Evaluation of SANS data from doublecrystal diffractometers Jan aroun Nuclear Physics Institute Re, C

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Evaluation of SANS data from double-crystal
diffractometersJan arounNuclear Physics
Institute Re, Czech Republic
Motivation Need of a data fitting program
addressing some specific features of
double-crystal SANS diffractometers.
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Project aims

• Problems to be solved
• Fitting of slit-smeared scattering cross-sections
  
• calculate slit-smeared scattering function
  directly from real-space model rather then by
  smearing of S(Q)
• no conversion to pin-hole geometry
• Multiple scattering - common problem in USANS
  experiments (large objects, high contrast, e.g.
  porous materials)
• Simultaneous fitting of multiple data sets taken
  at different Q-ranges/instruments
• Treatment of anisotropic scattering in slit
  geometry

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Approximations justified for double-crystal
instruments
Infinite-slit geometry dqvertical/qmax gt 10 for
measurements with neutrons
Ewald sphere plane at Qz0 angular range 10-5
.. 10-3 rad gt error dQz/Qlt 10-3
Narrow wavelength distribution wavelength spread
is given by beam divergence Dq Dl/l Dq cotgqB
( 2 for the instrument at NPI)
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Infinite-slit SANS and real-space model
Structural information from the horizontal cut
only ! The formula holds for anisotropic systems.
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Multiple scattering
Formula by Schelten Schmatz transformed to
infinite-slit geometry
Secondary extinction only !
FFT instead of Henkel transformation
non-linear even for gmodel(x)pigi(x)
Includes unscattered component t ... sample
thickness
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SASProFit ver. 5.0
Fitting program based on indirect Fourier
transformation method and multiple scattering
theory in diffraction regime

• User-friendly graphical interface
  (Win9x/2000/XP)
• Interactive work (fitting control plotting
  tools)
• Batch file processing ( simple batch file
  editor)
• Simulation of SANS data
• Levenberg-Marquart fitting with numerically
  calculated derivatives

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Complete SAS model
Basic model defines Particle, Structure factor
and Size distribution
Incoherent scattering as free parameter
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Fitting multiple data sets
Resolution function and wavelength are assigned
to each spectrum
Scale, background and centre position adjustable
by fitting procedure
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Test on simulated data
Bimodal distribution of spheres
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Combination with 2-dimensional SANS
measured data

• anisotropic system (simulated data)
• USANS at 2 orientations
• SANS, 2-dimensional

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Cavities in Superplastically Deformed Tetragonal
Zirconia PolycrystalsS. Harjo, N.Kojima,
Y.Motohashi, J.aroun, V.Ryukhtin, P.Strunz,
M.Baron, R.Loidl Mater. Trans. 43 (2002) 2480.
undeformed
deformed to e200
3Y-TZP strain e0 .. 200 temperature 1723
K strain-rate e0.33 10-4 s-1
s

• Measurements were carried out at double-crystal
  diffractometers
• Bonse-Hart (S18, ILL)
• bent crystals (DN2, NPI Re)

Sample preparation and SEM analysis by S.
Harjo, Y. Motohashi, Ibaraki University, The
Research Center for Superplasticity
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DC-SANS diffractometer at NPI Re
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.
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Double-crystal SANS measurements
f0o
Six measurements were fitted simultaneously by a
single model.
Samples were measured with tension axis parallel
(f0o) and perpendicular (f90o) to the
scattering vector.
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Double-crystal SANS measurements
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Comparison with SEM
Larger cavities formed by coalescence of the
smaller ones
Smaller cavities are more symmetric
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 mm.
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Comparison with SEM
Comparison of measured data with scattering
function calculated from SEM images.
V. Ryukhtin et al., J. Appl. Cryst. 36 (2003)
478-483.
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Concluding remarks
Plans for further development involve

• Implementation of other particle shapes (shells,
  cyllinders, ...), structure factors (fractals,
  ...) and size distributions (log-normal).
• An interface for user-provided scattering models
  (in DLLs)
• Documentation extended users guide
• ... and other features depending on users
  feedback.

SASProFit is freeware! Download at
http//omega.ujf.cas.cz/saroun/SAS
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Scattering model in SASProFit
Size distribution of particles
1) monodisperse 2) free size distribution
represented by splines
gN(x) corresponding to the N-th spline is stored
in lookup tables
Set of cubic splines on logarithmic scale
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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.
Hard sphere radius (in units of R)
excluded volume
volume size distribution
HS structure factor
form factor
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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.
Resolution and Q-range can be adjusted by varying
crystal curvatures.