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+<body lang=EN-US link=blue vlink=purple>
+
+<div class=Section1>
+
+<p class=MsoNormal><b><span style='font-size:16.0pt'>Model Functions</span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<ul style='margin-top:0in' type=disc>
+ <li class=MsoNormal style='line-height:115%'><a href="#Introduction"><b>Introduction</b></a>
+     </li>
+ <li class=MsoNormal style='line-height:115%'><a href="#Shapes"><b>Shapes</b></a>:
+     <a href="#SphereModel">SphereModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>,
+     <a href="#CoreShellModel">CoreShellModel</a>, <a href="#CoreFourShellModel">CoreFourShellModel</a>,
+     <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>,
+     <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#CylinderModel">CylinderModel</a>,
+     <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a
+     href="#HollowCylinderModel">HollowCylinderModel</a>, <a
+     href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a
+     href="#FlexCylEllipXModel">FlexCylEllipXModel</a>, <a
+     href="#StackedDisksModel">StackedDisksModel</a>, <a
+     href="#ParallelepipedModel">ParallelepipedModel</a>, <a
+     href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a
+     href="#EllipsoidModel">EllipsoidModel</a>, <a
+     href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a
+     href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a
+     href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>,
+     <a href="#LamellarPSModel">LamellarPSModel</a>, <a
+     href="#LamellarPSHGModel">LamellarPSHGModel</a>.</li>
+ <li class=MsoNormal style='line-height:115%'><a href="#Shape-Independent"><b>Shape-Independent</b></a>:
+     <a href="#Debye">Debye</a>, <a href="#Lorentz">Lorentz</a>, <a
+     href="#DAB_Model">DAB_Model</a>, <a href="#Absolute Power_Law">Power_Law</a>, <a
+     href="#Teubner Strey">Teubner Strey</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>,
+     <a href="#FractalModel">FractalModel</a>, <a href="#Guinier">Guinier</a>,
+     <a href="#PorodModel">PorodModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>,
+     <a href="#Peak Gauss Model">Peak Gauss Model</a>, <a
+     href="#Peak Lorentz Model">Peak Lorentz Model</a>, <a href="#LineModel">LineModel</a>.</li>
+ <li class=MsoNormal style='line-height:115%'><a href="#Customized_Models"><b>Customized
+     Models</b></a>: <a href="#A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a>, <a
+     href="#sinpoly_poly">sin(poly)/poly</a>.</li>
+ <li class=MsoNormal style='line-height:115%'><a href="#Structure_Factors"><b>Structure
+     Factors</b></a>: <a href="#HardsphereStructure">HardSphereStructure</a>, <a
+     href="#SquareWellStructure">SquareWellStructure</a>, <a
+     href="#HayterMSAStructure">HayterMSAStructure</a>, <a
+     href="#StickyHSStructure">StickyHSStructure</a>. </li>
+ <li class=MsoNormal style='line-height:115%'><a href="#References"><b>References</b></a>
+     </li>
+</ul>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:16.0pt'>1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;
+</span></b><a name=Introduction><b><span style='font-size:16.0pt'>Introduction </span></b></a></p>
+
+<p class=MsoNormal>The present text documents the modules made available by the
+DANSE software for SANS. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Readers are also referred to the SANS/DANSE wiki page:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='text-indent:.25in'><i>http://danse.us/trac/sans</i></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Users can report a bug at the following URL:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='text-indent:.25in'><i>http://danse.us/trac/sans/newticket</i></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;Note: Our model uses the form factor calculations
+implemented in a c-library provided by the NIST Center for Neutron Research and
+thus some content and figures in this document are originated from or shared
+with the NIST Igor analysis package by permission (S. Kline, NIST, 2006).</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:16.0pt'>2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;
+</span></b><a name=Shapes><b><span style='font-size:16.0pt'>Shapes (Scattering
+Intensity Models)</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This software provides form factors for various particle
+shapes. After giving a mathematical definition of each model, we draw the list
+of parameters available to the user. Validation plots for each model are also
+presented. Instructions on how to use the software is available with the source
+code, available from SVN:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='text-indent:.25in'><i>svn co svn://danse.us/sans/releases/sansmodels-x.x</i></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To easily compare to the scattering intensity measured in
+experiments, we normalize the form factors by the volume of the particle:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=192 height=43 src="image001.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>with</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=159 height=29 src="image002.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <i>P</i><sub>0</sub><i>(<b>q</b>)</i> is the
+un-normalized form factor, <i>&#961;(<b>r</b>)</i> is the scattering length
+density at a given point in space and the integration is done over the volume <i>V</i>
+of the scatterer. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For systems without inter-particle interference, the form
+factors we provide can be related to the scattering intensity by the particle
+volume fraction:<img border=0 width=66 height=15
+src="image003.png">.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Our so-called 1D scattering intensity functions provide <i>P(q)
+</i>for the case where the scatterer is randomly oriented. In that case, the
+scattering intensity only depends on the length of q. The intensity measured on
+the plane of the SANS detector will have an azimuthal symmetry around <i>q</i>=0.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Our so-called 2D scattering intensity functions provide <i>P(q,
+</i><i><span style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> for an
+oriented system as a function of a q-vector in the plane of the detector. We
+define the angle <span style='font-family:"Arial","sans-serif"'>&#966;</span>
+as the angle between the q vector and the horizontal (x) axis of the plane of
+the detector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=SphereModel><b><span style='font-size:14.0pt'>Sphere Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(q), for a
+monodisperse spherical particle with uniform scattering length density. The
+form factor is normalized by the particle volume as described below.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 1D scattering intensity is calculated in the following
+way (Guinier, 1955):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=337 height=53 src="image004.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <i>scale</i> is a scale factor* volume fraction, V is
+the volume of the scatterer, <i>r</i> is the radius of the sphere, <i>bkg</i>
+is the background level and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i>
+(contrast)&nbsp; is the scattering length density (SLD) &nbsp;difference
+between the scatterer and the solvent it is in.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above,
+regardless of the orientation of the q vector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the sphere model are the following:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:13.5pt'>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:13.5pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:13.5pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:13.5pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
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+ <tr style='height:12.8pt'>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:12.8pt'>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>60</p>
+  </td>
+ </tr>
+ <tr style='height:12.8pt'>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>sldSph</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>2.0e-6</p>
+  </td>
+ </tr>
+ <tr style='height:12.8pt'>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>1.0e-6</p>
+  </td>
+ </tr>
+ <tr style='height:12.8pt'>
+  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the
+sphere model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D model to the output of the software provided by the NIST (Kline, 2006).
+Figure 1 shows a comparison of the output of our model and the output of the
+NIST software.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText align=center style='text-align:center;page-break-after:
+avoid'><img border=0 width=439 height=287
+src="image005.jpg" alt="sphere_1D_validation"></p>
+
+<p class=MsoBodyText align=center style='text-align:center;page-break-after:
+avoid'>&nbsp;</p>
+
+<p class=MsoCaption>Figure 1: Comparison of the DANSE scattering intensity for
+a sphere with the output of the NIST SANS analysis software. The parameters
+were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>, and
+Background=0.01 cm<sup> -1</sup>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=StickyHSStructure></a><a name=FuzzySphereModel></a><b><span
+style='font-size:14.0pt'>FuzzySphereModel</span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b>This model is to calculate the scattering from
+spherical particles with a &quot;fuzzy&quot; interface. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 1D scattering intensity is calculated in the following
+way (Guinier, 1955):</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm-1 sr-1], absolute
+scale.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The scattering intensity I(q) is calculated as:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='position:relative;top:5pt'><img border=0 width=12 height=23
+src="image101.gif"><img border=0 width=233 height=41
+src="image102.gif"></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>where the amplitude A(q) is given as the typical sphere
+scattering convoluted with a Gaussian to get a gradual drop-off in the
+scattering length density:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='position:relative;top:18pt'><img border=0 width=315 height=56
+src="image103.gif"></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Here A<sup>2</sup>(q) is the form factor, P(q). The scale
+is equivalent to the volume fraction of spheres, each of volume, V. Contrast (<i><span
+style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> ) is the
+difference of scattering length densities of the sphere and the surrounding
+solvent.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The poly-dispersion in radius and in fuzziness is provided.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>(direct from the reference)</p>
+
+<p class=MsoNormal>The &quot;fuzziness&quot; of the interface is defined by the
+parameter (sigma)<sub>fuzzy</sub>. The particle radius R represents the radius
+of the particle where the scattering length density profile decreased to 1/2 of
+the core density. The (sigma)<sub>fuzzy</sub>&nbsp; is the width of the smeared
+particle surface: i.e., the standard deviation from the average height of the
+fuzzy interface. The inner regions [] that display a higher density are
+described by the radial box profile extending to a radius of approximately Rbox
+~ R - 2(sigma). the profile approaches zero as Rsans ~ R + 2(sigma).</p>
+
+<p class=MsoNormal>For 2D data: The 2D scattering intensity is calculated in
+the same way as 1D, where the <i>q</i> vector is defined as<span
+style='font-size:14.0pt'><img border=0 width=82 height=26
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>M. Stieger, J. S. Pedersen, P. Lindner, W. Richtering,
+Langmuir 20 (2004) 7283-7292.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>TEST DATASET</p>
+
+<p class=MsoNormal>This example dataset is produced by running the
+FuzzySphereModel, using 200 data points, qmin = 0.001 Å<sup>-1</sup>,&nbsp;
+qmax = 0.7 A<sup>-1</sup> and the default coef_fuzz values:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>60</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>fuzziness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldSph</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=442 height=275 src="image104.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=CoreShellModel><b><span style='font-size:14.0pt'>Core Shell
+(Sphere) Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a spherical
+particle with a core-shell structure. The form factor is normalized by the
+particle volume.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 1D scattering intensity is calculated in the following
+way (Guinier, 1955):</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=387 height=55 src="image006.png">&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=right style='text-align:right'>&nbsp;</p>
+
+<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>s</sub></i>
+is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the
+core, <i>r<sub>s</sub></i> is the radius of the shell, <i>r<sub>c</sub></i> is
+the radius of the core, <i>&#961;<sub>c</sub></i> is the scattering length
+density of the core, <i>&#961;<sub>s</sub></i> is the scattering length density
+of the shell, &#961;<sub>solv</sub> is the scattering length density of the
+solvent, and <i>bkg</i> is the background level. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above,
+regardless of the orientation of the q vector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The outer most radius (= radius + thickness) is
+used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the core-shell sphere model are the following:</p>
+
+<p class=MsoNormal>Here, radius = the radius of the core and thickness = the
+thickness of the shell.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>(core) radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>60</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>thickness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>shell_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>2e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>solvent_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.5pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Guinier, A. and G. Fournet, &quot;Small-Angle Scattering of
+X-Rays&quot;, John Wiley and Sons, New York, (1955).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the
+core-shell sphere model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D model to the output of the software provided by the NIST (Kline, 2006).
+Figure 1 shows a comparison of the output of our model and the output of the
+NIST software.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText align=center style='text-align:center;page-break-after:
+avoid'><img border=0 width=466 height=289
+src="image007.jpg" alt="core_shell_sphere_1D_validation"></p>
+
+<p class=MsoCaption>Figure 7: Comparison of the DANSE scattering intensity for
+a core-shell sphere with the output of the NIST SANS analysis software. The
+parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>,
+and Background=0.001 cm<sup> -1</sup>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=CoreFourShellModel><b><span style='font-size:14.0pt'>CoreFourShell(Sphere)Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the scattering from monodisperse core 4
+shell structures. It &nbsp;has a core of a specified radius, with four shells.
+The SLDs of the core and each shell are individually specified.&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm-1sr-1],
+absolute scale.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model is a trivial extension of the CoreShell function
+to a larger number of shells. See the CoreShell function for a diagram and
+documentation.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Be careful that the SLDs and scale can be highly correlated.
+Hold as many of these fixed as possible.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as P(q) of 1D,
+regardless of the orientation of the q vector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The outer most radius (= radius + 4 thicknesses) is
+used as the effective radius toward S(Q) if P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the CoreFourshell sphere model are the following:</p>
+
+<p class=MsoNormal>Here, rad_core = the radius of the core, &nbsp;thick_shelli
+= the thickness of the shell i and sld_shelli = the SLD of the shell i.</p>
+
+<p class=MsoNormal>And the sld_core and the sld_solv are the SLD of the core
+and the solvent, respectively.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>rad_core</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>60</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_core</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.4e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_shell1</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_shell2</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>2e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_shell3</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_shell4</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>4e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_solv</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.4e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>thick_shell1</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>thick_shell2</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>thick_shell3</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>thick_shell4</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:19.5pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>See the CoreShell documentation.&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>TEST DATASET</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This example dataset is produced by running the
+CoreFourShellModel using 200 data points, qmin = 0.001 Å-1,&nbsp; qmax = 0.7
+Å-1 and the above default values.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=412 height=287 src="image105.jpg"></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.5.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=VesicleModel><b><span style='font-size:14.0pt'>VesicleModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for an
+unilamellar vesicle. The form factor is normalized by the volume of the shell.</p>
+
+<p class=MsoNormal>The 1D scattering intensity is calculated in the following
+way (Guinier, 1955):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=377 height=54 src="image008.png">&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=right style='text-align:right'>&nbsp;</p>
+
+<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>shell</sub></i>
+is the volume of the shell, <i>V<sub>1</sub></i> is the volume of the core, <i>V<sub>2</sub></i>
+is the total volume, <i>R<sub>1</sub></i> is the radius of the core, <i>r<sub>2</sub></i>
+is the outer radius of the shell, <i>&#961;<sub>1</sub></i> is the scattering
+length density of the core and the solvent, <i>&#961;<sub>2</sub></i> is the
+scattering length density of the shell, and <i>bkg</i> is the background level.
+And <i>J<sub>1</sub></i> = (sin<i>x </i>- <i>x</i>cos<i>x</i>)/<i>x</i><sup>2</sup>.
+The functional form is identical to a &quot;typical&quot; core-shell structure,
+except that the scattering is normalized by the volume that is contributing to
+the scattering, namely the volume of the shell alone. Also, the vesicle is best
+defined in terms of a core radius (= R1) and a shell thickness, t. &nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=149 height=143 src="image009.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as <i>P</i>(<i>q</i>)
+above, regardless of the orientation of the <i>q</i> vector&nbsp; which is
+defined as<span style='font-size:14.0pt'><img border=0 width=69 height=22
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>For P*S: The outer most radius (= radius + thickness) is
+used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the vesicle model are the following:</p>
+
+<p class=MsoNormal>In the parameters, the radius represents the core radius
+(R<sub>1</sub>) and the thickness (R<sub>2</sub>  R<sub>1</sub>) is the
+shell thickness.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>100</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>thickness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>30</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>shell_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=454 height=275 src="image011.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>Guinier, A. and G. Fournet, &quot;Small-Angle Scattering of
+X-Rays&quot;, John Wiley and Sons, New York, (1955).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.6.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=MultiShellModel><b><span style='font-size:14.0pt'>MultiShellModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a
+multi-lamellar vesicle with N shells where the core is filled with solvent and the
+shells are interleaved with layers of solvent. For N = 1, this return to the
+vesicle model (above).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=255 height=224 src="image012.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of
+the orientation of the <i>q</i> vector which is defined as<span
+style='font-size:14.0pt'><img border=0 width=59 height=19
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>For P*S: The outer most radius (= core_radius +
+n_pairs&nbsp; * s_thickness&nbsp; + (n_pairs -1) * w_thickness) is used as the
+effective radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the multi-shell model are the following:</p>
+
+<p class=MsoNormal>In the parameters, the s_thickness is the shell thickness
+while the w_thickness is the solvent thickness, and the n_pair is the number
+of shells.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>60.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>n_pairs</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>2.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>shell_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>s_thickness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>w_thickness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=447 height=289 src="image013.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>Cabane, B., Small Angle Scattering Methods, Surfactant
+Solutions: New Methods of Investigation, Ch.2, Surfactant Science Series Vol.
+22, Ed. R. Zana, M. Dekker, New York, 1987.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.7.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=BinaryHSModel><b><span style='font-size:14.0pt'>BinaryHSModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model (binary hard sphere model) provides the
+scattering intensity, for binary mixture of spheres including hard sphere
+interaction between those particles. Using Percus-Yevick closure, the
+calculation is an exact multi-component solution:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=349 height=25 src="image014.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where S<sub>ij</sub> are the partial structure factors and f<sub>i</sub>
+are the scattering amplitudes of the particles. And the subscript 1 is for the
+smaller particle and 2 is for the larger. The number fraction of the larger
+particle, (<i>x</i> = &nbsp;n<sub>2</sub>/(n<sub>1</sub>+n<sub>2</sub>), n =
+the number density) is internally calculated based on:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=248 height=98 src="image015.png">.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of
+the orientation of the <i>q</i> vector which is defined as<span
+style='font-size:14.0pt'><img border=0 width=69 height=22
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The parameters of the binary hard sphere are the following
+(in the names, l (or ls) stands for larger spheres while s (or ss) for the
+smaller spheres):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>l_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>100.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>ss_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>ls_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>solvent_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>s_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>25.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>vol_frac_ls</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>vol_frac_ss</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.2</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=447 height=313 src="image016.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values above (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>See the reference for details.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>N. W. Ashcroft and D. C. Langreth, Physical Review, v. 156
+(1967) 685-692.</p>
+
+<p class=MsoNormal>[Errata found in Phys. Rev. 166 (1968) 934.]</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.8.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=CylinderModel><b><span style='font-size:14.0pt'>Cylinder
+Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor for a right circular
+cylinder with uniform scattering length density. The form factor is normalized
+by the particle volume.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The output of the 2D scattering intensity function for
+oriented cylinders is given by (Guinier, 1955):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=184 height=41 src="image017.png">&nbsp; (4)</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=359 height=45 src="image018.png">&nbsp;&nbsp; </p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span>
+is the angle between the axis of the cylinder and the q-vector, V is the volume
+of the cylinder, L is the length of the cylinder, r is the radius of the cylinder,
+and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i>
+(contrast) is the scattering length density difference between the scatterer
+and the solvent. J<sub>1</sub> is the first order Bessel function.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the cylinder,
+we define the axis of the cylinder using two angles theta and phi. Those angles
+are defined on Figure 2.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+border=0 width=478 height=258 src="image019.jpg"
+alt=cylinderangles.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'><a
+name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure 2a. Definition of
+the angles for oriented cylinders.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=409 height=276 src="image020.jpg"
+alt=cylinderangles2.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'>Figure 2b. Examples
+of the angles for oriented cylinders against the detector plane.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the
+cylinder is calculate based on the radius and length values, and used as the
+effective radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the cylinder model are the following:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>20.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>400.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>contrast</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>3.0e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>cyl_theta</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:20.6pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
+  <p class=MsoBodyText>cyl_phi</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
+  <p class=MsoBodyText>Radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The output of the 1D scattering intensity function for
+randomly oriented cylinders is then given by:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><a
+name="_Ref173306528"></a><a name="_Ref173306479"></a><img border=0 width=253
+height=56 src="image021.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The <i>cyl_theta</i> and <i>cyl_phi</i> parameter are not
+used for the 1D output. Our implementation of the scattering kernel and the 1D
+scattering intensity use the c-library from NIST.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the
+cylinder model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D model to the output of the software provided by the NIST (Kline, 2006).
+Figure 3 shows a comparison of the 1D output of our model and the output of the
+NIST software.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>In general, averaging over a distribution of orientations is
+done by evaluating the following:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=244 height=51 src="image022.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <i>p(&#952;,</i><i><span style='font-family:"Arial","sans-serif"'>&#966;</span>)</i>
+is the probability distribution for the orientation and <i>P<sub>0</sub>(q,</i><i><span
+style='font-family:"Arial","sans-serif"'>&#945;</span>)</i> is the scattering
+intensity for the fully oriented system. Since we have no other software to
+compare the implementation of the intensity for fully oriented cylinders, we
+can compare the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span
+style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.&nbsp; Figure
+4 shows the result of such a cross-check.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+border=0 width=512 height=282 src="image023.jpg"
+alt="cylinder_1D_validation"></p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p>
+
+<p class=MsoCaption><a name="_Ref173211066"></a><a name="_Ref173211049"></a>Figure
+3: Comparison of the DANSE scattering intensity for a cylinder with the output
+of the NIST SANS analysis software. The parameters were set to: Scale=1.0,
+Radius=20 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01
+cm<sup> -1</sup>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoCaption>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+border=0 width=467 height=287 src="image024.jpg"
+alt="cylinder_2D_average"></p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p>
+
+<p class=MsoCaption><a name="_Ref173213305">Figure </a>4: Comparison of the intensity
+for uniformly distributed cylinders calculated from our 2D model and the
+intensity from the NIST SANS analysis software. The parameters used were:
+Scale=1.0, Radius=20 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and
+Background=0.0 cm<sup> -1</sup>.</p>
+
+<p class=MsoCaption>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.9.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=CoreShellCylinderModel><b><span style='font-size:14.0pt'>Core-Shell
+Cylinder Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor for a circular cylinder
+with a core-shell scattering length density profile. The form factor is
+normalized by the particle volume.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The output of the 2D scattering intensity function for
+oriented core-shell cylinders is given by (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=184 height=45 src="image025.png">&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=357 height=93 src="image026.png">&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span>
+is the angle between the axis of the cylinder and the q-vector, <i>V<sub>s</sub></i>
+is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the
+core, <i>L</i> is the length of the core, <i>r</i> is the radius of the core, <i>t</i>
+is the thickness of the shell, <i>&#961;<sub>c</sub></i> is the scattering
+length density of the core, <i>&#961;<sub>s</sub></i> is the scattering length
+density of the shell, &#961;<sub>solv</sub> is the scattering length density of
+the solvent, and <i>bkg</i> is the background level. The outer radius of the
+shell is given by <i>r+t</i> and the total length of the outer shell is given
+by <i>L+2t</i>. J<sub>1</sub> is the first order Bessel function.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=384 height=196 src="image027.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the core-shell
+cylinder, we define the axis of the cylinder using two angles &#952; and <span
+style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
+of the cylinder, those angles are defined on Figure 2 in Cylinder Model.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid
+cylinder is calculate based on the (radius+thickness) and 2(length +thickness)
+values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.
+</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the core-shell cylinder model are the following:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>20.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>thickness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>10.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>400.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>core_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>shell_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>4e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>solvent_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>axis_theta</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.57</p>
+  </td>
+ </tr>
+ <tr style='height:20.0pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.0pt'>
+  <p class=MsoBodyText>axis_phi</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:20.0pt'>
+  <p class=MsoBodyText>Radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:20.0pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The output of the 1D scattering intensity function for randomly
+oriented cylinders is then given by the equation above.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not
+used for the 1D output. Our implementation of the scattering kernel and the 1D
+scattering intensity use the c-library from NIST.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the
+core-shell cylinder model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D model to the output of the software provided by the NIST (Kline, 2006).
+Figure 8 shows a comparison of the 1D output of our model and the output of the
+NIST software.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Averaging over a distribution of orientation is done by
+evaluating the equation above. Since we have no other software to compare the
+implementation of the intensity for fully oriented core-shell cylinders, we can
+compare the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span
+style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.&nbsp; Figure
+9 shows the result of such a cross-check.</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+border=0 width=454 height=272 src="image028.jpg"
+alt="core_shell_cylinder_1D_validation"></p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p>
+
+<p class=MsoCaption><a name="_Ref173307149">Figure </a>8: Comparison of the
+DANSE scattering intensity for a core-shell cylinder with the output of the
+NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20
+Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6
+Å<sup> -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup>
+-1</sup>.</p>
+
+<p class=MsoCaption>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+border=0 width=458 height=285 src="image029.jpg"
+alt="core_shell_cylinder_2D_average"></p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p>
+
+<p class=MsoCaption><a name="_Ref173307204">Figure </a>9: Comparison of the
+intensity for uniformly distributed core-shell cylinders calculated from our 2D
+model and the intensity from the NIST SANS analysis software. The parameters
+used were: Scale=1.0, Radius=20 Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6
+Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup> -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>,
+and Background=0.0 cm<sup> -1</sup>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.10.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=HollowCylinderModel><b><span style='font-size:14.0pt'>HollowCylinderModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a
+monodisperse hollow right angle circular cylinder (tube) where the form factor
+is normalized by the volume of the tube:</p>
+
+<p class=MsoNormal>P(q) = scale*&lt;f^2&gt;/V<sub>shell</sub>+background where
+the averaging &lt; &gt; id applied only for the 1D calculation. &nbsp;The
+inside and outside of the hollow cylinder have the same SLD.</p>
+
+<p class=MsoNormal>The 1D scattering intensity is calculated in the following
+way (Guinier, 1955):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=406 height=176 src="image030.png">&nbsp;&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=right style='text-align:right'>&nbsp;</p>
+
+<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>J<sub>1</sub></i>
+is the 1<sup>st</sup> order Bessel function, <i>J<sub>1</sub></i> (x)= (sin<i>x
+</i>- <i>x</i>cos<i>x</i>)/<i>x</i><sup>2</sup>. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=393 height=217 src="image031.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the core-shell
+cylinder, we define the axis of the cylinder using two angles &#952; and <span
+style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
+of the cylinder, those angles are defined on Figure 2 in Cylinder Model.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid
+cylinder is calculate based on the (radius) and 2(length) values, and used as
+the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>In the parameters, the contrast represents SLD (shell) -
+SLD (solvent) and the radius = R<sub>hell</sub> while core_radius = R<sub>core</sub>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>30</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>400</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>20</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldCyl</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>5e-06</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.01</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=443 height=259 src="image106.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoBodyText>Feigin, L. A, and D. I. Svergun, &quot;Structure Analysis
+by Small-Angle X-Ray and Neutron Scattering&quot;, Plenum Press, New York,
+(1987).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.11.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=FlexibleCylinderModel></a><b><span style='font-size:14.0pt'>FlexibleCylinderModel</span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a
+flexible cylinder where the form factor is normalized by the volume of the
+cylinder: Inter-cylinder interactions are NOT included. P(q) =
+scale*&lt;f^2&gt;/V+background where the averaging &lt; &gt;&nbsp; is applied
+over all orientation for 1D. &nbsp;The 2D scattering intensity is the same as
+1D, regardless of the orientation of the <i>q</i> vector which is defined as<span
+style='font-size:14.0pt'><img border=0 width=71 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=329 height=150 src="image033.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The chain of contour length, L, (the total length) can be
+described a chain of some number of locally stiff segments of length lp. The
+persistence length,lp, is the length along the cylinder over which the flexible
+cylinder can be considered a rigid rod. The Kuhn length (b = 2*lp) is also used
+to describe the stiffness of a chain. The returned value is in units of [cm-1],
+on absolute scale. In the parameters, the sldCyl and sldSolv&nbsp; represent
+SLD (chain/cylinder) &nbsp;and SLD (solvent) respectively.</p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>20</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1000</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldCyl</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-06</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.3e-06</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.01</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>kuhn_length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>100</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=434 height=291 src="image034.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>From the reference, &quot;Method 3 With Excluded
+Volume&quot; is used. The model is a parametrization of simulations of a
+discrete representation of the worm-like chain model of Kratky and Porod
+applied in the pseudocontinuous limit.&nbsp; See equations (13,26-27) in the original
+reference for the details.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Pedersen, J. S. and P. Schurtenberger (1996). Scattering
+functions of semiflexible polymers with and without excluded volume effects.
+Macromolecules 29: 7602-7612. </p>
+
+<p class=MsoNormal>Correction of the formula can be found in:</p>
+
+<p class=MsoNormal>Wei-Ren Chen, Paul D. Butler, and Linda J. Magid,
+&quot;Incorporating Intermicellar Interactions in the Fitting of SANS Data from
+Cationic Wormlike Micelles&quot; Langmuir, August 2006.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.12.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=FlexCylEllipXModel><b><span style='font-size:14.0pt'>FlexCylEllipXModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>Flexible Cylinder with Elliptical Cross-Section: &nbsp;</b>Calculates
+the form factor for a flexible cylinder with an elliptical cross section and a
+uniform scattering length density. The non-negligible diameter of the cylinder
+is included by accounting for excluded volume interactions within the walk of a
+single cylinder. The form factor is normalized by the particle volume such that
+P(q) = scale*&lt;f^2&gt;/Vol + bkg, where &lt; &gt; is an average over all
+possible orientations of the flexible cylinder. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The function calculated is from the reference given below.
+From that paper, &quot;Method 3 With Excluded Volume&quot; is used. The model
+is a parameterization of simulations of a discrete representation of the
+worm-like chain model of Kratky and Porod applied in the pseudo-continuous limit.&nbsp;
+See equations (13, 26-27) in the original reference for the details.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>NOTE: there are several typos in the original reference that
+have been corrected by WRC. Details of the corrections are in the reference
+below.</p>
+
+<p class=MsoNormal>&nbsp;- Equation (13): the term (1-w(QR)) should swap
+position with w(QR)</p>
+
+<p class=MsoNormal>&nbsp;- Equations (23) and (24) are incorrect. WRC has
+entered these into Mathematica and solved analytically. The results were
+converted to code.</p>
+
+<p class=MsoNormal>&nbsp;- Equation (27) should be q0 = max(a3/sqrt(RgSquare),3)
+instead of max(a3*b/sqrt(RgSquare),3)</p>
+
+<p class=MsoNormal>&nbsp;- The scattering function is negative for a range of
+parameter values and q-values that are experimentally accessible. A correction
+function has been added to give the proper behavior.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=329 height=150 src="image033.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The chain of contour length, L, (the total length) can be
+described a chain of some number of locally stiff segments of length lp. The
+persistence length, lp, is the length along the cylinder over which the
+flexible cylinder can be considered a rigid rod. The Kuhn length (b) used in
+the model is also used to describe the stiffness of a chain, and is simply b =
+2*lp.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The cross section of the cylinder is elliptical, with minor
+radius a. The major radius is larger, so of course, the 'axis_ratio' 
+must be greater than one. Simple constraints should be applied during curve
+fitting to maintain this inequality.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is in units of [cm-1], on absolute scale.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The sldCyl&nbsp; = SLD (chain), sldSolv = SLD (solvent). The
+scale, and the contrast are both multiplicative factors in the model and are
+perfectly correlated. One or both of these parameters must be held fixed during
+model fitting.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>If the scale is set equal to the particle volume fraction,
+<span style='font-family:Symbol'>f</span>, the returned value is the scattered
+intensity per unit volume, I(q) = <span style='font-family:Symbol'>f</span>*P(q).
+However, no inter-particle interference effects are included in this
+calculation.</p>
+
+<p class=MsoNormal>For 2D data: The 2D scattering intensity is calculated in
+the same way as 1D, where the <i>q</i> vector is defined as<span
+style='font-size:14.0pt'><img border=0 width=82 height=26
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>Pedersen, J. S. and P. Schurtenberger (1996). Scattering
+functions of semiflexible polymers with and without excluded volume effects.
+Macromolecules 29: 7602-7612. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Corrections are in:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Wei-Ren Chen, Paul D. Butler, and Linda J. Magid,
+&quot;Incorporating Intermicellar Interactions in the Fitting of SANS Data from
+Cationic Wormlike Micelles&quot; Langmuir, August 2006.</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal>TEST DATASET</p>
+
+<p class=MsoNormal>This example dataset is produced by running the Macro FlexCylEllipXModel,
+using 200 data points, qmin = 0.001 Å-1,&nbsp; qmax = 0.7 Å-1 and the default
+values below.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>axis_ratio</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'></td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.5</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0001</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Kuhn_length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>100</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>(Contour) length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e+3</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>20.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'></td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldCyl</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-6</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.3e-6</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=440 height=300 src="image107.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.13.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=StackedDisksModel><b><span style='font-size:14.0pt'>StackedDisksModel
+</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for
+stacked discs (tactoids) with a core/layer structure where the form factor is
+normalized by the volume of the cylinder. &nbsp;Assuming the next neighbor
+distance (d-spacing) in a stack of parallel discs obeys a Gaussian
+distribution, a structure factor S(q) proposed by Kratky and Porod in 1949 is
+used in this function. Note that the resolution smearing calculation uses 76
+Gauss quadrature points to properly smear the model since the function is
+HIGHLY oscillatory, especially around the q-values that correspond to the
+repeat distance of the layers.</p>
+
+<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of
+the orientation of the <i>q</i> vector which is defined as<span
+style='font-size:14.0pt'><img border=0 width=68 height=22
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+<img border=0 width=214 height=93 src="image035.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=173 height=136 src="image036.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup> sr<sup>-1</sup>],
+on absolute scale. </p>
+
+<p class=MsoNormal>The scattering intensity I(q) is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=325 height=38 src="image037.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where the contrast,</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=115 height=24 src="image038.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>N is the number of discs per unit volume, <span
+style='font-family:Symbol'>a</span>&nbsp;is the angle between the axis of the
+disc and q, and Vt and Vc are the total volume and the core volume of a single
+disc, respectively.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=374 height=113 src="image039.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where d = thickness of the layer (layer_thick), 2h= core
+thickness (core_thick),&nbsp; and R = radius of the disc (radius).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=348 height=46 src="image040.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>where n = the total number of the disc stacked (n_stacking),
+D=the next neighbor center to cent distance (d-spacing), and <span
+style='font-family:Symbol'>s</span><sub>D</sub>&nbsp;= the Gaussian standard
+deviation of the d-spacing (sigma_d).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the
+stackeddisks, we define the axis of the cylinder using two angles &#952; and <span
+style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
+of the cylinder, those angles are defined on Figure 2 of CylinderModel.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;For P*S: The 2<sup>nd</sup> virial coefficient of the
+solid cylinder is calculate based on the (radius) and length =
+n_stacking*(core_thick +2*layer_thick) values, and used as the effective radius
+toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>4e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>core_thick</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>layer_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>layer_thick</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>15</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>n_stacking</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>3e+003</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.01</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sigma_d</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>solvent_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>5e-006</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=451 height=297 src="image041.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=377 height=215 src="image042.jpg"
+alt=stackdiskangles.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of
+the angles for oriented stackeddisks against the detector plane.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Guinier, A. and Fournet, G., &quot;Small-Angle Scattering of
+X-Rays&quot;, John Wiley and Sons, New York, 1955.</p>
+
+<p class=MsoNormal>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949. </p>
+
+<p class=MsoNormal>Higgins, J.S. and Benoit, H.C., &quot;Polymers and Neutron
+Scattering&quot;, Clarendon, Oxford, 1994.</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.14.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=ParallelepipedModel><b><span style='font-size:14.0pt'>ParallelepipedModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a rectangular
+cylinder (below) where the form factor is normalized by the volume of the
+cylinder. P(q) = scale*&lt;f^2&gt;/V+background where the volume V= ABC and the
+averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;</p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=239 height=181 src="image043.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The side of the solid must be satisfied the condition of
+A&lt;B&lt;C in order for the calculation to be correct: where A = short_a,
+B=short_b, and C=long_c for the fitting parameters.</p>
+
+<p class=MsoNormal>By this definition, assuming </p>
+
+<p class=MsoNormal>a = A/B&lt;1; b=B/B=1; c=C/B&gt;1, the form factor,</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=347 height=145 src="image044.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The contrast is defined as</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=117 height=24 src="image045.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The scattering intensity per unit volume is returned in the
+unit of [cm<sup>-1</sup>]; &nbsp;I(q) = <span style='font-family:Symbol'>f</span>P(q).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid
+cylinder is calculate based on the averaged radius (= sqrt(short_a*short_b/pi))
+and length( = long_c) values, and used as the effective radius toward S(Q) when
+P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the
+parallelepiped, we define the axis of the cylinder using two angles &#952; , <span
+style='font-family:"Arial","sans-serif"'>&#966; </span>and<span
+style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span
+style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the
+cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span>
+are defined on Figure 2 of CylinderModel. The angle <span style='font-family:
+Symbol'>Y </span>is the rotational angle around its own long_c axis against the
+q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the
+short_b axis is parallel to the x-axis of the detector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=352 height=264 src="image046.jpg"
+alt=pprangles.gif></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b>Figure. Definition
+of angels for 2D</b>.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=379 height=256 src="image047.jpg"
+alt=cylinderangles2.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of
+the angles for oriented elliptical cylinders against the detector plane.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>contrast</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>5e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>long_c</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>400</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>short_a</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>35</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>short_b</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>75</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=431 height=298 src="image048.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span
+style='font-family:Symbol'>·</span><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span><b>Validation of the parallelepiped 2D model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D calculation to the angular average of the output of 2 D calculation over
+all possible angles. The Figure below shows the comparison where the solid dot
+refers to averaged 2D while the line represents the result of 1D calculation
+(for the averaging, 76, 180, 76 points are taken over the angles of theta, phi,
+and psi respectively).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=441 height=299 src="image049.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+Comparison between 1D and averaged 2D.</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Mittelbach and Porod, Acta Physica Austriaca 14 (1961)
+185-211.</p>
+
+<p class=MsoNormal>Equations (1), (13-14). (in German)</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.15.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=EllipticalCylinderModel><b><span style='font-size:14.0pt'>Elliptical
+Cylinder Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This function calculates the scattering from an oriented
+elliptical cylinder. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>For 2D (orientated system):</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The angles theta and phi define the orientation of the axis
+of the cylinder. The angle psi is defined as the orientation of the major axis
+of the ellipse with respect to the vector Q. A gaussian poydispersity can be
+added to any of the orientation angles, and also for the minor radius and the
+ratio of the ellipse radii.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=297 height=145 src="image050.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><b>Figure. a= r_minor
+and </b><b><span style='font-family:Symbol'>n</span>= r_ratio (i.e.,
+r_major/r_minor).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The function calculated is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=346 height=46 src="image051.png">&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>with the functions:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=297 height=114 src="image052.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+(13)</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>and the angle psi is defined as the orientation of the major
+axis of the ellipse with respect to the vector Q.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>For 1D (no preferred orientation):</b></p>
+
+<p class=MsoNormal>The form factor is averaged over all possible orientation
+before normalized by the particle volume: P(q) = scale*&lt;f^2&gt;/V .</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the elliptical,
+we define the axis of the cylinder using two angles &#952; , <span
+style='font-family:"Arial","sans-serif"'>&#966; </span>and<span
+style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span
+style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the
+cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span>
+are defined on Figure 2 of CylinderModel. The angle <span style='font-family:
+Symbol'>Y </span>is the rotational angle around its own long_c axis against the
+q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the
+r_minor axis is parallel to the x-axis of the detector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>All angle parameters are valid and given only for 2D
+calculation (Oriented system).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=370 height=277 src="image053.jpg"
+alt=ellcylinderangles.gif></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b>Figure. Definition
+of angels for 2D</b>.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=379 height=256 src="image047.jpg"
+alt=cylinderangles2.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'><span
+style='font-size:12.0pt'>Figure. Examples of the angles for oriented elliptical
+cylinders </span></p>
+
+<p class=MsoCaption align=center style='text-align:center'><span
+style='font-size:12.0pt'>against the detector plane.</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>For P*S</b>: The 2<sup>nd</sup> virial coefficient of the
+solid cylinder is calculate based on the averaged radius
+(=sqrt(r_minor^2*r_ratio))&nbsp; and length values, and used as the effective
+radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>r_minor</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>20.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>r_ratio</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.5</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>400.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>contrast</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>3e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cyl_phi</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cyl_psi</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cyl_theta</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.57</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=443 height=293 src="image054.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span
+style='font-family:Symbol'>·</span><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span><b>Validation of the elliptical cylinder 2D model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D calculation to the angular average of the output of 2 D calculation over
+all possible angles. The Figure below shows the comparison where the solid dot
+refers to averaged 2D while the line represents the result of 1D calculation
+(for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi,
+and psi respectively).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=448 height=278 src="image055.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+Comparison between 1D and averaged 2D.</b></p>
+
+<p class=MsoNormal style='text-autospace:none'><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='text-autospace:none'>In the 2D average, more binning
+in the angle phi is necessary to get the proper result. The following figure shows
+the results of the averaging by varying the number of bin over angles.</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><img
+border=0 width=409 height=303 src="image056.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+The intensities averaged from 2D over different number </b></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>of
+points of binning of angles.</b></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>REFERENCE</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='text-indent:.25in'>L. A. Feigin and D. I. Svergun Structure
+Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York,
+(1987).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.16.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=EllipsoidModel><b><span style='font-size:14.0pt'>Ellipsoid
+Model</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor for an ellipsoid
+(ellipsoid of revolution) with uniform scattering length density. The form
+factor is normalized by the particle volume.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The output of the 2D scattering intensity function for
+oriented ellipsoids is given by (Feigin, 1987):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoCaption align=center style='text-align:center'><img border=0
+width=184 height=41 src="image017.png">&nbsp;&nbsp; </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoCaption align=center style='text-align:center'><img border=0
+width=340 height=47 src="image057.png">&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoCaption align=center style='text-align:center'><img border=0
+width=272 height=29 src="image058.png">&nbsp;&nbsp;&nbsp;&nbsp;
+</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span>
+is the angle between the axis of the ellipsoid and the q-vector, V is the
+volume of the ellipsoid, R<sub>a</sub> is the radius along the rotation axis of
+the ellipsoid, R<sub>b</sub> is the radius perpendicular to the rotation axis
+of the ellipsoid and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i>
+(contrast) is the scattering length density difference between the scatterer
+and the solvent. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the ellipsoid,
+we define the rotation axis of the ellipsoid using two angles &#952; and <span
+style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
+of the cylinder, those angles are defined on Figure 2. For the ellipsoid,
+&#952; is the angle between the rotation axis and the z-axis. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid
+ellipsoid is calculate based on the radius_a and radius_b values, and used as
+the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters of the ellipsoid model are the following:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>radius_a (polar)</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>20.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>radius_b (equatorial)</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>400.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>sldEll</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>4.0e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>1.0e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>0.0 </p>
+  </td>
+ </tr>
+ <tr style='height:19.85pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>axis_theta</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>Radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
+  <p class=MsoBodyText>1.57</p>
+  </td>
+ </tr>
+ <tr style='height:20.6pt'>
+  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
+  <p class=MsoBodyText>axis_phi</p>
+  </td>
+  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
+  <p class=MsoBodyText>Radian</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The output of the 1D scattering intensity function for
+randomly oriented ellipsoids is then given by the equation above.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not
+used for the 1D output. Our implementation of the scattering kernel and the 1D
+scattering intensity use the c-library from NIST.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=396 height=297 src="image059.jpg"
+alt=ellipsoangles.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'><span
+style='font-size:12.0pt'>Figure. Examples of the angles for oriented ellipsoid </span></p>
+
+<p class=MsoCaption align=center style='text-align:center'><span
+style='font-size:12.0pt'>against the detector plane</span>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span
+style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the
+ellipsoid model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of
+the 1D model to the output of the software provided by the NIST (Kline, 2006).
+Figure 5 shows a comparison of the 1D output of our model and the output of the
+NIST software.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Averaging over a distribution of orientation is done by
+evaluating the equation above. Since we have no other software to compare the
+implementation of the intensity for fully oriented ellipsoids, we can compare
+the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span
+style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.&nbsp; Figure
+6 shows the result of such a cross-check.</p>
+
+<p class=MsoNormal align=center style='text-align:center'><b><i><span
+style='color:#3366FF'>&nbsp;</span></i></b></p>
+
+<p class=MsoNormal>The discrepancy above q=0.3 Å<sup> -1</sup> is due to the
+way the form factors are calculated in the c-library provided by NIST. A
+numerical integration has to be performed to obtain P(q) for randomly oriented
+particles. The NIST software performs that integration with a 76-point Gaussian
+quadrature rule, which will become imprecise at high q where the amplitude
+varies quickly as a function of q. The DANSE result shown has been obtained by
+summing over 501 equidistant points in <span style='font-family:"Arial","sans-serif"'>&#945;</span>.
+Our result was found to be stable over the range of q shown for a number of
+points higher than 500.</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><b><img
+border=0 width=449 height=258 src="image060.jpg"
+alt="ellipsoid_1D_validation"></b></p>
+
+<p class=MsoCaption><a name="_Ref173222904">Figure </a>5: Comparison of the
+DANSE scattering intensity for an ellipsoid with the output of the NIST SANS
+analysis software. The parameters were set to: Scale=1.0, Radius_a=20 Å,
+Radius_b=400 Å, </p>
+
+<p class=MsoCaption>Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup>
+-1</sup>.</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+border=0 width=426 height=247 src="image061.jpg"
+alt="ellipsoid_2D_average"></p>
+
+<p class=MsoCaption><a name="_Ref173223004">Figure </a>6: Comparison of the
+intensity for uniformly distributed ellipsoids calculated from our 2D model and
+the intensity from the NIST SANS analysis software. The parameters used were:
+Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and
+Background=0.0 cm<sup> -1</sup>.</p>
+
+<p class=MsoCaption>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.17.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=CoreShellEllipsoidModel><b><span style='font-size:14.0pt'>CoreShellEllipsoidModel
+</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a core
+shell ellipsoid (below) where the form factor is normalized by the volume of
+the cylinder. P(q) = scale*&lt;f^2&gt;/V+background where the volume V= 4pi/3*r<sub>maj</sub>*r<sub>min</sub><sup>2</sup>
+and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D.
+&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp; <img
+border=0 width=246 height=131 src="image062.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on
+absolute scale. </p>
+
+<p class=MsoNormal>The form factor calculated is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=334 height=137 src="image063.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the coreshell
+ellipsoid, we define the axis of the solid ellipsoid using two angles &#952; , <span
+style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
+of the cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span>
+are defined on Figure 2 of CylinderModel. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The contrast is defined as SLD(core)  SLD(shell) or
+SLD(shell  solvent). In the parameters, equat_core = equatorial radius of the
+core, polar_core = polar radius of the core, equat_shell = r<sub>min</sub> (or
+equatorial radius of the shell), and polar_shell = = r<sub>maj</sub> (or polar
+radius of the shell).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid
+ellipsoid is calculate based on the radius_a (= polar_shell) and radius_b (=
+equat_shell) values, and used as the effective radius toward S(Q) when
+P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>equat_core</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>200</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>equat_shell</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>250</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_solvent</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>ploar_shell</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>30</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>ploar_core</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>20</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  solid windowtext 1.0pt;border-bottom:none;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_core</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>2e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_shell</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-006</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=424 height=243 src="image108.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>&nbsp;</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=396 height=297 src="image059.jpg"
+alt=ellipsoangles.gif></p>
+
+<p class=MsoNormal align=center style='text-align:center'>Figure. Examples of
+the angles for oriented coreshellellipsoid against the detector plane where a
+=polar axis.</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.</p>
+
+<p class=MsoNormal>Berr, S.&nbsp; J. Phys. Chem., 1987, 91, 4760.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.18.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=TriaxialEllipsoidModel><b><span style='font-size:14.0pt'>TriaxialEllipsoidModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for an
+ellipsoid (below) where all three axes are of different lengths, i.e.,&nbsp; R<sub>a</sub>
+=&lt; R<sub>b</sub> =&lt; R<sub>c</sub> (Note that users should maintains this
+inequality for the all calculations).&nbsp; P(q) =
+scale*&lt;f^2&gt;/V+background where the volume V= 4pi/3*R<sub>a</sub>*R<sub>b</sub>*R<sub>c</sub>,
+and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D.
+&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp; <img
+border=0 width=261 height=157 src="image065.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp;</p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on
+absolute scale. </p>
+
+<p class=MsoNormal>The form factor calculated is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=396 height=97 src="image066.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>To provide easy access to the orientation of the triaxial ellipsoid,
+we define the axis of the cylinder using the angles &#952; , <span
+style='font-family:"Arial","sans-serif"'>&#966; </span>and<span
+style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span
+style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the
+cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span>
+are defined on Figure 2 of CylinderModel. The angle <span style='font-family:
+Symbol'>Y </span>is the rotational angle around its own semi_axisC axis against
+the q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when
+the semi_axisA axis is parallel to the x-axis of the detector.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The radius of gyration for this system is R<sub>g</sub><sup>2</sup>
+= (R<sub>a</sub><sup>2</sup>*R<sub>b</sub><sup>2</sup>*R<sub>c</sub><sup>2</sup>)/5.&nbsp;
+The contrast is defined as SLD(ellipsoid)  SLD(solvent). In the parameters,
+semi_axisA = R<sub>a</sub> (or minor equatorial radius), semi_axisB = R<sub>b</sub>
+(or major equatorial radius), and semi_axisC = R<sub>c</sub> (or polar radius
+of the ellipsoid).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid
+ellipsoid is calculate based on the radius_a (=semi_axisC) and radius_b
+(=sqrt(semi_axisA* semi_axisB)) &nbsp;values, and used as the effective radius
+toward S(Q) when P(Q)*S(Q) is applied. </p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>semi_axisA</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>35</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>semi_axisB</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>100</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>semi_axisC</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> </sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>400</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  solid windowtext 1.0pt;border-bottom:none;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldEll</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1.0e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sldSolv</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6.3e-006</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=441 height=282 src="image109.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span
+style='font-family:Symbol'>·</span><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span><b>Validation of the triaxialellipsoid 2D model</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Validation of our code was done by comparing the output of the
+1D calculation to the angular average of the output of 2 D calculation over all
+possible angles. The Figure below shows the comparison where the solid dot
+refers to averaged 2D while the line represents the result of 1D calculation
+(for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi,
+and psi respectively).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=438 height=272 src="image068.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+Comparison between 1D and averaged 2D.</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=396 height=297 src="image069.jpg"
+alt=triellrangles.gif></p>
+
+<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of
+the angles for oriented ellipsoid against the detector plane.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>L. A. Feigin and D. I. Svergun Structure Analysis by
+Small-Angle X-Ray and Neutron Scattering, Plenum, New York, 1987.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.19.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=LamellarModel><b><span style='font-size:14.0pt'>LamellarModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the scattering intensity, I(<i>q</i>),
+for a lyotropic lamellar phase where a uniform SLD and random distribution in
+solution are assumed. &nbsp;The ploydispersion in the bilayer thickness can be
+applied from the GUI.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The scattering intensity I(q) is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=103 height=46 src="image070.png"></p>
+
+<p class=MsoNormal>The form factor is,</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=174 height=48 src="image071.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <span style='font-family:Symbol'>d</span> = bilayer
+thickness.</p>
+
+<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as
+1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img
+border=0 width=69 height=22 src="image010.png"></span><span
+style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on
+absolute scale. In the parameters, sld_bi = SLD of the bilayer, sld_sol = SLD
+of the solvent, and bi_thick = the thickness of the bilayer.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_bi</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>bi_thick</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>50</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_sol</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=457 height=275 src="image072.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993)
+487-502.</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.20.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=LamellarFFHGModel><b><span style='font-size:14.0pt'>LamellarFFHGModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the scattering intensity, I(<i>q</i>),
+for a lyotropic lamellar phase where a random distribution in solution are
+assumed. &nbsp;The SLD of the head region is taken to be different from the SLD
+of the tail region.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The scattering intensity I(q) is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=151 height=46 src="image073.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The form factor is,</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=398 height=46 src="image078.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where <span style='font-family:Symbol'>d<sub>T</sub></span>
+= tail length (or t_length), <span style='font-family:Symbol'>d<sub>H</sub></span>
+= heasd thickness (or h_thickness) , <span style='font-family:Symbol'>Dr</span><sub>H</sub>
+= SLD (headgroup) - SLD(solvent), and <span style='font-family:Symbol'>Dr</span><sub>T</sub>
+= SLD (tail) - SLD(headgroup).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as
+1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img
+border=0 width=71 height=23 src="image010.png"></span><span
+style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on
+absolute scale. In the parameters, sld_tail = SLD of the tail group, and
+sld_head = SLD of the head group.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_head</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>3e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_solvent</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>h_thickness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>t_length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>15</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_tail</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=444 height=279 src="image079.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/1000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993)
+487-502.</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.21.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=LamellarPSModel><b><span style='font-size:14.0pt'>LamellarPSModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the scattering intensity (<b>form factor</b>
+<b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase
+where a random distribution in solution are assumed.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The scattering intensity I(q) is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=133 height=46 src="image077.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The form factor is</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=174 height=48 src="image071.png"></p>
+
+<p class=MsoNormal>and the structure is </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=308 height=51 src="image074.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=252 height=133 src="image075.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Here d= (repeat) spacing, <span style='font-family:Symbol'>d</span>
+= bilayer thickness, the contrast <span style='font-family:Symbol'>Dr</span> =
+SLD (headgroup) - SLD(solvent), K=smectic bending elasticity, B=compression
+modulus, and N = number of lamellar plates (n_plates).</p>
+
+<p class=MsoNormal>Note: When the Caille parameter is greater than
+approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due
+to the complication of the model function, users are responsible to make sure
+that all the assumptions are handled accurately: see the original reference
+(below) for more details.</p>
+
+<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as
+1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img
+border=0 width=82 height=26 src="image010.png"></span><span
+style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on
+absolute scale. </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>contrast</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>5e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>delta</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>30</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>n_plates</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>20</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>spacing</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>400</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>caille</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.1</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=439 height=287 src="image076.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/6000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993)
+487-502.</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.22.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=LamellarPSHGModel><b><span style='font-size:14.0pt'>LamellarPSHGModel</span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the scattering intensity (<b>form factor</b>
+<b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase
+where a random distribution in solution are assumed. &nbsp;The SLD of the head
+region is taken to be different from the SLD of the tail region.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The scattering intensity I(q) is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=133 height=46 src="image077.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The form factor is,</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=343 height=46 src="image078.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>The structure factor is</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=334 height=51 src="image074.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>where </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=252 height=133 src="image075.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>where <span style='font-family:Symbol'>d<sub>T</sub></span>
+= tail length (or t_length), <span style='font-family:Symbol'>d<sub>H</sub></span>
+= heasd thickness (or h_thickness) , <span style='font-family:Symbol'>Dr</span><sub>H</sub>
+= SLD (headgroup) - SLD(solvent), and <span style='font-family:Symbol'>Dr</span><sub>T</sub>
+= SLD (tail) - SLD(headgroup). Here d= (repeat) spacing, K=smectic bending
+elasticity, B=compression modulus, and N = number of lamellar plates
+(n_plates).</p>
+
+<p class=MsoNormal>Note: When the Caille parameter is greater than
+approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due
+to the complication of the model function, users are responsible to make sure
+that all the assumptions are handled accurately: see the original reference
+(below) for more details.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as
+1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img
+border=0 width=70 height=22 src="image010.png"></span><span
+style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on
+absolute scale. In the parameters, sld_tail = SLD of the tail group, sld_head =
+SLD of the head group, and sld_solvent = SLD of the solvent.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><sub>&nbsp;</sub></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_head</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>2e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>1</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_solvent</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>6e-006</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>deltaH</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>2</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>deltaT</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>sld_tail</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>n_plates</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>30</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>spacing</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>40</p>
+  </td>
+ </tr>
+ <tr style='height:18.8pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>caille</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>Å<sup> -2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=453 height=298 src="image080.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/6000 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal>Our model uses the form factor calculations implemented in a
+c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>REFERENCE</p>
+
+<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993)
+487-502.</p>
+
+<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:16.0pt'>3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;
+</span></b><a name=Shape-Independent><b><span style='font-size:16.0pt'>Shape-Independent
+Models </span></b></a></p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoBodyText>The following are models used for shape-independent SANS
+analysis.</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=Debye>Debye (Model)</a></span></b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The Debye
+model is a form factor for a linear polymer chain. In addition to the radius of
+gyration, Rg, a scale factor &quot;scale&quot;, and a constant background term
+are included in the calculation. </p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=188 height=76
+src="image081.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is defined
+as <span style='font-size:14.0pt'><img border=0 width=73 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>rg</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>50.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><img
+border=0 width=423 height=273 src="image110.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>Reference: Roe, R.-J.,
+&quot;Methods of X-Ray and Neutron Scattering in Polymer Science&quot;, Oxford
+University Press, New York (2000).</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=Lorentz>(Ornstein-Zernicke)
+Lorentz (Model)</a></span></b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-indent:.25in'>The Ornstein-Zernicke model is
+defined by:</p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=212 height=24
+src="image082.png"></span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='text-indent:.25in'>The parameter L is referred to as the
+screening length.</p>
+
+<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=103 height=33
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>50.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b><span
+style='font-size:14.0pt'><img border=0 width=422 height=273
+src="image111.jpg"></span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b><span
+style='font-size:14.0pt'>&nbsp;</span>Figure. 1D plot using the default values
+(w/200 data point).</b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="DAB_Model">DAB
+(Debye-Anderson-Brumberger)_Model</a></span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Calculates the
+scattering from a randomly distributed, two-phase system based on the Debye-Anderson-Brumberger
+(DAB) model for such systems. The two-phase system is characterized by a single
+length scale, the correlation length, which is a measure of the average spacing
+between regions of phase 1 and phase 2. The model also assumes smooth interfaces
+between the phases and hence exhibits Porod behavior (I ~ Q<sup>-4</sup>) at
+large Q (Q*correlation length &gt;&gt; 1). </p>
+
+<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=221 height=28
+src="image083.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The parameter
+L is referred to as the correlation length.</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=75 height=24
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>50.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b><span
+style='font-size:14.0pt'><img border=0 width=450 height=285
+src="image112.jpg"></span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b><span
+style='font-size:14.0pt'>&nbsp;</span>Figure. 1D plot using the default values
+(w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='margin-left:.5in'>Debye, Anderson, Brumberger,
+&quot;Scattering by an Inhomogeneous Solid. II. The Correlation Function and
+its Application&quot;, J. Appl. Phys. 28 (6), 679 (1957).</p>
+
+<p class=MsoNormal style='margin-left:.5in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.5in'>Debye, Bueche, &quot;Scattering by
+an Inhomogeneous Solid&quot;, J. Appl. Phys. 20, 518 (1949).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="Absolute Power_Law">
+Power_Law </a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>This model describes a power law
+with background.</p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=149 height=24
+src="image084.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Note the minus
+sign in front of the exponent.</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>m</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>4</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=416 height=259 src="image113.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b>Figure. 1D plot
+using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.5.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="Teubner Strey">Teubner
+Strey (Model)</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>This function calculates the
+scattered intensity of a two-component system using the Teubner-Strey model.</p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=213 height=45
+src="image085.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.1</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>c1</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>-30.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>c2</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>5000.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=426 height=275 src="image114.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b>Figure. 1D plot
+using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='margin-left:.5in'>Teubner, M; Strey, R. J. Chem.
+Phys., 87, 3195 (1987).</p>
+
+<p class=MsoNormal style='margin-left:.5in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.5in'>Schubert, K-V., Strey, R., Kline,
+S. R. and E. W. Kaler, J. Chem. Phys., 101, 5343 (1994).</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.6.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a
+name="Number_Density_Fractal"> </a><a name=FractalModel>FractalModel</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>Calculates the scattering from
+fractal-like aggregates built from spherical building blocks following the
+Texiera reference. The value returned is in cm<sup>-1</sup>.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=332 height=188
+src="image086.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The scale
+parameter is the volume fraction of the building blocks, R<sub>0</sub> is the
+radius of the building block, D<sub>f</sub> is the fractal dimension, &#958; is
+the correlation length, <i>&#961;<sub>solvent</sub></i> is the scattering
+length density of the solvent, and <i>&#961;<sub>block</sub></i> is the
+scattering length density of the building blocks. </p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'><b>The
+polydispersion in radius is provided.</b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.05</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>5.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>fractal_dim</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>2</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>corr_length</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>100.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>block_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>2e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>solvent_sld</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>6e-6</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center;
+text-autospace:none'><img border=0 width=445 height=280
+src="image115.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center'><b>Figure. 1D plot
+using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center;
+text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center;
+text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='margin-left:.25in;text-indent:.25in;text-autospace:
+none'>J. Teixeira, (1988) J. Appl. Cryst., vol. 21, p781-785</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.7.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=BEPolyelectrolyte>
+BEPolyelectrolyte Model</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>Calculates the structure factor of
+a polyelectrolyte solution with the RPA expression derived by Borue and
+Erukhimovich. The value returned is in cm<sup>-1</sup>.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=346 height=125
+src="image087.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>K is a contrast factor of the polymer,
+L<sub>b</sub> is the Bjerrum length, h is the virial parameter, b is the
+monomer length, C<sub>s</sub> is the concentration of monovalent salt, &#945;
+is the ionization degree, C<sub>a</sub> is the polymer molar concentration, and
+background is the incoherent background.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=75 height=24
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>K</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Barns = 10<sup>-24</sup> cm<sup>2</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>L<sub>b</sub></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>7.1</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>h</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-3</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>12</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>b</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>10</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Cs</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Mol/L</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>alpha</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.05</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Ca</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Mol/L</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.7</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Borue, V. Y., Erukhimovich,
+I. Y. Macromolecules 21, 3240 (1988).</p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'><span lang=FR>Joanny,
+J.-F., Leibler, L. Journal de Physique 51, 545 (1990).</span></p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'><span lang=FR>Moussaid,
+A., Schosseler, F., Munch, J.-P., Candau, S. J. Journal de Physique II France </span></p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>3, 573 (1993).</p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Raphaël, E.,
+Joanny, J.-F., Europhysics Letters 11, 179 (1990).</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.8.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=Guinier>Guinier
+(Model)</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>A Guinier analysis is done by
+linearizing the data at low q by plotting it as log(I) versus Q<sup>2</sup>.
+The Guinier radius R<sub>g</sub> can be obtained by fitting the following
+model:</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=183 height=28
+src="image088.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
+style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
+style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Rg</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.1</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.9.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=PorodModel>PorodModel</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>A Porod analysis is done by linearizing
+the data at high q by plotting it as log(I) versus log(Q). In the high q region
+we can fit the following model: </p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=319 height=25
+src="image089.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>C is the scale factor and &nbsp;Sv
+is the specific surface area of the sample and <span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;
+is the contrast factor. </p>
+
+<p class=MsoNormal style='margin-left:.25in'>The background term is added for
+data analysis.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=65 height=21
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
+style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-4</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.1</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.10&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <a
+name="Poly_GaussCoil">Poly_GaussCoil</a> (Model)</span></b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Polydisperse Gaussian
+Coil: Calculate an empirical functional form for scattering from a polydisperse
+polymer chain ina good solvent. The polymer is polydisperse with a Schulz-Zimm
+polydispersity of the molecular weight distribution.&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The returned
+value is scaled to units of [cm-1sr-1], absolute scale.</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center;
+text-autospace:none'><span style='position:relative;top:15pt'><img border=0
+width=261 height=48 src="image116.gif"></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>where the
+dimensionless chain dimension is:</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center;
+text-autospace:none'><span style='position:relative;top:12pt'><img border=0
+width=73 height=47 src="image117.gif"></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>and the
+polydispersion is</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center;
+text-autospace:none'><span style='position:relative;top:15pt'><img border=0
+width=81 height=47 src="image118.gif">.</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The scattering
+intensity I(q) is calculated as:</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The
+polydispersion in rg is provided.</p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>TEST DATASET</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;This example dataset is
+produced by running the Poly_GaussCoil, using 200 data points, qmin = 0.001
+Å-1,&nbsp; qmax = 0.7 Å-1&nbsp;&nbsp; and the default values below.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>None</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>rg</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>60.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>poly_m</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Mw/Mn</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>2</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.001</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><img
+border=0 width=431 height=268 src="image119.jpg"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (w/200 data point).</b></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>Reference: </p>
+
+<p class=MsoNormal style='margin-left:.25in'>Glatter &amp; Kratky - pg.404.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>J.S. Higgins, and H.C. Benoit, Polymers
+and Neutron Scattering, Oxford Science</p>
+
+<p class=MsoNormal style='margin-left:.25in'>Publications (1996).</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.11.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name="Peak Gauss Model"><b><span style='font-size:14.0pt'>&nbsp;Peak
+Gauss Model</span></b></a></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>This Gaussian peak model with a
+flat background calculates:</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=334 height=25
+src="image090.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>where scale is the peak height centered
+at q<sub>0</sub>, and B refers to the standard deviation of the function
+(equivalently, the FWHM is 2.54*B).</p>
+
+<p class=MsoNormal style='margin-left:.25in'>The background term is added for
+data analysis. </p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as <span style='font-size:14.0pt'><img border=0 width=71 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
+style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>100</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>B</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.005</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>q<sub>0</sub></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.05</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.12.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="Peak Lorentz Model">Peak
+Lorentz Model</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>This Lorentzean peak model with a
+flat background calculates:</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=241 height=70
+src="image091.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>where scale is the peak height
+centered at q<sub>0</sub>, and B refers to the standard deviation of the
+function.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>The background term is added for
+data analysis. </p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as<span style='font-size:14.0pt'><img border=0 width=77 height=25
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
+style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>scale</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>100</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>B</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.005</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>q<sub>0</sub></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.05</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>background</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.13.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=LineModel> LineModel</a></span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in'>This is a linear function that
+calculates:</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=92 height=21
+src="image092.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal style='margin-left:.25in'>where A and B are the coefficients
+of the first and second order terms.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'><b>Note:</b> For 2D plot, I(q) =
+I(q<sub>x</sub>)*I(q<sub>y</sub>)&nbsp; which is defined differently from other
+shape independent models.</p>
+
+<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>A</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>cm<sup>-1</sup></p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>B</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.0</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:16.0pt'>4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;
+</span></b><a name="Customized_Models"><b><span style='font-size:16.0pt'>Customized
+Models </span></b></a></p>
+
+<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoBodyText style='margin-left:.25in'>These model functions can be
+redefined by users (See SansView tutorial for details). </p>
+
+<p class=MsoBodyText style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>4.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name="A+Bcos(2x)+Csin(2x)"><b><span style='font-size:14.0pt'>A+Bcos(2x)+Csin(2x)</span></b></a></p>
+
+<p class=MsoListParagraph style='margin-left:.55in'><b><span style='font-size:
+14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal>This function, as a sample function, calculates the
+intensity = A + Bcos(2q) + Csin(2q).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>4.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name="sinpoly_poly"><b><span style='font-size:14.0pt'>sin(poly)/poly
+</span></b></a></p>
+
+<p class=MsoNormal><b>&nbsp;</b></p>
+
+<p class=MsoNormal>This function calculates the intensity =&nbsp; scale *
+sin(f)/f, where f = A + Bq + Cq<sup>2</sup> + Dq<sup>3 </sup>+ Eq<sup>4</sup> +
+Fq<sup>5</sup>.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:16.0pt'>5.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;
+</span></b><a name="Structure_Factors"><b><span style='font-size:16.0pt'>Structure
+Factors</span></b></a></p>
+
+<p class=MsoListParagraph style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.25in'>The information in this
+section is originated from NIST SANS IgorPro package.</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>5.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=HardsphereStructure><b><span style='font-size:14.0pt'>HardSphere
+Structure </span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This calculates the interparticle structure factor for monodisperse
+spherical particles interacting through hard sphere (excluded volume)
+interactions. The calculation uses the Percus-Yevick closure where the
+interparticle potential is: </p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=122 height=48
+src="image093.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>where r is the distance from the center of the sphere of a
+radius R.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as<span style='font-size:14.0pt'><img border=0 width=72 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>effect_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>50.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>volfraction</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.2</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=454 height=295 src="image094.png"></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (in linear scale).</b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Percus, J. K.;
+Yevick, J. Phys. Rev. 110, 1. (1958).</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>5.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=SquareWellStructure><b><span style='font-size:14.0pt'>&nbsp;SquareWell
+Structure </span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This calculates the interparticle structure factor for a
+squar well fluid spherical particles The mean spherical approximation (MSA)
+closure was used for this calculation, and is not the most appropriate closure
+for an attractive interparticle potential. This solution has been compared to
+Monte Carlo simulations for a square well fluid, showing this calculation to be
+limited in applicability to well depths e &lt; 1.5 kT and volume fractions f
+&lt; 0.08.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>Positive well depths correspond to an attractive potential
+well. Negative well depths correspond to a potential &quot;shoulder&quot;,
+which may or may not be physically reasonable.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The well width (l) is defined as multiples of the particle
+diameter (2*R)</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The interaction potential is:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=174 height=67
+src="image095.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>where r is the distance from the center of the sphere of a
+radius R.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as<span style='font-size:14.0pt'><img border=0 width=71 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='margin-left:-1.5pt;border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>effect_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>50.0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>volfraction</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.04</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>welldepth</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>kT</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.5</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>wellwidth</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>diameters</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>1.2</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=453 height=286 src="image096.png"></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (in linear scale).</b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Sharma, R. V.;
+Sharma, K. C. Physica, 89A, 213. (1977).</p>
+
+<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>5.3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><a name=HayterMSAStructure><b><span style='font-size:14.0pt'>&nbsp;HayterMSA
+Structure </span></b></a></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This calculates the Structure factor (the Fourier transform
+of the pair correlation function g(r)) for a system of charged, spheroidal
+objects in a dielectric medium.&nbsp; When combined with an appropriate form
+factor (such as sphere, core+shell, ellipsoid etc
+), this allows for inclusion
+of the interparticle interference effects due to screened coulomb repulsion
+between charged particles. This routine only works for charged particles.&nbsp;
+If the charge is set to zero the routine will self destruct.&nbsp; For
+non-charged particles use a hard sphere potential.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The salt concentration is used to compute the ionic strength
+of the solution which in turn is used to compute the Debye screening
+length.&nbsp; At present there is no provision for entering the ionic strength
+directly nor for use of any multivalent salts.&nbsp; The counterions are also
+assumed to be monovalent.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as<span style='font-size:14.0pt'><img border=0 width=71 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='margin-left:-1.5pt;border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>effect_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>20.8</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>charge</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>19</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>volfraction</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.2</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>temperature</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>K</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>318</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>salt conc</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>M</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>dielectconst</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>71.1</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=451 height=273 src="image097.png"></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (in linear scale).</b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='text-indent:.5in;text-autospace:none'>JP Hansen and
+JB Hayter, Molecular Physics 46, 651-656 (1982).</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+JB Hayter and J Penfold, Molecular Physics 42, 109-118 (1981).</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>5.4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></b><b><span style='font-size:14.0pt'>&nbsp;StickyHS Structure </span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This calculates the interparticle structure factor for a
+hard sphere fluid with a narrow attractive well. A perturbative solution of the
+Percus-Yevick closure is used. The strength of the attractive well is described
+in terms of &quot;stickiness&quot; as defined below. The returned value is a
+dimensionless structure factor, S(q).</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The perturb (perturbation parameter), epsilon, should be
+held between 0.01 and 0.1. It is best to hold the perturbation parameter fixed
+and let the &quot;stickiness&quot; vary to adjust the interaction strength. The
+stickiness, tau, is defined in the equation below and is a function of both the
+perturbation parameter and the interaction strength. Tau and epsilon are
+defined in terms of the hard sphere diameter (sigma = 2R), the width of the
+square well, delta (same units as R), and the depth of the well, u<sub>o</sub>,
+in units of kT. From the definition, it is clear that smaller tau mean stronger
+attraction.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=135 height=67
+src="image098.png"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>where the interaction potential is</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><img border=0 width=186 height=69
+src="image099.png"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>The Percus-Yevick (PY) closure was used for this
+calculation, and is an adequate closure for an attractive interparticle
+potential. This solution has been compared to Monte Carlo simulations for a
+square well fluid, with good agreement.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>The true particle volume fraction, f, is not equal to h,
+which appears in most of the reference. The two are related in equation (24) of
+the reference. The reference also describes the relationship between this perturbation
+solution and the original sticky hard sphere (or adhesive sphere) model by
+Baxter.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>NOTES: The calculation can go haywire for certain
+combinations of the input parameters, producing unphysical solutions - in this
+case errors are reported to the command window and the S(q) is set to -1 (it
+will disappear on a log-log plot). Use tight bounds to keep the parameters to
+values that you know are physical (test them) and keep nudging them until the
+optimization does not hit the constraints.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is
+defined as<span style='font-size:14.0pt'><img border=0 width=73 height=23
+src="image010.png"></span><span style='font-size:14.0pt'>.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0
+ style='margin-left:-1.5pt;border-collapse:collapse'>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Parameter name</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Units</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
+  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Default value</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>effect_radius</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>Å</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>50</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>perturb </p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.05</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>volfraction</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>&nbsp;</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.1</p>
+  </td>
+ </tr>
+ <tr style='height:19.25pt'>
+  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
+  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>stickiness</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>K</p>
+  </td>
+  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
+  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
+  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
+  <p class=MsoBodyText>0.2</p>
+  </td>
+ </tr>
+</table>
+
+</div>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal align=center style='text-align:center'><img border=0
+width=463 height=273 src="image100.png"></p>
+
+<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure.
+1D plot using the default values (in linear scale).</b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='text-autospace:none'>References: </p>
+
+<p class=MsoNormal style='text-indent:.5in;text-autospace:none'>Menon, S. V.
+G., Manohar, C. and K. Srinivas Rao J. Chem. Phys., 95(12), 9186-9190 (1991).</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<p class=MsoBodyText><a name=References><b><span style='font-size:14.0pt'>References</span></b></a></p>
+
+<p class=MsoBodyText>Feigin, L. A, and D. I. Svergun (1987) &quot;Structure
+Analysis by Small-Angle X-Ray and Neutron Scattering&quot;, Plenum Press, New
+York.</p>
+
+<p class=MsoBodyText>Guinier, A. and G. Fournet (1955) &quot;Small-Angle
+Scattering of X-Rays&quot;, John Wiley and Sons, New York.</p>
+
+<p class=MsoBodyText>Kline, S. R. (2006) <i>J Appl. Cryst.</i> <b>39</b>(6),
+895.</p>
+
+<p class=MsoBodyText>Hansen, S., (1990)<i> J. Appl. Cryst. </i>23, 344-346.</p>
+
+<p class=MsoBodyText>Henderson, S.J. (1996) <i>Biophys. J. </i>70, 1618-1627</p>
+
+<p class=MsoBodyText>Stöckel, P., May, R., Strell, I., Cejka, Z., Hoppe, W.,
+Heumann, H., Zillig, W. and Crespi, H. (1980) <i>Eur. J. Biochem. </i>112,
+411-417.</p>
+
+<p class=MsoBodyText>McAlister, B.C. and Grady, B.P., (1998) J. Appl. Cryst.
+31, 594-599.</p>
+
+<p class=MsoBodyText>Porod, G. (1982) in Small Angle X-ray Scattering, editors
+Glatter, O. and Kratky, O., Academic Press.</p>
+
+<p class=MsoBodyText>*Also, see the references in the each model function
+descriptions.</p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+</div>
+
+</body>
+
+</html>
