It is shown that an approach based on the use of label-free operators, which has been used to evaluate the interaction energies of pairs, can be adapted to evaluating the induced moments on each atom. Application of the method to H-H and H-He systems is given.

A calculation of the self-energy radiative correction to the energy level of an electron in the $2S_{\frac{1}{2}}$, $2P_{\frac{1}{2}}$, or $2P_{\frac{3}{2}}$ state in a Coulomb field, with nuclear charge $Z$ in the range 10-110, is described.

A total of 146 measured muonic Coulomb capture ratios of anhydrous solid binary compounds has been used to derive average muonic Coulomb capture probabilities for 65 elements. The measured capture ratios can be well reproduced by ratios of these capture probabilities. This is also true for ratios in ternary compounds, in glass, and in alloys. These muonic capture probabilities can serve as a basis for the application of muonic x rays in elemental analysis.

Various criteria for orbital contraction, most of them independent of the variational principle and based rather on formal properties of the exact wave function, all indicate that the formation of the chemical bond in ${\mathrm{H}}_2^{+}$ or ${\mathrm{H}}_2$ is accompanied by orbital contraction. Partially basing his argument on previous work by Pettifor, Heine recently criticized Ruedenberg's analysis of the chemical bond, especially the concept of orbital contraction. We show this to be due to interpretative problems of the linear-combination-of-atomic-orbital (LCAO) concept. The consequences for LCAO band-structure calculations are discussed.

The following test can distinguish regular from irregular orbits, even in marginal cases. A small sphere in phase space, initially centered at the starting point of the orbit, evolves into an ellipsoid by virtue of the Hamiltonian equations of motion. This ellipsoid rotates and vibrates "erratically" as its center moves along the classical trajectory. However, the proposed test involves only consecutive shapes of the ellipsoid as the orbit happens to pass in the vicinity of its initial point. The major axis of the ellipsoid then increases with time in a way which is roughly linear for regular orbits, and roughly exponential for irregular ones. Moreover, if the orbit is regular, the behavior of the angle between the major axis and the tangent to the trajectory indicates the smoothness of the invariant torus containing the orbit. This angle tends to a constant if the torus is very smooth, and oscillates if the torus has a complicated shape in the vicinity of the initial point.

The accurate third-order correlation energy ($E_3$) for Ne is evaluated within the framework of the Rayleigh-Schrödinger Hartree-Fock perturbation theory. The variational-perturbation method is applied. The $N$-electron first-order wave function is defined in terms of orbital configurations, of one-electron functions, and of symmetry-adapted pair functions in the form of partial-wave (PW) expansions. It is demonstrated that the value of $E_3$ critically depends on the accuracy of the approximate first-order wave function. In order to obtain accurate values of $E_3$, the pair functions have to be approximated in terms of extensive basis sets both in the radial and angular sense. A calculation based on all PW's up to 1′, 1″ ≤9 yields the value $E_3=0.00349\mathrm{}$ hartree. Careful radial and angular extrapolations result in $E_3=0.0044\mathrm{}$ hartree. The third-order energy is analyzed in terms of third-order pair energies and third-order PW increments to the pair energies. It is found that the third-order values modify the individual second-order PW increments and second-order pair energies by about 10%, whereas the modification caused by $E_3$ on the values given by $E_2$ is of the order of 1%. The stability of the third-order results with respect to the accuracies of the solution of the Hartree-Fock equations is also discussed.

Within the framework of the many-body Green's-function method we present a new approach to the polarization propagator for finite Fermi systems. This approach makes explicit use of the diagrammatic perturbation expansion for the polarization propagator, and reformulates the exact summation in terms of a simple algebraic scheme, referred to as the algebraic diagrammatic construction (ADC). The ADC defines in a natural way a set of approximation schemes ($n$th-order ADC schemes) which represent infinite partial summations exact up to $n$th order of perturbation theory. In contrast to the random-phase-approximation (RPA)-like schemes, the corresponding mathematical procedures are essentially Hermitian eigenvalue problems in limited configuration spaces of unperturbed excited configurations. Explicit equations for the first- and second-order ADC schemes are derived. These schemes are thoroughly discussed and compared with the Tamm-Dancoff approximation and RPA schemes.

The $L$-shell x-ray production in indium due to proton bombardment was studied using thin indium targets (19 μg/${\mathrm{cm}}^2$) over the proton energy range 120-400 keV. The total x-ray cross sections and five subshell cross sections were determined at 20-keV intervals in the energy range. The x rays were observed with a high-purity Ge low-energy-photon spectroscopy detector. The experimental cross sections were compared to the perturbed stationary-state theory with relativistic and Coulomb deflection corrections (CPSSR). The total $L$-shell x-ray cross sections were found to be in good agreement with the CPSSR theory. For the $L{\alpha }_{1,2}$ and $L{\beta }_{2,15}$ x-ray cross sections there was also reasonable agreement with the CPSSR theory. For the $L\beta$, $L{\gamma }_1$, and $L{\gamma }_{2,3}$ x-ray cross sections there is an effect that depends upon the proton energy. The ratios of these experimental and theoretical cross sections vary from about 1 at higher proton energies to near 2 for the lowest proton energies.

The results of a comprehensive study of projectile excitation and ionization cross sections for fast lithium atoms and positive ions impacting on low-$Z$ (≤11) targets are presented. This theoretical study is based on an asymptotic (high-speed) form of the Born approximation and the emphasis is placed on establishing the principal collision processes for ${\mathrm{Li}}^0$ at high energies, including inner-shell and valence-shell transitions to both discrete and continuum final states. An extensive comparison with experimental results for ${\mathrm{H}}_2$, He, and N targets is included. Recent experimental data for the ${\mathrm{Li}}^0$ $2P-2S$ emission cross sections for He targets in the 10—100-keV/u energy regime are in good agreement with the theoretical results, although significant departures from the trend of those data are predicted for higher energies due to the onset of doubly inelastic processes. Some discrepancies with recently reported ${\mathrm{Li}}^0$ electron-loss cross sections at $\frac{6}{7}$ MeV/u are noted for several targets.

An earlier paper by the same title recast the treatment of an electron in the field of an ionic core [quantum-defect theory (QDT)] into a form largely independent of the character of the long-range field. Analytical complexities and limitations of that paper are eliminated here by further reformulation. Qualitative aspects are highlighted. A WKB analysis interprets the QDT parameters as phase integrals over certain ranges of the radial coordinate; the limitations of the WKB analysis are then removed through the Milne approach to wave equations. The QDT analysis is connected to Jost's formulation of scattering, extended to long-range fields by explicit treatment of the relevant singularities. The effects of interactions confined to an internal many-body core are represented by QDT parameters wholly independent of the external region.

The stopping powers of aluminum and copper have been measured for electrons between 11 and 127 keV. Evaporated foils of about 200-nm thickness were stretched across rings and inserted into a short cylinder which was supported by thermocouple wires. Peltier heating and cooling by one thermopile were used to control foil temperature, while another thermopile measured its temperature. Calibration was accomplished by reducing beam energy until complete absorption was obtained. Delta rays produced in the foil were returned to it by repelling electrodes. Corrections were made for increased path length in the foil due to multiple elastic small-angle scattering and for spurious heating by scattered electrons striking the ring holding the foil. After correcting for these effects, good agreement with the Bethe-Bloch formula was found.

Beam-foil excitation of chlorine projectiles has been studied by means of optical spectrometry, at projectile energies ranging from 2 to 10 MeV. The relative populations of various levels in ClV through ClVIII have been measured as functions of the projectile energy. It is concluded that for ClVII and ClVIII, the $3p$ and $3d$ terms are populated entirely through molecular-orbital (MO) electron promotions. High-lying levels of these charge states are excited by means of pickup of electrons from the valence band of the solid. Intermediate levels are populated through multiple inner-shell processes as well as from valence-band electron pickup. For ClV and ClVI, most of the levels are excited predominantly by pickup of electrons from the valence band. However, for some of the valence-shell levels in ClVI, fairly strong contributions from MO processes are found at higher projectile energies. No evidence has been observed for direct Coulomb excitation.

The first measurements of elastic differential cross sections have been carried out for 25-, 50-, and 100-keV protons scattered through very small angles by helium atoms. The University of Missouri—Rolla energy-loss spectrometer provided the required high angular resolution and also separated the elastically scattered ions from the inelastically scattered ions. The data are compared to our Born, Glauber, and classical calculations as well as a four-state calculation. All of the measured elastic differential cross sections are more sharply peaked than theory for the smallest scattering angles. At the larger scattering angles all of the measured elastic differential cross sections are below the theoretical calculations. However, if the classical calculation is interpreted as a total differential scattering cross section, it compares well with our estimate of the sum of the elastic, charge-transfer, and inelastic differential cross sections.

A detailed derivation of a first-order analytic approximation to the electron-hydrogen charge-exchange amplitude in both the Glauber approximation and eikonal approximations in both post and prior forms is presented. The results are applicable to arbitrary initial states. The approximation is applied to elastic scattering by hydrogen in its ground state, and numerical results for both the modulus and phase of the exchange amplitude are obtained. The analytic approximation converges rapidly to the exact eikonal result with increasing energy. The techniques used in approximating the contribution to the exchange amplitude arising from the electron-nucleus part of the interaction potential are discussed in detail.

Expression have been derived that relate the stopping power and energy-loss straggling in a medium with internal motion for penetrating charged particles to the corresponding quantities applying to the equivalent medium at rest. These expressions have been based on a general binary-encounter picture and apply to nonrelativistic velocities and arbitrary mass ratios. Convenient expansions have been found in the limits of very high and very low projectile velocity. The results are applied to both nuclear and electronic stopping of charged particles. The capability of the scheme is tested upon the degenerate free-electron gas, for which accurate expansions at high and low projectile speed are known, with regard to both stopping and straggling. The scheme allows evaluation of shell corrections to stopping power and straggling of atomic and molecular gases beyond the range of validity of the leading terms in an expansion in inverse powers of electron velocity. A seeming disparity between high-speed straggling parameters calculated for the Fermi gas on the one hand, and an atomic target on the other hand, is attributed to different ground-state properties of the two systems in zero order. An essential difference is pointed out between the predictions of the dielectric theory and the present scheme with regard to the velocity dependence of energy-loss straggling in the low-speed limit.

A method of calculating the longitudinal range distribution function for ions incident at energies much greater than that of the stopping-power maximum is presented. The effects of electronic straggling and nuclear straggling and scattering are included. Straightforward analytical results are obtained from which the longitudinal distribution is determined by simple root finding. Results are given for the illustrative cases of ${}_{}{}^1{}_{}{}^{}\mathrm{H}$ in ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, and ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$ in ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$. The full width at half maximum of the distribution is found to result almost entirely from electronic straggling in all cases. At high ion energies ion-nucleus collisions affect the range distribution by producing a long, low-intensity tail on the distribution, stretching towards the target surface. The shape of this tail depends principally on the relative masses of ion and target nuclei. Comparison with limited available data on proton range distributions provides support for these calculations.

Using the basic muon-spin-rotation technique, the fractions of energetic positive muons thermalizing in diamagnetic environments ($f_{\mu }$) or as the paramagnetic muonium atom ($f_{\mathrm{Mu}}$) have been measured in low-pressure pure gases (He, Ne, Ar, Kr, Xe, ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, N${\mathrm{H}}_3$, and C${\mathrm{H}}_4$) as well as in several gas mixtures (Ne-Xe, Ne-Ar, Ne-N${\mathrm{H}}_3$, and Ne-C${\mathrm{H}}_4$). In the pure gases, the muonium fractions $f_{\mathrm{Mu}}$ are generally found to be smaller than expected from analogous proton-charge-exchange studies, particularly in the molecular gases. This is probably due to hot-atom reactions of muonium following the charge-exchange regime. Comparisons with muonium formation in condensed matter as well as positronium formation in gases are also presented. In the gas mixtures, the addition of only a few hundred parts per million of a dopant gas (e.g., Xe) which is exothermic for muonium formation gives rise to an $f_{\mathrm{Mu}}$ characteristic of the pure dopant gas itself, demonstrating the importance of the neutralization process right down to thermal energies. In all cases, the experimental signal amplitudes are found to be strongly pressure dependent, which is interpreted in terms of the time spent by the muon as neutral muonium in the charge-exchange regime, $t_n<0.2\mathrm{}$ ns. This time is generally shorter in the case of molecular gases than in rare gases.

The contributions of excitation autoionization to electron-impact ionization in the Nalike ${\mathrm{Al}}^{2+}$ and ${\mathrm{Si}}^{3+}$ are calculated using a two-state close-coupling approximation. Inner-shell excitations of the type $2p^{6}3s\to 2p^{5}3s3l\mathrm{}(l=0,1,2\mathrm{})\mathrm{}$ are included. The contributions of resonant-excitation double autoionization are estimated from a three-state close-coupling approximation with the aid of quantum-defect-theory analysis. These contributions explain the absence of the largest predicted excitation-autoionization step in the measurements of Crandall et al. Predictions for ${\mathrm{Mg}}^{+}$ also lead to similar agreement with experiment. We conclude that the experiments of Crandall et al. for the total ionization cross sections for ${\mathrm{Mg}}^{+}$, ${\mathrm{Al}}^{2+}$, and ${\mathrm{Si}}^{3+}$ by electron impact include measurements of the physical processes of direct-ionization, inner-shell excitation autoionization, and inner-shell resonant-excitation double autoionization.

A method for the calculation of cross sections for electron capture into arbitrary $n$, $l$, and $m$ states of fast bare projectiles in collisions with fully stripped ions and ground-state hydrogenic targets has been proposed in the framework of the continuum distorted-wave approximation. The charge-exchange amplitude has been reduced to a one-dimensional integral in real space which is very convenient for numerical evaluation. The present method is applied in ${\mathrm{Li}}^{3+}$-$\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}$ collisions and the calculated cross sections are in excellent agreement with the experimental data for the incident energies $E\ge 700$ keV.

A new method has been developed in order to calculate the total cross section for particle-atom collisions. For hydrogen we have found the explicit expression of the first three terms of the cross section for large incident momentum. The calculation is performed in momentum space. The propagator of the scattering amplitude is expanded in powers of the atomic energy levels, and use is made of the optical theorem. The polarization and distortion effects have been taken into account by calculating the contributions of the scattering amplitude up to the fourth order. The latter was evaluated using the Glauber amplitude. Thus the explicit dependence of the cross section on $Z^2$, $Z^3$, and $Z^4$, where $\mathrm{Ze}$ is the charge of the particle, has been obtained. An estimate of the precision of our result is given. The total elastic cross section has also been calculated.