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Lesson 4Rietveld Refinement
Nicola DöbelinRMS Foundation, Bettlach, Switzerland
February 11 – 14, 2013, Riga, Latvia
2
Repetition: Powder XRD Patterns
10 20 30 40 50 60
0
200
400
600
800
1000
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
3
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)
Rietveld Refinement
5
Known structuremodel
10 20 30 40 50 600
1000
2000
3000
4000
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
4000
Inte
nsity
[cou
nts]
Diffraction Angle [°2θ]
Minimize differences between calculated and observedpattern by least-squares method
Rietveld Refinement
6
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)
Beginning of the refinement:
- Phase was identified correctly (peaksat the right position)
- But differences exist:
- Peak width
- Peak positions slightly shifted
- Intensities
Rietveld Refinement
7
10 20 30 40 50 60
0
1000
2000
3000
4000
5000
Inte
nsity
[cts
]
Diffraction Angle [°2theta]
Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)
After the refinement:
- Straight difference curve (only noise)
Modelling the Peak Profile
8
27 28 29 30 31 320
2000
4000
6000
8000
10000
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.
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
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
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
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
2000
4000
6000
8000
10000
Inte
nsity
[cts
]
Diffraction Angle [°2θ]
0 20 40 60 80 100 120 140
0
2000
4000
6000
8000
10000
Inte
nsity
[cts
]
Diffraction Angle [°2θ]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
FW
HM
[°2θ
]
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
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
Fundamental Parameters Approach
15
www.bruker.com
Fundamental Parameters Approach
16
2θ=10° 2θ=30° 2θ=80°
Calculated Peak Profiles
Fundamental Parameters Approach
17
FPA needs:
- Very detailed and complete description ofthe instrument configuration
- Very well aligned instrument
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
Fundamental Parameters Approach
19
http://www.bgmn.de
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!
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
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
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
Graphical User Interfaces for BGMN
24
BGMNwin (shipped with BGMN software package)
Graphical User Interfaces for BGMN
25
Profex (developed by Nicola Döbelin)
- Open source
- Multi-Platform
- Device database
- Structure database
- Batch processing
Example Refinement
26
1.
2.
3.
Example Refinement
27
1.
2.
3.
4.
5.
Example Refinement
28
Start Refinement
Example Refinement
29
1.
2.
3.
Example Refinement
30
Good fit
Summary
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
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
Refinement Strategies
33
Wrong peak positions:
- Unit cell dimensions
- Sample height displacement
- Zero-shift (instrumentmisalignment)
Refinement Strategies
34
Wrong absolute intensities:
- Weight fraction (scaling)
Refinement Strategies
35
Wrong relative intensities:
- Preferred orientation
- Graininess
- Atomic species
- Atomic coordinates
- Site occupancies
- Thermal displacementparameters
Refinement Strategies
36
Wrong peak width:
- Crystallite size
- Micro-strain in crystal structure
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