X-ray Form Factors Research Articles

Resolution of discrepancies of atomic form factors and attenuation coefficients in the near-edge soft X-ray regime

Abstract X-rays interactions with atoms should be well understood quantitatively. Reliable knowledge of the complex X-ray form factor (Re( f ) and f ″) and the photoelectric attenuation coefficient σ PE 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. This paper addresses key discrepancies and derives new theoretical results in near-edge soft X-ray regions of recent interest.

Precision X-ray optics for fundamental interactions in atomic physics, resolving discrepancies in the X-ray regime

Reliable knowledge of the complex X-ray form factor (Re(f) and f″) is required for many fields including crystallography, medical diagnosis and XAFS studies. However, there are discrepancies between theory and theory, experiment and experiment and theory and experiment of 10% and more, over central X-ray energies. Discrepancies exist for most elements, despite claimed experimental accuracies of 1%. This paper summarises the current variation between experimental and theoretical results, and outlines key issues for obtaining experimental accuracies of 1% in critical wavelength ranges for selected elements to address these issues. This paper critically surveys available experimental data for attenuation coefficients and suggests a procedure for obtaining significantly higher accuracy measurements in the future.

Ternary Model of an Al-Cu-Co Decagonal Quasicrystal

We propose a structural model for decagonal Al-Cu-Co quasicrystals based upon existing experimental diffraction data supplemented by total energy calculations. The model is a decoration of tiles related to the Penrose rhombus tiling. Strong chemical ordering between Cu and Co leads to chains of alternating Cu and Co atoms. These chains project onto tile edges, where they define arrows. At low temperatures, the interatomic potentials lead to rules fixing the relative orientation of a majority of the arrows. These rules are a subset of the Penrose rhombus matching rules. [S0031-9007(98)06685-X] PACS numbers: 61.44.Br, 61.50.Lt, 61.43.Bn, 61.66.Dk More than a decade after the discovery of quasicrystals [1] answers are emerging to the fundamental problem of structure determination [2]. X-ray and neutron diffraction experiments successfully determine the location of the majority of atoms in quasicrystals [3]. Still, important questions remain. Quasicrystal structure refinements based on a finite number of diffraction peaks exhibit split atoms and atoms whose chemical identity cannot be determined [4,5]. Species determination is especially difficult for compounds such as Al-Cu-Co that contain elements near each other in a row of the periodic table, since their x-ray form factors are nearly equal. Furthermore, chemical disorder and partial occupancy are not well described. For example, a partially occupied site of high atomic number may mimic a fully occupied site of low atomic number. The atomicity condition is insufficient to resolve these issues; additional methods are required to investigate quasicrystal structure in greater detail. Techniques have been developed to supplement experimental diffraction data with total energy considerations to resolve the above difficulties [6 ‐ 8]. Here, we apply these techniques to create a model of the decagonal quasicrystal phase of Al-Cu-Co [9]. This phase exhibits quasiperiodic atomic layers stacked with a net 4 A periodicity in the perpendicular direction. We focus on this compound because its stable quasicrystalline phase has been well studied by diffraction, but experimental information suffers difficulties of the type described above. Furthermore, the existence of electronic-structure-based pair potentials for Al-Co alloys [10] allows total energy calculations to discriminate among candidate structures. We deduce the ternary alloy structure using only Al-Co binary potentials. After we review the current experimental knowledge of Al-Cu-Co decagonal phases, we discuss our computational method and then our results. Our results include an ideal hyperspace model and a corresponding real space structure. We describe the real space structure from two perspectives: linked clusters and space-filling tilings. We show that chemical ordering between Cu and Co enforces a subset of the Penrose matching rules. The temperature- and composition-dependent phase diagram of Al-Cu-Co alloys is well studied experimentally [11]. While the full composition range of thermodynamic stability of the decagonal phase shifts and widens as temperature increases, the principal domain of phase stability surrounds a straight line in the composition plane Al73.520.5xCuxCo26.520.5x ,

Energy Filtered Imaging and Convergent Beam Electron Diffraction (CBED) in the Transmission Electron Microscope (TEM)

Over the last ten years TEMs have been developed that are capable of HREM, EDX, PEELS and diffraction using a single objective pole piece. More recently these TEMs have been equipped with the capability of energy filtering the electron beam after it has passed through the sample so that energy filtered images and electron diffraction patterns can be obtained. In this work the use of a Topcon 002B TEM equipped with a GATAN PEELS imaging filter (GIF) to generate zero-loss energy filtered zone axis CBED patterns and elemental images from inelastically scattered electrons will be described. An analysis of this energy filtered data indicates that elemental imaging using the GIF is an informative, but semiquantitative technique, whereas zero-loss energy filtered zone axis CBED patterns can be accurately quantified so that the two lowest-angle x-ray form factors of cubic elements can be measured with errors of the order of 0.1% or less.

Crystalline Ropes of Metallic Carbon Nanotubes

Fullerene single-wall nanotubes (SWNTs) were produced in yields of more than 70 percent by condensation of a laser-vaporized carbon-nickel-cobalt mixture at 1200degreesC. X-ray diffraction and electron microscopy showed that these SWNTs are nearly uniform in diameter and that they self-organize into "ropes," which consist of 100 to 500 SWNTs in a two-dimensional triangular lattice with a lattice constant of 17 angstroms. The x-ray form factor is consistent with that of uniformly charged cylinders 13.8 +/- 0.2 angstroms in diameter. The ropes were metallic, with a single-rope resistivity of <10(-4) ohm-centimeters at 300 kelvin. The uniformity of SWNT diameter is attributed to the efficient annealing of an initial fullerene tubelet kept open by a few metal atoms; the optimum diameter is determined by competition between the strain energy of curvature of the graphene sheet and the dangling-bond energy of the open edge, where growth occurs. These factors strongly favor the metallic (10,10) tube with C5v symmetry and an open edge stabilized by triple bonds.

Structure of the ripple phase in lecithin bilayers.

The phases of the x-ray form factors are derived for the ripple (Pbeta') thermodynamic phase in the lecithin bilayer system. By combining these phases with experimental intensity data, the electron density map of the ripple phase of dimyristoyl-phosphatidylcholine is constructed. The phases are derived by fitting the intensity data to two-dimensional electron density models, which are created by convolving an asymmetric triangular ripple profile with a transbilayer electron density profile. The robustness of the model method is indicated by the result that many different models of the transbilayer profile yield essentially the same phases, except for the weaker, purely ripple (0,k) peaks. Even with this residual ambiguity, the ripple profile is well determined, resulting in 19 angstroms for the ripple amplitude and 10 degrees and 26 degrees for the slopes of the major and the minor sides, respectively. Estimates for the bilayer head-head spacings show that the major side of the ripple is consistent with gel-like structure, and the minor side appears to be thinner with lower electron density.

Electron Density Distributions and Atomic Charges

Abstract Accurate electron densities and X-ray form factors of Li, Be, F and their ions have been calculated. Electron correlation, crystal fields and ionic charge transfer change the form factors by up to a few percent, mainly in the range of sin θ/ λ &lt; 1/3 Â -1 . Although electron correlation and crystal fields are small perturbations, their effects on the density and form factor are not additive. Densities or form factors of atomic and ionic systems are very similar; [Li0F0] and [Li+F-] procrystals differ by an effective charge transfer of not more than 0.4 e. Charge transfer and charge overlap in crystals cannot be distinguished uniquely. When the experimental data on Li2BeF4 (approximately reproduced by 3/4 atomic plus 1/4 ionic procrystal) are interpreted from the atomic partial charges are as low as 0.1 e (Li+ 0.1 2Be+ 0.2F- 0.1 4); when interpreted from the ionic viewpoint, the charges are much higher, namely 0.7 e. Intermediate viewpoints are also possible.

X-ray form factors and Compton profiles for some solids derived using an atom-in-jellium-vacancy model

X-ray form factors f(k), Compton profiles J(q) and momentum expectation values (pn) are calculated for Mg,Al,Si,Ge and Ag using the local-density approximation with the spherical atom-in-jellium-vacancy model. These theoretical results show satisfactory agreement with experiment, thus indicating the success of the present method for these solids in the position (r) as well as the momentum (p) space.

Identification of a growth defect in solid C60 by electron diffraction

The origin of an anomalous sawtooth-shaped feature in x-ray powder diffraction of solid C60 is explained via electron diffraction analysis. Films sublimed on holey carbon crystallize with close-packed (111) planes parallel to the surface. Rods of diffuse scattering are found along the 〈111〉 axis normal to the surface but not along other 〈111〉 axes. Powder averaging of these rods, coupled with the x-ray form factor of spherical shells with 3.5 Å radius, accounts for the sawtooth feature. We attribute this phenomenon to planar defects parallel to close-packed layers, which form during the growth of solid C60 by sublimation. A possibly related consequence of the growth mode is the observation of strong macroscopic (111) preferred orientation in films sublimed on a variety of substrates.

Computer simulation of aromatic polyesters including molecular dynamics

A theoretical analysis of the possible conformations of poly( p-phenylene terephthalate) (PPTA) and poly( p-phenylene isophthalate) (PPIA) is performed on the basis of molecular mechanics and molecular dynamics trajectories. An algorithm for creating long polymer chains up to 150 monomers used for the calculation of the small-angle X-ray form factor is introduced. Three different methods to determine the experimentally accessible property, persistence length, are presented. The results of all three methods agree very well with the experimental result for the persistence length of PPTA; in addition, the relationship between stiffness and persistence length is elucidated by comparison of the two different conformations of PPTA and PPIA.

Variable Step-Counting Times for Rietveld Analysis, or, Getting the Most Out of Your Experiment Time

AbstractThe magnitudes of the peak intensities in a powder diffraction pattern are not uniformly distributed as a function of diffraction angle. The Lorentz-polarization factor, X-ray form-factor and the thermal vibration parameters all conspire to progressively decrease the intensities of the peaks as sinθ/λ increases. In spite of this, diffraction data for Rietveld analysis is universally collected using the same counting time for each step in the pattern. As a result, peaks at high angles are collected with lower counting precision than those at low angles, despite the fact that the high-angle region has a higher density of peaks and therefore contains more information than the low-angle part of the pattern. Indeed, the intensity, position and profile of reflections at medium and high sinθ/λ are often more easily modelled since they are subject to less interference from systematic effects such as extinction, specimen transparency and α1/α2 overlap errors. In the present work a novel variable step-counting-time regime has been devised that increases the counting times with diffraction angle in a manner inversely proportional to the intensity fall-off calculated from the Lorentz-polarization, form-factor and thermal parameters. The effect of this new data collection regime on the results of Rietveld refinement of a number of materials of varying structural complexity is discussed.

Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles

An enlarged set (atomic number Z = 2 to 98) of free-atom X-ray atomic scattering (form) factors for high angles [2 ≤ (sin θ)/λ ≤ 6Å-1] has been calculated based on those of Doyle & Turner [Acta Cryst. (1968). A24, 390-397]. Four-parameter 'exponential polynomial' fits for these are presented which give far more accurate estimates of the scattering factors at high angles than the Gaussian fits normally used. The use of the Mott formula in conjunction with these new high-angle X-ray form factors allows the calculation of improved-accuracy high-angle electron scattering factors. The use of these high-accuracy high-angle scattering factors for important applications such as Fourier charge-density analysis and computer simulation of high-resolution electron microscope (HREM) images is discussed.

Band structure and related properties of bcc niobium.

We report self-consistent, all-electron calculations of the band structure of paramagnetic bcc niobium. Density-functional theory and the linear combination of Gaussian orbitals method were employed. Results are presented and discussed in the light of the existing theoretical and experimental data for the energy bands, density of states, Fermi surface, optical conductivity, x-ray form factors, and Compton profile. In addition, spin-polarized band-structure calculations were carried out to investigate a possible magnetically ordered state for large values of the lattice constant. A composite transition from a nonmagnetic to a low-spin ferromagnetic state and from a low to a high-spin state was found.

Bond functions in calculating X-ray structure factors for diamond-structure crystals

Much attention is now given to electron--distribution models that are fairly simple from the computational viewpoint but capable of describing the main solid-state bond features. We construct electron distributions for crystals having the diamond structure by means of localized bond functions, whose performance is evaluated by comparing the Fourier transforms with the observed structure factors for x-ray Bragg reflections for diamond and silicon. That approach to distribution description is of interest because wave-function calibration of x-ray form factors gives excellent agreement with variational calculations for the atomic density and atomic orbitals.

A full-potential LAPW study of structural and electronic properties of beryllium

The authors present self-consistent band-structure calculations of HCP Be made by means of the full-potential linearised-augmented-plane-wave (LAPW) method and on the basis of the local density approximation. In contrast to earlier calculations no shape approximations for the charge density or the potential are made and all electrons are taken into account. From the full charge density, X-ray form factors are calculated that agree very well with recent experimental data. This indicates that the significant deviations between experiment and earlier calculations are due to strong muffin-tin- effects. The electric field gradient (EFG) is very sensitive to the quality of the theoretical calculation. In earlier theoretical work not even the sign of the EFG could be determined unambiguously. Within the authors formalism the EFG is determined directly from a given charge density without any approximations. Starting with a muffin-tin potential the EFG is found to be 15 times larger than experiment. The present full-potential LAPW calculation, however, yields an EFG in good agreement with experiment. From calculations at different lattice constants it is found that the EFG depends strongly on the c/a ratio but only weakly on the volume. The equilibrium lattice parameters are determined by total-energy minimisation and agree with experiment within 2%.

Calculated against experimental charge density distribution of Be metal

The authors have performed self-consistent linearised augmented plane wave calculations for the charge density of metallic beryllium, with and without warping of the interstitial potential, using the local density approximation for the exchange and correlation energy functional. Overall agreement with the newest X-ray form factors is good, and it is argued that the remaining discrepancies have their source in the adequacy of the local density approximation and, for some reflections, in neglecting the non-sphericity of the core charge density.

Electronic structure and Fermi surface of calcium.

The electronic structure of calcium is calculated by means of the self-consistent linear combination of Gaussian orbitals method and the local-density approximation (LDA). No shape approximations to the charge density or the potential are made. We obtain a band structure and a Fermi surface which is in reasonable agreement with experiment. For that reason we disagree with the conclusions drawn in a recent paper by Jan and Skriver, who employed the linear muffin-tin orbital method and stated that the LDA cannot give a reasonable Fermi surface. Our empty d-band width is somewhat narrower in comparison with other calculations or experiment, as one can also see from our calculated optical conductivity. X-ray form factors, which deviate only slightly from the free-atom values, and Compton profiles, are also given.

The effect of configuration interaction on atomic scattering factors for transition metals

Spin-only magnetic form factors and X-ray form factors for Mn, Mn+, Co, Co2+, Ni and Ni2+ are calculated using configuration interaction wavefunctions. The magnetic form factors for Co2+ are used in the analysis of polarized neutron diffraction data for Cs3CoCl5, to give spin populations. The inclusion of correlation functions in the wavefunction leads to changes in Hartree-Fock form factors of transition metal atoms, which maximize at K/4π ~ 0.4 Å-1, where they amount to approximately 5% of the form factor at that value of K.

X-ray form factors and the electronic structure of graphite

The bonding and spectroscopic properties of graphite are investigated by carrying out first-principles, self-consistent electronic structure calculations, and by comparing the results with high-resolution data from recent x-ray diffraction and angle-resolved photoemission measurements. The theoretical valence-charge density is in excellent agreement with values derived from experimental x-ray from factors. Unlike other group-IV covalent materials, the bonding charge exhibits a prominent double-humped structure due to the lack of $p$ core states. The energy band structure is also in good agreement with experimental measurements and previous calculations.

Internally folded densities

The 3 q-dimensional Fourier transforms of the q-electron charge density and momentum density respectively are defined to be the q-electron form factor and internally folded density. For q = 1, these quantities are just the usual X-ray form factor and the B( r) function recently introduced for the analysis of Compton profiles. Numerous properties of the one-electron internally folded density B( r) are derived. B ( r) functions for the atoms from He through Ne are calculated and examined for chemical content with the help of the concept of reduced Compton profiles.