# Fractal (Mildner)

## Description:

Calculates the scattering from fractal-like clustering of form factor normalised data. Equation and deriveation from Mildner reference.

Definition
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The functional form of the structure factor is defined below. It is important to note that this structure factor calculates the the scattering in the mass fractal regime.

$$S(q) = \frac{\Gamma(D_m-1)\xi^{D_m-1}}{\left[1+(q\xi)^2 \right]^{(D_m-1)/2}} \frac{sin\left[(D_m - 1) tan^{-1}(q\xi) \right]}{q}$$

Where D_m is the mass fractal dimension and \xi is the correlation length. In this case the correlation length refers to the cutoff length i.e. the cutoff length at which the clustering is no longer fractal-like.

#Note:

Addresses Sasmodels Issue #130, where this S(q) is derived from fractal and mass_fractal's S(q) term.

Has radius_effective and volfraction as they are required by sasmodels/product.py (182-186) but are unused in the actual calculation.

We would potentially like to add this to sasmodels permanently.

References
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D Mildner and P Hall, *J. Phys. D: Appl. Phys.*,
19 (1986) 1535-1545 Equation(9)

Wong, P. (1999). Methods in the physics of porous media . San Diego; London: Academic.

Authorship and Verification
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* **Author:** Ziggy Attala and Matt D G Hughes **Date:** 09/09/2019