The dynamics of disordered lattices

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Abstract
This article reviews the theory of atomic vibrations in disordered solids, ranging from almost perfect crystal lattices to glassy materials where geometrical regularity is entirely absent. As a preliminary, the basic notations and equations of conventional lattice dynamics are briefly outlined; the essential equivalence of quantal and classical formulations within the harmonic approximation is indicated, as in the considerable simplification which occurs for periodic systems. The classical time-independent Green's function formalism is then introduced and applied to lattices with isolated defects. Extension of the Green's function method to more grossly disordered systems is described, together with other relevant analytical techniques, such as the phase theory and the Anderson approach. Next, the time-independent numerical method, based on the negative eigenvalue theorem, is introduced and applied to two-component mass disordered lattices; here the numerical calculations are related, where possible, to existing analytical results. The most recent applications of the negative eigenvalue approach concern glasses, polymers and orientationally disordered crystals, and the calculated properties of these systems are reviewed in some detail. Finally, attention is drawn to several areas of the vibrational problem which seem ripe for further detailed study.