.. _mass-fractal:

mass_fractal
=======================================================

Mass Fractal model

================ =============================== ======= =============
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_mass Mass fractal dimension          None              1.9
cutoff_length    Cut-off length                  |Ang|             100
================ =============================== ======= =============

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


Calculates the scattering from fractal-like aggregates based on
the Mildner reference.

**Definition**

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

.. math::

    I(q) = scale \times P(q)S(q) + background

.. math::

    P(q) = F(qR)^2

.. math::

    F(x) = \frac{3\left[sin(x)-xcos(x)\right]}{x^3}

.. math::

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

.. math::

    scale = scale\_factor \times NV^2(\rho_\text{particle} - \rho_\text{solvent})^2

.. math::

    V = \frac{4}{3}\pi R^3

where $R$ is the radius of the building block, $D_m$ is the **mass** fractal
dimension, $\zeta$ 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 mass fractal dimension ( $D_m$ ) is only
    valid if $1 < mass\_dim < 6$. It is also only valid over a limited
    $q$ range (see the reference for details).



.. figure:: img/mass_fractal_autogenfig.png

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


**Source**

:download:`mass_fractal.py <src/mass_fractal.py>`
$\ \star\ $ :download:`mass_fractal.c <src/mass_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 Equation(9)

**Authorship and Verification**

* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**

