.. _core-shell-ellipsoid:

core_shell_ellipsoid
=======================================================

Form factor for an spheroid ellipsoid particle with a core shell structure.

================= ====================================================== ============ =============
Parameter         Description                                            Units        Default value
================= ====================================================== ============ =============
scale             Source intensity                                       None                     1
background        Source background                                      |cm^-1|              0.001
radius_equat_core Equatorial radius of core                              |Ang|                   20
x_core            axial ratio of core, X = r_polar/r_equatorial          None                     3
thick_shell       thickness of shell at equator                          |Ang|                   30
x_polar_shell     ratio of thickness of shell at pole to that at equator None                     1
sld_core          Core scattering length density                         |1e-6Ang^-2|             2
sld_shell         Shell scattering length density                        |1e-6Ang^-2|             1
sld_solvent       Solvent scattering length density                      |1e-6Ang^-2|           6.3
theta             Oblate orientation wrt incoming beam                   degree                   0
phi               Oblate orientation in the plane of the detector        degree                   0
================= ====================================================== ============ =============

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


**Definition**

Parameters for this model are the core axial ratio X and a shell thickness,
which are more often what we would like to determine and makes the model
better behaved, particularly when polydispersity is applied than the four
independent radii used in the original parameterization of this model.


.. figure:: img/core_shell_ellipsoid_geometry.png

The geometric parameters of this model are shown in the diagram above, which
shows (a) a cut through at the circular equator and (b) a cross section through
the poles, of a prolate ellipsoid.

When *X_core < 1* the core is oblate; when *X_core > 1* it is prolate.
*X_core = 1* is a spherical core.

For a fixed shell thickness *XpolarShell = 1*, to scale the shell thickness
pro-rata with the radius set or constrain *XpolarShell = X_core*.

When including an $S(q)$, the radius in $S(q)$ is calculated to be that of
a sphere with the same 2nd virial coefficient of the outer surface of the
ellipsoid. This may have some undesirable effects if the aspect ratio of the
ellipsoid is large (ie, if $X << 1$ or $X >> 1$ ), when the $S(q)$
- which assumes spheres - will not in any case be valid.  Generating a 
custom product model will enable separate effective volume fraction and effective 
radius in the $S(q)$.

If SAS data are in absolute units, and the SLDs are correct, then scale should
be the total volume fraction of the "outer particle". When $S(q)$ is introduced
this moves to the $S(q)$ volume fraction, and scale should then be 1.0,
or contain some other units conversion factor (for example, if you have SAXS data).

The calculation of intensity follows that for the solid ellipsoid, but with separate
terms for the core-shell and shell-solvent boundaries.

.. math::

    P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background}

where

.. math::
    \begin{align}    
    F(q,\alpha) = &f(q,radius\_equat\_core,radius\_equat\_core.x\_core,\alpha) \\
    &+ f(q,radius\_equat\_core + thick\_shell,radius\_equat\_core.x\_core + thick\_shell.x\_polar\_shell,\alpha)
    \end{align} 

where
 
.. math::

    f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)]
                - \cos[qr(R_p,R_e,\alpha)])}
                {[qr(R_p,R_e,\alpha)]^3}

and

.. math::

    r(R_e,R_p,\alpha) = \left[ R_e^2 \sin^2 \alpha
        + R_p^2 \cos^2 \alpha \right]^{1/2}


$\alpha$ is the angle between the axis of the ellipsoid and $\vec q$,
$V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the polar radius along the
rotational axis of the ellipsoid, $R_e$ is the equatorial radius perpendicular
to the rotational axis of the ellipsoid and $\Delta \rho$ (contrast) is the
scattering length density difference, either $(sld\_core - sld\_shell)$ or $(sld\_shell - sld\_solvent)$.

For randomly oriented particles:

.. math::

   F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}



.. figure:: img/core_shell_ellipsoid_autogenfig.png

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

**References**
see for example:
Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.
Berr, S.  J. Phys. Chem., 1987, 91, 4760.

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015
* **Last Reviewed by:** Richard Heenan **Date:** October 6, 2016