Phonons

Eckold, G.

doi:10.1107/97809553602060000638

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 266-293
https://doi.org/10.1107/97809553602060000638

Chapter 2.1. Phonons

G. Eckold^a ^*

^a Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany
Correspondence e-mail: geckold@gwdg.de

Footnotes

¹ The superscripts ^T and * are used to denote the transposed and the complex conjugate matrix, respectively.
² If for a given wavevector q the dynamical matrix exhibits degenerate eigenvalues, the most one can strictly infer from equation (2.1.3.2)

is that the eigenvector $[{\bf e}({\bf q}+{\bf g},j)]$ may be represented by some linear combination of those eigenvectors $[{\bf e}({\bf q},j')]$ that correspond to the same eigenvalue. One always can choose, however, an appropriate labelling of the degenerate phonon modes and appropriate phase factors for the eigenvectors in order to guarantee that the simple relation (2.1.3.3)

holds.
³ We use the Seitz notation for symmetry operations: $[{\bf S}]$ denotes a rigid rotation of the lattice, $[{\bf v}({\bf S}) ]$ is the corresponding vector of a fractional translation in the case of screw axes, glide planes etc. and $[{\bf x}(m)]$ is a lattice vector.
⁴ Since the rotation S uniquely defines the fractional translation $[{\bf v}({\bf S})]$ and since a lattice translation $[{\bf x}(m)]$ never changes the label of an atom within the primitive cell, the function $[F_{o}]$ depends only on S.
⁵ According to (2.1.3.10)

, the vector $[\{{\bf S}| {\bf v}({\bf S}) + {\bf x}(m)\}^{-1} \, {\bf r}_{K}^o - {\bf r}_\kappa ^o ]$ is always a lattice vector. Hence, the transformation matrix remains invariant when the wavevector is shifted by a reciprocal-lattice vector. If wavevectors within the first Brillouin zone are considered, g(q, S) is always zero. For wavevectors on the Brillouin-zone boundary, however, there may be symmetry operations like the inversion that transform q into another equivalent but not identical vector $[{\bf q}'={\bf q} +{\bf g} ]$ .
⁶ The choice of the left coset is arbitrary. We could also consider the right coset $[G({\bf q})\circ \{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\}]$ . The same enlarged group and the same representations are obtained.
⁷ $[{\bf T}^+]$ denotes the Hermitian conjugate matrix.
⁸ Note that the lower half of the Hermitian matrix in each case is omitted for clarity.
⁹ Note that we always choose the positive root $[\omega({\bf q})=+\sqrt{\omega^{2} ({\bf q})} ]$ for the phonon frequency.
¹⁰ Accidental degeneracies that are due to the specific strength of interatomic forces are not considered here.

International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 266-293
https://doi.org/10.1107/97809553602060000638