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Lattice Dynamics Theory

The theory of the lattice dynamics comes from Born and Huang. It assumes that the interatomic potential energy is a function of instantaneous position of atomic nuclei. The potential energy is expanded in terms of displacements of atoms from equilibrium positions. The expansion coefficients are called force constants. Linear expansion terms vanish because at equilibrium positions atomic forces vanish. Hence,

V = V(.... U(n, μ),...)  +  (½) Σn,μ,m    Φ(n,μ,m,ν)  U(n,μ) U(m,ν)  +  ...      (1)

Phonon Software is limited to harmonic interaction, although some anharmonic interaction could be handled by it.

The equation of atomic motion of vibrating atoms are exactly solvable, and the solution is provided by the eigenvalue problem

ω2(k,j)  E(k,j) =  D(k)  E(k,j)       (2)

Here, D(k) is defined later by Eq.(4) in direct method. D(k) is an hermitian matrix of 3r x 3r dimensions, where r is the number of atoms in the primitive unit cell. The eigenvalue equation provides the eigenfrequencies square ω2(k,j), and eigenvectors E(k,j), also called polarization vectors. The eigenvectors satisfy the orthonormality conditions. For a given wave vector k, there exist 3r values of phonon frequencies, each of which describes a normal mode vibration. To each frequency ω2(k,j) is associated an eigenvector E(k,j) having 3r components, as many as the number of degree of freedom in the primitive unit cell. The eigenvector controls the participation and displacements of atoms in the vibrational mode. The eigenvector components are complex numbers.