source: sasmodels/sasmodels/models/fuzzy_sphere.py @ d57b06c

Last change on this file since d57b06c was d57b06c, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

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Line 
1r"""
2For information about polarised and magnetic scattering, see
3the :ref:`magnetism` documentation.
4
5Definition
6----------
7
8The scattering intensity $I(q)$ is calculated as:
9
10.. math::
11
12    I(q) = \frac{\text{scale}}{V}(\Delta \rho)^2 A^2(q) S(q)
13           + \text{background}
14
15
16where the amplitude $A(q)$ is given as the typical sphere scattering convoluted
17with a Gaussian to get a gradual drop-off in the scattering length density:
18
19.. math::
20
21    A(q) = \frac{3\left[\sin(qR) - qR \cos(qR)\right]}{(qR)^3}
22           \exp\left(\frac{-(\sigma_\text{fuzzy}q)^2}{2}\right)
23
24Here $A(q)^2$ is the form factor, $P(q)$. The scale is equivalent to the
25volume fraction of spheres, each of volume, $V$. Contrast $(\Delta \rho)$
26is the difference of scattering length densities of the sphere and the
27surrounding solvent.
28
29Poly-dispersion in radius and in fuzziness is provided for, though the
30fuzziness must be kept much smaller than the sphere radius for meaningful
31results.
32
33From the reference:
34
35  The "fuzziness" of the interface is defined by the parameter
36  $\sigma_\text{fuzzy}$. The particle radius $R$ represents the radius of the
37  particle where the scattering length density profile decreased to 1/2 of the
38  core density. $\sigma_\text{fuzzy}$ is the width of the smeared particle
39  surface; i.e., the standard deviation from the average height of the fuzzy
40  interface. The inner regions of the microgel that display a higher density
41  are described by the radial box profile extending to a radius of
42  approximately $R_\text{box} \sim R - 2 \sigma$. The profile approaches
43  zero as $R_\text{sans} \sim R + 2\sigma$.
44
45For 2D data: The 2D scattering intensity is calculated in the same way as 1D,
46where the $q$ vector is defined as
47
48.. math:: q = \sqrt{{q_x}^2 + {q_y}^2}
49
50References
51----------
52
53.. [#] M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*, 20 (2004) 7283-7292
54
55Authorship and Verification
56----------------------------
57
58* **Author:**
59* **Last Modified by:**
60* **Last Reviewed by:**
61"""
62
63import numpy as np
64from numpy import inf
65
66name = "fuzzy_sphere"
67title = "Scattering from spherical particles with a fuzzy surface."
68description = """\
69scale: scale factor times volume fraction,
70or just volume fraction for absolute scale data
71radius: radius of the solid sphere
72fuzziness = the standard deviation of the fuzzy interfacial
73thickness (ie., so-called interfacial roughness)
74sld: the SLD of the sphere
75solvend_sld: the SLD of the solvent
76background: incoherent background
77Note: By definition, this function works only when fuzziness << radius.
78"""
79category = "shape:sphere"
80
81# pylint: disable=bad-whitespace,line-too-long
82# ["name", "units", default, [lower, upper], "type","description"],
83parameters = [["sld",         "1e-6/Ang^2",  1, [-inf, inf], "sld",    "Particle scattering length density"],
84              ["sld_solvent", "1e-6/Ang^2",  3, [-inf, inf], "sld",    "Solvent scattering length density"],
85              ["radius",      "Ang",        60, [0, inf],    "volume", "Sphere radius"],
86              ["fuzziness",   "Ang",        10, [0, inf],    "volume",       "std deviation of Gaussian convolution for interface (must be << radius)"],
87             ]
88# pylint: enable=bad-whitespace,line-too-long
89
90source = ["lib/sas_3j1x_x.c", "fuzzy_sphere.c"]
91have_Fq = True
92radius_effective_modes = ["radius", "radius + fuzziness"]
93
94def random():
95    """Return a random parameter set for the model."""
96    radius = 10**np.random.uniform(1, 4.7)
97    fuzziness = 10**np.random.uniform(-2, -0.5)*radius  # 1% to 31% fuzziness
98    pars = dict(
99        radius=radius,
100        fuzziness=fuzziness,
101    )
102    return pars
103
104demo = dict(scale=1, background=0.001,
105            sld=1, sld_solvent=3,
106            radius=60,
107            fuzziness=10,
108            radius_pd=.2, radius_pd_n=45,
109            fuzziness_pd=.2, fuzziness_pd_n=0)
110
111tests = [
112    # Accuracy tests based on content in test/utest_models_new1_3.py
113    #[{'background': 0.001}, 1.0, 0.001],
114
115    [{}, 0.00301005, 359.2315],
116
117    ]
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