THE QUANTUM PHYSICS OF SOLIDS 



695 



The specific heat of the crystal is just the sum of the specific heats 

 of its oscillators. Since the oscillators have different frequencies they 

 have different specific heats and in order to add up the specific heats of 

 all of them it is necessary to know how the various frequencies of the 



2.0 



kT 



TEMPERATURE, 7—- 



Fig. 20 — Thermal behavior of an oscillator according to quantum mechanics. 



(a) Energy versus temperature. 



(b) Specific heat versus temperature. 



oscillators are distributed. Once this distribution in frequencies is 

 known it is merely a matter of summation to find total specific heat. 

 The problem of finding the distribution in frequency of the oscillators 

 was first solved by Debye. The low-frequency vibrations are very 

 simply found for they are merely the acoustic vibrations of the crystal ; 

 they are very similar to the normal modes shown for the square mem- 

 brane of Fig. Ig. For these low-frequency vibrations it can be shown 

 by a straightforward argument, which is too long to give here, that 

 the number dNoi oscillators whose frequencies lie between v and v + dv 

 is 



dN=VA.{' + ' 



v^dv, 



(7) 



where Ct and Cl are the velocities of transverse and longitudinal 

 waves in the solid and V is its volume. ^^ Debye assumed that this 

 distribution held for all the normal modes. There is of course a highest 

 frequency of vibration, i^max, and the total number of normal modes 

 must be 3N; hence Debye concluded that 



1 Ct' ^ Cl'\ ■ 



Cj}\ 3 

 26 For a derivation see P. Debye, Ann. d. Physik, 39, 789 (1912). 



(8) 



