.. _gauss-lorentz-gel:

gauss_lorentz_gel
=======================================================

Gauss Lorentz Gel model of scattering from a gel structure

================== =============================== ======= =============
Parameter          Description                     Units   Default value
================== =============================== ======= =============
scale              Scale factor or Volume fraction None                1
background         Source background               |cm^-1|         0.001
gauss_scale        Gauss scale factor              None              100
cor_length_static  Static correlation length       |Ang|             100
lorentz_scale      Lorentzian scale factor         None               50
cor_length_dynamic Dynamic correlation length      |Ang|              20
================== =============================== ======= =============

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


This model calculates the scattering from a gel structure,
but typically a physical rather than chemical network.
It is modeled as a sum of a low-q exponential decay (which happens to
give a functional form similar to Guinier scattering, so interpret with
care) plus a Lorentzian at higher-q values. See also the gel_fit model.

**Definition**

The scattering intensity $I(q)$ is calculated as (Eqn. 5 from the reference)

.. math:: I(q) = I_G(0) \exp(-q^2\Xi ^2/2) + I_L(0)/(1+q^2\xi^2)

$\Xi$ is the length scale of the static correlations in the gel, which can
be attributed to the "frozen-in" crosslinks. $\xi$ is the dynamic correlation
length, which can be attributed to the fluctuating polymer chains between
crosslinks. $I_G(0)$ and $I_L(0)$ are the scaling factors for each of these
structures. Think carefully about how these map to your particular system!

.. note::
    The peaked structure at higher $q$ values (Figure 2 from the reference)
    is not reproduced by the model. Peaks can be introduced into the model
    by summing this model with the :ref:`gaussian-peak` model.

For 2D data the 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/gauss_lorentz_gel_autogenfig.png

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


**Source**

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

**References**

#. G Evmenenko, E Theunissen, K Mortensen, H Reynaers,
   *Polymer*, 42 (2001) 2907-2913

**Authorship and Verification**

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