.. _hardsphere:

hardsphere
=======================================================

Hard sphere structure factor, with Percus-Yevick closure

================ =============================== ======= =============
Parameter        Description                     Units   Default value
================ =============================== ======= =============
scale            Source intensity                None                1
background       Source background               |cm^-1|         0.001
radius_effective effective radius of hard sphere |Ang|              50
volfraction      volume fraction of hard spheres None              0.2
================ =============================== ======= =============

The returned value is a dimensionless structure factor, $S(q)$.

Calculate the interparticle structure factor for monodisperse
spherical particles interacting through hard sphere (excluded volume)
interactions.
May be a reasonable approximation for other shapes of particles that
freely rotate, and for moderately polydisperse systems. Though strictly
the maths needs to be modified (no \Beta(Q) correction yet in sasview).

radius_effective is the effective hard sphere radius.
volfraction is the volume fraction occupied by the spheres.

In sasview the effective radius may be calculated from the parameters
used in the form factor $P(q)$ that this $S(q)$ is combined with.

For numerical stability the computation uses a Taylor series expansion
at very small $qR$, there may be a very minor glitch at the transition point
in some circumstances.

The S(Q) uses the Percus-Yevick closure where the interparticle
potential is

.. math::

    U(r) = \begin{cases}
    \infty & r < 2R \\
    0 & r \geq 2R
    \end{cases}

where $r$ is the distance from the center of the sphere of a radius $R$.

For a 2D plot, the wave transfer is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}



.. figure:: img/hardsphere_autogenfig.png

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

**References**

J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1

