RECENT STATISTICAL THEORIES 707 



great as many hundreds of degrees. Probably no one who did not 

 often lament the defeat of the old and so very desirable electron-gas 

 theory by that hard fact about the specific heat will ever quite realize 

 the rejoicing caused by this victory of the new, which by this achieve- 

 ment succeeded a se faire pardonner many deficiencies in other fields- 



Features of the Fermi Distribution . 



I will now mention some of the features of the Fermi distribution 

 which has thus justified itself by passing its first test. 



The most startling of these may be inferred from the distribution- 

 function (62) or (61), by inserting the first-approximation formula 

 for A presented in equation (67), and a new symbol Wi: 



r _G 1 w=^l^\" nu 



At the absolute zero the exponential term is either infinity or zero, 

 according as the variable e is greater or less than Wi. Therefore the 

 density of the electrons in phase-space is constant and equal to G/Ji^ 

 for all energy-values less than Wi, zero for all values of energy greater 

 than Wi. 



This striking result can easily be deduced from Fermi's basic 

 assumption, without any statistics at all. Absolute zero is by defini- 

 tion the temperature of the state, being in which the assemblage 

 can give away no energy whatever. If not more than one electron 

 may occupy any compartment of the phase-space, absolute zero is 

 attained when there is an electron in every compartment from the 

 origin outwards to a sphere which is centered at the origin, and which 

 has just the volume needful to contain as many compartments as 

 there are electrons. The number of electrons in unit volume of the 

 phase-space is and remains equal to the number of compartments in 

 unit volume, i.e. to the reciprocal of the volume of the elementary 

 compartment, from the origin outward to this sphere; there it suddenly 

 sinks to zero, and so continues. Cooling-down of an assemblage is 

 settling-down of the particles into this the most condensed of all 

 permissible arrangements; it is like crystallization upon a lattice, 

 only the lattice is in the phase-space. 



The foregoing statements may all be repeated, with the words phase- 

 space replaced by momentum-space. In the momentum-space, a sphere 

 of radius p, consequently of volume 4:Tp^J3, contains (47r/>''/3)/(//^/F) 

 of the elementary compartments. If we set this number equal to the 

 total number of electrons N, and solve the resulting equation for p, 

 we get the radius of the sphere which would just contain all the 



