.. _correlation-length:

correlation_length
=======================================================

Calculates an empirical functional form for SAS data characterized
by a low-Q signal and a high-Q signal.

============= ============================================ ======= =============
Parameter     Description                                  Units   Default value
============= ============================================ ======= =============
scale         Scale factor or Volume fraction              None                1
background    Source background                            |cm^-1|         0.001
lorentz_scale Lorentzian Scaling Factor                    None               10
porod_scale   Porod Scaling Factor                         None            1e-06
cor_length    Correlation length, xi, in Lorentzian        |Ang|              50
porod_exp     Porod Exponent, n, in q^-n                   None                3
lorentz_exp   Lorentzian Exponent, m, in 1/( 1 + (q.xi)^m) 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}{Q^n} + \frac{C}{1 + (Q\xi)^m} + \text{background}

The first term describes Porod scattering from clusters (exponent = $n$) and
the second term is a Lorentzian function describing scattering from
polymer chains (exponent = $m$). This second term characterizes the
polymer/solvent interactions and therefore the thermodynamics. The two
multiplicative factors $A$ and $C$, and the two exponents $n$ and $m$ are
used as fitting parameters. (Respectively *porod_scale*, *lorentz_scale*,
*porod_exp* and *lorentz_exp* in the parameter list.) The remaining
parameter $\xi$ (*cor_length* in the parameter list) is a correlation
length for the polymer chains. Note that when $m=2$ this functional form
becomes the familiar Lorentzian function. Some interpretation of the
values of $A$ and $C$ may be possible depending on the values of $m$ and $n$.

For 2D data: The 2D 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/correlation_length_autogenfig.png

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


**Source**

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

**References**

#. B Hammouda, D L Ho and S R Kline, Insight into Clustering in
   Poly(ethylene oxide) Solutions, Macromolecules, 37 (2004) 6932-6937

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Steve King **Date:** September 24, 2019
* **Last Reviewed by:**