Phonons in Bravais lattices

This work establishes some exact results in lattice dynamics. For arbitrary temperatures, an elastic sum rule is derived which expresses the minus-first moment of the displacement correlation function with respect to the frequency in terms of the isothermal elastic constants, and yields the asymptotic form of the structure factor of the lattice for small wave numbers. It is shown that at zero temperature the low-lying excitations of the crystal are sound waves, the velocity of which can be expressed by the second derivatives of the ground-state energy with respect to homogeneous deformations. This implies that the specific heat in the limit of temperatures tending to zero follows a Debye law of the same form as for a gas of noninteracting phonons. These results are derived for an ideal Bravais lattice by taking into account the entire anharmonicity of the dynamics. As a mathematical tool, a diagram technique is developed which avoids the concept of the harmonic approximation as a zeroth-order step.