A self-consistent method is proposed to treat the small-angle scattering data from polydispersed self-assembled systems. This method presumes a particle-size distribution to fit the scattering data, then calculates the averaged scattering contrast according to the presumed particle-size distribution. The scattering contrast should be independent of concentration, unless the presumed particle-size distribution function is not appropriate. Therefore performing this analysis as a function of concentration provides a way to self-consistently check the presumed distribution function. This method, although indirect, allows one to select a proper particle-size distribution and evaluate the system polydispersity. Seven commonly used distribution functions are discussed. In addition, the explicit forms of the intraparticle structure factor for these seven size-distribution functions were derived.

The mechanism of the normally incident photoinduced current transients in homogeneous nematic samples in an applied dc voltage has been studied theoretically and experimentally. Explicit expressions for the photovoltaic emf are derived using the nonlinear optical Pockel’s effect induced by the Fréedericksz transition. The current transient is elucidated by the effect of dc field-induced ion drift to electrodes based on an equivalent circuit. The experimental results [briefly reported in Phys. Rev. Lett. 63, 555 (1989)] are completely explained by the present theory. According to the theoretical prediction, two methods to enhance the photoinduced current in liquid crystals, light exposure and dye doping, have been found and implemented.

Light-scattering measurements using circularly polarized light in a backscattering geometry are employed to measure the amplitude of fluctuations in two of the five structural modes present in the isotropic phase of chiral liquid crystals. From these measurements, the second-order transition temperatures for all five modes are then calculated. In order to investigate the effect of chirality on the fluctuations, the experiments are performed in various mixtures of the chiral liquid crystal 4’’-(2-methylbutylphenyl)-4’-(2-methylbutyl)-4-biphenylcarboxylate (CE2) and the nonchiral liquid crystal 4-n-pentylbenzenethio-4’-n-heptyloxybenzoate(7¯S5). The results show that fluctuations in the five modes are independent in low-chirality mixtures, and, as predicted by theory, the second-order transition temperatures grow farther apart as the chirality increases. In highly chiral mixtures, fluctuations in the mode with the highest second-order transition temperature deviate from the normal temperature dependence, resulting in a lower second-order transition temperature for this mode. The probable explanation for this is that fluctuations in the structural modes are coupled in high-chirality systems, but at present no theoretical calculations exist.

A controlled perturbation is introduced into the Saffman-Taylor flow problem by adding a gradient to the gap of a Hele-Shaw cell. The stability of the single-finger steady state was found to be strongly affected by such a perturbation. Compared with patterns in a standard Hele-Shaw cell, the single Saffman-Taylor finger was stabilized or destabilized according to the sign of the gap gradient. While a linear stability analysis shows that this perturbation should have a negligible effect on the early-stage pattern formation, the experimental data indicate that the characteristic length for the initial breakup of a flat interface has been changed by the perturbation.

The stability of uniaxial nematic-liquid-crystalline structures in supra-μm-size spherical cavities that impose a weak homeotropic anchoring is studied theoretically. The equilibrium equations are obtained with the minimization of the deformation, surface, and field contributions to the free energy and are solved numerically. The dependencies of the solutions on the ratio of elastic constants ${\mathit{K}}_{33}$/${\mathit{K}}_{11}$, ${\mathit{K}}_{24}$/${\mathit{K}}_{11}$, anchoring strength, and external field strength are discussed, and the stability diagrams with lines of structural (Fréedericksz) transitions are constructed. In the region of strong anchoring and large external field strengths, a triple point, where radial, nonsingular axial, and axial structure with the line defect, is predicted. Particular attention is paid to the inversion point corresponding to the critical-field strength above which radial structure is no longer stable. Two possible methods for saddle-splay elastic constant ${\mathit{K}}_{24}$ determination are suggested.

We study the patterns formed when a reactive fluid with viscosity μ is injected into a two-dimensional porous medium filled with a nonreactive fluid with unit viscosity. We consider the ‘‘mass-transfer limit,’’ where the time scale of chemical reaction between the injected fluid and porous medium is much smaller than the time scale of reactant transport. Also, the surface tension between two fluids is ignored. We formulate three-parameter position-space-renormalization-group equations for this system. We find two crossovers: (i) from the first diffusion-limited-aggregation (DLA) fixed point to the Eden fixed point due to finite viscosity, and (ii) from the Eden to the second DLA due to chemical dissolution. The time evolution between patterns is independent of the injection rate following a trivial rescaling of time. These results are checked by direct numerical simulations. The second crossover is characterized by the crossover radius ${\mathit{r}}_{\mathit{c}}$∼${\mathit{v}}^{\mathrm{-}1/\mathrm{\beta }\mathrm{\phi }}$, where v is the total volume of the injected fluid, and β=3/2, φ≃1.63. We also study the effect of consumption of the reactant, and find that it stabilizes the pattern from a DLA fractal to a compact shape.

A growth model is proposed to simulate the pit formation in pitting corrosion of binary alloys. The metal is modeled by a random binary alloy where one species is easy to dissolve and the other species is hard to dissolve. The pit growth model is described by the anti-diffusion-limited aggregation model. A computer simulation is performed to form the corrosion pit in two dimensions for various composition ratios. It is shown that at the percolation threshold, a structure transition of the corrosion pit occurs between the diffusion-limited corrosion and the dissolution-limited corrosion. Above the percolation threshold of easily dissolving species, the pitting corrosion is controlled by the diffusion of anions and the width w of the pit scales as w≊${\mathit{h}}^{1.86\pm 0.05}$ where h is the depth of the pit. Below the percolation threshold, the pitting corrosion is controlled by dissolution of metal and the width w of the pit is proportional to the depth h of the pit.

Bounds on diffusion-limited processes, including chemical reactions in two-phase media and permeation of fluid through porous media, depend in general upon the entire set of correlation functions of the material. We show that the three-point bounds of this kind are expressed economically by writing them in terms of a set of material correlation functions defined previously [J. Given and G. Stell, J. Chem. Phys. 94, 3060 (1991)]. We develop a linear, three-point hypernetted-chain integral equation satisfied by the material correlation functions; this is an extension of an equation applied by Pinski and Campbell [Phys. Rev. A 33, 4232 (1986); J. Phys. (Paris) Colloq. 39, C6-233(1979)] to problems in atomic physics. This equation applies to a random material with quite general correlations; it is also easy to solve. We obtain the three-point bounds on the diffusion-limited trapping rate for two commonly studied models of random aggregates; these give a substantial improvement over the corresponding two-point bounds. For the diffusion-limited-reaction rate, we find that the three-point extension of a bounding process first used by Doi [J. Phys. Soc. Jpn. 40, 507 (1976)] gives the tightest lower bound yet attained on the trapping rate.

In situ observations using interference contrast microscopy have been carried out in an electrochemical deposition system of a ${\mathrm{ZnSO}}_4$ solution film. Concentration oscillation around a growing dendrite tip, which is illustrated by local contrast oscillation, has been found to correspond to sidebranching of the dendrite. A model concerning the competition between diffusion of cations and growth of dendrites has been used to explain this oscillation phenomenon. It appears that this concentration field oscillation in front of the dendrite tip plays an important role in sidebranching.

An approximate treatment of the Boltzmann collision integral for electrons in a gas, valid for small fractional average energy loss, is studied. It is essentially a Fokker-Planck expansion in energy space, including mean energy loss (dynamical friction) and energy straggling (coefficient of diffusion). When applied to electron swarms in weakly ionized gases, treating angle variables in the two-term Legendre series, there results a useful, physically meaningful, differential equation for the time evolution of the energy spectrum in a time-dependent electric field. Elastic scattering, and inelastic and superelastic energy-transfer collisions are included. The time-independent solution in a constant field is a simple approximate expression for the steady-state energy spectrum of swarm electrons. The physical meaning of its functional form is made clear by showing its relation to ordinary diffusion-convection theory. Previous spectra by Pidduck [Proc. R. Soc. London Ser. A 88, 296 (1913); Proc. London Math. Soc. 15, 89 (1916); Q. J. Math. 7, 199 (1936)]; Druyvesteyn [Physica 10, 61 (1930)]; Davydov [Phys. Z. Sowjetunion 8, 59 (1935)]; Morse, Allis, and Lamar [Phys. Rev. 48, 412 (1935)]; Chapman and Cowling [The Mathematical Theory of Non-Uniform Gases, 2nd ed. (Cambridge University Press, Cambridge, 1952), p. 350]; and Wannier [Am. J. Phys. 39, 281 (1971)] are special cases.

The reasons for the inadequacy of the continuous-slowing-down approximation (CSDA) become apparent. The new spectrum is exact in the limit of small quantum transition energies. It is further shown that the CSDA violates detailed balance. Consequences of detailed balance on the loss function and related functions are investigated, and the Boltzmann H theorem is studied. During non-steady-state behavior, collisions may increase or decrease the swarm entropy, but the effect of the electric field is always to increase entropy. The spectrum is used with experimental cross sections to compute transport coefficients in ${\mathrm{O}}_2$ and ${\mathrm{N}}_2$, in both of which fractional average energy loss is acceptably small over most energy ranges. Agreement with compiled swarm data is excellent over more than four orders of magnitude in E/N for most coefficients, except at certain energies in ${\mathrm{N}}_2$ that strain the approximation’s validity. In the absence of an electric field, the inclusion of energy straggling provides a treatment of spectral relaxation valid for arbitrary energies that is an improvement over common mean-stopping-power formulas.

The general expression for the second-order wave energy of a Vlasov-Maxwell system derived by Morrison and Pfirsch [Phys. Rev. A 40, 3898 (1989); Phys. Fluids B 2, 1105 (1990)] is evaluated here for the case of electrostatic perturbations in a magnetized, homogeneous plasma. It is again shown that negative-energy waves (which could become nonlinearly unstable and cause anomalous transport) exist for any deviation from monotonicity and/or any (however small) anisotropy in the equilibrium distribution function of any of the particle species. The partly unexpected and particularly interesting feature of the results is that, contrary to the proof of Morrison and Pfirsch, no restricting condition has to be imposed on the perpendicular wave number ${\mathit{k}}_{\mathrm{\perp }}$ of the perturbation (i.e., large ${\mathit{k}}_{\mathrm{\perp }}$ is not required). Finite-gyroradius effects are therefore not expected to improve the situation. Anisotropy alone would, however, impose a restriction on ${\mathit{k}}_{\mathit{z}}$, the parallel wave number, relating it to the gyroradius. As far as distribution functions with ${\mathit{v}}_{\mathit{z}}$(∂${\mathit{f}}_{\nu }^{\mathrm{}\left(0\right)\mathrm{}}$/∂${\mathit{v}}_{\mathit{z}}$)>0 in some region of v space are concerned, however, this result agrees with a result found by Pfirsch and Morrison within the framework of drift-kinetic theory.

The structure of rf glow discharges in Ar at 13.56 MHz is described, making use of a relaxation continuum model. The model includes consideration of the relaxation kinetics for the momentum and energy of charged particles. The discharge structure and plasma property are modeled using molecular quantities (i.e., collision cross sections, radiative lifetimes, etc.) and macroscopic transport quantities for charged particles. Spatiotemporal distributions for the field and the net production rate and density of particles are studied in detail in the bulk plasma as well as the ion sheath under periodic steady-state conditions. In particular, the temporal profiles including phase shift and nonlinear variation with respect to the applied-voltage wave form are discussed. Each type of excited atom shows a different spatiotemporal density distribution due to different loss mechanisms, although the production profiles are similar. That is, Ar${(}^3$${\mathit{p}}_5$) is controlled by a radiative decay and quenching in collisions with Ar${(}^1$${\mathit{S}}_0$), while Ar${(}^3$${\mathit{P}}_{0,2}$) is controlled by diffusion. The validity of the relaxation continuum model is investigated by the observation of external electric properties and the measurement of spatiotemporally resolved optical emission from the rf glow. Quantitative comparison of the results from theory and experiment supports the validity of the relaxation continuum model. The relaxation continuum model is simple and physically reasonable.

Several types of x-ray diagnostics, such as x-ray energy spectrum analyses, x-ray absorption methods, and x-ray tomographic reconstructions using various types of x-ray detectors (i.e., a Si(Li), a pure Ge, a NaI(Tl), Si surface-barrier detectors, as well as microchannel plates) have been employed for obtaining various shapes of electron-velocity distribution functions as well as their spatial profiles: (i) a plateau-shaped electron-velocity distribution function in the plug region, supporting a scaling theory between thermal-barrier potentials and ion-confining potentials (Cohen’s strong electron-cyclotron heating theory); (ii) mirror-trapped 60-keV relativistic Maxwellian electrons in the thermal-barrier region; and (iii) two-component Maxwellian electrons in the central cell have been observed in the tandem-mirror GAMMA 10. During experiments with thermal barriers, it has been observed that the bulk-electron temperature and its evolution with time in the central cell are quite different from the electron energy and its temporal evolution in the plug region. These differences as well as the different shape of the electron-velocity distribution function in each region have clearly demonstrated the existence of a thermal-isolation effect due to the thermal barriers between the central cell and the plug region. X-ray tomography data in these three regions have shown good axisymmetric radial profiles peaked on the magnetic axis; this axisymmetric shape is useful for reducing nonambipolar radial particle losses. Furthermore, the validity of the Pastukhov theory for electron-energy confinement that is enhanced due to the formation of thermal-barrier potentials has been demonstrated using the electron-energy balance analyses. These data have represented a good electron-energy confinement capability due to the formation of thermal barriers.

The evolution of spatially periodic patterns lateral to the current flow in a dc gas-discharge system consisting of a semiconductor and a metal electrode is reported. To explain these experiments a previously introduced two-layer model is used, which can be described by a system of reaction-diffusion equations. The discretized form of this system of equations can be interpreted as an equivalent electric circuit. Applying the center manifold theory to the system of equations, the experiments can be explained in terms of a bifurcation in that the system is undergoing a Turing diffusion instability. For the model, as well as in the experiments, the bifurcation from a homogeneous state into a spatially periodic structure is obtained by varying the externally applied voltage, the load resistance, or the width of the discharge slit. In connection with the application of the model to the gas-discharge system, the j(U) characteristic is measured for the whole discharge space and the positive column.

Laser energy absorption and transmission by the anomalous skin effect has been studied numerically in hot plasmas, assuming a step-density profile and normal incidence. Our calculations are compared both to the normal skin effect and to that of Weibel [Phys. Fluids 10, 741 (1967)], who neglected the variation of the collision frequency with velocity. We find that the penetration of the field is considerably enhanced compared to that of Weibel, which is already far above that of the normal skin effect. This enhanced penetration means that fields of amplitudes ${10}^{\mathrm{-}3}$–${10}^{\mathrm{-}2}$ of the incident amplitude exist at several normal skin depths beyond what would be expected from normal skin-effect calculations. Only in very hot (${\mathit{T}}_{\mathit{e}}$>1 keV) and low-Z plasma does this absorption mechanism dominate, and it is then at most several percent. Field transmission through a thin target (about ten skin depths) is far higher than predicted on the basis of the normal skin effect: this could provide a possible means of observing the effect.

An efficient procedure for the elimination of transverse beam instabilities is derived using a simple model for the beam-ion interaction. The method consists in sweeping periodically the beam-shaking frequency near the hysteresis associated to a strongly nonlinear resonance. Efficient shaking is then obtained, as arbitrarily large ion amplitudes can be induced.

This paper studies the behavior of the moments of a particle distribution as it is transported through a Hamiltonian system. Functions of moments that remain invariant for an arbitrary linear Hamiltonian system are constructed. These functions remain approximately invariant for Hamiltonian systems that are not strongly nonlinear. Consequently, they can be used to characterize the degree of nonlinearity of the system.

A generic nonintegrable Hamiltonian system is characterized by a critical energy above which chaotic motion sets in. A general method of finding a lower bound to this critical energy that does not require a solution of the equations of motion is discussed. Below the critical energy, the motion is regular everywhere on the energy surface and no instabilities can develop. The method is applied to a practical situation encountered in modern large accelerators, where the transverse motion of the particles in an arrangement of quadrupole, sextupole, and octupole magnetic elements may become chaotic. The chaotic dynamical aperture of the beam is calculated as a function of a dimensionless strength parameter. The estimated critical energy is compared with that obtained from detailed studies of the Poincaré sections of the above system at various energies.

The electrodynamics of a relativistic electron beam passing through a nonrelativistic plasma embedded in an axial-wiggler magnetic field has been worked out in the framework of nonequilibrium statistical mechanics as developed by Prigogine and co-workers. The emerging radiation has a synergic frequency that is a combination of the original frequency, the plasma frequency, and the cyclotron frequencies due to the axial and wiggler magnetic fields. An exact summation of the relevant diagrams in the Dyson series corresponding to the self-consistent-field approximation has brought out non-Markovian nonlinear aspects of optics. The change in polarization has been evaluated due to the presence of magnetic fields. The axial and wiggler fields remove the degeneracy in the electronic states, and the summation over the Landau levels discriminates between the spin-up and spin-down species. The measurement of the energy loss enables one to make an estimate of the population of these species.

The amplitude dependence of the phase shift originating in an overtaking soliton-soliton collision is investigated for solitons that can be described with a Korteweg–de Vries equation and a nonlinear Schrödinger equation. The size dependence of the interaction regime is also amplitude dependent. Laboratory and numerical experiments are compared with a simple model for the elastic collision.