THE QUANTUM PHYSICS OF SOLIDS 693 



Thermal Properties of Crystals 



In this section, as in the last, we shall digress from a straightforward 

 exposition of the theory of energy bands and discuss the theories of 

 specific heat and thermal expansion. These theories are well worth 

 discussing on their own merits and furthermore their results and 

 methods can be applied later to other topics. Thus the thermal vibra- 

 tions that account for the specific heat will be shown in the second 

 paper of this series to account for the resistance of metals. The dis- 

 cussion of thermal expansion given here will in the next section on 

 magnetism be extended to an explanation of the unusual expansion 

 properties of magnetic materials, in particular to an explanation of 

 the very small expansion of invar. We shall, however, make use of 

 the band theory once in this section by showing why the free electrons 

 in a metal do not normally make an appreciable contribution to the 

 specific heat. 



In the introduction to this paper we pointed out that the specific 

 heat per gram atom of a solid should be by classical theory 2)R — 

 coming half from the kinetic energy and half from the potential energy 

 of the atoms. This prediction is in reasonable agreement with ex- 

 periment for many crystals at high temperatures. As the temperature 

 is lowered, however, the observed specific heat decreases in such a way 

 as to approach zero when the absolute zero of temperature is ap- 

 proached. This decrease in the specific heat at low temperatures, as 

 well as the value 3i? at high temperatures, is readily explained by 

 quantum mechanics. In order to understand the explanation we must 

 inquire into the atomic vibrations of a crystal. 



In considering atomic vibrations we are really concerned with the 

 motions of the nuclei. The electrons act as a cement to hold the nuclei 

 in their equilibrium positions and exert restoring forces on them when 

 they are displaced. (We shall see below why the electrons do not par- 

 take of the thermal energy.) The nuclei are eff"ectively mass points in 

 this theory and for quantum mechanical reasons, which we shall not 

 discuss, they are incapable of acquiring thermal energy of rotation; 

 hence so far as the crystal vibrations are concerned, we need consider 

 only their translational or rectilinear motions. A crystal containing 

 N atoms has 3iV degrees of freedom since each nucleus can move 

 in three dimensions. In order to find the specific heat of a crystal 

 we must find the normal modes of vibration. The system of coupled 

 oscillators in Fig. 11 represents reasonably well the normal modes 

 of vibration for a one dimensional crystal whose atoms have only 

 one degree of freedom. There is a similar set of normal modes for 



