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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.4.1. Bcc paracrystal</span></h2>
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39        <p>
40        «&#160;&#160;<a href="../ref/models/shape-paracrystal.html">2.1.4. Paracrystal Functions</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="fcc.html">2.1.4.2. Fcc paracrystal</a>&#160;&#160;»
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50       
51  <div class="section" id="bcc-paracrystal">
52<span id="id1"></span><h1>2.1.4.1. Bcc paracrystal<a class="headerlink" href="#bcc-paracrystal" title="Permalink to this headline">¶</a></h1>
53<p>Body-centred cubic lattic with paracrystalline distortion</p>
54<table border="1" class="docutils">
55<colgroup>
56<col width="16%" />
57<col width="49%" />
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>dnn</td>
80<td>Nearest neighbour distance</td>
81<td>Å</td>
82<td>220</td>
83</tr>
84<tr class="row-odd"><td>d_factor</td>
85<td>Paracrystal distortion factor</td>
86<td>None</td>
87<td>0.06</td>
88</tr>
89<tr class="row-even"><td>radius</td>
90<td>Particle radius</td>
91<td>Å</td>
92<td>40</td>
93</tr>
94<tr class="row-odd"><td>sld</td>
95<td>Particle scattering length density</td>
96<td>10<sup>-6</sup>Å<sup>-2</sup></td>
97<td>4</td>
98</tr>
99<tr class="row-even"><td>solvent_sld</td>
100<td>Solvent scattering length density</td>
101<td>10<sup>-6</sup>Å<sup>-2</sup></td>
102<td>1</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>Calculates the scattering from a <strong>body-centered cubic lattice</strong> with
123paracrystalline distortion. Thermal vibrations are considered to be negligible,
124and the size of the paracrystal is infinitely large. Paracrystalline distortion
125is assumed to be isotropic and characterized by a Gaussian distribution.</p>
126<p>The returned value is scaled to units of cm<sup>-1</sup>sr<sup>-1</sup>, absolute scale.</p>
127<div class="section" id="definition">
128<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
129<p>The scattering intensity <span class="math">\(I(q)\)</span> is calculated as</p>
130<p>where <em>scale</em> is the volume fraction of spheres, <em>Vp</em> is the volume of the
131primary particle, <em>V(lattice)</em> is a volume correction for the crystal
132structure, <span class="math">\(P(q)\)</span> is the form factor of the sphere (normalized), and <span class="math">\(Z(q)\)</span>
133is the paracrystalline structure factor for a body-centered cubic structure.</p>
134<p>Equation (1) of the 1990 reference is used to calculate <span class="math">\(Z(q)\)</span>, using
135equations (29)-(31) from the 1987 paper for <em>Z1</em>, <em>Z2</em>, and <em>Z3</em>.</p>
136<p>The lattice correction (the occupied volume of the lattice) for a
137body-centered cubic structure of particles of radius <span class="math">\(R\)</span> and nearest neighbor
138separation <span class="math">\(D\)</span> is</p>
139<p>The distortion factor (one standard deviation) of the paracrystal is included
140in the calculation of <span class="math">\(Z(q)\)</span></p>
141<p>where <span class="math">\(g\)</span> is a fractional distortion based on the nearest neighbor distance.</p>
142<p>The body-centered cubic lattice is</p>
143<img alt="../_images/bcc_lattice.jpg" src="../_images/bcc_lattice.jpg" />
144<p>For a crystal, diffraction peaks appear at reduced q-values given by</p>
145<p>where for a body-centered cubic lattice, only reflections where
146<span class="math">\((h + k + l) = \text{even}\)</span> are allowed and reflections where
147<span class="math">\((h + k + l) = \text{odd}\)</span> are forbidden. Thus the peak positions
148correspond to (just the first 5)</p>
149<p><strong>NB: The calculation of :math:`Z(q)` is a double numerical integral that must
150be carried out with a high density of points to properly capture the sharp
151peaks of the paracrystalline scattering.</strong> So be warned that the calculation
152is SLOW. Go get some coffee. Fitting of any experimental data must be
153resolution smeared for any meaningful fit. This makes a triple integral.
154Very, very slow. Go get lunch!</p>
155<p>This example dataset is produced using 200 data points,
156<em>qmin</em> = 0.001 Å<sup>-1</sup>, <em>qmax</em> = 0.1 Å<sup>-1</sup> and the above default values.</p>
157<img alt="../_images/bcc_1d.jpg" src="../_images/bcc_1d.jpg" />
158<p><em>Figure. 1D plot in the linear scale using the default values
159(w/200 data point).</em></p>
160<p>The 2D (Anisotropic model) is based on the reference below where <span class="math">\(I(q)\)</span> is
161approximated for 1d scattering. Thus the scattering pattern for 2D may not
162be accurate. Note that we are not responsible for any incorrectness of the 2D
163model computation.</p>
164<img alt="../_images/bcc_orientation.gif" src="../_images/bcc_orientation.gif" />
165<img alt="../_images/bcc_2d.jpg" src="../_images/bcc_2d.jpg" />
166<p><em>Figure. 2D plot using the default values (w/200X200 pixels).</em></p>
167</div>
168<div class="section" id="reference">
169<h2>REFERENCE<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h2>
170<p>Hideki Matsuoka et. al. <em>Physical Review B</em>, 36 (1987) 1754-1765
171(Original Paper)</p>
172<p>Hideki Matsuoka et. al. <em>Physical Review B</em>, 41 (1990) 3854 -3856
173(Corrections to FCC and BCC lattice structure calculation)</p>
174</div>
175</div>
176
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181        <p>
182        «&#160;&#160;<a href="../ref/models/shape-paracrystal.html">2.1.4. Paracrystal Functions</a>
183        &#160;&#160;::&#160;&#160;
184        <a class="uplink" href="../index.html">Contents</a>
185        &#160;&#160;::&#160;&#160;
186        <a href="fcc.html">2.1.4.2. Fcc paracrystal</a>&#160;&#160;»
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