source:sasmodels/sasmodels/models/fractal.py@0507e09

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Last change on this file since 0507e09 was 0507e09, checked in by smk78, 7 months ago

Added link to source code to each model. Closes #883

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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    F(q)&= \frac{3 (\sin x - x \cos x)}{x^3} \\
24    V_\text{particle} &= \frac{4}{3}\ \pi R_0 \\
25    S(q) &= 1 + \frac{D_f\  \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\  ]^{(D_f -1)/2}}
26    \frac{\sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}}
27
28where $\xi$ is the correlation length representing the cluster size and $D_f$
29is the fractal dimension, representing the self similarity of the structure.
30Note that S(q) here goes negative if $D_f$ is too large, and the Gamma function
31diverges at $D_f=0$ and $D_f=1$.
32
33**Polydispersity on the radius is provided for.**
34
35For 2D data: The 2D scattering intensity is calculated in the same way as
361D, where the *q* vector is defined as
37
38.. math::
39
40    q = \sqrt{q_x^2 + q_y^2}
41
42
43References
44----------
45
46.. [#] J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785
47
48Source
49------
50
51fractal.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/fractal.py>_
52
53fractal.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/fractal.c>_
54
55Authorship and Verification
56----------------------------
57
58* **Author:** NIST IGOR/DANSE **Date:** pre 2010
59* **Converted to sasmodels by:** Paul Butler **Date:** March 19, 2016
60* **Last Modified by:** Paul Butler **Date:** March 12, 2017
61* **Last Reviewed by:** Paul Butler **Date:** March 12, 2017
62* **Source added by :** Steve King **Date:** March 25, 2019
63"""
64from __future__ import division
65
66import numpy as np
67from numpy import inf
68
69name = "fractal"
70title = "Calculates the scattering from fractal-like aggregates of spheres \
71following theTexiera reference."
72description = """
73        The scattering intensity is given by
74        I(q) = scale * V * delta^(2) * P(q) * S(q) + background, where
76        F(x) = 3*[sin(x)-x cos(x)]/x**3
77        delta = sld_block -sld_solv
78        scale        =  scale * volfraction
80        sld_block    =  SDL block
81        sld_solv  =  SDL solvent
82        background   =  background
83        and S(q) is the interference term between building blocks given
84        in the full documentation and depending on the parameters
85        fractal_dim  =  Fractal dimension
86        cor_length  =  Correlation Length    """
87
88category = "shape-independent"
89
90# pylint: disable=bad-whitespace, line-too-long
91#             ["name", "units", default, [lower, upper], "type","description"],
92parameters = [["volfraction", "", 0.05, [0.0, 1], "",
93               "volume fraction of blocks"],
94              ["radius",    "Ang",  5.0, [0.0, inf], "volume",
95               "radius of particles"],
96              ["fractal_dim",      "",  2.0, [0.0, 6.0], "",
97               "fractal dimension"],
98              ["cor_length", "Ang", 100.0, [0.0, inf], "",
99               "cluster correlation length"],
100              ["sld_block", "1e-6/Ang^2", 2.0, [-inf, inf], "sld",
101               "scattering length density of particles"],
102              ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld",
103               "scattering length density of solvent"],
104             ]
105# pylint: enable=bad-whitespace, line-too-long
106
107source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "lib/fractal_sq.c", "fractal.c"]
108
109def random():
110    """Return a random parameter set for the model."""
111    radius = 10**np.random.uniform(0.7, 4)
112    #radius = 5
113    cor_length = 10**np.random.uniform(0.7, 2)*radius
114    #cor_length = 20*radius
115    volfraction = 10**np.random.uniform(-3, -1)
116    #volfraction = 0.05
117    fractal_dim = 2*np.random.beta(3, 4) + 1
118    #fractal_dim = 2
119    pars = dict(
120        #background=0, sld_block=1, sld_solvent=0,
121        volfraction=volfraction,
123        cor_length=cor_length,
124        fractal_dim=fractal_dim,
125    )
126    return pars
127
128demo = dict(volfraction=0.05,