998 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



attention. The second problem is more difficult mathematically. It is given 

 only an approximate treatment which is adequate, however, to indicate 

 that the solution to the first problem contains the necessary features to 

 explain the experiments. 



3b. The Phonons 



We must next consider how to describe the thermal vibrations and to 

 evaluate their interactions with the electrons. We shall present only the 

 principal results of the mathematical analysis here, leaving the details for 

 the app)endices. The earliest treatment of thermal vibration in a crystal 

 was that of Einstein, who considered each atom to be a separate harmonic 

 oscillator. This model was improved on by Debye who treated the crystal 

 as an elastic continuum that could support running waves. Debye's method 

 is regarded as essentially correct and we, therefore, resolve the atomic 

 motions into a set of running waves, or normal modes. There are three 

 times as many independent normal modes as there are atoms in the crystal, 

 or one per degree of freedom, and any possible atomic motion of the crystal 

 may be made up as a sort of Fourier series in these normal modes. 



Each normal mode must be treated as a Planck oscillator and has a 

 system of energy levels with values 



{n + l/2)hv (3.1) 



where v is its frequency of vibration. Each quantum of energy is referred 

 to as a phonon; if a normal mode makes a transition with 8n = -f-l, we say 

 a phonon has been emitted and ii dn = — 1, we say one has been absorbed. 



The description of the crystal in terms of phonons is in close analogy 

 with the description of electromagnetic waves in a cavity in terms of pho- 

 tons. For the case of light, the electromagnetic state of the cavity is deter- 

 mined by finding the normal modes, which are treated as quantized oscil- 

 lators, and transitions with 8n = zkl correspond to photon emission and 

 absorption. 



The normal modes for the crystal are unhke those for light. For low 

 frequencies the waves are essentially the microscopic transverse and longi- 

 tudinal waves of a solid. As the wave length becomes shorter, however, 

 the sound velocity varies and there is a limiting minimum wave length 

 which is about twice the spacing between atoms. In order to understand 

 the energy losses of electrons in high fields, we must consider the role of 

 this minimum wave length. For this purpose we shall describe the depend- 

 ence of frequency upon wave length for a longitudinal mode. 



Accordingly we consider the frequency of the normal modes correspond- 

 ing to a longitudinal wave propagating along a cube axis. Rather than 



