.. _core-shell-cylinder:

core_shell_cylinder
=======================================================

Right 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
sld_core    Cylinder core scattering length density  |1e-6Ang^-2|             4
sld_shell   Cylinder shell scattering length density |1e-6Ang^-2|             4
sld_solvent Solvent scattering length density        |1e-6Ang^-2|             1
radius      Cylinder core radius                     |Ang|                   20
thickness   Cylinder shell thickness                 |Ang|                   20
length      Cylinder length                          |Ang|                  400
theta       In plane angle                           degree                  60
phi         Out of plane angle                       degree                  60
=========== ======================================== ============ =============

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


**Definition**

The output of the 2D scattering intensity function for oriented core-shell
cylinders is given by (Kline, 2006 [#kline]_). The form factor is normalized
by the particle volume.

.. math::

    I(q,\alpha) = \frac{\text{scale}}{V_s} F^2(q,\alpha).sin(\alpha) + \text{background}

where

.. math::

    F(q,\alpha) = &\ (\rho_c - \rho_s) V_c
           \frac{\sin \left( q \tfrac12 L\cos\alpha \right)}
                {q \tfrac12 L\cos\alpha}
           \frac{2 J_1 \left( qR\sin\alpha \right)}
                {qR\sin\alpha} \\
         &\ + (\rho_s - \rho_\text{solv}) V_s
           \frac{\sin \left( q \left(\tfrac12 L+T\right) \cos\alpha \right)}
                {q \left(\tfrac12 L +T \right) \cos\alpha}
           \frac{ 2 J_1 \left( q(R+T)\sin\alpha \right)}
                {q(R+T)\sin\alpha}

and

.. math::

    V_s = \pi (R + T)^2 (L + 2T)

and $\alpha$ is the angle between the axis of the cylinder and $\vec q$,
$V_s$ is the volume of the outer shell (i.e. the total volume, including
the shell), $V_c$ is the volume of the core, $L$ is the length of the core,
$R$ is the radius of the core, $T$ is the thickness of the shell, $\rho_c$
is the scattering length density of the core, $\rho_s$ is the scattering
length density of the shell, $\rho_\text{solv}$ is the scattering length
density of the solvent, and *background* is the background level.  The outer
radius of the shell is given by $R+T$ and the total length of the outer
shell is given by $L+2T$. $J1$ is the first order Bessel function.

.. _core-shell-cylinder-geometry:

.. figure:: img/core_shell_cylinder_geometry.jpg

    Core shell cylinder schematic.

To provide easy access to the orientation of the core-shell cylinder, we
define the axis of the cylinder using two angles $\theta$ and $\phi$.
(see :ref:`cylinder model <cylinder-angle-definition>`)

NB: The 2nd virial coefficient of the cylinder is calculated based on
the radius and 2 length values, and used as the effective radius for
$S(q)$ when $P(q) \cdot S(q)$ is applied.

The $\theta$ and $\phi$ parameters are not used for the 1D output.


.. figure:: img/core_shell_cylinder_autogenfig.png

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

**Reference**

.. [#] see, for example, Ian Livsey  J. Chem. Soc., Faraday Trans. 2, 1987,83,
   1445-1452
.. [#kline] S R Kline, *J Appl. Cryst.*, 39 (2006) 895

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Kienzle **Date:** Aug 8, 2016
* **Last Reviewed by:** Richard Heenan **Date:** March 18, 2016