iconusas, oak ridge, 9-10 july 2003 sasprofit - program for usans data evaluation jan Šaroun...
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IConUSAS, Oak Ridge, 9-10 July 2003
SASProFit - program for USANS data evaluation
Jan ŠarounNuclear Physics Institute Řež, Czech Republic
The program SASProFit has been developed to address problems specific to double crystal SANS diffractometers. It is based on indirect Fourier transformation method and multiple scattering theory in diffraction regime.
IConUSAS, Oak Ridge, 9-10 July 2003
SASProFit - project aims
Problems to be addressed
• Fitting of slit-smeared scattering cross-sections => no conversion to pin-hole geometry
• Multiple scattering
• Combination of data taken at different Q-ranges/instruments
• Anisotropic scattering
Approximations justified for double-crystal instruments
• ‘infinite-slit’ geometry
• Ewald sphere ~ plane at Qz=0
IConUSAS, Oak Ridge, 9-10 July 2003
Vertically smeared scattering function
Elastic scattering (cut at Qz=0 => projection along z)
rrQrQ
2cos dgdd
S
dzyxg
r,where
rrQrQ 3cos ddd
V
Fourier transformation
of correlation function:
dxQxxgQ
dd
xxx
)cos()0,()(
Infinit-slit resolution (integration over Qy => cut at y=0)
Structural information from the horizontal cut only !
The formula holds for anisotropic systems.
IConUSAS, Oak Ridge, 9-10 July 2003
Multiple scattering
dxQxetQW xgxgt
x cos, )0()(1 2
Formula by Schelten & Schmatz transformed to infinite-slit geometry:
Includes unscattered component
t ... sample thickness
Convolution with measured instrumental curve:
dItWtI instrxx
,2
),(
… to be used in 2
Iinstr() is the rocking curve measured without sample
Secondary extinction only !
IConUSAS, Oak Ridge, 9-10 July 2003
Scattering model in SASProFit
Single particle
)(),( 22 yxgyxg S analytical formula
sphere:
ellipsoid of revolution: )(),( 2221 yxgyxg SE
magnetic spheres:
g(x,y) is calculated numerically and stored in look-up tables
includes magnetic form factor (only for 2-dim pin-hole SANS)
randomly oriented anisotropic particles
ellipsoid:prism:
IConUSAS, Oak Ridge, 9-10 July 2003
Scattering model in SASProFit
Size distribution of particles
Set of cubic splines on logarithmic scale
MAX
MAX
R
RSNN RdR
Rx
gRBxg
)()()(
0.01 0.1 10.00
0.01
0.02
0.03
0.04
0.05
... N21D
(R)
[m
-1]
radius [m]
i
MAXN
N
ii
MAX
i
RR
xgpR
Rxg
1
2
)(
gN(x) corresponding to the N-th spline is stored in lookup tables
1) monodisperse2) free size distribution represented by splines
IConUSAS, Oak Ridge, 9-10 July 2003
Scattering model in SASProFit
Structure factor
In the present version, choice is possible from either incoherent system (no structure factor) or Hard-sphere structure factor in local monodisperse approximation (Pedersen, J. S. (1994). J. Appl. Cryst. 27, 595-608). More options are likely to be added in future versions.
dRRQRSQRFRDQdd MAXR
HS
0
322 ,)(34
volume size distribution
excluded volume
form factor HS structure factor
Hard sphere radius (in units of R)
IConUSAS, Oak Ridge, 9-10 July 2003
Complete SAS model
Basic model defines Particle, Structure factor and Size distribution
Incoherent scattering as free parameter
IConUSAS, Oak Ridge, 9-10 July 2003
Fitting multiple data sets
Resolution function and wavelength are assigned to each spectrum
Scale, background and centre position adjustable by fitting procedure
IConUSAS, Oak Ridge, 9-10 July 2003
Combination with 2-dimensional SANS
anisotropic system (simulated data)• USANS at 2 orientations• SANS, 2-dimensional
measured datadifference
IConUSAS, Oak Ridge, 9-10 July 2003
Test on simulated data
Bimodal distribution of spheres
small large<R> [m] S/V [m-1] [%] <R> [m] S/V [cm-1] [%]
Model parameters 0.100 0.314 1.00 1.00 0.138 4.00
Fitted values 0.102(2) 0.32(2) 1.03(3) 1.03(2) 0.131(2) 4.01(5)
-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003101
102
103
104
105
106
simulated data <N>2 mm 1.74 mm 3.46 mm 5.1
thic
knes
s x
d/d
/ rad
0.1 10.00
0.05
0.10
0.15
0.20 Original model data
Result of data fitting 2 x 5 splines 20 splines
D(R
) / m
-1
sphere radius / m
IConUSAS, Oak Ridge, 9-10 July 2003
Cavities in Superplastically Deformed Tetragonal Zirconia Polycrystals
S. Harjo, N.Kojima, Y.Motohashi, J.Saroun, V.Ryukhtin, P.Strunz, M.Baron, R.Loidl
Mater. Trans. 43 (2002) 2480. undeformed deformed to e=200%
Measurements were carried out at double-crystal diffractometers:• Bonse-Hart (S18, ILL)• bent crystals (DN2, NPI Řež)
5 m
3Y-TZP: strain e=0 .. 200 %temperature = 1723 Kstrain-rate e‘=0.33 10-4 s-1
IConUSAS, Oak Ridge, 9-10 July 2003
position sensitivedetector
sam ples
bent S i 220
bent S i 111 analyzer(asym metric cut) PE + B
Pb
bent S i 111
diffraction planes 111
steel rods
x R LD A D S = ( s in(2 ) + )
S
xD
L D
DC-SANS diffractometer at NPI Řež
Bent analyzer in fully asymmetric diffraction geometry permits to convert angular distribution of scattered neutrons into the positional one.Mikula, P., Lukas, P. & Eichhorn, F. (1988). J. Appl. Cryst. 21, 33-37.
IConUSAS, Oak Ridge, 9-10 July 2003
Double-crystal SANS measurements
Qx
=0o
Qx
=90o
S18, ILL=1.9 Å
DN2, NPI Řež=2.1 Å
=0o
=90o
Six measurements were fitted simultaneously by a single model.
Samples were measured with tension axis parallel (=0o) and perpendicular (=90o) to the scattering vector.
IConUSAS, Oak Ridge, 9-10 July 2003
Fitting model
=1.12
=1.8
=1
The main peak corresponds to the cavities created due to the grain boundary sliding process. Its position is close to the mean grain radius of 0.19 m.
Large cavities formed by coalescence of the smaller ones
Shape irregularities ?
3 particle sets
IConUSAS, Oak Ridge, 9-10 July 2003
phase shift:=> possible problems with refraction on large cavities
Pin-hole SANS measurement (V4, HMI Berlin)
Scattering intensity Difference
MODELEXP II
Detector distance: 16 mWavelength: 0.6 nm
5.12 MAXR
Only anisotropic Porod scattering is observed ...
IConUSAS, Oak Ridge, 9-10 July 2003
Concluding remarks
The choice of scattering models is growing but still rather limited. Plans for further development involve:
SASProFit is freeware! Download at http://omega.ujf.cas.cz/~saroun/SAS
• Other models which don’t conform to the traditional scheme particle - cluster - size distribution (e.g. fractal models)
• An interface to externally provided scattering models (in DLLs)
• User’s guide
... and other features depending on users feedback.
IConUSAS, Oak Ridge, 9-10 July 2003
IConUSAS, Oak Ridge, 9-10 July 2003
Conversion factor between angular and positional coordinates is very accurately determined by measuring the dependence of diffraction peak position in on the analyzer rocking angle.
200 400 600 800 10000
1
2
3
4
5
6resolution:
high medium low
RA = -92.8 m
RA = -14.1 m
RA = -39.7 m
/ m
rad
position
200 400 600 800 10000
500
1000
medium resolution = 0.02 deg.
coun
ts
channel
Resolution and Q-range can be adjusted by varying crystal curvatures.