Thermal vibrations and a lattice mode in calcite and sodium nitrate

The observed direction of thermal motion of the oxygen atoms in calcite can be explained if the environment of the calcium atom is taken into consideration as well as the CO₃ group. Rotation of Ca octahedra accompanied by an alternate steepening and flattening which retains the point-symmetry and the Ca-O bond-length unchanged is associated with a screw-like movement of CO₃ in which rotation in its own plane is geared to translation perpendicular to its plane. Concerted movements of all atoms in the structure satisfying these requirements can be described in terms of a single-oscillating parameter; they represent a single lattice mode. They are predictable using the assumption that small departures, in the `static' structure, of particular position parameters from ideal values indicate weaknesses of a kind that allow a large amplitude to vibrations in which the same parameters oscillate about their `static' value. Comparison of CaCo₃ and the isomorphous NaNO₃ with LaAlO₃ and LiNbO₃, which have very similar formal geometry but great differences in relative bond strength, helps to illustrate the principle involved.