The long-range behavior of symmetry-restricted multiconfigurational self-consistent-field orbitals is given by ${\chi }_{i\alpha a}^{\mathrm{}\left(1\right)\mathrm{}}\sim (k_{i\alpha a}+{\mathrm{r}}^{-1\mathrm{}}{A}_{i\alpha a}) r^{{\eta }_{\alpha }}{e}^{-\zeta r}$, where the common orbital exponent $\zeta$ comes from the eigenvalue $\frac{{\zeta }^2}{2}$ of an "ionization potential" matrix ${\mathcal{I}}^{\overline{\alpha }}$, which may be thought of as being calculated with mixed-symmetry ($N-1\mathrm{}$)-electron functions extracted from the $N$-electron wave function. Here $\alpha$ is a "symmetry" index, $a$ is a "component" index, and $i$ is a "serial" index. The $k_{i\overline{\alpha }a}$ and $A_{i\overline{\alpha }a}$ from the "key" orbitals ${\chi }_{i\overline{\alpha }a}^{\mathrm{}\left(1\right)\mathrm{}}$ are derived from eigenvectors of ${\mathcal{I}}^{\overline{\alpha }}$. The corresponding key ${\eta }_{\overline{\alpha }}$ is obtained from the screened nuclear charge and from $\zeta$ by ${\eta }_{\overline{\alpha }}=\frac{(Z-N+1-\zeta )}{\zeta }$, while the other ${\eta }_{\alpha }$ satisfy ${\eta }_{\alpha }\le {\eta }_{\overline{\alpha }}-l_{\mathrm{GT}}(\overline{\alpha },\alpha )-1\mathrm{}$, where $l_{\mathrm{GT}}(\overline{\alpha },\alpha )$ is the smallest $l$ such that $Y_l^m(\theta ,\phi )$ couples $\overline{\alpha }$ to $\alpha$. The Carlson-Keller "theorem"—the diagonal expansion of the $N$-electron wave function on its $p$- and ($N-p$)-electron natural eigenfunctions—is generalized for symmetry-restricted natural eigenfunctions. Distinct symmetry-coupled parts of the wave function can separately but not simultaneously be put in diagonal form.

A many-body theory is presented based on an electronic Hamiltonian appropriate for describing core-valence excitation processes. It is shown that the core polarization propagator satisfies a factorization theorem relating this propagator to the product of two one-particle Green's functions, namely, a core Green's function and a valence Green's function which is calculated in the field of a time-dependent core potential. A diagrammatic theory for this valence Green's function is developed. The core-valence excitation energy is decomposed into relaxation and correlation contributions up to second order. A physical interpretation of this decomposition is given and the relationship between the analytical expressions and the various diagrams is pointed out. A general order analysis of relaxation and correlation terms is performed making use of the factorization theorem. An illustrative model is presented which allows an exact diagrammatic calculation of the valence Green's function with the time-dependent core potential via the Dyson equation. The analogy between the spectral function of our model and the spectral function of x-ray emission spectra of solids is discussed.

Differential cross sections for 15-100 keV ${\mathrm{He}}^{+}$ excitation of atomic hydrogen to its $n=2\mathrm{}$ level have been determined for c.m. angles from 0 to 8 mrad. The differential cross sections are obtained from an analysis of the angular distribution of the scattered ions which have lost an energy corresponding to the excitation of the target to its $n=2\mathrm{}$ level. The shape of the differential cross section changes rapidly with increasing incident energy. At 15 keV, the differential cross section falls off by a factor of 5 in 6 mrad. At 100 keV, the differential cross section decreases by nearly six orders of magnitude in the same angular range. The higher-energy results are in fair agreement with a recent symmetrized first-order Glauber approximation calculation of the process. Total cross section results are given for the same process in the 15-200 keV range.

Single- and multiple-electron-loss cross sections have been measured with 20-MeV ${\mathrm{Fe}}^{4+}$ on thin He, Ne, Ar, Kr, Xe, ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, C${\mathrm{H}}_4$, C${\mathrm{O}}_2$, CH${\mathrm{F}}_3$, C${\mathrm{F}}_4$, and S${\mathrm{F}}_6$ gaseous targets; cross-section-per-atom data for each system are presented. Cross-section-per-atom data from targets with approximately the same average atomic number indicate that both "nonadditivity" and "density" effects, depending on the final charge state of the ion, play roles in electronloss processes during projectile interactions with molecular targets. A modified Bohr formula is developed which reproduces single-electron-loss-per-atom data for atomic targets.

By using a saddle-point technique, the $1s2s2p{}_{}{}^2{}_{}{}^{}P$ resonance of ${\mathrm{He}}^{-}$ is found to be a Feshbach resonance. It is formed by the strong coupling of [$\mathrm{}\left(1s2s\right)\mathrm{}{}_{}{}^1{}_{}{}^{}S$,$2p$] and [$\mathrm{}\left(1s2p\right)\mathrm{}{}_{}{}^1{}_{}{}^{}P$,$3d$] configurations. By using a seven-partial-wave 54-term wave function and neglecting the open channels, the energy is -2.149 045 a.u. This is higher than the -2.175 23 a.u. $\mathrm{}\left(1s2s\right)\mathrm{}{}_{}{}^3{}_{}{}^{}S$ threshold, but significantly below the -2.145 97 a.u. $\mathrm{}\left(1s2s\right)\mathrm{}{}_{}{}^1{}_{}{}^{}S$ threshold. A quasi-projection-operator method is also carried out. Using the same wave function, the energy is -2.150 54 a.u. These results generally agree with that of experiment.

We have measured the electron-loss peak as a function of angle for the following collisions at 0.5 MeV/amu: ${\mathrm{H}}_2^{+}$-Ar, ${\mathrm{H}}_2^{+}$-Ne, ${\mathrm{H}}_2^{+}$-Kr, ${\mathrm{He}}^{+}$-Kr, and ${\mathrm{H}}^{-}$-Kr. Electron energy spectra at an angle of 6° and complete angular distributions are shown for the last three systems listed above. The energy at which the electron-loss peak has its maximum and the width at half-maximum of the peak are compared for all five systems in the angular range $1.5\le \theta \le 10°$. Our results suggest that, as previously indicated, the angular distribution of the electron-loss peak seems to be determined by the target. When the projectile is a positive ion the energy of the electron-loss peak seems to be independent of target or projectile and therefore is a general feature of the process. The width of the electron-loss peak and its dependence on angle seems to depend primarily on the projectile. Electron-scattering-model calculations do no predict features of the electron-loss peaks which are angular dependent.

We report the results of a theoretical study of resonant vibrational-excitation of ${\mathrm{N}}_2$ by low-energy electrons. The vibrational-excitation cross sections were calculated using ab initio fixed-nuclei resonance parameters in the complex-potential or "boomerang" model of Dubé and Herzenberg. The electronic resonance energy and decay width were extracted from several ab initio calculations of the ${}_{}{}^2{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$ shape resonance of ${\mathrm{N}}_2^{-}$. The good agreement between the present cross sections and those obtained recently with the $R$-matrix technique indicates that the assumptions underlying the complex-potential model are valid for the case of ${{\mathrm{N}}_2}^{-}\left({}_{}{}^2{}_{}{}^{}\Pi _g^{}{}_{}{}^{}\right)$. The present results also show the sensitivity of the computed vibrational-excitation cross sections to the fixed-nuclei resonance parameters employed in the calculations. The differential vibrational-excitation cross sections at 90° and the total integrated cross section which we have obtained agree resonably well with the available experimental data.

The charge transfer of ${\mathrm{N}}^{2+}$ and ${\mathrm{C}}^{3+}$ in collisions with atomic hydrogen at thermal energies is studied. Adiabatic molecular eigenfunctions, potential-energy surfaces, and nuclear coupling matrix elements are calculated, and a transformation is effected to a diabatic basis in terms of which the collision problems are solved. The emission characteristics of the spectrum resulting from charge transfer are presented.

A two-step laser excitation technique has been used to investigate the collisional broadening and shift of excited-state potassium transitions. Values for the argon collisional broadening and shift constants for the potassium $4p-7s$ and $4p-5d$ transitions were determined from line shapes for argon pressures up to 100 Torr. The values of these constants (in units of ${10}^{-9\mathrm{}}$ rad ${\mathrm{s}}^{-1\mathrm{}}$ ${\mathrm{atom}}^{-1\mathrm{}}$ ${\mathrm{cm}}^3$) are $(4\mathrm{}{P}_{\frac{1}{2}}-7\mathrm{}{S}_{\frac{1}{2}}):\gamma =11.60\mathrm{}\pm 0.07,\beta =-6.68\mathrm{}\pm 0.11;$ $(4\mathrm{}{P}_{\frac{3}{2}}-7\mathrm{}{S}_{\frac{1}{2}}):\gamma =11.49\mathrm{}\pm 0.15,\beta =-6.82\mathrm{}\pm 0.14;$ $(4\mathrm{}{P}_{\frac{1}{2}}-5\mathrm{}{D}_{\frac{3}{2}}):\gamma =8.64\mathrm{}\pm 0.07,\beta =-4.62\mathrm{}\pm 0.04;$ $(4\mathrm{}{P}_{\frac{3}{2}}-5\mathrm{}{D}_{\frac{3}{2}}):\gamma =8.58\mathrm{}\pm 0.10,\beta =-3.49\mathrm{}\pm 0.32;$ $(4\mathrm{}{P}_{\frac{3}{2}}-5\mathrm{}{D}_{\frac{5}{2}}):\gamma =9.13\mathrm{}\pm 0.10,\beta =-4.73\mathrm{}\pm \mathrm{0.09.}$ These broadening and shift constants are interpreted in terms of a Lennard-Jones interaction potential.

It is shown, with the help of model calculations, that a laser field can be used to suppress the background in electron scattering cross sections and to pick out only those parts which are rapidly varying as functions of the incoming electron energy. Therefore it becomes possible to measure the pure Breit-Wigner peaks and threshold effects, independent of the fact that interference between the resonances and the background in radiationless electron scattering is strong. The results are interpreted within the low-frequency limit for free-free transitions. The ac Stark shift of the resonance is also observed.

The excited products of collisions between xenon atoms and $\mathrm{Na}\mathrm{}\left(\mathrm{nd}\right)\mathrm{}$ atoms with $n=37\mathrm{}$ and $n=42\mathrm{}$ have been investigated by use of selective field ionization (SFI). It is observed that, at an ionizing-field slew rate of ∼ ${10}^9$ V/cm sec, most of the collision products follow predominantly diabatic paths to ionization which occurs at field strengths consistent with those predicted by quantum mechanical tunneling calculations. Detailed analysis of the SFI data indicates that the collision products have a wide (possibly complete) range of values of $l$ and $m_j$, although it is not clear whether the observed distribution results from single or multiple collisions. No evidence of $n$-changing collisions was seen. Similar data for collisions involving $\mathrm{Xe}\mathrm{}\left(\mathrm{nf}\right)\mathrm{}$ atoms are briefly discussed.

We discuss the difficulties inherent in the application of the separable approximation to scattering problems involving long-range forces. We propose a generalization of the usual separable expansion in which the long-range forces are treated exactly and only the short-range exchange forces are projected onto a set of $L^2$ functions. Application is made to the problems of $s$-wave electron-hydrogen- and electron-helium-atom scattering in the static-exchange approximation and a model problem involving a long-range ($\frac{1}{r^3}$) potential. In all cases, the separable approximation using distorted waves is found to converge much more rapidly than the standard separable expansion.

Electron-impact-ionization cross sections have been computed for several neonlike ions in a distorted-wave Born exchange approximation. For lower ionization stages the total cross section was found to be sensitive to the details of the distorted potential in which the partial waves were calculated. Ionization from the $2s^2$ core subshell increased in importance for higher charge states. Isoelectronic curves are presented which allow the interpolation of nonrelativistic electron-impact-ionization cross sections for any ion in the neon sequence in the incident electron energy range of one to three times threshold. A comparison to available experimental and semiempirical formulas is made.

$K$-shell photoelectric cross sections in Fe, Ni, Cu, Zn, As, Se, Br, Zr, Nb, Mo, Ag, In, Sn, and I have been determined using two different versions of Sood's method of measuring the absolute yield of fluorescent x rays when a target is irradiated with a known flux of photons. The results obtained by both of the methods agree with each other showing that no hidden systematic errors are involved in the measurements. The present results are found to compare well with the theoretical calculations of Scofield.

We have searched for long-lived $2S$ muonic hydrogen (${\mu }^{-}p$) in ${\mathrm{H}}_2$ gas at 0.8 atm. The state could be detected by observing delayed 1.9-keV $2P\to 1S$ photons from collisionally quenched ${\mu }^{-}p\left(2S\right)\mathrm{}$. The observed upper limit (90% confidence level) for the fraction,$f_{2S}$, of ${\mu }^{-}p\left(2S\right)\mathrm{}$ with lifetime ${\tau }_{2S}$ in the gas in the range, $100<\mathrm{}{\tau }_{2S}<1000\mathrm{}$ ns, is $f_{2S}<2\mathrm{}$%. Our results are consistent with the view that most of the ${\mu }^{-}p\left(2S\right)\mathrm{}$ have kinetic energies greater than the 0.3-eV threshold for the inelastic collision ${\mu }^{-}p\left(2S\right)+{\mathrm{H}}_2\to {\mu }^{-}p\left(2P\right)+{\mathrm{H}}_2$ and hence are rapidly quenched. We have also measured the relative Lyman line intensities in muonic hydrogen and muonic helium at 0.8 atm with solid-state detectors [Ge for ${\mu }^{-}{}_{}{}^4{}_{}{}^{}\mathrm{He}$ and Si(Li) for ${\mu }^{-}p$]. The results are ${\left(\frac{K_{\alpha }}{K_{\mathrm{total}}}\right)}_{\mu -p}=0.46\mathrm{}\pm 0.06$, ${\frac{K_{\alpha }}{K_{\mathrm{total}}})}_{\mu -{}_{}{}^4{}_{}{}^{}\mathrm{He}}=0.76\mathrm{}\pm 0.04$, ${\left(\frac{K_{\beta }}{K_{\mathrm{total}}}\right)}_{\mu -{}_{}{}^4{}_{}{}^{}\mathrm{He}}=0.04\mathrm{}\pm 0.02$. These results imply that about 7.5% of ${\mu }^{-}p$ atoms cascade to the $2S$ state.

This paper presents a modification to the Spencer-Attix theory, which allows application of the theory to larger cavity sizes. The modified theory is in better agreement with the actual process of energy deposition by delta rays. In the first part of the paper it is recalled how the Spencer-Attix theory can be derived from basic principles, which allows a physical interpretation of the theory in terms of a function describing the space and direction average of the deposited energy. A realistic model for the computation of this function is described and the resulting expression for the stopping-power ratio is calculated. For the comparison between the Spencer-Attix theory and this modified expression a correction factor to the "Bragg-Gray inhomogeneous term" has been defined. This factor has been computed as a function of cavity size for different source energies and mean excitation energies; thus, general properties of this factor have been elucidated. The computations have been extended to include the density effect. It has been shown that the computation of the inhomogeneous term can be performed for any expression describing the energy loss per unit distance of the electrons as a function of their energy. Thus an expression has been calculated which is in agreement with a quadratic range-energy relationship. In conclusion, the concrete procedure for computing the stopping-power ratio is reviewed.

High-resolution measurements of Mg $K\alpha$ x-ray spectra excited by 1-3-MeV/amu ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{+}$, ${\mathrm{O}}^{2+}$, ${\mathrm{Ne}}^{2+}$, and ${\mathrm{S}}^{4+}$ ions have been performed for Mg metal and some of its compounds. A detailed examination of the multiplet structure of the $K{L}^1$, $K{L}^2$, and $K{L}^3$ satellite groups revealed substantial variations in the intensities of various components as a function of projectile velocity and atomic number, and as a function of the chemical composition of the target. A theoretical spectrum was constructed using transition energies obtained from "corrected" single-configuration Hartree-Fock results and assuming a statistical population of all initial states belonging to the same electron configuration. This theoretical spectrum agreed quite favorably with the experimental spectrum for Mg metal obtained with 5.4-MeV ${\mathrm{He}}^{+}$ ions.

An analytic expression is derived for the scattering amplitudes, in the Glauber approximation, for a Coulomb plus corresponding spin-orbit interaction, $V=V_c\mathrm{}\left(r\right)+\left(\frac{\lambda }{r}\right)\left(\frac{d{V}_c}{\mathrm{dr}}\right)\overrightarrow{\sigma }·\overrightarrow{\mathrm{L}}$, with $V_c=\frac{{\mathrm{Ze}}^2}{r}$. The limiting cases of small and of large momentum transfers and of weak spin-orbit interactions are also presented.

The classical-trajectory impact-parameter method is used as the basis for deriving an equation of motion for the electronic state of two colliding, composite particles, one a projectile atom or ion consisting of a single valence electron outside a closed inner shell and the other, a target species comprised of two electrons plus a closed inner shell and/or atomic nucleus. The three outer-shell electrons are treated as being indistinguishable from one another, whereas the inner-shell, core electrons serve as sources of static, local pseudopotentials. We adopt for the electronic basis, products of isolated target states and projectile states which are distorted by exclusion forces emanating from the target. The associated set of close-coupled equations are written in the same form as that specific to the quasi-one-electron model of a projectile, such as Li or ${\mathrm{Be}}^{+}$, colliding with a target such as an He or Mg atom, modeled as a structureless scattering center. We then extract from these equations an effective, one-electron Hamiltonian comprised of parts identifiable as Coulombic and exchange interactions, suitably modified by projection operators which take account of the exclusion forces stemming from the indistinguishability of the projectile and target electrons, i.e., from the Pauli principle. Although the formulas for these interactions may be too complicated for direct use in scattering calculations, they are essentially exact and so can serve as useful guides in the search for approximate, model interactions.

The transmission of positrons from ${}_{}{}^{65}{}_{}{}^{}\mathrm{Zn}\left(E_{max}0.324 \mathrm{MeV}\right)$ and ${}_{}{}^{22}{}_{}{}^{}\mathrm{Na}\left(E_{max}0.544 \mathrm{MeV}\right)$ in Be, Al, Cu, Ag, and Pb were compared with that of electrons of identical energies. The comparison of their mass absorption coefficients ($\frac{\mu }{\rho }$) indicates a small but finite difference, implying greater transmission of positrons than electrons, with the exception of 0.324-MeV particles in Be and Al. These measurements were performed at atmospheric pressure. The results obtained clarify the existing ambiguity in the literature and are in agreement with multiple-scattering theory.