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Lesson 5
Rietveld Refinement
Nicola DöbelinRMS Foundation, Bettlach, Switzerland
January 14 – 16, 2015, Bern, Switzerland
2
Repetition: Phase Identification
10 20 30 40 50 60
0
200
400
600
800
1000
Inte
nsity [counts
]
Diffraction Angle [°2θ]
Phases are identified by their peak positions in 2θ
Search-Match software and database withpeak positions required
Phases must be identified before doingRietveld refinement
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 structure
model
10 20 30 40 50 60
0
1000
2000
3000
4000
Inte
nsity [co
un
ts]
Diffraction Angle [°2θ]
Calculate theoretical
diffraction pattern
Compare with
measured pattern
Optimize structure model, repeat calculation
10 20 30 40 50 60
0
1000
2000
3000
4000
Inte
nsity [co
un
ts]
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 32
0
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-x0
ω
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 31.0
0
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
[°2
θ]
Alternatives to Pseudo-Voigt Function
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.co
m
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
- Some fundamental parameters arenot documented
- Complete configuration can be hardto obtain
Fundamental Parameters Approach
18
http://www.bgmn.de
Fundamental Parameters Approach
19
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
Summary: Rietveld Basics
20
- 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
Refinement Strategies
21
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
Pattern Features – Structural Features
Refinement Strategy: Mismatches
22
- Peak Position
- Absolute Intensities
- Relative Intensities
- Peak WidthHow to fix this?
Refinement Strategies
23
Wrong peak positions:
- Unit cell dimensions- Sample height displacement
- Zero-shift (instrument misalignment)
Refinement Strategies
24
Refined unit cell dimensions:
Peak positions matched!
Refinement Strategies
25
Wrong absolute intensities:
- Weight fraction (scaling)
Refinement Strategies
26
Refined scale factor:
Intensities improved (but not fixed)!
Refinement Strategies
27
Wrong relative intensities:
- Preferred orientation
- Graininess
- Atomic species
- Atomic coordinates
- Site occupancies
- Thermal displacementparameters
Let’s try this first
Refinement Strategies
28
Refined texture:
Intensities fixed!
Refinement Strategies
29
Wrong peak width:
- Crystallite size
- Micro-strain in crystal structure- Surface roughness
Refinement Strategies
30
Refined crystallite sizes and micro-strain:
Peak shape fixed!
Refined Crystal Structure
31
Phase composition: 100% Al2O3 Corundum
Starting Model Refined
Parameter Value
Unit cell a 0.4760127 +-
0.0000028 nm
Unit cell c 1.2995974 +-
0.0000077 nm
Crystallite Size 1267 +- 138 nm
Atomic Coordinates Al 0.0 / 0.0 / 0.3522
Atomic Coordinates O 0.3062 / 0.0 / 0.25
Parameter Value
Unit cell a 0.4775 nm
Unit cell c 1.2993 nm
Crystallite Size Inf.
Atomic Coordinates Al 0.0 / 0.0 / 0.3522
Atomic Coordinates O 0.3062 / 0.0 / 0.25
Summary: Refinement Strategy
32
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
Grainy sample
Atomic species / Substitutions / Vacancies
Atomic coordinates
Site occupancies
Thermal displacement parameters
Wrong peak width Crystallite size
Micro-strain
Surface roughness
Transparency
Rietveld Refinement
33
Known structure
model
10 20 30 40 50 60
0
1000
2000
3000
4000
Inte
nsity [co
un
ts]
Diffraction Angle [°2θ]
Calculate theoretical
diffraction pattern
Compare with
measured pattern
Optimize structure model, repeat calculation
10 20 30 40 50 60
0
1000
2000
3000
4000
Inte
nsity [co
un
ts]
Diffraction Angle [°2θ]
Minimize differences between calculated and observedpattern by least-squares method
Rietveld Software Packages
34
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
Lesson 6: BGMN and Profex
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