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When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-spherical? The assumption that this particular term is very close to zero is generally made in non-spherical refinements (for example, implementations of Hirshfeld atom refinement or transferable aspherical atom models), unless the form factors are refinable parameters (for example multipole modelling). To evaluate this general approximation for one specific method, a numerical differentiation was implemented within the NoSpherA2 framework to calculate the derivatives of the structure factors in a Hirshfeld atom refinement directly as accurately as possible, thus bypassing the approximation altogether. Comparing wR 2 factors and atomic parameters, along with their uncertainties from the approximate and numerically differentiating refinements, it turns out that the impact of this approximation on the final crystallographic model is indeed negligible.

Electron scattering on liquid samples has been enabled recently by the development of ultrathin liquid sheet technologies. The data treatment of liquid-phase electron scattering has been mostly reliant on methodologies developed for gas electron diffraction, in which theoretical inputs and empirical fittings are often needed to account for the atomic form factor and remove the inelastic scattering background. The accuracy and impact of these theoretical and empirical inputs has not been benchmarked for liquid-phase electron scattering data. In this work, we present an alternative data treatment method that requires neither theoretical inputs nor empirical fittings. The merits of this new method are illustrated through the retrieval of real-space molecular structure from experimental electron scattering patterns of liquid water, carbon tetrachloride, chloroform, and dichloromethane.

High-accuracy X-ray mass attenuation coefficients were measured from the first X-ray Extended Range Technique (XERT)-like experiment at the Australian Synchrotron. Experimentally measured mass attenuation coefficients deviate by ∼50% from the theoretical values near the zinc absorption edge, suggesting that improvements in theoretical tabulations of mass attenuation coefficients are required to bring them into better agreement with experiment. Using these values the imaginary component of the atomic form factor of zinc was determined for all the measured photon energies. The zinc K-edge jump ratio and jump factor are determined and results raise significant questions regarding the definitions of quantities used and best practice for background subtraction prior to X-ray absorption fine-structure (XAFS) analysis. The XAFS analysis shows excellent agreement between the measured and tabulated values and yields bond lengths and nanostructure of zinc with uncertainties of from 0.1% to 0.3% or 0.003 Å to 0.008 Å. Significant variation from the reported crystal structure was observed, suggesting local dynamic motion of the standard crystal lattice. XAFS is sensitive to dynamic correlated motion and in principle is capable of observing local dynamic motion beyond the reach of conventional crystallography. These results for the zinc absorption coefficient, XAFS and structure are the most accurate structural refinements of zinc at room temperature.

In the present study, we calculated the imaginary part of the x-ray scattering factor of nickel based on the principles of quantum mechanics to find a wave function that describes the electronic state of atoms by approximate methods, observed the study suggested that in both low energy values , and at high energy values , the imaginary part is approximately zero, this means that the electrons are intensely connected to the atom, where in the spectrum the photon energies are approximately equal to the electron bonding energy we note the study pointed out that the imaginary part of the atomic scattering factor become prominent and the electron becomes highly absorbent, the relative accuracy varies within range (0.03-0.22)%, and there was also a good agreement between the behavior we obtained for the imaginary part of the atomic scattering factor and the behavior that was calculated using other models. http://dx.doi.org/10.25130/tjps.23.2018.171

A Fourier component of the potential energy of interaction between two atoms has been presented as a polynomial of the biquadratic atomic form factor. A numerical calculation has been performed in the screened Coulomb potential approximation. It is demonstrated that the account for the Pauli principle leads to the occurrence of a potential barrier and an additional region of attraction of two atoms. This model is shown to agree qualitatively with the results of the density-functional-theory calculation.

More than 80 years have passed since the first calculations of electron–nucleus bremsstrahlung cross sections were published by Sommerfeld, for non-relativistic electrons, and, independently, by Sauter, Bethe and Heitler, and Racah, for relativistic electrons. The Bethe–Heitler expression, that is based on the first Born approximation and includes the screening of the Coulomb field of the nucleus by the atomic electrons, has proven to work well at moderately high energies where the Landau–Pomeranchuk–Migdal effect is negligible. We review the current theoretical and experimental status with a highlight on electrons with kinetic energies between 1 and 10 MeV. The choice is motivated by the peculiar difficulties present in this energy region, where it is necessary to treat simultaneously the interaction with the Coulomb field beyond the first Born approximation and the effect of screening. A fully numerical approach within the S–matrix formalism has proven to be extremely difficult above a few MeV, because the number of partial waves needed for an accurate evaluation is prohibitively large. Here we focus on analytic results, including the more complex ones employing the Furry–Sommerfeld–Maue wave functions and taking into account the next-to-leading order, and discuss the advantages and limitations in light of the best available data. The influence of multiple scattering in the target is investigated under the actual experimental conditions. A comparison with the widely used cross section tabulations by Seltzer and Berger is also presented.

We prove some trigonometric identities involving Chebyshev polynomials of second kind. The identities were inspired by atomic form factor calculations. Generalizations of these identities, if found, will help to increase the numerical stability of atomic form factors calculations for highly excited states., Comment: 6 pages, no figures, to be published in Georgian Mathematical Journal

Total scattering measurements enable understanding of the structural disorder in crystalline materials by Fourier transformation of the total structure factor, S(Q), where Q is the magnitude of the scattering vector. In this work, the direct calculation of total scattering from a crystalline structural model is proposed. To calculate the total scattering intensity, a suitable Q-broadening function for the diffraction profile is needed because the intensity and the width depend on the optical parameters of the diffraction apparatus, such as the X-ray energy resolution and divergence, and the intrinsic parameters. X-ray total scattering measurements for CeO2 powder were performed at beamline BL04B2 of the SPring-8 synchrotron radiation facility in Japan for comparison with the calculated S(Q) under various optical conditions. The evaluated Q-broadening function was comparable to the full width at half-maximum of the Bragg peaks in the experimental total scattering pattern. The proposed calculation method correctly accounts for parameters with Q dependence such as the atomic form factor and resolution function, enables estimation of the total scattering factor, and facilitates determination of the reduced pair distribution function for both crystalline and amorphous materials.

Although the determination of site occupancies is often a major goal in Rietveld refinement studies, the accurate refinement of site occupancies is exceptionally challenging due to many correlations and systematic errors that have a hidden impact on the final refined occupancy parameters. Through the comparison of results independently obtained from neutron and synchrotron powder diffraction, improved approaches capable of detecting occupancy defects with an exceptional sensitivity of 0.1% (absolute) in the class of layered NMC (Li[NixMnyCoz]O2) Li-ion battery cathode materials have been developed. A new method of visualizing the diffraction parameter space associated with crystallographic site scattering power through the use of f* diagrams is described, and this method is broadly applicable to ternary compounds. The f* diagrams allow the global minimum fit to be easily identified and also permit a robust determination of the number and types of occupancy defects within a structure. Through a comparison of neutron and X-ray diffraction results, a systematic error in the synchrotron results was identified using f* diagrams for a series of NMC compounds. Using neutron diffraction data as a reference, this error was shown to specifically result from problems associated with the neutral oxygen X-ray atomic form factor and could be eliminated by using the ionic O2- form factor for this anion while retaining the neutral form factors for cationic species. The f* diagram method offers a new opportunity to experimentally assess the quality of atomic form factors through powder diffraction studies on chemically related multi-component compounds.

We have derived the universal eikonal-Glauber Thomas–Fermi model for atomic collision cross-sections with many-electron atoms, such as iron and tungsten atoms, including the influence of atomic screening in fusion devices and plasma technologies. The eikonal-Glauber method is employed to obtain the analytic expressions for the effective atomic charge, the scattering phase shift and the atomic cross-section in terms of the atomic form factor and the Mott–Massey screening parameter. The result shows that the effective atomic charge would be the same as the case of the net nuclear charge for the large momentum transfer domain and becomes zero without momentum transfer due to the influence of bound atomic electrons. It is shown that the eikonal scattering phase shift and the total eikonal-Glauber scattering cross-section increase with increasing charge number$Z$of the nucleus of the target atom. It is also found that the charge dependence of the total eikonal-Glauber scattering cross-section decreases with an increase of the scaled collision energy since the atomic form factor is small for large collision energies.

Rayleigh to Compton scattering ratio will takes a major and most significant role in the non-destructive nuclear radiation technique. This non destructive technique is feasible in the industry, particularly in the drug manufacturing field. The R/C ratio could be theoretically estimated by using the atomic form factor (F) and incoherent (Compton) scattering function (S) for some pharmaceutical ingredients at different scattering angle from 20 o to 180 o with defined photon energy from 8 keV to 32 keV X-rays. The opted samples are low Z compounds and are inspecting an interaction with x-rays and studied the variation of Z eff , R/C, and electron density of respective samples with the function of scattering angle.

To investigate the behaviour of poly(propylene imine) dendrimers -- and urea-adamantyl functionalized ones -- in solution using molecular dynamics simulations, we developed a coarse grained model to tackle the relatively large system sizes and time scales needed. Harmonic bond and angle potentials were derived from atomistic simulations using an iterative Boltzmann inversion scheme, modified to incorporate Gaussian fits of the bond and angle distributions. With the coarse grained model and accompanying force field simulations of generations 1 to 7 of both dendrimer types in water were performed. They compare favourably with atomistic simulations and experimental results on the basis of size, shape, monomer density, spacer back-folding and atomic form factor measurements. These results show that the structural dynamics of these dendrimers originate from flexible chains constrained by configurational and spatial requirements. Large dendrimers are more rigid and spherical, while small ones are flexible, alternatively rod-like and globular.

The study of charge transfer has been carried out on Thulium Sulphide (TmS) compound from x-ray powder diffraction data. The theoretical x-ray powder diffraction data (XRPD) has been simulated and found to be in good agreement with experimental x-ray powder diffraction data. The structure factor values have been determined. The atomic form factor curves deduce charge transfer of 0.03 electrons from Thulium (Tm) to Sulphur (S) in TmS which proves the trivalent state of the system.

A new method of elemental analysis of solid crystals by the X-ray diffraction method is proposed. The possibility of application of this method to multicomponent polymorphic systems (including a wide range of elements being analyzed and quantitative determination of their mass concentrations without using an external or internal standard) is demonstrated for natural aluminum silicate and titanium oxide samples as examples.

For photon energies below 1 MeV the predominant phenomenon involved is photoelectric absorption, coherent scattering (Rayleigh), and incoherent scattering (Compton). Biological samples with small t ...

We show that the Bayesian inference of recently measured x-ray diffraction spectra from laser-shocked aluminum [L. B. Fletcher et al., Nat. Photon. 9, 274 (2015)10.1038/nphoton.2015.41] with the one-component-plasma (OCP) model performs remarkably well at estimating the ionic density and temperature. This statistical approach requires many evaluations of the OCP static structure factor, which were done using a recently derived analytic fit. The atomic form factor is approximated by an exponential function in the diffraction window of the first peak. The electronic temperature is then estimated from a comparison of this approximated form factor with the electronic structure of an average atom model. Out-of-equilibrium states, with electrons hotter than ions, are diagnosed for the spectra obtained early after the pump, whereas at a late time delay the plasma is at thermal equilibrium. Apart from the present findings, this OCP-based modeling of warm dense matter has an important role to play in the interpretation of x-ray Thomson scattering measurements currently performed at large laser facilities.

The behavior of poly(propylene imine) (PPI) dendrimers in concentrated solutions has been investigated using molecular dynamics simulations containing up to a thousand PPI dendrimers of generation 4 or 5 in explicit water. To deal with large system sizes and time scales required to study the solutions over a wide range of dendrimer concentrations, a previously published coarse-grained model was applied. Simulation results on the radius of gyration, structure factor, intermolecular spacing, dendrimer interpenetration and water penetration are compared with available experimental data, providing a clear concentration dependent molecular picture of PPI dendrimers. It is shown that with increasing concentration the dendrimer volume diminishes accompanied by a reduction of internalized water, ultimately resulting in solvent filled cavities between stacked dendrimers. Concurrently dendrimer interpenetration increases only slightly, leaving each dendrimer a separate entity also at high concentrations. Moreover, we compare apparent structure factors, as calculated in experimental studies relying on the decoupling approximation and the constant atomic form factor assumption, with directly computed structure factors. We demonstrate that these already diverge at rather low concentrations, not because of small changes in form factor, but rather because the decoupling approximation fails as monomer positions of separate dendrimers become correlated at concentrations well below the overlap concentration, The behavior of poly(propylene imine) (PPI) dendrimers in concentrated solutions has been investigated using molecular dynamics simulations containing up to a thousand PPI dendrimers of generation 4 or 5 in explicit water. To deal with large system sizes and time scales required to study the solutions over a wide range of dendrimer concentrations, a previously published coarse-grained model was applied. Simulation results on the radius of gyration, structure factor, intermolecular spacing, dendrimer interpenetration and water penetration are compared with available experimental data, providing a clear concentration dependent molecular picture of PPI dendrimers. It is shown that with increasing concentration the dendrimer volume diminishes accompanied by a reduction of internalized water, ultimately resulting in solvent filled cavities between stacked dendrimers. Concurrently dendrimer interpenetration increases only slightly, leaving each dendrimer a separate entity also at high concentrations. Moreover, we compare apparent structure factors, as calculated in experimental studies relying on the decoupling approximation and the constant atomic form factor assumption, with directly computed structure factors. We demonstrate that these already diverge at rather low concentrations, not because of small changes in form factor, but rather because the decoupling approximation fails as monomer positions of separate dendrimers become correlated at concentrations well below the overlap concentration.

A new approach for approximating the continuum wave functions for hydrogenic atoms with Gaussians basis sets is developed and tested. In this the plane wave is left unchanged and the distorting factor, represented by the Confluent Hypergeometric function, is expanded as a sum of Spherical Harmonics multiplied by a series of Gaussians. In this way the number of spherical waves and Gaussians will be significantly reduced and the plane wave will be responsible for the momentum conservation. As compared with previous methods that expand the full continuum, including the plane wave, our strategy does not require a great quantity of partial waves for convergence. Dense oscillations which are characteristic of the plane wave, are avoided. To test the performance of this approximation to describe a free-bound atomic form factor, the ionization cross section of hydrogen by impact of protons in first Born approximation is calculated. Compared with the exact results, a good agreement with just 4 spherical waves and ten Gaussians each is obtained. The method looks very interesting, especially to speed up atomic and molecular collision calculations involving the continuum Fil: Fiori, Marcelo Raúl. Universidad Nacional de Salta; Argentina Fil: Miraglia, Jorge Esteban. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina

The limiting uncertainty in recent high accuracy measurements of the mass attenuation coefficient is the measurement of the integrated column density. An improvement in the design of the absorption foil holder is described which reduces the integrated column density uncertainty. The new design allows the edges of the foil to be more accurately mapped by the X-ray beam by reducing the largest source of uncertainty in the foil mapping: the uncertainty in the points along the foil edge. The method is shown to reduce the uncertainty in measurements of the mass attenuation coefficient of zinc foils. The reduced uncertainty in the full-foil mapping will allow the X-ray Extended Range Technique (XERT) to be applied to small non-metallic absorption foils more accurately.

Using analytical formulae which are exact in Born approximation, the doubly differential bremsstrahlung cross-section with form-factor screening is calculated. For the atomic form factor parameters are applied which approximate self-consistent Dirac–Hartree–Fock–Slater calculations. The evaluation of the bremsstrahlung spectrum requires a single numerical integration. The results are superior to the customary Bethe approximation, in particular at the high-energy part of the spectrum. At high energies the screening correction can be added to any Coulomb-corrected cross-section without screening. In the present work, we are using a cross-section calculated by means of Sommerfeld–Maue functions with additional higher-order terms.

Coherent to incoherent differential cross-section ratios of Au have been measured for several polar scattering angles (90°, 100°, 110°, 120° and 130°) and azimuthal angles (30°, 20°, 10°, 0°, −10°, and −20°) at 59.54 keV photon energy by using high purity germanium (HPGe) detector, which has a resolution of 199.6 eV at the 5.9 keV. The samples were excited with 59.54 keV gamma rays emitted from 3.7×10 9 Bq (100 mCi) Am 241 point source. The intensity ratios were corrected due to the photopeak efficiency of gamma detector and absorption of photons in the target and air. The experimental values obtained in this study were compared with those estimated on the basis of the non-relativistic form factors and relativistic form factors.

We use the x-ray extended range technique (XERT) to experimentally determine the mass attenuation coefficient of silver in the x-ray energy range 11 kev-28 kev including the silver K absorption edge. The results are accurate to better than 0.1%, permitting critical tests of atomic and solid state theory. This is one of the most accurate demonstrations of cross-platform accuracy in synchrotron studies thus far. We derive the mass absorption coefficients and the imaginary component of the form factor over this range. We apply conventional XAFS analytic techniques, extended to include error propagation and uncertainty, yielding bond lengths accurate to approximately 0.24% and thermal Debye-Waller parameters accurate to 30%. We then introduce the FDMX technique for accurate analysis of such data across the full XAFS spectrum, built on full-potential theory, yielding a bond length accuracy of order 0.1% and the demonstration that a single Debye parameter is inadequate and inconsistent across the XAFS range. Two effective Debye-Waller parameters are determined: a high-energy value based on the highly-correlated motion of bonded atoms (σ(DW) = 0.1413(21) A), and an uncorrelated bulk value (σ(DW) = 0.1766(9) A) in good agreement with that derived from (room-temperature) crystallography.

Atomic form factors are widely used for the characterization of targets and specimens, from crystallography to biology. By using recent mathematical results, here we derive an analytical expression for the atomic form factor within the independent particle model constructed from nonrelativistic screened hydrogenic wavefunctions. The range of validity of this analytical expression is checked by comparing the analytically obtained form factors with the ones obtained within the Hartee-Fock method. As an example, we apply our analytical expression for the atomic form factor to evaluate the differential cross section for Rayleigh scattering off neutral atoms., 7 pages, 1 figure

We hypothesize that in a non-metallic crystalline structure under extreme pressures, atomic wavefunctions deform to adopt a reduced rotational symmetry consistent with minimizing interstitial space in the crystal. We exemplify with a simple numeric variational calculation that yields the energy cost of this deformation for Helium to 25%. Balancing this with the free energy gained by tighter packing we obtain the pressures required to effect such deformation. The consequent modification of the structure suggests a decrease in the resistance to tangential stress, and an associated decrease of the crystal's shear modulus. The atomic form factor is also modified. We also compare with neutron matter in the interior of compact stars., 6 pages, 13 figure files

Many natural surface-environment samples, such as soils and marine and lacustrine sediments, are complex mixtures involving several mineral phases (both petrogenic and authigenic), several amorphous and nanoparticulate inorganic phases (usually authigenic or pedogenic), and a heterogeneous amorphous component of “organic matter” (OM). The main inorganic amorphous and crystalline components are often operationally separated and quantified by various selective or sequential chemical extraction methods that are subject to various artifacts and that can be significantly affected by the presence of OM. Here we develop a general method of absolute quantification based on powder X-ray diffraction (pXRD) measurements that are obtained using standard 𝛉- 𝛉 or 𝛉-2𝛉 type diffractometers. The method does not require calibration or the use of standards and does not require instrument parameter determinations but relies instead on exact normalization conditions that we prove. In particular, we develop a “general integrated intensity formula” (GIIF) for X-ray diffraction. All relevant sample-radiation interaction phenomena are considered, including polarization, mass absorption, Compton scattering, and resonance absorption re-emission. We show that the mole fraction of any given phase (crystalline, amorphous, quasicrystalline, or nanophase) is exactly given by the collection sphere integrated intensity of the resolved phase-specific contribution to the Compton corrected and electron unit normalized diffraction pattern, T (𝛉), divided by the wavevector ( q ) integrated average squared atomic form factor of the phase. Electron unit calibration is achieved by a global normalization that directly gives the product A ⊥ I o of the effective cross sectional area ( A ⊥ ) of the incident (and outgoing) beam and the effective incident beam intensity ( I o ), including counter efficiency, beam path losses, etc. The problem of incomplete collection sphere integration (including the q = 0 region) is resolved by showing that all the results hold for a given Bragg angle range of measurement for a sufficiently large range. We evaluate the accuracy of the method by application to synthetic binary amorphous-crystalline mixtures of (1) a rock standard, that provides an assembly of crystalline phases, (2) a certified OM standard, and (3) a synthetic inorganic amorphous phase (silica gel). We expect that the method will be particularly useful in surface sediment and environmental geochemistry applications.

Symmetry and physical aspects of 'forbidden' reflections excited by a local polarization anisotropy of the X-ray susceptibility are surveyed. Such reflections are observed near absorption edges where the anisotropy is caused by distortions of the atomic electronic states owing to interaction with neighbouring atoms. As a consequence, they allow for extracting nontrivial information about the resonant atom's local environment and their physical conditions. The unusual polarization properties of the considered reflections are helpful to distinguish them from other types of 'forbidden' reflections. When such reflections are excited, it is, for example, possible to determine not only the intrinsic anisotropy of an atomic form factor but also additional anisotropy induced by thermal motion, point defects and/or incommensurate modulations. Even the local ;chirality' of atoms in centrosymmetric crystals is accessible. Unsolved key problems and possible future developments are addressed.

New small angle coherent scattering data are presented in order to emphasize the need to update available open-access databases. The first aim is to integrate the present atomic form factor tabulations, which permit us to obtain the linear differential scattering coefficient of a compound from a simple weighted sum of the elemental components, by including the effect of the large-scale arrangement of the sample structure beside the molecular interference effect. These features, which can lead the intensity of scattered photons to present huge upturns at small values of the momentum transfer, are presented for some biological tissues and poly(methyl methacrylate) and their influence on the calculation of both the coherent mass attenuation coefficient and the total value is investigated.

We used the x-ray-extended range technique to measure the x-ray mass attenuation coefficients of silver in the 15–50 keV energy range with a level of uncertainty between 0.27% and 0.4% away from the K-edge. The imaginary part of the atomic form factor of silver was derived by subtracting the scattering component from the measured total mass attenuation coefficients. Discrepancies between the measured mass attenuation coefficients and alternative theoretical predictions are discussed.

This article, considers in detail P. Gombás's idea of grouping electrons into n- and nl-shells in the Thomas–Fermi theory of free atoms briefly, the TFG n- and TFG nl-models respectively). Using these models, exact analytical expressions for the total energy E and the atomic form factor F(κ) are obtained. All integrals of the TFG nl-model are computed by means of the hypergeometric functions 2F1(x), 3F2(x), F2(x,y) and FA(x1,…,x6) for the first time. In particular, Weizsäcker's gradient correction to the kinetic energy of the nl-th shell [Formula: see text] generates a new numerical triangle [Formula: see text] with coefficients bw=n+2l(n-l-1).