Elastic properties of a class of solids with negative thermal expansion

We consider the thermal expansion, change in sound velocity with pressure and temperature, and the Poisson ratio of lattices which have rigid units (polyhedra very large stiffness to change in bond-length and to bond-angle variations) connected to other such units through relatively compressible polyhedra. We show that in such a model, the potential energy for rotations of the rigid units can occur only as a function of the combination Theta(i) equivalent to [theta(i)-(del x u(i))/2], where theta(i) are the orthogonal rotation angles of the rigid unit i and u(i) is its displacement. Given such invariants in the theory of elasticity and the hierarchy of force constants of the model, a negative thermal-expansion coefficient as well as a decrease in the elastic constants of the solid with temperature and pressure is shown to follow. These are consistent with the observations.