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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>2.1.2.2. Triaxial ellipsoid</span></h2>
36      </div>
37      <div class="topnav">
38     
39        <p>
40        «&#160;&#160;<a href="ellipsoid.html">2.1.2.1. Ellipsoid</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="../ref/models/shape-lamellae.html">2.1.3. Lamellae Functions</a>&#160;&#160;»
45        </p>
46
47      </div>
48      <div class="content">
49       
50       
51  <div class="section" id="triaxial-ellipsoid">
52<span id="id1"></span><h1>2.1.2.2. Triaxial ellipsoid<a class="headerlink" href="#triaxial-ellipsoid" title="Permalink to this headline">¶</a></h1>
53<p>Ellipsoid of uniform scattering length density with three independent axes.</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>req_minor</td>
90<td>Minor equitorial radius</td>
91<td>Å</td>
92<td>20</td>
93</tr>
94<tr class="row-odd"><td>req_major</td>
95<td>Major equatorial radius</td>
96<td>Å</td>
97<td>400</td>
98</tr>
99<tr class="row-even"><td>rpolar</td>
100<td>Polar radius</td>
101<td>Å</td>
102<td>10</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<tr class="row-odd"><td>psi</td>
115<td>Out of plane angle</td>
116<td>degree</td>
117<td>60</td>
118</tr>
119</tbody>
120</table>
121<p>The returned value is scaled to units of cm<sup>-1</sup>.</p>
122<p>All three axes are of different lengths with <span class="math">\(R_a \le R_b &lt;= R_c\)</span>
123<strong>Users should maintain this inequality for all calculations</strong>.</p>
124<div class="math">
125\[\begin{split}P(Q) = \text{scale} V \left&lt; F^2(Q) \right&gt; + \text{background}\end{split}\]</div>
126<p>where the volume <span class="math">\(V = 4/3 \pi R_a R_b R_c\)</span>, and the averaging
127<span class="math">\(\left&lt; \cdots \right&gt;\)</span> is applied over all orientations for 1D.</p>
128<div class="figure">
129<img alt="../_images/triaxial_ellipsoid_geometry.jpg" src="../_images/triaxial_ellipsoid_geometry.jpg" />
130<p class="caption">Figure 1: Ellipsoid schematic.</p>
131</div>
132<p>The returned value is in units of cm<sup>-1</sup>, on absolute scale.</p>
133<div class="section" id="definition">
134<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
135<p>The form factor calculated is</p>
136<div class="math">
137\[P(Q) = \frac{\text{scale}}{V}\int_0^1\int_0^1
138    \Phi^2(QR_a^2\cos^2( \pi x/2) + QR_b^2\sin^2(\pi y/2)(1-y^2) + c^2y^2)
139    dx dy\]</div>
140<p>where</p>
141<div class="math">
142\[\Phi(u) = 3 u^{-3} (\sin u - u \cos u)\]</div>
143<p>To provide easy access to the orientation of the triaxial ellipsoid,
144we define the axis of the cylinder using the angles <span class="math">\(\theta\)</span>, <span class="math">\(\phi\)</span>
145and <span class="math">\(\psi\)</span>. These angles are defined on
146<a class="pageref" href="#triaxial-ellipsoid-angles">figure  2</a>.
147The angle <span class="math">\(\psi\)</span> is the rotational angle around its own <span class="math">\(c\)</span> axis
148against the <span class="math">\(Q\)</span> plane. For example, <span class="math">\(\psi = 0\)</span> when the
149<span class="math">\(a\)</span> axis is parallel to the <span class="math">\(x\)</span> axis of the detector.</p>
150<div class="figure" id="triaxial-ellipsoid-angles">
151<img alt="../_images/triaxial_ellipsoid_angles.jpg" src="../_images/triaxial_ellipsoid_angles.jpg" />
152<p class="caption">Figure 2: The angles for oriented ellipsoid.</p>
153</div>
154<p>The radius-of-gyration for this system is  <span class="math">\(R_g^2 = (R_a R_b R_c)^2/5\)</span>.</p>
155<p>The contrast is defined as SLD(ellipsoid) - SLD(solvent).  In the
156parameters, <em>a</em> is the minor equatorial radius, <em>b</em> is the major
157equatorial radius, and c is the polar radius of the ellipsoid.</p>
158<p>NB: The 2nd virial coefficient of the triaxial solid ellipsoid is
159calculated based on the polar radius <span class="math">\(R_p = R_c\)</span> and equatorial
160radius <span class="math">\(R_e = \sqrt{R_a R_b}\)</span>, and used as the effective radius for
161<span class="math">\(S(Q)\)</span> when <span class="math">\(P(Q) \cdot S(Q)\)</span> is applied.</p>
162<div class="figure">
163<img alt="../_images/triaxial_ellipsoid_1d.jpg" src="../_images/triaxial_ellipsoid_1d.jpg" />
164<p class="caption">Figure 3: 1D plot using the default values (w/1000 data point).</p>
165</div>
166</div>
167<div class="section" id="validation">
168<h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2>
169<p>Validation of our code was done by comparing the output of the
1701D calculation to the angular average of the output of 2D calculation
171over all possible angles.
172<a class="pageref" href="#triaxial-ellipsoid-comparison">Figure  4</a> shows the comparison where
173the solid dot refers to averaged 2D while the line represents the
174result of 1D calculation (for 2D averaging, 76, 180, and 76 points
175are taken for the angles of <span class="math">\(\theta\)</span>, <span class="math">\(\phi\)</span>, and <span class="math">\(\psi\)</span> respectively).</p>
176<div class="figure" id="triaxial-ellipsoid-comparison">
177<img alt="../_images/triaxial_ellipsoid_comparison.png" src="../_images/triaxial_ellipsoid_comparison.png" />
178<p class="caption">Figure 4: Comparison between 1D and averaged 2D.</p>
179</div>
180<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research
181(Kline, 2006)</p>
182<p>REFERENCE</p>
183<p>L A Feigin and D I Svergun, <em>Structure Analysis by Small-Angle X-Ray and Neutron Scattering</em>, Plenum,
184New York, 1987.</p>
185</div>
186</div>
187
188
189      </div>
190      <div class="bottomnav">
191     
192        <p>
193        «&#160;&#160;<a href="ellipsoid.html">2.1.2.1. Ellipsoid</a>
194        &#160;&#160;::&#160;&#160;
195        <a class="uplink" href="../index.html">Contents</a>
196        &#160;&#160;::&#160;&#160;
197        <a href="../ref/models/shape-lamellae.html">2.1.3. Lamellae Functions</a>&#160;&#160;»
198        </p>
199
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