Rigid unit modes in tetrahedral crystals

TL;DR

This paper analytically determines the wave-vectors of rigid unit modes in tetrahedral crystals like SiO_2, revealing how lattice symmetry influences the RUM surfaces or lines in reciprocal space.

Contribution

It provides an analytical framework for identifying RUM wave-vectors in tetrahedral crystals, highlighting the role of inversion symmetry in their reciprocal space structures.

Findings

Lattices with inversion symmetry have surfaces of RUMs in reciprocal space.

Lattices without inversion symmetry typically have lines of RUMs.

Explicit calculations for five SiO_2 modifications illustrate the theoretical results.

Abstract

The 'rigid unit mode' (RUM) model requires unit blocks, in our case tetrahedra of SiO_4 groups, to be rigid within first order of the displacements of the O-ions. The wave-vectors of the lattice vibrations, which obey this rigidity, are determined analytically. Lattices with inversion symmetry yield generically surfaces of RUMs in reciprocal space, whereas lattices without this symmetry yield generically lines of RUMs. Only in exceptional cases as in beta-quartz a surface of RUMs appears, if inversion symmetry is lacking. The occurence of planes and bending surfaces, straight and bent lines is discussed. Explicit calculations are performed for five modifications of SiO_2 crystals.