Ab initio determination of the framework structure of the heavy-metal oxide Cs(x)Nb2.54W2.46O14 from 100 kV precession electron diffraction data

The present work deals with the ab initio determination of the heavy metal framework in Cs(x)(Nb, W)(5)O(14) from precession electron diffraction intensities. The target structure was first discovered by Lundberg and Sundberg [Ultramicroscopy 52 (1993) 429-435], who succeeded in deriving a tentative structural model from high-resolution electron microsopy (HREM) images. The metal framework of the compound was solved in this investigation via direct methods from hk0 precession electron diffraction intensities recorded with a Philips EM400 at 100 kV. A subsequent (kinematical) least-squares refinement with electron intensities yielded slightly improved co-ordinates for the 11 heavy atoms in the structure. Chemical analysis of several crystallites by EDX is in agreement with the formula Cs(0.44)Nb(2.54)W(2.46)O(14). Moreover, the structure was independently determined by Rietveld refinement from X-ray powder data obtained from a multi-phasic sample. The compound crystallises in the orthorhombic space group Pbam with refined lattice parameters a=27.145(2), b=21.603(2), and c=3.9463(3)A. Comparison of the framework structure from electron diffraction with the result from Rietveld refinement shows an average agreement for the heavy atoms within 0.09 A.     

Rapid structure determination of a metal oxide from pseudo-kinematical electron diffraction data

The electron precession diffraction technique is employed to provide quasi-kinematical data for determination of atom positions in the (Ga,In)2SnO5m-phase. Precession data are compared with conventional diffraction data captured under identical conditions and show a distinct superiority because they exhibit kinematical characteristics in the structure-defining reflections. Precessed data are not usable within a kinematical interpretation in all cases, and a simple basis is presented for omission of errant reflections to improve adherence to kinematical behavior. A second approach is demonstrated where intensities are used with direct methods instead of amplitudes, enhancing the contrast between strong and weak beams. The unrefined atom positions recovered a priori via direct methods are consistent between the two approaches and fall on average within 4 picometers of positions in the previously refined structure.     

“Ab initio” structure solution from electron diffraction data obtained by a combination of automated diffraction tomography and precession technique

Using a combination of our recently developed automated diffraction tomography (ADT) module with precession electron technique (PED), quasi-kinematical 3D diffraction data sets of an inorganic salt (BaSO₄) were collected. The lattice cell parameters and their orientation within the data sets were found automatically. The extracted intensities were used for “ab initio” structure analysis by direct methods. The data set covered almost the complete set of possible symmetrically equivalent reflections for an orthorhombic structure. The structure solution in one step delivered all heavy (Ba, S) as well as light atoms (O). Results of the structure solution using direct methods, charge flipping and maximum entropy algorithms as well as structure refinement for three different 3D electron diffraction data sets were presented.     

Crystal structure solution via precession electron diffraction data: The BEA algorithm

The statistical features of the amplitudes obtained via precession electron diffraction have been studied, with particular concern with their effects on direct phasing procedures. A new algorithm, denoted by BEA, is described: according to it, the average amplitude of the symmetry equivalent reflections is used in the Direct Methods step. Once an even imperfect structural model is available, the best amplitude among the equivalent reflections is used to improve the model. It is shown that BEA is able to provide more complete structural models, to make the phasing process more straightforward and to end with crystallographic residual much better than those usually obtained by electron diffraction.     

Crystal structure solution via precession electron diffraction data: the BEA algorithm

The statistical features of the amplitudes obtained via precession electron diffraction have been studied, with particular concern with their effects on direct phasing procedures. A new algorithm, denoted by BEA, is described: according to it, the average amplitude of the symmetry equivalent reflections is used in the Direct Methods step. Once an even imperfect structural model is available, the best amplitude among the equivalent reflections is used to improve the model. It is shown that BEA is able to provide more complete structural models, to make the phasing process more straightforward and to end with crystallographic residual much better than those usually obtained by electron diffraction.     

Direct phase determination for quasi-kinematical electron diffraction intensity data from organic microcrystals.

Direct phasing techniques using cosine estimates for three-phase and four-phase structure invariants in space group P1 are applied to zonal hk0 electron diffraction intensity data from an orthorhombic paraffin and a cephalin. Ab initio use of these kinematical phases is shown to derive the correct structures of the methylene subcell packings for these compounds in agreement with earlier determinations from Patterson maps.     

An efficient approach to characterize pseudo-merohedral twins by precession electron diffraction: Application to the LaGaO3 perovskite

Pseudo-merohedral twins are frequently observed in crystals displaying pseudo-symmetry. In these crystals, many [u v w] zone axis electron diffraction patterns are very close and can only be distinguished from intensity considerations. On conventional diffraction patterns (selected-area electron diffraction or microdiffraction), a strong dynamical behaviour averages the diffracted intensities so that only the positions of the reflections on a pattern can be considered. On precession electron diffraction patterns, the diffracted beams display an integrated intensity and a “few-beam” or “systematic row” behaviour prevails which strongly reduces the dynamical interactions. Therefore the diffracted intensity can be taken into account. A procedure based on observation of the weak extra-reflections connected with the pseudo-symmetry is given to identify without ambiguity any zone axis. It is successfully applied to the identification and characterization of {1 2 1} reflection twins present in the LaGaO₃ perovskite.     

Automatic twin statistics from electron backscattered diffraction data

A new computer code has been developed to automatically extract quantitative twin statistics from electron backscatter diffraction data. The new code is an improvement upon previous codes in that it handles materials of any crystal symmetry, type I, Type II and compound twins, and general stress states. Moreover, accuracy of the results has been greatly improved. In addition, twin statistics including number, area fraction, twin thickness and twinning dependencies on orientation, grain size and neighbourhood effects can be routinely analysed. The new code has been applied to scan data from deformed magnesium, zirconium and uranium, and can potentially be used for any twinning material for which reliable electron backscatter diffraction results can be obtained.     

A quantitative analysis of the cone-angle dependence in precession electron diffraction

Precession electron diffraction (PED) is a technique which is gaining increasing interest due to its ease of use and reduction of the dynamical scattering problem in electron diffraction. To further investigate the usefulness of this technique, we have performed a systematic study of the effect of precession angle on the mineral andalusite where the semiangle was varied from 6.5 to 32 mrad in five discrete steps. The purpose of this study was to determine the optimal conditions for the amelioration of kinematically forbidden reflections, and the measurement of valence charge density. We show that the intensities of kinematically forbidden reflections decay exponentially as the precession semiangle () is increased. We have also determined that charge density effects are best observed at moderately low angles (6.5–13 mrad) even though PED patterns become more kinematical in nature as the precession angle is increased further.     

Direct space structure solution from precession electron diffraction data: Resolving heavy and light scatterers in Pb13Mn9O25

The crystal structure of a novel compound Pb₁₃Mn₉O₂₅ has been determined through a direct space structure solution with a Monte-Carlo-based global optimization using precession electron diffraction data (a=14.177(3) Å, c=3.9320(7) Å, SG P4/m, R_F=0.239) and compositional information obtained from energy dispersive X-ray analysis and electron energy loss spectroscopy. This allowed to obtain a reliable structural model even despite the simultaneous presence of both heavy (Pb) and light (O) scattering elements and to validate the accuracy of the electron diffraction-based structure refinement. This provides an important benchmark for further studies of complex structural problems with electron diffraction techniques. Pb₁₃Mn₉O₂₅ has an anion- and cation-deficient perovskite-based structure with the A-positions filled by the Pb atoms and 9/13 of the B positions filled by the Mn atoms in an ordered manner. MnO₆ octahedra and MnO₅ tetragonal pyramids form a network by sharing common corners. Tunnels are formed in the network due to an ordered arrangement of vacancies at the B-sublattice. These tunnels provide sufficient space for localization of the lone 6s² electron pairs of the Pb²⁺ cations, suggested as the driving force for the structural difference between Pb₁₃Mn₉O₂₅ and the manganites of alkali-earth elements with similar compositions.     

Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions.

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix, one orientation for each column in the matrix. At each orientation the corresponding column of the scattering-matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg-Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the scattering-matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.     

Application of a 2-beam model for improving the structure factors from precession electron diffraction intensities

A 2-beam model is used to simulate precession electron diffraction (PED) intensities. It is shown that this model can be inverted with minimal knowledge of the underlying crystal structure, permitting structure factor amplitudes to be deduced directly from measured intensities within the 2-beam approximation. This approach may be used in conjunction with direct methods to obtain correct, kinematically interpretable structure indications for data sets from relatively thin crystals (less than approximately 400A), and an experimental example based on (Ga,In)(2)SnO(5) is presented. The failure of this approach at large thickness is illustrated by an additional data set for MFI zeolite. The 2-beam approximation provides a simple model for PED intensities, and inversion using this model shows advantages over a kinematical approximation. It is however too rough approximation to be of general use and ultimately it is to be hoped that more accurate models with similar ease of use can be derived to treat PED data.     

Orientation averaging of electron backscattered diffraction data

The use of data averaging to improve the angular precision of electron backscattered diffraction (EBSD) maps is discussed. It is shown that orientations may be conveniently and rapidly averaged using the four Euler-symmetric parameters which are coefficients of a quaternion representation. The processing of EBSD data requires the use of an edge preserving filter and a modified Kuwahara filter has been successfully implemented and tested. Three passes of such a filter have been shown to reduce orientation noise by a factor of ∼10. Application of the method to deformed and recovered aluminium alloys has shown that such data processing enables small subgrain misorientation (< 0.5°) to be detected reliably.     

Rietveld analysis of electron powder diffraction data from nanocrystalline anatase, TiO2

The structure of nanocrystalline anatase (TiO2) was successfully refined from electron powder diffraction data using the Rietveld technique. A polycrystalline sample (average crystal size about 70 A) was characterised by selected area electron diffraction in a conventional transmission electron microscope operated at 300 kV. Radially integrated intensities were extracted from digitised photographic films and used in the course of structure refinements by a standard program for Rietveld analysis. The structure was refined in space group I4(1)/amd (#141) with lattice parameters a = 3.7710(9) A and c = 9.430(2) A. The reliability factors of the refinement are Rwp = 5.2% and R(B) = 2.6%. The close agreement of the refined structural parameters with previous results obtained from neutron diffraction on coarse-grained powders proves the applicability of the method for characterising nanocrystalline powders. The present study shows that Rietveld analysis on electron powder data is a good compliment to the existing methods for accurate structural investigations on nanocrystalline materials and thin films.     

Direct electron crystallographic determination of zeolite zonal structures

The prospect for improving the success of ab initio zeolite structure investigations with electron diffraction data is evaluated. First of all, the quality of intensities obtained by precession electron diffraction at small hollow cone illumination angles is evaluated for seven representative materials: ITQ-1, ITQ-7, ITQ-29, ZSM-5, ZSM-10, mordenite, and MCM-68. It is clear that, for most examples, an appreciable fraction of a secondary scattering perturbation is removed by precession at small angles. In one case, ZSM-10, it can also be argued that precession diffraction produces a dramatically improved 'kinematical' data set. There seems to no real support for application of a Lorentz correction to these data and there is no reason to expect for any of these samples that a two-beam dynamical scattering relationship between structure factor amplitude and observed intensity should be valid. Removal of secondary scattering by the precession mode appears to facilitate ab initio structure analysis. Most zeolite structures investigated could be solved by maximum entropy and likelihood phasing via error-correcting codes when precession data were used. Examples include the projected structure of mordenite that could not be determined from selected area data alone. One anomaly is the case of ZSM-5, where the best structure determination in projection is made from selected area diffraction data. In a control study, the zonal structure of SSZ-48 could be determined from selected area diffraction data by either maximum entropy and likelihood or traditional direct methods. While the maximum entropy and likelihood approach enjoys some advantages over traditional direct methods (non-dependence on predicted phase invariant sums), some effort must be made to improve the figures of merit used to identify potential structure solutions.