THE QUANTUM PHYSICS OF SOLIDS 677 



of the band. We shall find in the next paper that the quantum states 

 in a band represent electrons traversing the crystal with various aver- 

 age speeds. The state Eo has an average speed of zero. The sub- 

 script " M" has been assigned with these ideas in mind and stands for 

 "motion," implying that an electron with energy greater than Eo has 

 an energy of motion Em. In general both Eo and Em are actually 

 composite energies containing both kinetic and potential energy; only 

 in the case of free electrons is Em purely an energy of motion. We 

 shall not use in this paper the property of motion connected with the 

 values of Em; however, we shall use the division of the energy into 

 two parts, Eo and Em, and we shall for convenience refer to the latter 

 as an "energy of motion." 



We shall next apply the concept of the energy band to a determina- 

 tion of the binding energies of several types of crystals. It is one of 

 the principal merits of the theory of energy bands in crystals that we 

 can treat many different crystal types on the basis of the same set of 

 ideas. As we shall point out later, however, the band theory is most 

 appropriate for metals: for ionic and valence crystals other theories 

 are better suited. 



Energy Bands and Binding Energies of Metals 



For several metals the wave functions and distribution of states in 

 energy have been found by solving Schroedinger's equation for the 

 electrons in the metal. We shall discuss sodium since it constitutes 

 one of the simplest cases and is the first metal for which good calcula- 

 tions were carried out. 



A sodium atom, Na, contains ten electrons in filled K and L shells 

 and one valence electron in the M shell; its electron configuration is 

 Is^ 2s^ 2p^ 3s. When the atoms are assembled together as in the 

 metal, the 3s atomic state gives a wide band which overlaps the 3p 

 band while the lower levels widen only very slightly. 



The formation of the energy bands ^ is shown in Fig. 14. Since the 

 K and L bands are very narrow, it is possible to neglect the changes 

 in the wave functions of the electrons occupying them and to concen- 

 trate upon the valence electrons. The valence electrons then move 

 in a potential field produced by the Na+ ions and the other electrons. 



It can be shown by a lengthy argument that for the case of a mono- 

 valent metal, the energy of the metal as a whole is very nearly equal 

 to the sum of the energies of the valence electrons.^" It is rather 



» J. C. Slater, P/jy^. i?ez;., 45, 794 (1934). 



J" An exact statement of the situation is too involved for this paper. The reader 

 can find a more complete discussion in Mott and Jones "The Properties of Metals 

 and Alloys," Chapter IV. 



