A continuous-time random walk on a spatial lattice described by the Kramers-Moyal expansion has a continuum limit described by a Fokker-Planck equation. It is often desirable to know corrections to quantities computed in the continuum limit, but truncation of the Kramers-Moyal expansion at any level other than the Fokker-Planck either breaks down or yields unphysical results. Here we introduce an alternative approximation to the Kramers-Moyal expansion which circumvents the problems of a naive truncation and correctly incorporates the first-order corrections due to the discrete lattice.

The formal equivalence and algorithmic differences between the Leverrier-Bateman resolvent method and the recursive residue generation method for constructing the evolution operator of time-independent Hamiltonian operators are discussed.

A lattice model is used to investigate the surface behavior of smectic fluids close to the onset of smectic ordering. Enhanced surface nematic order favors smectic wetting of a liquid-vapor interface, but decreased density at the interface favors isotropic wetting. The competition is resolved by quasiwetting: finite smectic layer growth with identifiable smectic-wall and isotropic-smectic components. Finite-ranged interactions lead in addition to smectic layering transitions, in line with recent experimental observations.

Ab initio self-consistent spin-polarized calculations have been performed numerically for the total energy of an aluminum atom, to assess the various approximate forms of the exchange-correlation functionals used frequently in local-spin-density approximation. Transition-state calculations of all the ${\mathrm{Al}}^{\mathrm{n}+}$ ions are reported and compared with the restricted Hartree-Fock results and the experimental values. Calculations of the total energy of the ionic ground state confirms the accuracy of the transition-state method. The total energy is calculated employing an indirect path via information obtained from the ionization potentials and is found to be better than that given by the conventional method.

The summability of the Birkhoff-Gustavson normal form for the quartic anharmonic oscillator is studied with an order-dependent mapping. The transformation is determined by taking into account the asymptotic expansion of the semiclassical energy. A renormalized sequence is obtained which is rigorously convergent for all couplings and quantum numbers. We show that, when the proper asymptotic behavior is considered, the JWKB limit of eigenvalues is obtained by summing the power-series expansion about the low-quantum-number regime. Very accurate results are obtained, using the 21st-order expansion for low coupling, not only for the energy but also for the semiclassical periods of motion. The convergent renormalized sequences show clearly that the Birkhoff-Gustavson normal form has a nonzero radius of convergence and we determine a lower bound for it that is larger than a previous estimate. The whole set of critical constants describing the behavior of the semiclassical energy near the closest-to-the-origin singularity is also obtained.

The magnetic fine-structure intervals separating the four members of the 10G and 10H manifolds of helium have been measured with a precision of 0.1% using a fast-beam microwave-optical resonance technique. The results are found to be in good agreement with theory, illustrating that the two-electron wave function is very nearly hydrogenic. The measurements are also used to determine the small exchange energies of the 10G and 10H states.

The screening constant in the double Rydberg formula is estimated using the intrashell configuration-interaction states. Its explicit dependence on the bending rovibrational quantum numbers is included. Comparison of the energy levels so obtained with elaborate numerical results indicates its utility for doubly excited states of highly charged ions and also points to its geometrical origin. Procedures are introduced to generalize the formula to intershell doubly excited states and to doubly excited states of multielectron atoms. A number of specific numerical results are presented.

The K${\alpha }_2^h$ ($L^{\mathrm{-}n}$; n=0–5) hypersatellite spectrum of sodium has been detected for various electron-impact energies in the range of 15≤$E_0$≤200 keV. The measured energies of the transitions are in agreement with results of previous experiments at low electron energies and theoretical predictions. For the strongest hypersatellite, which corresponds to a $K^{\mathrm{-}2}$→$K^{\mathrm{-}1}$$L^{\mathrm{-}1}$ transition, the natural width has been determined. The natural linewidth was found to be ${\Gamma }_K^h$=2.2±0.4 eV, which is much larger than predicted. Furthermore, the electron double-ionization cross section of the K shell has been obtained. Within the investigated energy region the cross section decreases by about a factor of 4 with increasing electron-impact energy. This energy dependence suggests the dominance of direct double ionization in comparison to the shake-off process.

Dissociative electron attachment to ${\mathrm{F}}_2$ has been observed in the energy range 0≤ɛ≤140 meV, at a resolution of 6 meV (full width at half maximum). Results show conclusively a sharp, resolution-limited threshold behavior consistent with an s-wave cross section varying as ${\varepsilon }^{\mathrm{-}1/2}$. Two accurate theoretical calculations predict only p-wave behavior varying as ${\varepsilon }^{1/2}$. Several nonadiabatic coupling effects leading to s-wave behavior are outlined.

Measurements of inelastic collisions of ${\mathrm{F}}^{+}$ with Ne have been made. Transitions between ${}_{}{}^3{}_{}{}^{}\mathrm{P}$ and ${}_{}{}^1{}_{}{}^{}\mathrm{D}$ terms of ${\mathrm{F}}^{+}$ are seen, with the inelastically scattered ions sharply focused in the forward direction. Potential energy curves of (FNe${)}^{+}$ have been calculated. The ${}_{}{}^3{}_{}{}^{}\mathrm{\Sigma }$ and ${}_{}{}^3{}_{}{}^{}\mathrm{\Pi }$ curves correlating to ${\mathrm{F}}^{+}$${(\mathrm{}}^3$P) are repulsive, while the ${}_{}{}^1{}_{}{}^{}\mathrm{\Sigma }$ correlating to ${\mathrm{F}}^{+}$${(\mathrm{}}^1$D) is attractive. Several curve crossings are identified, where transitions occur through spin-orbit coupling. Scattering angles and differential cross sections have been calculated, and they show the presence of a ‘‘glory’’ (or halo) effect, which accounts for the forward scattering of ions.

We propose how to rewrite the scattering solution of the Schrödinger equation in terms of positive matrix elements of positive operators. These can be interpreted as probability distributions and we suggest applying a Monte Carlo method for its calculation. We outline the relation between the probability distributions and S-matrix elements for scattering reactions.

The differential and total cross sections for the formation of positronium in its ground state from Li and Na atoms by the impact of intermediate-energy positrons are calculated in the first Born and distorted-wave Born approximations. Hellmann-type pseudopotentials are used to represent the alkali-metal ion cores. The difference in the use of pseudopotentials and the static potential for the core representation for evaluating various rearrangement cross sections is discussed.

Absolute differential cross sections for He+He elastic scattering at laboratory angles in the range 0.018°–0.5° have been measured for projectile energies of 0.5, 1.5, and 5.0 keV. The experiment employs a position-sensitive detector for determining the angular distribution of scattered particles. The differential cross sections exhibit a transition from classical to quantum behavior within the range of angles studied, and excellent agreement is observed with partial-wave-theory calculations using phase shifts derived from the He-He interaction potentials of Ceperley and Partridge [J. Chem. Phys. 84, 820 (1986)] and Foreman, Rol, and Coffin [J. Chem. Phys. 61, 1658 (1974)].

The electron-impact double-ionization cross section for ${\mathrm{Fr}}^{+}$ is calculated in the distorted-wave Born approximation. A giant resonance in the 5d subshell ionization-autoionization contribution to the cross section is found to be quite sensitive to changes in the double-well potential caused by relativistic effects on bound-state wave functions.

Absolute cross sections for the emission of Lyman-α radiation have been measured for low-energy ${\mathrm{H}}^{+}$ impact on rare-gas-atom targets. Data were obtained for ${\mathrm{H}}^{+}$ energies from 2.0 keV down to 1.0 keV for He targets, to 0.5 keV for Ne targets, to 0.05 keV for Ar and Kr targets, and to 0.013 keV for Xe targets. The polarizations of the emitted Lyman-α radiation were also measured. Using previously measured cross sections for Balmer-α and Balmer-β emission for the same collisions, it was possible to make an approximate determination of the cascade contributions to the total Lyman-α emission, and to extract the cross sections for direct population of the 2p state of hydrogen during the collisions. The results are compared with the work of other investigators at higher ${\mathrm{H}}^{+}$ energies in an attempt to deduce the best working values of these cross sections for ${\mathrm{H}}^{+}$ energies up to 100 keV.

In a recent calculation [C. A. Weatherford, K. Onda, and A. Temkin, Phys. Rev. A 31, 3620 (1985)] an exact exchange method was developed for use in the partial-differential-equation approach to electron-molecule scattering, and was applied to e-${\mathrm{N}}_2$ scattering in the fixed-nuclei approximation with an adiabatic polarization potential at low energies (0–10 eV). Integrated elastic cross sections were calculated and found to be lower than experiment at energies both below and above the ${\Pi }_g$ resonance. It was speculated at that time that improved experimental agreement could be obtained if a correlated target representation were used in place of the uncorrelated one. The present paper implements this suggestion and demonstrates the improved agreement. These calculations are also extended to higher energies (0–30 eV) so as to include the ${\Sigma }_u$ resonance. Some discrepancies among the experiments and between experiment and the various calculations at very low energy (<1 eV) are noted.

Giant resonances in the electron-impact ionization cross section of the 4d subshell in heavy atoms and ions are studied as a function of the nuclear charge, the ionicity of the atom, and the presence of localized excited-state orbitals. The results of distorted-wave Born-exchange-approximation calculations are presented for the singly charged ions ${\mathrm{Ag}}^{+}$–${\mathrm{Ce}}^{+}$ and for the first four ions of the barium isonuclear sequence and the iodine isoelectronic sequence. Giant resonances were found to occur only for neutral atoms and singly charged ions with Z≳53. The cross sections for ionization of several ions were found to be very sensitive to the scattering exchange approximation. The effect of valence shell excitations into localized 4f states in ${\mathrm{Sb}}^{+}$, ${\mathrm{Ba}}^{2+}$, and ${\mathrm{La}}^{+}$ is studied. Such controlled changes in target ion charge density can have significant effects on the formation and structure of giant resonances in electron scattering from heavy ions.

Minima in cross sections (Cooper minima) are known to occur for all types of transitions in atoms. These minima are caused by zeros in the transition matrix elements which are the result of a zero value for the radial integral of a particular energy. These minima can occur in both the bound-bound and bound-free transition matrix elements. In this paper the approximate position (energy) of the zeros in the bound-bound quadrupole and octupole matrix elements of the point Coulomb field are presented. The positions are found to change in the same manner as those found in the continuum, even to the extent that a zero first occurring in a bound-bound matrix element will move into the corresponding bound-free matrix element.

Wavelengths, radiative transition probabilities, Auger energies, and Auger rates have been calculated relativistically for 1s2$s^2$2$p^2$, 1s2s2$p^3$, and 1s2$p^4$ configurations of the boron isoelectronic sequence, in terms of the multiconfiguration Dirac-Fock model. Ions with atomic numbers from Z=6 to 54 are included. The generalized Breit interaction, quantum-electrodynamic corrections, and relaxation are taken into account in the calculations of transition energies. Transition rates are computed in intermediate coupling, including configuration interaction within the same complex. It is found that the effects of relativity play a major role in the forbidden transitions, and also become quite important for allowed transitions for Z≥20. Configuration interaction is found to affect transition rates by as much as a factor of 2 for light ions.

The state-to-state photoionization cross sections for ionizing an L-subshell electron from the 1$s^2$2$s^2$2$p^2$ configuration of the carbonlike ions have been calculated for seven ions with atomic number 6≤Z≤54 for the photon energy in the range 6≤E≤16 keV by using the newly developed multiconfiguration Dirac-Fock photoionization code. The calculations were performed in the intermediate coupling including the initial-state and final-ionic-state configuration interaction from the same complex. The spin-orbit interaction has been found to drastically alter the branching ratios for 10<Z<45. For Z≤10, the branching ratios agree with the LSJ coupling limit while for Z≥45, the branching ratios approach the j-j coupling limit. The effects of relativity change the total cross section by an amount less than 14% for the ${\mathrm{Xe}}^{48+}$ ion.