The Brownian-motion Fokker-Planck equation describing the velocity-distribution function of a collection of Brownian particles suspended in a rarefied carrier gas in equilibrium is generalized to the case of a nonuniform suspending gas. The gas-particle collision integral appearing in the Boltzmann equation is expanded in powers of the ratio of masses between gas molecules and particles, and the gas-distribution function is taken as the first approximation to the Chapman-Enskog expression. Also, the characteristic width of the particle-distribution function is assumed to be of the order of its value in equilibrium. The modified collision term obtained involves a new force proportional to the local temperature gradient, and closely related to the Chapman-Enskog thermal-diffusion effect. The contributions from the nonhomogeneity of the gas-velocity field is small, but introduces new "shear forces" proportional to the fluid-particle relative velocity and the local gas traceless symmetrized velocity-gradient tensor. The new equation can be used to describe the non-equilibrium dynamics of gas mixtures with disparity of masses when the heavy species is dilute and the light is not too far from equilibrium.

In this paper we present a hydrodynamic model to study the features of the behavior of a fluid which is brought to a stationary state by the action of an external gradient. Two cases are considered, namely, the action of a thermal gradient and a constant shear rate. The hydrodynamic equations used are based on constitutive equations constructed from Curie's principle. The external gradients imposing the constraints as the steady state are regarded as quantities of zeroth order. We examine two features of the fluid's behavior, namely, the sound absorption and the light scattering. In the former case we show how the Stokes-Kirchhoff formula is modified by the presence of the gradients. This modification suggests an experimental verification of this model which is independent of the magnitude of the real wave vector. In the latter case we compute the Brillouin-Rayleigh spectra and obtain in both cases the same shift in the Brillouin peaks already predicted by others. However, we also predict a small charge in the intensity of the peaks because of the modification of the sound-absorption coefficient. The essential result in this calculation is the shift of the Rayleigh peak arising from the entropy flow in the case of the thermal gradient and an entropy-production term in the case of the constant rate of shear. In both cases the order of magnitude of this correction in terms of the wave vector is the same as the terms responsible for the shift in the Brillouin peaks. Finally, we also compare our results with those obtained by others.

A theory of the smectic-$A-C$ transition in which fluctuations in both the elastic and director degrees of freedom are treated is developed. It is argued that in zero magnetic field the asymptotic critical behavior belongs in the universality class of the threedimensional $\mathrm{XY}$ model, in agreement with earlier studies which did not treat fully both elastic and director fluctuations. For most experimental situations the true critical region is unobservably narrow whereupon a simple-harmonic approximation adequately describes the transition; in this approximation the static structure factor exhibits a crossover from algebraic decay characterized by a nonuniversal exponent ${\eta }_A$ in the $A$ phase to algebraic decay characterized by a different nonuniversal exponent ${\eta }_{\mathrm{AC}}$ at the critical temperature $T_c$. The splay elastic constant $K_1$ is $<\infty$ and $=\infty$ in the $A$ and $C$ phases, respectively, but does not exhibit diverging pretransitional fluctuations in the $A$ phase even arbitrarily near $T_c$; it jumps discontinuously at $T_c$. A strong magnetic field stabilizes long-range positional order in both the $A$ and $C$ phases. There is, nevertheless, no long-range order at $T_c$, where the density correlation function $〈m\left(\overrightarrow{\mathrm{x}}\right)m\left(\overrightarrow{\mathrm{o}}\right)〉$ decays algebraically; schematically, $〈m\left(\overrightarrow{\mathrm{x}}\right)m\left(\overrightarrow{\mathrm{o}}\right)〉\sim x^{-(2+\eta )}$ at $T_c$ in the harmonic theory, while $m\left(\overrightarrow{\mathrm{x}}\right)$ vanishes like $t^{\beta }$, where $t$ is the reduced temperature. The exponents $\beta$ and $\eta$, though nonuniversal, satisfy the scaling law $4\beta =2+\eta$. These finite-field results are modified at large distances (say $x\gtrsim {10}^5$ Å) by the inclusion of anharmonic terms. We find $〈m\left(\overrightarrow{\mathrm{x}}\right)m\left(\overrightarrow{\mathrm{o}}\right)〉\sim x\exp [-(2+\tilde{\eta }\left){\left|\ln x\right|}^{\frac{1}{37}}\right]$ at $T_c$, while $〈m\left(\overrightarrow{\mathrm{x}}\right)〉\sim t^{\tilde{\beta }{\left|\ln t\right|}^{\frac{1}{37}}}$ as $t\to 0$. Here $\tilde{\beta }$ and $\tilde{\eta }$ are nonuniversal but satisfy $4\tilde{\beta }=2+\tilde{\eta }$. At $T_c$, $K_1$ and $B$ (the compressional elastic constant), respectively, diverge and vanish like ${\left(\ln k\right)}^{\frac{2}{37}}$ and ${\left(\ln k\right)}^{-\frac{4}{37}}$ as the momentum $k\to 0$. The exponents $\frac{1}{37}$, $\frac{2}{37}$, and $\frac{4}{37}$ are universal.

Published carbon dioxide specific-heat data are analyzed to determine the behavior of ${\left(\frac{{\partial }^{2}P}{\partial T^2}\right)}_V$ and ${\left(\frac{{\partial }^2\mu }{\partial T^2}\right)}_{\rho }$ near the critical point and to compare this behavior to the predictions of the linear model. In the two-phase region $\frac{d^{2}P}{d{T}^2}$ is found to diverge as the critical point is approached and $\frac{d^2\mu }{d{T}^2}$ is found not to diverge. This experimental behavior is consistent with the linear model. In the homogeneous region it is not possible to determine from available data whether ${\left(\frac{{\partial }^{2}P}{\partial T^2}\right)}_V$ or ${\left(\frac{{\partial }^2\mu }{\partial T^2}\right)}_{\rho }$ or both diverge on the critical isochore. However, it is possible to qualitatively describe the thermodynamic paths where these quantities are zero. Again the experimental behavior of ${\left(\frac{{\partial }^{2}P}{\partial T^2}\right)}_V$ and ${\left(\frac{{\partial }^2\mu }{\partial T^2}\right)}_{\rho }$ so described agrees with linear-model predictions provided appropriate analytic backgrounds are used. The analytic background found for $\frac{d^2\mu }{d{T}^2}$ in the inhomogeneous region is not the same as the one found for ${\left(\frac{{\partial }^2\mu }{\partial T^2}\right)}_{\rho }$ in the homogeneous region.

We study the relaxation of nonequilibrium concentration fluctuations in binary-liquid mixtures near but above the critical consolute temperature. We show that if the strength of the fluctuations is far enough above the thermal level, hydrodynamic effects cause these fluctuations to migrate towards the smallest wave numbers. They are dissipated only in an exceedingly long time at the largest available scale. The structure factor of the fluctuations obeys a scaling law similar to the one observed in experiments of spinodal decomposition. Experimental tests of our result are suggested and their possible relevance to the theory of spinodal decomposition briefly discussed.

With illumination by ultraviolet light in the 280-390-nm wavelength region, copious nucleation is observed in supersaturated vapors containing carbon disulfide. This photoinduced nucleation cannot be explained by conventional nucleation theories nor by the photochemistry of C${\mathrm{S}}_2$. We suggest that the nucleation is initiated by the clustering of photoaffected C${\mathrm{S}}_2$ molecules, and that these molecules may be stable complexes formed by an interaction of the photoexcited C${\mathrm{S}}_2$ and host vapor molecules.

A time-dependent Kohn-Sham theory is presented for obtaining the time-dependent density which has a periodic dependence on time. A set of coupled single-particle equations $-\frac{1}{2}{\nabla }^2{\chi }_i+v_{\mathrm{eff}}{\chi }_i={\epsilon }_i{\chi }_i$ and $\frac{\partial {\chi }_i^2}{\partial t}+\overrightarrow{\nabla }·\left({\chi }_i^2\overrightarrow{\nabla }{S}_i\right)=0\mathrm{}$ are obtained. The ${\chi }_i(\overrightarrow{\mathrm{r}},t)$ and $S_i(\overrightarrow{\mathrm{r}},t)$ are the phase and amplitude, respectively, of the time-dependent Kohn-Sham orbitals, $v_{\mathrm{eff}}(\overrightarrow{\mathrm{r}},t)$ is the time-dependent Kohn-Sham effective potential, and ${\epsilon }_i(\overrightarrow{\mathrm{r}},t)=-\frac{\partial S_i(\overrightarrow{\mathrm{r}},t)}{\partial t}$. The density $\rho (\overrightarrow{\mathrm{r}},t)$ is equal to the sum of the squares of the ${\chi }_i(\overrightarrow{\mathrm{r}},t)$.

The energy eigenvalues of the exponential cosine screened Coulomb potential for various eigenstates are accurately determined within the framework of the hypervirial Padé scheme.

Expect for very low velocities of incoming particles, ionizing collisions produced by a wide variety of charged particles (electrons, protons, $\alpha$ particles,...) are shown to be treatable by use of the same formalism. Based upon a slight modification to common impulse assumptions, the proposed treatment makes use of the target Compton profile, which further provides a simple method for the normalization of experimental results.

In a two-photon resonant three-level system, population can be trapped in a noninteracting superposition state. We demonstrate that this phenomenon persists in a multilevel system in which the final state is replaced by a number of nondegenerate states. The magnitude of the trapped population depends only on the matrix elements between the pair of two-photon resonant levels and the intermediate level. A dressed-state approach is also considered.

We calculate the internal energy of the classical one-component plasma using a Monte Carlo technique for 128, 250, 432, 686, and 1024 particles for $1<\mathrm{}\Gamma <300\mathrm{}$ in order to determine the effect of a differing number of particles on the thermodynamics. By fitting the internal energy to a function of $\Gamma$ and $N$ (the particle number), we find the free energy for both the liquid and solid for an infinite number of particles.

It is shown that the non-Maxwellian heating by inverse bremsstrahlung which results in a truncation of the tail of the electron-distribution function is responsible for a reduction of the thermal conductivity in a smooth temperature gradient by a factor of 3 to 4 from the Spitzer-Härm conductivity. In a steep temperature gradient, the reduction is still a factor of about 1.5 from the recent nonlinear theories based on a Maxwellian electron-distribution source.

The conventional picture of chaos consists of the coupling of a cascade of Fourier modes. Instead, we find that deep in the chaotic region of the driven, damped anharmonic oscillator, the states can be characterized very simply as the spontaneous generation of Fourier modes of random phases and small frequencies. This leads to a new and relatively simple way to calculate a measure to convert time averages to ensemble averages, as is normally done in statistical mechanics.

The steady-state equations are discussed. It is pointed out that the equality of paired probability flows is not a necessary condition for steady-state operation.

Electron energy-loss spectra and differential cross sections are reported for inelastic scattering from Zn II. Measurements were carried out in a crossed electron-beam—ion-beam apparatus, at incident electron energies of 30, 40, 50, 60, 75, 85, and 100 eV, and at a scattering angle of 14°. The present results are the first reported measurements of inelastic electron scattering from an ion.

Angular distribution and spin polarization of Auger electrons following photoionization and photoexcitation are presented in concise parametrized expressions. The striking similarity with corresponding expressions for photoelectrons enables one to apply kinematic properties known for photoelectrons in analyzing Auger electrons.

Photoemission spectra from autoionizing states in the presence of a strong laser field are calculated. Such spectra have very sharp features near "confluence" (at the Fano minimum) and are very similar, in structure, to photoelectron spectra. The total intensity, as a function of laser intensity or detuning, has a peak near confluence, which can be used, among other things, for the accurate determination of Fano asymmetry parameter $q$.

Three-photon ionization of xenon, resonantly enhanced by the $\mathrm{Xe}6p{\left[\frac{1}{2}\right]}_0$ state at 80 119 ${\mathrm{cm}}^{-1\mathrm{}}$, is observed when a xenon atom simultaneously absorbs one ArF (51733-${\mathrm{cm}}^{-1\mathrm{}}$) and two XeF (28 482-${\mathrm{cm}}^{-1\mathrm{}}$) photons. For ArF and XeF laser intensities in the 1-100-MW ${\mathrm{cm}}^{-2\mathrm{}}$ range, the peak electron density is more than an order-of-magnitude larger than that obtainable by the nonresonant, two-photon ionization of Xe by ArF. The spontaneous radiative lifetime of the $6p{\left[\frac{1}{2}\right]}_0$ level and the rate constant for quenching of the state by background Xe atoms were measured to be 29±4 ns and 4±1 × ${10}^{-10\mathrm{}}$ ${\mathrm{cm}}^3$${\mathrm{s}}^{-1\mathrm{}}$, respectively. Excitation of the $6p{\left[\frac{1}{2}\right]}_0$ state is both selective and intense and has resulted in lasing on the $6p{\left[\frac{1}{2}\right]}_0\to 6s{\left[\frac{3}{2}\right]}_1$ transition at 828.0 nm, which represents the first optically pumped rare-gas laser.