Elastic scattering experiments have been performed on the systems ${\mathrm{H}}^{+}$ + Kr, ${\mathrm{H}}^{+}$ + Ar, ${\mathrm{H}}^{+}$ + Ne, and ${\mathrm{H}}^{+}$ + He with collision energies between 3 and 60 eV. For each of the above systems the experimental differential cross section $\sigma \left(\theta \right)$ at one of the lower energies (typically 4 or 6 eV) has been compared with the results of a partial-wave calculation where the JWKB method was used to find the phase shifts and consequently the differential cross section. For the systems Kr${\mathrm{H}}^{+}$, Ne${\mathrm{H}}^{+}$, and Ar${\mathrm{H}}^{+}$an analyticalform was chosen for $V\left(r\right)\mathrm{}$, and parameters (e.g., the well depth $U$ and the value $r_m$ of the internuclear separation for which the potential is a minimum) in this model were varied until the calculated $\sigma \left(\theta \right)$ agreed quite well with the experimental results. In the case of ${\mathrm{H}}^{+}$ + He two ab initio calculations of the interatomic potential $V\left(r\right)\mathrm{}$ are available, and each has been used in the JWKB expression to determine the differential cross section. For one of the potentials the predicted $\sigma \left(\theta \right)$ is in striking agreement with the experiment. This method is seen, therefore, to provide a sensitive test of such calculations when they exist.

Expressions for the stored energy, energy flow, and power dissipated are derived for electromagnetic waves in terms of the complex permittivity ${\epsilon }^{\prime }$ and permeability ${\mu }^{\prime }$ for a frequency-dispersive absorbing medium. This is shown to be possible when ${\epsilon }^{\prime }$ and ${\mu }^{\prime }$ are known functions not only of frequency but also of all the loss factors —e.g., collision frequencies, etc. The derivation is not restricted to media with small losses. An oscillating particle system and an electric circuit are used as illustrative examples of the application of the energy expressions derived.

We use the Coulomb-Born approximation to calculate the total cross section for the electron-impact excitation of ionized atoms which have one electron outside their closed shells. When the incident energy is high and the atom is highly ionized, we can use the eikonal and classical approximations in conjunction with the Coulomb-Born approximation. Then, the scattering amplitude reduces to a one-dimensional path integral. The path integral, in turn, can be reduced, to any desired accuracy, to a sum over integrals which can be evaluated analytically. Using this procedure, we have calculated the total cross sections for the heavily ionized atoms ${\mathrm{Fe}}^{15+}$, ${\mathrm{Co}}^{16+}$, ${\mathrm{Ni}}^{17+}$, and ${\mathrm{Cu}}^{18+}$.

A study has been made of the charge-transfer processes whereby fast neutral atoms of hydrogen are formed in the $3s$, $3p$, and $3d$ excited states as a result of the impact of protons on targets of helium and of nitrogen. Impact energies ranged from 75 to 400 keV. The experimental procedure involved quantitative measurement of the Balmer $\alpha$ radiation emitted by the spontaneous decay of atoms in these three states. The three contributions to the emission were separately identified according to the different lifetimes of the $3s$, $3p$, and $3d$ states by means of a time-of-flight technique. A study was also made of the processes whereby the excited atoms were collisionally destroyed before undergoing spontaneous radiative decay. Detailed measurements are presented of the cross sections for capture into the $3s$ state, which are by far the largest of these capture cross sections. Cross sections for capture into the $3p$ and $3d$ states are one to two orders of magnitude smaller, and preliminary measurements are discussed. All of these cross sections decrease rapidly with increasing impact energy. There is general agreement between the present measurements and the predictions of the Born approximation for capture into the $3s$ and $3d$ states from a target of helium, but the calculation for the $3p$ state appears to overestimate the cross section by a factor of at least 4. The cross sections for collisional destruction of atoms in the $3s$ state are several orders of magnitude larger than for electron capture into this state, and the magnitude of the measured values are in agreement with theoretical predictions.

Translational-vibrational energy transfer in collinear collisions of an atom and an harmonic oscillator has been treated in the impulse approximation of Chew. Although transition probabilities for a soft exponential interaction potential compared poorly with exact quantum-mechanical results, for sufficiently small incident masses good results were obtained for a hard impulsive interaction. This is understood in terms of the ratio of the collision time to the oscillator period; the impulse approximation is expected to be valid when this ratio is small. For the smaller incident masses, qualitative agreement with exact results was obtained over a wide energy range for the "hard-sphere" collisions. This represents the first successful approximate treatment of impulsive collisions for this model. In contrast to the familiar perturbative methods, the quality of the approximation does not decrease for multiple-quantum transitions and improves as the energy increases. The one-dimensional impulse-approximation transition probabilities $P_{\mathrm{if}}$ are ill behaved at threshold; however, most computed probabilities remained well behaved to within about $\frac{1}{4}\hslash \omega$ of threshold. The approximation fails to conserve probability. It was found that the sums of probabilities $\Sigma f^{}{P}_{\mathrm{if}}$ for the various initial states $i$ provided reliable measures of confidence in the approximation. For a given $i$, if the probability sum was nearly constant with changing energy, the $P_{\mathrm{if}}$ curves were qualitatively good; on the contrary, the curves were in less good agreement with exact results if the probability sums were less stable with changing energy. When the curves for a given $i$ were qualitatively good, a simple renormalization yielded quantitative agreement with exact results and minimized the effects of threshold misbehavior.

The third law of thermodynamics is proved for a large class of Ising models with generalized ferromagnetic many-body interactions. A sufficient condition for the third law to hold is that the model have nearest-neighbor couplings which are bounded from below by a positive constant. The proof is based on a spin-correlation inequality of Griffiths which implies a corresponding inequality for the bulk entropy per spin. Ground-state degeneracy considerations are completely avoided.

The so-called quasiclassical approximations are shown to give, in a certain restricted sense, lower bounds to energy eigenvalues. Variational calculations of the energies of hydrogenlike atoms are carried out as an illustration.

By adding a small amount of absorption to many liquids, it is shown that the stimulated Raman emission and self-phase modulated emission pulse trains are radically shortened from that of a long pulse train of a mode-locked picosecond laser. The first pulses in a picosecond pulse train are more efficient in producing self-focusing because of self-absorption.

Density-versus-height profiles have been measured in the critical region of oxygen by means of capacitance techniques. Results are given for the liquid and vapor densities at coexistence, for compressibilities along the coexistence curve to within $t\equiv \frac{(T-T_c)}{T_c}=-6\mathrm{}\times {10}^{-5\mathrm{}}$, for compressibilities along the critical isotherm to within $\frac{(\rho -{\rho }_c)}{{\rho }_c}=5\mathrm{}\times {10}^{-2\mathrm{}}$, and for compressibilities along the critical isochore to within $t=2\mathrm{}\times {10}^{-4\mathrm{}}$. The data are analyzed in terms of power-law descriptions and are shown to be in excellent agreement with recent scaling-law analyses of data for other fluids.

The grand partition function for a degenerate electron gas with relativistic corrections of order ${\left(\frac{v}{c}\right)}^2$ in ring approximation is given. The technique used is the method of Montroll and Ward; in this problem independent field degrees of freedom are not required. The correlation energy, which represents a generalization of the Gell-Mann-Brueckner result, is obtained. Comparison with the field-theoretic results of Akhiezer and Peletminskii is made, with the same results. However, in view of the absence of independent field degrees of freedom in our treatment, renormalization is not required here (to obtain their result Akhiezer and Peletminskii must resort to charge, mass, and vacuum renormalization).

We study the van der Waals interaction $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ arising from two-photon exchange between neutral spinless systems $A$ and $B$. By using the analytic properties of the two-photon contribution to the scattering amplitude for $A+B\to A+B$ and of the full amplitudes for $\gamma +A\to \gamma +A$ and $\gamma +B\to \gamma +B$, we show that it is possible to express $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ entirely in terms of measurable quantities, the elastic scattering amplitudes for photons of various frequencies $\omega$. This approach includes relativistic corrections, higher multipoles, and retardation effects from the outset and thus avoids any $\frac{v}{c}$ expansion or any direct reference to the detailed structure of the systems involved. We obtain a generalized form of the Casimir-Polder potential, which includes both electric and magnetic effects, and, correspondingly, a generalized asympotic form $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(r\right)\mathrm{}\sim -\frac{D}{R^7}$, where $D=\frac{\left[23\right({\alpha }_E^A{\alpha }_E^B+{\alpha }_M^A{\alpha }_M^B)-7({\alpha }_E^A{\alpha }_M^B+{\alpha }_M^A{\alpha }_E^B\left)\right]}{4\pi }$ and the $\alpha$'s denote static polarizabilities. In addition, we show that the potential may be written as a single integral over $\omega$, involving products of the dynamical polarizabilities ${\alpha }_X\left(\omega \right)$ evaluated at real frequencies, in contrast to the familiar integral over imaginary frequencies; for the case of interacting atoms, the domain of applicability of the various formulas is clarified, and the problem of evaluating $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ from present experimental information is discussed. Some simple interpolation formulas are presented, which may accurately describe $V_{2\gamma }^{\mathrm{AB}}\mathrm{}\left(R\right)\mathrm{}$ in terms of a few constants.

A simple theory of neutron scattering from liquid He II at zero temperature for large momentum transfers is described. The theory is based on a generalized mean-field approximation involving the polarization potential and the screened response function. The latter, instead of having a free-particle form, is assumed to be a sum of Gaussian functions, weighted by the momentum distribution function appropriate for liquid helium. The zero and third moments of the scattering law determine the polarization potential and the width of the Gaussians. Numerical calculations have been made for the dynamical structure function $S(q,\omega )$ for different values of the condensate fraction. Both the width and the peak position of $S(q,\omega )$ in the range of momentum transfers 2.5-9 ${\mathrm{\AA }}^{-1\mathrm{}}$ are in fair agreement with experiment for a 6% condensate fraction. Calculations have also been done for $q$ values as large as 20 ${\mathrm{\AA }}^{-1\mathrm{}}$ for three different values of the condensate fraction, including 0. For a condensate fraction of 6% or larger, the contribution of the condensate to $S(q,\omega )$ becomes separated from that of the noncondensed particles at these large -$q$ values. The separation is distinct enough that it should be experimentally observable.

An improved version of the theory of Singwi et al. for the collective motions in classical liquids is described. The theory is based on a generalized mean-field approximation involving the polarization potential and the screened response function. The latter, instead of having a free-particle form or that corresponding to the true self-motion of the particle in a liquid, is assumed to be a sum of Gaussian functions weighted by the momentum distribution of the particles. The zero and fourth moments of the scattering law determine the polarization potential and the width of the Gaussians. Numerical calculations have been made for the spectral function of the longitudinal current correlations, quasielastic scattering, and absolute intensities for liquid argon in the range of momentum transfer 0.5-6.3 ${\mathrm{\AA }}^{-1\mathrm{}}$. The results are in excellent agreement with the molecular-dynamics calculations and recent experiments on a purely coherent sample of liquid argon.

We present a theory of pressure effects in which the atomic center-of-mass motion is treated quantum mechanically. The quantum-mechanical calculation treats the perturber-induced energy-level variations and the velocity changes of the emitter caused by collisions on an equal basis. Specifically, we shall treat the problem of a laser to first order in the laser field, allowing for the fact that the laser atoms are undergoing collisions, but our results will also be applicable to the cases of stimulated emission or absorption. It will be sufficient to carry out a perturbation solution of the problem assuming that the laser atoms undergo at most one collision in their lifetime, since such a restricted calculation reveals the salient features of the theory. We shall find that, in general, there is no classical limit for our results. Thus, previous treatments employing a classical Boltzmann-equation approach for the atomic center-of-mass motion are invalid, and a quantum-mechanical description is necessary to correctly treat cases where both the modified Doppler effect (modified by collisions) and perturber-induced energy-level variations are present.

The asymptotic pair correlations between charged particles, ions and electrons, in a hot dilute gas are found through the use of a perturbation theory for a system of molecules, where a molecule is specified by the totality of its bound and scattering states. The derived screened electrostatic potential (as well as the charge distribution) around an ion is given by ${\varphi }_i\mathrm{}\left(r\right)=\frac{(C_S{e}^{-{\kappa }_{S}r}+C_{O}cos{\kappa }_{O}r)}{r}$, where ${\kappa }_S>{\kappa }_O$, $\left|\frac{C_S}{C_O}\right|\gg 1$, thus leading to a main contribution from the Debye-Hückel—type term at moderately large distances, and from the oscillating Coulomb term mainly at very large distances. Since ${\kappa }_S$, ${\kappa }_O\propto \kappa ={\left(4\mathrm{}\pi \beta e^{2}n\right)}^{\frac{1}{2}}$, where $n$ is the molecular density, the oscillatory term is more pronounced at increasing densities.

The triple-collision integrals which determine the first density correction to the transport coefficients are derived for a gas of hard spheres using the binary-collision expansion. This expansion provides a convenient technique for classifying the contributions in terms of sequences of successive binary collisions between three molecules. Such sequences contain both interacting and noninteracting collisions. It is demonstrated that for three hard spheres all sequences terminate after four successive collisions, independent of the interacting or noninteracting nature of the collisions. As a consequence, the collision integrals are related to a limited number of sequences with three and four collisions only. It is shown that equivalent results are obtained from the surface-integral form of the triple-collision operator, derived earlier by Green and Sengers.

The osmotic pressure of ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures has been measured at temperatures 0.027-0.65 K, for concentrations up to 10-mole% ${\mathrm{He}}^3$, and for hydrostatic pressures of 0.26, 10, and 20 atm. The osmotic pressure was measured directly with a sensitive specially designed diaphragm pressure gauge. The temperature and concentration dependence of the osmotic pressure is in fair agreement with the effective interaction theory proposed by Bardeen, Baym, and Pines (BBP). It also agrees with a simple empirical model which avoids the complicated calculations involved in obtaining the thermodynamic properties at finite temperatures from the BBP theory. The model fits the temperature dependence of other thermodynamic properties of solutions. The osmotic pressure at absolute zero at 10 and 20 atm is used to determine the ${\mathrm{He}}^3$ effective mass and somewhat speculative values of the BBP quasiparticle interaction $V\left(k\right)\mathrm{}$ under pressure. The interaction under pressure is found to have a minimum at a nonzero value of $k$, and it may give rise to a "supermobile" transition at comparatively high temperatures.

The normal modes of an incompressible cholesteric liquid crystal are determined from the hydrodynamical equations. There are two important modes which should be readily observable by light scattering. The first mode is a twisting and untwisting of the helical structure. The second mode is a combination viscous-splay mode. Both modes are overdamped. These modes scatter light most strongly for momentum transfers exactly equal to and exactly one-half of those leading to Bragg scattering, respectively. Thus the modes should be easily separated experimentally. The normal modes are also evaluated in the presence of a transverse magnetic field, and the damping constants turn out to be quite sensitive functions of the magnetic field.

Narrow-beam mass-attenuation coefficients were measured for air, neon, argon, krypton, and xenon. Radioisotope sources were used to provide photons whose energies were 4.508, 5.895, 9.243, 27.380, 44.229, 88.09, and 145.43 keV. Experimental errors were less than 2% for all measurements. The experimental results are compared with theoretical predictions of the attenuation coefficients.

Liquid crystals are considered as gases of long barlike molecules. Variables describing the position and orientation of individual molecules are introduced. These are used to arrive at a microscopic definition of the order in liquid crystals in terms of a symmetric traceless tensor. The Frank and Leslie-Ericksen directors are shown to be equivalent to the unit eigenvector associated with the largest eigenvalue of this symmetric traceless tensor. Attention is then restricted to nematics. It is shown that the Frank free energy gives a valid description of equilibrium properties of nematics to order $\frac{{\chi }^a{H}^2}{\rho \kappa T{m}^{-1\mathrm{}}}$, where ${\chi }^a$ is the anisotropic magnetic susceptibility, $H$ the external magnetic field, $\rho$ the mass density, $T$ the temperature, $k$ the Boltzmann constant, and $m$ the molecular mass. Microscopic justification is given for the linearized Leslie-Ericksen hydrodynamic theory for nematics. A microscopic response-function description of nematics is then presented, and contact is made between the general frequency- and wave-number-dependent response functions and those calculated from the phenomenological hydrodynamic theory. Applications to light scattering and nuclear magnetic resonance are considered. $\frac{1}{T_1}$ calculated here differs from that calculated from the Pincus theory by a factor of $S^2$, where $S$ is the parameter measuring the degree of nematic order.