.. _sphere:

Sphere
=======================================================

Spheres with uniform scattering length density

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Source intensity                  None                     1
background  Source background                 |cm^-1|                  0
sld         Layer scattering length density   |1e-6Ang^-2|             1
solvent_sld Solvent scattering length density |1e-6Ang^-2|             6
radius      Sphere radius                     |Ang|                   50
=========== ================================= ============ =============

The returned value is scaled to units of |cm^-1|.


For information about polarised and magnetic scattering, click here_.

.. _here: polar_mag_help.html

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, *background* is the background level and
*sld* and *solvent_sld* are the scattering length densities (SLDs) of the
scatterer and the solvent respectively.

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 $\vec q$.

Our model uses the form factor calculations as defined in the IGOR
package provided by the NIST Center for Neutron Research (Kline, 2006).

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 :num:`figure #sphere-comparison` shows a comparison of the output
of our model and the output of the NIST software.

.. _sphere-comparison:

.. figure:: img/sphere_comparison.jpg

    Comparison of the DANSE scattering intensity for a sphere with the
    output of the NIST SANS analysis software. The parameters were set to:
    *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and
    *background* = 0.01 |cm^-1|.


Reference
---------

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

*2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.*