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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>2.1.1.3. Core shell 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="capped_cylinder.html">2.1.1.2. Capped 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="cylinder.html">2.1.1.4. Cylinder</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="core-shell-cylinder"> |
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52 | <span id="id1"></span><h1>2.1.1.3. Core shell cylinder<a class="headerlink" href="#core-shell-cylinder" title="Permalink to this headline">¶</a></h1> |
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53 | <p>Right circular cylinder with a core-shell scattering length density profile.</p> |
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54 | <table border="1" class="docutils"> |
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55 | <colgroup> |
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56 | <col width="14%" /> |
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57 | <col width="53%" /> |
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58 | <col width="16%" /> |
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59 | <col width="17%" /> |
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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>core_sld</td> |
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80 | <td>Cylinder core 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>shell_sld</td> |
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85 | <td>Cylinder shell 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>4</td> |
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88 | </tr> |
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89 | <tr class="row-even"><td>solvent_sld</td> |
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90 | <td>Solvent scattering length density</td> |
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91 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
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92 | <td>1</td> |
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93 | </tr> |
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94 | <tr class="row-odd"><td>radius</td> |
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95 | <td>Cylinder core radius</td> |
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96 | <td>â«</td> |
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97 | <td>20</td> |
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98 | </tr> |
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99 | <tr class="row-even"><td>thickness</td> |
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100 | <td>Cylinder shell thickness</td> |
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101 | <td>â«</td> |
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102 | <td>20</td> |
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103 | </tr> |
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104 | <tr class="row-odd"><td>length</td> |
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105 | <td>Cylinder length</td> |
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106 | <td>â«</td> |
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107 | <td>400</td> |
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108 | </tr> |
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109 | <tr class="row-even"><td>theta</td> |
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110 | <td>In plane angle</td> |
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111 | <td>degree</td> |
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112 | <td>60</td> |
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113 | </tr> |
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114 | <tr class="row-odd"><td>phi</td> |
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115 | <td>Out of plane angle</td> |
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116 | <td>degree</td> |
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117 | <td>60</td> |
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118 | </tr> |
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119 | </tbody> |
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120 | </table> |
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121 | <p>The returned value is scaled to units of cm<sup>-1</sup>.</p> |
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122 | <p>The form factor is normalized by the particle volume.</p> |
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123 | <div class="section" id="definition"> |
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124 | <h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2> |
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125 | <p>The output of the 2D scattering intensity function for oriented core-shell |
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126 | cylinders is given by (Kline, 2006)</p> |
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127 | <div class="math"> |
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128 | \[P(Q,\alpha) = {\text{scale} \over V_s} F^2(Q) + \text{background}\]</div> |
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129 | <p>where</p> |
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130 | <div class="math"> |
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131 | \[\begin{split}F(Q) = &\ (\rho_c - \rho_s) V_c |
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132 | {\sin \left( Q \tfrac12 L\cos\alpha \right) |
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133 | \over Q \tfrac12 L\cos\alpha } |
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134 | {2 J_1 \left( QR\sin\alpha \right) |
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135 | \over QR\sin\alpha } \\ |
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136 | &\ + (\rho_s - \rho_\text{solv}) V_s |
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137 | {\sin \left( Q \left(\tfrac12 L+T\right) \cos\alpha \right) |
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138 | \over Q \left(\tfrac12 L +T \right) \cos\alpha } |
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139 | { 2 J_1 \left( Q(R+T)\sin\alpha \right) |
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140 | \over Q(R+T)\sin\alpha }\end{split}\]</div> |
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141 | <p>and</p> |
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142 | <div class="math"> |
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143 | \[V_s = \pi (R + T)^2 (L + 2T)\]</div> |
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144 | <p>and <span class="math">\(\alpha\)</span> is the angle between the axis of the cylinder and <span class="math">\(\vec q\)</span>, |
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145 | <span class="math">\(V_s\)</span> is the volume of the outer shell (i.e. the total volume, including |
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146 | the shell), <span class="math">\(V_c\)</span> is the volume of the core, <span class="math">\(L\)</span> is the length of the core, |
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147 | <span class="math">\(R\)</span> is the radius of the core, <span class="math">\(T\)</span> is the thickness of the shell, <span class="math">\(\rho_c\)</span> |
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148 | is the scattering length density of the core, <span class="math">\(\rho_s\)</span> is the scattering |
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149 | length density of the shell, <span class="math">\(\rho_\text{solv}\)</span> is the scattering length |
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150 | density of the solvent, and <em>background</em> is the background level. The outer |
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151 | radius of the shell is given by <span class="math">\(R+T\)</span> and the total length of the outer |
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152 | shell is given by <span class="math">\(L+2T\)</span>. <span class="math">\(J1\)</span> is the first order Bessel function.</p> |
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153 | <div class="figure" id="core-shell-cylinder-geometry"> |
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154 | <img alt="../_images/core_shell_cylinder_geometry.jpg" src="../_images/core_shell_cylinder_geometry.jpg" /> |
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155 | <p class="caption">Figure 1: Core shell cylinder schematic.</p> |
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156 | </div> |
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157 | <p>To provide easy access to the orientation of the core-shell cylinder, we |
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158 | define the axis of the cylinder using two angles <span class="math">\(\theta\)</span> and <span class="math">\(\phi\)</span>. As |
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159 | for the case of the cylinder, those angles are defined in |
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160 | .</p> |
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161 | <p>NB: The 2nd virial coefficient of the cylinder is calculated based on |
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162 | the radius and 2 length values, and used as the effective radius for |
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163 | <span class="math">\(S(Q)\)</span> when <span class="math">\(P(Q) \cdot S(Q)\)</span> is applied.</p> |
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164 | <p>The <span class="math">\(\theta\)</span> and <span class="math">\(\phi\)</span> parameters are not used for the 1D output. Our |
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165 | implementation of the scattering kernel and the 1D scattering intensity |
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166 | use the c-library from NIST.</p> |
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167 | </div> |
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168 | <div class="section" id="validation"> |
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169 | <h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2> |
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170 | <p>Validation of our code was done by comparing the output of the 1D model to |
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171 | the output of the software provided by the NIST (Kline, 2006). |
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172 | <a class="pageref" href="#core-shell-cylinder-1d">Figure 2</a> shows a comparison |
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173 | of the 1D output of our model and the output of the NIST software.</p> |
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174 | <div class="figure" id="core-shell-cylinder-1d"> |
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175 | <img alt="../_images/core_shell_cylinder_1d.jpg" src="../_images/core_shell_cylinder_1d.jpg" /> |
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176 | <p class="caption">Figure 2: Comparison of the SasView scattering intensity for a core-shell cylinder |
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177 | with the output of the NIST SANS analysis software. The parameters were |
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178 | set to: <em>scale</em> = 1.0 â«, <em>radius</em> = 20 â«, <em>thickness</em> = 10 â«, |
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179 | <em>length</em> =400 â«, <em>core_sld</em> =1e-6 â«<sup>-2</sup>, <em>shell_sld</em> = 4e-6 â«<sup>-2</sup>, |
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180 | <em>solvent_sld</em> = 1e-6 â«<sup>-2</sup>, and <em>background</em> = 0.01 cm<sup>-1</sup>.</p> |
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181 | </div> |
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182 | <p>Averaging over a distribution of orientation is done by evaluating the |
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183 | equation above. Since we have no other software to compare the |
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184 | implementation of the intensity for fully oriented cylinders, we can |
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185 | compare the result of averaging our 2D output using a uniform |
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186 | distribution <span class="math">\(p(\theta,\phi) = 1.0\)</span>. |
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187 | <a class="pageref" href="#core-shell-cylinder-2d">Figure 3</a> shows the result |
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188 | of such a cross-check.</p> |
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189 | <div class="figure" id="core-shell-cylinder-2d"> |
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190 | <img alt="../_images/core_shell_cylinder_2d.jpg" src="../_images/core_shell_cylinder_2d.jpg" /> |
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191 | <p class="caption">Figure 3: Comparison of the intensity for uniformly distributed core-shell |
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192 | cylinders calculated from our 2D model and the intensity from the |
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193 | NIST SANS analysis software. The parameters used were: <em>scale</em> = 1.0, |
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194 | <em>radius</em> = 20 â«, <em>thickness</em> = 10 â«, <em>length</em> = 400 â«, |
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195 | <em>core_sld</em> = 1e-6 â«<sup>-2</sup>, <em>shell_sld</em> = 4e-6 â«<sup>-2</sup>, |
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196 | <em>solvent_sld</em> = 1e-6 â«<sup>-2</sup>, and <em>background</em> = 0.0 cm<sup>-1</sup>.</p> |
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197 | </div> |
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198 | <p>2013/11/26 - Description reviewed by Heenan, R.</p> |
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199 | </div> |
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206 | <p> |
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207 | «  <a href="capped_cylinder.html">2.1.1.2. Capped cylinder</a> |
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208 |   ::   |
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209 | <a class="uplink" href="../index.html">Contents</a> |
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210 |   ::   |
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211 | <a href="cylinder.html">2.1.1.4. Cylinder</a>  Â» |
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212 | </p> |
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