Surface Fractal

This model calculates the scattering from fractal-like aggregates based on the Mildner reference.

Definition

The scattering intensity $I(q)$ is calculated as

$$ \begin{align*} I(q) = \text{scale} \times P(q)S(q) + \text{background} \\ P(q) = F(qR)^2 \\ F(x) = \frac{3\left[\sin(x)-x\cos(x)\right]}{x^3} \\ S(q) = \Gamma(5-D_S)\xi^{\,5-D_S}\left[1+(q\xi)^2 \right]^{-(5-D_S)/2} \sin\left[-(5-D_S) \tan^{-1}(q\xi) \right] q^{-1} \\ \text{scale} = \text{scale factor}\, N V^1(\rho_\text{particle} - \rho_\text{solvent})^2 \\ V = \frac{4}{3}\pi R^3 \end{align*}
$$
where $R$ is the radius of the building block, $D_S$ is the **surface** fractal dimension, $\xi$ is the cut-off length, $\rho_\text{solvent}$ is the scattering length density of the solvent and $\rho_\text{particle}$ is the scattering length density of particles.

.. note::

The surface fractal dimension is only valid if $1<D_S<3$. The result is only valid over a limited $q$ range, $\tfrac{5}{3-D_S}\xi^{\,-1} < q < R^{-1}$. See the reference for details.

References

D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545

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