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
8000
10000
12000
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
2000
4000
6000
8000
10000
12000
14000
16000
18000
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
The pre-Rietveld life
hkl Intensity
101 23110 34002 10121 120112+220 450022+130 1000
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Inte
nsity
[arb
. uni
ts]
2000
002 121
112/220
022/130
11.02.2009
The Rietveld method• Developed as a technique for
structure refinement.
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
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?
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Inte
nsity
[arb
. uni
ts]
2000
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
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 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 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 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 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 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 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 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 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 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 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
2000
4000
6000
8000
10000
12000
14000
16000
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
4000
6000
8000
10000
12000
14000
16000
18000
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
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
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
11.02.2009
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
0
2000
4000
6000
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
Top Related