Finite Mean Free Path Research Articles

Coagulation of monodisperse aerosol particles by isotropic turbulence

The rate of coagulation of initially monodisperse aerosols due to isotropic turbulence is studied with particular emphasis on the effects of noncontinuum hydrodynamics and particle inertia. The prevalence of these two factors distinguishes aerosol coagulation from the coagulation of colloidal particles. The turbulent flow seen by an interacting pair of particles is modelled as a stochastically varying flow field that is a linear function of position. This approximation is valid because the 1–10 micron diameter particles for which turbulence dominates coagulation are much smaller than the smallest eddies of a typical turbulent flow field. It is shown that the finite mean-free path of the gas enhances the rate of coagulation and leads to a finite coagulation rate even in the absence of van der Waals attractions. The coupled effects of turbulent shear and Brownian motion are treated. As in the case of laminar shear flows, it is found that Brownian motion plays an important role in the coagulation process even when the Peclet number is moderately large. It is shown that particle inertia increases the coagulation rate in two ways. First, preferential concentration increases the radial distribution function on length scales intermediate between the Kolmogorov length scale and the particle diameter. Second, the greater persistence of particles’ relative motion during their local interaction leads to an increase in coagulation rate with increasing particle Stokes number.

Disordered RKKY lattice mean field theory for ferromagnetism in diluted magnetic semiconductors.

We develop a lattice mean field theory for ferromagnetic ordering in diluted magnetic semiconductors by taking into account the spatial fluctuations associated with random disorder in the magnetic impurity locations and the finite mean free path associated with low carrier mobilities. Assuming a carrier-mediated indirect RKKY exchange interaction among the magnetic impurities, we find substantial deviation from the extensively used continuum Zener model Weiss mean field predictions. Our theory allows accurate analytic predictions for Tc and provides simple explanations for a number of observed anomalies, including the non-Brillouin function magnetization curves, the suppressed low-temperature magnetization saturation, and the dependence of Tc on conductivity.

Electronic Transport Properties of Liquid Less-Simple Metals

Electronic transport properties, namely the electrical resistivity and the thermoelectric power, of liquid less-simple metals, Zn, Cd, Hg, In, Tl, Sn, Pb, Sb, and Bi, are calculated using Ziman's theory. The effective electron–ion and ion–ion interactions are described by the Bretonnet-Silbert model. The static structure factors are evaluated by the self-consistent Variational Modified Hypernetted Chain (VMHNC) integral equation of liquids. The results for the electrical resistivity are fairly good when compared with experimental data, but the agreement improves significantly when the blurring of the Fermi surface due to the finite electron mean free path is accounted for. The values of the thermoelectric power, for most systems, have the same sign and order of magnitude as the experimental ones. The results for the thermopower also show that the contribution of the energy dependent term of the electron–ion potential is significant for the concerned systems.

Dynamics of cosmological perturbations in position space

We show that the linear dynamics of cosmological perturbations can be described by coupled wave equations, allowing their efficient numerical and, in certain limits, analytical integration directly in position space. The linear evolution of any perturbation can then be analyzed with the Green's function method. Prior to hydrogen recombination, assuming tight coupling between photons and baryons, neglecting neutrino perturbations, and taking isentropic (adiabatic) initial conditions, the obtained Green's functions for all metric, density, and velocity perturbations vanish beyond the acoustic horizon. A localized primordial cosmological perturbation expands as an acoustic wave of photon-baryon density perturbation with narrow spikes at its acoustic wave fronts. These spikes provide one of the main contributions to the cosmic microwave background radiation anisotropy on all experimentally accessible scales. The gravitational interaction between cold dark matter and baryons causes a dip in the observed temperature of the radiation at the center of the initial perturbation. We first model the radiation by a perfect fluid and then extend our analysis to account for finite photon mean free path. The resulting diffusive corrections smear the sharp features in the photon and baryon density Green's functions over the scale of Silk damping.

From chaos to disorder: Statistics of the eigenfunctions of microwave cavities

We study the statistics of the experimental eigenfunctions of chaotic and disordered microwave billiards in terms of the moments of their spatial distributions, such as the Inverse Participation Ratio (IPR) and density-density auto-correlation. A path from chaos to disorder is described in terms of increasing IPR. In the chaotic, ballistic limit, the data correspond well with universal results from random matrix theory. Deviations from universal distributions are observed due to disorder induced localization, and for the weakly disordered case the data are well-described by including finite conductance and mean free path contributions in the framework of nonlinear sigma models of supersymetry.

Application of random walk theory to the first-order Fermi acceleration in shock waves

We formulate the first-order Fermi acceleration in shock waves in terms of the random walk theory. The formulation is applicable to any value of the shock speed and the particle speed – in particular, to the acceleration in relativistic shocks and to the injection problem, where the particle speed is comparable to the fluid speed – as long as large-angle scattering is suitable for the scattering process of particles. We first show that the trajectory of a particle suffering from large-angle scattering can be treated as a random walk in a moving medium with an absorbing boundary (e.g., the shock front). We derive an integral equation to determine the density of scattering points of the random walk, and by solving it approximately we obtain approximate solutions of the probability density of pitch angle at, and the return probability after, the shock crossing in analytic form. These approximate solutions include corrections of several non-diffusive effects to the conventional diffusion approximation, and we show that they agree well with Monte Carlo results for the isotropic scattering model for any shock speed and particle speed. When we neglect effects of ‘a few-step return’, we obtain ‘the multistep approximation’ which includes only the effect of finite mean free path and which is equivalent to ‘the relativistic diffusion approximation’ used by Peacock if the correct diffusion length is used in his expression. We find that the multistep approximation is not appropriate to describe the probability densities of individual particles for relativistic shocks, but that the pitch angle distribution at the shock front in steady state is in practice quite well approximated by that given by the multistep approximation, because the effects of a finite mean free path and a few-step return compensate each other when averaged over the pitch angle distribution. Finally, we give an analytical expression for the spectral index of accelerated particles in parallel shocks valid for arbitrary shock speed using this approximation.

Landau-Pomeranchuk-Migdal effect in thermal field theory

In recent studies, the production rate of photons or lepton pairs by a quark gluon plasma has been found to be enhanced due to collinear singularities. This enhancement pattern is very dependent on rather strict collinearity conditions between the photon and the quark momenta. It was estimated by neglecting the collisional width of quasi-particles. In this paper, we study the modifications of this collinear enhancement when we take into account the possibility for the quarks to have a finite mean free path. Assuming a mean free path of order $(g^2T\ln(1/g))^{-1}$, we find that only low invariant mass photons are affected. The region where collision effects are important can be interpreted as the region where the Landau-Pomeranchuk-Migdal effect plays a role in thermal photon production by bremsstrahlung. It is found that this effect modifies the spectrum of very energetic photons as well. Based on these results and on a previous work on infrared singularities, we end this paper by a reasonable physical picture for photon production by a quark gluon plasma, that should be useful to set directions for future technical developments.

The Rigidity Dependence of Solar Particle Scattering Mean Free Paths

The rigidity dependence of solar particle scattering mean free paths in the range from 0.1 to a few GV is a key parameter to distinguish between different models for the nature of wave-particle interactions and of interplanetary magnetic turbulence. Here we present an analysis of electron and ion data, covering a large portion of the above range, obtained from modeling intensity and anisotropy profiles of a series of solar particle events in 1982 August that show exceptionally large but finite mean free paths. Combined with particle measurements from previously analyzed events that exhibit mean free paths in the range of 0.02 to 0.5 AU, we find that a uniform shape of the functional form of the rigidity dependence, which varies only in absolute height for different events, can explain all observations. These results place severe constraints on any propagation theory and rule out a number of recently suggested models to improve the standard quasilinear theory of particle scattering. Our findings may indicate that the component of the turbulence the particles interact with does not vary much with respect to its decomposition into different modes and their respective spectral and angular wavenumber dependencies, and basically the total power of this component is responsible for the level of the mean free paths.

Electrical Resistivity of Alkali-Lead Binary Alloys

The electrical resistivity of liquid Li-Pb, Pb-K and Pb-Na binary alloys as a function of compassion has been calculated using Faber-Ziman formalism modified for finite mean free-path. the partial structure factors described by hard sphere model of Ashcroft and Langreth are used in the calculations. the nonlocal pseudopotentials of Heine-Abarenkov have been used also. the calculated resistivity values are in reasonably good agreement with the experiment.

Diamagnetic response of cylindrical normal-metal–superconductor proximity structures with low concentration of scattering centers

We have investigated the diamagnetic response of composite NS proximity wires, consisting of a clean silver or copper coating, in good electrical contact to a superconducting niobium or tantalum core. The samples show strong induced diamagnetism in the normal layer, resulting in a nearly complete Meissner screening at low temperatures. The temperature dependence of the linear diamagnetic susceptibility data is successfully described by the quasiclassical Eilenberger theory including elastic scattering characterised by a mean free path l. Using the mean free path as the only fit parameter we found values of l in the range 0.1-1 of the normal metal layer thickness d_N, which are in rough agreement with the ones obtained from residual resistivity measurements. The fits are satisfactory over the whole temperature range between 5 mK and 7 K for values of d_N varying between 1.6 my m and 30 my m. Although a finite mean free path is necessary to correctly describe the temperature dependence of the linear response diamagnetic susceptibility, the measured breakdown fields in the nonlinear regime follow the temperature and thickness dependence given by the clean limit theory. However, there is a discrepancy in the absolute values. We argue that in order to reach quantitative agreement one needs to take into account the mean free path from the fits of the linear response. [PACS numbers: 74.50.+r, 74.80.-g]

Diamagnetic response of a normal-metal–superconductor proximity system at arbitrary impurity concentration

We investigate the magnetic response of normal-metal--superconductor proximity systems for arbitrary concentrations of impurities and at arbitrary temperatures. Using the quasiclassical theory of superconductivity a general linear-response formula is derived which yields a nonlocal current-field relation in terms of the zero-field Green's functions. Various regimes between clean-limit and dirty-limit response are investigated by analytical methods and by solving the general formula numerically. In the ballistic regime, a finite mean free path reduces the nonlocality and leads to a stronger screening than in the clean limit even for a mean free path much larger than the system size. Additionally, the range of the kernel describing the nonlocality is strongly temperature dependent in this case. In the diffusive limit we find a crossover from local to nonlocal screening, which restricts the applicability of the dirty-limit theory.

Probes for the early reaction dynamics of heavy ion collisions at AGS and SPS

We discuss the early evolution of ultra-relativistic heavy ion collisions within a multi-fluid dynamical model. In particular, we show that due to the finite mean-free path of the particles compression shock waves are smeared out considerably as compared to the one-fluid limit. Also, the maximal energy density of the baryons is much lower. We discuss the time scale of kinetic equilibration of the baryons in the central region and its relevance for directed flow. Finally, thermal emission of direct photons from the fluid of produced particles is calculated within the three-fluid model and two other simple expansion models. It is shown that the transverse momentum and rapidity spectra of photons give clue to the cooling law and the early rapidity distribution of the photon source.

Variation of multilayer magnetoresistance with ferromagnetic layer sequence: Spin-memory loss effects

We find the magnetoresistance (MR) with current flow perpendicular to the layer planes (CPP-MR) of multilayers made of the ferromagnetic metals Co and Py (permalloy=Ni84Fe16) and the nonmagnetic metal Ag to vary with the Co and Py layer sequencing. That is, [Co/Ag/Py/Ag]N (type I) multilayers have larger MRs than [Co/Ag]N[Py/Ag]N (type II) ones, and the difference grows with the number of repeats N. We ascribe the differences to an apparent short spin-diffusion length in Py. Interestingly, similar differences appear also in the current in plane MR, probably due to finite mean-free paths in all three metals, plus interface scattering.

Direct transitions, indirect transitions, and surface photoemission in the prototypical system Ag(111)

The sp band of Ag(111) is nearly free-electron like near the Fermi level. Photoemission from this band in the surface-normal direction gives rise to a direct-transition peak which disperses with varying photon energies. Contrary to predictions of the standard three-step model, the line shape is significantly asymmetric. The lower binding-energy side of this peak has a higher intensity, and is joined by a broad indirect-transition plateau extending to a rounded cut-off at the sp band edge. This study is an analysis of the asymmetry of the direct-transition peak and the intensity and line shape of the indirect-transition region. The optical transition-matrix element consists of two terms. One is the usual momentum-matrix element dominated by the direct band-to-band transitions in the bulk, with a small contribution from the surface due to the finite mean-free path. The other is a ∇· A term, which dominates the surface contribution and gives rise to indirect transitions involving all valence states without any restrictions on the crystal momentum. Interference between the bulk and surface contributions results in an asymmetric line shape for the direct-transition peak. The contribution from ∇· A is shown to be well correlated with the difference in dielectric function between Ag and vacuum. The effects discussed in this paper are fairly general in nature, and should be taken into account in high-resolution photoemission work.

Electrical conductivity in thin metallic films with self-consistent level broadening

We study the subband structure in a smooth film where isotropic volume scattering gives rise to a finite mean free path. The intersubband transitions into both the propagating and evanescent modes are described by an infinite system of equations. Its solution determines self-consistently the scattering-induced level broadening. The thickness dependence of the density of states and the conductivity are discussed in detail and compared with results which follow from the neglect of level broadening. For typical mean free paths, level broadening strongly smears out the steps of the density of states and suppresses size-induced oscillations of the conductivity. These results are in agreement with the general trend in experimental observations.

Radiation-hydrodynamic waves and global solar oscillations

We investigate nonadiabatic hydrodynamic waves in a nongrey, radiating, thermally conducting, homogeneous atmosphere in LTE with a finite mean free path of photons. Avoiding the Eddington approximation the remaining simplifications in the basic equations are discussed, the generalized dispersion relation is analysed, and some wave properties in a grey model are studied. The properties of waves in a stratified atmosphere are analysed as well. In connection with the predicted properties of the nonadiabatic waves we discuss observations of p-modes by measuring brightness fluctuations.

Magnetic penetration depth and fluxon-line dynamics in the organic superconductor kappa -

The muon-spin-relaxation (\ensuremath{\mu}SR) technique has been used to study the internal field modulation dynamics of fluxons in crystals of \ensuremath{\kappa}-[BEDT-TTF${]}_{2}$Cu[NCS${]}_{2}$ (${\mathit{T}}_{\mathit{c}}$\ensuremath{\approxeq}10.5 K). The data exhibit a strong field dependence for applied fields in the range 129 Oe\ensuremath{\le}\ensuremath{\Vert}${\mathbf{H}}_{\mathrm{ext}}$\ensuremath{\Vert}\ensuremath{\le}4 kOe, which is attributed to a combination of vortex motion, finite core size effects, and pinning. In low fields (\ensuremath{\Vert}${\mathbf{H}}_{\mathrm{ext}}$\ensuremath{\Vert}=129 Oe), the data exhibit a flux-pinning transition at ${\mathit{T}}_{\mathit{x}}$\ensuremath{\sim}5 K, as evidenced by a peak in the second moment of the local field distribution at \ensuremath{\sim}5 K accompanied by an enhanced minimum in the first moment at the same temperature. In higher fields (\ensuremath{\Vert}${\mathbf{H}}_{\mathrm{ext}}$\ensuremath{\Vert}=750 Oe, 3 kOe, and 4 kOe), where the effects of motion and pinning are minimized, the symmetry of the underlying gap function is confirmed as s wave. Corrections for finite fluxon core, finite mean free path, and flux-lattice disorder were applied in determining the London penetration depth, ${\ensuremath{\lambda}}_{\mathit{L}}^{(\mathit{b}\mathit{c})}$(0)=7680\ifmmode\pm\else\textpm\fi{}700 \AA{}.

Resonant tunneling in a resistive wire.

Using a recently developed superposition method, we consider resonant tunneling through a double-barrier structure in a resistive wire. The wire with finite mean free path permits phase-randomizing scattering; the barriers are described by elastic scatterers. We analyze in detail the evolving residual-resistivity dipole surrounding the barriers. We investigate the additional resistance due to the barriers and study the transition from completely coherent to completely incoherent transmission. The transport regime is governed by the ratio of the mean free path to the well width. We discuss the effect of phase-breaking scattering on the tunneling process leading to a decrease in the resonant transmission probability and a rise in the off-resonant one. A formula for the density of states in the well is also obtained.

Fitting of Bardeen-Cooper-Schrieffer theory to measured microwave conductivity of YBa2Cu3O7 films.

The weak-coupling BCS expression for the complex conductivity, in the local limit and with a finite carrier mean free path l, is fitted to data from recent experiments that accurately measured both the real and imaginary parts of the microwave (58.9-GHz) conductivity on thin ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ films. The fitting, for temperatures up to a few degrees below ${\mathit{T}}_{\mathit{c}}$, is achieved by adjusting the temperature dependence of the l and of the effective mass, within a model for a pure superconductor. Just below ${\mathit{T}}_{\mathit{c}}$, an effective-medium model is invoked to describe a mixture of superconducting- and normal-state regions. This phenomenological procedure fits, in particular, the narrow peak that was measured in the real part of the conductivity near ${\mathit{T}}_{\mathit{c}}$.

Generalized hydrodynamic transport in lattice-gas automata.

The generalized hydrodynamics of two-dimensional lattice-gas automata is solved analytically in the linearized Boltzmann approximation. The dependence of the transport coefficients (kinematic viscosity, bulk viscosity, and sound speed) upon wave number k is obtained analytically. Anisotropy of these coefficients due to the lattice symmetry is studied for the entire range of wave number, k. Boundary effects due to a finite mean free path (Knudsen layer) are analyzed, and accurate comparisons are made with lattice-gas simulations.