Rigid unit modes in tetrahedral crystals
Abstract
The 'rigid unit mode' (RUM) model requires unit blocks, in our case tetrahedra of SiO4 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 SiO2 crystals.
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