Mass Fractal - mass_fractal.py
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_\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).
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:**
"""
import numpy as np
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
fractal_dim_mass = 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<fractal_dim_mass<6.
Note II: This model is not in absolute scale.
"""
category = "shape-independent"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [
["radius", "Ang", 10.0, [0.0, inf], "", "Particle radius"],
["fractal_dim_mass", "", 1.9, [1.0, 6.0], "", "Mass fractal dimension"],
["cutoff_length", "Ang", 100.0, [0.0, inf], "", "Cut-off length"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "mass_fractal.c"]
def random():
"""Return a random parameter set for the model."""
radius = 10**np.random.uniform(0.7, 4)
cutoff_length = 10**np.random.uniform(0.7, 2)*radius
# TODO: fractal dimension should range from 1 to 6
fractal_dim_mass = 2*np.random.beta(3, 4) + 1
#volfrac = 10**np.random.uniform(-4, -1)
pars = dict(
#background=0,
scale=1, #1e5*volfrac/radius**(fractal_dim_mass),
radius=radius,
cutoff_length=cutoff_length,
fractal_dim_mass=fractal_dim_mass,
)
return pars
demo = dict(scale=1, background=0,
radius=10.0,
fractal_dim_mass=1.9,
cutoff_length=100.0)
tests = [
# Accuracy tests based on content in test/utest_other_models.py
[{'radius': 10.0,
'fractal_dim_mass': 1.9,
'cutoff_length': 100.0,
}, 0.05, 279.59422],
# Additional tests with larger range of parameters
[{'radius': 2.0,
'fractal_dim_mass': 3.3,
'cutoff_length': 1.0,
}, 0.5, 1.29116774904],
[{'radius': 1.0,
'fractal_dim_mass': 1.3,
'cutoff_length': 1.0,
'background': 0.8,
}, 0.001, 1.69747015932],
[{'radius': 1.0,
'fractal_dim_mass': 2.3,
'cutoff_length': 1.0,
'scale': 10.0,
}, 0.051, 11.6237966145],
]
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