The solution of the inverse problem for the time-dependent density matrix means finding the time-dependent Hamiltonian operator $\hat{H}$ corresponding to a given time-dependent density matrix $\hat{\rho }$. We point out that the standard requirements for the time-dependent density operator imply a number of compatibility relations. Accordingly, a given $N\times N$ density matrix might result from a whole family of Hamiltonians, characterized by $N$ free, real parameters. The cases of a 2×2 and 3×3 density matrix and of the free-fermion chain are worked out explicitly.

It is shown that a statistical angular-correlation coefficient $\tau$ may be obtained from the small-momentum-transfer behavior of x-ray and high-energy electron-scattering intensities. Published tables of these quantities are exploited to obtain values of $\tau$ for correlated wave functions of first-row atoms and ions. $\tau$ is negligibly small in lithium, about -0.09 in beryllium, and its magnitude decreases monotonically as one goes across the first row from Be to Ne.

Stimulated emission in the high-gain regime from a cold, relativistic beam of electrons gyrating in a combined solenoidal and longitudinally polarized periodic wiggler magnetic field is considered as a source of short-wavelength radiation. The emitted wave frequency is Doppler upshifted in proportion to the wave number of the wiggler magnetic field. Amplification is due to a ponderomotive bunching force acting on the electrons in either the transverse or axial directions. Expressions for the linear growth rate are obtained; conditions for their validity and estimates for the saturated efficiency are given.

A recursion theory for the determination of binding energies and expectation values of $r^{-1\mathrm{}}$ is presented and discussed for neutral atoms. The derived recursion relations are parameter free and provide accurate estimates of $E$ and $〈r^{-1\mathrm{}}〉$ in terms of just the atomic number $Z$ and the corresponding zero-order perturbation energy ${\epsilon }_0$. By construction the formulas yield exact values for $Z=1\mathrm{}$ and for $Z\to \infty$ (in a relative sense). It is shown that for large $Z$ the model energy corresponds to a $Z^{-\frac{1}{3}}$ series; i.e., $E=C_0{Z}^{\frac{7}{3}}+C_1{Z}^2+\cdots$, where $C_1=0.4701\mathrm{}$. Various modifications of the theory are considered; the most accurate one is manifested in a single-parameter formula that yields energies of all atoms, in the range $1\le Z\le 86$, with an average deviation of 0.09% compared to the corresponding Hartree-Fock values. Similarly, the single-parameter counterpart for $〈r^{-1\mathrm{}}〉$ gives results, for these atoms, with an average deviation of 0.2% compared to the corresponding Hartree-Fock values. Finally, it is noticed that the recursion theory is based to some extent on the local properties of a hypothetical two-dimensional surface, $E(Z,N)\mathrm{}$, that contains the energies of neutral atoms as a particular subset.

A generalization of the Feshbach-Rubinow method for variationally determining the ground-state energy of a multiparticle system is described and applied to the ${\mathrm{H}}^{-}$ ion. The method finds a variationally optimized function of a contour that characterizes the system. The contour can be quite complicated, being itself determined by some approximation scheme for solving the multiparticle Schrödinger equation. The application to the ${\mathrm{H}}^{-}$ ion uses an already optimized three-parameter contour due to Bethe, and a significant improvement in the ground-state energy is noted.

We consider the result that transient moments of a Fokker-Planck equation may have a long-time decay constant which is smaller than the smallest eigenvalue of the Fokker-Planck operator. We find the general mechanism by which such decay constants can be generated, and show how to obtain such decay constants from the expansion in eigenfunctions.

Bloch's technique of joining the nonrelativistic Bethe and Bohr stopping-power expressions by taking into account wave-packet effects for close collisions is extended to the relativistic case. It is found that Bloch's nonrelativistic correction term must be modified and that charge asymmetric terms appear. Excellent agreement is observed by comparing the results of these calculations to recent data on the stopping power of relativistic heavy ions.

We have measured the energy loss of several heavy ions (C, Si, Cl, Ti, Fe, Ni, Ge, Br, Nb, and I) in elemental targets (C, Al, Cu, Ag, and Au) at energies near the maximum in the stopping power versus energy curve. These measurements have been used to examine the charge dependence of heavy-ion stopping powers, as well as the magnitude of the ion's effective charge inside the target material. Inclusion of the Lindhard $Z_1^3$,$Z_1^4$ corrections provides the best fit to the data, and the use of these corrections results in a simple expression for the effective charge which is valid for all our measured projectile-target combinations ($Z_1=6-53\mathrm{}$ and $Z_2=6-79\mathrm{}$). Both the magnitude and the target dependence of this effective-charge expression are consistent with average equilibrium charge-state measurements made in gases. This suggests that the high charge states observed for ions leaving solid materials are due mainly to charge-changing effects (such as Auger deexcitation) at the exit surface.

The anisotropy of the resolved $\mathrm{Ll}$ x-ray line emitted from proton-excited Pb atoms has been measured. The anisotropy coefficient is strongly dependent on the projectile energy within the proton energy range studied (from 0.4 to 3.0 MeV). A detailed explanation of this energy dependence is suggested in terms of the Coulomb interaction of the projectile with the target nucleus. The deflection effect is discussed in earlier publications; the retardation effect is introduced here for the first time in the context of the alignment of $L_3$-subshell vacancies. Both effects are treated with appropriate weight functions for the different magnetic substates of the $L_3$ subshell.

We have used selective excitation and monitoring of individual ($J,m_J$) Zeeman sublevels of the $4{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ and $4{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ states of K at 22 and 131 kG to measure ($J,m_J\to J^{\prime },m_{J^{\prime }}$) transition cross sections for K colliding with He, Ne, and Kr, at 380 K. The results, translated into irreducible tensor cross sections for depolarization are, in units of ${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$:

$\left\{{\Lambda }_{\frac{3}{2}}^1\right(\mathrm{He}),=97\mathrm{}\pm 13; {\Lambda }_{\frac{3}{2}}^2(\mathrm{He})=130\mathrm{}\pm 17; {\Lambda }_{\frac{3}{2}}^3(\mathrm{He})=99\mathrm{}\pm 23\}\{{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}; {\Lambda }_{\frac{3}{2}}^1(\mathrm{Ne}),=103\mathrm{}\pm 10; {\Lambda }_{\frac{3}{2}}^2(\mathrm{Ne})=146\mathrm{}\pm 14; {\Lambda }_{\frac{3}{2}}^3(\mathrm{Ne})=111\mathrm{}\pm 14\}\left\{{\Lambda }_{\frac{3}{2}}^1\right(\mathrm{Kr}),=270\mathrm{}\pm 24; {\Lambda }_{\frac{3}{2}}^2(\mathrm{Kr})=341\mathrm{}\pm 29; {\Lambda }_{\frac{3}{2}}^3(\mathrm{Kr})=276\mathrm{}\pm 30\}$

$\left\{{\Lambda }_{\frac{1}{2}}^1\right(\mathrm{He}),=45\mathrm{}\pm 5\}\{{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}; {\Lambda }_{\frac{1}{2}}^1(\mathrm{Ne}),=56\mathrm{}\pm 6\}\left\{{\Lambda }_{\frac{1}{2}}^1\right(\mathrm{Kr}),=81\mathrm{}\pm 6.\}$

The measured total cross sections for intramultiplet transitions, ${\sigma }_{\frac{1}{2}}^0({}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{})$ and ${\sigma }_{\frac{3}{2}}^0({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{})$, are, in units of ${10}^{-16\mathrm{}}$${\mathrm{cm}}^2$:

$\left\{{\sigma }_{\frac{1}{2}}^0\right(\mathrm{He})=36.5\mathrm{}\pm 9.5; {\sigma }_{\frac{3}{2}}^0(\mathrm{He}),=18.7\mathrm{}\pm 7\}\left\{{\sigma }_{\frac{3}{2}}^0\right(\mathrm{Ne}),=5.9\mathrm{}\pm 2.6\}\left\{{\sigma }_{\frac{3}{2}}^0\right(\mathrm{Kr}),=25\mathrm{}\pm 7.\}$

The ${\Lambda }_J^{\mathrm{}\left(x\right)\mathrm{}}$ and ${\sigma }_J^0$ are related to the multipole relaxation cross sections ${\sigma }_J^{\mathrm{}\left(x\right)\mathrm{}}$ by the equation ${\sigma }_J^{\mathrm{}\left(x\right)\mathrm{}}={\Lambda }_J^{\mathrm{}\left(x\right)\mathrm{}}+{\sigma }_J^0(J\to J^{\prime }\ne J)$. The measurements of ${\Lambda }_{\frac{3}{2}}^1$ and ${\Lambda }_{\frac{3}{2}}^2$ agree well with earlier work; the ${\Lambda }_{\frac{3}{2}}^3$ have been measured for the first time. Measurements of ${\Lambda }_{\frac{1}{2}}^1$ and ${\sigma }_{\frac{1}{2}}^0$ yield cross sections smaller than some of those reported by others; the new values correlate well with other work and resolve some long-standing puzzles. A previously reported increase in ${\Lambda }_{\frac{1}{2}}^1$ at high magnetic field is verified. No significant further change of ${\Lambda }_{\frac{1}{2}}^1$ was observed in the range of magnetic fields from 22 to 131 kG, however. Calculations of ${\Lambda }_{\frac{3}{2}}^{\mathrm{}\left(x\right)\mathrm{}}$, ${\Lambda }_{\frac{1}{2}}^1$, and ${\sigma }_{\frac{1}{2}}^0$ for K-He by Pascale and Perrin based on the Baylis potential are in excellent agreement with the experimental values measured in the present experiment.

The first realistic systematic correction to the Glauber approximation for scattering of charged particles by atoms is made, and it leads to substantial improvements for electron-hydrogen ($e^{-}-\mathrm{H}$) elastic scattering at all energies equal to or greater than 30 eV and angles (10° - 140°) for which data exist. At the lower energies and larger angles disagreement with the data persists because of limitations in the approximation. The calculations also yield $e^{+}-\mathrm{H}$ and $e^{-}-\mathrm{H}$ cross sections which differ from each other even when exchange is neglected.

The experimental investigation of the resonant collision process $\mathrm{Na}ns+\mathrm{Na}\mathrm{ns}\to \mathrm{Na}np+\mathrm{Na}\mathrm{}(n-1\mathrm{})p$ which occurs when the levels are tuned with an electric field so that the $\mathrm{ns}$ level lies midway between the two $p$ levels is described in detail. The observations are shown to be in good agreement with a long-range resonant dipole-dipole interaction.

The impact-parameter dependence of Ne $K$-Auger emission probability has been studied for 4.4- and 10-MeV F projectiles in incoming charge states $6^{+}$ and $8^{+}$, i.e., without and with a $K$ vacancy in the entrance channel of the collisions. The strong enhancement in the emission probability for collisions with an initial $K$ vacancy is attributed to $K-K$ quasiresonant charge transfer. For the smaller collision velocity, i.e., for 4.4-MeV collision energy and a projectile charge state $q=8\mathrm{}$, the $K-K$ quasiresonant transfer is dominated by a destructive interference of contributions to the excitation amplitude from the incoming and outgoing parts of the trajectory.

The single-electron charge-transfer cross sections for ground-state ${\mathrm{C}}^{+}$, ${\mathrm{N}}^{+}$, and ${\mathrm{O}}^{+}$ ions colliding with ${\mathrm{N}}_2$ and ${\mathrm{H}}_2$ are reported. Comparison with results from a mixed-state beam (i.e., an ion beam containing both ground-state and metastable-state ions) of ${\mathrm{C}}^{+}$ and ${\mathrm{O}}^{+}$ ions indicate that single-electron charge-transfer cross sections are equal for ground-state and metastable-state ${\mathrm{C}}^{+}$ and ${\mathrm{O}}^{+}$ ions for velocities greater than ∼5×${10}^7$ cm ${\sec }^{-1\mathrm{}}$.

$K$ x-ray spectra from the nearly symmetric collision system ${\mathrm{F}}^{8+}$ plus Ne were obtained over the bombarding energy range from 10 to 40 MeV with sufficient spectral resolution to observe separately the $K$ x-ray transitions from states formed by $K$-shell electron excitation and electron capture to excited states of fluorine. The cross sections for the two processes were obtained by normalization to total $K$ x-ray production cross sections. The $K$ x-ray production cross section for electron capture to excited states falls off by a factor of 40 over the energy range of 10 to 40 MeV, and the $K$-shell electron excitation cross section increases by a factor of 4 over the energy range 20 to 40 MeV. The $K$-shell excitation cross section has an energy dependence quite different from that observed for the asymmetric systems.

The excitation of electronic transitions in matter by a group of swift ions traveling close together and at nearly the same velocity may depend on the spatial configuration of ions making up the cluster. We have studied theoretically the effect of configuration on the interactions of clusters with an electron gas and in collision with single atoms (vicinage effect). The latter case is explored with a classical harmonic-oscillator model and with quantum-mechanical perturbation theory. We discuss similarities between the vicinage function for energy loss of a swift cluster in an electron gas and that for the same cluster colliding with a system of noninteracting atoms at condensed-matter density. The aligning effect of the wake potential on the trailing ion of a discluster penetrating a solid target, as first observed by Gemmell et al., is not expected to occur in cluster collisions with single atoms, and does not occur in collisions with gases at ordinary pressures. Aligning forces comparable with those in solids require target densities of the same order of magnitude as those occurring in condensed matter. The data of Lurio, Anderson, and Feldman taken in a search for vicinage effects in inner-shell excitation are discussed. The effect of wake fluctuations on cluster energy loss is shown to be negligible under ordinary conditions. We evaluate the effect of residual molecular ionic structure on cluster energy loss.

We present results for the scattering of low-energy electrons by C${\mathrm{O}}_2$ at the static-exchange level. The collision equations were solved by the iterative Schwinger method. All necessary partial-wave expansions have been carried out to high order so as to assure convergence of the scattering solutions. Of special interest are our calculated position and width for the ${}_{}{}^2{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ resonance of 5.39 and 0.64 eV, respectively, which differ significantly from previously published results.

We have observed $\mathrm{Al} L_{2,3}$ Auger electrons from clean aluminum surfaces bombarded by mass-analyzed beams of noble-gas ions at 45° incidence, and in the energy range 600 eV - 15 keV. The electron energy spectra show a wide structure similar, though not identical, to that observed under electron impact, and sharp lines. These lines are identified as resulting from the Auger decay of $2p$-excited Al and ${\mathrm{Al}}^{+}$. The decay occurs in vacuum from sputtered particles, as suggested by the asymmetric Doppler broadening of the Auger lines. The Auger yields can be scaled when plotted against the maximum energy transfer in a projectile-Al collision, indicating that the excitation occurs mainly in symmetric collisions between target atoms. Exceptions are the cases of impact with ${\mathrm{He}}^{+}$ and low-energy ${\mathrm{Ne}}^{+}$ ions where projectile-Al collisions are responsible for the excitation, possibly by two-electron transitions.

We have calculated the cross sections for one- and two-photon absorption by an electron scattering from the Krüger-Schulz model atom in the presence of a low-frequency (linearly or circularly polarized) laser. Let $\xi \equiv \frac{e{A}_0\left(\hat{\epsilon }·\overrightarrow{\Delta }\right)}{\left(mc\omega \right)}$, where $A_0$, $\hat{\epsilon }$ and $\omega$ are the amplitude, polarization, and frequency of the laser field, and where $\hslash \overrightarrow{\Delta }$ is the momentum transfer to the electron of mass $m$ and charge $e$. If $\left|\xi \right|\lesssim 1$, one-photon dominates over two-photon absorption except perhaps over some range of angles, and except over a range of energies near a resonance. We examined in detail the applicability of the soft-photon approximation.

The whole-atom integrated incoherent-scattering cross sections have been obtained from the total attenuation cross sections in the elements La, Ce, Pr, Nd, Sm, Gd, Dy, Ho, Er, Ta, Pb, and Bi at photon energies of 279.2, 514, 661.6, and 1115.5 keV. The total attenuation cross sections for the elements were derived with the aid of the mixture rule from the total attenuation cross sections of the compounds measured by performing transmission experiments in a good-geometry setup. The effect of multiple scattering of photons on the measured values has been corrected by using an extrapolation technique. The present values of the incoherent-scattering cross sections are found to agree fairly well with the recent theoretical values of Hubbell et al.