Refinement Of Structure Factors Research Articles

Improvement of precision in refinements of structure factors using convergent-beam electron diffraction patterns taken at Bragg-excited conditions

A local structure analysis method based on convergent-beam electron diffraction (CBED) has been used for refining isotropic atomic displacement parameters and five low-order structure factors with sin θ/λ ≤ 0.28 Å-1 of potassium tantalate (KTaO3). Comparison between structure factors determined from CBED patterns taken at the zone-axis (ZA) and Bragg-excited conditions is made in order to discuss their precision and sensitivities. Bragg-excited CBED patterns showed higher precision in the refinement of structure factors than ZA patterns. Consistency between higher precision and sensitivity of the Bragg-excited CBED patterns has been found only for structure factors of the outer zeroth-order Laue-zone reflections with larger reciprocal-lattice vectors. Correlation coefficients among the refined structure factors in the refinement of Bragg-excited patterns are smaller than those of the ZA ones. Such smaller correlation coefficients lead to higher precision in the refinement of structure factors.

Charge density determination in icosahedral AlPdMn quasicrystal using quantitative convergent beam electron diffraction

Charge density distribution in icosahedral AlPdMn quasicrystal has been studied on a single-crystal specimen by using quantitative convergent beam electron diffraction (QCBED) technique. The QCBED systematic row method was used in the refinement of structure factors. To refine the low-order structure factors, the wave-mechanical formulation of electron diffraction dynamical theory was used in the calculation of electron diffraction intensities for the quasicrystal in fitting the experimental intensity line scan profiles. The shapes of atomic surfaces (occupation domains) were described with symmetry-adapted series of surface harmonics. An iterative procedure was used in determination of structure factors of the quasicrystal. The structure factors of nine strongest symmetry inequivalent reflections according to X-ray diffraction experiment were refined with QCBED technique. The average of refinement results for a given reflection performed on several CBED patterns, which were slightly different in orientation and sample thickness, and on different line scans, was taken as the value of structure factor for the reflection. The obtained structure factors for electrons were transformed into X-ray structure factors with Mott formula. The bonding charge density map for the quasicrystal was constructed with the obtained nine structure factors. Assuming that the atoms are spheres, the gain or loss of electrons for different atoms were calculated. It shows that identical atoms can have different valences at different kinds of positions. The bonding charge is localized along certain directions.

Quantitative Zone-Axis Convergent Beam Electron Diffraction: Current Status and Future Prospects

Quantitative zone-axis convergent beam electron diffraction (CBED) is now an established technique. Over the past decade it has been developed into a tested method for the accurate refinement of structure factors, allowing the details of the charge density and bonding effects to be studied in crystalline materials. Strategies for obtaining the most accurate results have evolved, and the most important influences on the accuracy have been determined. Initial applications of the technique to bond charge density determination have led to the extension of the method to the refinement of other important parameters influencing the experimental data, such as Debye-Waller factors and the absorption potential. The development and current status of quantitative zone-axis CBED are discussed. Prospects for the future development and application of the technique are also considered.

How Accurate is Qcbed ? -- the Case of Rutile (TiO2)

Abstract An accuracy of better than 1% is needed to measure the changes in charge density due to bonding. Here we report an accuracy up to 0.025% (random error) obtained in rutile crystal structure factors measurement by QCBED. This error is the standard deviation in the mean value obtained from ten data sets. Systematic errors may be present. Figure 1 gives an example of the (200) refinement results. Table 1 lists several low order structure factor refinement results. The accuracy of the measured electron structure factors was 0.1-0.2% but after conversion to x-ray structure factors, the accuracy for low orders improved due to the Mott formula [1] For (110) and (101) reflections, the accuracy in x-ray structure factors became 0.025% and 0.048% respectively. This accuracy is equivalent to that of the X-ray single crystal Pendellosung method on silicon crystals [2]. The experiments were done on a Leo 912 Omega TEM.

Effects of Debye-Waller factors and compositional uncertainties on the 200 structure factor refinement eta in-TiAl

The effects of uncertainties in Debye-Waller factors and chemical composition on 200 structure factor refinement in eta-TiAl have been investigated. Energy-filtered convergent-beam electron diffraction has been used to record 200 rocking curves from two different thicknesses of a thin foil taken from a Ti-51.3 1Al alloy (all compositions in atomic per cent). The structure factor has been refined by matching rocking curves determined by experiment to those theoretically calculated using the Bloch wave formalism of the dynamical theory of electron diffraction. The influence of uncertainties in the Debye-Waller factors and sample composition have been assessed: for samples deviating from stoichiometry, the need to use accurate site-specific Debye-Waller factors has been demonstrated. Also, to extract accurate structure factors, it is necessary to make use of homogeneous samples, the compositions of which must be known to better than 0.2%, a rather restrictive limitation.

Efficient beam-selection criteria in quantitative convergent beam electron diffraction

New criteria to identify significant beams in many beam dynamical calculations are introduced. The new beam selection criteria are used in a structure factor refinement procedure where the correct identification of the most significant beams is crucial. Beams are divided into two categories. Strong beams are included in the dynamical calculations through diagonalisation. The effects of weak beams are successfully described by Bethe-perturbation of the dynamical matrix. The new method is compared to existing algorithms and found favourable, both in terms of computing time and stability in calculations.

Structure factor measurement in TiAl and silicon

The accurate measurement of low order structure factors is required for the determination of the electron charge density distribution in crystals. In this work the energy-filtered convergent beam electron diffraction (CBED) rocking curve method has been used for accurate structure factor measurements. This CBED method for structure factor refinement involves matching of the experimental CBED intensities to those calculated using dynamical electron diffraction theory. The CBED experiments were conducted with a Philips EM420 Transmission Electron Microscope coupled with a custom built energy-filtering attachment enabling single electron counting. The theoretical pattern matching was performed using FORTRAN programs which were developed by Swaminathan. Initially the experimental plan involved an attempt to refine structure factors of TiAl by two dimensional Bloch wave calculations. The results of this project have been reported elsewhere. Subsequently it proved impossible to obtain results with sufficient precision for TiAl reproducibly, i.e. less than 0.1%, from samples of different thicknesses.

Crystal structures and cation distributions in simple spinels from powder XRD structural refinements: MgCr2O4, ZnCr2O4, Fe3O4 and the temperature dependence of the cation distribution in ZnAl2O4

The crystal structure and cation distributions in the spinels MgCr2O4, ZnCr2O4, Fe3O4 and a suite of ZnAl2O4 samples annealed at 900 to 1400° C and then rapidly quenched, have been determined by powder X-ray diffraction, using several different X-ray procedures and both conventional structure-factor refinement and whole-pattern (or Rietveld) refinement methods. The chromite spinels are expected from crystal chemical considerations to have an almost completely normal cation distribution (inversion parameter, x, equal to zero). In agreement with this expectation, three samples of MgCr2O4 annealed at 900, 1100 and 1300° C, and ZnCr2O4 were all found to have x=0 within two estimated standard deviations (esd), suggesting that the accuracy with which cation distributions in spinels may be determined by powder XRD is close to the estimated precision. Slightly better results are obtained assuming neutral-atom scattering curves rather than half-ionized or fully ionized, but the differences are small (within the esd). The results from the Rietveld refinements are similarly in good agreement with those using the conventional structure factor refinement approach (agreement within the combined esd's), although in detail the Rietveld procedure sometimes produces small systematic differences in refined parameters. The suite of ZnAl2O4 spinels show a smooth increase in x from 0.01 at 900° C to 0.05 at 1300° C, and this behaviour is well described by the simple thermodynamic model for disordering in spinels with αZn-Al=89 kJ/mol, assuming β=−20 kJ/mol. The oxygen positional parameters for Fe3O4 are similar to those from published single crystal studies, indicating that the powder method also yields accurate interatomic distances in spinels.

Structure factor refinement by least-squares inversion of 2D convergent-beam patterns

While electron microscopy and diffraction for decades mainly have been a qualitative tool, there has been an increasing trend over the last years for more quantitative work. This trend applies in particular to convergent beam electron diffraction (CBED) where different methods for the retrieval of quantitative structure information from CBED intensities have been proposed. In our group we have aimed at developing a general and automatic method based on the strong parameter dependencies that govern the complicated two-dimensional (2-D) intensity variations in the CBED-disks particularly in the nonsystematic many-beam case. The method is based on least squares fitting between digitized experimental and computed patterns. The advantage of this method as compared with alternative methods is many turning points in the intensity variations, which should give a high parameter sensitivity, and the fact that non-systematic many-beam effects are particularly sensitive to structure factor phases, which should make the method especially suitable to non-centric crystals.

Measurements of Debye-Waller factors in TiAl from energy-filtered HOLZ line intensities

This paper reports a new method to measure Debye Waller factors. The measurements are required for subsequent structure factor refinement in TiAl. The procedure is to obtain a large angle CBED pattern near a sparse zone axis, extract line scans along HOLZ lines from this pattern and use two beam intensity calculations to fit experiment and simulation by varying the Debye Waller factors. Unlike earlier work, we have used non-systematic reflections. The material studied is γ−TiAl (50-50), made from 25g ingots produced by arc-melting in an argon atmosphere, and homogenized for 1 day at 1473 K. Specimens were thinned by twin-jet electropolishing.Elastic energy filtered patterns are required for accurate quantitative comparisons between experiment and theory. A Zeiss 912 Omega TEM/STEM electron microscope was used to obtain energy filtered CBED patterns, fitted with an Omega imaging energy filter, which forms images or diffraction patterns within any desired energy loss range.

Automated structure-factor refinement from convergent-beam electron diffraction patterns

An improved algorithm is described for automated structure-factor refinement using convergent-beam electron diffraction patterns. In addition to refinement of structure factors, absorption coefficients and sample thickness, the new algorithm includes beam-direction refinement, the inclusion of weak-beam effects using the Bethe potential and the inclusion of finite-sized detector effects. The use of these new features is illustrated through the refinement of the MgO 200 systematic reflections.

A structural study of poly( p-phenylene vinylene)

X-ray diffraction profiles of highly oriented poly( p-phenylene vinylene) (PPV) samples have been acquired at temperatures ranging from 20 to 700 K. The resulting equatorial data have been analysed using a linked-atom-least-square structure factor refinement algorithm that allows for a more complete characterization of the polymer structure and of its temperature-dependent evolution. The resulting analysis identifies strong librational motions of the phenyl rings, a finding in agreement with recent n.m.r. studies. In additon, thin films of PPV are found to possess measurable equatorial anisotropy with [200] and [110] lattice planes preferentially oriented parallel to the film surface.

Automated structure factor refinement from convergent-beam patterns

An algorithm is described which automatically adjusts values of the low-order structure factors, crystal thickness and absorption coefficients for the best fit to experimental convergent-beam electron diffraction (CBED) patterns recorded in the systematics orientation. A fitting index is defined, by analogy with R factors used in neutron diffraction. A comparison of several least-squares optimization routines is made, and the “Simplex” method found to be most useful. An analysis of errors, background and noise is presented. Source-code listings of most of the program have been published elsewhere. This treats three-dimensional dynamical diffraction by the Bloch-wave method from non-centrosymmetric crystals with absorption. The use of perturbation methods for the final 33-beam refinement is found to reduce the computation time from 6 h to 30 min. An example of the use of the program is given. For MgO, we find U(200) = 0.05847 ± 0.00051 and U(400) = 0.02484 ± 0.00028. This algorithm may be used to measure and study any of the parameters (such as atomic coordinates, bonding effects or Debye-Waller factors) on which crystal structure factors depend. Similar methods allow lattice parameters to be refined to an accuracy of ∽ 0.05%.

Frequency-restrained structure-factor refinement. II. Comparison of methods

The frequency distribution of electron-density-function values encountered in a protein crystal has a characteristic shape and may be predicted for a protein with unknown spatial structure. It is shown that various methods of refinement of structure-factor phases (frequency-restrained refinement, histogram matching, density modification) may be regarded as various approaches to the same problem of obtaining the electron-density distribution which agrees with the X-ray experimental data and has a prescribed histogram. Test computations illustrate the relative efficiency of the methods analyzed.

Structure factor refinement by simulation of 2-D convergent beam electron diffraction patterns

Structure factor refinement by simulation of 2-D convergent beam electron diffraction patterns

A comparison of computer algorithms for structure factor refinement from convergent beam electron diffraction

A comparison of computer algorithms for structure factor refinement from convergent beam electron diffraction

Two-beam features in electron diffraction patterns – application to refinement of low-order structure factors in GaAs

Progress in theoretical approaches to the refinement of structure factors and structure determination by electron diffraction is briefly reviewed. A strategy based on the use of two-beam-like features in convergent-beam electron diffraction (CBED) patterns is advocated, based on an effective potential and a reduced excitation error. Model calculations suggest inversion of the Bethe potentials to be useful. New experimental measurements for CBED rocking curves in GaAs are reported and these give the refined values V(200) = 0.432 (9), V(400) = 4.53 (4) V at 90 K, including Debye-Waller factor. The results from the inverse Bethe potential method are in excellent agreement with full many-beam calculations. These experimental values are compared with the results of recent pseudopotential total-energy calculations.

A CCD parallel detection system for electron diffraction and imaging with large dynamic range

The quantitative recording of convergent-beam electron diffraction patterns (CBED), spot patterns. RHEED patterns and high resolution images requires a parallel detection system with large dynamic range. Systems based on YAG detectors and television tubes are ideal for recording rapidly changing events, however the integration of many such images provides a limited improvement in dynamic range since the read-out process itself introduces noise. Charge-coupled devices (CCDs) have been shown to provide a promising alternative for recording diffraction patterns. Our system is intended for recording the weak reflections from reconstructed surfaces in transmission on the U.H.V. Philips-Gatan EM430. and for structure-factor refinement by quantitative CBED (see these proceedings). Our CCD system, fitted to a Philips EM400, is based on the Photometries CC210 camera (see figure 1). A YAG single-crystal screen is bonded to fibre-optics, which transfers the image out of the vacuum and onto a Thomson-CSF TH7882CDA 576x384x14 bit CCD cooled by liquid nitrogen.