Logo of Phonon Home

Anharmonic Phonons

Anharmonic Potential

In conventional lattice dynamic theory the potential energy V between crystal atoms is expanded over the instanteneous displacements U(n,μ) of the atom nuclei. Since the 2-d order term is names harmonic contribution, all the remaining terms of N-order, where N=3, 4, 5, 6, ⋅⋅  are called anharmonic force constants.

V = V(.... U ( n , μ),...),
V =V(....0,...) +  (1/2!)Σ n ,μ, m ,ν Φ( n ,μ, m ,ν)  U ( n ,μ)  U ( m ,ν)
                      +  (1/3!)Σ n ,μ, m ,ν, l ,λ Φ(3)( n ,μ, m ,ν, l ,λ)  U ( n ,μ)  U ( m ,ν)  U ( l ,λ) 
                      + (1/4!)Σ n ,μ, m ,ν, l ,λ, j ,ξ Φ(4)( n ,μ, m ,ν, l ,λ, j ,ξ)  U ( n ,μ)  U ( m ,ν)  U ( l ,λ)  U ( j ,ξ) 
                      + Φ (5) +  Φ (6) + ⋅⋅⋅⋅⋅ + ⋅⋅⋅⋅⋅   etc.                    (6)

where Φ(5)  and Φ(5)  are alredy schematically written.

It should be emphesised that there is rather vast number of anharmonic force constants. For interesting systems it may be of order of 104 to 106 coefficients, which numerical values should be computed. Symmetry can reduce this number. But in practice only anharmonic effects arising from the Φ(3)(n,μ,m,ν,l,λ) terms are computed. Consequently, weak anharmonic effects studied in not too complex crystals, preferable of simple cubic structure, can only be analyzed within reasonable computer time.

Indirectly PhononA Anharmonic Software, takes into account many anharmonic terms. It creates displacements of atoms of supercell with amplitudes comparable to real atom displacements at a given temperature. Then, it performs computation of phonons according to loop indicated in Ab initio phonons.