source: sasmodels/sasmodels/models/mass_fractal.py @ 9c461c7

r"""
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_1d.jpg

    1D plot using the default values.

Reference
---------

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


"""

from numpy import inf

name = "mass_fractal"
title = "Mass Fractal model"
description = """
        The scattering intensity  I(x) = scale*P(x)*S(x) + background, where
        scale = scale_factor  * V * delta^(2)
        p(x)=  F(x*radius)^(2)
        F(x) = 3*[sin(x)-x cos(x)]/x**3
        S(x) = [(gamma(Dm-1)*colength^(Dm-1)*[1+(x^2*colength^2)]^((1-Dm)/2)
        * sin[(Dm-1)*arctan(x*colength)])/x]
        where delta = sldParticle -sldSolv.
        radius       =  Particle radius
        mass_dim  =  Mass fractal dimension
        cutoff_length  =  Cut-off length
        background   =  background
        Ref.:Mildner, Hall,J Phys D Appl Phys(1986), 9, 1535-1545
        Note I: This model is valid for 1<mass_dim<6.
        Note II: This model is not in absolute scale.
        """
category = "shape-independent"

#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["radius",        "Ang",  10.0, [0.0, inf], "", "Particle radius"],
              ["mass_dim",      "",      1.9, [1.0, 6.0], "", "Mass fractal dimension"],
              ["cutoff_length", "Ang", 100.0, [0.0, inf], "", "Cut-off length"],
              ]


source = ["lib/sph_j1c.c", "lib/lanczos_gamma.c", "mass_fractal.c"]

demo = dict(scale=1, background=0,
            radius=10.0,
            mass_dim=1.9,
            cutoff_length=2.3)

oldname = 'MassFractalModel'
oldpars = dict(radius='radius',
               mass_dim='mass_dim',
               cutoff_length='co_length')

tests = [[{'radius':        2.0,
           'mass_dim':      3.3,
           'cutoff_length': 1.0,
           }, 0.5, 1.29016774904],

         [{'radius':        1.0,
           'mass_dim':      1.3,
           'cutoff_length': 1.0,
           'background':    0.8,
           }, 0.001, 1.69747015932],

         [{'radius':        1.0,
           'mass_dim':      2.3,
           'cutoff_length': 1.0,
           'scale':        10.0,
           }, 0.051, 11.6227966145],
         ]