694 BELL SYSTEM TECHNICAL JOURNAL 



a three dimensional array of atoms and, once the forces between the 

 atoms are known, the frequency of vibration of each of the modes can 

 be found. This means that so far as thermal vibrations are concerned, 

 we can consider the crystal as equivalent to a set of 3iV oscillators 

 whose frequencies are those of the normal modes. We must next dis- 

 cuss the specific heat of a single oscillator. 



According to classical statistical mechanics, a harmonic oscillator 

 in a temperature bath at absolute temperature T will have an average 

 thermal energy equal to kT, where k is Boltzmann's constant. The 

 value kT is only an average value, we emphasize, and the oscillator 

 will have other energies some of the time, the probability of each 

 energy being given by known equations. The probability is very 

 small, however, that the oscillator acquires more than two or three 

 times kT of thermal energy. In a very large system of oscillators, 

 the fluctuations of energy of the oscillators tend to cancel out and the 

 probability of any appreciable fractional deviation of the total energy 

 from its mean value is very small. If N is the number of mole- 

 cules in a gram molecule {N = 6.06 X 10-^), then Nk = R, the gas 

 constant, = 1.99 cal. per gm. molecule per degree C. Hence the 

 energy of 2>N oscillators is E = 3NkT = 3RT and the specific heat is 

 C = dEjdT = 3R] this classical result that the specific heat of one 

 gram atom of solid is 3R is known as the DuLong-Petit Law. 



According to quantum mechanics, an oscillator of frequency v has a 

 set of quantum states whose energies are }4hv, (1 + /4)hv, (2 -|- y2)hv, 

 etc. The oscillator can take on only these energies. If it is in a heat 

 bath of temperature T, however, it will sometimes have one allowed 

 energy and sometimes another and as for the classical case we shall 

 be concerned with its average energy. At absolute zero, the average 

 energy is, of course, }4hv. Now the probability of the oscillator gain- 

 ing much more than kT of thermal energy is very slight. Hence the 

 average energy of the oscillator remains at y2hv until thermal energy 

 becomes large enough to excite it to the next state which is hv higher, 

 and consequently so long as kT is much less than hv the quantum 

 oscillator acquires much less thermal energy than would a classical 

 oscillator. For kT much greater than hv, the oscillator will spend an 

 appreciable fraction of its time in many of the quantum states and, 

 as may be shown mathematically, the quantum restriction is no longer 

 of importance so far as the average energy is concerned and the value 

 kT is obtained just as in the classical case. In Fig. 20 the dependence 

 upon temperature of the average energy and the specific heat for a 

 quantum oscillator are shown. 



