We present a general theory of electron degradation and of the yields of initial species such as ions and excited states that appear in matter as a result of irradiation with electrons. We give variational expressions for observable quantities such as the degradation spectrum, the mean yields, and the yield fluctuations. A systematic analysis using variational principles reveals, in greater detail than ever before, relations among major analytical methods including the Fowler method, the Spencer-Fano method, and the method of Knipp et al. Each of these methods represents a different angle of approach to the same problem and leads to the same prediction for the mean yield of any initial species, for a fixed set of electron cross-section data. Among our findings it is noteworthy that the knowledge of the Spencer-Fano degradation spectra for various source-electron energies enables one to calculate by quadrature not only the mean but also the statistical fluctuations of the yield of any initial species. Furthermore, when one introduces small changes in the cross-section data (because of new information or upon consideration of chemical impurities in the medium), one may express the ensuing modifications of any observable quantity in the form of a perturbation expansion. Then, every term in the perturbation series is calculable again from the Spencer-Fano degradation spectra for various source energies for the unperturbed problem. In this respect and many others, the Spencer-Fano degradation spectrum is the most basic element in the solution of the degradation problem. Finally, our mathematical analysis of various difference-integral equations (including nonlinear equations) may be of interest in areas other than electron degradation. Therefore, the present paper includes remarks on the physical meaning of adjoint operators and on other points belonging to general mathematical physics.

The mobilities of the doubly charged rare-gas ions ${\mathrm{He}}^{2+}$, ${\mathrm{Ne}}^{2+}$, ${\mathrm{Ar}}^{2+}$, ${\mathrm{Kr}}^{2+}$, and ${\mathrm{Xe}}^{2+}$ in their parent gases have been determined in a drift-tube mass-spectrometer apparatus. The finding of more than one mobility for each of the doubly charged ions (with the exception of ${\mathrm{He}}^{2+}$) is attributed to the presence of the ${}_{}{}^3{}_{}{}^{}P$, ${}_{}{}^1{}_{}{}^{}D$, and ${}_{}{}^1{}_{}{}^{}S$ long-lived, lowest-energy states of these ions. At 300 K the reduced mobilities (in units of ${\mathrm{cm}}^2$/V sec) extrapolated to zero field have the values ${\mu }_0\left({\mathrm{He}}^{2+} \mathrm{in}\mathrm{He}\right)=18.3\mathrm{}$; ${\mu }_0\left({\mathrm{Ne}}^{2+} \mathrm{in}\mathrm{Ne}\right)=6.5\left({}_{}{}^1{}_{}{}^{}D\right), 7.0\left({}_{}{}^3{}_{}{}^{}P\right), 8.5\left({}_{}{}^1{}_{}{}^{}S\right)$; ${\mu }_0\left({\mathrm{Ar}}^{2+} \mathrm{in}\mathrm{Ar}\right)=2.7\left({}_{}{}^3{}_{}{}^{}P\right), \sim 2.2\left({}_{}{}^1{}_{}{}^{}D\right)$; ${\mu }_0\left({\mathrm{Kr}}^{2+} \mathrm{in}\mathrm{Kr}\right)=1.11\mathrm{} \mathrm{and} 1.28$; ${\mu }_0\left({\mathrm{Xe}}^{2+} \mathrm{in}\mathrm{Xe}\right)=0.65\mathrm{} \mathrm{and} 0.75$. The measured values are accurate to ±3%.

Radiative charge-transfer coefficients for single-electron transfer between doubly or singly charged noble-gas ions and noble-gas atoms have been determined at thermal energy using a drift-tube mass-spectrometer apparatus. At 300 K, $k({\mathrm{He}}^{2+}+\mathrm{He})=(4.8\mathrm{}\pm 0.5)\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^3$/sec, with slightly smaller coefficients for the reactions ${\mathrm{Ne}}^{2+}$ + Ne, ${\mathrm{Ar}}^{2+}$ + Ar, ${\mathrm{Kr}}^{2+}$ + Kr, and ${\mathrm{Xe}}^{2+}$ + Xe. Differences in reactivity are found for the different, low-lying ${}_{}{}^1{}_{}{}^{}S$, ${}_{}{}^1{}_{}{}^{}D$, and ${}_{}{}^3{}_{}{}^{}P$ states of the doubly charged ions. A three-body contribution to the radiative process is noted, with rate coefficients ranging between ${10}^{-31\mathrm{}}$ and ${10}^{-30\mathrm{}}$ ${\mathrm{cm}}^6$/sec at 300 K for the different noble-gas systems. The radiative charge-transfer coefficients for singly ionized ions are somewhat smaller, $k({\mathrm{He}}^{+}+\mathrm{Ne})\sim 1\times {10}^{-15\mathrm{}}$ ${\mathrm{cm}}^3$/sec and $k({\mathrm{Ne}}^{+}+\mathrm{Ar})\sim 6\times {10}^{-15\mathrm{}}$ ${\mathrm{cm}}^3$/sec at 300 K. The measured value of $k({\mathrm{He}}^{2+}+\mathrm{He})$ is in reasonable agreement with the very recent theoretical calculations of Cohen and Bardsley.

The functional forms of the wave functions for the bound states of hydrogenic systems and for free particles as described by the complex-coordinate method are presented. Writing the wave function for an electron resonance as the sum of a "boundlike" or "$Q$-space" part and a "scatteringlike" or "$P$-space" part, we suggest functional forms or bases for these two parts based on the solutions of the hydrogenic and freeparticle systems. We present an argument suggesting that when the rotation angle in the complex-coordinate method is greater than $\left|\arg \right(k_r\left)\right|$ for the electron resonance, this method is identical to calculations based on a Siegert resonance. This assumed structure of the wave function should yield a rate of convergence similar to other methods. The advantages of this method are that the basis functions are all square integrable, a single calculation yields both the position and width, only a solution of a straightforward eigenvalue problem is required, arbitrarily accurate target states are easily incorporated, and polarization terms can be explicitly included. Variational calculations for the position and width of the lowest ${}_{}{}^2{}_{}{}^{}S$ resonance in the negative helium ion are reported using trial wave functions containing 39, 43, 55, 59, and 67 configurations. These wave functions contain 8, 8, 20, 24, and 32 "$P$-space" configurations, respectively. Values of 19.387 eV and 12.1 meV are obtained for the position and width, respectively, of the resonance. One also finds that inclusion of free-particle-like basis functions improves the representation of the scattering states.

Several small, previously uncalculated contributions to the $2s_{\frac{1}{2}}-2\mathrm{}{p}_{\frac{3}{2}}$ splitting $S_1$ in muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are calculated, and previously calculated contributions are verified. The improved theoretical result is $1817.5\pm 0.6+0.4{〈r^2〉}^{\frac{1}{2}}-106.2\mathrm{}〈r^2〉+1.4\mathrm{}{〈r^2〉}^{\frac{3}{2}}\simeq 1527.3-102.6(〈r^2〉=2.80)$ meV, where $〈r^2〉$ is the mean-square charge radius of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ in ${\mathrm{fm}}^2$. Using the most recent value of the rms charge radius we get $S_{1\mathrm{th}}=1527.1\mathrm{}\pm 4.2$ meV, in excellent agreement with the experimental result $S_{1\exp }=1527.5\mathrm{}\pm 0.3$ meV.

Upper bounds are derived to the many-electron density ${\rho }_k(x_1, \dots , x_k)$ ($2\le k\le n$) of an $n$-electron atomic or molecular wave function. The derivation is based on former results on the decay of the one-electron density and on the "Schrödinger inequality" for ${\left({\rho }_k\right)}^{\frac{1}{2}}$ which, by means of a comparison theorem, allows the deduction of upper bounds for ${\rho }_k$ in the region $G^k=\left\{\right(x_1, \dots , x_k\left)\right|r_i\ge c, 1\le i\le k\}$, $c$ being a constant and $r_i=\left|x_i\right|$, provided an upper bound to ${\rho }_k$ is available on the boundary $\partial G^k$. The latter can be obtained from bounds to ${\rho }_{k-1\mathrm{}}$. A recurrence procedure leads to the final result which for the case of an atom is given by ${\rho }_k\le \mathrm{dS}\Pi {i=1\mathrm{}}^k\left[r_i^{2[\frac{Z}{2}{\left(2\mathrm{}{\epsilon }_i\right)}^{\frac{1}{2}}-1]} e^{-{\left(2\mathrm{}{\epsilon }_i\right)}^{\frac{1}{2}}}{r}_i\right]$ ($d$ is a constant, $S$ acts as a symmetrizer, $Z$ denotes the nuclear charge, and the ${\epsilon }_i$ ($1\le i\le k$) denote the successive ionization potentials of the state under consideration). We further report an improved bound for ${\rho }_2$ which depends explicitly on the interelectronic distance.

A variational form of the cellular method is proposed as a new model to solve the one-electron Schrödinger equation for molecules and crystals. The model keeps such good features of the traditional cellular method as the arbitrary partition of space, and eliminates its main drawback, the slow convergence of the cellular expansion series. With the aid of a criterion of precision on the trial wave functions, we discuss the possibilities offered by the method for more accurate calculations of the electronic structures of molecules and solids. As an example of the accuracy and fast convergence of the model, computation of the energy spectrum of the molecular hydrogen ion ${\mathrm{H}}_2^{+}$ is presented.

Variational principles have been used extensively for estimating some given functional $F(\phi , {\phi }^{†})$ where the functions $\phi$ and ${\phi }^{†}$ are well defined by a set of differential equations and boundary conditions but cannot be determined exactly. The variational principle for the estimation of a matrix element of an arbitrary Hermitian operator $W$ involves not only the trial wave functions ${\phi }_t$ but also trial auxiliary Lagrange functions $L_t$; the $L_t$ depend on the ${\phi }_t$ and on $W$. To determine the parameters in the $L_t$ efficiently, a functional $M\left(L_{\mathrm{tt}}\right)$ is constructed which is an extremum for $L_{\mathrm{tt}}=L_t$. The technique was recently used successfully in the variational estimation of two diagonal matrix elements. We here use this technique for the variational estimation of an off-diagonal matrix element, the generalized oscillator strengths of helium for the transition between the ground state and the excited $2{}_{}{}^1{}_{}{}^{}P$ state. Two $L_t\text{'}\mathrm{s}$ must be determined. Our results on helium indicate that variational estimates are a significant improvement over the first-order estimates. The results are also compared with those obtained nonvariationally using more elaborate ground-and excited-state wave functions; the comparison represents a check on the method. It is not yet clear which of the two approaches is more efficient.

The fine-structure interval $\Delta \nu$ of ground-state positronium formed in Si${\mathrm{O}}_2$ and MgO powder pellets has been determined through measurements of microwave-induced transitions between Zeeman substates of orthopositronium. The $\Delta \nu$ values for Ps in the powders are less than $\Delta \nu$ for free Ps by the order of 1 part in ${10}^4$, suggesting an attractive interaction between Ps and the powder grain surfaces within the pellet. No Zeeman transition was found for positrons stopping in a silicon single crystal or in fused quartz.

We derive a new form for the matrix element of the two-particle mutual spin-orbit interaction in the Breit-Pauli approximation. The complexity of standard tensor-operator expansion techniques is obviated by suitably decomposing the mutual spin-orbit interaction into products of spin and orbital angular-momentum operators, whose matrix elements in the $l{m}_{i}s{m}_s$ scheme are obtained in a straightforward manner. Graphical representations are given for both the one- and two-particle spin-orbit interactions, thereby permitting the analytical evaluation by graphical techniques of the spin-orbit interaction between arbitrary configurations in any coupling scheme. The graphical procedure for evaluating these matrix elements is demonstrated for $\mathrm{LS}$-coupled configurations having two nonequivalent electrons outside closed shells. Various properties of the two-particle mutual spin-orbit operator are easily illustrated using its graphical representation.

$K$-shell x-ray production cross sections, $\frac{K\beta }{K\alpha }$ x-ray intensity ratios and $K\alpha$ and $K\beta$ x-ray energy shifts have been determined for thin solid targets of ${}_{15}{}^{}{}_{}{}^{}\mathrm{P}$, ${}_{19}{}^{}{}_{}{}^{}\mathrm{K}$, ${}_{20}{}^{}{}_{}{}^{}\mathrm{Ca}$, ${}_{21}{}^{}{}_{}{}^{}\mathrm{Sc}$, ${}_{22}{}^{}{}_{}{}^{}\mathrm{Ti}$, ${}_{23}{}^{}{}_{}{}^{}\mathrm{V}$, ${}_{25}{}^{}{}_{}{}^{}\mathrm{Mn}$, ${}_{26}{}^{}{}_{}{}^{}\mathrm{Fe}$, ${}_{27}{}^{}{}_{}{}^{}\mathrm{Co}$, and ${}_{28}{}^{}{}_{}{}^{}\mathrm{Ni}$ for 4-38-MeV ${}_5{}^{10}{}_{}{}^{}\mathrm{B}$ ion bombardment. Comparisons of the cross sections were made to direct Coulomb ionization and electron-capture theories by means of fluorescence yields corrected for multiple-ionization effects. The dominant contribution to $K$-vacancy production for these projectile-target combinations is believed to be direct ionization. Electron capture is expected to be important only for the lighter target elements and then primarily at the higher velocities. The direct-ionization theories employed were the binary-encounter approximation and the plane-wave Born approximation (PWBA) both of which overestimated the experimental data especially at the lower incident ion velocities. The PWBA was modified for increased target-electron binding, Coulomb deflection of the incident ion, polarization of the target-electron wave functions due to the passage of the incident ion, and relativistic target-electron velocities. The experimental data were found to agree quite well with the sum of the theoretical predictions of the modified PWBA and electron capture.

Beginning with a Feshbach projection-operator analysis, formulas for the local complex potential in the entrance-channel radial wave equation are reviewed and developed in detail for inelastic atom-atom collisional processes that involve an electronic transition between Born-Oppenheimer states. Penning and associative ionization are used as illustrative examples in the presentation. A complete derivation of $T$-matrix elements involving radial wave functions is then presented in the context of transition rates and inelastic differential cross sections (both for energy and angle). Two approximate formulas for the energy spectrum of the particle ejected in the electronic transition are developed in which quantities obtained from exact, complex radial wave functions are replaced by those obtained from approximate, totally real radial wave functions. Numerical computations for the electron energy spectrum of the Penning process $\mathrm{He}(1s2s, 2{}_{}{}^3{}_{}{}^{}S)+\mathrm{H}(1s, 1{}_{}{}^2{}_{}{}^{}S)\to \mathrm{He}(1\mathrm{}{s}^2, 1{}_{}{}^1{}_{}{}^{}S)+{\mathrm{H}}^{+}+e^{-}$ are then presented to probe the effects of the imaginary width of the complex potential. It is found that the real component of the entrance-channel radial wave function is not much affected by the imaginary width when the width is small compared to other energy terms. In such cases, the imaginary component of the wave functions is then $\frac{\pi }{2}$ out of phase with respect to the real component, even at fairly close separations, owing to their asymptotic boundary conditions. This result also leads to a method of estimating the contribution of the imaginary radial wave functions to matrix elements using information obtained only from approximate, totally real wave functions. Finally, the eigenenergies of all 149 rotational-vibrational states of $\mathrm{He}{\mathrm{H}}^{+}\left({}_{}{}^1{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}\right)$ are reported, along with the computed cross sections of associative ionization to each state.

As suggested previously, in a heavy-ion collision the sum of the total $L$-vacancy-production cross section of the higher-$Z$ partner and of the $K$-vacancy-production cross section of the lower-$Z$ partner, close to $K-L$ level-matching conditions, represents the vacancy-formation cross section in the $3d\sigma$ molecular orbital. It is found empirically that at a given projectile velocity $v_1$ the $3d\sigma$ cross section depends mainly on the united-atom $Z$, independent of the projectile $Z\left(Z_1\right)$, between $Z_1=28\mathrm{} \mathrm{and} 53$. A united-atom ionization model of Briggs is evaluated in the semiclassical approximation. The results provide an upper-limit estimate for the $3d\sigma$ cross section, which reproduces quantitatively its empirical $v_1$ and $Z$ dependence and its order of magnitude. The results do not imply, though, that other vacancy-production mechanisms may not be as, or more, important than ionization.

Absolute measurements of stopping powers and energy straggling for hydrogen- and helium-ion beams in Ge, Se, Pd, Ag, Sb, and Bi in the energy range $20\le E\le 260$ keV were performed by the method of determining the energy loss suffered by an ion beam which has traversed a thin film. The measured stopping powers fit reasonably well with those of other authors obtained at higher and lower ion energies. Agreement with theoretical predictions for energies >100 keV/amu is found to be better for hydrogen than it is for helium beams. The $Z_2$ dependence of the stopping power for hydrogen projectiles is discussed including measurements of other authors performed on other elements, and is found to agree in shape with that for helium for ion velocities $v>\mathrm{}2\mathrm{}$ a.u., but there is evidence that it begins to differ with decreasing $v$. It is deduced from the present measurements that for hydrogen as well as helium ions the amplitude of oscillations of the $Z_2$ dependence increases with decreasing ion velocity, reaching a maximum at $v=1.4\mathrm{}$ a.u. The Lindhard-Scharff theory is found to be in satisfactory agreement with the measured energy straggling.

Inelastic processes leading to excited-state formation have been studied by spectral analysis of collision-produced uv-visible-near-ir radiation for (2-800 eV) ${\mathrm{He}}^{+}$ and ${\mathrm{He}}_2^{+}$ collisions with Mg and Zn atoms. Although observation of MgII radiation must result from exothermic charge transfer in the ${\mathrm{He}}^{+}$-Mg system, the emission cross sections for the two observed MgII lines exhibit threshold behavior; it is suggested that production of ground-state MgIII dominates at low energy. ${\mathrm{He}}_2^{+}$-Mg collisions yield two prominent MgII lines, the emission cross sections for which are relatively large at low energy and decrease rapidly with increasing kinetic energy. ${\mathrm{He}}^{+}$-Zn collisions produce rich spectra of ZnII lines at all energies and HeI lines at elevated energies. In contrast, ${\mathrm{He}}_2^{+}$-Zn collisions yield only a few lines with large cross sections at low energy.

Born cross sections have been calculated for electron-impact excitation of metastable levels of argon and krypton to higher-lying excited states. It is shown that the intermediate-coupling representation must be used to obtain reliable results. An approximate treatment of strong-coupling effects is included for the dominant $ns-np$ transitions, and the applicable range and validity of the Born calculations are considered.

The energy and oxygen-pressure dependence of Balmer radiation photon yield due to ion-surface collisions on polycrystalline molybdenum using hydrogen atomic and molecular ions were measured in the energy range from 200 to 10 keV. These results, obtained under ultrahigh-vacuum conditions, provide new information on the ways in which the photon yield is produced by electronic interactions of the emerging particles with the surface. Radiation yield, normalized to current and backscatter yield, varied in a consistently exponential manner over the entire incident-ion energy range. The molecular state and the charge state of the incident particles were measured to have no discernable effect on the final excitation for clean surfaces. At partial pressures of oxygen below ${10}^{-11\mathrm{}}$ Torr, surface concentrations of oxygen below 2% of a monolayer on molybdenum could be maintained during the experiments. Increased surface coverages (up to 0.5 to 1 monolayer) raised the photon yield for molybdenum by a factor of 7. These oxygen-dependent effects are tentatively interpreted as owing mainly to changes in the deexcitation probability in the vicinity of the surface.

The Coulomb-modified form of the Glauber approximation is applied to the $n=2\mathrm{}$ and $n=3\mathrm{}$ excitation of hydrogenlike ions by incident electrons for various values of the target-ion nuclear charge $Z_n$. The properly computed $e^{-}$-${\mathrm{He}}^{+}$ $1s\to 2s, 2p$ Glauber predictions, including appropriate cascade effects, are compared with available experiment. The $Z_n$ dependence of the scaled integrated (over scattering angles) cross section is discussed, including the limit as $Z_n\to \infty$.

Calculations of cross sections for (vibrationally and electronically) elastic collisions of electrons with several alkali-metal halides were performed for energies in the range of 0.13 to 20.0 eV. The applicability of the adiabatic (fixed-nuclei) approximation for strongly polar systems is investigated by model calculations on CsF, KI, and LiF. We demonstrate that integrated, momentum transfer, and differential cross sections for polar systems can be reliably generated entirely within the body-frame, adiabatic approximation. We also suggest resolutions of several discrepancies between the results of earlier calculations and between these results and measurements. Close-coupling calculations, based on the adiabatic approximation and an alternative form of the frame transformation, were performed for electron-LiF collisions using the full static and static model-exchange surface. Reasonable agreement was found with measured differential cross sections at 5.44 and 20.0 eV. Shape resonances in the $\Sigma$ and $\Pi$ body-frame symmetries, centered near 1.8 and 1.5 eV, respectively, were observed. Similar features appeared in static-exchange calculation for NaF and NaCl. We also compare the results of the static and static-exchange calculations with the results of calculations using simpler model potentials and other approaches to the collision problem.

Differential cross sections for electron capture in fast-proton-multielectron-atom collisions are obtained by using the eikonal approximation for the internuclear interaction and the two-state, two-center atomic-expansion method for the electron-capture amplitude in the impact-parameter formulation. The importance of the internuclear potential and the various approximations for the electron-capture probability for the differential cross section is examined. It is found that when the projectile deflection is included, the two-state, two-center atomic-expansion method can predict differential cross sections with good agreement with experimental data of Cocke et al. and of Bratton et al.