source: sasmodels/sasmodels/models/spinodal.py

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

This model calculates the SAS signal of a phase separating system
undergoing 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 $x=q/q_0$, $q_0$ is the peak position, $I_{max}$ is the intensity
at $q_0$ (parameterised as the $scale$ parameter), and $B$ is a flat
background. The spinodal wavelength, $\Lambda$, is given by $2\pi/q_0$.

The definition of $I_{max}$ in the literature varies. Hashimoto *et al* (1991)
define it as

.. math::
    I_{max} = \Lambda^3\Delta\rho^2

whereas Meier & Strobl (1987) give

.. math::
    I_{max} = V_z\Delta\rho^2

where $V_z$ is the volume per monomer unit.

The exponent $\gamma$ is equal to $d+1$ for off-critical concentration
mixtures (smooth interfaces) and $2d$ for critical concentration mixtures
(entangled interfaces), where $d$ is the dimensionality (ie, 1, 2, 3) of the
system. Thus 2 <= $\gamma$ <= 6. A transition from $\gamma=d+1$ to $\gamma=2d$
is expected near the percolation threshold.

As this function tends to zero as $q$ tends to zero, in practice it may be
necessary to combine it with another function describing the low-angle
scattering, or to simply omit the low-angle scattering from the fit.

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).
.. [#] H. Meier & G. Strobl. Small-Angle X-ray Scattering Study of Spinodal Decomposition in Polystyrene/Poly(styrene-co-bromostyrene) Blends. Macromolecules 20, 649-654 (1987).
.. [#] T. Hashimoto, M. Takenaka & H. Jinnai. Scattering Studies of Self-Assembling Processes of Polymer Blends in Spinodal Decomposition. J. Appl. Cryst. 24, 457-466 (1991).

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

* **Author:** Dirk Honecker **Date:** Oct 7, 2016
* **Last Modified by:** Steve King **Date:** Oct 25, 2018
* **Last Reviewed by:** Steve King **Date:** Oct 25, 2018
"""

import numpy as np
from numpy import inf, errstate

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

      List of default parameters:

      Imax = correlation peak intensity at q_0
      background = incoherent background
      gamma = exponent (see model documentation)
      q_0 = correlation peak position [1/A]
      x = q/q_0"""

category = "shape-independent"

# pylint: disable=bad-whitespace, line-too-long
#             ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["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,
       gamma=3.0,
       q_0=0.1):
    """
    :param q:              Input q-value
    :param gamma:          Exponent
    :param q_0:            Correlation peak position
    :return:               Calculated intensity
    """
    with errstate(divide='ignore'):
        x = q/q_0
        inten = ((1 + gamma / 2) * x ** 2) / (gamma / 2 + x ** (2 + gamma))
    return inten
Iq.vectorized = True  # Iq accepts an array of q values

def random():
    """Return a random parameter set for the model."""
    pars = dict(
        scale=10**np.random.uniform(1, 3),
        gamma=np.random.uniform(0, 6),
        q_0=10**np.random.uniform(-3, -1),
    )
    return pars

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