Self-avoiding-walk (SAW) configurations are considered by incorporating local bridges, i.e., connecting any two nearest-neighbor sites visited by the SAW by massless cross links. Introducing a new elegant method of estimating the resistance of an arbitrary network, we find the resistance exponent δ=0.920±0.005 in d=2 by enumerating the random samples of SAW’s using a Monte Carlo method. We also find the shortest-connecting-path-length exponent t to be equal to 0.975±0.005 using a simulation technique in two dimension. Random walks on SAW networks with local bridges are studied using a scaling formalism in the ‘‘grand canonical ensemble’’ picture of SAW’s. We fit the mean end-to-end distance 〈$R_t$〉 of random walks of t steps with a scaling form 〈$R_t$〉∼$t^1$/$d_w$F((f-$f_c$)$t^x$) (f being the fugacity of SAW’s) and find $d_w$=$d_F$(1+δ). In two dimensions this predicts the spectral dimension $d_S$ (=2$d_F$/$d_w$) of SAW networks with bridges to be equal to 1.042.

One can easily obtain a good estimate of the total energy E(Z) of a ground-state neutral atom with Z (≫1) electrons, primarily because an effective central potential $V_{\mathrm{eff}\left(\mathrm{r}\right)}$ provides a very good starting point. A sufficient, if not necessary, condition for there to be such a $V_{\mathrm{eff}}$ is for the ratio ρ(Z)≡$V_{e\mathrm{-}e}$(Z)/$V_{e\mathrm{-}\nu }$(Z) of the electron-electron and electron-nuclear contributions to E(Z) to be small, and it is found to be only about (1/7) for large Z. In the (statistical) nonrelativistic Thomas-Fermi (TF) model, which becomes exact as Z∼∞, E(Z) (in rydbergs) is approximated by $E_{\mathrm{TF}\left(\mathrm{Z}\right)=\mathrm{-}{c}_7{Z}^{7/3}}$ for all Z, with $c_7$ a known constant, while ρ(Z) is approximated by ${\rho }_{\mathrm{TF}\left(\mathrm{Z}\right)=(1/7)\mathrm{}}$ for all Z. A simple proof that ρ(Z)∼(1/7) as Z∼∞, by Rau (unpublished) and by Thirring [Quantum Mechanics of Large Systems (Springer-Verlag, New York, 1983)], used only the assumption (built into TF theory) that the binding energy of the last electron in a neutral atom is negligible.

We use that fact, the (recently developed) corrected TF result E(Z) ≊-($c_7$$Z^{7/3}$+$c_6$$Z^{6/3}$+$c_5$$X^{5/3}$) (with $c_7$, $c_6$, and $c_5$ theoretically determined), and the virial theorem, and we show that $V_{e\mathrm{-}\nu }$(Z)≊Z(dE/dZ) with an error that is at most of order $Z^{4/3}$; we thereby obtain an expression for ρ(Z) which approaches (1/7) as Z∼∞, the error for large Z being of order 1/Z, and is reasonably accurate even for moderate values of Z. One finds ρ(Z)≊J(n-6)$s_n$/J${\mathrm{ns}}_n$, where each sum is over n=7, 6, and 5, and where $s_n$=$c_n$$Z^{\left(n\mathrm{-}7\right)/3}$. The value of ρ(Z) for heavy atoms in ‘‘strong’’ magnetic fields, those describable by a one-dimensional (statistical) TF-like theory, is discussed briefly. Some numerical studies are included. The results obtained are of interest in atomic theory itself, and in low-energy β-decay processes for which the difference in atomic binding energies is important.

The properties of the level set boundaries in ${}_{}{}^{\mathrm{N}}{}_{}{}^{}$ Z space are used to determine bounds to the interatomic distances at which the electronic energy of a diatomic molecule reaches a specified value. Bounds to the molecular electronic energy curve are obtained from that result. The procedure is very simple and requires having only an approximate representation for the electronic density function of a single atom (the one taken as a reference to define the level set). In the case of one-electron diatomic molecules we derive a function of the internuclear separation which is a rigorous, nonvariational, upper bound to the electronic energy. The method provides complementary results to those global constraints on electronic energy hypersurfaces previously derived in the context of level set topology.

We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the ${}_{}{}^1{}_{}{}^{}\mathrm{S}_{}^{\mathrm{e}}{}_{}{}^{}$ ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term $H_{\mathrm{mp}=\mathrm{-}(2/\mathrm{M}){\nabla }_1\mathrm{\cdot }{\nabla }_2}$ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited ${}_{}{}^3{}_{}{}^{}\mathrm{P}_{}^{\mathrm{e}}{}_{}{}^{}$ state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion ($e^{\mathrm{-}}$${\mu }^{+}$$e^{\mathrm{-}}$).

Exact analytical expressions are derived for the matrix elements of the Uehling potential in a basis of explicitly correlated two-particle wave functions (geminals) used for the approximate solution of the quantum-mechanical three-body problem within the framework of the integral transform method. The derived formulas may be used to include corrections due to polarization of the vacuum in calculations of the binding energies of muonic molecules.

As a continuation of parts I and II of this series [A. Riera, Phys. Rev. A 30, 2304 (1984) and L. F. Errea, J. M. Gómez-Llorente, L. Méndez, and A. Riera, Phys. Rev. A 32, 2158 (1985)], and for ${\mathrm{He}}^{2+}$-H(1s) collisions as a benchmark case, we use the criterion based on the minimization of the Euclidean (N) and weighted ($N_w$) norms, to obtain a common translation factor with optimal asymptotic form. Mathematical and intuitive conditions for this translation factor are developed, and pitfalls of the use of these conditions without an objective criterion, such as provided by N or $N_w$, are pointed out.

The interaction of free muons with hydrogen atoms is discussed within the framework of time-dependent Hartree-Fock theory. Both the muon and the electron are treated quantum mechanically. The incident muon energies considered are 2.7 and 0.27 eV, and we discuss the results for fixed muon wave-packet widths in this paper. At the lower energy, for our packet widths of 1.5 Å, the probability for muon capture is high (0.7), and the probability distribution as a function of principal quantum number peaks at n=14≃($m_{\mu }$/$m_e$${)}^{1/2}$, i.e., at the radius corresponding to the electron’s initial ground-state radius. At the higher energy, there is a sharp drop in the capture probability. The time development in this case is qualitatively similar to that for elastic scattering of the same muon wave function from the ‘‘frozen’’ ground-state hydrogen-atom potential. Radial probability distributions, principal quantum number distributions, probability density contours, etc., are examined and discussed, as are total cross sections and comparisons with other theoretical results.

Directional x-ray scattering factors and Compton profiles have been evaluated in several directions relevant to the molecular structure of ${\mathrm{CH}}_4$ and compared to their respective isotropic components for four different self-consistent-field wave functions and one single-determinant configuration-interaction wave function constructed especially for this study with Cartesian Gaussian-type basis orbitals. The internally folded density (‘‘reciprocal form factor’’) B(r) is calculated and discussed as are various momentum-space expectation values. A comparison is made with available experimental and other theoretical results.

Quantal close-coupling calculations of the impact excitation of the n=2 fine-structure levels in the hydrogenlike ions ${\mathrm{Ar}}^{17+}$, ${\mathrm{S}}^{15+}$, ${\mathrm{Mg}}^{11+}$, ${\mathrm{C}}^{5+}$, and ${\mathrm{He}}^{+}$ by protons and electrons are presented, and the resulting cross sections and rate coefficients are compared with previous values. The effects of relativity on the proton-impact excitation cross sections for ions with high nuclear charge are investigated. Plasma-density effects on the transition cross sections are explored using a Debye-screening model, and it is shown that the screening of the ion alters the excitation cross sections substantially and brings the ratio of the cross sections towards the statistical value. The calculated transition rates are used to model the ratio of the intensities of the two components of Ly-α radiation emitted by impurity hydrogenlike ions in an optically thin, hydrogen plasma. The results confirm that at intermediate plasma densities proton-impact-induced transitions redistribute the n=2 fine-structure-level population ratio away from the statistical value.

The generalized oscillator strength (GOS) of ${\mathrm{H}}_2$${\mathrm{}}^{+}$ is calculated comprehensively for both discrete-discrete and discrete-continuum transitions using exact Born-Oppenheimer wave functions. The calculation presumes the Born-Oppenheimer separation, and thus gives the GOS dependence upon the internuclear distance and the angle between the molecular axis and the momentum transfer, in addition to the dependence upon the energy and momentum-transfer magnitude. The theoretical study of the molecular GOS is extremely rare even for discrete excitation and is virtually absent for continuum transitions. The present study serves as a guideline for full understanding of the molecular GOS in general, although no experimental results on the key aspect, i.e., the dependence of the GOS upon the momentum transfer, are available for comparison with theory.

Three-particle radiative electron-capture cross sections with retardation are calculated within the nonrelativistic treatment. The formalism is carried out in the laboratory frame. It is found that the center of mass genuinely radiates, and it plays an important role in restoring the backward-forward symmetry for angular distributions. Comparison with experiments are made. The formalism also describes the nonrelativistic Doppler effect which shifts and spreads the photon energy distributions. Total, single-, and double-differential cross sections are reported, and comparisons with the dipole approximation are presented. The range of validity of the results obtained is discussed.

High-resolution measurements of oxygen K Auger electrons have been performed using multiply charged 10-MeV ${\mathrm{O}}^{\mathrm{n}+}$ (n=2, 3, and 4) projectiles in collisions with He. Line-broadening effects are strongly reduced so that high-resolution spectroscopy is possible for Auger electrons ejected from fast projectiles. The spectra obtained for ${\mathrm{O}}^{4+}$ and ${\mathrm{O}}^{3+}$ impact show relatively simple structures dominated by lines arising from three-electron (Li-like) and four-electron (Be-like) states, respectively. Using configuration-interaction wave functions, the relativistic energies and fine structures of Li-like states are studied. The relativistic correction, Breit-Pauli operator, and mass-polarization effect are calculated for numerous resonances and metastable autoionizing states in O vi. Excellent agreement with experimental data is obtained. For example, the calculated transition energy of the (1s2s2p${)}^4$P° state is 416.02 eV. It is to be compared with the corresponding experimental data of 416.0±0.2 eV.

With the use of functional integration, an exact analytic expression valid for the whole temperature range is derived for the partition function of a charged quantum oscillator coupled to a non-self-interacting (with the $A^2$ term neglected) radiation field. To derive this result, special care is taken with renormalization which is carried out at two stages. The temperature-dependent free-energy shift due to blackbody radiation is studied and compared with recent work. It is shown that the ground state is the only stable state and that the higher states have a finite lifetime.

Dielectronic recombination (DR) coefficients for the ground-state ion of Ni-like ${}_{64}{}^{}{}_{}{}^{}\mathrm{Gd}$ have been calculated in the isolated-resonance approximation for temperatures in the range of 0.05≤T≤3 keV. The Auger and radiative rates of each doubly excited state were evaluated explicitly by using the multiconfiguration Dirac-Fock model. The calculations were carried out in the intermediate coupling. The effect of the configuration interaction was also included for the low-lying doubly excited states. The intermediate autoionizing states included in the present work are the states from the 3$d^{\mathrm{-}1}$4lnl’ (n≤10), 3$p^{\mathrm{-}1}$4lnl’ (n≤5), and 3$s^{\mathrm{-}1}$4lnl’ (n≤5) configurations. The dielectronic recombination coefficients for Ni-like ${}_{64}{}^{}{}_{}{}^{}\mathrm{Gd}$ from the present calculations are 1.33×${10}^{\mathrm{-}10}$ and 1.08×${10}^{\mathrm{-}10}$ ${\mathrm{cm}}^3$/sec for T=500 and 800 eV, respectively. The semiempirical Burgess-Merts formula has been found to overestimate the dielectronic recombination coefficients for the Ni-like ${}_{64}{}^{}{}_{}{}^{}\mathrm{Gd}$ by a factor of 5 for T≥250 eV.

We show that molecules and nuclei with well-expressed rotational degrees of freedom can be excited by means of a multistep process, initiated by a short and sufficiently strong electromagnetic pulse, into a coherent superposition of rotational states. Properties of a molecule or a nucleus in such a nonstationary state change (quasi-) periodically in time, due to the fact that the energies of the rotational states satisfy (approximately) the rule ωJ(J+1). Accordingly, their interaction with an electromagnetic field or with other particles changes (quasi-) periodically in time. A gas of molecules in a coherent state exhibits a (quasi-) periodic change of the refractive index, an effect which has been observed for the case of ${\mathrm{CS}}_2$ gas at 325 K. The nuclear case does not lead to direct experimental detection, because of its short coherence period.

A laser-microwave double-resonance experiment on electrodynamically trapped ${}_{}{}^{173}{}_{}{}^{}\mathrm{Yb}_{}^{+}{}_{}{}^{}$ ions has been performed and a value of Δ${\nu }_{\mathrm{h}\mathrm{f}\mathrm{s}=10491720239.55}$±0.09 Hz for the ground-state hyperfine separation has been determined. This value is corrected for small Zeeman and second-order Doppler shifts. Combined with a previous similar measurement on ${}_{}{}^{171}{}_{}{}^{}\mathrm{Yb}_{}^{+}{}_{}{}^{}$ and with the known $g_I$ values of both isotopes, we obtain a value of -0.004 25(6) for the differential hyperfine anomaly ${}_{}{}^{171}{}_{}{}^{}\mathrm{\Delta }_{}^{173}{}_{}{}^{}$.

The secondary-electron density in electron-beam-irradiated Ar-${\mathrm{F}}_2$ gas mixtures has been measured for the first time using time-resolved interferometry at 9.6 μm. Measurements were performed for gas mixtures with various ${\mathrm{F}}_2$ concentrations between 0.0025% and 2% and total pressures between 0.5 and 2 atm. The e-beam pulse was 350 nsec long at an energy of 150 keV and current density ranging between 0.025 and 6.3 A/${\mathrm{cm}}^2$. From the dependence of the measured steady-state electron density on fluorine concentration, the rate constant for electron attachment to fluorine at zero electric field has been determined for the first time. For a given e-beam current density, the attachment rate constant is found to be a decreasing function of the fluorine concentration, while for a given ${\mathrm{F}}_2$ concentration it is found to be an increasing function of the e-beam current density. The results are interpreted as a manifestation of the variation of the electron energy distribution as a function of ${\mathrm{F}}_2$ concentration and e-beam current density. The predictions of a theoretical kinetic model for calculation of the electron distribution function, taking into account the removal of slow electrons by attachment, are compared with the experimental results. It is concluded that for the range of the experimental parameters presented above, the net effect of adding fluorine to the gas mixture is an increase in the average electron energy, caused by removal of low-energy electrons by attachment.

The problem of a two-dimensional isotropic oscillator coupled through a variable parameter is considered. The problem is reduced to uncoupled form with two different variable frequencies. The solution has been obtained in both the Schrödinger and Heisenberg pictures. Applications in quantum optics are indicated.

An improved computational two-cavity model of the soliton laser proposed and designed by Mollenauer and Stolen [Opt. Lett. 9, 13 (1984)] is obtained through refinements of (i) the laser cavity model, (ii) the pulse propagation in the fiber cavity, and (iii) the coupling between the two cavities. As a result of the coupling to the fiber cavity, stable output pulses of temporal width 7.5 psec from the laser cavity are narrowed by a factor of approximately 10.

We discuss a method to obtain analytical solutions of the evolution equation of the optical signal in long-pulse free-electron-laser oscillators in the low-gain and small-signal regime. Supermodes are identified with the eigenstates, harmonic-oscillator orthonormal functions, of a non-Hermitian Hamiltonian. Inhomogeneous broadening effects due to energy spread and emittance are included. The space-time characteristics of the laser field are obtained with a significant reduction in the computer time.