Form Factor Approximation Research Articles

Coulomb drag study in electron-electron bilayer system with a dielectric medium

Abstract We have theoretically investigated the drag resistivity in electron-electron bilayer system by using the random phase approximation (RPA) for weakly interacting regime, at low temperature, high density, and large interlayer separation limit. We show analytical expressions by using long wavelength (low frequency regime) expansion of response function in Boltzmann regime and bare Coulomb interaction for non homogeneous dielectric environment which make the system interested and different compare to others. The influence of electron-electron bare inter and intra layer interaction are considered static cause of static local form factor (LFF) approximation in coupled layer system of non homogeneous dielectric background. The effective dielectric function and LFF are obtained from the solution of the Poisson equation of a three-layer dielectric medium. The effects of local form factors are included in bare inter and intralayer potential, yields the dependency of layer separation and dielectric constants of the materials. Drag resistivity dependency is measured numerically with respect to temperature T and density n and compared to other 2DES, which yields to possess the same qualitative and quantitative behaviour compare to bilayer system of homogeneous dielectric environment.

Probing uncertainties of nuclear structure corrections in light muonic atoms

Recent calculations of nuclear structure corrections to the Lamb shift in light muonic atoms are based on an expansion in a parameter eta, where only terms up to second order are retained. The parameter eta can be shown to be proportional to the square root of the muon/proton mass ratio, so that it is small and the expansion is expected to converge. However, practical implementations show that the eta convergence may be slower than expected. In this work we probe the uncertainties due to this expansion using a different formalism, which is based on a multipole expansion of the longitudinal and transverse response functions and was first introduced by Leidemann and Rosenfelder. We refer to this alternative expansion as the eta-less formalism. We generalize this formalism to account for the cancellation of elastic terms such as the third Zemach moment (or Friar moment) and embed it in a computationally efficient framework. We implement and test this approach in the case of muonic deuterium. The comparison of results in the point nucleon limit for both methods achieve sub-percent agreement. When nucleon form factors are introduced we find a 4% and 2% difference in the third Zemach moment and nuclear polarizability, respectively, compared to the eta-less expansion, indicating that the nucleon form factor approximations should be improved. However, we find that the sum of these terms removes this dependence and the uncertainty due to the eta-expansion and the related second-order approximation in the nucleon form factors amounts only to 0.2% and thus is fully justified in muonic deuterium. This computationally efficient framework paves the way to further studies in light muonic systems with more than two nucleons, where controlling and reducing uncertainties in nuclear structure corrections is key to the experimental efforts of the CREMA collaboration.

Scattering of twisted light from a crystal

Recent years have seen significant progress in the generation and application of twisted beams carrying orbital angular momentum. Here we study the elastic scattering of twisted Bessel light from a crystal and compare our predictions with the results for incident plane-wave radiation. Based on form-factor approximation our numerical calculations of the differential scattering cross sections have been carried out for a crystal of lithium at x-ray energies. It is shown that the use of twisted light can lead to a measurable change in the scattering cross section for the nanocrystals approaching a few nm in size.

Including Delbrück scattering in GEANT4

Elastic scattering of γ-rays is a significant interaction among γ-ray interactions with matter. Therefore, the planning of experiments involving measurements of γ-rays using Monte Carlo simulations usually includes elastic scattering. However, current simulation tools do not provide a complete picture of elastic scattering. The majority of these tools assume Rayleigh scattering is the primary contributor to elastic scattering and neglect other elastic scattering processes, such as nuclear Thomson and Delbrück scattering.Here, we develop a tabulation-based method to simulate elastic scattering in one of the most common open-source Monte Carlo simulation toolkits, GEANT4. We collectively include three processes, Rayleigh scattering, nuclear Thomson scattering, and Delbrück scattering. Our simulation more appropriately uses differential cross sections based on the second-order scattering matrix instead of current data, which are based on the form factor approximation. Moreover, the superposition of these processes is carefully taken into account emphasizing the complex nature of the scattering amplitudes. The simulation covers an energy range of 0.01MeV≤E≤3MeV and all elements with atomic numbers of 1≤Z≤99.In addition, we validated our simulation by comparing the differential cross sections measured in earlier experiments with those extracted from the simulations. We find that the simulations are in good agreement with the experimental measurements. Differences between the experiments and the simulations are 21% for uranium, 24% for lead, 3% for tantalum, and 8% for cerium at 2.754MeV. Coulomb corrections to the Delbrück amplitudes may account for the relatively large differences that appear at higher Z values.

Theoretical treatments of the bound-free contribution and experimental best practice in X-ray Thomson scattering from warm dense matter

By comparison with high-resolution synchrotron x-ray experimental results, we assess several theoretical treatments for the bound-free (core-electron) contribution to x-ray Thomson scattering (i.e., also known as nonresonant inelastic x-ray scattering). We identify an often overlooked source of systematic error in the plane-wave form factor approximation (PWFFA) used in the inference of temperature, ionization state, and free electron density in some laser-driven compression studies of warm dense matter. This error is due to a direct violation of energy conservation in the PWFFA. We propose an improved practice for the bound-free term that will be particularly relevant for XRTS experiments performed with somewhat improved energy resolution at the National Ignition Facility or the Linac Coherent Light Source. Our results raise important questions about the accuracy of state variable determination in XRTS studies, given that the limited information content in low-resolution XRTS spectra does not strongly constrain the models of electronic structure being used to fit the spectra.

Photon Elastic Scattering Simulation: Validation and Improvements to Geant4

Several models for the simulation of photon elastic scattering are quantitatively evaluated with respect to a large collection of experimental data retrieved from the literature. They include models based on the form factor approximation, on S-matrix calculations and on analytical parameterizations; they exploit publicly available data libraries and tabulations of theoretical calculations. Some of these models are currently implemented in general purpose Monte Carlo systems; some have been implemented and evaluated for the first time in this paper for possible use in Monte Carlo particle transport. The analysis mainly concerns the energy range between 5 keV and a few MeV. The validation process identifies the newly implemented model based on second order S-matrix calculations as the one best reproducing experimental measurements. The validation results show that, along with Rayleigh scattering, additional processes, not yet implemented in Geant4 nor in other major Monte Carlo systems, should be taken into account to realistically describe photon elastic scattering with matter above 1 MeV. Evaluations of the computational performance of the various simulation algorithms are reported along with the analysis of their physics capabilities.

Compton scatter profiles for warm dense matter

In this paper, we discuss the possibility of using x-ray Compton scattering as a probe of the outer electronic structure of ions immersed in warm dense matter. It is proposed that the x-ray free-electron lasers currently under construction will provide an ideal tool for this, with the main pulse being used to create a uniform well-defined sample and the third harmonic providing a clean monochromatic probe. We model the plasma photon scatter spectrum by combining self-consistent finite-temperature electronic structure calculations with molecular dynamics simulations of the ion-ion structure factor. In particular, we present bound-free Compton profiles that are more accurate that those obtained using form factor or impulse approximations.

Method of separated form factors for polydisperse vesicles

Use of the Schulz or Gamma distribution in the description of particle sizes facilitates calculation of analytic polydisperse form factors using Laplace transforms, {\cal L}[f(u)]. Here, the Laplace transform approach is combined with the separated form factor (SFF) approximation [Kiselevet al.(2002).Appl. Phys. A,74, S1654–S1656] to obtain expressions for form factors,P(q), for polydisperse spherical vesicles with various forms of membrane scattering length density (SLD) profile. The SFF approximation is tested against exact form factors that have been numerically integrated over the size distribution, and is shown to represent the vesicle form factor accurately for typical vesicle sizes and membrane thicknesses. Finally, various model SLD profiles are used with the SFF approximation to fit experimental small-angle neutron scattering (SANS) curves from extruded unilamellar vesicles.

Recent experimental studies of photon–atom scattering in the X-ray energy region

We present a review of our recent measurements of large angle elastic and inelastic scattering differential cross-sections in the photon energy range 14–88 keV (momentum transfer ranging 1.135 to 6.310 Å −1 covering large number of elements in the atomic region Z=1–92, with special emphasis on the elements having K/Li shell/subshell binding energy in the vicinity of the incident photon energy. These measurements were performed using energy dispersive X-ray fluorescence (EDXRF) setup involving radioisotope as a photon source and a solid-state photon detector arranged in an annular geometry. The measured scattering differential cross-sections were compared with the theoretical values based on the state-of-art relativistic second order S-matrix calculations and those based on the form factor approximations in order to check their reliability. The KL and KM resonant Raman scattering (RRS) at 59.54 keV incident photon energy was also investigated for some heavy elements.

Electron density distribution in paramagnetic chromium: Aγ-ray diffraction study

High-accuracy single-crystal structure factors, complete up to $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\theta}∕\ensuremath{\lambda}=1.78\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$ have been measured from paramagnetic chromium at $333\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ using $316.5\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$ $\ensuremath{\gamma}$ radiation. A detailed description of the electron density distribution is derived in terms of a multipolar atomic deformation model. There is pronounced charge asphericity in the valence region arising from preferential occupancy of the ${t}_{2g}$ subshell. The $3d$ charge distribution is contracted by 12.6% relative to the free atom, in accordance with magnetic synchrotron x ray and neutron measurements. By contrast, the atomic crystal scattering factor deduced from $\ensuremath{\gamma}$-ray diffraction is found to be in contradiction with earlier experimental and theoretical work. Achievement of a reliable Debye-Waller factor is of vital importance in this context. There is no evidence for an anharmonic term in the atomic potential. Real space and energetic features of the charge density topology are used to characterize the directed metallic bonds. Special attention is paid to the form factor approximation in diffraction data analysis.

Elastic scattering of gamma rays and X-rays

Studies of elastic gamma ray scattering were pursued independently by the groups of Prof. Ghose and the author for several decades in spite of somewhat meagre support. Several techniques for such studies developed by the two groups and some of the results obtained in the energy range from tens of keV to about 1.5 MeV are described briefly. The theoretical background necessary for understanding these results is also outlined. The results showed the importance of Modified Relativistic Form Factor (MRFF) approximation in the explanation of atomic Rayleigh scattering cross sections in the small angle regime and the necessity for an inclusion of real Delbrück scattering amplitudes at large scattering angles. Dispersion corrections to form factor amplitudes or the so-called anomalous scattering factors are shown to be needed at photon energies close to electron binding energy thresholds. A few promising future extensions of such studies are indicated at the end.

Search for contact interactions, large extra dimensions and finite quark radius in ep collisions at HERA

A search for physics beyond the Standard Model has been performed with high-Q2 neutral current deep inelastic scattering events recorded with the ZEUS detector at HERA. Two data sets, e+p→e+X and e−p→e−X, with respective integrated luminosities of 112 pb−1 and 16 pb−1, were analyzed. The data reach Q2 values as high as 40000 GeV2. No significant deviations from Standard Model predictions were observed. Limits were derived on the effective mass scale in eeqq contact interactions, the ratio of leptoquark mass to the Yukawa coupling for heavy leptoquark models and the mass scale parameter in models with large extra dimensions. The limit on the quark charge radius, in the classical form factor approximation, is 0.85×10−16 cm.

Search for contact interactions, large extra dimensions and finite quark radius in ep collisions at HERA

A search for physics beyond the Standard Model has been performed with high- Q 2 neutral current deep inelastic scattering events recorded with the ZEUS detector at HERA. Two data sets, e + p → e + X and e − p → e − X , with respective integrated luminosities of 112 pb −1 and 16 pb −1 , were analyzed. The data reach Q 2 values as high as 40 000 GeV 2 . No significant deviations from Standard Model predictions were observed. Limits were derived on the effective mass scale in eeqq contact interactions, the ratio of leptoquark mass to the Yukawa coupling for heavy leptoquark models and the mass scale parameter in models with large extra dimensions. The limit on the quark charge radius, in the classical form factor approximation, is 0.85×10 −16 cm.

Nuclear isospin diffusivity

The isospin diffusion and other irreversible phenomena are discussed for a two-component nuclear Fermi system. The set of Boltzmann transport equations, such as employed for reactions, are linearized, for weak deviations of a system from uniformity, in order to arrive at nonreversible fluxes linear in the nonuniformities. Besides the diffusion driven by a concentration gradient, also the diffusion driven by temperature and pressure gradients is considered. Diffusivity, conductivity, heat conduction and shear viscosity coefficients are formally expressed in terms of the responses of distribution functions to the nonuniformities. The linearized Boltzmann-equation set is solved, under the approximation of constant form-factors in the distribution-function responses, to find concrete expressions for the transport coefficients in terms of weighted collision integrals. The coefficients are calculated numerically for nuclear matter, using experimental nucleon-nucleon cross sections. The isospin diffusivity is inversely proportional to the neutron-proton cross section and is also sensitive to the symmetry energy. At low temperatures in symmetric matter, the diffusivity is directly proportional to the symmetry energy.

Anomalous Rayleigh scattering with dilute concentrations of elements of biological importance

The anomalous scattering factor (ASF) correction to the relativistic form-factor approximation for Rayleigh scattering is examined in support of its utilization in radiographic imaging. ASF corrected total cross-section data have been generated for a low resolution grid for the Monte Carlo code EGS4 for the biologically important elements, K, Ca, Mn, Fe, Cu and Zn. Points in the fixed energy grid used by EGS4 as well as 8 other points in the vicinity of the K-edge have been chosen to achieve an uncertainty in the ASF component of 20% according to the Thomas–Reiche–Kuhn sum rule and an energy resolution of 20 eV. Such data is useful for analysis of imaging with a quasi-monoenergetic source. Corrections to the sampled distribution of outgoing photons, due to ASF, are given and new total cross-section data including that of the photoelectric effect have been computed using the Slater exchange self-consistent potential with the Latter tail. A measurement of Rayleigh scattering in a dilute aqueous solution of manganese (II) was performed, this system enabling determination of the absolute cross-section, although background subtraction was necessary to remove K β fluorescence and resonant Raman scattering occurring within several 100 eV of the edge. Measurements confirm the presence of below edge bound–bound structure and variation in the structure due to the ionic state that are not currently included in tabulations.

Approximation of Hartree-Fock atomic form factors within the shell model

On the basis of the shell model for the electron distribution in an atom, analytic expressions and approximations are constructed for relativistic Hartree-Fock atomic form factors. The proposed approach ensures a precision that is two to three times higher than that of approximations based on existing models. This approach makes it possible to obtain simple expressions for a screening function of the Thomas-Fermi type.

Anomalous Rayleigh scatter in dilute media

Anomalous Rayleigh scatter is examined for dilute concentrations of the biomedically relevant element iodine in aqueous media including measurements with monochromatic synchrotron radiation in the vicinity of the iodine K-edge. The measurements agree with anomalous scatter-factor corrections to the form-factor approximation which has been shown to have good agreement with higher precision S-matrix calculations for small angle scatter over a wide range of energies but has not been adequately tested at the edge. Monte Carlo modelling, including the modelling of polarized Compton and Rayleigh scattered x-rays, is used to determine the relative contributions of the scatter and fluorescent components at the detector as well as the modelling of self-absorption and relative dose in the determination of detection limits. A Rayleigh scatter minimum of 28 barns/sr was observed at an energy 10 ± 5 eV below the K-edge of iodine at a position predicted from an evaluation of the dispersion integral that includes bound–bound resonance contributions. Minimum detectable concentrations for observation of the anomalous Rayleigh scatter feature at an exposure of 10 mSv, predicted for iodine and iron, are 1 mg ml−1 and 10 mg ml−1, respectively. Upper limits to detection of the feature imposed by degradation of the signal by self-absorption are 0.021 g cm−2 and 0.0029 g cm−2 radiation lengths, respectively.

Measurements of Rayleigh scattering and relativistic calculations of incoherent scattering functions

Recent measurements of Rayleigh scattering employing neutron capture γ-rays are presented. Experimental conditions are achieved such that the Rayleigh contribution is dominant and much larger than the other competing coherent processes. A detailed comparison with the modified relativistic form factor (MRFF) approximation is made and it is concluded that the latter overestimates the cross-section by 3–4%. New calculations of S, the incoherent scattering function, are presented in the relativistic treatment of Ribberfors and Berggren, using multiconfiguration Dirac–Fock relativistic wavefunctions. Tables of S, for Z=1–110, are shown on a momentum transfer mesh identical to previous non-relativistic calculations. S has been calculated at a representative angle θ=60° and energies compatible with the presentation mesh. For other scattering angles, the values presented in the tables are accurate to within 1–2% for momentum transfers larger than 0.1 Å −1. In the region below 0.1 Å −1 the accuracy worsens with decreasing momentum transfer, reaching 6% at 0.01 Å −1 and 10% at 0.005 Å −1. The same multiconfiguration wave functions were used to evaluate new MRFFs. The new elastic scattering cross sections differ by 3–6% compared with calculations based on single configuration wave functions.

Corrections to the X-ray incoherent scattering factor to obtain the total inelastic X-ray scattering

Corrections to the form factor approximation required to obtain the A 2 contribution to the non-relativistic X-ray scattering are considered. The previously proposed expansion method [Phys. Rev. A 23 (1981) 2950], based on sum rules of the Generalized Oscillator Strength (GOS), is compared to a direct calculation based on a hydrogenic wave function and the exact analytical form of the GOS for single ionization of a hydrogen atom. It is shown that the results of the expansion method are in good agreement with the exact hydrogen atom result for the continuum contribution to the GOS. For the case of He the results of the two approaches agree less well with the best agreement occurring at high momentum transfer. The problem can be traced to the lack of orthogonality between the bound state and continuum functions used. It is concluded that, for complex atoms, the expansion method is to be preferred over corrections based on direct calculations of the GOS from approximate wave functions unless orthogonality between bound state and continuum functions is taken in to account.

Nonlocal-exchange effects in elastic photon scattering from low- Z elements

We investigate the importance of using nonlocal versus local exchange in describing the elastic scattering of photons from low- Z elements in the range 2≤ Z≤18. Correction factors are given that can be used to account for nonlocal exchange effects starting from a cross section evaluated using local exchange, as, for example, in local independent-particle-approximation S-matrix calculations. The correction factors are derived using the form-factor approximation, and are therefore only applicable in the above-threshold regime where the beyond-form-factor-approximation anomalous scattering amplitudes are comparatively small. However, it is seen that even well above threshold, the nonlocal correction can be as large as ≈10% or more for low- Z elements, as has recently been demonstrated experimentally for the case of neon.