Teubner Strey - teubner_strey.py
r"""
Definition
----------
This model calculates the scattered intensity of a two-component system
using the Teubner-Strey model. Unlike :ref:`dab` this function generates
a peak. A two-phase material can be characterised by two length scales -
a correlation length and a domain size (periodicity).
The original paper by Teubner and Strey defined the function as:
.. math::
I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background}
where the parameters $a_2$, $c_1$ and $c_2$ are defined in terms of the
periodicity, $d$, and correlation length $\xi$ as:
.. math::
a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]^2\\
c_1 &= -2\xi^2\bigl(\frac{2\pi\xi}{d}\bigr)^2+2\xi^2\\
c_2 &= \xi^4
and thus, the periodicity, $d$ is given by
.. math::
d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2}
- \frac14\frac{c_1}{c_2}\right]^{-1/2}
and the correlation length, $\xi$, is given by
.. math::
\xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2}
+ \frac14\frac{c_1}{c_2}\right]^{-1/2}
Here the model is parameterised in terms of $d$ and $\xi$ and with an explicit
volume fraction for one phase, $\phi_a$, and contrast,
$\delta\rho^2 = (\rho_a - \rho_b)^2$ :
.. math::
I(q) = \frac{8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi}
{a_2 + c_1q^2 + c_2q^4}
where :math:`8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi` is the constant of
proportionality from the first equation above.
In the case of a microemulsion, $a_2 > 0$, $c_1 < 0$, and $c_2 >0$.
For 2D data, 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}
References
----------
.. [#] M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195
.. [#] K V Schubert, R Strey, S R Kline and E W Kaler, *J. Chem. Phys.*, 101 (1994) 5343
.. [#] H Endo, M Mihailescu, M. Monkenbusch, J Allgaier, G Gompper, D Richter, B Jakobs, T Sottmann, R Strey, and I Grillo, *J. Chem. Phys.*, 115 (2001), 580
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
from __future__ import division
import numpy as np
from numpy import inf, pi
name = "teubner_strey"
title = "Teubner-Strey model of microemulsions"
description = """\
Calculates scattering according to the Teubner-Strey model
"""
category = "shape-independent"
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [
["volfraction_a", "", 0.5, [0, 1.0], "", "Volume fraction of phase a"],
["sld_a", "1e-6/Ang^2", 0.3, [-inf, inf], "", "SLD of phase a"],
["sld_b", "1e-6/Ang^2", 6.3, [-inf, inf], "", "SLD of phase b"],
["d", "Ang", 100.0, [0, inf], "", "Domain size (periodicity)"],
["xi", "Ang", 30.0, [0, inf], "", "Correlation length"],
]
def Iq(q, volfraction_a, sld_a, sld_b, d, xi):
"""SAS form"""
drho = sld_a - sld_b
k = 2.0*pi*xi/d
a2 = (1.0 + k**2)**2
c1 = 2.0*xi**2 * (1.0 - k**2)
c2 = xi**4
prefactor = 8.0*pi * volfraction_a*(1.0 - volfraction_a) * drho**2 * c2/xi
return 1.0e-4*prefactor / np.polyval([c2, c1, a2], q**2)
Iq.vectorized = True # Iq accepts an array of q values
def random():
"""Return a random parameter set for the model."""
d = 10**np.random.uniform(1, 4)
xi = 10**np.random.uniform(-0.3, 2)*d
pars = dict(
#background=0,
scale=100,
volfraction_a=10**np.random.uniform(-3, 0),
sld_a=np.random.uniform(-0.5, 12),
sld_b=np.random.uniform(-0.5, 12),
d=d,
xi=xi,
)
return pars
demo = dict(scale=1, background=0, volfraction_a=0.5,
sld_a=0.3, sld_b=6.3,
d=100.0, xi=30.0)
tests = [[{}, 0.06, 41.5918888453]]
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