Heat capacity of monoatomic crystals

1 Heat capacity of monoatomic crystals Consider N atoms of mass m in a solid crystal. We can assume the size of the system to be large enough and the geometry to be such that the surface to volume ratio of the system is vanishingly small. The crystal can be considered as a lattice with regularly spaced atoms (Fig. 9.3). The configuration of the solid is described by the 3A-cUmensional vector of the positions of the atoms 

At equilibrium each atom i is positioned at lattice site ro,. By definition, the potential energy becomes minimum at equilibrium, i.e.. 

We can assume that atoms vibrate independently around their equilibrium positions (Fig. 9.4). We then define the displacement 8r, for each atom i as follows  

In general, the vibrations can be assumed small for temperatures lower than the melting point of the solid. A Taylor series expansion can then be constructed for the potential energy up to quadratic terms of 8r, as follows  

So is the first derivative of the potential energy around the lattice site positions, 

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