# Relationship Between XRD and Autocorrelation¶

X-ray Scattering plots provide the intensity of X-ray signal defined as $I(\pmb k)$. The intensity can be related to the Fourier transform of the electron density denoted as $A$ (the electron density will be represented as $\rho(\pmb x)$).

$$A(\pmb{k})= \int \rho(\pmb x) e^{i\pmb{kx}} d\pmb x$$$$I(\pmb k) = |A(\pmb k)| = A(\pmb k) (A(\pmb k))^*$$

where $(A(\pmb k ))^*$ is the complex conjugate of $A(\pmb k)$. $I(\pmb k)$ and $\rho(\pmb x)$ have the following relationship.

$$I(\pmb k) = \int \rho(\pmb x) e^{i\pmb{kx}} dx \int \rho(\pmb y) e^{-i\pmb {ky}} d\pmb y$$$$I(\pmb k) = \int \int \rho(\pmb x) \rho(\pmb y) e^{i\pmb k(\pmb x - \pmb y)} d\pmb x d\pmb y$$

Now lets make a change of variable and let

$$\pmb z = \pmb x - \pmb y$$

Substituting this relationship back in the the equation we get.

$$I(\pmb k) = \int \int \rho(\pmb z + \pmb y) \rho(\pmb y) e^{i\pmb k\pmb z} d\pmb z d\pmb y = \int \Gamma_{\rho}(\pmb z) e^{i\pmb k\pmb z} d\pmb z$$$$I(\pmb k) = \mathscr{F} \Big \{ \Gamma_{\rho}(\pmb z)\Big \}$$

where

$$\Gamma_{\rho}(\pmb z) = \int \rho( \pmb x) \rho( \pmb x + \pmb z) d \pmb x$$

Therefore the autocorrelaiton of the electron density $\rho(\pmb x)$ equal to the inverse Fourier transform of the intensity $I(\pmb k)$.

$$\Gamma_{\rho}(\pmb z) = \mathscr{F}^{-1} \{ I(\pmb k) \}$$$$\Gamma_{\rho}(\pmb z)= \int I(\pmb k) e^{-i\pmb k\pmb z} d\pmb k \space \space \space \space \pmb{(1)}$$

$$\Gamma_{\rho}(\pmb z) = \int I(\pmb k) [cos(\pmb k \pmb z) - i sin(\pmb k \pmb z)] d\pmb k$$$$\Gamma_{\rho}(\pmb z) = \int I(\pmb k) cos(\pmb k\pmb z)d\pmb z - i \int I sin(\pmb k\pmb z)d\pmb k$$

In the case that $I(\pmb k)$ is symetric integral containing $sin(\pmb k \pmb z)$ will always be equal to zero.

$$\Gamma_{\rho}(\pmb z) = \int I(\pmb k) cos(\pmb k\pmb z)d\pmb k$$

# Autocorrelation of Local State One from SAXS DATA¶

SAXS data provides the difference in electron density in the range of 1 to 1,000 nm. This difference can be represented as deviation from the mean electron density, $\bar{\rho}$ as shown below.

$$\eta( \pmb z) = \rho( \pmb z) - \bar \rho$$

Let $\Gamma_{\eta} (\pmb x)$ represent the autocorrelation of $\eta ( \pmb x)$.

$$\Gamma_{\eta} (\pmb x) = \int \eta ( \pmb y) \eta ( \pmb y + \pmb x) d \pmb y$$

Let $\gamma (\pmb x)$ represent the normalized autocorrelation of $\eta ( \pmb x)$ and is referred to as the Debye correlation function .

$$\gamma (\pmb x) = \frac{\Gamma_{\eta} (\pmb x)}{\Gamma_{\eta} (0)}$$

The normalization constant is

$$\Gamma_{\eta}(0) = \int \eta ( \pmb y) \eta ( \pmb y + \pmb 0) d \pmb y = V \eta_o^2$$

where $V$ is the scattering volume and $\eta_o^2$ is the mean square perturbation of the scattering length density throughout the system .

The relationship between the Debye correlation function and the correlation function of $\eta( \pmb x)$ can be expressed as follows.

$$\Gamma_{\eta} (\pmb x) = V \eta_o^2 \gamma (\pmb x)$$

This new definition allows us to write equation $(1)$ in terms of $\gamma (\pmb x)$.

$$\gamma (\pmb x) = \frac{1}{V \eta_o^2} \int I ( \pmb k) e^{-i\pmb k\pmb z} d\pmb k \space \space \space \space (2)$$

The $\gamma (\pmb x)$ is related to the autocorrelation for local state one was shown below .

$$\gamma (\pmb x) = f^{11}( \pmb x) - V_1^2 \space \space \space \space (3)$$

where $f^{11}( \pmb x)$ is the autocorrelation of local state one and $V_1$ is the volume fraction of local state one.

By combining of equations $(2)$ and $(3)$ provides the relationship between the 2-point statistics, X-ray scattering intensity and volume fraction.

$$f^{11}( \pmb x) = \frac{1}{V \eta_o^2} \int I ( \pmb k) e^{-i\pmb k\pmb x} d\pmb k + V_1^2$$

 R. J. Roe, Methods of X-Ray and Neutron Scattering in Polymer Science, 174-176, Oxford University Press 2000.

 M. Baniassadi, F. Addiego, F. Hassouna, A. Laachachi, A. Makradi, S. Ahzi, V. Toniazzo, H. Garmestani, D. Ruch Using SAXS approach to estimate thermal conductivity of polystyrene/zirconia nanocomposite by exploiting strong contrast technique, Acta Mater (2011) doi:10.1016/j.actamat.2011.01.013.