ORIGINAL PAPER M. Gambhir  M.T. Dove  V. Heine Rigid unit modes and dynamic disorder: SiO 2 cristobalite and quartz Received: 10 March 1998 / Revised, accepted 15 January 1999 Abstract The high temperature (b) phases of SiO 2 cristobalite and quartz are studied by performing mo- lecular dynamics simulations using a model which al- lows easy analysis of tetrahedral motions. The dynamic nature of the disordered high-temperature phase of cristobalite is attributed to rigid unit mode (RUM) ex- citations, and it is found that the entire spectrum of RUMs is responsible for the disorder. Comparisons of the results of b-cristobalite with those of b-quartz lead to the conclusion that framework structures with high de- grees of geometric ¯exibility, and hence many RUMs, are free to deform through cooperative tetrahedral ro- tations even in the limit of extremely large tetrahedral stinesses. 1 Introduction Many framework mineral structures consist of sti polyhedral units, made up of two or more atoms, such as SiO 4 tetrahedra. These sti units are loosely jointed with one another and the characteristic inter-polyhedral force constant (stiness) is often 25±100 times smaller than the intra-polyhedral stiness. It has been found that there exist a few motions of the system which only involve coherent rotations and/or translations of the polyhedra with zero distortion of the sti units. These motions are, therefore, of very low energies. They are called rigid unit modes (RUMs) and are determined geometrically by moving the constituent units without distortion (Dove et al. 1993, 1995; Dove 1997b; Hammonds et al. 1994). It should be stressed that the units are deformable and practically all phonon modes involve their deformation, but there are also motions that may be accomplished without deformation which are especially important because of the stiness of the polyhedra. In the structure of Fig. 1, showing the view down a h111i direction in the high-temperature b-phase of cristobalite, such motions occur when adjacent tetrahedra rotate in opposite di- rections through the same angle. The exploration of the RUM model has illuminated several phenomena, particularly the reasons for the widespread occurrence of displacive phase transitions in framework structures (Hammonds et al. 1996), the phase transition temperatures (Dove et al. 1995), nega- tive coecients of thermal expansion (Pryde et al. 1996, 1998; Welche et al. 1998), possible adsorption sites for catalyst ions in zeolites (Hammonds et al. 1997, 1998), the origins of thermal diuse X-ray and electron scat- tering patterns (Hammonds et al. 1996; Dove et al. 1996b) and, recently, a possible explanation of the low frequency dynamics in silica glasses (Dove et al. 1997b; Trachenko et al. 1998). It has been found that the number of geometrically allowed RUMs varies greatly from structure to structure (Hammonds et al. 1996). Even among the family of tetrahedral framework structures in which all corners of all tetrahedra are linked to other tetrahedra, there are great variations. For example, zeolites such as faujasite have very many RUMs (Hammonds et al., 1997, 1998), whereas the low-temperature a-phase of cristobalite has very few (Hammonds et al., 1996). The aim of the present work is to use this model to provide an understanding of the atomic structure of the high-temperature b-phase of SiO 2 cristobalite. Cristo- balite undergoes a displacive phase transition from the high-symmetry b-phase to the lower-symmetry a-phase on cooling below 548 K. The a-phase is well understood and the RUM instability at the X-point of the Brillouin zone which is responsible for the rotations and transla- tions from the b ! a forms has been identi®ed and documented (Swainson and Dove 1993a, b; Hammonds et al. 1996). Phys Chem Minerals (1999) 26: 484±495 Ó Springer-Verlag 1999 M. Gambhir  M.T. Dove (&) Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom V. Heine Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom