Accounting for unknown systematic errors in Rietveld refinements: a Bayesian statistics approach

2015 ◽  
Vol 48 (4) ◽  
pp. 1201-1211 ◽  
Author(s):  
Anton Gagin ◽  
Igor Levin
Keyword(s):  
Bayesian Statistics ◽  
Diffraction Data ◽  
Structural Parameters ◽  
Simulated Data ◽  
Systematic Errors ◽  
Data Sets ◽  
X Ray Diffraction ◽  
Rietveld Refinements ◽  
X Ray ◽  
Minimization Algorithms

A method has been developed to address the effects of systematic errors in Rietveld refinements using powder diffraction data. Relevant errors were categorized into multiplicative, additive and peak-shape types. Corrections for these errors were incorporated into structural refinements using a Bayesian statistics approach, with the corrections themselves treated as nuisance parameters and marginalized out of the analysis. Structural parameters refined using the proposed method represent probability-weighted averages over all possible error corrections. The developed formalism has been adapted to least-squares minimization algorithms and implemented as an extension to the Rietveld software packageGSAS-II. The technique was first tested using neutron and X-ray diffraction data simulated for PbSO4and then applied to the equivalent experimental data sets for the same compound. The results obtained using the simulated data confirmed that the proposed method yields significantly more accurate estimates of structural parameters and their uncertainties than standard refinements. The benefits were particularly significant for joint refinements using neutron and X-ray diffraction data because accounting for systematic errors enabled more adequate weighting of the individual data sets.

Applications of Bayesian corrections for systematic errors in Rietveld refinements

2016 ◽  
Vol 49 (3) ◽  
pp. 814-822 ◽  
Author(s):  
Anton Gagin ◽  
Igor Levin
Keyword(s):  
Monte Carlo ◽  
Structural Parameters ◽  
Systematic Errors ◽  
Monte Carlo Algorithm ◽  
Data Sets ◽  
Correction Procedure ◽  
Bayesian Procedure ◽  
Rietveld Refinements ◽  
X Ray ◽  
Least Squares Minimization

Recently, a Bayesian statistics approach for correction of systematic errors in Rietveld refinements has been developed and implemented as a patch toGSAS-II. This paper demonstrates the benefits of the proposed method in a series of structural refinements from diffraction data collected for one sample using four different powder diffractometers,i.e.synchrotron and laboratory X-ray and two time-of-flight neutron instruments. Differences between the parameters estimated while fitting these four data sets provided magnitudes of the systematic errors while also highlighting the efficacy of the Bayesian procedure for their correction. Structural parameters estimated from the standard Rietveld refinements using the four data sets differed significantly. In all cases, the agreement improved markedly after applying the Bayesian error-correction procedure. The Bayesian refinements were paired with a Markov chain Monte Carlo algorithm, which was implemented as part of the same patch toGSAS-II, to confirm that uncertainties in the refined parameters obtained using the much faster least-squares minimization were realistic.

scholarly journals AUSPEX: a graphical tool for X-ray diffraction data analysis

2017 ◽  
Vol 73 (9) ◽  
pp. 729-737 ◽  
Author(s):  
Andrea Thorn ◽  
James Parkhurst ◽  
Paul Emsley ◽  
Robert A. Nicholls ◽  
Melanie Vollmar ◽  
...  
Keyword(s):  
Data Analysis ◽  
Diffraction Data ◽  
Software Tool ◽  
Data Conversion ◽  
Data Sets ◽  
X Ray Diffraction ◽  
X Ray ◽  
Pixel Detectors ◽  
Graphical Tool ◽  
Data Acquisition And Processing

In this paper,AUSPEX, a new software tool for experimental X-ray data analysis, is presented. Exploring the behaviour of diffraction intensities and the associated estimated uncertainties facilitates the discovery of underlying problems and can help users to improve their data acquisition and processing in order to obtain better structural models. The program enables users to inspect the distribution of observed intensities (or amplitudes) against resolution as well as the associated estimated uncertainties (sigmas). It is demonstrated howAUSPEXcan be used to visually and automatically detect ice-ring artefacts in integrated X-ray diffraction data. Such artefacts can hamper structure determination, but may be difficult to identify from the raw diffraction images produced by modern pixel detectors. The analysis suggests that a significant portion of the data sets deposited in the PDB contain ice-ring artefacts. Furthermore, it is demonstrated how other problems in experimental X-ray data caused, for example, by scaling and data-conversion procedures can be detected byAUSPEX.

scholarly journals Experience with exchange and archiving of raw data: comparison of data from two diffractometers and four software packages on a series of lysozyme crystals

2012 ◽  
Vol 46 (1) ◽  
pp. 108-119 ◽  
Author(s):  
Simon W. M. Tanley ◽  
Antoine M. M. Schreurs ◽  
John R. Helliwell ◽  
Loes M. J. Kroon-Batenburg
Keyword(s):  
Diffraction Data ◽  
Data Sets ◽  
X Ray Diffraction ◽  
Noticeable Effect ◽  
Raw Data ◽  
X Ray ◽  
Data Comparison ◽  
Data Processing Software ◽  
Software Packages ◽  
Lysozyme Crystals

The International Union of Crystallography has for many years been advocating archiving of raw data to accompany structural papers. Recently, it initiated the formation of the Diffraction Data Deposition Working Group with the aim of developing standards for the representation of these data. A means of studying this issue is to submit exemplar publications with associated raw data and metadata. A recent study on the effects of dimethyl sulfoxide on the binding of cisplatin and carboplatin to histidine in 11 different lysozyme crystals from two diffractometers led to an investigation of the possible effects of the equipment and X-ray diffraction data processing software on the calculated occupancies andBfactors of the bound Pt compounds. 35.3 Gb of data were transferred from Manchester to Utrecht to be processed withEVAL. A systematic comparison shows that the largest differences in the occupancies andBfactors of the bound Pt compounds are due to the software, but the equipment also has a noticeable effect. A detailed description of and discussion on the availability of metadata is given. By making these raw diffraction data sets availableviaa local depository, it is possible for the diffraction community to make their own evaluation as they may wish.

XRD Instrument Sensitivity Results from a Round Robin Study

1991 ◽  
Vol 35 (A) ◽  
pp. 333-340 ◽  
Author(s):  
W.N. Schreiner ◽  
R. Jenkins ◽  
P.F. Dismore
Keyword(s):  
Diffraction Data ◽  
Round Robin ◽  
Data Sets ◽  
X Ray Diffraction ◽  
Diffraction Intensity ◽  
Generic Type ◽  
X Ray ◽  
The Past ◽  
Instrument Sensitivity

During the course of the past ten years the International Centre for Diffraction Data has sponsored a number of “Round Robin” tests to evaluate the quality of experimental X-ray diffraction data [1-5]. The latest of this series, called the Instrument Parameter Round Robin, was designed to evaluate, among other things, relative angularly-dependent sensitivity differences between diffractometers. Previous experiments have indicated that even perfectly aligned diffractometers of the same generic type, do not necessarily give the same set of relative intensities. One objective of the round robin was to quantify the magnitude of the experimental differences between data sets, and to demonstrate a means for external calibration of diffractometers, so that digitized diffraction intensity data obtained from different instruments could be directly compared.

Preparation and Crystal Structures of the Isotypic Compounds CdXO4 · 2 HgO (X = S, Se)

2004 ◽  
Vol 59 (3) ◽  
pp. 281-285 ◽  
Author(s):  
Matthias Weil
Keyword(s):  
Single Crystals ◽  
Crystal Structures ◽  
Diffraction Data ◽  
Building Blocks ◽  
Data Sets ◽  
X Ray Diffraction ◽  
Hydrothermal Conditions ◽  
Space Group ◽  
X Ray ◽  
Structure Factors

Colourless single crystals of the compounds CdXO4 · 2 HgO (X = S, Se) were obtained under hydrothermal conditions (250 °C, 5 d), starting from stoichiometric amounts of HgO, CdSO4 ·7H2O and CdSeO4 ·2H2O, respectively. The crystal structures were determined from X-ray diffraction data sets. The CdXO4 · 2HgO compounds crystallise isotypically with two formula units in space group P1̅ (# 2) [CdSO4 · 2HgO (CdSeO4 · 2HgO): a = 6.793(2) (6.9097(5)) Å , b = 7.205(2) (7.1786(6)) Å , c=7.359(2) (7.4556(6)) Å ,α =73.224(6) (74.586(2))°, β =66.505(6) (68.229(1))°, γ =63.054(5) (63.886(1))°, 1670 (1786) structure factors, 92 parameters, R[F2 > 2σ(F2)] = 0.0379 (0.0244)] and are made up from zig-zag [O-Hg-O]∞ chains with very short bonds of d̅(Hg-O) 2.025 Å , distorted [CdO6] octahedra (d̅(Cd-O)= 2.297 Å ), and XO4 tetrahedra (d̅(S-O)= 1.458 Å , d̅(Se-O)= 1.633 Å ) as the main building blocks. The CdXO4 ·2HgO compounds reveal no structural relationship with the corresponding HgXO4 ·2HgO phases

Crystal structure of low-cristobalite-type Al0.5Ga0.5PO4: A combined Rietveld refinement of neutron and X-ray diffraction data

2005 ◽  
Vol 20 (3) ◽  
pp. 207-211 ◽  
Author(s):  
S. N. Achary ◽  
A. K. Tyagi ◽  
S. K. Kulshreshtha ◽  
O. D. Jayakumar ◽  
P. S. R. Krishna ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Diffraction Data ◽  
Three Dimensional ◽  
Crystal Structure Analysis ◽  
Tetrahedral Coordination ◽  
X Ray Diffraction ◽  
Rietveld Refinements ◽  
X Ray ◽  
Cell Parameters ◽  
Bond Lengths

The low-cristobalite-type modification of Al0.5Ga0.5PO4 is prepared by annealing the amorphous precipitate of stoichiometric phosphate at 1300 °C. The phase purity of the sample is ascertained by powder X-ray diffraction. The crystal structure is refined by Rietveld refinements of the neutron and X-ray diffraction data of the polycrystalline powder. This compound crystallizes in an orthorhombic lattice with unit cell parameters, a=7.0295(8), b=7.0132(8), and c=6.9187(4) Å, V=341.08(6) Å3, Z=4 (Space group C 2221, No. 20). The crystal structure analysis reveals the random distribution of the Al3+ and Ga3+ having tetrahedral coordination with typical M–O (M=Al3+:Ga3+) bond lengths as 1.74 Å. Similarly, the P5+ have tetrahedral coordination with typical P–O bond lengths 1.52–1.54 Å. The Mo4 and PO4 tetraheda are linked by common corners forming a three-dimensional framework lattice. The details of the crystal structure are presented in this paper.

On the quality of the continuous rotation electron diffraction data for accurate atomic structure determination of inorganic compounds

2018 ◽  
Vol 51 (4) ◽  
pp. 1094-1101 ◽  
Author(s):  
Yunchen Wang ◽  
Taimin Yang ◽  
Hongyi Xu ◽  
Xiaodong Zou ◽  
Wei Wan
Keyword(s):  
Electron Diffraction ◽  
Single Crystal ◽  
Diffraction Data ◽  
Structural Models ◽  
Electron Diffraction Data ◽  
Data Sets ◽  
X Ray Diffraction ◽  
X Ray ◽  
Continuous Rotation

The continuous rotation electron diffraction (cRED) method has the capability of providing fast three-dimensional electron diffraction data collection on existing and future transmission electron microscopes; unknown structures could be potentially solved and refined using cRED data collected from nano- and submicrometre-sized crystals. However, structure refinements of cRED data using SHELXL often lead to relatively high R1 values when compared with those refined against single-crystal X-ray diffraction data. It is therefore necessary to analyse the quality of the structural models refined against cRED data. In this work, multiple cRED data sets collected from different crystals of an oxofluoride (FeSeO3F) and a zeolite (ZSM-5) with known structures are used to assess the data consistency and quality and, more importantly, the accuracy of the structural models refined against these data sets. An evaluation of the precision and consistency of the cRED data by examination of the statistics obtained from the data processing software DIALS is presented. It is shown that, despite the high R1 values caused by dynamical scattering and other factors, the refined atomic positions obtained from the cRED data collected for different crystals are consistent with those of the reference models refined against single-crystal X-ray diffraction data. The results serve as a reference for the quality of the cRED data and the achievable accuracy of the structural parameters.

The direct determination of X-ray diffraction data from specific depths

2005 ◽  
Vol 20 (3) ◽  
pp. 233-240
Author(s):  
A. Broadhurst ◽  
K. D. Rogers ◽  
D. W. Lane ◽  
T. W. Lowe
Keyword(s):  
Diffraction Data ◽  
Direct Method ◽  
Direct Determination ◽  
Simulated Data ◽  
X Ray Diffraction ◽  
Linear Absorption ◽  
X Ray ◽  
Measured Angle ◽  
Data Collections

A direct method for determining powder diffraction data from a range of depths is described, where the linear absorption coefficient may vary with depth. A series of traditional data collections with varying angles of incidence are required, and the X-ray diffraction data arising from specific depths will be calculated by the transformation of these measured, angle-dependent spectra. These may then be analysed using any conventional method in order to gain information about characteristics of the sample in question at specific depths. Regularisation techniques have been used to solve the governing Fredholm integral equation to determine the depth-dependent diffractograms. The method has been validated by the use of simulated data having known model profiles, and has also been applied to experimental data from polycrystalline thin film samples.

On reproducibility of Rietveld analysis of reference Portland cement clinkers

2003 ◽  
Vol 18 (1) ◽  
pp. 16-22 ◽  
Author(s):  
O. Pritula ◽  
Lˇ. Smrcˇok ◽  
B. Baumgartner
Keyword(s):  
Reference Values ◽  
Structural Data ◽  
Rietveld Analysis ◽  
Data Sets ◽  
X Ray Diffraction ◽  
Rietveld Refinements ◽  
X Ray ◽  
Numerical Instabilities ◽  
Relative Errors ◽  
Cement Clinkers

Weight fractions of four dominant phases (C3S, C2S, C4AF and C3A) present in the NIST Reference Portland clinkers 8486, 8487 and 8488 were estimated by a series of Rietveld refinements. Calculated powder patterns were derived from the structural data for monoclinic C3S and C2S, orthorhombic C4AF and cubic C3A. X-ray diffraction data were collected in two laboratories with two diffractometers, a reflection and a transmission one. There were no significant differences between the results of the refinements based on the data sets collected on the machines with different experimental arrangements. Estimated phase compositions were compared to the reference values found by optical microscopy (MPC). Median agreement between refined and reference values within ±5% (absolute) was found only for 8488 clinker; for 8486 and C3A-rich 8487 it was within ±10% (absolute). In the majority of the refinements numerical instabilities were detected, leading to large correlations between FWHM and temperature parameters of some phases. The results obtained for C4AF were probably influenced by the presence of possible solid solutions with the structures close to that of C4AF. Weight fractions of low abundant C3A were estimated with the largest relative errors reaching in several cases ∼100%.

Sign in / Sign up

Export Citation Format

Share Document