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

core_shell_microgelscostrafo411magnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4This model calculates the scattering from fractal-like aggregates of spherical
5building blocks according the following equation:
6
7.. math::
8
9    I(q) = \phi\ V_\text{block} (\rho_\text{block}
10          - \rho_\text{solvent})^2 P(q)S(q) + \text{background}
11
12where $\phi$ is The volume fraction of the spherical "building block" particles
13of radius $R_0$, $V_{block}$ is the volume of a single building block,
14$\rho_{solvent}$ is the scattering length density of the solvent, and
15$\rho_{block}$ is the scattering length density of the building blocks, and
16P(q), S(q) are the scattering from randomly distributed spherical particles
17(the building blocks) and the interference from such building blocks organized
18in a fractal-like clusters.  P(q) and S(q) are calculated as:
19
20.. math::
21
22    P(q)&= F(qR_0)^2
23
24    F(q)&= \frac{3 (\sin x - x \cos x)}{x^3}
25
26    V_\text{particle} &= \frac{4}{3}\ \pi R_0
27
28    S(q) &= 1 + \frac{D_f\  \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\  ]^{(D_f -1)/2}}
29    \frac{\sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}}
30
31where $\xi$ is the correlation length representing the cluster size and $D_f$
32is the fractal dimension, representing the self similarity of the structure.
33Note that S(q) here goes negative if $D_f$ is too large, and the Gamma function
34diverges at $D_f$=0 and $D_f$=1. 
35
36**Polydispersity on the radius is provided for.**
37
38For 2D data: The 2D scattering intensity is calculated in the same way as
391D, where the *q* vector is defined as
40
41.. math::
42
43    q = \sqrt{q_x^2 + q_y^2}
44
45
46References
47----------
48
49J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785
50
51**Author:** NIST IGOR/DANSE **on:** pre 2010
52
53**Last Modified by:** Paul Butler **on:** March 20, 2016
54
55**Last Reviewed by:** Paul Butler **on:** March 20, 2016
56
57"""
58
59from numpy import inf
60
61name = "fractal"
62title = "Calculates the scattering from fractal-like aggregates of spheres \
63following theTexiera reference."
64description = """
65        The scattering intensity is given by
66        I(q) = scale * V * delta^(2) * P(q) * S(q) + background, where
67        p(q)= F(q*radius)^(2)
68        F(x) = 3*[sin(x)-x cos(x)]/x**3
69        delta = sld_block -sld_solv
70        scale        =  scale * volfraction
71        radius       =  Block radius
72        sld_block    =  SDL block
73        sld_solv  =  SDL solvent
74        background   =  background
75        and S(q) is the interference term between building blocks given
76        in the full documentation and depending on the parameters
77        fractal_dim  =  Fractal dimension
78        cor_length  =  Correlation Length    """
79
80category = "shape-independent"
81
82# pylint: disable=bad-whitespace, line-too-long
83#             ["name", "units", default, [lower, upper], "type","description"],
84parameters = [["volfraction", "", 0.05, [0.0, 1], "",
85               "volume fraction of blocks"],
86              ["radius",    "Ang",  5.0, [0.0, inf], "",
87               "radius of particles"],
88              ["fractal_dim",      "",  2.0, [0.0, 6.0], "",
89               "fractal dimension"],
90              ["cor_length", "Ang", 100.0, [0.0, inf], "",
91               "cluster correlation length"],
92              ["sld_block", "1e-6/Ang^2", 2.0, [-inf, inf], "sld",
93               "scattering length density of particles"],
94              ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld",
95               "scattering length density of solvent"],
96             ]
97# pylint: enable=bad-whitespace, line-too-long
98
99source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "lib/fractal_sq.c", "fractal.c"]
100
101demo = dict(volfraction=0.05,
102            radius=5.0,
103            fractal_dim=2.0,
104            cor_length=100.0,
105            sld_block=2.0,
106            sld_solvent=6.4)
107
108# NOTE: test results taken from values returned by SasView 3.1.2
109tests = [
110    [{}, 0.0005, 40.4980069872],
111    [{}, 0.234734468938, 0.0947143166058],
112    [{}, 0.5, 0.0176878183458],
113    ]
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