Single Scattering Cross Section Research Articles

Strategies for removing multiple scattering effects revisited

The multiple scattering problem of small angle neutron scattering (SANS) is revisited using differential equations from which different contributions are derived. The coherent scattering is connected to multiple scattering events and the ideal single scattering cross sections can be related to the apparent scattering cross section. The multiple scattering problem of the incoherent scattering is more demanding because the sample geometry – I assumed a slab – matters. I tried to solve this problem analytically, and used Monte Carlo simulations. From all concepts I derived a strategy for a computer program that is capable to remove multiple scattering effects. As a side aspect, I could also remove resolution effects.

Production of γγ+2 jets from double parton scattering in proton-proton collisions at the LHC**Supported by National Natural Science Foundation of China (11061140514, 11505208), China Ministry of Science and Technology (2013CB838700) and CAS Center for Excellence in Particle Physics (CCEPP)

Cross sections for the production of pairs of photons plus two additional jets produced from double parton scattering in high-energy proton-proton collisions at the LHC are calculated for the first time. The estimates are based on the theoretical perturbative QCD predictions for the productions of γγ at next-to-next-to-leading-order, jet+jet and γ+jet at next-to-leading-order, for their corresponding single-scattering cross sections. The cross sections and expected event rates for γγ+2 jets from double parton scattering, after typical acceptance and selections, are given for proton-proton collisions with the collision energy and integrated luminosity of 100 fb−1 planned for the following years, and also with 3000 fb−1 of integrated luminosity as the LHC design.

Pair production of quarkonia and electroweak bosons from double-parton scatterings in nuclear collisions at the LHC

Cross sections for the concurrent production of pairs of quarkonia (J/psi, Upsilon) and/or gauge bosons (W, Z) from double-parton scatterings (DPS) in high-energy proton-nucleus and nucleus-nucleus collisions at the LHC are calculated. The estimates are based on next-to-leading-order perturbative QCD predictions, including nuclear modifications of the parton densities, for the corresponding single-scattering cross sections. Expected event rates for J/psi+J/psi, J/psi+Upsilon, J/psi+W, J/psi+Z, Upsilon+Upsilon, Upsilon+W, Upsilon+Z, and same-sign W+W production in their (di)leptonic decay modes, after typical acceptance and efficiency losses, are given for pPb and PbPb collisions.

A Monte Carlo study of absorbed dose distributions in both the vapor and liquid phases of water by intermediate energy electrons based on different condensed-history transport schemes

Monte Carlo transport calculations of dose point kernels (DPKs) and depth dose profiles (DDPs) in both the vapor and liquid phases of water are presented for electrons with initial energy between 10 keV and 1 MeV. The results are obtained by the MC4 code using three different implementations of the condensed-history technique for inelastic collisions, namely the continuous slowing down approximation, the mixed-simulation with δ-ray transport and the addition of straggling distributions for soft collisions derived from accurate relativistic Born cross sections. In all schemes, elastic collisions are simulated individually based on single-scattering cross sections. Electron transport below 10 keV is performed in an event-by-event mode. Differences on inelastic interactions between the vapor and liquid phase are treated explicitly using our recently developed dielectric response function which is supplemented by relativistic corrections and the transverse contribution. On the whole, the interaction coefficients used agree to better than ∼5% with NIST/ICRU values. It is shown that condensed phase effects in both DPKs and DDPs practically vanish above 100 keV. The effect of δ-rays, although decreases with energy, is sizeable leading to more diffused distributions, especially for DPKs. The addition of straggling for soft collisions is practically inconsequential above a few hundred keV. An extensive benchmarking with other condensed-history codes is provided.

Dielectric properties of pure and Nb-doped SrTiO3 surfaces: a reflection electron energy loss spectroscopy study

Reflection Electron Energy Loss Spectroscopy (REELS) is a valuable tool to characterize the electronic structure of insulating surfaces and nanometric films. The REELS inelastic single scattering cross section obtained after removing multiple losses is related to the complex dielectric function in the surface layers. Moreover the probing depth can be varied by changing the primary energy of electrons. One can expect in essence to obtain non-destructive REELS depth profiles, where the evolution of the REELS spectrum translates to the in-depth evolution of the dielectric constant. We present a REELS study of pure and Nb-doped SrTiO3 single crystals, with primary energies ranging from 200 eV to 1500 eV so as to vary the probing depth between 0.7 and 3 nm. The resulting variations in the shape of the spectra are qualitatively interpreted by considering the energy loss functions corresponding to a double interface SrTiO3/carbon monolayer/vacuum, on the basis of the experimental dielectric function derived from TEELS measurements on strontium titanate and amorphous carbon.

Analysis of neutron double-crystal diffractometer scattering curves including multiple scattering

Double-crystal diffractometers (DCD) of the Bonse–Hart type are used for small-angle neutron scattering (SANS) experiments with very high resolution. Scattering from inhomogeneities in the size range of typically 100 nm to 10 µm can be detected with a DCD. The evaluation of DCD scattering curves is more complicated than for conventional SANS instruments with point collimation for two reasons: first, a DCD operates with a long-slit geometry, and second, owing to the high scattering power of large inhomogeneities, multiple small-angle scattering is likely to play a significant role in many DCD experiments. A numerical procedure is proposed for the determination of particle size distributions from DCD scattering curves influenced by multiple scattering. This procedure is based on ann-fold convolution of single-scattering cross sections, wherendepends on the degree of multiple scattering. It makes use of the indirect Fourier transformation method and thus, firstly, size distributions of arbitrary shape can be determined, and secondly, the calculation is fast, allowing an interactive analysis. The procedure is demonstrated with scattering from latex particles with different degrees of multiple scattering. The results show that the size and volume fraction of the latex spheres can be determined with a reasonable error even from scattering curves that are strongly influenced by multiple small-angle scattering.

CPP calculation of multiple scattering distributions for charged particles penetrating compounds or mixtures.

Charged particle multiple scattering distributions may be constructed from individual atomic scattering events on the basis of compound Poisson process (CPP) theory. We present a CPP method for computing multiple scattering transition probability densities from charged particles penetrating compounds and mixtures. Water as a scattering medium provides here an example of the calculation method which is applicable to compounds or mixtures. Electrons are chosen as examples of charged particle beams. The Rutherford single scattering cross section and a partial wave analysis single scattering cross section are chosen as example cross sections. Transition probability densities predicted on the basis of CPP theory can be calculated with great accuracy for the improvement of radiation dose calculations. The advantages of the CPP method are (a) an effective atomic number need not be defined for the scattering medium, (b) it can be applied in both spherical and planar coordinate systems, and (c) it does not require any specific form for the single scattering cross section.

Generalized Landauer equation: absorption-controlled diffusion processes.

The exact expression of the one-dimensional Boltzmann multiple-scattering coefficients, for the passage of particles through a slab of a given material, is obtained in terms of the single-scattering cross section of the material, including absorption. The remarkable feature of the result is that for multiple scattering in a metal, free from absorption, one recovers the well-known Landauer result for conduction electrons. In the case of particles, such as neutrons, moving through a weak absorbing media, Landuer's formula is modified due to the absorption cross section. For photons, in a strong absorbing media, one recovers the Lambert-Beer equation. In this latter case one may therefore speak of absorption-controlled diffusive processes.

Asymptotic behavior of neutron small-angle multiple scattering spectra

The asymptotic behavior of the spectra for large values of the scattering vector for the case of elastic multiple small-angle neutron scattering (SANS) is investigated theoretically and experimentally. An expansion of the spectrum in terms of the reciprocal of the magnitude of the momentum transfer is obtained taking account of the influence of the instrumental line. It is shown that, to within some factor, the leading term of the expansion is identical to the differential single-scattering cross section averaged over a statistical ensemble of particles; several subsequent terms in the expansion are calculated and the range of applicability of the resulting expressions is determined. The asymptotic behavior of the multiple SANS spectrum is measured, using a two-crystal neutron spectrometer, for samples of an HTSC ceramic, the alloy Fe-Ni, and Al powder. The agreement between the experimental results and the theoretical predictions is analyzed.

The concept of mass angular scattering power and its relation to the diffusion constant

An understanding of the scattering of high energy charged particle beams by tissue is required in radiotherapy since the particle trajectories determine the pattern of radiation dose deposition in patients. Numerical calculations of radiation dose often utilize energy dependent values of the angular scattering power. However, the physics literature is replete with confused interpretations of the concept of angular scattering power and its relation to the single scattering cross section for the medium or the diffusion constant in the diffusional limit. The purpose of this article is to clarify these notions.

Elastic wave band gaps and single scattering

The scattering cross section of an elastic plane wave incident at right angle to a cricular cylinder was calculated. The frequency dependence of this single scattering cross section is connected to the existence (or not) of spectral gaps for elastic waves propagating in a periodic array of parallel circular cylinders. The present work confirms the picture of propagation along two distinct channels: one employing mainly the host material and the other exploiting the resonance modes of each cylinder.

Radiative energy loss of high energy quarks and gluons in a finite-volume quark-gluon plasma

The medium induced energy loss spectrum of a quark or gluon of high energy E traversing a hot QCD medium of finite volume is studied. We model the interaction by a simple picture of static scattering centres. The total induced energy loss is found to grow as L 2, where L is the extent of the medium. This quadratic behaviour holds for L smaller than a critical length L cr α √ E. The solution of the energy loss problem is reduced to the solution of a Schrödinger-like equation whose “potential” is given by the single-scattering cross section of the high energy parton in the medium. These results should be directly applicable to a quark-gluon plasma.

Enhanced backscatter from one-dimensional random rough surfaces: stationary-phase approximations to full-wave solutions

We use the full-wave approach to obtain the solutions for the diffuse single- and double-scatter radar cross sections for one-dimensional rough surfaces. The solutions are expressed in terms of two- and six-dimensional integrals (not integral equations) for the single- and double-scatter cross sections. High-frequency, stationary-phase approximations are applied to these multidimensional expressions. The full-wave, high-frequency approximate solutions are obtained in closed forms for the single-scatter cross sections and in terms of two-dimensional integrals for the double-scatter cross sections. The sharp peak in the backscatter direction is due to the contributions from the double-scatter quasi-antiparallel paths. The level and the angular width of the enhanced backscatter peak are frequency dependent. The high-frequency approximations provide physical insight into the multiple-scatter problem. However, they cannot be used to predict accurately the angular width of the backscatter peak and the polarization dependence of the scatter cross sections.

Incorporation of single elastic scattering in the EGS4 Monte Carlo code system: tests of Molière theory

To avoid prohibitively long computation times, conventional Monte Carlo e − transport algorithms (e.g. EGS4, ETRAN, ITS) employ multiple scattering theories and “condensed history” methods to model e − transport. Although highly successful for many calculations, these techniques do not model backscatter very well, particularly for high- Z materials. In an attempt to correct for this shortcoming, we have extended the EGS4 Monte Carlo code to allow for the simulation of single elastic scattering. The single scattering method also allows quantities to be scored in submicrometer dimension geometries where the Molière multiple scattering theory fails and the Ooudsmit-Saunderson multiple scattering equations converge very slowly. Two single scattering schemes have been implemented: (i) Screened Rutherford cross sections which form the basis of Molière's multiple scattering theory, (ii) Single scattering cross sections based upon phase-shift data. In this work we describe the implementation of single elastic scattering in the EGS4 Monte Carlo code system and employ it to verify the Molière multiple scattering theory in its range of validity. We demonstrate that the Molière multiple scattering formalism provides a good description of multiple scattering despite its use of a relatively crude cross section and that it may be employed with semiquantitative accuracy in the plural scattering regime, where electron step-lengths are so short that only as few as five atoms participate in the angular deflection. However, the remaining differences of the Molière distributions with the phase-shift data motivate the use of more accurate fundamental data, in particular, for applications involving high- Z elements.

On the theory and simulation of multiple elastic scattering of electrons

Multiple elastic scattering of electrons in matter is analyzed on the basis of accurate single scattering differential cross sections obtained from partial wave calculations. We give a brief derivation of Molière's multiple scattering theory that clarifies its physical content and points out its limitations. In particular, it is shown that transport mean free paths calculated from the Molière single scattering cross section differ significantly from the values obtained from partial wave calculations. We present a mixed simulation algorithm that overcomes most of the limitations of the currently available condensed Monte Carlo codes. This algorithm takes advantage of the fact that most of the collisions experienced by a high-energy electron along a given path length are soft, i.e. the scattering angle is less than a selected small value χs. The global effect of these soft collisions is described by using a multiple scattering approximation. Hard collisions, with scattering angle larger than χs, occur in a moderately small number and are described as in detailed simulations. This mixed algorithm can be applied to any single scattering differential cross section, it leads to the correct spatial distributions and it completely avoids problems related to boundary crossing. Moreover, when the single scattering law underlying Molière's theory is adopted, the algorithm can be formulated in a completely analytical way.

Super-resonant scatter under ice sheets and in full spaces

Systems of monopole scatterers interacting through multiple scatter may develop normal modes, i.e., true resonant behavior at certain frequencies (the equivalent source coefficients of each scatterer have real poles). This occurs when the single scatter cross sections are large enough to give significant interaction. This happens for highly tuned scatterers, e.g., gas-filled balloons, bubbles, and thin shells in water, insonified at frequencies near their characteristic “bubble” frequencies. This kind of super-resonance (SR) occurs when the system is at or near the interface with an elastic medium, e.g., a thin elastic plate [I. Tolstoy, J. Acoust. Soc. Am. 80, 282–294 (1986)]. Energy transport between scatterers is then mediated by boundary modes excited by the scatterers acting as point sources (e.g., the flexure waves of a thin plate). The case of two and three bubble systems at an ice-water or lucite-water interface is explored, with particular reference to the parameters ka, kl (k is the acoustic wavenumber, a the scatterer radius, l the spacing between scatterers) and the plate elastic constants. For a pair (doublet) the pressure amplification re: the free-field value is ≅ 2 × 103 for kl values as small as 0.5, with a directionality factor of 30: This is a passive, highly tuned, high magnification, strongly directional compact array (≅λ/12 size). A 30-Hz system would consist of two ballons of 10-cm radius near an ice boundary, with a discrete set of permissible l values ranging from about 5–20 m. Optimum spacings and magnifications depend upon the ice thickness (≅5 m). In an acoustic full space, energy exchange between scatterers is mediated less effectively by volume waves, and the scatter coefficient poles vanish: Bona-fide SR does not take place. Nevertheless, strong peaks may occur, giving magnifications ≅103. This quasiresonant behavior is discussed for simple two-, three-, four-, and six-element plane and polyhedral configurations. However, true SR behavior is reintroduced in full spaces by partially blocking selected pathways between scatterers. [Work supported by ONR.]

Monte Carlo Simulation and Analysis of Elastic and Inelastic Scattering of Electrons in Insulators

An electron transport Monte Carlo program, using single scatter cross sections and the latest phonon scattering theory, has been developed to obtain electron and hole distributions in insulators subjected to electron bombardment.

Multiple scattering in neutron polarization analysis experiments

A simple analytic method to correct the experimentally observed spin-flip and non-spin-flip scattering cross sections in neutron polarization analysis experiments for the effects of multiple scattering is presented. From known parameters of the constituent elements of a specimen and from the measured experimental cross sections the single scattering cross sections can readily be determined. This is particularly useful in situations where the scattering is isotropic or exhibits only slight angular dependence.

Energy dependent 4-dimensional multiple scattering distributions

A complete analytic solution in Fourier space is presented of the four dimensional small angle, multiple scattering distribution MSD in angle and space, produced by an energy dependent single scattering cross section from an initial pencil beam of heavy particles. Independently, simple integrals are derived for the central moments of the energy dependent MSD in the continuous-slowing-down approximation. The distributions of the projections t and x of the scattering angle and displacement into a plane through the axis of propagation are evaluated numerically for a truncated Rutherford scattering cross section using a fast Fourier transform. The resulting MSDs for a wide range of particles, initial and final momenta, and scattering materials are found to be approximately represented by one-dimensional set of standard distributions symmetrized by a linear transformation in t− x-space.

Use of realistic single scattering cross sections in computing interproton separations in the dissociation of fast H 2+ ions traversing thin carbon foils

In previous work we have demonstrated that the internucleon separation which results from the combined Coulomb explosion and multiple scattering of fast H 2 + ions traversing thin carbon foils can be very different for different single scattering cross sections, even though these cross sections may - in fact - give rise to identical multiple scattering distributions. This makes imperative the use of realistic single scattering cross sections when internucleon separations are computed. We have generated carbon atom wave functions using the Froese Fischer Hartree-Fock program and from the resulting potential have obtained scattering cross sections using the Born approximation, the classical approximation and the WKBJ method. The effect upon the separation distribution of using these cross sections as well as the cross section of Lindhart (classical solution to a Thomas-Fermi screened potential) have been investigated and will be discussed here.