By Prasanta Kumar Misra

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1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantization of Lattice Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 References . . . . . . . . . . . . . .

12. 12 (a) Debye spectrum; (b) actual lattice spectrum. 3 The Einstein Model of Specific Heat As we noted earlier, in a lattice with a basis, there are both acoustic modes and optical modes. The three acoustic modes can be easily treated by the Debye model. The optical modes can be treated by the Einstein model in which each mode has the same frequency, ωE , which is independent of q. Therefore, from Eq. 67) It should be noted that Einstein was the first to derive the theory of specific heat of solids by using quantum statistics instead of classical statistics.

1 Theory In a crystalline solid, at finite temperatures, the ions or the atoms are not stationary but vibrate around an equilibrium position. In 1907, Einstein proposed a theory of the heat capacity of a solid based on Planck’s quantum hypothesis. He assumed that each atom of the solid vibrates around its equilibrium position with a frequency νE , known as the Einstein frequency.

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