Theoretical studies of the $n=3,4,\mathrm{and} 5$ Rydberg series in ${\mathrm{O}}_2$, converging to ${\mathrm{O}}_2^{+}\left(X {}_{}{}^2{}_{}{}^{}\Pi _g^{}{}_{}{}^{}\right)$, are presented and correlated with electron energy-loss spectra of molecular oxygen, in the region from 7.0 to 10.5 eV. The energy-loss spectra have been studied as a function of the scattering angle for a incident electron energy of 45 eV. The angular dependence determined for various features in the energy-loss spectra have been correlated with the known behavior for transitions between electronic states of certain symmetry, and with the theoretical results, to identify the low-lying Rydberg states. Four vibrational levels of the $\left(3s {\sigma }_g\right) {}_{}{}^3{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$ Rydberg state have been observed superposed on top of the maximum intensity portion of the Schumann-Range continuum with $T_{00}=8.145\mathrm{}\pm 0.020$ eV. The theoretical results, combined with the energy-loss measurements and the high-resolution photon-absorption work, lead to the assignment of the lowest dipole-allowed ${}_{}{}^3{}_{}{}^{}\Sigma _u^{-}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ Rydberg states at $T_{00}\le 9.31$ eV and $T_{00}=9.97\mathrm{}$ eV, respectively.

Angular distributions of electrons ejected from helium by proton impact are calculated in the plane-wave Born approximation using bound and continuum electron wave functions obtained from a Hartree-Fock potential. The results of this calculation are compared with experimental data and good agreement is found when the relative electron-proton velocity is larger than the proton velocity. The good agreement observed for large angles of the ejected electron is to be contrasted with previous calculations, which fell lower than the large-angle-scattering experimental data.

Some $s$-wave scattering parameters are calculated using a close-coupling wave function similar to that of Drachman's but with the explicit addition of the positronium channel and its polarization. The elastic phase shifts are in close agreement with those of Bhatia et al. and the inelastic $R$-matrix diagonal elements show a better phase-shift bound than previous calculations.

The electron-impact cross section and radiative lifetime of the 10-eV metastable state in CO has been investigated from threshold (9.5±0.4 eV) to 45 eV using a time-of-flight apparatus. The absolute cross section at the peak of the excitation function (15.5 eV) was estimated to be 3×${10}^{-18\mathrm{}}$ ${\mathrm{cm}}^2$ to within a factor of 3. The radiative lifetime at 15 eV was found to be 97±15 μsec. Measurements at 45 eV resulted in a smaller lifetime, suggesting the existence of more than one state. The assignment of the state as $b {}_{}{}^3{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ is rejected based on cross section and lifetime arguments, and the $D{}_{}{}^1{}_{}{}^{}\Delta$ and $I {}_{}{}^1{}_{}{}^{}\Sigma _{}^{-}{}_{}{}^{}$ states are suggested as alternate candidtates based on threshold considerations.

The $K$ -shell ionization cross sections for Fe, Co, Cu, Zn, Ga, Ge, and As were measured for proton impact in the range 0.5 to 2.0 MeV. The cross sections were found to be 242±23, 173±15, 94.1±7.8, 70.9±6.9, 56.4±4.6, 36.5±2.7, and 27.1±2.1 b, respectively, at 2.0 MeV. These measurements agree to within less than 17% with predictions of the binary-encounter approximation. Predictions of the plane-wave Born approximation exceed the measured values by approximately a factor of 2 at the maximum deviation. Corrections suggested by Brandt et al. account for this descrepancy. These measurements also indicate a trend for the measured ionization cross sections to drop below the theoretical predictions for low projectile energies.

The close-coupling approximation (c.c.), the Vainshtein-Presnyakov-Sobelman approximation (VPS), and various first-order approximations, including the first Born approximation (B), are compared to accurate normalized experimental differential cross sections (estimated error less than 20%) for excitation of the $2{}_{}{}^1{}_{}{}^{}P$ state of helium for impact energies 29.6, 34, and 40.1 eV, and scattering angles 3°-138°. The most accurate new measurements are those at 29.6 and 40.1 eV. These measurements were normalized to the experimental integral cross sections of Donaldson et al. The experimental differential cross sections of Hall et al. are in good agreement with these present measurements. All the calculations except B include electron exchange. The VPS has the most accurate magnitude of the present calculations at small scattering angles but it is still too large (by a factor which at 45° is about 2.5 at 29.6 and 34 eV and is about 1.5 at 40.1 eV). The c.c. has the most accurate magnitude at large scattering angles and the most accurate angular dependence at all scattering angles, but it is not in good agreement with the measurements at the largest scattering angles.

The angle and energy dependence of the differential cross sections are reported for resolved vibrational transitions in low-energy collisions of ${\mathrm{H}}^{+}$ with ${\mathrm{H}}_2$, HD, and ${\mathrm{D}}_2$. The specific transition probabilities for the different isotopic molecules scale in the order ${\mathrm{D}}_2$ > HD > ${\mathrm{H}}_2$ at the same initial kinetic energy and scattering angle, but when compared at the same reduced kinetic energy, $E_r=\frac{E}{\hslash \omega }$, are approximately equal, suggesting that the interaction leading to vibrational excitation in this system is primarily the dilution of the molecular bond as the proton passes. We discuss the results in terms of a semiclassical model for vibrational excitation based on an oscillator driven by a time-dependent force.

The angular correlation for ($e,2e$) experiments involving knock out of the least-bound electrons in helium and argon is compared with theory. A distorted-wave off-shell impulse approximation is derived and shown to fit the data within experimental error. Outstanding features are that the shape of the angular correlation at energies from 200 to 800 eV depends only on the momentum transfer. Hence the angular correlation is directly related to the momentum wave function of the struck electron. The reaction is sensitive enough to distinguish between different approximations for the wave function.

Using the heat-pulse technique, we have measured the velocity spectrum of atoms evaporating from a free liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ surface. In the ballistic regime (at sufficiently low ambient temperatures) this spectrum is found to be Maxwell-Boltzmann in nature with no effects evident owing to the quasiparticle nature of the liquid. At higher vapor pressures (higher temperatures) the transition from ballistic atomic propagation to collective adiabatic sound in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ has been studied.

The recently measured associative ionization rates in helium plasmas indicate an extremely rapid vibrational relaxation of the resulting molecular ions. Autoionization of weakly bound (high-$n$) molecular states, in near Saha equilibrium with the molecular ions in the corresponding vibrational level $v$, is shown to yield the required rates of vibrational deexcitation.

Exact results are presented for one-dimensional models of a band problem, an elastic solid, and a classical relativistic field theory with broken symmetry.

The thermodynamic properties of several systems of multilevel atoms interacting with a quantized radiation field are investigated. We allow a quantum-mechanical treatment of the translational degrees of freedom and do not require the rotating-wave approximation. In the finite-photon-mode case one can calculate the free energy per atom in the thermodynamic limit exactly and rigorously. In the infinite-mode case we only get upper and lower bounds, but these are sufficient to give conditions for thermodynamic stability and instability. The kind of phase transition previously found by us for the one-mode Dicke model with the rotating-wave approximation persists in the general multimode case.

We consider the correlation function $G_R$ for two vertical arrows in the same column. We calculate the critical index $\nu$ of the correlation length, and we find the scaling relation $\nu =1-\frac{\alpha }{2}$ is satisfied. In the decoupling limit, we prove that the correlation length is not determined by the next-largest eigenvalue. In order to obtain the correct correlation length, it is necessary to integrate over the entire band of complex next-largest eigenvalues. We argue that this is also the situation in the general case of the eight-vertex model. Under certain well-defined assumptions, we compute the correlation length of $G_R$. We also calculate the low-lying excitation energies of the $X-Y-Z$ Hamiltonian. In addition to free states existing for $0<\mathrm{}\mu <\pi$, there are bound states appearing in the spectrum for $\mu >\frac{\pi }{2}$. In the course of our work, we have rewritten the results of Cheng and Wu for the Ising-model correlation functions in an elegant form, using Baxter's elliptic-function parametrization.

Separation of an arbitrary potential $\varphi$ into a short-range, repulsive part ${\varphi }_0$ and a weak correction ${\varphi }_1$ affords the possibility of describing the $\varphi$-system properties as corrections to the assumed-known ${\varphi }_0$ reference system. We derive here an expression for such a correction of the classical Helmholtz free energy that is the analog of a result familiar from the development of the hypernetted-chain integral equation. Other corrections are obtained therefrom, including a corrected pair-distribution function proposed earlier. All results are easily adapted for numerical calculation.

It is of interest to solve the master equation of a one-dimensional Ising model in a magnetic field (Glauber's equation) not only as a mathematical problem, but also for some practical applications. While an exact treatment of Glauber's equation seems impossible, we consider its approximate solutions. We review the applications of the perturbation-expansion method and applications of the mean-field theory to this problem. We also introduce a local-equilibrium method that improves the mean-field theory by taking the short-range correlation into account. The new method yields an analytically soluble differential equation, which, contrary to the preceding two methods, has an exact steady-state solution.

A drift-tube—mass-spectrometer apparatus has been used to determine the mobilities of mass-identified atomic ions in the noble gases as a function of $\frac{E}{p}$ (drift field/gas pressure) at $T=295\mathrm{}$ K. The measured values of the reduced mobilities ${\mu }_0$ of ${\mathrm{He}}^{+}$ in He, ${\mathrm{Ne}}^{+}$ in Ne, and ${\mathrm{Ar}}^{+}$ in Ar are in agreement with earlier investigations that did not employ mass identification. The measured value at $\frac{E}{p}=O$, ${\mu }_0\left(H^{+} \mathrm{in}\mathrm{He}\right)=\left({31.0}_{-1.0\mathrm{}}^{+0.5\mathrm{}}\right)$ ${\mathrm{cm}}^2$ ${\mathrm{V}}^{-1\mathrm{}}$ ${\sec }^{-1\mathrm{}}$, is in good agreement with quantum-mechanical calculations, but the observed decrease of ${\mu }_0$ with ion energy is somewhat slower than that predicted by theory. The measured mobility of each of the noble-gas ions ${\mathrm{Ne}}^{+}$, ${\mathrm{Ar}}^{+}$, ${\mathrm{Kr}}^{+}$, and ${\mathrm{Xe}}^{+}$ in He is found to be $\sim \mathrm{}(6-12)\%$ lower than that of the corresponding alkali-metal ion (as determined by Tyndall's group).

The Jastrow wave function with relative angular-momentum-dependent correlation functions, is shown to give the back flow of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ around the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ impurity. Variational calculations with these wave functions and the Lennard-Jones (6, 12), and Bruch-McGee-2 potentials respectively give $m^{*}=2.1\mathrm{}{m}_3 \mathrm{and} 2.25m_3$. A simple density dependence of $m^{*}$ is discussed.

The thermodynamics of fermions in one dimension with a repulsive $\delta$-function interaction is analyzed in some detail. The excitation spectrum at finite temperature is also derived. It is shown that the spectrum can be classified into two types of excitation.

A general mathematical form of a surface condition is derived for any extensive physical quantity by considering a global balance equation. Because of the validity of both the conservation laws and the second law of thermodynamics, a distinct set of surface conditions must hold for any special discontinuous thermohydrodynamic system. This set of surface conditions is established for the classical liquid-vapor system to show that the whole set of surface conditions must be taken into account if mass and energy transport across the interface takes place. In particular, the surface density of entropy production is determined quantitatively and is written as the bilinear expression of mass and energy flux and their conjugated forces. It is shown that the form of the expression is independent of whether or not the surface tension is taken into account. The method presented for finding the production of entropy on an interface is of importance not only in connection with the classical liquid-vapor system but also in other branches of physics, for example in the physics of liquid helium II, in atmospheric physics, and in the theory of shock waves.

The decay rate of the order-parameter fluctuations in fluids near the critical point can be determined by measuring the linewidth of the central component in the spectrum of the scattered light, and many such experiments have been reported in the past eight years. In the past two years the dynamical theories developed by Kawasaki and Ferrell to describe the decay rate have been modificed to take into account the anomaly in the shear viscosity and the effect of departures of the correlation function from the Ornstein-Zernike form, and at the same time new accurate measurements of the parameters which enter the theory (the correlation length $\xi$ and the shear viscosity ${\eta }_s$) have been reported. Thus we are able to present here an absolute comparison of the refined mode-mode coupling and decoupled-mode theories with the linewidth data for seven fluids: carbon dioxide, xenon, sulfur hexafluoride, isobutyric acid-water, 3 methylpentane-nitroethane, aniline-cyclohexane, and 2,6 lutidine-water. These linewidth data were obtained in many different laboratories over the past few years; however, in addition to linewidth data previously reported, we also include in our analysis new data which we have obtained for carbon dioxide and xenon and a tabulation and detailed error analysis for all our data for these two fluids. The theories describe only the "critical part" of the decay rate ${\Gamma }^C$; hence the nonanomalous background contributions are first substracted from the measured linewidths to obtained ${\Gamma }^C$. Then it is shown that the resultant values for a quantity we call the "scaled linewidth," ${\Gamma }^{*}\equiv \left(\frac{6\pi {\eta }_s{\Gamma }^C}{k_{B}T{q}^3}\right)$, are described by a single universal curve as a function of $q\xi$, for all fluids and every thermodynamic path that has been investigated near the critical point. This universal curve is described remarkably well by the modified mode-mode-coupling expression of Kawasaki and Lo and the similar decoupled-mode-theory expression of Perl and Ferrell. The accuracy of this comparison, which involves no adjustable parameters, is limited to $\sim 10\%$ by the uncertainties in the background corrections, linewidths, viscosities, and correlation lengths, and by the uncertainties in the various modifications to the theories. The two theories differ significantly only in the extreme nonhydrodynamic region ($q\xi \gg 1$), where the decoupled-mode values for ${\Gamma }^{*}$ are $\sim 10\%$ smaller than those predicted by the mode-mode-coupling theory. Although the available data in the extreme nonhydrodynamic region appear to be described somewhat better by the decoupled-mode theory than the mode-mode-coupling theory, this result is suggestive rather than conclusive since the data in this region are sparse and exhibit considerable scatter.