In the free-electron laser the resonant interaction between the radiation field and electron beam can result in radiation focusing (optical guiding). If the centroid of the electron beam is transversely displaced off axis, the radiation field, under certain conditions, will follow and be steered by the electron beam. The effect of a spatial modulation on the electron-beam envelope can also modify the propagation characteristics of the radiation. These and other phenomena are analytically and numerically studied using a novel source-dependent Laguerre-Gaussian modal representation of the fully three-dimensional radiation field. Unlike the vacuum Laguerre-Gaussian modal expansion, the longitudinal spatial dependence of the radiation waist and curvature are determined and characterized by the source term in the wave equation. Among the advantages of this general source-dependent expansion approach is that few modes are needed to accurately describe the radiation. Hence, fast and accurate numerical solutions of the fully three-dimensional free-electron laser problem can be obtained over distances of many Rayleigh lengths. Furthermore, this expansion enables us to drive an envelope equation for the radiation beam as well as an expression for the centroid of the beam.

Theoretical predictions are given of the spatiotemporal distribution of charged-particle densities and electric field in a radio-frequency glow discharge. The model is based on a numerical solution of the continuity equations for electrons, positive ions, and negative ions coupled with Poisson’s equation. The influence of frequency and gas composition (electropositive versus electronegative gas) on the discharge properties is analyzed and compared (qualitatively) with experimental results. In the case of electronegative gas mixtures, and at low frequencies (of the order and less than the ion plasma frequency), the numerical model predicts the formation of double layers at the plasma-sheath boundaries, in agreement with recent experimental measurements based on spectroscopic plasma-diagnostic techniques.

Results are reported of extensive molecular-dynamics simulations of a two-dimensional Coulomb gas made up of finite-size ions held fixed on the sites of a triangular lattice, and of classical electrons moving in the periodic field of the ions. The fixed-ion model maps the dielectric-plasma (or Kosterlitz-Thouless) transition of the Coulomb gas onto a delocalization transition of the electrons. The transition is characterized by a number of static and dynamic ‘‘diagnostics.’’ As the temperature is increased in the dielectric phase, electron self-diffusion and electrical conductivity set in at a density-dependent threshold temperature $T_1$. The breakup of ion-electron pairs is signaled by a sharp peak in the specific heat at a temperature $T_2$>$T_1$. As $T_1$ is approached from above in the plasma phase, the screening length diverges. In the high-temperature plasma, the frequency of the long-wavelength collective charge oscillation (plasmon) mode decreases with T and the mode becomes overdamped by ion-electron recombination well before the threshold $T_1$ is reached. The dispersion ω(k) of the mode exhibits an unexpected oscillatory behavior. The temperatures $T_1$ and $T_2$ increase as the density decreases and there is strong evidence that the low-density limit of the reduced temperatures $T_1^{\mathrm{*}}$=$k_B$T/$e^2$ and $T_2^{\mathrm{*}}$ is 1/2 as compared with the value (1/4 expected for the mobile-ion case.

We examine the McLean-Saffman equations for viscous fingering in the limit where the finger fills almost completely the Hele-Shaw channel (λ≃1). We find an infinite countable set of solutions. For each branch of solutions, λ increases toward 1 as (U-$U_n^{\mathrm{*}}$${)}^{3/2}$, when the velocity U of the finger approaches a lower value $U_n^{\mathrm{*}}$ that we calculate. We then discuss the connections of these results with directional solidification at small Péclet numbers. Our analysis does not reveal any sign of wavelength selection for steady-state cells by a solvability condition, contrary to recent numerical findings.

We have measured the electric field distribution across the high-voltage gap of a magnetically insulated intense ion beam diode by observing the Stark shift of line emission from ions accelerating in the gap. The measured distribution yielded the time dependence of the actual diode gap. We observed rapid gap closure early in time resulting from expansion of the electric-field-excluding electrode plasmas in the diode. This can contribute substantially to the previously observed ion-current-density enhancements over the values calculated based upon the mechanical diode gap. Assuming planar geometry the electric field distribution was used to determine the electron density across the diode gap. This was compared with one-dimensional Brillouin-flow model calculations using the actual acceleration gap. The electrons were observed to spread towards the anode beyond the region of the theoretical electron sheath. The electron presence near the anode was further enhanced when a cathode vane was used. In addition, an increase in the total electron number in the gap relative to analytic theory was inferred. This serves to explain the enhancement of the measured ion current density over the one calculated with use of the actual diode gap that we observed for the configurations tested.

A simple model for the displacement of a viscous fluid by a nonviscous fluid in a porous medium has been developed. This model is based on the Witten-Sander model for diffusion-limited aggregation and employs a multifractal lattice to represent both short- and long-range heterogeneities in the porous medium. It is shown that such heterogeneities can have important effects on both the local and global structure of the displacement pattern. Long-range heterogeneities enhance the effects of the outer boundary on the overall shape of the cluster and shorter-range heterogeneities reduce the thickness of the ‘‘viscous finger’’ generated by the displacement process. Our results indicate that the effective fractal dimensionality of the displacement patterns is decreased as the magnitude of the heterogeneity is increased.

A prediction was made recently that a quantum-mechanical correction must be added to the Einstein relation for electrons in some dense gases. While we are awaiting a definitive measurement of De/μ to test this prediction, some empirical support has been found for the prediction by extrapolating the Milloy-Crompton results [Aust. J. Phys. 30, 51 (1977)] in dense argon to E/N=0. The extrapolation is done by first deriving a two-term expansion of De/μ as a function of (E/N${)}^2$. Significantly, the coefficient of the second term is not a constant but depends (indirectly) on E/N, decreasing for small values because of the varying slope of the collision cross section. The result of fitting this expansion to the measurements is an extrapolation which tends to favor the prediction. Where a straightforward (but incorrect) quadratic fitting of De/μ tends inexorably toward the accepted value of kT=25 meV, thus apparently disproving the prediction, the presently derived curve bends upward for decreasing E/N and extrapolates to a higher value, much closer to the predicted 32 meV. Direct measurements are still much needed, but I believe this result foreshadows their direction.

Explicit expressions for the heat and momentum transport are given for a gas in a stationary state with temperature and velocity gradients. The results are obtained from a formally exact analysis of the normal solution to the Bhatnagar-Gross-Krook model for the nonlinear Boltzmann equation, and are not restricted to small gradients. The irreversible momentum and heat fluxes are found to be nonanalytic functions of the velocity gradients, indicating that the Chapman-Enskog expansion does not converge for this state. However, these fluxes are analytic in the temperature gradients; in particular, the heat flux is simply proportional to the temperature gradient so that Fourier’s law applies even for large gradients. The space dependence of the thermodynamic and velocity fields is determined as a function of the interaction potential, and the results for Maxwell molecules and hard spheres are compared.

We introduce the definitions of generalized dimensions for fractal sets characterized by logarithmic and additive corrections to the measure. The definitions are simple to compute. This framework is used to analyze several classes of sets with partial scaling symmetry. This includes an important class of sets having scaling symmetry with respect to a fixed point. We give two important physical examples of the above class where our analysis is useful.

Numerical experiments on two-dimensional thermosolutal convection reveal oscillations in the form of traveling, standing, modulated, and chaotic waves. Transitions between these wave forms and steady convection are investigated and compared with theory. Such rich nonlinear behavior is possible in fluid layers of wide horizontal extent, and provides an explanation for waves observed in recent laboratory experiments with binary fluid mixtures.

The molecular reorientation in nematic phases formed by rigid molecules with arbitrary shape is considered theoretically. In the framework of the small-step rotational diffusion model, the orientational correlation functions are obtained as fast converging series of exponentials. For cylindrical molecules our solutions are reduced to the results already reported in the literature. For molecules with effective $D_{2h}$ symmetry we demonstrate that at reasonable values of the order parameter 〈$D_{02}^2$〉 the influence of the biaxiality on some correlation functions is significant and cannot be neglected.

In this paper we apply the theoretical results presented in the preceding paper to the case of infrared spectroscopy. A theoretical evaluation of the correlation functions connected with spinning and tumbling molecular fluctuations and comparison with the experimental correlation functions obtained from band-shape analysis of infrared dichroic spectra in ordered nematic phases allows us to evaluate the effect of molecular biaxiality (i.e., deviations from cylindrical symmetry) on polarized ir absorption spectra. From the fitting of our data we are able to recover the ‘‘true’’ behavior of the spinning diffusion coefficients (the tumbling coefficients are not affected by molecular biaxiality as predicted by the theory) and obtain a semiquantitative estimate of the strength of the biaxial term in the nematic potential.

In the case of nonlinear dielectric effect (NDE), temperature measurements of doped binary solutions of critical concentration do not enable a definitive determination of the influence of Fisher’s renormalization on the form of the critical effect. For this reason it was attempted to achieve this from isothermal studies of changes in value of the NDE as a function of a third constituent added to the critical solution. It was ascertained that its value is largely independent of the direction of approach to the critical point. Results of an initial investigation of this type are presented here.

A comprehensive study of the thickness dependence of the phase diagram of freely suspended films of the liquid crystal 4-n-heptyloxybenzylidene-4-n-heptylaniline (7O.7) between 50 and 69 °C is reported. In thick films (thicker than about 300 layers and characteristic of bulk samples) there is a low-temperature crystalline-G phase followed by five crystalline-B phases with different stacking arrangements at higher temperatures. In thinner films there are two additional crystalline-B phases and two tilted hexatic phases, smectic-F and smectic-I, which do not appear in bulk samples. The in-plane and interlayer correlations in the tilted hexatic phases are anisotropic with a clear dependence on the molecular tilt direction; the in-plane correlations are more developed (longer range) perpendicular to the molecular tilt direction and the interlayer correlations are more developed parallel to the tilt direction.

Detailed x-ray scattering studies of the modulated phases of 4-n-heptyloxybenzylidene-4-n-heptylaniline (7O.7) are reported. In addition to the previously identified one-dimensional modulation that appears at the transition from the crystalline-B (CrB) phase with ABAB stacking to the CrB phase with orthorhombic stacking, there are two other modulation structures each of which contains two noncolinear modulations. The modulations are explained in terms of a phenomenological model in which domains are separated by walls for which the intralayer packing is identical to the lower-temperature crystalline-G phase. This model yields the measured values for both the modulation wave vector and the lattice distortion.

A method based on the ideas of collective motion is applied to Fokker-Planck dynamics. The usual diagrammatic techniques used in stochastic dynamics are enlarged in order to deal with the nonpolynomial Lagrangians that appear in the theory. The technique is tested and applied to the case of a self-sustained sinusoidal oscillator whose statistical behavior is well understood: the Poincaré oscillator.

The phase diagram of a system of hard parallel spherocylinders with length-to-width ratios L/D between 0 and ∞ is investigated by Monte Carlo simulation. In addition to a low-density nematic phase and a high-density crystalline phase, two phases with partial translational order are observed, viz., a smectic phase and a columnar phase. By computing the absolute free energy of all phases, their range of stability and coexistence points are determined. For L/D ratios exceeding 0.5, a stable smectic phase is formed at densities well below the thermodynamic melting point. The nematic-to-smectic transition appears to be continuous. A columnar phase is only observed for L/D>3 at densities intermediate between the smectic and crystalline phases. The smectic-to-columnar transition is first order for L/D≤5. In the limit L/D→∞ this transition becomes continuous.

The interaction of solitary waves in hydrogen-bonded chains is investigated numerically. The physical model consists of two interacting sublattices of harmonically coupled protons and heavy ions. The protons can move in symmetric double-well potentials created by the pair of nearest-neighbor heavy ions. The proton displacement fields admit solitary kink and antikink solutions and the corresponding solitary displacements of the heavy ions are bell shaped. It is shown here that during head-on collisions the coupled-solitary-wave solutions do not preserve their shapes. Therefore, solitary waves in hydrogen-bonded chains are, in general, not solitons.

A new formula is presented for computing Maslov indices in integrable and near-integrable Hamiltonian systems. For several kinds of applications the new formula is particularly easy to use. It does not rely on counting caustics or other kinds of discontinuities. Its theoretical justification calls on wave-packet concepts and the topological properties of the group of symplectic matrices. Techniques are also presented for manipulating the Maslov index in analytical expressions.

In several perturbative calculations, such as the Lamb shift and decay of positronium into two photons, it is necessary to calculate the wave function of the S state, or the first derivative of the P state, etc. at the origin. This value diverges for the Dirac-Coulomb wave function. A method suggested by Bethe and Salpeter is used and the result is obtained in analytically closed form. For the hydrogen atom in the ground state (1$S_{1/2}$) our result for $u^2$(0) is equal to the traditional result, calculated by using the nonrelativistic Schrödinger wave function, multiplied by a factor slightly greater than unity, 1.000 118. For the 2$S_{1/2}$ state, the multiplication factor is 1.000 004 1.