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Experimental Setup
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Slit 1Slit 2
QZ
Log I Specular Scanθ2 = 2θ
QZ
Background Scanθ2 ≠ 2θ
Log I
White Beam
θ
I Rocking Curveθ or θ2 fixed
Data Reduction
Slit 3
Detector
Slit 4θ2
Sample
A=Repeat each curve for:
D=−−++
B=+−C=−+
Polarizer andFlipper (+/−) Polarizer and
Flipper (+/−)
Detector
Monochromator
QZ
I
Slit Scanθ2 = 0
Fixed slits
θ
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What is it good for?
Subsurface structure up to 1μmPolymers, biofilms, magnetic surfaces, ...Determines average density at depth z
0
2e-05
4e-05
6e-05
8e-05
0.0001
0.00012
-100 0 100 200 300 400 500 600 700 800
rho
(num
ber d
ensi
ty)
Depth (Ang)
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Optical Matrix Formalism
jQS ijii
8)(162
1 2
)cosh()sinh(/)sinh()cosh(
iiiii
iiiiii SdSdS
SSdSdM
nMMMr 21
Oscillations in reflectivity R(Q) of periodd2
d i
z
translates reflectivity into lab frame
)()( rfQR where f
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Fitted Data
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Ref
lect
ivity
Q (inverse Angstroms)
data 0fit 0
0
2e-05
4e-05
6e-05
8e-05
0.0001
0.00012
-100 0 100 200 300 400 500 600 700 800
rho (num
ber density)
Depth (Ang)
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χ2 Landscape (ρ2 vs d2)
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χ2 Landscape (d2 vs d3)
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Heuristics
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Ref
lect
ivity
Q (inverse Angstroms)
data 0fit 0
0
2e-05
4e-05
6e-05
8e-05
0.0001
0.00012
-100 0 100 200 300 400 500 600 700 800
rho (num
ber density)
Depth (Ang)
170
7100.0085 ≈2π/740
0.035 ≈2π/180
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Prior Knowledge
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-1e-06
0
1e-06
2e-06
3e-06
4e-06
5e-06
6e-06
7e-06
-20 0 20 40 60 80 100 120 140 160 180
rho
(num
ber d
ensi
ty)
Depth (Ang)
Current best profile for each model
rho 0rho 1rho 2
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Ref
lect
ivity
Q (inverse Angstroms)
Current best fit for each model
data 0fit 0
data 1fit 1
data 2fit 2
Simultaneous Fitting
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Our Problem
Many local minima'Garden Path' fit spaceExpensive objective functionContinuous but no analytic derivativeSignificant number of parameters... but many priors
E.g., known material, known sputtering time, information from other measurements, theoretical models, bounds constraints
There is hope for ye who enter.