.. _squarewell:

squarewell
=======================================================

Square well structure factor, with MSA 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 spheres                          None               0.04
welldepth        depth of well, epsilon                              kT                  1.5
wellwidth        width of well in diameters (=2R) units, must be > 1 diameters           1.2
================ =================================================== ========= =============

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


This calculates the interparticle structure factor for a square well fluid
spherical particles. The mean spherical approximation (MSA) closure was
used for this calculation, and is not the most appropriate closure for
an attractive interparticle potential. This solution has been compared
to Monte Carlo simulations for a square well fluid, showing this calculation
to be limited in applicability to well depths $\epsilon < 1.5$ kT and
volume fractions $\phi < 0.08$.

Positive well depths correspond to an attractive potential well. Negative
well depths correspond to a potential "shoulder", which may or may not be
physically reasonable. The stickyhardsphere model may be a better choice in
some circumstances. Computed values may behave badly at extremely small $qR$.

The well width $(\lambda)$ is defined as multiples of the particle diameter
$(2 R)$.

The interaction potential is:

  .. image:: img/squarewell.png

.. math::

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

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

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 2D data: The 2D scattering intensity is calculated in the same way as 1D,
where the $q$ vector is defined as

.. math::

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


.. figure:: img/squarewell_autogenfig.png

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

**References**

R V Sharma, K C Sharma, *Physica*, 89A (1977) 213.


