.. _sphere:

sphere
=======================================================

Spheres with uniform scattering length density

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Scale factor or Volume fraction   None                     1
background  Source background                 |cm^-1|              0.001
sld         Layer scattering length density   |1e-6Ang^-2|             1
sld_solvent Solvent scattering length density |1e-6Ang^-2|             6
radius      Sphere radius                     |Ang|                   50
=========== ================================= ============ =============

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


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::

    I(q) = \frac{\text{scale}}{V} \cdot \left[
        3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3}
        \right]^2 + \text{background}

where *scale* is a volume fraction, $V$ is the volume of the scatterer,
$r$ is the radius of the sphere and *background* is the background level.
*sld* and *sld_solvent* are the scattering length densities (SLDs) of the
scatterer and the solvent respectively, whose difference is $\Delta\rho$.

Note that if your data is in absolute scale, the *scale* should represent
the volume fraction (which is unitless) if you have a good fit. If not,
it should represent the volume fraction times a factor (by which your data
might need to be rescaled).

The 2D scattering intensity is the same as above, regardless of the
orientation of $\vec q$.

**Validation**

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



.. figure:: img/sphere_autogenfig.png

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


**Source**

:download:`sphere.py <src/sphere.py>`
$\ \star\ $ :download:`sphere.c <src/sphere.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:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06