.. _rectangular-prism:

rectangular_prism
=======================================================

Rectangular parallelepiped with uniform scattering length density.

=========== ======================================== ============ =============
Parameter   Description                              Units        Default value
=========== ======================================== ============ =============
scale       Scale factor or Volume fraction          None                     1
background  Source background                        |cm^-1|              0.001
sld         Parallelepiped scattering length density |1e-6Ang^-2|           6.3
sld_solvent Solvent scattering length density        |1e-6Ang^-2|             1
length_a    Shorter side of the parallelepiped       |Ang|                   35
b2a_ratio   Ratio sides b/a                          None                     1
c2a_ratio   Ratio sides c/a                          None                     1
theta       c axis to beam angle                     degree                   0
phi         rotation about beam                      degree                   0
psi         rotation about c axis                    degree                   0
=========== ======================================== ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.



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, 1961 [#Mittelbach1961]_), but the implementation here is closer to
the equations given by (Nayuk and Huber, 2012 [#Nayuk2012]_).
Note also that the angle definitions used in the code and the present
documentation correspond to those used in Nayuk and Huber (see Fig. 1 of
that reference), with $\theta$ 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 $\theta$
and $\phi$

.. math::

    A_P\,(q) =
      \frac{\sin \left( \tfrac{1}{2}qC \cos\theta \right) }{\tfrac{1}{2} qC \cos\theta}
      \,\times\,
      \frac{\sin \left( \tfrac{1}{2}qA \cos\theta \right) }{\tfrac{1}{2} qA \cos\theta}
      \,\times\ ,
      \frac{\sin \left( \tfrac{1}{2}qB \cos\theta \right) }{\tfrac{1}{2} qB \cos\theta}

where $A$, $B$ and $C$ are the sides of the parallelepiped and must fulfill
$A \le B \le C$, $\theta$ is the angle between the $z$ axis and the
$C$ axis of the parallelepiped, 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\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 rectangular prism, $\rho_\text{p}$
is the scattering length of the parallelepiped, $\rho_\text{solvent}$
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 was conducted by comparing the output of the 1D model
to the output of the existing :ref:`parallelepiped` model.



.. figure:: img/rectangular_prism_autogenfig.png

    1D and 2D plots corresponding to the default parameters of the model.


**Source**

:download:`rectangular_prism.py <src/rectangular_prism.py>`
$\ \star\ $ :download:`rectangular_prism.c <src/rectangular_prism.c>`
$\ \star\ $ :download:`gauss76.c <src/gauss76.c>`

**References**

See also Onsager [#Onsager1949]_.

.. [#Mittelbach1961] P Mittelbach and G Porod, *Acta Physica Austriaca*,
   14 (1961) 185-211
.. [#Nayuk2012] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854
.. [#Onsager1949] L Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

**Authorship and Verification**

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