The level spacing of the Bassichis-Foldy model of an interacting boson system is studied as a function of the coupling F between the two levels of opposite momenta. For a specific value of this coupling F=$F_T$, the ground state of the system becomes degenerate, and the histogram of the energy differences between neighboring levels shows a sharp peak near the origin. The dependence of the energy levels corresponding to the minimum spacing on F follows the classical separatrix energy for a wide range of the parameters of the system, except when F approaches $F_T$. Around $F_T$ the ground-state energy changes very rapidly as a function of F unlike the slow variation of the classical separatrix energy. By studying the classical description of the motion in terms of the action-angle variables, one can show an additional symmetry of the Hamiltonian for F very close to $F_T$. Thus, in this model the nonlinear classical Hamiltonian can be used as an indicator of the degeneracy of the system, even though the quantum-mechanical system does not have the symmetry of the classical problem.

The spin-adapted reduced Hamiltonian theory has been applied to calculate the energies and the first- and second-order reduced density matrices for the isoelectronic sequence of beryllium up to argon in the singlet symmetry. The projection on the spin symmetry has been performed in two ways: by considering the space spanned by the eigenfunctions of Ŝ ${}_{}{}^2{}_{}{}^{}$ and S${^}_z$ or by considering the space spanned by the S${^}_z$ eigenfunctions. Several approximations for calculating the reduced density matrices are analyzed and discussed in detail. The results obtained with two of the approximations considered are highly encouraging when compared both to experimental data and to the total energy values obtained with the full configuration-interaction method.

In 1931 Schrödinger tried unsuccessfully to probabilistically derive his wave equation. We show that, after some improvements, his program yields two versions of stochastic mechanics. The first one coincides with Nelson stochastic mechanics, the second with the imaginary-time version of this theory, whose existence was unknown. The construction is founded on a variational approach inspired by Yasue.

The physics of elementary atoms, i.e., Coulomb bound states such as positronium, pionium, etc., is briefly presented. The formulas for cross sections of elementary-atom interactions with an external field are evaluated in the nonrelativistic and Born approximations. The numerical calculations of the total, elastic, and excitation cross sections are performed for seven elementary atoms and five typical targets. Atomic form factors are discussed in detail.

This paper examines the processes affecting relativistic heavy ions in solid targets. Three pieces of experimental information, the charge states of emergent projectiles, Kα x-ray production cross sections, and K radiative electron-capture photon cross sections for 82- and 197-MeV/amu Xe projectiles incident on solid targets between Be and U, are examined. A theory incorporating the ground states and a minimal number of excited states of zero-, one-, and two-electron projectiles is formulated which includes capture, ionization, excitation, and decay processes. The requisite cross sections are obtained from calculations and measurements. The present charge-state, x-ray, and radiative capture photon measurements are in good agreement with theory. The relationship between charge-changing cross sections measured using thick and thin targets is examined.

The first-Born-approximation expression for elastic scattering by molecules in the gas phase is expanded in a power series in the momentum transfer. It is shown that this expansion can be fitted to high-energy electron scattering measurements to obtain accurate estimates of the mean-square radius of molecules. In addition, by use of information on the molecular quadrupole moment and the molecular structure, the electronic moments 〈$x^2$〉 and 〈$z^2$〉, and the diamagnetic susceptibility may be calculated. It also appears possible to obtain information on some of the higher electronic moments such as 〈$r^4$〉. Results are given for ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, and ${\mathrm{O}}_2$.

A unified description for anisotropic near-threshold excitation in terms of statistical O(4) multipoles is presented. It is shown that a single set of multipole parameters can characterize both the angular distribution of low-energy electron emission and the nonstatistical population of high Rydberg states. Relations between multipole moments of the bound-state charge cloud and of multipole moments of the velocity distributions are derived. The formalism is applied to electron capture to the continuum and into Rydberg states for ${\mathrm{H}}^{+}$+He using the continuum distorted-wave approximation. Comparison is made with recent experimental data.

The classical-trajectory Monte Carlo method has been used to determine the cross section for stripping of ${\mathrm{H}}^{\mathrm{-}}$ in collisions with p¯. The calculations were done for relative velocities from 0.015αc to 1.0αc. A maximum cross section of about 40 A${̊}^2$ is found, somewhat smaller than previous estimates.

Cross sections for all significant channels leading to positive-ion and free-electron production in ${\mathrm{He}}^{2+}$–rare-gas collisions are presented. The energy range studied is 15–200 keV. It is shown that the electron production is dominated by the single charge transfer plus ionization channels—even for a helium target. It is also shown that charge transfer plus an additional ionization can be more probable than charge transfer by itself. For double capture from argon and krypton, this is observed to occur throughout the entire energy range studied.

Based on the dielectric-function method, together with the use of local-electron-density models (LEDM’s), the stopping powers of solid materials for the proton and helium-ion beams are investigated over a wide specific energy range from 1 keV/amu to 2.5×${10}^3$ keV/amu. The screening effect due to free electrons on ions with bound electrons is taken into account within the framework of the linear response. For a ${\mathrm{H}}^0$ atom in alkali metals, the screening effect increases significantly the stopping powers, while for a ${\mathrm{He}}^{+}$ ion it does not increase so much. The mean excitation energy is also calculated using the solid and the atomic LEDM’s. For a Ag target, not being a simple metal, we report how much the change in the number of free electrons affects the stopping cross section for a ${\mathrm{H}}^{+}$ ion. The charge-state distribution (CSD) suggested from the master curves of compiled stopping data is in agreement with the observed CSD. The calculated energy dependence of the effective charge of the helium-ion beams reproduces experimental data well.

We report the analysis of vibrational and rotational energy of ${\mathrm{Na}}_2$${\mathrm{}}^{+}$ ions produced by associative ionization (AI) in two processes: Na(3p)+Na(3p)→${\mathrm{Na}}_2$${\mathrm{}}^{+}$+e and Na(3s)+Na(6p)→${\mathrm{Na}}_2$${\mathrm{}}^{+}$+e. The nascent distributions are measured using photofragment translational spectroscopy. The results show that AI produces vibrationally excited ${\mathrm{Na}}_2$${\mathrm{}}^{+}$ with a distribution increasing approximately linearly with vibrational level. Rotational distributions indicate a short-range opacity function for AI with a maximum impact parameter of about 2 Å in both processes listed above.

Electron capture in ${\mathrm{Ar}}^{+}$${(\mathrm{}}^2$P)${+\mathrm{H}}_2$(X ${\mathrm{}}^1$${\mathrm{\Sigma }}_{\mathrm{g}}$) collisions in the keV energy regime has been studied theoretically. The molecular-orbital expansion method was used within a semiclassical formalism and an electron translation factor correction was incorporated to the first order in the magnitude of the relative velocity V. The molecular wave function and eigenenergy were obtained using the diatoms-in-molecules (DIM) method. We have examined the effect of the orientation of the target ${\mathrm{H}}_2$ molecule on the electron-capture mechanism within the sudden adiabatic approximation. Since π symmetry arising from the p orbital of the ${\mathrm{Ar}}^{+}$ ion is involved in this system, a strong influence on the probability of the molecular orientation was found in all energies studied. As the collision energy increases, the Π-symmetry state in the initial channel becomes more important through the rotational coupling to the electron-capture mechanism, while at lower energies the Σ-symmetry state in the initial channel is the dominant source for the electron capture through strong radial coupling. Agreement of the present theory with measurements is good, but marked disagreement is seen with the atomic-orbital calculation.

We have compared variational functionals for multichannel scattering. The functionals considered were Schwinger-type functionals based on the close-coupling equations, the Schwinger-type variational functionals of Takatsuka and McKoy [Phys. Rev. A 24, 2473 (1981)], and a Kohn-type variational functional. The results for a simple Huck-model potential containing both open and closed channels indicate that the Schwinger-type variational functionals yielded very similar results and that all the methods considered converged at a similar rate with respect to the closed-channel expansion. To obtain good convergence with any of these methods it was essential to include separate trial functions outside the range of the Huck square-well potential in the channels where a correct asymptotic form of the wave function was required, those being only the closed channels for the Schwinger-type functionals and both open and closed channels for the Kohn-type functionals. These asymptotic functions were needed to reproduce the discontinuity in the second derivative of the wave function due to the discontinuities in the model potential. Convergence characteristics of these methods with respect to target-state expansions were also considered.

Cross sections are calculated for excitation of Cu for elastic scattering of electrons from the 4s ground state and for excitation to excited states 3$d^9$4$s^2$, 4p, and 4d. A three-state close-coupling approximation is used for the energy range 3.8<E<10 eV, and a four-state one for 6<E<100 eV. Accurate target functions are used in the expansion. Poor agreement is obtained with renormalized experimental data of Trajmar et al. .AE

Differential cross sections for excitation of Mg ii, Zn ii, and Cu i from their ground states to the first excited p and d states are calculated for the electron-impact energy range up to 100 eV. In addition, elastic angular distributions for Cu are given at 10 and 100 eV. A five-state close-coupling approximation in which 3s, 3p, 4s, 3d, and 4p states are included is used in the Mg ii calculation. The 4s, 4p, 3$d^9$4$s^2$, 5s, and 4d states are included in the close-coupling calculation for Zn ii, and the 4s, 3$d^9$4$s^2$, 4p, and 4d states for Cu i. For all targets, s→p transitions exhibit strong forward scattering. The same is true for the ions for s→d transitions, but for Cu the 6-eV results are nearly isotropic, and a maximum at 6° is found for higher energies.

Two-photon Doppler-free spectroscopy has been used to measure Stark shifts of n ${\mathrm{}}^2$D Rydberg states of ${}_{}{}^{85}{}_{}{}^{}\mathrm{Rb}$ (for n=13–55). Scalar and tensor polarizabilities were determined and found to depend on ($n^{\mathrm{*}}$${)}^6$ and ($n^{\mathrm{*}}$${)}^7$, in agreement with theory. The resulting polarizabilities were used to obtain estimates of radial integrals for D-P transitions.

We show how the electronic stopping power for slow protons is influenced by the deceleration and deflection of the projectile in the field of the target nucleus (Coulomb effect) and by the fact that in insulators a finite energy is also required for excitation of the outermost electrons (threshold effect). Estimates are derived from the Fermi-Teller description of the stopping process, from a modified local-density approximation, and from measured inner-shell ionization cross sections. It is found that the introduction of an energy threshold reduces at low energies the stopping cross section by a large factor and hence leads to an appreciable deviation from $v_1$ proportionality.

The dielectric function method is applied to investigate the energy straggling of protons and helium ions using the atomic and the solid local-electron-density models. A partially stripped ion is treated as well as point charges. At low energies, the conventional stragglings of a proton (${\Omega }_{{\mathrm{H}{}^{+}}^2)}$ and of helium ions (${\Omega }_{{\mathrm{He}{}^{+}}^2}$,${\Omega }_{\mathrm{He}}^{2+}$ ${}_{}{}^2{}_{}{}^{}$) due to the fluctuation in electronic excitations are proportional to the kinetic energies of the ions even when the local-electron-density models are adopted, and at high energies they approach the values predicted by Bohr. Here we do not consider the bunching term since we assume the probability of exciting an electron is small. The straggling ratio ${\Omega }_{{\mathrm{He}{}^{+}}^2/{\Omega }_{{\mathrm{H}{}^{+}}^2}}$ shows a remarkable feature that it is nearly constant up to ∼50 keV/amu, and increases gradually beyond this energy for solid targets. The estimation of the collisional straggling of helium-ion beams is performed using the charge-state fractions, resulting in displaying $Z_2$ (target atomic number) oscillations similar to those of ${\Omega }_{{\mathrm{He}{}^{+}}^2}$ and ${\Omega }_{{\mathrm{He}{}^{2+}}^2}$. The straggling caused by charge-state fluctuations, which enhance the $Z_2$ oscillations of the total straggling of helium-ion beams more sharply at the energies considered, is also estimated.

We present a semiclassical analysis of ultrahigh-resolution two-photon optical Ramsey spectroscopy of cold neutral atoms falling freely in a fountain. Considering atoms which interact with the same standing-wave laser field twice on their parabolic trajectories and averaging over a broad atomic velocity distribution, we predict a nearly Lorentzian line shape whose width is just the natural linewidth. We have investigated a number of systematic corrections to the atomic resonance frequency, including first-order and second-order Doppler shifts, gravitational red shifts, and ac Stark shifts. First-order Doppler shifts due to vertical motion cancel even if the counterpropagating beams are slightly misaligned, and resolutions below 1 Hz appear feasible with a fountain of modest dimensions.

A quantum version of the Drude-Lorentz model of the atom is used to study abnormally large energy transfers found in recent ionization experiments by strong laser fields. It is argued that collective excitations greatly enhanced by parametric resonance may lead to an exponential growth of the atomic energy with time. It is shown that the repulsion between the electrons plays an essential role, reducing the frequency at which parametric resonance occurs.