THERMIONIC ELECTRON EMISSION " 417 



that there is Httle point in examining the temperature dependence of 

 the quantity P — TdP/dT more closely. On the other hand, the 

 quantity K and its variation with temperature does depend on the 

 particular assumption that is made with regard to the energy dis- 

 tribution of the free electrons in the metal. In particular K is very 

 nearly independent of T if the electrons in the metal have kinetic 

 energies given by the Fermi-Dirac function.* That this is the correct 

 distribution function is quite well established by the numerous suc- 

 cesses which this theory has had in explaining experimental facts in 

 connection with metals. ^^~^^ For the Fermi-Dirac distribution K is 

 practically a constant term in the expression for the heat function h. 

 There is then no reason for changing the form of equation (6) which 

 contains the term 2 log T. This is equivalent to an exponent of 2 in 

 equation (la). 



The case was somewhat different before the advent of the quantum 

 theory. The electrons in the metal were then assumed to act like a 

 perfect gas. Hence the energy K was taken to be {3/2)RT. It was 

 thus natural to subtract this from the (5/2)jRr for the electron vapor. 

 In this way one is led to an expression for log i similar to equation 

 (6), but instead of 2 log T there now appears | log T. So that the 

 exponent of T in equation (la) was taken to be |. 



It is well to note that on the basis of this thermodynamic argument, 

 there is no good reason why the heat function should be independent 

 of T and why the Richardson lines should be straight. Experiment 

 shows, however, that for nearly all surfaces which are not close to 

 their melting point, the heat function is independent of T to within 

 experimental error. In the neighborhood of the melting point, the 

 heat function varies with T. It should also be noted that thermo- 

 dynamics does not predict that all Richardson lines should have a 

 common intercept on the y axis. This prediction which is true only 

 for special classes of surfaces has been made on the basis of a statistical 

 theory which we will now discuss. 



The Statistical Equations 

 a. Classical treatment. If we knew the velocity distribution and 

 density of the electrons inside a metal at various temperatures and the 

 difference in potential energy between an electron at rest inside and 

 outside the metal, it would be a comparatively easy task to determine 

 statistically how many electrons could escape from a square centimeter 

 of surface in one second. It was at first assumed that the electrons 

 inside the metal acted like a perfect gas; the velocity distribution is 



* This function will be discussed later in this paper. 



