# source:sasmodels/sasmodels/models/vesicle.py

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1r"""
2Definition
3----------
4
5This model provides the form factor, *P(q)*, for an unilamellar vesicle and is
6effectively identical to the hollow sphere reparameterized to be
7more intuitive for a vesicle and normalizing the form factor by the volume of
8the shell. The 1D scattering intensity is calculated in the following way
9(Guinier,1955\ [#Guinier1955]_)
10
11.. math::
12
13    P(q) = \frac{\phi}{V_\text{shell}} \left[
14           \frac{3V_{\text{core}}({\rho_{\text{solvent}}
15           - \rho_{\text{shell}})j_1(qR_{\text{core}})}}{qR_{\text{core}}}
16           + \frac{3V_{\text{tot}}(\rho_{\text{shell}}
17           - \rho_{\text{solvent}}) j_1(qR_{\text{tot}})}{qR_{\text{tot}}}
18           \right]^2 + \text{background}
19
20
21where $\phi$ is the volume fraction of shell material, $V_{shell}$ is the volume
22of the shell, $V_{\text{cor}}$ is the volume of the core, $V_{\text{tot}}$ is
23the total volume, $R_{\text{core}}$ is the radius of the core, $R_{\text{tot}}$
24is the outer radius of the shell, $\rho_{\text{solvent}}$ is the scattering
25length density of the solvent (which is the same as for the core in this case),
26$\rho_{\text{scale}}$ is the scattering length density of the shell, background
27is a flat background level (due for example to incoherent scattering in the
28case of neutrons), and $j_1$ is the spherical bessel function
29$j_1 = (\sin(x) - x \cos(x))/ x^2$.
30
31The functional form is identical to a "typical" core-shell structure, except
32that the scattering is normalized by the volume that is contributing to the
33scattering, namely the volume of the shell alone, the scattering length density
34of the core is fixed the same as that of the solvent, the scale factor when the
35data are on an absolute scale is equivalent to the volume fraction of material
36in the shell rather than the entire core+shell sphere, and the parameterization
37is done in terms of the core radius = $R_{\text{core}}$ and the shell
38thickness = $R_{\text{tot}} - R_{\text{core}}$.
39
40.. figure:: img/vesicle_geometry.jpg
41
42    Vesicle geometry.
43
44The 2D scattering intensity is the same as *P(q)* above, regardless of the
45orientation of the *q* vector which is defined as
46
47.. math::
48
49    q = \sqrt{q_x^2 + q_y^2}
50
51
52NB: The outer most radius (= *radius* + *thickness*) is used as the effective
53radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied.
54
55
56References
57----------
58
59.. [#Guinier1955] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and
60   Sons, New York, (1955)
61
62Authorship and Verification
63----------------------------
64
65* **Author:** NIST IGOR/DANSE **Date:** pre 2010
67* **Last Reviewed by:** Paul Butler **Date:** September 7, 2018
68"""
69
70import numpy as np
71from numpy import inf
72
73name = "vesicle"
74title = "Vesicle model representing a hollow sphere"
75description = """
76    Model parameters:
78        thickness: the shell thickness
79        sld: the shell SLD
80        sld_solvent: the solvent (and core) SLD
81        background: incoherent background
82        volfraction: shell volume fraction
83        scale : scale factor = 1 if on absolute scale"""
84category = "shape:sphere"
85
86#             [ "name", "units", default, [lower, upper], "type", "description"],
87parameters = [["sld", "1e-6/Ang^2", 0.5, [-inf, inf], "sld",
88               "vesicle shell scattering length density"],
89              ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld",
90               "solvent scattering length density"],
91              ["volfraction", "", 0.05, [0, 1.0], "",
92               "volume fraction of shell"],
93              ["radius", "Ang", 100, [0, inf], "volume",
95              ["thickness", "Ang", 30, [0, inf], "volume",
96               "vesicle shell thickness"],
97             ]
98
99source = ["lib/sas_3j1x_x.c", "vesicle.c"]
100have_Fq = True
102
103def random():
104    """Return a random parameter set for the model."""
108    volfraction = 10**np.random.uniform(-3, -1)
109    pars = dict(
110        #background=0,
111        scale=1,  # volfraction is part of the model, so scale=1
113        thickness=thickness,
114        volfraction=volfraction,
115    )
116    return pars
117
118# parameters for demo
119demo = dict(sld=0.5, sld_solvent=6.36,
120            volfraction=0.05,