RECENT STATISTICAL THEORIES 711 



Evidently any thermionic emission must distort the distribution of 

 the electron-gas inside the metal, as it is an unbalanced outflow of 

 electrons. The situation in which the efflux is balanced by a corre- 

 sponding influx from an electron-gas outside the metal is much 

 regarded in thermodynamic theory, but one cannot measure currents 

 in that situation any more than one can measure heat-flow between 

 two bodies at equal temperature. In assuming, then, that the 

 distribution of the internal electrons is that of Fermi or that of Max- 

 well, we shall probably be invalidating our conclusions except for the 

 limiting case of an infinitesimal emission. It seems probable, however, 

 that with the thermionic currents of practice the approximation is 

 good enough. 



The simplest theory of the thermionic current, then, consists 

 entirely of the equation: 



' = ^ ( 1 )' '^ .£./" X. *1. *■" A^J'^r + 1 ■ (-3) 



This is a restatement of the first sentence of this section, plus the 

 assertion that even when electrons are leaking out through the wall of 

 the metal the distribution within remains practically that of Fermi. 

 The factor e stands for the electron-charge; the symbol i thus for the 

 thermionic current-density in electrostatic units. The factor m^ 

 enters because, in conformity with usage, I have translated from the 

 momenta into the velocity-components u, v, w as independent variables. 

 The quantities Uo and Wa are related by the equation: 



Wa = \mu^\ (74) 



In the integrand we are of course to put \m{i.i^ + z;^ -+- w"-) for e. 



Setting for A the first-approximation value from (107) with the 

 symbol Wi defined in (71), we obtain: 



I = e 



(f)''X" ££".'•-»■■''- +!'"'"'''"• ^"' 



The integration is perfectly straightforward if the second term in 

 the denominator may be neglected relatively to the first. This seems 

 unnatural, for we have just been noticing that over part of the energy- 

 range, from € = W i downwards, the second term is larger than the 

 first. But if Wa is considerably larger than W ., — and by this I mean, 

 if {Wa — Wi)lkT is positive and considerably larger than unity — - 

 then the electrons which escape are those which belong to the extreme 



