source: sasmodels/sasmodels/models/mass_surface_fractal.py @ b297ba9

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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
57Authorship and Verification
58----------------------------
59
60* **Converted to sasmodels by:** Piotr Rozyczko **Date:** Jan 20, 2016
61* **Last Reviewed by:** Richard Heenan **Date:** May 30, 2018
62"""
63
64import numpy as np
65from numpy import inf
66
67name = "mass_surface_fractal"
68title = "Mass Surface Fractal model"
69description = """
70        The scattering intensity  I(x) = scale*P(x)*S(x) + background, where
71        p(x)= {[1+(x^2*a)]^(Dm/2) * [1+(x^2*b)]^(6-Ds-Dm)/2}^(-1)
72        a = Rg^2/(3*Dm/2)
73        b = rg^2/(3*(6-Ds-Dm)/2)
74        scale        =  scale factor * N*Volume^2*contrast^2
75        fractal_dim_mass       =  Dm (mass fractal dimension)
76        fractal_dim_surf  =  Ds
77        rg_cluster  =  Rg
78        rg_primary    =  rg
79        background   =  background
80        Hurd, Schaefer, Martin, Phys Rev A, eq(2),(1987),35, 2361-2364
81        Note that 0 < Ds< 6 and 0 < Dm < 6.
82        """
83category = "shape-independent"
84
85# pylint: disable=bad-whitespace, line-too-long
86#   ["name", "units", default, [lower, upper], "type","description"],
87parameters = [
88    ["fractal_dim_mass", "",      1.8, [0.0, 6.0], "", "Mass fractal dimension"],
89    ["fractal_dim_surf", "",      2.3, [0.0, 6.0], "", "Surface fractal dimension"],
90    ["rg_cluster",       "Ang", 4000., [0.0, inf], "", "Cluster radius of gyration"],
91    ["rg_primary",       "Ang",  86.7, [0.0, inf], "", "Primary particle radius of gyration"],
92]
93# pylint: enable=bad-whitespace, line-too-long
94
95source = ["mass_surface_fractal.c"]
96
97def random():
98    """Return a random parameter set for the model."""
99    fractal_dim = np.random.uniform(0, 6)
100    surface_portion = np.random.uniform(0, 1)
101    fractal_dim_surf = fractal_dim*surface_portion
102    fractal_dim_mass = fractal_dim - fractal_dim_surf
103    rg_cluster = 10**np.random.uniform(1, 5)
104    rg_primary = rg_cluster*10**np.random.uniform(-4, -1)
105    scale = 10**np.random.uniform(2, 5)
106    pars = dict(
107        #background=0,
108        scale=scale,
109        fractal_dim_mass=fractal_dim_mass,
110        fractal_dim_surf=fractal_dim_surf,
111        rg_cluster=rg_cluster,
112        rg_primary=rg_primary,
113    )
114    return pars
115
116
117demo = dict(scale=1, background=0,
118            fractal_dim_mass=1.8,
119            fractal_dim_surf=2.3,
120            rg_cluster=4000.0,
121            rg_primary=86.7)
122
123tests = [
124
125    # Accuracy tests based on content in test/utest_other_models.py  All
126    # except first, changed so rg_cluster is the larger, RKH 30 May 2018
127    [{'fractal_dim_mass':   1.8,
128      'fractal_dim_surf':   2.3,
129      'rg_cluster':   86.7,
130      'rg_primary': 4000.0,
131      'background':    0.0,
132     }, 0.05, 1.77537e-05],
133
134    # Additional tests with larger range of parameters
135    [{'fractal_dim_mass':      3.3,
136      'fractal_dim_surf':   1.0,
137      'rg_cluster': 4000.0,
138      'rg_primary':   90.0,
139     }, 0.001, 0.0932516614456],
140
141    [{'fractal_dim_mass':      1.3,
142      'fractal_dim_surf':   2.0,
143      'rg_cluster': 2000.0,
144      'rg_primary':   90.0,
145      'background':    0.8,
146     }, 0.001, 1.28296431786],
147
148    [{'fractal_dim_mass':      2.3,
149      'fractal_dim_surf':   3.1,
150      'rg_cluster':  1000.0,
151      'rg_primary':  30.0,
152      'scale':        10.0,
153      'background':    0.0,
154     }, 0.051, 0.00333804044899],
155    ]
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