Phonon dispersion dynamic matrix

In order to determine the phonon dispersion of CuZn and FeaNi we made use of an expanded tight binding theory from Varma and Weber . In the framework of a second order perturbation theory the dynamical matrix splits in two parts. The short range part can be treated by a force constant model, while the T>2 arising from second order perturbation theory is given by... 

The expressions used in calculating the properties referred to above from these derivatives are discussed in greater detail in Reference 9. For more detailed discussions of the calculation of phonon dispersion curves from the second derivative or dynamical matrix W, the reader should consult References 41 and 42. Finally, we note that by the term lattice stability we refer to the equilibrium conditions both for the atoms within the unit cell, and for the unit cell as a whole. The former are available from the gradient vector g, while the latter are described in terms of the six components ei- ee, which define the strain matrix e, where... 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

This difficulty could be avoided by applying linear response theory, which is widely used in solid-state physics to determine directly the dynamic matrix, polarization, and frequency-dependent dielectric functions, as well as phonon dispersion curves in the harmonic approximation. This method has the great advantage that it requires only a band structure at the equilibrium geometry of the solid (chain), i.e., one does not have to determine a potential hypersurface. However, since this theory has not yet been applied to polymers and involves a rather complicated formalism, we cannot enter into details here but refer the reader to standard solid-state physical works and an application to a simple solid (Si). ... 

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