Lesson 1

XRD and Rietveld Refinement

Nicola DöbelinRMS Foundation, Bettlach, Switzerland

March 2, 2015, Lyon, France

Powder Diffraction

2

n · λ = 2 · d · sin(θ)

d

λ

θθ 2θ

(120)

(100)

(010)

Powder sample

X-ray Diffractometer

3

Sample

X-ray Tube

Detector

Start

Scan

End

2θ angle

Digital Diffractometers

4

Transmission Geometry

Glass Capillary

Foil

Fluid Cell

Capillaries are ideal for:

• Light atoms (Polymers, Pharmaceuticals)

• Small amounts

• Hazardous materials

• Air-sensitive materials

Use characteristic radiation with low absorption coefficient

Flat powder sample

Reflective Geometry

Reflective Geometry is ideal for:

• Absorbing materials (Ceramics, Metals)

• Thin films

• Texture analysis

Use characteristic radiation with high absorption coefficient

Diffraction Pattern

5

10 20 30 40 50 60

0

200

400

600

800

1000

Inte

nsity [counts

]

Diffraction Angle [°2θ]

Features:

Peak Positions (°2theta)

Peak Intensities (counts)

Peak Width (°2theta)

Phases are identified frompeak positions only

6

Search-Match software:

- Extract peak positions

- Compare with database

Databases:

- PDF-2 (commercial)

- PDF-4+ (commercial)

- COD (free online resource)

Rietveld Refinement

7

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

8

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

9

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

10

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

11

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]

Kα1

Kα2

Asymmetry

Mathematical modelfor peak shape required

Modelling the Peak Profile

12

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

13

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

14

Kα1

Kα2

Kβ

Fitting n, ω to peaksof a reference material

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]

Pseudo-Voigt: Problems

15

Asymmetry

Peaks at low 2θ anglesare asymmetric.

Pseudo-Voigt curvesare symmetric.

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

16

Alternatives to PV function:

- Pearson VII

- Thompson-Cox-Hastings PV

- Split PV

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

Fundamental Parameters Approach FPA

17

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

18

www.bruker.com

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 2: BGMN and Profex