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 | All three axes are of different lengths with $R_a \leq R_b \leq R_c$ |
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5 | **Users should maintain this inequality for all calculations**. |
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6 | |
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7 | .. math:: |
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8 | |
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9 | P(q) = \text{scale} V \left< F^2(q) \right> + \text{background} |
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10 | |
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11 | where the volume $V = 4/3 \pi R_a R_b R_c$, and the averaging |
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12 | $\left<\ldots\right>$ is applied over all orientations for 1D. |
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13 | |
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14 | .. figure:: img/triaxial_ellipsoid_geometry.jpg |
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15 | |
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16 | Ellipsoid schematic. |
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17 | |
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18 | Definition |
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19 | ---------- |
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20 | |
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21 | The form factor calculated is |
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22 | |
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23 | .. math:: |
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24 | |
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25 | P(q) = \frac{\text{scale}}{V}\int_0^1\int_0^1 |
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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) |
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27 | dx dy |
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28 | |
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29 | where |
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30 | |
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31 | .. math:: |
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32 | |
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33 | \Phi(u) = 3 u^{-3} (\sin u - u \cos u) |
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34 | |
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35 | To provide easy access to the orientation of the triaxial ellipsoid, |
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36 | we define the axis of the cylinder using the angles $\theta$, $\phi$ |
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37 | and $\psi$. These angles are defined on |
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38 | :num:`figure #triaxial-ellipsoid-angles`. |
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39 | The angle $\psi$ is the rotational angle around its own $c$ axis |
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40 | against the $q$ plane. For example, $\psi = 0$ when the |
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41 | $a$ axis is parallel to the $x$ axis of the detector. |
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42 | |
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43 | .. _triaxial-ellipsoid-angles: |
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44 | |
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45 | .. figure:: img/triaxial_ellipsoid_angles.jpg |
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46 | |
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47 | The angles for oriented ellipsoid. |
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48 | |
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49 | The radius-of-gyration for this system is $R_g^2 = (R_a R_b R_c)^2/5$. |
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50 | |
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51 | The contrast is defined as SLD(ellipsoid) - SLD(solvent). In the |
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52 | parameters, $R_a$ is the minor equatorial radius, $R_b$ is the major |
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53 | equatorial radius, and $R_c$ is the polar radius of the ellipsoid. |
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54 | |
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55 | NB: The 2nd virial coefficient of the triaxial solid ellipsoid is |
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56 | calculated based on the polar radius $R_p = R_c$ and equatorial |
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57 | radius $R_e = \sqrt{R_a R_b}$, and used as the effective radius for |
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58 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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59 | |
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60 | .. figure:: img/triaxial_ellipsoid_1d.jpg |
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61 | |
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62 | 1D plot using the default values (w/1000 data point). |
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63 | |
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64 | Validation |
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65 | ---------- |
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66 | |
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67 | Validation of our code was done by comparing the output of the |
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68 | 1D calculation to the angular average of the output of 2D calculation |
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69 | over all possible angles. |
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70 | :num:`Figure #triaxial-ellipsoid-comparison` shows the comparison where |
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71 | the solid dot refers to averaged 2D while the line represents the |
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72 | result of 1D calculation (for 2D averaging, 76, 180, and 76 points |
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73 | are taken for the angles of $\theta$, $\phi$, and $\psi$ respectively). |
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74 | |
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75 | .. _triaxial-ellipsoid-comparison: |
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76 | |
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77 | .. figure:: img/triaxial_ellipsoid_comparison.png |
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78 | |
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79 | Comparison between 1D and averaged 2D. |
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80 | |
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81 | References |
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82 | ---------- |
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83 | |
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84 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray |
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85 | and Neutron Scattering*, Plenum, New York, 1987. |
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86 | """ |
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87 | |
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88 | from numpy import inf |
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89 | |
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90 | name = "triaxial_ellipsoid" |
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91 | title = "Ellipsoid of uniform scattering length density with three independent axes." |
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92 | |
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93 | description = """\ |
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94 | Note: During fitting ensure that the inequality ra<rb<rc is not |
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95 | violated. Otherwise the calculation will |
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96 | not be correct. |
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97 | """ |
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98 | category = "shape:ellipsoid" |
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99 | |
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100 | # ["name", "units", default, [lower, upper], "type","description"], |
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101 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "", |
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102 | "Ellipsoid scattering length density"], |
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103 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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104 | "Solvent scattering length density"], |
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105 | ["req_minor", "Ang", 20, [0, inf], "volume", |
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106 | "Minor equitorial radius"], |
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107 | ["req_major", "Ang", 400, [0, inf], "volume", |
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108 | "Major equatorial radius"], |
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109 | ["rpolar", "Ang", 10, [0, inf], "volume", |
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110 | "Polar radius"], |
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111 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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112 | "In plane angle"], |
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113 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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114 | "Out of plane angle"], |
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115 | ["psi", "degrees", 60, [-inf, inf], "orientation", |
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116 | "Out of plane angle"], |
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117 | ] |
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118 | |
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119 | source = ["lib/J1.c", "lib/sph_j1c.c", "lib/gauss76.c", "triaxial_ellipsoid.c"] |
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120 | |
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121 | def ER(req_minor, req_major, rpolar): |
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122 | """ |
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123 | Returns the effective radius used in the S*P calculation |
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124 | """ |
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125 | import numpy as np |
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126 | from .ellipsoid import ER as ellipsoid_ER |
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127 | return ellipsoid_ER(rpolar, np.sqrt(req_minor * req_major)) |
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128 | |
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129 | demo = dict(scale=1, background=0, |
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130 | sld=6, solvent_sld=1, |
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131 | theta=30, phi=15, psi=5, |
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132 | req_minor=25, req_major=36, rpolar=50, |
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133 | req_minor_pd=0, req_minor_pd_n=1, |
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134 | req_major_pd=0, req_major_pd_n=1, |
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135 | rpolar_pd=.2, rpolar_pd_n=30, |
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136 | theta_pd=15, theta_pd_n=45, |
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137 | phi_pd=15, phi_pd_n=1, |
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138 | psi_pd=15, psi_pd_n=1) |
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139 | oldname = 'TriaxialEllipsoidModel' |
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140 | oldpars = dict(theta='axis_theta', phi='axis_phi', psi='axis_psi', |
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141 | sld='sldEll', solvent_sld='sldSolv', |
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142 | req_minor='semi_axisA', req_major='semi_axisB', |
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143 | rpolar='semi_axisC') |
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