The cross sections for $L$ electron ionization by incident heavy, but slow, charged particles are evaluated. Incident particles are described in the plane-wave Born approximation, while relativistic wave functions are used for the atomic electrons. Numerical results are given for holmium and mercury and compared with experimental data for holmium. The results are discussed.

A general formalism for the computation of low-energy inelastic and elastic electron scattering cross sections within the context of the particle-hole Bethe-Salpeter equation is presented and shown to reduce to the random-phase approximation (RPA) in lowest order. The theory is then applied to triplet elastic electron-${\mathrm{He}}^{+}$ scattering. A short discussion of the differences between the RPA for scattering processes and for low-lying bound states is given, and the numerical methods used to solve the equations are considered in some detail in an appendix.

Close-coupling calculations are reported for low-energy H-${\mathrm{H}}_2$ collisions. These results are compared with Tang's distorted-wave calculations of the differential and total cross sections for the $j=0\mathrm{}$ to $j=2\mathrm{}$ rotational transition. The close-coupling results for the total cross section are found to agree quantitatively with the distorted-wave results. Furthermore, the relative transition probabilities for various partial waves determined by Tang agree qualitatively with the close-coupling results reported here. The strong forward peaking in the cross section $\sigma \left(\theta \right)$ reported by Tang is also found in the present study. However, there appear to be significant differences in the magnitudes and positions of oscillations in the differential cross sections as determined by the close-coupling and distorted-wave methods. Finally, close-coupling results for the $j=0\mathrm{}$ elastic scattering differential cross section are presented and discussed briefly.

Low-energy electron-ejection cross sections of atoms in collisions with relativistic electrons have been calculated. The calculations are similar to those of Weber et al. In the approximate integration of the differential cross section over certain angles, Weber et al. considered $Z$ small and obtained a ${\left(\alpha Z\right)}^4$-dependent result. We have followed a different type of approximation, valid for $Z\gg 1$, and obtained the explicit ${\left(\alpha Z\right)}^5$-dependent result. The result shows a much stronger dominance of the distribution of ejected electrons over the "Parzen distribution" in the low-energy region than the results of Weber et al. when applied to atoms with $Z\gg 1$.

In the collision ${\mathrm{He}}^{+}$ + He→${\mathrm{He}}^{+}$ + ${\mathrm{He}}^{*}$, excitation at low incident ion energies (≲ 1 keV) proceeds via a pseudocrossing of the elastic ${}_{}{}^2{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}$ diabatic potential with excited-state ${}_{}{}^2{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}$ potentials. As previously reported, this excitation mechanism fails to explain diverse experimental data concerning total excitation cross sections. A new physical mechanism is hypothesized, its existence verified, and it is shown to provide good qualitative and semiquantitative interpretation of the observations. Nonadiabatic couplings among some excited states occur at pseudocrossings of the respective inelastic-channel molecular potentials at large internuclear separations, resulting in coherent phase interference in the inelastic-scattering amplitudes. A linear-combination-of-atomic-orbitals (LCAO) calculation of 18 excited-state ${}_{}{}^2{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}$ potentials of the intermediate ${\left(\mathrm{He}_2^{}{}_{}{}^{+}\right)}^{*}$ system verifies the presence of these outer pseudocrossing. Such a mechanism is shown to be likely in many ion-atom collisions.

The drift velocity and rates of formation of the $\mathrm{O}_4^{}{}_{}{}^{-}$ ion have been investigated in oxygen at 310 °K and pressures between 0.5 and 4 Torr. This paper reports the first determination of the rate of formation of the $\mathrm{O}_4^{}{}_{}{}^{-}$ ion. The rates for the forward and reverse reactions $\mathrm{O}_2^{}{}_{}{}^{-}+{\mathrm{O}}_2+{\mathrm{O}}_2\rightleftharpoons \mathrm{O}_4^{}{}_{}{}^{-}+{\mathrm{O}}_2$ were found to be 3×${10}^{31}$ cm${}_{}{}^6{}_{}{}^{}/\mathrm{s}\mathrm{e}\mathrm{c}$ and 2×${10}^{-14\mathrm{}}$ ${\mathrm{cm}}^3$/sec, respectively. The drift velocity of $\mathrm{O}_4^{}{}_{}{}^{-}$ is essentially identical to that of $\mathrm{O}_2^{}{}_{}{}^{-}$ with a low-field reduced mobility ($\mu N$) of 5.9×${10}^{19}$ ${\mathrm{}\left(\mathrm{V}\mathrm{cm}\sec \right)\mathrm{}}^{-1\mathrm{}}$. These measurements indicate that not only can $\mathrm{O}_4^{}{}_{}{}^{-}$ play an important role in experiments involving negative ions in oxygen, but also that any future attempts to model the negative-ion chemistry of the atmosphere must include the reactions of the $\mathrm{O}_4^{}{}_{}{}^{-}$ ion.

The "area" and the "slope" determination of reaction rates from the ion arrival-time spectrum of Tyndall drift tubes are generally thought to yield identical results. They are shown to be different when lateral diffusion and the difference in ion temperature between the parent and the daughter species are taken into account. Both methods yield only an apparent, instead of the real, rate of the ion-molecule reactions. The apparent rate determined from the area method is usually much closer to the real rate than that determined from the slope method. Formulas and diagrams are provided in order to estimate whether the area method of rate determination is sufficiently accurate for the purpose of an experiment. The dependences of the apparent rate on the radius of the drift tube, $\frac{E}{P_0}$, the initial ion distribution, and the collector size are discussed. One important conclusion is that the exact solutions from the transport equations are not necessary in an appreciable number of situations for fairly accurate determination of reaction rates, provided that the area method is used. The relevance of this analysis to other methods of data analysis in drift tubes is discussed. A method for a relative-abundance calibration of a mass spectrometer is suggested.

A detailed critical analysis is made of Stueckelberg's treatment of inelastic transitions at a crossing of two potential curves. Using an asymptotic method analogous to the WKB approximation, Stueckelberg obtained the well-known Landau-Zener-Stueckelberg (LZS) formula for the inelastic transition probability. His method involved the determination of "connection formulas" linking amplitudes associated with his asymptotic approximants on either side of the crossing-point region. Here we show that (a) Stueckelberg's asymptotic approximants are just the WKB approximants for elastic scattering on the adiabatic (noncrossing) potential curves; (b) Stueckelberg's method for obtaining the connection formulas can be put on a rigorous footing, including sufficient conditions for its validity, using the classical trajectory equations derivable from a general semiclassical theory of inelastic atomic collisions; (c) there is an undetermined phase in the $S$ matrix, which Stueckelberg incorrectly assumed to be zero, and which has the value $\frac{1}{4}\pi$ in the distorted-wave approximation; (d) Stueckelberg's derivation is not valid whenever the inelastic transition probability is small, either in the rapid-passage (diabatic) case or the near-adiabatic limit; (e) for realistic model parameters, the conditions needed for Stueckelberg's derivation to be valid are almost never satisfied. Since the LZS formula is known from numerical computations to be valid under some conditions when the Stueckelberg derivation is not valid, we conclude that analysis via connection-formula methods is not a useful technique for treating the crossing problem. In an appendix we derive an analytical result for the Stokes's constants determining the Stueckelberg connection formulas. The result is an absolutely convergent, infinite series whose numerical evaluation would yield exactly the unknown phase associated with the LZS formula.

Calculations of $\alpha$-particle stopping cross sections based on the theories of Lindhard et al. using the Hartree-Fock-Slater atomic-charge distribution provide generally good agreement with previously obtained experimental results. In particular, the calculations are helpful in understanding the structure which results when, for fixed energy of the $\alpha$ particle, the stopping cross section is plotted against the atomic number of the stopping medium.

A spin-1 Ising model, which simulates the thermodynamic behavior of ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures along the $\lambda$ line and near the critical mixing point, is introduced and solved in the mean-field approximation. For reasonable values of the parameters of the model the phase diagram is qualitatively similar to that observed experimentally and the phase separation appears as a consequence of the superfluid ordering. Changing the parameters produces many different types of phase diagram, including as features $\lambda$ lines, critical points, tricritical points, and triple points. Certain thermodynamic features which differ from the ${\mathrm{He}}^3$-${\mathrm{He}}^4$ experiments may be artifacts of the mean-field theory.

A theory is proposed to explain the Raman spectra of inert solutions of diatomic molecules; this theory can also be considered as an approximate theory of Raman spectra of pure liquids. The theory is a stochastic-type theory related to similar theories of infrared and NMR spectra. The vapor-solution band shifts are shown to depend on the difference between the solvent-solute interaction energies of the two vibrational states involved in the transition. The analysis of the band profiles is made separately for the isotropic and anisotropic components, respectively, of a Raman spectrum. The former spectrum is produced by vibrational relaxation mechanisms alone; the bands are all asymmetric although, in certain cases, the asymmetric perturbation is small enough to be neglected. The latter spectrum is produced by both vibrational and reorientational relaxation mechanisms. The theory predicts the existence of a continuous sequence of band forms comprising, among others, the profile with an $O-Q-S-$ type structure, the Lorentzian profile, the Gaussian profile, the Voigt profile, and several sorts of asymmetric profiles. The resulting Raman spectrum appears as a superposition, with appropriate coefficients, of an isotropic and anisotropic spectrum. A procedure is indicated permitting a separate study of vibrational and reorientational relaxation effects.

The energy dependence of the size of quantized vortex rings in He II is measured by studying the transmission of charged vortex rings through grids with holes of the order of a micron. A maximum-energy ring which can be transmitted is found for each hole size. These cutoffs are interpreted geometrically and lead to a relationship between ring size and energy which supports the quantization of circulation and the applicability of classical hydrodynamics to this problem. The circulation $\kappa$ and core radius $a$ of the rings are determined to be $\kappa =(1.002\mathrm{}\pm 0.03)\times {10}^{-3}$ ${\mathrm{cm}}^2$ ${\sec }^{-1\mathrm{}}$ and $a=(0.90\mathrm{}\pm 0.50)\times {10}^{-8}$ cm.

On the basis of our previous theory used to calculate the static properties of crystalline ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ at $T=0\mathrm{}$, we compute the exchange interaction of the nuclear spins in bcc ${\mathrm{He}}^3$ as a function of molar volume, obtaining good agreement with experiment. Comparison with the theories of Nosanow and co-workers and of Guyer and Zane shows that the success of the theory is mainly due to the effective-force constant used in the computations.

We have measured in a drift-tube mass spectrometer the mobilities of N${\mathrm{O}}^{+}$, N${\mathrm{O}}^{+}$ · NO, and ${\mathrm{K}}^{+}$ ions in NO at 300 °K. The measurements were made over a substantial range of $\frac{E}{N}$, where $E$ is the drift field intensity and $N$ is the gas number density. The zero-field reduced mobilities were found to be 1.91 ± 0.06, 1.78 ± 0.05, and 2.245 ± 0.067 ${\mathrm{cm}}^2$/V sec, respectively. Measurements were also made of the longitudinal diffusion coefficients of N${\mathrm{O}}^{+}$ and ${\mathrm{K}}^{+}$ ions in nitric oxide at 300 °K as a function of $\frac{E}{N}$. Our results are in fair agreement with those predicted by the Einstein equation from the experimental zero-field mobilities.

Approximate formulas are derived for the correlation function $U\left(r\right)\mathrm{}$ contained in the Bijl-Dingle-Jastrow wave function describing the ground state of a many-body boson system. Abe's formal iteration procedure is followed, but terms linear in the hole function $h\left(r\right)\mathrm{}$ are included to give an additional term to the hypernetted-chain equation. Ground-state energies of liquid ${\mathrm{He}}^4$ and the charged-boson gas are evaluated using the derived expressions for $U\left(r\right)\mathrm{}$, and the resulting energies are compared with those obtained from other approximate expressions for $U\left(r\right)\mathrm{}$.

The rigorous evolution equations for the one-body Green's functions ($g^{>},g^{<}$) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, $N$ (particle number) $\to \infty$, $\Omega$ (volume) $\to \infty$ while $\frac{N}{\Omega }$ remains finite, these equations can rigorously be transformed into closed equations with respect to ($g^{>},g^{<}$).

A derivation of exact expressions, given in terms of thermodynamic quantities, for the $s$-wave parts of the ${\mathrm{He}}^3$-${\mathrm{He}}^4$ and ${\mathrm{He}}^4$-${\mathrm{He}}^4$ quasiparticle forward-scattering amplitudes in very dilute solutions of ${\mathrm{He}}^4$ in liquid ${\mathrm{He}}^3$ is given. The result for the $p$-wave part of the ${\mathrm{He}}^3$-${\mathrm{He}}^4$ amplitude is related to the ${\mathrm{He}}^4$ quasiparticle effective mass. The associated Fermi-liquid parameters are also derived.