A theory is presented of the damping and frequency shift of phonons and of the ground-state energy corrections due to interactions between phonons in quantum crystals with singular forces. The technique begins with the adoption of a trial ground-state wave function of the Jastrow form, together with trial excited-state wave functions constructed to represent one-, two-, and three-phonon excitations. The Hamiltonian matrix in this restricted basis is diagonalized, and the basis is optimized by minimizing the lowest eigenvalue with respect to variational phonon parameters. Using a lowest-order cluster expansion, the unambiguous prescription is obtained that a specific effective potential, softened by the Jastrow correlation function, replaces everywhere the true potential in the existing self-consistent theory of phonon damping applicable to nonsingular forces. Close analogies are drawn with the correlated basis function treatment, of superfluid liquid helium.

Reduced pressure, volume, and temperature variables $P^{*}=\frac{{\sigma }^{\prime }{}_{}{}^3{}_{}{}^{}P}{{\epsilon }^{\prime }}$, $V^{*}=\frac{V}{N{\sigma }^{\prime 3}}$, and $T^{*}=\frac{\mathrm{kT}}{{\epsilon }^{\prime }}$, in which ${\sigma }^{\prime }$ is the atomic radius and ${\epsilon }^{\prime }$ is the heat of sublimation per atom at 0 °K, are introduced as the basis for a new law of corresponding states. This law of corresponding states is applied to all experimental $P\left(T\right)\mathrm{}$ vapor and sublimation pressure data for the group-I and -II elements. The $P^{*}\left(T^{*}\right)$ data for these elements fall on a common curve of the form $\ln P^{*}=\frac{a^{*}-b^{*}}{T^{*}}$, with an average deviation of about 40%, over ten decades in $P^{*}$ from ${10}^{-15\mathrm{}}$ to ${10}^{-5\mathrm{}}$. For group-$\mathrm{II}b$ elements alone the corresponding curve shows an average deviation of only 9.3% over almost seven decades in $P^{*}$. Expected extensions and limitations of this law of corresponding states are investigated. It is shown that the correspondence may extend to vapor- and sublimation-pressure curves of many substances over wide $P$ and $T$ ranges, but probably does not extend generally to other properties. This law of corresponding states is shown to have a theoretical basis in thermodynamics and the Clapeyron equation. Theoretical expressions for $\frac{d{P}^{*}}{d{T}^{*}}$ and, approximately, for $P^{*}\left(T^{*}\right)$ are derived which show correspondence and agree with experiment. For the case of heavier rare gases, the present law of corresponding states is approximately identical to the usual law of corresponding states for van der Waals substances based on potential parameters. These two laws of corresponding states are compared.

We study the classical lattice gas with hard cores, nearest-neighbor repulsive forces, and next-nearest-neighbor attractive forces. We find several thermodynamic phases (vapor, liquid, and solid) and unusually shaped critical curves, yet no critical point or triple point. We find and discuss discontinuities in $m\left(h\right)\mathrm{}$ in the analogous magnetic systems. The principal calculational tool is molecular field theory but accurate numerical solutions of the transfer matrix are also obtained as a check on some of the results. It is conjectured that the critical line separating vapor and liquid phases (bounded by the triple point on one end and the critical point on the other) exists in the exact solution of a lattice gas with structured, relatively long-ranged interactions, but cannot be found in the molecular field approximation.

A simple theory is given to explain the many different features of small-scale filaments under various experimental conditions. It is shown that the observed filaments are the consequences of moving foci. Under appropriate conditions, a focal spot can move in the forward direction with a speed faster than light. As a result, the light pulse emitted from the filament at the end of the nonlinear medium has a much shorter duration than the input pulse, and shows a characteristic spectral broadening. Because of its high intensity, a moving focal spot also leaves behind it a temporary track of field-induced refractive index. Light trailing after the focal spot can then be partially trapped in the track. Trapping over an appreciable distance is possible if the velocity of the forward-moving focal spot is not much higher than the light velocity.

Measurements of the longitudinal and transverse sound velocities in x-ray-oriented single crystals of hcp ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ (19.0-20.5 ${\mathrm{cm}}^3$/mole) and bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ (21.6 ${\mathrm{cm}}^3$/mole) are reported. These data are used to compute the respective second-order adiabatic elastic moduli. The moduli for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ imply an extremely soft transverse mode in the <101> directions and thus demonstrate that the low-temperature bcc-${}_{}{}^3{}_{}{}^{}\mathrm{He}$ specific-heat anomaly is due to an anomalous phonon spectrum. The Debye temperatures and volume compressibilities determined by the elastic moduli for both bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and hcp ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are in agreement with values determined by other methods.

It is shown how the method of thermodynamic Green's functions can be used to approximate the memory function associated with the equilibrium-fluctuation function $S_c(\overrightarrow{\mathrm{r}}-{\overrightarrow{\mathrm{r}}}^{\prime },\overrightarrow{\mathrm{p}}{\overrightarrow{\mathrm{p}}}^{\prime },t-t^{\prime })\equiv 〈\left[f\right(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t)-〈f(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\left)〉\right]\left[f\right({\overrightarrow{\mathrm{r}}}^{\prime }{\overrightarrow{\mathrm{p}}}^{\prime }{t}^{\prime })-〈f({\overrightarrow{\mathrm{r}}}^{\prime }{\overrightarrow{\mathrm{p}}}^{\prime }{t}^{\prime }\left)〉\right]〉$, where $f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)$ is the phase-space distribution operator. We obtain an approximation for the memory function for a gas, in the low-density limit, that is valid for all distances and times, satisfied various relevant symmetry conditions and sum rules, reduces for long times and distances to the Boltzmann collision operator, and gives results completely consistent with the conservation laws governing the system. We also indicate how these methods can be extended to treat other types of systems.

We continue the considerations of the first paper in this series by studying the time dependence of the spin-correlation functions in response to a step-function change in the external magnetic field. We find that these correlation functions exhibit nonergodic behavior.

The Kirkwood superposition approximation (KSA), which approximates three-particle distribution functions by symmetrical products of two-particle distribution functions, is examined quantitatively for ${\mathrm{He}}^3$-${\mathrm{He}}^4$ solutions in the zero-concentration limit. Numerical tests are devised for this study, which as a result indicates that errors caused by the KSA are not significant except for quantities involving small differences between large numbers, such as the chemical potential of ${\mathrm{He}}^3$ in solution. This note serves as an appendix to a series of earlier publications on a microscopic theory for dilute ${\mathrm{He}}^3$-${\mathrm{He}}^4$ solutions.

Bender's theory of frequency shifts in spin exchange is applied to two illustrations of coherence transfer. The effects of coherence transfer include changes in the shift-to-broadening ratio and a frequency shift which does not reverse with polarization.

We reanalyze in greater generality the recently predicted phenomena associated with the liquid-solid phase transition.

Far-infrared radiation generated from the difference frequency between two temperature-tuned ruby lasers operated on the $R_1$ and $R_2$ transitions has been observed. This radiation is continuously tunable over the frequency range 20-38 ${\mathrm{cm}}^{-1\mathrm{}}$. Lithium niobate was used as a mixing crystal. The expected frequencies of the far-infrared radiation were measured using a Fabry-Perot interferometer. The phase-matching conditions were also verified.

Electron correlation in three-electron atoms and ions is studied in two important regions of configuration space: the vicinity of the nucleus, and the vicinity of three-electron coalescence away from the nucleus. In both cases, weakly singular logarithmic behavior of the wave function obtains. A study is made of the nodal characteristics of the wave function near the nucleus. Model systems are introduced which reflect some of the salient features present in the actual atomic system. Suggestions are made as to the utility of the present type of analysis with special attention to the recent transcorrelated approach to the Schrödinger equation.

Approximations to the Hartree-Fock (HF) exchange energy density have been derived directly from the HF exchange energy density. The approximations involving two free parameters take the form of a Kohn-Sham-Gaspar (KSG) exchange energy density plus inhomogeneity corrections. The approximation including lowest-order inhomogeneity corrections only is equivalent to the approximation recently proposed by Herman, Van Dyke, and Ortenburger. An approximation including the next-higher-order inhomogeneity corrections with no additional parameters is also derived. The approximations have been evaluated by direct comparison with the actual HF exchange energy density for Ar and ${\mathrm{Cu}}^{+}$. The optimum choices of parameters in the case of the approximation including lowest-order inhomogeneity corrections are in excellent agreement with the corresponding values found by Herman et al. and are independent of the atomic system considered. The approximation including higher-order inhomogeneity corrections did not significantly improve the approximation including lowest-order inhomogeneity corrections. However, the higher-order approximation is of interest in that it does lead to an angular-momentum-dependent exchange potential.

Using a previously developed unified theory of spectral line broadening, the effects of time ordering over the complete line profile are investigated. The results of calculations for the Lyman-$\alpha$ line are presented.