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Solid State Effects So far, I’ve used only the theoretical scattering factors because of the absence of measured factors for Ge.
Citation preview
Anomalous Diffraction Project
Status Update1-22-09
Completed So Far
• Incorporating Daniel Haskel’s solid-state effects into the model
• Incorporating experimental uncertainties into the data and fits
• Inverting the ratios taken, so I fit S1/F instead of F/S1• Creating and testing a full model of the Bragg
reflections and slit function• Systematically fitting the data, varying the Debye-
Waller factor
Solid State Effects
-10
-9
-8
-7
-6
-5
-4
11.311.211.111.010.9x103
Ge_f1 GeTheo_f1
6
5
4
3
2
1
11.311.211.111.010.9x103
Ge_f2 GeTheo_f2
-12
-10
-8
-6
-4
680067006600650064006300
Mn_f1 MnTheo_f1 MnExp_f1
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
680067006600650064006300
Mn_f2 MnTheo_f2 MnExp_f2
4
3
2
1
79007800770076007500
Co_f2 CoTheo_f2 CoExp_f2
-9
-8
-7
-6
-5
-4
-3
79007800770076007500
Co_f1 CoTheo_f1 CoExp_f1
So far, I’ve used only the theoretical scattering factors because of the absence of measured factors for Ge.
Uncertainties• Since we originally took the data with DAFS in mind, the
uncertainty from the amplitudes are fairly low• The larger source of uncertainty comes from the L-scan
intensities, which were used to scale all the amplitudes– the superlattice peaks were relatively weak and a peak shape
suffered– Could have been avoided if I wasn’t trying to get more
compositions than I had time for• Of note is that some data sets lack the (111) amplitude
data.– Again, could have been avoided if I hadn’t tried to get too many
compositions (took lots of data at the Fundamental)
Inverting the Data
• Since one reason for fitting intensity ratios rather than intensity is to ‘normalize’ to the fundamental, Yong suggested fitting S1/F and S2/F rather than the reverse.
• As expected, this did not noticeably affect the fits or results.
Slit Function Correction
• Several models were attempted taking various short-cuts and making various assumptions
• None gave a correction even close to that of the Full Model:– 3D Lorentian Bragg Reflection– 2D Slit Window in 3-space, sectioning the Ewald
Sphere– Correct reflection widths confirmed by comparing
modeled scans to various data sets (see next slides)
Deteco
r
L-direction
Ewald Sphere
BraggReflection
q
ki
kf
Another BraggReflection
kf
Blending Lab with Reciprocal Space
Whatever intersects the Ewald sphere in reciprocal
space will become scattered photons in lab space.
Sample
Slits
• This is only a 2D representation. • The Bragg reflection is really a
3D volume and the Ewald sphere is a surface.
• The slits limit the amount of the Ewald sphere for the detector to see.
L-direction
Ewald Sphere
Slits limit how much solid angle of the Ewald Sphere the detector can see.
The diffractometer manipulates the Ewald Sphere so that it travels through the Bragg reflection along L.
BraggReflection
Illustrating an L-Scan
The Model• Right: A 2D representation of the
final Lorentian used for all three reflections and four compositions
• Below: an example of the modeled slit window and the intensity it captured from the Ewald Sphere. This was summed as one data point in a diffractometer scan such as an L-scan.
0.10
0.08
0.06
0.04
0.02
0.00
-0.0
2-0
.04
-0.0
6-0
.08
-0.1
0 -40 -20 0 20 40x10-3
-0.5
0.0
0.5
V P
os [m
m]
-2.0 -1.0 0.0 1.0 2.0Horizontal Position [mm]
15
10
5
x10-3
Confirming Model Accuracy• With known slit sizes, all
I had to do was model the three reflection widths.
• I had all three directions (L, In-Plane, and Phi) scanned for the Fundamental and the (111)-type to test the model
• One set of widths clearly work for all three directions, both reflections and most compositions.
• L-scan’s tails are off due to interface fringes -1.0 -0.5 0.0 0.5 1.0
Phi [Deg]
160
140
120
100
80
60
40
20
Inte
nsity
[arb
. uni
ts] Ge30
Ge35 Ge40 Ge45 Model
1.03rlu1.021.011.000.990.980.97K [rlu]
160
140
120
100
80
60
40
20
Inte
nsity
[arb
. uni
ts]
Data (offset) Model
4.2rlu4.14.03.93.8L [rlu]
160
140
120
100
80
60
40
20
Inte
nsity
[arb
. uni
ts]
Data (offset) Model
Comparison(014) Scans
at E=10.58 keV(1,5)mm Slits
pNames pw
Amp 1
H-FWHM 0.005
H-Pos 1
K-FWHM 0.005
K-Pos 0
L-FWHM 0.055
L-Pos 4
-1.0 -0.5 0.0 0.5 1.0Phi [Deg]
250
200
150
100
50
0In
tens
ity [a
rb. u
nits
]
1.03rlu1.021.011.000.990.980.97K [rlu]
250
200
150
100
50
0
Inte
nsity
[arb
. uni
ts]
1.15rlu1.101.051.000.950.900.85L [rlu]
250
200
150
100
50
0
Inte
nsity
[arb
. uni
ts]
Ge30 Ge35 Ge40 Ge45 Model
Comparison(011) Scans
Final CorrectionThe same model also worked for a data series with varying slit sizes.
40
30
20
10
0
NaI
[103 ]
4.1rlu4.03.93.83.7 L [rlu]
250
200
150
100
50
0
Modeled Intensity [arb. units]
4.14.03.93.8 L [rlu]
S#1414 in 's137a' Det Slits (2x5mm)
S#1415 in 's137a' Det Slits (1x5mm)
S#1416 in 's137a' Det Slits (0.5x5mm)
Model Slit=(2x5) Model Slit=(1x5) Model Slit=(0.5x5)
0.1
2
3
4
5678
1
Rat
io
4.1rlu4.03.93.83.7 L [rlu]
4.14.03.93.8 L [rlu]
Data Model M/L M/L S/L S/L S/M/ S/M
1.8
1.7
1.6
1.5
S2/
F In
tens
ity R
atio
1211109876Energy [keV]
Gauss L Lorentz L fit_F2int
values ± stdevK0=0.32873 ± 0.00445K1=0.27133 ± 0.00154K2=-0.01697 ± 0.000174K3=0.00042096 ± 6.45e-006
1.30
1.28
1.26
1.24
1.22
1.20
1.18
S1/
F In
tens
ity R
atio
1211109876Energy [keV]
Gauss L Lorentz L fit_F1int
values ± stdevK0=0.63427 ± 0.00474K1=0.14546 ± 0.00164K2=-0.011079 ± 0.000186K3=0.00030874 ± 6.87e-006
6000
5000
4000
3000
Inte
grat
ed In
tens
ity [a
rb]
1211109876Energy [keV]
Fint S1int S2int
Right: The final Corrections used in the anomalous diffraction fits.
Systematic Fitting• The Debye-Waller factor measured (including the
latest Slit correction) gave a value of σ=0.13 → Imeasured = I0 exp(- σqi
2), where i represents different reflections
• This value is still probably not accurate because σ is actually different for different directions
• The measured value allows a range, which are systematically set and fit.
• The proper value is determined based on the fit outputting the measured composition.
Fits with All Corrections but Sold State
• Next slide shows results from these fits• We desire for an acceptable fit:
– low nchisq value (first param)– Co-Mn ratio near 2 (panel 2, blue)– Ge comp near 30, 35, 40 and 45 respectively
(panel 2, red)– Co-Mn swapping = 0 (panel 3, red)
35
30
25
20
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
30
25
20
15
10
5
0G
e in C-S
ites Corr30_bG Corr30_cG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
6.0
5.5
5.0
4.5
4.0
3.5
C-S
ite Vacancies
Corr30_bV Corr30_cV
10
8
6
4
2
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
22
20
18
16
14
% M
n-Ge S
wapping
Corr30_CM Corr30_GC Corr30_MG
44
40
36
32
% G
e
0.400.300.200.10Debye-Waller Factor
2.6
2.5
2.4
2.3
2.2
Co-M
n Ratio
Corr30_GeComp Corr30_CMcomp
4.4
4.2
4.0
3.8
3.6
3.4
Nch
isq
0.400.300.200.10Debye-Waller Factor
12
10
8
6
Nch
isq
0.400.300.200.10Debye-Waller Factor
40
38
36
34
32
% G
e0.400.300.200.10
Debye-Waller Factor
4.0
3.5
3.0
2.5
Co-M
n Ratio
Corr35_GeComp Corr35_CMcomp
12
8
4
0
% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
35
30
25
20
15
% M
n-Ge S
wapping
Corr35_CM Corr35_GC Corr35_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
7.0
6.5
6.0
5.5
5.0
4.54.0
C-S
ite Vacancies
Corr35_bV Corr35_cV
60
50
40
30
20
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
12
8
4
0
Ge in C
-Sites
Corr35_bG Corr35_cG
All Corrections but Solid StateGe=30% Data Set
All Corrections but Solid StateGe=35% Data Set
8
7
6
5
Nch
isq
0.400.300.200.10Debye-Waller Factor
48
46
44
42
40
38
% G
e
0.400.300.200.10Debye-Waller Factor
10
8
6
4
Co-M
n Ratio
Corr45_GeComp Corr45_CMcomp
12
8
4
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
20
15
10
5
0
% M
n-Ge S
wapping
Corr45_CM Corr45_GC Corr45_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
11
10
9
8
7
6
C-S
ite Vacancies
Corr45_bV Corr45_cV
80
70
60
50
40
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
15
10
5
0
Ge in C
-Sites
Corr45_bG Corr45_cG
5.45.2
5.0
4.8
4.6
4.4
4.2
Nch
isq
0.400.300.200.10Debye-Waller Factor
46
44
42
40
38
36
34
% G
e
0.400.300.200.10Debye-Waller Factor
3.0
2.8
2.6
2.4
Co-M
n Ratio Corr40_GeComp
Corr40_CMcomp
10
8
6
4
2
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
24
23
22
21
20
19
% M
n-Ge S
wapping
Corr40_CM Corr40_GC Corr40_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
9
8
7
6
5
C-S
ite Vacancies
Corr40_bV Corr40_cV
44
40
36
32
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
25
20
15
10
5
0
Ge in C
-Sites
Corr40_bG Corr40_cG
All Corrections but Solid StateGe=40% Data Set
All Corrections but Solid StateGe=45% Data Set
Compositions Fi
t Par
amet
ers
Adding Solid State Effects
• Problem: I don’t get compositions that are correct for the Ge-level
• It seems as though the spike in the Ge scattering factor is the cause as residuals go down everywhere else when adding solid state effects.
80.001
2
4
6
80.01
2
4
6
8
S1/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
4.0
3.8
3.6
3.4
3.2
Nch
isq
0.400.300.200.10Debye-Waller Factor
30
29
28
27
% G
e
0.400.300.200.10Debye-Waller Factor
2.25
2.20
2.15
2.10
Co-M
n Ratio
SS30_GeComp SS30_CMcomp
25
20
15
10
5
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
26
24
22
20
% M
n-Ge S
wapping
SS30_CM SS30_GC SS30_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
7
6
5
4
C-S
ite Vacancies
SS30_bV SS30_cV
16
12
8Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
2.0
1.5
1.0
0.5
0.0G
e in C-S
ites
SS30_bG SS30_cG
With Solid State CorrectionsGe=30% Data Set
12
10
8
6
Nch
isq
0.400.300.200.10Debye-Waller Factor
36
34
32
30
28
26%
Ge
0.400.300.200.10Debye-Waller Factor
3.0
2.8
2.6
2.4
2.2
2.0
Co-M
n Ratio
SS35_GeComp SS35_CMcomp
20
15
10
5
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
40
35
30
25
20
% M
n-Ge S
wapping
SS35_CM SS35_GC SS35_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
8
7
6
5
C-S
ite Vacancies
SS35_bV SS35_cV
40
30
20
10
0
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
2.0
1.5
1.0
0.5
0.0
Ge in C
-Sites
SS35_bG SS35_cG
With Solid State CorrectionsGe=35% Data Set
5.0
4.8
4.6
4.4
4.2
4.0
Nch
isq
0.400.300.200.10Debye-Waller Factor
33
32
31
30
29
28
% G
e
0.400.300.200.10Debye-Waller Factor
2.40
2.35
2.30
2.25
2.20
2.15
Co-M
n Ratio
SS40_GeComp SS40_CMcomp
25
20
15
10
5
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
31
30
29
28
27
% M
n-Ge S
wapping
SS40_CM SS40_GC SS40_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
10
9
8
7
6
5
C-S
ite Vacancies SS40_bV
SS40_cV
25
20
15
10
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
2.0
1.5
1.0
0.5
0.0
Ge in C
-Sites
SS40_bG SS40_cG
With Solid State CorrectionsGe=40% Data Set
8
7
6
5
Nch
isq
0.400.300.200.10Debye-Waller Factor
40
36
32
% G
e
0.400.300.200.10Debye-Waller Factor
4.5
4.0
3.5
3.0
2.5
Co-M
n Ratio
SS45_GeComp SS45_CMcomp
20
15
10
5
0% C
M a
nd G
C S
wap
ping
0.400.300.200.10Debye-Waller Factor
30
25
20
15
10
% M
n-Ge S
wapping
SS45_CM SS45_GC SS45_MG
1.0
0.8
0.6
0.4
0.2
0.0
B-S
ite V
acan
cies
0.400.300.200.10Debye-Waller Factor
12
11
10
9
8
7
6
C-S
ite Vacancies
SS45_bV SS45_cV
60
50
40
30
20
Ge
in B
-Site
s
0.400.300.200.10Debye-Waller Factor
2.0
1.5
1.0
0.5
0.0
Ge in C
-Sites
SS45_bG SS45_cG
With Solid State CorrectionsGe=45% Data Set
Compositions Fi
t Par
amet
ers
Chosen Fits (no SS): Ge30 Data
0.01
0.1
1
10R
esid
uals
1110987Energy [keV]
0.001
0.01
0.1
S1/
F In
tens
ity R
atio
Data: Final30Fit: Corr30Model GeShift: -0.006Nchisq: 3.492Comp: [48.2, 21.1, 30.7]
0.01
0.1
1
10
Res
idua
ls
1110987Energy [keV]
0.001
0.01
0.1
S2/
F In
tens
ity R
atio
NamesParams uParams
DW 0.10 0.00
CM 0.00 0.55
MG 22.41 0.23
GC 0.52 0.58
aV 0.00 0.00
bV 0.00 0.04
cV 5.94 0.04
bG 18.18 0.78
cG 0.72 0.88
0.001
2
4
6
0.01
2
4
6
S1/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
2
4
6
0.001
2
4
6
0.01
2
S2/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
Chosen Fits (no SS): Ge35 Data
0.01
0.1
1
10R
esid
uals
1110987Energy [keV]
0.001
0.01
0.1
1
S1/
F In
tens
ity R
atio
Data: Final35Fit: Corr35Model GeShift: -0.006Nchisq: 4.423Comp: [45.2, 19.9, 34.8]
0.01
0.1
1
10
Res
idua
ls
1110987Energy [keV]
10-4
10-3
10-2
S2/
F In
tens
ity R
atio
NamesParams uParams
DW 0.22 0.00
CM 0.00 0.73
MG 42.96 0.82
GC 0.00 1.02
aV 0.00 0.00
bV 0.23 0.18
cV 5.72 0.09
bG 24.17 1.34
cG 8.84 0.79
0.001
2
4
6
0.01
2
4
6
0.1
S1/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
6
10-4
2
4
6
10-3
2
4
S2/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
Chosen Fits (no SS): Ge40 Data
0.01
0.1
1
10R
esid
uals
1110987Energy [keV]
0.001
0.01
0.1
1
S1/
F In
tens
ity R
atio
Data: Final40Fit: Corr40Model GeShift: -0.006Nchisq: 4.111Comp: [42.8, 16.8, 40.3]
0.01
0.1
1
10
Res
idua
ls
1110987Energy [keV]
10-4
10-3
10-2
S2/
F In
tens
ity R
atio
NamesParams uParams
DW 0.18 0.00
CM 0.00 0.44
MG 22.67 0.21
GC 2.17 0.49
aV 0.00 0.00
bV 0.00 0.03
cV 7.12 0.03
bG 37.24 0.65
cG 14.17 0.74
0.001
2
4
6
0.01
2
4
6
0.1
S1/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
6
10-4
2
4
6
10-3
2
4
S2/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
Chosen Fits (no SS): Ge45 Data
0.01
0.1
1
10R
esid
uals
1110987Energy [keV]
0.001
0.01
0.1
1
S1/
F In
tens
ity R
atio
Data: Final45Fit: Corr45Model GeShift: -0.006Nchisq: 4.469Comp: [43.1, 13.8, 43.1]
0.01
0.1
1
10
Res
idua
ls
1110987Energy [keV]
10-4
10-3
10-2
S2/
F In
tens
ity R
atio
NamesParams uParams
DW 0.22 0.00
CM 0.00 0.47
MG 21.00 0.26
GC 0.54 0.64
aV 0.00 0.00
bV 0.00 0.17
cV 8.18 0.08
bG 48.76 0.83
cG 12.70 0.63
0.001
2
4
6
0.01
2
4
6
0.1
S1/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]
6
10-4
2
4
6
10-3
2
4
S2/
F In
tens
ity R
atio
11.611.511.411.311.211.111.0Energy [keV]