Calculation of g(r) Functions from Small-Angle Scattering Data by ...

**Calculation** **of** g(r) **Functions** **from**

**Small**-**Angle** **Scattering** **Data** **by**

Means **of** Indirect Fourier

Transformation

Gerhard FRITZ

Institute **of** Chemistry

Karl-Franzens University Graz, Austria

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 1/22

Introduction

**Scattering** generally for particle size, shape, internal

structure. Typically models or model-free description **of**

size and shape, but always models for interaction effects.

Sometimes, however, interest in interaction effects.

Interaction effects can be described **by** pair correlation

function g(r): Probability to find a neighbour at distance r

relative mean particle density.

New technique largely model free for g(r) function. Needs

model assumption for particle size and shape.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 2/22

Theory

Intensity product **of** form factor P(q) and structure factor

S(q)

I(q) = n∆ρ 2 mM 2 P(q)S(q) (1)

S(q) is the Fourier transform **of** g(r)

∞

S(q) = 1 + 4πn (g(r) − 1)r 2sin(qr)

dr (2)

qr

0

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 3/22

IFT Basics I

g(r) can be represented as a sum **of** cubic B-splines:

(g(r) − 1)r 2 =

N

ciϕi(r) (3)

Fourier transformation leads to:

∞

ψi(q) = 4π ϕi(r) sin(qr)

dr (4)

qr

0

S(q) = 1 + n

i=1

N

ciψi(q) (5)

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 4/22

i=1

IFT Basics II

The intensity I(q) is therefore

I(q) = n∆ρ 2 mM 2 P(q)

1 + n

N

ciψi(q)

i=1

Which is a simple linear equation system with

ξi(q) = P(q)ψi(q). It can be solved for the coefficients

f0 = n∆ρ 2 mM 2 and fi = n 2 ∆ρ 2 mM 2 ci

I(q) = f0P(q) +

(6)

N

fiξi(q) (7)

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 5/22

i=1

Forcing g(r) = 0 for **Small** r-values I

Equation system more stable if g(r) = 0 for r < l.

Adding a function

ζ(r) =

⎧

⎨

⎩

−r 2 if r ≤ l;

0 if r > l.

and defining the splines only in the regime r > l leads to

(g(r) − 1)r 2 = ζ(r) +

(8)

N

ciϕi(r) (9)

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 6/22

i=1

Forcing g(r) = 0 for **Small** r-values II

The equivalent in reciprocal space is

I(q) = f0 (P(q) + nP(q)κ(q)) +

N

fiξi(q) (10)

i=1

With ξ0(q) = (P(q) + nP(q)κ(q)) one gets the linear

equation system

I(q) =

N

fiξi(q) (11)

i=0

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 7/22

Instrumental Broadening

Instrumental broadening effects are linear operations. It is

sufficient to apply convolutions with beam pr**of**iles to the

Fourier transformed basis functions:

Iexp(q) ≈

Ĩ(q) =

N

i=0

fi ˜ ξi(q) (12)

The equation system can be solved, but a stabilisation is

taken into account.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 8/22

Overview Method

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 9/22

Test on Simulated **Data**

Spheres, radius = 5 nm,

slit smearing, statistical

noise

Charged spheres, 50

electron charges per

particle in water at

298K, volume

fraction 0.1

Hard spheres, volu-

me fraction 0.01.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 10/22

Resulting g(r) Function

Good estimation for g(r), even if P(q) is not perfect.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 11/22

Effect **of** g(r) = 0 Condition

Forcing g(r) = 0 necessary for small r, but only minor

effect on large r.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 12/22

Experimental Test

PEO-PPO-PEO triblock copolymer P94. Forms spherical

micelles at 40 ◦ C. Structures can be described **by** effective

hard sphere model. Effective volume fraction 2.05 times

weight fraction in H2O

SANS experiments on solutions **from** 0.33% up to 15%

(Corresponding to effective volume fractions **of** 0.7% up to

31.5%).

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 13/22

Determination **of** P(q)

P(q) estimated **from** dilute( 0.33%) solution.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 14/22

Determination **of** n

n estimated **from** particle diameter (**from** P(q)), and

effective volume fraction determined **by** viscosimetry.

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 15/22

Approximations

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 16/22

Resulting g(r) functions

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 17/22

Resulting g(r) functions

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 17/22

Conclusions

Pair correlation function g(r) can be determined

without model for interactions.

Simulated and experimental data show viability **of**

approach

No deconvolution **of** scattering curve and beam pr**of**iles

necessary

No extrapolations needed, basis functions are defined

analytically

G. Fritz, J. Chem. Phys. (2006) 124 214707(1-6)

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 18/22

Solving the Equation System I

Expansion coefficients:

dν =

Co-variance matrix:

Bµν =

qmax

qmin

qmax

qmin

ψν(q)Iexp(q)

σ2 dq (13)

(q)

ψµ(q)ψν(q)

σ2 dq (14)

(q)

Bf = d (15)

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 19/22

Solving the Equation System II

Main condition: Minimise mean deviation MD

qmax Iexp(q) −

MD =

Ĩ(q)

2 σ2 (q)

qmin

dq (16)

Side condition: Minimise difference between neighbouring

splines:

Nf ′ =

N−1

i=1

(fi+1 − fi) 2

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 20/22

(17)

Solving the Equation System III

K =

⎛

⎜

⎝

0 0 0 0 · · · 0 0

0 1 −1 0 · · · 0 0

0 −1 2 −1 · · · 0 0

0 0 −1 2 · · · 0 0

. . . .

. .. . .

0 0 0 0 · · · 2 −1

0 0 0 0 · · · −1 1

⎞

⎟

⎠

(18)

(B + λK)f = d (19)

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 21/22

Determining l

**Calculation** **of** g(r) **Functions** **from** **Small**-**Angle** **Scattering** **Data** **by** Means **of** Indirect Fourier Transformation – p. 22/22