.. _core-shell-sphere:

core_shell_sphere
=======================================================

Form factor for a monodisperse spherical particle with particle with a core-shell structure.

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Scale factor or Volume fraction   None                     1
background  Source background                 |cm^-1|              0.001
radius      Sphere core radius                |Ang|                   60
thickness   Sphere shell thickness            |Ang|                   10
sld_core    core scattering length density    |1e-6Ang^-2|             1
sld_shell   shell scattering length density   |1e-6Ang^-2|             2
sld_solvent Solvent scattering length density |1e-6Ang^-2|             3
=========== ================================= ============ =============

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


.. _core_shell_sphere:

This model provides the form factor, $P(q)$, for a spherical particle with
a core-shell structure. The form factor is normalized by the particle volume.

For information about polarised and magnetic scattering, see
the :ref:`magnetism` documentation.

**Definition**

The 1D scattering intensity is calculated in the following way (Guinier, 1955)

.. math::

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

where

.. math::

    F(q) = \frac{3}{V_s}\left[
       V_c(\rho_c-\rho_s)\frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3} +
       V_s(\rho_s-\rho_\text{solv})\frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}
       \right]

where $V_s$ is the volume of the whole particle, $V_c$ is the volume of the
core, $r_s$ = $radius$ + $thickness$ is the radius of the particle, $r_c$
is the radius of the core, $\rho_c$ 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.

The 2D scattering intensity is the same as $P(q)$ above, regardless of the
orientation of the $q$ vector.

NB: The outer most radius (ie, = radius + thickness) is used as the
effective radius for $S(Q)$ when $P(Q) \cdot S(Q)$ is applied.

**Validation**

Validation of our code was done by comparing the output of the 1D model to
the output of the software provided by NIST (Kline, 2006). Figure 1 shows a
comparison of the output of our model and the output of the NIST software.


.. figure:: img/core_shell_sphere_autogenfig.png

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


**Source**

:download:`core_shell_sphere.py <src/core_shell_sphere.py>`
$\ \star\ $ :download:`core_shell_sphere.c <src/core_shell_sphere.c>`
$\ \star\ $ :download:`core_shell.c <src/core_shell.c>`
$\ \star\ $ :download:`sas_3j1x_x.c <src/sas_3j1x_x.c>`

**References**

#. A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*,
   John Wiley and Sons, New York, (1955)

**Authorship and Verification**

* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**