Origin of the negative thermal expansion in and

The negative thermal expansion recently observed over a wide range of temperatures in may be attributed to the existence of low-frequency phonon modes which can propagate with no distortions of the tetrahedra and octahedra, the so-called `rigid unit modes'. Using methods developed for the study of similar modes in silicates we have located the surfaces of these modes in wave-vector space. The rigid-unit mode interpretation accounts for the weak effect of the 430 K structural phase transition on the negative thermal expansion, provided that the disordered phase does not involve formation of and complexes. On the other hand, the crystal structure of the related material is cross braced by pairs of linked tetrahedra and is therefore significantly less flexible. In this case a qualitatively different mechanism may be responsible for the negative thermal expansion observed in .