J. E. Fernández

Polarization effects on multiple scattering gamma transport

Abstract The scattering of X-rays and γ-rays are events that have strong dependencies on the polarization of the incident and scattered photons. Because of this, scattering problems that can be solved without explicit reference to the state of polarization of the incident and scattered radiation are exceptional. This article reviews available information on polarization effects arising when photons in the X-ray and γ-ray regime undergo photoelectric effect, coherent (Rayleigh) scattering and incoherent (Compton) scattering by atomic electrons. In addition to descriptions and discussion of these effects, we study the backscattering of γ-rays from an infinite thickness target excited with a plane slant monodirectional and monochromatic source, using the Boltzmann transport theory and the mathematical representation of polarization introduced by Stokes. Results from this model, for both unpolarized and polarized γ-ray sources, are compared with computations performed neglecting or averaging polarization effects, showing the limitations of such approximations.

Mathematical Modelling of 3D Electron-Photon Transport in Microbeam Analysis

Abstract. In electron microbeam techniques, the particle beam is focused on the material to be analysed. When the electron beam enters the target, the electrons give rise to ionization processes producing secondary electrons and photons, the latter being used to characterize the material. As a consequence, a detailed description of the photon diffusion requires the solution of two coupled equations describing respectively electron and photon diffusion. The approach considering two transport equations, even if formally correct, is almost unaffordable because of the high mathematical complexity of the electron transport equation. In this article, an alternative approach is suggested which is based on the use of an approximate solution for the electron transport using the Fokker-Planck equation [5]. The resulting electron distribution, computed analytically as a solution of the above equation, is very similar to the ionization distribution and is used as the source term in the Boltzmann transport equation describing the photon diffusion in the material. The 3D photon transport equation for unpolarised photons with this source term is solved to obtain a detailed description of the photon fluorescence from a homogeneous slab.

On the angular dependence of L x-ray production cross sections following photoionization at an energy of 59.54 keV

The anisotropy of L x-ray fluorescence induced by 59.54 keV unpolarized photons is investigated by means of an experimental procedure which allows the relative L x-ray production cross section to be evaluated without taking account of the angular set-up and the instrumental efficiency. Thick targets of Yb, Hf, Ta, W and Pb are considered, and the angular trend of the relative experimental ratios, IL?/IL?, is calculated by simple evaluations of the peak area alone. Within the experimental uncertainties, which are found to be of the order of 1.6% in the worst cases, the results do not show any significant angular dependence of the L? emission lines if the isotropy of the L? lines is assumed. This fact contrasts with the results of some groups, which report evidence of strong angular dependence at this level of photon energy.

Diffusion of polarized photons in the frame of the transport theory

Abstract In the frame of the multiple applications of synchrotron radiation X-ray spectrometry (SRXRS), a detailed description of the transport of polarized photons in condensed media is of utmost importance. The effect of polarization requires four parameters to describe an X-ray beam in place of the single intensity of the models without polarization. Further, the state of polarization changes every time the photon undergoes a scattering event. Accordingly, a proper description of photon transport including polarization effects, here represented with the Stokes parameters, involves a system of four coupled equations of transfer. In this paper, the system of transport equations, describing the diffusion of X-ray photons (including polarization effects) in a homogeneous target of infinite thickness, is solved. An iterative solution, universally valid for all the interactions in the X-ray regime, is obtained. The solution is applied, e.g. to study Rayleigh scattering of (initially) unpolarized X-rays. The first and second order Rayleigh intensities are compared with similar terms computed with a scalar model using a kernel with averaged polarization, in order to quantify the influence of including, rigorously, polarization in the prediction of multiple scattering intensities of the Rayleigh effect.

Photon backscattering tissue characterization by energy dispersive spectroscopy evaluations

Techniques for in vivo tissue characterization based on scattered photons have usually been confined to evaluating coherent and Compton peaks. However, information can also be obtained from the energy analysis of the Compton scattered distribution. This paper looks at the extension of a technique validated by the authors for characterizing tissues composed of low-atomic-number elements. To this end, an EDXRS (energy dispersive x-ray spectrometry) computer simulation procedure was performed and applied to test the validity of a figure of merit able to characterize binary compounds. This figure of merit is based on the photon fluence values in a restricted energy interval of the measured distribution of incoherently scattered photons. After careful experimental tests with 59.54 keV incident photons at scattering angles down to 60degrees, the simulation procedure was applied to quasi-monochromatic and polychromatic high-radiance sources. The results show that the characterization by the figure of merit, which operates satisfactorily with monochromatic sources, is unsatisfactory in the latter cases, which seem to favour a different parameter for compound characterization.

X-Ray Photon Spectroscopy Calculations

X-ray photons - as many other particles - interact with matter producing secondary radiation that carries useful information about the atoms comprising the target. The availability of intense sources of highly monochromatic X-rays and the great improvement in detector technology intensified research in X-ray spectrometry in the last twenty years. New techniques allowed the attenuation coefficients and the physics of the atom to be better known: Extended X-ray Absorption Fine Structure (EXAFS), X-ray Absorption Near Edge Structure (XANES), and Inelastic X-ray Scattering Spectroscopy (IXSS). Old techniques, like X-ray Fluorescence (XRF), gained in precision thus extending the horizon of applicability to new elements and energy ranges, and consequently Energy Dispersive X-ray Fluorescence (EDXRF) and Synchrotron Radiation X-ray Fluorescence (SRXRF) were evolved. Particle induced X-ray emission spectroscopy also benefited from this improvement. The field of application of X-ray pectrometry has grown from atomic, to nuclear, to plasma physics, to astrophysics.

Effect of the X-ray scattering anisotropy on the diffusion of photons in the frame of the transport theory

Abstract In the frame of the multiple applications of X-ray techniques a detailed description of the photon transport under several boundary conditions in condensed media is of utmost importance. In this work the photon transport equation for a homogeneous specimen of infinite thickness is considered and an exact iterative solution is reported, which is universally valid for all types of interactions because of its independence of the shape of the interaction kernel. As a test probe we use a specially simple elastic scattering expression that renders possible the exact calculation of the first two orders of the solution. It is shown that the second order does not produce any significant improvement over the first one. Due to its particular characteristics, the first-order solution for the simplified kernel can be extended to include the form factor, thus giving a more realistic description of the coherent scattering of monochromatic radiation by bound electrons. The relevant effects of the scattering anisotropy are also placed in evidence when they are contrasted with the isotropic solution calculated in the same way.

Diffusion of photons with arbitrary state of polarization: the Monte Carlo code MCSHAPE

Abstract When X-rays penetrate in the matter, they interact with the atoms, producing secondary radiation that carries important information about the composition of the target. The polarization state is one of the properties of the incoming photons which changes as a consequence of the number and the type of the undergone interaction. Therefore, to study properly the atomic properties of a material, it is necessary to consider the evolution of the polarization state of radiation. It is presented MCSHAPE, a Monte Carlo code developed to describe the evolution of the polarization state of X-ray photons as a consequence of the multiple scattering collisions undergone during the diffusion into the sample. In order to study properly the transport of photons with an arbitrary state of polarization, the model adopted in this code is derived from the so called ‘vector’ transport equation [Radiative Transfer, Chapter 1, Section 15, Clarendon, Oxford, 1950; Nucl. Instr. and Meth. B 73 (1993) 341]. Using the Stokes parameters I, Q, U and V, having the dimension of an intensity and containing all the physical information about the polarization state, MCSHAPE simulates the full state of polarization of the photons at any given position, wavelength and solid angle.

Vector Monte Carlo for simulation of polarized photons

Based on a recent analytical solution of vector transport equations in plane geometry including polarization effects, a vector scheme for a Monte Carlo code was worked out. The new code, called MC-SHAPE, was developed both to emphasize the miscalculation in using the scalar approach with average polarization kernels and to extend results obtained with the analytical solution of the transport equation to higher orders of interaction. The code includes the Doppler broadening giving the momentum distribution due to electron pre-collision motion. To improve the comparison with analytical results, the code takes some feature from this model, which considers the scattering of a monochromatic and monodirectional collimated beam of x-rays on a semi-infinite and homogeneous target. Results of the evolution of the four components of the Stokes vector describing the back-scattered beam, are presented which allow the analysis of the complete polarization characteristics of the multiple scattering of Rayleigh and Compton effects. The good agreement found with both deterministic calculations and experimental data indicates the validity of the new code.

Polarisation effects in multiple scattering photon calculations using the Boltzmann vector equation

Abstract Studies on radiative transfer lead to a clean formulation of polarised photon transport in terms of the vector Boltzmann equation whose solution gives the four Stokes components of the flux, from which the full polarisation state of the photons can be determined at any given position, wavelength (energy) and solid angle. One of the relevant results observed during the formulation of the vector transport equation is the partial coverage of the wave properties of the photons with this model. In fact, even if the Boltzmann vector equation is an important step forward for the description of radiative transfer with respect to the scalar approach used to describe “particle”-like photons, it is still insufficient to provide a whole description of an important phase-related property like coherence. In this sense, the above vector equation seems to be appropriate for describing photon beams which add incoherently among them, but not for describing coherent interference. Up to now, coherence has been described independently to the transport equation, as an additive term to the vector solution valid for diffuse incoherent radiation. New approaches are being investigated in order to include the concept of coherence into the vector transport equation. In this article, a summary view of the evolution of the Boltzmann transport equation will be provided, first from scalar to vector, to take care of the description of the evolution of the polarization state. Secondly the present state of the treatment of coherence will be considered. Finally, the state-of-the-art description of multiple scattering involving Rayleigh scattering will be discussed, in the framework of both reflection and transmission experiments.