A treatment of transient monochromatic perturbations for the determination of fast relaxation times in gas spectra is presented. In a corresponding experiment, the radiative lifetimes of the $3s_2$ and $2p_4$ states of neon I were determined by direct perturbation of these levels with nanosecond bursts of intense 6328-Å monochromatic radiation ($\approx 50$ ${\mathrm{W}/\mathrm{c}\mathrm{m}}^2$). The effective inelastic-destruction cross sections of these states for collisions with neutral helium atoms were also evaluated. The respective radiative lifetimes and cross sections obtained are 54.0 ± 4 nsec, (1.49 ± 0.2) × ${10}^{-15\mathrm{}}$ ${\mathrm{cm}}^2$ and 15.9 ± 0.7 nsec, (0.72 ± 0.3) × ${10}^{-15\mathrm{}}$ ${\mathrm{cm}}^2$. A value of 9.4 ± 0.6 nsec was determined for the radiative lifetime of the argon II $4p{}_{}{}^2{}_{}{}^{}D_{\frac{5}{2}}^{}{}_{}{}^{}$ level from direct perturbation of this level with picosecond mode-locked pulses at 4880 Å. The relatively intense optical pulses were generated by acousto-optically diffracting a part of the circulating radiation from within He-Ne and argon ion-laser cavities. The method provides significant improvements in resolution for optically initiated direct lifetime measurements in gas spectra.

Detailed measurements of the absolute photoabsorption cross section of atomic calcium between 2028 and 1753 Å and 1589 and 1424 Å are presented. Structure is displayed in the absorption cross section below 1589 Å, confirming previous theoretical predictions of autoionization in this region. However, the observed peak magnitudes are an order of magnitude smaller than suggested by theory. A theoretical analysis has succeeded in achieving better agreement between experiment and the two-channel quantum-defect theory for the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}\to {}_{}{}^1{}_{}{}^{}P_1^0{}_{}{}^{}$ transitions in the 2028-1753-Å region, both in the magnitude and in the general shape of the absorption profile.

The long-range interaction in alkali-noble-gas systems is studied in the multiconfiguration-self-consistent-field (MCSCF) formalism. Owing to the diffuse valence shell of the alkali, there are important contributions to the interaction potential from the higher-order multipole moments of the alkali (although the dipole moment states are by far the most important). The MCSCF formalism is shown to lead to a simple and effective way of handling the above effect, as well as evaluating intra-atomic correlation contributions and the coupling between interatomic and intra-atomic correlations when they are important. The nature of the intra-atomic-interatomic correlation coupling is analyzed and the relationship of the MCSCF results to perturbation-theory considerations clarified. Computed interaction potentials are presented for the systems HeH and LiHe. The results for HeH are in good agreement with the corresponding experimental results, while those for the system LiHe, agreeing with Dalgarno's recommended value of $C_6$, differ markedly from Baylis's semiempirical calculations.

Using an electromagnetically driven $T$ tube as a source, the profiles of the He II 304-Å ($n=2\mathrm{} \mathrm{to} 1$), He II 1640-Å ($n=3\mathrm{} \mathrm{to} 2$), He II 4686-Å ($n=4\mathrm{} \mathrm{to} 3$), and He II 10 123-Å ($n=5\mathrm{} \mathrm{to} 4$) lines were scanned and compared with Stark broadening theory. The plasma conditions correspond to electron temperature from 4 to 5 eV and electron densities from 1 × ${10}^{17}$ to 3 × ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$. Agreement of measured relative intensities with theory is typically within ∼ 30%, provided a distinction is made between absorption and emission line shapes of the resonance line in a singly ionized helium plasma.

The isotope shifts of ${}_{}{}^{41}{}_{}{}^{}\mathrm{K}$ and 12.4-h ${}_{}{}^{42}{}_{}{}^{}\mathrm{K}$ relative to ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ were studied in the $D_1$ line ($4{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\to 4{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$) 7699 Å by the atomic beam method and found to be +7.4 ± 0.3 and + 16 ± 4, respectively, in units of ${10}^{-3\mathrm{}}$${\mathrm{cm}}^{-1\mathrm{}}$.

One needs a necessary condition (rather than a sufficient condition, such as that of Rayleigh-Ritz) to even attempt to prove that a given bound state cannot exist. We use an adiabaticlike method to show that a positron ($e^{+}$) and a hydrogen atom (H) cannot form a bound state. The separation $r$ (but not $\overrightarrow{\mathrm{r}}$) of the proton ($p$) and $e^{+}$ is fixed; working in a subspace of zero total angular momentum, we calculate the lowest energy $E_{\mathrm{e}1}\mathrm{}\left(r\right)\mathrm{}$ of the electron in the field of $e^{+}$ and $p$. An effective one-body $p+e^{+}$ potential is then defined by $V^{\mathrm{}\left(1\right)\mathrm{}}\mathrm{}\left(r\right)=E_{\mathrm{e}1}\mathrm{}\left(r\right)+\frac{e^2}{r}+\frac{e^2}{2a_0}$. The necessary condition for the existence of a bound state of the true $\mathrm{H}+e^{+}$ system is the existence of a bound state of an artificial $p+e^{+}$ system with an interaction $V^{\mathrm{}\left(1\right)\mathrm{}}\mathrm{}\left(r\right)\mathrm{}$. This one-body problem is readily solved, and we find that it (and therefore the true problem) has no bound state. The proof is not rigorous since $E_{\mathrm{e}1\mathrm{}}\mathrm{}\left(r\right)\mathrm{}$ is not determined exactly, but the accuracy attained is such as to make the existence of an $\mathrm{H}+e^{+}$ bound state extremely unlikely. A by-product of the calculation is the determination of an improved lower bound on the ground-state energy of ${\mathrm{H}}^{-}$.

The radiative magnetic dipole transition rate between $2p_{\frac{3}{2}}$ and $2p_{\frac{1}{2}}$ single-particle electron states has been calculated with relativistic screened hydrogenic wave functions for seven elements with $70<~Z<~93$ and is found to contribute only ∼${10}^{-5\mathrm{}}$ of the total $L_2$ level width. The result has been corroborated by an experimental study of the $\mathrm{Pb} M$ x-ray spectrum in coincidence with $K{\alpha }_1$ and $K{\alpha }_2$ x rays, establishing a limit ${\omega }_{23}\lesssim (1.4\mathrm{}\pm 3.0)\times {10}^{-3\mathrm{}}$ for the radiative part of the $L_2-L_{3}X$ Coster-Kronig transition probability in Pb.

Hanle-effect measurements yield values for the collisional depolarization rates of ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ excited atomic states that are about three times too small unless one explicitly includes the effects of nuclear spin. The broadening also becomes a nonlinear function of foreign-gas pressure. We present experimental evidence for nuclear-spin effects on pressure-broadened Hanle-effect signals of ${\mathrm{Rb}}^{85}$ and ${\mathrm{Rb}}^{87}$ in helium.

We discuss the solution of the "superradiance master equation" derived in a preceding paper. During the first few photon transient times the cooperative atomic decay goes through a non-adiabatic oscillatory regime. For later times the decay takes place monotonically in time with the electromagnetic field following it adiabatically. The emitted light pulse has different statistical properties for an incoherently and a coherently prepared "superradiant" atomic initial state. The former case is characterized by large quantum fluctuations and strong atom-atom and atom-field correlations. In the latter case quantum fluctuations are small and the system behaves essentially classically. By also solving for a class of coherently prepared intermediate initial states we show that large quantum fluctuations occur only if the initial total occupancy of the excited state differs from the total number of atoms at most by a number of order unity.

A detailed calculation based on a perturbed stationary-state description is presented for associative ionization of hydrogen atoms. Specific rate constants $k_{\mathrm{AI}} \mathrm{}(v,K,T)\mathrm{}$ are evaluated for the set of electronic states correlating with $\mathrm{H}\mathrm{}(n=1\mathrm{})+\mathrm{H}\mathrm{}(n=3\mathrm{})\mathrm{}$ for associative ionization into vibrational ($v=0-8\mathrm{}$) and rotational ($K=0-21\mathrm{}$) states of $\mathrm{H}_2^{}{}_{}{}^{+}\left(X^2{\Sigma }_g^{+}\right)$ at $T=300\mathrm{}$ °K and at $T=600\mathrm{}$ °K. Cross sections $Q_{\mathrm{AI}}\mathrm{}(v,K,E)\mathrm{}$ are given for the most significant electronic state, the ($1s\sigma ,4s\sigma$) united atom configuration, and, by microscopic reversibility, cross sections for the reverse reaction, dissociative recombination, are obtained. Cross sections for associative ionization at thermal energies, $Q_{\mathrm{AI}}(v,K,{10}^{-3\mathrm{}}\mathrm{a}.\mathrm{u}.\mathrm{})$, are at most of a magnitude about ${10}^{-18\mathrm{}}$ ${\mathrm{cm}}^2$. Corresponding specific rate constants, $k_{\mathrm{AI}}$ ($v,K,300\mathrm{}$ °K), fall in the range ${10}^{-14\mathrm{}}$-${10}^{-13\mathrm{}}$ ${\mathrm{cm}}^3$/sec and display a temperature dependence $k_{\mathrm{AI}}\mathrm{}\left(T\right)\mathrm{}\propto T^{-0.33\mathrm{}}$. Dissociative recombination cross sections, $Q_{\mathrm{DR}}(v,K,{10}^{-3\mathrm{}}\mathrm{a}.\mathrm{u}.\mathrm{})$, are obtained for the $v=4-8\mathrm{}$ vibrational states, with magnitudes about ${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$. The resulting specific rate constants, $k_{\mathrm{Dr}}$ ($v,K,300\mathrm{}$ °K), are at most ${10}^{-10\mathrm{}}$ ${\mathrm{cm}}^3$/sec and display a temperature dependence $k_{\mathrm{DR}}\left(T_e\right)\propto T_e^{}{}_{}{}^{\frac{-1\mathrm{}}{2}}$ for $T_e\lesssim 80$ °K, $k_{\mathrm{DR}}\left(T_e\right)\propto T_e^{}{}_{}{}^{-0.75\mathrm{}}$ for $T_e\approx 300$ °K, and a still larger-power decrease for increasing $T_e$ at $T_e>300\mathrm{}$ °K.

$f$ values have been computed for the transitions $m{}_{}{}^1{}_{}{}^{}S-n{}_{}{}^1{}_{}{}^{}P$, $m=1-5\mathrm{}$, $n=2-5\mathrm{}$ and $m{}_{}{}^3{}_{}{}^{}S-n{}_{}{}^3{}_{}{}^{}P$, $m,n=2-5\mathrm{}$ for members of the helium isoelectronic sequence up to $Z=10\mathrm{}$. The agreement between the results obtained using the dipole length, and velocity formulas, together with the convergence of the results as an increasing number of terms are included in the expansions of the wave functions, indicate that the values obtained are accurate to within 1% or better for the large majority of the transitions.

A discussion of the problem of the construction of a configuration-space Hamiltonian describing a many-electron atom with relativistic effects included is given. It appears that the total three-body energy in such an atom must be small if the problem of the construction of this Hamiltonian is to be simply solvable. To that end, a relativistic three-body potential is constructed for a three-electron system. It has novel terms which arise from a pathology of the two-body Hamiltonian previously noted. It is shown that a speculative generalization to the many-electron system results in a total three-body energy which may be many rydbergs but which is still a small perturbation on the total result.

Papers I-III of this series describe a theory of direct-impact ionization in slow ${\mathrm{H}}^{+}-\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}$ collisions, and the first steps in the calculation of the cross sections ${\sigma }_I\left(\epsilon \right)$. It was found that electronic matrix elements for direct-impact ionization are extremely sensitive to the form of the kinematic factor representing radial transport of the electron by a moving heavy particle. In III a heuristic criterion for systematic improvement of the factor and a variational trial form for it were proposed. Using this proposal we have recomputed radial electronic transition matrix elements for ionization. There is an optimum choice of variational parameters which greatly reduces the over-all size of transition matrix elements. In particular, contributions at large internuclear distances (judged to be spurious in the results of Paper II) are now eliminated, and (except at high electron energies) the contributions to higher partial waves in the electronic continuum are reduced by orders of magnitude. The present results give reasonable confidence that convergence on a "best" first-order transition matrix for the effects of radial motion has been obtained. The existence of a similar optimization problem for the Coriolis-coupling ionization matrix elements is pointed out.

A method is proposed for the accurate determination of atomic wave functions and energies by the explicit introduction of interelectronic coordinates into a configuration-interaction wave function. This is accomplished by choosing the configurations in the wave function to be antisymmetrized projected products of one-electron functions with powers of interelectronic coordinates. A 107-configuration wave function for the ground state of the beryllium atom was constructed from a basis set consisting of $s$ and $p$ Slater-type orbitals and powers of interelectronic coordinates: $r_{\mathrm{ij}}^v \mathrm{}(v.=0,1,2\mathrm{})\mathrm{}$. The energy obtained from this wave function ($E=-14.666\mathrm{}54$ a.u.) is an upper bound to the "exact" nonrelativistic energy of this state and it is believed to be within 0.0002a. u. of the "exact" value. The advantages that the present method offers for extending accurate Hylleraas-method calculations to atomic systems with $N>\mathrm{}3\mathrm{}$ are discussed.

The cross section for the double Compton effect is integrated numerically in the laboratory system, and the effect of the radiative tail to the Compton line is discussed. The double Compton effect is also studied in the c. m. system for the cases in which the energy of one of the emitted photons is much smaller than the electron rest energy or much smaller than the energy of the incident photon. Expressions for the radiative corrections to the total cross section for Compton scattering are obtained in the high-energy approximation to first order in the fine-structure constant. The total cross section for the double Compton effect is computed by numerical integration. The contribution to the absorption coefficient for high-energy photons from double Compton scattering and Compton scattering with the emission and reabsorption of a virtual photon is obtained.

The spectral distributions of light scattered under various polarization conditions in liquids composed of optically anisotropic molecules are analyzed. Macroscopic equations are derived using analogous electric circuits which simulate the viscoelastic behavior due to shear stress, reorientations, and coupling with internal degrees of freedom. The equations describe the long-wavelength thermal fluctuations in the liquid and form the basis of a unified calculation of depolarized and Brillouin scattering. Under the condition that shear relaxation takes place more rapidly than reorientational processes, the sharp component and broad background in the depolarized spectrum are shown to depend primarily on the fluctuations in orientations and shear stress, respectively. Our results also provide a simple interpretation of the recently observed doublet structure. The Brillouin spectrum, on the other hand, shows the well-known effects of thermal relaxation as well as the effects of coupling with shear stress and reorientations. Numerical results on quinoline and nitrobenzene are presented, and the computed depolarized spectra are in good agreement with the experimental data of Stegeman and Stoicheff.

Two alternative expressions for the two-photon decay rate of the $2{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ metastable state of helium are evaluated by performing the summations over intermediate ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ states. The values obtained from the "length" and "velocity" expressions for the decay rate are 50.85 ${\sec }^{-1\mathrm{}}$ and 50.89 ${\sec }^{-1\mathrm{}}$, respectively, in good agreement with a recent experimental value and with previous calculations done by different methods. Incidental results are given for the photoionization cross sections of the $1{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and $2{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ states of helium in the energy region 0-0.2 Ry.

The Glauber approximation is applied to elastic scattering of protons by hydrogen atoms and to excitation of the $2s$, $2p$, $3s$, and $3p$ levels of the hydrogen atom by proton impact. The predicted differential and integrated elastic and excitation cross sections are compared with other calculations and with the available experimental data. A minimum appears in both the $1s-2s$ and $1s-3\mathrm{s}$ integrated cross sections near 15 keV. The corresponding differential cross sections near 15 keV exhibit a minimum and a maximum at very small scattering angles. The Glauber approximation is a considerable improvement over the Born approximation at energies < ∼ 200 keV and the distortion approximation at energies < ∼ 100 keV. At energies > ∼ 200 keV, our calculated total excitation cross sections are close to the Born approximation. Nevertheless, over most of the angular range our calculated differential cross sections are very different from the Born-approximation values. Above 10 keV our results are in good agreement with the only available experimental data ($1s-2p$), indicating that above ∼ 10 keV the Glauber approximation should yield reasonable estimates for proton-hydrogen-atom collision cross sections.

Recent observations of characteristic x rays produced by higher-energy (30-80 MeV) $\alpha$ particles are compared to previous measurements and to theoretical calculations based on the impulse approximation. These data are found to be in agreement with theory. The effects of choice of fluorescence yield are discussed.