# Fractal

## Description:

Definition

This model calculates the scattering from fractal-like aggregates of spherical building blocks according the following equation:

$$I(q) = \phi\ V_\text{block} (\rho_\text{block} - \rho_\text{solvent})^2 P(q)S(q) + \text{background}$$
where $\phi$ is The volume fraction of the spherical "building block" particles of radius $R_0$, $V_{block}$ is the volume of a single building block, $\rho_{solvent}$ is the scattering length density of the solvent, and $\rho_{block}$ is the scattering length density of the building blocks, and P(q), S(q) are the scattering from randomly distributed spherical particles (the building blocks) and the interference from such building blocks organized in a fractal-like clusters. P(q) and S(q) are calculated as:

$$P(q)= F(qR_0)^2 \\ F(q)= \frac{3 (\sin x - x \cos x)}{x^3} \\ V_\text{particle} = \frac{4}{3}\ \pi R_0 \\ S(q) = 1 + \frac{D_f\ \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\ ]^{(D_f -1)/2}} \frac{\sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}}$$
where $\xi$ is the correlation length representing the cluster size and $D_f$ is the fractal dimension, representing the self similarity of the structure. Note that S(q) here goes negative if $D_f$ is too large, and the Gamma function diverges at $D_f=0$ and $D_f=1$.

**Polydispersity on the radius is provided for.**

For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as

$$q = \sqrt{q_x^2 + q_y^2}$$

References

J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785

Authorship and Verification

**Author:** NIST IGOR/DANSE **Date:** pre 2010
**Converted to sasmodels by:** Paul Butler **Date:** March 19, 2016