424 BELL SYSTEM TECHNICAL JOURNAL 



free electron per atom. For curve 3 the value of n has been so chosen 

 that this curve is shifted with respect to curve 2 by K/e — 3kT/2e 

 or 5.52 volts. The value of n which does this is 16.6 X lO-^/'cm.^ 

 To account for the observed emission from tungsten we have previously 

 deduced a value | as great or 8.4 X lO^^/cm.' The factor of 2 is due 

 to the fact that the intercept of the observed Richardson plot for 

 tungsten corresponds to 60 amp./cm.^ ° K- while the theoretical 

 intercept corresponds to 120 amp. /cm.- ° K.' 



At first sight it might appear that the shift between curves 2 and 3 

 should be K/e rather than K/e — 3kT/2e. The additional term is 

 accounted for by comparing the classical or T^ equation (11a) with 

 the quantum-mechanical or T^ equation (22). It is well known that 

 the experimental results can be made to fit either the 7"' or the 7"- 

 equation and that the constants in the two equations are related by 



W{or Pm - K) ^ p - {3/2)kT, (30) 



and 



A = A'/^^TK (31) 



From equation (30) it follows that the classical work function p is 

 larger than the quantum-mechanical work function IF or P,„ — K 

 by {3/2)kT and that to obtain the same emission from the two dis- 

 tributions the curves must be shifted by K/e — 3kT/2e, 



The Temperature Dependence of the Work Function 

 Thus far little has been said about the temperature dependence of 

 the work function. While there is no good theoretical reason for 

 expecting a large temperature dependence, there is also no good reason 

 to expect that the work function is accurately independent of T. 

 Experiments on contact potential and photoelectric eftect indicate 

 that there is indeed a small temperature effect.* In investigating the 

 effect of the temperature dependence we shall limit ourselves to the 

 quantum-mechanical equations. However, a similar treatment would 

 be applicable to the classical or T^ equation. 



If in equation (22), w or its equivalents W or ^p are independent of 

 T, then the slope of a Richardson line is — w/2.3 or — lF/2.3^ or 

 — (fe/2.3k; the intercept is log U. So that 



h = w= W/k = ^e/k and log A = log U. (32) 



If the work function varies linearly with temperature, 



w = W't -+- aT or IF = IFo + akT 

 * For a detailed discussion see reference 8. 



