Critical conditions are obtained for bifurcation phenomena in nonequilibrium systems (chemical instabilities) which are appropriate for transitions between homogeneous steady states as well as for symmetry-breaking transitions to static structures. In the case of symmetry-breaking instabilities these criteria enable the theory to be applied to systems in any number of spatial dimensions, eliminating a restriction to one-dimensional systems encountered in other treatments. These critical conditions allow for the derivation of time-dependent Ginzburg-Landau (TDGL)-type equations for the critical-mode amplitude (the order parameter) that grows into the new macrostate beyond the critical point. For homogeneous transitions the usual TDGL equation is obtained. For the case of intrinsic symmetry breaking, TDGL equations are found for coupled order parameters corresponding to different directions in $k$ space. In both the intrinsic and the extrinsic cases the TDGL equations are found to have nonlinear transport terms. When the TDGL equations are turned into Langevin equations, Ginzburg criteria (defining the region where meanfield theory breaks down) are derived. The critical dimensionality thus determined is 4 for homogeneous and intrinsic symmetry-breaking transitions, and 6 for the extrinsic symmetry-breaking case (under given mild technical conditions). Expressions for the size of the nonclassical critical regions are obtained for the different transitions in terms of characteristic parameters. For chemical instabilities these regions are in principle accessible.

Two new nonequilibrium methods for determining the bulk viscosity via molecular dynamics are described. These methods are applied to a soft-sphere dense fluid. Viscosities obtained from the equilibrium Green-Kubo-correlation-function method agree with the nonequilibrium results. The viscosities found are considerably smaller than those predicted by the Enskog theory of dense fluids.

The asymptotic behavior of the population probabilities in the regime of high electron density is calculated. The population probabilities in this case follow a simple formula characterized by a parameter ${\mu }_{j,m}$, for which an explicit formula is given. The results are illustrated by means of a numerical example. Finally, a simple algebraic function is proposed which has the correct asymptotic behavior in both the low-and high-electron-density plasmas.

$K$-shell ionization cross sections are measured for Dy atoms produced in ($\alpha$,$\mathrm{xn}$) reactions induced by $\alpha$ particles in the energy range of 47-130 MeV on several separated Gd isotopes. The ratios of these cross sections to those of the corresponding ($\alpha$,$\mathrm{xn}$) nuclear reactions are found to be practically independent of the $\alpha$-particle bombarding energy and to vary strongly from one final nucleus to another. A recently proposed general relationship between the nuclear reaction and $K$-x-ray production must be considered only a crude order-of-magnitude estimate.

By using the method of multiconfiguration self-consistent field, the electronic wave functions for the $X{}_{}{}^3{}_{}{}^{}\Sigma _g^{-}{}_{}{}^{}$ and $B{}_{}{}^3{}_{}{}^{}\Sigma _u^{-}{}_{}{}^{}$ states of ${\mathrm{O}}_2$ are determined for twelve values of internuclear distances. From the appropriate electronic-vibrational functions, the oscillator strengths for the $X{}_{}{}^3{}_{}{}^{}\Sigma _g^{-}{}_{}{}^{}\to B{}_{}{}^3{}_{}{}^{}\Sigma _u^{-}{}_{}{}^{}$ absorption (Schumann-Runge) continuum are calculated over the energy range of 7-9.5 eV. The results are in good agreement with experiments except at the high-energy edge (above 9 eV) which corresponds to the very steeply varying part of the potential curve. A calculation of the electron-impact excitation cross sections for the $B{}_{}{}^3{}_{}{}^{}\Sigma _u^{-}{}_{}{}^{}$ state with the Born-Ochkur approximation is also reported for incident electron energy up to 1000 eV.

We study the parity-violating electric-dipole transitions $\gamma +2\mathrm{}{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\to 2{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and $\gamma +2\mathrm{}{}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}\to 2{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ in He in order to gain some insight into the reliability of approximate calculations which have been carried out for similar transitions in many-electron atoms. The contributions of the nearest-lying states are computed with a variety of wave functions, including very simple product wave functions, Hartree-Fock functions, and Hylleraas-type wave functions with up to eighty-four parameters. We find that the values for the matrix elements of the parity-violating interaction which are given by the fairly simple wave functions can differ considerably from the values obtained from the very good wave functions, even when these simple wave functions give accurate values for energies and dipole matrix elements. An identity derived in a previous paper, which converts a delta-function matrix element to that of a global operator, is used to obtain alternative values for the matrix elements in question. It is found that use of this identity can substantially improve the results obtained with less accurate wave functions. We discuss the implications of our results for calculations of parity mixing in many-electron atoms.

It is shown that the positions of atomic shape resonances may be identified by adding to the one-electron Hamiltonian operator an artificial "shift potential" that differentiates between scattering and resonance states by exploiting their different small-$r$ and large-$r$ behavior. We explored the use of shift potentials which are either repulsive for large $r$ or attractive for small $r$. In the first case, resonances correspond to virtual orbitals that are relatively stable in energy. In the latter case the scattering states are stable in energy, and the resonance orbitals are identified by their relatively strong response to the attractive shift potential. The second method is found to be more sensitive and reliable in the application to ${}_{}{}^2{}_{}{}^{}P$ ${\mathrm{Be}}^{-}$ and ${\mathrm{Mg}}^{-}$ presented here. Moreover, it is found that, although the rather simple methods developed here are not capable of giving quantitatively correct resonance energies, they are useful in separating the resonance states from the underlying continuum.

We study the problem of screened Coulomb potentials, using an alternative approach to perturbation theory. Assuming that the radial wave functions $R_{\mathrm{nl}}\mathrm{}\left(r\right)\mathrm{}$ can be written in exponential forms, we calculate the wave functions of the $K$-, $L$-, and $M$-shell electrons, through the fifth order in the perturbation parameter, by solving directly the set of resultant differential equations. The binding energies and wave functions obtained with this simple approach agree with previous work.

Diagrammatic many-body perturbation theory is applied to the calculation of potential-energy curves for ${\mathrm{He}}_2$. Both model and shifted perturbative procedures are described. The shifted scheme produces a spurious $R$ dependence for the long-range interatomic potential in both second and third order. Variational-perturbative upper-bound methods produce energies that approach an incorrect asymptotic limit at large internuclear distances. In contrast, the model-perturbative scheme in both second and third order achieves a proper distance dependence for the ${\mathrm{He}}_2$ potential curve. Four basis sets are employed to illustrate basis-set effects.

We show that the existing and widely used results for an approximation to eikonal exchange amplitudes, which contain an indeterminate phase, are not valid. We derive a new analytic result, containing a well-defined phase, which can be used unambiguously with direct amplitudes to calculate full differential cross sections. The modulus of our result also differs from that of the previous formulations. Comparisons are made with the earlier $e^{-}$-H approximate exchange calculations and with exact eikonal exchange calculations.

We have made comprehensive measurements of the noble-gas-induced (He, Ne, Ar, Kr, and Xe) broadening cross sections of the Na $3S-nS$ ($3S-nD$) transitions, from $n=5\mathrm{} \mathrm{to} 38$ ($n=4\mathrm{} \mathrm{to} 40$). These measurements were made using the trilevel-echo technique and are an extension and refinement of our earlier work. We find that the cross sections increase as the geometric cross section at low values of $n$, and then, at a value of $n$ which depends on the particular noble gas, begin to decrease and eventually reach an asymptotic value at high $n$. The broadening cross sections for the nearly degenerate $D$ states are significantly larger than those for the $S$ states and clearly show the effect of $l$-changing collisions. We also compare our measured $S$-state broadening cross sections with the cross sections calculated using the theory of Omont and find good qualitative agreement.

Absolute differential, integral, and momentum-transfer cross sections for electrons elastically scattered from helium are reported for the impact energy range of 5 to 200 eV. Angular distributions for elastically scattered electrons are measured in a crossed-beam geometry using a collimated, differentially pumped atomic-beam source which requires no effective-path-length correction. Below the first inelastic threshold the angular distributions were placed on an absolute scale by use of a phase-shift analysis. Above this threshold, the angular distributions from 10° to 140° were fitted using the phase-shift technique, and the resulting integral cross sections were normalized to a semiempirically derived integral elastic cross section. Depending on the impact energy, the data are estimated to be accurate to within 5 to 9%.

The author presents results of quasiclassical close-coupling-approximation calculations for differential and integrated cross sections and multipole moments for electron-hydrogen impact excitation. These results are compared with calculations using other scattering-theory approximations and with experiment. Discrepancies between the various theoretical results and between theory and experiment are found. Possible sources for some of the discrepancies are mentioned.

The classical double-scattering model of Thomas for the capture of electrons from atoms by fast ions yields a cross section $\sigma$ which dominates over the single scattering contribution for sufficiently fast ions. The magnitude of the classical double-scattering $\sigma$ differs, however, from its quantum-mechanical (second-Born) analog by an order of magnitude. Further, a "fast ion" means an ion of some MeV, and at those energies the cross sections are very low. On the other hand, as noted by Bates, Cook, and Smith, the double-scattering cross section for the capture of atoms from molecules by fast ions dominates over the single-scattering contribution for incident ions of very much lower energy; roughly, one must have the velocity of the incident projectile much larger than a characteristic internal velocity of the particles in the target. It follows that we are in the asymptotic domain not at about 10 MeV but at about 100 eV. For the reaction ${\mathrm{H}}^{+}$ + C${\mathrm{H}}_4$→${\mathrm{H}}_2^{+}$ + C${\mathrm{H}}_3$ with incident proton energies of 70 to 150 eV, the peak in the angular distribution as determined experimentally is at almost precisely the value predicted by the classical model, but the theoretical total cross section is about 30 times too large. Using a quantum version of the classical model, which involves the same kinematics and therefore preserves the agreement with the angular distribution, we obtain somewhat better agreement with the experimental total cross section, by a factor of about 5. (To obtain very good agreement, one may have to perform a really accurate calculation of large-angle elastic scattering of protons and H atoms by C${\mathrm{H}}_3$, and take into account interference effects.) In the center-of-mass frame, for sufficiently high incident energy, the first of the two scatterings involves the scattering of ${\mathrm{H}}^{+}$ by H through an angle of very close to 90°, and it follows that the nuclei of the emergent ${\mathrm{H}}_2^{+}$ ion will almost all be in the singlet state. We have also calculated the cross section for the reaction ${\mathrm{D}}^{+}$ + C${\mathrm{H}}_4$→${\mathrm{}\left(\mathrm{HD}\right)\mathrm{}}^{+}$ + C${\mathrm{H}}_3$.

The stopping po$\stackrel{\mathrm{´}}{\mathrm{w}}$er of ${}_{}{}^{107}{}_{}{}^{}\mathrm{Ag}$, ${}_{}{}^{109}{}_{}{}^{}\mathrm{Ag}$, and ${}_{}{}^{150}{}_{}{}^{}\mathrm{Sm}$ in nickel and gold was measured as a preliminary test of a new technique for measuring energy loss based on the $\gamma$-ray Doppler shift. The analysis of the data was based on the theories of Lindhard, Scharff, and Schiott for nuclear and electronic stopping. The results are compared with the semiempirical predictions of Northcliffe and Schilling and the Lindhard-Scharff-Schiott theory.

Qualitative arguments suggest that diffraction effects might be important for the interpretation of the experiments of Shih et al. We derive a tractable quantum theory to describe the scattering of neutral atoms at long range by van der Waals forces from solid cylinders. The theory is based on an angular-momentum expansion and evaluates the scattered intensity close to the cylinder in the geometrical shadow region. The formal quantum theory indicates that one is far from the conventional classical limit, in the sense that WKB and stationary-phase approximations are not valid. Yet an essentially exact evaluation appears to agree with the classical prediction, at least over the range of deflections that have been experimentally studied. This surprising classical-quantum correspondence is clarified by a Regge-pole analysis. Finally we also set limits on the extent of specular reflection from the cylinder surfaces. The discrepancy between theory and experiment remains unresolved.

The angular and the energy distributions of fast ionized recoils produced from copper irradiation by noble-gas ions have been investigated. The distributions have been found to be strongly determined by regular arrangement of the target atoms. The results obtained are discussed in terms of the blocking-effect model.

The density model that was successfully used to obtain electronic stopping cross sections for slow-moving heavy ions traversing solid media was applied to estimate the well-known $Z_2$ oscillations in these cross sections. Good agreement with experimental results was generally found. Some of the possible sources of discrepancy are discussed. Relations between the results of Compton-profile measurements which depend on the bulk properties of the solid and the electronic stopping in thin foil targets assist in resolving ambiguities in the $Z_2$ dependence of the experimental results. It is concluded that the electronic stopping cross sections are not a universal function of $E$, $Z_1$, and $Z_2$. They depend strongly on the method of the preparation of the target foil and on its history.

The continuum-distorted-wave (CDW) method is used to determine total electron-capture cross sections $Q=\Sigma {\mathrm{nl}}^{}\sigma \mathrm{}\left[\mathrm{nl}\right]\mathrm{}$ for proton and $\alpha$-particle projectiles incident on ${\mathrm{Li}}^{+}$ in energy ranges of $100\le E\le 3000$ keV and $100\le E\le 10000$ keV, respectively. A configuration-interaction (CI) wave function is used to describe the ${\mathrm{Li}}^{+}$ electrons; for comparison, capture cross sections for ${\mathrm{H}}^{-}$ and He are examined. For each system the percentage change in $Q$, $\Delta (\mathrm{HF}\to \mathrm{CI})$, is given for progression from a Hartree-Fock (HF) to a CI description of the target electrons. The main emphasis in this work is devoted to a rationalization of the trends observed in the ordering of $Q$ for these three targets. This was achieved by an analysis of the CDW expression for an individual capture cross section $\sigma [nl, n^{\prime }{l}^{\prime }]$, where $\mathrm{nl}$ and $n^{\prime }{l}^{\prime }$ are the states of the "active" (captured) and "passive" electrons, respectively.