Atomic Form Factors Research Articles

Eikonal-Glauber Thomas–Fermi model for atomic collisions with many-electron atoms for plasma applications

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.

DiSCaMB: a software library for aspherical atom model X-ray scattering factor calculations with CPUs and GPUs.

It has been recently established that the accuracy of structural parameters from X-ray refinement of crystal structures can be improved by using a bank of aspherical pseudoatoms instead of the classical spherical model of atomic form factors. This comes, however, at the cost of increased complexity of the underlying calculations. In order to facilitate the adoption of this more advanced electron density model by the broader community of crystallographers, a new software implementation called DiSCaMB, 'densities in structural chemistry and molecular biology', has been developed. It addresses the challenge of providing for high performance on modern computing architectures. With parallelization options for both multi-core processors and graphics processing units (using CUDA), the library features calculation of X-ray scattering factors and their derivatives with respect to structural parameters, gives access to intermediate steps of the scattering factor calculations (thus allowing for experimentation with modifications of the underlying electron density model), and provides tools for basic structural crystallographic operations. Permissively (MIT) licensed, DiSCaMB is an open-source C++ library that can be embedded in both academic and commercial tools for X-ray structure refinement.

Новый подход к вычислению потенциальной энергии взаимодействия двух атомов

AbstractA 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.

Bremsstrahlung from few-electron ions

We have developed the new approach which can be used for accurate evaluations of the bremsstrahlung cross-sections in the case of few-electron atoms and ions. Our approach is based on the explicit formula for the electron density distribution in such systems. We derive the closed analytical formula for the matrix elements which are needed for highly accurate computations of atomic form-factors of two-electron atoms and ions. Multiple bremsstrahlung from few-electron ions is considered. We also discuss the energy loss due to bremsstrahlung in a plasma which contains multi-charged ions and free electrons. Bremsstrahlung from a high-temperature plasma is considered as well as its role in the high-temperature burn-up of deuterium plasma.

Channeling and electromagnetic radiation of relativistic charged particles in metal-organic frameworks

We have developed the theory of electromagnetic interaction of relativistic charged particles with metal-organic frameworks (MOFs). The electrostatic potential and electron number density distribution in MOFs were calculated using the most accurate data for the atomic form factors. Peculiarities of axial channeling of fast charged particles and various types of electromagnetic radiation from relativistic particles has been discussed.

An algorithm for Monte Carlo simulation of bremsstrahlung emission by electrons

An algorithm for Monte Carlo simulation of bremsstrahlung emission by electrons based on the framework of SuperMC is presented in this paper with efficient and accurate methods to sample the angular distribution and energy of bremsstrahlung photons. The photon energy is sampled according to scaled energy-loss differential cross sections tabulated by Seltzer and Berger. A novel hybrid model for photon angular distribution by low- and high-energy incident electrons is developed. The model uses Tsai’s full form of angular distribution function with atomic form factors for high-energy incident electrons. For electrons of <500 keV, a simple efficient and accurate analytical distribution function is developed, using adjustable parameters determined from the fitting of numerical values of the shape functions tabulated by Kissel et al. The efficiency of sampling photon energy is 80%. Our angular sampling algorithm for high-energy electron bremsstrahlung based on Tsai distribution function is very efficient (sampling efficiency ~70%) in the useful photon energy range.

Implicit Hydration Modelling in Small-Angle X-Ray Scattering for Protein Structure Determination

Small angle X-ray scattering (SAXS) data are being increasingly used in structure modeling of biomacromolecular systems such as proteins. While some great progresses have been made in modelling theoretical scattering intensity from a given structure, what is still challenging is the molecular structure optimization based on experimental intensity data. To address this issue, we propose a new method to evaluate the theoretical scattering intensity of an input protein structure by accounting for hydration in an accurate, but efficient fashion. The method is designed specifically for performing efficient structure optimizations. In our approach, form factors of protein atoms or coarse-grained beads are modified by incorporating an implicit hydration term scaled by their solvent accessible surface areas to represent the hydration-layer scattering. A key advantage of the approach is the availability of analytical derivatives of the scattering intensity with respect to the protein coordinates, making our method very suitable for performing structure modelling using gradient-based optimization schemes. Another technical advance is that the implicit hydration term is parameterized by fitting the theoretical scattering intensities computed from all-atom protein models with explicit hydration water molecules for a high resolution non-redundant protein structure dataset. We show that the obtained implicit hydration term is of good accuracy and applicable to both atomistic and coarse-grained structure representations. By using a pseudo energy function containing a SAXS-fitting score and a structure-restraint term, together with the conjugate gradient method to perform structure optimization, we can obtain final structures very close to the known target structures for four benchmark protein molecules. Taken together, this newly developed implicit hydration approach for SAXS-intensity evaluation is capable of accurate SAXS computing, thereby can be of great usage in the structure modelling based on experimental SAXS profiles.

Bayesian inference of x-ray diffraction spectra from warm dense matter with the one-component-plasma model.

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.

Coarse-grained simulations of poly(propylene imine) dendrimers in solution.

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.

Modeling Solution X-Ray Scattering with Knowledge-Based Coarse-Grained form Factors

In molecular simulation, the atomic interaction sites of a biomolecule are often grouped into a fewer number of coarse-grained sites for higher computational efficiency. Coarse-grained protein models are also widely used in computing solution X-ray scattering intensities. Traditionally, coarse-grained scattering form factors are constructed based on the Debye formula, which assumes free rotation of each coarse-grained site, such as an amino acid residue. Nevertheless, this approach results in significant discrepancy between all-atom and coarse-grained scattering profiles, due to the structure correlations between different parts of a protein.We implemented a bioinformatics approach to tackle this problem, and a set of coarse-grained form factors were developed on the basis of a protein structure database. In our numerical approach, the form factor parameterization is converted to a linear least square problem, which can efficiently construct form factors with arbitrary coarse-grained mapping. We systematically compared the scattering intensities of various proteins from atomic form factors, Debye formula-based coarse-grained form factors and our developed form factors. The results show that the chi-square values from our developed form factors are substantially improved, using intensities from all-atom models as the standard. Moreover, we computed the hydration effect using an explicit water model for a few proteins with experimental small angle X-ray scattering data, and fitted the simulated scattering profiles to the experimental counterparts. We found that the intensities modeled using our coarse-grained form factors can fit the experimental intensities with small chi-square values, indicating the capability of our form factors in modeling experimental data. The developed form factors have potential usage in scattering-guided biomacromolecule structure optimization and high throughput scattering modeling, where computational efficiency plays an important role.

Study of Rayleigh to Compton scattering ratio for pharmaceutical ingredients at 8 keV to 32 keV X-ray energy

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 20o to 180o 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 Zeff, R/C, and electron density of respective samples with the function of scattering angle.

Neutron diffraction and the electronic properties of BaFe2Se3

It is argued on the basis of previously published experimental data that, the magnetic space-group Cac (#9.41) is the correct description of magnetically ordered BaFe2Se3. The corresponding crystal class m1′ allows axial and polar dipoles and forbids bulk ferromagnetism. Magneto–electric multipoles that are both time-odd and parity-odd are allowed, e.g., a magnetic charge (monopole) and an anapole (magnetic toroidal dipole). The experimental observation of magneto–electric multipoles must shed light on valence electrons involved in bonding, including charge transfer using 3d(Fe) and p-states of ligand ions. We provide the appropriate structure factors for the Bragg diffraction neutrons, together with estimates of atomic form factors. Structure factors for resonant x-ray Bragg diffraction are also considered, because the analysis of successful experiments will yield complementary information about electronic properties. Magneto–electric multipoles, over and above those that contribute to magnetic neutron diffraction, include the magnetic monopole. A time-odd, parity-even monopole created from the magnetic dipole and an electric toroidal dipole, which is a manifestation of a structural rotation, is allowed in BaFe2Se3 but it is not visible in diffraction, nor is the corresponding dipole.

Modeling solution X-ray scattering of biomacromolecules using an explicit solvent model and the fast Fourier transform

A novel computational method based on atomic form factors and the fast Fourier transform (FFT) is developed to compute small- and near-wide-angle X-ray scattering profiles of biomacromolecules from explicit solvent modeling. The method is validated by comparing the results with those from non-FFT approaches and experiments, and good agreement with experimental data is observed for both small and near-wide angles. In terms of computational efficiency, the FFT-based method is advantageous for protein solution systems of more than 3000 atoms. Furthermore, the computational cost remains nearly constant for a wide range of system sizes. The FFT-based approach can potentially handle much larger molecular systems compared with popular existing methods.

Coarse-grained modelling of urea-adamantyl functionalised poly(propylene imine) dendrimers

To investigate the behaviour of poly(propylene imine) dendrimers – and urea–adamantyl functionalised 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 1–7 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.

Effective atomic number of some sugars and amino acids for scattering of 241Am and 137Cs gamma rays at low momentum transfer

In this paper, we report the effective atomic number of some H, C, N and O based sugars and amino acids. These have been determined by using a handy expression which is based on the theoretical angle integrated small angle (coherent+incoherent) scattering cross sections of seven elements of Z≤13 in four angular ranges of (0–4°), (0–6°), (0–8°) and (0–10°)for 241Am (59.54keV) and 137Cs (661.6keV) gamma rays. The theoretical scattering cross sections were computed by a suitable numerical integration of the atomic form factor and incoherent scattering function compilations of Hubbell et al. (1975) which make use of the non-relativistic Hartree–Fock (NRHF) model for the atomic charge distribution of the elements in the angular ranges of interest. The angle integrated small angle scattering cross sections of the H, C, N and O based sugars and amino acids measured by a new method reported recently by the authors were used in the handy expression to derive their effective atomic number. The results are compared with the other available data and discussed. Possible conclusions are drawn based on the present study.

Analytical evaluation of atomic form factors: Application to Rayleigh scattering

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 wave functions. 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.

Cubic Wavefunction Deformation of Compressed Atoms

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.

Studying the x- ray atomic Scattering form Factor and Nuclear Magnetic Shielding Constant for Be atom in its Excited state (1s2 2s 3s) and the Be - like ions

The division partitioning technique has been used to analyze the four electron systems into six-pairs electronic wave functions for ( for the Beryllium atom in its excited state (1s2 2s 3s ) and like ions ( B+1 ,C+2 ) using Hartree-Fock wave functions . The aim of this work is to study atomic scattering form factor f(s) for and nuclear magnetic shielding constant. The results are obtained numerically by using the computer software (Mathcad).

Studying the x- ray atomic Scattering form Factor and Nuclear Magnetic Shielding Constant for Be atom in its Excited state (1s2 2s 3s) and the Be - like ions

The division partitioning technique has been used to analyze the four electron systems into six-pairs electronic wave functions for ( for the Beryllium atom in its excited state (1s2 2s 3s ) and like ions ( B+1 ,C+2 ) using Hartree-Fock wave functions . The aim of this work is to study atomic scattering form factor f(s) for and nuclear magnetic shielding constant. The results are obtained numerically by using the computer software (Mathcad).

Fast and accurate conversion of atomic models into electron density maps

New image processing methodologies and algorithms have greatly contributed to the signi cant progress in three-dimensional electron microscopy (3DEM) of biological complexes we have seen over the last decades. Naturally, the availability of accurate procedures for the objective testing of new algorithms is a crucial requirement for the further advancement of the eld. A good and accepted testing work ow involves the generation of realistic 3DEM-like maps of biological macromolecules from which some measure of ground truth can be derived, ideally because their 3D atomic structure is already known. In this work we propose a very accurate generation of maps using atomic form factors for electron scattering. We thoroughly review current approaches in the eld, quantitatively demonstrating the bene ts of the new methodology. Additionally, we study a concrete example of the use of this approach for hypothesis testing in 3D Electron Microscopy.