.. _broad-peak:

broad_peak
=======================================================

Broad Lorentzian type peak on top of a power law decay

============== ====================================== ======== =============
Parameter      Description                            Units    Default value
============== ====================================== ======== =============
scale          Scale factor or Volume fraction        None                 1
background     Source background                      |cm^-1|          0.001
porod_scale    Power law scale factor                 None             1e-05
porod_exp      Exponent of power law                  None                 3
lorentz_scale  Scale factor for broad Lorentzian peak None                10
lorentz_length Lorentzian screening length            |Ang|               50
peak_pos       Peak position in q                     |Ang^-1|           0.1
lorentz_exp    Exponent of Lorentz function           None                 2
============== ====================================== ======== =============

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


**Definition**

This model calculates an empirical functional form for SAS data characterized
by a broad scattering peak. Many SAS spectra are characterized by a broad peak
even though they are from amorphous soft materials. For example, soft systems
that show a SAS peak include copolymers, polyelectrolytes, multiphase systems,
layered structures, etc.

The d-spacing corresponding to the broad peak is a characteristic distance
between the scattering inhomogeneities (such as in lamellar, cylindrical, or
spherical morphologies, or for bicontinuous structures).

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

.. math:: I(q) = \frac{A}{q^n} + \frac{C}{1 + (|q - q_0|\xi)^m} + B

Here the peak position is related to the d-spacing as $q_0 = 2\pi / d_0$.

$A$ is the Porod law scale factor, $n$ the Porod exponent, $C$ is the
Lorentzian scale factor, $m$ the exponent of $q$, $\xi$ the screening length,
and $B$ the flat background.

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/broad_peak_autogenfig.png

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


**Source**

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

**References**

None.

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul kienle **Date:** July 24, 2016
* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016