Direct method

PHONON Software

An ab initio program optimizes the structure of the crystalline supercell within constraints imposed by a crystallographic space group. In optimized configuration the forces acting on all atoms of the supercell vanish. Displacing one atom from its equilibium position, one generates non-zero Hellmann-Feynman forces acting on all atoms of the supercell.

The lattice dynamics requires to know the force constants, which are the second derivatives of the potential energy. In the direct method one calculates the force constants from the Hellmann-Feynman forces. The Hellmann-Feynman forces, in turn, are found when a single atom is displaced. In this way all force constants of the supercell can be found.

To obtain the reliable phonon dispersion relations the supercell diameter should be about 8 - 10 angsterms. The magnitude of the force constants beyond that distnce is usually negligible, and in such a case the phonon dispersion curve can be calculated exactly. But even if the supercell size is small, exact phonon frequencies can be obtained for the wave vectors commensurate with the supercell size. Then, the phonon dispersion curves are the symmetry controlled interpolations between the exact points.

For high-symmetry supercells treated by an ab initio program, Phonon calculates the phonon dispersion curves within minutes.

Within the direct method approach and according to the symmetry imposed by the crystal space group Phonon calculates the following items:

Required atomic displacements for generating the Hellmann-Feynman forces
Wave vectors at which exact phonon frequencies can be calculated
Neighbor list of atoms in the supercell
Symmetry of force constant's matrices
Number of independent parameters for each force constant matrix
Modification of the force constants due to translational-rotational invariances
Phonon dispersion relations along any line of reciprocal space
Form factors of phonon dispersion relations
LO/TO splitting from known effective charges and dielectric constant
Polarization vectors for any mode
Irreducible representations of all phonon modes at the Brillouin zone center
Total density of phonon states
Partial density of phonon states for each degree of freedom and each atom
Thermodynamic functions: internal energy, free energy, entropy, heat capacity
Thermal displacement tensors (Debey-Waller fator)
Dynamical structure factor giving intensity of coherently scattering neutrons i different Brillouin zones.
Doubly differential incoherent inelastic scattering cross section on monocrystals and polycrystals, including multi-phonon processes

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