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r"""
Definition
----------
Parameters for this model are the core axial ratio $X_{core}$ and a shell
thickness $t_{shell}$, which are more often what we would like to determine
and make the model better behaved, particularly when polydispersity is
applied, than the four independent radii used in the original parameterization
of this model.
.. figure:: img/core_shell_ellipsoid_geometry.png
The geometric parameters of this model are shown in the diagram above, which
shows (a) a cut through at the circular equator and (b) a cross section through
the poles, of a prolate ellipsoid.
When $X_{core}$ < 1 the core is oblate; when $X_{core}$ > 1 it is prolate.
$X_{core}$ = 1 is a spherical core.
For a fixed shell thickness $X_{polar shell}$ = 1, to scale $t_{shell}$
pro-rata with the radius set or constrain $X_{polar shell}$ = $X_{core}$.
.. note::
When including an $S(q)$, the radius in $S(q)$ is calculated to be that of
a sphere with the same 2nd virial coefficient of the outer surface of the
ellipsoid. This may have some undesirable effects if the aspect ratio of the
ellipsoid is large (ie, if $X << 1$ or $X >> 1$), when the $S(q)$
- which assumes spheres - will not in any case be valid. Generating a
custom product model will enable separate effective volume fraction and
effective radius in the $S(q)$.
If SAS data are in absolute units, and the SLDs are correct, then scale should
be the total volume fraction of the "outer particle". When $S(q)$ is introduced
this moves to the $S(q)$ volume fraction, and scale should then be 1.0, or
contain some other units conversion factor (for example, if you have SAXS data).
The calculation of intensity follows that for the solid ellipsoid, but
with separate terms for the core-shell and shell-solvent boundaries.
.. math::
P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background}
where
.. In following equation SK changed radius\_equat\_core to R_e
.. math::
:nowrap:
\begin{align*}
F(q,\alpha) = &f(q,R_e,R_e.x_{core},\alpha) \\
&+ f(q,R_e + t_{shell},
R_e.x_{core} + t_{shell}.x_{polar shell},\alpha)
\end{align*}
where
.. math::
f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin(qr)
- qr\cos(qr)}
{(qr)^3}
for
.. math::
r = \left[ R_e^2 \sin^2 \alpha + R_p^2 \cos^2 \alpha \right]^{1/2}
$\alpha$ is the angle between the axis of the ellipsoid and $\vec q$,
$V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the
polar radius along the rotational axis of the ellipsoid, $R_e$ is the
equatorial radius perpendicular to the rotational axis of the ellipsoid,
$t_{shell}$ is the thickness of the shell at the equator,
and $\Delta \rho$ (the contrast) is the scattering length density difference,
either $(\rho_{core} - \rho_{shell})$ or $(\rho_{shell} - \rho_{solvent})$.
For randomly oriented particles:
.. math::
F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}
For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters
will appear when fitting 2D data, see the :ref:`elliptical-cylinder` model
for further information.
References
----------
see for example:
#. Kotlarchyk, M.; Chen, S.-H. *J. Chem. Phys.*, 1983, 79, 2461
#. Berr, S. *J. Phys. Chem.*, 1987, 91, 4760
Authorship and Verification
----------------------------
* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015
* **Last Reviewed by:** Steve King **Date:** March 27, 2019
"""
import numpy as np
from numpy import inf, sin, cos, pi
name = "core_shell_ellipsoid"
title = "Form factor for an spheroid ellipsoid particle with a core shell structure."
description = """
[core_shell_ellipsoid] Calculates the form factor for an spheroid
ellipsoid particle with a core_shell structure.
The form factor is averaged over all possible
orientations of the ellipsoid such that P(q)
= scale*<f^2>/Vol + bkg, where f is the
single particle scattering amplitude.
[Parameters]:
radius_equat_core = equatorial radius of core,
x_core = ratio of core polar/equatorial radii,
thick_shell = equatorial radius of outer surface,
x_polar_shell = ratio of polar shell thickness to equatorial shell thickness,
sld_core = SLD_core
sld_shell = SLD_shell
sld_solvent = SLD_solvent
background = Incoherent bkg
scale =scale
Note:It is the users' responsibility to ensure
that shell radii are larger than core radii.
oblate: polar radius < equatorial radius
prolate : polar radius > equatorial radius - this new model will make this easier
and polydispersity integrals more logical (as previously the shell could disappear).
"""
category = "shape:ellipsoid"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
["radius_equat_core","Ang", 20, [0, inf], "volume", "Equatorial radius of core"],
["x_core", "None", 3, [0, inf], "volume", "axial ratio of core, X = r_polar/r_equatorial"],
["thick_shell", "Ang", 30, [0, inf], "volume", "thickness of shell at equator"],
["x_polar_shell", "", 1, [0, inf], "volume", "ratio of thickness of shell at pole to that at equator"],
["sld_core", "1e-6/Ang^2", 2, [-inf, inf], "sld", "Core scattering length density"],
["sld_shell", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Shell scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"],
["theta", "degrees", 0, [-360, 360], "orientation", "elipsoid axis to beam angle"],
["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "core_shell_ellipsoid.c"]
have_Fq = True
radius_effective_modes = [
"average outer curvature", "equivalent volume sphere",
"min outer radius", "max outer radius",
]
def random():
"""Return a random parameter set for the model."""
volume = 10**np.random.uniform(5, 12)
outer_polar = 10**np.random.uniform(1.3, 4)
outer_equatorial = np.sqrt(volume/outer_polar) # ignore 4/3 pi
# Use a distribution with a preference for thin shell or thin core
# Avoid core,shell radii < 1
thickness_polar = np.random.beta(0.5, 0.5)*(outer_polar-2) + 1
thickness_equatorial = np.random.beta(0.5, 0.5)*(outer_equatorial-2) + 1
radius_polar = outer_polar - thickness_polar
radius_equatorial = outer_equatorial - thickness_equatorial
x_core = radius_polar/radius_equatorial
x_polar_shell = thickness_polar/thickness_equatorial
pars = dict(
#background=0, sld=0, sld_solvent=1,
radius_equat_core=radius_equatorial,
x_core=x_core,
thick_shell=thickness_equatorial,
x_polar_shell=x_polar_shell,
)
return pars
q = 0.1
# tests had in old coords theta=0, phi=0; new coords theta=90, phi=0
qx = q*cos(pi/6.0)
qy = q*sin(pi/6.0)
# 11Jan2017 RKH sorted tests after redefinition of angles
tests = [
# Accuracy tests based on content in test/utest_coreshellellipsoidXTmodel.py
[{'radius_equat_core': 200.0,
'x_core': 0.1,
'thick_shell': 50.0,
'x_polar_shell': 0.2,
'sld_core': 2.0,
'sld_shell': 1.0,
'sld_solvent': 6.3,
'background': 0.001,
'scale': 1.0,
}, 1.0, 0.00189402],
# Additional tests with larger range of parameters
[{'background': 0.01}, 0.1, 11.6915],
[{'radius_equat_core': 20.0,
'x_core': 200.0,
'thick_shell': 54.0,
'x_polar_shell': 3.0,
'sld_core': 20.0,
'sld_shell': 10.0,
'sld_solvent': 6.0,
'background': 0.0,
'scale': 1.0,
}, 0.01, 8688.53],
# 2D tests
[{'background': 0.001,
'theta': 90.0,
'phi': 0.0,
}, (0.4, 0.5), 0.00690673],
[{'radius_equat_core': 20.0,
'x_core': 200.0,
'thick_shell': 54.0,
'x_polar_shell': 3.0,
'sld_core': 20.0,
'sld_shell': 10.0,
'sld_solvent': 6.0,
'background': 0.01,
'scale': 0.01,
'theta': 90.0,
'phi': 0.0,
}, (qx, qy), 0.01000025],
]
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