THERMIONIC ELECTRON EMISSION 419 



If p is independent of T and if log i — h log T is plotted versus \/T, 

 the slope = — pjl-Zk and the y intercept = log A' or log ne{k j l-wm)^ . 

 For clean tungsten it is found by experiment that the current density 

 can be represented by 



i = 2.06 X Wr^ exp. (- 55,300/r). 



From this it follows that ne{k/2Tmy^ = 2.06 X 10^ and that n = 8.4 

 X 10-" electrons/cm.^ This is to be compared with 635 X 10-" 

 atoms/cm.^ So that if we postulate one " free electron " for every 

 75 atoms, we can account for the observed thermionic emission 

 classically. 



Such a concentration of free electrons may be considered to be in 

 quite good accord with the first of two possible deductions from 

 experiments on specific heats. From these it follows that: (1) Either 

 the number of free electrons must be small compared to the number of 

 atoms and the mean kinetic energy per electron is {3/2)kT; or else, 

 (2) the number of free electrons is of the order of the number of atoms 

 but the kinetic energy increase per degree rise in temperature is much 

 smaller for electrons than it is for atoms. The correlation of experi- 

 ment and classical theory in the case of the optical properties, electrical 

 conductivity, thermoelectricity, Thomson and Peltier effects lead to 

 certain inconsistencies. These disappear when theories based on the 

 Fermi-Dirac distribution are used for these effects and it is postulated 

 that the number of free electrons in metals is of the same order as the 

 number of atoms. The classical theory for Richardson's equation 

 thus leads to values of n which are incompatible with values deduced 

 from these effects. The newer theory has also made progress in 

 explaining ferromagnetism. It is thus a better basis for a statistical 

 theory of electron emission. Such a theory was developed by Som- 

 merfeld ^^ and Nordheim.^® 



b. Quantum-mechanical treatment. The Fermi-Dirac theory gives 

 the velocity distribution as 



u{u, V, w)dudvdw = —r^ 



X x?-[ r / ■' 1 — n n/of,-ri I 1 dudvdw, (13) 



M ^ exp. \_?n{u^ + y^ -f w')/2kl J -f 1 



G is the statistical weight ; for electrons its value is 2. h is Planck's 

 constant. The quantity M is so adjusted that the integral of n{u, v, w) 

 gives the total number of electrons/cm.^ This integration is so difficult 

 that no relatively simple and exact expression for M can be found. 



