Electron-induced Cu-, Ge-, $\mathrm{A}\mathrm{g}-K$ shells, and $\mathrm{A}\mathrm{u}-L_3$-shell ionization cross sections have been obtained from the measurement of $K\alpha$ and $L\alpha$ x-ray emission cross sections around the threshold energy. $K$- and $L_3$-shell ionization cross sections have been observed to rise as a function of $\ln \frac{\left(U_i\right)}{U_i}$, where $U_i$ is the ratio of electron impact energy to ionization energy. In the increasing behavior of $\mathrm{A}\mathrm{u}-L\alpha$ and $-L\beta$ x-ray emission cross sections with increasing impact energies, the onset of $L_2$- and $L_1$-shell ionizations have been observed at $L_2$- and $L_1$-shell ionization energies. Absolute values of Cu-, Ge-, $\mathrm{A}\mathrm{g}-K$-shell ionization cross sections are well reproduced by the empirical formula of Green and Cosslett down to the vicinity of threshold energy.

Doubly differential cross sections of secondary electrons ejected from ${\mathrm{H}}_2$ by electron impact have been measured by a crossed-beam method. The incident energies used were 25, 40, 60, 100, 150, and 250 eV. The energy and angular range of secondary electrons measured were from one half of the difference between incident energy and ionization potential to 1.0 eV and from 12° to 156°, respectively. The present results do not agree with those of DuBois and Rudd for slow secondary electrons (<20 eV) ejected by 100-eV primary electrons. The present data lead to total ionization cross sections within 10% of other measurements.

We discuss the choice of the basis set to calculate excitations in time-dependent problems. Starting from a variational principle which minimizes the time-integrated coupling strength out of a given initial state, an equation determining the "optimal" basis is derived. We discuss the application of this concept to the harmonic oscillator of variable frequency and to electronic excitations in heavy-ion collisions. We also comment on particle production in time-dependent gravitational fields.

The scattering amplitude for a projectile of nuclear charge $Z_1$ carrying $N$ electrons colliding with an atomic target with charge $Z_2$, evaluated in the first Born approximation using hydrogenic wave functions, is compared with recent experimental results. In the present approximation where the minimum momentum transfer $t_{min}$ is considered to be approximately independent of the final state of the projectile, the differential cross sections separate into a product of one term that depends only on the target times the square $Z_1^2\mathrm{}\left(t\right)\mathrm{}$ of an effective projectile charge. Here an analytic expression for $Z_1^2\mathrm{}\left(t\right)\mathrm{}$ is given for 2, 1, and 0 electrons. Some total ionization cross-section ratios are also given.

Multichannel quantum-defect theory provides an attractive framework for compact, a priori calculations of the binding energies or scattering resonances of highly excited, multielectron atoms. However, in addition to obtaining a good representation of interchannel interactions, it is difficult in practice to find a sufficiently accurate, selfconsistent, simple, one-electron potential for describing the average motion of a Rydberg electron in the multielectron core. Moreover, when there is a nonspherical core there is a significant non-Coulomb tail even at fairly large distances from the origin, and due to exchange and correlations, such a potential will be nonlocal. We describe a self-consistent, local-density approximation to calculate a single-particle potential with proper self-interaction corrections, which represents quite accurately the motion of the outer electron in the presence of a spherically averaged core. This is derived from the well-known solid-state calculational technique based on the Hedin-Lundqvist approximation. The channel interactions and nonspherical contributions to intrachannel potentials are then calculated by explicitly considering the motion of two electrons outside the outermost closed-shell configuration of the atom. This procedure permits greatly increased accuracy in the prediction of excited-state energies of the entire spectral series. Explicit numerical results for quantum-defect parameters are presented for many Rydberg series in several alkali-metal and alkaline-earth atoms.

Some one-dimensional dissipative systems can be found to admit canonical-quantization procedure, contrary to opinions expressed occassionally in the literature. Also, a natural way of quantizing an equation $\ddot{q}+\dot{V}\mathrm{}\left(q\right)=0\mathrm{}$ for a function $V\left(q\right)\mathrm{}$ is found to use a nonstandard commutation relation $[q,\pi ]=i\hslash \pi$ together with Hamiltonian $H=\pi +V\left(q\right)\mathrm{}$. Here, $\pi =\dot{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.

A simple and accurate analytic model is derived for the correlation energy of any atom or ion within three densityfunctional approximations based on the uniform electron gas: the local-spin-density approximation (LSD), the selfinteraction corrected (SIC) version of LSD, and the antiparallel-spin LSD of Stoll et al. The last two approximations give good results for the correlation energies of neutral atoms, in contrast to LSD which overestimates these energies by a factor of 2. However, all three approximations show an incorrect $\ln Z$ leading behavior when the nuclear charge $Z$ tends to infinity at fixed electron number $N$. It is hard to see how any a priori electron-gas approximation can reproduce the exact leading behavior, which is constant or linear in $Z$ according to the value of $N$.

A model $S$-wave crossing of a bound state into the continuum is investigated using nonlinear variational calculations performed with the exact wave function and a trial function which has incorrect asymptotic behavior. The variational estimates of the energy-branch structure become less accurate near threshold, as expected, but the calculations indicate a sequence of branches which are followed in order to obtain estimates of the energy and width of a decaying-state resonance above threshold.

Hartree-Fock-Bogoliubov (HFB) theory is generalized by eliminating the restriction to quasiparticle vacuum states. However, the fundamental HFB transformation to quasiparticles is retained. The theory is formulated in terms of the quasiparticle density matrix $R$, but, in contrast to conventional HFB, there is no restriction to vacuum densities $R^2=R$.

Stemming from the classic work of Planck, classical radiative transfer theory works with pencils of light rays streaming in straight lines in any homogeneous medium. But light is described by Maxwell's equations in classical and quantum electrodynamics, and so we need to find a bundle of rays equivalent to the electromagnetic waves. In this paper we realize such a pencils of light rays. Their properties and the equations obeyed by the ray density function are deduced. We need to generalize radiative transfer theory for this purpose appropriately. For homogeneous statistical wave fields radiative transfer theory obtains rigorously. Particular attention is paid to the polarization properties of light rays. These concepts have some relation with the phase-space picture of quantum mechanics. Also, other general questions including possible further developments are discussed.

We show that in order to satisfy the condition of relativistic invariance for the one-photon radiative transition amplitudes of a composite system to relative order $\frac{1}{c^2}$ in an approximately relativistic canonical theory, the position operator in the theory should satisfy a condition which encompasses the world-line condition of Currie, Jordan, and Sudarshan. We also obtain the expression for the $E1\mathrm{}$ and the $M1\mathrm{}$ single-photon transition amplitudes of the composite system in the center of momentum frame, including the relativistic corrections of leading order. These expressions show that the relativistic corrections are qualitatively different for the electrically charged and neutral composite systems. They also depend, especially in the case of the $M1\mathrm{}$ amplitudes, rather sensitively on the binding interaction among the constituent particles. There are also relativistic corrections which depend simultaneously on the recoil momentum of the composite system and on the use of the relativistic center-of-mass variables in the problem and which so far have not been considered in the literature.

The accuracy of two different semiclassical methods of evaluating $T$-matrix elements by stationary-phase techniques are reported. Their predictions are compared to quantal results for the far-red-wing emission spectrum of Rb collisionally broadened by Xe perturbers. One method is based on the direct use of uniform semiclassical wave functions in the overlap integral of the $T$-matrix elements. It is shown to give very accurate results (generally to within 3% of quantal values) for both classically allowed and classically forbidden transitions, whether they are near to or far from the classical singularity associated with the collisional turning points. The other semiclassical method is based on a cubic expansion of the phases of the semiclassical wave functions. If simple semiclassical amplitudes and phases are used in the resulting expression, individual $T$-matrix elements can have significant errors. However, the sum over angular momenta tends to compensate for the individual discrepancies so that the final differential intensity is generally within 10% of quantal values, except for narrow spikes due to the turning-point singularities. The use of uniform semiclassical amplitudes and phases removes these spikes and reduces the error in the total differential intensity given by the cubic expression to only a few percent.

The capture of $1s$ electrons into partially stripped ions has been calculated rigorously within the Oppenheimer-Brinkman-Kramers approximation taking into account both the departure from the Coulomb potential and the effect of occupied projectile states. Numerical calculations have been performed for ${\mathrm{Li}}^{q+}$, ${\mathrm{C}}^{q+}$, ${\mathrm{N}}^{q+}$, and ${\mathrm{O}}^{q+}$ colliding with H atoms. After applying a reduction factor derived from the eikonal approximation excellent agreement between theory and experiment is obtained for $q=1\mathrm{}$ and slightly decreasing agreement for $q=2\mathrm{}$. In all cases considered the partially stripped projectile behaves like a Coulomb particle with the bare charge $q$ if $q\ge 3$ except at very high energies. Within the present treatment it is not possible to reproduce the oscillations in the charge-state dependence of the cross section for the reaction ${\mathrm{Ta}}^{q+}+\mathrm{H}\to {\mathrm{H}}^{+}+{\mathrm{Ta}}^{\mathrm{}(q-1\mathrm{})+}$ observed by Kim et al.

Spin relaxation of Rb atoms induced by collisional modification of the hyperfine interaction has been measured for ${}_{}{}^{85}{}_{}{}^{}\mathrm{Rb}$ and ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ in He, Ne, and Ar buffer gases at high magnetic field. The relaxation rates at 40 kG are 0.028 ${\sec }^{-1\mathrm{}}$ ${\mathrm{Torr}}^{-1\mathrm{}}$ for ${}_{}{}^{85}{}_{}{}^{}\mathrm{Rb}$ in He, 0.029 ${\sec }^{-1\mathrm{}}$ ${\mathrm{Torr}}^{-1\mathrm{}}$ for ${}_{}{}^{85}{}_{}{}^{}\mathrm{Rb}$ in Ne, and 0.041 ${\sec }^{-1\mathrm{}}$ ${\mathrm{Torr}}^{-1\mathrm{}}$ for ${}_{}{}^{85}{}_{}{}^{}\mathrm{Rb}$ in Ar. The corresponding relaxation rates for ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ are approximately 4.9 times larger, in accord with theory. Cross sections for the binary $\delta a\left(\overrightarrow{\mathrm{S}}·\overrightarrow{\mathrm{I}}\right)$ collisional interaction are estimated to be $\sigma \left(\delta a\right(\overrightarrow{\mathrm{S}}·\overrightarrow{\mathrm{I}}\left)\right)(\mathrm{Rb}-\mathrm{He})=2.3\mathrm{}\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$ and $\sigma \left(\delta a\right(\overrightarrow{\mathrm{S}}·\overrightarrow{\mathrm{I}}\left)\right)(\mathrm{Rb}-\mathrm{Ne})=1.4\mathrm{}\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$. Anomalous relaxation, $R^{*}$, is present in the same degree at 40 kG as at low magnetic field, lending strong support to earlier attributions of $R^{*}$ to the $\delta a\left(\overrightarrow{\mathrm{S}}·\overrightarrow{\mathrm{I}}\right)$ interaction in Rb-atom-noble-gas-atom van der Waals molecules.

In a nonrelativistic formulation employing noncovariant perturbation theory in the Born approximation and including exchange effects, the differential cross section for bremsstrahlung is derived for scattering by identical charged particles of arbitrary spin. The problem is first treated in a general manner, giving the expression for the total direct amplitude for the complete process (photon emission in scattering) for the case where the scattering potential is arbitrary. The derivation by means of perturbation theory demonstrates the critical role played by the small momentum associated with the emitted photon. Graviton bremsstrahlung is discussed briefly and the corresponding quadrupole cross section is given by a simple comparison with the photon bremsstrahlung formula.

Experimentally and theoretically determined differential cross sections are reported for electron capture in collisions of protons with helium atoms for incident proton energies of 25, 30, 50, and 100 keV and for center-of-mass scattering angles of 0.0 to 2.0 mrad. The magnitudes of the experimentally determined differential cross sections decrease from ${10}^{-10\mathrm{}}$ to ${10}^{-12\mathrm{}}$ ${\mathrm{cm}}^2$/sr within the 0.0-0.8-mrad range of the center-of-mass scattering angle. At approximately 0.8 mrad a distinct change in the slope of the differential cross section is observed. The experimental results which are for capture into all bound states of hydrogen are compared with the theoretical results of a calculation for capture into the ground state using the two-state two-center atomic expansion method in the eikonal approximation. Good agreement between the theoretical and the experimental results is obtained with a static potential which accounts for screening of the helium nucleus by a single passive electron.

The zero-core-contribution model allows photodetachment cross sections to be computed using a few readily accessible input parameters. We apply this model to the computation of photodetachment cross sections for the homonuclear diatomic ions ${\mathrm{O}}_2^{-}$ and ${\mathrm{S}}_2^{-}$, which have ${\pi }_g$ outer orbitals. Values are calculated for the total cross sections, partial cross sections for each vibrational channel, and angular distributions for each vibrational channel. The results are in good agreement with experimental values. Inclusion of vibrational effects allows the prediction of total photodetachment cross sections as a function of ion vibrational temperature. This allows comparison with both swarm and beam-type experiments. The simplicity of the model makes it of use in the interpretation of experimental results. This is illustrated by applying the model to infer equilibrium internuclear distances. We obtain $r_e^{\prime \prime }=1.348\mathrm{}$ Å for ${\mathrm{O}}_2^{-}$ and $r_e^{\prime \prime }=2.004\mathrm{}$ Å for ${\mathrm{S}}_2^{-}$. We anticipate that the simplicity of the model will allow its application to heteronuclear diatomic and polyatomic negative ions.

Using the method of Bleistein and of Ursell, a uniform asymptotic expansion for an important class of so-called generalized Bessel functions is derived, which permits a fast and efficient evaluation of these functions in the interesting domain of large values of their parameters where conventional methods fail. With the help of this asymptotic representation, a whole number of cross sections that have been given in the literature in terms of generalized Bessel functions for a variety of fundamental scattering processes in intense linearly polarized plane-wave laser fields can now be converted into actual numbers, thus permitting a quantitative assessment of the investigated effects at a given intensity of the applied laser field.

We report cross-section measurements for collisions of multicharged iron ions with a molecular-hydrogen target: (a) electron capture and electron loss of the multicharged ions, and (b) impact ionization of the hydrogen target. Iron ions, ${\mathrm{Fe}}^{+q}$, were used with charge states $q=3-13\mathrm{}$ at 103 keV/amu, $q=9-16\mathrm{}$ at 294 keV/amu, $q=11-22\mathrm{}$ at 1160 keV/amu, and $q=20-25\mathrm{}$ at 3400 keV/amu. We find that an empirically determined expression for the electron-capture cross section, $\sigma =(1.2\mathrm{}\times {10}^{-8\mathrm{}})q^{3.15}E{(\mathrm{keV}/\mathrm{amu})}^{-4.48\mathrm{}}$ ${\mathrm{cm}}^2$/molecule, describes all the data at and above 275 keV/amu. These measurements are compared with recent theoretical calculations.

Several local approximations to exchange are investigated. These are compared against exact static-exchange results for $e$,${\mathrm{Li}}^{+}$ scattering and for the photoionization of the $2S$ and $2P$ state of Li. All except one are found to be fairly accurate for these cases. Calculations are also performed for photoionization of the $5D$ state of Cs, which exhibits an $f$-wave resonance. All exchange models studied for this case (including the Slater $X\alpha$) are found to give widely differing results on the position and width of this resonance. Calculations are also performed for the photoionization of the Cs $6P$ state and compared against experimental results.