5 Layer Core Shell Disc

This model calculates the form factor for a core-shell circular cylinder. The core includes
three layers, two methylene and one methyl, which creates a five-layer model when combined
with the two headgroup layers. The shell on the walls and ends of the model can be of
different thicknesses and scattering length densities, but is only a single shell (see the figure in
the second reference below). Normalization of the form factor is done using the particle volume.
Cylindrical symmetry is assumed for this model.

Given the scattering length densities (sld) $\rho_{mlene}$, the methylene sld, $\rho_{myl}$,
the methyl sld, $\rho_f$, the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld,
the scattering length density variation along the cylinder axis is:

\begin{align}
\rho(r) =
\begin{cases}
&\rho_{mlene} \text{ for } 0 \lt r \lt R; -L_c/2 \lt z\lt L_c/2 \\[1.5ex]
&\rho_{myl} \text{ for } 0 \lt r \lt R; -L_{myl}/2 \lt z\lt L_{myl}/2 \\[1.5ex]
&\rho_f \text{ for } 0 \lt r \lt R; -(L_c+2t_f)/2 \lt z\lt -L; L \lt z\lt (L_c+2t_f)/2 \\[1.5ex]
&\rho_r\text{ for } R \lt r \lt R+t_r; -(L_c+2t_f)/2 \lt z\lt -L; L \lt z\lt (L_c+2t_f)/2
\end{cases}
\end{align}

Cylindrical coordinates are used for this model, where $\alpha$ is the angle between the
$Q$ vector and the cylinder axis, to give:

\begin{align}
I(Q,\alpha) = \frac{\text{scale}}{V_t} \cdot
F(Q,\alpha)^2.sin(\alpha) + \text{background}
\end{align}

where

\begin{align}
F(Q,\alpha) = &\bigg[
(\rho_{myl} - \rho_{mlene}) V_{myl} \frac{2J_1(QRsin\alpha)}{QRsin\alpha}\frac{sin(Q(L_{myl})cos\alpha/2)}{Q(L_{myl}/2)cos\alpha} \\
&+(\rho_{mlene} - \rho_f) V_c \frac{2J_1(QRsin\alpha)}{QRsin\alpha}\frac{sin(QL_ccos\alpha/2)}{Q(L_c/2)cos\alpha} \\
&+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QRsin\alpha)}{QRsin\alpha}\frac{sin(Q(L_c/2+t_f)cos\alpha)}{Q(L_c/2+t_f)cos\alpha} \\
&+(\rho_r - \rho_s) V_t \frac{2J_1(Q(R+t_r)sin\alpha)}{Q(R+t_r)sin\alpha}\frac{sin(Q(L_c/2+t_f)cos\alpha)}{Q(L_c/2+t_f)cos\alpha}
\bigg]
\end{align}
\begin{align}
L_c = &\bigg[
2L_{mlene}+L_{myl}
\bigg]
\end{align}

where $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core,
$V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius
of the core, $L_c$ the length of the core, $L_{mlene}$ is the length of the methylene
layer, $L_{myl}$ the length of the methyl layer, $t_f$ the thickness of the face,
$t_r$ the thickness of the rim and $J_1$ the usual first order bessel function.

The 1D scattering intensity for randomly oriented cylinders is calculated by
integrating over all possible $\theta$ and $\phi$.

The 1D output does not use the *theta* and *phi* parameters. The scattering kernel
and the 1D scattering intensity use the c-library from NIST.

References
----------

.. [#] D Singh (2009). *Small angle scattering studies of self assembly in
lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available
from Proquest <http://search.proquest.com/docview/304915826?accountid
=26379>`_
[#] C Cheu, L Yang, M P Nieh (2020). *Refining internal bilayer structure of bicelles resolved
by extended-q small angle X-ray scattering*, Chemistry and Physics of Lipids, 231
(2020) 104945.