  We consider the thermal expansion, change of 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 ${\boldsymbol
\Theta}_i \equiv ({\boldsymbol \theta}_i - (\nabla \times {\bf u}_i)/2)$, where
${\boldsymbol \theta_i} $ are the orthogonal rotation angles of the rigid unit
$i$ and ${\bf u}_i$ is its displacement. Given such new 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.
