RECENT STATISTICAL THEORIES 701 



instead of "not more than one." The affinity of this to one of Fermi's 

 assumptions will soon be manifest. It would take much too long 

 to give an idea of the successes of the Pauli principle; they are however 

 so great as to increase the inherent plausibility of Fermi's idea very 

 much — or perhaps I should say, so great as to render the idea plausible, 

 which otherwise it might not seem. 



The reasoning follows exactly the same course as when we were 

 deriving the distribution-law (56), except that all the summations 

 over the variable i are now summations of two terms only, the term 

 for i = and the term for i = \. For each of the shells there are 

 only two numbers Z.s required to describe the distribution: viz. Zqs 

 the number of empty compartments and Zis the number of compart- 

 ments containing one particle apiece. We try the distribution (54) : 



Zo. = a.; Zu = a^e^-'^'^^, (58) 



and easily find that it is the distribution of maximum probability, 

 by comparison with all the others compatible with the same total 

 number of particles and the same total energy. We arrive then at 

 the following expression for the number of particles in the shell s: 



^^-g^ ,.wi+i - (59) 



There is no point in putting for Qs the value appropriate to radiation- 

 gas, since the Bose formula has already proved adequate for that case. 

 Fermi put the value appropriate to material gases, and obtained: 



^^ = ^4^ (2-"0'^'^^i^^ = FieMes. (60) 



This formula is the point of departure for the theory of the electron- 

 gas in metals revived and remodelled by Pauli and Sommerfeld, to 

 the experimental tests of which most of the rest of this article will 

 be devoted. 



Application of the Fermi Statistics to the Electrons 



IN Metals 

 We are asked to conceive of a piece of metal as a region populated 

 with "free" electrons, and surrounded by a wall; the electrons being 

 distributed according to the formula of Fermi. 



The Fermi distribution-function involves the total number N of 

 the electrons, which is a disposable constant. It also involves the 

 volume V which this assemblage of N particles pervades. For this 



