.. _core-shell-cylinder:

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

Right circular 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
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       cylinder axis to beam angle              degree                  60
phi         rotation about beam                      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 [#Kline2006]_. The form factor is normalized
by the particle volume. Note that in this model the shell envelops the entire
core so that besides a "sleeve" around the core, the shell also provides two
flat end caps of thickness = shell thickness. In other words the length of the
total cylinder is the length of the core cylinder plus twice the thickness of
the shell. If no end caps are desired one should use the
:ref:`core-shell-bicelle` and set the thickness of the end caps (in this case
the "thick_face") to zero.

.. 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 total volume (i.e. including both the core and the outer 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 $L+2T$. $J_1$ 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.


**Source**

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

**Reference**

See also Livsey [#Livsey1987]_ and Onsager [#Onsager1949]_.

.. [#Livsey1987] I Livsey, *J. Chem. Soc., Faraday Trans. 2*, 83 (1987) 1445-1452

.. [#Kline2006] S R Kline, *J Appl. Cryst.*, 39 (2006) 895

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

**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