X-ray Structure Factors Research Articles

Pressure dependence of the electronic structure in germanium

Based on the pseudopotential formalism of Ge, which can describe the electronic energy structure over a wide energy range of more than 20 eV from the bottom of the valence band, calculations of the pressure dependence of the energy gaps and charge densities have been performed. The predictions for the pressure coefficient are consistent with the experiment. The valence charge densities of Ge under high pressure are studied. The forbidden X-ray structure factor F(222) is stable under pressure and varies less than 3% under volume changes as big as 5%.

Topological definition of crystal structure: determination of the bonded interactions in solid molecular chlorine

The heavier halide molecules form layered crystals indicative of the presence of a specific directed intermolecular interaction. It is shown that this interaction within the crystal can be defined and characterized using the topology of the electron density within the theory of atoms in crystals. It is also shown that its presence in the crystal and the resulting geometry of the layered structure can be predicted in terms of the topology of the Laplacian distribution of an isolated Cl2 molecule, as it relates to the definition of Lewis acid and base sites within the valence shell of an atom. The generality of the definition of both primary and secondary interactions in terms of the topology of the electron density is demonstrated for all types of crystal. The electron density of solid molecular chlorine was determined by fitting the experimental X-ray structure factors and by theoretical calculation and its topology determined. Each Cl atom is found to be linked by bond paths, lines of maximum electron density, to twelve other atoms in the crystal: to four atoms in the same layer parallel to the bc plane, one of which defines the intramolecular bond of the Cl2 group, to six atoms in the four neighbouring molecules lying in the same stack parallel to the b axis and to two atoms in molecules situated in a neighbouring stack.

特集―分子性結晶のダイナミックス Ｘ線精密結晶構造解析と化学反応

The role of the researches on the X-ray electron densities in the studies of chemical reactions in the crystalline state is discussed. And the application of the X-ray AO analysis to PtP2 and NiTiO3 crystals were described in conjunction with the future possible application of the method to the studies of chemical reactions in the crystalline state. In both measurements the effect of multiple diffraction on structure factors was significant. Structure factors were measured avoiding the effect with the Ψ-scan technique in PtP2, while in the case of NiTiO3, the structure factors measured in the past were examined and those apparently affected by the effect were discarded. 5d- and 3d-orbital functions of Pt and Ni were deter-mined by the X-ray AO analysis. However the residual density maps implies that better model which explains the electron density in the whole unit cell is necessary. Hence the basic problems inherent in the X-ray MO analysis, which analyze the molecular orbitals from the measured X-ray structure factors, is also discussed.

Discrete Harmonic Model for Stacked Membranes: Theory and Experiment

― A discrete harmonic (DH) model has been developed which describes the static structure factor of stacked membranes. The (DH) model was used to analyze a synchrotron small-angle X-ray scattering study in stacked membranes. We studied lyotropic lamellar L α phase samples in a quaternary mixture consisting of thin water layers coated with surfactant sodium dodecyl sulfate (SDS) and cosurfactant (pentanol) molecules, separated by oil. The experiments on highly oriented L α phase samples covered a large interlayer spacing range from d = 49.1 to 255.8 A produced by dodecane dilution, which considerably exceeded those of previous high resolution synchrotron scattering studies of powder samples. Two significant differences emerge between the (DH) model and the continuum Caille model description of smectic-A liquid crystals and multilayer membranes. First, whereas the continuum model is necessarily restricted to the vicinity of the Bragg peaks of the structure factor, the discrete nature of the (DH) model allowed us to fit the experimentally measured X-ray structure factor over the full range of wave-vectors and dilutions. This enabled measurements of the membrane bending and multilayer compressibility elastic constants κ and B separately, in contrast to the continuum model which gives a reliable measurement of the product κB. Second, the (DH) model is able to account for the universally observed anomalously large small angle scattering (SAS) in strongly fluctuating dilute fluid multilayer membranes. The (SAS) is shown to contain contributions both due to concentration fluctuations described previously by Porte et al. and Nallet et al., and unexpectedly from a divergent thermal-coherent diffraction effect which dominates in single crystal multilayers.

Comparison of experimental and theoretical electronic charge distribution in γ-TiAl

Using critical voltage electron diffraction, Fox has recently determined the lowest seven X-ray structure factors of γ-TiAl (L1 0 structure). We present here a comparison of these accurately measured (0.15%) structure factors with first-principles local density calculations, finding an agreement within 0.7% and an r.m.s. error of 0.013 e/atom. While such measurements are limited to the first few structure factors ϱ( G) (where G is the crystal momentum), theory is able to obtain ϱ( G) for arbitrarily high G. If we construct charge density deformation maps by Fourier summations up to the lowest measured G, the calculated and experimental density deformation maps agree very closely. However, if we include in the theoretical density deformation map high G values (outside the range accessible to experiment), qualitatively different bonding patterns appear, in particular between Ti atoms. Systematic study of the total, valence, and deformation charge densities as well as comparison with results for NiAl in the hypothetical L1 0 structure elucidate the bonding patterns in these transition metal aluminides.

X-Ray Investigations on Molten Cu-Sb Alloys

Abstract Using X-ray diffraction, structure factors and pair correlation functions of several molten Cu-Sb alloys and pure antimony were determined and compared with published structural, thermodynamic and electronic properties. The eutectic concentration Cu37Sb63 was investigated in dependence on temperature, and a model structure factor was calculated applying a segregation model.

Density-matrix refinement for molecular crystals.

The fitting of an idempotent density matrix to x-ray structure factors is discussed. If the density matrix is identified with a single Slater determinant of orbitals, then a model wave function can be recovered, enabling the computation of two-electron properties as well as the usual one-electron properties derived from the electron density. Hence this approach is potentially superior to the usual multipole expansion used in charge-density analysis, although the current formalisms are only useful for molecular crystals of very small molecules. A procedure is described, and applications to one theoretical example (methylamine) and one experimental example (formamide) are presented.

Measurement of Debye-Waller factors of silicon as a function of temperature using energy-filtered CBED rocking-curve technique

The advantages of quantitative Convergent Beam Electron Diffraction (CBED) method for x-ray structure factor determination have been reviewed by Spence. The CBED method requires accurate values of Debye-Waller (D-W) factors for the estimation of the coefficients of crystal potential of the higher order beams, Vg, the calculation of the absorption potential, V′gusing the Einstein model for phonons, and finally the conversion of the fitted values of the coefficients of crystal potential, V″, to x-ray structure factors. Debye-Waller factors are conventionally determined by neutron or x-ray diffraction methods. Because of the difficulties in conducting high temperature neutron and x-ray diffraction experiments, D-W factors are rarely measured at temperatures above room temperature. Debye-Waller factors at high temperatures can be determined by Convergent Beam Electron diffraction (CBED) method using Transmission Electron Microscopy (TEM) employed with a hot stage attachment. Recently Holmestad et al. have attempted to measure the D-W factors by matching the energy-filtered Higher Order Laue Zone (HOLZ) line intensities near liquid nitrogen temperature.

Site Occupation of Excess Al Atoms in Off-Stoichiometric γ-Tial: Measurement of Debye-Waller Factors

AbstractIn recent years, the determination of x-ray structure factors and hence the electron charge density distribution in TiAl has received considerable attention. Experimental assessment of electron charge densities requires accurate values of the Debye-Waller (D-W) factors. This work attempts to determine the Debye-Waller factors from measurements of an Al rich off-stoichiometric single crystal of TiAl using the four circle x-ray diffraction method. The results suggest an ordered substitution of the excess Al atoms on the Ti-sublattice. This result is consistent with previous site occupancy studies.

Experimental determination of low order structure factors in the intermetallic compound TiAl

Rocking curves from several low order reflections in TiAl were recorded in digitized form using energy-filtered convergent beam electron diffraction in a transmission electron microscope. The low order reflections studied were 001, 002, 200, 111, 110, 220, 202, 113, and 311. These experimental rocking curves were simulated using computations based on the dynamic theory of electron diffraction. From these simulations, values of the X-ray structure factors for these reflections were deduced, and then used to determine charge difference maps for varioius plane sections in TiAl. These charge difference maps were compared with theoretical predictions, and it was found that there is a considerable degree of agreement between theory and experiment.

Amorphous ZrF 4-a molecular dynamics study

Using an effective two-body interaction potential, a molecular dynamics study of the structural properties of amorphous ZrF 4 phase is presented. The effective pair potential includes steric repulsion, Coulomb interaction due to charge transfer, and charge-dipole interaction due to the large electronic polarizability of anions. The results for structural correlations, such as pair distribution functions, coordination numbers, and bond angle distributions are presented. Excellent agreement is obtained by comparing experimental X-ray diffraction and the simulated static X-ray structure factor.

Intensity statistics and normalized structure factors for crystals with an incommensurate one-dimensional modulation

An analytical expression is derived for the probability density function (p.d.f.) of X-ray structure-factor amplitudes of a crystal with an incommensurate one-dimensional modulation. The influence of the (3+1)-dimensional superspace symmetry is taken into account. It is shown that, in first-order approximation, this p.d.f. has the same functional form as the p.d.f. for a nonmodulated crystal, with a suitable modification of the atomic form factor. For main reflections and satellite reflections, an expression for the average intensity is derived. This leads to a definition of normalized structure factors for a crystal with an incommensurate one-dimensional modulation. In the same first-order approximation, the p.d.f. for the amplitudes of these normalized structure factors is identical to the p.d.f, for a nonmodulated crystal and does not distinguish between main reflections and satellite reflections. The theoretical p.d.f.s are compared to p.d.f.s obtained from X-ray diffraction data of some incommensurate one-dimensionally modulated crystals.

Structure determination of inorganic structures by HREM, Cip, and electron diffraction

High resolution electron microscopy (HREM) combined with crystallographic image processing (CIP) is becoming a powerful technique for solving inorganic structures. With the image processing systems CRISP and ELD, running on a personal computer, this technique can be easily established in other laboratories. HREM images and electron diffraction patterns are digitized by a CCD camera and transferred into a PC. Phases and amplitudes are extracted from the Fourier transform of the HREM images. For thin crystals of metal oxides, the phases obtained by HREM and CIP inside the Scherzer resolution of the microscope are identical to the x-ray structure factor phases.Electron diffraction extends to much higher resolution than EM images (beyond 1 Å). The quality of the amplitudes is also higher than that from images, since ED data is not affected by the contract transfer function (CTF). Amplitudes extracted by ELD are close to x-ray diffraction amplitudes (within 30%).

Determination of the defocus value of micrographs of ice-embedded specimen without carbon support film

It is well recognized that the contrast transfer function (CTF) of an electron microscope modulates the image contrast The effects of this CTF are to reverse the sign of the phases and to alter the amplitudes at different spatial frequencies. These changes are dependent on the defocus of the objective lens in a given microscope setting. Therefore, it is necessary to determine the defocus experimentally in order to correct the phase reversal and the amplitudes due to the CTF for attaining a high resolution reconstruction. The most straightforward way of determining the defocus value is to determine the positions of the Thon rings in the CTF by optical or computed transforms. In a crystalline specimen, the defocus value of an image can be refined against the electron diffraction amplitude. For specimen of which the x-ray structure is known, one can also use the x-ray structure factor to determine the CTF parameters.

Extinction corrections from equivalent reflections

Corrections for secondary extinction evaluated from the diffraction intensities for equivalent reflections with different path lengths provide an independent check on values that minimize differences between observed and calculated structure factors. Comparison of equivalent intensities also avoids any extinction-parameter bias, which originates in correlation of the extinction corrections with bonding-electron contributions to X-ray structure factors. Corrections from the comparison of equivalent reflections for several X-ray diffraction studies on small crystals of ionic compounds are markedly less than those that minimize differences between observed and calculated structure factors. The discrepancies that originate in extinction-parameter bias are exacerbated by the unfavourable form of the statistical distribution function for the residuals when differences between observed and calculated structure factors are minimized. Analysis of intensities for equivalent reflections, although more demanding experimentally, provides least-squares residuals closer to the normal distribution required for reliability in nonlinear least-squares processes.

Measurement of individual structure-factor phases with tenth-degree accuracy: the 00.2 reflection in BeO studied by electron and X-ray diffraction

Measurements of the phase (and amplitude) of the 00.2 structure factor of BeO are reported. These were obtained by applying an automated least-squares refinement method to experimental convergent-beam transmission electron diffraction data, collected from sub-micrometer single-crystal regions of two different thicknesses. Multiple-scattering effects were included, using the Bloch-wave method. Perturbation methods were used to include weak beams and to estimate errors. After conversion to X-ray structure factors, it was found that ϕx (00.2) = −1.190 (9) rad. The same value was found, within experimental error, using data collected from a region of different thickness. New single-crystal X-ray results are also reported, which are in good agreement with the electron diffraction results. Phases derived from multipole refinement using X-ray data are about fifty times less accurate than the electron results. This electron diffraction technique may be the most accurate method for the measurement of low-order structure-factor phases from acentric crystals with small unit cells. The method may therefore allow the extension of previous highly accurate studies of charge density in silicon to acentric materials from which large single crystals cannot be grown.

On the accurate measurement of structure-factor amplitudes and phases by electron diffraction

A review is given of research into the measurement of crystal structure-factor amplitudes and phases by transmission electron diffraction. Accuracies for amplitudes are commonly a fraction of a percent (after conversion to X-ray structure factors) while phases may now be measured in favorable cases using three-beam electron diffraction to an accuracy of much better than 1°. Following a brief review of theory, the main techniques are outlined. These include quantitative convergent-beam electron diffraction, the critical-voltage method, the intersecting Kikuchi- and HOLZ-Iine methods and methods based on weak high-order reflections in wide-angle patterns. Enantiomorphs and polarity are discussed. Summaries are given of measurements of the mean inner potential and of structure factors generally. A brief review of the implications of this work for studies of crystal bonding and ordering in alloys is given, and its use to test ab initio computations of charge density. The mean inner potential is found to be the quantity most sensitive to the bonding distribution of valence electrons and sensitively constrains accurate X-ray structure-factor measurements. Electron diffraction techniques are to be preferred for studies of individual microcrystals or artificially formed structures and for the very accurate measurement of structure-factor phases in acentric crystals with small unit cells.

On the reconstruction of i(AlCuLi) X-ray structure factors

The structure factors of i(AlCuLi) reconstructed from two independent X-ray diffraction data sets using a recently developed method by Jarić and Qiu are analyzed. The two data sets were found to lead to consistent results. The validity of the method is evaluated with respect to non-negativity of the resulting electron densities. In order to identify negativity due to the intensity cut-off, the negativity is evaluated as a function of the cut-off and compared with the exact and reconstructed quasiperiodic Ammann tilings as well as with the experimental and model crystal data of R(Al 0.564Cu 0.116Li 0.32). It is concluded that the negativity of the reconstructed quasiperiodic electron density of i(AlCuLi) is consistent with a cut-off effect.

Dependence of the one-body density matrix on the superconducting gap parameter Delta (T).

We show how the one-electron thermally averaged density matrix \ensuremath{\rho}(1',1,T)==〈${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$(1')\ensuremath{\Psi}(1)〉==N〈Fd(2,...,N)${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$ (1',...,n)\ensuremath{\Psi}(1,...,N)〉 varies with the superconducting gap parameter \ensuremath{\Delta}(T). It is suggested how these results can be applied to experiments that probe the density matrix. In particular it is suggested that the x-ray structure factor F(G) should be an ideal probe of the change in \ensuremath{\rho}(r) as a material goes superconducting. We also discuss how our results can be applied to recent analysis of atomic pair distribution functions (PDF) coming from measurements of diffuse scattering of pulsed neutrons on the thallium high-${\mathit{T}}_{\mathit{c}}$ materials.

High-pressure x-ray diffraction of SiO2 glass.

We have measured the x-ray structure factor S(Q) for ${\mathrm{SiO}}_{2}$ glass to 42 GPa in the diamond-anvil cell using new experimetnal techniques for studying amorphous materials at high pressures. Large changes in the first sharp diffraction peak of S(Q) with increasing pressure provide evidence for changes in the medium-range order of the glass (length scales of \ensuremath{\sim}4\char21{}10 \AA{}). Average pair correlation functions at high pressure reveal significant changes in the nearest-neighbor geometry of the glass with compression. Between 8 and 28 GPa the coordination of Si increases, and at 42 GPa the value is close to six.