Elliptical Cylinder - elliptical_cylinder.py
# pylint: disable=line-too-long
r"""
.. figure:: img/elliptical_cylinder_geometry.png
Elliptical cylinder geometry $a = r_\text{minor}$
and $\nu = r_\text{major} / r_\text{minor}$ is the *axis_ratio*.
The function calculated is
.. math::
I(\vec q)=\frac{1}{V_\text{cyl}}\int{d\psi}\int{d\phi}\int{
p(\theta,\phi,\psi)F^2(\vec q,\alpha,\psi)\sin(\alpha)d\alpha}
with the functions
.. math::
F(q,\alpha,\psi) = 2\frac{J_1(a)\sin(b)}{ab}
where
.. math::
a = qr'\sin(\alpha)
b = q\frac{L}{2}\cos(\alpha)
r'=\frac{r_{minor}}{\sqrt{2}}\sqrt{(1+\nu^{2}) + (1-\nu^{2})cos(\psi)}
and the angle $\psi$ is defined as the orientation of the major axis of the
ellipse with respect to the vector $\vec q$. The angle $\alpha$ is the angle
between the axis of the cylinder and $\vec q$.
For 1D scattering, with no preferred orientation, the form factor is averaged over all possible orientations and normalized
by the particle volume
.. math::
P(q) = \text{scale} <F^2> / V
For 2d data the orientation of the particle is required, described using a different set
of angles as in the diagrams below, for further details of the calculation and angular
dispersions see :ref:`orientation` .
.. figure:: img/elliptical_cylinder_angle_definition.png
Note that the angles here are not the same as in the equations for the scattering function.
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 cylinder.
The neutron or X-ray beam is along the $z$ axis.
.. figure:: img/elliptical_cylinder_angle_projection.png
Examples of the angles for oriented elliptical cylinders against the
detector plane, with $\Psi$ = 0.
The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data.
NB: The 2nd virial coefficient of the cylinder is calculated based on the
averaged radius $(=\sqrt{r_\text{minor}^2 * \text{axis ratio}})$ and length
values, and used as the effective radius for $S(Q)$ when $P(Q)*S(Q)$ is applied.
Validation
----------
Validation of our code was done by comparing the output of the 1D calculation
to the angular average of the output of the 2D calculation over all possible
angles.
In the 2D average, more binning in the angle $\phi$ is necessary to get the
proper result. The following figure shows the results of the averaging by
varying the number of angular bins.
.. figure:: img/elliptical_cylinder_averaging.png
The intensities averaged from 2D over different numbers of bins and angles.
References
----------
.. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, New York, (1987) [see table 3.4]
.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:** Richard Heenan - corrected equation in docs **Date:** December 21, 2016
"""
import numpy as np
from numpy import pi, inf, sqrt, sin, cos
name = "elliptical_cylinder"
title = "Form factor for an elliptical cylinder."
description = """
Form factor for an elliptical cylinder.
See L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987).
"""
category = "shape:cylinder"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["radius_minor", "Ang", 20.0, [0, inf], "volume", "Ellipse minor radius"],
["axis_ratio", "", 1.5, [1, inf], "volume", "Ratio of major radius over minor radius"],
["length", "Ang", 400.0, [1, inf], "volume", "Length of the cylinder"],
["sld", "1e-6/Ang^2", 4.0, [-inf, inf], "sld", "Cylinder scattering length density"],
["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Solvent scattering length density"],
["theta", "degrees", 90.0, [-360, 360], "orientation", "cylinder axis to beam angle"],
["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"],
["psi", "degrees", 0, [-360, 360], "orientation", "rotation about cylinder axis"]]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "elliptical_cylinder.c"]
have_Fq = True
radius_effective_modes = [
"equivalent cylinder excluded volume",
"equivalent volume sphere", "average radius", "min radius", "max radius",
"equivalent circular cross-section",
"half length", "half min dimension", "half max dimension", "half diagonal",
]
demo = dict(scale=1, background=0, radius_minor=100, axis_ratio=1.5, length=400.0,
sld=4.0, sld_solvent=1.0, theta=10.0, phi=20, psi=30,
theta_pd=10, phi_pd=2, psi_pd=3)
def random():
"""Return a random parameter set for the model."""
# V = pi * radius_major * radius_minor * length;
volume = 10**np.random.uniform(3, 9)
length = 10**np.random.uniform(1, 3)
axis_ratio = 10**np.random.uniform(0, 2)
radius_minor = sqrt(volume/length/axis_ratio)
volfrac = 10**np.random.uniform(-4, -2)
pars = dict(
#background=0, sld=0, sld_solvent=1,
scale=1e9*volfrac/volume,
length=length,
radius_minor=radius_minor,
axis_ratio=axis_ratio,
)
return pars
q = 0.1
# april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct!
qx = q*cos(pi/6.0)
qy = q*sin(pi/6.0)
tests = [
#[{'radius_minor': 20.0, 'axis_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024],
#[{'radius_minor': 20.0, 'axis_ratio': 1.2, 'length':300.0}, 'VR', 1],
# The SasView test result was 0.00169, with a background of 0.001
[{'radius_minor': 20.0, 'axis_ratio': 1.5, 'sld': 4.0, 'length':400.0,
'sld_solvent':1.0, 'background':0.0},
0.001, 675.504402],
#[{'theta':80., 'phi':10.}, (qx, qy), 7.88866563001 ],
]
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