[1f65db5] | 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 | |
---|
[afd4692] | 18 | the scattering for randomly oriented particles is defined by the average over |
---|
| 19 | all orientations $\Omega$ of: |
---|
[1f65db5] | 20 | |
---|
| 21 | .. math:: |
---|
| 22 | |
---|
[afd4692] | 23 | P(q) = \text{scale}(\Delta\rho)^2\frac{V}{4 \pi}\int_\Omega\Phi^2(qr)\,d\Omega |
---|
| 24 | + \text{background} |
---|
[1f65db5] | 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 | |
---|
[afd4692] | 42 | \langle\Phi^2\rangle = \int_0^{2\pi} \int_{-\pi/2}^{\pi/2} \Phi^2(qr) |
---|
| 43 | \cos \gamma\,d\gamma d\phi |
---|
[1f65db5] | 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 | |
---|
[34a9e4e] | 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. |
---|
[31df0c9] | 73 | |
---|
[1f65db5] | 74 | To provide easy access to the orientation of the triaxial ellipsoid, |
---|
| 75 | we define the axis of the cylinder using the angles $\theta$, $\phi$ |
---|
[34a9e4e] | 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. |
---|
[1f65db5] | 78 | |
---|
| 79 | .. figure:: img/elliptical_cylinder_angle_definition.png |
---|
| 80 | |
---|
[31df0c9] | 81 | Definition of angles for oriented triaxial ellipsoid, where radii are for illustration here |
---|
[34a9e4e] | 82 | $a < b << c$ and angle $\Psi$ is a rotation around the axis of the particle. |
---|
[1f65db5] | 83 | |
---|
[31df0c9] | 84 | For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters will appear when fitting 2D data, |
---|
[9802ab3] | 85 | see the :ref:`elliptical-cylinder` model for further information. |
---|
[1f65db5] | 86 | |
---|
| 87 | .. _triaxial-ellipsoid-angles: |
---|
| 88 | |
---|
| 89 | .. figure:: img/triaxial_ellipsoid_angle_projection.png |
---|
| 90 | |
---|
[34a9e4e] | 91 | Some examples for an oriented triaxial ellipsoid. |
---|
[1f65db5] | 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 |
---|
[34a9e4e] | 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 |
---|
[1f65db5] | 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 | [1] Finnigan, J.A., Jacobs, D.J., 1971. |
---|
| 117 | *Light scattering by ellipsoidal particles in solution*, |
---|
| 118 | J. Phys. D: Appl. Phys. 4, 72-77. doi:10.1088/0022-3727/4/1/310 |
---|
| 119 | |
---|
| 120 | Authorship and Verification |
---|
| 121 | ---------------------------- |
---|
| 122 | |
---|
| 123 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
---|
| 124 | * **Last Modified by:** Paul Kienzle (improved calculation) **Date:** April 4, 2017 |
---|
[3fd0499] | 125 | * **Last Reviewed by:** Paul Kienzle & Richard Heenan **Date:** April 4, 2017 |
---|
[3401a7a] | 126 | """ |
---|
[1f65db5] | 127 | |
---|
| 128 | from numpy import inf, sin, cos, pi |
---|
| 129 | |
---|
| 130 | name = "triaxial_ellipsoid" |
---|
| 131 | title = "Ellipsoid of uniform scattering length density with three independent axes." |
---|
| 132 | |
---|
[afd4692] | 133 | description = """ |
---|
[34a9e4e] | 134 | Triaxial ellipsoid - see main documentation. |
---|
[1f65db5] | 135 | """ |
---|
| 136 | category = "shape:ellipsoid" |
---|
| 137 | |
---|
| 138 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
| 139 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
---|
| 140 | "Ellipsoid scattering length density"], |
---|
| 141 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
| 142 | "Solvent scattering length density"], |
---|
| 143 | ["radius_equat_minor", "Ang", 20, [0, inf], "volume", |
---|
| 144 | "Minor equatorial radius, Ra"], |
---|
| 145 | ["radius_equat_major", "Ang", 400, [0, inf], "volume", |
---|
| 146 | "Major equatorial radius, Rb"], |
---|
| 147 | ["radius_polar", "Ang", 10, [0, inf], "volume", |
---|
| 148 | "Polar radius, Rc"], |
---|
[9b79f29] | 149 | ["theta", "degrees", 60, [-360, 360], "orientation", |
---|
| 150 | "polar axis to beam angle"], |
---|
| 151 | ["phi", "degrees", 60, [-360, 360], "orientation", |
---|
| 152 | "rotation about beam"], |
---|
| 153 | ["psi", "degrees", 60, [-360, 360], "orientation", |
---|
| 154 | "rotation about polar axis"], |
---|
[1f65db5] | 155 | ] |
---|
| 156 | |
---|
| 157 | source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "triaxial_ellipsoid.c"] |
---|
| 158 | |
---|
| 159 | def ER(radius_equat_minor, radius_equat_major, radius_polar): |
---|
| 160 | """ |
---|
| 161 | Returns the effective radius used in the S*P calculation |
---|
| 162 | """ |
---|
| 163 | import numpy as np |
---|
| 164 | from .ellipsoid import ER as ellipsoid_ER |
---|
| 165 | |
---|
[afd4692] | 166 | # now that radii can be in any size order, radii need sorting a,b,c |
---|
| 167 | # where a~b and c is either much smaller or much larger |
---|
[1f65db5] | 168 | radii = np.vstack((radius_equat_major, radius_equat_minor, radius_polar)) |
---|
| 169 | radii = np.sort(radii, axis=0) |
---|
| 170 | selector = (radii[1] - radii[0]) > (radii[2] - radii[1]) |
---|
| 171 | polar = np.where(selector, radii[0], radii[2]) |
---|
| 172 | equatorial = np.sqrt(np.where(~selector, radii[0]*radii[1], radii[1]*radii[2])) |
---|
| 173 | return ellipsoid_ER(polar, equatorial) |
---|
| 174 | |
---|
[31df0c9] | 175 | def random(): |
---|
| 176 | import numpy as np |
---|
| 177 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
---|
| 178 | pars = dict( |
---|
| 179 | radius_equat_minor=a, |
---|
| 180 | radius_equat_major=b, |
---|
| 181 | radius_polar=c, |
---|
| 182 | ) |
---|
| 183 | return pars |
---|
| 184 | |
---|
| 185 | |
---|
[1f65db5] | 186 | demo = dict(scale=1, background=0, |
---|
| 187 | sld=6, sld_solvent=1, |
---|
| 188 | theta=30, phi=15, psi=5, |
---|
| 189 | radius_equat_minor=25, radius_equat_major=36, radius_polar=50, |
---|
| 190 | radius_equat_minor_pd=0, radius_equat_minor_pd_n=1, |
---|
| 191 | radius_equat_major_pd=0, radius_equat_major_pd_n=1, |
---|
| 192 | radius_polar_pd=.2, radius_polar_pd_n=30, |
---|
| 193 | theta_pd=15, theta_pd_n=45, |
---|
| 194 | phi_pd=15, phi_pd_n=1, |
---|
| 195 | psi_pd=15, psi_pd_n=1) |
---|
| 196 | |
---|
| 197 | q = 0.1 |
---|
[afd4692] | 198 | # april 6 2017, rkh add unit tests |
---|
| 199 | # NOT compared with any other calc method, assume correct! |
---|
[e645373] | 200 | # check 2d test after pull #890 |
---|
[1f65db5] | 201 | qx = q*cos(pi/6.0) |
---|
| 202 | qy = q*sin(pi/6.0) |
---|
| 203 | tests = [[{}, 0.05, 24.8839548033], |
---|
[e645373] | 204 | [{'theta':80., 'phi':10.}, (qx, qy), 166.712060266 ], |
---|
[1f65db5] | 205 | ] |
---|
| 206 | del qx, qy # not necessary to delete, but cleaner |
---|