Average Number Of Scatterings Research Articles

A new spectrally-invariant approach to the remote sensing of inhomogeneous clouds

Current operational satellite retrievals of cloud optical and microphysical properties go back to the Nakajima-King technique developed at the end of the 1980 s. This technique is based on library calculations for plane-parallel homogeneous clouds. It often works well for overcast skies but leads to substantial errors for inhomogeneous and broken cloud fields where the plane-parallel-geometry assumption is no longer valid. The basic concept of a new technique was first introduced in nuclear physics to quantify the critical conditions of reactors. Based on the eigenvalues of the radiative transfer equation, their approach provides a powerful means to parameterize the structure of 3D media. This parameterization was later successfully applied to relate surface reflectance spectra to 3D canopy structure and is known as the spectrally-invariant approximation. The proposed approach adapts this technique to the remote sensing of cloud properties such as droplet single scattering albedo and the average number of scatterings (which are the fundamental parameters in radiative transfer theory), with an emphasis on quantifying the associated errors and uncertainties. This retrieval is free from the plane-parallel homogeneous cloud assumption.

Influence of turbulence on Ly α scattering

ABSTRACT We develop a Monte Carlo radiative transfer code to study the effect of turbulence with a finite correlation length on scattering of Ly α photons propagating through neutral atomic hydrogen gas. We investigate how the effective mean free path, the emergent spectrum, and the average number of scatterings that Ly α photons experience change in the presence of turbulence. We find that the correlation length is an important and sensitive parameter that has an influence on physically relevant properties of Ly α radiative transfer. In particular, it can significantly, by orders of magnitude, reduce the number of scattering events that the average Ly α photon undergoes before it escapes the turbulent cloud.

Gluon Spectrum to Any Order in Opacity for Jets at the EIC

Jets produced at a future electron-ion collider (EIC) are predominantly initiated by energetic quarks, and the subsequent branching of these quarks into systems of quarks and gluons is heavily modified by multiple scattering in the nuclear medium. In cold nuclear matter and in the quark-gluon plasma, this feature allows jet cross sections and jet substructure to be used as a probe of the medium's properties. Here we present a formalism for computing the gluon substructure of a quark jet to an arbitrary order in opacity, the average number of scatterings in the medium, while maintaining the exact kinematics of the splitting. We present explicit results at second order in opacity for the first time, along with a recursion relation which can be used to construct the solution for arbitrarily high orders in opacity. The theoretical framework demonstrated here applies equally well to light and heavy quark branching, and is easily generalizable to other systems.

Quark branching in QCD matter to any order in opacity beyond the soft gluon emission limit

Cold nuclear matter effects in reactions with nuclei at a future electron-ion collider (EIC) lead to a modification of semi-inclusive hadron production, jet cross sections, and jet substructure when compared to the vacuum. At leading order in the strong coupling, a jet produced at an EIC is initiated as an energetic quark, and the process of this quark splitting into a quark-gluon system underlies experimental observables. The spectrum of gluons associated with the branching of this quark jet is heavily modified by multiple scattering in a medium, allowing jet cross sections and jet substructure to be used as a probe of the medium's properties. We present a formalism that allows us to compute the gluon spectrum of a quark jet to an arbitrary order in opacity, the average number of scatterings in the medium. This calculation goes beyond the simplifying limit in which the gluon radiation is soft and can be interpreted as energy loss of the quark, and it significantly extends previous work which computes the full gluon spectrum only to first order in opacity. The theoretical framework demonstrated here applies equally well to light parton and heavy quark branching, and is easily generalizable to all in-medium splitting processes.

Transport coefficients from energy loss studies in an expanding QGP

We use linear viscous hydrodynamics to describe the energy and momentum deposited by a fast moving parton in a quark gluon plasma. This energy-momentum is used to compute the probability density for the production of soft partons by means of the Cooper-Frye formula. Using this probability density, we render manifest a relation between the average transverse momentum given to the fast moving parton from the medium, q̂, the shear viscosity to entropy density ratio, η/s, and the energy lost by the fast moving parton, ΔE, in an expanding medium under similar conditions to those generated in nucleus-nucleus collisions at the LHC and RHIC energies. We find that q̂ increases with ΔE. On the other hand, η/s is more stable with ΔE. The behavior of q̂, with ΔE is understood as arising from the length of medium the parton traverses from the point where it is produced. However, since η/s is proportional to the ratio of the length of medium traversed by the fast parton and the average number of scatterings it experiences, it has a milder dependence on the energy it loses. This study represents a tool to obtain a direct connection between transport coefficients and the description of in-medium energy loss within a linear viscous hydrodynamical evolution of the bulk.

Relatingq̂, η/s, andΔEin an expanding quark-gluon plasma

We use linear viscous hydrodynamics to describe the energy and momentum deposited by a fast moving parton in a quark gluon plasma. This energy-momentum is in turn used to compute the probability density for the production of soft partons by means of the Cooper-Frye formula. We use this probability density to render manifest a relation between the average transverse momentum given to the fast moving parton from the medium $\hat{q}$, the entropy density to shear viscosity ratio $\eta/s$ and the energy lost by the fast moving parton $\Delta E$ in an expanding medium under similar conditions to those generated in nucleus-nucleus collisions at the LHC. We find that $\hat{q}$ increases linearly with $\Delta E$ for both trigger and away side partons that have been produced throughout the medium. On the other hand, $\eta/s$ is more stable with $\Delta E$. We also study how these transport coefficients vary with the geometrical location of the hard scattering that produces the fast moving partons. The behavior of $\hat{q}$, with $\Delta E$ is understood as arising from the length of medium the parton traverses from the point where it is produced. However, since $\eta/s$ is proportional to the ratio of the length of medium traversed by the fast parton and the average number of scatterings it experiences, it has a milder dependence on the energy it loses. This study represents a tool to obtain a direct connection between transport coefficients and the description of in-medium energy loss within a linear viscous hydrodynamical evolution of the bulk.

Monte Carlo Study of Dopant-Segregated Schottky Barrier SoI MOSFETs: Enhancement of the RF Performance

This paper presents a detailed Monte Carlo study of the optimization of the dopant segregation (DS) layer in n-type Schottky barrier (SB)-MOSFET. It is shown that with a careful control of the DS layer parameters, dopant concentration (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dop</sub> ), and length (L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dop</sub> ), the performance of the devices is significantly enhanced. The presence of the DS layer induces crucial effects in the injection processes at the Schottky contacts. The benefits of increasing the length and the doping level of the DS layer are studied from the microscopic point of view (transit times, average number of scatterings). The effect of varying these parameters is also analyzed through the nonquasi-static parameters of the small signal equivalent circuit, which can be useful for designers to improve the reliability of the SB-MOSFET technology.

THE IMPACT OF GAS BULK ROTATION ON THE Lyα LINE

We present results of radiative transfer calculations to measure the impact of gas bulk rotation on the morphology of the Lyman $\alpha$ emission line in distant galaxies. We model a galaxy as a sphere with an homogeneous mixture of dust and hydrogen at a constant temperature. These spheres undergo solid-body rotation with maximum velocities in the range 0-300 \kms and neutral hydrogen optical depths in the range $\tau_{\rm H}=10^{5}-10^{7}$. We consider two types of source distributions in the sphere: central and homogeneous. Our main result is that rotation introduces a dependence of the line morphology with viewing angle and rotational velocity. Observations with a line of sight parallel to the rotation axis yield line morphologies similar to the static case. For lines of sight perpendicular to the rotation axis both the intensity at the line center and the line width increase with rotational velocity. Along the same line of sight, the line becomes single peaked at rotational velocities close to half the line width in the static case. Notably, we find that rotation does not induce any spatial anisotropy in the integrated line flux, the escape fraction or the average number of scatterings. This is because Lyman {\alpha} scattering through a rotating solid-body proceeds identical as in the static case. The only difference is the doppler shift from the different regions in the sphere that move with respect to the observer. This allows us to derive an analytic approximation for the viewing-angle dependence of the emerging spectrum, as a function of rotational velocity.

Approximations for horizontal photon transport in cloud remote sensing problems

The effect of horizontal photon transport within real-world clouds can be of consequence to remote sensing problems based on plane-parallel cloud models. In this paper, analytic approximations for the root-mean-square horizontal displacement of reflected and transmitted photons, relative to the incident cloud-top location, are derived for plane-parallel cloud layers. With anisotropic scattering, separate approximations are needed depending on the order of scattering. When sufficient numbers of photon scatterings occur, an approximation based on random walk theory (photon diffusion) is applicable; when scattering numbers are relatively small, a modification to the diffusion result is used. The resulting formulae are a function of the average number of photon scatterings, as well as particle asymmetry parameter and single scattering albedo. In turn, the average number of scatterings from plane-parallel, vertically inhomogeneous cloud layers can be determined from efficient adding/doubling radiative transfer procedures. The transport approximations are applied to liquid water clouds for typical remote sensing solar spectral bands, involving both conservative and non-conservative scattering. Results compare well with Monte Carlo calculations. Though the emphasis is on horizontal photon transport in terrestrial clouds, the derived approximations are applicable to general anisotropic, multiple scattering, plane-parallel radiative transfer problems. Approximations useful for three-dimensional transport are also given. The complete horizontal transport probability distribution can be described with an analytic distribution specified by the root-mean-square and average radial displacement values. However, it is shown empirically that the average displacement can be reasonably inferred from the root-mean-square value. An estimate for the horizontal transport distribution can then be made from the root-mean-square photon displacement alone.

Nonlinear equations for statistical averages describing processes of multiple scattering

The Lagrangian formalism is used to obtain nonlinear equations for frequently encountered statistical characteristics of radiative transfer. Under the assumption of complete frequency redistribution, the average number of scatterings and the average time of stay of a photon in a one-dimensional, semi-infinite atmosphere are analyzed. Two typical problems of multiple scattering are solved to illustrate the possible applications of the equations obtained.

Modeling low-frequency surface scatter intensity statistics

Low-frequency surface backscattering experiments (CST-7, ASREX) show that the surface scattered field statistics are non-Gaussian, which means the intensity statistics will be nonexponential. These data show the intensity distribution to be closer to log normal. In such cases the statistical distribution depends on the scattering area on the surface or, equivalently, on the number of scatterers. An analytical model has been developed for the intensity distribution that results when the scattering area contains multiple, spatially independent, identically distributed scatterers. The individual scatterers, e.g., bubble clouds, are assumed to yield a log-normal intensity distribution. The assumption that the scatterers are spatially independent leads to a Poisson distribution for the number of scatterers in a given surface area. The parameters to be specified are the standard deviation of the log-normal distribution and the average number of scatterers per unit area. Monte Carlo simulations yield estimated intensity distributions in very good agreement with the analytical model. Preliminary comparisons will be made with ASREX data. [Work supported by ONR.]

New aspects of Holstein's treatment of radiation trapping

We compare two equivalent solutions to the Holstein-Biberman equation of radiative tranfer, and transform Holstein's diffusion modes to a representation that considers consecutive generations of scattered photons. As an example for this general transformation experimental results obtained by pulse excitation of the first reasonance line of rubidium by a diode laser are compared with time dependent resonance fluorescence calculated by means of a simple model. The probability for multiple photon scattering and the average number of scatterings are calculated for the experimental situation.

Multiple scattering — A different approach for any cross section and any scatterer thickness

A new and simple method is shown for obtaining the small-angle multiple (or plural) scattering distribution of an incident beam of charged particles. The same method works for any physically reasonable shape of single-scattering cross section and for any thickness of scatterer (i.e., average number of scatterings anywhere from ⪡1 to ⪢1). Results are given for two forms of single scattering. One is σ a ∼ (η 2 + η a 2) −3 2 , used by Snyder and Scott. The resulting scattering distribution agrees with theirs within 1% for 100 collisions. The other (hypothetical) is σ b∼exp(− η 2/ η b 2). For σ a the tails of the multiple scattering distribution asymptotically approach the single-scattering distribution, as is known. For σ b that does not happen. Graphs and tables show quantitatively how the multiple-scattering distributions depart from a Gaussian in the central region and from single scattering in the tails. The application to atom-atom scattering is considered, and the strong forward peaking in σ is shown not to be an important problem. The method starts with an infinitesimal thickness of scatterer and produces, by successive foldings, the projected scattering distributions for thickness of 2 m times the original thickness, m = 1,2,3,…. The inversion of the projected distribution is considered.

On certain theories of multiple scattering in random media of discrete scatterers

The paper presents new information on the accuracy of the heuristic approximations in two important theories of multiple scattering in random media of discrete scatterers: Twersky's “freespace” and “two‐space scatterer” formalisms. Two complementary approaches, based primarily on a one‐dimensional model and the one‐dimensional forms of the theories, are used. For scatterer distributions of low average density, the “heuristic” asymptotic forms for the coherent field and the incoherent intensity are compared with asymptotic forms derived from a systematic analysis of the multiple scattering processes. For distributions of higher density, both in the average number of scatterers per wavelength and in the degree of packing of finite‐size scatterers, the analysis is carried out “experimentally” by means of a Monte Carlo computer simulation. Approximate series expressions based on the systematic approach are numerically evaluated along with the heuristic expressions. The comparison (for both forward‐ and back‐scattered field moments) is made for the worst‐case conditions of strong multiple scattering for which the theories have not previously been evaluated. Several significant conclusions are drawn which have certain practical implications: (a) in application of the theories to describe some of the scattering phenomena which occur in the troposphere, and (b) in the further evaluation of the theories using experiments on physical models.

A Mathematical Model for Volume Reverberation. I. Experiment

An experiment was conducted to measure parameters for a Poisson model for volume scattering. Using a stable platform suspended beneath a ship, echoes from discrete scatterers were observed and analyzed. A cumulative probability distribution of scatterer reflectivity (ratio of scattered to incident pressure) was obtained by examining more than 1200 individual echoes. The average number of scatterers per unit volume was estimated from experimental data. The envelopes of echoes from scatterers were similar to the envelope of the transmitted signal.

Sea-wave spectrum by echo sounder

Nanda (1957) suggested a method of making a relative estimate of the standing crop in the ocean by studying the fluctuation of the echoes received on an echo-sounder. Taking the number of scatterers to be obeying Poisson distribution, it was shown that the average number of scatterers N is equal to 4/a2 times the mean square deviation of a series of values of log a/a0 where a is the average scattering coefficient per organism and a and ao are the received and transmitted amplitudes. Besides the density of standing crop in the ocean, the other factors which contribute to the fluctuation of the echoes are the changes in reflectivity of the ocean bottom, and the oscillation of the vessel due to state of the sea.

On Scattering and Reflection of Sound by Rough Surfaces

We consider random distributions of arbitrary identical protuberances on free or rigid base planes, and derive the approximate reflection coefficients R = |1 + Z1 − Z|2 and differential scattering cross sections per unit area σ(s,ki) = ρk2|f(s,ki)1 − Z|2. Here Z = πρf(k0,ki)k2n ⋅ k0, where f(s,ki) is the scattering amplitude of an isolated protuberance. ki, k0, and s are the directions of incidence, specular reflection (with respect to the base plane), and observation; n is the normal of the base; ρ is the average number of scatterers in unit area, and k = 2π/λ. (For a two-dimensional “striated” surface we divide Z by π/k, and σ by π/2k.) We find for example, that if the horizon angle β approaches zero (i.e., near grazing incidence), then the reflection coefficients approach unity minus terms of the order β, while the back scattering vanishes like β1, β2 for a free, rigid base. Explicit forms are given for arbitrary hemispheres and circular semicylinders, and for the limiting cases of free and rigid surfaces with scatterer radius very small and very large compared to wavelength. The analysis is based on a Green's function formulation of the problem of a single configuration; R and σ follow from an approximation of the ensemble averaged energy flux which takes account of multiple coherent scattering. (The transmission problem of a “random screen” is treated simultaneously.) An elementary derivation is also given, and extensions to distributions of nonidentical scatterers are made.