# Rectangular Prism - rectangular_prism.py

```    ```# rectangular_prism model
# Note: model title and parameter table are inserted automatically
r"""

This model provides the form factor, \$P(q)\$, for a rectangular prism.

Note that this model is almost totally equivalent to the existing
:ref:`parallelepiped` model.
The only difference is that the way the relevant
parameters are defined here (\$a\$, \$b/a\$, \$c/a\$ instead of \$a\$, \$b\$, \$c\$)
which allows use of polydispersity with this model while keeping the shape of
the prism (e.g. setting \$b/a = 1\$ and \$c/a = 1\$ and applying polydispersity
to *a* will generate a distribution of cubes of different sizes).

Definition
----------

The 1D scattering intensity for this model was calculated by Mittelbach and
Porod (Mittelbach, 1961), but the implementation here is closer to the
equations given by Nayuk and Huber (Nayuk, 2012).
Note also that the angle definitions used in the code and the present
documentation correspond to those used in (Nayuk, 2012) (see Fig. 1 of
that reference), with \$	heta\$ corresponding to \$alpha\$ in that paper,
and not to the usual convention used for example in the
:ref:`parallelepiped` model.

In this model the scattering from a massive parallelepiped with an
orientation with respect to the scattering vector given by \$	heta\$
and \$phi\$

.. math::

A_P,(q) =
frac{sin left( 	frac{1}{2}qC cos	heta
ight) }{	frac{1}{2} qC cos	heta}
,	imes,
frac{sin left( 	frac{1}{2}qA cos	heta
ight) }{	frac{1}{2} qA cos	heta}
,	imes ,
frac{sin left( 	frac{1}{2}qB cos	heta
ight) }{	frac{1}{2} qB cos	heta}

where \$A\$, \$B\$ and \$C\$ are the sides of the parallelepiped and must fulfill
\$A le B le C\$, \$	heta\$ is the angle between the \$z\$ axis and the
longest axis of the parallelepiped \$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

.. math::
P(q) =  frac{2}{pi} int_0^{frac{pi}{2}} ,
int_0^{frac{pi}{2}} A_P^2(q) , sin	heta , d	heta , dphi

And the 1D scattering intensity is calculated as

.. math::
I(q) = 	ext{scale} 	imes V 	imes (
ho_	ext{p} -

ho_	ext{solvent})^2 	imes P(q)

where \$V\$ is the volume of the rectangular prism, \$
ho_	ext{p}\$
is the scattering length of the parallelepiped, \$
ho_	ext{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 \$	heta\$, \$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 \$	heta\$, 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 was conducted by comparing the output of the 1D model
to the output of the existing :ref:`parallelepiped` model.

References
----------

.. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211
.. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854
.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

Authorship and Verification
----------------------------

* **Author:**
* **Last Reviewed by:**
"""

import numpy as np
from numpy import inf

name = "rectangular_prism"
title = "Rectangular parallelepiped 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 = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2)
S(x) = sin(x)/x
"""
category = "shape:parallelepiped"

#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld",
"Parallelepiped scattering length density"],
["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
"Solvent scattering length density"],
["length_a", "Ang", 35, [0, inf], "volume",
"Shorter side of the parallelepiped"],
["b2a_ratio", "", 1, [0, inf], "volume",
"Ratio sides b/a"],
["c2a_ratio", "", 1, [0, inf], "volume",
"Ratio sides c/a"],
["theta", "degrees", 0, [-360, 360], "orientation",
"c axis to beam angle"],
["phi", "degrees", 0, [-360, 360], "orientation",
["psi", "degrees", 0, [-360, 360], "orientation",
]

source = ["lib/gauss76.c", "rectangular_prism.c"]
have_Fq = True
"equivalent cylinder excluded volume", "equivalent volume sphere",
"half length_a", "half length_b", "half length_c",
"equivalent circular cross-section", "half ab diagonal", "half diagonal",
]

def random():
"""Return a random parameter set for the model."""
a, b, c = 10**np.random.uniform(1, 4.7, size=3)
pars = dict(
length_a=a,
b2a_ratio=b/a,
c2a_ratio=c/a,
)
return pars

# parameters for demo
demo = dict(scale=1, background=0,
sld=6.3, sld_solvent=1.0,
length_a=35, b2a_ratio=1, c2a_ratio=1,
length_a_pd=0.1, length_a_pd_n=10,
b2a_ratio_pd=0.1, b2a_ratio_pd_n=1,
c2a_ratio_pd=0.1, c2a_ratio_pd_n=1)

tests = [[{}, 0.2, 0.375248406825],
[{}, [0.2], [0.375248406825]],
]
```
```