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

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Last change on this file since 0507e09 was 0507e09, checked in by smk78, 6 months ago

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

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1r"""
2
3A number of natural and commercial processes form high-surface area materials
4as a result of the vapour-phase aggregation of primary particles.
5Examples of such materials include soots, aerosols, and fume or pyrogenic
6silicas. These are all characterised by cluster mass distributions (sometimes
7also cluster size distributions) and internal surfaces that are fractal in
8nature. The scattering from such materials displays two distinct breaks in
9log-log representation, corresponding to the radius-of-gyration of the primary
10particles, $rg$, and the radius-of-gyration of the clusters (aggregates),
11$Rg$. Between these boundaries the scattering follows a power law related to
12the mass fractal dimension, $Dm$, whilst above the high-Q boundary the
13scattering follows a power law related to the surface fractal dimension of
14the primary particles, $Ds$.
15
16Definition
17----------
18
19The scattered intensity I(q) is calculated using a modified
20Ornstein-Zernicke equation
21
22.. math::
23
24    I(q) = scale \times P(q) + background \\
25    P(q) = \left\{ \left[ 1+(q^2a)\right]^{D_m/2} \times
26                   \left[ 1+(q^2b)\right]^{(6-D_s-D_m)/2}
27           \right\}^{-1} \\
28    a = R_{g}^2/(3D_m/2) \\
29    b = r_{g}^2/[-3(D_s+D_m-6)/2] \\
30    scale = scale\_factor \times NV^2 (\rho_{particle} - \rho_{solvent})^2
31
32where $R_g$ is the size of the cluster, $r_g$ is the size of the primary
33particle, $D_s$ is the surface fractal dimension, $D_m$ is the mass fractal
34dimension, $\rho_{solvent}$ is the scattering length density of the solvent,
35and $\rho_{particle}$ is the scattering length density of particles.
36
37.. note::
38
39    The surface ( $D_s$ ) and mass ( $D_m$ ) fractal dimensions are only
40    valid if $0 < surface\_dim < 6$, $0 < mass\_dim < 6$, and
41    $(surface\_dim + mass\_dim ) < 6$.
42    Older versions of sasview may have the default primary particle radius
43    larger than the cluster radius, this was an error, also present in the
44    Schmidt review paper below. The primary particle should be the smaller
45    as described in the original Hurd, et al., who also point out that
46    polydispersity in the primary particle sizes may affect their
47    apparent surface fractal dimension.
48
49
50References
51----------
52
53.. [#] P Schmidt, *J Appl. Cryst.*, 24 (1991) 414-435 Equation(19)
54.. [#] A J Hurd, D W Schaefer, J E Martin, *Phys. Rev. A*,
55   35 (1987) 2361-2364 Equation(2)
56
57Source
58------
59
60`mass_surface_fractal.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/mass_surface_fractal.py>`_
61
62`mass_surface_fractal.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/mass_surface_fractal.c>`_
63
64Authorship and Verification
65----------------------------
66
67* **Author:** Piotr Rozyczko **Date:** Jan 20, 2016
68* **Last Modified by:**
69* **Last Reviewed by:** Richard Heenan **Date:** May 30, 2018
70* **Source added by :** Steve King **Date:** March 25, 2019
71"""
72
73import numpy as np
74from numpy import inf
75
76name = "mass_surface_fractal"
77title = "Mass Surface Fractal model"
78description = """
79        The scattering intensity  I(x) = scale*P(x)*S(x) + background, where
80        p(x)= {[1+(x^2*a)]^(Dm/2) * [1+(x^2*b)]^(6-Ds-Dm)/2}^(-1)
81        a = Rg^2/(3*Dm/2)
82        b = rg^2/(3*(6-Ds-Dm)/2)
83        scale        =  scale factor * N*Volume^2*contrast^2
84        fractal_dim_mass       =  Dm (mass fractal dimension)
85        fractal_dim_surf  =  Ds
86        rg_cluster  =  Rg
87        rg_primary    =  rg
88        background   =  background
89        Hurd, Schaefer, Martin, Phys Rev A, eq(2),(1987),35, 2361-2364
90        Note that 0 < Ds< 6 and 0 < Dm < 6.
91        """
92category = "shape-independent"
93
94# pylint: disable=bad-whitespace, line-too-long
95#   ["name", "units", default, [lower, upper], "type","description"],
96parameters = [
97    ["fractal_dim_mass", "",      1.8, [0.0, 6.0], "", "Mass fractal dimension"],
98    ["fractal_dim_surf", "",      2.3, [0.0, 6.0], "", "Surface fractal dimension"],
99    ["rg_cluster",       "Ang", 4000., [0.0, inf], "", "Cluster radius of gyration"],
100    ["rg_primary",       "Ang",  86.7, [0.0, inf], "", "Primary particle radius of gyration"],
101]
102# pylint: enable=bad-whitespace, line-too-long
103
104source = ["mass_surface_fractal.c"]
105
106def random():
107    """Return a random parameter set for the model."""
108    fractal_dim = np.random.uniform(0, 6)
109    surface_portion = np.random.uniform(0, 1)
110    fractal_dim_surf = fractal_dim*surface_portion
111    fractal_dim_mass = fractal_dim - fractal_dim_surf
112    rg_cluster = 10**np.random.uniform(1, 5)
113    rg_primary = rg_cluster*10**np.random.uniform(-4, -1)
114    scale = 10**np.random.uniform(2, 5)
115    pars = dict(
116        #background=0,
117        scale=scale,
118        fractal_dim_mass=fractal_dim_mass,
119        fractal_dim_surf=fractal_dim_surf,
120        rg_cluster=rg_cluster,
121        rg_primary=rg_primary,
122    )
123    return pars
124
125
126demo = dict(scale=1, background=0,
127            fractal_dim_mass=1.8,
128            fractal_dim_surf=2.3,
129            rg_cluster=4000.0,
130            rg_primary=86.7)
131
132tests = [
133
134    # Accuracy tests based on content in test/utest_other_models.py  All
135    # except first, changed so rg_cluster is the larger, RKH 30 May 2018
136    [{'fractal_dim_mass':   1.8,
137      'fractal_dim_surf':   2.3,
138      'rg_cluster':   86.7,
139      'rg_primary': 4000.0,
140      'background':    0.0,
141     }, 0.05, 1.77537e-05],
142
143    # Additional tests with larger range of parameters
144    [{'fractal_dim_mass':      3.3,
145      'fractal_dim_surf':   1.0,
146      'rg_cluster': 4000.0,
147      'rg_primary':   90.0,
148     }, 0.001, 0.0932516614456],
149
150    [{'fractal_dim_mass':      1.3,
151      'fractal_dim_surf':   2.0,
152      'rg_cluster': 2000.0,
153      'rg_primary':   90.0,
154      'background':    0.8,
155     }, 0.001, 1.28296431786],
156
157    [{'fractal_dim_mass':      2.3,
158      'fractal_dim_surf':   3.1,
159      'rg_cluster':  1000.0,
160      'rg_primary':  30.0,
161      'scale':        10.0,
162      'background':    0.0,
163     }, 0.051, 0.00333804044899],
164    ]
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