source: sasmodels/sasmodels/models/spinodal.py @ bb73096

r"""
Definition
----------

This model calculates the SAS signal of a phase separating solution under spinodal decomposition. 
The scattering intensity $I(q)$ is calculated as

.. math:: 
    I(q) = I_{max}\frac{(1+\gamma/2)x^2}{\gamma/2+x^{2+\gamma}}+B

where $x=q/q_0$ and $B$ is a flat background. The characteristic structure length 
scales with the correlation peak at $q_0$. The exponent $\gamma$ is equal to 
$d+1$ with d the dimensionality of the off-critical concentration mixtures.
A transition to $\gamma=2d$ is seen near the percolation threshold into the 
critical concentration regime.

References
----------

H. Furukawa. Dynamics-scaling theory for phase-separating unmixing mixtures: 
Growth rates of droplets and scaling properties of autocorrelation functions. Physica A 123,497 (1984).

Authorship and Verification
----------------------------

* **Author:** Dirk Honecker **Date:** Oct 7, 2016
* **Last Modified by:** 
* **Last Reviewed by:** 
"""

from numpy import inf, errstate

name = "spinodal"
title = "Spinodal decomposition model"
description = """\
      I(q) = scale ((1+gamma/2)x^2)/(gamma/2+x^(2+gamma))+background

      List of default parameters:
      scale = scaling
      gamma = exponent
      x = q/q_0
      q_0 = correlation peak position [1/A]
      background = Incoherent background"""
category = "shape-independent"

# pylint: disable=bad-whitespace, line-too-long
#             ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["scale",    "",      1.0, [-inf, inf], "", "Scale factor"],
              ["gamma",      "",    3.0, [-inf, inf], "", "Exponent"],
              ["q_0",  "1/Ang",     0.1, [-inf, inf], "", "Correlation peak position"]
             ]
# pylint: enable=bad-whitespace, line-too-long

def Iq(q,
       scale=1.0,
       gamma=3.0,
       q_0=0.1):
    """
    :param q:              Input q-value
    :param scale:          Scale factor
    :param gamma:          Exponent
    :param q_0:            Correlation peak position
    :return:               Calculated intensity
    """
    
    with errstate(divide='ignore'):
        x = q/q_0
        inten = scale * ((1 + gamma / 2) * x ** 2) / (gamma / 2 + x ** (2 + gamma)) 
    return inten
Iq.vectorized = True  # Iq accepts an array of q values

demo = dict(scale=1, background=0,
            gamma=1, q_0=0.1)