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32 | <body> |
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33 | <div class="header"><h1 class="heading"><a href="../index.html"> |
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34 | <span>Home</span></a></h1> |
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35 | <h2 class="heading"><span>2.1.1.4. Cylinder</span></h2> |
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36 | </div> |
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37 | <div class="topnav"> |
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38 | |
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39 | <p> |
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40 | «  <a href="core_shell_cylinder.html">2.1.1.3. Core shell cylinder</a> |
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41 |   ::   |
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42 | <a class="uplink" href="../index.html">Contents</a> |
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43 |   ::   |
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44 | <a href="../ref/models/shape-ellipsoid.html">2.1.2. Ellipsoid Functions</a>  Â» |
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45 | </p> |
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46 | |
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47 | </div> |
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48 | <div class="content"> |
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49 | |
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50 | |
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51 | <div class="section" id="cylinder"> |
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52 | <span id="id1"></span><h1>2.1.1.4. Cylinder<a class="headerlink" href="#cylinder" title="Permalink to this headline">¶</a></h1> |
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53 | <p>Right circular cylinder with uniform scattering length density.</p> |
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54 | <table border="1" class="docutils"> |
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55 | <colgroup> |
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56 | <col width="16%" /> |
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57 | <col width="49%" /> |
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58 | <col width="17%" /> |
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59 | <col width="19%" /> |
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60 | </colgroup> |
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61 | <thead valign="bottom"> |
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62 | <tr class="row-odd"><th class="head">Parameter</th> |
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63 | <th class="head">Description</th> |
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64 | <th class="head">Units</th> |
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65 | <th class="head">Default value</th> |
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66 | </tr> |
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67 | </thead> |
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68 | <tbody valign="top"> |
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69 | <tr class="row-even"><td>scale</td> |
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70 | <td>Source intensity</td> |
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71 | <td>None</td> |
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72 | <td>1</td> |
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73 | </tr> |
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74 | <tr class="row-odd"><td>background</td> |
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75 | <td>Source background</td> |
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76 | <td>cm<sup>-1</sup></td> |
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77 | <td>0</td> |
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78 | </tr> |
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79 | <tr class="row-even"><td>sld</td> |
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80 | <td>Cylinder scattering length density</td> |
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81 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
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82 | <td>4</td> |
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83 | </tr> |
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84 | <tr class="row-odd"><td>solvent_sld</td> |
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85 | <td>Solvent scattering length density</td> |
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86 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
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87 | <td>1</td> |
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88 | </tr> |
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89 | <tr class="row-even"><td>radius</td> |
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90 | <td>Cylinder radius</td> |
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91 | <td>â«</td> |
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92 | <td>20</td> |
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93 | </tr> |
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94 | <tr class="row-odd"><td>length</td> |
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95 | <td>Cylinder length</td> |
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96 | <td>â«</td> |
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97 | <td>400</td> |
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98 | </tr> |
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99 | <tr class="row-even"><td>theta</td> |
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100 | <td>In plane angle</td> |
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101 | <td>degree</td> |
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102 | <td>60</td> |
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103 | </tr> |
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104 | <tr class="row-odd"><td>phi</td> |
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105 | <td>Out of plane angle</td> |
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106 | <td>degree</td> |
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107 | <td>60</td> |
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108 | </tr> |
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109 | </tbody> |
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110 | </table> |
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111 | <p>The returned value is scaled to units of cm<sup>-1</sup>.</p> |
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112 | <p>The form factor is normalized by the particle volume.</p> |
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113 | <p>For information about polarised and magnetic scattering, click <a href="#id2"><span class="problematic" id="id3">here_</span></a>.</p> |
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114 | <div class="section" id="definition"> |
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115 | <h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2> |
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116 | <p>The output of the 2D scattering intensity function for oriented cylinders is |
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117 | given by (Guinier, 1955)</p> |
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118 | <div class="math"> |
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119 | \[P(Q,\alpha) = {\text{scale} \over V} F^2(Q) + \text{background}\]</div> |
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120 | <p>where</p> |
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121 | <div class="math"> |
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122 | \[F(Q) = 2 (\Delta \rho) V |
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123 | {\sin \left(Q\tfrac12 L\cos\alpha \right) |
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124 | \over Q\tfrac12 L \cos \alpha} |
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125 | {J_1 \left(Q R \sin \alpha\right) \over Q R \sin \alpha}\]</div> |
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126 | <p>and <span class="math">\(\alpha\)</span> is the angle between the axis of the cylinder and <span class="math">\(\vec q\)</span>, <span class="math">\(V\)</span> |
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127 | is the volume of the cylinder, <span class="math">\(L\)</span> is the length of the cylinder, <span class="math">\(R\)</span> is the |
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128 | radius of the cylinder, and <span class="math">\(\Delta\rho\)</span> (contrast) is the scattering length |
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129 | density difference between the scatterer and the solvent. <span class="math">\(J_1\)</span> is the |
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130 | first order Bessel function.</p> |
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131 | <p>To provide easy access to the orientation of the cylinder, we define the |
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132 | axis of the cylinder using two angles <span class="math">\(\theta\)</span> and <span class="math">\(\phi\)</span>. Those angles |
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133 | are defined in <a class="pageref" href="#cylinder-orientation">figure 1</a>.</p> |
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134 | <div class="figure" id="cylinder-orientation"> |
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135 | <img alt="../_images/orientation.jpg" src="../_images/orientation.jpg" /> |
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136 | <p class="caption">Figure 1: Definition of the angles for oriented cylinders.</p> |
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137 | </div> |
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138 | <div class="figure"> |
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139 | <img alt="../_images/orientation2.jpg" src="../_images/orientation2.jpg" /> |
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140 | <p class="caption">Figure 2: Examples of the angles for oriented cylinders against the detector plane.</p> |
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141 | </div> |
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142 | <p>NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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143 | radius and length values, and used as the effective radius for <span class="math">\(S(Q)\)</span> |
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144 | when <span class="math">\(P(Q) \cdot S(Q)\)</span> is applied.</p> |
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145 | <p>The output of the 1D scattering intensity function for randomly oriented |
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146 | cylinders is then given by</p> |
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147 | <div class="math"> |
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148 | \[P(Q) = {\text{scale} \over V} |
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149 | \int_0^{\pi/2} F^2(Q,\alpha) \sin \alpha\ d\alpha + \text{background}\]</div> |
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150 | <p>The <em>theta</em> and <em>phi</em> parameters are not used for the 1D output. Our |
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151 | implementation of the scattering kernel and the 1D scattering intensity |
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152 | use the c-library from NIST.</p> |
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153 | </div> |
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154 | <div class="section" id="validation"> |
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155 | <h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2> |
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156 | <p>Validation of our code was done by comparing the output of the 1D model |
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157 | to the output of the software provided by the NIST (Kline, 2006). |
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158 | <a class="pageref" href="#cylinder-compare">Figure 3</a> shows a comparison of |
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159 | the 1D output of our model and the output of the NIST software.</p> |
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160 | <div class="figure" id="cylinder-compare"> |
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161 | <img alt="../_images/cylinder_compare.jpg" src="../_images/cylinder_compare.jpg" /> |
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162 | <p class="caption">Figure 3: Comparison of the SasView scattering intensity for a cylinder with the |
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163 | output of the NIST SANS analysis software. |
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164 | The parameters were set to: <em>scale</em> = 1.0, <em>radius</em> = 20 â«, |
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165 | <em>length</em> = 400 â«, <em>contrast</em> = 3e-6 â«<sup>-2</sup>, and |
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166 | <em>background</em> = 0.01 cm<sup>-1</sup>.</p> |
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167 | </div> |
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168 | <p>In general, averaging over a distribution of orientations is done by |
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169 | evaluating the following</p> |
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170 | <div class="math"> |
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171 | \[P(Q) = \int_0^{\pi/2} d\phi |
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172 | \int_0^\pi p(\theta, \phi) P_0(Q,\alpha) \sin \theta\ d\theta\]</div> |
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173 | <p>where <span class="math">\(p(\theta,\phi)\)</span> is the probability distribution for the orientation |
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174 | and <span class="math">\(P_0(Q,\alpha)\)</span> is the scattering intensity for the fully oriented |
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175 | system. Since we have no other software to compare the implementation of |
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176 | the intensity for fully oriented cylinders, we can compare the result of |
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177 | averaging our 2D output using a uniform distribution <span class="math">\(p(\theta, \phi) = 1.0\)</span>. |
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178 | <a class="pageref" href="#cylinder-crosscheck">Figure 4</a> shows the result of |
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179 | such a cross-check.</p> |
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180 | <div class="figure" id="cylinder-crosscheck"> |
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181 | <img alt="../_images/cylinder_crosscheck.jpg" src="../_images/cylinder_crosscheck.jpg" /> |
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182 | <p class="caption">Figure 4: Comparison of the intensity for uniformly distributed cylinders |
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183 | calculated from our 2D model and the intensity from the NIST SANS |
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184 | analysis software. |
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185 | The parameters used were: <em>scale</em> = 1.0, <em>radius</em> = 20 â«, |
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186 | <em>length</em> = 400 â«, <em>contrast</em> = 3e-6 â«<sup>-2</sup>, and |
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187 | <em>background</em> = 0.0 cm<sup>-1</sup>.</p> |
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188 | </div> |
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189 | </div> |
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190 | </div> |
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191 | |
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192 | |
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193 | </div> |
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194 | <div class="bottomnav"> |
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195 | |
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196 | <p> |
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197 | «  <a href="core_shell_cylinder.html">2.1.1.3. Core shell cylinder</a> |
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198 |   ::   |
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199 | <a class="uplink" href="../index.html">Contents</a> |
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200 |   ::   |
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201 | <a href="../ref/models/shape-ellipsoid.html">2.1.2. Ellipsoid Functions</a>  Â» |
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202 | </p> |
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203 | |
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