Total elastic cross sections for electron scattering by ground-state C, N, and O atoms have been computed by the matrix-variational method for energies up to the first $n=3\mathrm{}$ excitation. Two levels of approximation are considered: single configuration (SC) and near-degenerate configuration interaction (CI). Both approximations neglect the principal effects of target-atom electric-dipole polarizability. The SC calculations are the variational equivalent of earlier close-coupling calculations including all states of the ground configuration $2s^{2}2p^m$. The CI calculations augment this with configurations $2s2p^{m+1\mathrm{}}$ and $2p^{m+2\mathrm{}}$. The short-range correlations introduced by the CI calculations are found to have dramatic effects on resonance structures found at low energies in SC calculations. In addition to the well-established negative-ion bound states ${\mathrm{C}}^{-}\left({}_{}{}^4{}_{}{}^{}S_{}^o{}_{}{}^{}\right)$ bound and ${\mathrm{O}}^{-}\left({}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}\right)$, the CI calculations give qualitative predictions of the true location of other $2s^{2}2p^{m+1\mathrm{}}$ states: ${\mathrm{C}}^{-}\left({}_{}{}^2{}_{}{}^{}D_{}^o{}_{}{}^{}\right)$ bound; ${\mathrm{C}}^{-}\left({}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}\right)$ resonance near 1 eV; ${\mathrm{N}}^{-}\left({}_{}{}^3{}_{}{}^{}P\right)$ very near threshold, either weakly bound or a very narrow resonance; no ${\mathrm{O}}^{-}$ resonance. A method for precise location of these negative-ion states is proposed.

The position and width of the lowest ${}_{}{}^1{}_{}{}^{}S$ resonance in $e^{-}\mathrm{H}$ scattering are calculated using the complex-coordinate-rotation method and compared with the results of other methods.

Coupled integral equations linking the direct and exchange (rearrangement) $K$ operators are proposed in analogy to similar ones linking the direct and exchange $T$ operators derived in this paper. It is shown that these pairs of coupled equations lead to the damping equation which was used in previous work on identical-particle scattering and which expresses the unitarity condition. Other formulations of integral equations for the $K$ operator are also discussed.

The elastic scattering by positive ions of electrons with energies below 2 Ry was investigated theoretically using the independent-particle-model (IPM) potentials of Green, Sellin, and Zachor to represent the electron-ion interaction. Parameters were determined from minimization of the total energy of the bound system consisting of the electron plus target ion, or in some cases from fits to the single-electron eigenvalues of this same system. Potentials of the former type are generally the stronger of the two. The IPM phase shifts versus energy are qualitatively similar to, but usually larger than, those obtained using Hartree-Fock-Slater potentials. For energies below 1 Ry there is excellent agreement between the differential cross sections for ${\mathrm{Na}}^{+}$ obtained from the IPM and those computed from semiempirical quantum-defect methods. For this same ion the IPM differential cross sections likewise resemble those computed from Hartree-Fock potentials and lie between the Hartree-Fock results obtained with and without local exchange contributions. Phase shifts are computed using the independent-particle model for all positive ions of nitrogen and oxygen, species of atmospheric interest.

The electron energy dependence of the angular distribution of ${\mathrm{O}}^{-}$ produced by dissociative electron attachment to NO has been measured in the energy range 8.0-11.0 eV. The distributions are observed to be anisotropic and of a form that is nearly ${sin}^2\theta$ at all energies. Deviations from a ${sin}^2\theta$ dependence are discussed in terms of the relative contributions of different partial waves to the differential cross section. The results indicate that the final repulsive negative-ion resonances involved must be either $\Sigma$ or $\Delta$ states.

The differences between atom-atom differential scattering cross sections calculated by integration of the classical orbit equations, and those obtained from the universal curve of Lindhard, have been used to calculate corrections to the specific nuclear energy loss, ${\left(\frac{d\epsilon }{d\rho }\right)}_n$, as a function of the reduced energy $\epsilon$. The calculations have been performed both for the Thomas-Fermi (TF) interaction potential, which appears to be appropriate for gas scatterers, and for a finite-range Thomas-Fermi potential which may be more appropriate for scattering centers in a solid. For values of $\epsilon >0.1\mathrm{}$ the correction factor can be as small as 0.9. At lower values of $\epsilon$ the correction factor can be greater than unity for the infinite-range TF potential, but continues to decrease for all the finite-range calculations.

Photodetachment cross sections of the negative ions of the three lightest alkali-metal atoms have been calculated using configuration-interaction wave functions for the initial state and close-coupling scattering wave functions for the final state. Pronounced structure obtained at the threshold for leaving the neutral atom in the first excited state is due to a combination of real resonance behavior and the threshold law for collisional excitation. The range of validity of this law is, however, found to be very narrow. The shape of the cross sections in this region is in excellent agreement with experimental measurements. Results are also presented for the angular distribution of the photoelectrons following photodetachment with simultaneous neutral excitation, for the dipole polarizabilities of the negative ions, and simple analytic formulas for the radiative attachment coefficients.

Photodetachment of alkali-metal negative ions by an Ar-ion laser (4880 Å) has been studied in a crossed-beam geometry. Energy analysis of the photodetached electrons yields the following electron affinities: EA(Li)=(0.620±0.007) eV, EA(Na)=(0.548±0.004) eV, EA(Rb)=(0.486±0.003) eV, and EA(Cs)=(0.470±0.003) eV. These values are obtained relative to an independent measurement of the potassium electron affinity: EA(K)=(0.5012±0.0005). In addition, the angular distribution anisotropy parameter $\beta$ has been measured at 4880 Å for photodetachment processes leaving the neutral in its ground state and in its first excited state. Relative photodetachment cross sections for these two processes are presented for K, Rb, and Cs.

The Lagrangian theory of random chains with excluded volume is used to study $Z_N\left(\overrightarrow{\mathrm{r}}\right)$, the number of chains with $N$ links, starting from the origin and arriving at a point $\overrightarrow{\mathrm{r}}$. Its asymptotic expression ($N\to \infty$) is $Z_N\mathrm{}\left(r\right)\mathrm{}\simeq N^{\gamma -1-\nu d} F\left(r{N}^{-\nu }\right)$, where $\gamma$ and $\nu$ are critical indices. The short- and long-range behaviors of $F\left(x\right)\mathrm{}$ are calculated in terms of $\gamma$ and $\nu$. In particular, it is shown that for $x\ll 1$, we have $F\left(x\right)\mathrm{}\simeq F{x}^{\theta }$ with $F=\mathrm{const}$ and $\theta =\frac{(\gamma -1)}{\nu }$.

A detailed analysis is presented for second-sound velocity ($u_2$) measurements in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures near $T_{\lambda }$ in terms of thermodynamic parameters and the superfluid fraction $\frac{{\rho }_s}{\rho }$. Several incorrect conclusions in the literature are corrected. The hydrodynamic expressions for the velocities of the propagating modes are obtained for the case where the thermal expansion coefficient does not vanish. The relevant thermodynamics near the $\lambda$ line is discussed, and expressions for the necessary parameters in terms of measured quantities are given. The importance of various contributions to $u_2$ is considered. The experimental data for $u_2$ are reviewed, and the corresponding results for the superfluid fraction are calculated. Special attention is given throughout to the propagation of errors from the uncertainty in thermodynamic parameters. The results for $\frac{{\rho }_s}{\rho }$ are found to be consistent with those obtained by the oscillating-disk method, as expected from two-fluid hydrodynamics. The difference in behavior of ${\rho }_s$ along paths of constant concentration and constant chemical potential is considered. The exponent of ${\rho }_s$ is found to be consistent with scaling and universality. The superfluid density is compared with the measured superfluid healing length $l$, and the parameter $\frac{l{\rho }_s}{T}$ is found to be independent of the concentration $X$. The data for ${\rho }_s$ are also found to be consistent with a concentration-independent free energy per correlation volume, as expected on the basis of universality arguments.

A measurement of the tensor polarizability of the $2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state of atomic helium has been made by the electric-field level-crossing technique using an atomic-beam apparatus. The experimental procedure is as follows: An atomic beam containing metastable $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and $2 {}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ helium atoms is produced by electron bombardment of ground-state helium atoms. The beam then passes through a region in which the atoms are subjected to a uniform, variable electric field. In this region the $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ atoms are excited to the $2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state by linearly polarized 20 582-Å ($2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}-2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$) resonance radiation. The extreme uv photons emitted in the direction of the incident resonance light when the $2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state decays to the ground state are counted. The sum intensity of the 584-Å ($2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}-1 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$) photons and the 601-Å ($2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}-1 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$) photons, due to the electric-field quenching of the $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ atoms, is measured as a function of the applied electric field for the incident light polarized at 45° and 90° to the direction of the static electric field. From these measurements the tensor polarizability ${\alpha }_2\left(2\mathrm{} {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\right)$ of the $2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state is found to be (3.32 ± 0.10) × ${10}^{-23\mathrm{}}$ ${\mathrm{cm}}^3$, in good agreement with the theoretically calculated value of 3.34 × ${10}^{-23\mathrm{}}$ ${\mathrm{cm}}^3$.

Spin waves in superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ are studied in the collisionless regime by introducing a canonical transformation in the kinetic equation for the distribution-function matrix. This transformation is a space-time—dependent spin rotation, which takes into account the fluctuations of the direction of the order parameter which are coupled to the spin-density fluctuations. After the transformation, the order parameter is invariant. The kinetic equation is then expanded to lowest order in $\omega$ and $\overrightarrow{\mathrm{q}}$ and diagonalized. Fermi-liquid effects are taken into account. The result is a set of equations describing the coupled motion of the superfluid parameters and the quasiparticle distribution. In the axial state, only two degenerate spin-wave modes exist. These have their spin polarization perpendicular to the direction of the order parameter in spin space. In the isotropic state, one longitudinal and two transverse modes are obtained. Equations giving the velocity of these modes are written.

The transverse component (${\sigma }_T$) of the tensor electric conductivity of a weakly ionized magnetized plasma has been measured under near-thermal equilibrium conditions. Care has been taken to minimize the effects of $\overrightarrow{\mathrm{E}}\times \overrightarrow{\mathrm{B}}$ drifts and finite apparatus size. For magnetic fields that are large enough, it is found that the magnetic field dependence of ${\sigma }_T$ is $\frac{1}{B^2}$, in agreement with theory. It is also found that the overall density dependence of ${\sigma }_T$ agrees with standard conductivity theory. At the highest values of the magnetic field used, the data show some deviation from theoretical predictions which include only electron transport. This deviation is attributed to ion transport, and opens up the possibility of measuring transport properties of the ions alone.

It is found that, in liquid—helium-II pure counterflow, the dimensionless number $\mathcal{A}={\rho }_{s}A\overline{w}l$ (where $A$ is the Gorter-Mellink constant, $\overline{w}$ is the mean relative velocity between the two fluids, and $l$ is a characteristic dimension of the geometry) scales the mutual friction between the two fluids. Critical relative velocities in cylindrical channels are shown to be correlated by the rule ${\mathcal{A}}_c\equiv {\rho }_{s}A{\overline{w}}_c\pi d\simeq 1$.

A straightforward dynamical analysis of the local velocity field around a classical vortex ring yields the well-known translational velocity $\left(\frac{\kappa }{4\pi r_0}\right) \left[\ln \right(\frac{8r_0}{a})-\frac{1}{2}+\delta ]$, where $\kappa$ is the circulation, $r_0$ the radius, $a$ the core radius, and $\delta =0\mathrm{} or \frac{1}{4}$ for a hollow core or a core with solid-body rotation, respectively.

We discuss the problem of calculating the superradiant behavior in samples small compared to a radiation wavelength. Difficulties associated with the analysis of Dicke are considered in the framework of quantum electrodynamics and equivalently from a classical point of view. The evolution of superradiance in a small sphere is followed numerically with attention to spatial variation of the Bloch dipole density. Coherence is maintained except in the region ${\mathrm{}\left(\mathrm{kR}\right)\mathrm{}}^{-1\mathrm{}}\left|cos\theta \right|\lesssim 2$ which is at the Bloch equator, where it suddenly disappears. Away from the equator, the spatial variations of dipole density oscillate with amplitude $\sim {\mathrm{}\left(\mathrm{kR}\right)\mathrm{}}^{2}sin\theta$ and frequency $\sim {\mathrm{}\left(\mathrm{kR}\right)\mathrm{}}^{-3\mathrm{}}\left|cos\theta \right|$. All these results agree with the predictions of a theoretical discussion previously published.

A class of projective adiabatic approximations is described that enables a variety of lower bounds to be determined for the partition function of a system and, hence, upper bounds for its Helmholtz function. In one extreme, this class of approximations embraces the well-known Peierls inequality, while, in another extreme, it yields the statistical thermodynamic analog of the upper-bound property exhibited by the Born-Huang approximation for the lowest-energy eigenvalue of a system. An examination is made of the variational characteristics of the present class of approximations, for the Hamiltonians ordinarily encountered, and for those adiabatic projections that are restricted to be a real, complete, and irreducible set that depends only parametrically on certain of the degrees of freedom of the system having classical counterparts. It is established, in these circumstances, that any greatest lower bounds for the partition function that may be generated by these approximations cannot generally be associated with a stationary value of that quantity.

Extensive inelastic-neutron-scattering experiments have been performed on superfluid helium over a wide range of energy and momentum transfers. A high-resolution study has been made of the pressure dependence of the single-excitation scattering at the first maximum of the dispersion curve over the range 1-24.4 atm and at 1 atm of the wave-vector dependence of the single-excitation scattering beyond the roton minimum. The latter data, but not the former, are consistent with the idea of an instability threshold suggested by Pitaevskii and by Enz. The pressure and wave-vector dependence of the multiexcitation scattering was also studied. It is shown that the multiphonon spectrum of a simple Debye solid with the phonon dispersion and single-excitation cross section of superfluid helium qualitatively reproduces these data.

Hydrodynamic fluctuations of a horizontal liquid layer heated from below are considered in the vicinity of the point where convection sets in because of buoyancy. It is assumed that convection occurs in the form of nearly two-dimensional rolls. Close to the instability, the hydrodynamics (described in the Boussinesq approximation) is simplified considerably by the appearance of a slow mode which dominates the motion of all hydrodynamic variables. It is described by a slowly varying complex amplitude whose absolute value and phase describe the strength and the position of the convection rolls, respectively. Generalizing previous work by several authors, an approximate equation of motion is derived, which is satisfied by the slow variable. New in this analysis is the inclusion of fluctuating terms, which leads to a Langevin equation. The fluctuations are shown to satisfy a detailed-balance principle. Consequently, a generalized thermodynamic potential can be defined, which was discussed briefly in an earlier paper. It depends as a functional on the slow variable, which thereby assumes the role of an order parameter of the transition. I give a further evaluation of the hydrodynamic fluctuations for a horizontally unbounded liquid layer on both sides of and at the Bénard point by using my potential and applying various approximations. For strictly two-dimensional flows (i.e., independent of one horizontal coordinate) I calculate the time-independent steady-state properties (coherence lengths) without any further approximation by relying on published numerical data obtained for one-dimensional Ginzburg-Landau fields. Dynamic steady-state properties (coherence times) for that case and fluctuations in the three-dimensional case are calculated in a quasilinear approximation which reproduces the time-independent results for two-dimensional flows reasonably well. In the purely heat-conducting region my results contain some earlier results of Zaitsev and Shliomis in lowest order. Large and long-lived fluctuations of velocity and temperature are shown to appear at the critical wave number as the liquid is brought near the convection instability. They are due to the random appearance and disappearance of convection cells. Their size and lifetime at the Bénard point are only limited by nonlinear coupling of the critical modes to other passive modes. In the heat-convection region, the coupling to passive modes stabilizes the amplitude of the convection cells; only slow fluctuations of the positions of the rolls remain (for unbounded layers) and destroy the long-range order of the one-dimensional roll lattice, in agreement with well-known general theorems. If the Bénard point is approached from this side, the stabilizing influence of the passive modes decreases and is efficient only for the large fluctuations at the Bénard point. Approached from either side, the Bénard point resembles the critical point of a Landau phase transition. The width of the region around the Bénard point where the Landau description breaks down is calculated and found to be unobservably small in realistic liquids. An experimental check of these results, though very tedious, seems possible and very worthwhile.

A number of properties of the exchange operator are discussed in detail. (i) The problem of a harmonic crystal with a one- and two-phonon exchange operator is solved exactly. The matrix elements of the exchange operator are related to experimental quantities. The one- and two-phonon exchange operator does not provide an explanation of the experiment of Kirk and Adams [Phys. Rev. Lett. 24, 392 (1971)]. (ii) The surface integral formulas for the one- and two-phonon exchange operators are shown to give the wrong magnitude and sign. It is argued that the exchange operator must undergo the same kind of renormalization as the spring constants of the phonons. (iii) A four-spin contribution to the generalized exchange operator is identified that has the qualitative features required to explain the anomalous low-temperature properties of solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$.