1 | # triaxial ellipsoid model |
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2 | # Note: model title and parameter table are inserted automatically |
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3 | r""" |
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4 | Definition |
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5 | ---------- |
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6 | |
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7 | .. figure:: img/triaxial_ellipsoid_geometry.jpg |
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8 | |
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9 | Ellipsoid with $R_a$ as *radius_equat_minor*, $R_b$ as *radius_equat_major* |
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10 | and $R_c$ as *radius_polar*. |
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11 | |
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12 | Given an ellipsoid |
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13 | |
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14 | .. math:: |
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15 | |
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16 | \frac{X^2}{R_a^2} + \frac{Y^2}{R_b^2} + \frac{Z^2}{R_c^2} = 1 |
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17 | |
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18 | the scattering for randomly oriented particles is defined by the average over all orientations $\Omega$ of: |
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19 | |
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20 | .. math:: |
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21 | |
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22 | P(q) = \text{scale}(\Delta\rho)^2\frac{V}{4 \pi}\int_\Omega \Phi^2(qr) d\Omega + \text{background} |
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23 | |
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24 | where |
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25 | |
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26 | .. math:: |
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27 | |
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28 | \Phi(qr) &= 3 j_1(qr)/qr = 3 (\sin qr - qr \cos qr)/(qr)^3 \\ |
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29 | r^2 &= R_a^2e^2 + R_b^2f^2 + R_c^2g^2 \\ |
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30 | V &= \tfrac{4}{3} \pi R_a R_b R_c |
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31 | |
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32 | The $e$, $f$ and $g$ terms are the projections of the orientation vector on $X$, |
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33 | $Y$ and $Z$ respectively. Keeping the orientation fixed at the canonical |
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34 | axes, we can integrate over the incident direction using polar angle |
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35 | $-\pi/2 \le \gamma \le \pi/2$ and equatorial angle $0 \le \phi \le 2\pi$ |
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36 | (as defined in ref [1]), |
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37 | |
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38 | .. math:: |
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39 | |
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40 | \langle\Phi^2\rangle = \int_0^{2\pi} \int_{-\pi/2}^{\pi/2} \Phi^2(qr) \cos \gamma\,d\gamma d\phi |
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41 | |
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42 | with $e = \cos\gamma \sin\phi$, $f = \cos\gamma \cos\phi$ and $g = \sin\gamma$. |
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43 | A little algebra yields |
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44 | |
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45 | .. math:: |
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46 | |
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47 | r^2 = b^2(p_a \sin^2 \phi \cos^2 \gamma + 1 + p_c \sin^2 \gamma) |
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48 | |
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49 | for |
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50 | |
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51 | .. math:: |
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52 | |
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53 | p_a = \frac{a^2}{b^2} - 1 \text{ and } p_c = \frac{c^2}{b^2} - 1 |
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54 | |
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55 | Due to symmetry, the ranges can be restricted to a single quadrant |
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56 | $0 \le \gamma \le \pi/2$ and $0 \le \phi \le \pi/2$, scaling the resulting |
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57 | integral by 8. The computation is done using the substitution $u = \sin\gamma$, |
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58 | $du = \cos\gamma\,d\gamma$, giving |
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59 | |
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60 | .. math:: |
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61 | |
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62 | \langle\Phi^2\rangle &= 8 \int_0^{\pi/2} \int_0^1 \Phi^2(qr) du d\phi \\ |
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63 | r^2 &= b^2(p_a \sin^2(\phi)(1 - u^2) + 1 + p_c u^2) |
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64 | |
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65 | To provide easy access to the orientation of the triaxial ellipsoid, |
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66 | we define the axis of the cylinder using the angles $\theta$, $\phi$ |
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67 | and $\psi$. These angles are defined on |
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68 | :numref:`triaxial-ellipsoid-angles` . |
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69 | The angle $\psi$ is the rotational angle around its own $c$ axis |
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70 | against the $q$ plane. For example, $\psi = 0$ when the |
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71 | $a$ axis is parallel to the $x$ axis of the detector. |
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72 | |
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73 | .. _triaxial-ellipsoid-angles: |
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74 | |
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75 | .. figure:: img/triaxial_ellipsoid_angle_projection.jpg |
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76 | |
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77 | The angles for oriented ellipsoid. |
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78 | |
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79 | The radius-of-gyration for this system is $R_g^2 = (R_a R_b R_c)^2/5$. |
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80 | |
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81 | The contrast $\Delta\rho$ is defined as SLD(ellipsoid) - SLD(solvent). In the |
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82 | parameters, $R_a$ is the minor equatorial radius, $R_b$ is the major |
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83 | equatorial radius, and $R_c$ is the polar radius of the ellipsoid. |
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84 | |
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85 | NB: The 2nd virial coefficient of the triaxial solid ellipsoid is |
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86 | calculated based on the polar radius $R_p = R_c$ and equatorial |
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87 | radius $R_e = \sqrt{R_a R_b}$, and used as the effective radius for |
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88 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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89 | |
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90 | Validation |
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91 | ---------- |
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92 | |
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93 | Validation of our code was done by comparing the output of the |
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94 | 1D calculation to the angular average of the output of 2D calculation |
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95 | over all possible angles. |
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96 | |
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97 | |
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98 | References |
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99 | ---------- |
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100 | |
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101 | [1] Finnigan, J.A., Jacobs, D.J., 1971. |
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102 | *Light scattering by ellipsoidal particles in solution*, |
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103 | J. Phys. D: Appl. Phys. 4, 72-77. doi:10.1088/0022-3727/4/1/310 |
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104 | |
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105 | Authorship and Verification |
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106 | ---------------------------- |
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107 | |
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108 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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109 | * **Last Modified by:** Paul Kienzle (improved calculation) **Date:** April 4, 2017 |
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110 | * **Last Reviewed by:** Paul Kienzle &Richard Heenan **Date:** April 4, 2017 |
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111 | |
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112 | """ |
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113 | |
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114 | from numpy import inf |
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115 | |
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116 | name = "triaxial_ellipsoid" |
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117 | title = "Ellipsoid of uniform scattering length density with three independent axes." |
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118 | |
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119 | description = """\ |
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120 | Note: During fitting ensure that the inequality ra<rb<rc is not |
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121 | violated. Otherwise the calculation will |
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122 | not be correct. |
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123 | """ |
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124 | category = "shape:ellipsoid" |
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125 | |
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126 | # ["name", "units", default, [lower, upper], "type","description"], |
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127 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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128 | "Ellipsoid scattering length density"], |
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129 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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130 | "Solvent scattering length density"], |
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131 | ["radius_equat_minor", "Ang", 20, [0, inf], "volume", |
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132 | "Minor equatorial radius, Ra"], |
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133 | ["radius_equat_major", "Ang", 400, [0, inf], "volume", |
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134 | "Major equatorial radius, Rb"], |
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135 | ["radius_polar", "Ang", 10, [0, inf], "volume", |
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136 | "Polar radius, Rc"], |
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137 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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138 | "In plane angle"], |
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139 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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140 | "Out of plane angle"], |
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141 | ["psi", "degrees", 60, [-inf, inf], "orientation", |
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142 | "Out of plane angle"], |
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143 | ] |
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144 | |
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145 | source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "triaxial_ellipsoid.c"] |
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146 | |
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147 | def ER(radius_equat_minor, radius_equat_major, radius_polar): |
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148 | """ |
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149 | Returns the effective radius used in the S*P calculation |
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150 | """ |
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151 | import numpy as np |
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152 | from .ellipsoid import ER as ellipsoid_ER |
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153 | # now that radii can be in any size order, radii need sorting a,b,c where a~b and c is either much smaller or much larger |
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154 | # also need some unit tests! |
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155 | |
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156 | return ellipsoid_ER(radius_polar, np.sqrt(radius_equat_minor * radius_equat_major)) |
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157 | |
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158 | demo = dict(scale=1, background=0, |
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159 | sld=6, sld_solvent=1, |
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160 | theta=30, phi=15, psi=5, |
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161 | radius_equat_minor=25, radius_equat_major=36, radius_polar=50, |
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162 | radius_equat_minor_pd=0, radius_equat_minor_pd_n=1, |
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163 | radius_equat_major_pd=0, radius_equat_major_pd_n=1, |
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164 | radius_polar_pd=.2, radius_polar_pd_n=30, |
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165 | theta_pd=15, theta_pd_n=45, |
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166 | phi_pd=15, phi_pd_n=1, |
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167 | psi_pd=15, psi_pd_n=1) |
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