A simple series-expansion form for the direct-scattering amplitude (involving real quantities) $T_{n_{1}s-n_{2}s}^d$ is given in the case of the $n_{1}s-n_{2}s$ transition in the electron—hydrogen-atom collision. Another form for $T_{n_{1}s-n_{2}s}^d$, consisting of few terms, is derived when the initial-state quantum number $n_1$ is small. When the quantum numbers $n_1$ and $n_2$ are very large, the dominant part of $T_{n_{1}s-n_{2}s}^d$ can be reduced to a single oscillatory trigonometric term, and this leads to some interesting features of the differential cross section. Simple integral forms for $T_{n_{1}s-n_{2}s}^d$ in the above-mentioned cases are also given. Expressions are further obtained for the exchange-scattering amplitude $T_{n_{1}s-n_{2}s}^{\mathrm{ex}}$, which includes terms of orders $O\left({\beta }_0^{-1\mathrm{}}\right)$ and $O\left({\beta }_0^{-2\mathrm{}}\right)$, with ${\beta }_0=k_1^2+n_2^{-2\mathrm{}}=k_2^2+n_1^{-2\mathrm{}}$, where $k_1 \left(k_2\right)$ is the initial (final) electron momentum. We also derive exact expressions for $T_{2s-2s}^{\mathrm{ex}}$ and $T_{1s-3s}^{\mathrm{ex}}$. All calculations are done in the first Born approximation.

Molecule-surface collisions are analyzed classically using a two-dimensional hard-core ellipsoid—flat-surface model. Closed expressions are derived for the maximal excitations and deexcitations of the rotational transitions for an initially rotating molecule in the state $j_0$. These bounds are not symmetric with respect to $j_0$. We make comparisons with the predictions of the infiniteorder sudden approximation, and show that it is not suitable for analyzing quantitatively the problem of transition bounds. A classical condition for the onset of multiple molecule-surface collisions is also derived, which is applicable to both the initially rotating and nonrotating molecules. Quantum two-dimensional calculations fully support the results of the classical analysis.

To assess the accuracy with which polarization is represented in the Schwinger multichannel formulation, we have calculated elastic integral and differential scattering cross sections for $e$-${\mathrm{H}}_2$ collisions from 1 to 10 eV. Our ab initio results are found to be in good agreement with other recent theoretical studies and with a variety of experimental data including measured differential cross sections. Our method is designed to be applicable to electronically inelastic collisions and to closedshell targent molecules of arbitrary geometry.

A concise overview of the present situation of the near-threshold energy distribution investigations for electron-atom (-ion) ionizing collisions is given. By making use of an essentially perturbative approach, the distribution function has been calculated for two extreme regimes of the twoelectron final state: (i) ${\epsilon }_1\approx {\epsilon }_2$ and (ii) ${\epsilon }_{1,2}\ll {\epsilon }_{2,1}$, where ${\epsilon }_{1,2}$ are the escaping-electron final energies. It is shown that within the classical Wannier theory (linear configuration) one obtains the flat distribution function in both cases. The calculations for case (ii) are compared with recent Coulomb-dipole results. A brief discussion of some conceptual differences between Wannier and Coulomb-dipole theories is also given.

The $2P$ core region of atomic potassium has been studied with Al $K\alpha$ (1486.6-eV) radiation. Core binding energies of 300.56(6) eV for $2p_{\frac{3}{2}}$ and 303.31(6) eV for $2p_{\frac{1}{2}}$ electrons have been measured and compared with previous measurements and with theory. A satellite at 6.3 eV higher binding energy than the $2p_{\frac{3}{2}}$ peak with a relative intensity of (17.8±1.0)% has been observed; an analogous satellite has also been observed for $2p_{\frac{1}{2}}$. Good agreement was found with calculations which assign the satellites to $4s\to 5s$ excitation accompanying core ionization. Analogous satellites were identified in the $L_{2,3}{M}_{2,3}{M}_{2,3}$ and $L_{2,3}{M}_{2,3}{N}_1$ Auger spectra, and it was found that the background in the Auger spectra should be taken to be lower than previously assumed in order to get good agreement between the relative intensities obtained in the photoelectron and Auger studies. The photoelectron intensity ratio $\frac{I_{{2p}_{\frac{3}{2}}}}{I_{{2p}_{\frac{1}{2}}}}$ was found to be 1.75(7); this confirms the prediction of Dirac-Slater calculations that this ratio is less than the statistical value of 2.

Closed-form expressions for the first two moments of the wave-number distribution of unresolved transition arrays (UTA's) observed in highly ionized atoms have been published previously. However, this is not sufficient in many observed cases of asymmetrical UTA's. This paper describes a method for the evaluation of the third moment for arrays of frequently observed type $l^{N+1\mathrm{}}-l^N{l}^{\prime }$. Distributions are represented by skewed Gaussians. The maximum is thus shifted with respect to the average wave number, and the full width at half maximum is smaller than that of a symmetrical Gaussian. Applications to Pr xvi and Xe xxviii-xxx are presented.

An analysis is presented of the redistribution of scattered light in resonant electronic Raman scattering from localized scattering centers. Results are obtained for the integrated cross sections characterizing the coherently scattered and the redistributed radiation. A statistical $S$-matrix theory is outlined which relates the cross section for the redistributed radiation to the coherent-scattering cross section. For the theory to be applicable it is necessary that the absorption lines from the various initial and final states to the resonant intermediate state have identical shapes. The conditions for this to occur are discussed.

The photoionization of atoms in a strong electromagnetic field leads to the effect of "above-threshold" ionization, displaying a number of peaks in the energy spectrum of the photoelectrons. In the case of a one-electron model atom in a circularly polarized field, we derive expressions that describe the position as well as the intensity of those peaks as a function of the field parameters by means of a nonperturbative method. It is immediately clear from our formulas that the lowest-energy peaks in general disappear with increasing field intensity, for Coulomb-like as well as shortrange potentials. For a zero-range potential, a numerical evaluation is easily performed, leading to results that are in every respect qualitatively similar to the experimental results obtained thus far on these kinds of processes.

Magnetic dipole ($M1\mathrm{}$) and electric quadrupole ($E2\mathrm{}$) lines have been observed from trapped metastable Ar III and Cu II ions. The decay of these lines following excitation has yielded the lifetimes (or total radiative decay rates) for the upper levels. The lines observed are, from Ar III, the $M1\mathrm{}$ transition at 3109 Å from ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ to ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ in the ground state $3p^4$ configuration, and from Cu II, the $E2\mathrm{}$ lines at 4377 and 3807 Å from $3d^{9}4s {}_{}{}^1{}_{}{}^{}D_2^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}D_2^{}{}_{}{}^{}$ to the ground state $3d^{10}{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$. The results for the total radiative rates are $\gamma \left({}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}\right)=9.2\mathrm{}\pm 2.3$ ${\sec }^{-1\mathrm{}}$ for Ar III and $\gamma \left({}_{}{}^3{}_{}{}^{}D_2^{}{}_{}{}^{}\right)=0.14\mathrm{}\pm 0.28$ ${\sec }^{-1\mathrm{}}$ and $\gamma \left({}_{}{}^1{}_{}{}^{}D_2^{}{}_{}{}^{}\right)=1.95\mathrm{}\pm 0.24$ ${\sec }^{-1\mathrm{}}$ for Cu II. These results are compared to existing theoretical values.

A "complex" resonance, due to the autoionization of a Rydberg series close to its ionization limit perturbed by an interloper, appears frequently in photoionization spectra. This resonance structure is studied by means of an analytical formula derived from a multichannel quantum-defect treatment involving one open and two closed channels. Model calculations demonstrate the effect of the interaction between the two closed channels. Examples taken from the Ba and ${\mathrm{H}}_2$ spectra are discussed. Finally, we reproduce the general feature of a complex resonance observed in the 785-Å region of the ${\mathrm{N}}_2$ photoionization spectrum, using ab initio electronic parameters and taking account of the rotational structure.

Experimental data are presented for the spectral shapes for beta-electron—nucleus bremsstrahlung produced by beta particles from ${}_{38}{}^{90}{}_{}{}^{}\mathrm{Sr}\to {}_{39}{}^{90}{}_{}{}^{}\mathrm{Y}_{}^{+}{}_{}{}^{}+e^{-}$ in thick targets of Cu, Cd, Ta, and Pb over the photon-energy range from 40 to 500 keV. The experimental data are compared with the numerical calculations based on the recent formulation of Pratt et al and with the Born-approximation theory of Bethe and Heitler. Though small discrepancies exist in certain restricted energy region, the overall agreement between the theory of Pratt et al and experiment appears to be good. The experimental data agree with Bethe-Heitler theory within experimental uncertainties for Cu, and a systematic positive deviation is found for Cd, Ta, and Pb.

We describe the results of our calculation of dielectronic-recombination rate coefficients, ${\alpha }^{\mathrm{DR}}$, for Be-sequence target ions ($\Delta n=0\mathrm{}$ excitation). The most suitable $\mathrm{LS}$-coupling order is employed to construct the intermediate state, i.e., core coupled first and high-Rydberg-state electron coupled last. Comparison is made with previous results in which the active electron pair was coupled first. The effect upon ${\alpha }^{\mathrm{DR}}$ of configuration interaction among the target states, as well as among the final states, is considered. The role played by the exchange interaction is discussed.

The influence of finite radial geometry on the longitudinal wiggler free-electron-laser instability is investigated for TE-mode perturbations about a uniform-density electron beam with radius ${\hat{R}}_b$. The equilibrium and stability analysis is carried out for a thin, tenuous electron beam propagating down the axis of a multiple-mirror (undulator) magnetic field ${\overrightarrow{\mathrm{B}}}_0\left(\overrightarrow{\mathrm{x}}\right)\simeq B_0[1+(\frac{\delta B}{B_0}\left)sin\right(k_{0}z\left)\right]{\overrightarrow{\mathrm{e}}}_z$, where ${\lambda }_0=\frac{2\pi }{k_0}=\mathrm{const}$ is the wiggler wavelength. It is assumed that $k_0^2{\hat{R}}_b^2\ll 1$, and that perturbations are about the self-consistent Vlasov equilibrium $f_b^0(\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{p}})=\left(\frac{{\hat{n}}_b}{\pi }\right)\delta (p_{\perp }^2-{2\gamma }_{b}m{\omega }_b{P}_{\theta }-{2\gamma }_{b}m{\hat{T}}_{\perp b})\delta (p_z-{\gamma }_{b}m{V}_b)$, where $p_{\perp }^2=p_r^2+p_{\theta }^2$, $P_{\theta }$ is the canonical angular momentum, and ${\hat{n}}_b$, ${\gamma }_b$, ${\omega }_b$, ${\hat{T}}_{\perp b}$, and $V_b$ are positive constants. For $\frac{\delta B}{B_0}\ll 1$ and slow beam rotation (${\omega }_b\ll {\omega }_{\mathrm{cb}}=\frac{e{B}_0}{{\gamma }_b\mathrm{mc}}$), the equilibrium density is uniform (${\hat{n}}_b$) out to the beam radius ${\hat{R}}_b={\left(\frac{2{\hat{T}}_{\perp b}}{{\gamma }_{b}m{\omega }_b{\omega }_{\mathrm{cb}}}\right)}^{\frac{1}{2}}$. Detailed free-electron-laser stability properties are investigated for the case where the amplifying radiation field has TE-mode polarization with perturbed field components ($\delta {\hat{E}}_{\theta }$,$\delta {\hat{B}}_r$,$\delta {\hat{B}}_z$). The matrix dispersion equation is analyzed in the diagonal approximation, and it is shown that the positioning of the beam radius (${\hat{R}}_b$) relative to the conducting wall radius ($R_c$) can have a large influence on the growth rate and detailed stability properties. Analytic and numerical studies show that the growth rate increases as $\frac{{\hat{R}}_b}{R_c}$ is increased.

Maxwell's equations are solved for the self-reproducing TEM Gaussian-Hermite modes of a stable resonator formed by two curved mirrors. The boundary condition is satisfied that the electric field vector of a mode is everywhere perpendicular to the mirror surfaces, which are regarded as perfect conductors. The vector solution for a semiconfocal or confocal resonator geometry is compared with the standard scalar solution, and a correction to the scalar representation of the field is discussed.

The problem of the dynamical evolution of an atomic system excited from state 1 to state $N$ in $N-1\mathrm{}$ steps by using one or more lasers is studied. The $N^2$-dimensional dynamical space is shown to factorize into $N$ independent subspaces under certain experimental conditions by using Racah tensorial representation. Different classes of multiple optical soliton and multilevel excitation solutions obtained previously are shown to arise from a unified scheme. A new class of multiphoton excitation solutions is also given. General equations of motion in the higher-dimensional subspaces are given but have not been solved.

We present the theory of resonant backward degenerate four-wave mixing in gaseous media for Doppler-broadened degenerate two-level systems. The degeneracy of the atomic levels as well as the influence of the polarization of the incident fields are taken into account within the framework of a tensorial formalism. The amplitude, line shape, and polarization of the reemitted phase-conjugate field are calculated up to the third order in the incident fields. The velocity average is performed in the approximation of large Doppler linewidths. One considers in detail the influence of depolarizing relaxation processes and the effects of pump-probe angular separation. In particular, the importance of the residual Doppler effect (shortening of the lifetime of the optically induced gratings by the atomic motion) associated with the pump-probe angle is emphasized. We show that the polarization of the reemitted fields is governed by very simple laws as soon as the residual Doppler effect overcomes the lifetimes of the atomic levels. Phase-conjugate mirrors can be simply characterized by a linear dichroism (parallel pump polarizations) or a birefringence (cross-polarized pumps). Vectorial phase conjugation is also analyzed, in the case of counter-rotating circular polarizations of the pumps. Finally, the case of two-photon transition is studied and compared with the case of a resonant transition.

The stability of the critical solutions of Jones's linear differential equation against the addition of a nonlinear term is studied. The class of correlation functions considered include those which do not possess a power-law decay at criticality, but which are of scaling form.

We have reanalyzed published data on the conductance as a function of temperature near the liquid-liquid critical points of isobutyric acid + water [A. Stein and G. F. Allen, J. Chem. Phys. 59, 6097 (1973)] and of phenol + water [C.-H. Shaw and W. I. Goldburg, J. Chem. Phys. 65, 4906 (1976)]. The aim was to test whether the anomalous decrease in the conductance near the critical point could be ascribed to a critical anomaly in the extent of acid dissociation and/or in the rate of proton transfer. In this new analysis, background contributions due to the normal temperature dependence of the dissociation constant and due to confluent critical singularities are given consideration. We find that the leading critical anomaly in the conductance is characterized by a critical exponent $1-\alpha$, where $\alpha$ is the exponent associated with the specific heat at constant pressure and composition. The exponent $1-\alpha$ is expected for an anomaly in the extent of the dissociation reaction and also for an anomaly in the proton-transfer rate.

A model for the surface tension and density profile of simple liquid metals is presented. It is based on second-order perturbation theory in the electron-ion pseudopotential about the inhomogeneous electron gas at a jellium surface. The pair correlations in the ion fluid are computed using hard-sphere perturbation theory. The model yields good agreement with the experimental surface tensions of the alkali metals. For the polyvalent metals it is shown that it is essential to include the position-dependent self-energy of the ions, a quantity that is fortuitously small in the monovalent systems. To obtain such self-energies, very accurate values of the response function of the inhomogeneous electron gas are required.

We have performed a quantitative calculation of growth rates and tip radii for dendrites growing in undercooled dilute solutions. Included in our calculations are the capillary corrections to the steady-state Ivantsov needle-crystal solutions. Our results show good agreement with available experimental data and support the validity of the marginal-stability theory of dendritic growth.