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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.3.2. Lamellarps</span></h2>
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39        <p>
40        «&#160;&#160;<a href="lamellar.html">2.1.3.1. Lamellar</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="lamellarCailleHG.html">2.1.3.3. Lamellarcaillehg</a>&#160;&#160;»
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51  <div class="section" id="lamellarps">
52<span id="id1"></span><h1>2.1.3.2. Lamellarps<a class="headerlink" href="#lamellarps" title="Permalink to this headline">¶</a></h1>
53<p>Random lamellar sheet with Caille structure factor</p>
54<table border="1" class="docutils">
55<colgroup>
56<col width="22%" />
57<col width="45%" />
58<col width="16%" />
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>thickness</td>
80<td>sheet thickness</td>
81<td>Å</td>
82<td>30</td>
83</tr>
84<tr class="row-odd"><td>Nlayers</td>
85<td>Number of layers</td>
86<td>None</td>
87<td>20</td>
88</tr>
89<tr class="row-even"><td>spacing</td>
90<td>d-spacing of Caille S(Q)</td>
91<td>Å</td>
92<td>400</td>
93</tr>
94<tr class="row-odd"><td>Caille_parameter</td>
95<td>Caille parameter</td>
96<td>Å<sup>-2</sup></td>
97<td>0.1</td>
98</tr>
99<tr class="row-even"><td>sld</td>
100<td>layer scattering length density</td>
101<td>10<sup>-6</sup>Å<sup>-2</sup></td>
102<td>6.3</td>
103</tr>
104<tr class="row-odd"><td>solvent_sld</td>
105<td>Solvent scattering length density</td>
106<td>10<sup>-6</sup>Å<sup>-2</sup></td>
107<td>1</td>
108</tr>
109</tbody>
110</table>
111<p>The returned value is scaled to units of cm<sup>-1</sup>.</p>
112<p>This model provides the scattering intensity, <span class="math">\(I(q) = P(q) S(q)\)</span>, for a
113lamellar phase where a random distribution in solution are assumed.
114Here a Caille <span class="math">\(S(Q)\)</span> is used for the lamellar stacks.</p>
115<p>The scattering intensity <span class="math">\(I(q)\)</span> is</p>
116<p>The form factor is</p>
117<p>and the structure factor is</p>
118<p>where</p>
119<p>Here <span class="math">\(d\)</span> = (repeat) spacing, <span class="math">\(\delta\)</span> = bilayer thickness,
120the contrast <span class="math">\(\Delta\rho\)</span> = SLD(headgroup) - SLD(solvent),
121<span class="math">\(K\)</span> = smectic bending elasticity, <span class="math">\(B\)</span> = compression modulus, and
122<span class="math">\(N\)</span> = number of lamellar plates (<em>n_plates</em>).</p>
123<p>NB: <strong>When the Caille parameter is greater than approximately 0.8 to 1.0, the
124assumptions of the model are incorrect.</strong> And due to a complication of the
125model function, users are responsible for making sure that all the assumptions
126are handled accurately (see the original reference below for more details).</p>
127<p>Non-integer numbers of stacks are calculated as a linear combination of
128results for the next lower and higher values.</p>
129<p>The 2D scattering intensity is calculated in the same way as 1D, where the
130<span class="math">\(q\)</span> vector is defined as</p>
131<div class="math">
132\[q = \sqrt{q_x^2 + q_y^2}\]</div>
133<p>The returned value is in units of cm<sup>-1</sup>, on absolute scale.</p>
134<img alt="../_images/lamellarCaille_1d.jpg" src="../_images/lamellarCaille_1d.jpg" />
135<p><em>Figure. 1D plot using the default values (w/6000 data point).</em></p>
136<p>Our model uses the form factor calculations as implemented in a c library
137provided by the NIST Center for Neutron Research (Kline, 2006).</p>
138<div class="section" id="reference">
139<h2>REFERENCE<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h2>
140<p>F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502</p>
141<p>also in J. Phys. Chem. B, 105, (2001) 11081-11088</p>
142</div>
143</div>
144
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150        «&#160;&#160;<a href="lamellar.html">2.1.3.1. Lamellar</a>
151        &#160;&#160;::&#160;&#160;
152        <a class="uplink" href="../index.html">Contents</a>
153        &#160;&#160;::&#160;&#160;
154        <a href="lamellarCailleHG.html">2.1.3.3. Lamellarcaillehg</a>&#160;&#160;»
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