Theory of the structure and lattice dynamics of incommensurate solids: A classical treatment

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Abstract
A classical theory of the static strains and lattice dynamics for solids possessing incommensurate structures is developed using a spatial Fourier representation of the static strains and a space-time Fourier representation of the dynamical modes. Although the theory is presented for one-dimensional systems, the general technique is not restricted to such systems and can be applied in a straightforward manner to problems in two or three dimensions. A set of coupled, nonlinear equations is derived for the amplitudes and phases of the strains and a controllable sequence of approximations is described for finding the solutions to these equations. Excellent results are obtained using only low-order terms in various truncated expansions even for those cases where the domain walls are quite narrow. It is shown that provided the amplitudes of the dynamical oscillations are small enough, normal modes do exist even in the limit of large static strains. However, these modes are not characterized by a single wave vector but rather have important contributions from density fluctuations of various wavelengths. Strong evidence is given for the energy at zero kelvin of the one-dimensional system being a nonanalytic function of the mean lattice spacing of the type described by Aubry. This means that the pressure is a discontinuous function of the mean spacing with discontinuities existing at rational values of this spacing. The sizes of these discontinuities decrease rapidly as the order of the commensurate state increases. Each commensurate state would thus exist over a finite pressure range and the system will exhibit a "devil's-stair" behavior at zero kelvin.