By measuring the pressure dependence of the velocity of sound, we have determined both the pressure dependence of the density and the Grüneisen constant $u$ of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. Measurements were made below 0.1 K and in the vicinity of 0.5 K. Our determinations of the pressure dependence of the density agree quite well with that determined by Boghosian and Meyer, who used a capacitance bridge. Since the latter results rely on the validity of the Clausius-Mossotti relation and a pressure-independent electric polarizability, the present work can be interpreted as supporting both of these assumptions. We found that $u\left({\rho }_0\right)\equiv \left(\frac{\rho }{c}\right)\frac{\mathrm{dc}}{d\rho }=2.84\mathrm{}$ under the vapor pressure at 0.1 K. Using this value of $u$ to calculate the attenuation of sound according to a three-phonon mechanism, we obtain an attenuation of less than half the measured value. Thus, the present theory of sound attenuation must be incomplete.

Electronic multipolar polarizabilities of closed-shell isolated atomic systems are calculated by a variational method using, as trial functions, the perturbed wave functions given by the uncoupled Hartree-Fock method. The external field is replaced by an effective parametric field taking into account the coupling between the deformations of the various orbitals. Numerical results obtained for ${\mathrm{F}}^{-}$ and ${\mathrm{Cl}}^{-}$ show a much better agreement with the experimental data.

The absolute electron-impact ionization cross section of the metastable $\mathrm{He}\left(2s{}_{}{}^3{}_{}{}^{}S\right)$ state of helium has been measured by the application of a double-electron-beam technique. The cross section at the plateau is about 6.5×${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$ ± 30%. The ionization of the metastable $\mathrm{He}\left(2s{}_{}{}^1{}_{}{}^{}S\right)$ state of helium has also been observed. This absolute ionization cross-section measurement is in good agreement with the experimental results for lithium and with the theories for $\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$ when compensation for the differing ionization potentials is made.

Measurements of the volume recombination coefficient of a highly ionized potassium plasma demonstrate the strong inverse temperature dependence predicted theoretically, and provide further confirmation of the three-body nature of the recombination process.

The beam-foil light source was used for a study of the Stark effect on He II ($\lambda 4686$Å) and ${\mathrm{H}}_{\beta }$. For the former, the $p$ and $s$ components were detected for transverse and longitudinal fields. The transverse-field data suggest that levels of high $j$ are underpopulated relative to their equilibrium numbers. The longitudinal-field data are anomalous. For $H_{\beta }$, splitting was observed, but the arrangement precluded the detection of polarization. The data also imply underpopulation of levels of high $j$.

Many-body perturbation theory is used to calculate the parallel and perpendicular dynamic polarizabilities of molecular hydrogen. High-order terms in the perturbation expansion are included and the poles of $\alpha \left(\omega \right)$ are calculated. Results for the dynamic dipole polarizabilities and for the molecular anisotropy are in good agreement with experiment.

Cross sections for production of inner-shell vacancies in atoms by proton impact have been examined in an impulse approximation. The model used includes modifications to account for the nuclear repulsion of the proton. Comparisons with experiment and with other calculations are made. It is found that the model compares favorably with existing Born approximations. If the most recent fluorescent-yield data are used, the present results agree well with experiment for proton energies larger than ∼ 300 times the electron binding energies. Determination of fluorescent yields by means of such experiments is also discussed.

We report here measurements of the absorption coefficient and velocity of ultrasound in the critical liquid-mixture aniline-cyclohexane ($T_c=29.8\mathrm{}°$C) as a function of frequency and temperature. The standard pulse technique was used for the absorption measurements over the frequency range between 1 and 75 Mc/sec. In this range, measurements have been performed at a minimum of 8 and a maximum of 11 frequencies. The temperature range covered is between 30.5 and 36.2°C. Velocity measurements have been performed in the same temperature range but only at 2 and 15 Mc/sec. The experimental data have been analyzed in terms of Fixman's theory. We also include a discussion of this theory, together with a comparison with the more recent works by Romanov and Solov'ev, Kawasaki and Tanaka, and Kadanoff and Swift. Our results are not conclusive, but we have provided further indications that the sound-diffusion coupling, originally suggested by Fixman, is the only mechanism responsible for the excess absorption in a region not too close to the critical one. Some limitation appears for the temperature closest to the critical point.

The exchange constant $J$ is calculated in helium mixtures, using the single-particle theory of Nosanow for atomic configurations likely to yield large enhancements of $J$ over the pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ value. ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atoms near a pair of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atoms are first considered. Little relaxation (the same as around 1 ${}_{}{}^4{}_{}{}^{}\mathrm{He}$) and change in the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ functions are found. Nonspherical Gaussian functions ${\phi }_{\mathrm{NS}}$ are then used to describe ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ surrounded by 6 ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ as first neighbors in otherwise pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The ${\phi }_{\mathrm{NS}}$ obtained on energy minimization is nearly spherical and little changed from its pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ value. From these examples, no enhanced $J$ is predicted for mixtures. This is in agreement with recent high-temperature measurements on ${}_{}{}^4{}_{}{}^{}\mathrm{He}$-rich mixtures, but disagrees with low-temperature measurements of the exchange specific heat which suggest a significantly enhanced $J$.

Time-dependent perturbation theory is used to evaluate the angular distribution of electrons emitted from heavy atoms in collisions with protons and other heavy charged particles. The incident particle is assumed to move on a straight-line trajectory. Hydrogenic wave functions are used for the initial electron state, and continuum Coulomb wave functions for the final electrons. The angular distributions of electrons emitted from $K$ and $L$ shells by a given projectile are compared. The separate contributions of the $L$ subshells to the angular distributions are also calculated.

The propagation of a Gaussian light pulse through a medium having a positive or negative absorption line is examined. Analytical approximations are obtained for the case where the spectral width of the pulse is much smaller than that of the line. It is shown that the pulse remains substantially Gaussian and unchanged in width for many exponential absorption depths, and that the locus of instants of maximum amplitude follows the classical expression for the group velocity, even if this is greater than the velocity of light, or negative. Numerical calculations have been used to examine what happens beyond the limit of usefulness of the analytical approximations.

Metastable He atoms produced by charge exchange of slow ${\mathrm{He}}^{+}$ with He were detected by secondary electron emission from a tungsten target. The cross section for metastable production rose from a threshold near 22 eV to a peak value of 2.2×${10}^{-18\mathrm{}}$ ${\mathrm{cm}}^2$ ±50% at 27 eV and decreased to a minimum at 32 eV (center-of-mass energy). A subsequent rise was measured up to 110 eV and was attributed to the increased production of ${\mathrm{He}}^m$ and, to a smaller extent, to kinetic emission of secondary electrons at the higher energies.

The diamagnetic susceptibility of an electron gas is reduced by Coulomb interactions below the Landau value in proportion to $\epsilon \beta {{\kappa }_0}^2$, where $\epsilon$ is the plasma parameter, ${\kappa }_0$ is the Debye screening constant, and $\beta =\frac{1}{\mathrm{kT}}$.

The quadrupole polarizability ${\alpha }_q$ which measures the induced quadrupole moment produced by an external field gradient has been calculated for three ions: ${\mathrm{Rb}}^{+}$, ${\mathrm{Cs}}^{+}$, and ${\mathrm{Bi}}^{5+}$, and for the neutral alkali atoms. The present calculations for ${\mathrm{Rb}}^{+}$ and ${\mathrm{Cs}}^{+}$, using the appropriate Hartree-Fock wave functions, represent a considerable improvement over our previous calculations, in which Hartree wave functions (without exchange) were employed. The values obtained for ${\alpha }_q$ of the alkali atoms are very large, ranging from 60.0 ${\mathrm{\AA }}^5$ for Li to 440.8 ${\mathrm{\AA }}^5$ for Cs. In Sec. III, we have collected the values of ${\alpha }_q$ for 11 ions, as obtained using the author's method of direct solution of the inhomogeneous wave equation. Wherever possible, these values have been compared to those obtained by other methods. The agreement is generally very good.

The two-dimensional Magnus formula ${\overrightarrow{\mathrm{f}}}_m=\rho ({\overrightarrow{\mathrm{v}}}_s-{\overrightarrow{\mathrm{V}}}_v)\times \overrightarrow{\kappa }$ can be extended to three-dimensional fluid-core vortices of arbitrary core structure by a precise redefinition of the stream velocity vector ${\overrightarrow{\mathrm{v}}}_s$, vortex velocity vector ${\overrightarrow{\mathrm{V}}}_v$, and circulation vector $\overrightarrow{\kappa }$. For example, ${\overrightarrow{\mathrm{V}}}_v$ becomes the velocity, in a plane slicing the vortex core, of the "center of mass" of the vorticity. The exact three-dimensional formula is derived by analyzing the motion of vorticity relative to the fluid particles - a relative motion caused by the action of nonconservative forces on the vortex core (e.g., the electric force on a charged vortex ring in superfluid helium). The analysis deals with the vorticity field $\overrightarrow{\omega }\equiv \overrightarrow{\nabla }\times \overrightarrow{\mathrm{v}}$ and, thus, can lead to simple formulas and insights in cases where the velocity field $\overrightarrow{\mathrm{v}}$ may be impossibly complicated. Examples introduced are (i) the concept of a "conserved vorticity current," (ii) a possible classical mechanism for the creation or destruction of vortex lines in superfluid helium, and (iii) a simple technique for analyzing the effect of viscosity on the structure of a vortex core.

The dissipation of energy in the flow of an ideal incompressible fluid is described by a theorem whose derivation relies upon the exact three-dimensional Magnus formula discussed in the previous paper. The theorem, which explicitly demonstrates the role of vortex motion in the process of energy dissipation, can be used to calculate the trajectories of vortices. Also derived is a detailed Josephson equation - an extension of Anderson's "new corollary in classical hydrodynamics" - which provides an exact non-time-average relation between chemical potentials and vortex motion.

The classical binary-encounter model for the ionization of ions by impact of electrons given by Thomas and Garcia has been extended to calculate the excitation cross section of ions. The electron-impact excitation cross sections for the $3s-3p$ and the $3s-3d$ transitions in ${\mathrm{Mg}}^{+}$, the $2s-2p$ transition in ${\mathrm{Be}}^{+}$, and the $4s-4p$ transition in ${\mathrm{Ca}}^{+}$ have been calculated. The results are compared with calculations based on the close-coupling and the Coulomb-Born approximations. Our results agree better with close-coupling calculations than with calculations based on the Coulomb-Born approximation.

A general method is presented for calculating the $f$ values for all permitted dipole transitions involving simultaneous photoionization and photoexcitation of helium. The cross section for ions excited into the $n=2\mathrm{}$ state is compared with experiment and with a similar calculation by Salpeter and Zaidi; their result, that the ejected electron is in a $p$ state is confirmed. The probability for the residual ${\mathrm{He}}^{+}$ ion to be excited to a state of principle quantum number $n$ is evaluated for $n=2-10\mathrm{}$, and the probability for the photoelectron to be in a $p$ state is compared with the probability for it to be emitted in an $s$ state. It is found that the inclusion of contributions from simultaneous photoionization and photoexcitation, and from double ionization in the total helium photoionization cross section, improves the agreement between theory and the experimental data at high energies.

We study the coupling of the surface oscillations of a free ${\mathrm{He}}^3$ film to the ${\mathrm{He}}^3$ particles of which the film is composed. A check on the self-consistency of the lowest-order theory of this coupling fails because of an infrared divergence. We, therefore, are led to use exact dynamical equations and find that the infrared difficulty then disappears because the exact theory takes the classical Maxwellian distribution of surface vibration properly into account. The resulting form of the coupling may be used to study the possibility of a superfluid pairing transition in a free ${\mathrm{He}}^3$ film.

A method is provided whereby all the possible results of sequences of three-particle binary collisions may be calculated, given the initial velocities of the three particles. All the one-dimensional bounds on the number of collisions given by Doolen are confirmed. It is further found that only sequences of certain types are possible, the longest of which arises when the lighest particle rebounds between the two heavier particles. If in this instance, we let $\theta$ be the angle between the relative velocity of the first and second pairs of colliding particles and remove the one-dimensional restriction, we find that an upper bound on the maximum number of collisions $n_c$ is given by $\frac{{\pi }_c}{{n_c}^{-1\mathrm{}}}>invcos\left[{\left(\frac{m_B{m}_C}{(m_A+m_B)(m_A+m_C)}\right)}^{\frac{1}{2}}cos\theta \right],$ where $n_c$ is the largest integer which satisfies the inequality and $m_A$ is the mass of the lightest particle.