Displacement Autocorrelation Function Research Articles

Crystallization and Instabilities in Highly Anharmonic Crystals

A unified treatment of crystalline order and instabilities in highly anharmonic crystals at all temperatures is presented. This treatment is based on the study of singularities in the atomic-displacement correlation function or its Fourier transform (structure factor). As a result, it is rigorously shown that in the thermodynamic limit the mean-square fluctuations of the equilibrium position of a lattice particle are infinite in one and two dimensions at nonzero temperatures and in one dimension at zero temperature. This result, which is proved for an interacting many-body system without assuming the harmonic approximation, is obtained by using an exact Dyson equation for the displacement-response function. At finite temperatures, the demonstration of similar instabilities in a variety of condensed many-body systems of one and two dimensions is usually based on inequalities originally due to Bogoliubov. Since an analogous inequality can readily be extracted from the Dyson equation of the present approach, our method allows the extension of these results in anharmonic crystals to zero temperature. Finally, it is shown that additional dynamical information in this Dyson equality can be used to derive the relationship between elastic anomalies and sound absorption in the vicinity of critical points from the anomalous increase of the second derivative of the displacement-autocorrelation function.

Dynamical motion of a substitutional impurity in a crystal within harmonic approximations

The displacement autocorrelation function and its Fourier transform of a substitutional impurity atom in a crystal lattice are calculated for a specific model of a crystal taking into account the change in force constants as well as the change in mass. The results are expressed in terms of the frequency spectrum of the pure crystal and the perturbation parameters characterizing the effect of the impurity. As an application of the calculations the Mössbauer one-phonon absorption cross section is calculated by supposing that only the impurity is Mössbauer-active. Particular attention is paid to the case in which a sharp resonant mode appears in the low frequency region. Using the Debye model, a simple expression is obtained for the ratio of the peak value of the one-phonon line to that of the zero-phonon line.

Thermodynamic Green's Function Methods in Neutron Scattering by Crystals

Formulas were derived for the transition probabilities per unit time for both inelastic coherent scattering of neutrons by crystals and resonant emission of photons and neutrons by nuclei bound in crystals, without making the assumption that the crystal is hamnonic. In deriving these transition probabilities, the analytic structure of thermodynamic correlation or Green's functions, considered as functions of complex temperatures and times, was developed and used. In particu1ar a spectral form was found for the phonon Green's function. Only one assumption was made about the crystal, namely that the displacement of the nuclei due to the forces exerted by the neutron in scattering are linear functions of these forces. This led to an evaluation of the transition probabilities in terms of the exact thermodynamic displacement autocorrelation function. This evaluation obeyed the detailed balancing condition, and Placzek's sum rule. A consequence of this evaluation is that the widths of the one-phonon peaks in the neutron scattering are exactly equal to the widths of the corresponding phonon states of the crystal. (auth)