{ "metadata": { "name": "", "signature": "sha256:9b5acbd8a84cd2f0b67ae101b543e9826a738aec5424398ec4ce06557c7dc15a" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "#Relationship Between XRD and Autocorrelation\n", "\n", "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 $\\rho(\\pmb x)$).\n", "\n", "$$A(\\pmb{k})= \\int \\rho(\\pmb x) e^{i\\pmb{kx}} d\\pmb x$$\n", "\n", "$$I(\\pmb k) = |A(\\pmb k)| = A(\\pmb k) (A(\\pmb k))^*$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $(A(\\pmb k ))^*$ is the complex conjugate of $A(\\pmb k)$. $I(\\pmb k)$ and $\\rho(\\pmb x)$ have the following relationship." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$I(\\pmb k) = \\int \\rho(\\pmb x) e^{i\\pmb{kx}} dx \\int \\rho(\\pmb y) e^{-i\\pmb {ky}} d\\pmb y$$\n", "\n", "$$I(\\pmb k) = \\int \\int \\rho(\\pmb x) \\rho(\\pmb y) e^{i\\pmb k(\\pmb x - \\pmb y)} d\\pmb x d\\pmb y$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now lets make a change of variable and let\n", "\n", "$$\\pmb z = \\pmb x - \\pmb y$$\n", "\n", "Substituting this relationship back in the the equation we get.\n", "\n", "$$I(\\pmb k) = \\int \\int \\rho(\\pmb z + \\pmb y) \\rho(\\pmb y) e^{i\\pmb k\\pmb z} d\\pmb z d\\pmb y = \\int \\Gamma_{\\rho}(\\pmb z) e^{i\\pmb k\\pmb z} d\\pmb z$$\n", "$$I(\\pmb k) = \\mathscr{F} \\Big \\{ \\Gamma_{\\rho}(\\pmb z)\\Big \\}$$\n", "\n", "where\n", "\n", "$$\\Gamma_{\\rho}(\\pmb z) = \\int \\rho( \\pmb x) \\rho( \\pmb x + \\pmb z) d \\pmb x$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Therefore the autocorrelaiton of the electron density $\\rho(\\pmb x)$ equal to the inverse Fourier transform of the intensity $I(\\pmb k)$.\n", "\n", "$$\\Gamma_{\\rho}(\\pmb z) = \\mathscr{F}^{-1} \\{ I(\\pmb k) \\}$$\n", "\n", "$$\\Gamma_{\\rho}(\\pmb z)= \\int I(\\pmb k) e^{-i\\pmb k\\pmb z} d\\pmb k \\space \\space \\space \\space \\pmb{(1)}$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\\Gamma_{\\rho}(\\pmb z) = \\int I(\\pmb k) [cos(\\pmb k \\pmb z) - i sin(\\pmb k \\pmb z)] d\\pmb k$$\n", "\n", "$$\\Gamma_{\\rho}(\\pmb z) = \\int I(\\pmb k) cos(\\pmb k\\pmb z)d\\pmb z - i \\int I sin(\\pmb k\\pmb z)d\\pmb k$$\n", "\n", "In the case that $I(\\pmb k)$ is symetric integral containing $sin(\\pmb k \\pmb z)$ will always be equal to zero.\n", "\n", "$$\\Gamma_{\\rho}(\\pmb z) = \\int I(\\pmb k) cos(\\pmb k\\pmb z)d\\pmb k$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Autocorrelation of Local State One from SAXS DATA\n", "\n", "SAXS data provides the difference in electron density in the range of 1 to 1,000 nm. \n", "This difference can be represented as deviation from the mean electron density, $\\bar{\\rho}$ as \n", "shown below.\n", "\n", "$$\\eta( \\pmb z) = \\rho( \\pmb z) - \\bar \\rho$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let $\\Gamma_{\\eta} (\\pmb x)$ represent the autocorrelation of $\\eta ( \\pmb x)$.\n", "\n", "$$\\Gamma_{\\eta} (\\pmb x) = \\int \\eta ( \\pmb y) \\eta ( \\pmb y + \\pmb x) d \\pmb y$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let $\\gamma (\\pmb x)$ represent the normalized autocorrelation of $\\eta ( \\pmb x)$ and is referred to as the Debye correlation function [1].\n", "\n", "$$\\gamma (\\pmb x) = \\frac{\\Gamma_{\\eta} (\\pmb x)}{\\Gamma_{\\eta} (0)}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The normalization constant is\n", "\n", "$$\\Gamma_{\\eta}(0) = \\int \\eta ( \\pmb y) \\eta ( \\pmb y + \\pmb 0) d \\pmb y = V \\eta_o^2$$\n", "\n", "where $V$ is the scattering volume and $\\eta_o^2$ is the mean square perturbation of the scattering length density throughout the system [1]." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The relationship between the Debye correlation function and the correlation function of $\\eta( \\pmb x)$ can be expressed as follows.\n", "\n", "$$\\Gamma_{\\eta} (\\pmb x) = V \\eta_o^2 \\gamma (\\pmb x)$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This new definition allows us to write equation $(1)$ in terms of $\\gamma (\\pmb x)$.\n", "\n", "$$\\gamma (\\pmb x) = \\frac{1}{V \\eta_o^2} \\int I ( \\pmb k) e^{-i\\pmb k\\pmb z} d\\pmb k \\space \\space \\space \\space (2)$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The $\\gamma (\\pmb x)$ is related to the autocorrelation for local state one was shown below [2].\n", "\n", "$$\\gamma (\\pmb x) = f^{11}( \\pmb x) - V_1^2 \\space \\space \\space \\space (3)$$\n", "\n", "where $f^{11}( \\pmb x)$ is the autocorrelation of local state one and $V_1$ is the volume fraction of local state one." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By combining of equations $(2)$ and $(3)$ provides the relationship between the 2-point statistics, X-ray scattering intensity and volume fraction.\n", "\n", "$$f^{11}( \\pmb x) = \\frac{1}{V \\eta_o^2} \\int I ( \\pmb k) e^{-i\\pmb k\\pmb x} d\\pmb k + V_1^2$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#References\n", "\n", "[1] R. J. Roe, *Methods of X-Ray and Neutron Scattering in Polymer Science*, 174-176, Oxford University Press 2000.\n", "\n", "[2] 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](http://dx.doi.org/doi:10.1016/j.actamat.2011.01.013)." ] } ], "metadata": {} } ] }