Showing papers in "Physical Review A in 1975"

From Maxwell to paraxial wave optics

Abstract: In this paper we are concerned with the propagation of a light beam through an inhomogeneous, isotropic medium with a possibly nonlinear index of refraction. The customary paraxial approximations of neglecting grad $\mathrm{div}\mathcal{E}$ and seeking a plane-polarized solution are shown to be incompatible with the exact Maxwell equations. By starting from Maxwell's equations, and scaling transverse and longitudinal distances by the beam waist ${w}_{0}$ and diffraction length $l$, respectively, an expansion procedure in powers of $\frac{{w}_{0}}{l}$ is developed. The exact equations obeyed by the zeroth-order fields are not Maxwell's equations but the customary paraxial approximation to Maxwell's equations. Equations for the first-, second-, and third-order fields are developed. The first-order field is found to be a longitudinal field. It is solved for explicitly in terms of the zeroth-order field which is transverse. Thus a precise knowledge of the meaning and accuracy of paraxial wave optics is obtained.

New computational method in the theory of spinodal decomposition

Abstract: We present a new series of calculations in the theory of spinodal decomposition. The computational scheme is based on a simple ansatz for the two-point distribution function which leads to closure of the hierarchy of equations of motion for the high-order correlation functions. The resulting theory is accurate throughout the spinodal region of the phase diagram, including at the boundaries of this region where the spinodal mechanism is difficult to distinguish from nucleation and growth. The computational scheme is worked out in detail for parameters approximating those of the three-dimensional, kinetic, spin-exchange Ising model with nearest-neighbor interactions. Numerical agreement with recent Monte Carlo data appears to be satisfactory.

Bound states in the continuum

Abstract: Quantum-mechanical examples have been constructed of local potentials with bound eigenstates embedded in the dense continuum of scattering states. The method employed corrects and extends a procedure invented by von Neumann and Wigner. Cases are cited whereby deformation of the local potential causes the continuum bound state to move downward through the bottom of the continuum, and to connect analytically to a nodeless ground state. A doubly excited model atom is also displayed, with interactions between its two "electrons," having an infinite lifetime (in the Schr\"odinger equation regime). In the light of these examples, attention is focused on quantitative interpretation of real tunneling phenomena, and on the existence of continuum bound states in atoms and molecules.

Cooperative radiation processes in two-level systems: Superfluorescence

Abstract: We give a general nonperturbative treatment of cooperative emission in systems of $N$ two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to ${N}^{2}$, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length $L$ of the active volume is much larger than a suitable threshold length ${L}_{T}$ (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length $L$ is smaller or of the same order of a suitable cooperation length ${L}_{c}$ (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to $N$). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to $N$, so that superfluorescence effects disappear.

Dense-fluid shear viscosity via nonequilibrium molecular dynamics

Abstract: A novel fluid-transport calculation by computer simulation, via nonequilibrium molecular dynamics, of laboratory methods of transport measurement is described. Shear viscosity of soft-sphere (${r}^{\ensuremath{-}12}$ potential) and Lennard-Jones particles (${r}^{\ensuremath{-}12}\ensuremath{-}{r}^{\ensuremath{-}6}$ potential) has been obtained from molecular dynamic modeling of Couette flow. Soft-sphere deviations from Enskog theory are similar to those found for hard spheres by Alder, Gass, and Wainwright, using time-correlations of equilibrium molecular dynamic system fluctuations. For the Lennard-Jones shear viscosity near the triple-point region, there is agreement between the equilibrium calculation of Levesque, Verlet, and Kurkijarvi and the nonequilibrium results using 108 atoms in a cube. However, systems two and three cubes wide give lower results, which, when extrapolated with inverse width, yield close agreement with the experimental argon shear viscosity. Comparison of the Lennard-Jones shear viscosity with experimental argon data along the saturated vapor-pressure line of argon confirms our successful simulation of macroscopic viscous flow with few-particle nonequilibrium molecular dynamic systems. A new result of the nonequilibrium molecular dynamics is the characterization of nonequilibrium distribution functions, which might provide the basis for a perturbation theory of transport. Since momentum transport is primarily accomplished by the repulsive potential core for high temperatures, the Lennard-Jones shear viscosity must behave like the soft-sphere system for high temperatures [viscosity divided by ${(\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e})}^{\frac{2}{3}}$ is a function of density divided by ${(\mathrm{temperature})}^{\mathrm{\textonequarter{}}}$]. In fact, the calculated excess shear viscosity (that part above the zero-density temperature dependence) has been successfully correlated in terms of the 12th-power scaling variables for temperatures as low as the critical value (along the freezing line). The utilization of soft-sphere scaling variables yields relatively simple functions for describing both the excess shear viscosity and the thermal-conductivity behavior throughout the fluid phase. The introduction of these scaling variables also clearly reveals two features: (i) weak temperature dependence, and (ii) the sign of the temperature derivative at constant density (negative for shear viscosity and positive for thermal conductivity). While both of these features have been experimentally observed in simple fluid experimental data, their cause has not been previously traced to the dominance of the core potential. Thus, the soft-sphere scaling variables should be useful for correlating experimental data.

Comparison of doubly-excited helium energy levels, isoelectronic series, autoionization lifetimes, and group-theoretical configuration-mixing predictions with large-configuration-interaction calculations and experimental spectra

Abstract: For comparison with our recent group-theoretical predictions of configuration-mixing coefficients, we report extensive configuration-interaction calculations for doubly excited states in the helium isoelectronic sequence below the $N=2$ ($S, P, D$ states), $N=3$ ($S, P, D, F, G$ states), $N=4$ ($S, P, D$ states), and $N=5$ ($^{1}P^{\mathrm{o}}$ states) ionization thresholds. Two new quantum numbers label the Rydberg series, and are used to predict three "selection rules" for dipole excitation and autoionization processes which agree with experiment. New autoionization widths are reported for the helium states and confirm our group-theoretically predicted selection rules. Our width of 0.151 eV for the $^{1}P^{\mathrm{o}}$ state at 62.92 eV is in agreement with the recent experimental value 0.132 \ifmmode\pm\else\textpm\fi{} 0.014 eV. Calculations of helium energies and autoionization widths are strongly affected by "avoided crossings" of the doubly excited states as $Z$ is varied continuously. The new quantum numbers project out states in ${\mathrm{H}}^{\ensuremath{-}}$ which correspond to observed $^{1}P^{\mathrm{o}}$ shape resonances above the $N=2 \mathrm{and} 3$ ionization thresholds.

Theory and simulation of resonant absorption in a hot plasma

Abstract: A detailed theoretical and simulation study of resonant absorption in a hot plasma is presented which isolates the behavior of the plasma for times short compared to an ion response time. The extent to which an electron fluid model can describe the absorption process in the kinetic regime is discussed. At high intensities the absorbed energy is observed to be deposited in a suprathermal tail of electrons whose energy varies approximately as the square root of the incident power. The density profile modification due to the ion response to the pondermotive force is also discussed.

Statistical mechanics of dense ionized matter. IV. Density and charge fluctuations in a simple molten salt

Abstract: The results of a molecular-dynamics study of a simple model of a molten salt are reported. The interionic pair potential which is used consists of the Coulombic term an an inverse-power repulsion which is assumed to be the same for all ions. The structure of the liquid is found to be dominated by charge-ordering effects and the calculated equilibrium properties are in good agreement with the predictions of the hypernetted-chain approximation. The relation between the self-diffusion coefficient and the electrical conductivity is discussed, and the observed deviations from the Nernst-Einstein relation in real molten salts are shown to have a natural explanation in terms of short-lived cross correlations. Data on the spectra of charge and particle density fluctuations are presented. At small wave numbers there is a propagating optic-type mode which shows a strong negative dispersion, but no Brillouin peak is seen even at the lowest wave number which is accessible. The data are analyzed in terms of a single-relaxation-time model incorporating the low-order spectral moments, for which we give explicit formulas. The fit achieved is fair, but the low-frequency behavior of the charge fluctuations at small wave numbers is incorrectly reproduced, and there is evidence for the necessity of introducing a second relaxation time. Comparison is made with results previously obtained for the classical one-component plasma.

Pure-state analysis of resonant light scattering: Radiative damping, saturation, and multiphoton effects

Abstract: A fully quantum-mechanical treatment of resonant light scattering is presented. The incident field is assumed to be described by a coherent state, and is allowed to be intense enough to cause saturation. Complete solutions are obtained for the correlated atom-field pure state vector, including multiphoton contributions of arbitrary order. The frequency spectrum of the scattered field is evaluated and is found to agree exactly with the result previously obtained by means of the quantum fluctuation-regression theorem. A derivation of the fluctuation-regression theorem and of the optical Bloch equations is given which is fully quantum mechanical and which relies upon no assumption of statistical factorization of atom and field states. The accuracy of the result found for the scattered - field spectrum is thus shown to be limited only by the assumption of the smallness of the saturated linewidth compared to the (optical) atomic resonance frequency. The one-photon approximation is analyzed in some detail. The method of adding an imaginary term to the upper-atomic-state energy is clarified, and it is shown how the vacuum and one-photon amplitudes thereby obtained may be used, within a simple and plausible iteration scheme, to construct the complete multiphoton spectrum. A variety of commonly used injection schemes and methods of representing atomic relaxation are discussed, and comparisons are made with results found by other authors. The entire analysis is performed with the aid of a canonical transformation which replaces the applied field by a $c$ number. It is thus proved quite rigorously and generally that the use of a $c$-number applied field is a fully quantum-mechanical procedure, provided only that radiation-reaction terms are retained.

Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries

Abstract: A form of quantum electrodynamics is developed which allows us to treat a number of problems involving dielectric and conducting surfaces, the presence of which leads to a number of new observable effects. A number of suitably defined response functions play a basic role in the present approach, as these in conjunction with the fluctuation-dissipation theorem lead to electromagnetic field correlation functions, which describe physical effects such as lifetimes, frequency shifts of the excited states, dispersion forces, etc. The quantization of the electromagnetic field is only implicitly used. A large part of the present paper is devoted to the calculation of the response functions involving different geometries and various types of dielectrics. Both spatially dispersive and spatially nondispersive dielectrics are considered. The response functions are calculated using Maxwell's equations and the usual boundary conditions at the interface adjoining the two mediums. As a first application of the present approach, the black-body fluctuations in finite geometries and the influence of surfaces on its temporal and spatial coherence are studied. An interesting theorem is also proved which enables us to calculate the normally ordered (antinormally ordered) correlation functions from the symmetrized correlation functions.

Coherent two-photon processes: Transient and steady-state cases

Abstract: A class of molecular two-photon processes, occurring under transient or steady-state conditions, is analyzed exactly for certain cases using a density-matrix formalism. Our results are relevant to the recently observed phenomena of transient coherent Raman beats and steady-state two-photon absorption of oppositely directed laser beams. Lowest-order perturbation theories have shown that these processes are insensitive to the Doppler effect, and consequently the linewidth is unaffected by Doppler broadening and by elastic collisions that change molecular velocity. The solutions presented here are of the same form for the Raman and two-photon problems and reveal significant power-dependent frequency shifts and line broadening that correct the ideal Doppler-free solutions. For the transient case, we assume a molecular three-level quantum structure that can be switched in or out of optical resonance with cw laser radiation by means of a pulsed dc Stark field, but the solutions also apply when resonant optical pulses are introduced. During the resonant condition when the Stark pulse is on, the three levels are prepared in coherent superposition, and during the nonresonant condition following the Stark pulse, the laser field(s) stimulates the coherently prepared sample in a transient two-photon process. One of these processes, the Ramanbeat effect, has been observed in forward scattering, but the second one, transient two-photon emission, should be observable in backward scattering. Bloch-like equations are derived for this three-level problem that facilitate an exact pulse solution for state preparation. Following the pulse, transient decay is well-approximated by a perturbation calculation. For the steady-state case, an exact solution is also obtained and is of interest for continuous-wave spectroscopy or for transient experiments requiring an initial preparation of quantum states, prior to Stark switching. These solutions exhibit a power-dependent line broadening and a frequency shift of magnitude $\ensuremath{\sim}\frac{\ensuremath{\Delta}({\ensuremath{\alpha}}^{2}\ensuremath{-}{\ensuremath{\beta}}^{2})}{({\ensuremath{\Delta}}^{2}+\frac{1}{{T}_{2}^{2}})}$, in agreement with our earlier estimate, where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are the Rabi flopping frequencies for the two intermediate transitions and $\ensuremath{\Delta}$ is the off-resonance tuning behavior for one of them.

Inadequacy of entropy and entropy derivatives in characterizing the steady state

Abstract: In bistable systems the transition kinetics between the two locally stable states can be altered without changing the behavior in the immediate vicinity of the two favored steady states. It follows that quantities which characterize only the vicinity of the favored states cannot determine the most probable state. A second point: Even in monostable linear circuits the steady state need not correspond to minimum entropy production.

Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation

Abstract: Two-photon resonantly enhanced parametric generation processes have generally been described using time-dependent perturbation theory In this paper we show that a theory of two-photon coherent effects can be used to derive and explain these nonlinear mixing processes Our technique makes use of the adiabatic following (AF) approximation to obtain solutions to a vector model describing the two-photon resonance We show that the usual results for the nonlinear susceptibilities correspond to the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ vector of Feynman, Vernon, and Hellwarth adiabatically following the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\gamma}}}$ vector in the small-angle limit Consequently, the theory allows a natural extension to large angles, and power-dependent nonlinear susceptibilities are obtained We then use these AF results for the polarization to study the propagation of pulses nearly resonant with a two-photon transition, and we demonstrate that the pulse reshaping is due to the two related effects of a nonlinear pulse velocity and self-phase modulation

Fluctuation-dissipation theorems for classical processes

Abstract: For certain types of classical processes a fluctuation-dissipation theorem (FDT) holds. The validity of such a theorem is preserved in each order of mass renormalization in the perturbation scheme recently developed by Martin, Siggia, and Rose (MSR). As a consequence, whenever the response function can be expressed in terms of the two-point correlation function via a FDT, the perturbation scheme of MSR simplifies considerably. Especially, it reduces to the scheme constructed by Kawasaki for random processes obeying detailed balance which are (i) linearly damped and (ii) linearly driven by Gaussian white noise.

Spin-1 lattice-gas model. I. Condensation and solidification of a simple fluid

Abstract: A spin-1 lattice-gas model, similar to the Blume-Emery-Griffiths model for $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures, is shown to describe condensation and solidification of a simple fluid. The Ising-like Hamiltonian of the system involves quadrupolar and dipolar interactions, which are responsible for condensation and solidification, respectively. The molecular-field approximation is used, and the ordinary phase diagram of a simple fluid is reproduced. However, for some range of the parameters, the liquid-gas equilibrium curve disappears. Also, the melting curve may exhibit a tricritical point: For pressures larger than the tricritical pressure, critical melting is found. Other physical applications of the model are briefly discussed.

Independent-particle-model potentials for atoms and ions with 36 < Z?54 and a modified Thomas-Fermi atomic energy formula

Abstract: Using the ab initio energy-minimization procedure of Bass, Green, and Wood, we determine two potential parameters, $\ensuremath{\xi}$ and $\ensuremath{\eta}$, characterizing the independent-particle-model potential of Green, Sellin, and Zachor (GSZ) for atoms and positive ions with $36lZ\ensuremath{\le}54$. This extends earlier modified-Hartree-Fock (MHF) calculations of Szydlik and Green and of Green, Garvey, and Jackman. We find that both of the parameters in question display, to a good approximation, a linear dependence on the degree of ionization $Z\ensuremath{-}N$ for fixed numbers of electrons $N$. The slopes and $y$ intercepts associated with the linear dependence of $\ensuremath{\xi}$ display marked shell-like behavior, while those associated with $\ensuremath{\eta}$ vary rather smoothly with $N$. Our determinations of total energies are usually within 50 ppm of earlier Hartree-Fock calculations for those cases in which such calculations exist. Using the entire collection of energies and GSZ minimization parameters now available, we reexamine a modified version of the Thomas-Fermi statistical model (MTF) due to Green, Sellin, and Darewych. We show that this model is capable of yielding the linear $Z\ensuremath{-}N$ dependence of the GSZ parameters which we found empirically in the MHF work. By numerical adjustment of the coefficients of our MTF model, we obtain energies of stable atoms and ions, as well as GSZ potential parameters which are in good agreement with the MHF calculations.

Quantum electrodynamics in the presence of dielectrics and conductors. IV. General theory for spontaneous emission in finite geometries

Abstract: A quantum-electrodynamic theory of spontaneous emission in presence of dielectrics and conductors is developed. The theory makes use of the master-equation techniques and the response-function formalism of part I of this series of papers. Various observable entities such as damping coefficients (lifetimes), Lamb shifts, and frequency shifts are related to the appropriate surface-dependent response functions. The results are valid for arbitrary geometries (involving linear dielectrics) and naturally contain, as a special case, the usual results of spontaneous emission in free space. As explicit examples we consider a two-level atom (also the multilevel atom) in presence of a plane dielectric interface and between two conducting plates. Formulas for the shifts and widths are given and their asymptotic behavior for large and short distances is discussed. The behavior when the atom is embedded inside the dielectric is different than when the atom is outside the dielectric. The origin of coherence in the present model is discussed and the coherence effects in this model are contrasted with those in Dicke's model. The results are also compared with those obtained by the image method. Exact expressions for the operator radiation-reaction fields are obtained in terms of the atomic polarization operators and response functions. Approximate results for such fields are also given. The far-zone behavior of the radiation fields is obtained in terms of response functions and the polarization operators. Some of the normally ordered correlation functions are also calculated. The connection of some of the theoretical results with a recent experimental work of Carniglia, Mandel, and Drexhage is discussed. The effect of anisotropy of the dielectric function on the lifetimes as well as on the far-field correlation functions is also considered. Finally the contribution to the shifts and widths due to the excitation of surface polariton modes is computed and the results are compared with those obtained by the quantization of surface polariton field. It is found that such surface modes contribute significantly to widths.

Systematics of moments of dipole oscillator-strength distributions for atoms of the first and second row

Abstract: The moments $S(\ensuremath{\mu})$ and $L(\ensuremath{\mu})=\frac{dS(\ensuremath{\mu})}{d\ensuremath{\mu}}$ for $\ensuremath{-}6\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1$ are derived from comprehensive Hartree-Slater oscillator-stength distributions for He through Ar. For $\ensuremath{\mu}\ensuremath{\le}\ensuremath{-}2$, these moments are governed by valence excitations only, and therefore exhibit a pronounced periodic variation that repeats in each row. Inner shells begin to contribute appreciably to $S(\ensuremath{-}1)$, which retains a periodic variation superimposed upon an over - all increase with increasing atomic number $Z$. For $\ensuremath{\mu}\ensuremath{\ge}0$, the $Z$ dependence of the moments becomes dominated by inner-shell contributions; as $\ensuremath{\mu}$ increases, the over-all increase with $Z$ becomes more rapid. Another perspective of the voluminous data is gained by plotting $logS(\ensuremath{\mu})$ vs $\ensuremath{\mu}$. The plot reveals three classes of behavior---"tight," "intermediate," and "loose" atoms. Comparisons with experiment and more detailed calculations are made where possible.

Statistical mechanics of dense ionized matter. V. Hydrodynamic limit and transport coefficients of the classical one-component plasma

Abstract: We study the static and dynamical correlations of density fluctuations in a classical one-component plasma (OCP), in the framework of thermodynamic fluctuation theory and linearized hydrodynamics. First we show that the fluctuations of the local electric field stabilize the OCP against density fluctuations, even when the compressibility becomes negative for $\ensuremath{\Gamma}\ensuremath{\gtrsim}3$, where $\ensuremath{\Gamma}$ is the usual plasma coupling parameter. Then, following closely the work of Mountain, we compute the dynamical structure factor in the hydrodynamic limit and show that the thermal Rayleigh peak vanishes in the long-wavelength limit, in agreement with recent molecular dynamics results. The shear and bulk viscosity coefficients $\ensuremath{\eta}$ and $\ensuremath{\zeta}$ are calculated for large $\ensuremath{\Gamma}$ in the framework of the "generalized hydrodynamics" formalism, using the known short-time expansion of the correlation functions. The coefficient $\ensuremath{\eta}$ is found to exhibit a minimum as a function of $\ensuremath{\Gamma}$ and $\ensuremath{\zeta}$ is found to be negligible compared to $\ensuremath{\eta}$ in the OCP.

Calculation of autoionization of He and H/-/ using the projection-operator formalism

Abstract: Improved calculations are reported for the first several autoionization states of the lower symmetries of He and H(-). Unshifted energies are calculated by diagonalizing QHQ using a Hylleraes basis with more terms than previously used; shifts, widths, and photoabsorption shape parameters are obtained with the additional use of exchange-approximate nonresonant continuum functions. Previous calculations of H(-) resonances are reviewed and slightly augmented by applying various nonresonant continua and correcting small errors. A comparison is made between the calculations and experimental results and is found to be very satisfactory except for the lowest 1P autoionization state of He, which is shown to need a more accurate experimental determination.

Average energy expended per ion pair in liquid xenon

Abstract: Measurements were made of the average energy $W$ per ion pair formed in liquid xenon by internal-conversion electrons from $^{207}\mathrm{Bi}$. We observed voltage pulses resulting from electron collection either in liquid xenon or in a gaseous mixture of argon (95%) and methane (5%). The relative pulse heights for the two materials determine the ratio of the $W$ values. Using the known $W$ for the gaseous mixture, we obtained a liquid-xenon $W$ of 15.6 \ifmmode\pm\else\textpm\fi{} 0.3 eV. This value is considerably smaller than the gas-phase values, 21.5 or 21.9 eV. For interpretation, we adapted Platzman's energy-balance equation to liquids, assuming a conduction-band picture. Theoretical values thus calculated agree well with experiment.

Collisional-radiative recombination in cold plasmas

Abstract: We have investigated the electron-ion three-body collisional-radiative recombination for electron temperatures below 4000 \ifmmode^\circ\else\textdegree\fi{}K using the Mansbach-Keck rates of electron-impact-induced transitions between hydrogenic energy levels of high principal quantum number. At such temperatures and for a wide range of electron densities, the statistical recombination process is simultaneously governed by collisional and radiative transitions between excited levels near the ionization limit, where many atomic and molecular systems possess a hydrogenic energy-level structure. In order to also take into account radiative transitions, we have numerically solved a system of coupled equations, describing the quasi-steady-state populations of 100 bound levels. These equations are expressed in, and solved for, the first differences between the reduced population densities $\ensuremath{\rho}(p)$, which improves the computing precision and establishes the location of the "bottleneck" in the recombination sequence. Our results are consistent with the following approximation for the collisional-radiative recombination rate coefficient (in ${\mathrm{cm}}^{3}$ ${\mathrm{sec}}^{\ensuremath{-}1}$): ${\ensuremath{\alpha}}_{\mathrm{cr}}=1.55\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}{T}^{\ensuremath{-}0.63}+6.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}{T}^{\ensuremath{-}2.18}{[e]}^{0.37}+3.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}{T}^{\ensuremath{-}4.5}[e]$, where the electron temperature $T$ is in \ifmmode^\circ\else\textdegree\fi{}K and the electron density [$e$] is in ${\mathrm{cm}}^{\ensuremath{-}3}$; the first and last terms describe purely radiative and collisional recombination, respectively, and the second term results from the complex interplay of collisional and radiative processes. Agreement with experimental data is reasonable.

Thermodynamic properties of small aggregates of rare-gas atoms

Abstract: The present work reports on the equilibrium thermodynamic properties of small clusters of xenon, krypton, and argon atoms, determined from a biased random-walk Monte Carlo procedure. Cluster sizes ranged from 3 to 13 atoms. Each cluster was found to have an abrupt liquid-gas phase transition at a temperature much less than for the bulk material. An abrupt solid-liquid transition is observed for thirteen- and eleven-particle clusters. For cluster sizes smaller than 11, a gradual transition from solid to liquid occurred over a fairly broad range of temperatures. Distribution of number of bond lengths as a function of bond length was calculated for several systems at various temperatures. The effects of box boundary conditions are discussed. Results show the importance of a correct description of boundary conditions. A surprising result is the slow rate at which system properties approach bulk behavior as cluster size is increased.

Energy and Angular Distribution of Electrons Ejected from Helium by Fast Protons and Electrons: Theory and Experiment

Abstract: A comprehensive study of the angular and energy distributions of electrons ejected in collisions of fast electrons and protons with He is presented. New experimental results for 300-keV, 1-MeV, and 5-MeV proton impact are reported along with theoretical results for 2-keV electron impact and 100-keV, 300-keV, 1-MeV, and 5-MeV proton impact. The theoretical results, based upon Born approximation with Hartree-Slater initial discrete and final continuum wave functions, show excellent agreement with experimental electron-impact results. Serious discrepancies are found between theory and experiment in the angular distribution of ejected electrons for forward angles for 100- and 300-keV proton impact; the discrepancies decrease markedly for 1-MeV proton impact and are absent for 5-MeV protons. The agreement between theory and experiment for intermediate and backward angles of electron ejection, on the other hand, is uniformly good for all proton impact energies. The reasons for this behavior in terms of a charge-exchange process to a continuum state contributing to electron ejection at forward angles is discussed, and the energy dependence of the data is shown to be consistent with this explanation.

Modified Van der Waals theory of fluid interfaces

Abstract: A modified version of the Van der Waals theory of fluid interfaces is presented. The modified theory is shown to retain the qualitative simplicity of the original theory while yielding a much more quantitative description of fluid properties.

Steric model for the smectic- C phase

Abstract: A steric model of the smectic-$C$ phase is presented, with molecules assumed to have a symmetric zig-zag shape (this kind of molecular gross shape may arise from obliquely placed end chains). The interaction between such molecules depends on mixed tensors ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{3}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{2}$, etc., as well as on the usual tensors ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{3}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{3}$, ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{2}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{2}$, and similar higher-order tensors, where ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{3}$ and ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{2}$ are unit vectors denoting the directions of the long molecular axis and a transverse axis, respectively. A simple interaction, which includes a term that mimics the effect of the zig-zag shape, is constructed in terms of these tensors. Using quite plausible values for the magnitude of the zig-zag part of the interaction, a second-order phase transition to a smectic-$C$ phase is found, characterized by nonvanishing values of biaxial order parameters $〈{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{3}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ u}}}^{2}〉$, etc. In the limit of perfect nematic order, the biaxial order parameters of the model are essentially equivalent to the vector order parameters used by de Gennes and by McMillan. The smectic planar structure is accounted for in the simple density-wave approximation of Kobayashi and McMillan. Only one phase with more orientational order than the smectic-$A$ phase is obtained; this is in contrast to McMillan's dipole model, which shows three different ordered phases (one of which corresponds to the smectic-$C$ phase). The present model predicts that the characteristic biaxialities in the smectic-$C$ phase may easily grow up to values of $O({10}^{\ensuremath{-}1})$.

Variable dimensionality in atoms and its effect on the ground state of the helium isoelectronic sequence

Abstract: %'e calculate binding energies for heliumlike ions of variable dimensionality (D) with the wave functions A = e&-+&i-«.&+e~-«i-0&. & and 8 =A(1+ cA»). The binding energy decreases with increasing D. Functions A and 8 predict \"critical binding dimensionalities\" at D= 3.99 and 4.89, respectively, above which there is no binding in the hydride anion. The exact groundstate binding energy at D = 5 is shown to be equal to that of the doubly excited 2p' 'I\" state into in three dimensions. By \"dimensionality scaling\" of atomic units the D = 1 atom is transformed the 5-function model for which exact energies are known. In the infinite dimensional limit, function A predicts no exchange contribution to binding for nuclear charge Z & Q2, with a. +P only for Z & g2.

Classical-quantum correspondence for multilevel systems

Abstract: The properties of $r$-mode harmonic-oscillator coherent states are reviewed. In particular, the $\mathcal{D}$-algebra differential-operator realization of the creation and annihilation operators on the coherent states and their diagonal projectors is constructed. A homomorphism between the algebra describing $r$-field modes and the algebra describing $r$-level systems is exhibited explicitly. This homomorphism allows the projection of the multimode calculus onto the multilevel calculus. In particular, multimode coherent states and projectors can be used as generating functions for multilevel coherent states and projectors. In addition, the multilevel $\mathcal{D}$ algebra is constructed directly from the multimode $\mathcal{D}$ algebra under this homomorphism. For illustrative purposes, the $\mathcal{D}$ algebra for the diagonal coherent-state projectors for two-level atomic systems is presented explicitly in terms of a parametrization in the Bloch angles $\ensuremath{\theta}$ and $\ensuremath{\phi}$. Two classes of applications are treated: (a) the mapping of atomic-density-operator equations of motion into phase-space equations of motion for the quasiprobability weighting function $P$; (b) the construction of equations of motion for the diagonal elements $Q$ of the density operator in the coherent states. It is shown that the solution to either equation with the appropriate initial condition gives complete statistical information for the atomic system. It is shown explicitly that the functions $P$ and $Q$ are related by a convolution integral.