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

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4Calculates the scattering from a fractal structure with a primary building
5block of core-shell spheres, as opposed to just homogeneous spheres in
6the fractal model. It is an extension of the well known Teixeira\ [#teixeira]_
7fractal model replacing the $P(q)$ of a solid sphere with that of a core-shell
8sphere. This model could find use for aggregates of coated particles, or
9aggregates of vesicles for example.
10
11.. math::
12
13    I(q) = P(q)S(q) + \text{background}
14
15Where $P(q)$ is the core-shell form factor and $S(q)$ is the
16Teixeira\ [#teixeira]_ fractal structure factor both of which are given again
17below:
18
19.. math::
20
21    P(q) &= \frac{\phi}{V_s}\left[3V_c(\rho_c-\rho_s)
22    \frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3}+
23    3V_s(\rho_s-\rho_{solv})
24    \frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}\right]^2 \\
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)]}{(qr_s)^{D_f}}
27
28where $\phi$ is the volume fraction of particles, $V_s$ is the volume of the
29whole particle, $V_c$ is the volume of the core, $\rho_c$, $\rho_s$, and
30$\rho_{solv}$ are the scattering length densities of the core, shell, and
31solvent respectively, $r_c$ and $r_s$ are the radius of the core and the radius
32of the whole particle respectively, $D_f$ is the fractal dimension, and $\xi$ the
33correlation length.
34
35Polydispersity of radius and thickness are also provided for.
36
37This model does not allow for anisotropy and thus the 2D scattering intensity
38is calculated in the same way as 1D, where the $q$ vector is defined as
39
40.. math::
41
42    q = \sqrt{q_x^2 + q_y^2}
43
44Our model is derived from the form factor calculations implemented in IGOR
45macros by the NIST Center for Neutron Research\ [#Kline]_
46
47References
48----------
49
50.. [#teixeira] J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785
51.. [#Kline]  S R Kline, *J Appl. Cryst.*, 39 (2006) 895
52
53Source
54------
55
56fractal_core_shell.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/fractal_core_shell.py>_
57
58fractal_core_shell.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/fractal_core_shell.c>_
59
60Authorship and Verification
61----------------------------
62
63* **Author:** NIST IGOR/DANSE **Date:** pre 2010
64* **Last Modified by:** Paul Butler and Paul Kienzle **Date:** November 27, 2016
65* **Last Reviewed by:** Paul Butler and Paul Kienzle **Date:** November 27, 2016
66* **Source added by :** Steve King **Date:** March 25, 2019
67"""
68
69import numpy as np
70from numpy import inf
71
72name = "fractal_core_shell"
73title = "Scattering from a fractal structure formed from core shell spheres"
74description = """\
75    Model for fractal aggregates of core-shell primary particles. It is based on
76    the Teixeira model for the S(q) of a fractal * P(q) for a core-shell sphere
77
79    thickness = thickness of the shell
80    thick_layer = thickness of a layer
81    sld_core = the SLD of the core
82    sld_shell = the SLD of the shell
83    sld_solvent = the SLD of the solvent
84    volfraction = volume fraction of core-shell particles
85    fractal_dim = fractal dimension
86    cor_length = correlation length of the fractal like aggretates
87    """
88category = "shape-independent"
89
91#   ["name", "units", default, [lower, upper], "type","description"],
92parameters = [
94    ["thickness",   "Ang",        10.0, [0.0, inf],  "volume", "Sphere shell thickness"],
95    ["sld_core",    "1e-6/Ang^2", 1.0,  [-inf, inf], "sld",    "Sphere core scattering length density"],
96    ["sld_shell",   "1e-6/Ang^2", 2.0,  [-inf, inf], "sld",    "Sphere shell scattering length density"],
97    ["sld_solvent", "1e-6/Ang^2", 3.0,  [-inf, inf], "sld",    "Solvent scattering length density"],
98    ["volfraction", "",           0.05,  [0.0, inf],  "",       "Volume fraction of building block spheres"],
99    ["fractal_dim",    "",        2.0,  [0.0, 6.0],  "",       "Fractal dimension"],
100    ["cor_length",  "Ang",      100.0,  [0.0, inf],  "",       "Correlation length of fractal-like aggregates"],
101]
103
104source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "lib/core_shell.c",
105          "lib/fractal_sq.c", "fractal_core_shell.c"]
106
107def random():
108    """Return a random parameter set for the model."""
110    # Use a distribution with a preference for thin shell or thin core
111    # Avoid core,shell radii < 1
112    thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1
115    volfraction = 10**np.random.uniform(-3, -1)
116    fractal_dim = 2*np.random.beta(3, 4) + 1
117    pars = dict(
118        #background=0, sld_block=1, sld_solvent=0,
119        volfraction=volfraction,
121        cor_length=cor_length,
122        fractal_dim=fractal_dim,
123    )
124    return pars
125
126demo = dict(scale=0.05,
127            background=0,
129            thickness=5,
130            sld_core=3.5,
131            sld_shell=1.0,
132            sld_solvent=6.35,
133            volfraction=0.05,
134            fractal_dim=2.0,
135            cor_length=100.0)
136
137# TODO: why is there an ER function here?
139    """
142        @param thickness: shell thickness
143    """
145
146#tests = [[{'radius': 20.0, 'thickness': 10.0}, 'ER', 30.0],
147tests = [
148    # At some point the SasView 3.x test result was deemed incorrect.  The
149    # following tests were verified against NIST IGOR macros ver 7.850.
150    # NOTE: NIST macros do only provide for a polydispers core (no option
151    # for a poly shell or for a monodisperse core.  The results seemed
152    # extremely sensitive to the core PD, varying non monotonically all
153    # the way to a PD of 1e-6. From 1e-6 to 1e-9 no changes in the
154    # results were observed and the values below were taken using PD=1e-9.
155    # Non-monotonically = I(0.001)=188 to 140 to 177 back to 160 etc.