.. _mass-fractal:

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

Mass Fractal model

================ ====================== ======= =============
Parameter        Description            Units   Default value
================ ====================== ======= =============
scale            Source intensity       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_{particle} - \rho_{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, $\rho_{solvent}$ is the scattering
length density of the solvent,
and $\rho_{particle}$ is the scattering length density of particles.

.. note::

    The mass fractal dimension ( $D_m$ ) is only
    valid if $0 < mass\_dim < 6$. 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.

**References**

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