source: sasmodels/sasmodels/models/vesicle.py @ 2d81cfe

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4
5The 1D scattering intensity is calculated in the following way (Guinier, 1955)
6
7.. math::
8
9    P(q) = \frac{\phi}{V_\text{shell}} \left[
10           \frac{3V_{\text{core}}({\rho_{\text{solvent}}
11           - \rho_{\text{shell}})j_1(qR_{\text{core}})}}{qR_{\text{core}}}
12           + \frac{3V_{\text{tot}}(\rho_{\text{shell}}
13           - \rho_{\text{solvent}}) j_1(qR_{\text{tot}})}{qR_{\text{tot}}}
14           \right]^2 + \text{background}
15
16
17where $\phi$ is the volume fraction of shell material, $V_{shell}$ is the volume
18of the shell, $V_{\text{cor}}$ is the volume of the core, $V_{\text{tot}}$ is
19the total volume, $R_{\text{core}}$ is the radius of the core, $R_{\text{tot}}$
20is the outer radius of the shell, $\rho_{\text{solvent}}$ is the scattering
21length density of the solvent (which is the same as for the core in this case),
22$\rho_{\text{scale}}$ is the scattering length density of the shell, background
23is a flat background level (due for example to incoherent scattering in the
24case of neutrons), and $j_1$ is the spherical bessel function
25$j_1 = (\sin(x) - x \cos(x))/ x^2$.
26
27The functional form is identical to a "typical" core-shell structure, except
28that the scattering is normalized by the volume that is contributing to the
29scattering, namely the volume of the shell alone, the scattering length density
30of the core is fixed the same as that of the solvent, the scale factor when the
31data are on an absolute scale is equivalent to the volume fraction of material
32in the shell rather than the entire core+shell sphere, and the parameterization
33is done in terms of the core radius = $R_{\text{core}}$ and the shell
34thickness = $R_{\text{tot}} - R_{\text{core}}$.
35
36.. figure:: img/vesicle_geometry.jpg
37
38    Vesicle geometry.
39
40The 2D scattering intensity is the same as *P(q)* above, regardless of the
41orientation of the *q* vector which is defined as
42
43.. math::
44
45    q = \sqrt{q_x^2 + q_y^2}
46
47
48NB: The outer most radius (= *radius* + *thickness*) is used as the effective
49radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied.
50
51
52References
53----------
54
55A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and
56Sons, New York, (1955)
57
58**Author:** NIST IGOR/DANSE **on:** pre 2010
59
60**Last Modified by:** Paul Butler **on:** March 20, 2016
61
62**Last Reviewed by:** Paul Butler **on:** March 20, 2016
63"""
64
65import numpy as np
66from numpy import pi, inf
67
68name = "vesicle"
69title = "This model provides the form factor, *P(q)*, for an unilamellar \
70    vesicle. This is model is effectively identical to the hollow sphere \
71    reparameterized to be more intuitive for a vesicle and normalizing the \
72    form factor by the volume of the shell."
73description = """
74    Model parameters:
75        radius : the core radius of the vesicle
76        thickness: the shell thickness
77        sld: the shell SLD
78        sld_solvent: the solvent (and core) SLD
79        background: incoherent background
80        volfraction: shell volume fraction
81        scale : scale factor = 1 if on absolute scale"""
82category = "shape:sphere"
83
84#             [ "name", "units", default, [lower, upper], "type", "description"],
85parameters = [["sld", "1e-6/Ang^2", 0.5, [-inf, inf], "sld",
86               "vesicle shell scattering length density"],
87              ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld",
88               "solvent scattering length density"],
89              ["volfraction", "", 0.05, [0, 1.0], "",
90               "volume fraction of shell"],
91              ["radius", "Ang", 100, [0, inf], "volume",
92               "vesicle core radius"],
93              ["thickness", "Ang", 30, [0, inf], "volume",
94               "vesicle shell thickness"],
95             ]
96
97source = ["lib/sas_3j1x_x.c", "vesicle.c"]
98
99def ER(radius, thickness):
100    '''
101    returns the effective radius used in the S*P calculation
102
103    :param radius: core radius
104    :param thickness: shell thickness
105    '''
106    return radius + thickness
107
108def VR(radius, thickness):
109    '''
110    returns the volumes of the shell and of the whole sphere including the
111    core plus shell - is used to normalize when including polydispersity.
112
113    :param radius: core radius
114    :param thickness: shell thickness
115    :return whole: volume of core and shell
116    :return whole-core: volume of the shell
117    '''
118
119    whole = 4./3. * pi * (radius + thickness)**3
120    core = 4./3. * pi * radius**3
121    return whole, whole - core
122
123def random():
124    total_radius = 10**np.random.uniform(1.3, 5)
125    radius = total_radius * np.random.uniform(0, 1)
126    thickness = total_radius - radius
127    volfraction = 10**np.random.uniform(-3, -1)
128    pars = dict(
129        #background=0,
130        scale=1,  # volfraction is part of the model, so scale=1
131        radius=radius,
132        thickness=thickness,
133        volfraction=volfraction,
134    )
135    return pars
136
137# parameters for demo
138demo = dict(sld=0.5, sld_solvent=6.36,
139            volfraction=0.05,
140            radius=100, thickness=30,
141            radius_pd=.2, radius_pd_n=10,
142            thickness_pd=.2, thickness_pd_n=10)
143
144# NOTE: test results taken from values returned by SasView 3.1.2, with
145# 0.001 added for a non-zero default background.
146tests = [[{}, 0.0005, 859.916526646],
147         [{}, 0.100600200401, 1.77063682331],
148         [{}, 0.5, 0.00355351388906],
149         [{}, 'ER', 130.],
150         [{}, 'VR', 0.54483386436],
151        ]
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