Lesson 4: Rietveld Refinement

Lesson 4Rietveld Refinement

Nicola DöbelinRMS Foundation, Bettlach, Switzerland

October 16 – 17, 2013, Uppsala, Sweden

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Repetition: Powder XRD Patterns

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Diffraction Angle [°2θ]

Generation of X-rays

Diffraction at crystal structures

Powder samples (Debye rings)

Diffractometers (Types, optical elements)

Sample preparation (errors to avoid)

Phase identification

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Rietveld Refinement

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For more than just identification:

Rietveld refinement

Prof. Hugo Rietveld

Extracts much more information from powder XRD data:

- Unit cell dimensions

- Phase quantities

- Crystallite sizes / shapes

- Atomic coordinates / Bond lengths

- Micro-strain in crystal lattice

- Texture effects

- Substitutions / Vacancies

No phase identification!

Identify your phases first(unknown phase � no Rietveld refinement)

Needs excellent data quality!

No structure solution(just structure refinement)

Rietveld Refinement

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Known structuremodel

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Calculate theoreticaldiffraction pattern

Compare withmeasured pattern

Optimize structure model, repeat calculation

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Minimize differences between calculated and observedpattern by least-squares method

Rietveld Refinement

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Diffraction Angle [°2theta]

Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)

Beginning of the refinement:

- Phase was identified correctly (peaksat the right position)

- But differences exist:

- Peak width

- Peak positions slightly shifted

- Intensities

Rietveld Refinement

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Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)

After the refinement:

- Straight difference curve (only noise)

Modelling the Peak Profile

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Diffraction Angle (°2θ)

CuKβ

CuKα1 & CuKα2 duplet

Remaining Bremsstrahlung

Absorption Edge

CuKα Satellites

(= CuKα3)

All these signals are generatedby one d-spacing.

Mathematical model to describethe profile is needed.

Modelling the Peak Profile

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Traditional («Rietveld») Approach:

Pseudo Voigt curves for Kα1, Kα2 and Kβ

VP(x) = n * L(x) + (1-n) * G(x)

Gaussian curveLorentzian curve

Lorentzian (ω = 1.0) Gaussian (ω = 1.0) Pseudo-Voigt (n = 0.5)

L(x) = 1

1+( )2x-x0ω

G(x) = exp[-ln(2)·( )2]x-x0ω

Pseudo-Voigt Curves

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n = 0.5

n = 0.25

n = 0.75

n = 0.0

n = 1.0

ω = 1.0

ω = 0.5

ω = 0.25

n = 0.5

Pseudo-Voigt Curves

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Kα1

Kα2

Kβ

Fitting n, ω to peaksof a reference material

ω2 = f(θ) = U · tan2(θ) + V · tan(θ) + W

Actually fitting n, U, V, W

Pseudo-Voigt: Problems

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Asymmetry

Peaks at low 2θ anglesare asymmetric.

Pseudo-Voigt curvesare symmetric.

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Alternative Pseudo-Voigt Functions

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Alternatives to PV function:

- Pearson VII

- Thompson-Cox-Hastings PV

- Split PV

- PV with axial divergence(Finger-Cox-Jephcoat PV)

FWHM of all these functionsmust be fitted to peaksof a reference material.

Common reference materialsdo not have peaks < 20° 2θ

Fundamental Parameters Approach FPA

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Calculate the peak profile from the device configuration

Take into account the contributions of:

- Source emission profile (X-ray wavelength distribution from Tube)

- Every optical element in the beam path (position, size, etc.)

- Sample contributions (peak broadening due to crystallite size & strain)

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w.b

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Tube Device Configuration Sample

Fundamental Parameters Approach

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www.bruker.com

Fundamental Parameters Approach

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2θ=10° 2θ=30° 2θ=80°

Calculated Peak Profiles

Fundamental Parameters Approach

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FPA needs:

- Very detailed and complete description ofthe instrument configuration

- Very well aligned instrument

Fundamental Parameters Approach

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http://www.bgmn.de

Fundamental Parameters Approach

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°2θ

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If done properly:

Very good description of the peak profile

Fundamental Parameters Approach

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- Some fundamental parameters are not documented

- Complete configuration can be hard to obtain

Good news:

We created and verified configurations for all instruments involved in the course!

Summary: Rietveld Basics

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- Calculate XRD pattern from model structure

- Minimize differences between calculated and measured pattern

- Accurate mathematical description of peak profile required:

- Classical Rietveld approach: Fit a peak shape function (PVor similar) to reference pattern

- Fundamental Parameters Approach: Calculate peakprofile from device configuration

Rietveld Software Packages

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Academic Software:

- Fullprof

- GSAS

- BGMN

- Maud

- Brass

- … many more1)

Commercial Software:

- HighScore+ (PANalytical)

- Topas (Bruker)

- Autoquan (GE)

- PDXL (Rigaku)

- Jade (MDI)

- WinXPOW (Stoe)

1) http://www.ccp14.ac.uk/solution/rietveld_software/index.html

FPACommercial UI

for BGMN

BGMN

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BGMN:

- Fundamental Parameters Approach

- Open source (GPL v2 since mid 2013, not officially announced yet)

- Device independent

- Very robust automatic refinement strategy

- Good usability (at least better than most other academic programs)

- Slightly less steep learning curve

- Powerful scripting language

- Multi-Platform

- Multi-threaded

Visit: http://www.bgmn.de for tutorials and documentation

Graphical User Interfaces for BGMN

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BGMNwin (shipped with BGMN software package)

Graphical User Interfaces for BGMN

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Profex (developed by Nicola Döbelin)

- Open source

- Multi-Platform

- Device database

- Structure database

- Batch processing

- File conversion

- …

Example Refinement

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Example Refinement

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Example Refinement

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Start Refinement

Example Refinement

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Example Refinement

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Good fit

Summary

Summary: BGMN and Profex

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- BGMN is the Rietveld kernel

- Fundamental Parameters Approach

- Profex is an editor / graphical user interface to BGMN

- Best-case scenario (example):

- Load scan file

- Specify instrument configuration from internal database

- Load structure files from internal database

- Start refinement

- … done

Refinement Strategies

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Diffraction Angle [°2theta]

Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)

Relation

Mismatch – Structural Feature

Refinement Strategies

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Wrong peak positions:

- Unit cell dimensions

- Sample height displacement

- Zero-shift (instrumentmisalignment)

Refinement Strategies

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Wrong absolute intensities:

- Weight fraction (scaling)

Refinement Strategies

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Wrong relative intensities:

- Preferred orientation

- Graininess

- Atomic species

- Atomic coordinates

- Site occupancies

- Thermal displacementparameters

Refinement Strategies

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Wrong peak width:

- Crystallite size

- Micro-strain in crystal structure

Summary: Refinement Strategy

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Effect in diffraction pattern Origin in crystal structure model

Wrong peak positions Unit cell dimensions

Sample height displacement

Zero-shift

Wrong absolute intensities Weight fraction (scaling)

Wrong relative intensities Preferred orientation

Atomic species / Substitutions

Atomic coordinates

Site occupancies

Thermal displacement parameters

Wrong peak width Crystallite size

Lattice strain