1 | # triaxial ellipsoid model |
---|
2 | # Note: model title and parameter table are inserted automatically |
---|
3 | r""" |
---|
4 | All 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 | |
---|
11 | where 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 | |
---|
18 | Definition |
---|
19 | ---------- |
---|
20 | |
---|
21 | The 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 | |
---|
29 | where |
---|
30 | |
---|
31 | .. math:: |
---|
32 | |
---|
33 | \Phi(u) = 3 u^{-3} (\sin u - u \cos u) |
---|
34 | |
---|
35 | To provide easy access to the orientation of the triaxial ellipsoid, |
---|
36 | we define the axis of the cylinder using the angles $\theta$, $\phi$ |
---|
37 | and $\psi$. These angles are defined on |
---|
38 | :numref:`triaxial-ellipsoid-angles`. |
---|
39 | The angle $\psi$ is the rotational angle around its own $c$ axis |
---|
40 | against 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 | |
---|
49 | The radius-of-gyration for this system is $R_g^2 = (R_a R_b R_c)^2/5$. |
---|
50 | |
---|
51 | The contrast is defined as SLD(ellipsoid) - SLD(solvent). In the |
---|
52 | parameters, $R_a$ is the minor equatorial radius, $R_b$ is the major |
---|
53 | equatorial radius, and $R_c$ is the polar radius of the ellipsoid. |
---|
54 | |
---|
55 | NB: The 2nd virial coefficient of the triaxial solid ellipsoid is |
---|
56 | calculated based on the polar radius $R_p = R_c$ and equatorial |
---|
57 | radius $R_e = \sqrt{R_a R_b}$, and used as the effective radius for |
---|
58 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
---|
59 | |
---|
60 | Validation |
---|
61 | ---------- |
---|
62 | |
---|
63 | Validation of our code was done by comparing the output of the |
---|
64 | 1D calculation to the angular average of the output of 2D calculation |
---|
65 | over all possible angles. |
---|
66 | |
---|
67 | |
---|
68 | References |
---|
69 | ---------- |
---|
70 | |
---|
71 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray |
---|
72 | and Neutron Scattering*, Plenum, New York, 1987. |
---|
73 | """ |
---|
74 | |
---|
75 | from numpy import inf |
---|
76 | |
---|
77 | name = "triaxial_ellipsoid" |
---|
78 | title = "Ellipsoid of uniform scattering length density with three independent axes." |
---|
79 | |
---|
80 | description = """\ |
---|
81 | Note: During fitting ensure that the inequality ra<rb<rc is not |
---|
82 | violated. Otherwise the calculation will |
---|
83 | not be correct. |
---|
84 | """ |
---|
85 | category = "shape:ellipsoid" |
---|
86 | |
---|
87 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
88 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
---|
89 | "Ellipsoid scattering length density"], |
---|
90 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
91 | "Solvent scattering length density"], |
---|
92 | ["req_minor", "Ang", 20, [0, inf], "volume", |
---|
93 | "Minor equatorial radius"], |
---|
94 | ["req_major", "Ang", 400, [0, inf], "volume", |
---|
95 | "Major equatorial radius"], |
---|
96 | ["rpolar", "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 | |
---|
106 | source = ["lib/sph_j1c.c", "lib/gauss76.c", "triaxial_ellipsoid.c"] |
---|
107 | |
---|
108 | def ER(req_minor, req_major, rpolar): |
---|
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(rpolar, np.sqrt(req_minor * req_major)) |
---|
115 | |
---|
116 | demo = dict(scale=1, background=0, |
---|
117 | sld=6, solvent_sld=1, |
---|
118 | theta=30, phi=15, psi=5, |
---|
119 | req_minor=25, req_major=36, rpolar=50, |
---|
120 | req_minor_pd=0, req_minor_pd_n=1, |
---|
121 | req_major_pd=0, req_major_pd_n=1, |
---|
122 | rpolar_pd=.2, rpolar_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) |
---|