Numerical results for finite order X and Y functions of radiative transfer

It is well known that, in the theory of radiative transfer, Chandrasekhar's X and Y functions play an important role in the diffuse reflection and transmission problem (cf. Chandrashekhar⁽¹⁾). In a preceding paper (cf. Bellmanet al.⁽¹⁰⁾), graphs and selected tables of these functions covering wide ranges of slab thickness and albedos for single scattering have been provided. In this paper, making use of a system of coupled integral recurrence relations for finite order X and Y functions (cf. Bellmanet al.⁽¹⁴⁾), numerical results for these basic functions are tabulated up to optical thickness τ = 2.0 from τ = 0.1, assuming the conservative case of isotropic scattering. The maximum order of these functions is taken to be fifteenth. It is shown that the accuracy obtained is satisfactory in the domain under consideration. Furthermore, numerical results for Chandrasekhar's approximation for X and Y functions are also tabulated for stabs of small optical thickness.     

Scaling functions for critical surface scattering

Scattering of x-rays at grazing angles is studied for semi-infinite systems which exhibit a second order phase transition. We calculate the scaling functions for the diffuse scattering intensity aboveT _c and for the Bragg-intensity belowT _c, using -expansion techniques. Corrections to the leading critical surface behaviour and the size of the asymptotic scaling region are discussed. In addition, we obtain the crossover function for the excess susceptibility _s.This work was supported by Bundesministerium für Forschung und Technologie     

Numerical simulation methods for rough surface scattering

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.     

Numerical simulation and experimental results of ultrasonic waves scattering on a model of the artery

The aim of this paper is to compare the results of the mathematical modeling and experimental results of the ultrasonic waves scattering in the inhomogeneous dissipative medium. The research was carried out for an artery model (a pipe made of a latex), with internal diameter of 5 mm and wall thickness of 1.25 mm. The numerical solver was created for calculation of the fields of ultrasonic beams and scattered fields under different boundary conditions, different angles and transversal displacement of ultrasonic beams with respect to the position of the arterial wall. The investigations employed the VED ultrasonic apparatus. The good agreement between the numerical calculation and experimental results was obtained.     

Numerical simulation methods for rough surface scattering

Abstract

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.     

Algebraic recurrence relations for the finite-order scattering and transmission functions

In preceding papers (cf. Bellmanet al.(^1,2,3,), integral recurrence for the finite-order scattering and transmission functions have been given in connection with integral recurrence relations for the finite order X- and Y-functions with the aid of an initial-value method. In the present paper, it is shown how to find algebraic recurrence relations for the finite-order scattering and transmission functions in terms of finite-order X- and Y-functions without referring to an initial-value method. These recurrence relations are suitable for the numerical computation of the finite-order scattering and transmission functions by use of a digital computer. Furthermore, in Tables 1–3, a numerical example of the finite-order reflection functions for optical thickness 0·2 with albedo = 1 is listed. Table 4 furnishes an example of the cumulative reflection functions for optical thickness 0·2 with three different albedos.     

An electrical model for the elastic scattering of thermal neutrons

In slow neutron physics, the true neutron-nucleus potential is replaced by the Fermi pseudo-potential. We discuss the adequacy of applying this potential for various elements and various values of the wave vectork. In order to do this, we form an electrical model for the Schrödinger equation that describes the elastic scattering of thermal neutrons. It is possible to obtain an analytical solution for the wave function at a sufficiently large distance from the origin. Using our model, we estimate this distance for the Fermi pseudo-potential. This serves as a criterion for discussing the adequacy of replacing the true potential with the Fermi pseudo-potential. A network approach to the numerical problem is shown to have some advantages over a purely numerical one.     

Numerical simulation for all-optical Thomson scattering X-ray source

Energy spectra, angular distributions, and temporal profiles of the photons produced by an all-optical Thomson scat- tering X-ray source are explored through numerical simulations based on the parameters of the SILEX-I laser system （800 nm, 30 fs, 300 TW） and the previous wakefield acceleration experimental results. The simulation results show that X-ray pulses with a duration of 30 fs and an emission angle of 50 mrad can be produced from such a source. Using the optimized electron parameters, X-ray pulses with better directivity and narrower energy spectra can be obtained. Besides the electron parameters, the laser parameters such as the wavelength, pulse duration, and spot size also affect the X-ray yield, the angular distribution, and the maximum photon energy, except the X-ray pulse duration which is slightly changed for the case of ultrafast laser-electron interaction.     

Numerical solution of inverse scattering for near-field optics

A novel regularized recursive linearization method is developed for a two-dimensional inverse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium located on a substrate from data accessible through photon scanning tunneling microscopy experiments. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to weak scattering at a low frequency, and each update is obtained by continuation on the wavenumber from solutions of one forward problem and one adjoint problem of the Helmholtz equation.     

Completeness of the Coulomb scattering wave functions

The completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is the basic ingredient of quantum mechanics, plays an important role in nuclear reaction and nuclear-structure theory. Here we present the first formal proof of the completeness of the two-body Coulomb scattering wave functions for a repulsive unscreened Coulomb potential using Newton’s method (R. Newton, J. Math. Phys. 1, 319 (1960)). The proof allows us to claim that the eigenfunctions of the two-body Hamiltonian, with the potential given by the sum of the repulsive Coulomb plus short-range (nuclear) potentials, form a complete set. It also allows one to extend Berggren’s approach for the modification of the complete set of eigenfunctions by including the resonances for charged particles. We also demonstrate that the resonant Gamow functions with Coulomb tail can be regularized using Zel’dovich’s regularization method. Communicated by U.-G. Meiβner For the continuum spectrum the eigenfunctions are not square-integrable, strictly speaking we need to use a rigged Hilbert space which extends the normal Hilbert space by bringing together the discrete and continuum spectrum eigenstates.     

Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method

A general algorithm is introduced for the analysis of near-field radiative heat transfer in one-dimensional multi-layered structures. The method is based on the solution of dyadic Green's functions, where the amplitude of the fields in each layer is calculated via a scattering matrix approach. Several tests are presented where cubic boron nitride is used in the simulations. It is shown that a film emitter thicker than 1 μm provides the same spectral distribution of near-field radiative flux as obtained from a bulk emitter. Further simulations have pointed out that the presence of a body in close proximity to an emitter can alter the near-field spectrum emitted. This algorithm can be employed to study thermal one-dimensional layered media and photonic crystals in the near-field in order to design radiators optimizing the performances of nanoscale-gap thermophotovoltaic power generators.     

Regge-pole model for the invariant functions of the nucleon-nucleon and meson-nucleon scattering

Unlike Toller poles, Regge type poles in the invariant functions of the nucleon-nucleon and meson-nucleon scattering produce finite conspiracy families which have the remarkable property that the residues of the parent and daughter trajectories automatically satisfy the factorization hypothesis. Possible interpretations of these families in terms of known resonances are discussed as well as consequences for the high energy behaviour of the nucleon-nucleon and meson-nucleon scattering.     

Fast weighting functions for retrievals from limb scattering measurements

The satellite-borne UV-visible-NIR spectrometers SCIAMACHY and OSIRIS perform operational measurements of the Earth's limb radiance with global coverage. The vertically resolved atmospheric composition is retrieved from the measurements. We have computed synthetic limb measurements and weighting functions (WFs) with several orders of scattering and surface reflection. Comparisons reveal the wavelength-dependent contributions of single scattering and the second orders of scattering and surface reflection. We have also performed test retrievals of the and profiles with the program package SCIARAYS. They prove that the single scattering approximation is sufficient for the calculation of the WFs during the retrieval process. We conclude that algorithms for the analysis of limb scattering measurements can be accelerated by neglecting higher orders of scattering in the WF calculations.     

Transition probability functions for applications of inelastic electron scattering

In this work, the transition matrix elements for inelastic electron scattering are investigated which are the central quantity for interpreting experiments. The angular part is given by spherical harmonics. For the weighted radial wave function overlap, analytic expressions are derived in the Slater-type and the hydrogen-like orbital models. These expressions are shown to be composed of a finite sum of polynomials and elementary trigonometric functions. Hence, they are easy to use, require little computation time, and are significantly more accurate than commonly used approximations.     

Discrete hydrodynamics: Numerical results for the soft-sphere viscosity

Numerical calculations have been done of the viscosity of the soft-sphere liquid, using a new molecular dynamics technique. It is based on a formulation of hydrodynamics which is discrete in space and time, and exactly renormalizable. The present data turn out to be sufficient to estimate the viscosity, but determination of the full equations of motion (and therefore renormalization) requires further calculations using a smaller discrete time interval; these are presently under way. The present results indicate that this method is more than 100 times more efficient than previous (Green-Kubo or nonequilibrium molecular dynamics) methods. This suggests that the discrete formulation is the most natural way to approach hydrodynamics.