source: sasmodels/sasmodels/models/fuzzy_sphere.py @ 925ad6e

core_shell_microgelscostrafo411magnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 925ad6e was 925ad6e, checked in by wojciech, 6 years ago

sph_j1c translated to sas_3j1x_x

  • Property mode set to 100644
File size: 3.9 KB
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
53M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*,
5420 (2004) 7283-7292
55"""
56
57from numpy import inf
58
59name = "fuzzy_sphere"
60title = "Scattering from spherical particles with a fuzzy surface."
61description = """\
62scale: scale factor times volume fraction,
63or just volume fraction for absolute scale data
64radius: radius of the solid sphere
65fuzziness = the standard deviation of the fuzzy interfacial
66thickness (ie., so-called interfacial roughness)
67sld: the SLD of the sphere
68solvend_sld: the SLD of the solvent
69background: incoherent background
70Note: By definition, this function works only when fuzziness << radius.
71"""
72category = "shape:sphere"
73
74# pylint: disable=bad-whitespace,line-too-long
75# ["name", "units", default, [lower, upper], "type","description"],
76parameters = [["sld",         "1e-6/Ang^2",  1, [-inf, inf], "sld",    "Particle scattering length density"],
77              ["sld_solvent", "1e-6/Ang^2",  3, [-inf, inf], "sld",    "Solvent scattering length density"],
78              ["radius",      "Ang",        60, [0, inf],    "volume", "Sphere radius"],
79              ["fuzziness",   "Ang",        10, [0, inf],    "",       "std deviation of Gaussian convolution for interface (must be << radius)"],
80             ]
81# pylint: enable=bad-whitespace,line-too-long
82
83source = ["lib/sas_3j1x_x.c"]
84
85# No volume normalization despite having a volume parameter
86# This should perhaps be volume normalized?
87form_volume = """
88    return M_4PI_3*cube(radius);
89    """
90
91Iq = """
92    const double qr = q*radius;
93    const double bes = sas_3j1x_x(qr);
94    const double qf = q*fuzziness;
95    const double fq = bes * (sld - sld_solvent) * form_volume(radius) * exp(-0.5*qf*qf);
96    return 1.0e-4*fq*fq;
97    """
98
99def ER(radius):
100    """
101    Return radius
102    """
103    return radius
104
105# VR defaults to 1.0
106
107demo = dict(scale=1, background=0.001,
108            sld=1, sld_solvent=3,
109            radius=60,
110            fuzziness=10,
111            radius_pd=.2, radius_pd_n=45,
112            fuzziness_pd=.2, fuzziness_pd_n=0)
113
114tests = [
115    # Accuracy tests based on content in test/utest_models_new1_3.py
116    #[{'background': 0.001}, 1.0, 0.001],
117
118    [{}, 0.00301005, 359.2315],
119
120    ]
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