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

1999 ◽  
Vol 32 (6) ◽  
pp. 1069-1083 ◽  
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).

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

1994 ◽  
Vol 38 ◽  
pp. 387-395 ◽  
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.

scholarly journals Location of Cu2+in CHA zeolite investigated by X-ray diffraction using the Rietveld/maximum entropy method

IUCrJ ◽  
2014 ◽  
Vol 1 (6) ◽  
pp. 382-386 ◽  
Author(s):  
Casper Welzel Andersen ◽  
Martin Bremholm ◽  
Peter Nicolai Ravnborg Vennestrøm ◽  
Anders Bank Blichfeld ◽  
Lars Fahl Lundegaard ◽  
...  
Keyword(s):  
Maximum Entropy ◽  
Density Functional ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Membered Ring ◽  
Zeolite Catalysts ◽  
Structural Description ◽  
X Ray Diffraction ◽  
X Ray ◽  
Catalytically Active

Accurate structural models of reaction centres in zeolite catalysts are a prerequisite for mechanistic studies and further improvements to the catalytic performance. The Rietveld/maximum entropy method is applied to synchrotron powder X-ray diffraction data on fully dehydrated CHA-type zeolites with and without loading of catalytically active Cu2+for the selective catalytic reduction of NOxwith NH3. The method identifies the known Cu2+sites in the six-membered ring and a not previously observed site in the eight-membered ring. The sum of the refined Cu occupancies for these two sites matches the chemical analysis and thus all the Cu is accounted for. It is furthermore shown that approximately 80% of the Cu2+is located in the new 8-ring site for an industrially relevant CHA zeolite with Si/Al = 15.5 and Cu/Al = 0.45. Density functional theory calculations are used to corroborate the positions and identity of the two Cu sites, leading to the most complete structural description of dehydrated silicoaluminate CHA loaded with catalytically active Cu2+cations.

Maximum Entropy Method Charge Density Distributions of Novel Thermoelectric Clathrates

1999 ◽  
Vol 590 ◽  
Author(s):  
B. Iversen ◽  
A. Bentien ◽  
A. Palmqvist ◽  
D. Bryan ◽  
S. Latturner ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Charge Density ◽  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Mixed Valence ◽  
Entropy Method ◽  
X Ray Diffraction ◽  
X Ray ◽  
Density Distributions ◽  
Positively Charged

ABSTRACTRecently materials with promising thermoelectric properties were discovered among the clathrates. Transport data has indicated that these materials have some of the characteristics of a good thermoelectric, namely a low thermal conductivity and a high electrical conductivity. Based on synchrotron powder and conventional single crystal x-ray diffraction data we have determined the charge density distribution in Sr8Ga16Ge3O using the Maximum Entropy Method. The MEM density shows clear evidence of guest atom rattling, and this contributes to the reduction of the thermal conductivity. Analysis of the charge distribution reveals that Sr8Ga16Ge30 contains mixed valence alkaline earth guest atoms. The Sr atoms in the small cavities are, as expected, doubly positively charged, whereas the Sr atoms in the large cavities appear negatively charged. The MEM density furthermore suggests that the Ga and Ge atoms may not be randomly disordered on the framework sites as found in the conventional leastsquares refinements.

Small-Angle X-ray Scattering Measurements and Image Reconstruction by the Maximum Entropy Method

1997 ◽  
Vol 69 (4) ◽  
pp. 794-800 ◽  
Author(s):  
Yuji Sasanuma ◽  
Robert V. Law ◽  
Yuji Kobayashi ◽  
Katsunari Sasaki
Keyword(s):  
Image Reconstruction ◽  
Maximum Entropy ◽  
Small Angle ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
X Ray ◽  
X Ray Scattering ◽  
Scattering Measurements ◽  
Ray Scattering

Nuclear and charge density distributions in ferroelectric PbTiO3: maximum entropy method analysis of neutron and X-ray diffraction data

2013 ◽  
Vol 28 (4) ◽  
pp. 276-280 ◽  
Author(s):  
Jinlong Zhu ◽  
Wei Han ◽  
Jianzhong Zhang ◽  
Hongwu Xu ◽  
Sven C. Vogel ◽  
...  
Keyword(s):  
Charge Density ◽  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Rietveld Analysis ◽  
Entropy Method ◽  
X Ray Diffraction ◽  
X Ray ◽  
Density Distributions ◽  
Increasing Temperature ◽  
Ionic States

We conducted in-situ high-temperature neutron and X-ray diffraction studies on tetragonal PbTiO3. Using a combination of Rietveld analysis and Maximum Entropy Method, the nuclear and charge density distributions were determined as a function of temperature up to 460 °C. The ionic states obtained from charge density distributions reveal that the covalency of Pb–O2 bonds gradually weakens with increasing temperature. The spontaneous polarizations calculated from the contributions of ionic state, ionic displacement, and nuclear polarization, are in good agreement with the experimental measurements. This method provides an effective approach to determine spontaneous polarizations in multiferroics with high-current leakage and low resistance.

Modulation functions of incommensurately modulated Cr2P2O7 studied by the maximum entropy method (MEM)

2010 ◽  
Vol 66 (2) ◽  
pp. 130-140 ◽  
Author(s):  
Liang Li ◽  
Andreas Schönleber ◽  
Sander van Smaalen
Keyword(s):  
Maximum Entropy ◽  
Diffraction Data ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Chromium Atom ◽  
Structure Model ◽  
X Ray Diffraction ◽  
Acta Cryst ◽  
X Ray

The maximum entropy method (MEM) has been used to determine electron density in superspace of incommensurately modulated chromium pyrophosphate from X-ray diffraction data measured by Palatinus et al. [(2006), Acta Cryst. B62, 556–566]. Chromium pyrophosphate, Cr2P2O7, contains ordered regions (83% of the volume) and regions with disorder. Analysis of the MEM density has allowed the determination of the displacive modulation functions within ordered regions. The disordered regions can be described as the alternate occupation of two conformations of the pyrophosphate group and two positions of the chromium atom, with occupational probabilities that depend continuously on the phase of modulation t. A structure model based on the interpretation of the MEM density provides a fit to the diffraction data of the same quality as the model given by Palatinus et al. (2006). The failure to find a model that better fits the data is attributed to the intrinsic inaccuracy of ∼ 0.01 Å for positions derived from the MEM and to the difficulties in constructing an appropriate model for the anharmonic ADPs and their modulation functions from electron densities.

Sign in / Sign up

Export Citation Format

Share Document