1. IConUSAS, Oak Ridge, 9-10 July 2003 SASProFit - program for USANS data evaluation Jan Šaroun Nuclear 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.

2. 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 Q z =0

3. IConUSAS, Oak Ridge, 9-10 July 2003 Vertically smeared scattering function Elastic scattering (cut at Q z =0 => projection along z) where Fourier transformation of correlation function: Infinit-slit resolution (integration over Q y => cut at y=0) Structural information from the horizontal cut only ! The formula holds for anisotropic systems.

4. IConUSAS, Oak Ridge, 9-10 July 2003 Multiple scattering Formula by Schelten & Schmatz transformed to infinite-slit geometry: Includes unscattered component t... sample thickness Convolution with measured instrumental curve: … to be used in  2 I instr (  ) is the rocking curve measured without sample Secondary extinction only !

5. IConUSAS, Oak Ridge, 9-10 July 2003 Scattering model in SASProFit Single particle analytical formula sphere: ellipsoid of revolution: 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:

6. IConUSAS, Oak Ridge, 9-10 July 2003 Scattering model in SASProFit Size distribution of particles Set of cubic splines on logarithmic scale g N (x) corresponding to the N-th spline is stored in lookup tables 1) monodisperse 2) free size distribution represented by splines

7. 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. volume size distribution excluded volume form factor HS structure factor Hard sphere radius (in units of R)

8. IConUSAS, Oak Ridge, 9-10 July 2003 Complete SAS model Basic model defines Particle, Structure factor and Size distribution Incoherent scattering as free parameter

9. 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

10. IConUSAS, Oak Ridge, 9-10 July 2003 Combination with 2-dimensional SANS anisotropic system (simulated data) USANS at 2 orientations SANS, 2-dimensional measured data difference

11. IConUSAS, Oak Ridge, 9-10 July 2003 Test on simulated data Bimodal distribution of spheres

12. 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. undeformeddeformed 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 K strain-rate e‘=0.33 10 -4 s -1

13. IConUSAS, Oak Ridge, 9-10 July 2003 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.

14. IConUSAS, Oak Ridge, 9-10 July 2003 Double-crystal SANS measurements  =0 o  =90 o S18, ILL =1.9 Å DN2, NPI Řež =2.1 Å  =0 o  =90 o Six measurements were fitted simultaneously by a single model. Samples were measured with tension axis parallel (  =0 o ) and perpendicular (  =90 o ) to the scattering vector.

15. 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

16. IConUSAS, Oak Ridge, 9-10 July 2003 phase shift: => possible problems with refraction on large cavities Pin-hole SANS measurement (V4, HMI Berlin) Scattering intensityDifference Detector distance: 16 m Wavelength: 0.6 nm Only anisotropic Porod scattering is observed...

17. 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.

18. IConUSAS, Oak Ridge, 9-10 July 2003

19. 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.