Analysis of powder (neutron) powder diffraction data....• Introduced by Hugo Rietveld in 1967 H....

$: Analysis of powder (neutron) powder diffraction data....• Introduced by Hugo Rietveld in 1967 H. M. Rietveld, Acta Cryst. 22 (1967) 151 H. M. Rietveld, J. App. Cryst. 2 (1969) 65$
11.02.2009

Analysis of powder neutron diffraction data

• The Rietveld method• Examples

Magnus H. Sørby

11.02.2009

Analysis of powder (neutron) 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 6516 times.

• Developed as a technique for structure refinement.

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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)

2 (deg)

VD0.8a = 3.16 Å

The pre-Rietveld life

hkl Intensity

110200211220310....

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0 20 40 60 80 100 120 1402000

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Inte

nsity

(cou

nts)

2 (deg)

VD0.8a = 3.16 Å


hkl Intensity

110 1000200 82211 80220 120310 285.... ....

2)( 2

i

lzklhxiihkl

iiiebI

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2 [deg]

Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

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0

2000

Zr2 NiD4.5 (300 oC)a = 6.79 Åc = 5.61 Å


hkl Intensity

101 23110 34002 10121 120112+220 450022+130 1000

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Inte

nsity

[arb

. uni

ts]

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002 121

112/220

022/130

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Inte

nsity

[arb

. uni

ts]

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0

2000

Zr2 NiD4.5 (300 oC)a = 6.79 Åc = 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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nsity

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022/130

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The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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180In

tens

ity

2

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The Rietveld methodThe calculated profile

K

backgroundiKKiKK

calci yAPFLsy )22(2

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nsity

[arb

. uni

ts]

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2000

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calculated intensity in point i

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. uni

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2000

K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor

K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor

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

K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor

Lorentz factor and multiplicity


K

backgroundiKKiKK

calci yAPFLsy )22(2

= 1/(sin

sin2) for powders

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scale factor


The square modulus of the structure factor for Bragg peak K


K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor




The profile function

K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor





preferred orientation

K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor






absorption

K

backgroundiKKiKK

calci yAPFLsy )22(2

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scale factor






absorption

background

K

backgroundiKKiKK

calci yAPFLsy )22(2

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The Rietveld method

The parameters in yicalc undergo a

least-square refinement to minimize Rwp

K

backgroundiKKiKK

calci yAPFLsy )22(2

0

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2 [deg]


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nsity

[arb

. uni

ts]

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0

2000 2

2

i

obsii

i

calci

obsii

wpyw

yywR

Fitting the profile

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The Rietveld method – structure refinement

The structure factor K

backgroundiKKiKK

calci yAPFLsy )22(2

2)( 2

2)( 22

i

lzkyhxii

i

KriiK

iiii ebebF

a

ir

2)sin()(

B

ebb

The displacement factor

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

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The profile function K

backgroundiKKiKK

calci yAPFLsy )22(2

Gaussian: 2

2)22()22( xH

C

gaussKi eeAKi

Lorentzian: x

HE

DKi

LorentzKi

1

1)22(1

1)22(

2

2

pseudo-Voigt: gausslorentzVoigtpseudoKi )1()22(

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An example

Approximate model of Al2 O3 :• Trigonal, space group R –3 c• a ~ 4.75 Å, c ~ 12.99 Å• Al in 0 0 ~0.35

O in ~0.30 0 ¼

We want a more accurate structure model!

PND (PUS)

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An example K

backgroundiKKiKK

calci yAPFLsy )22(2

Refining:• Scale factor

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An example K

backgroundiKKiKK

calci yAPFLsy )22(2

Refining:• Scale factor• Unit cell parameters

(a = 4.75 Å 4.7587 Å c = 12.99 Å 12.988Å)

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An example K

backgroundiKKiKK

calci yAPFLsy )22(2


(a = 4.75 Å 4.7587 Å c = 12.99 Å 12.988Å)

• Background (6 parameters)

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An example K

backgroundiKKiKK

calci yAPFLsy )22(2


(a = 4.75 Å 4.7587 Å c = 12.99 Å 12.988Å)


• Profile shape

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An example K

backgroundiKKiKK

calci yAPFLsy )22(2


(a = 4.75 Å 4.7587 Å c = 12.99 Å 12.988Å)


• Profile shape• Atomic positions

z(Al) = 0.35 0.3524 x(O) = 0.30 0.3068

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An example K

backgroundiKKiKK

calci yAPFLsy )22(2


(a = 4.75 Å 4.7587 Å c = 12.99 Å 12.988Å)


• Profile shape• Atomic positions

z(Al) = 0.35 0.3524 x(O) = 0.30 0.3068

• Displacement factors

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Case 1: Titanium hydride

• What we know:• Ti absorb hydrogen (deuterium).• The product has a fcc lattice of Ti with a = 4.44 Å.

• What we want to know:• Where is the hydrogen (deuterium) located?

a

b

c

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10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

34601

17300

0

P o w d e r C e l l 2 . 2

TID2 50.0%"TID" 50.0%

X-ray diffraction

a

b

a

b

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12818

6409

0


TID2 50.0%"TID" 50.0%


a

b

a

b

Neutron diffraction

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Case 2: Th2 AlD4

• Very few metal hydrides with H-H distances below 2 Å are reported in the literature.

• Th2 AlD4 is a rare exception.ab

c

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Case 2: Th2 AlD4



c

Structure data (Bergsma et al. 1961)a = 7.629 Å c = 6.517 Å s.g. I4/mcmTh: 0.162 0.662 0Al: 0 0 ¼D: 0.368 0.868 0.137

D-D = 2*z(D)*c

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Case 2: Th2 AlD4



c

1.79 Å

Structure data (Bergsma et al. 1961)a = 7.629 Å c = 6.517 Å s.g. I4/mcmTh: 0.162 0.662 0Al: 0 0 ¼D: 0.368 0.868 0.137

D-D = 2*z(D)*c

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Case 2: Th2 AlD4

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Inte

nsite

t

2 [o]

PND anno 1961

PND anno 1999

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Case 2: Th2 AlD4

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265789

132895

0


TH2ALD4_BERGSMA 50.0%TH2AL(D4 ) 50.0%

X-ray diffraction

ab

c

ab

c

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10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

2738

1369

0


TH2ALD4_BERGSMA 9.1%TH2AL(D4 ) 90.9%

Case 2: Th2 AlD4ab

c

ab

c

Neutron diffraction

Analysis of powder (neutron) powder diffraction data....• Introduced by Hugo Rietveld in 1967 H....

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