source: sasmodels/sasmodels/models/spinodal.py @ 07646b6

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Last change on this file since 07646b6 was 07646b6, checked in by Paul Kienzle <pkienzle@…>, 12 months ago

Merge branch 'cuda-test' into beta_approx

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Line 
1r"""
2Definition
3----------
4
5This model calculates the SAS signal of a phase separating system
6undergoing spinodal decomposition. The scattering intensity $I(q)$ is calculated
7as
8
9.. math::
10    I(q) = I_{max}\frac{(1+\gamma/2)x^2}{\gamma/2+x^{2+\gamma}}+B
11
12where $x=q/q_0$, $q_0$ is the peak position, $I_{max}$ is the intensity
13at $q_0$ (parameterised as the $scale$ parameter), and $B$ is a flat
14background. The spinodal wavelength, $\Lambda$, is given by $2\pi/q_0$.
15
16The definition of $I_{max}$ in the literature varies. Hashimoto *et al* (1991)
17define it as
18
19.. math::
20    I_{max} = \Lambda^3\Delta\rho^2
21   
22whereas Meier & Strobl (1987) give
23
24.. math::
25    I_{max} = V_z\Delta\rho^2
26   
27where $V_z$ is the volume per monomer unit.
28
29The exponent $\gamma$ is equal to $d+1$ for off-critical concentration
30mixtures (smooth interfaces) and $2d$ for critical concentration mixtures
31(entangled interfaces), where $d$ is the dimensionality (ie, 1, 2, 3) of the
32system. Thus 2 <= $\gamma$ <= 6. A transition from $\gamma=d+1$ to $\gamma=2d$
33is expected near the percolation threshold.
34
35As this function tends to zero as $q$ tends to zero, in practice it may be
36necessary to combine it with another function describing the low-angle
37scattering, or to simply omit the low-angle scattering from the fit.
38
39References
40----------
41
42H. Furukawa. Dynamics-scaling theory for phase-separating unmixing mixtures:
43Growth rates of droplets and scaling properties of autocorrelation functions.
44Physica A 123, 497 (1984).
45
46H. Meier & G. Strobl. Small-Angle X-ray Scattering Study of Spinodal
47Decomposition in Polystyrene/Poly(styrene-co-bromostyrene) Blends.
48Macromolecules 20, 649-654 (1987).
49
50T. Hashimoto, M. Takenaka & H. Jinnai. Scattering Studies of Self-Assembling
51Processes of Polymer Blends in Spinodal Decomposition.
52J. Appl. Cryst. 24, 457-466 (1991).
53
54Revision History
55----------------
56
57* **Author:**  Dirk Honecker **Date:** Oct 7, 2016
58* **Revised:** Steve King    **Date:** Oct 25, 2018
59"""
60
61import numpy as np
62from numpy import inf, errstate
63
64name = "spinodal"
65title = "Spinodal decomposition model"
66description = """\
67      I(q) = Imax ((1+gamma/2)x^2)/(gamma/2+x^(2+gamma)) + background
68
69      List of default parameters:
70     
71      Imax = correlation peak intensity at q_0
72      background = incoherent background
73      gamma = exponent (see model documentation)
74      q_0 = correlation peak position [1/A]
75      x = q/q_0"""
76     
77category = "shape-independent"
78
79# pylint: disable=bad-whitespace, line-too-long
80#             ["name", "units", default, [lower, upper], "type", "description"],
81parameters = [["gamma",      "",    3.0, [-inf, inf], "", "Exponent"],
82              ["q_0",  "1/Ang",     0.1, [-inf, inf], "", "Correlation peak position"]
83             ]
84# pylint: enable=bad-whitespace, line-too-long
85
86def Iq(q,
87       gamma=3.0,
88       q_0=0.1):
89    """
90    :param q:              Input q-value
91    :param gamma:          Exponent
92    :param q_0:            Correlation peak position
93    :return:               Calculated intensity
94    """
95    with errstate(divide='ignore'):
96        x = q/q_0
97        inten = ((1 + gamma / 2) * x ** 2) / (gamma / 2 + x ** (2 + gamma))
98    return inten
99Iq.vectorized = True  # Iq accepts an array of q values
100
101def random():
102    pars = dict(
103        scale=10**np.random.uniform(1, 3),
104        gamma=np.random.uniform(0, 6),
105        q_0=10**np.random.uniform(-3, -1),
106    )
107    return pars
108
109demo = dict(scale=1, background=0,
110            gamma=1, q_0=0.1)
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