RECENT STATISTICAL THEORIES 737 



Sommerfeld adopted the Fermi function, and obtained for the 

 degenerate case : ^^ 



The coefficient of dT/dx in these expressions for W is by definition 

 the thermal conductivity, usually denoted by k. One notices that 

 these expressions for k like those for a, involve the more or less dis- 

 posable constants n and /. This however is not true of the ratio of 

 the conductivities. 



The Wiedemann- Franz Ratio 

 For the ratio of thermal to electric conductivity, the old statistics 

 and the new supply expressions involving nothing but T and the ratio 

 of the universal constants k and e, and difTering only by a slight 

 numerical factor: 



/c/o- = 2{kle)-T{= 4.2 X IQ-" at T = 291° K) (121) 



by the old statistics, and 



/c/o- = i7r2(yfe/e)2r(= 7.1 X 10-11 at T - 291° K) (122) 



by the new. 



This "Wiedemann-Franz ratio" seems to have been predestined to 

 encourage the devotees of the electron-gas theory. Every other 

 formula offered by the theory contained either w or / or both, and 

 therefore could not serve as an ultimate critical test; for any discrep- 

 ancy with the data could be removed by adjusting these constants. 

 True, the ensemble of the formulae provided by the classical theory 

 ran counter to the data in so many different ways, that the net result 

 was quite unfavourable; but one could not point out any single predic- 

 tion which was certainly wrong. If however the Wiedemann-Franz 

 ratio had departed by an order of magnitude or more from the value 

 of 2(k/eyT, the electron-gas theory could hardly have survived the 

 blow. But in this one case where disagreement would have been 

 fatal, there was agreement; not perfect, but rather too good to be 

 discarded as fortuitous. For many of the familiar metals the ratio, 

 when measured at room-temperature, turned out to be around 6 or 

 7-10~"ii. This more than any other one fact was what kept alive the 

 feeling, that in spite of all its difficulties the electron-gas theory must 

 be fundamentally right. 



2* To derive this formula it was necessary to proceed to the second-approximation 

 expression for the Fermi distribution-function; the first approximation merely 

 yielded zero for W. 



