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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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