.. _star-polymer:

star_polymer
=======================================================

Star polymer model with Gaussian statistics

========== ============================================= ======= =============
Parameter  Description                                   Units   Default value
========== ============================================= ======= =============
scale      Source intensity                              None                1
background Source background                             |cm^-1|         0.001
rg_squared Ensemble radius of gyration SQUARED of an arm Ang^2             100
arms       Number of arms in the model                   None                3
========== ============================================= ======= =============

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


The Benoit model for a simple star polymer, with Gaussian coils arms from
a common point.

**Definition**

For a star with $f$ arms the scattering intensity $I(q)$ is calculated as

.. math::

    I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2}
           \left[ 1-\exp(-v)\right]^2\right]

where

.. math:: v=\frac{u^2f}{(3f-2)}

and

.. math:: u = \left\langle R_{g}^2\right\rangle q^2

contains the square of the ensemble average radius-of-gyration of an arm.
Note that when there is only one arm, $f = 1$, the Debye Gaussian coil
equation is recovered. Star polymers in solutions tend to have strong
interparticle and osmotic effects, so the Benoit equation may not work well.
At small $q$ the Guinier term and hence $I(q=0)$ is the same as for $f$ arms
of radius of gyration $R_g$, as described for the :ref:`mono-gauss-coil` model.


.. figure:: img/star_polymer_autogenfig.png

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

**References**

H Benoit *J. Polymer Science*, 11, 596-599 (1953)