1 | # rectangular_prism 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 | |
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5 | This model provides the form factor, $P(q)$, for a rectangular prism. |
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6 | |
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7 | Note that this model is almost totally equivalent to the existing |
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8 | :ref:`parallelepiped` model. |
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9 | The only difference is that the way the relevant |
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10 | parameters are defined here ($a$, $b/a$, $c/a$ instead of $a$, $b$, $c$) |
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11 | which allows use of polydispersity with this model while keeping the shape of |
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12 | the prism (e.g. setting $b/a = 1$ and $c/a = 1$ and applying polydispersity |
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13 | to *a* will generate a distribution of cubes of different sizes). |
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14 | |
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15 | |
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16 | Definition |
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17 | ---------- |
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18 | |
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19 | The 1D scattering intensity for this model was calculated by Mittelbach and |
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20 | Porod (Mittelbach, 1961), but the implementation here is closer to the |
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21 | equations given by Nayuk and Huber (Nayuk, 2012). |
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22 | Note also that the angle definitions used in the code and the present |
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23 | documentation correspond to those used in (Nayuk, 2012) (see Fig. 1 of |
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24 | that reference), with $\theta$ corresponding to $\alpha$ in that paper, |
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25 | and not to the usual convention used for example in the |
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26 | :ref:`parallelepiped` model. |
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27 | |
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28 | In this model the scattering from a massive parallelepiped with an |
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29 | orientation with respect to the scattering vector given by $\theta$ |
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30 | and $\phi$ |
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31 | |
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32 | .. math:: |
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33 | |
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34 | A_P\,(q) = |
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35 | \frac{\sin \left( \tfrac{1}{2}qC \cos\theta \right) }{\tfrac{1}{2} qC \cos\theta} |
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36 | \,\times\, |
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37 | \frac{\sin \left( \tfrac{1}{2}qA \cos\theta \right) }{\tfrac{1}{2} qA \cos\theta} |
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38 | \,\times\ , |
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39 | \frac{\sin \left( \tfrac{1}{2}qB \cos\theta \right) }{\tfrac{1}{2} qB \cos\theta} |
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40 | |
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41 | where $A$, $B$ and $C$ are the sides of the parallelepiped and must fulfill |
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42 | $A \le B \le C$, $\theta$ is the angle between the $z$ axis and the |
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43 | longest axis of the parallelepiped $C$, and $\phi$ is the angle between the |
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44 | scattering vector (lying in the $xy$ plane) and the $y$ axis. |
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45 | |
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46 | The normalized form factor in 1D is obtained averaging over all possible |
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47 | orientations |
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48 | |
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49 | .. math:: |
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50 | P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \, |
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51 | \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi |
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52 | |
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53 | And the 1D scattering intensity is calculated as |
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54 | |
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55 | .. math:: |
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56 | I(q) = \text{scale} \times V \times (\rho_\text{p} - |
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57 | \rho_\text{solvent})^2 \times P(q) |
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58 | |
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59 | where $V$ is the volume of the rectangular prism, $\rho_\text{p}$ |
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60 | is the scattering length of the parallelepiped, $\rho_\text{solvent}$ |
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61 | is the scattering length of the solvent, and (if the data are in absolute |
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62 | units) *scale* represents the volume fraction (which is unitless). |
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63 | |
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64 | For 2d data the orientation of the particle is required, described using |
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65 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details |
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66 | of the calculation and angular dispersions see :ref:`orientation` . |
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67 | The angle $\Psi$ is the rotational angle around the long *C* axis. For example, |
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68 | $\Psi = 0$ when the *B* axis is parallel to the *x*-axis of the detector. |
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69 | |
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70 | For 2d, constraints must be applied during fitting to ensure that the inequality |
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71 | $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, |
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72 | but the results may be not correct. |
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73 | |
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74 | .. figure:: img/parallelepiped_angle_definition.png |
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75 | |
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76 | Definition of the angles for oriented core-shell parallelepipeds. |
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77 | Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then |
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78 | rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the cylinder. |
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79 | The neutron or X-ray beam is along the $z$ axis. |
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80 | |
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81 | .. figure:: img/parallelepiped_angle_projection.png |
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82 | |
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83 | Examples of the angles for oriented rectangular prisms against the |
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84 | detector plane. |
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85 | |
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86 | |
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87 | |
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88 | Validation |
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89 | ---------- |
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90 | |
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91 | Validation of the code was conducted by comparing the output of the 1D model |
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92 | to the output of the existing :ref:`parallelepiped` model. |
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93 | |
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94 | |
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95 | References |
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96 | ---------- |
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97 | |
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98 | P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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99 | |
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100 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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101 | |
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102 | """ |
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103 | |
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104 | from numpy import pi, inf, sqrt |
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105 | |
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106 | name = "rectangular_prism" |
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107 | title = "Rectangular parallelepiped with uniform scattering length density." |
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108 | description = """ |
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109 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background |
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110 | P(q,theta,phi) = (2/pi) * double integral from 0 to pi/2 of ... |
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111 | AP^2(q)*sin(theta)*dtheta*dphi |
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112 | AP = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2) |
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113 | S(x) = sin(x)/x |
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114 | """ |
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115 | category = "shape:parallelepiped" |
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116 | |
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117 | # ["name", "units", default, [lower, upper], "type","description"], |
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118 | parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
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119 | "Parallelepiped scattering length density"], |
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120 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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121 | "Solvent scattering length density"], |
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122 | ["length_a", "Ang", 35, [0, inf], "volume", |
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123 | "Shorter side of the parallelepiped"], |
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124 | ["b2a_ratio", "", 1, [0, inf], "volume", |
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125 | "Ratio sides b/a"], |
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126 | ["c2a_ratio", "", 1, [0, inf], "volume", |
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127 | "Ratio sides c/a"], |
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128 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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129 | "c axis to beam angle"], |
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130 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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131 | "rotation about beam"], |
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132 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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133 | "rotation about c axis"], |
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134 | ] |
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135 | |
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136 | source = ["lib/gauss76.c", "rectangular_prism.c"] |
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137 | |
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138 | def ER(length_a, b2a_ratio, c2a_ratio): |
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139 | """ |
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140 | Return equivalent radius (ER) |
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141 | """ |
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142 | b_side = length_a * b2a_ratio |
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143 | c_side = length_a * c2a_ratio |
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144 | |
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145 | # surface average radius (rough approximation) |
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146 | surf_rad = sqrt(length_a * b_side / pi) |
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147 | |
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148 | ddd = 0.75 * surf_rad * (2 * surf_rad * c_side + (c_side + surf_rad) * (c_side + pi * surf_rad)) |
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149 | return 0.5 * (ddd) ** (1. / 3.) |
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150 | |
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151 | def random(): |
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152 | import numpy as np |
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153 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
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154 | pars = dict( |
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155 | length_a=a, |
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156 | b2a_ratio=b/a, |
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157 | c2a_ratio=c/a, |
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158 | ) |
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159 | return pars |
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160 | |
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161 | # parameters for demo |
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162 | demo = dict(scale=1, background=0, |
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163 | sld=6.3, sld_solvent=1.0, |
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164 | length_a=35, b2a_ratio=1, c2a_ratio=1, |
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165 | length_a_pd=0.1, length_a_pd_n=10, |
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166 | b2a_ratio_pd=0.1, b2a_ratio_pd_n=1, |
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167 | c2a_ratio_pd=0.1, c2a_ratio_pd_n=1) |
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168 | |
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169 | tests = [[{}, 0.2, 0.375248406825], |
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170 | [{}, [0.2], [0.375248406825]], |
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171 | ] |
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