.. _surface-fractal:

surface_fractal
=======================================================

Fractal-like aggregates based on the Mildner reference

================ =============================== ======= =============
Parameter        Description                     Units   Default value
================ =============================== ======= =============
scale            Scale factor or Volume fraction None                1
background       Source background               |cm^-1|         0.001
radius           Particle radius                 |Ang|              10
fractal_dim_surf Surface fractal dimension       None                2
cutoff_length    Cut-off Length                  |Ang|             500
================ =============================== ======= =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


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

**Definition**

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

.. math::
    :nowrap:

    \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.



.. figure:: img/surface_fractal_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`surface_fractal.py <src/surface_fractal.py>`
$\ \star\ $ :download:`surface_fractal.c <src/surface_fractal.c>`
$\ \star\ $ :download:`sas_gamma.c <src/sas_gamma.c>`
$\ \star\ $ :download:`sas_3j1x_x.c <src/sas_3j1x_x.c>`

**References**

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

**Authorship and Verification**

* **Author:**
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