.. _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
=========== ======================================== ============ =============

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).
Note also that, contrary to :ref:`parallelepiped`, it does not compute
the 2D scattering.


**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. As the present model does not compute
the 2D scattering, this has no further consequences.

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).

**The 2D scattering intensity is not computed by this model.**


**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 plot 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


