Rayleigh-Gans-Debye Approximation Research Articles

The amplitude of light scattering by a hexagonal column with hollow ends in the Rayleigh-Gans-Debye approximation

The formulas for light scattering amplitude and phase function [or element of scattering matrixf11] of hexagonal column with concave ends in the Rayleigh-Gans-Debye (RGD) approximation are derived. Numerical results for the phase function of light scattering of hexagonal column with pyramidal hollow ends in the RGD approximation and in the method of Purcell-Pennypacker (or discrete dipole method, or coupled dipole method) are compared. The good agreement for hexagonal columns having small phase shifts has been found.

Генерация второй гармоники в поверхностном слое диэлектрической сфероидальной частицы. I. Аналитическое решение

The problem of second-harmonic generation by a plane elliptically polarized electromagnetic wave in a thin optically nonlinear surface layer of a dielectric particle shaped as an ellipsoid of revolution is solved. The generalized Rayleigh–Gans–Debye approximation is used for an analytical description with taking into account the difference in refractive indices of the medium corresponding to the frequencies of the exciting and generated radiation. The limiting forms of functions are obtained, with the use of which the electric field strength of the generated radiation is expressed. The order of dependence of these functions on the linear dimensions is found, when the lengths of the semiaxes of the particle are small compared to the wavelength of the exciting radiation and their ratio remains constant. It was found that the power density of the generated radiation in this case is determined to a greater extent by the chiral components of the nonlinear dielectric susceptibility tensor and is proportional to the fourth power of the length of the semiaxis of the particle, if the shape of the spheroidal particle differs significantly from the spherical one. The solution of this problem, obtained by other authors, is supplemented for the possibility of applying to the description of generation in the surface layer of a dielectric particle not only in the form of a prolate, but also in the form of an oblate spheroid. Corrections of inaccuracies and misprints made in similar works by other authors are proposed. The relationships between the formulas used in these works are found, taking into account the corrections and the formulas used in this work.

Absorption and scattering of light by soot aggregates with uniform and pendular ring coatings

The magnitude of radiative forcing by soot is strongly dependent on its morphology and mixing state, which continuously evolve during atmospheric aging. One of the major soot aging mechanisms is via interaction with condensable trace-gas chemicals, and our previous study has shown the formation of two distinct coating distributions on fractal soot aggregates, primarily depending on the radius of monomer spheres and supersaturation of the condensing vapor. The two coating distributions, spherical shells around monomers and pendular rings in junctions between monomers, differ profoundly in their ability to restructure soot aggregates. Here, we investigate the impact of these two distributions on the optical properties of soot, using Discrete Dipole Approximation (DDA) calculations, which are also compared against the Rayleigh Debye Gans (RDG) approximation and the core-shell model. For most optical parameters, the difference between the two coating distributions was minor, except for the asymmetry parameter, where it was significant (5–13%). We found that the placement of condensate in junctions significantly depressed the asymmetry parameter for most coating volumes, thus promoting isotropic scattering by soot. Compared to a uniform condensate distribution, the presence of coating between monomers always resulted in slightly larger absorption, scattering and single scattering albedo (SSA), with the difference reaching ~10% for scattering at high coating volumes. We found that the RDG approximation underestimated absorption, but was able to predict scattering to within ~3%. On the other hand, the core-shell model significantly overestimated absorption, scattering, and SSA, especially for thinly coated soot.

Refractive Indices of Biomass Burning Aerosols Obtained from African Biomass Fuels Using RDG Approximation

Biomass burning (BB) aerosols contribute to climate forcing, but much is still unknown about the extent of this forcing, owing partially to the high level of uncertainty regarding BB aerosol optical properties. A key optical parameter is the refractive index (RI), which influences the absorbing and scattering properties of aerosols. This quantity is not measured directly, but it is obtained by fitting the measured scattering cross section and extinction cross section to a theoretical model using the RI as a fitting parameter. We used the Rayleigh–Debye–Gans (RDG) approximation to retrieve the complex RI of freshly emitted BB aerosol from two fuels (eucalyptus and olive) from Africa in the spectral range of 500–580 nm. Experimental measurements were carried out using cavity ring-down spectroscopy to measure extinction over the range of wavelengths of 500–580 nm and nephelometry to measure scattering at three wavelengths of 450, 550, and 700 nm for size-selected BB aerosol particles. The fuels were combusted in a tube furnace at a temperature of 800 °C, which is representative of the flaming stage of burning. Filter samples were collected and imaged using tunneling electron microscopy to obtain information on the morphology and size of the particles, which was used in the RDG calculations. The mean radii of the monomers were 27.8 and 31.5 nm for the eucalyptus and the olive fuels, respectively. The components of the retrieved complex RI were in the range of 1.31 ≤ n ≤ 1.56 and 0.045 ≤ k ≤ 0.468. The real and complex parts of the RI increase with increasing particle mobility diameter. The real part of the RI is lower, and the imaginary part is higher than what was recommended in literature for black carbon generated by propane or field measurements from fires of mixed wood samples. Fuel dependent results from controlled laboratory experiments can be used in climate modeling efforts and to constrain field measurements from biomass burning.

Second-Harmonic Generation from a Thin Cylindrical Layer: II. Analysis of Solution

The problem of second-harmonic generation by a plane elliptically polarized electromagnetic wave from a thin optically nonlinear layer on the surface of a cylindrical dielectric particle of finite dimensions placed in a dielectric medium has been solved analytically in the Rayleigh–Gans–Debye approximation. The result has been presented in the tensor and vector forms in the general case, in which the nonlinear dielectric susceptibility tensor has four independent components (one chiral component and three nonchiral components). It has been shown for the first time that the contribution of chiral components to the generation from the butt-end surfaces of a cylindrical particle differs in phase from the contribution of nonchiral components. It has also been revealed that, if the linear dimensions of a cylindrical particle (height and base radius) are small, the radiation determined by the chiral component of the second-order nonlinear dielectric susceptibility tensor makes a predominant contribution to the second-harmonic generation from a nonlinear cylindrical layer (butt-end and lateral surfaces).

Генерация второй гармоники от тонкого цилиндрического слоя. I. Аналитическое решение

In the Rayleigh–Gans–Debye approximation an analytical solution is obtained for the problem of second harmonic generation by a plane electromagnetic wave with elliptical polarization from a thin optically nonlinear layer on the surface of a cylindrical dielectric particle of finite size placed in a dielectric. The result is presented in tensor and vector forms in general case, in which the nonlinear dielectric susceptibility tensor has four independent components (one chiral and three non-chiral). For the first time it is shown that at generation from the end plane surfaces of a cylindrical particle the contribution of chiral components differs in phase from that of non-chiral components. It is also found that for small linear dimensions of the cylindrical particle (height and radius of the base), the radiation due to the chiral component of the second-order nonlinear dielectric susceptibility tensor makes a dominant contribution to the second harmonic generation from a non-linear cylindrical layer (end and side surfaces).

Sum-Frequency Generation from a Thin Cylindrical Layer

In the Rayleigh–Gans–Debye approximation, we have solved the problem of the sum-frequency generation by two plane elliptically polarized electromagnetic waves from the surface of a dielectric particle of a cylindrical shape that is coated by a thin layer possessing nonlinear optical properties. The formulas that describe the sum-frequency field have been presented in the tensor and vector forms for the second-order nonlinear dielectric susceptibility tensor, which was chosen in the general form, containing chiral components. Expressions describing the sum-frequency field from the cylindrical particle ends have been obtained for the case of a nonlinear layer possessing chiral properties. Three-dimensional directivity patterns of the sum-frequency radiation have been analyzed for different combinations of parameters (angles of incidence, degrees of ellipticity, orientations of polarization ellipses, cylindrical particle dimensions). The mathematical properties of the spatial distribution functions of the sum-frequency field, which characterize the symmetry of directivity patterns, have been revealed.

Second-harmonic generation from a thin spherical layer and No-generation conditions

In the Rayleigh–Gans–Debye approximation, we solve the problem of second-harmonic generation by an elliptically polarized electromagnetic wave incident on the surface of a spherical particle that is coated by an optically nonlinear layer and is placed in a dielectric. The formulas obtained characterize the spatial distribution of the electric field of the second harmonic in the far-field zone. The most general form of the second-order dielectric susceptibility tensor is considered, which contains four independent components, with three of them being nonchiral and one, chiral. Consistency and inconsistencies between the obtained solution and formulas from works of other authors are found. We analyze the directivity patterns that characterize the spatial distribution of the generated radiation for the nonchiral layer and their dependences on the anisotropy and ellipticity coefficients of the incident wave. It is found that, with increasing radius of the nonlinear layer, the generated radiation becomes more directional. Combinations of parameters for which no radiation is generated are revealed. Based on this, we propose methods for experimental determination of the anisotropy coefficients.

Light scattering by hexagonal ice crystal in the Wentzel–Kramers–Brillouin approximation

The Wentzel–Kramers–Brillouin (WKB) method is applied for hexagonal ice crystal. A closed form expression of the form factor is derived for an hexagonal ice crystal. The other approximations, such as the Rayleigh–Gans–Debye approximation (RGD) and the anomalous diffraction (AD), are deducted as particular cases from our formula. Furthermore, the coefficient of extinction is also given. This study is made for two special cases of the incidence (flat incident and edge-on incidence). To illustrate our analytical results, some numerical examples are analyzed.

A modified Rayleigh-Gans-Debye formula for small angle X-ray scattering by interstellar dust grains

A widely used approximation in studies relating to small angle differential scattering cross-section of X-rays scattered by interstellar dust grains is the well known Rayleigh-Gans-Debye approximation (RGDA). The validity of this approximation, however, is limited only to X-ray energies greater than about 1 keV. At lower energies, this approximation overestimates the exact results. In this paper a modification to the RGDA is suggested. It is shown that a combination of the RGDA with Ramsauer approximation retains the formal simplicity of the RGDA and also yields good agreement with Mie computations at all X-ray energies.

Absorption and scattering by fractal aggregates and by their equivalent coated spheres

This paper demonstrates that the absorption and scattering cross-sections and the asymmetry factor of randomly oriented fractal aggregates of spherical monomers can be rapidly estimated as those of coated spheres with equivalent volume and average projected area. This was established for fractal aggregates with fractal dimension ranging from 2.0 to 3.0 and composed of up to 1000 monodisperse or polydisperse monomers with a wide range of size parameter and relative complex index of refraction. This equivalent coated sphere approximation was able to capture the effects of both multiple scattering and shading among constituent monomers on the integral radiation characteristics of the aggregates. It was shown to be superior to the Rayleigh–Debye–Gans approximation and to the equivalent coated sphere approximation proposed by Latimer. However, the scattering matrix element ratios of equivalent coated spheres featured large angular oscillations caused by internal reflection in the coating which were not observed in those of the corresponding fractal aggregates. Finally, the scattering phase function and the scattering matrix elements of aggregates with large monomer size parameter were found to have unique features that could be used in remote sensing applications.

Light scattering by cylindrical nanoparticles: Limits of applicability of the Rayleigh-Gans-Debye approximation

Applicability of the Rayleigh-Gans-Debye (RGD) approximation for describing light scattering by nanoparticles with large dielectric losses (such as carbon nanotubes) is analyzed. By a comparison of the approximate results with exact ones, it is shown that the presence of dielectric losses expands the range of applicability of the RGD approximation. This conclusion is illustrated by a differential cross-section diagram of scattering by a multiwall carbon nanotube.

The single scattering properties of soot aggregates with concentric core–shell spherical monomers

Anthropogenic soot aerosols are shown as complex, fractal-like aggregated structures with high light absorption efficiency. In atmospheric environment, soot monomers may tend to acquire a weakly absorbing coating, such as an organic coating, which introduces further complexity to the optical properties of the aggregates. The single scattering properties of soot aggregates can be significantly influenced by the coated status of these kinds of aerosols. In this article, the monomers of fractal soot aggregates are modelled as semi-external mixtures (physical contact) with constant radius of soot core and variable sizes of the coating for specific soot volume fractions. The single scattering properties of these coated soot particles, such as phase function, the cross sections of extinction and absorption, single scattering albedo (SSA) and asymmetry parameter (ASY), are calculated using the numerically exact superposition T-matrix method. The random-orientation averaging results have shown that the single scattering properties of these coated soot aggregates are significantly different from the single volume-equivalent core–shell sphere approximation using the Mie theory and the homogeneous aggregates with uncoated monomers using the effective medium theory, such as Maxwell-Garnett and Bruggemann approximations, which overestimate backscattering of coated soot. It is found that the SSA and cross sections of extinction and absorption are increased for soot aggregates with thicker weakly absorbing coating on the monomers. Especially, the SSA values of these simulated aggregates with less soot core volume fractions are remarkably (~50% for core volume fraction of soot aggregates of 0.5, ~100% for a core volume fraction of 0.2, at 0.67μm) larger than for uncoated soot particles without consideration of coating. Moreover, the cross sections of extinction and absorption are underestimated by the computation of equivalent homogeneous fractal aggregate approximation (within 5% for the T-matrix method and 10–25% for the Rayleigh–Debye–Gans approximation due to different soot volume fractions). Further understanding of the optical properties of these coated soot aggregates would be helpful for both environment monitoring and climate studies.

Investigation of Absorption and Scattering Properties of Soot Aggregates of Different Fractal Dimension at 532 nm Using RDG and GMM

Radiative properties of numerically generated fractal soot aggregates of different fractal dimensions were studied using the numerically accurate generalized Mie-solution method (GMM) and the Rayleigh-Debye-Gans (RDG) approximate theory. Fractal aggregates of identical prefactor but different fractal dimensions, namely, 1.4, 1.78, and 2.1, were generated numerically using a tunable algorithm of cluster–cluster aggregation for aggregates containing up to 800 primary particles. Radiative properties of these aggregates were calculated at a wavelength of 532 nm assuming a soot refractive index of 1.6 + 0.6i. Four commonly used structure factors in the RDG approximation were used to investigate the effect of structure factor on the differential and total scattering cross-sections and the asymmetry factor. The differential and total scattering properties calculated using the RDG approximation become increasingly sensitive to the structure factor with increasing the fractal dimension. Primary particle interactions are the fundamental mechanism for the aggregate absorption enhancement for small aggregates and the shielding effect for larger aggregates. The extent of these two competing factors is dependent on the fractal dimension and aggregate size. RDG reasonably predicts the effect of fractal dimension on the scattering properties, but fails to account for the effect of aggregation or fractal morphology on the absorption property of fractal soot aggregates, though the error is in general less than 15%.Copyright 2013 American Association for Aerosol Research

Light Scattering by a Prism and Pyramid in the Rayleigh-Gans-Debye Approximation

The general approach to finding the form factor of a compound particle and a system of particles in the Ray-leigh-Gans-Debye (RGD) approximation is considered. The rotational-translational properties of light scattering amplitude in the RGD approximation are formulated. Using such properties, the analytical expressions for the amplitude of light scattering by a prism and pyramid with an arbitrary polygonal base in the RGD approximation are obtained. The phase functions of light scattering by a prism and pyramid in the RGD approximation are computed.

Relationship between the morphology and the light scattering cross section of optically soft aggregates

The purpose of this paper is to establish the suitable analytical expressions for the light scattering cross section of aggregates constituted by optically soft primary particles. The used optical approximation is the well-known Rayleigh–Debye–Gans approximation. The geometrical description of the aggregate is based on the pair, or distance, distribution density function describing at once the two spatial scales (primary particle and aggregate). Firstly, a study of the two-particle set is presented. An exact and an approximated formulations are proposed for the distance distribution density and the scattering cross section. Then the corresponding results are extended to the aggregates.

Turbidity Spectroscopy for Characterization of Submicroscopic Drug Carriers, Such as Nanoparticles and Lipid Vesicles: Size Determination

To apply UV/Vis spectrometry for characterization of submicroscopic drug carriers, such as nanoparticles and lipid vesicles. We first investigated theoretically, within the framework of the Rayleigh-Gans-Debye approximation (RGDA), parameters affecting turbidity spectrum, τ(λ), of nanosized light scatterers. We then analyzed, within the framework of the RGDA, experimental turbidity spectra (λ = 400-600 nm) of extruded unilamellar vesicle (70 nm ≤ 2r ≤ 110 nm) suspensions to derive vesicle size, using dynamic light scattering results for comparison. We similarly studied the preparations polydispersity and lamellarity and monitored vesicle size changes. Turbidimetry suffices for accurate, fast, and viscosity-independent characterization of submicroscopic particles. Analysis of turbidity spectra, or more precisely wavelength exponent spectra (derivatives of logarithmic turbidity spectra), yielded similar average radii (r = 54.2 ± 0.2 nm; 46.0 ± 0.2 nm; 35.5 ± 0.1 nm) as dynamic light scattering (r = 55.9 ± 1.5 nm; 46.1 ± 0.4 nm; 36.1 ± 0.4 nm). Both methods also revealed similar suspension polydispersity and cholate-induced vesicle size changes in a few nanometer range. Despite its experimental simplicity, the widely accessible turbidimetric method provides accurate size values and is suitable for (continuous) monitoring size stability, or sameness, of submicroscopic drug carriers.

Light scattering by particles of toroidal shape in the Rayleigh-Gans-Debye approximation

Expressions for the calculation of the amplitude and indicatrix of light scattering by optically soft toroidal and tubular particles are obtained in the Rayleigh-Gans-Debye (RGD) approximation. Light scattering cross sections of toroidal particles in the RGD approximation and cross sections obtained by the method of discrete dipoles are numerically compared.

Spectrally resolved light absorption properties of cooled soot from a methane flame

The optical properties of combustion-generated soot, crucial information for quantitative soot emission diagnostics and for climate modeling, have been determined for the particular case of cooled soot from a methane flame. Optical extinction measurements were performed over a wavelength range of 450–750 nm using a novel diffuse-light, spectrally resolved line-of-sight attenuation experiment, and quantified using extractive methods coupled with scanning and transmission electron microscopy in conjunction with a detailed uncertainty analysis. The absorption component of the total measured extinction was isolated by calculating the expected scattering contribution, according to the Rayleigh–Debye–Gans approximation for polydisperse fractal aggregates. In contrast to the large degree of scatter seen in data previously reported in the literature, a consistent trend of negligible variation of the soot absorption refractive index function E(m) with wavelength over the visible was observed (E(m)=0.35±0.03 at wavelengths of 450–750 nm). These new data are also cast in the form of dimensionless extinction, which is independent of the scatter correction, as well as mass absorption cross section, which is independent of the mass density of soot and is commonly used by atmospheric modelers.

The complemented system approach: A novel method for calculating the x-ray scattering from computer simulations

In this paper, we review the main problem concerning the calculation of x-ray scattering of simulated model systems, namely, their finite size. A novel method based on the Rayleigh-Debye-Gans approximation was derived, which allows sidestepping this issue by complementing the missing surroundings of each particle with an average image of the system. The method was designed to operate directly on particle configurations without an intermediate step (e.g., calculation of pair distribution functions): in this way, all information contained in the configurations was preserved. A comparison of the results against those of other known methods showed that the new method combined several favorable properties: an arbitrary q-scale, scattering curves free of truncation artifacts, and good behavior down to the theoretical lower limit of the q-scale. A test of computational efficiency was also performed to establish a relative scale between the speeds of all known methods: the reciprocal lattice approach, the brute force method, the Fourier transform approach, and the newly presented complemented system approach.