4.7.1 Specific Heat Capacity

The lattice (or phonon) specific heat capacity c _{?, p}, is the total value for all phonon modes. The specific heat capacity is given in J/kg-K, J/m ³-K, or J/K (per atom), depending on relevance. Using the Debye model of the phonon DOS, c _{?, p} is derived for monatomic crystalline solids. It has a temperature dependence such that, as expected, it vanishes at 0 K, has a steep dependence as T = 0 K is approached, and the temperature dependence disappears at high temperatures (at high temperatures, it reaches a value of 3 k _B per atom, associated with 3 degrees of freedom, and equipartition of energy, which gives crystal kinetic and potential energies). Debye, noting these trends, suggested a cut-off frequency beyond which the high-energy phonons do not contribute to the lattice specific heat capacity (Section 4.2.2). His formulation is subsequently given.

For a monatomic, isotropic crystal, the energy per unit volume and atomic mass M/N _A = m for N _p phonons, in a volume V, is

where the i summation is over the phonon modes, which are N _p in volume V or n _p per unit volume. Note that we have used the integral of the phonon DOS (1/m ³-rad/s) in place of the summation. The lattice specific heat capacity of a...