We point out that the well-known second-order time-dependent perturbation theory result for the level shift of an atomic system in a plane electromagnetic wave is appropriate for calculating the change in ionization potential due to the presence of the wave. The calculation is valid when the electric field strength of the wave is small compared with atomic field strengths (intensity ≪${10}^{16}$ W/${\mathrm{cm}}^2$). We use this result to show that corresponding calculations carried out using the Kramers-Henneberger transformation yield a result which is valid only when the frequency of the plane wave is high compared with the atomic frequencies.

Rational approximations are presented for the second-order (Uehling) and fourth-order (Källén-Sabry) vacuum-polarization potentials in configuration space. These approximations may be applied to point-charge or finite-inducing-charge distributions. For the second-order potential, two approximations are given, with nominal accuracies of nine and four figures for the range $0\le r\le \infty$. For the fourth-order potential, one approximation of about three-figure accuracy is given for the range $0\le r\le {\lambda }_e$.

Hylleraas-type (HT) expansions, containing correlation factors of the form $r_{\mathrm{ij}}^p$, have been used to calculate ground states of the Li isoelectronic sequence. A maximum of 30 expansion terms was used with a common set of exponential parameters for all terms in a particular calculation. The ordering of expansion terms was roughly optimized, so that the expansions can be truncated at various lengths to give suitable approximations. About 15 terms of the present HT expansions give results comparable to those of Weiss's 45-term configuration-interaction calculations. For the Li ground state the present 30-term result is poorer than results of other more extensive calculations which include HT expansion terms.

Arbitrary inelastic transition form factors in the H atom have been exactly evaluated in closed form by a group-theoretical method. The final expression contains exactly known $O\left(4\right)\mathrm{}$ and $O(2,1)\mathrm{}$ representation functions and a single trivial finite sum. We also give the square of the form factors summed over $l$ and $l^{\prime }$ and obtain as a special case the result of Massey and Mohr.

Starting with Schrödinger's time-independent equation for $N$ indistinguishable particles, an exact equation is derived for the reduced density matrix of order $s$, ${\rho }_s$. The equation contains terms dependent on ${\rho }_{s+1\mathrm{}}$ and ${\rho }_{s+2\mathrm{}}$. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that ${\rho }_2$ and ${\rho }_3$ are particular functionals of ${\rho }_1$.

The independent-particle model of Green, Sellin, and Zachor (GSZ) with parameters fixed by an ab initio variational procedure is used to obtain approximate atomic form factors for all atoms and ions with $Z\le 54$. For convenience of application we replace the electron density obtained using Poisson's equation from the GSZ potential by an approximating sum of Debye or Yukawa functions. This sum implies a corresponding approximating sum of Debye or Yukawa potentials for the GSZ potential. These steps lead to convenient analytic form factors whose parameters may all be fixed terms of the two GSZ parameters. In this way, we can define approximate form factors for any of the 1485 species with $Z\le 54$. For most of these (∼ 1300 ionic species), no previous form-factor calculations have been reported.

Differential and total cross sections for elastic scattering of electrons from atomic hydrogen have been calculated using the full eikonal approximation. The eikonal results for 50-, 100-, and 200-eV incident electrons are compared with other theoretical calculations and with recent experimental data. It is found that the full eikonal approximation predicts both small-angle differential cross sections and total cross sections, which are in better agreement with experiment than either the Born or Glauber results.

The $L$ x-ray transition ratios $L\left(\frac{{\alpha }_{12}}{l}\right)$, $L\left(\frac{{\gamma }_1}{\eta }\right)$, $L\left(\frac{\eta }{{\gamma }_5}\right)$, $L\left(\frac{{\gamma }_{236}}{{\gamma }_1}\right)$, and $L\left(\frac{{\gamma }_{{44}^{\prime }}}{{\gamma }_1}\right)$ have been measured for the elements ${}_{}{}^{73}{}_{}{}^{}\mathrm{T}\mathrm{a},$ ${}_{}{}^{78}{}_{}{}^{}\mathrm{P}\mathrm{t},$ ${}_{}{}^{79}{}_{}{}^{}\mathrm{A}\mathrm{u},$ ${}_{}{}^{80}{}_{}{}^{}\mathrm{H}\mathrm{g},$ and ${}_{}{}^{82}{}_{}{}^{}\mathrm{Pb}$ by ionization of the $L$-subshell electrons with 0.4-2.0-MeV proton bombardment. Transition-rate ratios $L\left(\frac{{\gamma }_{23}}{{\gamma }_{{44}^{\prime }}}\right)$ to the $L_1$ subshell, $L\left(\frac{{\gamma }_6}{{\gamma }_1}\right)$, $L\left(\frac{{\gamma }_1}{\eta }\right)$, and $L\left(\frac{\eta }{{\gamma }_5}\right)$ to the $L_2$ subshell, and $L\left(\frac{{\alpha }_{12}}{l}\right)$ to the $L_3$ subshell have been extracted. Comparison of the extracted ratios with the relativistic Hartree-Fock model calculations of Scofield shows agreement within experimental errors for the $L_1$, $L_2$, and $L_3$ subshell ratios. A previously reported minimum in the Pb $L\left(\frac{{\alpha }_{12}}{l}\right)$ ratio between proton energies 1.0 to 1.5 MeV was not observed. Total $L$-shell production cross sections have also been measured for Pt and Au over the energy range 0.4-2.0 MeV and are found to be in good agreement with the predictions of the nonrelativistic plane-wave Born approximation and the constrained binary-encounter approximation, but not with the binary-encounter approximation.

The relative optical excitation functions and polarizations of the Sr resonance line (4607 Å) and of the ${\mathrm{Sr}}^{+}$ resonance lines (4078 and 4216 Å) have been measured, using crossed electron and strontium-atom beams, for electron energies from the excitation thresholds to 1497 eV. The electron-beam energy resolution was ∼ 0.22 eV for energies below 13 eV, and the atom beam was optically thin. The excitation functions of the ionic lines were measured relative to that of the atomic resonance line at fixed energies near the maxima of the excitation functions. Using spontaneous-emission branching ratios, this yields the ratios of total excitation cross sections for the atomic and ionic resonance levels (including cascades). The Sr $5{}_{}{}^1{}_{}{}^{}P$ excitation function has been normalized to Born theory taking account of cascade at 1497 eV, where the energy dependence of the excitation function has converged to the theoretical behavior. The resulting normalized cross sections have been compared with available theoretical calculations and other measurements. The 4607-Å polarization function is consistent with the theoretical threshold limit within experimental uncertainity.

$K$-shell x-ray production cross sections for 1.0-4.4 MeV $\alpha$ particles have been measured. The elements selected were Ti, V, Mn, Fe, Co, Ni, Cu, Zn, Ge, and Se, and all data were for thin targets evaporated onto thin carbon backings. The measured cross sections are compared with predictions of the plane-wave Born approximation (PWBA), the PWBA with binding-energy and trajectory corrections, and the binary-encounter approximation. It is found that the PWBA with both corrections fits the data best for all elements and energies considered.

Absolute $K$-shell x-ray production cross sections for thin Ca, Sc, Ti, V, Fe, Ni, Cu, Zn, Ge, Rb, Sr, Y, Zr, and Pd targets have been measured for incident carbon ions in the energy range from 8 to 36 MeV. Ionization cross sections are calculated from atomic $K$-shell fluorescence yields and compared to the theoretical predictions of the plane-wave Born approximation including binding energy, Coulomb deflection, and relativistic corrections. $\frac{K\alpha }{K\beta }$ ratios and the shifts in the characteristic $K\alpha$ and $K\beta$ energies indicate the existence of multiple ionization in the low-$Z$ targets.

The reaction ${\mathrm{He}}^{+}$ + Hg→He + (${\mathrm{Hg}}^{+}$)* has been studied at thermal energy in a drift-tube-mass-spectrometer apparatus fitted with a sapphire window for observing the light emitted from the electronically excited ions. Over the wavelength range 300-900 nm, the only emissions observed from the (${\mathrm{Hg}}^{+}$)* product ions are the 614.9-nm ($7p{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\to 7s{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$) and 794.4-nm ($7p{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\to 7s{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$) transitions, in approximately equal amounts. This equal branching in the case of reaction paths having substantially different energy defects, $\Delta \epsilon \left(\infty \right)=0.27\mathrm{} \mathrm{and} 0.72$ eV, respectively, is not explained by present theoretical models of the charge-transfer process.

Pronounced oscillatory structure and strong optical polarization effects have been observed in emission cross sections for the production of optical radiation measured as a function of energy in low-energy (100 eV to 6 keV) ${\mathrm{Na}}^{+}$-Ne atomic collisions. Highly regular oscillatory structure found in the energy dependence of the emission cross sections for ten $\mathrm{NeI}(3p\to 3s)$ optical transitions is observed to be in antiphase with similar structure in two $\mathrm{NaI}(3p\to 3s)$ optical emission cross sections. In addition, large polarization effects have been observed in optical radiation from excited $J\ne 0$ states. In one particular case involving a $J=1\mathrm{}$ to $J=0\mathrm{}$ transition, oscillations in the excitation function appear to arise exclusively from the polarization component of the radiation perpendicular to the beam direction. These observations indicate strong sublevel-state selection processes associated with collisional quantum-mechanical phase-interference phenomena. A simple model is presented which allows us to uniquely determine for the first time the quasimolecular states responsible for such oscillatory structure and consequently associated with each polarization component. Analysis based on the model for the case of the ${\mathrm{Na}}^{*}\mathrm{}\left(3p\right)\mathrm{}$ radiation data indicates that the oscillatory structure is not attributable to interactions between simple single-electron diabatic states as believed previously. Similar analysis of the ${\mathrm{Ne}}^{*}\mathrm{}\left(3p\right)\mathrm{}$ radiation data uniquely identifies the ${}_{}{}^1{}_{}{}^{}\Pi (\Omega =\pm 1)$ and ${}_{}{}^3{}_{}{}^{}\Pi (\Omega =\pm 2)$ molecular states as the major contributors.

The charge-transfer cross sections ${\sigma }_{10}$ for ${\mathrm{He}}^{+}$ ions on vapors of lithium and cadmium have been measured using the method of charge analysis of the fast products of the collisions in a magnetic field. The cross section for lithium falls with energy and that for cadmium rises with energy in the measured energy range 5-44 keV.

The process of formation of ${\mathrm{H}}^{-}$ as a result of double electron capture by protons from helium is investigated using an atomic-state expansion in which three traveling atomic orbitals have been retained. The cross sections have been presented for incident proton energy ranging from 10 to 200 keV. The results are compared with the previous experimental and theoretical values. Our results show a maximum in the cross-section energy curve in qualitative agreement with the experimental findings.

The $K$ and $L$ x rays from ten target elements from Cr to Bi bombarded by 5-MeV/amu $\alpha$ particles and N ions were measured by a Si(Li) detector. Energy shifts of $K$ x rays and intensity ratios of $\frac{K\beta }{K\alpha }$, $\frac{L\alpha }{\mathrm{Ll}}$, $\frac{L\alpha }{L\beta }$, $\frac{L\alpha }{L\gamma }$, and $\frac{L}{K}$ were determined by analyzing the experimental results. The ratio of the reduced $K$-shell ionization cross sections, $R_{2K}$, showed a systematic deviation from the $Z_1^2$ dependence.

The system of observed energy levels of CsII of Wheatley and Sawyer has been reinterpreted through the use of Hartree-Fock calculations and published Zeeman effect and hyperfine-structure measurements. Of the 80 known levels, 30 have been rejected as being not real. A revised list of 54 observed levels of CsII is presented that includes the $5p^{5}5d{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ level recently found by Reader and Epstein and three levels of the $5p^{5}6d$ configuration that were located with the aid of Sommer's line list. Percentage compositions and Slater parameters are given for the $5p^5\mathrm{}(5d+6s)\mathrm{}$, $6p$, and ($6d+7s$) configurations, Severe configuration mixing is found for levels of the $5p^5\mathrm{}(5d+6s)\mathrm{}$ group. The measured $g$ values of the $5p^5\mathrm{}(5d+6s)\mathrm{}$ levels provide evidence of a strong term dependence for the parameter $G^1\mathrm{}\left(5p5d\right)\mathrm{}$. By taking into account displacements of the $5p^5\mathrm{ns}$ levels due to configuration interaction, the ionization energy is revised to be 186 600 ± 150 ${\mathrm{cm}}^{-1\mathrm{}}$ (23.14 ± 0.02 eV).

Following a method analogous to one pointed out earlier, a simplified expression for the expansion of a general function about a point displaced from its center is described. Utilizing this, a general and closed expression useful for multicenter integrals in quantum mechanics is derived for the coefficients in the expansion of a general Slater-type orbital (including special cases). The derived expression for the Slater orbital contains terms only of the form $r^{k-1\mathrm{}}\exp (\pm \eta r)$ (where $k$ is an integer ⩾ 0; $l$ is the order of the coefficient and $\eta$ is the exponent in the Slater orbital)—very convenient and useful for the analytical evaluation of multicenter integrals. The expression is equivalent to the ones obtained by Silverstone and by Rakauskas and Bolotin but based on a completely different approach. The asymptotic forms for small as well as large values of $r$ are also presented. The importance of the expression is demonstrated by undertaking an example of overlap matrix elements involving Slater orbitals and deriving easily a simple and closed form applicable for all quantum numbers concerned. The ease with which one can write readily the overlap formulas in various cases starting from the general formula is indicated, and some numerical examples are given to support the usefulness of the expressions. The advantages associated with the expression (for the expansion coefficients) for large-scale calculations of multicenter integrals are discussed.