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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.4.2. Fcc paracrystal</span></h2>
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39        <p>
40        «&#160;&#160;<a href="bcc.html">2.1.4.1. Bcc paracrystal</a>
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51  <div class="section" id="fcc-paracrystal">
52<span id="id1"></span><h1>2.1.4.2. Fcc paracrystal<a class="headerlink" href="#fcc-paracrystal" title="Permalink to this headline">¶</a></h1>
53<p>Face-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>face-centered cubic lattice</strong> with
123paracrystalline distortion. Thermal vibrations are considered to be
124negligible, and the size of the paracrystal is infinitely large.
125Paracrystalline distortion is assumed to be isotropic and characterized by
126a Gaussian distribution.</p>
127<p>The returned value is scaled to units of cm<sup>-1</sup>sr<sup>-1</sup>, absolute scale.</p>
128<div class="section" id="definition">
129<h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2>
130<p>The scattering intensity <em>I(q)</em> is calculated as</p>
131<img alt="model/img/image158.jpg" src="model/img/image158.jpg" />
132<p>where <em>scale</em> is the volume fraction of spheres, <em>Vp</em> is the volume of
133the primary particle, <em>V(lattice)</em> is a volume correction for the crystal
134structure, <em>P(q)</em> is the form factor of the sphere (normalized), and <em>Z(q)</em>
135is the paracrystalline structure factor for a face-centered cubic structure.</p>
136<p>Equation (1) of the 1990 reference is used to calculate <em>Z(q)</em>, using
137equations (23)-(25) from the 1987 paper for <em>Z1</em>, <em>Z2</em>, and <em>Z3</em>.</p>
138<p>The lattice correction (the occupied volume of the lattice) for a
139face-centered cubic structure of particles of radius <em>R</em> and nearest
140neighbor separation <em>D</em> is</p>
141<img alt="model/img/image159.jpg" src="model/img/image159.jpg" />
142<p>The distortion factor (one standard deviation) of the paracrystal is
143included in the calculation of <em>Z(q)</em></p>
144<img alt="model/img/image160.jpg" src="model/img/image160.jpg" />
145<p>where <em>g</em> is a fractional distortion based on the nearest neighbor distance.</p>
146<p>The face-centered cubic lattice is</p>
147<img alt="model/img/image161.jpg" src="model/img/image161.jpg" />
148<p>For a crystal, diffraction peaks appear at reduced q-values given by</p>
149<img alt="model/img/image162.jpg" src="model/img/image162.jpg" />
150<p>where for a face-centered cubic lattice <em>h</em>, <em>k</em>, <em>l</em> all odd or all
151even are allowed and reflections where <em>h</em>, <em>k</em>, <em>l</em> are mixed odd/even
152are forbidden. Thus the peak positions correspond to (just the first 5)</p>
153<img alt="model/img/image163.jpg" src="model/img/image163.jpg" />
154<p><strong>NB: The calculation of</strong> <em>Z(q)</em> <strong>is a double numerical integral that
155must be carried out with a high density of</strong> <strong>points to properly capture
156the sharp peaks of the paracrystalline scattering.</strong> So be warned that the
157calculation is SLOW. Go get some coffee. Fitting of any experimental data
158must be resolution smeared for any meaningful fit. This makes a triple
159integral. Very, very slow. Go get lunch!</p>
160<p>This example dataset is produced using 200 data points, <em>qmin</em> = 0.01 Å<sup>-1</sup>,
161<em>qmax</em> = 0.1 Å<sup>-1</sup> and the above default values.</p>
162<img alt="model/img/image164.jpg" src="model/img/image164.jpg" />
163<p><em>Figure. 1D plot in the linear scale using the default values (w/200 data point).</em></p>
164<p>The 2D (Anisotropic model) is based on the reference below where <em>I(q)</em> is
165approximated for 1d scattering. Thus the scattering pattern for 2D may not
166be accurate. Note that we are not responsible for any incorrectness of the
1672D model computation.</p>
168<img alt="model/img/image165.gif" src="model/img/image165.gif" />
169<img alt="model/img/image166.jpg" src="model/img/image166.jpg" />
170<p><em>Figure. 2D plot using the default values (w/200X200 pixels).</em></p>
171<p>REFERENCE</p>
172<p>Hideki Matsuoka et. al. <em>Physical Review B</em>, 36 (1987) 1754-1765
173(Original Paper)</p>
174<p>Hideki Matsuoka et. al. <em>Physical Review B</em>, 41 (1990) 3854 -3856
175(Corrections to FCC and BCC lattice structure calculation)</p>
176</div>
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