source: sasmodels/sasmodels/models/mass_surface_fractal.py @ c1e44e5

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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* **Author:** Piotr Rozyczko **Date:** Jan 20, 2016
61* **Last Modified by:**
62* **Last Reviewed by:** Richard Heenan **Date:** May 30, 2018
63"""
64
65import numpy as np
66from numpy import inf
67
68name = "mass_surface_fractal"
69title = "Mass Surface Fractal model"
70description = """
71        The scattering intensity  I(x) = scale*P(x)*S(x) + background, where
72        p(x)= {[1+(x^2*a)]^(Dm/2) * [1+(x^2*b)]^(6-Ds-Dm)/2}^(-1)
73        a = Rg^2/(3*Dm/2)
74        b = rg^2/(3*(6-Ds-Dm)/2)
75        scale        =  scale factor * N*Volume^2*contrast^2
76        fractal_dim_mass       =  Dm (mass fractal dimension)
77        fractal_dim_surf  =  Ds
78        rg_cluster  =  Rg
79        rg_primary    =  rg
80        background   =  background
81        Hurd, Schaefer, Martin, Phys Rev A, eq(2),(1987),35, 2361-2364
82        Note that 0 < Ds< 6 and 0 < Dm < 6.
83        """
84category = "shape-independent"
85
86# pylint: disable=bad-whitespace, line-too-long
87#   ["name", "units", default, [lower, upper], "type","description"],
88parameters = [
89    ["fractal_dim_mass", "",      1.8, [0.0, 6.0], "", "Mass fractal dimension"],
90    ["fractal_dim_surf", "",      2.3, [0.0, 6.0], "", "Surface fractal dimension"],
91    ["rg_cluster",       "Ang", 4000., [0.0, inf], "", "Cluster radius of gyration"],
92    ["rg_primary",       "Ang",  86.7, [0.0, inf], "", "Primary particle radius of gyration"],
93]
94# pylint: enable=bad-whitespace, line-too-long
95
96source = ["mass_surface_fractal.c"]
97
98def random():
99    """Return a random parameter set for the model."""
100    fractal_dim = np.random.uniform(0, 6)
101    surface_portion = np.random.uniform(0, 1)
102    fractal_dim_surf = fractal_dim*surface_portion
103    fractal_dim_mass = fractal_dim - fractal_dim_surf
104    rg_cluster = 10**np.random.uniform(1, 5)
105    rg_primary = rg_cluster*10**np.random.uniform(-4, -1)
106    scale = 10**np.random.uniform(2, 5)
107    pars = dict(
108        #background=0,
109        scale=scale,
110        fractal_dim_mass=fractal_dim_mass,
111        fractal_dim_surf=fractal_dim_surf,
112        rg_cluster=rg_cluster,
113        rg_primary=rg_primary,
114    )
115    return pars
116
117
118demo = dict(scale=1, background=0,
119            fractal_dim_mass=1.8,
120            fractal_dim_surf=2.3,
121            rg_cluster=4000.0,
122            rg_primary=86.7)
123
124tests = [
125
126    # Accuracy tests based on content in test/utest_other_models.py  All
127    # except first, changed so rg_cluster is the larger, RKH 30 May 2018
128    [{'fractal_dim_mass':   1.8,
129      'fractal_dim_surf':   2.3,
130      'rg_cluster':   86.7,
131      'rg_primary': 4000.0,
132      'background':    0.0,
133     }, 0.05, 1.77537e-05],
134
135    # Additional tests with larger range of parameters
136    [{'fractal_dim_mass':      3.3,
137      'fractal_dim_surf':   1.0,
138      'rg_cluster': 4000.0,
139      'rg_primary':   90.0,
140     }, 0.001, 0.0932516614456],
141
142    [{'fractal_dim_mass':      1.3,
143      'fractal_dim_surf':   2.0,
144      'rg_cluster': 2000.0,
145      'rg_primary':   90.0,
146      'background':    0.8,
147     }, 0.001, 1.28296431786],
148
149    [{'fractal_dim_mass':      2.3,
150      'fractal_dim_surf':   3.1,
151      'rg_cluster':  1000.0,
152      'rg_primary':  30.0,
153      'scale':        10.0,
154      'background':    0.0,
155     }, 0.051, 0.00333804044899],
156    ]
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