ELECTRON STREAMS IN A DIODE 849 



.f -Q= -/ (69) 



The con 'uction current density Q is 



Q^-'I^-V'-^^-U^'^ (70) 



m m dx 



an(i its substitution in (69) gives 



OX ot e 



which is the analogue of (5) . 



The mechanical equation for the physical stream is obtained from the 

 Liouville theorem. In the diode regions with which we shall be concerned, 

 the individual electrons are so far apart that their microscopic forces are 

 negligible, the electrons flow freely under the action of the macroscopic 

 forces, and they therefore obey the Liouville theorem for particle motion. 

 This theorem states that 



J = (72) 



at 



that is, n remains constant as we travel along with any particular electron. 

 This equation may also be written in terms of partial derivatives of n 



dn dx ,8ndv, bn _ .^ . 



Yxdi'^ b'vdt'^ U "^ ^' ^ 



and the substitution of the values of the total derivatives then gives 



bn ^bn , bn ^ .^.^ 



u - - ryE - + r- = (74) 



ox bv bt 



The mechanical relations are obtained by integrating this equation with 

 respect to u. It is first multiplied by dv and then integrated as follows: 



/ - vdv - riE r^i^ + / 77 ^^ = ^ (75) 



, J— 00 ox J-oo tU J—ao ot 



The second integral reduces to the difference in the values of w at u = +00, 

 and V = — 00 . It vanishes because there are no electrons with infinite 

 velocities. The differential operators may also be moved outside the other 

 integrals, to give 



nvdv + - / ndv = (76) 



00 5/ J— 00 



b_ 

 bx 



