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.

11.02.2009

The Rietveld method• Developed as a technique for

structure refinement.

0 20 40 60 80 100 120 1402000

4000

6000

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Inte

nsity

(cou

nts)

2 (deg)

VD0.8a = 3.16 Å

The pre-Rietveld life

hkl Intensity

110200211220310....

11.02.2009

The Rietveld method• Developed as a technique for

structure refinement.

0 20 40 60 80 100 120 1402000

4000

6000

8000

10000

12000

Inte

nsity

(cou

nts)

2 (deg)

VD0.8a = 3.16 Å

The pre-Rietveld life

hkl Intensity

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

2)( 2

i

lzklhxiihkl

iiiebI

11.02.2009

The Rietveld method• Developed as a technique for

structure refinement.

0

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2 [deg]

Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

20 40 60 80 100-2000

0

2000

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

The pre-Rietveld life

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

11.02.2009

The Rietveld method• Developed as a technique for

structure refinement.

0

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20000

2 [deg]

Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

20 40 60 80 100-2000

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

[arb

. uni

ts]

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002 121

112/220

022/130

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

40

60

80

100

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140

160

180In

tens

ity

2

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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

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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60

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100

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160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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60

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100

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180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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60

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100

120

140

160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

40

60

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100

120

140

160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

40

60

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100

120

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160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

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160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

40

60

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100

120

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160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

40

60

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100

120

140

160

180In

tens

ity

2

11.02.2009

The Rietveld method

20.0 20.5 21.0 21.5 22.0 22.520

40

60

80

100

120

140

160

180In

tens

ity

2

11.02.2009

The Rietveld methodThe calculated profile

K

backgroundiKKiKK

calci yAPFLsy )22(2

0

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10000

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14000

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18000

20000

2 [deg]

Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

20 40 60 80 100-2000

0

2000

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

0

2000

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10000

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20000

2 [deg]

Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

20 40 60 80 100-2000

0

2000

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

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

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

Lorentz factor and multiplicity

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

K

backgroundiKKiKK

calci yAPFLsy )22(2

= 1/(sin

sin2) for powders

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

Lorentz factor and multiplicity

The square modulus of the structure factor for Bragg peak K

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

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

Lorentz factor and multiplicity

The square modulus of the structure factor for Bragg peak K

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

The profile function

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

Lorentz factor and multiplicity

The square modulus of the structure factor for Bragg peak K

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

The profile function

preferred orientation

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

Lorentz factor and multiplicity

The square modulus of the structure factor for Bragg peak K

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

The profile function

preferred orientation

absorption

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld methodThe calculated profile

calculated intensity in point i

scale factor

Lorentz factor and multiplicity

The square modulus of the structure factor for Bragg peak K

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

The profile function

preferred orientation

absorption

background

K

backgroundiKKiKK

calci yAPFLsy )22(2

11.02.2009

The Rietveld method

The parameters in yicalc undergo a

least-square refinement to minimize Rwp

K

backgroundiKKiKK

calci yAPFLsy )22(2

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10000

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2 [deg]

Zr2NiD3.9 at 573 KPND, SLAD

Inte

nsity

[arb

. uni

ts]

20 40 60 80 100-2000

0

2000 2

2

i

obsii

i

calci

obsii

wpyw

yywR

Fitting the profile

11.02.2009

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.

11.02.2009

The Rietveld method – structure refinement

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(

11.02.2009

The Rietveld method – structure refinement

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)

11.02.2009

The Rietveld method – structure refinement

An example K

backgroundiKKiKK

calci yAPFLsy )22(2

Refining:• Scale factor

11.02.2009

The Rietveld method – structure refinement

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)

11.02.2009

The Rietveld method – structure refinement

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)

• Background (6 parameters)

11.02.2009

The Rietveld method – structure refinement

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)

• Background (6 parameters)

• Profile shape

11.02.2009

The Rietveld method – structure refinement

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)

• Background (6 parameters)

• Profile shape• Atomic positions

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

11.02.2009

The Rietveld method – structure refinement

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)

• Background (6 parameters)

• Profile shape• Atomic positions

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

• Displacement factors

11.02.2009

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

11.02.2009

Case 1: Titanium hydride

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

12818

6409

0

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

TID2 50.0%"TID" 50.0%

Case 1: Titanium hydride

a

b

a

b

Neutron diffraction

11.02.2009

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

11.02.2009

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

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

11.02.2009

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

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

11.02.2009

Case 2: Th2 AlD4

20 40 60 80 100 120

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2 4

Inte

nsite

t

2 [o]

PND anno 1961

PND anno 1999

11.02.2009

Case 2: Th2 AlD4

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

265789

132895

0

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

TH2ALD4_BERGSMA 50.0%TH2AL(D4 ) 50.0%

X-ray diffraction

ab

c

ab

c

11.02.2009

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

2738

1369

0

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

TH2ALD4_BERGSMA 9.1%TH2AL(D4 ) 90.9%

Case 2: Th2 AlD4ab

c

ab

c

Neutron diffraction