source: sasmodels/sasmodels/models/teubner_strey.py @ 0507e09

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 0507e09 was 0507e09, checked in by smk78, 6 months ago

Added link to source code to each model. Closes #883

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1r"""
2Definition
3----------
4
5This model calculates the scattered intensity of a two-component system
6using the Teubner-Strey model. Unlike :ref:`dab` this function generates
7a peak. A two-phase material can be characterised by two length scales -
8a correlation length and a domain size (periodicity).
9
10The original paper by Teubner and Strey defined the function as:
11
12.. math::
13
14    I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background}
15
16where the parameters $a_2$, $c_1$ and $c_2$ are defined in terms of the
17periodicity, $d$, and correlation length $\xi$ as:
18
19.. math::
20
21    a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]^2\\
22    c_1 &= -2\xi^2\bigl(\frac{2\pi\xi}{d}\bigr)^2+2\xi^2\\
23    c_2 &= \xi^4
24
25and thus, the periodicity, $d$ is given by
26
27.. math::
28
29    d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2}
30                  - \frac14\frac{c_1}{c_2}\right]^{-1/2}
31
32and the correlation length, $\xi$, is given by
33
34.. math::
35
36    \xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2}
37                  + \frac14\frac{c_1}{c_2}\right]^{-1/2}
38
39Here the model is parameterised in terms of  $d$ and $\xi$ and with an explicit
40volume fraction for one phase, $\phi_a$, and contrast,
41$\delta\rho^2 = (\rho_a - \rho_b)^2$ :
42
43.. math::
44
45    I(q) = \frac{8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi}
46        {a_2 + c_1q^2 + c_2q^4}
47
48where :math:`8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi` is the constant of
49proportionality from the first equation above.
50
51In the case of a microemulsion, $a_2 > 0$, $c_1 < 0$, and $c_2 >0$.
52
53For 2D data, scattering intensity is calculated in the same way as 1D,
54where the $q$ vector is defined as
55
56.. math::
57
58    q = \sqrt{q_x^2 + q_y^2}
59
60References
61----------
62
63.. [#] M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195
64.. [#] K V Schubert, R Strey, S R Kline and E W Kaler, *J. Chem. Phys.*, 101 (1994) 5343
65.. [#] 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
66
67Source
68------
69
70`teubner_strey.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/teubner_strey.py>`_
71
72`teubner_strey.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/teubner_strey.c>`_
73
74Authorship and Verification
75----------------------------
76
77* **Author:**
78* **Last Modified by:**
79* **Last Reviewed by:**
80* **Source added by :** Steve King **Date:** March 25, 2019
81"""
82from __future__ import division
83
84import numpy as np
85from numpy import inf, pi
86
87name = "teubner_strey"
88title = "Teubner-Strey model of microemulsions"
89description = """\
90    Calculates scattering according to the Teubner-Strey model
91"""
92category = "shape-independent"
93
94#   ["name", "units", default, [lower, upper], "type","description"],
95parameters = [
96    ["volfraction_a", "", 0.5, [0, 1.0], "", "Volume fraction of phase a"],
97    ["sld_a", "1e-6/Ang^2", 0.3, [-inf, inf], "", "SLD of phase a"],
98    ["sld_b", "1e-6/Ang^2", 6.3, [-inf, inf], "", "SLD of phase b"],
99    ["d", "Ang", 100.0, [0, inf], "", "Domain size (periodicity)"],
100    ["xi", "Ang", 30.0, [0, inf], "", "Correlation length"],
101    ]
102
103def Iq(q, volfraction_a, sld_a, sld_b, d, xi):
104    """SAS form"""
105    drho = sld_a - sld_b
106    k = 2.0*pi*xi/d
107    a2 = (1.0 + k**2)**2
108    c1 = 2.0*xi**2 * (1.0 - k**2)
109    c2 = xi**4
110    prefactor = 8.0*pi * volfraction_a*(1.0 - volfraction_a) * drho**2 * c2/xi
111    return 1.0e-4*prefactor / np.polyval([c2, c1, a2], q**2)
112
113Iq.vectorized = True  # Iq accepts an array of q values
114
115def random():
116    """Return a random parameter set for the model."""
117    d = 10**np.random.uniform(1, 4)
118    xi = 10**np.random.uniform(-0.3, 2)*d
119    pars = dict(
120        #background=0,
121        scale=100,
122        volfraction_a=10**np.random.uniform(-3, 0),
123        sld_a=np.random.uniform(-0.5, 12),
124        sld_b=np.random.uniform(-0.5, 12),
125        d=d,
126        xi=xi,
127    )
128    return pars
129
130demo = dict(scale=1, background=0, volfraction_a=0.5,
131            sld_a=0.3, sld_b=6.3,
132            d=100.0, xi=30.0)
133tests = [[{}, 0.06, 41.5918888453]]
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