Indirect Fourier Transformation
Introduction
Pair Distance Distribution Function PDDF
Indirect Fourier Transformation
Symmetries, Polydispersity
Examples
Deconvolution of the PDDF
The Scattered Field Es(q)
The scattering amplitudes of all
coherently scattered waves have
to be added according to their
amplitude and relative phase ÿ.
Es(q)
ϕ
The phase difference depends
on the relative location of the
scattering centers.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
1
The Scattered Field Es(q)
In order to find the total scattered field we have to integrate over the whole
illuminated scattering volume V
Es (q) = const ÿ ρ (r) e−iqr dr
V
We can now express the density ÿ (r) by its mean ρ and its fluctuations ÿ (r):
ρ ( r ) = ρ + ∆ρ ( r )
The Fourier integral is linear, so we can rewrite the above equation:
Es ( q ) = const
ÿρ ⋅e
V
− iqr
dr + ÿ ∆ρ ( r ) eiqr dr
V
Taking into account the large dimension of the scattering volume we get:
Es ( q ) = const ÿ ∆ρ ( r ) eiqr dr
V
Institute of Chemistry, University of Graz, Austria
From Scattering Amplitudes to Scattering Intensities
For monodisperse dilute systems we can write:
I s ( q ) = N < | F (q ) |2 > = NI ( q )
We have introduced the particle scattering amplitude F(q) which is the scattered
field resulting from integration over the particle volume only.
F (q) = ÿ ∆ρ (r ) e − iqr dr
V
| F (q) | = F ( q ) ⋅ F ∗ (q ) = ÿ ÿ ∆ρ (r1 ) ∆ρ (r2 ) e − iq (r1 −r2 ) dr1 dr2
2
V
We put r1 - r2 = r and use r2 = r1 - r and introduce the convolution square of the
density fluctuations:
γ (r ) ≡ ∆ ρÿ 2 (r ) = ÿ ∆ρ (r1 ) ∆ρ (r1 − r ) dr1
V
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
2
The Convolution Square of the Density Fluctuations ÿ(r)
and ÿ(r):
The function ÿ(r) is calculated by shifting the “ghost”
particle a vector r and integrating the overlapping
volume.
This function is also called spatial autocorrelation
function (ACF).
The spatially averaged convolution square ÿ(r) results
from the same process, the ghost is shifted by a
distance r = |r|, but we have to average over all possible
directions in space.
γ (r ) = ρÿ 2 (r ) − V ( ρ )2 = < ∆ρÿ 2 (r ) > = < ÿ ∆ρ (r1 ) ∆ρ (r1 − r ) dr1 >
V
Institute of Chemistry, University of Graz, Austria
Spatially Averaged Intensity I(q)
The spatially averaged intensity I(q) is given by:
I (q ) = <| F (q) |2 > = < ÿ ∆ρÿ 2 ( r ) e −iqr dr >
V
∞
= 4π ÿ γ ( r ) r 2
0
sin qr
dr
qr
by introducing the pair distance distribution function (PDDF) p(r) with
p ( r ) = γ ( r ) ⋅ r 2 = ∆ρÿ 2 ( r ) ⋅ r 2
we finally get
I ( q) = 4π
∞
ÿ p (r )
0
sin ( qr )
dr
qr
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
3
Definition of the Pair Distance Distribution Function (PDDF)
p(r)
We can relate the meaning of a distance histogram
to the PDDF p(r) if the particles are homogeneous.
The height of p(r) is proportional to the number of
distances that can be found inside the particle within
the interval r and r+dr
The p(r) function of inhomogeneous particles is
proportional to the product of the difference
scattering lengths nink [ ni = ∆ρ ( ri )dV( ri ) ] of two
volume elements i and k with a center-to-center
distance between r and r+dr and we sum over all
pairs with this distance.
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 PDDF
form the experimental scattering curve with minimum termination effect.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
4
SAXS Cameras - Slit Collimation („Kratky Camera“)
The block camera, designed by O. Kratky, uses blocks to define the size of the
primary beam. Contrary to the slit system it does not allow measurements above
and below the direct beam. The system is built by a U-shaped middle part M, a
bridge B and an entrance slit (or block) E. The main idea is to allow full parasitic
scattering below the primary beam but to have negligible parasitic scattering
above the beam, the half-plane used for the measurement.
Institute of Chemistry, University of Graz, Austria
SAXS Cameras - Slit Collimation with X-ray mirror
l
Goebe
ebe l
Goror
Mir
Mir ror
Parabolic
Goebel mirror
X-ray
X-ray
Tube
Tube
Collimation
system
Sample
X-ray tube
ns
dete ensitiv
e
ctor
ageePlPlatatee
Imag
Im
or
Pos
itio
D
PS
PS
or D
Beam stop
In this new, modified slit collimation
system the divergent primary beam is
collimated by a Goebel mirror increasing
the flux by a factor of 5. At the same time
the radiation becomes monochromatic.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
5
SAXS Cameras - Slit Collimation with X-ray mirror
The intensity can be increased by another factor of 4 with a focusing
optic, the total increase in intensity a factor of 20, at the same time
having monochromatic radiation!
Institute of Chemistry, University of Graz, Austria
SAXS Cameras - Slit Collimation with X-ray mirror
-1
Auflösung:
π/qmin=125 nm
Auflösung:22
π/qmin=125 nm
Intensität[PSL/s]
[PSL/s]
Intensität
qq
=0.05 nm -1
min =0.05 nm
min
10
10
11
0.1
0.10
0
Typical scan of an image plate
Hydroxy Nitril Lyase (64mg/mL)
Hydroxy Nitril Lyase (64mg/mL)
Puffer
Puffer
HNL-Puffer
HNL-Puffer
1
1
2
2
-1
3
3
-1
qq[nm
[nm ] ]
4
4
5
5
Typical result for a protein solution. The blue curve is
the difference pattern after subtraction of the buffer. (A.
Bergmann, Thesis).
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
6
Absolute Intensity - Calibration with Water
The horizontal part of the larger q-range corresponds to the isothermal compressibility
of water, therefore the constant scattering intensity of water is 1.648*10-2 cm-1 at 20°C.
Orthaber, D., Bergmann, A. and Glatter, O. J. Appl. Cryst. (2000) 33, 218-225. “SAXS experiments
on 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 scattering
and bring the sample scattering data on absolute scale, the forward intensity I(0) of
lysozyme is 0.202 cm-1. With this value of I(0) it is possible to estimate the molecular
weight 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.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
7
Inverse Problem in Scattering – Artists View*
sample
primary beam
design of the
experiment
* “Asterix in Belgium”
associated by Anna Stradner & Gerhard Fritz
result in
q-space
?
structure
of the scattering particle
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 PDDF
form the experimental scattering curve with minimum termination effect.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
8
From experimental data to the PDDF
All Transformations T1 to T4 are linear and are mathematically well defined, this does
not hold for their inverse transformations.
Institute of Chemistry, University of Graz, Austria
The Principles of the Indirect Fourier Transformation I
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
9
The Principles of the Indirect Fourier Transformation II
We start with the following “Ansatz”:
pa ( r ) =
N
i =1
ciϕ i ( r )
0 ≤ r ≤ Dmax
for
Here we have used the essential assumption that we can estimate a
maximum dimension Dmax for the particle.
Now we transform this series into the reciprocal space using the linear
transformation T1:
N
I a (q)= T1 p a(r)= T1 [
c i ϕ i(r)] =
i=1
N
N
c i T1 ϕ i(r)=
i=1
c i ψ i(q)
i=1
Here we have introduced the functions ψi(q) defined by:
ψ i (q)= T1 ϕ i (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 them
may be negligible) and get:
I a ( q) = T4 T3 T2 I a ( q) =
N
ci χ i ( q)
i =l
where we find again the same coefficients ci and the set of functions χi(q) in the
experimental space
χ i ( q) = T4 T3 T2 ψ i ( q ) = T4 T3 T2 T1 ϕ i ( r )
With this operation we have created the three systems of functions ϕi(r), ψi(q) and
χi(q) which are optimized for the representation of the scattering functions from a
particle (scattering object) with maximum dimension smaller or equal to Dmax. In
order find these functions we must calculate the expansion coefficients ci by a
weighted least squares operation:
M
L=
k=1
M
[ I exp( q k ) − I a( q k ) ]
=
2
σ ( qk )
k=1
2
[ I exp( q k ) −
N
c i χ i( q k ) ]
2
i=1
2
σ ( qk )
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
10
The Stability Problem
If we apply the basic idea described above we find a good fit
to the data, but a solution in real space which shows strong
oscillations around the correct solution.
We can reduce or even eliminate these artificial oscillations
by adding the following condition to the least squares
condition:
N c' =
N −1
i =1
( ci+1 − ci )
2
This condition is coupled to the least squares condition L
by a so-called Lagrange-Multiplier λ:
(L + λ N c′ )= Min
Here we are left with the problem to find the right value for !
This figure: x-x-x-x: exact solution, ——— unstable solution.
Institute of Chemistry, University of Graz, Austria
Stability Plot - Selection of Lagrange Multiplier
Stability plot – point of inflection
Typical influence of
real space
on the solution in
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
11
Selection of Parameters
With the point of inflection method we can find the optimum value for . Still open is the
problem how to select Dmax and the number of spline functions N.
Choice of Dmax:
The sampling theorem of Fourier transformation gives a clear answer to the question of
largest possible particle size. If the scattering curve is sampled at increments ∆q ≤ qmin
starting at qmin, the scattering data contain full information for all particles with maximum
dimension Dmax
Dmax =
π
qmin
In practice on will try to stay below this limit, i.e.
qmin <
π
D
and ∆q
qmin
Number of Spline Functions N:
The stabilizing routine makes the procedure nearly independent of N , in practice N is chosen
in the limits 10 ≤ N ≤ 40 depending on the measured q – range and on the statistical accuracy
of the data
Institute of Chemistry, University of Graz, Austria
Application of IFT to simulated slit collimation data
Fit and desmeared scattering curve of
a sphere simulated with slit collimation
Example of an extreme cut-off
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
12
Other IFT Applications - Overview
The IFT technique can also be applied to data from cylindrical or lamellar particles as
well as to polydisperse systems
Institute of Chemistry, University of Graz, Austria
Other IFT Applications - Equations
Summary of the different transforms T1 used in IFT:
Arbitrary shape:
I ( q) = 4π
∞
ÿ p (r)
0
Cylindrical Symmetry:
2π 2 L
I (q) =
q
Lamellar Symmetry:
∞
ÿ p ( r ) J ( qr ) dr
c
sin (qr )
dr
qr
0
I plane ( q ) =
0
∞
4π A
pt ( r ) cos ( qr ) dr
q 2 ÿ0
Polydisperse Systemsnumber (volume distribution):
∞
I ( q ) = cv ÿ Dv ( R ) ⋅ R 3 ⋅ P0 ( q, R ) dR
0
The structure is the same for all equations, just the kernels of the integrals differ!
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
13
Deconvolution of the PDDF (Convolution Square Root)
We have seen how inhomogeneities influence the scattering functions I(q) and
p(r), but until now we have no solution for the inverse problem, i.e. how to
determine the structure of the particles from these functions.
There is no general solution for this problem for arbitrary three-dimensional
structures.
Such methods do exist, however, for:
•spherical symmetry,
•circular cylinders with centro-symmetric radial density distributions (no
angular or axial dependence of the density) and for
•centro-symmetric lamellae without in-plane inhomogeneities.
It is obvious that we do not lose any information by spatial averaging in the
case of spherical symmetry. So we may hope that there is a chance to
determine the one-dimensional information ∆ρ(r) from the measured onedimensional function I(q). We discuss the solution of this problem in the
following section.
Institute of Chemistry, University of Graz, Austria
Deconvolution of the PDDF – The Magic Square
The Magic square of small-angle scattering: The correlations between the radial
density ∆ρ(r) and the PDDF p(r) and their Fourier transforms, the scattering amplitude
F(q) and scattering intensity I(q) under the assumption of spherical symmetry.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
14
Deconvolution of the PDDF – Principles I
Here we are facing a similar situation as in the IFT method: for a given density
distribution ρ(r) we can calculate the exact p(r)-function for all three cases
(spherical, cylindrical and lamellar symmetry) by a convolution square
operation but we do not have a useful description of the inverse problem, the
so-called convolution square root.
As an additional problem we have to keep in mind the fact, that the
convolution square operation is a nonlinear transformation which will not
allow an inversion by the solution of a simple linear least squares technique
like in the case of the indirect Fourier transformation.
We start again with a series expansion of the radial density function ρ(r) in the
usual way:
ρ (r) =
N
i =1
ci ϕ i ( r )
Institute of Chemistry, University of Graz, Austria
Deconvolution of the PDDF – Principles II
The approximation for the density profile corresponds to an approximation to the PDDF:
p (r) =
N
i =1
Vii ( r ) ci2 + 2
i >k
Vik ( r ) ci ck
The overlap integrals Vik(r) describe the overlapping of the i-th with the k-th step or shell
where one function has been shifted an arbitrary distance r . These overlap or
convolution integrals are very simple for the planar case (one-dimensional convolution
of two step function leads simply to a triangle) but are a bit more complicated for the
cylindrical and spherical case:
Illustration of the five sub-regions for the
calculation of the overlap integrals Vik(r).
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
15
Deconvolution of the PDDF – Iterative Solution
The above equation for the PDDF is nonlinear in its coefficients ci. The
corresponding least squares problem has to be linearized by a series expansion
where higher order terms are omitted.
Such linearized systems must be solved iteratively. In addition one needs
starting values ci(0) for the first iteration. Here we set all coefficients equal to a
constant.
We then calculate the difference function
∆p ( r ) = p ( r ) − p (
o)
(r )
which would be zero only if we would know the exact coefficients ci.
Now we calculate correction terms ∆ci in order to minimize ∆p(r) in a least square
sense.
N
i =1
Vii ( r )
( ci + ∆ci )
2
+2
i >k
Vik ( r ) ( ci + ∆ci )( ck + ∆ck ) − ci ck = ∆p ( r )
Institute of Chemistry, University of Graz, Austria
Deconvolution of the PDDF – Iterative Solution II
We linearize this equation by omitting the second order terms ∆ci2 and ∆ci∆ck
and we get
N
N
2
k =1 i =1
ci Vik ( rj ) ∆ck = ∆p ( rj )
for j = 1,2,3,... M and M > N. These equations can be written in matrix notation
Ajk ∆ck = ∆p j or A ∆c ( ) = ∆p(
0
0)
where the matrix elements Ajk are given by
Ajk = 2
N
i =1
ci Vik ( rj )
This system is solved with a weighted least squares condition considering the
standard deviations of the function ∆p(r) and we get the correction terms ∆c.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
16
Deconvolution of the PDDF – Iterative Solution III
They allow the calculation of improved coefficients ci(1):
ci( ) = ci( ) + ∆ci
1
0
and with these coefficients we start the next iteration, get further improvements
and if this iterative procedure converges we have solved the problem.
This problem is, however, again an ill-posed problem so that we have to add again
a stabilization criterion and we have to solve the nonlinear problem by iteration for
every Lagrange multiplier.
Many applications performed in the meantime have shown that the deconvolution
technique works well in combination with the indirect transformation method, also
in cases where the conditions of symmetry are not perfectly fulfilled.
Institute of Chemistry, University of Graz, Austria
Deconvolution of the PDDF – Application
Example of an inhomogeneous oblate spheroid with an axial ratio of 1:1.2:1.2.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
17
Surfactant Systems - Spherical micelles
Scattering pattern of spherical aggregates with radius R. The scattering curve I(q) is the
Fourier transform of the PDDF p(r). This PDDF is the convolution square of the radial
density distribution ∆ρ(r) .
Institute of Chemistry, University of Graz, Austria
Surfactant Systems - Rod-like micelles I
Scattering function I(q) and PDDF p(r) for cylindrical aggregates
with radius R and length L.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
18
Surfactant Systems - Rod-like micelles. II: Cross-section
Cross-section functions I(q)q, pc(r) and ∆ρc(r) for cylindrical aggregates.
Institute of Chemistry, University of Graz, Austria
Surfactant Systems - Lamellar Systems I
Scattering function I(q) and PDDF p(r) for vesicles and planar aggregates (L3 phase).
Full lines: typical results for lamellar structures; dashed lines: functions for
monodisperse vesicles.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
19
Surfactant Systems - Lamellar Systems II
Thickness functions I(q)q2, pt(r) and ∆ρt(r) other planar aggregates with the
thickness T.
Institute of Chemistry, University of Graz, Austria
IFT Application: Lipid IVA Vesicles
Lipid IV A is a bioactive precursor of
Lipid IV, which is most important for the
physiological activity of the cells. Lipid
IVA has a disaccharide as hydrophilic
head group and four hydrophobic side
chains.
Lipid IVA: experimental data points (ooo) and
fitting function (——).
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
20
IFT Application: Lipid IVA Vesicles
PDDF of the Lipid IVA vesicles.
Thickness Guinier plot for the
desmeared data from Lipid IVA.
Institute of Chemistry, University of Graz, Austria
IFT Application: Lipid IVA Vesicles
Scattering curve I(q) desmeared under
the assumption of an extended lamellar
structure.
Thickness-PDDF, calculated for Dmax of 6
nm (——) and 9 nm (-----).
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
21
IFT Application: Lipid IVA Vesicles
Electron density distribution ρ(r) calculated
with 7 equidistant steps (——) and the
optimized 2-step model (-----).
Thickness PDDF pt(r) calculated from
Lipid IV A data (ooo) and fit by the
convolution square root technique (—)
model with 7 steps.
Institute of Chemistry, University of Graz, Austria
Institute of Chemistry, University of Graz,
Austria
22