.. _core-shell-bicelle-elliptical-belt-rough:

core_shell_bicelle_elliptical_belt_rough
=======================================================

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 or belt 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
sigma       Interfacial roughness                    |Ang|                    0
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.
In this version the "rim" or "belt" does NOT extend the full length of
the particle, but has the same length as 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.
This version includes an approximate "interfacial roughness".


.. figure:: img/core_shell_bicelle_belt_geometry.png

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

.. figure:: img/core_shell_bicelle_belt_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/2 \lt z\lt L/2 \\[1.5ex]
      &\rho_f \text{ for } 0 \lt r \lt R;   -(L/2 +t_\text{face}) \lt z\lt -L/2;
      L/2 \lt z\lt (L/2+t_\text{face}) \\[1.5ex]
      &\rho_r\text{ for } R \lt r \lt R+t_\text{rim}; -L/2 \lt z\lt L/2
      \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)
        \cdot\exp\left\{ -\frac{1}{2}Q^2\sigma^2 \right\} + \text{background}

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

.. math::

    F(Q,\alpha,\psi) = &\bigg[
      (\rho_c -\rho_r - \rho_f + \rho_s) V_c
      \frac{2J_1(QR'\sin \alpha)}{QR'\sin\alpha}
      \frac{\sin(QL\cos\alpha/2)}{Q(L/2)\cos\alpha} \\
    &+(\rho_f - \rho_s) 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_{c+r}
      \frac{2J_1(Q(R'+t_r)\sin\alpha)}{Q(R'+t_r)\sin\alpha}
      \frac{\sin(Q(L/2)\cos\alpha)}{Q(L/2)\cos\alpha}
    \bigg]

where

.. math::

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


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

An approximation for the effects of "Gaussian interfacial roughness" $\sigma$
is included, by multiplying $I(Q)$ by
$\exp\left \{ -\frac{1}{2}Q^2\sigma^2 \right \}$. This applies, in some way, to
all interfaces in the model not just the external ones. (Note that for a one
dimensional system convolution of the scattering length density profile with
a Gaussian of standard deviation $\sigma$ does exactly this multiplication.)
Leave $\sigma$ set to zero for the usual sharp interfaces.

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, for further details of the calculation
and angular dispersions  see :ref:`orientation` .

.. figure:: img/elliptical_cylinder_angle_definition.png

    Definition of the angles for the oriented core_shell_bicelle_elliptical
    particles.




.. figure:: img/core_shell_bicelle_elliptical_belt_rough_autogenfig.png

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


**Source**

:download:`core_shell_bicelle_elliptical_belt_rough.py <src/core_shell_bicelle_elliptical_belt_rough.py>`
$\ \star\ $ :download:`core_shell_bicelle_elliptical_belt_rough.c <src/core_shell_bicelle_elliptical_belt_rough.c>`
$\ \star\ $ :download:`adaptive.c <src/adaptive.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:** October 5, 2017
* **Last Modified by:**  Richard Heenan new 2d orientation **Date:** October 5, 2017
* **Last Reviewed by:**  Richard Heenan 2d calc seems agree with 1d **Date:** Nov 2, 2017

