Phonon Dispersion Relations and Phonon Models

Abstract

The lattice potential ϕ of a crystal in thermal equilibrium can be derived from the free energy F(T,V). ϕ reads in the classical harmonic approximation (ϕ ≃ ϕ₂),

$${\phi _2} = \frac{1}{2}\,\,\sum\limits_{L,L} , \,\underline u \left( L \right)\underline \phi(LL')\underline u (L')$$

(2.1)

where L = (ℓ, κ) denotes the κ^th particle in the ℓ^th cell. The u(ℓ, κ) denote the displacement vectors of the ions with equilibrium positions, X(ℓ, κ) = X(ℓ) + X(κ), where X(ℓ) is pointing to the £ cell of the crystal and X(ℓ) describes the relative position of the κ^th ion in the cell. The 3 × 3 two-ion force constant matrices ϕ(L, L’) are subject to the conservation laws of energy, momentum, and angular momentum and to the symmetry restrictions of the space group [2.2].