In this paper a many-variable form of the Mori continued-fraction technique is employed to investigate the relaxation function of simple liquids. A general relationship between a truncated-continued-fraction expression and the Zwanzig and Nossal variational result for the relaxation function is presented. In addition, a specific example of this relationship is explored to show the connection between the eigenvalues found from high-frequency hydrodynamic equations and those obtained from a variational method. The structure of the relaxation function in the low-wave-vector limit is examined in considerable detail, and some comments are made concerning the applicability of the variational method in the low-wave-vector and high-frequency regime.

Single-particle wave functions and energies are calculated for ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ adsorbed on an argon-plated substrate. Because of the two-dimensional periodic potential, the energy spectrum has a band structure. The heat capacities of submonolayer films are calculated for both isotopes taking account of the statistics but excluding He-He interactions. Substantial band-structure effects are found. The heat capacities are compared with available experimental results. Posssible reasons for the gross disagreement are discussed and the prospects for a more favorable comparison with other feasible experimental work are considered briefly.

Phase shifts and their weighted sums over angular momentum states have been obtained for the static screened Coulomb potential (SSCP). Results are given for both an attractive and a repulsive SSCP. A procedure is given for obtaining the phase shifts which uses direct numerical integration near the origin and the first- and second-order WKB approximation at larger distances. At high energy a simple asymptotic fit to the weighted phase-shift sums is obtained. The results are applied to the calculation of the second virial coefficient of plasmas that have no bound states and of those that do have bound states. Very good agreement with the Wigner-Kirkwood expansion is found for a Boltzmann gas interacting through a repulsive SSCP, except that at high temperature, in agreement with DeWitt, the Wigner-Kirkwood expansion is shown to diverge for the SSCP.

The classical and first quantum correction terms in a high-energy expansion of the viscosity cross section $Q^{\mathrm{}\left(2\right)\mathrm{}}$ for a Boltzmann gas of hard spheres is derived. The first correction is found to be proportional to ${\left(\frac{1}{k\sigma }\right)}^{\frac{4}{3}}$, which is a term nonanalytic in $\hslash$ (i.e., ${\hslash }^{\frac{4}{3}}$), and results from scattering near the edge of the sphere. A bound is established showing the remainder of the asymptotic series to be of $O\left[\frac{\ln \left(k\sigma \right)}{{\left(k\sigma \right)}^2}\right]$. This asymptotic formula is compared with calculations based on the exact phase-shift expressions and its range of validity is established. The next correction terms are deduced to be proportional to $\frac{\left(\ln k\sigma \right)}{{\left(k\sigma \right)}^2}$ and $\frac{1}{{\left(k\sigma \right)}^2}$ which involve ${\hslash }^2\ln \hslash$ and ${\hslash }^2$, respectively.

The linearized Vlasov-Maxwell equations for a plasma described by a Fermi-Dirac distribution are reduced to an integrodifferential equation that may be solved for the resonant frequencies and Landau-damping rates. A special uniform-plasma analysis is given reproducing the quantum results in the low-temperature limit.

Expresssions for the $l=0\mathrm{}$ forward scattering amplitudes for ${\mathrm{He}}^3$-${\mathrm{He}}^4$ and ${\mathrm{He}}^4$-${\mathrm{He}}^4$ quasiparticle pairs in dilute solutions of ${\mathrm{He}}^4$ in liquid ${\mathrm{He}}^3$ are derived in terms of thermodynamic quantities. A Fermi-liquid theory of the solutions is developed, and various thermodynamic quantities are derived. The theory agrees well with the available experimental data.

The equations describing the transfer of intensity from a pump beam simultaneously to forward and backward stimulated scattered beams in an infinite medium are shown to lead to pulsations similar to relaxation oscillations in the scattered intensities. The period of the pulsations is simply related to the gain length for the forward-scattered beam, and may be much longer than the ordinary transient time (related to the damping of material excitations) usually associated with stimulated scattering. Similar oscillations occur for backward scattering alone (stimulated Brillouin scattering) if the scattering medium is finite. In this case the period equals the photon round-trip time in the medium if the latter exceeds the backward gain length. Both phenomena should be observable in scattering from gases such as ${\mathrm{N}}_2$ and ${\mathrm{H}}_2$ at high pressure, and may also play a role in determining the temporal structure of light scattered from the region of a self-focus in liquids.

The permanent damage in optical glass investigated here is characterized by filamentary tracks of fine fractures. These tracks are a few micrometers in diameter, straight to within 0.7 μm, up to 9 cm long, and collinear with the incident laser beam. There may also be damage stars (regions of fracture gross compared with the track diameter). These damage stars are usually near the upstream ends of the tracks. Track formation is characterized by a flash of side-scattered white light from the track, laser light side-scattered from the damage stars, a marked increase in the exit divergence angle of the laser beam, and a weak back-scattered pulse of laser light. The back-scattered pulse preserves the polarization of the incident beam, is of shorter duration than the incident laser pulse, and has a frequency shift corresponding to Brillouin scattering from a free compressional sound wave in the glass. Track formation is accompanied by a detectable cylindrical sound wave. The track formation threshold is repeatable at different locations in the glass sample. Both the power threshold and the energy-density threshold are rapidly varying functions of the incident beam radius at focus. The threshold power is as low as 10 kW for a ruby-laser beam sharply focused in dense flint glass, and more than 2 MW for an unfocused beam in fused silica.

An electrostriction mechanism for laser-beam self-focusing and track formation in transparent optical glass is analyzed theoretically. Electrostrictive self-focusing occurs when a laser pulse of sufficiently high power and rapid rise time passes through a transparent medium. For a pulse duration of 50 nsec, trapping thresholds vary from 20 kW to 2 MW. During self-focusing, the beam collapses to a small radius. In solid dielectrics, self-focusing causes permanent damage in the form of isolated regions of gross fracture, termed "damage stars" and long straight tracks of very fine fractures. Typical tracks have a diameter of a few wave-lengths of light and extend up to several centimeters. Self-focusing occurs because of an interaction between light and sound. The laser beam electrostrictively excites an ultrasonic cylindrical disturbance or sound wave. The sound wave initially increases the refractive index along the beam axis. This focuses the beam into a waveguide channel called a filament. In the filament, the intensity is so high that many nonlinear mechanisms may occur, leading to damage and track formation. This paper analyzes only the self-focusing mechanism, not the various damage mechanisms. We assume the beam always remains Gaussian. The propagation of the beam is described by the quasioptics beam-tracing equation, which includes the effects of diffraction. The sound wave and beam-tracing equations are solved in several approximate models. A trapping threshold is derived for three pulse shapes, covering the steady-state, transitional, and transient regimes of pulse duration and beam size. There is a trapping-power coefficient $K$ for each material, calculable from the density, speed of sound for a compression wave or elastic moduli, and refractive index at the laser wavelength. A formula for computing the power to achieve a given constant maximum intensity $I$ as a function of beam size, and pulse duration, and trapping-power coefficient is derived. Values of this given constant intensity have been selected so the constant-intensity curve closely matches experimental track-formation thresholds for three optical glasses. The results are $K=221\mathrm{}$ kW and $I=2.5\mathrm{}$ ${\mathrm{G}\mathrm{W}/\mathrm{c}\mathrm{m}}^2$ for dense flint glass, $K=937\mathrm{}$ kW and $I=60\mathrm{}$ ${\mathrm{G}\mathrm{W}/\mathrm{c}\mathrm{m}}^2$ for borosilicate crown glass, and $K=1119\mathrm{}$ kW and $I=180\mathrm{}$ ${\mathrm{G}\mathrm{W}/\mathrm{c}\mathrm{m}}^2$ for fused silica, at a fixed pulse duration of 55 nsec and a laser wavelength of 694.3 nm. A computer movie of beam trapping shows the collapse of the beam to a relatively constant small radius, which causes track formation. It also shows the extremely rapid upstream motion of the focal points at speeds greater than 100 times the speed of sound. The long period of time they dwell at the upstream end of their motion explains the appearance of damage stars at the upstream ends of the tracks.

The statistical mechanics of an interacting electron gas in a magnetic field is developed based on a propagator method for weak fields and low temperatures. Detailed analytical considerations are given of the eigenvalues of relevant propagators, through which the grand partition function and the correlation energy are evaluated. The energy and the susceptibility are expressed in powers of $r_s$ and the magnetic field. Corrections to the Landau susceptibility are obtained taking the finiteness of the density into consideration. In the absence of the magnetic field, the correlation energy reduces to the correct limit. The paramagnetic susceptibility and the polarization energy agree with what Brueckner and Sawada obtained by a different method in which a polarization parameter is assumed. An explicit expression for this parameter is also given. Thus, the evaluation of the correlation energy by Gell-Mann and Brueckner and that of the susceptibility by Brueckner and Sawada, Pines, and others are generalized.

The Maier-Saupe model of the nematic phase with an orientational order parameter is extended to the smectic $A$ phase by introducing a new order parameter, the amplitude of a density wave in the direction of the nematic preferred axis. Self-consistent equations for the two order parameters are derived from an anisotropic model interaction and are solved numerically. We calculate the order parameters, the entropy, and the specific heat as a function of temperature for several values of dimensionless interaction strength $\alpha$ for the smectic $A$ phase. The transition temperatures plotted versus $\alpha$ provide a theoretical phase diagram which resembles experimental plots of transition temperature versus alkyl chain length for homologous series of compounds. The model qualitatively reproduces chemical trends in transition entropies. Experiments are suggested to measure the order parameters in the smectic $A$ phase.

A second-quantized cell-model Hamiltonian is derived to provide a model for quantum crystals and liquids. The boson crystal is treated in this article. The cells are divided into two sublattices: (i) regular, which are usually occupied, and (ii) interstitial, which are usually empty. The particle hard cores are simulated by assuming Fermi commutation relations for operators referring to a single cell; this allows a discussion in terms of a spin-analog Hamiltonian which is diagonalized in the spin-wave approximation. Because the Hamiltonian includes a term which allows tunneling between regular and interstitial sites, the ground state includes a description of zero-point motion and exchange via virtual intermediate interstitial occupation. Excited states include nonlocalized vacancies, interstitials, and vacancy-interstitial pairs. Phonon states are not included in this analysis. The model exhibits a phase transition to the Bose condensed state which is examined briefly.

We present a new, completely rigorous way of proving the relation between the second virial coefficient and the $S$ matrix. The method involves the use of the $U_{2,\infty }$ function introduced by Lee and Yang and proceeds in the most straightforward manner. Although difficulties are encountered for virial coefficients of higher order, there is hope that this type of approach might shed some light on the question of the connection between the virial series and the scattering matrix.

The stability of drift waves in sheared magnetic fields has been studied using two approaches. One method follows a highly localized wave packet which moves across the plasma, and the other solves for the normal modes of oscillation with the boundary condition of outward energy flow. The two methods lead to stability conditions which are virtually identical. In this paper we provide an explanation for this correspondence, and further show that the same results can be obtained by treating the problem by the eikonal approach considering non-normal-mode WKB solutions of the differential equations.

A variational principle, analogous to Hamilton's principle of mechanics, is proposed which leads to the equations of both nonequilibrium thermodynamics and hydrodynamics. The Onsager relations are equivalent to the use of a heat production function ${\dot{u}}_d$, analogous to the Rayleigh dissipation function. ${\mathrm{du}}_c$, where ${\mathrm{du}}_c=du-{\mathrm{du}}_d$, is used as the total differential of the potential energy. $\mathrm{du}$ represents the total change in the internal energy. Limitations of the theory are briefly considered. The effect of an applied magnetic field is briefly considered in an Appendix. The Onsager relations are derived for a system held in a magnetic field.

We report measurements of the optical Kerr effect induced by $Q$-switched laser waves in some liquids, using an original experimental arrangement allowing easy measurements in weak anisotropic liquids. We compared the results obtained by this method to those deduced from our direct measurements of depolarized Rayleigh scattering. It is the first time that a good agreement has been found between these two effects, allowing precise determinations of apparent optical anisotropies.

An approximate theory of the motion of a ${\mathrm{He}}^3$ atom in the vicinity of a superfluid ${\mathrm{He}}^4$ surface is presented. The theory predicts a bound state of the ${\mathrm{He}}^3$ atom in the vicinity of the surface, in accord with experimental results. The theoretical value for the energy of this state (measured with respect to the ${\mathrm{He}}^4$ chemical potential) is within 35% of the experimentally determined value.

The generalized Ornstein-Zernike formalism developed in previous papers is extended to systems specified by an arbitrary number of spatially varying extensive variables. This extension exhibits the critical eigenvector previously introduced as the first member of a more general structure of nested null spaces of the $T$ matrix which is invariant under an affine transformation of the extensive variables. A $k$-dependent version of Tisza's method of diagonalization of the stiffness and compliance matrices by completion of squares is shown to provide a natural coordinate system for critical fluctuations. This coordinate system defines a sequence of nested null spaces of the stiffness matrix which leads to the structure of null spaces of the $T$ matrix referred to above. The nature of critical fluctuations revealed in this way is used to derive an asymptotic expression for the $G\left(\right[m],[n\left]\right)$ matrix when [$m$] and [$n$] are far apart, as well as an expression for the short-range critical behavior of the molecular distribution functions. The conclusions of the paper are illustrated for simple and binary fluids. The fundamental invariance theorem for the structure of nested null spaces of the stiffness matrix is proved in an appendix.

Assuming a two-state model for ethanol, Hall's theory of structural relaxation has been used to explain the excess ultrasonic absorption at different pressures (up to 2000 ${\mathrm{k}\mathrm{g}/\mathrm{c}\mathrm{m}}^2$) and temperatures (25-55 °C). The theoretically calculated structural and experimentally observed excess-absorption results are in close agreement with each other at all the presures and temperatures. The values of the structural relaxation compressibility and structural relaxation times for ethanol have been determined by correlating its acoustical properties to those of methanol, which are known.

The density of excited atoms and the diffusely reemitted radiation is calculated for a homogeneous radiation field incident on a slab. The solutions of the Biberman-Holstein integral equation are used in the calculations. The optical thickness is assumed to be large. The line shapes of both the absorption line and the incident radiation are arbitrary.