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33      <div class="header"><h1 class="heading"><a href="../index.html">
34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.1.1. Barbell</span></h2>
36      </div>
37      <div class="topnav">
38     
39        <p>
40        «&#160;&#160;<a href="../ref/models/shape-cylinder.html">2.1.1. Cylinder Functions</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="capped_cylinder.html">2.1.1.2. Capped cylinder</a>&#160;&#160;»
45        </p>
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48      <div class="content">
49       
50       
51  <div class="section" id="barbell">
52<span id="id1"></span><h1>2.1.1.1. Barbell<a class="headerlink" href="#barbell" title="Permalink to this headline">¶</a></h1>
53<p>Cylinder with spherical end caps</p>
54<table border="1" class="docutils">
55<colgroup>
56<col width="16%" />
57<col width="48%" />
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>Barbell scattering length density</td>
81<td>10<sup>-6</sup>Å<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>Å<sup>-2</sup></td>
87<td>1</td>
88</tr>
89<tr class="row-even"><td>bell_radius</td>
90<td>Spherical bell radius</td>
91<td>Å</td>
92<td>40</td>
93</tr>
94<tr class="row-odd"><td>radius</td>
95<td>Cylindrical bar radius</td>
96<td>Å</td>
97<td>20</td>
98</tr>
99<tr class="row-even"><td>length</td>
100<td>Cylinder bar length</td>
101<td>Å</td>
102<td>400</td>
103</tr>
104<tr class="row-odd"><td>theta</td>
105<td>In plane angle</td>
106<td>degree</td>
107<td>60</td>
108</tr>
109<tr class="row-even"><td>phi</td>
110<td>Out of plane angle</td>
111<td>degree</td>
112<td>60</td>
113</tr>
114</tbody>
115</table>
116<p>The returned value is scaled to units of cm<sup>-1</sup>.</p>
117<p>Calculates the scattering from a barbell-shaped cylinder (This model simply
118becomes the DumBellModel when the length of the cylinder, <em>L</em>, is set to zero).
119That is, a sphereocylinder with spherical end caps that have a radius larger
120than that of the cylinder and the center of the end cap radius lies outside
121of the cylinder. All dimensions of the BarBell are considered to be
122monodisperse. See the diagram for the details of the geometry and restrictions
123on parameter values.</p>
124<div class="section" id="definition">
125<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
126<p>The returned value is scaled to units of cm<sup>-1</sup>sr<sup>-1</sup>, absolute scale.</p>
127<p>The barbell geometry is defined as</p>
128<img alt="../_images/barbell_geometry.jpg" src="../_images/barbell_geometry.jpg" />
129<p>where <em>r</em> is the radius of the cylinder. All other parameters are as defined
130in the diagram.</p>
131<p>Since the end cap radius
132<em>R</em> &gt;= <em>r</em> and by definition for this geometry <em>h</em> &lt; 0, <em>h</em> is then
133defined by <em>r</em> and <em>R</em> as</p>
134<p><em>h</em> = -1 * sqrt(<em>R</em><sup>2</sup> - <em>r</em><sup>2</sup>)</p>
135<p>The scattered intensity <em>I(q)</em> is calculated as</p>
136<div class="math">
137\[\begin{split}I(Q) = \frac{(\Delta \rho)^2}{V} \left&lt; A^2(Q)\right&gt;\end{split}\]</div>
138<p>where the amplitude <em>A(q)</em> is given as</p>
139<div class="math">
140\[\begin{split}A(Q) =&amp;\ \pi r^2L
141    {\sin\left(\tfrac12 QL\cos\theta\right)
142        \over \tfrac12 QL\cos\theta}
143    {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\
144    &amp;\ + 4 \pi R^3 \int_{-h/R}^1 dt
145    \cos\left[ Q\cos\theta
146        \left(Rt + h + {\tfrac12} L\right)\right]
147    \times (1-t^2)
148    {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right]
149         \over QR\sin\theta \left(1-t^2\right)^{1/2}}\end{split}\]</div>
150<p>The &lt; &gt; brackets denote an average of the structure over all orientations.
151&lt;<em>A</em> <sup>2</sup><em>(q)</em>&gt; is then the form factor, <em>P(q)</em>. The scale factor is
152equivalent to the volume fraction of cylinders, each of volume, <em>V</em>. Contrast
153is the difference of scattering length densities of the cylinder and the
154surrounding solvent.</p>
155<p>The volume of the barbell is</p>
156<div class="math">
157\[V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right)\]</div>
158<p>and its radius-of-gyration is</p>
159<div class="math">
160\[\begin{split}R_g^2 =&amp;\ \left[ \tfrac{12}{5}R^5
161    + R^4\left(6h+\tfrac32 L\right)
162    + R^2\left(4h^2 + L^2 + 4Lh\right)
163    + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\
164    &amp;\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3
165    + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right]
166    \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1}\end{split}\]</div>
167<p><strong>The requirement that</strong> <em>R</em> &gt;= <em>r</em> <strong>is not enforced in the model!</strong> It is
168up to you to restrict this during analysis.</p>
169<p>This example dataset is produced by running the Macro PlotBarbell(),
170using 200 data points, <em>qmin</em> = 0.001 Å<sup>-1</sup>, <em>qmax</em> = 0.7 Å<sup>-1</sup>,
171<em>sld</em> = 4e-6 Å<sup>-2</sup> and the default model values.</p>
172<img alt="../_images/barbell_1d.jpg" src="../_images/barbell_1d.jpg" />
173<p><em>Figure. 1D plot using the default values (w/256 data point).</em></p>
174<p>For 2D data: The 2D scattering intensity is calculated similar to the 2D
175cylinder model. For example, for Ξ = 45 deg and φ = 0 deg with
176default values for other parameters</p>
177<img alt="../_images/barbell_2d.jpg" src="../_images/barbell_2d.jpg" />
178<p><em>Figure. 2D plot (w/(256X265) data points).</em></p>
179<img alt="../_images/orientation.jpg" src="../_images/orientation.jpg" />
180<p>Figure. Definition of the angles for oriented 2D barbells.</p>
181<img alt="../_images/orientation2.jpg" src="../_images/orientation2.jpg" />
182<p><em>Figure. Examples of the angles for oriented pp against the detector plane.</em></p>
183</div>
184<div class="section" id="reference">
185<h2>REFERENCE<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h2>
186<p>H Kaya, <em>J. Appl. Cryst.</em>, 37 (2004) 37 223-230</p>
187<p>H Kaya and N R deSouza, <em>J. Appl. Cryst.</em>, 37 (2004) 508-509 (addenda and errata)</p>
188</div>
189</div>
190
191
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194     
195        <p>
196        «&#160;&#160;<a href="../ref/models/shape-cylinder.html">2.1.1. Cylinder Functions</a>
197        &#160;&#160;::&#160;&#160;
198        <a class="uplink" href="../index.html">Contents</a>
199        &#160;&#160;::&#160;&#160;
200        <a href="capped_cylinder.html">2.1.1.2. Capped cylinder</a>&#160;&#160;»
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