Measurements were made of the profiles of the He I 4471.5- and 4922-Å lines in a dc arc for electron densities in the range of 0.4 to 2 × ${10}^{16}$ ${\mathrm{cm}}^{-3\mathrm{}}$. Concurrent determinations of the electron densities and temperatures by various methods could not be fully reconciled and are critically discussed. The normalized experimental and theoretical profiles, when compared on the basis of equal peak separation, show increased broadening of the forbidden component and filling in of the dip in the observed profile.

Two-photon absorption of light has been investigated by a technique involving the amplification of mode-locked dye-laser pulses in a cell excited by two-photon absorption. From the experimental measurements it is possible to calculate an absolute value for the two-photon molecular absorption cross section.

The classical van der Waals model of fluids is modified by a more accurate equation of state for hard spheres. The hard-sphere diameter and the van der Waals constant $a$ are obtained from experimental data. The model is used to predict the critical constants of metals, as well as the equation of state, cohesive energy, and coexistence curves near the critical point. The model is also applied to rare gases and ionic salts. The semiquantitative predictions of the model are at least as accurate as those of other theories. Of the three critical constants, the critical temperature is most accurately predicted, being within 11% of experiment. More information about the interatomic potential in metals is needed before the theory can be substantially improved.

We summarize the previous treatments of the question of superfluidity in ${\mathrm{He}}^3$, which have not shown conclusively whether this phase should or should not occur. The most promising calculation was by Pitaevskii, whose determination of the screened quasiparticle transition amplitude ${\tilde{\mathcal{T}}}_j$ was carried out in the limiting case of large angular momentum $j$. In estimating $T_c$ this was extrapolated to $j=2\mathrm{}$ by Gor'kov and Pitaevskii (GP). The work of GP was found to contain an inadequate approximation (the $D$-wave Born approximation) plus also an algebraic error. When these corrections are made, their formula for $T_c$ then yields the unattainably low value $T_c\sim {10}^{-100\mathrm{}}$ deg. We have projected precisely ${\tilde{\mathcal{T}}}_j\left(0\mathrm{}\le j\le 10\right)$ in second-order perturbation theory. We used as the bare vertex function the off-energy-shell $\mathcal{K}$ matrix, the particular form of which was chosen so that $\mathcal{K}$ would satisfy two exact identities (Ward identities). It was found that GP's extrapolation underestimates the effective $D$-wave interaction. In fact, the polarization (particle-hole) diagram is an attractive correction to ${\mathcal{K}}_j$, for $j=2\mathrm{}$ and $j\ge 6$, in contrast to Pitaevskii's result that the second-order diagram for ${\tilde{\mathcal{T}}}_j$ is repulsive. We study carefully the three approximations involved in Pitaevskii's large-$j$ calculation. Theoretical arguments are given, and numerical evidence is furnished, which indicate that as $j$ increases, Pitaevskii's limit is being approached asymptotically, but so slowly that his analytical result does not hold for angular momenta of interest (i.e., for $j<\mathrm{}20\mathrm{}$). The possibility of $D$-state superfluidity is discussed. The transition temperature is such a sensitive function of the calculated theoretical parameters that it is difficult to furnish a precise estimate for $T_c$.

Stimulated Raman scattering of linearly polarized light from degenerate and selectively populated atomic sublevels of potassium have been observed. The scattered radiation has a polarization perpendicular to that of the incident beam. Such transition can take place between substates of different or of the same atomic level. In the latter case, the frequencies of the incident and scattered beams are equal and the polarizations are perpendicular. The induced transitions are associated with parametric four-wave coupling possessing also characteristic polarization properties.

Hall's theory of structural relaxation is applied to explain the excess ultrasonic absorption in methyl alcohol in the pressure range 1-2000 kg/${\mathrm{cm}}^2$. Two states, one characterized by a higher volume (icelike structure) and the other by a lower volume (close-packed structure), are assumed to be available to each molecule of methanol, just as in water. It is found that Hall's theory for this case must be modified in order to be in agreement with the experimental results. The modification consists in assuming that the variation of the volume difference $\Delta V$ and the free-energy difference $\Delta F$ is nonlinear with pressure, in contrast with water. This leads to excellent agreement with theory, and also to the conclusion that the variation of $\Delta F$ with pressure is similar to that of ultrasonic velocity in methanol. Assumed values of the relaxational compressibility ${\beta }_r$ and the structural relaxation time $\tau$ (at $P=1000\mathrm{}$ kg/${\mathrm{cm}}^2$) are also justified because the other parameters calculated are as expected.

The ground-state energy of a mixture of spin-½ fermions and bosons in one dimension, interacting with a repulsive $\delta$-function potential, is analyzed. The wave function is given by repeated use of a generalized Bethe hypothesis. The "momenta" in the hypothesis are determined by coupled Fredholm integral equations. Numerical solutions are given. No phase separation is found.

The macroscopic dynamics of classical many-component plasmas in electric and magnetic fields is formulated in terms of scalar complex wave equations which contain pressure and electromagnetic potentials. The complex wave equation of each particle component is shown to be mathematically equivalent to the nonlinear conservation equations of mass, vector momentum, and energy. Thus, the description of a plasma by wave equations leads to a considerable mathematical simplification compared to the conventional many-fluid electromagnetohydrodynamics. As an elementary illustration of the wave-mechanical formalism, the dispersion of longitudinal and transverse electromagnetic waves in an electron plasma is treated.

From the general formulation of the coupling of a laser wave and a scattered light wave with density, temperature, and concentration fluctuations in a binary fluid mixture, an expression for the gain $g_c$ of stimulated concentration scattering is derived. It is shown that this gain is usually much smaller than the gain for stimulated Brillouin scattering $g_B$. However, in gaseous mixtures at relatively low pressures with a large difference in polarizability as well as in mass between the two components, $g_c$ can become larger than $g_B$. A large difference in polarizabilities increases the coupling of light to the concentration fluctuations, while a large difference in mass produces a pronounced increase in damping of a sound wave. The calculated values for $\frac{g_c}{g_B}$ are compared with experimental results.

It is well established that a totally reflected light beam of finite diameter undergoes a lateral displacement, known as the Goos-Haenchen shift. The theory for the corresponding effect in nonlinear optics is presented. The special phase-matched case, in which both the fundamental and the second harmonic are at critical total reflection, is shown to have a characteristic radiation pattern. Since the finite beam diameter is taken into account, divergencies of earlier theories are eliminated.

The consideration of the resonant interaction between a linear harmonic oscillator (LO) and a (nonlinear) angular momentum oscillator (NO) is motivated by its relation to the interaction between a field mode and a number of atomic systems, its relation to the interaction between three field modes, and the appearance of qualitatively different results quantum mechanically and classically even in cases that can be regarded as macroscopic. The analysis is carried out both classically and quantum mechanically in the rotating-wave approximation, and the time dependence of the energies of the two oscillators is examined, mainly by means of a nonlinear differential equation for the NO energy. In the classical analysis, the problem turns out to be identical to that of a spherical pendulum, where the NO energy $L_3$ corresponds to the pendulum potential energy, the LO energy $n$ corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\gamma K}{\omega }$ is proportional to the angular momentum $K$ of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for $L_3\mathrm{}\left(t\right)\mathrm{}$ is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\equiv L_3+n)$ and $K$. $L_3$ oscillates periodically, except for the "unstable-equilibrium" solution and the "conical-pendulum" solution. The quantum-mechanical problem is solved exactly for the cases in which the NO is a two-level system and a three-level system, and certain aspects of the exact solution are derived in the case of a four-level system. The constants $E$ and $K$ become a complete set of quantum numbers. It is found that unstable equilibrium, in the sense that $〈L_3〉$ is constant, does not exist, because of spontaneous emission; it is also found that $〈L_3〉$ oscillates sinusoidally for the two-level system, oscillates periodically but not sinusoidally for the three-level system, and oscillates aperiodically for a four- or higher-level system, conclusions about the higher-level systems coming from general considerations and numerical solutions found in the literature. Approximate solutions for NO' s with arbitrarily large numbers of levels are obtained. One method of approximation displays spontaneous emission but not aperiodicity. Another approximate method, which contains the statistical spread of the initial phase difference between the oscillations of the two oscillators inherent in the uncertainty principle, displays both spontaneous emission and aperiodicity. It is shown that the aperiodicity - a property which does not exist classically for precise initial conditions - does not disappear even in the macroscopic limit and is due to the averaging performed, in taking the quantum-mechanical expectation value, over an ensemble of periodic oscillations with a spread in frequencies.

A new thermodynamic Green's-function method is developed to handle interacting many-body systems containing a small subsystem with a fixed number of particles. The new method involves the use of the canonical ensemble and is more complicated than the usual Green's-function method. But, by means of the functional derivative technique, it is shown that systematic perturbative procedures are available and can be carried out in an entirely straight-forward manner. The example given in the present paper is the system of phonons interacting with a fixed number of fermions. It serves to illustrate the general formulation as well as perturbation calculation in the new method.

In single crystals with the large anisotropy of bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ the wave vector of a sound wave deviates as much as 60° from the direction of energy flow. This effect permits one to find single-crystal elastic constants from previous measurements of the transverse velocity, the Debye temperature and the compressibility: $c_{11}=2.35\mathrm{}$, $c_{12}=1.97\mathrm{}$, $c_{44}=1.085\mathrm{}$ (in ${10}^8$ dyn/${\mathrm{cm}}^2$) at a molar volume of 24 ${\mathrm{cm}}^3$/mole. The constants agree well with measured longitudinal sound velocities and with theoretical constants of Horner, less well with constants obtained from the single-particle theories of Nosanow et al.

Hartree-Fock calculations have been made for the configurations $3d^{2}4s$ of Sc I and $5d^{2}6s$, $5d6s^2$, and $5d^3$ of La I. Attention is drawn to the variation of atomic quantities which influence hyperfine structures owing to the LS-term dependence of the radial wave functions. Theoretical calculations of the level structure of ${\mathrm{}(5d+6s)\mathrm{}}^3$ of La I are also presented.

The solution of the Liouville equation is expressed in the interaction representation. We show that the early-time behavior of the single-particle distribution function $F_1$ for a classical gas consisting of particles interacting through weak two-body central forces is governed by the Vlassov equation. The two-body potentials are assumed to be "good" functions. The derivation fails if initial correlations are present. The argument is carried out in configuration space. The analysis is extended to the derivation of differential equations governing the time evolution of the multiparticle distribution functions $F_k$. In this "Vlasov approximation" to the solution of the Liouville equation, we find that the Vlassov equation for $F_1$, together with the differential equations for the $F_k$ ($k<\mathrm{}1\mathrm{}$), amounts to the statement that the multiparticle distribution functions are factorable into products of single-particle distribution functions.

A nonlinear damping of Tonks-Dattner resonances has been observed in active and afterglow plasmas. The damping is found to decrease with increasing strength of the exciting microwave electric field with no change in resonant frequency. A decrease in the Landau damping due to a flattening of the steady-state electron velocity distribution appears to be responsible. We believe that a balance between the wave-particle interaction and electron-particle collisions is important in determining the steady-state slope of the velocity distribution, and therefore the Landau damping. Order-of-magnitude comparisons are made with a simplified quasilinear analysis based upon this idea.

Measurements at room temperature of the mobility and longitudinal diffusion coefficient have been made for low-energy mass-identified potassium ions and positive and negative oxygen ions in oxygen gas in a drift-tube mass spectrometer. The zero-field reduced mobilities of ${\mathrm{K}}^{+}$, ${\mathrm{O}}_2^{+}$, and ${\mathrm{O}}_4^{+}$ in ${\mathrm{O}}_2$ were found to be 2.68 ± 0.07, 2.24 ± 0.07, and 2.16 ± 0.08 ${\mathrm{cm}}^2$/V sec, respectively. ${\mathrm{O}}^{-}$, ${\mathrm{O}}_2^{-}$, ${\mathrm{O}}_3^{-}$, and ${\mathrm{O}}_4^{-}$ were determined to have zero-field reduced mobilities of 3.20 ± 0.09, 2.16 ± 0.07, 2.55 ± 0.08, and 2.14 ± 0.08 ${\mathrm{cm}}^2$/V sec, respectively. The longitudinal diffusion coefficients were measured for ${\mathrm{O}}_2^{+}$, ${\mathrm{K}}^{+}$, ${\mathrm{O}}^{-}$, and ${\mathrm{O}}_2^{-}$ in ${\mathrm{O}}_2$. The longitudinal diffusion coefficients of all investigated ionic species at low $\frac{E}{N}$ were at least within 16% of the values predicted from the mobilities by the Einstein relation. At higher $\frac{E}{N}$, the coefficients increased, in some cases quite rapidly, as $\frac{E}{N}$ was increased.

The rate coefficients for the reactions of ${\mathrm{O}}^{-}$ ions with ${\mathrm{O}}_2$ molecules have been measured in a drift-tube mass spectrometer at 300 °K as a function of $\frac{E}{N}$, the ratio of the electric drift field to the gas number density. Two different measurement techniques were employed. One technique was based on the observation of the attenuation of the parent ${\mathrm{O}}^{-}$ ion species with increasing drift distance; the other involved examination of the shape of the product ion arrival-time spectra. The rate coefficient for the reaction ${\mathrm{O}}^{-}$ + ${2\mathrm{O}}_2$ → ${\mathrm{O}}_3^{-}$ + ${\mathrm{O}}_2$ was measured on the $\frac{E}{N}$ range 6-29 × ${10}^{-17\mathrm{}}$ V ${\mathrm{cm}}^2$. The rate coefficient was found to be (1.0 ± 0.2) × ${10}^{-30\mathrm{}}$ ${\mathrm{cm}}^6$/sec over the entire $\frac{E}{N}$ range. The reaction ${\mathrm{O}}^{-}$ + ${\mathrm{O}}_2$ → ${\mathrm{O}}_2^{-}$ + O was observed with a rate that increased sharply with $\frac{E}{N}$. The rate coefficient was found to increase from (2.5 ± 1.5) × ${10}^{-14\mathrm{}}$ ${\mathrm{cm}}^3$/sec at an $\frac{E}{N}$ of 63 × ${10}^{-17\mathrm{}}$ V ${\mathrm{cm}}^2$ to a value of (3.4 ± 1.0) × ${10}^{-12\mathrm{}}$ ${\mathrm{cm}}^3$/sec at an $\frac{E}{N}$ of 140 × ${10}^{-17\mathrm{}}$ V/${\mathrm{cm}}^2$.

Spectra of laser fields phase modulated by pulses in Kerr liquids have been calculated, retaining all phase information. For short and long pulses, spectra like those observed in C${\mathrm{S}}_2$ and bromobenzene, respectively, are predicted. The C${\mathrm{S}}_2$ spectra are quite sensitive to a phase parameter.

Hamiltonians to describe vacancies and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ impurities in solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ derived from a Hubbard-like-model starting point are presented and discussed.