Electronic motion in a model solid of short range potentials

The motion of an electron moving in a model solid which consists of an array of very short range scatterers is considered. This system introduces simplifications because the detailed nature of the individual scatterers enters in only a very simple fashion. The theory is developed for an arbitrary array of short range potentials and is then specialized to a periodic system. (Both infinite and semiinfinite [surface] cases are discussed.) The case of a system of periodically positioned scatterers, the inexactly known strengths of which are described by a Lorentzian distribution, is also considered. Finally, the theory is applied to the case of an electron, under the influence of a rational magnetic field, moving amid a periodic array of short range potentials.     

Density of states for high landau levels and random potential

The density of states of a two-dimensional, non-interacting electron gas under the influence of a strong perpendicular magnetic field and random impurities is examined. In the strong magnetic field the ensemble averaged Green's function can be restricted to a single Landau leveln. Using 1/n as expansion parameter we calculate first order corrections. The results for two models of disorder are compared: white-noise potential and uniformly distributed zero-range scatterers with a Lorentzian distribution of strength.     

Hall Conductivity in the presence of repulsive magnetic impurities

The Hall conductivity of disordered magnetic systems consisting of hard-core point vortices randomly dropped on the plane with a Poissonian distribution, has a behavior analogous to the one observed experimentally by Haug, Gerhardts, Klitzling and Ploog, with repulsive scatterers [#!1!#]. We also argue that models of homogeneous magnetic field with disordered potential, have necessarily vanishing Hall conductivities when their Hilbert space is restricted to a given Landau level subspace. Received: 29 September 1998 / Accepted: 21 December 1998     

Scattering of longitudinal waves (sound) by defects in fluids. Rough surface

The classical theory of scattering of longitudinal waves (sound) by small inhomogeneities (scatterers) in an ideal fluid is generalized to a distribution of scatterers and such as to include the effect of the inhomogeneities on the elastic properties of the fluid. The results are obtained by a new method of solving the wave equation with spatial restrictions (caused by the presence of the scatterers), which can also be applied to other types of inhomogeneities (like surface roughness, for instance). A coherent forward scattering is identified for a uniform distribution of scatterers (practically equivalent with a mean-field approach), which is due to the fact that our treatment does not include multiple scattering. The reflected wave is obtained for a half-space (semi-infinite fluid) of uniformly distributed scatterers, as well as the field diffracted by a perfect lattice of scatterers. The same method is applied to a (inhomogeneous) rough surface of a semi-infinite ideal fluid. A perturbation-theoretical scheme is devised, with the roughness function as a perturbation parameter, for computing the waves scattered by the surface roughness. The waves scattered by the rough surface are both waves localized (and propagating only) on the surface (two-dimensional waves) and waves reflected back in the fluid. They exhibit directional effects, slowness, attenuation or resonance phenomena, depending on the spatial characteristics of the roughness function. The reflection coefficients and the energy carried on by these waves are calculated both for fixed and free surfaces. In some cases, the surface roughness may generate waves confined to the surface (damped, rough-surface waves).     

Photogalvanic current in a parabolic well

We study the in-plane stationary photocurrent in a parabolic potential well. The well has vertical asymmetry due to inhomogeneous distribution of scatterers. The electric field of light has both vertical and in-plane components. The photogalvanic effect originates from the periodic oscillation of electrons in a vertical direction caused by the normal component of the alternating electric field with simultaneous in-plane acceleration/deceleration by the in-plane component of electric field. The problem is considered in classical approximation assuming inhomogeneously-distributed friction. Photocurrent has a resonance character. Resonance occurs at light frequencies close to a characteristic well frequency. The effect of in-plane magnetic field is also studied.     

Effect of impurities on the low-temperature behavior of the specific heat of anisotropic superconductors in a mixed state

In the presence of zeros of the order parameter in an anisotropic superconductor, the combined effect of a magnetic field and impurities leads to two different limiting magnetic-field dependences of the specific heat. These dependences are studied both for Born scatterers and in the unitary limit for several specific examples of anisotropic pairing. A estimate is given for the crossover field. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 8, 606–611 (25 April 1997)     

Application of the mutual-interaction method to a class of two-scatterer systems: I. Two discrete scatterers

In a recent paper we developed a formalism that fully accommodates the mutual interactions among scatterers separable by parallel planes. The total fields propagating away from these planes are the unknowns of a system of difference equations. Each scatterer is characterized by a scattering function that expresses the scattered wave amplitude as a function of the incident and scattered wavevectors for a unit-amplitude plane wave scattered from the object in isolation. This function can be derived completely from the scattered far field with the help of analytic continuation. For a two-scatterer system the mutual-interaction equations reduce to a single Fredholm integral equation of the second kind. It turns out that analytic solutions are tractable for those scattering functions that are Dirac deltas or a sum of products of separable functions of the incident and scattered wavevectors. Scattering functions for planes and isotropic scatterers, as well as electric and magnetic dipoles all possess this property and are considered. The exact scattering functions agree with results obtained by analytic continuation. This paper consists of two parts. Part I derives analytic solutions for two discrete scatterers (isotropic scatterers. electric dipoles, magnetic dipoles). Part II is devoted to scattering from an object (isotropic or dipole scatterer) near an interface separating two semi-infinite uniforn-media. Because the results in this paper are exact, the effects of near-field interactions can be assessed. The forms of the scattering solutions can be adapted to objects that are both radiating and scattering.     

Method for solving the problems of multiple scattering by several bodies in a homogeneous unbounded medium

A method is developed for solving problems of multiple scattering by an aggregate of bodies in a homogeneous unbounded medium. For this purpose, the problem on the multiple scattering produced by two bodies in the field of a plane wave is first considered under the assumption that the initial unperturbed scattering amplitudes of both scatterers are known. The solution is constructed by considering plane waves multiply rescattered by the scatterers. Integral equations are obtained that allow one to calculate the resulting scattering amplitude of each scatterer and the combined scattering amplitude of the system of two scatterers. It is shown that knowledge of the solution to this problem is sufficient to solve the problem on the scattering field of a system consisting of an arbitrary number of scatterers. Expressions for the scattering amplitude in the case of an arbitrary primary field are presented. The relationship between the integral equations describing the multiple scattering in a homogeneous space and the multiple scattering by a single scatterer located near an interface is demonstrated. Approximate expressions are given for calculating the scattering amplitude in the case of multiple scattering.     

From the Liouville Equation to the Generalized Boltzmann Equation for Magnetotransport in the 2D Lorentz Model

We consider a system of non-interacting charged particles moving in two dimensions among fixed hard scatterers, and acted upon by a perpendicular magnetic field. Recollisions between charged particles and scatterers are unavoidable in this case. We derive from the Liouville equation for this system a generalized Boltzmann equation with infinitely long memory, but which still is analytically solvable. This kinetic equation has been earlier written down from intuitive arguments.     

Zur Quantentheorie der Transportprozesse im Magnetfeld. II

Applying Kubo's formulae and damping theoretical arguments the electric, thermoelectric and thermal transport coefficients for a system of free and independent electrons and fixed scatterers in the presence of an external magnetic field are calculated. Assumingδ-type scatterers it is shown that in the quantum regionkT?ω?ζ the Wiedemann-Franz law holds for the oscillating symmetric conductivities, whereas in the case of the antisymmetric ones quantum mechanical deviations from this law occur. For strong magnetic fields and arbitrary scattering potential the symmetric transversal transport coefficients can be expressed by Titeica type formulae.     

Application of the mutual-interaction method to a class of two-scatterer systems: I. Two discrete scatterers

Abstract

In a recent paper we developed a formalism that fully accommodates the mutual interactions among scatterers separable by parallel planes. The total fields propagating away from these planes are the unknowns of a system of difference equations. Each scatterer is characterized by a scattering function that expresses the scattered wave amplitude as a function of the incident and scattered wavevectors for a unit-amplitude plane wave scattered from the object in isolation. This function can be derived completely from the scattered far field with the help of analytic continuation. For a two-scatterer system the mutual-interaction equations reduce to a single Fredholm integral equation of the second kind. It turns out that analytic solutions are tractable for those scattering functions that are Dirac deltas or a sum of products of separable functions of the incident and scattered wavevectors. Scattering functions for planes and isotropic scatterers, as well as electric and magnetic dipoles all possess this property and are considered. The exact scattering functions agree with results obtained by analytic continuation. This paper consists of two parts. Part I derives analytic solutions for two discrete scatterers (isotropic scatterers. electric dipoles, magnetic dipoles). Part II is devoted to scattering from an object (isotropic or dipole scatterer) near an interface separating two semi-infinite uniforn-media. Because the results in this paper are exact, the effects of near-field interactions can be assessed. The forms of the scattering solutions can be adapted to objects that are both radiating and scattering.     

Semi-empirical bone model for determination of trabecular structure properties from backscattered ultrasound

A novel semi-empirical scattering model of trabecular bone facilitating its characterization and allowing optimization of the interrogating pulse-echo transducer performance was developed. The model accounts for spatial density distribution of the trabeculae and includes measurement conditions such as pressure–time waveform of the probing ultrasound wave, the emitted field structure, and the transfer function and limited bandwidth of the acoustic source operating in pulse-echo mode. These measurement conditions are of importance as they modify the scattered echoes, which in turn are linked to the micro-architecture of the bone. The bone was modeled by a random distribution of long and thin cylindrical scatterers having randomly varying diameters and mechanical properties, and oriented perpendicularly to the ultrasound beam axis. To mimic clinically encountered conditions the relevant empirical data obtained at 1 MHz were input to the model. The data included pulse-echo source pressure field distribution in the focal zone and the above mentioned transfer function. With these data the model allowed frequency dependent backscattering coefficient of the simulated bone structure and its statistical properties to be determined. The results obtained indicated that the computer simulation is of particular relevance in studying scattering properties of the cancellous bone and holds promise as a tool to determine the relationship between the physical dimensions and shape of the scatterers and for monitoring of osteoporosis. The results of simulations also indicated that the new bone model proposed is well suited to mimic clinically relevant conditions. In contrast to the existing bone models, which usually assume scatterers to be randomly distributed as infinitely long identical cylinders with a cross-section much smaller than the probing ultrasound wave, the new model includes two populations of scatterers having different physical dimensions and also allows the mechanical properties of the scatterers to be varied.     

Effective scatterer approximation for mean field modeling with application to vegetation

A new electromagnetic method is developed for the mean field analysis of discrete random media in which particles are distributed in disjoint regions in space. This type of distribution can be observed in organized vegetation canopies, photonic band gap materials and disordered crystals. The method is based on an ‘effective scatterer’ concept. As far as the mean field is concerned, the random problem is approximated by a deterministic one composed of effective scatterers. Application of the effective scatterer approach to two-dimensional mean field propagation through a cylinder distribution that is periodic on average is verified by Monte Carlo simulations. Comparisons with other known methods such as the Foldy approximation are also presented.     

Effective scatterer approximation for mean field modeling with application to vegetation

A new electromagnetic method is developed for the mean field analysis of discrete random media in which particles are distributed in disjoint regions in space. This type of distribution can be observed in organized vegetation canopies, photonic band gap materials and disordered crystals. The method is based on an 'effective scatterer' concept. As far as the mean field is concerned, the random problem is approximated by a deterministic one composed of effective scatterers. Application of the effective scatterer approach to two-dimensional mean field propagation through a cylinder distribution that is periodic on average is verified by Monte Carlo simulations. Comparisons with other known methods such as the Foldy approximation are also presented.     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.     

Magneto-transport in the two dimensional Lorentz model

A recently developed theory for the motion of a classical particle in a random array of scatterers is improved and extended to discuss the effects of weak and intermediate magnetic fields. By deriving expressions for the general relaxation kernels it is shown that only the current relaxation kernel is the physical relevant one diverging at the percolation edge. The percolation density and localization length turn out to be independent of the magnetic field. A negative magneto resistance at low scatterer density, a positive magneto resistance at larger density and a non classical Hall coefficient are obtained. For the velocity correlation spectrum a shift of the cyclotron resonance to higher frequency and a new low frequency side peak is predicted.     

Zur Quantentheorie der Transportprozesse im Magnetfeld. III

In the presence of a transversal magnetic field the electric thermoelectric and thermal transportcoefficients are calculated, taking into account phonon drag. The calculations are based on the model of free electrons, phonons, and impurity scatterers. Starting from Kubo's formulae, and using truncation technique for Green's functions an integral equation (generalized transport equation) is developed. It is solved in the case of strong magnetic field. If phonon drag is neglected Titeica type formulae hold for all transportcoefficients. Phonon drag reduces the values in the sense that the electron-phonon-relaxation time is replaced by the sum of the electron-phonon- and phonon-scatterer-relaxation time.     

Localization of a small change in a multiple scattering environment without modeling of the actual medium

A method to actively localize a small perturbation in a multiple scattering medium using a collection of remote acoustic sensors is presented. The approach requires only minimal modeling and no knowledge of the scatterer distribution and properties of the scattering medium and the perturbation. The medium is ensonified before and after a perturbation is introduced. The coherent difference between the measured signals then reveals all field components that have interacted with the perturbation. A simple single scatter filter (that ignores the presence of the medium scatterers) is matched to the earliest change of the coherent difference to localize the perturbation. Using a multi-source/receiver laboratory setup in air, the technique has been successfully tested with experimental data at frequencies varying from 30 to 60 kHz (wavelength ranging from 0.5 to 1 cm) for cm-scale scatterers in a scattering medium with a size two to five times bigger than its transport mean free path.