.. _poly-gauss-coil:

poly_gauss_coil
=======================================================

Scattering from polydisperse polymer coils

============== =============================== ======= =============
Parameter      Description                     Units   Default value
============== =============================== ======= =============
scale          Scale factor or Volume fraction None                1
background     Source background               |cm^-1|         0.001
i_zero         Intensity at q=0                |cm^-1|            70
rg             Radius of gyration              |Ang|              75
polydispersity Polymer Mw/Mn                   None                2
============== =============================== ======= =============

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


This empirical model describes the scattering from *polydisperse* polymer
chains in theta solvents or polymer melts, assuming a Schulz-Zimm type
molecular weight distribution.

To describe the scattering from *monodisperse* polymer chains, see the
:ref:`mono-gauss-coil` model.

**Definition**

.. math::

     I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background}

where

.. math::

     I_0 &= \phi_\text{poly} \cdot V \cdot (\rho_\text{poly}-\rho_\text{solv})^2 \\
     P(q) &= 2 [(1 + UZ)^{-1/U} + Z - 1] / [(1 + U) Z^2] \\
     Z &= [(q R_g)^2] / (1 + 2U) \\
     U &= (Mw / Mn) - 1 = \text{polydispersity ratio} - 1 \\
     V &= M / (N_A \delta)

Here, $\phi_\text{poly}$, is the volume fraction of polymer, $V$ is the
volume of a polymer coil, $M$ is the molecular weight of the polymer,
$N_A$ is Avogadro's Number, $\delta$ is the bulk density of the polymer,
$\rho_\text{poly}$ is the sld of the polymer, $\rho_\text{solv}$ is the
sld of the solvent, and $R_g$ is the radius of gyration of the polymer coil.

The 2D scattering intensity is calculated in the same way as the 1D,
but where the $q$ vector is redefined as

.. math::

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


.. figure:: img/poly_gauss_coil_autogenfig.png

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


**Source**

:download:`poly_gauss_coil.py <src/poly_gauss_coil.py>`

**References**

#. O Glatter and O Kratky (editors), *Small Angle X-ray Scattering*,
   Academic Press, (1982) Page 404
#. J S Higgins, H C Benoit, *Polymers and Neutron Scattering*,
   Oxford Science Publications, (1996)
#. S M King, *Small Angle Neutron Scattering*
   in *Modern Techniques for Polymer Characterisation*, Wiley, (1999)
#. http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf

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