.. _core-shell-bicelle-elliptical:

core_shell_bicelle_elliptical
=======================================================

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

=========== ======================================== ============ =============
Parameter   Description                              Units        Default value
=========== ======================================== ============ =============
scale       Scale factor or Volume fraction          None                     1
background  Source background                        |cm^-1|              0.001
radius      Cylinder core radius r_minor             |Ang|                   30
x_core      Axial ratio of core, X = r_major/r_minor None                     3
thick_rim   Rim shell thickness                      |Ang|                    8
thick_face  Cylinder face thickness                  |Ang|                   14
length      Cylinder length                          |Ang|                   50
sld_core    Cylinder core scattering length density  |1e-6Ang^-2|             4
sld_face    Cylinder face scattering length density  |1e-6Ang^-2|             7
sld_rim     Cylinder rim scattering length density   |1e-6Ang^-2|             1
sld_solvent Solvent scattering length density        |1e-6Ang^-2|             6
theta       Cylinder axis to beam angle              degree                  90
phi         Rotation about beam                      degree                   0
psi         Rotation about cylinder axis             degree                   0
=========== ======================================== ============ =============

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


**Definition**

This model provides the form factor for an elliptical cylinder with a
core-shell scattering length density profile [#Onsager1949]_.
Thus this is a variation
of the core-shell bicelle model, but with an elliptical cylinder for the core.
Outer shells on the rims and flat ends may be of different thicknesses and
scattering length densities. The form factor is normalized by the total
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 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 bicelle 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, and $\psi$
is the angle for the ellipsoidal cross section core, to give:

.. math::

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

where a numerical integration of $F(Q,\alpha, \psi)^2 \cdot sin(\alpha)$
is carried out over \alpha and \psi for:

.. math::
    :nowrap:

    \begin{align*}
    F(Q,\alpha,\psi) = &\bigg[
    (\rho_c - \rho_f) V_c
     \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha}
     \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\
    &+(\rho_f - \rho_r) V_{c+f}
     \frac{2J_1(QR'sin\alpha)}{QR'sin\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

.. math::

    R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)}


and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of
the bicelle, $V_c = \pi.Xcore.R^2.L$ the volume of the core,
$V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$ the volume of the core plus the volume
of the faces, $R$ is the radius of the core, $Xcore$ is the axial ratio 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 core has radii $R$ and $Xcore.R$ so is circular, as for the
core_shell_bicelle model, for $Xcore$ =1. Note that you may need to
limit the range of $Xcore$, especially if using the Monte-Carlo algorithm,
as setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an
equivalent solution!

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

For oriented bicelles the *theta*, *phi* and *psi* orientation parameters will
appear when fitting 2D data, see the :ref:`elliptical-cylinder` model
for further information.

.. figure:: img/elliptical_cylinder_angle_definition.png

    Definition of the angles for the oriented core_shell_bicelle_elliptical particles.

Model verified using Monte Carlo simulation for 1D and 2D scattering.


.. figure:: img/core_shell_bicelle_elliptical_autogenfig.png

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


**Source**

:download:`core_shell_bicelle_elliptical.py <src/core_shell_bicelle_elliptical.py>`
$\ \star\ $ :download:`core_shell_bicelle_elliptical.c <src/core_shell_bicelle_elliptical.c>`
$\ \star\ $ :download:`gauss76.c <src/gauss76.c>`
$\ \star\ $ :download:`sas_J1.c <src/sas_J1.c>`
$\ \star\ $ :download:`polevl.c <src/polevl.c>`
$\ \star\ $ :download:`sas_Si.c <src/sas_Si.c>`

**References**

.. [#Onsager1949] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

**Authorship and Verification**

* **Author:** Richard Heenan **Date:** December 14, 2016
* **Last Modified by:**  Richard Heenan **Date:** December 14, 2016
* **Last Reviewed by:**  Paul Kienzle **Date:** Feb 28, 2018

