746 BELL SYSTEM TECHNICAL JOURNAL 



electrons are about as numerous as the atoms. For the Thomson 

 heat as for the specific heat, the new statistics sharply reduces the 

 amounts demanded — to about one per cent of those on which the 

 classical theory insists, at room-temperature that is to say and as- 

 suming always that the free electrons are equal in number to the atoms. 

 Agreement in order of magnitude is now attained, and for some metals 

 the advantage is possibly greater; there are indications, too, that the 

 Thomson heat is proportional to T over wide ranges of temperature. 

 Finally we consider the "Peltier heat" — a term proportional to the 

 current and changing sign when the current changes sense, observed 

 when there is a flow of electricity across a weld or area of contact 

 between two metals. This is clearly to be interpreted as a term in 

 the first power of /, occurring when into the combination J{ma/e) — - 

 all that remains of the expression (138) for r, when the gradient of 

 temperature is annulled — we substitute the value of a derived from 

 (143) with the assumption that n varies continuously across the weld 

 from the value appropriate to the one metal to the value appropriate 

 to the other. Using the classical statistics, we find that there is such 

 a term; denoting by n and Wo the electron-concentrations in the two 

 metals, we find for its value: 



(kT/e) log (w/«o)/. (147) 



Its value for unit current is obviously the intrinsic potential-difference 

 between the metals. Using instead the new statistics, we find that 

 at the absolute zero there is no such term; we must proceed to the 

 next approximation, doing which, Sommerfeld obtained the expression: 



2t^ m(kTy \(±n^Y" _ /4tG\'-" 

 3 e/r \\3n/ \ 3»o 



(148) 



Putting the current equal to unity, we find a value very considerably 



smaller than the intrinsic potential-difference between the metals — 



a fraction of a millivolt. This is the order of magnitude of the Peltier 



heat as it is actually observed in many cases. Curiously enough, 



this fact by itself is in accord with both the theories. By the classical 



statistics, the intrinsic potential-difference between two metals is 



generally small, and the Peltier heat for unit current gives its value 



directly; by the Fermi statistics, the intrinsic potential-difference is 



generally large, but the Peltier heat for unit current is only a small 



fraction of it. 



Omissions 



Among the subjects omitted from this article there are several of 

 much interest, which the reader may trace from the annexed bibli- 

 ography; in particular: 



