THE QUANTUM PHYSICS OF SOLIDS 689 



larly there are compounds, in particular intermetallic compounds, 

 which are more like metals than like either ionic crystals or valence 

 crystals. Thus there is an intermediate field which connects all three 

 of the simple types of binding. Good computations are lacking for 

 these intermediate cases; we shall return to a discussion of some aspects 

 of them in connection with semiconductors in the next paper. 



Concerning a Classification of Crystals 

 In the last section we saw how the concept of the energy band can 

 explain the binding energies of a number of different types of crystals. 

 Although the band theory has the merit of being very general it has 

 the disadvantage of being at the same time rather abstract. Other 

 theories have been developed to explain the cohesion of particular types 

 of crystals; and, while lacking the generality of the band theory, they 

 have the advantage of a more immediate physical interpretation in 

 their own particular fields. In this section we shall digress from the 

 exposition of the band theory in order to describe briefly some of the 

 simpler viewpoints of the other theories. 



We have discussed in the last section three types of binding. Sodium 

 exemplified the metallic type; diamond, the homopolar or valence type; 

 and sodium chloride, the ionic type. The distinction between the 

 valence bond and the metallic bond is not very clearly indicated in the 

 band theory; the only difference there had to do with the degree of 

 filling of the bands. There is another difference, however, which has 

 been long familiar to chemists. The homopolar compounds are usually 

 characterized by "directed valence." Thus the "tetrahedral carbon 

 atom" is a familiar concept of organic chemistry. In crystals in 

 which homopolar binding is dominant the atoms are arranged so that 

 each atom has the proper valence bonds with its neighbors. In dia- 

 mond each carbon atom is tetrahedrally surrounded by four other 

 carbon atoms. In silicon carbide, carborundum, a similar situation 

 prevails: each carbon is tetrahedrally surrounded by four silicons and 

 vice versa. These crystals are said to have a "coordination number" 

 of four, or z = 4, meaning that each atom has four nearest neighbors. In 

 crystals of the divalent elements— sulphur, selenium and tellurium — 

 each atom has two near neighbors and the valence condition is satis- 

 fied; these crystals have a coordination number of two. The mono- 

 valent halogens form crystals in which each atom has one near neigh- 

 bor, coordination number one. In the metals, however, the neighbors 

 of a given atom are as many as eight or twelve — do these large coordi- 

 nation numbers imply that the metals have eight or twelve electron 

 pair bonds with their neighbors? 



