1 | # cylinder 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 | The form factor is normalized by the particle volume. |
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5 | |
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6 | Definition |
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7 | ---------- |
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8 | |
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9 | The output of the 2D scattering intensity function for oriented cylinders is |
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10 | given by (Guinier, 1955) |
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11 | |
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12 | .. math:: |
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13 | |
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14 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q) + \text{background} |
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15 | |
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16 | where |
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17 | |
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18 | .. math:: |
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19 | |
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20 | F(q) = 2 (\Delta \rho) V |
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21 | \frac{\sin \left(q\tfrac12 L\cos\alpha \right)} |
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22 | {q\tfrac12 L \cos \alpha} |
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23 | \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha} |
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24 | |
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25 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V$ |
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26 | is the volume of the cylinder, $L$ is the length of the cylinder, $R$ is the |
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27 | radius of the cylinder, and $\Delta\rho$ (contrast) is the scattering length |
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28 | density difference between the scatterer and the solvent. $J_1$ is the |
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29 | first order Bessel function. |
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30 | |
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31 | To provide easy access to the orientation of the cylinder, we define the |
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32 | axis of the cylinder using two angles $\theta$ and $\phi$. Those angles |
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33 | are defined in :num:`figure #cylinder-orientation`. |
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34 | |
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35 | .. _cylinder-orientation: |
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36 | |
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37 | .. figure:: img/orientation.jpg |
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38 | |
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39 | Definition of the angles for oriented cylinders. |
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40 | |
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41 | .. figure:: img/orientation2.jpg |
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42 | |
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43 | Examples of the angles for oriented cylinders against the detector plane. |
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44 | |
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45 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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46 | radius and length values, and used as the effective radius for $S(q)$ |
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47 | when $P(q) \cdot S(q)$ is applied. |
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48 | |
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49 | The output of the 1D scattering intensity function for randomly oriented |
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50 | cylinders is then given by |
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51 | |
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52 | .. math:: |
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53 | |
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54 | P(q) = \frac{\text{scale}}{V} |
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55 | \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background} |
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56 | |
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57 | The $\theta$ and $\phi$ parameters are not used for the 1D output. |
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58 | |
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59 | Validation |
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60 | ---------- |
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61 | |
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62 | Validation of our code was done by comparing the output of the 1D model |
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63 | to the output of the software provided by the NIST (Kline, 2006). |
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64 | :num:`Figure #cylinder-compare` shows a comparison of |
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65 | the 1D output of our model and the output of the NIST software. |
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66 | |
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67 | .. _cylinder-compare: |
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68 | |
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69 | .. figure:: img/cylinder_compare.jpg |
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70 | |
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71 | Comparison of the SasView scattering intensity for a cylinder with the |
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72 | output of the NIST SANS analysis software. |
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73 | The parameters were set to: *scale* = 1.0, *radius* = 20 |Ang|, |
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74 | *length* = 400 |Ang|, *contrast* = 3e-6 |Ang^-2|, and |
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75 | *background* = 0.01 |cm^-1|. |
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76 | |
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77 | In general, averaging over a distribution of orientations is done by |
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78 | evaluating the following |
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79 | |
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80 | .. math:: |
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81 | |
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82 | P(q) = \int_0^{\pi/2} d\phi |
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83 | \int_0^\pi p(\theta, \phi) P_0(q,\alpha) \sin \theta\ d\theta |
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84 | |
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85 | |
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86 | where $p(\theta,\phi)$ is the probability distribution for the orientation |
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87 | and $P_0(q,\alpha)$ is the scattering intensity for the fully oriented |
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88 | system. Since we have no other software to compare the implementation of |
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89 | the intensity for fully oriented cylinders, we can compare the result of |
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90 | averaging our 2D output using a uniform distribution $p(\theta, \phi) = 1.0$. |
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91 | :num:`Figure #cylinder-crosscheck` shows the result of |
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92 | such a cross-check. |
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93 | |
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94 | .. _cylinder-crosscheck: |
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95 | |
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96 | .. figure:: img/cylinder_crosscheck.jpg |
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97 | |
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98 | Comparison of the intensity for uniformly distributed cylinders |
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99 | calculated from our 2D model and the intensity from the NIST SANS |
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100 | analysis software. |
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101 | The parameters used were: *scale* = 1.0, *radius* = 20 |Ang|, |
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102 | *length* = 400 |Ang|, *contrast* = 3e-6 |Ang^-2|, and |
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103 | *background* = 0.0 |cm^-1|. |
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104 | """ |
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105 | |
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106 | import numpy as np |
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107 | from numpy import pi, inf |
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108 | |
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109 | name = "cylinder" |
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110 | title = "Right circular cylinder with uniform scattering length density." |
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111 | description = """ |
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112 | f(q,alpha) = 2*(sld - solvent_sld)*V*sin(qLcos(alpha/2)) |
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113 | /[qLcos(alpha/2)]*J1(qRsin(alpha/2))/[qRsin(alpha)] |
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114 | |
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115 | P(q,alpha)= scale/V*f(q,alpha)^(2)+background |
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116 | V: Volume of the cylinder |
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117 | R: Radius of the cylinder |
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118 | L: Length of the cylinder |
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119 | J1: The bessel function |
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120 | alpha: angle between the axis of the |
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121 | cylinder and the q-vector for 1D |
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122 | :the ouput is P(q)=scale/V*integral |
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123 | from pi/2 to zero of... |
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124 | f(q,alpha)^(2)*sin(alpha)*dalpha + background |
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125 | """ |
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126 | category = "shape:cylinder" |
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127 | |
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128 | # [ "name", "units", default, [lower, upper], "type", "description"], |
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129 | parameters = [["sld", "4e-6/Ang^2", 4, [-inf, inf], "", |
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130 | "Cylinder scattering length density"], |
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131 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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132 | "Solvent scattering length density"], |
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133 | ["radius", "Ang", 20, [0, inf], "volume", |
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134 | "Cylinder radius"], |
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135 | ["length", "Ang", 400, [0, inf], "volume", |
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136 | "Cylinder length"], |
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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 | ] |
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142 | |
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143 | source = ["lib/J1c.c", "lib/gauss76.c", "cylinder.c"] |
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144 | |
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145 | def ER(radius, length): |
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146 | """ |
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147 | Return equivalent radius (ER) |
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148 | """ |
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149 | ddd = 0.75 * radius * (2 * radius * length + (length + radius) * (length + pi * radius)) |
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150 | return 0.5 * (ddd) ** (1. / 3.) |
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151 | |
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152 | # parameters for demo |
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153 | demo = dict(scale=1, background=0, |
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154 | sld=6, solvent_sld=1, |
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155 | radius=20, length=300, |
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156 | theta=60, phi=60, |
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157 | radius_pd=.2, radius_pd_n=9, |
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158 | length_pd=.2, length_pd_n=10, |
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159 | theta_pd=10, theta_pd_n=5, |
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160 | phi_pd=10, phi_pd_n=5) |
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161 | |
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162 | # For testing against the old sasview models, include the converted parameter |
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163 | # names and the target sasview model name. |
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164 | oldname = 'CylinderModel' |
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165 | oldpars = dict(theta='cyl_theta', phi='cyl_phi', sld='sldCyl', solvent_sld='sldSolv') |
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166 | |
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167 | |
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168 | qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) |
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169 | tests = [[{}, 0.2, 0.041761386790780453], |
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170 | [{}, [0.2], [0.041761386790780453]], |
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171 | [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03414647218513852], |
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172 | [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03414647218513852]], |
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173 | ] |
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174 | del qx, qy # not necessary to delete, but cleaner |
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