source: sasmodels/sasmodels/models/gauss_lorentz_gel.py @ c1e44e5

Last change on this file since c1e44e5 was c1e44e5, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

Add local link to source files. Refs #1263.

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1r"""
2This model calculates the scattering from a gel structure,
3but typically a physical rather than chemical network.
4It is modeled as a sum of a low-q exponential decay (which happens to
5give a functional form similar to Guinier scattering, so interpret with
6care) plus a Lorentzian at higher-q values. See also the gel_fit model.
7
8Definition
9----------
10
11The scattering intensity $I(q)$ is calculated as (Eqn. 5 from the reference)
12
13.. math:: I(q) = I_G(0) \exp(-q^2\Xi ^2/2) + I_L(0)/(1+q^2\xi^2)
14
15$\Xi$ is the length scale of the static correlations in the gel, which can
16be attributed to the "frozen-in" crosslinks. $\xi$ is the dynamic correlation
17length, which can be attributed to the fluctuating polymer chains between
18crosslinks. $I_G(0)$ and $I_L(0)$ are the scaling factors for each of these
19structures. Think carefully about how these map to your particular system!
20
21.. note::
22    The peaked structure at higher $q$ values (Figure 2 from the reference)
23    is not reproduced by the model. Peaks can be introduced into the model
24    by summing this model with the :ref:`gaussian-peak` model.
25
26For 2D data the scattering intensity is calculated in the same way as 1D,
27where the $q$ vector is defined as
28
29.. math:: q = \sqrt{q_x^2 + q_y^2}
30
31References
32----------
33
34.. [#] G Evmenenko, E Theunissen, K Mortensen, H Reynaers, *Polymer*, 42 (2001) 2907-2913
35
36Authorship and Verification
37----------------------------
38
39* **Author:**
40* **Last Modified by:**
41* **Last Reviewed by:**
42"""
43
44import numpy as np
45from numpy import inf, exp
46
47name = "gauss_lorentz_gel"
48title = "Gauss Lorentz Gel model of scattering from a gel structure"
49description = """
50            Class that evaluates a GaussLorentzGel model.
51
52            I(q) = scale_g*exp(- q^2*Z^2 / 2)+scale_l/(1+q^2*z^2)
53                    + background
54            List of default parameters:
55                scale_g = Gauss scale factor
56                Z = Static correlation length
57                scale_l = Lorentzian scale factor
58                z = Dynamic correlation length
59                background = Incoherent background
60            """
61category = "shape-independent"
62# pylint: disable=bad-whitespace, line-too-long
63#            ["name", "units", default, [lower, upper], "type", "description"],
64parameters = [["gauss_scale",   "",    100.0,  [-inf, inf], "", "Gauss scale factor"],
65              ["cor_length_static",    "Ang", 100.0,  [0, inf],    "", "Static correlation length"],
66              ["lorentz_scale", "",     50.0,  [-inf, inf], "", "Lorentzian scale factor"],
67              ["cor_length_dynamic",   "Ang",  20.0,  [0, inf],    "", "Dynamic correlation length"],
68             ]
69# pylint: enable=bad-whitespace, line-too-long
70
71def Iq(q,
72       gauss_scale=100.0,
73       cor_length_static=100.0,
74       lorentz_scale=50.0,
75       cor_length_dynamic=20.0):
76    """
77
78    :param q:                    Input q-value
79    :param gauss_scale:   Gauss scale factor
80    :param cor_length_static:    Static correlation length
81    :param lorentz_scale: Lorentzian scale factor
82    :param cor_length_dynamic:   Dynamic correlation length
83    :return:                     1-D intensity
84    """
85
86    term1 = gauss_scale *\
87            exp(-1.0*q*q*cor_length_static*cor_length_static/2.0)
88    term2 = lorentz_scale /\
89            (1.0+(q*cor_length_dynamic)*(q*cor_length_dynamic))
90
91    return term1 + term2
92
93Iq.vectorized = True  # Iq accepts an array of q values
94
95
96def random():
97    """Return a random parameter set for the model."""
98    gauss_scale = 10**np.random.uniform(1, 3)
99    lorentz_scale = 10**np.random.uniform(1, 3)
100    cor_length_static = 10**np.random.uniform(0, 3)
101    cor_length_dynamic = 10**np.random.uniform(0, 3)
102    pars = dict(
103        #background=0,
104        scale=1,
105        gauss_scale=gauss_scale,
106        lorentz_scale=lorentz_scale,
107        cor_length_static=cor_length_static,
108        cor_length_dynamic=cor_length_dynamic,
109    )
110    return pars
111
112
113demo = dict(scale=1, background=0.1,
114            gauss_scale=100.0,
115            cor_length_static=100.0,
116            lorentz_scale=50.0,
117            cor_length_dynamic=20.0)
118
119tests = [
120
121    # Accuracy tests based on content in test/utest_extra_models.py
122    [{'gauss_scale':  100.0,
123      'cor_length_static':   100.0,
124      'lorentz_scale': 50.0,
125      'cor_length_dynamic':   20.0,
126     }, 0.001, 149.482],
127
128    [{'gauss_scale':  100.0,
129      'cor_length_static':   100.0,
130      'lorentz_scale': 50.0,
131      'cor_length_dynamic':   20.0,
132     }, 0.105363, 9.1913],
133
134    [{'gauss_scale':  100.0,
135      'cor_length_static':   100.0,
136      'lorentz_scale': 50.0,
137      'cor_length_dynamic':   20.0,
138     }, 0.441623, 0.633811],
139
140    # Additional tests with larger range of parameters
141    [{'gauss_scale':  10.0,
142      'cor_length_static':  100.0,
143      'lorentz_scale': 3.0,
144      'cor_length_dynamic':   1.0,
145     }, 0.1, 2.9712970297],
146
147    [{'gauss_scale':  10.0,
148      'cor_length_static':  100.0,
149      'lorentz_scale': 3.0,
150      'cor_length_dynamic':   1.0,
151      'background':         100.0
152     }, 5.0, 100.116384615],
153
154    [{'gauss_scale':  10.0,
155      'cor_length_static':  100.0,
156      'lorentz_scale': 3.0,
157      'cor_length_dynamic':   1.0,
158      'background':           0.0,
159     }, 200., 7.49981250469e-05],
160    ]
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