Geometrical origin and theory of negative thermal expansion in framework structures

By framework structures are meant materials consisting of relatively stiff units such as octahedra or tetrahedra, joined by shared oxygen (or other) atoms at the corners. Examples are ZrW2O8 and many aluminosilicates, Rigid rotation of the units often gives a reduction of the volume or of some lattice constant as a purely geometrical effect. The theory of this effect is developed and shown to give a negative contribution to the thermal expansion coefficient. This is in addition to the usual positive contribution from anharmonicity of the interatomic forces. The negative effect varies through the phonon spectrum, being strongest for low frequencies, but the sign of the temperature coefficient may be reversed above a soft mode phase transition.