` ````
# octahedron model
# Note: model title and parameter table are inserted automatically
r"""
This model provides the form factor, $P(q)$, for an octahedron.
This model is constructed in a similar way as the rectangular prism model.
Definition
----------
The octahedron is defined by three dimensions along the two-fold axis which contain the 6 vertices.
length_a, length_b and length_c are the distances from the center of the octahedron to its vertices.
Coordinates of the six vertices are :
(length_a,0,0)
(-length_a,0,0)
(0,length_b,0)
(0,-length_b,0)
(0,0,length_c)
(0,0,-length_c)
.. math::
The model is using length_a and the two ratios b2a_ratio and c2a_ratio :
b2a_ratio = length_b/length_a
c2a_ratio = length_c/length_a
Volume of the octahedron is:
V = (4/3) * length_a * (length_a*b2a_ratio) * (length_a*c2a_ratio)
Lengths of edges are equal to :
A_edge^2 = length_a^2+length_b^2
B_edge^2 = length_a^2+length_c^2
C_edge^2 = length_b^2+length_c^2
For a regular octahedron :
b2a_ratio = c2a_ratio = 1
A_edge = B_edge = C_edge = length_a*sqrt(2)
length_a = length_b = length_c = A_edge/sqrt(2)
V = (4/3) * length_a^3 = (sqrt(2)/3) * A_edge^3
Amplitude of the form factor AP is calculated with a scaled scattering vector (qx,qy,qz) :
AP = (3./4.)*(A+B)/(qx*qx-qy*qy)
with :
A = 8.*(qy*sin(qy)-qz*sin(qz))/(qy*qy-qz*qz)
B = 8.*(qz*sin(qz)-qx*sin(qx))/(qx*qx-qz*qz)
and :
Qx = q * sin_theta * cos_phi;
Qy = q * sin_theta * sin_phi;
Qz = q * cos_theta;
qx = Qx * length_a;
qy = Qy * length_b;
qz = Qz * length_c;
$\theta$ is the angle between the $z$ axis and the
c axis of the octahedron ($length_c$), and $\phi$ is the angle between the
scattering vector (lying in the $xy$ plane) and the $y$ axis.
The normalized form factor in 1D is obtained averaging over all possible
orientations. Same code as for a rectangular prism.
.. math::
P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \,
\int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi
And the 1D scattering intensity is calculated as
.. math::
I(q) = \text{scale} \times V \times (\rho_\text{p} -
\rho_\text{solvent})^2 \times P(q)
where $V$ is the volume of the octahedron, $\rho_\text{p}$
is the scattering length inside the volume, $\rho_\text{solvent}$
is the scattering length of the solvent, and (if the data are in absolute
units) *scale* represents the volume fraction (which is unitless).
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 long *C* axis. For example,
$\Psi = 0$ when the *B* axis is parallel to the *x*-axis of the detector.
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 cylinder.
The neutron or X-ray beam is along the $z$ axis.
.. figure:: img/parallelepiped_angle_projection.png
Examples of the angles for oriented rectangular prisms against the
detector plane.
Validation
----------
Validation of the code is made using numerical checks.
References
----------
1. Wei-Ren Chen et al. “Scattering functions of Platonic solids”.
In:Journal of AppliedCrystallography - J APPL CRYST44 (June 2011).
DOI:10.1107/S0021889811011691
2. Croset, Bernard, "Form factor of any polyhedron: a general compact
formula and its singularities" In: J. Appl. Cryst. (2017). 50, 1245–1255
https://doi.org/10.1107/S1600576717010147
Authorship and Verification
----------------------------
* **Authors: Marianne Imperor-Clerc (marianne.imperor@universite-paris-saclay.fr)
Alexandra Beikert (abeikert@gmx.de)**
* **Last Modified by MI: 9 October 2020**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf
name = "octahedron_h_mi"
title = "Octahedron with uniform scattering length density."
description = """
I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background
P(q,theta,phi) = (2/pi) * double integral from 0 to pi/2 of ...
AP^2(q)*sin(theta)*dtheta*dphi
AP = (3./4.)*(A+B)/(qx*qx-qy*qy)
A = 8.*(qy*sin(qy)-qz*sin(qz))/(qy*qy-qz*qz)
B = 8.*(qz*sin(qz)-qx*sin(qx))/(qx*qx-qz*qz)
normalisation to 1. of AP at q = 0. Division by a Factor 4/3.
Qx = q * sin_theta * cos_phi;
Qy = q * sin_theta * sin_phi;
Qz = q * cos_theta;
qx = Qx * length_a;
qy = Qy * length_b;
qz = Qz * length_c;
"""
category = "shape:parallelepiped"
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld", "1e-6/Ang^2", 128, [-inf, inf], "sld",
"Octahedron scattering length density"],
["sld_solvent", "1e-6/Ang^2", 9.4, [-inf, inf], "sld",
"Solvent scattering length density"],
["length_a", "Ang", 35, [0, inf], "volume",
"half height along a axis"],
["b2a_ratio", "", 1, [0, inf], "volume",
"Ratio b/a"],
["c2a_ratio", "", 1, [0, inf], "volume",
"Ratio c/a"],
["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", "octahedron_h_mi.c"]
have_Fq = True
```

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