D. L. Price

The self-consistent treatment of correlations in the electron liquid recently given by Singwi et al. is applied to calculations of the lattice dynamics of alkali metals. With the Ashcroft form for the pseudopotential, in which the only parameter is the core radius, good agreement with the measured dispersion curves is obtained for the four metals for which such data exist. For Na and K the fitted values of this parameter are close to those derived from Fermi-surface and liquid-resistivity data; for Li and Rb the fitted values lie between those obtained from the other physical properties. Phonon lifetimes due to the electron-phonon interaction are calculated for K. Finally, the cohesive energy, lattice parameter, and compressibility are derived; agreement with the measured values of all three quantities is obtained with a not unreasonable adjustment of the Hartree energy. The relation between the compressibility sum rule and the long-wavelength limit of terms containing third and fourth powers of the electron-phonon matrix element is evaluated for the case of Na within the same framework.

We have measured the phonon-dispersion relations in symmetry directions for $\ensuremath{\alpha}\ensuremath{-}\mathrm{S}\mathrm{n}$ at 90 \ifmmode^\circ\else\textdegree\fi{}K and InSb at 300 \ifmmode^\circ\else\textdegree\fi{}K. A limited number of measurements were made at 240 \ifmmode^\circ\else\textdegree\fi{}K in $\ensuremath{\alpha}\ensuremath{-}\mathrm{S}\mathrm{n}$ to look for effects associated with the phase transition, but none were found. The two dispersion relations are qualitatively similar, the main differences arising from the lower symmetry and a slight general drop in frequency in going from $\ensuremath{\alpha}\ensuremath{-}\mathrm{S}\mathrm{n}$ to InSb. Systematic trends in the comparison with other group-IV and III-V semiconductors can be seen. We have found second-neighbor shell models which give a reasonable description of the dispersion relations but in which the parameters are poorly defined and apparently devoid of physical meaning. The microscopic implications of this fact are discussed.

Calculations of cohesive energy, pressure, and compressibility have been made for liquid Na at 393\ifmmode^\circ\else\textdegree\fi{}K. They are based on the Ashcroft form for the pseudopotential and the self-consistent dielectric function recently given by Singwi et al., which have been found to give good agreement with the measured phonon dispersion relations in crystalline Na, and a liquid strucutre derived from a recent molecular-dynamics calculation by Rahman. The theory is formulated in terms of a volume-dependent pair potential and additional volume-dependent terms. A model which considers only volume-independent pairwise interactions gives incorrect results for the cohesive energy and pressure. It also gives a different result for the compressibility from the full theory, owing to the neglect of certain three- and four-particle interactions which are considered in the full theory; this result is, however, in reasonable agreement with experiment. The full theory gives good agreement for the cohesive energy, pressure, and compressibility if and only if an adjustment is made to the Hartree energy, similar to that made in a previous calculation for the crystal.

We show how local-field corrections in solids may be treated by a very general factorization scheme for $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}}^{\ensuremath{'}})$ from which practically all existing models of dielectric screening and lattice dynamics may be derived as special cases, including the shell model, the breathing-shell model, and the bond-charge model, as well as generalizations of these models which result from the introduction of a "screening medium." The latter arise naturally in our formalism from a portion of $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}}^{\ensuremath{'}})$ which is purely diagonal. It is shown that the formalism also allows for charge-transfer and multipole effects. In this first paper we derive explicit expressions for the elements of $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\stackrel{\ensuremath{\rightarrow}}{\mathrm{G}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+{\stackrel{\ensuremath{\rightarrow}}{\mathrm{G}}}^{\ensuremath{'}})$ and its inverse and show that they have the correct analytic behavior as $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremath{\rightarrow}0$. For insulators of cubic and tetrahedral symmetry, explicit microscopic expressions are derived for the high-frequency dielectric constant ${\ensuremath{\epsilon}}_{\ensuremath{\infty}}$ thus realizing a generalization of the Lorentz-Lorenz formula, and for the local field produced by an applied field.