.. _hollow-rectangular-prism:

hollow_rectangular_prism
=======================================================

Hollow 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                          |Ang|                    1
c2a_ratio   Ratio sides c/a                          |Ang|                    1
thickness   Thickness of parallelepiped              |Ang|                    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 hollow rectangular
parallelepiped with a wall of thickness $\Delta$.
It computes only the 1D scattering, not the 2D.

**Definition**

The 1D scattering intensity for this model is calculated by forming
the difference of the amplitudes of two massive parallelepipeds
differing in their outermost dimensions in each direction by the
same length increment $2\Delta$ (Nayuk, 2012).

As in the case of the massive parallelepiped model (:ref:`rectangular-prism`),
the scattering amplitude is computed for a particular orientation of the
parallelepiped with respect to the scattering vector and then averaged over all
possible orientations, giving

.. math::
  P(q) =  \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \,
  \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, \sin\theta \, d\theta \, d\phi

where $\theta$ is the angle between the $z$ axis and the longest axis
of the parallelepiped, $\phi$ is the angle between the scattering vector
(lying in the $xy$ plane) and the $y$ axis, and

.. math::
  :nowrap:

  \begin{align}
  A_{P\Delta}(q) & =  A B C
    \left[\frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)}
    {\left( q \frac{C}{2} \cos\theta \right)} \right]
    \left[\frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)}
    {\left( q \frac{A}{2} \sin\theta \sin\phi \right)}\right]
    \left[\frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)}
    {\left( q \frac{B}{2} \sin\theta \cos\phi \right)}\right] \\
    & - 8
    \left(\frac{A}{2}-\Delta\right) \left(\frac{B}{2}-\Delta\right) \left(\frac{C}{2}-\Delta\right)
    \left[ \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]}
    {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \right]
    \left[ \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]}
    {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \right]
    \left[ \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]}
    {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \right]
  \end{align}

where $A$, $B$ and $C$ are the external sides of the parallelepiped fulfilling
$A \le B \le C$, and the volume $V$ of the parallelepiped is

.. math::
  V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta)

The 1D scattering intensity is then calculated as

.. math::
  I(q) = \text{scale} \times V \times (\rho_\text{p} -
  \rho_\text{solvent})^2 \times P(q) + \text{background}

where $\rho_\text{p}$ 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 qualitatively comparing the output
of the 1D model to the curves shown in (Nayuk, 2012).



.. figure:: img/hollow_rectangular_prism_autogenfig.png

    1D plot corresponding to the default parameters of the model.

**References**

R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854