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This paper describes a computational method for the determination of all possible phonon modes in framework crystal structures that leave the fundamental structural units (tetrahedra and octahedra) undistorted. Such rigid-unit modes (RUMs) are prime candidates as soft modes for displacive phase transitions, such as in the perovskite structure, and this computational method can be used to rationalize the phase transitions in any framework structure. The method has been programmed for general use. The RUM approach is illustrated by consideration of the perovskite, quartz and cristobalite structures.

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Landau free-energy expansions are commonly used to describe systems undergoing structural phase transitions. The Landau free energy of a model solid with an anharmonic potential (often a double well) at each site and mean-field-like inter-site coupling has been calculated both analytically and from molecular dynamics simulation.The calculated free-energy function is not well described by a simple polynomial in the order parameter. This result is not due to critical fluctuations in the Ginzburg interval. If, however, such a polynomial is used the coefficient of the fourth-order term is found to be highly temperature dependent. For a certain range of model parameters this coefficient is small relative to that of the second-order term.These observations help to explain the occurrence of 'non-critical, non-standard' values of the exponent / 3 in the variation of the order parameter, X, with temperature:. f a (T, -T)O.More importantly they also help to explain why so many natural systems behave in a tricritical-type manner, with /3 = t .

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