THE QUANTUM PHYSICS OF SOLIDS 687 



been carried out for the alkali halides. In it an analytical expression 

 suggested by theory and containing adjustable constants has been used 

 for the closed shell repulsions. The adjustable constants have been 

 determined from certain data and then used for predictions which can 

 be compared with other data. Using a relatively small number of 

 adjustable constants, Born and Mayer.^" Mayer and Helmholz,^^ and 

 Huggins and Mayer ^^ have calculated a much larger number of values 

 for lattice constant and binding energy for many alkali halides with 

 an agreement with experiment of the order of one per cent. 



Let us now consider NaCl using the band picture. We shall reach 

 the rather surprising conclusion that there is no fundamental differ- 

 ence between the results obtained from it and those just deduced from 

 the ionic picture described above. 



In Fig. 18 we show qualitatively the behavior of the bands for NaCl.^^ 

 In the ionic state, an electron is transferred from the Na Zs to the CI Zp. 

 The general shifting of the bands is explained as follows. The wave 

 functions corresponding to the Cl~ Zp band, like all energy band wave 

 functions, are distributed over the whole crystal. They are not, 

 however, equally intense at Na+ and at Cl~ ions; instead they are defi- 

 nitely concentrated about the Cl~ ions. The electrostatic potential 

 at a Cl~ ion, due to the remainder of the crystal, has the same value 

 (6) as was found in discussing the ionic method. Since the charge on 

 the electron is —e, the energy of each of the states in the Cl~ 3p band 

 varies with a in the same manner as does —ecf). A similar argument 

 shows that the Na+ energy bands vary as +e</). At a certain lattice 

 constant, the Cl~ d>p and Ss bands and the Na+ 2p and 2s bands begin 

 to widen. Since these bands are full, this widening gives the cus- 

 tomary encroachment energy just as it was obtained in the ionic 

 picture. The shifting of the bands similarly gives the coulomb energy. 

 To see this we note that per NaCl molecule there are 18 electrons 

 in the Cl~ bands where energies vary as — 18e0 and that there is also 

 one chlorine nucleus with charge +17e whose energy varies as -\-17 ecj). 

 This leaves a net effect of —ecj) for the Cl~ ions. Similarly a net effect 

 of —e(f> comes from the electrons and nuclei of the Na+ ions. As in 

 the case of the ionic method the sum, — 2e</>, of these energies really 

 contains each ionic energy twice and the total electrostatic energy per 

 NaCl molecule is —ecf). So far as calculating energies is concerned, 

 the two methods give equivalent results; the advantage, if any, lies 



^''Zeits.f. Phys., 75, 1 (1932). 

 ^^Zeits.f. Phys., 75, 19 (1932). 



22 Jour. Chem. Phys., i, 643 (1933). 



23 J, C. Slater and W. Shockley, Phys. Rev., 50, 705 (1936). 



