.. _mass-surface-fractal:

mass_surface_fractal
=======================================================

Mass Surface Fractal model

================ =================================== ======= =============
Parameter        Description                         Units   Default value
================ =================================== ======= =============
scale            Scale factor or Volume fraction     None                1
background       Source background                   |cm^-1|         0.001
fractal_dim_mass Mass fractal dimension              None              1.8
fractal_dim_surf Surface fractal dimension           None              2.3
rg_cluster       Cluster radius of gyration          |Ang|            4000
rg_primary       Primary particle radius of gyration |Ang|            86.7
================ =================================== ======= =============

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



A number of natural and commercial processes form high-surface area materials
as a result of the vapour-phase aggregation of primary particles.
Examples of such materials include soots, aerosols, and fume or pyrogenic
silicas. These are all characterised by cluster mass distributions (sometimes
also cluster size distributions) and internal surfaces that are fractal in
nature. The scattering from such materials displays two distinct breaks in
log-log representation, corresponding to the radius-of-gyration of the primary
particles, $rg$, and the radius-of-gyration of the clusters (aggregates),
$Rg$. Between these boundaries the scattering follows a power law related to
the mass fractal dimension, $Dm$, whilst above the high-Q boundary the
scattering follows a power law related to the surface fractal dimension of
the primary particles, $Ds$.

**Definition**

The scattered intensity I(q) is calculated using a modified
Ornstein-Zernicke equation

.. math::

    I(q) &= \text{scale} \times P(q) + \text{background} \\
    P(q) &= \left\{ \left[ 1+(q^2a)\right]^{D_m/2} \times
                   \left[ 1+(q^2b)\right]^{(6-D_s-D_m)/2}
           \right\}^{-1} \\
    a &= R_{g}^2/(3D_m/2) \\
    b &= r_{g}^2/[-3(D_s+D_m-6)/2] \\
    \text{scale} &= \text{scale factor}
        \times NV^2 (\rho_\text{particle} - \rho_\text{solvent})^2

where $R_g$ is the size of the cluster, $r_g$ is the size of the primary
particle, $D_s$ is the surface fractal dimension, $D_m$ is the mass fractal
dimension, $\rho_\text{solvent}$ is the scattering length density of the
solvent, and $\rho_\text{particle}$ is the scattering length density of
particles.

.. note::

    The surface ( $D_s$ ) and mass ( $D_m$ ) fractal dimensions are only
    valid if $0 < surface\_dim < 6$, $0 < mass\_dim < 6$, and
    $(surface\_dim + mass\_dim ) < 6$.
    Older versions of sasview may have the default primary particle radius
    larger than the cluster radius, this was an error, also present in the
    Schmidt review paper below. The primary particle should be the smaller
    as described in the original Hurd, et al., who also point out that
    polydispersity in the primary particle sizes may affect their
    apparent surface fractal dimension.



.. figure:: img/mass_surface_fractal_autogenfig.png

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


**Source**

:download:`mass_surface_fractal.py <src/mass_surface_fractal.py>`
$\ \star\ $ :download:`mass_surface_fractal.c <src/mass_surface_fractal.c>`

**References**

#. P Schmidt, *J Appl. Cryst.*, 24 (1991) 414-435 Equation(19)
#. A J Hurd, D W Schaefer, J E Martin, *Phys. Rev. A*,
   35 (1987) 2361-2364 Equation(2)

**Authorship and Verification**

* **Author:** Piotr Rozyczko **Date:** Jan 20, 2016
* **Last Modified by:**
* **Last Reviewed by:** Richard Heenan **Date:** May 30, 2018