Powder diffraction and the Rietveld method - The Rietveld... · 2017-09-14 · The Rietveld method • Developed as a technique for structure refinement. 0 20 40 60 80 100 120 140

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Powder diffraction and the Rietveld method

• Principles • Example

Magnus H. Sørby

Institute for Energy Technology, Norway

NNSP School Tartu 2017

Analysis of powder diffraction data The fingerprint method • The position and relative intensities of

Bragg peaks are unique for a crystalline phase a «fingerprint».

• «Automatic» identification of known phases from databases (powder X-ray diffraction!)


Analysis of powder diffraction data Crystal structure determination


Indexing

Space group determination

Structure solution

Structure refinement

Find the size and shape of the unit cell

Find the space group or possible space groups

Find the approximate atomic arrangement within the unit cell

Make the structure model as accurate as possible

Analysis of powder diffraction data. The Rietveld method • Introduced by Hugo Rietveld in 1967

H. M. Rietveld, Acta Cryst. 22 (1967) 151 H. M. Rietveld, J. App. Cryst. 2 (1969) 65

• Revolutionized analysis of powder diffraction data. Cited 12324 times.

• Developed as a technique for structure refinement.


The Rietveld method • Developed as a technique for

structure refinement.

0 20 40 60 80 100 120 1402000

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Inte

nsity

(cou

nts)

2θ (deg)

VD0.8 a = 3.16 Å

The pre-Rietveld way

hkl Intensity

110 200 211 220 310 ....




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nsity

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nts)

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VD0.8 a = 3.16 Å


hkl Intensity

110 1000 200 82 211 80 220 120 310 285 .... ....

2)( 2∑ ++⋅∝

j

lzkyhxijhkl

jjjebI π




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Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

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0

2000

Zr2NiD4.5 (300 oC) a = 6.79 Å c = 5.61 Å


hkl Intensity

101 23 110 34 002 10 121 120 112+220 450 022+130 1000

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The Rietveld method Rietveld’s observations:

Well-resolved Bragg- peaks in his PND data had the shape of Gaussian curves.

𝑦𝑦 = 𝑒𝑒−𝑥𝑥2

The Gaussian curve 20 40 60

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nstiy

(cou

nts)

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The Gaussian curve 0 20 40 60 80 100 120 140

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Inte

nstiy

(cou

nts)

CaWO4





The Gaussian curve 84 86 88 90 92

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Inte

nstiy

(cou

nts)





The Gaussian curve 84 86 88 90 92

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Inte

nstiy

(cou

nts)




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Inte

nsity

[arb

. uni

ts]

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Zr2NiD4.5 (300 oC) a = 6.79 Å c = 5.61 Å

Rietveld’s idea: Why not fit the entire calculated profile from the model to the data, instead of just the integrated intensities?

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The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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tens

ity

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The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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tens

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The Rietveld method The calculated profile

calculated intensity in point i

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nsity

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∑ +−=K

backgroundiKKiKK

calci yAPFLsy )22(2 θθφ




scale factor

∑ +−=K

backgroundiKKiKK





scale factor

Sum over all Bragg peaks, K, that contribute with intensity to point i

∑ +−=K

backgroundiKKiKK


20.0 20.5 21.0 21.5 22.0 22.520

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nsity

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scale factor

Lorentz factor and multiplicity


∑ +−=K

backgroundiKKiKK


= 1/(sinθ sin2θ) for powders




scale factor


The square modulus of the structure factor for Bragg peak K


∑ +−=K

backgroundiKKiKK





scale factor




The profile function

∑ +−=K

backgroundiKKiKK





scale factor





preferred orientation

∑ +−=K

backgroundiKKiKK





scale factor






absorption

∑ +−=K

backgroundiKKiKK





scale factor






absorption

background

∑ +−=K

backgroundiKKiKK



The Rietveld method

The parameters in yicalc undergo a

least-square refinement to minimize Rwp

∑ +−=K

backgroundiKKiKK


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nsity

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. uni

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0

2000( )( )2

2

∑∑ −

=

i

obsii

i

calci

obsii

wpyw

yywR

Fitting the profile


The Rietveld method The structure factor ∑ +−=

K

backgroundiKKiKK


2)( 2

2)( 22 ∑∑ ++⋅ ⋅=⋅=

i

lzkyhxii

i

KriiK

iiii ebebF ππ

a

ir

2)sin()( λ

θ

θB

ebb−

⋅=The displacement factor

UB 28π= The square mean shift of the atom from its equilibrium position.


The Rietveld method The profile function ∑ +−=

K

backgroundiKKiKK


Gaussian: 𝜙𝜙(2𝜃𝜃𝑖𝑖 − 2𝜃𝜃𝐾𝐾)𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 =2 𝑙𝑙𝑙𝑙𝑙𝐻𝐻 𝜋𝜋

𝑒𝑒−4ln2𝐻𝐻2 (2𝜃𝜃𝑖𝑖− 2𝜃𝜃𝐾𝐾)2

2θK

H (or FWHM)

H varies with 2θ! Caglioti equiation: 𝐻𝐻 = √(𝑈𝑈 tan2 𝜃𝜃 + 𝑉𝑉 tan𝜃𝜃 + 𝑊𝑊)

20 40 60 80 100 120

0.3

0.4

0.5

FWHM

(deg

)

2θ

Refineable parameters


The Rietveld method The profile function ∑ +−=

K

backgroundiKKiKK


Lorentzian:

pseudo-Voigt: gausslorentzVoigtpseudoKi φηφηθθφ ⋅−+⋅=− − )1()22(

𝜙𝜙(2𝜃𝜃𝑖𝑖 − 2𝜃𝜃𝐾𝐾)𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 =2 𝑙𝑙𝑙𝑙𝑙𝐻𝐻𝜋𝜋

1

1 + 4𝐻𝐻2 (2𝜃𝜃𝑖𝑖 − 2𝜃𝜃𝐾𝐾)2

2θK

H (or FWHM)

𝐻𝐻 = √(𝑈𝑈 tan2 𝜃𝜃 + 𝑉𝑉 tan𝜃𝜃 + 𝑊𝑊) Refineable parameter

Fullprof: Npr = 5 GSAS: N.A.


The Rietveld method Sample contributions to the profile function

Broadening due to particle size: 𝛽𝛽𝑔𝑔𝑖𝑖𝑙𝑙𝑙𝑙 = 𝜆𝜆𝜆𝜆

𝐷𝐷 cos 𝜃𝜃 ∝

1cos 𝜃𝜃

Scherrer equation

Broadening due to strain: 𝛽𝛽𝑔𝑔𝑙𝑙𝑙𝑙𝑔𝑔𝑖𝑖𝑙𝑙 = 𝐵𝐵𝐵𝐵 tan𝜃𝜃 ∝ tan𝜃𝜃

«Particle size» (volume-weighted average size of coherent domains)

~1

Mostly gaussian

Mostly lorentzian

strain

𝐻𝐻 = √(𝑈𝑈 tan2 𝜃𝜃 + 𝑉𝑉 tan𝜃𝜃 + 𝑊𝑊)

gausslorentzVoigtpseudoKi φηφηθθφ ⋅−+⋅=− − )1()22(


The Rietveld method The Thompson-Cox-Hastings pseudo-Voigt

𝛽𝛽𝑔𝑔𝑖𝑖𝑙𝑙𝑙𝑙 ∝1

cos 𝜃𝜃

𝛽𝛽𝑔𝑔𝑙𝑙𝑙𝑙𝑔𝑔𝑖𝑖𝑙𝑙 ∝ tan𝜃𝜃 Mostly gaussian

Mostly lorentzian 𝐻𝐻𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 = √(𝑈𝑈 tan2 𝜃𝜃 + 𝑉𝑉 tan𝜃𝜃 + 𝑊𝑊 +

𝑍𝑍cos2 𝜃𝜃)

𝐻𝐻𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑋𝑋 tan θ +𝑌𝑌

cos𝜃𝜃

Strain-dependent

Size-dependent

Sample independent!

Fullprof: Npr = 7 GSAS: type 2


The (Thompson-Cox-Hastings) pseudo-Voigt Fullprof GSAS vs.

𝐻𝐻𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 = √(𝑈𝑈 tan2 𝜃𝜃 + 𝑉𝑉 tan𝜃𝜃 + 𝑊𝑊 +𝑍𝑍

cos2 𝜃𝜃)

𝐻𝐻𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑋𝑋 tan θ +𝑌𝑌

cos𝜃𝜃

Strain-dependent

Size-dependent

Sample independent!

Z

*10000/(2.352)


The Rietveld method Joint exercise

Approximate model of Al2O3: • Trigonal, space group R –3 c • a ~ 4.75 Å, c ~ 12.99 Å • Al in 0 0 ~0.35

O in ~0.29 0 ¼

We want a more accurate structure model!

PND (PUS)


Le Bail refinements «structureless Rietveld» - Bragg peak posititions are caluclated from unit cell and space group,

but all intensities can vary freely. - Good for checking the unit cell - … or to refine the profile parameter if the structure model is bad.


Jbt = 0 : Rietveld Jbt = 2 : Le Bail («profile matching)

Evaluating the fit


( )( )2

2

∑∑ −

=

i

obsii

i

calci

obsii

wpyw

yywR

( )∑=

i

obsii yw

NRexp

exp

2

RRwp=χ

( ) 2/1""

2/12/1"" )()(

∑

∑ −=

i

obsK

K

calcK

obsK

FI

IIR

Single crystal-like. But we don’t observe independent intensities for overlapping reflections!

Most often used. But can be highly influenced by factors that do not imply a poor structure model.

Should be close to 1 when there are no systematic errors in the model, but in many cases that’s not possible.

Evaluating the fit


( )( )2

2

∑∑ −

=

i

obsii

i

calci

obsii

wpyw

yywR

( )∑=

i

obsii yw

NRexp

exp

2

RRwp=χ

( ) 2/1""

2/12/1"" )()(

∑

∑ −=

i

obsK

K

calcK

obsK

FI

IIR

Single crystal-like. But we don’t observe independent intensities for overlapping reflections!

Most often used. But can be highly influenced by factors that do not imply a poor structure model.

Should be close to 1 when there are no systematic errors in the model, but in many cases that’s not possible.

Evaluating the fit


Conclusion

• Rietveld refinement is an effective technique to refine crystal structure models against powder diffraction data.

• It can also be used for quantitative phase analysis and to extract microstructural information.

• Except for simple cases, it requires active participation from the user. It’s not a black-box technique!


Documents

Powder diffraction and the Rietveld method - The Rietveld... · 2017-09-14 · The Rietveld method • Developed as a technique for structure refinement. 0 20 40 60 80 100 120 140