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Lesson 5 Rietveld Refinement Nicola Döbelin RMS Foundation, Bettlach, Switzerland January 14 – 16, 2015, Bern, Switzerland
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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

- 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

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

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

- Fullprof

- GSAS

- BGMN

- Maud

- Brass

- … many more1)

Commercial Software:

- HighScore+ (PANalytical)

- Topas (Bruker)

- Autoquan (GE)

- PDXL (Rigaku)