682 BELL SYSTEM TECHNICAL JOURNAL 



definition, the average energy per electron of a degenerate electron 

 gas. In a degenerate electron gas the electrons have the least possible 

 energy consistent with Pauli's principle and with the distribution of 

 quantum states in energy. For reasons associated with the origin 

 of the statistical mechanics of electrons — that is, with the Fermi-Dirac 

 statistics — the energy Ep is called the "Fermi energy" and given the 

 subscript F. The energy Ef is far greater than the average energy 

 per particle of an ordinary classical gas. We shall see below how this 

 fact accounts for the very small specific heat of the electron gas. 

 From the dependence of the energy upon volume, the pressure of the 

 electron gas can be calculated. It is usually very large, for sodium it 

 is about 50,000 atmospheres. The force that prevents this pressure 

 from blowing the metal apart is represented by the Eq curve, which 

 gives decreasing energy with decreasing lattice constant and corre- 

 sponds to a force pulling the atoms together. A more detailed dis- 

 cussion of these forces will be taken up in the third paper of this series. 



Other Metals 



Calculations similar to those for sodium can be carried out for other 

 metals. The band structure as calculated for copper by Krutter i* is 

 shown in Fig. 16. Ten electrons per atom can be accommodated in 

 the M band and two per atom in the 45. For copper the 2>d band is 

 filled — in keeping with the fact that the Cu+ ion consists of filled 

 K, L, and M shells. From the discussion of molecules given above 



is the same as the volume per atom. Denoting by fl the value of F/w, we find 



/ 2 \ 2/3 hi 

 £max. = f^) ^12-=/" = Se.lflo-^/'eV (iv) 



where fio is the volume per atom in cubic Angstroms. For a body-centered cubic 

 lattice with lattice constant a Angstroms, Oo = a^l2. Substituting the expression 

 for N{E) into the equation for Ep gives 



Ef = \ Sraax. = 21.6fio-2/3ev. (v) 



Expressing Ef in atomic units and fto in terms of the lattice constant, we find 



Ef = 2.54a-2. (vi) 



This is the equation of the curve for Figure 15. The values of Emax. calculated from 

 the above equations for a series of metals are 



Metal Li Na K Rb Cs Cu Ag Au 



£max.(ev) 4.74 3.16 2.06 1.79 1.53 7.10 5.52 5.56 



" H. M. Krutter, Phys. Rev., 48, 664 (1935). 



