a bayesian approach to crystal structure refinement...gauss-newton algorithm local minimum absolute...
TRANSCRIPT
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A Bayesian Approach to Crystal Structure Refinement
Dylan Quintana
Carnegie Mellon University / NIST Center for
Neutron Research
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Preview
Crystal Structure
Refinement
A Bayesian Approach
Image: topdefinitions.com
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Crystal Structure
Unit Cell
Symmetry
Atoms
Image: saimport.com
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Monoclinic Triclinic
Cubic Tetragonal Orthorhombic
Hexagonal Rhombohedral
Images: allaboutgemstones.com
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Image: organicsoul.com
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Adding atoms
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Diffraction
d
λ
Bragg’s Law: 2𝑑𝑑 sin𝜃𝜃 = 𝜆𝜆
θ
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Peaks from Bragg Angles 2θ Peaks from Bragg Angles 2θ
2θ
Inte
nsity
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Peaks from Bragg Angles 2θ Superposition of Peaks
2θ
Inte
nsity
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Peaks from Bragg Angles 2θ Diffraction Pattern
2θ
Inte
nsity
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Images: ncnr.nist.gov, theochem.unito.it
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2θ
Inte
nsity
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2θ
Inte
nsity
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2θ
Inte
nsity
Rietveld Refinement: minimize the value χ2
𝜒𝜒2 = ∑(𝐼𝐼𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐−𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜)2
𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜
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Diffraction Pattern
Cell Parameters a b c α β γ
Atomic Parameters
x y z
Peak Parameters u v w scale
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(Juan Rodríguez-Carvajal)
Fitting algorithm: Gauss-Newton See also: GSAS, TOPAS, etc.
PresenterPresentation NotesGauss-Newton: approximates function with Taylor series
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Gauss-Newton Algorithm
Local Minimum
Absolute Minimum
χ2
Variation in Parameters
PresenterPresentation NotesStuck in local minimaNeed good initial parameters, refinement orderNo guarantee of convergence
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Refinement for Lazy People
Dylan Quintana
Carnegie Mellon University / NIST Center for
Neutron Research
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(Paul Kienzle)
Fitting algorithm: DREAM
Bumps Bayesian Uncertainty Modeling of Parametric Systems
PresenterPresentation NotesBumps is a generic fitting package – not applied specifically to diffraction
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DREAM Algorithm
Markov Chain
Monte Carlo
Differential Evolution
Bayesian Analysis
Image: lumick.buroncp.com
PresenterPresentation NotesGlobal fitting!Markov Chain Monte Carlo – error analysis on parameters
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BLAND
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Bumps Library for Analysis of Neutron Diffraction
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FullProf Library
Bumps BLAND
Image: successfullivinginstitute.com
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BLAND Goals
Global fitting Eliminate need for intuition User-friendly interface Take over the world
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Refinement at the push of a button!
Image: graphicsfuel.com
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Fitting Example: Al2O3
Image: alumina-ceramic.net
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Pre-fit
χ2 = 363.927
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Post-fit: Parameters
Parameter Value 95% Interval Al1 B 0 [0, 0.063] Al1 z 0.352282 [0.35188, 0.35279] O1 B 0.0001 [0, 0.064] O1 x 0.69347 [0.6929, 0.6941] a 4.760787 [4.76068, 4.76100] c 13.00072 [13.0000, 13.0015] scale 1.4711 [1.457, 1.519] u 0.1721 [0.161, 0.185] v -0.1188 [-0.139, -0.092] w 0.0772 [0.069, 0.086]
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Post-fit: Diffraction Pattern
2θ
Inte
nsity
Observed
Calculated
χ2 = 9.486
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Post-fit: Correlation Plot
Al B
Al z O B
O x
a
c
scale
u
v
w
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BLAND Goals
Global fitting Eliminate need for intuition User-friendly interface Extensive testing Feature completeness Take over the world
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Summary
Crystal structures and diffraction
The refinement process
The old ways (FullProf)
The new ways (BLAND)
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Special Thanks
William Ratcliff
Paul Kienzle
Alex Zhang
Julie Borchers
SURF directors
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?
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Appendix
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Why use neutrons?
-4
-2
0
2
4
6
8
10
12 Neutron X-ray
Cr Mn
Fe
Co Ni Cu Zn
Scaled scattering length densities:
Neutrons can differentiate between similar elements, and even isotopes of the same element
Sc Ti
V
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COMPATIBLE INCOMPATIBLE
(Awkward)
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Cell constants: a, b, c, α, β, γ
Refinement Parameters
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Refinement Parameters
Atomic positions: x, y, z
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Refinement Parameters
Peak width
and scale
A Bayesian Approach to Crystal Structure RefinementPreviewSlide Number 3Slide Number 4Slide Number 5Adding atomsDiffractionPeaks from Bragg Angles 2θPeaks from Bragg Angles 2θPeaks from Bragg Angles 2θSlide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Gauss-Newton AlgorithmSlide Number 19Refinement for Lazy PeopleSlide Number 21DREAM AlgorithmSlide Number 23Slide Number 24Slide Number 25BLAND GoalsRefinement at the push of a button!Fitting Example: Al2O3Pre-fitSlide Number 30Post-fit: ParametersPost-fit: Diffraction PatternPost-fit: Correlation PlotBLAND GoalsSummarySpecial Thanks?AppendixWhy use neutrons?Slide Number 40Slide Number 41Refinement ParametersRefinement ParametersRefinement Parameters