.. _fractal:

fractal
=======================================================

Calculates the scattering from fractal-like aggregates of spheres following theTexiera reference.

=========== ====================================== ============ =============
Parameter   Description                            Units        Default value
=========== ====================================== ============ =============
scale       Scale factor or Volume fraction        None                     1
background  Source background                      |cm^-1|              0.001
volfraction volume fraction of blocks              None                  0.05
radius      radius of particles                    |Ang|                    5
fractal_dim fractal dimension                      None                     2
cor_length  cluster correlation length             |Ang|                  100
sld_block   scattering length density of particles |1e-6Ang^-2|             2
sld_solvent scattering length density of solvent   |1e-6Ang^-2|           6.4
=========== ====================================== ============ =============

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


**Definition**
This model calculates the scattering from fractal-like aggregates of spherical
building blocks according the following equation:

.. math::

    I(q) = \phi\ V_\text{block} (\rho_\text{block}
          - \rho_\text{solvent})^2 P(q)S(q) + \text{background}

where $\phi$ is The volume fraction of the spherical "building block" particles
of radius $R_0$, $V_{block}$ is the volume of a single building block,
$\rho_{solvent}$ is the scattering length density of the solvent, and
$\rho_{block}$ is the scattering length density of the building blocks, and
P(q), S(q) are the scattering from randomly distributed spherical particles
(the building blocks) and the interference from such building blocks organized
in a fractal-like clusters.  P(q) and S(q) are calculated as:

.. math::

    P(q)&= F(qR_0)^2 \\
    F(q)&= \frac{3 (\sin x - x \cos x)}{x^3} \\
    V_\text{particle} &= \frac{4}{3}\ \pi R_0 \\
    S(q) &= 1 + \frac{D_f\  \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\  ]^{(D_f -1)/2}}
    \frac{\sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}}

where $\xi$ is the correlation length representing the cluster size and $D_f$
is the fractal dimension, representing the self similarity of the structure.
Note that S(q) here goes negative if $D_f$ is too large, and the Gamma function
diverges at $D_f=0$ and $D_f=1$.

**Polydispersity on the radius is provided for.**

For 2D data: The 2D scattering intensity is calculated in the same way as
1D, where the *q* vector is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}



.. figure:: img/fractal_autogenfig.png

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


**Source**

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

**References**

#.  J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Converted to sasmodels by:** Paul Butler **Date:** March 19, 2016
* **Last Modified by:** Paul Butler **Date:** March 12, 2017
* **Last Reviewed by:** Paul Butler **Date:** March 12, 2017