source: sasmodels/sasmodels/models/triaxial_ellipsoid.py @ 416f5c7

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Last change on this file since 416f5c7 was 416f5c7, checked in by richardh, 7 years ago

fixes for numref warnings in docu, new equations core_shell_bicelle core_shell_ellipsoid

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1# triaxial ellipsoid model
2# Note: model title and parameter table are inserted automatically
3r"""
4All three axes are of different lengths with $R_a \leq R_b \leq R_c$
5**Users should maintain this inequality for all calculations**.
6
7.. math::
8
9    P(q) = \text{scale} V \left< F^2(q) \right> + \text{background}
10
11where the volume $V = 4/3 \pi R_a R_b R_c$, and the averaging
12$\left<\ldots\right>$ is applied over all orientations for 1D.
13
14.. figure:: img/triaxial_ellipsoid_geometry.jpg
15
16    Ellipsoid schematic.
17
18Definition
19----------
20
21The form factor calculated is
22
23.. math::
24
25    P(q) = \frac{\text{scale}}{V}\int_0^1\int_0^1
26        \Phi^2(qR_a^2\cos^2( \pi x/2) + qR_b^2\sin^2(\pi y/2)(1-y^2) + R_c^2y^2)
27        dx dy
28
29where
30
31.. math::
32
33    \Phi(u) = 3 u^{-3} (\sin u - u \cos u)
34
35To provide easy access to the orientation of the triaxial ellipsoid,
36we define the axis of the cylinder using the angles $\theta$, $\phi$
37and $\psi$. These angles are defined on
38:numref:`triaxial-ellipsoid-angles` .
39The angle $\psi$ is the rotational angle around its own $c$ axis
40against the $q$ plane. For example, $\psi = 0$ when the
41$a$ axis is parallel to the $x$ axis of the detector.
42
43.. _triaxial-ellipsoid-angles:
44
45.. figure:: img/triaxial_ellipsoid_angle_projection.jpg
46
47    The angles for oriented ellipsoid.
48
49The radius-of-gyration for this system is  $R_g^2 = (R_a R_b R_c)^2/5$.
50
51The contrast is defined as SLD(ellipsoid) - SLD(solvent).  In the
52parameters, $R_a$ is the minor equatorial radius, $R_b$ is the major
53equatorial radius, and $R_c$ is the polar radius of the ellipsoid.
54
55NB: The 2nd virial coefficient of the triaxial solid ellipsoid is
56calculated based on the polar radius $R_p = R_c$ and equatorial
57radius $R_e = \sqrt{R_a R_b}$, and used as the effective radius for
58$S(q)$ when $P(q) \cdot S(q)$ is applied.
59
60Validation
61----------
62
63Validation of our code was done by comparing the output of the
641D calculation to the angular average of the output of 2D calculation
65over all possible angles.
66
67
68References
69----------
70
71L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray
72and Neutron Scattering*, Plenum, New York, 1987.
73"""
74
75from numpy import inf
76
77name = "triaxial_ellipsoid"
78title = "Ellipsoid of uniform scattering length density with three independent axes."
79
80description = """\
81Note: During fitting ensure that the inequality ra<rb<rc is not
82        violated. Otherwise the calculation will
83        not be correct.
84"""
85category = "shape:ellipsoid"
86
87#             ["name", "units", default, [lower, upper], "type","description"],
88parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld",
89               "Ellipsoid scattering length density"],
90              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
91               "Solvent scattering length density"],
92              ["radius_equat_minor", "Ang", 20, [0, inf], "volume",
93               "Minor equatorial radius"],
94              ["radius_equat_major", "Ang", 400, [0, inf], "volume",
95               "Major equatorial radius"],
96              ["radius_polar", "Ang", 10, [0, inf], "volume",
97               "Polar radius"],
98              ["theta", "degrees", 60, [-inf, inf], "orientation",
99               "In plane angle"],
100              ["phi", "degrees", 60, [-inf, inf], "orientation",
101               "Out of plane angle"],
102              ["psi", "degrees", 60, [-inf, inf], "orientation",
103               "Out of plane angle"],
104             ]
105
106source = ["lib/sph_j1c.c", "lib/gauss76.c", "triaxial_ellipsoid.c"]
107
108def ER(radius_equat_minor, radius_equat_major, radius_polar):
109    """
110        Returns the effective radius used in the S*P calculation
111    """
112    import numpy as np
113    from .ellipsoid import ER as ellipsoid_ER
114    return ellipsoid_ER(radius_polar, np.sqrt(radius_equat_minor * radius_equat_major))
115
116demo = dict(scale=1, background=0,
117            sld=6, sld_solvent=1,
118            theta=30, phi=15, psi=5,
119            radius_equat_minor=25, radius_equat_major=36, radius_polar=50,
120            radius_equat_minor_pd=0, radius_equat_minor_pd_n=1,
121            radius_equat_major_pd=0, radius_equat_major_pd_n=1,
122            radius_polar_pd=.2, radius_polar_pd_n=30,
123            theta_pd=15, theta_pd_n=45,
124            phi_pd=15, phi_pd_n=1,
125            psi_pd=15, psi_pd_n=1)
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