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## Rigid Unit Modes
## Additional recommended knowledge## The interest in Rigid Unit ModesThe idea of Rigid Unit Modes was developed for crystalline materials to enable an understanding of the origin of displacive phase transitions in materials such as silicates, which can be described as infinite three-dimensional networks of corner-lined SiO The original work in silicates showed that many of the phase transitions in silicates could be understood in terms of soft modes that are RUMs. After the original work on displacive phase transitions, the RUM model was also applied to understanding the nature of the disordered high-temperature phases of materials such as cristobalite, the dynamics and localised structural distortions in zeolites, and negative thermal expansion. ## Why Rigid Unit Modes can existThe simplest way to understand the origin of RUMs is to consider the balance between the numbers of constraints and degrees of freedom of the network, an engineering analysis that dates back to James Clerk Maxwell and which was introduced to amorphous materials by Jim Phillips and Mike Thorpe. If the number of constraints exceeds the number of degrees of freedom, the structure will be rigid. On the other hand, if the number of degrees of freedom exceeds the number of constraints, the structure will be floppy. For a structure that consists of corner-linked tetrahedra (such as the SiO (Note that we can get an identical result by considering the atoms to be the basic units. There are 5 atoms in the structural tetrahedron, but 4 of there are shared by two tetrahedra, so that there are 3 + 4*3/2 = 9 degrees of freedom per tetrahedron. The number of constraints to hold together such a tetrahedron is 9 (4 distances and 5 angles)). What this balance means is that a structure composed of structural tetrahedra joined at corners is exactly on the border between being rigid and floppy. What appears to happen is that symmetry reduces the number of constraints so that structures such as quartz and cristobalite are slightly floppy and thus support some RUMs. The above analysis can be applied to any network structure composed of polyhedral groups of atoms. One example is the perovskite family of structures, which consist of corner linked TiO ## References- Kenton D. Hammonds, Martin T. Dove, Andrew P. Giddy, Volker Heine, and Björn Winkler. Rigid-unit phonon modes and structural phase transitions in framework silicates.
- Martin T. Dove. Theory of displacive phase transitions in minerals.
Categories: Crystallography | Materials science |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Rigid_Unit_Modes". A list of authors is available in Wikipedia. |