698 BELL SYSTEM TECHNICAL JOURNAL 



"Electrons in Crystals" and extensions of them* to be discussed in the 

 third paper of the series. Using the theoretical values one obtains 

 a value of 143° K for 6d, whereas the value that fits experiment best 

 is 172° K. 



Recently calculations have been made from a model of the crystal 

 as an assemblage of atoms rather than as a continuum as postulated 

 in deriving Eq. (7) — that is, a model like the coupled oscillators, rather 

 than like the stretched membrane, is used. These calculations, prin- 

 cipally by Blackman, have explained some discrepancies between the 

 Debye theory and experiment. 



The Specific Heat of the Electrons 



We must now see why the electrons contribute only slightly to the 

 specific heat. Let us consider a case like that of sodium where we have 

 a partially filled band. At the absolute zero of temperature, the elec- 

 trons will fill all the levels below a certain energy £i and all the higher 

 levels in the band will be empty (Fig. 22a). Now at temperature T 

 some of the electrons will be excited to higher states; since, however, an 

 electron cannot gain more than about kT of energy thermally, only 

 those electrons whose energies lie in a range kT below Ei can be excited. 

 Electrons occupying states farther down in the band cannot acquire 

 kT of thermal energy for, if they did so, they would have to move to 

 states already occupied and such an act is forbidden by Pauli's prin- 

 ciple. In order to demonstrate what a small fraction of the electrons 

 can gain energy thermally, we point out that the width of the energy 

 band is usually 4 or 5 ev while the value of kT in electron volts is 

 r/1 1,600 and room temperature corresponds to a kT of about .03 ev. 

 The electrons which do gain thermal energy have a normal value for 

 the specific heat but constitute only about one per cent of all the 

 valence electrons. 



It might be maintained that the above argument is specious and 

 that the electrons could all gain energy kT; this would not violate 

 Pauli's principle because the electrons would move upward in the 

 band as a unit, each moving into a state vacated by another elec- 

 tron. This contention is found to be wrong; one finds by using the 

 statistical mechanics appropriate to electrons that the distribution of 

 the electrons among the energy levels is given by the Fermi-Dirac 

 distribution function. ^^ According to this, the distribution of the elec- 

 trons among the levels would be as indicated in Fig. 22b. The proba- 



* K. Fuchs, Proc. Roy. Soc. 157, 444 (1936). 



2* For a discussion of the Fermi-Dirac statistics see K. K. Darrow, The Bell System 

 Technical Journal, Vol. VIII, p. 672, 1929, or The Physical Review Supplement, Vol. I, 

 p. 90, 1929. 



