724 BELL SYSTEM TECHNICAL JOURNAL 



paths of the electrons from impact to impact. This statement is 

 true with the classical statistics, a fortiori with the new. Denote by 

 / the average distance traversed by an electron between impacts. 

 Then /o is Ijv, and the average drift-speed is ^{eEl/mi').'^^ The corre- 

 sponding current-density is the product of this by the numer of 

 electrons in unit volume multiplied by the charge of each. So, for 

 the current-density produced by unit-field-strength, which is by 

 definition the conductivity a, we obtain the formula: 



a = hneHjmv. (91) 



The constant /, the mean free path, is the third disposable constant 

 of the theory of electrons in metals. 



I fear that the foregoing passage sounds very old-fashioned; but 

 nevertheless it expresses the corpuscle-theory of conduction. The 

 notion of elastic spheres is only accessory — an image which may or 

 may not be the best to represent the central idea, the idea that the 

 life-history of a corpuscle in a metal pervaded by a field is an alter- 

 nation of gradual gain and sudden loss. The mean free path is the 

 average distance of uninterrupted gain. 



The common test of the formula (91) is the test by the temperature- 

 variation. The result of this, incidentally, was regarded as quite as 

 grave a demerit of the old electron-gas theory as the difficulty with 

 the specific heat. 



It is a fact of experience that the resistivity p = l/cr of any metal 

 varies rapidly with temperature. For many metals it varies directly 

 as T over quite a wide range; at low temperatures even more swiftly, 

 not to speak of the strange phenomenon of supraconductivity. Now 

 in equation (151) we have p set equal to a combination of two universal 

 constants with three quantities v, n, I between which last the responsi- 

 bility for these great variations must be divided. 



According to the classical statistics v is proportional to T^'~. This 

 is a variation in the right sense, but not fast enough. To make p 

 vary as T we must then make nl vary as 1/7"^'-. With the Fermi 

 statistics the requirement is harder. The mean speed of thermal 

 agitation is almost independent of temperature, and the burden of 

 the whole responsibility for making p proportional to T must be 

 loaded upon nl. The first step with the new statistics is a step 

 backward. 



Can we reasonably assume n to be the cause of the variation? 



^' It is the mean of the reciprocal of the speed, not the reciprocal of the mean 

 speed, which should figure here; but with so rough a formula the distinction is 

 scarcely worth making. 



