Transport equations of amplitudes in the eikonal approximation of dynamical diffraction equations

An approach of the eikonal approximation of the dynamical diffraction equations of X-rays in deformed crystals, based on the second-order differential equations for the transmitted and diffracted waves, is presented. By analogy with usual optics, this approach allows one not only to obtain the eikonal equation and to study the behavior of the amplitude in zero-order approximation, which usually is performed in the eikonal dynamical diffraction theory, but also to establish for all orders of the amplitude asymptotic expansion the corresponding transport equations and to present their solutions as integrals over the amplitude propagation trajectory. Summarizing the transport equations, an equation for the total amplitude, analogous with the parabolic diffraction equation in optics, is obtained.     

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.     

A Novel Envelope Soliton Solution to the Granular Crystal Model

A theoretical work on envelope solitons to a one-dimensional granular chain model is reported. In the small amplitude approximation, we analytically solve the equation of motion with the help of the semidiscrete multiple-scale method. Our results show that the granular chain model can support an asymmetric high-order envelope soliton under the certain condition. It is found that the second-harmonic term of this high-order envelope soliton has an additional phase. In addition, the influence of both the material parameter and the static load on the localized features of the high-order envelope soliton is discussed.     

Enhancement of spectral-angular density of parametric X-rays in Laue geometry due to change in the angle between a target surface and reflecting atomic planes

Coherent X-rays of a relativistic electron crossing a single crystal with a uniform velocity in the Laue scattering geometry are considered in the two-wave approximation of dynamic diffraction theory [1]. Analytical expressions for the spectral-angular distribution of parametric X-rays (PXR) and diffracted transition radiation (DTR) have been obtained. The case when the system of diffracting atomic planes of a crystal is located at an arbitrary angle δ to a crystal surface (asymmetric reflection) is considered. The value δ = π/2 corresponds to the symmetric reflection in the given scattering geometry. The dependence of the PXR and DTR spectral-angular density on the angle δ has been investigated. It has been shown that the PXR spectrum width depends substantially on the given angle, which, in particular, allows one to increase significantly the PXR angular density by decreasing the angle δ.     

Kinetics of crystalline configurations: I. General theory; Kinetics of homogeneous short range order

A simple and exact way of coarse graining the master equation for a Markov process in configuration space is shown to exist. The coarse grained master equation is applied to the Ising model of a homogeneous binary interstitial alloy and to a “magnetic” Ising model. Using an approximation analogous to the quasi-chemical approximation, for both models the macroscopic rate equations for the establishment of short range order and the Fokker-Planck equation governing the fluctuations are derived.     

Diffraction equations for weak scattering reflective surface

In order to investigate the surface roughness dependence of speckle patterns, the complex-amplitude distribution of the speckle field should be obtained first. In previous studies, most investigators have treated this problem using the Fresnel or Fraunhofer diffraction equation. But for a weakly scattering reflective surface, when the observation plane is not parallel to the object plane, the Fresnel and Fraunhofer diffraction equations become inapplicable. Therefore, a reflective surface diffraction model (RSDM) is formed. When the difference between the RSDM and the transmission aperture diffraction model (TADM) is considered, then a general diffraction equation is presented. Considering the variations of the near-field approximation caused by coordinate system rotation, the near-field diffraction equation is derived. By introducing the far-field approximation, the far-field diffraction equation is obtained. The physical meanings of factors in the new equations are interpreted. Comparisons between the Fresnel and Fraunhofer diffraction equations and the newly derived ones show that the former are just the special cases of the latter. Finally, an application of these new diffraction equations is proposed.     

Behavior of the photoelectron boundary layer with sinusoidal timedependence

At high-incident photon intensities (X-rays, UV, or visible), a photoelectron boundary layer can form. This layer's properties, and especially its electric dipole moment, change in a nonlinear way as the light intensity varies in time. We show here that the dipole moment obeys an exact dynamical equation, which can be used as the basis for analytic approximation schemes for a driver of arbitrary time dependence. An approximate analytic solution for a sinusoidally oscillating light source is presented, and compared with a particle-pusher numerical simulation. Scaling laws are extracted from the analytic solution     

Borman effect in resonant diffraction of X-rays

A dynamic theory of resonant diffraction (occurring when the energy of incident radiation is close to the energy of the absorption edge of an element in the composition of a given substance) of synchronous X-rays is developed in the two-wave approximation in the coplanar Laue geometry for large grazing angles in perfect crystals. A sharp decrease in the absorption coefficient in the substance with simultaneously satisfied diffraction conditions (Borman effect) is demonstrated, and the theoretical and first experimental results are compared. The calculations reveal the possibility of applying this approach in analyzing the quadrupole-quadrupole contribution to the absorption coefficient.     

The validity of the average t-matrix approximation for low-energy electron diffraction from random alloys

We explore the validity of the average t-matrix approximation for the calculation of low-energy electron diffraction IV spectra from substitutionally disordered alloy surfaces. The accuracy of this approximation is assessed by comparison with the results of calculations for Ni_xPt_100−x(100) surfaces obtained using the more accurate coherent-potential approximation. We find excellent agreement the two approaches, demonstrating the validity of the average t-matrix approximation for interpreting low-energy electron diffraction from alloys. The physical origin of this agreement is discussed by reference to the scattering properties of the electrons in alloy surfaces.     

Bethe-Peierls approximation for the disordered Ising model

We study the Ising model for an alloy with an arbitrary number of components. We develop an approximation which reduces to that of Bethe and Peierls when the concentration of one of the components is unity. We investigate within this approximation the dependence of the various thermodynamic quantities, in particularT _c, on the composition of the alloy and the magnetic properties of its constituents. Comparison with the only exact calculation available, that of F. T. Leeet al., for a linear chain, shows extremely satisfactory agreement.Research supported by ARO (D). It has also benefited from the general support of Materials Science at the University of Chicago by the NSF.     

Ab initio calculation of the Gilbert damping parameter via the linear response formalism

A Kubo-Greenwood-like equation for the Gilbert damping parameter α is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker band structure method in combination with coherent potential approximation alloy theory allows it to be applied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy system Fe(1-x)Co(x) as well as for a series of alloys of Permalloy with 5d transition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding calculations for Ni correctly describe the rapid change of α when small amounts of substitutional Cu are introduced.     

Approximate solutions of the Liouville equation. IV. The two-body additive approximation

The two-body additive approximation on the time-dependent Liouville distribution, first introduced in part I of this series, is put into the conventional form of a self-contained kinetic equation for the doublet distribution. From this point of view the approximation consists in truncating the BBGKY chain by expressing the triplet distribution as a functional of lower distributions at the same value of the time variable. To accomplish this, it is necessary to study two associated purely spatial integral equations. The doublet kinetic equation can then be written in terms of solutions of these integral equations and comparison with conventional methods of truncating the BBGKY chain can then be made. For the purpose of comparison a method of truncating the chain based on the Kirkwood superposition approximation is introduced and discussed briefly. The momentum structure of the resulting doublet kinetic equation is similar, but the nonlocality in space of our truncation introduces distinct differences in the spatial structure. The inconsistency between conventional truncations and the exact initial conditions used for the calculation of time-dependent correlation functions is pointed out. This inconsistency is not shared by the two-body additive approximation.     

Approximation of the parabolic equation and the wavefield of a point source in a layered random medium

This paper studies the wavefield of a source in a multidimensional randomly layered medium. They obtained asymptotical expressions of the wave statistical characteristics for different boundary conditions both in the framework of the parabolic equation approximation and the exact formulation of the boundary problem for the Helmholtz equation. It is shown that the presence of a small but finite absorption γ is most important for the statistics. The diffraction effects turn out to be like those of absorption, but γ cannot tend to zero in this problem. In an appendix they give the factorization formulae of the wave equation solution in a layered medium.     

On the Approximation of the Stochastic Burgers Equation

 We prove mathematical approximation results for the (hyperviscous) Burgers equation driven by additive Gaussian noise. In particular we show that solutions of ``approximating equations' driven by a discretized noise converge towards the solution of the original equation when the discretization parameter gets small. The convergence takes place in the expected value of arbitrary powers of certain norms; i.e., all moments of the difference of the solutions tend to zero in certain function spaces. For the hyperviscous Burgers equation, these results are applied to justify the approximation of certain correlation functions that play a major role in statistical turbulence theory. Received: 10 October 2001 / Accepted: 21 May 2002 Published online: 6 August 2002     

Nonlinear diffraction in a quantum-dot system with allowance for the Hubbard interaction

The propagation of monochromatic laser radiation in a volume system of quantum dots (QDs) that are tunnel-coupled along one axis is considered. The electron energy spectrum of the QD system is modeled in the tight-binding approximation with allowance for the Coulomb interaction of electrons in the Hubbard model. The electromagnetic field of laser radiation in a QD system is described quasi-classically by Maxwell equations; as applied to this problem, they are reduced to a non-one-dimensional wave equation for the vector potential. As a result of the analysis of the wave equation in the approximation of varying amplitudes and phases, an effective equation describing the electromagnetic field in a QD system is obtained and numerically solved. The influence of the parameters of the system and the amplitude and frequency of the field of incident laser radiation on the character of its propagation is investigated. Nonmonotonic dependences of the factor characterizing the laser beam diffraction spread on the parameters of the electron energy spectrum of the system are obtained.     

The second-order Rytov approximation and residual error in dual-frequency satellite navigation systems

The second-order Rytov approximation has been used to determine ionospheric corrections for the phase path up to third order. We show the transition of the derived expressions to previous results obtained within the ray approximation using the second-order approximation of perturbation theory by solving the eikonal equation. The resulting equation for the phase path is used to determine the residual ionospheric first-, second- and third-order errors of a dual-frequency navigation system, with diffraction effects taken into account. Formulas are derived for the biases and variances of these errors, and these formulas are analyzed and modeled for a turbulent ionosphere. The modeling results show that the third-order error that is determined by random irregularities can be dominant in the residual errors. In particular, the role of random irregularities is enhanced for small elevation angles. Furthermore, in the case of small angles the role of diffraction effects increases. It is pointed out that a need to pass on to diffraction formulas arises when the Fresnel radius exceeds the inner scale of turbulence.     

Nonlinear diffraction in inhomogeneous superlattice

We considered the propagation of laser monochromatic radiation in a superlattice that contains regions with an elevated concentration of carriers. The model of the energy spectrum of electrons is chosen in the strong coupling approximation. The electromagnetic field is described quasiclassically with Maxwell equations, which, as applied to the problem under study, are reduced to a non-one-dimensional sine-Gordon wave equation for the vector-potential. We analyzed the wave equation in the approximation of slowly varying amplitudes and phases and obtained and numerically solved an effective equation that describes the electromagnetic field in the superlattice. We studied different regimes of propagation of laser radiation, analyzed diffraction by regions with an elevated electron concentration.     

Dynamic response of a disordered ferromagnetic chain: Alloy transfer matrix approximation

The alloy transfer matrix approximation is used to study the uniform dynamic susceptibility of a disordered ferromagnetic chain. This approximation allows for a consistent treatment of diagonal and off-diagonal disorder. The results, in the limit of low concentrations, are in agreement with the exact single impurity ones. Intensities and lineshapes for infrared absorption are calculated for finite impurity concentrations and different values of the relative anisotropy parameter of a model alloy.