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Institute of Chemistry, University of Graz,Austria

Indirect Fourier Transformation

IntroductionPair Distance Distribution Function PDDF

Indirect Fourier TransformationSymmetries, Polydispersity

ExamplesDeconvolution of the PDDF

Institute of Chemistry, University of Graz, Austria

The Scattered Field Es(q)

The scattering amplitudes of allcoherently scattered waves haveto be added according to theiramplitude and relative phase .

The phase difference dependson the relative location of thescattering centers.

Es(q)

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

The Scattered Field Es(q)

( ) ( ) isV

E const e d = qrq r r

In order to find the total scattered field we have to integrate over the wholeilluminated scattering volume V

We can now express the density (r) by its mean and its fluctuations (r):

( ) ( ) = + r r

( ) ( )i isV V

E const e d e d

= + qr qrq r r r

( ) ( ) isV

E const e d= qrq r r

The Fourier integral is linear, so we can rewrite the above equation:

Taking into account the large dimension of the scattering volume we get:

Institute of Chemistry, University of Graz, Austria

From Scattering Amplitudes to Scattering Intensities

For monodisperse dilute systems we can write:

( ) ( )2| ( ) |sI q N F NI q= < > =q

We have introduced the particle scattering amplitude F(q) which is the scatteredfield resulting from integration over the particle volume only.

( ) ( ) i

V

F e d = qrq r r

( ) 1 2( )2 1 2 1 2| ( ) | ( ) ( ) ( ) i rV

F F F e d d = = q rq q q r r r rWe put r1 - r2 = r and use r2 = r1 - r and introduce the convolution square of thedensity fluctuations:

21 1 1( ) ( ) ( ) ( )

V

d = r r r r r r

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

The Convolution Square of the Density Fluctuations (r)and (r):

The function (r) is calculated by shifting the ghostparticle a vector r and integrating the overlappingvolume.

This function is also called spatial autocorrelationfunction (ACF).

The spatially averaged convolution square (r) resultsfrom the same process, the ghost is shifted by adistance r = |r|, but we have to average over all possibledirections in space.

2 2 21 1 1( ) ( ) ( ) ( ) ( ) ( )

V

r r V d = = < > = < >r r r r r

Institute of Chemistry, University of Graz, Austria

Spatially Averaged Intensity I(q)

( )2 2( ) | ( ) | iV

I q F e d = < > = < > qrq r rThe spatially averaged intensity I(q) is given by:

( ) 20

sin4

qrr r dr

qr

=

by introducing the pair distance distribution function (PDDF) p(r) with

( ) ( ) ( )2 2 2p r r r r r = =

we finally get

( )0

sin ( )( ) 4

qrI q p r dr

qr

=

4

Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

Definition of the Pair Distance Distribution Function (PDDF)p(r)

We can relate the meaning of a distance histogramto the PDDF p(r) if the particles are homogeneous.The height of p(r) is proportional to the number ofdistances that can be found inside the particle withinthe interval r and r+dr

The p(r) function of inhomogeneous particles isproportional to the product of the differencescattering lengths nink [ ] of twovolume elements i and k with a center-to-centerdistance between r and r+dr and we sum over allpairs with this distance.

i i in = ( )dV( ) r r

Institute of Chemistry, University of Graz, Austria

The Scattering Problem and the Inverse Scattering Problem

For the solution of the inverse Problem it is essential to be able to calculate the PDDFform the experimental scattering curve with minimum termination effect.

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

SAXS Cameras - Slit Collimation (Kratky Camera)

The block camera, designed by O. Kratky, uses blocks to define the size of theprimary beam. Contrary to the slit system it does not allow measurements aboveand below the direct beam. The system is built by a U-shaped middle part M, abridge B and an entrance slit (or block) E. The main idea is to allow full parasiticscattering below the primary beam but to have negligible parasitic scatteringabove the beam, the half-plane used for the measurement.

Institute of Chemistry, University of Graz, Austria

SAXS Cameras - Slit Collimation with X-ray mirror

X-rayTube

GoebelMirrorPSDIm

ageP

late

or

X-rayTube

GoebelMirrorPSDIm

ageP

late

or

In this new, modified slit collimationsystem the divergent primary beam iscollimated by a Goebel mirror increasingthe flux by a factor of 5. At the same timethe radiation becomes monochromatic.

ParabolicGoebel mirror

X-ray tube

Collimationsystem

Pos

ition

sens

itive

dete

ctor

Sample

Beam stop

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

SAXS Cameras - Slit Collimation with X-ray mirror

The intensity can be increased by another factor of 4 with a focusingoptic, the total increase in intensity a factor of 20, at the same timehaving monochromatic radiation!

Institute of Chemistry, University of Graz, Austria

SAXS Cameras - Slit Collimation with X-ray mirror

0 1 2 3 4 50.1

1

10

Auflsung: 2/qmin=125 nmqmin=0.05 nm-1

Hydroxy Nitril Lyase (64mg/mL)PufferHNL-Puffer

Inte

nsi

tt

[PS

L/s

]

q [nm-1]0 1 2 3 4 5

0.1

1

10

Auflsung: 2/qmin=125 nmqmin=0.05 nm-1

Hydroxy Nitril Lyase (64mg/mL)PufferHNL-Puffer

Inte

nsit

t[P

SL/

s]

q [nm-1]Typical scan of an image plate

Typical result for a protein solution. The blue curve isthe difference pattern after subtraction of the buffer. (A.Bergmann, Thesis).

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

Absolute Intensity - Calibration with Water

The horizontal part of the larger q-range corresponds to the isothermal compressibilityof water, therefore the constant scattering intensity of water is 1.648*10-2 cm-1 at 20C.

Orthaber, D., Bergmann, A. and Glatter, O. J. Appl. Cryst. (2000) 33, 218-225. SAXS experimentson absolute scale with Kratky systems using water as a secondary standard

Institute of Chemistry, University of Graz, Austria

Application Absolute Intensity - Lysozyme

It is possible to put the scattering of any sample in relation to the water scatteringand bring the sample scattering data on absolute scale, the forward intensity I(0) oflysozyme is 0.202 cm-1. With this value of I(0) it is possible to estimate the molecularweight for lysozyme to 13300 g/mol. The effect of the finit concentration of 20 mg/mL(decreasing of the forward intensity) is taken into account.

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

Inverse Problem in Scattering Artists View*

primary beam

sample design of theexperiment

result inq-space

?structure

of the scattering particle

* Asterix in Belgium

associated by Anna Stradner & Gerhard Fritz

Institute of Chemistry, University of Graz, Austria

The Scattering Problem and the Inverse Scattering Problem

For the solution of the inverse Problem it is essential to be able to calculate the PDDFform the experimental scattering curve with minimum termination effect.

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

From experimental data to the PDDF

All Transformations T1 to T4 are linear and are mathematically well defined, this doesnot hold for their inverse transformations.

Institute of Chemistry, University of Graz, Austria

The Principles of the Indirect Fourier Transformation I

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Institute of Chemistry, University of Graz,Austria

Institute of Chemistry, University of Graz, Austria

The Principles of the Indirect Fourier Transformation II

We start with the following Ansatz:

( ) ( )1

N

a i ii

p r c r=

= for 0 r Dmax

Here we have used the essential assumption that we can estimate amaximum dimension Dmax for the particle.

Now we transform this series into the reciprocal space using the lineartransformation T1:

1 1 11

N N N

a i i ia i i ii=1 i=1 i=

(q)= T (r)= T [ (r)] = T (r)= (q)p c c cI

Here we have introduced the functions i(q) defined by:

i 1 i(q)= T (r)

Institute of Chemistry, University of Graz, Austria

The Principles of the Indirect Fourier Transformation III

Now we transform according to the instrumental broadening effects T2 - T4 (some of themmay be negligible) and get:

4 3 2( ) ( ) ( )N

a a i ii l

I q T T T I q c q=

= = where we find again the same coefficients ci and the set of functions i(q) in theexperimental space

( ) ( )4 3 2 4 3 2 1( )i i iq T T T q T T T T r = =

With this operation we have created the three systems of functions i(r), i(q) andi(q) which are optimized for the representation of the scattering functions from