source: sasmodels/sasmodels/models/triaxial_ellipsoid.py @ 19dcb933

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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 $R_a \le R_b <= R_c$
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$\left< \cdots \right>$ is applied over all orientations for 1D.
13
14.. figure:: img/triaxial_ellipsoid_geometry.jpg
15
16    Ellipsoid schematic.
17
18The returned value is in units of |cm^-1|, on absolute scale.
19
20Definition
21----------
22
23The form factor calculated is
24
25.. math::
26
27    P(Q) = \frac{\text{scale}}{V}\int_0^1\int_0^1
28        \Phi^2(QR_a^2\cos^2( \pi x/2) + QR_b^2\sin^2(\pi y/2)(1-y^2) + c^2y^2)
29        dx dy
30
31where
32
33.. math::
34
35    \Phi(u) = 3 u^{-3} (\sin u - u \cos u)
36
37To provide easy access to the orientation of the triaxial ellipsoid,
38we define the axis of the cylinder using the angles $\theta$, $\phi$
39and $\psi$. These angles are defined on
40:num:`figure #triaxial-ellipsoid-angles`.
41The angle $\psi$ is the rotational angle around its own $c$ axis
42against the $Q$ plane. For example, $\psi = 0$ when the
43$a$ axis is parallel to the $x$ axis of the detector.
44
45.. _triaxial-ellipsoid-angles:
46
47.. figure:: img/triaxial_ellipsoid_angles.jpg
48
49    The angles for oriented ellipsoid.
50
51The radius-of-gyration for this system is  $R_g^2 = (R_a R_b R_c)^2/5$.
52
53The contrast is defined as SLD(ellipsoid) - SLD(solvent).  In the
54parameters, *a* is the minor equatorial radius, *b* is the major
55equatorial radius, and c is the polar radius of the ellipsoid.
56
57NB: The 2nd virial coefficient of the triaxial solid ellipsoid is
58calculated based on the polar radius $R_p = R_c$ and equatorial
59radius $R_e = \sqrt{R_a R_b}$, and used as the effective radius for
60$S(Q)$ when $P(Q) \cdot S(Q)$ is applied.
61
62.. figure:: img/triaxial_ellipsoid_1d.jpg
63
64    1D plot using the default values (w/1000 data point).
65
66Validation
67----------
68
69Validation of our code was done by comparing the output of the
701D calculation to the angular average of the output of 2D calculation
71over all possible angles.
72:num:`Figure #triaxial-ellipsoid-comparison` shows the comparison where
73the solid dot refers to averaged 2D while the line represents the
74result of 1D calculation (for 2D averaging, 76, 180, and 76 points
75are taken for the angles of $\theta$, $\phi$, and $\psi$ respectively).
76
77.. _triaxial-ellipsoid-comparison:
78
79.. figure:: img/triaxial_ellipsoid_comparison.png
80
81    Comparison between 1D and averaged 2D.
82
83Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research
84(Kline, 2006)
85
86REFERENCE
87
88L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum,
89New York, 1987.
90"""
91
92from numpy import pi, inf
93
94name = "triaxial_ellipsoid"
95title = "Ellipsoid of uniform scattering length density with three independent axes."
96
97description = """\
98Note: During fitting ensure that the inequality ra<rb<rc is not
99        violated. Otherwise the calculation will
100        not be correct.
101"""
102
103parameters = [
104#   [ "name", "units", default, [lower, upper], "type",
105#     "description" ],
106    [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "",
107      "Ellipsoid scattering length density" ],
108    [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "",
109      "Solvent scattering length density" ],
110    [ "req_minor", "Ang",  20, [0, inf], "volume",
111      "Minor equitorial radius" ],
112    [ "req_major", "Ang",  400, [0, inf], "volume",
113      "Major equatorial radius" ],
114    [ "rpolar", "Ang",  10, [0, inf], "volume",
115      "Polar radius" ],
116    [ "theta", "degrees", 60, [-inf, inf], "orientation",
117      "In plane angle" ],
118    [ "phi", "degrees", 60, [-inf, inf], "orientation",
119      "Out of plane angle" ],
120    [ "psi", "degrees", 60, [-inf, inf], "orientation",
121      "Out of plane angle" ],
122    ]
123
124source = [ "lib/J1.c", "lib/gauss76.c", "triaxial_ellipsoid.c"]
125
126def ER(req_minor, req_major, rpolar):
127    import numpy as np
128    from .ellipsoid import ER as ellipsoid_ER
129    return ellipsoid_ER(rpolar, np.sqrt(req_minor*req_major))
130
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