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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.5.1. Parallelepiped</span></h2>
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39        <p>
40        «&#160;&#160;<a href="../ref/models/shape-parallelpiped.html">2.1.5. Parallelpiped Functions</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-sphere.html">2.1.6. Sphere Functions</a>&#160;&#160;»
45        </p>
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49       
50       
51  <div class="section" id="parallelepiped">
52<span id="id1"></span><h1>2.1.5.1. Parallelepiped<a class="headerlink" href="#parallelepiped" title="Permalink to this headline">¶</a></h1>
53<p>Rectangular parallelepiped with uniform scattering length density.</p>
54<table border="1" class="docutils">
55<colgroup>
56<col width="13%" />
57<col width="59%" />
58<col width="14%" />
59<col width="15%" />
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>Parallelepiped 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>a_side</td>
90<td>Shorter side of the parallelepiped</td>
91<td>Å</td>
92<td>35</td>
93</tr>
94<tr class="row-odd"><td>b_side</td>
95<td>Second side of the parallelepiped</td>
96<td>Å</td>
97<td>75</td>
98</tr>
99<tr class="row-even"><td>c_side</td>
100<td>Larger side of the parallelepiped</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<tr class="row-odd"><td>psi</td>
115<td>Rotation angle around its own c axis against q plane</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>The form factor is normalized by the particle volume.</p>
123<p>For information about polarised and magnetic scattering, click <a href="#id4"><span class="problematic" id="id5">here_</span></a>.</p>
124<div class="section" id="definition">
125<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
126<p>This model provides the form factor, <em>P(q)</em>, for a rectangular parallelepiped
127(below) where the form factor is normalized by the volume of the
128parallelepiped. If you need to apply polydispersity, see also the
129<a href="#id6"><span class="problematic" id="id7">RectangularPrismModel_</span></a>.</p>
130<p>The calculated form factor is:</p>
131<div class="math">
132\[P(Q) = {\text{scale} \over V} F^2(Q) + \text{background}\]</div>
133<p>where the volume <em>V</em> = <em>A B C</em> and the averaging &lt; &gt; is applied over all
134orientations for 1D.</p>
135<img alt="model/img/parallelepiped.jpg" src="model/img/parallelepiped.jpg" />
136<p><a href="#id2"><span class="problematic" id="id3">*</span></a>Figure. Parallelepiped with the corresponding Definition of sides.</p>
137<p>The edge of the solid must satisfy the condition that** <em>A</em> &lt; <em>B</em> &lt; <em>C</em>.
138Then, assuming <em>a</em> = <em>A</em> / <em>B</em> &lt; 1, <em>b</em> = <em>B</em> / <em>B</em> = 1, and
139<em>c</em> = <em>C</em> / <em>B</em> &gt; 1, the form factor is</p>
140<div class="math">
141\[P(q) = \frac{\textstyle{scale}}{V}\int_0^1 \phi(\mu \sqrt{1-\sigma^2},a)
142[S(\mu c \sigma/2)]^2 d\sigma\]</div>
143<p>with</p>
144<div class="math">
145\[\phi(\mu,a) = \int_0^1 \{S[\frac{\mu}{2}\cos(\frac{\pi}{2}u)]
146S[\frac{\mu a}{2}\sin(\frac{\pi}{2}u)]\}^2 du\]\[S(x) = \frac{\sin x}{x}\]\[\mu = qB\]</div>
147<p>and the contrast is defined as</p>
148<div class="math">
149\[\Delta\rho = \rho_{\textstyle p} - \rho_{\textstyle solvent}\]</div>
150<p>The scattering intensity per unit volume is returned in units of cm<sup>-1</sup>;
151ie, <em>I(q)</em> = φ <em>P(q)</em>.</p>
152<p>NB: The 2nd virial coefficient of the parallelpiped is calculated based on
153the averaged effective radius (= sqrt(<em>short_a</em> * <em>short_b</em> / π)) and
154length(= <em>long_c</em>) values, and used as the effective radius for
155<em>S(Q)</em> when <em>P(Q)</em> * <em>S(Q)</em> is applied.</p>
156<p>To provide easy access to the orientation of the parallelepiped, we define
157three angles Ξ, φ and Κ. The definition of Ξ and φ
158is the same as for the cylinder model (see also figures below).
159The angle Κ is the rotational angle around the <em>long_c</em> axis against
160the <em>q</em> plane. For example, Κ = 0 when the <em>short_b</em> axis is parallel
161to the <em>x</em>-axis of the detector.</p>
162<div class="figure" id="parallelepiped-orientation">
163<img alt="../_images/orientation.jpg" src="../_images/orientation.jpg" />
164<p class="caption">Figure 1: Definition of the angles for oriented parallelepipeds.</p>
165</div>
166<div class="figure">
167<img alt="../_images/orientation2.jpg" src="../_images/orientation2.jpg" />
168<p class="caption">Figure 2: Examples of the angles for oriented parallelepipeds against the detector plane.</p>
169</div>
170</div>
171<div class="section" id="validation">
172<h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2>
173<p>Validation of the code was done by comparing the output of the 1D calculation
174to the angular average of the output of a 2D calculation over all possible
175angles. The Figure below shows the comparison where the solid dot refers to
176averaged 2D while the line represents the result of the 1D calculation (for
177the averaging, 76, 180, 76 points are taken for the angles of Ξ, φ,
178and ψ respectively).</p>
179<div class="figure" id="parallelepiped-compare">
180<img alt="model/img/parallelepiped_compare.jpg" src="model/img/parallelepiped_compare.jpg" />
181</div>
182<p><em>Figure. Comparison between 1D and averaged 2D.</em></p>
183<p>This model reimplements the form factor calculations implemented in a c-library
184provided by the NIST Center for Neutron Research (Kline, 2006).</p>
185</div>
186</div>
187
188
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193        «&#160;&#160;<a href="../ref/models/shape-parallelpiped.html">2.1.5. Parallelpiped Functions</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-sphere.html">2.1.6. Sphere Functions</a>&#160;&#160;»
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