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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>2.1.2.1. Ellipsoid</span></h2>
36      </div>
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38     
39        <p>
40        «&#160;&#160;<a href="../ref/models/shape-ellipsoid.html">2.1.2. Ellipsoid Functions</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="triaxial_ellipsoid.html">2.1.2.2. Triaxial ellipsoid</a>&#160;&#160;»
45        </p>
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47      </div>
48      <div class="content">
49       
50       
51  <div class="section" id="ellipsoid">
52<span id="id1"></span><h1>2.1.2.1. Ellipsoid<a class="headerlink" href="#ellipsoid" title="Permalink to this headline">¶</a></h1>
53<p>Ellipsoid of revolution with uniform scattering length density.</p>
54<table border="1" class="docutils">
55<colgroup>
56<col width="15%" />
57<col width="49%" />
58<col width="17%" />
59<col width="18%" />
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>Ellipsoid 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>rpolar</td>
90<td>Polar radius</td>
91<td>Å</td>
92<td>20</td>
93</tr>
94<tr class="row-odd"><td>requatorial</td>
95<td>Equatorial radius</td>
96<td>Å</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<div class="section" id="definition">
114<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
115<p>The output of the 2D scattering intensity function for oriented ellipsoids
116is given by (Feigin, 1987)</p>
117<div class="math">
118\[P(Q,\alpha) = {\text{scale} \over V} F^2(Q) + \text{background}\]</div>
119<p>where</p>
120<div class="math">
121\[F(Q) = {3 (\Delta rho)) V (\sin[Qr(R_p,R_e,\alpha)]
122            - \cos[Qr(R_p,R_e,\alpha)])
123        \over [Qr(R_p,R_e,\alpha)]^3 }\]</div>
124<p>and</p>
125<div class="math">
126\[r(R_p,R_e,\alpha) = \left[ R_e^2 \sin^2 \alpha
127    + R_p^2 \cos^2 \alpha \right]^{1/2}\]</div>
128<p><span class="math">\(\alpha\)</span> is the angle between the axis of the ellipsoid and <span class="math">\(\vec q\)</span>,
129<span class="math">\(V\)</span> is the volume of the ellipsoid, <span class="math">\(R_p\)</span> is the polar radius along the
130rotational axis of the ellipsoid, <span class="math">\(R_e\)</span> is the equatorial radius perpendicular
131to the rotational axis of the ellipsoid and <span class="math">\(\Delta \rho\)</span> (contrast) is the
132scattering length density difference between the scatterer and the solvent.</p>
133<p>To provide easy access to the orientation of the ellipsoid, we define
134the rotation axis of the ellipsoid using two angles <span class="math">\(\theta\)</span> and <span class="math">\(\phi\)</span>.
135These angles are defined in the
136<a class="reference internal" href="cylinder.html#cylinder-orientation"><em>cylinder orientation figure</em></a>.
137For the ellipsoid, <span class="math">\(\theta\)</span> is the angle between the rotational axis
138and the $z$-axis.</p>
139<p>NB: The 2nd virial coefficient of the solid ellipsoid is calculated based
140on the <span class="math">\(R_p\)</span> and <span class="math">\(R_e\)</span> values, and used as the effective radius for
141<span class="math">\(S(Q)\)</span> when <span class="math">\(P(Q) \cdot S(Q)\)</span> is applied.</p>
142<div class="figure" id="ellipsoid-1d">
143<img alt="../_images/ellipsoid_1d.JPG" src="../_images/ellipsoid_1d.JPG" />
144<p class="caption">Figure 1: The output of the 1D scattering intensity function for randomly oriented
145ellipsoids given by the equation above.</p>
146</div>
147<p>The <span class="math">\(\theta\)</span> and <span class="math">\(\phi\)</span> parameters are not used for the 1D output. Our
148implementation of the scattering kernel and the 1D scattering intensity
149use the c-library from NIST.</p>
150<div class="figure" id="ellipsoid-geometry">
151<img alt="../_images/ellipsoid_geometry.JPG" src="../_images/ellipsoid_geometry.JPG" />
152<p class="caption">Figure 2: The angles for oriented ellipsoid.</p>
153</div>
154</div>
155<div class="section" id="validation">
156<h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2>
157<p>Validation of our code was done by comparing the output of the 1D model
158to the output of the software provided by the NIST (Kline, 2006).
159 below shows a comparison of
160the 1D output of our model and the output of the NIST software.</p>
161<div class="figure" id="ellipsoid-comparison-1d">
162<img alt="../_images/ellipsoid_comparison_1d.jpg" src="../_images/ellipsoid_comparison_1d.jpg" />
163<p class="caption">Figure 3: Comparison of the SasView scattering intensity for an ellipsoid
164with the output of the NIST SANS analysis software.  The parameters
165were set to: <em>scale</em> = 1.0, <em>rpolar</em> = 20 Å,
166<em>requatorial</em> =400 Å, <em>contrast</em> = 3e-6 Å<sup>-2</sup>,
167and <em>background</em> = 0.01 cm<sup>-1</sup>.</p>
168</div>
169<p>Averaging over a distribution of orientation is done by evaluating the
170equation above. Since we have no other software to compare the
171implementation of the intensity for fully oriented ellipsoids, we can
172compare the result of averaging our 2D output using a uniform distribution
173<span class="math">\(p(\theta,\phi) = 1.0\)</span>.  <a class="pageref" href="#ellipsoid-comparison-2d">Figure  4</a>
174shows the result of such a cross-check.</p>
175<div class="figure" id="ellipsoid-comparison-2d">
176<img alt="../_images/ellipsoid_comparison_2d.jpg" src="../_images/ellipsoid_comparison_2d.jpg" />
177<p class="caption">Figure 4: Comparison of the intensity for uniformly distributed ellipsoids
178calculated from our 2D model and the intensity from the NIST SANS
179analysis software. The parameters used were: <em>scale</em> = 1.0,
180<em>rpolar</em> = 20 Å, <em>requatorial</em> = 400 Å,
181<em>contrast</em> = 3e-6 Å<sup>-2</sup>, and <em>background</em> = 0.0 cm<sup>-1</sup>.</p>
182</div>
183<p>The discrepancy above <em>q</em> = 0.3 cm<sup>-1</sup> is due to the way the form factors
184are calculated in the c-library provided by NIST. A numerical integration
185has to be performed to obtain <span class="math">\(P(Q)\)</span> for randomly oriented particles.
186The NIST software performs that integration with a 76-point Gaussian
187quadrature rule, which will become imprecise at high <span class="math">\(Q\)</span> where the amplitude
188varies quickly as a function of <span class="math">\(Q\)</span>. The SasView result shown has been
189obtained by summing over 501 equidistant points. Our result was found
190to be stable over the range of <span class="math">\(Q\)</span> shown for a number of points higher
191than 500.</p>
192<p>REFERENCE</p>
193<p>L A Feigin and D I Svergun. <em>Structure Analysis by Small-Angle X-Ray and Neutron Scattering</em>, Plenum,
194New York, 1987.</p>
195</div>
196</div>
197
198
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200      <div class="bottomnav">
201     
202        <p>
203        «&#160;&#160;<a href="../ref/models/shape-ellipsoid.html">2.1.2. Ellipsoid Functions</a>
204        &#160;&#160;::&#160;&#160;
205        <a class="uplink" href="../index.html">Contents</a>
206        &#160;&#160;::&#160;&#160;
207        <a href="triaxial_ellipsoid.html">2.1.2.2. Triaxial ellipsoid</a>&#160;&#160;»
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