iconusas, oak ridge, 9-10 july 2003 sasprofit - program for usans data evaluation jan Šaroun...

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.


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


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.


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 !


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:



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



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)


Complete SAS model

Basic model defines Particle, Structure factor and Size distribution

Incoherent scattering as free parameter


Fitting multiple data sets

Resolution function and wavelength are assigned to each spectrum

Scale, background and centre position adjustable by fitting procedure


Combination with 2-dimensional SANS

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

measured datadifference


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


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


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.


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.


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


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 ...


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.

http://omega.ujf.cas.cz/~saroun/SAS.html


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.

iconusas, oak ridge, 9-10 july 2003 sasprofit - program for usans data evaluation jan Šaroun...

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