.. _two-power-law:

two_power_law
=======================================================

This model calculates an empirical functional form for SAS data characterized by two power laws.

============ =============================== ======== =============
Parameter    Description                     Units    Default value
============ =============================== ======== =============
scale        Scale factor or Volume fraction None                 1
background   Source background               |cm^-1|          0.001
coefficent_1 coefficent A in low Q region    None                 1
crossover    crossover location              |Ang^-1|          0.04
power_1      power law exponent at low Q     None                 1
power_2      power law exponent at high Q    None                 4
============ =============================== ======== =============

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) = \begin{cases}
    A q^{-m1} + \text{background} & q <= q_c \\
    C q^{-m2} + \text{background} & q > q_c
    \end{cases}

where $q_c$ = the location of the crossover from one slope to the other,
$A$ = the scaling coefficient that sets the overall intensity of the lower Q
power law region, $m1$ = power law exponent at low Q, and $m2$ = power law
exponent at high Q.  The scaling of the second power law region (coefficient C)
is then automatically scaled to match the first by following formula:

.. math::
    C = \frac{A q_c^{m2}}{q_c^{m1}}

.. note::
    Be sure to enter the power law exponents as positive values!

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_power_law_autogenfig.png

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


**Source**

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

**References**

None.

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Wojciech Wpotrzebowski **Date:** February 18, 2016
* **Last Reviewed by:** Paul Butler **Date:** March 21, 2016