X-ray Form Factors Research Articles

Analytical models representing X-ray form factors of ions.

Parameters in analytical models for X-ray form factors of ions f0(s), based on the inverse Mott-Bethe formula involving a variable number of Gaussians, are determined for a wide range of published data sets {s, f0(s)}. The models reproduce the calculated form-factor values close to what is expected from a uniform statistical distribution with limits determined by their precision. For different ions associated with the same atom, the number of Gaussians in the models decreases with increasing net positive charge.

Variational Calculation of Lithium-Like Ions from B+2 to N+4 Using β-Type Roothaan–Hartree–Fock Wavefunction

Within the KaKB, KaLa, and KBLa shells in the position space, the properties of a series of three-electron systems, for instance, B+2, C+3, and N+4 ions, have been studied. This required the partitioning of the two-particle space-spin density and was explicit for the Hartree–Fock description which have been proposed by considering a basis set based on single-zeta B-type orbitals (BTOs). The one- and two-body radial electronic densities R(r1), R(r1, r2), moments ⟨rn1⟩, X-ray form factor F(s), nucleus density R(0), nuclear magnetic shielding constant qd, and the diamagnetic susceptibility бs in the position space are reported. Our results are realized via the Mathematica program and compared with previous theoretical values in the literature.

Exact and fast calculation of the X-ray pair distribution function.

A fast and exact algorithm to calculate the powder pair distribution function (PDF) for the case of periodic structures is presented. The new algorithm calculates the PDF by a detour via reciprocal space. The calculated normalized total powder diffraction pattern is transferred into the PDF via the sine Fourier transform. The calculation of the PDF via the powder pattern avoids the conventional simplification of X-ray and electron atomic form factors. It is thus exact for these types of radiation, as is the conventional calculation for the case of neutron diffraction. The new algorithm further improves the calculation speed. Additional advantages are the improved detection of errors in the primary data, the handling of preferred orientation, the ease of treatment of magnetic scattering and a large improvement to accommodate more complex instrumental resolution functions.

Simulations of lipid bilayers using the CHARMM36 force field with the TIP3P-FB and TIP4P-FB water models.

The CHARMM36 force field for lipids is widely used in simulations of lipid bilayers. The CHARMM family of force fields were developed for use with the mTIP3P water model. This water model has an anomalously high dielectric constant and low viscosity, which limits its accuracy in the calculation of quantities like permeability coefficients. The TIP3P-FB and TIP4P-FB water models are more accurate in terms of the dielectric constant and transport properties, which could allow more accurate simulations of systems containing water and lipids. To test whether the CHARMM36 lipid force field is compatible with the TIP3P-FB and TIP4P-FB water models, we have performed simulations of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine bilayers. The calculated headgroup area, compressibility, order parameters, and X-ray form factors are in good agreement with the experimental values, indicating that these improved water models can be used with the CHARMM36 lipid force field without modification when calculating membrane physical properties. The water permeability predicted by these models is significantly different; the mTIP3P-model diffusion in solution and at the lipid–water interface is anomalously fast due to the spuriously low viscosity of mTIP3P-model water, but the potential of mean force of permeation is higher for the TIP3P-FB and TIP4P-FB models due to their high excess chemical potentials. As a result, the rates of water permeation calculated the FB water models are slower than the experimental value by a factor of 15–17, while simulations with the mTIP3P model only underestimate the water permeability by a factor of 3.

Probing the Ripple Phase of Lipid Bilayers using Molecular Simulations

Lipid bilayers without sterols can exist in the gel, ripple, and liquid-crystalline phase depending on temperature. The ripple phase is the mysterious transition phase between gel and liquid-crystalline phases. For this work, a set of molecular dynamics (MD) simulations were carried out for pure 1,2-dimyristoyl-3-sn-phosphatidylcholines (DMPC), pure 1,2-dipalmitoyl-3-sn-phosphatidylcholine (DPPC), and their mixture at different temperatures. For some mixture and pure DMPC conditions, simulations were also performed in larger system size (144 lipids for each leaflet), which was four times bigger than small system (36 lipids for each leaflet), to determine its influence on ripple phase structure. Simulations were performed for 300ns, and the CHARMM 36 force field was used (J. Phys. Chem. B., 114, 7830-7843). For pure DMPC and DPCC (small system), ripple phase was obtained at 275 and 293K, respectively. For small system size of the DMPC-DPPC mixture, which had %75 DMPC, the ripple phase was obtained at slightly higher temperature at 298K. Ripple conformations were fairly stable at the last 100 ns of simulations. Interestingly, for large system of pure DMPC simulation, ripple phase was obtained at 291K, which is the temperature that ripple phase was observed based on X-ray scattering technique (Soft Matter, 11, 918-926). In case of DMPC-DPPC mixture with large system size, ripple phase was still obtained in a fairly stable conformation. Moreover, in large system set up for both pure DMPC and DMPC-DPPC mixture, the ripple length was increased, which is more in agreement with experiment. Form factor from simulations will be compared with 2D X-ray form factors from experiment (Soft Matter, 11, 918-926). In overall, CHARMM 36 force field can predict the ripple phase within reasonable range of temperatures and provide interpretation of experimental form factors.

Form factor dispersion at La M5,4 edges and average density of resonant atoms

Resonant soft x-ray scattering on complex oxide superlattices shows very large variations in the superlattice reflection position and intensity near La M5,4 edges. Resonant dispersion of the La x-ray form factor describes the observations well. We determine the average density of resonant La atoms and the thickness of superlattice layers.

Biomolecular Simulations with the Transferable Potentials for Phase Equilibria: Extension to Phospholipids

The Transferable Potentials for Phase Equilibria (TraPPE) is extended to zwitterionic and charged lipids including phosphatidylcholine (PC), phosphatidylethanolamine (PE), phosphatidylserine (PS), and phosphatidylglycerol (PG). The performance of the force field is validated through isothermal-isobaric ensemble (NPT) molecular dynamics simulations of hydrated lipid bilayers performed with the aforementioned head groups combined with saturated and unsaturated alkyl tails containing 12-18 carbon atoms. The effects of water model and sodium ion parameters on the performance of the lipid force field are determined. The predictions of the TraPPE force field for the area per lipid, bilayer thickness, and volume per lipid are within 1-5% of experimental values. Key structural properties of the bilayer, such as order parameter splitting in the sn-2 chain and X-ray form factors, are found to be in close agreement with experimental data.

Anions, the Reporters of Structure in Ionic Liquids.

In this work we compare the role that different anions play in the structure function S(q) for a set of liquids with the same cation. It is well established that because of their amphiphilic nature and their often larger size, cations play a fundamental role in the structural landscape of ionic liquids. On the other hand, it is often atoms in the anions that display the largest X-ray form factors and therefore play a very significant role as reporters of structure in small- and wide-angle X-ray scattering (SAXS/WAXS)-type experiments. For a set of liquids with similar topological landscape, how does S(q) change when the anionic scattering is deemphasized? Also, how do we computationally recover the typical length scale of important and perhaps universal ionic liquid structural features such as charge alternation when these are experimentally inaccessible from S(q) because of interference cancellations? We answer these questions by studying three different tetrapentylammonium-based liquids with the I(-), PF6(-) and N(CN)2(-) anions.

Improving the CHARMM Force Field for Polyunsaturated Fatty Acid Chains

CHARMM36 (C36) is the most up-to-date pairwise additive all-atom lipid force field and is able to accurately represent bilayer properties of saturated and monounsaturated lipid molecules in the natural constant particle, pressure, and temperature (NPT) ensemble. However, molecular dynamics (MD) simulations on 1-stearoyl-2-docosahexaenoyl-sn-glycerco-3-phosphocholine (SDPC) bilayers of the polyunsaturated fatty acid (PUFA) chains result in inaccuracies of the surface area per lipid (SA), deuterium order parameters (S(CD)), and X-ray form factors. Therefore, in this study, high-level quantum mechanical calculations are used to improve the dihedral potential of neighboring double bonds, and the corresponding force field is referred to as C36p. The SA for SDPC at 303 K increases from 63.2 ± 0.2 (C36) to 70.8 ± 0.2 (C36p) Å(2) and agrees favorably with X-ray diffraction results at 297 K. The resulting S(CD) are in excellent agreement with experimental values of both the sn-1 and sn-2 chains. Calculated NMR (13)C relaxation times and X-ray form factors from MD simulations of SDPC bilayers also agree with experiments. MD simulations of 1,2-diarachidonyl-phosphatidylcholine (DAPC) bilayers are used to further validate our force field parameters on a lipid with both chains containing PUFAs. As expected, the thickness of DAPC bilayers is reduced, and the SA is increased compared to the SDPC bilayers. This update in the PUFA force field should allow for accurate MD simulations of PUFA-containing bilayers in the NPT ensemble.

X-ray mass attenuation coefficients and imaginary components of the atomic form factor of zinc over the energy range of 7.2–15.2 keV

The x-ray mass attenuation coefficients of zinc are measured in a high-accuracy experiment between 7.2 and 15.2 keV with an absolute accuracy of 0.044% and 0.197%. This is the most accurate determination of any attenuation coefficient on a bending-magnet beamline and reduces the absolute uncertainty by a factor of 3 compared to earlier work by advances in integrated column density determination and the full-foil mapping technique described herein. We define a relative accuracy of $0.006%$, which is not the same as either the precision or the absolute accuracy. Relative accuracy is the appropriate parameter for standard implementation of analysis of near-edge spectra. Values of the imaginary components $f\ensuremath{''}$ of the x-ray form factor of zinc are derived. Observed differences between the measured mass attenuation coefficients and various theoretical calculations reach a maximum of about 5% at the absorption edge and up to 2% further than 1 keV away from the edge. The measurements invite improvements in the theoretical calculations of mass attenuation coefficients of zinc.

Exact and asymptotic X-ray intensity values from 20-parameter hylleraas-type wave functions. He and He-like ions

X-ray form factors and total scattered X-ray intensity values, computed analytically from radial electron-nuclear and electron-electron distribution functions, D0(r)'s and P0(r12)'s, are presented for the 1S ground states of H−, He, Li+, B3+, O6+, Ne8+, and Mg10+. The distribution functions are those obtained algebraically from the 20-parameter Hylleraas-type wave functions of Hart and Herzberg. Asymptotic total intensity values, which have been shown to depend on the derivatives of even order of g(r12) = P0(r12)/r12, are in excellent agreement with those computed directly from P0(r12). The asymptotic values are listed only for He; numerical values of the derivatives g(2)(0), g(4)(0), g(6)(0) and g(8)(0) are given for He and the 2-electron ions. None of the wave functions studied here satisfy the Kato electron-nucleus cusp condition.

Size effects of semiempirical large unit cell method in comparison with nanoclusters properties of diamond-structured covalent semiconductors

Self-consistent Hartree–Fock method within the framework of large unit cell (LUC) formalism using complete neglect of differential overlap (CNDO) is used to simulate size effects on nanoclusters of covalent crystalline diamond-structured semiconductor C, Si, Ge, and Sn using k=0 approximation. Three sizes are investigated namely 8, 64, and 216 atom LUCs. Cohesive energy, energy gap, valence band width, and hybridization orbitals are obtained from electronic structure calculations. Charge distribution, density of states, and orbital wave functions are also reported. Sensitivity analysis of the dependence of some of the calculated properties on model parameters is performed. Results revealed that electronic properties converge to some limit as the size of the LUC increases and that the 216 atoms LUC is very near to the bulk of these materials. Increasing LUC size or atomic number of covalent semiconductor removed some electronic cloud from bonding region to the spherical region around the atom. The same is true in going to higher energy levels. Increasing the size of LUC also resulted in a decrease of energy gap, increasing valance band width, and generally slightly increasing the cohesive energy. The model predicts energy gap reduction in going from ultra-small nanoclusters to the bulk of the four elements to be around 2 eV. The approximations imposed on the present model calculations leading to resemblance with nanoclusters properties in addition to the evidences of such resemblance such as orbital shapes and trends of data are discussed. Small nanoclusters are expected to have stronger directional bonds than in their bulk structure. The smallest cells (8 atoms) are found to be of a slightly longer lattice constant of the order 0.01, 0.1, 0.04, and 0.02 a.u. for diamond, silicon, germanium, and tin, respectively. X-ray form factors show a slight decrease of low-angle lines when we go to larger cells that was also found experimentally. To the best of my knowledge, this is the first systematic determination of covalent nanoclusters lattice constant and X-ray form factors variation with ultra-small nanocluster size.

An Improved United Atom Force Field for Simulation of Mixed Lipid Bilayers

We introduce a new force field (43A1-S3) for simulation of membranes by the Gromacs simulation package. Construction of the force fields is by standard methods of electronic structure computations for bond parameters and charge distribution and specific volumes and heats of vaporization for small-molecule components of the larger lipid molecules for van der Waals parameters. Some parameters from the earlier 43A1 force field are found to be correct in the context of these calculations, while others are modified. The validity of the force fields is demonstrated by correct replication of X-ray form factors and NMR order parameters over a wide range of membrane compositions in semi-isotropic NTP 1 atm simulations. 43-A1-S3 compares favorably with other force fields used in conjunction with the Gromacs simulation package with respect to the breadth of phenomena that it accurately reproduces.

Improved Coarse Grained Force-Field Parameters for Biomembranes

Coarse-grained (CG) simulation method provides an important step in understanding cell fusion process. Based on the CG Martini force-field model (S. J. Marrink et al., J. Phys. Chem. B 2007, 111, 7812-7824), we are refining and reparameterizing the force-field parameters for the smaller molecules that are components of phospholipid membranes. For united-atom model, we have recently shown this from-the-ground-up derived force fields for the hydrocarbons and the smaller organic compounds are completely transferable to simulation of phospholipid bilayers, as evidenced by the recreation of x-ray form factors with a high degree of fidelity (S.-W. Chiu et al., manuscript submitted to J. Phys. Chem B.)

Generalized x-ray scattering cross section from nonequilibrium plasmas

We propose a modified x-ray form factor that describes the scattering cross section in warm dense matter valid for both the plasma and the solid (crystalline) state. Our model accounts for the effect of lattice correlations on the electron-electron dynamic structure, as well as provides a smooth transition between the solid and the plasma scattering cross sections. In addition, we generalize the expression of the dynamic structure in the case of a two-temperature system (with different electron and ion temperatures). This work provides a unified description of the x-ray scattering processes in warm and dense matter, as the one encountered in inertial confinement fusion, laboratory astrophysics, material science, and high-energy density physics and it can be used to verify temperature relaxation mechanisms in such environments.

Ab initiocalculations of the electronic structure and bonding characteristics ofLaB6

Lanthanum hexaboride (${\mathrm{LaB}}_{6}$, NIST SRM--660a) is widely used as a standard reference material for calibrating the line position and line shape parameters of powder diffraction instruments. The accuracy of this calibration technique is highly dependent on how completely the reference material is characterized. Critical to x-ray diffraction, this understanding must include the valence of the La atomic position, which in turn will influence the x-ray form factor $(f)$ and hence the diffracted intensities. The electronic structure and bonding properties of ${\mathrm{LaB}}_{6}$ have been investigated using ab initio plane-wave pseudopotential total energy calculations. The electronic properties and atomic bonding characteristics were analyzed by estimating the energy band structure and the density of states around the Fermi energy level. The calculated energy band structure is consistent with previously reported experimental findings; de Haas--van Alphen and two-dimensional angular correlation of electron-positron annihilation radiation. In addition, the bond strengths and types of atomic bonds in the ${\mathrm{LaB}}_{6}$ compound were estimated by analyzing the Mulliken charge density population. The calculated result revealed the coexistence of covalent, ionic, and metallic bonding in the ${\mathrm{LaB}}_{6}$ system and partially explains its high efficiency as a thermionic emitter.

Measurement of carbon ionization balance in high-temperature plasma mixtures by temporally resolved X-ray scattering

We have measured carbon ionization balance in a multi-component plasma in the high-temperature, up to fully ionized, regime by spectrally resolved X-ray scattering. In particular, the measurements have been performed in an underdense ( n e ≈ 10 21 cm - 3 ) 0.35- μ m laser-produced plasma, containing a mixture of C, H with Al and Ar impurities, by using time-resolved back-scattered spectra from a 9.0 keV Zn He- α X-ray probe detected with a high-efficiency graphite Bragg crystal coupled to a framing camera. Measured values for the plasma temperature and carbon ionization state as well as impurity concentrations were obtained by fitting the Doppler-broadened and Compton-shifted scattered spectra at various times after the plasma heating with a modified X-ray form factor that includes the full effects of cross-correlation between different species. These data test collisional-radiative and radiation hydrodynamics modeling from cold ( T e ≲ 5 eV) to fully ionized carbon ( T e ∼ 280 eV).

Scattering factor calculations and dispersion corrections for heavy atoms

Reliable knowledge of the complex X-ray form factor (Re( f) and Im( f)) is required for crystallography, deformation density bonding studies and refractive index studies. Dispersion corrections are important in DAFS, MAD analyses and XAFS studies. Discrepancies between currently used theoretical approaches of 200% exist for numerous elements from 1 to 3 keV X-ray energies, and at higher energies these discrepancies can persist at the 10% level or more. This has suggested that new formulations and new experimental approaches are necessary. Recent tabulations improve upon the theoretical uncertainty in some of these regions by a factor of 10 and reduce the error of this approach to below one standard deviation. Recent experimental developments have begun to probe alternate theory, providing a tool to investigate the significance of relativistic corrections and convergence criteria across a range of atomic number. In part, this has revealed that more theoretical work will be required in future years. Recent advances permit new probes and insights into atomic and solid state physics, as well as to many X-ray optical applications. This paper discusses significant improvements upon the theoretical uncertainty in near-edge regions, reducing the error of this approach to less than one standard deviation.

New theoretical investigation resolving discrepancies of atomic form factors and attenuation coefficients in the near-edge soft X-ray regime

Reliable knowledge of the complex X-ray form factor and the photoelectric attenuation coefficient is required for crystallography, medical diagnosis, radiation safety and XAFS studies. Discrepancies between currently used theoretical approaches of 200% exist for numerous elements for X-ray energies from 1 to 3 keV. This work addresses key discrepancies and derives theoretical results in near-edge soft X-ray regions. DHF wave functions are employed and computational and convergence issues are of direct concern. Comparisons with simpler wave functions, including additional relativistic corrections to the form factors, are insightful. The current result improves upon the theoretical uncertainty in these regions to an estimated standard deviation of 20–30%.

Detailed Tabulation of Atomic Form Factors, Photoelectric Absorption and Scattering Cross Section, and Mass Attenuation Coefficients in the Vicinity of Absorption Edges in the Soft X-Ray (Z=30–36, Z=60–89, E=0.1 keV–10 keV), Addressing Convergence Issues of Earlier Work

Reliable knowledge of the complex x-ray form factor [Re(f ) and f″] and the photoelectric attenuation coefficient (σPE) is required for crystallography, medical diagnosis, radiation safety, and XAFS studies. Discrepancies between currently used theoretical approaches of 200% exist for numerous elements from 1 to 3 keV x-ray energies. The key discrepancies are due to the smoothing of edge structure, the use of nonrelativistic wave functions, and the lack of appropriate convergence of wave functions. This paper addresses these key discrepancies and derives new theoretical results of substantially higher accuracy in near-edge soft x-ray regions. The high-energy limitations of the current approach are also illustrated. The energy range covered is 0.1 to 10 keV. The associated figures and tabulation demonstrate the current comparison with alternate theory and with available experimental data. In general, experimental data are not sufficiently accurate to establish the errors and inadequacies of theory at this level. However, the best experimental data and the observed experimental structure as a function of energy are strong indicators of the validity of the current approach. New developments in experimental measurement hold great promise in making critical comparisons with theory in the near future.