Origin of the negative thermal expansion in ? and ?

The negative thermal expansion recently observed over a wide range of temperatures in 0953-8984/8/50/023/img13 may be attributed to the existence of low-frequency phonon modes which can propagate with no distortions of the 0953-8984/8/50/023/img14 tetrahedra and 0953-8984/8/50/023/img15 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 0953-8984/8/50/023/img16 and 0953-8984/8/50/023/img17 complexes. On the other hand, the crystal structure of the related material 0953-8984/8/50/023/img18 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 0953-8984/8/50/023/img18.