An analysis of extended random-phase approximations is made with emphasis on the justification of the simplifying assumptions used in applications. It is demonstrated that in a complete operator space, the equations of motion in expectation value form, supplemented by the ground-state annihilation condition, yield the exact result for excitation operators and are equivalent to the operator equation of motion. Certain operators that are normally neglected have to be included to obtain the exact solution; some of these additional operators are shown to be important while others are expected to have small effects. The random-phase approximation and its self-consistent variants are analyzed, and a number of improvements that can be introduced without increasing the difficulty of the calculations are suggested.

The ratio of the $g_J$ factor of ${}_{}{}^4{}_{}{}^{}\mathrm{He}\left(2\mathrm{}{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\right)$ to that of ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\left(5\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)$ has been measured to a fractional accuracy of ±5×${10}^{-8\mathrm{}}$. The Rb atoms were optically pumped, and the ${}_{}{}^4{}_{}{}^{}\mathrm{He}\left(2\mathrm{}{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\right)$ Zeeman resonance was observed through a combination of spin-exchange collisions and spin-dependent Penning ionization. Linewidths as narrow as 220 Hz were observed for the ${}_{}{}^4{}_{}{}^{}\mathrm{He}\left(2\mathrm{}{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\right)$ Zeeman resonance. The accuracy of this experiment was limited by a systematic shift in the He $g_J$ factor that depended upon the intensity of the discharge used to excite the He atoms to the $2{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ state. Calculations indicate that transfer of coherence in spin-exchange collisions with free electrons contributes to this shift. The measured ratio $\frac{g_J({}_{}{}^4{}_{}{}^{}\mathrm{He}, 2{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{})}{g_J({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}, 5{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{})}=1-46.798\mathrm{}\left(50\right)\mathrm{}\times {10}^{-6\mathrm{}}$. Using the previously measured ratio of the $g_J$ factors of the ground state of rubidium to the ground state of hydrogen, we determine the ratio $\frac{g_J({}_{}{}^4{}_{}{}^{}\mathrm{He}, 2{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{})}{g_J({}_{}{}^1{}_{}{}^{}\mathrm{H}, 1{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{})}=1-23.214\mathrm{}\left(50\right)\mathrm{}\times {10}^{-6\mathrm{}}$. This result agrees with recent theoretical calculations and experimental determinations using atomic-beam techniques, but disagrees with an earlier experimental determination using optical-pumping techniques.

The dissociation energy $D_e$ and the equilibrium proton-proton separation $D_{\mathrm{eq}}$ of ${\mathrm{H}}_2$ are calculated using the methods of arrangement-channel quantum mechanics. This theory is the channel component version of the channel-coupling array approach to many-body scattering, applied to bound-state problems. In the approximation used herein, the wave function is identical to that of the classic Heitler-London-Sugiura valence-bond calculation, which gave $D_e=3.14\mathrm{}$ eV and $R_{\mathrm{eq}}=1.65\mathrm{}{a}_0$, values accurate to 34% and 17.8%, respectively. The present method yields $D_e=4.437\mathrm{}$ eV and $R_{\mathrm{eq}}\simeq 1.42a_0$, accurate to 6.5% and 1%, respectively. Some implications of these results are discussed.

Calculations are presented of the $E1\mathrm{}$ amplitude expected in the $6{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}-7\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ forbidden $M1\mathrm{}$ transition in Tl if parity conservation is violated in the neutral weak $e-N$ interaction, as proposed in a number of gauge models, including that of Weinberg and Salam. Valence-electron wave functions are generated as numerical solutions to the Dirac equation in a modified Tietz central potential. These wave functions are used to calculate allowed $E1\mathrm{}$ oscillator strengths, hfs splittings, and Stark $E1\mathrm{}$ transition amplitudes. These results are compared with experiment and the agreement is generally good. The relativistic $6{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}-7\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ $M1\mathrm{}$ transition amplitude $\mathcal{m}$ is also calculated, and corrections due to interconfiguration mixing, Breit interaction, and hfs mixing are included. The result, ${\mathcal{m}}_{\mathrm{theor}}=(-3.2\mathrm{}\pm 1.0)\times {10}^{-5\mathrm{}}\frac{\mathrm{}\left|e\right|\mathrm{}\hslash }{2m_{e}c}$, is in agreement with the experimental value, ${\mathcal{m}}_{\mathrm{expt}}=(-2.11\mathrm{}\pm 0.30)\times {10}^{-5\mathrm{}}\frac{\mathrm{}\left|e\right|\mathrm{}\hslash }{2m_{e}c}$. The parity-nonconserving $E1\mathrm{}$ amplitude ${\mathcal{E}}_{\mathrm{PN}}$ is calculated, and a value for the circular dichroism, $\delta \simeq \frac{2\mathrm{Im}\left({\mathcal{E}}_{\mathrm{P}\mathrm{N},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}\right)}{{\mathcal{m}}_{\mathrm{expt}}}=2.6\mathrm{}\times {10}^{-3\mathrm{}}$, is obtained. Parity-nonconserving effects in other Tl transitions are discussed.

This work advocates the use in atomic physics of a new relativistic two-body formalism equal in rigor to the Bethe-Salpeter formalism and clearly superior to it in several respects. Outstanding among these is the existence of a Coulomb-like kernel for which the exact analytic solutions of the bound-state equations are known. These solutions are derived and applied in a calculation of the $O\left({\alpha }^{6}m\ln {\alpha }^{-1\mathrm{}}\right)$ contributions to hfs in muonium and positronium. Three previously unknown contributions are found. Theory and experiment are compared.

The theory of the field ionization of the hydrogen atom is developed analytically. The leading term of an asymptotic expansion for the ionization rate—the reciprocal lifetime—is derived. In the weak-field limit, the formula for the ionization rate reduces to $\frac{1}{\tau }=n^{-3\mathrm{}}{[n_2!(n_2+\left|m\right|)!]}^{-1\mathrm{}} {\left(n\frac{{}_{}{}^3{}_{}{}^{}F}{4}\right)}^{-2\mathrm{}{n}_2-{\mathrm{}\left|m\right|\mathrm{}}_{-1\mathrm{}}}\times \exp \left[3\right(n_1-n_2\left)\frac{-2\mathrm{}}{\left(3n{}_{}{}^3{}_{}{}^{}F\right)}\right]$, where $n$, $m$, and $n_1$ and $n_2$ are the usual principal, magnetic, and parabolic quantum numbers, respectively, and $F$ is the field strength in atomic units. For the ground state, this formula agrees with that of Landau and Lifshitz. For all $m^2=1\mathrm{}$ states, the formula agrees asymptotically for large $n_2$ with that of Lanczos after correction of the latter for an error. It is in disagreement with the result of Oppenheimer and with the low-field result of Rice and Good. Significantly better agreement with numerical calculations of Alexander, of Hehenberger, McIntosh, and Brändas, of Damburg and Kolosov, and of Bailey, Hiskes, and Riviere is obtained with a formula for not quite such small $F$, $\frac{1}{\tau }={(-2\mathrm{}\mathrm{Re}E)}^{\frac{3}{2}}{[n_2!(n_2+\left|m\right|)!]}^{-1\mathrm{}}{f}^{-B}\exp [-\frac{1}{\mathrm{}\left(6f\right)\mathrm{}}]$, where $E$ is the perturbed energy, where $f=\frac{\left[{\mathrm{}(-2E)\mathrm{}}^{-\frac{3}{2}}F\right]}{4}$, and where $\frac{B}{2}={\beta }_{2,n_2}$ is the usual perturbed separation constant [${\beta }_2\to \frac{n_2+\left|m\right|\mathrm{}}{2}+\frac{1}{2}$, as $F\to 0$].

If it is assumed that the electron density of an atom in its ground state is piecewise exponentially decreasing as a function of the distance from the nucleus, then it is shown that much improved values for the energy, and electron densities exhibiting shell structure, are obtained from optimization of the density functionals of Thomas-Fermi, Thomas-Fermi-Dirac, and Thomas-Fermi-Dirac with inhomogeneity correction. An inhomogeneity correction which is one-ninth of the original Weizsacker correction is favored. Numerical results are presented for all first-row atoms and selected second-row atoms, and comparisons are made with results of other methods.

$K$-shell x-ray production cross sections and $\frac{K\beta }{K\alpha }$ ratios are presented for 0.6- to 2.4-MeV protons and alpha particles incident on thin targets of selected elements from Nb to Gd. The $\frac{K\beta }{K\alpha }$ ratios are compared to the theoretical predictions of Scofield and agree within 10%. The experimental cross sections are compared to the theoretical predictions of the plane-wave Born approximation (PWBA) and the PWBA modified to include binding energy, Coulomb deflection, and relativistic effects. It is seen that the PWBA modified for binding energy and Coulomb deflection effects agrees with the experimental data for Nb to within uncertainties, but its predictions are lower than the data by larger amounts as the target $Z$ increases. Inclusion of an ad hoc semiclassical relativistic correction to the theoretical cross section, as suggested by Hansen, improves the agreement with the data, but overestimates the data for the highest-$Z$ elements.

This paper examines the spectral yield of molecular-orbital (MO) x rays observed in asymmetric 470-MeV Xe collisions. According to a proposal by Heinig et al., x-ray transitions to $2p\sigma$ vacancies occurring during the collision give an intense continuum spectrum which falls off rapidly above the $K$ x ray lines emitted by the heavier collision partner. Dynamical calculations of the $2p\sigma$ MO x-ray thick-target yield indeed agree within a factor of 3 with the observed intensity. The velocity and impact-parameter dependence of $2p\sigma$ MO x-ray production are discussed.

We examine possible interactions between neutral atoms and metal surfaces at separations greater than 500 Å, treating the Lifshitz dispersion force, dynamic forces, and fringe fields due to surface roughness. Using the derived interaction energies we compute the classical trajectories of atoms passing over metal cylinders and compare them with the results of a recent series of experiments. There is a significant discrepancy between theory and experiment, with theory predicting more scattering than is observed.

Semiclassical expressions of two-photon ionization of two colliding atoms are derived for a wide range of electromagnetic field intensity and detunings from the isolated atom line. The dependence of the ionization yield on the details of the interaction potential of the system is derived. This process promises an extremely sensitive method for studying line broadening on the far wing, especially when absorption or fluorescence becomes very weak.

Cross sections for production of the metastable $\mathrm{H}\mathrm{}\left(2S\right)\mathrm{}$ state in collisions of 6-25-keV protons with atomic hydrogen have been measured in a modulated crossed-beam experiment. The metastable states excited through the process ${\mathrm{H}}^{+}+\mathrm{H}\to {\mathrm{H}}^{+}+\mathrm{H}\left(2\mathrm{S}\right)\mathrm{}$ (direct excitation) and ${\mathrm{H}}^{+}+\mathrm{H}\to \mathrm{H}\mathrm{}\left(2S\right)+{\mathrm{H}}^{+}$ (charge transfer) were quenched in a weak electric field applied at the collision region. The emitted Lyman-$\alpha$ radiation was detected in a direction perpendicular to the plane of the crossed beams. The processes were separable since the direct excitation signals saturated well before the charge transfer signals did as the quench field was increased. The present values were normalized to the absolute total cross section for $\mathrm{H}\mathrm{}\left(2P\right)\mathrm{}$ production measured by Kondow et al. The charge-transfer cross sections are in satisfactory agreement with the measurements of Morgan et al. and Bayfield except at 6 keV, where their values also begin to differ. The direct-excitation cross section differs in shape from the only other measurement of Morgan et al., but agrees in absolute magnitude when normalization uncertainties are included. The coupled-state calculations of Cheshire et al. using a pseudostate expansion show good agreement with the present results, indicating the importance of the inclusion of molecular features at small internuclear separations.

The double-closure Born approximation developed in previous work is used to calculate the asymptotic cross sections for electron loss from ${\mathrm{H}}^{-}$ ions incident on a variety of atomic targets. Corrections to the closure technique for the next-order terms in heavy-ion-atom cross sections are examined. The corrections for energy conservation are shown to be unimportant, but the Bethe ridge in the inelastic form factors does yield terms which contribute to the next leading order. The total asymptotic electron-loss cross section, and the next-order contribution including the Bethe ridge correction, are calculated using various models for the target atoms obtained from the literature. Comparative results based on Hartree-Fock, configuration-interaction, and relativistic Hartee-Fock wave functions are presented. The results for the electron-loss cross sections for N, O, and Ar target atoms are compared with the available experimental data. Results are also given for parameters which are required for the asymptotic elastic and nondetachment inelastic cross sections. Some comments regarding other applications of these methods are included.

We consider the effect of spin-orbit coupling between the molecular orbitals correlated to the united-atom $2p_{\frac{1}{2}}$ and $2p_{\frac{3}{2}}$ states on rotational-coupling transition probabilities in heavy-ion collisions. In low-$Z$, lowvelocity encounters, where the splitting is small, calculations of the rotational-coupling cross section have been shown to be in agreement with experiment. We compute a correction factor to the previous calculations in terms of a general scaling parameter $\xi =\frac{\Delta E}{{\left(k{v}_1^2\right)}^{\frac{1}{3}}}$, where $\Delta E$ is the spin-orbit splitting in the united atom, $v_1$ is the ion velocity, and $k=\frac{Z_1{Z}_2{(Z_1+Z_2)}^2}{40}$ a.u. The correction is generally small, but is observed in high-$Z$ encounters at low velocities.

Single- and double-electron-transfer cross sections have been measured for ${\mathrm{B}}^{3+}$, ${\mathrm{C}}^{4+}$, ${\mathrm{N}}^{5+}$, and ${\mathrm{O}}^{6+}$ in collision with He at velocities between 0.5 and 1.2×${10}^8$ cm/sec. The single-electron-capture cross sections peak near 15×${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$ for each case except ${\mathrm{C}}^{4+}$, where single capture is anomalously low. The double-capture cross sections are about 3×${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$, except for ${\mathrm{B}}^{3+}$, where the highest observed value is 1.5×${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$. These measurements for ${\mathrm{B}}^{3+}$ and ${\mathrm{C}}^{4+}$ compare well with existing experiments and theory, except for ${\mathrm{C}}^{4+}$ single capture. Within the narrow range tested, ${\mathrm{C}}^{4+}$ and ${\mathrm{B}}^{3+}$ cross sections exhibit variation with velocity, but ${\mathrm{N}}^{5+}$ and ${\mathrm{O}}^{6+}$ cross sections remain constant.

The energy dependence of the cross sections for both the $K\alpha$ (one-electron, one-photon) and $K\alpha \alpha$ (two-electron, one-photon) x-ray emissions for Ni-Ni collisions have been measured in the energy range 17.6-91.5 MeV. To account for these observations, we have extended the electron promotion model to the quantitative, ab initio calculation of double $K$-shell vacancies. The results furnish, for the first time, an independent test of both the Briggs-Macek formulation of the electron promotion model and the concept of velocity-dependent exit channels suggested by Fastrup et al. We also present, for the first time, a theory of the vacancy-sharing mechanism for double $K$-shell excitation in symmetric systems. The theoretical results are in quantitative agreement with experiment.

Rotational excitation cross sections for $e$-C${\mathrm{O}}_2$ scattering in the energy range from 0.07 to 10.0 eV have been calculated using a coupled-channels procedure together with the adiabatic-nuclei (impulse) approximation. Results are presented for excitation from $j=0\mathrm{}$ to $j^{\prime }=0, 2, \mathrm{and} 4$. Cross sections for $j=0\mathrm{}$ to $j^{\prime }=2\mathrm{}$ in $e$-C${\mathrm{O}}_2$ scattering are compared with corresponding results for $e$-${\mathrm{H}}_2$ and $e$-${\mathrm{N}}_2$ collisions. The Born approximation for rotational excitation is examined for this problem and found to be unsatisfactory even at the lowest energies studied. Results for large initial rotational quantum number $j$ are presented and discussed.

Theoretical angular distributions for $p$-H charge transfer to the $1s$ state for energies of 1 keV and above have been examined and compared for three approximation schemes: The plane-wave Born approximation of Jackson and Schiff (JS), the Coulomb projected Born approximation of Geltman (G), and the distorted-wave eikonal approximation of one of the authors (D). The sharp dip in the forward distribution characteristic of JS is found to exist in G and D as well. As expected, G and D give identical results for all but the lowest energies. In the cases of G and D the dip, which is located close to that of JS, disappears and then reappears as the energy rises. Analytic high-energy limits for the angular dependence in both the JS and G cases have been found and are discussed.

Collision strengths $\Omega$ for electron-impact excitation of Civ and Arxvi are calculated in a five-state close-coupling approximation for energies up to 3 times threshold energy. These results are combined with previous calculations for Nv and Neviii and isoelectronic fits are given for $\Omega (2s, 2p)$ and $\Omega (2s, 3l)$, where $l=0, 1, 2$. In addition, polarization of the resonance $2s$, $2p$ line is calculated for Li-like C, N, Ne, and Ar up to 12 times threshold. A common feature is that the polarization changes from positive to negative at about 9 times threshold energy.