source: sasmodels/sasmodels/models/mass_surface_fractal.py @ 7994359

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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    fractal_dim = np.random.uniform(0, 6)
99    surface_portion = np.random.uniform(0, 1)
100    fractal_dim_surf = fractal_dim*surface_portion
101    fractal_dim_mass = fractal_dim - fractal_dim_surf
102    rg_cluster = 10**np.random.uniform(1, 5)
103    rg_primary = rg_cluster*10**np.random.uniform(-4, -1)
104    scale = 10**np.random.uniform(2, 5)
105    pars = dict(
106        #background=0,
107        scale=scale,
108        fractal_dim_mass=fractal_dim_mass,
109        fractal_dim_surf=fractal_dim_surf,
110        rg_cluster=rg_cluster,
111        rg_primary=rg_primary,
112    )
113    return pars
114
115
116demo = dict(scale=1, background=0,
117            fractal_dim_mass=1.8,
118            fractal_dim_surf=2.3,
119            rg_cluster=4000.0,
120            rg_primary=86.7)
121
122tests = [
123
124    # Accuracy tests based on content in test/utest_other_models.py  All except first, changed so rg_cluster is the larger, RKH 30 May 2018
125    [{'fractal_dim_mass':   1.8,
126      'fractal_dim_surf':   2.3,
127      'rg_cluster':   86.7,
128      'rg_primary': 4000.0,
129      'background':    0.0,
130     }, 0.05, 1.77537e-05],
131
132    # Additional tests with larger range of parameters
133    [{'fractal_dim_mass':      3.3,
134      'fractal_dim_surf':   1.0,
135      'rg_cluster': 4000.0,
136      'rg_primary':   90.0,
137     }, 0.001, 0.0932516614456],
138
139    [{'fractal_dim_mass':      1.3,
140      'fractal_dim_surf':   2.0,
141      'rg_cluster': 2000.0,
142      'rg_primary':   90.0,
143      'background':    0.8,
144     }, 0.001, 1.28296431786],
145
146    [{'fractal_dim_mass':      2.3,
147      'fractal_dim_surf':   3.1,
148      'rg_cluster':  1000.0,
149      'rg_primary':  30.0,
150      'scale':        10.0,
151      'background':    0.0,
152     }, 0.051, 0.00333804044899],
153    ]
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