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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.6.2. Sphere</span></h2>
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38     
39        <p>
40        «&#160;&#160;<a href="sphere.html">2.1.6.1. Sphere</a>
41        &#160;&#160;::&#160;&#160;
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43        &#160;&#160;::&#160;&#160;
44        <a href="../ref/models/shape-independent.html">2.2. Shape-Independent Functions</a>&#160;&#160;»
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48      <div class="content">
49       
50       
51  <div class="section" id="sphere">
52<span id="id1"></span><h1>2.1.6.2. Sphere<a class="headerlink" href="#sphere" title="Permalink to this headline">¶</a></h1>
53<p>Spheres with uniform scattering length density</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>Layer scattering length density</td>
81<td>10<sup>-6</sup>Å<sup>-2</sup></td>
82<td>1</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>6</td>
88</tr>
89<tr class="row-even"><td>radius</td>
90<td>Sphere radius</td>
91<td>Å</td>
92<td>50</td>
93</tr>
94</tbody>
95</table>
96<p>The returned value is scaled to units of cm<sup>-1</sup>.</p>
97<p>For information about polarised and magnetic scattering, click <a class="reference external" href="polar_mag_help.html">here</a>.</p>
98<div class="section" id="definition">
99<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
100<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955)</p>
101<div class="math">
102\[I(Q) = \frac{\text{scale}}{V} \cdot \left[ \
103    3V(\Delta\rho) \cdot \frac{\sin(QR) - QR\cos(QR))}{(QR)^3} \
104    \right]^2 + \text{background}\]</div>
105<p>where <em>scale</em> is a volume fraction, <span class="math">\(V\)</span> is the volume of the scatterer,
106<span class="math">\(R\)</span> is the radius of the sphere, <em>background</em> is the background level and
107<em>sld</em> and <em>solvent_sld</em> are the scattering length densities (SLDs) of the
108scatterer and the solvent respectively.</p>
109<p>Note that if your data is in absolute scale, the <em>scale</em> should represent
110the volume fraction (which is unitless) if you have a good fit. If not,
111it should represent the volume fraction times a factor (by which your data
112might need to be rescaled).</p>
113<p>The 2D scattering intensity is the same as above, regardless of the
114orientation of <span class="math">\(\vec q\)</span>.</p>
115<p>Our model uses the form factor calculations as defined in the IGOR
116package provided by the NIST Center for Neutron Research (Kline, 2006).</p>
117</div>
118<div class="section" id="validation">
119<h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2>
120<p>Validation of our code was done by comparing the output of the 1D model
121to the output of the software provided by the NIST (Kline, 2006).
122Figure <a class="pageref" href="#sphere-comparison">figure  1</a> shows a comparison of the output
123of our model and the output of the NIST software.</p>
124<div class="figure" id="sphere-comparison">
125<img alt="../_images/sphere_comparison.jpg" src="../_images/sphere_comparison.jpg" />
126<p class="caption">Figure 1: Comparison of the DANSE scattering intensity for a sphere with the
127output of the NIST SANS analysis software. The parameters were set to:
128<em>scale</em> = 1.0, <em>radius</em> = 60 Å, <em>contrast</em> = 1e-6 Å<sup>-2</sup>, and
129<em>background</em> = 0.01 cm<sup>-1</sup>.</p>
130</div>
131</div>
132<div class="section" id="reference">
133<h2>Reference<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h2>
134<p>A Guinier and G. Fournet, <em>Small-Angle Scattering of X-Rays</em>,
135John Wiley and Sons, New York, (1955)</p>
136<p><em>2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.</em></p>
137</div>
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139
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149        <a href="../ref/models/shape-independent.html">2.2. Shape-Independent Functions</a>&#160;&#160;»
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