THE QUANTUM PHYSICS OF SOLIDS 



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cubic lattice. They did not, however, assume that the lattice con- 

 stant was that given by experiment but instead carried out calculations 

 for each of several assumed values for the lattice constant lying on 

 both sides of the experimental value. The results of their calculations 

 are shown in Fig. 15. The curve marked Eq is the energy of the lowest 



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LATTICE CONSTANT IN ANGSTROMS 



Fig. 15 — Energy for sodium versus lattice constant. 



level in the valence band. Only two electrons, one with each spin, 

 can occupy this energy level and all others must occupy states of 

 higher energy — that is, only two electrons can have zero value for the 

 "energy of motion" Em and all others must have larger values. By a 

 method of calculation described below, it can be shown that the 

 average energy of motion of a valence electron is given by the curve 

 marked Ef in the figure. Hence the total energy per valence electron 

 in the metal, which for a monovalent metal is equal to the energy per 

 atom, is represented by the curve marked E in the figure; £ = £o + Ep. 

 Figure 15 exhibits the dependence of this energy upon the lattice 

 constant. The abscissa of the minimum in the E curve gives the 

 theoretically predicted value for the equilibrium lattice constant. The 

 binding energy or heat of sublimation is defined as the energy required 

 to separate the metal into isolated atoms; it is the difference in energy 

 between the minimum of the curve and the value of E for infinite 

 lattice constant — that is, for free atoms. Finally, the curvature of the 

 curve at its minimum is a measure of the energy required to compress 

 or expand the crystal and from it a value for the compressibility can 



