The polarizability tensor of a pair of interacting He atoms has been calculated as a function of internuclear separation $r$ using the fully self-consistent Hartree-Fock theory. It was found that the trace of the polarizability tensor, $\alpha \mathrm{}\left(r\right)\mathrm{}$, to which the second dielectric virial coefficient $B_{\epsilon }$ is directly proportional, decreases with decreasing $r$, giving a theoretical value of $B_{\epsilon }=-0.093\mathrm{}$ a.u. at room temperature, compared with the experimental result $B_{\epsilon }=-0.06\mathrm{}\pm 0.04$ a.u., measured by Orcutt and Cole [J. Chem. Phys. 46, 697 (1967)]. This is the first calculation that predicts the correct sign of $B_{\epsilon }$. We conclude that for He the effects of overlap are of opposite sign from and of sufficient magnitude to overcome the contributions of the van der Waals interaction to $\alpha \mathrm{}\left(r\right)\mathrm{}$. Furthermore, the anisotropy of the pair polarizability $\beta \mathrm{}\left(r\right)\mathrm{}$ can be represented by a simple form: $\beta \mathrm{}\left(r\right)=6\mathrm{}{\alpha }^2{r}^{-3\mathrm{}}-\lambda e^{\frac{-r}{r_0}}$,, where $r_0=0.74\mathrm{}$ a.u., and the collision-induced light-scattering spectrum predicted by this form has an essentially exponential line shape. These results are in qualitative agreement with recent work on collision-induced light-scattering spectra from rare gases.

In this paper we show that for an intensity parameter much less than unity, the Reiss approximation for the multiphoton ionization of atoms is equivalent to the neglect of the mean-frequency term in the treatment of Bebb and Gold. We have also improved upon the result of Reiss by including a first-order term. As an example of the above result, numerical values for many-photon $1s\to 2s$ transitions in hydrogen atoms are presented and are compared with the exact values of Gontier and Trahin.

Results of low-order Hylleraas-type calculations of the Be ground state are given for expansions of 1-18 terms. The 18-term expansion gives an energy of - 14.6611 a.u., which is slightly better than results of a 25-term Hylleraas-type calculation by Gentner and Burke and is comparable to the 55-term configuration-interaction result of Weiss: it is substantially poorer than results of recent much more extensive calculations by Bunge and by Sims and Hagstrom. A brief discussion is given of the additivity properties of energy contributions from expansion terms contributing to various types of correlation effects and of the relative importance, in the inner and outer shells separately, of various types of correlation effects.

A detailed description is presented of an optical-pumping charge-exchange method for establishing and observing nuclear-spin orientation in atomic ions having a ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ electronic configuration. ${\mathrm{Rb}}^{+}$, ${\mathrm{Cs}}^{+}$, ${\mathrm{K}}^{+}$, and free protons have been studied thus far. Various applications of the method include precision measurements of the nuclear moments in the ions and the first measurements of resonant charge-exchange cross sections at thermal energies.

We present the details of our calculation of the $\left(\frac{m_e}{m_p}\right){\alpha }^2\ln {\alpha }^{-1\mathrm{}}$ corrections to the hyperfine splitting (${\nu }_{\mathrm{H}}$) of the hydrogen ground state. The result $\Delta {\nu }_{\mathrm{H}}=\left(\frac{8}{3}\right){\left(\frac{m_e}{m_p}\right)}^2{(1+\frac{m_e}{m_p})}^{-5\mathrm{}}\times c{\alpha }^4{R}_{\infty } \ln {\alpha }^{-1\mathrm{}}[9+7\mathrm{}\kappa (1+\kappa )-(\frac{m_e}{m_p}\left) \kappa \right(3+14\mathrm{}\kappa \left)\right]=0.0016\mathrm{}$ MHz (1.1 ppm), is combined with the corresponding formula for the muonium hyperfine interval to obtain an estimate of the proton polarizability ${{\delta }_p}^{\prime }$. This estimate of ${{\delta }_p}^{\prime }$ is compared with those obtained by several other methods and all seem to be in reasonable agreement.

The quantum calculation leading to the bremsstrahlung power density which varies as the square root of the plasma temperature $T$ is reasonably successful in predicting the x-ray production when electrons bombard metal targets. It is expected to be useful in other cases where interaction times are small, such as when electrons stream through neutral gases, or when plasmas are sufficiently dense so that the number of particles in a Debye sphere is small. In the case of rarefied plasmas, for which there is no corresponding quantum treatment, common practice is to begin with classical orbit theory and either cut off in impact parameter or in frequency in order to obtain a power density which also varies as $T^{\frac{1}{2}}$. We show that all such "semiclassical" treatments for rarefied plasmas are either physically or mathematically inconsistent; that the usual cutoffs in impact parameter do not lead to the expected $T^{\frac{1}{2}}$ power density in the temperature range from 0.01 eV to 1 keV; and that the spectrum of the power density obtained by cutting off at the de Broglie wavelength contains frequencies in excess of the usual one-photon emission limit.

The linked-cluster many-body perturbation theory has been applied to the study of the quadratic Stark shift of the hyperfine transition frequency of lithium atom. This procedure provides a method of quantitatively analyzing the relative importance of one- and many-electron contributions to the hyperfine Stark shift. The theoretical result $\delta \nu =-(5.2\mathrm{}\pm 0.3)\times {10}^{-8\mathrm{}}{\epsilon }^2$ Hz/${\mathrm{}(\mathrm{V}/\mathrm{c}\mathrm{m})\mathrm{}}^2$ obtained for lithium is in substantial agreement (86%) with the recently measured value $\delta \nu =-(6.1\mathrm{}\pm 0.2)\times {10}^{-8\mathrm{}}{\epsilon }^2$ approaching the situation in the hydrogen atom.

It is shown that the factorization method, when introducing accelerated ladder operators or accelerated ladder matrices, leads to a formula in closed form for the general off-diagonal ($n\ne n^{\prime }$) hydrogenic $r^k$ matrix elements. This explicit expression, which involves only factorials and binomial coefficients, is directly reducible to any particular case. The well-known selection rules follow from the formula. The method seems appropriate to other cases of factorizable equations. Its extension to the Dirac-Coulomb, generalized Kepler problems, and the discrete-continuous case is considered.

The values of the orbit-dependent hyperfine-structure constants are given for the ground-state terms of the atoms B, C, O, and F. These values are obtained from multiconfiguration Hartree-Fock wave functions using a very limited number of excited terms. The excited terms are chosen from guidelines based on second-order perturbation theory. The results are in satisfactory agreement with available experimental data and accurate many-body perturbation-theory calculations for O. A physical explanation, in terms of the configuration-interaction model, is presented for the essential importance of non-core-polarization contributions to the Fermi contact term for these atoms.

Calculations on the $2p{\sigma }_u-2p{\pi }_u$ excitation in proton-hydrogen collisions demonstrate that strongly coupled radial equations may be solved using simple semiclassical approximations at the turning points. A theoretical justification of the best of these methods is based on an extension of the large-mass high-energy approximation into the turning-point region. Other model problems are solved to illustrate properties of this new average approximation.

A uniformization technique is used to compute elastic scattering amplitudes at energies where arbitrarily many channels are open. The method takes knowledge of the Fredholm determinant $D\left(z\right)\mathrm{}$ on the physical sheet and uses rational-fraction analytic continuation to continue $D\left(z\right)\mathrm{}$ onto the appropriate Riemann sheet to construct the elastic amplitude and total inelasticity. Numerical examples for two-, three-, and four-channel problems are given, and preliminary results which indicate that the method may be of use in the breakup region are presented.

Photodetachment of ${\mathrm{Se}}^{-}$ ions is studied in a crossed-beam experiment, in which a 2-keV negative-ion beam is intersected by a pulsed tunable dye laser ($\Delta \lambda \approx 1-2$ Å). The threshold behavior of the photodetachment cross section $\sigma$ is compatible with Wigner's law ${\sigma }_L\propto k^{2L+1}$ ($k$ is the magnitude of the momentum of the outgoing electron and $L$ is its orbital angular momentum) over only about 5 meV above threshold in this case ($L=0\mathrm{}$). Some differences are observed for the threshold behavior of transitions terminating in different final states of $\mathrm{Se}\left({}_{}{}^3{}_{}{}^{}P_{2,1,0}^{}{}_{}{}^{}\right)$. These results are discussed in the light of existing theory. The observation of four of the six possible fine-structure transition thresholds allows us to determine unambiguously the electron affinity (EA) of $\mathrm{Se} \left\{\mathrm{EA}\right[\mathrm{Se}=(16297\mathrm{}\pm 2) {\mathrm{cm}}^{-1\mathrm{}}\cong (2.0206\mathrm{}\pm 0.0003) \mathrm{eV}\left]\right\}$ and the spin-orbit splitting between the ${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ states in ${\mathrm{Se}}^{-}\left[\right(2279\mathrm{}\pm 2\left) {\mathrm{cm}}^{-1\mathrm{}}\cong \right(282.6\mathrm{}\pm 0.3\left) \mathrm{MeV}\right]$. Comparison of experimental results for this splitting in ${\mathrm{O}}^{-}$, ${\mathrm{S}}^{-}$, and ${\mathrm{Se}}^{-}$ is made with predictions from isoelectronic extrapolation. The transition strengths for the various fine-structure transitions are determined and agree well with predictions by Rau and Fano. Finally, the absolute cross section for ${\mathrm{Se}}^{-}\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)$ detachment is reported.

The electron-impact-broadened profile of the resonance doublet of singly ionized beryllium ($2s-2p$, $\lambda =3130.4\mathrm{} \mathrm{and} 3131.1$ Å) has been measured using an electromagnetically driven shock tube and a rapid-scanning Fabry-Perot spectrometer. For the conditions $N_e={10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and $T$ = 19 000°K, we found the Lorentz half-width of each line in the doublet to be 0.035 Å ± 15%. For comparison with theory, we did both a quantum-mechanical (using the close-coupling method) and a semiclassical calculation, obtaining the values 0.015 and 0.042 Å, respectively. Possible reasons for the discrepancy are discussed.

The method of "comparison equations" for obtaining approximate solutions to the one-dimensional Schrödinger equation is discussed. In this method, one Schrödinger equation, ${\psi }^{\prime \prime }\mathrm{}\left(x\right)+k^2\mathrm{}\left(x\right)\mathrm{}\psi \mathrm{}\left(x\right)=0\mathrm{}$, is transformed into another, $v^{\prime \prime }\mathrm{}\left(z\right)+K^2\mathrm{}\left(z\right)v\left(z\right)=0\mathrm{}$, by a simultaneous change of independent and dependent variables, $x\to z$, $\psi \to v$. Then $\psi \mathrm{}\left(x\right)\mathrm{}$ and $v\left(z\right)\mathrm{}$ are related by $\psi \mathrm{}\left(x\right)=v\left(z\right)\mathrm{}{\left(\frac{\mathrm{dz}}{\mathrm{dx}}\right)}^{-\left(\frac{1}{2}\right)}$ whenever $k^2\mathrm{}\left(x\right)\mathrm{}$ and $K^2\mathrm{}\left(z\right)\mathrm{}$ satisfy the relation $K^2\mathrm{}\left(z\right)\mathrm{}{\left(\frac{\mathrm{dz}}{\mathrm{dx}}\right)}^2=k^2\mathrm{}\left(x\right)-\left(\frac{1}{2}\right) 〈z;x〉$. Here $〈z;x〉=\frac{z^{\prime \prime \prime }}{z^{\prime }}-\left(\frac{3}{2}\right){\left(\frac{z^{\prime \prime }}{z^{\prime }}\right)}^2$ is the Schwarzian derivative of $z$ with respect to $x$ and the prime indicates $\frac{d}{\mathrm{dx}}$. A set of "best" criteria for the transformed potential $K^2\mathrm{}\left(z\right)\mathrm{}$ is obtained, where by "best" is meant that we can completely neglect $〈z;x〉$ in first approximation and yet not have the turning-point problems that plague the WKB method (which is a special case of the comparison-equation method). The WKB method sets $K^2\mathrm{}\left(x\right)\mathrm{}\equiv 1$ and neglects $〈z;x〉$; the result is that the transformation $x\to z$, $\psi \to \nu$ is singular at the turning points, where $k^2\mathrm{}\left(x\right)=0\mathrm{}$. We choose $K^2\mathrm{}\left(z\right)\mathrm{}$ to have the proper asymptotic behavior far from the turning points, so that $z^{\prime }\approx 1$; hence $〈z;x〉\approx 0$ in these regions, and we match the zeroes of $k^2\mathrm{}\left(x\right)\mathrm{}$ and $K^2\mathrm{}\left(z\right)\mathrm{}$ in order to keep the transformation regular. This method is applied to various potentials with one and two turning points. Transmission and reflection coefficients $T$ and $R$ and transmitted and reflected phase shifts $\mu$ and $\nu$ are calculated for potentials with one turning point and potential barriers, and expressed in terms of the energy $E$ and the quantity $W=\left|\int {x_1}^{x_2}k\mathrm{dx}\right|$, where $x_{1,2}$ are the possibly complex turning points, $k^2\left(x_{1,2}\right)=0\mathrm{}$. Quantization rules, in terms of the classical action, are derived for various types of potential wells.

Metastable nitrogen atoms, produced by electron-impact dissociation of ${\mathrm{N}}_2$, have been detected in time-of-flight apparatuses located at Oregon State University and at University of California at Berkeley. Two different types of metastable atoms are detected; one type resembles Rydberg hydrogen atoms, while the other does not. The measured natural lifetime for the non-Rydberg atoms is 100 ± 25 μsec. Additional information regarding the identity of both types of metastable atoms is obtained by observing their quenching in a static electric field. The field-dependent decay constant for the Rydberg atoms is essentially a step function, and these atoms are entirely quenched by a field of 1.5 kV/cm. Both results are consistent with theory if the Rydberg atoms are in levels with principal quantum number greater than 25. The field-dependent decay constant for the non-Rydberg atoms varies quadratically with the strength of the applied electric field. In a field of 4 kV/cm applied along a 6-cm length of flight path, the value of this decay constant is 0.3 × ${10}^6$ ${\sec }^{-1\mathrm{}}$. The quenching results, together with the lifetime measurement, are a strong indication that the non-Rydberg atoms detected in the time-of-flight spectra are nitrogen atoms in the $2s{\mathrm{}\left(2p\right)\mathrm{}}^{3}3s{}_{}{}^6{}_{}{}^{}S_{\frac{5}{2}}^{}{}_{}{}^{}$ metastable autoionizing state.

A simple and convenient approximation is obtained for the multiphoton energy-transfer processes which accompany the scattering of a charged particle by a scattering potential in the presence of a strong external electromagnetic field. It is expressed in terms of the differential elastic-scattering cross section combined with known functions, and is valid when the scattering potential is weak or when the wave frequency is small. A detailed form of the classical limit is obtained.

The exchange excitation of the $4{}_{}{}^3{}_{}{}^{}S$ and $3{}_{}{}^3{}_{}{}^{}P$ levels of helium by electron impact has been studied by means of time-resolved spectroscopy of the $4{}_{}{}^3{}_{}{}^{}S\to 2{}_{}{}^3{}_{}{}^{}P$ ($\lambda =4713\mathrm{}$-Å) and $3{}_{}{}^3{}_{}{}^{}P\to 2{}_{}{}^3{}_{}{}^{}S$ ($\lambda =3889\mathrm{}$-Å) transitions. Measurements were carried out for incident electron energies in the range 50-400 eV at helium-gas pressures of 8 and 4 mtorr. Radiative cascade transitions from the $n{}_{}{}^3{}_{}{}^{}P$ levels and from the $n{}_{}{}^3{}_{}{}^{}S$ and $n{}_{}{}^3{}_{}{}^{}D$ levels contributed 12 and 10% to the total $4{}_{}{}^3{}_{}{}^{}S$ and $3{}_{}{}^3{}_{}{}^{}P$ populations, respectively, at 50 eV. These fractions increased to about 30 and 40%, respectively, at 400-eV electron-impact energy. The $4{}_{}{}^3{}_{}{}^{}S$ and $3{}_{}{}^3{}_{}{}^{}P$ direct-excitation cross sections were observed to decrease with increasing electron impact according to the relation ${\mathrm{}\left(\mathrm{energy}\right)\mathrm{}}^{-3\mathrm{}}$. This result is in agreement with the theoretical predictions of Ochkur and Bratsev.

A 250-keV electron accelerator was used to carry out time-dependent studies in neon gas at the wavelength of the ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ resonance line and the 850-Å continuum over a wide range of pressures, up to 1000 Torr. The resonance line decays exponentially with a lifetime that is governed predominantly by the escape of resonance radiation to the walls of the emission chamber and by three-body destruction of the resonance states. The main continuum near 859 Å comes to a maximum well after termination of the excitation pulse and then decays with a lifetime that is very long at low pressures. However, this lifetime decreases with increasing pressure at low and intermediate pressures and assumes a nearly constant value of 5.1 μsec at the highest pressures. From the time-dependent studies it is proposed that both ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ resonance states are converted to metastable molecular states by three-body collisions. The metastable molecules in turn may undergo two- and three-body collisions with ground-state atoms which convert them to radiative molecules that radiate the 850-Å continuum with a lifetime of about 5.1 μsec.

The binary-encounter approximation (BEA) is transformed from momentum space to configuration space. In this frame the impact-parameter representation allows one to calculate a variety of quantities pertinent to the general problem of ionization. Among these are cross sections for proton ionization of hydrogen and helium; in the latter case, cross sections for ejection of both electrons are also given. A number of tables and formulas are given, enabling one to correct the simple hydrogenlike-model predictions of the BEA for effects which arise in multielectron atoms. Multiple-ionization probabilities ($K+L$ shell) are calculated and compared to experimental results and to the predictions of the semiclassical approximation.

The Bethe theory for the calculation of the total inelastic cross section ${\sigma }_{\mathrm{tot}}$ for the scattering of fast charged particles by atoms and molecules is applied to neon. As part of the analysis, the spectral distribution of the oscillator strength has been determined by critical evaluation of all experimental data and by test of several sum rules. The result is ${\sigma }_{\mathrm{tot}}=z^2{\beta }^{-2\mathrm{}}\left\{3.524\right[\ln \left(\frac{{\beta }^2}{(1-{\beta }^2)}\right)-{\beta }^2]+36.06\}\times {10}^{-20\mathrm{}}$ ${\mathrm{cm}}^2$ for a particle of charge $\mathrm{ze}$ and speed $\beta c$. As an application, the ionization cross section is estimated by subtracting the sum of all the discrete excitation cross sections from ${\sigma }_{\mathrm{tot}}$ and is compared with experimental values.