Phonons, internal and external modes

A crystal with n atoms per unit cell has 3nN degrees of freedom, N being the number of unit cells in the crystal. Thus, subtracting the translations and rotations of the crystal as a whole, there are 3nJV — 6 ( 3nN) normal modes. Since the displacements of atoms in different cells are correlated, the normal modes are waves, or phonons, extending over the crystal, with force constants d , obtained from a sum over the interactions between atoms in all unit cells, and wavevector q. 

For a small change in magnitude of q, the change in frequency to is small, and oj is a continuous function of q ( — 2n/X). The dependence of co on q is referred to as the dispersion relation. The number of phonon branches with continuously varying co equals 3n, but some of these may be degenerate due to the symmetry of the crystal. 

For a molecular crystal, the description can be simplified considerably by differentiating between internal and external modes. If there are M molecules in the cell, each with nM atoms, the number of external translational phonon branches will be 3M, as will the number of external rotational branches. When the molecules are linear, only 2M external rotational modes exist. For each molecule, there are 3nM — 6 (3nM — 5 for a linear molecule) internal modes, the wavelength of which is independent of q. Summing all modes gives a total number of N M(3nM — 6) + 6M = 3nN, as required, because each of the modes that have been constructed is a combination of the displacements of the individual atoms. 

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