RECENT STATISTICAL THEORIES 



741 



distinguishes a system conforming to the new statistics, the second 

 term in (127) is no longer proportional to dn/dx. Instead, we have: 



3 3hn I 3n \~'^ 



nv =-nVrn = —i^^-^j ; v~^ = 3/2vm, (133) 



substituting which values into (237), we obtain: 



F / 3 Y'^ 

 eElni = a = - — : - — - n~^'^dnldx, 

 3ni" \ 4tG / 



and integrating: 



- E{x- xo) = + (F- Fo) =r^ 



2me 



3n 

 4^ 



3wo 

 AttG 



This is the formula which supplants Boltzmann's equation. 

 Consulting (71), we see that (135) may be rewritten thus: 



2(Fi - Fo) = {Wi)i - {Wi) 



(134) 



(135) 



(136) 



which is to say: if there is equilibrium between two samples of electron- 

 gas, both being at absolute zero and distributed according to the 

 Fermi law, and the fastest electrons of the two having values of kinetic 

 energy Wa and Wi^ respectively — then there is a potential-difference 

 between the two, such that if the fastest electron of either group 

 were to cross over to the other, its kinetic energy on arrival would 

 be equal to that of the fastest electron of the group which it joins. 

 So stated, the proposition is easy to remember, and one might even 

 come to think it obvious. 



Consider now a pair of pieces of different metals, in contact with 

 one another. One may conceive that they are welded together by an 

 alloy in which the proportion of either varies continuously from zero 

 to one hundred per cent, if one feels the need for a mathematical 

 continuity. If the two pieces were separate, the number of electrons 

 per unit volume would probably not be the same for the two; certainly 

 it is not the same if the number of electrons is equal to the number 

 of atoms per unit volume, for this varies from metal to metal. If 

 the process of welding the metals together does not alter the concen- 

 tration of the electrons in either at points remote from the junction, 

 then a potential-difference given by (132) or (135) — according as the 

 old or the new statistics is the proper one — must arise between the 

 metals. Taking Sommerfeld's example of potassium and silver: if in 

 unit volume of each of these metals there are as many electrons as 

 atoms, and if this state of affairs continued when the two are welded 

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