source: sasmodels/sasmodels/models/hollow_rectangular_prism_thin_walls.py

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1# rectangular_prism model
2# Note: model title and parameter table are inserted automatically
3r"""
4Definition
5----------
6
7
8This model provides the form factor, $P(q)$, for a hollow rectangular
9prism with infinitely thin walls. It computes only the 1D scattering, not the 2D.
10The 1D scattering intensity for this model is calculated according to the
11equations given by Nayuk and Huber\ [#Nayuk2012]_.
12
13Assuming a hollow parallelepiped with infinitely thin walls, edge lengths
14$A \le B \le C$ and presenting an orientation with respect to the
15scattering vector given by $\theta$ and $\phi$, where $\theta$ is the angle
16between the $z$ axis and the longest axis of the parallelepiped $C$, and
17$\phi$ is the angle between the scattering vector (lying in the $xy$ plane)
18and the $y$ axis, the form factor is given by
19
20.. math::
21
22    P(q) = \frac{1}{V^2} \frac{2}{\pi} \int_0^{\frac{\pi}{2}}
23           \int_0^{\frac{\pi}{2}} [A_L(q)+A_T(q)]^2 \sin\theta\,d\theta\,d\phi
24
25where
26
27.. math::
28
29    V &= 2AB + 2AC + 2BC \\
30    A_L(q) &=  8 \times \frac{
31            \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right)
32            \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right)
33            \cos \left( \tfrac{1}{2} q C \cos\theta \right)
34        }{q^2 \, \sin^2\theta \, \sin\phi \cos\phi} \\
35    A_T(q) &=  A_F(q) \times
36      \frac{2\,\sin \left( \tfrac{1}{2} q C \cos\theta \right)}{q\,\cos\theta}
37
38and
39
40.. math::
41
42  A_F(q) =  4 \frac{ \cos \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right)
43                       \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) }
44                     {q \, \cos\phi \, \sin\theta} +
45              4 \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right)
46                       \cos \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) }
47                     {q \, \sin\phi \, \sin\theta}
48
49The 1D scattering intensity is then calculated as
50
51.. math::
52
53  I(q) = \text{scale} \times V \times (\rho_\text{p} - \rho_\text{solvent})^2 \times P(q)
54
55where $V$ is the surface area of the rectangular prism, $\rho_\text{p}$
56is the scattering length density of the parallelepiped, $\rho_\text{solvent}$
57is the scattering length density of the solvent, and (if the data are in
58absolute units) *scale* is related to the total surface area.
59
60**The 2D scattering intensity is not computed by this model.**
61
62
63Validation
64----------
65
66Validation of the code was conducted  by qualitatively comparing the output
67of the 1D model to the curves shown in (Nayuk, 2012\ [#Nayuk2012]_).
68
69
70References
71----------
72
73.. [#Nayuk2012] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854
74.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
75
76Authorship and Verification
77----------------------------
78
79* **Author:** Miguel Gonzales **Date:** February 26, 2016
80* **Last Modified by:** Paul Kienzle **Date:** October 15, 2016
81* **Last Reviewed by:** Paul Butler **Date:** September 07, 2018
82"""
83
84import numpy as np
85from numpy import inf
86
87name = "hollow_rectangular_prism_thin_walls"
88title = "Hollow rectangular parallelepiped with thin walls."
89description = """
90    I(q)= scale*V*(sld - sld_solvent)^2*P(q)+background
91        with P(q) being the form factor corresponding to a hollow rectangular
92        parallelepiped with infinitely thin walls.
93"""
94category = "shape:parallelepiped"
95
96#             ["name", "units", default, [lower, upper], "type","description"],
97parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld",
98               "Parallelepiped scattering length density"],
99              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
100               "Solvent scattering length density"],
101              ["length_a", "Ang", 35, [0, inf], "volume",
102               "Shorter side of the parallelepiped"],
103              ["b2a_ratio", "Ang", 1, [0, inf], "volume",
104               "Ratio sides b/a"],
105              ["c2a_ratio", "Ang", 1, [0, inf], "volume",
106               "Ratio sides c/a"],
107             ]
108
109source = ["lib/gauss76.c", "hollow_rectangular_prism_thin_walls.c"]
110have_Fq = True
111radius_effective_modes = [
112    "equivalent cylinder excluded volume", "equivalent outer volume sphere",
113    "half length_a", "half length_b", "half length_c",
114    "equivalent outer circular cross-section",
115    "half ab diagonal", "half diagonal",
116    ]
117
118
119def random():
120    """Return a random parameter set for the model."""
121    a, b, c = 10**np.random.uniform(1, 4.7, size=3)
122    pars = dict(
123        length_a=a,
124        b2a_ratio=b/a,
125        c2a_ratio=c/a,
126    )
127    return pars
128
129
130# parameters for demo
131demo = dict(scale=1, background=0,
132            sld=6.3, sld_solvent=1.0,
133            length_a=35, b2a_ratio=1, c2a_ratio=1,
134            length_a_pd=0.1, length_a_pd_n=10,
135            b2a_ratio_pd=0.1, b2a_ratio_pd_n=1,
136            c2a_ratio_pd=0.1, c2a_ratio_pd_n=1)
137
138tests = [[{}, 0.2, 0.837719188592],
139         [{}, [0.2], [0.837719188592]],
140        ]
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