l - Pl/4mkT) = {4mkT/M) - {P\/M). (3.19) 



This is the correct expression for the average gain in energy per collision of 

 a light mass m colliding with a heavy mass M which is moving with the 

 thermal energy appropriate to temperature T. This corresponds to a thermal 

 velocity of 



Vtm = (IkT/My/'' = 2i/2c (3.20) 



