Coagulation processes may be balanced by very slow breakup reactions that lead to a self-similar stationary cluster size distribution, as recently described by Family, Meakin, and Deutch (FMD [Phys. Rev. Lett. 57, 727 (1986); 57, 2332 (1986)]. Here, a class of reversible coagulation-fragmentation models is presented that satisfies detailed balance, for which the scaling functions can be calculated explicitly. The FMD models do not have the property of detailed balance in general.

A simple theory describing the dynamics of two-dimensional arrays of soap bubbles is proposed, and compared with a recent experiment. The average area of bubbles scales linearly at late times. Agreement with experiment is satisfactory, although not conclusive.

It is shown that the projection approach offers a straightforward way to elucidate the connections among the current theories on the effects of colored noise. An error indicator with a simple analytical expression is built up which allows us to evaluate the error associated with each approximation.

We present the results of a multiconfiguration calculation of the oscillator strengths for the dipole transitions between 3snl ${}_{}{}^1{}_{}{}^{}$,3L states of a magnesium atom. The effect of the configuration interaction on the change of the calculated oscillator strengths in the length and velocity approximation is discussed in detail. In particular, we have investigated the large shift of the oscillator strengths between groups of transitions due to the strong configuration interaction between the 3$p^2$ configuration and the entire 3snd ${}_{}{}^1{}_{}{}^{}D$ configuration series. The ‘‘accidental’’ agreement between the length and velocity results is also examined.

We have carried out numerical calculations for the classical bremsstrahlung spectrum, angular distribution, and polarization resulting from electrons scattering in screened atomic potentials. The calculations utilize the classical theory of electromagnetic radiation from moving charges together with the classical mechanics of energy-loss-free orbits of charged particles in such potentials. The trajectories of incident electrons in the given central potential are calculated numerically, and the doubly differential bremsstrahlung cross section is obtained by taking the Fourier transform of the acceleration of the dipole moment of the projectile and integrating the radiation over the impact parameters of the beam. Results have been obtained for several neutral atoms for electron energies from 1 eV to 500 keV, using self-consistent Dirac-Slater atomic potentials. Comparisons with quantum-mechanical partial-wave results for the spectrum show that, as in the point Coulomb case, the classical method is generally good for screened potentials for low incident electron energies. The classical method therefore can be useful in low-energy situations for which the usual Elwert-Born form-factor method is not satisfactory. We have also obtained and examined the angular distributions of classical bremsstrahlung. Screening reduces the value of the asymmetry parameter which characterizes the angular distribution of dipole radiation. The difference of screened results from the point Coulomb results increases with decreasing incident electron energy. At very low energy the value of the asymmetry parameter oscillates with energy, and this also leads to oscillations in the degree of polarization.

The high-precision verification of a retardation effect on the interaction V(r) between a pair of systems probably demands that the systems be microscopic and form a bound state. The ideal pair would seem to be a helium ionic core, with the electron in the 1s state, interacting with an electron in a state with n and l not too small, possibly in a high Rydberg state. We extend a previous calculation for this pair which made use of time-ordered Feynman diagrams, with the electrons treated nonrelativistically, and obtain an expression for V(r) valid for r greater than several $a_0$. [The V(r) obtained is identical to the main result of a recent dispersion theoretic approach.] Our approach can be interpreted as an extension of the physical argument that for r≳137$a_0$ the retardation component of V(r) follows easily from considerations of the interactions between electromagnetic vacuum fluctuations and each member of the pair, with each member characterized solely by its frequency-dependent electric dipole polarizability, ${\alpha }_d$(ω); the extension from r≳137$a_0$ down to just several $a_0$ is achieved by including in the characterization of each member not only its ${\alpha }_d$(ω) but its frequency-dependent nonadiabatic dipole polarizability, β(ω). The simplicity of the approach, with the electrons again treated nonrelativistically and with physical insights relatively apparent, allows one, with some confidence, to guess at the form of V(r) for systems not yet studied; it strongly suggests that to include relativistic dynamical effects on the inner or outer electron, or to study the interaction of a Rydberg electron with either certain many-electron cores or a one-electron core in certain low-lying excited states, at least formally and to lowest order, one need merely use the corresponding forms of ${\alpha }_d$(ω) and of β(ω). Here we consider neither exchange nor higher multipole contributions.

An analytical approximation, depending on five parameters, for the atomic screening function is proposed. The corresponding electrostatic potential takes a simple analytical form (superposition of three Yukawa potentials) well suited to most practical applications. Parameters in the screening function, determined by an analytical fitting procedure to Dirac-Hartree-Fock-Slater (DHFS) self-consistent data, are given for Z=1–92. The reliability of this analytical approach is demonstrated by showing that (a) Born cross sections for elastic scattering of fast charged particles by the present analytical field and by the DHFS field practically coincide and (b) one-electron binding energies computed from the independent-particle model with our analytical field (corrected for exchange and electrostatic self-interaction) agree closely with the DHFS energy eigenvalues.

The time dependence of the dressing-undressing process, i.e., the acquiring or losing by a source of a boson field intensity and hence of a field energy density in its neighborhood, is considered by examining some simple soluble models. First, the loss of the virtual field is followed in time when a point source is suddenly decoupled from a neutral scalar meson field. Second, an initially bare point source acquires a virtual meson cloud as the coupling is switched on. The third example is that of an initially bare molecule interacting with the vacuum of the electromagnetic field to acquire a virtual photon cloud. In all three cases the dressing-undressing is shown to take place within an expanding sphere of radius r=ct centered at the source. At each point in space the energy density tends, for large times, to that of the ground state of the total system. Differences in the time dependence of the dressing between the massive scalar field and the massless electromagnetic field are discussed. The results are also briefly discussed in the light of Feinberg’s ideas on the nature of half-dressed states in quantum field theory.

Some recent works on the path-integral formulation of nonconservative forces quadratic in velocity are examined critically. It is argued that the ambiguity resulting from ordering the operators in the Lagrangian and the Hamiltonian for this force is more serious than it is usually believed, and that one can construct a number of Hermitian Hamiltonians all satisfying Ehrenfest’s theorem, but forming a set of noncommuting operators. A similar problem is encountered in the ordering of the Lagrangian for the path-integral approach. This is in addition to a number of well-known difficulties such as violation of the position-velocity uncertainty, and the differences between the invariances of the equation of motion, and the Lagrangian. An alternative way is suggested in which the damping force is replaced by a purely time-dependent term which produces the same classical motion, thus avoiding the above-mentioned difficulties. Using this formulation, wave functions and time-dependent energies are found for a quadratically damped harmonic oscillator, and for a particle moving in a dissipative medium when the friction is proportional to an arbitrary power of velocity.

If the neutral molecule core of a molecular negative ion has sufficient rotational energy, the weakly bound electron will autodetach by transfer of molecular rotational energy to this electron. Under certain circumstances, this process can be theoretically described by an analogy with the internal-conversion mechanism of nuclear physics. The internal-conversion mechanism is described and its applicability to the calculation of rotational autodetachment transition rates discussed. It is shown that under the conditions for applicability of the internal-conversion model, this model is equivalent to, but much simpler than, the conventional nonadiabatic couplings (both couplings, not just the gradient-gradient coupling) which result from the breakdown of the Born-Oppenheimer approximation. One state of ${\mathrm{FeO}}^{\mathrm{-}}$, which satisfies the criteria, has a very distinctive dependence of autodetachment transition rate on rotational quantum number. With the help of the internal-conversion model, this dependence is reasonably well described.

By continuing earlier investigations, generalized coherent states are introduced to describe the quasiclassical motion of electrons in a microwave and in a homogeneous magnetic field. The microwave is circularly polarized and propagates along the direction of the magnetic field. In view of possible applications to laboratory plasmas, the problem is treated nonrelativistically and the microwave is taken in the dipole approximation. The generalized coherent states emerge in polar coordinates (ρ,φ) from ordinary Landau states ‖l,m> by an appropriate choice of unitary and displacement operations. These coherent states are shown to form a complete and orthonormal set. Consequently, these states may be conveniently used for the description of scattering processes. As an example, scattering of electrons by a screened Coulomb potential is considered in the presence of the above two external fields. For simplicity, the scattering potential is treated in the first-order Born approximation. The quasiclassical features of the corresponding cross sections for the induced nonlinear processes are discussed.

Lie’s method of extended-group theory has been used to obtain explicit forms of the generators and the structure constants of the maximal symmetry group of point transformations for the nonrelativistic Schrödinger equation. For atoms and molecules this symmetry group is an infinite-parameter Lie group with an infinite-parameter invariant subgroup. The corresponding factor group is a 14-parameter Lie group containing a proper subgroup locally isomorphic to SO(4), the orthogonal group in four dimensions.

Energy levels of triply excited states of an atom with all three electrons in the same shell are obtained by a model calculation. The nodal pattern of an eigenfunction at the symmetry configuration plays a crucial role in the manifestation of moleculelike modes as in the case of doubly excited states. It is shown that the correlation of the three electrons leads to normal modes isomorphic to those of a molecule with $D_{3h}$ symmetry. The energy levels may be thus regrouped into manifolds, each revealing the rotational level structure of a symmetric top. The validity of the model is discussed in reference to known experimental and theoretical results.

The hyperspherical coordinate representation of N-electron atoms is developed and applied to simple systems. Algebraic expressions are given for Coulomb radial integrals, analogous to standard forms for spin and angular matrix elements. Numerical evaluation of the algebraic forms is tractable over a sizable range of the hyperspherical quantum numbers, as demonstrated by pilot calculations for beryllium and carbon. Correlation diagrams connecting hyperspherical and independent-electron classification schemes are presented and discussed.

A variational functional for transition amplitudes is introduced which is invariant under phase changes of trial wave functions, stationary around the exact transition amplitudes, and avoids divergences in numerical applications. Euler-Lagrange equations are derived and briefly considered for parametric and basis-set expansions of wave functions. A detailed analysis is presented of a procedure to improve upon time-dependent Hartree-Fock amplitudes. Variational corrections are derived in terms of one-electron transition density functions and also introducing atomic basis sets. The way in which the procedure can be used to correct results for electron transfer in atomic collisions is discussed in some detail.

A variational functional for transition amplitudes is applied to atom-atom collisions using a time-dependent formalism. The transition amplitude is a functional of two independent trial wave functions with specified initial and final conditions, respectively. Transition probabilities are obtained by evaluating the functional in terms of standard time-dependent Hartree-Fock (TDHF) trial functions. The TDHF method is implemented by expanding the molecular orbitals as linear combinations of traveling atomic orbitals. The procedure requires no other molecular integrals than those needed to construct the Fock operator. The present method is applied to He+${\mathrm{He}}^{2+}$ and ${\mathrm{H}}^{+}$+${\mathrm{He}}^{+}$ collisions with specified nuclear trajectories using a minimal atomic-orbital basis set. Within this model an exact, close-coupling-type solution is also feasible, and has been obtained for comparison purposes. Numerical values of probabilities and of cross sections obtained from the variational functionals for elastic, spin-flip, and one- and two-electron transfer processes show substantial improvements over the standard TDHF results.

Total elastic cross sections have been computed for electron-${\mathrm{SiH}}_4$ collisions in the range of energy between 0.1 and 20 eV. All contributions to the full interaction have been included by means of functional dependences on the total electron density of the bound orbitals and no parameters have been employed to generate the final potential. The experimental suggestions of a low-energy, Ramsauer-type minimum at approximately 0.4 eV and of a marked resonance between 2 and 4 eV are fully confirmed by the present calculations. Possible reasons for the differences found in previous calculations are discussed.

The second virial coefficient B(T) can be expressed in terms of two-body phase shifts and bound-state energies. The fact that B(T) must remain continuous as the potential strength is varied is used to deduce the relationship between the phase shifts at zero energy and the number of bound states supported by the potential (Levinson’s theorem). There still remains an ambiguity due to the possibility of zero-energy resonances; in one and three dimensions this can be removed by the additional information of whether the tangent of the phase shift goes to zero or to infinity at zero energy. In two dimensions, the problem is more subtle and a difficulty is encountered if the potential strength is such that a p-wave zero-energy resonance is present; the expression for B(T) as the inverse Laplace transform of the Jost function gives the correct contribution from this resonance, but the phase-shift formula derived using asymptotic wave functions fails to include it, and hence is in disagreement with Levinson’s theorem. The origin of this disagreement is traced to a noncommutivity in the wave function between the limits of large distance and low energy; this occurs only in the two-dimensional case, and needs to be handled carefully.

Dielectronic recombination cross sections for ${\mathrm{Mo}}^{32+}$,31+,30+ with 10, 11, and 12 electrons initially and involving 2s and 2p electron excitations are estimated. The first observations by Tanis and co-workers of the L-shell excitation in the resonant transfer excitation of ${\mathrm{Nb}}^{31+}$+${\mathrm{H}}_2$ were reported recently, and we analyze the data using the ${\mathrm{Mo}}^{32+}$ result and further predict the cross sections for ${\mathrm{Nb}}^{30+}$ and ${\mathrm{Nb}}^{31+}$.

The relativistically extended strong-potential Born (SPB) formalism is applied to the radiative electron capture process caused by the bombardment of a heavy and highly stripped charged particle with relativistically high velocity. The results are compared with those by use of nonrelativistic SPB calculations and with those by use of the relativistic Born calculation (Sauter’s formula), which includes no distortion effects between a heavy projectile ion and an active electron. Even if the strong distortion effects are taken into consideration, the shapes of photon angular distributions in the laboratory frame still nearly depend on ${\sin }^2$${\mathrm{\theta }}_{\mathrm{L}}$(${\mathrm{\theta }}_{\mathrm{L}}$ is the angle of the emitted photon) in the vicinity of the angle of 90°, which is the same as the results by use of Sauter’s formula. The higher the charge of a projectile ion becomes, however, the greater the discrepancy between the angular shape of our results and that of Sauter’s becomes at both smaller and larger angles than at 90°. As is expected, the magnitudes of the differential and the total cross sections are drastically influenced by the distortion effects ascribable to a large charge of a heavy projectile ion such as ${\mathrm{U}}^{92+}$. Our results are in good agreement with recent experiments. In addition, the Coulomb off-shell factor introduced by the SPB theory is found playing important roles in the case of the relativistic radiative electron capture process because the results calculated by using the relativistic impulse approximation are too underestimated.