source: sasmodels/sasmodels/models/hollow_rectangular_prism_thin_walls.py @ 5024a56

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Last change on this file since 5024a56 was ee60aa7, checked in by Paul Kienzle <pkienzle@…>, 2 years ago

clean up effective radius functions; improve mono_gauss_coil accuracy; start moving VR into C

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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
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 pi, inf, sqrt
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
111effective_radius_type = [
112    "equivalent sphere", "half length_a", "half length_b", "half length_c",
113    "equivalent outer circular cross-section",
114    "half ab diagonal", "half diagonal",
115    ]
116
117
118def random():
119    a, b, c = 10**np.random.uniform(1, 4.7, size=3)
120    pars = dict(
121        length_a=a,
122        b2a_ratio=b/a,
123        c2a_ratio=c/a,
124    )
125    return pars
126
127
128# parameters for demo
129demo = dict(scale=1, background=0,
130            sld=6.3, sld_solvent=1.0,
131            length_a=35, b2a_ratio=1, c2a_ratio=1,
132            length_a_pd=0.1, length_a_pd_n=10,
133            b2a_ratio_pd=0.1, b2a_ratio_pd_n=1,
134            c2a_ratio_pd=0.1, c2a_ratio_pd_n=1)
135
136tests = [[{}, 0.2, 0.837719188592],
137         [{}, [0.2], [0.837719188592]],
138        ]
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