.. _hollow-rectangular-prism-thin-walls:

hollow_rectangular_prism_thin_walls
=======================================================

Hollow rectangular parallelepiped with thin walls.

=========== ======================================== ============ =============
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                          |Ang|                    1
c2a_ratio   Ratio sides c/a                          |Ang|                    1
=========== ======================================== ============ =============

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


**Definition**
This model provides the form factor, $P(q)$, for a hollow rectangular prism
with infinitely thin walls. It computes only the 1D scattering, not the 2D.
The 1D scattering intensity for this model is calculated according to the
equations given by Nayuk and Huber\ [#Nayuk2012]_.

Assuming a hollow parallelepiped with infinitely thin walls, edge lengths
$A \le B \le C$ and presenting an orientation with respect to the
scattering vector given by $\theta$ and $\phi$, where $\theta$ 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 form factor is given by

.. math::

    P(q) = \frac{1}{V^2} \frac{2}{\pi} \int_0^{\frac{\pi}{2}}
           \int_0^{\frac{\pi}{2}} [A_L(q)+A_T(q)]^2 \sin\theta\,d\theta\,d\phi

where

.. math::

    V &= 2AB + 2AC + 2BC \\
    A_L(q) &=  8 \times \frac{
            \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right)
            \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right)
            \cos \left( \tfrac{1}{2} q C \cos\theta \right)
        }{q^2 \, \sin^2\theta \, \sin\phi \cos\phi} \\
    A_T(q) &=  A_F(q) \times
      \frac{2\,\sin \left( \tfrac{1}{2} q C \cos\theta \right)}{q\,\cos\theta}

and

.. math::

  A_F(q) =  4 \frac{ \cos \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right)
                       \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) }
                     {q \, \cos\phi \, \sin\theta} +
              4 \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right)
                       \cos \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) }
                     {q \, \sin\phi \, \sin\theta}

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)

where $V$ is the surface area of the rectangular prism, $\rho_\text{p}$
is the scattering length density of the parallelepiped, $\rho_\text{solvent}$
is the scattering length density of the solvent, and (if the data are in
absolute units) *scale* is related to the total surface area.

**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\ [#Nayuk2012]_).



.. figure:: img/hollow_rectangular_prism_thin_walls_autogenfig.png

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


**Source**

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

**References**

See also Onsager [#Onsager1949]_.

.. [#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**

* **Author:** Miguel Gonzales **Date:** February 26, 2016
* **Last Modified by:** Paul Kienzle **Date:** October 15, 2016
* **Last Reviewed by:** Paul Butler **Date:** September 07, 2018