.. _two-lorentzian:

two_lorentzian
=======================================================

This model calculates an empirical functional form for SAS data characterized by two Lorentzian-type functions.

================ ============================================= ======= =============
Parameter        Description                                   Units   Default value
================ ============================================= ======= =============
scale            Scale factor or Volume fraction               None                1
background       Source background                             |cm^-1|         0.001
lorentz_scale_1  First power law scale factor                  None               10
lorentz_length_1 First Lorentzian screening length             |Ang|             100
lorentz_exp_1    First exponent of power law                   None                3
lorentz_scale_2  Second scale factor for broad Lorentzian peak None                1
lorentz_length_2 Second Lorentzian screening length            |Ang|              10
lorentz_exp_2    Second exponent of power law                  None                2
================ ============================================= ======= =============

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


**Definition**

The scattering intensity $I(q)$ is calculated as

.. math::

    I(q) = \frac{A}{1 +(Q\xi_1)^n} + \frac{C}{1 +(Q\xi_2)^m} + \text{B}

where $A$ = Lorentzian scale factor #1, $C$ = Lorentzian scale #2,
$\xi_1$ and $\xi_2$ are the corresponding correlation lengths, and $n$ and
$m$ are the respective power law exponents (set $n = m = 2$ for
Ornstein-Zernicke behaviour).

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/two_lorentzian_autogenfig.png

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


**Source**

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

**References**

None.

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Piotr rozyczko **Date:** January 29, 2016
* **Last Reviewed by:** Paul Butler **Date:** March 21, 2016

