THE QUANTUM PHYSICS OF SOLIDS 701 



heat of iron near absolute zero.^^ The theoretical curve c, which is 

 seen to represent the experimental data quite well, is the sum of two 

 terms represented by curves a and b. a is linear in the temperature 

 and represents the electronic specific heat while h is cubic and repre- 

 sents that due to lattice vibrations. Numerical calculations from 

 theory of the slope of curve a which could be compared with the 

 observed slope are not available. Curve h, we have said, is just the 

 Debye curve and is drawn as if the Debye temperature were 462°, 

 a value which is in good agreement with 453°, the value deduced from 

 the specific heat at higher temperatures in connection with Fig. 21. 

 At very high temperatures the electronic specific heat will again be 

 of importance. But at high temperatures it is necessary to apply 

 corrections to the Debye theory and the writer is not acquainted with 

 any unambiguous evidence for electronic specific heat in that case. 



Thus we see that only a very small fraction of the electrons of a 

 partially filled band contribute to the specific heat. It is the Pauli 

 principle which restrains the remainder. We shall see in the next 

 paper why the Pauli principle does not interfere with the conduction 

 of electricity. For the case of an insulator — that is, a crystal each 

 of whose bands is either wholly filled or wholly empty — it is still harder 

 for electrons to arrive at empty states and the electronic specific heat 

 is quite negligible. Hence all of the specific heat for an insulator is of 

 the atomic vibration type discussed in the Debye theory. 



The Theory of Thermal Expansion 



In order to understand the theory of thermal expansion we must 

 study the curve representing energy versus lattice constant for the 

 solid. This is shown qualitatively in Fig. 24. We note that the 

 energy curve is unsymmetrical about its minimum. We may describe 

 its behavior by saying that it is harder to compress than to expand the 

 solid. This statement is illustrated by a comparison of the expansion 

 and the compression which can be produced by a given energy E; it is 

 seen that the asymmetry of the curve causes the expansion produced 

 by this energy to be greater than the compression. Now the origin of 

 thermal expansion is as follows: owing to thermal agitation — that is, 

 atomic vibrations — regions of the crystal are alternately expanding and 

 contracting; since the expansions occur more readily than the con- 

 tractions, there is on the average a net expansion. The greater the 

 temperature the greater this net expansion ; hence we find that the size 

 of the solid increases with increasing temperature. This explanation 

 of thermal expansion can be made clearer by considering, not a solid, 



29 W. H. Keesom and B. Kurrelmyer, Physica 6, 633 (1939). 



