High Resolution X-Ray Powder Diffraction by the Combination of Synchrotron Radiation and Imaging Plate to Observe Electron Distribution by the Maximum Entropy Method

AbstractA Debye-Scherrer camera (radius 572 mm) was designed for the use at the Photon Factory BL-6A2. It uses 4 pieces of the Imaging Plate to cover 0° ∼ 160° in 2θ. The profiles measured by the camera had 0.083° full width at half maximum (FWHM) and were rather symmetric compared with conventional X-ray source. The observed profiles were well represented by either the split Pearson VII or pseudo-Voigt function. In order to examine the performance of the new camera, the whole powder data of LiF was analyzed by the Maximum Entropy Method(MEM). The MEM analysis of LiF was successfully accomplished using 20 measured structure factors and gave R(Bragg) 0.25%. The density distribution obtained represented a characteristic feature of ionic crystals and was consistent with the theoretical result of the self-consistent LCAO method.

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Electron-density distribution from X-ray powder data by use of profile fits and the maximum-entropy method

Journal of Applied Crystallography ◽

10.1107/s0021889890008214 ◽

1990 ◽

Vol 23 (6) ◽

pp. 526-534 ◽

Cited By ~ 65

Author(s):

M. Sakata ◽

R. Mori ◽

S. Kumazawza ◽

M. Takata ◽

H. Toraya

Keyword(s):

Density Distribution ◽

Maximum Entropy ◽

Electron Density ◽

Electron Density Distribution ◽

Maximum Entropy Method ◽

Entropy Method ◽

Distribution Map ◽

X Ray ◽

Profile Decomposition ◽

Structure Factors

Following the profile decomposition of CeO2 X-ray powder data into individual structure factors, the maximum-entropy method (MEM) has been used to obtain an electron-density-distribution map. In the profile decomposition process, it is impossible to avoid the problems of overlapping peaks which have the same magnitude of reciprocal vectors, such as d*(511) and d*(333), for a cubic crystal, or very severely overlapping reflections. The formalism to treat such overlapping reflections in the MEM analysis is to introduce combined structure factors. The maximum value of the scattering vector, 4π(sinθ)/λ, which was used in the present analysis is small (about 7.8 Å−1) but the resulting electron-density-distribution map is of a high quality and much superior to the conventional map. As a consequence, the ionic charge of Ce and O ions can be obtained with reasonable accuracy from the MEM density map. Furthermore, the map reveals the existence of electrons around the supposedly vacant site surrounded by eight O atoms, which is probably related to the high ionic conductivity of this substance.

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The Electron Density Distribution in Be Metal Obtained from Synchrotron-Radiation Powder Data by the Maximum-Entropy Method

Zeitschrift für Naturforschung A ◽

10.1515/zna-1993-1-219 ◽

1993 ◽

Vol 48 (1-2) ◽

pp. 75-80 ◽

Cited By ~ 14

Author(s):

Masaki Takata ◽

Yoshiki Kubota ◽

Makoto Sakata

Keyword(s):

Synchrotron Radiation ◽

Density Distribution ◽

Maximum Entropy ◽

Electron Density ◽

Electron Density Distribution ◽

Maximum Entropy Method ◽

Entropy Method ◽

Imaging Plate ◽

Acta Cryst ◽

Structure Factors

Abstract The nature of the bonding in Be metal was studied by investigating the MEM map, which is the electron density distribution obtained by the Maximum-Entropy Method. In order to avoid extinc-tion effects, 19 Bragg reflections were measured by a new powder-diffraction experiment that utilizes Synchrotron Radiation as an incident X-ray and an Imaging Plate as detector. The experiment was carried out at the Photon Factory BL6A2. In spite of the limited number of reflections used in the MEM analysis, the electron density distribution of Be was obtained accurately and reliably. The structure factors for unmeasured reflections were calculated and compared with the values observed by Larsen and Hansen [Acta Cryst. B40, 169 (1984)]. The agreement is very good. Furthermore, the MEM map revealed that Be metal forms an electronic layer in the shape of a honeycomb that is parallel to the basal plane.

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Phase determination and extension using X-ray multiple diffraction and the maximum-entropy method

Acta Crystallographica Section A Foundations of Crystallography ◽

10.1107/s0108767300018869 ◽

2001 ◽

Vol 57 (4) ◽

pp. 420-428 ◽

Cited By ~ 3

Author(s):

Chien-Mei Wang ◽

Chun-Hsiung Chao ◽

Shih-Lin Chang

Keyword(s):

Maximum Entropy ◽

Maximum Entropy Method ◽

Entropy Method ◽

Multiple Diffraction ◽

Phase Determination ◽

X Ray

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A model-independent maximum-entropy method for the inversion of small-angle X-ray diffraction patterns

Journal of Applied Crystallography ◽

10.1107/s0021889899010560 ◽

1999 ◽

Vol 32 (6) ◽

pp. 1069-1083 ◽

Cited By ~ 10

Author(s):

J. A. Elliott ◽

S. Hanna

Keyword(s):

Maximum Entropy ◽

Small Angle ◽

Disordered Systems ◽

Maximum Entropy Method ◽

Structural Model ◽

Entropy Method ◽

X Ray Diffraction ◽

X Ray ◽

Model Independent ◽

Diffraction Patterns

A model-independent maximum-entropy method is presented which will produce a structural model from small-angle X-ray diffraction data of disordered systems using no other prior information. In this respect, it differs from conventional maximum-entropy methods which assume the form of scattering entitiesa priori. The method is demonstrated using a number of different simulated diffraction patterns, and applied to real data obtained from perfluorinated ionomer membranes, in particular Nafion™, and a liquid crystalline copolymer of 1,4-oxybenzoate and 2,6-oxynaphthoate (B–N).

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An Evaluation of Deconvolution Techniques in X-ray Profile Broadening Analysis and the Application of the Maximum Entropy Method to Alumina Data

Advances in X-ray Analysis ◽

10.1154/s0376030800018036 ◽

1994 ◽

Vol 38 ◽

pp. 387-395 ◽

Cited By ~ 1

Author(s):

Walter Kalceff ◽

Nicholas Armstrong ◽

James P. Cline

Keyword(s):

Maximum Entropy ◽

Maximum Entropy Method ◽

Profile Analysis ◽

Domain Size ◽

Entropy Method ◽

X Ray Diffraction ◽

Profile Function ◽

Diffraction Profile ◽

X Ray ◽

Deconvolution Techniques

Abstract This paper reviews several procedures for the removal of instrumental contributions from measured x-ray diffraction profiles, including: direct convolution, unconstrained and constrained deconvolution, an iterative technique, and a maximum entropy method (MEM) which we have adapted to x-ray diffraction profile analysis. Decorevolutions using the maximum entropy approach were found to be the most robust with simulated profiles which included Poisson-distributed noise and uncertainties in the instrument profile function (IPF). The MEM procedure is illustrated by application to the analysis for domain size and microstrain carried out on the four calcined α-alumina candidate materials for Standard Reference Material (SRM) 676 (a quantitative analysis standard for I/Ic determinations), along with the certified material. Williamson-Hall plots of these data were problematic with respect to interpretation of the microstrain, indicating that the line profile standard, SRM 660 (LaB6), exhibits a small amount of strain broadening, particularly at high 2θ angle. The domain sizes for all but one of the test materials were much smaller than the crystallite (particle) size; indicating the presence of low angle grain boundaries.

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Electron Density Distribution Obtained from X-Ray Powder Data by the Maximum Entropy Method

Advances in X-ray Analysis ◽

10.1154/s0376030800008697 ◽

1991 ◽

Vol 35 (A) ◽

pp. 77-83 ◽

Cited By ~ 1

Author(s):

Makoto Sakata ◽

Masaki Takata ◽

Yoshiki Kubota ◽

Tatsuya Uno ◽

Shintaro Kuhazawa ◽

...

Keyword(s):

Density Distribution ◽

Maximum Entropy ◽

Electron Density ◽

Diffraction Data ◽

Electron Density Distribution ◽

Maximum Entropy Method ◽

Entropy Method ◽

Nuclear Density ◽

X Ray ◽

Distribution Maps

AbstractThe electron density distribution maps for CaF2 and TiO2 (rutile) were obtained from profile fitting of powder diffraction data by a Maximum Entropy Method (MEM) analysis. The resultant electron density maps show clearly the nature of the chemical bonding. In order to interpret the results, the nuclear density distribution was also obtained for rutile from powder neutron diffraction data. In the electron density map for rutile obtained by HEM analysis from the X-ray data, both apical and equatorial bonding can be seen. On the other hand, the nuclear density of rutile Is very simple and shows the thermal vibration of nuclei.

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Charge density study with the Maximum Entropy Method on model data of silicon. A search for non-nuclear attractors

Canadian Journal of Chemistry ◽

10.1139/v96-118 ◽

1996 ◽

Vol 74 (6) ◽

pp. 1054-1058 ◽

Cited By ~ 12

Author(s):

R.Y. de Vries ◽

W.J. Briels ◽

D. Fell ◽

G. te Velde ◽

E.J. Baerends

Keyword(s):

Charge Density ◽

Maximum Entropy ◽

Density Functional ◽

Maximum Entropy Method ◽

Entropy Method ◽

X Ray Diffraction ◽

Functional Theory ◽

Structure Factors ◽

The Density Functional Theory ◽

Structure Calculations

In 1990 Sakata and Sato applied the maximum entropy method (MEM) to a set of structure factors measured earlier by Saka and Kato with the Pendellösung method. They found the presence of non-nuclear attractors, i.e., maxima in the density between two bonded atoms. We applied the MEM to a limited set of Fourier data calculated from a known electron density distribution (EDD) of silicon. The EDD of silicon was calculated with the program ADF-BAND. This program performs electronic structure calculations, including periodicity, based on the density functional theory of Hohenberg and Kohn. No non-nuclear attractor between two bonded silicon atoms was observed in this density. Structure factors were calculated from this density and the same set of structure factors that was measured by Saka and Kato was used in the MEM analysis. The EDD obtained with the MEM shows the same non-nuclear attractors that were later obtained by Sakata and Sato. This means that the non-nuclear attractors in silicon are really an artefact of the MEM. Key words: Maximum Entropy Method, non-nuclear attractors, charge density. X-ray diffraction.

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Nuclear Enhanced MEM Used to Analyze Local Distortions in Lead Chalcogenides

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314098945 ◽

2014 ◽

Vol 70 (a1) ◽

pp. C105-C105

Author(s):

Sebastian Christensen ◽

Niels Bindzus ◽

Mogens Christensen ◽

Bo Brummerstedt Iversen

Keyword(s):

Maximum Entropy ◽

Diffraction Data ◽

Thermoelectric Materials ◽

Maximum Entropy Method ◽

Functional Materials ◽

Entropy Method ◽

Green Energy ◽

Lead Chalcogenides ◽

X Ray ◽

Local Distortions

We introduce a novel method for reconstructing nuclear density distributions (NDDs): Nuclear Enhanced X-ray Maximum Entropy Method (NEXMEM). NEXMEM offers an alternative route to experimental NDDs, exploiting the superior quality of synchrotron X-ray data compared to neutron data. The method was conceived to analyse local distortions in the thermoelectric lead chalcogenides, PbX (X = S, Se, Te). Thermoelectric materials are functional materials with the unique ability to interconvert heat and electricity, holding much promise for green energy solutions such as waste heat recovery. The extraordinary thermoelectric performance of binary lead chalcogenides has caused huge research activity, but the mechanisms governing their unexpected low thermal conductivity still remain a controversial topic. It has been proposed to result from giant anharmonic phonon scattering or from local fluctuating dipoles on the Pb site.[1,2] No macroscopic symmetry change are associated with these effects, rendering them invisible to conventional crystallographic techniques. For this reason PbX was until recently believed to adopt the ideal, undistorted rock-salt structure. In the present study, we investigate PbX using multi-temperature synchrotron powder X-ray diffraction data in combination with the maximum entropy method (MEM) and NEXMEM. In addition NEXMEM has been validated by testing against simulated powder diffraction data of PbTe with known displacements of Pb. The increased resolution of NEXMEM proved essential for resolving Pb-displacement of 0.2 Å in simulated data. The figure below shows Pb in the (100) plane for MEM, NEXMEM and the actual NDD of the test structure. Our findings outline the extent of disorder in lead chalcogenides, promoting our understanding of this class of high-performance thermoelectric materials. Furthermore we introduce NEXMEM which can be used for widespread characterization of subtle atomic features in crystals with unusual properties.

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