The cross sections for elastic scattering and for excitation of hydrogen atoms by electron impact are calculated for energies in the range from 10.2 to 12.1 eV. The calculations are made variationally. A pseudostate expansion is employed to represent the target atom. Energies and widths of resonances are obtained. Results are compared with experiment.

A calculation of the elastic scattering cross section for the screened Coulomb potential of a charge $\mathrm{Ze}$ provides second-order terms, of relativistic and nonrelativistic origin, that are proportional to $Z^3$. Here we study the $Z^3$ effects in the elastic scattering of electrons and positrons in atoms and in the energy loss of fast charged particles in solids, for a wide range of energies. Owing to the $Z^3$ effect, the scattering of electrons in atoms becomes always larger than the scattering of positrons. In the relativistic limit this agrees with the result of McKinley and Feshbach for bare nuclei. The screening effect introduces a nonrelativistic enhancement of the $Z^3$ correction; this increases the asymmetry between electron and positron scattering. Using the same approach we calculate the contribution from close collisions to the $Z^3$ correction in the energy loss of fast particles in solids. For nonrelativistic velocities the $Z^3$ correction shows a behavior similar to the result of Ashley, Ritchie, and Brandt, including distant and close collisions; in the relativistic limit we retrieve the result of Jackson and McCarthy. The limitations of this approach in describing distant collisions, and the validity of the partition rule, are considered.

A new second-quantization representation for low-energy positron-hydrogen scattering is derived by starting with the standard Fock representation and carrying out two canonical transformations, one to introduce the field operator for bound positronium and the other to introduce one for the bound atomic electron. The transformed "Fock-Tani representation" Hamiltonian is obtained in closed form. Its various terms have simple physical interpretations and manifest the various scattering and reaction channels. The interaction-matrix elements obtained are automatically renormalized through inclusion of bound-state-continuum orthogonality corrections. The definition of the $S$-matrix elements is simpler in the new representation since bound states are exactly redescribed therein as elementary particles. Certain constraints necessary and sufficient for the one-one property of the mapping from Fock- to Fock-Tani-state space are shown to be automatically satisfied when the asymptotic initial and final states for scattering processes are defined by the standard wave-packet method. This representation is expected to be useful for inclusion of the intermediate-state positronium channel in the standard manybody Green's-function approach to evaluation of $S$-matrix elements.

The radiative corrections to the photoeffect from the $K$ shell were evaluated recently by McEnnan and Gavrila assuming a hydrogenlike atomic model. The result was expressed in the form of a corrective factor [$1+\left(\frac{\alpha }{\pi }\right)\delta$] multiplying the basic photoeffect differential cross section. An expression for $\delta$, correct to lowest order in $\alpha$ and $\alpha Z$, was derived in analytic form in terms of a large number of single and double integrals over Feynman parameters, requiring numerical integration. $\delta$ depends on the photon energy $\omega$, on the electron-ejection angle $\theta$, and on the energy threshold $\Delta E$ below which we allow the additional (soft) photon, emitted together with the ejected electron to go undetected. The $\Delta E$ dependence of $\delta$ reflects the inseparable connection between the $K$-shell photoeffect and the Compton effect from the $K$ shell with (soft) photons emitted in the range $\Delta E$. It was shown that the same expression for $\delta$ with an appropriate redefinition of the variables, also radiatively corrects the bremsstrahlung spectrum at its high-energy tip. In the present work we have carried out the computation of $\delta$. It covers the $\omega$ energy range from 0.5 keV to 50 MeV at all relevant angles. The relative error was kept below 0.001, except at high energies where it was only 0.01. We find that ($-\frac{\alpha \delta }{\pi }$) is always positive and increases with $\theta$ and $\omega$. At energies of about 500 keV, ($\frac{\alpha \delta }{\pi }$) becomes of the order of 1% and at 5 MeV it has grown to about 5%, when taking $\Delta E=0.01\mathrm{}$${\mathit{mc}}^2$. By integrating over the angles we have derived the quantity $\Delta$ which radiatively corrects the total photoelectric cross section, in the form of a factor [$1+\left(\frac{\alpha }{\pi }\right)\Delta$]. We also infer by exponentiation the correct form of $\delta$ and $\Delta$ in the limit $\Delta E\to 0$. Finally, we derive the expression for the cross section describing the energy spectrum of the electrons emitted in the vicinity of the photoelectric peak, which is the basic information coming from coincidence experiments.

We have systematically developed a new approximation for the calculation of the cross section for electron capture from a hydrogenlike ion of large nuclear charge $Z_{T}e$ by a bare ion of charge $Z_{P}e$ moving with speed $v$. The amplitude in the wave treatment is obobtained through consistent expansion in the small parameters $\frac{Z_P}{Z_T}$ and $\frac{Z_P{e}^2}{\hslash v}$; however, the ratio $\frac{Z_T{e}^2}{\hslash v}$ is not assumed small. Following a careful analysis of the approach to the energy shell, an off-shell factor is seen to arise which does not appear in the impulse approximation. The effects of this factor on the capture amplitude are explored. Using a peaking approximation, we derive a closed-form expression for the $1s\to \mathrm{ns}$ capture amplitude which includes the effects of the off-shell factor and is accurate to order ${\left(\frac{Z_P{e}^2}{\hslash v}\right)}^2$. We tabulate the peaking-approximation cross section. The derivation of the asymptotic form of the cross section when $\frac{Z_P}{Z_T}<<1\mathrm{}$ is justified. Using a realistic one-electron model, $K$-shell capture cross sections are calculated for protons on carbon, neon, and argon. These are seen to represent the data generally better than do the impulse approximation cross sections.

We develop an expansion for the absorption line shape of a two-level system in a strong radiation field being perturbed by dephasing collisions with a thermal bath of foreign atoms. We assume that the bath atoms do not interact with the radiation field and with each other. The microscopic information relevant for line broadening in a strong field is expressed in terms of a hierarchy of dipole correlation functions. The latter may be calculated with the use of the Franck-Condon factors for the nuclear wave functions. The present theory reduces in the weak-field limit to the well-known linear unified theory of line broadening and in the impact limit we recover the Karplus-Schwinger formula.

The theory of atom-ellipsoid scattering in two dimensions is developed. The model is typical of atom-molecular rotationally inelastic collisions. It is shown that the problem is not suitable for the use of the standard techniques because of the presence of the hard core in the potential and the semiclassical character of the system. In this article several points are discussed: the choice of the proper technique for solving the problem, the convergence problem of the perturbation schemes for the coupled multichannel equations, forbidden transitions and features of the cross sections, and finally, the static limit of the equations.

The Coulomb-projected Born approximation has been used to calculate the total charge-transfer cross sections for $\alpha$ particles (${\mathrm{He}}^{2+}$) colliding with hydrogenic systems at medium and high energies. The present results overestimate the recent experimental data of Hvelplund et al. for the system ${\mathrm{He}}^{2+}$ + H in the energy range 0.5 to 5 MeV. Further, it is seen that the present scaled results $Z_T^6\sigma$ for capture into the $n=1\mathrm{}$ state follow the universal curve in the energy range $\frac{E}{Z_T^2}\ge 100$ keV when plotted against $\frac{E}{Z_T^2}$ and are in agreement with the scaling formula of Nikolaev.

A general method is stated for obtaining functionals associated with the scattering operators. Four conditions are required on the functional to determine the quantities involved in it, which are related to the operator and its perturbative series. No asymptotic condition is specified on the trial functions. The method is applied to the $T$ matrix leading to a family of functionals in which the lowest order is Schwinger's functional. The applications to the $S$ matrix and other operators are also discussed.

New studies of ${\mathrm{Na}}_2$ hyperfine structure (hfs) by the laser-induced-fluorescence molecular-beam magnetic-resonance technique are described which show that state selection and detection by strong optical fields can produce rf-magnetic-resonance spectra which differ drastically from the Rabi shape, even though the optical and rf fields do not act simultaneously on the molecules. We suggest that in the case of optically unresolved ground-state sublevels these effects are due to the creation of ground-state coherence by absorption and stimulated-emission cycles in the laser field, and we present a model calculation which exhibits the essential features of the observed spectra. We also give a proof that no model based only upon populations can explain the negative signals we observe.

The Faddeev-Yakubovsky theory is applied to the system of ${\left({}_{}{}^4{}_{}{}^{}\mathrm{He}\right)}_4$ interacting pairwise through a realistic He-He potential. The $S$-wave two-body $t$ matrix is obtained using the unitary pole expansion method, while the $S$-wave [3+1] and [2+2] subamplitudes are obtained by the Hilbert-Schmidt and the energy-dependent pole expansion (EDPE) methods. The convergences of the latter subamplitudes are tested and the superiority of the EDPE over the Hilbert-Schmidt expansion (HSE) is confirmed. The present results also confirm the results of a previous calculation using the ATMS method (amalgamation of two-body correlations into the multiple-scattering process).

The $O\left(4\right)\mathrm{}$ symmetry of the hydrogen-atom degeneracy is exploited to obtain an exact separation of the quadratic Zeeman interaction on the Fock hypersphere in momentum space, and hence a complete classification of levels in the weak-field limit. The separation constant includes a double-minimum potential, and levels below the top of the barrier have an approximate $O\left(2\right)\mathrm{}\times O\left(2\right)\mathrm{}$ symmetry and vibrational structure. Levels above the top of the barrier have an approximate $O\left(3\right)\mathrm{}$ symmetry, and a structure similar to that of an oblate symmetric top. The crossover between the two types of levels becomes sharper at high $n$, and is related to two different types of collective coupling of angular momentum and the Runge-Lenz vector.

The lifetimes of 50 levels in Nb I are measured with the use of time-resolved laser-induced fluorescence on a Nb atomic beam. We describe the hollow-cathode atomic-beam source in detail. The source is capable of producing an atomic beam, which is rich in metastables, of most metallic elements including the highly refractory metals.

Selective field ionization (SFI) is frequently used to detect atoms in high Rydberg states and to analyze Rydberg population distributions. In order to realize the full potential of this technique, however, a detailed understanding of the behavior of Rydberg atoms during passage from low to ionizing fields is required. The factors influencing this passage are discussed quantitatively and are illustrated by means of detailed SFI data obtained for $\mathrm{Na}34d$ atoms.

We develop an expansion for the resonance Raman line shape of a three-level system $|a〉$, $|b〉$, and $|c〉$ in an arbitrary strong radiation field, being perturbed by dephasing collisions with foreign gas particles. The microscopic information relevant to the line broadening is expresssed in terms of a hierarchy of $n$-time correlation functions of the transition dipole operator ($n=2,4,6,\dots$). To lowest order ($n=2\mathrm{}$) the spectrum is expressed in terms of the three ordinary line-shape functions corresponding to the $\mathrm{ab}$, $\mathrm{bc}$, and $\mathrm{ac}$ transitions. The present theory is not restricted to the impact limit and, depending on the collision time scale, may yield very different results from the well-known impact spectra. Explicit calculations of the predicted spectra are presented for a model system.

Three-photon resonance enhancement of third-harmonic generation and multiphoton ionization is treated for a slab geometry configuration. Striking pressure effects which become important at concentrations $n\gtrsim {10}^{12}$ ${/\mathrm{c}\mathrm{m}}^3$ for narrow-bandwidth short-pulse lasers are predicted for both the third-harmonic signal and the multiphoton ionization. For example, part of the third-harmonic signal exists the slab simultaneously with the laser pulse while another part is time delayed for small detunings from resonance. At intermediate detuning from the three-photon resonance the two parts can interfere. Multiphoton ionization yields near three-photon resonance are strongly suppressed under certain conditions, and peaks may occur for proper detunings on both sides of the three-photon resonance. A detailed experimental verification is suggested for Xe.

A new investigation is presented of the $1s$ photoabsorption spectrum in sodium vapor with the use of monochromatized synchrotron radiation and a heat pipe in which the optical path of the sample was varied over a broad range. The observed transitions are well separated and their absolute energies are measured to within ±0.3 eV. A comparison between the $1s$ spectrum and the $2s$ spectrum recently reported shows that the coupling of outer electrons in the final state plays a strong part in the relative energies of excited levels which are similar in both spectra. The presence of the hole in the $1s$ or $2s$ shell influences the profiles of the lines. From the profile analysis of the nearly Lorentzian first resonance line, the natural width of the $1s$ level is estimated to be 0.30±0.05 eV.

Even-parity levels of neutral barium located in the vicinity of the first ionization limit $6s$ have been investigated by using a two-step pulsed-laser excitation with optogalvanic detection. From the metastable states $5d6s$ populated by a dc discharge in a heat-pipe vapor, we reach the $6snl$ even-Rydberg series with $J$ equal to 0 to 5 and all the levels of the $5d6d$ and $5d7d$ configurations via the intermediate $5d6p$ levels. The $6sns{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ ($13\le n\le 47$) and $6snd{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ ($12\le n\le 38$) series are observed for the first time with five perturbers assigned to $5d8s{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$, $5d7d{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$, and ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$. The $6snd{}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$ ($11\le n\le 24$) series is perturbed by five levels $5d8s{}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$, $5d7d{}_{}{}^3{}_{}{}^{}G_3^{}{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}F_3^{}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$, and ${}_{}{}^3{}_{}{}^{}F_3^{}{}_{}{}^{}$. The fine structure of the $6dng{}_{}{}^{1,3}{}_{}{}^{}G$ series is partially resolved in the vicinity of the perturbing $5d7d J=3,4, \mathrm{and} 5$ levels. Semiempirical calculations using Racah's method have allowed us to understand the observed energies of the levels belonging to $5d6d$ and $5d7d$ configurations and to remove the ambiguity which was subsisting in the literature for the energy position of the $6p^2{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ level.

A detailed analysis of the excitation spectrum of the $\mathrm{Ba} (6\mathrm{}{P}_{\frac{3}{2}}12s,J=1)$ autoionizing state is presented. From this it is deduced that the dominant decay channel for autoionization of the $\mathrm{Ba} (6\mathrm{}{P}_{\frac{3}{2}}ns,J=1)$ states ($n\ge 12$) is to the ${\mathrm{Ba}}^{+} \left(6\mathrm{}{P}_{\frac{1}{2}}\right)+e^{-}\epsilon d$ configuration. Quadrupole coupling is presumed to be the important interaction, and an estimate of the resulting autoionization rate is presented.

This is the final paper in a series dealing with the spectrum of the beat signal of a ring-laser gyro. The remaining region of operation of the gyro for which an approximate expression for the spectrum has not been obtained, i.e., when the laser rotation rate is close to the edge of the unlocked region, is examined here using an approach of a different character to those developed in Papers I and II. The method is based on solving the original Langevin equation for the phase by a variation-of-constants method. The need to adopt a consistent interpretation of such equations is emphasized; the Stratonovich interpretation is used here. The final results are obtained through an iterative procedure which converges much more rapidly than the continued-fractions method of Paper II. The lower-order result yields an approximate analytic expression for the beat signal spectrum which explicitly exhibits the harmonic structure of the spectrum for laser rotation rates close to the edge of the unlocked region. Numerical results are also presented showing the convergence of the overall procedure, and the accuracy of the approximate analytic results, for noise values appropriate to the ring-laser gyro.