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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.1.3.3. Lamellarcaillehg</span></h2>
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39        <p>
40        «&#160;&#160;<a href="lamellarCaille.html">2.1.3.2. Lamellarps</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="lamellarFFHG.html">2.1.3.4. Lamellar ffhg</a>&#160;&#160;»
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51  <div class="section" id="lamellarcaillehg">
52<span id="id1"></span><h1>2.1.3.3. Lamellarcaillehg<a class="headerlink" href="#lamellarcaillehg" 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>tail_length</td>
80<td>Tail thickness</td>
81<td>Å</td>
82<td>10</td>
83</tr>
84<tr class="row-odd"><td>head_length</td>
85<td>head thickness</td>
86<td>Å</td>
87<td>2</td>
88</tr>
89<tr class="row-even"><td>Nlayers</td>
90<td>Number of layers</td>
91<td>None</td>
92<td>30</td>
93</tr>
94<tr class="row-odd"><td>spacing</td>
95<td>d-spacing of Caille S(Q)</td>
96<td>Å</td>
97<td>40</td>
98</tr>
99<tr class="row-even"><td>Caille_parameter</td>
100<td>Caille parameter</td>
101<td>None</td>
102<td>0.001</td>
103</tr>
104<tr class="row-odd"><td>sld</td>
105<td>Tail scattering length density</td>
106<td>10<sup>-6</sup>Å<sup>-2</sup></td>
107<td>0.4</td>
108</tr>
109<tr class="row-even"><td>head_sld</td>
110<td>Head scattering length density</td>
111<td>10<sup>-6</sup>Å<sup>-2</sup></td>
112<td>2</td>
113</tr>
114<tr class="row-odd"><td>solvent_sld</td>
115<td>Solvent scattering length density</td>
116<td>10<sup>-6</sup>Å<sup>-2</sup></td>
117<td>6</td>
118</tr>
119</tbody>
120</table>
121<p>The returned value is scaled to units of cm<sup>-1</sup>.</p>
122<p>This model provides the scattering intensity, <span class="math">\(I(q) = P(q)S(q)\)</span>, for a lamellar
123phase where a random distribution in solution are assumed. Here a Caille <span class="math">\(S(Q)\)</span>
124is used for the lamellar stacks.</p>
125<p>The scattering intensity <span class="math">\(I(q)\)</span> is</p>
126<div class="math">
127\[I(q) = 2 \pi \frac{P(q)S(q)}{\delta q^2}\]</div>
128<p>The form factor <span class="math">\(P(q)\)</span> is</p>
129<div class="math">
130\[P(q) = \frac{4}{q^2}\big\{
131\Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right]
132    + \Delta\rho_T\sin(q\delta_T)\big\}^2\]</div>
133<p>and the structure factor <span class="math">\(S(q)\)</span> is</p>
134<div class="math">
135\[S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right)
136    \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right)\]</div>
137<p>where</p>
138<div class="math">
139\[\begin{split}\begin{eqnarray}
140\alpha(n) &amp;=&amp; \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right)  \\
141\gamma_E &amp;=&amp; 0.5772156649&amp;&amp;\text{Euler's constant} \\
142\eta_{cp} &amp;=&amp; \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} &amp;&amp; \text{Caille constant}
143\end{eqnarray}\end{split}\]</div>
144<p><span class="math">\(\delta_T\)</span> is the tail length (or <em>tail_length</em>), <span class="math">\(\delta_H\)</span> is the head
145thickness (or <em>head_length</em>), <span class="math">\(\Delta\rho_H\)</span> is SLD(headgroup) - SLD(solvent),
146and <span class="math">\(\Delta\rho_T\)</span> is SLD(tail) - SLD(headgroup). Here <span class="math">\(d\)</span> is (repeat) spacing,
147<span class="math">\(K\)</span> is smectic bending elasticity, <span class="math">\(B\)</span> is compression modulus, and <span class="math">\(N\)</span> is the
148number of lamellar plates (<em>Nlayers</em>).</p>
149<p>NB: <strong>When the Caille parameter is greater than approximately 0.8 to 1.0, the
150assumptions of the model are incorrect.</strong>  And due to a complication of the
151model function, users are responsible for making sure that all the assumptions
152are handled accurately (see the original reference below for more details).</p>
153<p>Non-integer numbers of stacks are calculated as a linear combination of
154results for the next lower and higher values.</p>
155<p>The 2D scattering intensity is calculated in the same way as 1D, where
156the <span class="math">\(q\)</span> vector is defined as</p>
157<div class="math">
158\[q = \sqrt{q_x^2 + q_y^2}\]</div>
159<p>The returned value is in units of cm<sup>-1</sup>, on absolute scale.</p>
160<img alt="../_images/lamellarCailleHG_1d.jpg" src="../_images/lamellarCailleHG_1d.jpg" />
161<p><em>Figure. 1D plot using the default values (w/6000 data point).</em></p>
162<p>Our model uses the form factor calculations implemented in a C library provided
163by the NIST Center for Neutron Research (Kline, 2006).</p>
164<p>REFERENCE</p>
165<p>F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502</p>
166<p>also in J. Phys. Chem. B, 105, (2001) 11081-11088</p>
167</div>
168
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174        «&#160;&#160;<a href="lamellarCaille.html">2.1.3.2. Lamellarps</a>
175        &#160;&#160;::&#160;&#160;
176        <a class="uplink" href="../index.html">Contents</a>
177        &#160;&#160;::&#160;&#160;
178        <a href="lamellarFFHG.html">2.1.3.4. Lamellar ffhg</a>&#160;&#160;»
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