source: sasmodels/sasmodels/models/fuzzy_sphere.py @ 2c74c11

core_shell_microgelscostrafo411magnetic_modelrelease_v0.94release_v0.95ticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 2c74c11 was 2c74c11, checked in by Paul Kienzle <pkienzle@…>, 8 years ago

implicit Iqxy; fix divide by 0 for q=0

  • Property mode set to 100644
File size: 3.9 KB
Line 
1r"""
2For information about polarised and magnetic scattering, see
3the :doc:`magnetic help <../sasgui/perspectives/fitting/mag_help>` documentation.
4
5Definition
6----------
7
8The scattering intensity *I(q)* is calculated as:
9
10.. math::
11    I(q) = \frac{scale}{V}(\Delta \rho)^2 A^2(q) S(q) +\text{background}
12
13
14where the amplitude *A(q)* is given as the typical sphere scattering convoluted
15with a Gaussian to get a gradual drop-off in the scattering length density:
16
17.. math::
18
19    A(q) = \frac{3\left[\sin(qR) - qR \cos(qR)\right]}{(qR)^3}
20           \exp\left(\frac{-(\sigma_{fuzzy}q)^2}{2}\right)
21
22Here *|A(q)|*:sup:`2`\  is the form factor, *P(q)*. The scale is equivalent to the
23volume fraction of spheres, each of volume, *V*\. Contrast (|drho|) is the
24difference of scattering length densities of the sphere and the surrounding
25solvent.
26
27Poly-dispersion in radius and in fuzziness is provided for, though the fuzziness
28must be kept much smaller than the sphere radius for meaningful results.
29
30
31
32From the reference:
33
34  The "fuzziness" of the interface is defined by the parameter
35  |sigma| :sub:`fuzzy`\ . The particle radius *R* represents the radius of the
36  particle where the scattering length density profile decreased to 1/2 of the
37  core density. The |sigma| :sub:`fuzzy`\ is the width of the smeared particle
38  surface; i.e., the standard deviation from the average height of the fuzzy
39  interface. The inner regions of the microgel that display a higher density
40  are described by the radial box profile extending to a radius of
41  approximately *Rbox* ~ *R* - 2\ |sigma|\ . The profile approaches zero as
42  *Rsans* ~ *R* + 2\ |sigma|\ .
43
44For 2D data: The 2D scattering intensity is calculated in the same way as 1D,
45where the *q* vector is defined as
46
47.. math::
48
49    q = \sqrt{{q_x}^2 + {q_y}^2}
50
51
52References
53----------
54
55M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*,
5620 (2004) 7283-7292
57"""
58
59from numpy import inf
60
61name = "fuzzy_sphere"
62title = "Scattering from spherical particles with a fuzzy surface."
63description = """\
64scale: scale factor times volume fraction,
65or just volume fraction for absolute scale data
66radius: radius of the solid sphere
67fuzziness = the standard deviation of the fuzzy interfacial
68thickness (ie., so-called interfacial roughness)
69sld: the SLD of the sphere
70solvend_sld: the SLD of the solvent
71background: incoherent background
72Note: By definition, this function works only when fuzziness << radius.
73"""
74category = "shape:sphere"
75
76# pylint: disable=bad-whitespace,line-too-long
77# ["name", "units", default, [lower, upper], "type","description"],
78parameters = [["sld",         "1e-6/Ang^2",  1, [-inf, inf], "sld",    "Particle scattering length density"],
79              ["sld_solvent", "1e-6/Ang^2",  3, [-inf, inf], "sld",    "Solvent scattering length density"],
80              ["radius",      "Ang",        60, [0, inf],    "volume", "Sphere radius"],
81              ["fuzziness",   "Ang",        10, [0, inf],    "",       "std deviation of Gaussian convolution for interface (must be << radius)"],
82             ]
83# pylint: enable=bad-whitespace,line-too-long
84
85source = ["lib/sph_j1c.c"]
86
87# No volume normalization despite having a volume parameter
88# This should perhaps be volume normalized?
89form_volume = """
90    return 1.333333333333333*M_PI*radius*radius*radius;
91    """
92
93Iq = """
94    const double qr = q*radius;
95    const double bes = sph_j1c(qr);
96    const double qf = q*fuzziness;
97    const double fq = bes * (sld - sld_solvent) * form_volume(radius) * exp(-0.5*qf*qf);
98    return 1.0e-4*fq*fq;
99    """
100
101def ER(radius):
102    """
103    Return radius
104    """
105    return radius
106
107# VR defaults to 1.0
108
109demo = dict(scale=1, background=0.001,
110            sld=1, sld_solvent=3,
111            radius=60,
112            fuzziness=10,
113            radius_pd=.2, radius_pd_n=45,
114            fuzziness_pd=.2, fuzziness_pd_n=0)
115
116tests = [
117    # Accuracy tests based on content in test/utest_models_new1_3.py
118    #[{'background': 0.001}, 1.0, 0.001],
119
120    [{}, 0.00301005, 359.2315],
121
122    ]
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