Motion of Electrons in Solids. 775 



loss o£ momentum, regarded as a vector, can have no direction 

 except that o£ the initial velocity u, v, w : its amount will 

 depend on the constants of the electron and molecules and 

 on ii 2 + v 2 + w 2 , but not on u } v, and to separately. We can 

 accordingly suppose that the original velocity u, v, w is 

 reduced by collision to 



(1 — a)u, (1 — <x)v, (1 — a) io, 



where a depends on u 2 + v 2 + iv 2 and constants only. 



In a lime dt which is large compared with the time o£ a 

 collision, the loss to the momentum of the N electrons is 

 therefore of the form 



~N<yu dt, l$yv dt } J8yw dt, 



where u , r , w are average values of u, v, w and y depends 

 on u 2 + v 2 + iv 2 and constants only. 

 We accordingly have the equation 



|(Nm«„) = NX e -N 7 « , .... (3) 



provided the interval of time dt is taken to be large compared 

 with the time of collision. 



5. Let us now carry out the corresponding calculation 

 without assuming the existence of free paths. We fix our 

 attention on all the electrons of which the velocity-components 

 at a given instant t = lie within a small range du dv dio 

 surrounding the values te, v, to. If there were no externally 

 applied electric force, the law of distribution of these electrons 

 in space would be 



TSAe-**dxdydz (4) 



where % is the potential energy of an electron at the point 

 .r, y } z and A is such that 



K^e-^Xdxdydz = 1, 



the integral being taken throughout a unit volume. With 



an electric force X acting, the law will be different from this 

 by terms of the order of - , but these terms may be neglected 

 whenever their retention would lead to ultimate terms of the 



order of '!, . 



ii- 



Assuming (I) to be the law of distribution of these 

 electrons in space, we calculate the values of i(, r, iv after 

 time t } corresponding to all initial positions of the electrons, 



3 G2 



