The widths of $K{\alpha }_1$ and $K{\alpha }_2$ x rays in Eu, Tm, and Ta have been measured to an accuracy of (2-4)% with a high-resolution spectrometer by means of third- and fourth-order diffraction from the (111) crystalline planes of a bent silicon crystal. The measured values confirm the results from calculations of the radiative, Auger, and Coster-Kronig widths. Possible effects from hyperfine interaction, Coulomb exchange interaction, and chemical shift are discussed.

This paper describes the dynamics of multilevel excitation by coherent radiatin, in the $N$-level rotating-wave approximation, when the levels have nonzero angular momentum $J$ and consequent spatial-orientation degeneracy $2J+1$. Observable populations average over $2J+1$ independent sublevel excitation sequences. The phenomenon of inaccessible sublevels is noted, as is the effect of polarization. The paper illustrates how averaging over orientation can, in some cases, render illegible the Rabi oscillations and the ac Stark effect, whereas in other cases ($J\to J$, circular polarization on $J\to J\pm 1$ linear polarization) appreciable structure remains.

The connection between the critical properties of the Dicke model, subject to equilibrium and nonequilibrium boundary conditions, is explicitly exhibited. The Langevin equations generated by the Dicke Hamiltonian lead to a single nonlinear order-parameter equation that characterizes the stationary states of the system. We prove the existence of two essentially identical manifolds of stationary states, one for equilibrium, the other for nonequilibrium boundary conditions. These manifolds can each be derived from a potential obtained from the nonlinear order-parameter equations. In the equilibrium case the potential is the free energy of the system; in the nonequilibrium case it can be identified with the so-called "laser potential." The reduced-density operator for the field and atomic subsystems factors into a geometric and a physical part. The geometric part is determined by the system signal; the physical part, by the system noise. This is exhibited explicitly with an example taken from the theory of photoelectron counting. The identification of the stationary-state manifolds for the Dicke model subject to equilibrium and nonequilibrium conditions with the same geometric (cusp) manifold allows a real analytic continuation of model properties from the equilibrium configuration to the nonequilibrium dissipative regime. Three types of stability are considered: static, dynamical, and structural. Structural-stability considerations lead to strong conclusions about the effects of additional perturbations on the Dicke model.

The self-induced transparency solution of the Bloch-Maxwell equations is found in the case of inhomogeneously broadened resonant atomic lines and a host medium with a Kerr-type nonlinearity in the index of refraction. The characteristic feature of the solution is that the atomic functions are not factorized. The time dependence of the pulse phase is different from that in the case of a sharp atomic line. In addition to the known term quadratic in the pulse envelope, it contains a term increasing in time, attaining a maximum at the trailing edge of the pulse, and rapidly decreasing afterwards. The average carrier frequency is proportional to the square of the spectral width of the atomic line divided by the squared width of the pulse. A satisfactory solution is found by modulation of the center of the Lorentz distribution function of the purely resonant solutions. It is shown that in the presence of the inhomogeneous broadening the problem cannot be solved by a linear perturbation expansion with respect to the Kerr constant. This method leads to divergencies near the trailing edge of the pulse.

Different approximate analytical theories for dealing with nonuniform classical fluids are compared as approximations to a formally exact theory. The focus is on the situation where the nonuniformities are due to the interaction of the fluid with a model solid substrate. Detailed calculations are made using a density-functional theory previously developed by the authors, and a theory of a type used in a number of recent calculations, in which the perturbation caused by the fluid is dealt with using the Percus-Yevick (PY) approximation. Both theories use the PY approximation to treat pair correlations in the fluid. The density-functional theory turns out to be markedly the superior of the two for the case where the substrate should produce an unsaturated liquid film near the fluid-substrate interface. The density-functional theory correctly predicts the presence of such a film, whereas the other theory, essentially because it treats the response to the substrate perturbation linearly, yields no film. Microscopically detailed fluid-density profiles are given for both theories.

A calculation of the current carried on the sine-Gordon chain in an external field with viscous damping is formulated in terms of the solution of a Smolouchowski equation. The Smolouchowski equation is solved by employing its relationship to the BBGKY hierarchy and by using transfer-integral techniques. The method of solution is not specific to the sine-Gordon chain and can be usefully employed in examining other nonlinear systems. The solution of the Smolouchowski equation takes the form of a self-consistent equation for the single-particle distribution function; the solution describes the system for all values of the external field. At low field, the current is proportional to the field and due to thermally activated phase solitons; at high field, the current is proportional to the field and the conductivity many orders of magnitude greater than the low-field conductivity. A region of very nonlinear conduction separates the low-field and high-field regimes. It is possible to determine the effects of various perturbations on the conductivity of the sine-Gordon chain.

The gain of a free-electron laser may be multiplied by a very large factor when a uniform magnetic field $B_0$ is superimposed on the extant helical field. The effect is due to cyclotron resonance and requires $B_0=\frac{10.6}{l{(1-\frac{v_{\parallel }^2}{c^2})}^{\frac{1}{2}}}$ kG, where $l$ is the period of the helical field and $v_{\parallel }$ is the streaming-electron velocity.

Electron affinities are computed in a modified variational approximation for the ground states of B, C, N, O, and F. The method augments the variational basis for the neutral-atom wave function to include all configurations defined by open-shell virtual excitations that are implicit in the basis used for the negative ion. This basis is generated, as in electron-scattering calculations, from a specified list of neutral-atom configurations, by adjoining all available unoccupied or partially occupied orbital functions. The neutral-atom basis includes configurations defined by the first-order perturbing wave function for electric dipole polarizability. When single-particle virtual excitations $2s\to \overline{s}$, $2s\to \overline{d}$, $2p\to \overline{p}$ are included, results are close to experiment, with errors from 0.02 to 0.20 eV.

Density matrices defined with respect to a finite basis set are considered as elements in a vector space. A basis set is introduced in the space whose elements are Hermitian matrices. By a suitable transformation of the basis, one component only will contribute to the trace, so the space of unit trace matrices is a translated linear subspace. The positivity constraint is then examined in terms of distance from a multiple of the unit matrix, which is in the interior of the set of density matrices. The boundary of this set is shown to lie between two concentric hyperspheres. The set whose elements are $N$-representable 1 matrices has similar properties.

The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of $p$ particles for each value of $p$. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the $N$-representability problem is given, and two sufficient conditions and one necessary condition for $N$ representability are given in terms of distances and norms.

Expressions for multipole polarizabilities and shielding of the hydrogen atom have been obtained by generalizing the scheme worked out by Weiner and Askar based on the hydrodynamic analogy to quantum mechanics. Our expressions, for the dipole case, reduce to those of Weiner and Askar and for the zero-frequency limit to those of Dalgarno. Our calculated resonance frequencies are in much better agreement with the exact values than those reported by the variational approach.

We obtain, in exact closed form, the wave-function normalization constants for the Schrödinger equation with potential $V=B_{0}tanhz-U_0{cosh}^{-2\mathrm{}}z$. These constants are derived in terms of a variety of formulations and solutions of the equation. We give discussions of both mathematical aspects and physical motivations of the problem. The main results are gathered in two appendixes. Wave-function normalization constants for the related Pöschl-Teller potential are given in a third appendix.

Selected total cross sections are calculated in the closure-Born approximation for the ions ${\mathrm{Li}}^{+}$, ${\mathrm{Cs}}^{+}$, and ${\mathrm{Au}}^{+}$ incident on several gas constituents. Targets included are H, ${\mathrm{H}}_2$, He, C, N, and O. Four general types of cross sections are considered depending on whether the incident ion or target atom is scattered elastically or inelastically. Expressions are given for the Born cross sections to the first two orders in an expansion in ${\beta }^{-2\mathrm{}}$, where $\beta =\frac{v}{c}$ and $v$ is the relative velocity. The atomic form factors and incoherent-scattering functions for the ions ${\mathrm{Cs}}^{+}$ and ${\mathrm{Au}}^{+}$, which are required in order to evaluate the leading-order cross-section parameters, are calculated from the relativistic and nonrelativistic Hartree-Fock wave functions. Results are also given for some of the energy moments of the dipole-oscillator-strength distributions for ${\mathrm{Cs}}^{+}$ and ${\mathrm{Au}}^{+}$, as determined from the ground-state wave functions. These parameters, together with atomic properties taken from the literature for ${\mathrm{Li}}^{+}$ and the various targets considered are then used to evaluate the cross sections to the first two orders in the expansion.

Photon-scattered-atom coincidence measurements, for the collision ${\mathrm{He}}^{+}+{\mathrm{H}}_2\to \mathrm{He}\left(3\mathrm{}{}_{}{}^3{}_{}{}^{}P\right)+{{\mathrm{H}}_2}^{+}$, both with and without linear polarization analysis, have been made for small-angle scattering of He at 1.5 and 3.0 keV. A very large polarization for the $3{}_{}{}^3{}_{}{}^{}P$ to $2{}_{}{}^3{}_{}{}^{}S$ transition is observed at 1.5 keV, 1.5° and 3.0 keV, 1.25°. Also, the excitation probability exhibits small but definite structure as a function of scattering angle.

The $K\alpha$ satellite spectra of Si in the gases Si${\mathrm{H}}_4$ and Si${\mathrm{F}}_4$, of S in the gases ${\mathrm{H}}_2$S, S${\mathrm{O}}_2$, and S${\mathrm{F}}_6$, of Cl in the gases HCl and ${\mathrm{Cl}}_2$ and in the liquid C${\mathrm{Cl}}_4$, and of gaseous Ar produced by bombardment with oxygen ions were examined in detail and compared with the spectra of solid compounds obtained in a previous study. It was found that the apparent average state of $L$-shell ionization at the time of $K\alpha$ x-ray emission does not depend upon the physical state, but upon the availability of electrons from nearest-neighbor atoms. The $K\alpha$ satellite energies in conjunction with Hartree-Fock calculations provide a measure of the average numbers of missing $M$-shell electrons, and this information further supports the conclusion that interatomic electron transfer dominates the fast rearrangement occurring prior to $K\alpha$ x-ray emission in atoms having highly depleted $M$ shells. Various $K\alpha$ satellite peak-broadening mechanisms were considered and it was found that broadening due to the distribution of $M$-shell vacancies accounts for a significant portion of the total peak width.

Cross sections for electron-N${\mathrm{H}}_4^{+}$ recombination have been measured for an electron energy range of $0.065\le E\le 2.00$ eV using a trapped-ion technique with an ion temperature of ∼340 K. The cross section $\sigma$ deduced from these measurements can be represented by $4.30\times {10}^{-16\mathrm{}}{E}^{-1.40\mathrm{}}$ for $0.050\le E\le 0.3$ eV and by $3.73\times {10}^{-17\mathrm{}}{E}^{-3.43\mathrm{}}$ for $E>\mathrm{}0.3\mathrm{}$ eV. There is a total uncertainty in the measurements of ±60%. Using a Maxwellian electron-velocity distribution, recombination-rate coefficients were calculated and compared with measurements of Huang, Biondi, and Johnsen.

Differential cross sections for the ${\mathrm{He}}^{+}$-He collisions have been measured at eleven laboratory energies in the range 0.4-30 eV and laboratory scattering angles from 2 to 90 deg using a crossed-beam apparatus. Data are presented as absolute differential cross sections in the center-of-mass coordinate system as functions of relative translational energy and center-of-mass scattering angle. The experimental data required to establish the instrumental characteristics accurately at each energy are presented. These data will allow theoretical differential cross sections to be convoluted with the distributions characteristic of our experiments and should allow a rather precise determination of the long-range portion of the ${\mathrm{He}}^{+}$-He interaction potentials. Preliminary interpretation of the rainbow scattering data gives a dissociation energy $D_e$ for the $X{}_{}{}^2{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ state of ${\mathrm{He}}_2^{+}$ of 2.58±0.05 eV. This value is in good agreement with an earlier experimental result and with the most recent ab initio calculations. At relative collision energies above 1 eV, total charge-exchange cross sections obtained by integrating differential cross sections over the appropriate angular range are in reasonable agreement with previous experimental and theoretical work.

Differential cross sections for ${\mathrm{Ar}}^{+}$-Ar collisions have been measured at four laboratory energies in the range 2.7-20 eV and laboratory scattering angles from 2 to 90 deg using a crossed-beam apparatus. Data are presented as absolute differential cross sections in the center-of-mass coordinate system as functions of relative translational energy and center-of-mass scattering angle. The data presented should be sufficient to allow rather precise determination of the long-range portion of the ${\mathrm{Ar}}^{+}$-Ar interaction potentials. Preliminary interpretation of the rainbow scattering data gives a dissociation energy $D_e$ for the ${}_{}{}^2{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ state of ${\mathrm{Ar}}_2^{+}$ of 1.40±0.05 eV.

A quantum-mechanical study of low-energy inelastic collisions of excited- and ground-state Ne atoms has been undertaken. Close-coupling calculations of cross sections for the transitions ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\to {}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\to {}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}\to {}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$, and ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}\to {}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$ have been performed at collision energies below 3 eV. All cross sections exhibit a rapid falloff at low energies, reflecting barriers and repulsive walls present in the potential curves. Spin-orbit coupling is assumed to be the transition mechanism. Comparison with experimental data suggests that neglected rotational coupling might be important for some of the quite weak transitions at energies below about 0.1 eV.

Oxygen atoms in the ${}_{}{}^5{}_{}{}^{}S_{}^o{}_{}{}^{}$ state and long-lived high-Rydberg states along with carbon atoms in long-lived high-Rydberg states produced by electron-impact dissociation of CO have been observed in a time-of-flight experiment. A number of dissociation limits were deduced and correlated with dissociation processes. Ranges of kinetic energies were established for the various fragments. The observed high-Rydberg features are discussed in terms of the ion-core model. The data are compared with other dissociative-excitation and analogous dissociative-ionization work. Correlation of observed features with molecular dissociative states was made where possible. The fact that dissociatively excited 1356-Å emissions are Doppler broadened and nonthermal in nature was deduced as a result of this work. This conclusion has import for radiative-transfer treatments of the upper atmospheres of Mars and Venus.