[68532f3] | 1 | <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" |
---|
| 2 | "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> |
---|
| 3 | |
---|
| 4 | |
---|
| 5 | <html xmlns="http://www.w3.org/1999/xhtml"> |
---|
| 6 | <head> |
---|
| 7 | <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> |
---|
| 8 | |
---|
| 9 | <title>2.1.1.4. Cylinder — Sasmodels</title> |
---|
| 10 | |
---|
| 11 | <link rel="stylesheet" href="../_static/haiku-site.css" type="text/css" /> |
---|
| 12 | <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> |
---|
| 13 | |
---|
| 14 | <script type="text/javascript"> |
---|
| 15 | var DOCUMENTATION_OPTIONS = { |
---|
| 16 | URL_ROOT: '../', |
---|
| 17 | VERSION: '0.9', |
---|
| 18 | COLLAPSE_INDEX: false, |
---|
| 19 | FILE_SUFFIX: '.html', |
---|
| 20 | HAS_SOURCE: true |
---|
| 21 | }; |
---|
| 22 | </script> |
---|
| 23 | <script type="text/javascript" src="../_static/jquery.js"></script> |
---|
| 24 | <script type="text/javascript" src="../_static/underscore.js"></script> |
---|
| 25 | <script type="text/javascript" src="../_static/doctools.js"></script> |
---|
| 26 | <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> |
---|
| 27 | <link rel="top" title="Sasmodels" href="../index.html" /> |
---|
| 28 | <link rel="up" title="2.1.1. Cylinder Functions" href="../ref/models/shape-cylinder.html" /> |
---|
| 29 | <link rel="next" title="2.1.2. Ellipsoid Functions" href="../ref/models/shape-ellipsoid.html" /> |
---|
| 30 | <link rel="prev" title="2.1.1.3. Core shell cylinder" href="core_shell_cylinder.html" /> |
---|
| 31 | </head> |
---|
| 32 | <body> |
---|
| 33 | <div class="header"><h1 class="heading"><a href="../index.html"> |
---|
| 34 | <span>Home</span></a></h1> |
---|
| 35 | <h2 class="heading"><span>2.1.1.4. Cylinder</span></h2> |
---|
| 36 | </div> |
---|
| 37 | <div class="topnav"> |
---|
| 38 | |
---|
| 39 | <p> |
---|
| 40 | «  <a href="core_shell_cylinder.html">2.1.1.3. Core shell cylinder</a> |
---|
| 41 |   ::   |
---|
| 42 | <a class="uplink" href="../index.html">Contents</a> |
---|
| 43 |   ::   |
---|
| 44 | <a href="../ref/models/shape-ellipsoid.html">2.1.2. Ellipsoid Functions</a>  Â» |
---|
| 45 | </p> |
---|
| 46 | |
---|
| 47 | </div> |
---|
| 48 | <div class="content"> |
---|
| 49 | |
---|
| 50 | |
---|
| 51 | <div class="section" id="cylinder"> |
---|
| 52 | <span id="id1"></span><h1>2.1.1.4. Cylinder<a class="headerlink" href="#cylinder" title="Permalink to this headline">¶</a></h1> |
---|
| 53 | <p>Right circular cylinder with uniform scattering length density.</p> |
---|
| 54 | <table border="1" class="docutils"> |
---|
| 55 | <colgroup> |
---|
| 56 | <col width="16%" /> |
---|
| 57 | <col width="49%" /> |
---|
| 58 | <col width="17%" /> |
---|
| 59 | <col width="19%" /> |
---|
| 60 | </colgroup> |
---|
| 61 | <thead valign="bottom"> |
---|
| 62 | <tr class="row-odd"><th class="head">Parameter</th> |
---|
| 63 | <th class="head">Description</th> |
---|
| 64 | <th class="head">Units</th> |
---|
| 65 | <th class="head">Default value</th> |
---|
| 66 | </tr> |
---|
| 67 | </thead> |
---|
| 68 | <tbody valign="top"> |
---|
| 69 | <tr class="row-even"><td>scale</td> |
---|
| 70 | <td>Source intensity</td> |
---|
| 71 | <td>None</td> |
---|
| 72 | <td>1</td> |
---|
| 73 | </tr> |
---|
| 74 | <tr class="row-odd"><td>background</td> |
---|
| 75 | <td>Source background</td> |
---|
| 76 | <td>cm<sup>-1</sup></td> |
---|
| 77 | <td>0</td> |
---|
| 78 | </tr> |
---|
| 79 | <tr class="row-even"><td>sld</td> |
---|
| 80 | <td>Cylinder scattering length density</td> |
---|
| 81 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
---|
| 82 | <td>4</td> |
---|
| 83 | </tr> |
---|
| 84 | <tr class="row-odd"><td>solvent_sld</td> |
---|
| 85 | <td>Solvent scattering length density</td> |
---|
| 86 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
---|
| 87 | <td>1</td> |
---|
| 88 | </tr> |
---|
| 89 | <tr class="row-even"><td>radius</td> |
---|
| 90 | <td>Cylinder radius</td> |
---|
| 91 | <td>â«</td> |
---|
| 92 | <td>20</td> |
---|
| 93 | </tr> |
---|
| 94 | <tr class="row-odd"><td>length</td> |
---|
| 95 | <td>Cylinder length</td> |
---|
| 96 | <td>â«</td> |
---|
| 97 | <td>400</td> |
---|
| 98 | </tr> |
---|
| 99 | <tr class="row-even"><td>theta</td> |
---|
| 100 | <td>In plane angle</td> |
---|
| 101 | <td>degree</td> |
---|
| 102 | <td>60</td> |
---|
| 103 | </tr> |
---|
| 104 | <tr class="row-odd"><td>phi</td> |
---|
| 105 | <td>Out of plane angle</td> |
---|
| 106 | <td>degree</td> |
---|
| 107 | <td>60</td> |
---|
| 108 | </tr> |
---|
| 109 | </tbody> |
---|
| 110 | </table> |
---|
| 111 | <p>The returned value is scaled to units of cm<sup>-1</sup>.</p> |
---|
| 112 | <p>The form factor is normalized by the particle volume.</p> |
---|
| 113 | <p>For information about polarised and magnetic scattering, click <a href="#id2"><span class="problematic" id="id3">here_</span></a>.</p> |
---|
| 114 | <div class="section" id="definition"> |
---|
| 115 | <h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2> |
---|
| 116 | <p>The output of the 2D scattering intensity function for oriented cylinders is |
---|
| 117 | given by (Guinier, 1955)</p> |
---|
| 118 | <div class="math"> |
---|
| 119 | \[P(Q,\alpha) = {\text{scale} \over V} F^2(Q) + \text{background}\]</div> |
---|
| 120 | <p>where</p> |
---|
| 121 | <div class="math"> |
---|
| 122 | \[F(Q) = 2 (\Delta \rho) V |
---|
| 123 | {\sin \left(Q\tfrac12 L\cos\alpha \right) |
---|
| 124 | \over Q\tfrac12 L \cos \alpha} |
---|
| 125 | {J_1 \left(Q R \sin \alpha\right) \over Q R \sin \alpha}\]</div> |
---|
| 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> |
---|
| 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 |
---|
| 128 | radius of the cylinder, and <span class="math">\(\Delta\rho\)</span> (contrast) is the scattering length |
---|
| 129 | density difference between the scatterer and the solvent. <span class="math">\(J_1\)</span> is the |
---|
| 130 | first order Bessel function.</p> |
---|
| 131 | <p>To provide easy access to the orientation of the cylinder, we define the |
---|
| 132 | axis of the cylinder using two angles <span class="math">\(\theta\)</span> and <span class="math">\(\phi\)</span>. Those angles |
---|
| 133 | are defined in <a class="pageref" href="#cylinder-orientation">figure 1</a>.</p> |
---|
| 134 | <div class="figure" id="cylinder-orientation"> |
---|
| 135 | <img alt="../_images/orientation.jpg" src="../_images/orientation.jpg" /> |
---|
| 136 | <p class="caption">Figure 1: Definition of the angles for oriented cylinders.</p> |
---|
| 137 | </div> |
---|
| 138 | <div class="figure"> |
---|
| 139 | <img alt="../_images/orientation2.jpg" src="../_images/orientation2.jpg" /> |
---|
| 140 | <p class="caption">Figure 2: Examples of the angles for oriented cylinders against the detector plane.</p> |
---|
| 141 | </div> |
---|
| 142 | <p>NB: The 2nd virial coefficient of the cylinder is calculated based on the |
---|
| 143 | radius and length values, and used as the effective radius for <span class="math">\(S(Q)\)</span> |
---|
| 144 | when <span class="math">\(P(Q) \cdot S(Q)\)</span> is applied.</p> |
---|
| 145 | <p>The output of the 1D scattering intensity function for randomly oriented |
---|
| 146 | cylinders is then given by</p> |
---|
| 147 | <div class="math"> |
---|
| 148 | \[P(Q) = {\text{scale} \over V} |
---|
| 149 | \int_0^{\pi/2} F^2(Q,\alpha) \sin \alpha\ d\alpha + \text{background}\]</div> |
---|
| 150 | <p>The <em>theta</em> and <em>phi</em> parameters are not used for the 1D output. Our |
---|
| 151 | implementation of the scattering kernel and the 1D scattering intensity |
---|
| 152 | use the c-library from NIST.</p> |
---|
| 153 | </div> |
---|
| 154 | <div class="section" id="validation"> |
---|
| 155 | <h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2> |
---|
| 156 | <p>Validation of our code was done by comparing the output of the 1D model |
---|
| 157 | to the output of the software provided by the NIST (Kline, 2006). |
---|
| 158 | <a class="pageref" href="#cylinder-compare">Figure 3</a> shows a comparison of |
---|
| 159 | the 1D output of our model and the output of the NIST software.</p> |
---|
| 160 | <div class="figure" id="cylinder-compare"> |
---|
| 161 | <img alt="../_images/cylinder_compare.jpg" src="../_images/cylinder_compare.jpg" /> |
---|
| 162 | <p class="caption">Figure 3: Comparison of the SasView scattering intensity for a cylinder with the |
---|
| 163 | output of the NIST SANS analysis software. |
---|
| 164 | The parameters were set to: <em>scale</em> = 1.0, <em>radius</em> = 20 â«, |
---|
| 165 | <em>length</em> = 400 â«, <em>contrast</em> = 3e-6 â«<sup>-2</sup>, and |
---|
| 166 | <em>background</em> = 0.01 cm<sup>-1</sup>.</p> |
---|
| 167 | </div> |
---|
| 168 | <p>In general, averaging over a distribution of orientations is done by |
---|
| 169 | evaluating the following</p> |
---|
| 170 | <div class="math"> |
---|
| 171 | \[P(Q) = \int_0^{\pi/2} d\phi |
---|
| 172 | \int_0^\pi p(\theta, \phi) P_0(Q,\alpha) \sin \theta\ d\theta\]</div> |
---|
| 173 | <p>where <span class="math">\(p(\theta,\phi)\)</span> is the probability distribution for the orientation |
---|
| 174 | and <span class="math">\(P_0(Q,\alpha)\)</span> is the scattering intensity for the fully oriented |
---|
| 175 | system. Since we have no other software to compare the implementation of |
---|
| 176 | the intensity for fully oriented cylinders, we can compare the result of |
---|
| 177 | averaging our 2D output using a uniform distribution <span class="math">\(p(\theta, \phi) = 1.0\)</span>. |
---|
| 178 | <a class="pageref" href="#cylinder-crosscheck">Figure 4</a> shows the result of |
---|
| 179 | such a cross-check.</p> |
---|
| 180 | <div class="figure" id="cylinder-crosscheck"> |
---|
| 181 | <img alt="../_images/cylinder_crosscheck.jpg" src="../_images/cylinder_crosscheck.jpg" /> |
---|
| 182 | <p class="caption">Figure 4: Comparison of the intensity for uniformly distributed cylinders |
---|
| 183 | calculated from our 2D model and the intensity from the NIST SANS |
---|
| 184 | analysis software. |
---|
| 185 | The parameters used were: <em>scale</em> = 1.0, <em>radius</em> = 20 â«, |
---|
| 186 | <em>length</em> = 400 â«, <em>contrast</em> = 3e-6 â«<sup>-2</sup>, and |
---|
| 187 | <em>background</em> = 0.0 cm<sup>-1</sup>.</p> |
---|
| 188 | </div> |
---|
| 189 | </div> |
---|
| 190 | </div> |
---|
| 191 | |
---|
| 192 | |
---|
| 193 | </div> |
---|
| 194 | <div class="bottomnav"> |
---|
| 195 | |
---|
| 196 | <p> |
---|
| 197 | «  <a href="core_shell_cylinder.html">2.1.1.3. Core shell cylinder</a> |
---|
| 198 |   ::   |
---|
| 199 | <a class="uplink" href="../index.html">Contents</a> |
---|
| 200 |   ::   |
---|
| 201 | <a href="../ref/models/shape-ellipsoid.html">2.1.2. Ellipsoid Functions</a>  Â» |
---|
| 202 | </p> |
---|
| 203 | |
---|
| 204 | </div> |
---|
| 205 | |
---|
| 206 | <div class="footer"> |
---|
| 207 | © Copyright 2014, sasview team. |
---|
| 208 | Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2.3. |
---|
| 209 | </div> |
---|
| 210 | </body> |
---|
| 211 | </html> |
---|