Rigid-unit modes in tetrahedral crystals.

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 oxygen ions. The wavevectors 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 such as in β-quartz does a surface of RUMs appear, if inversion symmetry is lacking. The occurrence of planes and bending surfaces, and straight and bent lines, is discussed. Explicit calculations are performed for five polymorphs of SiO(2) crystals.