.. _torus-elliptical-shell:

torus_elliptical_shell
=======================================================

core-shell torus with elliptical cross-section

=========== =================================== ============ =============
Parameter   Description                         Units        Default value
=========== =================================== ============ =============
scale       Scale factor or Volume fraction     None                     1
background  Source background                   |cm^-1|              0.001
radius      Torus major radius R                |Ang|                  100
radius_core Elliptical core minor radius a      |Ang|                    5
thickness   Shell thickness t                   |Ang|                    2
core_nu     Aspect ratio of core cross-section  None                     1
shell_nu    Aspect ratio of shell cross-section None                     1
sld_core    Core SLD                            |1e-6Ang^-2|             0
sld_shell   Shell SLD                           |1e-6Ang^-2|             1
sld_solvent Solvent SLD                         |1e-6Ang^-2|             0
=========== =================================== ============ =============

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


**Definition**

This model computes the scattering from a torus with an elliptical tube
cross-section and a concentric shell.

The geometric parameters are:

* $R$: major (ring) radius of the torus centerline
* $a$: core minor radius in the radial direction
* $\nu_{core}$: aspect ratio of the elliptical core cross-section
* $\nu_{shell}$: aspect ratio of the elliptical shell cross-section
* $t$: shell thickness

.. figure:: img/torus_elliptical_shell_geometry.png

    Schematic geometry of the torus with elliptical tube cross-section.

The core and shell have independent aspect ratios, so the core semi-axes are
$a$ and $\nu_{core} \cdot a$, and the outer semi-axes are $a+t$ and
$\nu_{core} \cdot a + \nu_{shell} \cdot t$.

For a given orientation angle $\theta$ between the torus symmetry axis and
$\vec q$, the kernel evaluates the amplitude using a numerical integral over
the tube section:

.. math::

    F(q,\theta; x, \Delta\rho, \nu)
    = 4\pi\,\Delta\rho\int_{R-x}^{R+x}
      r\,J_0\!\left(qr\sin\theta\right)
      \frac{\sin\!\left(q\,\gamma\cos\theta\right)}{q\cos\theta}\,dr


with

.. math::

    \gamma = \nu\sqrt{x^2-(r-R)^2}


The core-shell amplitude is formed from outer and inner contributions:

.. math::

    F_{cs}(q,\theta) =
    F\!\left(q,\theta;a + t,\rho_{shell}-\rho_{solvent},\nu_{outer}\right)
    -F\!\left(q,\theta;a,\rho_{shell}-\rho_{core},\nu_{core}\right)


where $\nu_{outer} = \frac{\nu_{shell}t + \nu_{core}a}{a+t}$ is the aspect ratio of the outer ellipse.

and the orientationally averaged intensity is

.. math::

    I(q) = \int_0^{\pi/2}\!\left|F_{cs}(q,\theta)\right|^2\sin\theta\,d\theta



.. figure:: img/torus_elliptical_shell_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`torus_elliptical_shell.py <src/torus_elliptical_shell.py>`
$\ \star\ $ :download:`torus_elliptical_shell.c <src/torus_elliptical_shell.c>`
$\ \star\ $ :download:`gauss76.c <src/gauss76.c>`
$\ \star\ $ :download:`sas_J0.c <src/sas_J0.c>`
$\ \star\ $ :download:`polevl.c <src/polevl.c>`

**References**
#.  T. Kawaguchi, *J. Appl. Crystallogr*, 34(2001) 580-584
#.  S. Förster, *J. Phys. Chem.*, 103(1999) 6657-6668

**Authorship and Verification**

* **Author:** Itsuki Tajima and Yuhei Yamada (Github user name: Indigo Carmine, https://orcid.org/0009-0003-9780-4135)
* **Last Modified by:**
* **Last Reviewed by:**