.. _core-shell-bicelle:

core_shell_bicelle
=======================================================

Circular cylinder with a core-shell scattering length density profile..

=========== ======================================= ============ =============
Parameter   Description                             Units        Default value
=========== ======================================= ============ =============
scale       Source intensity                        None                     1
background  Source background                       |cm^-1|              0.001
radius      Cylinder core radius                    |Ang|                   80
thick_rim   Rim shell thickness                     |Ang|                   10
thick_face  Cylinder face thickness                 |Ang|                   10
length      Cylinder length                         |Ang|                   50
sld_core    Cylinder core scattering length density |1e-6Ang^-2|             1
sld_face    Cylinder face scattering length density |1e-6Ang^-2|             4
sld_rim     Cylinder rim scattering length density  |1e-6Ang^-2|             4
sld_solvent Solvent scattering length density       |1e-6Ang^-2|             1
theta       In plane angle                          degree                  90
phi         Out of plane angle                      degree                   0
=========== ======================================= ============ =============

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


**Definition**

This model provides the form factor for a circular cylinder with a
core-shell scattering length density profile. Thus this is a variation
of a core-shell cylinder or disc where the shell on the walls and ends
may be of different thicknesses and scattering length densities. The form
factor is normalized by the particle volume.


.. figure:: img/core_shell_bicelle_geometry.png

    Schematic cross-section of bicelle. Note however that the model here
    calculates for rectangular, not curved, rims as shown below.

.. figure:: img/core_shell_bicelle_parameters.png

   Cross section of cylindrical symmetry model used here. Users will have 
   to decide how to distribute "heads" and "tails" between the rim, face 
   and core regions in order to estimate appropriate starting parameters.

Given the scattering length densities (sld) $\rho_c$, the core 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:

.. math::

    \rho(r) = 
      \begin{cases} 
      &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex]
      &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L;
      L \lt z\lt (L+2t) \\[1.5ex]
      &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t)
      \end{cases}

The form factor for the bicelle is calculated in cylindrical coordinates, where
$\alpha$ is the angle between the $Q$ vector and the cylinder axis, to give:

.. math::

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

where

.. math::

        \begin{align}    
    F(Q,\alpha) = &\bigg[ 
    (\rho_c - \rho_f) V_c \frac{2J_1(QRsin \alpha)}{QRsin\alpha}\frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\
    &+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QRsin\alpha)}{QRsin\alpha}\frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/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/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha}
    \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, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$
the thickness of the rim and $J_1$ the usual first order bessel function.

The output of the 1D scattering intensity function for randomly oriented
cylinders is then given by integrating over all possible $\theta$ and $\phi$.

The *theta* and *phi* parameters are not used for the 1D output.
Our implementation of the scattering kernel and the 1D scattering intensity
use the c-library from NIST.

.. figure:: img/cylinder_angle_definition.jpg

    Definition of the angles for the oriented core shell bicelle tmodel.



.. figure:: img/core_shell_bicelle_autogenfig.png

    1D and 2D plots corresponding to the default parameters of the model.

**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>`_

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** September 30, 2016
* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017