Single-scattering Approximation Research Articles

Linearized inverse scattering problems in acoustics and elasticity

Using the single-scattering approximation we invert for the material parameters of an acoustic two-parameter medium and then for a three-parameter isotropic elastic medium. Our procedure is related to various methods of depth migration in seismics, i.e. methods for locating major discontinuities in the subsurface material without specifying which quantities are discontinuous or by how much they jump. Our asymptotic multiparameter inversion makes use of amplitude information to reconstruct the size of the jumps in the parameters describing the medium. We allow spatially varying background parameters (both vertically and laterally) and an almost arbitrary source-receiver configuration. The computation is performed in the time domain and we use all available data even if it is redundant. This ability to incorporate the redundant information in a natural way is based upon a formula for double integrals over spheres. We solve for perturbations in different parameters treating separately P-to- P, P-to- S, S-to- P, and S-to- S data. It turns out that one may invert using subsets of the data, or all of it together. We also describe modifications to our scheme which allow us to use the Kirchhoff instead of the Born approximation for the forward problem when the scatterers are smooth surfaces of discontinuity.

A lower bound for the solution of the radiative transfer equation in finite clouds

An easily computable lower bound for the radiance within a cloud of arbitrary shape is derived in this paper. The bound is used to estimate a priori the error in the single-scattering approximation to the solution of the radiative transfer equation.

Porosity estimation using the frequency dependence of the ultrasonic attenuation

A new technique is reported for estimating the volume fraction of porosity in structural materials. The estimate for the volume fraction is proportional to the slope of the ultrasonic attenuation when plotted as a function of frequency. Both theory and experiment are considered. The theory, appropriate for dilute porosity, uses the uncorrelated, single-scatter approximation. An “attenuation slope” algorithm is derived within this approximation and its limits of validity are tested by computer simulation. Experimental tests consist of three parts. First, the method is compared with other existing techniques through estimates from the published data on gas porosity in aluminum casts. For the second test, cylindrical porosity is simulated by parallel through-holes drilled in aluminum blocks. Finally, the attenuation in porous graphite-epoxy samples is measured and compared with results predicted from theory.

Light extinction in a dispersion of small nematic droplets

A study is made of the light-transmission properties of materials in which submicrometer-sized nematic droplets are dispersed in a solid polymeric matrix. Theoretical evaluations of the differential and total cross sections for droplets with different nematic configurations are performed in the Rayleigh–Gans approximation. The corresponding attenuation coefficients are calculated in a single-scattering approximation in which the effect of interdroplet interference is taken into account. The spatial distribution of the droplets is introduced by means of a pair-correlation function. Theoretical calculations are in excellent agreement with experimental data obtained from thin films of materials consisting of cyanobiphenyl droplets dispersed in an epoxy resin.

Radiative transfer in stratified waters: a single-scattering approximation for irradiance

The singly scattered irradiance (SSI) model is an approximate radiative transfer model designed to describe optically shallow stratified waters. It is intended to be a tool to aid interpretation of remote (satellite or aircraft) observations of water color. The SSI model was derived in analogy to single-scattering radiance models. As it is based on irradiance (rather than radiance), multiple-scattering events are included implicitly, yielding a model which is expressed in terms of readily measurable parameters and which preserves mathematical simplicity without sacrificing many important subtleties of the radiative transfer process. The success of the SSI model relies on the underwater radiance distribution being nearly independent of sun angle, cloud cover and depth, and the resulting quasi-inherent property of the irradiance attenuation coefficient, irradiance reflectance, and distribution functions. A derivation of the model is presented, along with evidence for the qualitative and quantitative correctness of the model.

Theory of extended energy loss fine structure in the reflection mode

An analytic expression for the current detected in an extended energy loss fine structure experiment in the reflection mode is derived, which takes into account the diffraction by the atoms of the crystal of the two electrons participating in the excitation process. Within a single-scattering approximation, the authors show that the signal contains two oscillating factors, one of the EXAFS type and the other, which has been neglected in all studies until now, similar to a photo-electron diffraction signal. They perform a model computation in order to show the experimental conditions in which an EXAFS-type analysis still applies and the precision in the distances that may be expected. They also propose a new type of experiment in which the detection is angularly resolved and which would allow a bond angle determination in addition to the distance determination, under specified excitation conditions.

Theory of angle-resolved photoemission extended fine structure.

We present a theory for photoelectron scattering in the 100--1000 eV energy range designed to simulate experimental measurements of angle-resolved photoemission extended fine structure (ARPEFS) from ordered surfaces. The zero-order problem of photoabsorption in the solid is treated first, followed by a scattering problem which incorporates the scattering ion cores in a perturbation series (cluster expansion). The dynamics of core-hole relaxation are discussed, but the dynamical effects are shown to be small. The Taylor-series magnetic-quantum-number expansion is used for the curved-wave, multiple-scattering equations. We argue that a velocity-dependent surface barrier gives primarily an inner potential shift, with no clear evidence for surface electron refraction. Analytic formulas for aperture integration are derived and thermal averaging in a correlated Debye model is extended to multiple scattering. Reasonable values for nonstructural parameters in the theory are shown to give very good simulations of the experimental ARPEFS measurements from c(2\ifmmode\times\else\texttimes\fi{}2)S/Ni(001) in contrast to previous theoretical calculations. We find, in agreement with full multiple-scattering calculations, that forward focusing is a fundamental feature of ARPEFS and that curved-wave corrections are essential for quantitative results. Since the scattering path-length difference is not appreciably altered by forward scattering, the ARPEFS oscillation frequency is equal to the geometrical path-length difference plus a small potential phase shift, but the amplitude and constant phase of the oscillations cannot be predicted by theories based upon single-scattering or plane-wave approximations.

Effects of multiple scattering on time- and depth-resolved signals in airborne lidar systems

A semianalytic Monte Carlo radiative transfer model (SALMON) is employed to probe the effects of multiple-scattering events on the time- and depth-resolved lidar signals from homogeneous aqueous media. The effective total attenuation coefficients in the single-scattering approximation are determined as functions of dimensionless parameters characterizing the lidar system and the medium. Results show that single-scattering events dominate when these parameters are close to their lower bounds and that when their values exceed unity multiple-scattering events dominate.

The role of vibrational excitations in collision-induced dissociation using Faddeev–AGS theory

This paper examines the role of vibrational (and rotational) excitations in collision-induced dissociation in atom–diatom reactions. We treat a model system of identical, bosonic hydrogen atoms and investigate the total H+H2→H+H+H cross section as functions of total center-of-mass (c.m.) energy and vibrational–rotational quantum numbers v, j. The investigations are based on Faddeev–AGS theory and both three-dimensional (3D) and collinear (one-dimensional, 1D) geometries are considered. We derive both low- and high-energy relations between the total dissociation cross section, c.m. energy, and the vibrational–rotational wave functions, employing the single-scattering approximation of Faddeev theory. We apply these relations to the spectrum of the Kolos–Wolniewicz potential, both in three dimensions and one dimension. For collisions with a fixed total c.m. energy, our investigations predict considerable vibrational enhancement of the total cross section in the low-energy limit, with this enhancement much more pronounced in the true 3D dynamics than in the artificial collinear geometry, indicating that translational energy is less effective than vibrational in CID. As the c.m. (or translational) energy increases, approaching infinity, a transition occurs to either no enhancement or inhibition, or to slight vibrational inhibition, depending on the nature of the underlying interaction. This property mainly results from how the momentum distributions of the diatomic wave functions sense the available phase space of the dissociation reaction. In light of the anticipated failure of the single-scattering approximation at low collision energies, the predicted trends for vibrational enhancement are tested by calculating dissociation cross sections with exact Faddeev theory for weakened H–H potentials. While the single-scattering approximation does indeed break down badly, the ratio of cross sections for different vibrational states is largely unchanged from the exact. An explanation of this fact, using a modified single-scattering argument with radially cut-off diatomic wave functions, is discussed and developed physically.

Reflection and Emission Models for Deserts Derived from Nimbus-7 ERB Scanner Measurements

Broadband shortwave and longwave radiance measurements obtained from the Nimbus-7 Earth Radiation Budget scanner were used to develop reflectance and emittance models for the Sahara-Arabian, Gibson, and Saudi Deserts. The models were established by fitting the satellite measurements to analytic functions. For the shortwave, the model function is based on an approximate solution to the radiative transfer equation. The bidirectional-reflectance function was obtained from a single-scattering approximation with a Rayleigh-like phase function. The directional-reflectance model followed from integration of the bidirectional model and is a function of the sum and product of cosine solar and viewing zenith angles, thus satisfying reciprocity between these angles. The emittance model was based on a simple power-law of cosine viewing zenith angle.

Continuum spectra for 165 MeV pion-nucleus single charge exchange.

Continuum spectra for pion single charge-exchange scattering at 165 MeV kinetic energy are calculated for a number of nuclei within the single-scattering approximation over a wide range of nuclear excitation energy. The discrepancy between our calculations and the data suggests that multiple-scattering processes may contribute significantly to the charge-exchange spectra. This suggestion is consistent with the present understanding of the delta-nucleus interaction as given, for example, by the isobar-hole model.

X-ray-absorption spectra in KrF2 and FeCl2: Theory.

Theoretical x-ray-absorption near-edge structure (XANES) and extended x-ray-absorption fine structure (EXAFS) of molecular and crystalline ${\mathrm{KrF}}_{2}$ and gas-phase ${\mathrm{FeCl}}_{2}$ have been obtained by using a modified X\ensuremath{\alpha} scattered-wave procedure. EXAFS calculations are made in the single-scattering approximation. Effects of multiple scattering near the absorption edge have been investigated by comparing predicted XANES and EXAFS spectra in the near-edge region. Two exchange-correlation potentials, X\ensuremath{\alpha} and energy-dependent Dirac (ED), were incorporated in the EXAFS calculations. Comparison between experimental and theoretical results confirm that the ED potential is more valid for EXAFS spectra. Molecular field splitting as well as resonance coupling were demonstrated by comparing the theoretical XANES of the isolated ${\mathrm{KrF}}_{2}$ molecule to that of ${\mathrm{FeCl}}_{2}$. The single-particle XANES calculation of molecular ${\mathrm{FeCl}}_{2}$ is compared with the experimental crystalline ${\mathrm{FeCl}}_{2}$ results of Stern.

Wave scattering by an irregularity slab embedded in a stratitfied medium: Applications to ionospheric propagation

Plane wave scattering by an irregularity slab embedded in a linearly stratified isotropic medium is studied under the single scatter approximation. In addition to regular reflection from the stratified background, both the incident and the reflected waves are scattered by the random irregularities. The behavior of the unperturbed field in the neighborhood of the turning point is accurately taken into account by using the full wave solution. The scattered fields can be interpreted in terms of the Bragg condition similar to the case of the well‐known Booker‐Gordon form, but with four terms accounting for the reflection effect of the turning point on both the unperturbed and scattered waves. The case of elongated irregularities is studied in detail. Analytical expression for the angular spectrum of the average scattered field intensity is derived, and its physical meaning discussed. In comparison with the case where the background medium is homogeneous, the relation to the two‐dimensional Booker‐Gordon formula is clarified. Applications to high‐frequency propagation in the disturbed ionosphere will be discussed.

Radiation of two-phase media bounded by reflecting and radiating surfaces

Regularities of radiation propagation in anisotropically dissipating media bounded by surfaces with given optical properties are investigated in the single-scattering approximation.

Sound scattering in a turbulent medium

Tatarski [V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw‐Hill, New York, 1960)] has derived an expression for the effective scattering cross section of an acoustic wave incident on a remote volume of atomospheric turbulence. His result is subject to the single scatter approximation and the restriction that both transmitter and receiver be in the farfield of the large scale inhomogeneities within the volume. We derive an expression for the effective scattering cross section valid in both the near and farfield and extend these results to include the scatter of acoustic waves from underwater turbulence.

Allowance for multiple scattering in proton energy spectra

Abstract The energy spectra of 6.3 MeV energy protons backscattered by a thick Au target are studied experimentally for a broad range of varying backscattering angle. It has been shown in what manner the deviations of the forms of the energy spectra from the spectral forms calculated in the single-scattering approximation increase with decreasing scattering angle. It has been shown also that at not too high proton energy loss in the target the deviations may be described by a correction function which takes into account the contribution to the energy spectra of multiple particle scattering in the target.

Epitaxially grown paraffin: An electron crystal structure analysis

Epitaxial crystal growth proceeds in a manner entirely different from solution- growth. By a careful two-dimensional lattice matching in the substrate and the material of interest, crystallization in a desired orientation may be obtain. The area of each single microcrystal will, of course, depend upon the crystallinity of the substrate and the possibility of different growth directions thereon.Epitaxy offers considerable advantages over solution growth, since resultant crystals give electron diffraction intensity data suitable for total structure determination. Crystallographic phasing techniques such as Patterson maps and direct methods rely upon the kinematic or single-scattering approximation. Their usefulness thus requires minimizing perturbations to the intensity data due to dynamical scattering and crystal bending. Diffracted beam amplitudes are most insensitive to crystal bends when the electron beam is parallel to a short unit cell dimension. Dynamical scattering effects are reduced if there are few repeat units in the beam direction, which requires very thin crystals. The value of this approach is illustrated with paraffin.

Nuclear Excitation by Gamma Rays from Cobalt-60

A photoexcitation process by gamma rays from a 60Co source has been studied for the nuclei of 87Sr, 111Cd, 115In, and 176Lu. The induced isomeric activity was measured with a Ge(Li) detector. The flux of photons scattered into the target has been estimated with the Monte Carlo method using the single-scattering approximation. From the observed induced activities and the calculated photon flux, the integral cross sections for isomer production by photoexcitation were obtained and compared with other experimental data.

An Approximation to Multiple Scattering in the Earth's Atmosphere: Almucantar Radiance Formulation

The retrieval of aerosol size distributions from solar aureole measurements, in the past, have been made tractable by assuming the single-scattering approximation, since the inclusion of multiple scattering by solving the complete radiative transfer equation would make such retrievals prohibitively expensive. Recently, ways have been sought to improve the accuracy of such retrievals without increasing the computational effort afforded by the single-scattering approximation. This has led to the development of a perturbation approximation, which makes allowance for molecular multiple scattering; aerosol multiple-scattering events are largely ignored. In this paper, the derivation of an empirical expression to account for this molecular multiple scattering is discussed, and the accuracy of this approximation is illustrated by some examples of the direct problem for the solar aureole.

Theory of excess-electron mobility in compressed argon

The purpose of this paper is to calculate the mobility of excess electrons in compressed argon (20-100 atm) both in single-scattering and multiple-scattering approximations. Calculations in the single-scattering approximation are carried out with the aid of the Cohen-Lekner theory. A new formula for the mobility of excess electrons in the multiple-scattering approximation is derived from the Kubo-Greenwood relation for electric conductivity. Numerical results inferred from this formula and from the Cohen-Lekner theory are compared with the experimental data reported in the literature. The results obtained in this paper indicate that the Cohen-Lekner theory is applicable only at very low densities of gaseous argon and that inclusion of the long-range part of the electron-atom potential gives a significant improvement in the theoretical results. The multiple-scattering corrections have been estimated to be about (10-25)% of the value of mobility calculated in the single-scattering approximation.