676 BELL SYSTEM TECHNICAL JOURNAL 



the parent of each state in the band. This confusion is of no conse- 

 quence, however, for it does not interfere with using the distribution 

 in energy curves when they are obtained. Furthermore, the conserva- 

 tion of states holds when the bands overlap so that the total number of 

 states per atom in the combined bands is the sum of the number of 

 states per atom in the separated bands. In Fig. 136 we represent a 

 distribution qualitatively similar to that occurring in various metals 

 where 5 and p bands overlap. The number of states per atom in the 

 combined bands is eight, four for each spin. 



A very important distribution-in-energy curve is that of the case of 

 "free electrons." This is the distribution one obtains by imagining 

 that the electrons in a crystal are perfectly free — that is, subjected to 

 no electrostatic forces whatever — but that they are required to remain 

 within a certain prescribed volume. The distribution of quantum 

 states in energy for this case is represented in Fig. 13c. At low tem- 

 peratures the electrons tend to occupy the lowest states consistent with 

 Pauli's principle and the system is referred to as a "degenerate electron 

 gas." With the aid of the distribution curve, the energy and pressure 

 of this gas can be calculated. We shall require its energy for a discus- 

 sion of the binding energy of sodium, but we shall give here only 

 the equation of the curve, leaving the calculation of the energy until 

 later.* According to the theory, then, for the case of free electrons 

 the number of states per unit energy is given by 



iV(£) =^(2m)3/2£i/2, (4) 



where V is the volume of container, h is Planck's constant, ni the 

 mass and E the energy of the electron; for free electrons E is all 

 kinetic energy, there being no potential energy. For the case of 

 the alkali metals, calculations show that the wave functions for the 

 valence electrons are very similar to the wave functions for free elec- 

 trons. For these metals we can use Eq. (4) to calculate energies. 

 Before utilizing the concepts of energy bands in a discussion of the 

 binding energies of crystals, we must define two symbols to be used in 

 describing the energy of a state in the band. For this purpose we 

 arbitrarily separate the energy E of a crystal state into two parts: 

 one of these is denoted by Eo and stands for the energy of the lowest 

 state in the band and the other is Em which stands for the energy which 

 the state possesses in excess of Eq — that is, its energy above the bottom 



8 The reader will find a derivation of this curve given in K. K. Darrow's article 

 "Statistical Theories of Matter, Radiation and Electricity," Bell System Technical- 

 Journal, Vol. VIII, 672, 1929 or Physical Review Supplement, Vol. I, 90 (1929), and 

 in various texts on quantum statistics and the theory of metals. 



