.. _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         Source intensity                             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) |Ang^-2|             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.

**References**

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

