The velocity of second sound was measured in superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ between vapor pressure and the melting pressure for $T\ge 1.6$ K. Particular emphasis was given to the temperature region near $T_{\lambda }$ with $2\times {10}^{-5\mathrm{}}\lesssim \epsilon \equiv 1-\frac{T}{T_{\lambda }\mathrm{}\left(P\right)\mathrm{}}\lesssim {10}^{-2\mathrm{}}$. Using these results, other existing thermodynamic information, and linear two-fluid hydrodynamics, the superfluid fraction $\frac{{\rho }_s}{\rho }$ was determined. The higher-pressure data at small $\epsilon$ revealed that $\frac{{\rho }_s}{\rho }$ along isobars cannot be described by a pure power law in $\epsilon$. Instead, singular corrections to the leading power-law term exist which are much larger than of order $\epsilon$. Therefore, the results were fitted to the expression $\frac{{\rho }_s}{\rho }=k\left(P\right)\mathrm{}{\epsilon }^{\zeta }\mathrm{}[1+a(P\left)\mathrm{}{\epsilon }^y\right]$. They yielded $0.66\lesssim \zeta \lesssim 0.68$ and $0.4\lesssim y\lesssim 0.6$, independent of the pressure $P$. These results for the exponents, and the existence of singular corrections, are consistent with scaling, universality, and recent explicit calculations.

Ground states of two typical many-boson model systems are studied: (i) a weakly interacting boson system at low density, and (ii) a low-density hard-sphere boson gas in the low-energy limit. For both of these systems it is shown explicitly that the uniform-limit formalism based on the method of correlated basis functions can be used to obtain, in a relatively simple way, exact results for the ground-state energy per particle and other related properties in the weak-coupling limit. In the problem of the hard-sphere boson gas the $T$-matrix method is found to be particularly useful in dealing with the singular interaction potential.

Critical exponents are functions of $d$, the dimension, and $n$, the number of components of the order parameter involved in the second-order phase transition. They can be expanded in power series of $\frac{1}{n}$ when $n$ is large. We present details of calculating by perturbation theory the critical exponents above $T_c$ for arbitrary $d$ and to $O\left(\frac{1}{n}\right)$ for short-range interacting systems and also for long-range interacting systems.

The persistent flow of a superfluid in a ring is discussed in terms analogous to those previously used for superconductors. The existence of a phase memory around the ring is shown to be responsible for energy minima with a periodic dependence on the total momentum which is directly related to the quantization of circulation. The general features are illustrated by means of the ideal Bose gas and the model of quasiparticles as examples.

We study the collision kernels of two linearized equations for a moderately dense gas. These are the low-density Bogoliubov kinetic equation and the equation at the next order in the density (ternary-collision level), in the form given by Green and Cohen. We reduce the collision kernels of these equations to expressions which are nonlocal in space but local in time, and show that each of them breaks up into static and collisional parts. The former parts agree with the mean-field expressions. We compare the collisional parts with the fully dynamical expressions previously found by Mazenko and us, and find complete agreement only at zero wave vector and frequency. For nonzero wave vectors, the desired symmetry in the momentum variables breaks down. We show that this has no effect on the first-order transport coefficients; it would presumably be significant in the kinetic regime. We also find that the speed of sound predicted by the Bogoliubov equation agrees with the thermodynamic result truncated at first order in the density.

A generalization of the Ashcroft-Lekner (AL) hard-sphere mode for calculation of structure factor is proposed. The AL model is derived from the Wertheim-Thiele solution of the Percus-Yevick integral equation for the case of hard spheres. This generalization introduces a correct hard-sphere equation of state for dense fluids into the AL model. This allows us to determine one coefficient of the AL model exactly and gives a condition on the other two coefficients in terms of the packing fraction. However, in order to determine all the three coefficients uniquely, we need a third condition which is proposed to be obtained from a fit to the experimental data at one point. Our generalization of the AL model is expected to yield a structure factor for dense systems in better agreement with the experimental data, at least for machine calculations on hard-sphere systems.

A general formalism is presented for a system whose Hamiltonian is periodic in time. The formalism is intended to deal with the interactions between bond electrons and an external electromagnetic field, which can be treated semiclassically, such as electric and magnetic polarizations, optical rotation, and transitions among discrete levels. A particular bound-state solution of the Schrödinger equation which belongs to an irreducible representation of the time-translation symmetry group is defined as a steady state, and the characteristic number of the irreducible representation as a quasienergy. It is shown that the defined steady states and quasienergies behave in a newly constructed Hilbert space like stationary states and energies of a conservative system in many respects. It is also shown that for a resonant case the unperturbed quasienergy becomes degenerate and the transitions among discrete levels can be accounted for by the familiar degenerate perturbation procedure. Using a suitable Hilbert space, the steady states are established as firmly as the stationary states stand in the theory of a conservative system.

The total nuclear capture rates of muons by argon and neon have been measured. The experiment was performed by stopping negative muons in a target of ultrapure gaseous deuterium (at 6 atm pressure and 293°K) separately contaminated by small amounts of argon and neon, and observing the differential time distribution of the decay electrons coming from muons stopped within the gaseous mixture. The results are ${\lambda }_{c,\mathrm{Ar}}=(1.20\mathrm{}\pm 0.08)\times {10}^6$ ${\sec }^{-1\mathrm{}}$ and ${\lambda }_{c,\mathrm{Ne}}=(0.30\mathrm{}\pm 0.02)\times {10}^6$ ${\sec }^{-1\mathrm{}}$.

Two methods [D. G. Truhlar and J. K. Rice, J. Chem. Phys. 52, 4480 (1970)] and B. Ritchie, Phys. Rev. A 6, 1456 (1972)] for calculating cross sections for vibrational excitation of molecules by electron impact are compared. The former method is shown to have a greater range of applicability. Cross sections computed by the two methods differ by about a factor of 2 for excitation to the first vibrationally excited level of ${\mathrm{H}}_2$. The neglect of the short-range potential in Ritchie's method is shown to be an important source of error in that method.

We examine a three-parameter formula for the electronic stopping cross section at nonrelativistic velocities of Brice based upon the Firsov formalism. We show that it is substantially a special case of a simple formula applied earlier by Green and Peterson, and that the work of Brice provides an approximate rule relating two of their four parameters.

Soft-x-ray appearance-potential spectroscopy has been used to determine the binding energies of the $N_4-N_7$, $O_1-O_5$, and $P_1-P_3$ levels of clean uranium surfaces. This is the first direct determination of most of these levels. The measured energies of the $N$ and $O$ levels are consistently a few electron volts lower than tabulated x-ray binding energies based on photoelectron spectroscopy of oxidized uranium. Exposure of clean uranium to oxygen, however, did not appreciably shift the appearance potentials. It is suggested that the discrepancy is in part a consequence of a "surface chemical shift." The density of $5f$ states above the Fermi energy was found to be double peaked with a peak separation of 2.5 eV.

Theoretical line-shape calculations, which employ an approximation to the classical theory, are compared with experiment to demonstrate that the main features of the satellite-band problem-the line, the high-intensity red satellites, and the blue satellite—can be described simultaneously using simple intermolecular potentials. The comparisons focus on the cesium resonance transitions for which all of the characteristic satellite structures have been observed. In particular, illustrative calculations are detailed for cesium-xenon systems at relative densities of 5-41 and for cesium-argon at 61, and these results are correlated with experiments performed at lower relative densities. The results indicate that widely different potentials can give very similar line profiles. Ancillary conclusions are presented about the role of Lennard-Jones potentials in line-shape calculations.

The radiative decay of metastable nsnp ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ atomic states is found to occur only by odd-parity multiphoton modes such as $E1M1\mathrm{}$ and $3E1$, in the absence of hyperfine structure. A detailed calculation of the $E1M1\mathrm{}$ rates for several members of the Be sequence is presented, in the nonrelativistic approximation. Only the general properties of $3E1$ decay are presented. Among other results, the $3E1$ rate is found to be zero if any two photons have the same energy, and its spectrum, unlike the symmetrical two-photon spectra, is found to be irregular and asymmetric, being peaked somewhat below half the transition energy.

The hyperfine interaction in the ($1s^{2}2s2p{}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$) excited state of the beryllium atom is of interest since it is the lowest excited state in which hyperfine effects can occur, the ground state being a singlet. In the present work the magnetic hyperfine interaction for ${\mathrm{Be}}^9$ is studied theoretically using the linked cluster many-body perturbation theory including up to second order in the electron-electron interaction, certain classes of ladder diagrams, and selected third-order diagrams. Our theoretical value of $A$ in $A\overrightarrow{\mathrm{I}}·\overrightarrow{\mathrm{J}}$ is composed of - 107.41 MHz from the zero-order valence $2s$ and $2p$ electrons and - 16.80 MHz from higher-order diagrams the total value - 124.21 MHz being in good agreement with the experimental result of - 124.5368±0.0017 MHz. The contributions from various diagrams are interpreted in terms of physical effects such as exchange core-polarization and correlation effects, the influence of $2s2p$ interactions being significant in influencing the hyperfine constant.

We use the Laplace transform obtained by Wichmann and Kroll for the vacuum-polarization charge density of order ${\left(Z\alpha \right)}^3$ induced by a point nucleus to determine the first few terms in the expansion of the charge density for small distances from the nucleus. The result is used to estimate the effect of the polarization charge near the nucleus on the muonic x rays studied experimentally by Dixit et al. The effect is found to be somewhat smaller than an earlier estimate, and the discrepancy between theory and experiment is silightly reduced, but the discrepancy remains large for the high-$Z$ muonic atoms, as much as three or four standard devitations in lead and barium.

A general approximation method is developed to find the state of a system composed of a low-energy free electron and hydrogen atom in an intense electromagnetic field. The electron wave function includes two parts: One, of infinite extension, represents a free-electron state in an electromagnetic field; the other, localized on the atom, is a bound pseudostate. This method complements and takes its place among other known methods used for weak electromagnetic fields or high-energy electrons. It can be applied to interaction scalar potentials which decrease faster than $\frac{1}{r}$.

The pressure shift of the hyperfine splitting $\nu$ in the ground state of the hydrogen isotopes for an argon buffer gas has been measured with about 0.1% accuracy by an optical-pumping method with improvements in sample design, temperature control, and magnetic field stabilization. If the shift is expressed as $\left(\frac{1}{{\nu }_0}\right)\left(\frac{\delta \nu }{\delta \rho }\right)=A+B(T-T_0)$, the results for hydrogen, deuterium, and tritium are, respectively, $A_H=(-4.800\mathrm{}\pm 0.006)\times {10}^{-9\mathrm{}}$ ${\mathrm{torr}}^{-1\mathrm{}}$ (0°C); $B_H=(+0.956\mathrm{}\pm 0.011)\times {10}^{-11\mathrm{}}$ °${\mathrm{C}}^{-1\mathrm{}}$ ${\mathrm{torr}}^{-1\mathrm{}}$ (0°C); $A_D=(-4.802\mathrm{}\pm 0.022)\times {10}^{-9\mathrm{}}$ ${\mathrm{torr}}^{-1\mathrm{}}$ (0°C); $B_D=(+1.091\mathrm{}\pm 0.047)\times {10}^{-11\mathrm{}}$ ${\mathrm{torr}}^{-1\mathrm{}}$ (0°C); $A_T=(-4.795\mathrm{}\pm 0.012)\times {10}^{-9\mathrm{}}$ ${\mathrm{torr}}^{-1\mathrm{}}$ (0°C); $B_T=(+0.929\mathrm{}\pm 0.021)\times {10}^{-11\mathrm{}}$ ${\mathrm{torr}}^{-1\mathrm{}}$ (0°C); where the arbitrary reference temperature $T_0=26\mathrm{}°\mathrm{C}$ has been used in every case. The density $\rho$ is quoted in units of torr (0°C), the equivalent of 3.536×${10}^{16}$ atoms/${\mathrm{cm}}^3$. The absence of isotope dependence supports existing theories of the shift and encourages comparison with the measurements of muonium hyperfine splitting.