1 | # triaxial ellipsoid model |
---|
2 | # Note: model title and parameter table are inserted automatically |
---|
3 | r""" |
---|
4 | Definition |
---|
5 | ---------- |
---|
6 | |
---|
7 | .. figure:: img/triaxial_ellipsoid_geometry.jpg |
---|
8 | |
---|
9 | Ellipsoid with $R_a$ as *radius_equat_minor*, $R_b$ as *radius_equat_major* |
---|
10 | and $R_c$ as *radius_polar*. |
---|
11 | |
---|
12 | Given an ellipsoid |
---|
13 | |
---|
14 | .. math:: |
---|
15 | |
---|
16 | \frac{X^2}{R_a^2} + \frac{Y^2}{R_b^2} + \frac{Z^2}{R_c^2} = 1 |
---|
17 | |
---|
18 | the scattering for randomly oriented particles is defined by the average over |
---|
19 | all orientations $\Omega$ of: |
---|
20 | |
---|
21 | .. math:: |
---|
22 | |
---|
23 | P(q) = \text{scale}(\Delta\rho)^2\frac{V}{4 \pi}\int_\Omega\Phi^2(qr)\,d\Omega |
---|
24 | + \text{background} |
---|
25 | |
---|
26 | where |
---|
27 | |
---|
28 | .. math:: |
---|
29 | |
---|
30 | \Phi(qr) &= 3 j_1(qr)/qr = 3 (\sin qr - qr \cos qr)/(qr)^3 \\ |
---|
31 | r^2 &= R_a^2e^2 + R_b^2f^2 + R_c^2g^2 \\ |
---|
32 | V &= \tfrac{4}{3} \pi R_a R_b R_c |
---|
33 | |
---|
34 | The $e$, $f$ and $g$ terms are the projections of the orientation vector on $X$, |
---|
35 | $Y$ and $Z$ respectively. Keeping the orientation fixed at the canonical |
---|
36 | axes, we can integrate over the incident direction using polar angle |
---|
37 | $-\pi/2 \le \gamma \le \pi/2$ and equatorial angle $0 \le \phi \le 2\pi$ |
---|
38 | (as defined in ref [1]), |
---|
39 | |
---|
40 | .. math:: |
---|
41 | |
---|
42 | \langle\Phi^2\rangle = \int_0^{2\pi} \int_{-\pi/2}^{\pi/2} \Phi^2(qr) |
---|
43 | \cos \gamma\,d\gamma d\phi |
---|
44 | |
---|
45 | with $e = \cos\gamma \sin\phi$, $f = \cos\gamma \cos\phi$ and $g = \sin\gamma$. |
---|
46 | A little algebra yields |
---|
47 | |
---|
48 | .. math:: |
---|
49 | |
---|
50 | r^2 = b^2(p_a \sin^2 \phi \cos^2 \gamma + 1 + p_c \sin^2 \gamma) |
---|
51 | |
---|
52 | for |
---|
53 | |
---|
54 | .. math:: |
---|
55 | |
---|
56 | p_a = \frac{a^2}{b^2} - 1 \text{ and } p_c = \frac{c^2}{b^2} - 1 |
---|
57 | |
---|
58 | Due to symmetry, the ranges can be restricted to a single quadrant |
---|
59 | $0 \le \gamma \le \pi/2$ and $0 \le \phi \le \pi/2$, scaling the resulting |
---|
60 | integral by 8. The computation is done using the substitution $u = \sin\gamma$, |
---|
61 | $du = \cos\gamma\,d\gamma$, giving |
---|
62 | |
---|
63 | .. math:: |
---|
64 | |
---|
65 | \langle\Phi^2\rangle &= 8 \int_0^{\pi/2} \int_0^1 \Phi^2(qr) du d\phi \\ |
---|
66 | r^2 &= b^2(p_a \sin^2(\phi)(1 - u^2) + 1 + p_c u^2) |
---|
67 | |
---|
68 | Though for convenience we describe the three radii of the ellipsoid as equatorial |
---|
69 | and polar, they may be given in $any$ size order. To avoid multiple solutions, especially |
---|
70 | with Monte-Carlo fit methods, it may be advisable to restrict their ranges. For typical |
---|
71 | small angle diffraction situations there may be a number of closely similar "best fits", |
---|
72 | so some trial and error, or fixing of some radii at expected values, may help. |
---|
73 | |
---|
74 | To provide easy access to the orientation of the triaxial ellipsoid, |
---|
75 | we define the axis of the cylinder using the angles $\theta$, $\phi$ |
---|
76 | and $\psi$. These angles are defined analogously to the elliptical_cylinder below, note that |
---|
77 | angle $\phi$ is now NOT the same as in the equations above. |
---|
78 | |
---|
79 | .. figure:: img/elliptical_cylinder_angle_definition.png |
---|
80 | |
---|
81 | Definition of angles for oriented triaxial ellipsoid, where radii are for illustration here |
---|
82 | $a < b << c$ and angle $\Psi$ is a rotation around the axis of the particle. |
---|
83 | |
---|
84 | For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters will appear when fitting 2D data, |
---|
85 | see the :ref:`elliptical-cylinder` model for further information. |
---|
86 | |
---|
87 | .. _triaxial-ellipsoid-angles: |
---|
88 | |
---|
89 | .. figure:: img/triaxial_ellipsoid_angle_projection.png |
---|
90 | |
---|
91 | Some examples for an oriented triaxial ellipsoid. |
---|
92 | |
---|
93 | The radius-of-gyration for this system is $R_g^2 = (R_a R_b R_c)^2/5$. |
---|
94 | |
---|
95 | The contrast $\Delta\rho$ is defined as SLD(ellipsoid) - SLD(solvent). In the |
---|
96 | parameters, $R_a$ is the minor equatorial radius, $R_b$ is the major |
---|
97 | equatorial radius, and $R_c$ is the polar radius of the ellipsoid. |
---|
98 | |
---|
99 | NB: The 2nd virial coefficient of the triaxial solid ellipsoid is |
---|
100 | calculated after sorting the three radii to give the most appropriate |
---|
101 | prolate or oblate form, from the new polar radius $R_p = R_c$ and effective equatorial |
---|
102 | radius, $R_e = \sqrt{R_a R_b}$, to then be used as the effective radius for |
---|
103 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
---|
104 | |
---|
105 | Validation |
---|
106 | ---------- |
---|
107 | |
---|
108 | Validation of our code was done by comparing the output of the |
---|
109 | 1D calculation to the angular average of the output of 2D calculation |
---|
110 | over all possible angles. |
---|
111 | |
---|
112 | |
---|
113 | References |
---|
114 | ---------- |
---|
115 | |
---|
116 | .. [#] Finnigan, J.A., Jacobs, D.J., 1971. *Light scattering by ellipsoidal particles in solution*, J. Phys. D: Appl. Phys. 4, 72-77. doi:10.1088/0022-3727/4/1/310 |
---|
117 | |
---|
118 | Source |
---|
119 | ------ |
---|
120 | |
---|
121 | `triaxial_ellipsoid.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/triaxial_ellipsoid.py>`_ |
---|
122 | |
---|
123 | `triaxial_ellipsoid.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/triaxial_ellipsoid.c>`_ |
---|
124 | |
---|
125 | Authorship and Verification |
---|
126 | ---------------------------- |
---|
127 | |
---|
128 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
---|
129 | * **Last Modified by:** Paul Kienzle (improved calculation) **Date:** April 4, 2017 |
---|
130 | * **Last Reviewed by:** Paul Kienzle & Richard Heenan **Date:** April 4, 2017 |
---|
131 | * **Source added by :** Steve King **Date:** March 25, 2019 |
---|
132 | """ |
---|
133 | |
---|
134 | import numpy as np |
---|
135 | from numpy import inf, sin, cos, pi |
---|
136 | |
---|
137 | name = "triaxial_ellipsoid" |
---|
138 | title = "Ellipsoid of uniform scattering length density with three independent axes." |
---|
139 | |
---|
140 | description = """ |
---|
141 | Triaxial ellipsoid - see main documentation. |
---|
142 | """ |
---|
143 | category = "shape:ellipsoid" |
---|
144 | |
---|
145 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
146 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
---|
147 | "Ellipsoid scattering length density"], |
---|
148 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
149 | "Solvent scattering length density"], |
---|
150 | ["radius_equat_minor", "Ang", 20, [0, inf], "volume", |
---|
151 | "Minor equatorial radius, Ra"], |
---|
152 | ["radius_equat_major", "Ang", 400, [0, inf], "volume", |
---|
153 | "Major equatorial radius, Rb"], |
---|
154 | ["radius_polar", "Ang", 10, [0, inf], "volume", |
---|
155 | "Polar radius, Rc"], |
---|
156 | ["theta", "degrees", 60, [-360, 360], "orientation", |
---|
157 | "polar axis to beam angle"], |
---|
158 | ["phi", "degrees", 60, [-360, 360], "orientation", |
---|
159 | "rotation about beam"], |
---|
160 | ["psi", "degrees", 60, [-360, 360], "orientation", |
---|
161 | "rotation about polar axis"], |
---|
162 | ] |
---|
163 | |
---|
164 | source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "triaxial_ellipsoid.c"] |
---|
165 | have_Fq = True |
---|
166 | effective_radius_type = [ |
---|
167 | "equivalent biaxial ellipsoid average curvature", |
---|
168 | "equivalent volume sphere", "min radius", "max radius", |
---|
169 | ] |
---|
170 | |
---|
171 | def random(): |
---|
172 | """Return a random parameter set for the model.""" |
---|
173 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
---|
174 | pars = dict( |
---|
175 | radius_equat_minor=a, |
---|
176 | radius_equat_major=b, |
---|
177 | radius_polar=c, |
---|
178 | ) |
---|
179 | return pars |
---|
180 | |
---|
181 | |
---|
182 | demo = dict(scale=1, background=0, |
---|
183 | sld=6, sld_solvent=1, |
---|
184 | theta=30, phi=15, psi=5, |
---|
185 | radius_equat_minor=25, radius_equat_major=36, radius_polar=50, |
---|
186 | radius_equat_minor_pd=0, radius_equat_minor_pd_n=1, |
---|
187 | radius_equat_major_pd=0, radius_equat_major_pd_n=1, |
---|
188 | radius_polar_pd=.2, radius_polar_pd_n=30, |
---|
189 | theta_pd=15, theta_pd_n=45, |
---|
190 | phi_pd=15, phi_pd_n=1, |
---|
191 | psi_pd=15, psi_pd_n=1) |
---|
192 | |
---|
193 | q = 0.1 |
---|
194 | # april 6 2017, rkh add unit tests |
---|
195 | # NOT compared with any other calc method, assume correct! |
---|
196 | # check 2d test after pull #890 |
---|
197 | qx = q*cos(pi/6.0) |
---|
198 | qy = q*sin(pi/6.0) |
---|
199 | tests = [ |
---|
200 | [{}, 0.05, 24.8839548033], |
---|
201 | [{'theta':80., 'phi':10.}, (qx, qy), 166.712060266], |
---|
202 | ] |
---|
203 | del qx, qy # not necessary to delete, but cleaner |
---|