.. _triaxial-ellipsoid:
triaxial_ellipsoid
=======================================================
Ellipsoid of uniform scattering length density with three independent axes.
================== =================================== ============ =============
Parameter Description Units Default value
================== =================================== ============ =============
scale Scale factor or Volume fraction None 1
background Source background |cm^-1| 0.001
sld Ellipsoid scattering length density |1e-6Ang^-2| 4
sld_solvent Solvent scattering length density |1e-6Ang^-2| 1
radius_equat_minor Minor equatorial radius, Ra |Ang| 20
radius_equat_major Major equatorial radius, Rb |Ang| 400
radius_polar Polar radius, Rc |Ang| 10
theta polar axis to beam angle degree 60
phi rotation about beam degree 60
psi rotation about polar axis degree 60
================== =================================== ============ =============
The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.
**Definition**
.. figure:: img/triaxial_ellipsoid_geometry.jpg
Ellipsoid with $R_a$ as *radius_equat_minor*, $R_b$ as *radius_equat_major*
and $R_c$ as *radius_polar*.
Given an ellipsoid
.. math::
\frac{X^2}{R_a^2} + \frac{Y^2}{R_b^2} + \frac{Z^2}{R_c^2} = 1
the scattering for randomly oriented particles is defined by the average over
all orientations $\Omega$ of:
.. math::
P(q) = \text{scale}(\Delta\rho)^2\frac{V}{4 \pi}\int_\Omega\Phi^2(qr)\,d\Omega
+ \text{background}
where
.. math::
\Phi(qr) &= 3 j_1(qr)/qr = 3 (\sin qr - qr \cos qr)/(qr)^3 \\
r^2 &= R_a^2e^2 + R_b^2f^2 + R_c^2g^2 \\
V &= \tfrac{4}{3} \pi R_a R_b R_c
The $e$, $f$ and $g$ terms are the projections of the orientation vector on $X$,
$Y$ and $Z$ respectively. Keeping the orientation fixed at the canonical
axes, we can integrate over the incident direction using polar angle
$-\pi/2 \le \gamma \le \pi/2$ and equatorial angle $0 \le \phi \le 2\pi$
(as defined in ref [1]),
.. math::
\langle\Phi^2\rangle = \int_0^{2\pi} \int_{-\pi/2}^{\pi/2} \Phi^2(qr)
\cos \gamma\,d\gamma d\phi
with $e = \cos\gamma \sin\phi$, $f = \cos\gamma \cos\phi$ and $g = \sin\gamma$.
A little algebra yields
.. math::
r^2 = b^2(p_a \sin^2 \phi \cos^2 \gamma + 1 + p_c \sin^2 \gamma)
for
.. math::
p_a = \frac{a^2}{b^2} - 1 \text{ and } p_c = \frac{c^2}{b^2} - 1
Due to symmetry, the ranges can be restricted to a single quadrant
$0 \le \gamma \le \pi/2$ and $0 \le \phi \le \pi/2$, scaling the resulting
integral by 8. The computation is done using the substitution $u = \sin\gamma$,
$du = \cos\gamma\,d\gamma$, giving
.. math::
\langle\Phi^2\rangle &= 8 \int_0^{\pi/2} \int_0^1 \Phi^2(qr) du d\phi \\
r^2 &= b^2(p_a \sin^2(\phi)(1 - u^2) + 1 + p_c u^2)
Though for convenience we describe the three radii of the ellipsoid as
equatorial and polar, they may be given in $any$ size order. To avoid
multiple solutions, especially with Monte-Carlo fit methods, it may be
advisable to restrict their ranges. For typical small angle diffraction
situations there may be a number of closely similar "best fits", so some
trial and error, or fixing of some radii at expected values, may help.
To provide easy access to the orientation of the triaxial ellipsoid, we
define the axis of the cylinder using the angles $\theta$, $\phi$ and $\psi$.
These angles are defined analogously to the elliptical_cylinder below, note
that angle $\phi$ is now NOT the same as in the equations above.
.. figure:: img/elliptical_cylinder_angle_definition.png
Definition of angles for oriented triaxial ellipsoid, where radii are for
illustration here $a < b << c$ and angle $\Psi$ is a rotation around the
axis of the particle.
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.
.. _triaxial-ellipsoid-angles:
.. figure:: img/triaxial_ellipsoid_angle_projection.png
Some examples for an oriented triaxial ellipsoid.
The radius-of-gyration for this system is $R_g^2 = (R_a R_b R_c)^2/5$.
The contrast $\Delta\rho$ is defined as SLD(ellipsoid) - SLD(solvent). In the
parameters, $R_a$ is the minor equatorial radius, $R_b$ is the major
equatorial radius, and $R_c$ is the polar radius of the ellipsoid.
NB: The 2nd virial coefficient of the triaxial solid ellipsoid is calculated
after sorting the three radii to give the most appropriate prolate or oblate
form, from the new polar radius $R_p = R_c$ and effective equatorial radius,
$R_e = \sqrt{R_a R_b}$, to then be used as the effective radius for $S(q)$
when $P(q) \cdot S(q)$ 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 2D calculation
over all possible angles.
.. figure:: img/triaxial_ellipsoid_autogenfig.png
1D and 2D plots corresponding to the default parameters of the model.
**Source**
:download:`triaxial_ellipsoid.py `
$\ \star\ $ :download:`triaxial_ellipsoid.c `
$\ \star\ $ :download:`gauss76.c `
$\ \star\ $ :download:`sas_3j1x_x.c `
**References**
#. Finnigan, J.A., Jacobs, D.J., 1971. *Light scattering by ellipsoidal
particles in solution*, J. Phys. D: Appl. Phys. 4, 72-77.
doi:10.1088/0022-3727/4/1/310
**Authorship and Verification**
* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Kienzle (improved calculation) **Date:** April 4, 2017
* **Last Reviewed by:** Paul Kienzle & Richard Heenan **Date:** April 4, 2017