1 | r""" |
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2 | .. warning:: This model and this model description are under review following |
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3 | concerns raised by SasView users. If you need to use this model, |
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4 | please email help@sasview.org for the latest situation. *The |
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5 | SasView Developers. September 2018.* |
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6 | |
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7 | Definition |
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8 | ---------- |
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9 | |
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10 | Calculates the scattering from a **simple cubic lattice** with |
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11 | paracrystalline distortion. Thermal vibrations are considered to be |
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12 | negligible, and the size of the paracrystal is infinitely large. |
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13 | Paracrystalline distortion is assumed to be isotropic and characterized |
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14 | by a Gaussian distribution. |
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15 | |
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16 | The scattering intensity $I(q)$ is calculated as |
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17 | |
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18 | .. math:: |
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19 | |
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20 | I(q) = \text{scale}\frac{V_\text{lattice}P(q)Z(q)}{V_p} + \text{background} |
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21 | |
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22 | where scale is the volume fraction of spheres, $V_p$ is the volume of |
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23 | the primary particle, $V_\text{lattice}$ is a volume correction for the crystal |
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24 | structure, $P(q)$ is the form factor of the sphere (normalized), and |
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25 | $Z(q)$ is the paracrystalline structure factor for a simple cubic structure. |
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26 | |
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27 | Equation (16) of the 1987 reference\ [#CIT1987]_ is used to calculate $Z(q)$, |
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28 | using equations (13)-(15) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and |
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29 | $Z3$. |
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30 | |
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31 | The lattice correction (the occupied volume of the lattice) for a simple cubic |
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32 | structure of particles of radius *R* and nearest neighbor separation *D* is |
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33 | |
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34 | .. math:: |
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35 | |
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36 | V_\text{lattice}=\frac{4\pi}{3}\frac{R^3}{D^3} |
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37 | |
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38 | The distortion factor (one standard deviation) of the paracrystal is included |
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39 | in the calculation of $Z(q)$ |
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40 | |
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41 | .. math:: |
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42 | |
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43 | \Delta a = gD |
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44 | |
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45 | where *g* is a fractional distortion based on the nearest neighbor distance. |
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46 | |
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47 | The simple cubic lattice is |
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48 | |
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49 | .. figure:: img/sc_crystal_geometry.jpg |
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50 | |
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51 | For a crystal, diffraction peaks appear at reduced q-values given by |
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52 | |
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53 | .. math:: |
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54 | |
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55 | \frac{qD}{2\pi} = \sqrt{h^2+k^2+l^2} |
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56 | |
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57 | where for a simple cubic lattice any h, k, l are allowed and none are |
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58 | forbidden. Thus the peak positions correspond to (just the first 5) |
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59 | |
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60 | .. math:: |
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61 | :nowrap: |
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62 | |
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63 | \begin{align*} |
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64 | q/q_0 \quad & \quad 1 |
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65 | & \sqrt{2} \quad |
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66 | & \quad \sqrt{3} \quad |
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67 | & \sqrt{4} \quad |
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68 | & \quad \sqrt{5}\quad \\ |
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69 | Indices \quad & (100) |
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70 | & \quad (110) \quad |
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71 | & \quad (111) |
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72 | & (200) \quad |
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73 | & \quad (210) |
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74 | \end{align*} |
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75 | |
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76 | .. note:: |
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77 | |
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78 | The calculation of *Z(q)* is a double numerical integral that must be |
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79 | carried out with a high density of points to properly capture the sharp |
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80 | peaks of the paracrystalline scattering. |
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81 | So be warned that the calculation is slow. Fitting of any experimental data |
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82 | must be resolution smeared for any meaningful fit. This makes a triple |
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83 | integral which may be very slow. |
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84 | |
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85 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is |
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86 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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87 | be accurate particularly at low $q$. For general details of the calculation |
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88 | and angular dispersions for oriented particles see :ref:`orientation` . |
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89 | Note that we are not responsible for any incorrectness of the |
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90 | 2D model computation. |
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91 | |
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92 | .. figure:: img/parallelepiped_angle_definition.png |
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93 | |
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94 | Orientation of the crystal with respect to the scattering plane, when |
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95 | $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis). |
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96 | |
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97 | Reference |
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98 | --------- |
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99 | |
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100 | .. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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101 | (Original Paper) |
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102 | .. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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103 | (Corrections to FCC and BCC lattice structure calculation) |
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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 Butler **Date:** September 16, 2018 |
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110 | * **Last Reviewed by:** Paul Butler **Date:** September 16, 2018 |
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111 | """ |
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112 | |
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113 | import numpy as np |
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114 | from numpy import inf |
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115 | |
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116 | name = "sc_paracrystal" |
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117 | title = "Simple cubic lattice with paracrystalline distortion" |
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118 | description = """ |
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119 | P(q)=(scale/Vp)*V_lattice*P(q)*Z(q)+bkg where scale is the volume |
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120 | fraction of sphere, |
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121 | Vp = volume of the primary particle, |
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122 | V_lattice = volume correction for |
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123 | for the crystal structure, |
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124 | P(q)= form factor of the sphere (normalized), |
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125 | Z(q)= paracrystalline structure factor |
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126 | for a simple cubic structure. |
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127 | [Simple Cubic ParaCrystal Model] |
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128 | Parameters; |
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129 | scale: volume fraction of spheres |
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130 | bkg:background, R: radius of sphere |
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131 | lattice_spacing: Nearest neighbor distance |
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132 | lattice_distortion: Paracrystal distortion factor |
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133 | radius: radius of the spheres |
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134 | sldSph: SLD of the sphere |
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135 | sldSolv: SLD of the solvent |
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136 | """ |
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137 | category = "shape:paracrystal" |
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138 | single = False |
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139 | # pylint: disable=bad-whitespace, line-too-long |
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140 | # ["name", "units", default, [lower, upper], "type","description"], |
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141 | parameters = [["lattice_spacing", "Ang", 220.0, [0.0, inf], "", "Lattice spacing"], |
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142 | ["lattice_distortion", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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143 | ["radius", "Ang", 40.0, [0.0, inf], "volume", "Radius of sphere"], |
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144 | ["sld", "1e-6/Ang^2", 3.0, [0.0, inf], "sld", "Sphere scattering length density"], |
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145 | ["sld_solvent", "1e-6/Ang^2", 6.3, [0.0, inf], "sld", "Solvent scattering length density"], |
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146 | ["theta", "degrees", 0, [-360, 360], "orientation", "c axis to beam angle"], |
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147 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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148 | ["psi", "degrees", 0, [-360, 360], "orientation", "rotation about c axis"] |
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149 | ] |
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150 | # pylint: enable=bad-whitespace, line-too-long |
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151 | |
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152 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c", "lib/gauss150.c", "sc_paracrystal.c"] |
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153 | |
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154 | def random(): |
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155 | # copied from bcc_paracrystal |
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156 | radius = 10**np.random.uniform(1.3, 4) |
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157 | lattice_distortion = 10**np.random.uniform(-2, -0.7) # sigma_d in 0.01-0.7 |
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158 | lattice_spacing_fraction = np.random.beta(a=10, b=1) |
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159 | lattice_spacing = radius*4/np.sqrt(4)/lattice_spacing_fraction |
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160 | pars = dict( |
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161 | #sld=1, sld_solvent=0, scale=1, background=1e-32, |
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162 | lattice_spacing=lattice_spacing, |
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163 | lattice_distortion=lattice_distortion, |
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164 | radius=radius, |
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165 | ) |
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166 | return pars |
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167 | |
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168 | tests = [ |
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169 | # Accuracy tests based on content in test/utest_extra_models.py, 2d tests added April 10, 2017 |
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170 | [{}, 0.001, 10.3048], |
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171 | [{}, 0.215268, 0.00814889], |
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172 | [{}, 0.414467, 0.001313289], |
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173 | [{'theta': 10.0, 'phi': 20, 'psi': 30.0}, (0.045, -0.035), 18.0397138402], |
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174 | [{'theta': 10.0, 'phi': 20, 'psi': 30.0}, (0.023, 0.045), 0.0177333171285], |
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175 | ] |
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