r"""
Definition
----------
Calculates the form factor for a rectangular solid with a core-shell structure.
The thickness and the scattering length density of the shell or "rim" can be
different on each (pair) of faces. The three dimensions of the core of the
parallelepiped (strictly here a cuboid) may be given in *any* size order as
long as the particles are randomly oriented (i.e. take on all possible
orientations see notes on 2D below). To avoid multiple fit solutions, e
specially with Monte-Carlo fit methods, it may be advisable to restrict their
ranges. There may be a number of closely similar "best fits", so some trial and
error, or fixing of some dimensions at expected values, may help.
The form factor is normalized by the particle volume $V$ such that
.. math::
I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle
+ \text{background}
where $\langle \ldots \rangle$ is an average over all possible orientations
of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be
pulled out of the form factor term due to the multiple slds in the model.
The core of the solid is defined by the dimensions $A$, $B$, $C$ here shown
such that $A < B < C$.
.. figure:: img/parallelepiped_geometry.jpg
Core of the core shell parallelepiped with the corresponding definition
of sides.
There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension
(on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$
$(=t_C)$ faces. The projection in the $AB$ plane is
.. figure:: img/core_shell_parallelepiped_projection.jpg
AB cut through the core-shell parallelipiped showing the cross secion of
four of the six shell slabs. As can be seen, this model leaves **"gaps"**
at the corners of the solid.
The total volume of the solid is thus given as
.. math::
V = ABC + 2t_ABC + 2t_BAC + 2t_CAB
The intensity calculated follows the :ref:`parallelepiped` model, with the
core-shell intensity being calculated as the square of the sum of the
amplitudes of the core and the slabs on the edges. The scattering amplitude is
computed for a particular orientation of the core-shell parallelepiped with
respect to the scattering vector and then averaged over all possible
orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis
of the parallelepiped, and $\beta$ is the angle between the projection of the
particle in the $xy$ detector plane and the $y$ axis.
.. math::
P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha
\ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta}
and
.. math::
F(q,\alpha,\beta)
&= (\rho_\text{core}-\rho_\text{solvent})
S(Q_A, A) S(Q_B, B) S(Q_C, C) \\
&+ (\rho_\text{A}-\rho_\text{solvent})
\left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\
&+ (\rho_\text{B}-\rho_\text{solvent})
S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\
&+ (\rho_\text{C}-\rho_\text{solvent})
S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right]
with
.. math::
S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L}
and
.. math::
Q_A &= q \sin\alpha \sin\beta \\
Q_B &= q \sin\alpha \cos\beta \\
Q_C &= q \cos\alpha
where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$
are the scattering lengths of the parallelepiped core, and the rectangular
slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$
is the scattering length of the solvent.
.. note::
the code actually implements two substitutions: $d(cos\alpha)$ is
substituted for -$sin\alpha \ d\alpha$ (note that in the
:ref:`parallelepiped` code this is explicitly implemented with
$\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that
$du = \pi/2 \ d\beta$. Thus both integrals go from 0 to 1 rather than 0
to $\pi/2$.
FITTING NOTES
~~~~~~~~~~~~~
There are many parameters in this model. Hold as many fixed as possible with
known values, or you will certainly end up at a solution that is unphysical.
NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated
based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$
and length $(C+2t_C)$ values, after appropriately sorting the three dimensions
to give an oblate or prolate particle, to give an effective radius
for $S(q)$ when $P(q) * S(q)$ is applied.
For 2d data the orientation of the particle is required, described using
angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details
of the calculation and angular dispersions see :ref:`orientation` .
The angle $\Psi$ is the rotational angle around the $C$ axis.
For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis
oriented parallel to the y-axis of the detector with $A$ along the x-axis.
For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated
$\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis,
before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis
of the particle to obtain the final orientation of the parallelepiped.
.. note:: For 2d, constraints must be applied during fitting to ensure that the
inequality $A < B < C$ is not violated, and hence the correct definition
of angles is preserved. The calculation will not report an error,
but the results may be not correct.
.. figure:: img/parallelepiped_angle_definition.png
Definition of the angles for oriented core-shell parallelepipeds.
Note that rotation $\theta$, initially in the $xz$ plane, is carried
out first, then rotation $\phi$ about the $z$ axis, finally rotation
$\Psi$ is now around the axis of the particle. The neutron or X-ray
beam is along the $z$ axis.
.. figure:: img/parallelepiped_angle_projection.png
Examples of the angles for oriented core-shell parallelepipeds against the
detector plane.
Validation
----------
Cross-checked against hollow rectangular prism and rectangular prism for equal
thickness overlapping sides, and by Monte Carlo sampling of points within the
shape for non-uniform, non-overlapping sides.
References
----------
.. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211
Equations (1), (13-14). (in German)
.. [#] D Singh (2009). *Small angle scattering studies of self assembly in
lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available
from Proquest `_
Authorship and Verification
----------------------------
* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016
* **Last Modified by:** Paul Kienzle **Date:** October 17, 2017
"""
import numpy as np
from numpy import pi, inf, sqrt, cos, sin
name = "core_shell_parallelepiped"
title = "Rectangular solid with a core-shell structure."
description = """
P(q)=
"""
category = "shape:parallelepiped"
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld",
"Parallelepiped core scattering length density"],
["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld",
"Parallelepiped A rim scattering length density"],
["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld",
"Parallelepiped B rim scattering length density"],
["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld",
"Parallelepiped C rim scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
"Solvent scattering length density"],
["length_a", "Ang", 35, [0, inf], "volume",
"Shorter side of the parallelepiped"],
["length_b", "Ang", 75, [0, inf], "volume",
"Second side of the parallelepiped"],
["length_c", "Ang", 400, [0, inf], "volume",
"Larger side of the parallelepiped"],
["thick_rim_a", "Ang", 10, [0, inf], "volume",
"Thickness of A rim"],
["thick_rim_b", "Ang", 10, [0, inf], "volume",
"Thickness of B rim"],
["thick_rim_c", "Ang", 10, [0, inf], "volume",
"Thickness of C rim"],
["theta", "degrees", 0, [-360, 360], "orientation",
"c axis to beam angle"],
["phi", "degrees", 0, [-360, 360], "orientation",
"rotation about beam"],
["psi", "degrees", 0, [-360, 360], "orientation",
"rotation about c axis"],
]
source = ["lib/gauss76.c", "core_shell_parallelepiped.c"]
def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c):
"""
Return equivalent radius (ER)
"""
from .parallelepiped import ER as ER_p
a = length_a + 2*thick_rim_a
b = length_b + 2*thick_rim_b
c = length_c + 2*thick_rim_c
return ER_p(a, b, c)
# VR defaults to 1.0
def random():
outer = 10**np.random.uniform(1, 4.7, size=3)
thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1
length = outer - thick
pars = dict(
length_a=length[0],
length_b=length[1],
length_c=length[2],
thick_rim_a=thick[0],
thick_rim_b=thick[1],
thick_rim_c=thick[2],
)
return pars
# parameters for demo
demo = dict(scale=1, background=0.0,
sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6,
length_a=35, length_b=75, length_c=400,
thick_rim_a=10, thick_rim_b=10, thick_rim_c=10,
theta=0, phi=0, psi=0,
length_a_pd=0.1, length_a_pd_n=1,
length_b_pd=0.1, length_b_pd_n=1,
length_c_pd=0.1, length_c_pd_n=1,
thick_rim_a_pd=0.1, thick_rim_a_pd_n=1,
thick_rim_b_pd=0.1, thick_rim_b_pd_n=1,
thick_rim_c_pd=0.1, thick_rim_c_pd_n=1,
theta_pd=10, theta_pd_n=1,
phi_pd=10, phi_pd_n=1,
psi_pd=10, psi_pd_n=1)
# rkh 7/4/17 add random unit test for 2d, note make all params different,
# 2d values not tested against other codes or models
if 0: # pak: model rewrite; need to update tests
qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.)
tests = [[{}, 0.2, 0.533149288477],
[{}, [0.2], [0.533149288477]],
[{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222],
[{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]],
]
del qx, qy # not necessary to delete, but cleaner