A Bayesian Approach to Crystal Structure Refinement

Dylan Quintana

Carnegie Mellon University / NIST Center for

Neutron Research

Preview

Crystal Structure

Refinement

A Bayesian Approach

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Crystal Structure

Unit Cell

Symmetry

Atoms

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Monoclinic Triclinic

Cubic Tetragonal Orthorhombic

Hexagonal Rhombohedral

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Adding atoms

Diffraction

d

λ

Bragg’s Law: 2𝑑𝑑 sin𝜃𝜃 = 𝜆𝜆

θ

Peaks from Bragg Angles 2θ Peaks from Bragg Angles 2θ

2θ

Inte

nsity

Peaks from Bragg Angles 2θ Superposition of Peaks

2θ

Inte

nsity

Peaks from Bragg Angles 2θ Diffraction Pattern

2θ

Inte

nsity

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2θ

Inte

nsity

2θ

Inte

nsity

2θ

Inte

nsity

Rietveld Refinement: minimize the value χ2

𝜒𝜒2 = ∑(𝐼𝐼𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐−𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜)2

𝐼𝐼𝑜𝑜𝑜𝑜𝑜𝑜

Diffraction Pattern

Cell Parameters a b c α β γ

Atomic Parameters

x y z

Peak Parameters u v w scale

(Juan Rodríguez-Carvajal)

Fitting algorithm: Gauss-Newton See also: GSAS, TOPAS, etc.

PresenterPresentation NotesGauss-Newton: approximates function with Taylor series

Gauss-Newton Algorithm

Local Minimum

Absolute Minimum

χ2

Variation in Parameters

PresenterPresentation NotesStuck in local minimaNeed good initial parameters, refinement orderNo guarantee of convergence

Refinement for Lazy People

Dylan Quintana

Carnegie Mellon University / NIST Center for

Neutron Research

(Paul Kienzle)

Fitting algorithm: DREAM

Bumps Bayesian Uncertainty Modeling of Parametric Systems

PresenterPresentation NotesBumps is a generic fitting package – not applied specifically to diffraction

DREAM Algorithm

Markov Chain

Monte Carlo

Differential Evolution

Bayesian Analysis

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PresenterPresentation NotesGlobal fitting!Markov Chain Monte Carlo – error analysis on parameters

BLAND

Bumps Library for Analysis of Neutron Diffraction

FullProf Library

Bumps BLAND

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BLAND Goals

Global fitting Eliminate need for intuition User-friendly interface Take over the world

Refinement at the push of a button!

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Fitting Example: Al2O3

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Pre-fit

χ2 = 363.927

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]

Post-fit: Diffraction Pattern

2θ

Inte

nsity

Observed

Calculated

χ2 = 9.486

Post-fit: Correlation Plot

Al B

Al z O B

O x

a

c

scale

u

v

w

BLAND Goals

Global fitting Eliminate need for intuition User-friendly interface Extensive testing Feature completeness Take over the world

Summary

Crystal structures and diffraction

The refinement process

The old ways (FullProf)

The new ways (BLAND)

Special Thanks

William Ratcliff

Paul Kienzle

Alex Zhang

Julie Borchers

SURF directors

?

Appendix

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

COMPATIBLE INCOMPATIBLE

(Awkward)

Cell constants: a, b, c, α, β, γ

Refinement Parameters

Refinement Parameters

Atomic positions: x, y, z

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