The calculation of phonon dispersion for crystalline solids with r atoms in a unit cell requires solving a 3r-dimensional eigenvalue problem. In this paper we propose a simplified approach to lattice dynamics which yields approximate analytical expressions and accurate numerical solutions to phonon dispersion without solving the eigenvalue problem. This is accomplished by making coordinate transformations to the normal modes of the isolated unit cell, which are extended over the entire crystal by Fourier transformations, so each phonon branch is labelled by the irreducible representations of the symmetry group of the unit cell from which the atomic displacements can be readily identified. The resulting dynamical matrix can be analyzed perturbatively, with the diagonal elements as the zeroth order matrix and the off-diagonal elements as the perturbation. The zeroth-order matrix provides approximate analytical expressions for the phonon dispersions, the first-order terms vanish, and the higher-order terms converge to the exact solutions. We describe the application of this method to a one-dimensional diatomic chain, graphene, and hexagonal closepacked zirconium. In all cases, the zeroth-order solution provides reasonable approximations, while the second-order solutions are close to the exact dispersion curves. This approach provides insight into the lattice dynamics of crystals, molecular solids and Jahn-Teller systems, while significantly reducing the computational cost.

Non-linear molecules undergo distortions when the orbital degeneracy of the highest occupied level is lifted by the Jahn-Teller effect. If such molecules or clusters of atoms are coupled to one another, the system may experience a cooperative Jahn-Teller effect (CJTE). In this paper, we describe a model of how the CJTE leads to the crystallization of the disordered phase. The model Hamiltonian is based on a normal mode decomposition of the clusters in order to maintain the symmetry labels. We take account of the electron-strain and the electron-phonon couplings and, by displacing the coordinates of the oscillators, obtain a term that explicitly couples the Jahn-Teller centers, enabling us to perform a mean-field analysis. The calculation of the free energy then becomes straightforward, and obtaining phase diagrams in various regimes follows from the minimization of this free energy. The results show that the character of the phase transition may change from strong to weak first order and even to second-order, depending on the coupling to the vibrational modes. Taken together, these results may serve as a paradigm for crystallization near the transition temperature, where the atoms tend to form clusters of icosahedral symmetry.

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