A local-molecular-field description for the ground-state properties of the quantum solids is presented. The dynamical behavior of atoms contributing to the local field, which acts on an arbitrary pair of test particles, is incorporated by decoupling the pair correlations between these field atoms. The energy, pressure, compressibility, single-particle-distribution function, and the root-mean-square atomic deviations about the equilibrium lattice sites are calculated for ${\mathrm{H}}_2$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ over the volume range $5<V\le 24.5$ ${\mathrm{cm}}^3$/mole. The results are in close agreement with existing Monte Carlo calculations wherever comparisons are possible. At very high pressures the results agree with simplified descriptions which depend on negligible overlap of the system wave function between neighboring lattice sites.

A renormalization-group method of recursion relations is developed and applied to the general Landau-Ginzburg problem in one-dimensional statistical mechanics. In special cases, simple closed-form expressions are obtained. In one case, the group recursion relations completely determine the free energy, whereas in another the relations are barely stronger than dimensional analysis.

It is shown that a large class of approximations in statistical thermodynamics that are based on the free-energy variational principle for the density operator yield expressions for the macroscopic quantities of the system that are consistent from both the statistical-mechanical and thermodynamical points of view. This corrects some erroneous statements in the literature.

A time-dependent Landau theory is proposed for the smectic-$A$-nematic phase transition. It is shown that the nematic viscosity coefficients exhibit a pretransition effect similar to that for the elastic constants.

A Monte Carlo method is used to calculate the excitation spectrum of He II. The variational method and the trial wave function of Feynman and Cohen (FC) are used. No approximations, such as the Kirkwood superposition approximation, or estimates are employed. Important terms in the FC calculation are found to be in error by as much as 10-20% but the final answer, $\Delta =11.2\mathrm{}$ K, at the equilibrium density, is hardly different from the FC value $\Delta =11.5\mathrm{}$ K. Results of calculations for densities up to 1.2 times the equilibrium density (i.e., the freezing density) are also presented; we find that the FC wave function does not lead to the expected pressure dependence of $\Delta$. Some modifications to the usual Monte Carlo methods, necessary for this calculation, are discussed.

In a previous paper the present authors have formulated a generalized theory of the dielectric function of an interacting electron liquid which was found to yield an almost exact satisfaction of the compressibility sum rule and fair results of the pair-correlation function $g\left(r\right)\mathrm{}$. In the present paper we attempt to apply this generalized procedure to the case of a classical one-component plasma. Calculations are performed for the various physical quantities such as the pair-correlation function, compressibility, correlation energy, etc., and a substantial improvement upon the Debye-Huckel theory (classical random-phase approximation) is observed. The present study gives a remarkable improvement in the satisfaction of the compressibility sum rule in comparison to both the theory of Singwi et al. and of Ichimaru. The values of $g\left(r\right)\mathrm{}$ show a close resemblance to the previous estimates.

The utility of digital-correlation spectroscopy for investigating structural relaxation in liquids on time scales from 1 μsec to 1 sec is established. A digital correlator was used to analyze the Rayleigh scattering of light from undercooled glycerol. The correlation function obtained is related to the isothermal structural relaxation dynamics of the liquid. The data were found to join smoothly with those obtained ultrasonically at higher temperatures.

We prove the existence and discuss the uniqueness of the solution of the equations proposed by Gaudin for the free energy of the $\Delta \ge 1$ Heisenberg-Ising ring. A numerical technique for solving the equations is outlined, and some results of the numerical work are displayed. Comparisons are made with the previous numerical study of Bonner and Fisher with favorable results, giving support to Gaudin's formulation. The finite-temperature excitations of the system are derived, and some sample dispersion curves are given.

The second virial coefficients of two-dimensional systems of ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ have been calculated assuming that the atoms interact via the Beck potential. The virial coefficient and its contribution to the specific heat are almost the same as calculated previously with a Lennard-Jones potential. Therefore small discrepancies between experimental data and our previous calculation are ascribed to effects of the substrate.

The pressure dependence of the parallel three-phonon process is investigated and the expected pressure dependence of the wide-angle phonon scattering rate is discussed.

The ground state of a one-dimensional many-boson system with a repulsive $\delta$-function potential is reexamined in the weak-coupling limit. It is shown that addition of the third-order variational correction to a previously obtained ground-state energy series leads to an improved agreement with the exact numerical results evaluated by Lieb and Liniger.

The structure of the second-order matrix element describing two-photon emission of x rays is discussed. The point of departure is the gauge invariance of a general Hamiltonian in the formulation by Sachs and Austern.

Relaxation rates and frequency shifts of the 0-0 hyperfine transition in the ground state of ${\mathrm{Rb}}^{85}$ are studied in some detail at 3.03 GHz. Results on the relaxation rates of the populations and of the coherence introduced in the system by a short microwave pulse are given. Physical parameters are extracted from the data and are interpreted in terms of wall interactions, buffer-gas interactions, and spin-exchange interactions. The data are analyzed in terms of simple theoretical models and are also discussed in relation to published results on other isotopes or alkali-metal atoms.

Separate relativistic Hartree-Fock solutions for atoms in their initial and final states are used for calculating the radiative decay of a $K$-vacancy state. The matrix-element calculations include the exchange effects of the nonzero overlap of wave functions from different subshells. Results are presented for the filling of the vacancy by the $p$ electrons for a set of elements from $Z=10\mathrm{}\mathrm{to}98$. The Hartree-Fock theory is also used for calculating the rates of production of multiple-vacancy states in the decay process. In contrast to earlier single-potential calculations, the calculated values of ratios of the rates of the $K$ x-ray components are in good agreement with the experimental values.

An equation for the transport of resonance radiation at low and intermediate pressures is derived and shown to be in reasonable agreement with an experiment conducted in parallel with the theoretical development. Both dealt with the transport of 1048-Å resonance photons in argon in the pressure range 0.002-10 Torr. At pressures greater than 1 Torr where collisions are frequent, the present transport equation is equivalent to that of Holstein and Biberman. In the pressure region from 0.002 to 0.5 Torr there is substantial disagreement (as much as a factor of 3) between the present work and theory that assumes complete redistribution of frequency, i.e., takes the emission profile to be equal to the absorption profile.

Widths of $M$-level vacancy states in atoms with $29\le Z\le 48$ have been measured by x-ray photoelectron spectroscopy. The results are consistently smaller, sometimes by a factor of ≥3, than expected from previous theory, and do not show the pronounced effect of predicted "super-Coster-Kronig" transitions. The Cu and Zn $L_3-M_{4,5}{M}_{4,5}$ Auger and $M_{4,5}$ photoelectron spectra have been measured; the Auger spectra exhibit characteristics of free-atom spectra and do not reflect the structure of the band from which the $M$ electrons originate. Drawing upon this and other evidence suggesting that atoms in solids can behave as quasifree when undergoing radiationless transitions, free-atom $M$ Auger and Coster-Kronig energies have been computed. These are smaller than measured energies because the effect of extra-atomic relaxation is absent. $M$-level widths were calculated with these energies, in a neutral-atom potential. Good agreement with experiment is attained.

The third-order Stark effect is re-examined using an operator method presented by Schwinger. By observing an identity for the perturbed part of the Hamiltonian and by using commutation relations, we are able to obtain the energy spectra of hydrogenic atoms in a constant electric field without solving the Schrödinger equation. Our result confirms that of Doi and El-Sherbini.

The general formulas available in the literature for calculating the time average of powers of the orbital radius of the electron in hydrogenic atoms or ions, $〈r^s〉$, are of a degree about twice that of the resulting explicit expressions. In this paper, general formulas are given which are of the right degree and which are written in a suitable form for direct calculation of explicit formulas for specific values of s. Explicit expressions for $〈r^s〉$ are given for $-8\mathrm{}\le s\le 5$ It is shown that formulas for positive and negative powers are connected by the formula $〈r^{-(p+2\mathrm{})\mathrm{}}〉={\left(\frac{2Z}{n{a}_0}\right)}^{2p+1}\left[\frac{\mathrm{}(2l-p)!}{\mathrm{}(2l+p+1)!}\right]〈r^{p-1\mathrm{}}〉$, where $p$ is any integer or zero.

The intensity ratios of the multiplet-split $\mathrm{N}1s$ peak of NO and $\mathrm{O}1s$ peak of ${\mathrm{O}}_2$ in the x-ray photoemission spectra of these paramagnetic molecules were found to exceed the simple multiplet ratios $\frac{\mathrm{}(S+1\mathrm{})\mathrm{}}{S}$. Hartree-Fock calculations were performed on the ground state and the $1s$-hole states in each case, and total overlap probabilities for the passive orbitals were computed. Positive deviations from the multiplet ratios were obtained, but they were smaller than the experimental deviations. The $\mathrm{NO}(\mathrm{N}\overline{1s},\frac{{}_{}{}^3{}_{}{}^{}\Pi }{{}_{}{}^1{}_{}{}^{}\Pi })$ ratios were 3.43 (8) (expt.), 3.09 (calc.), and 3 (multiplet), while for ${\mathrm{O}}_2(\mathrm{O}\overline{1s},\frac{{}_{}{}^4{}_{}{}^{}\Sigma }{{}_{}{}^2{}_{}{}^{}\Sigma })$ the ratios were 2.47(3) (expt.), 2.28 (calc.), and 2 (multiplet). Thus incomplete overlap accounts for some, but not all, of the observed deviation. Another consequence of incomplete overlap is multiple excitation during photoemission. The total probabilities for multiple excitation were computed as $0.25(\mathrm{NO},\mathrm{N}\overline{1s})$, $0.28(\mathrm{NO},\mathrm{O}\overline{1s})$, and $0.42({\mathrm{O}}_2,\mathrm{O}\overline{1s})$, in agreement with the observation by Carlson et al. of large multiple-excitation probabilities.