.. _rectangular-prism:

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

Rectangular parallelepiped with uniform scattering length density.

=========== ======================================== ============ =============
Parameter   Description                              Units        Default value
=========== ======================================== ============ =============
scale       Source intensity                         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 (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 $\theta$ 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
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\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.

**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

