THE QUANTUM PHYSICS OF SOLIDS 



673 



number of states in the band is, of course, proportional to the number 

 of atoms; however, if the number of atoms is large, the width of the 

 band is independent of the number of atoms. This concept of allowed 

 bands of energies for the crystal states plays the same role in crystals 

 as the concept of energy levels in the atom. We shall refer to 3^ bands 



LATTICE CONSTANT (OR STRENGTH OF COUPLING) 



Fig. 12 — Dependence of energy levels upon lattice constant or frequency 

 of vibration upon strength of coupling. 



and M bands of energy levels in crystals in much the same way as we 

 refer to the Zs and Zd atomic energy levels from which these bands arise. 



We must emphasize that like the molecular states, the crystal states 

 do not belong to the atoms individually but instead belong to entire 

 system of atoms. 



Before proceeding with the application of these ideas to crystals 

 with large numbers of atoms, we shall digress by anticipating several 

 subjects to be taken up in the next paper. For the energy levels of 

 isolated atoms the quantum numbers n, I, m, and Ws were satisfactory. 

 For a crystal, however, there will be many crystal quantum states 

 in an energy band all arising from atomic levels having the same values 

 of n, I, m, and Ws. A new quantum number is therefore needed to dis- 

 tinguish the various crystal states in an energy band one from another. 

 In Fig. 11 we see that the wave function of each crystal state is asso- 



