738 BELL SYSTEM TECHNICAL JOURNAL 



For the twelve metals Al, Cu, Ag, Au, Ni, Zn, Cd, Pb, Sn, Pt, Pd 

 and Fe, the average of the values of /c/o- at 291° K is 7.11 -10"^^ 

 The agreement with the prediction of the new statistics is more than 

 good. It is so very good, that it must be partly accidental, especially 

 as the individual values from which the average is formed depart 

 from it by varying amounts. One may still doubt whether it is to 

 be admitted as one of the items which compel the adoption of the 

 new statistics. Drude, be it recalled, obtained the value 6.3 -10"^^ 

 out of the crude assumption that all of the electrons in any volume- 

 element have the same speed. ^^ It used to be regarded as rather 

 amusing that the elaborate calculations of Lorentz merely impaired 

 the agreement which Drude had attained in a naively simple way. 



Both theories require that the ratio be proportional to T', this is 

 fairly well satisfied over wide ranges of temperature, but at extreme 

 degrees of cold there is marked divergence, which is inconvenient. 

 It may be desirable to invoke other mechanisms of conduction to 

 supplement the free electrons — as for instance the passing-along of 

 electrons from atom directly to atom to assist in the conduction of 

 electricity, or the transmission of elastic vibrations to aid in the 

 transfer of heat. Indeed, when one reflects that insulators though 

 they lack free electrons yet have some device for the transmission 

 of heat, one wonders why this device should not be available to 

 metals also, and exalt their values of k and of KJa above the predictions 

 of the electron-gas theory. 



Intrinsic Potential Difference 



We have seen that in a metal where there is no electric current and 

 yet there is a current of heat, an internal electric field must be imagined. 

 We shall now see that in a metal where there is no electric current, 

 but the number of electrons per unit volume varies from point to 

 point, there must also be an internal electric field. This sounds 

 plausible to intuition, for one would expect the electrons to diffuse 

 from regions of higher to regions of lower density unless they were 

 impeded by some force. The equations (117) and (118) enable us 

 to evaluate this force. 



Returning to these equations, introduce polar coordinates v, 6, 

 in the velocity-space as we have formerly done; then ^ = v cos 6. 

 Multiply both sides of equation (117) by cos 9; the right-hand member 

 of the new equation is then proportional to ^-g. Integrate both 



2^ Drude of course could evaluate the ratio k/e without knowing either ^ or e 

 accurately or at all, since it is the same as the ratio Nok/Nfte — No standing for the 

 number of molecules in a gramme-molecule, the Loschmidt number — -and Nok is 

 the gas-constant R while Noe is the Faraday constant of electrolysis. 



