A7 - correlation function theory

introduction

SAS is assumed to be elastic, so the momentum transfer can be directly related to the wavelength, ie, spatial distance
the correlation function $Γ (\vec{r})$ relates the scattering intensity in terms this spatial distance
$Γ (\vec{r})$ is proportional to the pairwise product of scattering length densities (SLDs) for all points separated by $\vec{r}$ summed over all orientations and locations
it can be thought og as SLD without the phase information
correlation function take a 3D vector input, but the measurements from SAS are limited to 1/2D
$Γ_{1}$ looks at changes in a single direction perpendicular to the beam, averaging the other directions
$Γ_{3}$ is described by the Debye equation
lorentz correction is made

I (q) = q^{2} I_{m e a s u r e d} (q)

the correlation function arises from calculating the square magnitude of the 3D fourier transform:

\frac{d σ}{d Ω} \propto F (\vec{q}) F^{*} (\vec{q}) = \int [\int ρ (\vec{s}) ρ (\vec{s} + \vec{r}) d s^{3}] \exp (i \vec{r} \cdot \vec{q}) d r^{3}

where, $F (q) = \int ρ (r) \exp (i \vec{r} \cdot \vec{q}) d r^{3}$

γ (\vec{r}) = \int ρ (\vec{s}) ρ (\vec{s} + \vec{r}) d s^{3}

they contain no phase information, ie. its fourier transform is purely real and it is an even function that can be written solely in terms of cosines
- this is as $Z = a \cos θ + i b \sin θ$ , with $b = 0$ for real values

γ (\vec{r}) = \int ρ (\vec{s}) ρ (\vec{s} + \vec{r}) d s^{3} = \int (\vec{u} - \vec{r}) ρ (\vec{u}) d u^{3} = γ (- \vec{r})

\begin{aligned} \int_{- \infty}^{\infty} f (x) e^{i x ξ} d x & = \int_{- \infty}^{0} f (x) e^{i x ξ} d x + \int_{- 0}^{\infty} f (x) e^{i x ξ} d x \\ = \int_{0}^{\infty} f (u) e^{- i u ξ} d u + \int_{- 0}^{\infty} f (x) e^{i x ξ} d x \\ = 2 \int_{0}^{\infty} f (x) \frac{e^{i x ξ}) + e^{- i x ξ}}{2} d x \\ = \int_{- \infty}^{\infty} f (x) \cos (x ξ) d x \end{aligned}

∭ γ (\vec{r}) e^{i \vec{r} \cdot q} d x d y d z

I (q) = \int γ (\vec{r}) e^{i x q_{x}} d x d y d z = \int (\iint γ (\vec{r}) d y d z) e^{i x q_{x}} d x

I (q) = \int Γ_{1} (x) \cos (q x d x)

Γ_{1} (x) = \int I (q) \cos (q x) d x

I (q) = \int_{R^{3}} γ (\vec{)} \exp (i \vec{r} \cdot \vec{q}) d r^{3}

I (\vec{q}) = \frac{1}{4 π} \int_{ϕ = 0}^{2 π} \int_{θ = 0}^{π} \int_{\vec{r} \in R^{3}} γ (\vec{r}) \exp (i q r \cos θ) d {\vec{r}}^{3} \sin θ d θ d ϕ

I (q) = \int_{\vec{r} \in R^{3}} γ (\vec{r}) \frac{\sin (q r)}{q r} d r

I (q) = \int_{0}^{\infty} \int_{Ω} γ (\vec{r}) d Ω sinc (q r) d r

Γ_{3} = \int_{Ω} γ (\vec{r}) d Ω

\begin{aligned} x Γ_{3} & = \int I (q) \frac{\sin (q x)}{q} d q \\ \frac{d}{d x} (x Γ_{3}) & = \frac{d}{d x} \int I (q) \frac{\sin (q x)}{q} d q = \int I (q) \cos (q x) d q \\ ∴ Γ_{3} (x) & = \int_{0}^{x} \frac{Γ_{1} (r)}{r} d r \end{aligned}