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TRANSCRIPT
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Lesson 4Rietveld Refinement
Nicola DöbelinRMS Foundation, Bettlach, Switzerland
February 11 – 14, 2013, Riga, Latvia
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2
Repetition: Powder XRD Patterns
10 20 30 40 50 60
0
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Inte
nsity
[cou
nts]
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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3
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Rietveld Refinement
4
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)
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Rietveld Refinement
5
Known structuremodel
10 20 30 40 50 600
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Inte
nsity
[cou
nts]
Diffraction Angle [°2θ]
Calculate theoreticaldiffraction pattern
Compare withmeasured pattern
Optimize structure model, repeat calculation
10 20 30 40 50 600
1000
2000
3000
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Inte
nsity
[cou
nts]
Diffraction Angle [°2θ]
Minimize differences between calculated and observedpattern by least-squares method
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Rietveld Refinement
6
10 20 30 40 50 60
-1000
0
1000
2000
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5000
Inte
nsity
[cts
]
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
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Rietveld Refinement
7
10 20 30 40 50 60
0
1000
2000
3000
4000
5000
Inte
nsity
[cts
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Diffraction Angle [°2theta]
Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)
After the refinement:
- Straight difference curve (only noise)
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Modelling the Peak Profile
8
27 28 29 30 31 320
2000
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12000
14000
16000
18000
20000
Inte
nsity
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.
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Modelling the Peak Profile
9
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
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Pseudo-Voigt Curves
10
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
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Pseudo-Voigt Curves
11
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
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Pseudo-Voigt: Problems
12
Asymmetry
Peaks at low 2θ anglesare asymmetric.
Pseudo-Voigt curvesare symmetric.
29.8 30.0 30.2 30.4 30.6 30.8 31.00
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Inte
nsity
[cts
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Diffraction Angle [°2θ]
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0 20 40 60 80 100 120 140
0
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Inte
nsity
[cts
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Diffraction Angle [°2θ]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
FW
HM
[°2θ
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Alternative Pseudo-Voigt Functions
13
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
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Fundamental Parameters Approach FPA
14
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)
ww
w.b
ruke
r.com
Tube Device Configuration Sample
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Fundamental Parameters Approach
15
www.bruker.com
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Fundamental Parameters Approach
16
2θ=10° 2θ=30° 2θ=80°
Calculated Peak Profiles
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Fundamental Parameters Approach
17
FPA needs:
- Very detailed and complete description ofthe instrument configuration
- Very well aligned instrument
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Fundamental Parameters Approach
18
21.0 21.2 21.4 21.6 21.8
Inte
nsity
[a.u
.]
°2θ
63.0 63.2 63.4 63.6
°2θ
120 121 122
°2θ
If done properly:
Very good description of the peak profile
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Fundamental Parameters Approach
19
http://www.bgmn.de
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Fundamental Parameters Approach
20
- 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!
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Summary: Rietveld Basics
21
- Calculate XRD pattern from model structure
- Minimize differences between calculated and measured pattern
- Accurate mathematical description of peak profile required:
- Classical Rietveld approach: Fit PV functionto reference pattern
- Fundamental Parameters Approach: Calculate peakprofile from device configuration
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Rietveld Software Packages
22
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
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BGMN
23
BGMN:
- Fundamental Parameters Approach
- Free for academic use
- Device independent
- Very robust automatic refinement strategy
- Good usability
- Slightly less steep learning curve
- Powerful scripting language
- Multi-Platform
- Multi-threaded
Visit: http://www.bgmn.de for tutorials and documentation
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Graphical User Interfaces for BGMN
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BGMNwin (shipped with BGMN software package)
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Graphical User Interfaces for BGMN
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Profex (developed by Nicola Döbelin)
- Open source
- Multi-Platform
- Device database
- Structure database
- Batch processing
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Example Refinement
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1.
2.
3.
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Example Refinement
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1.
2.
3.
4.
5.
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Example Refinement
28
Start Refinement
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Example Refinement
29
1.
2.
3.
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Example Refinement
30
Good fit
Summary
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Summary: BGMN and Profex
31
- 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
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Refinement Strategies
32
10 20 30 40 50 60
-1000
0
1000
2000
3000
4000
5000
Inte
nsity
[cts
]
Diffraction Angle [°2theta]
Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)
Relation
Mismatch – Structural Feature
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Refinement Strategies
33
Wrong peak positions:
- Unit cell dimensions
- Sample height displacement
- Zero-shift (instrumentmisalignment)
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Refinement Strategies
34
Wrong absolute intensities:
- Weight fraction (scaling)
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Refinement Strategies
35
Wrong relative intensities:
- Preferred orientation
- Graininess
- Atomic species
- Atomic coordinates
- Site occupancies
- Thermal displacementparameters
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Refinement Strategies
36
Wrong peak width:
- Crystallite size
- Micro-strain in crystal structure
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Summary: Refinement Strategy
37
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