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+</head>
+
+<body lang=EN-US 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>
+<li><a href="#Introduction"><b>Introduction</b></a></li>
+<li><a href="#Shapes"><b>Shapes</b></a>:  
+<a href="#SphereModel">SphereModel</a>, 
+<a href="#CylinderModel">CylinderModel</a>, 
+<a href="#EllipsoidModel">EllipsoidModel</a>, 
+<a href="#CoreShellModel">CoreShellModel</a>, 
+<a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>,
+<a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>. 
+</li>
+<li><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="#Power_Law">Power_Law</a>,
+<a href="#Teubner Strey">Teubner Strey</a>,
+<a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>,
+<a href="#Number Density Fractal">Number Density Fractal</a>,
+<a href="#Guinier">Guinier</a>,
+<a href="#PorodModel">PorodModel</a>. </li>
+<li><a href="#Customized Models"><b>Customized Models</b></a>:  
+<a href="#A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a>. </li>
+<li><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><a href="#References"><b>References</b></a></li>
+</ul>
+
+<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:14.0pt'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="Introduction">Introduction</a></span></b></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 style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:14.0pt'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="Shapes">Shapes (Scattering intensity models)</a></span></b></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 align=center style='text-align:center'><sub><img width=192
+height=43 src="../images/html/image001.gif"></sub>            (1)</p>
+
+<p class=MsoNormal>with</p>
+
+<p class=MsoNormal align=center style='text-align:center'><sub><img width=159
+height=29 src="../images/html/image002.gif"></sub>        (2)</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:<sub><img width=92 height=21
+src="../images/html/image003.gif"></sub>.</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 style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="SphereModel">SphereModel</a></span></b></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>2.1.1.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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'><sub><img width=337
+height=53 src="../images/html/image004.gif"></sub>            (3)</p>
+
+<p class=MsoNormal>where <i>scale</i> is a scale factor, 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)  is the scattering length density 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>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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>
+ </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>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>1.0e-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>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>Contrast</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</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>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 style='margin-left:.85in;text-indent:-.35in'><b>2.1.2.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 width=573 height=315 id="Picture 5"
+src="../images/html/image005.jpg" alt="sphere_1D_validation"></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 style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="CylinderModel">CylinderModel</a></span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor for a monodisperse 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>2.2.1.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 align=center style='text-align:center'><sub><img width=184
+height=41 src="../images/html/image006.gif"></sub>  (4)</p>
+
+<p class=MsoNormal align=center style='text-align:center'><sub><img width=373
+height=45 src="../images/html/image007.gif"></sub>   (5)</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 &#952; and <span
+style='font-family:"Arial","sans-serif"'>&#966;</span>. Those angles are
+defined on Figure 2.</p>
+
+<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
+width=300 height=281 id="Picture 8" src="../images/html/image008.jpg"
+alt=angles></p>
+
+<p class=MsoCaption align=center style='text-align:center'><a
+name="_Ref173306040"></a><a name="_Ref173213915">Figure </a>2: Definition of the angles for oriented cylinders.</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>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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>The output of the 1D scattering intensity function for
+randomly oriented cylinders is then given by:</p>
+
+<p class=MsoNormal align=center style='text-align:center'><a
+name="_Ref173306479"></a><a name="_Ref173306528"><sub><img width=253 height=56
+src="../images/html/image009.gif"></sub>         (</a>6)</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 style='margin-left:.85in;text-indent:-.35in'><b>2.2.2.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 align=center style='text-align:center'><sub><img width=244
+height=51 src="../images/html/image010.gif"></sub>           (7)</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.  Figure 4
+shows the result of such a cross-check.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
+id="Picture 11" src="../images/html/image011.jpg"
+alt="cylinder_1D_validation"></p>
+
+<p class=MsoCaption><a name="_Ref173211049"></a><a name="_Ref173211066">Figure </a>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=MsoCaption>&nbsp;</p>
+
+<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
+id="Picture 12" src="../images/html/image012.jpg"
+alt="cylinder_2D_average"></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>&nbsp;</p>
+
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="EllipsoidModel">EllipsoidModel</a></span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor for a monodisperse
+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>2.3.1.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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=MsoCaption align=center style='text-align:center'><sub><img width=184
+height=41 src="../images/html/image006.gif"></sub>   <span
+style='font-size:12.0pt;font-weight:normal'>(8)</span></p>
+
+<p class=MsoCaption align=center style='text-align:center'><sub><img width=371
+height=47 src="../images/html/image013.gif"></sub>    <span
+style='font-size:12.0pt;font-weight:normal'>(9)</span></p>
+
+<p class=MsoCaption align=center style='text-align:center'><sub><img width=272
+height=29 src="../images/html/image014.gif"></sub>     <span
+style='font-size:12.0pt;font-weight:normal'>(10)</span></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>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>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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 (parallel)</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 (perpendicular)</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>Contrast</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>3.0e-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 equation 6.</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 style='margin-left:.85in;text-indent:-.35in'><b>2.3.2.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 equation 7. 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.  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
+(equation 6). 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 style='page-break-after:avoid'><b><img width=573 height=315
+id="Picture 16" src="../images/html/image015.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 style='page-break-after:avoid'><img width=573 height=315
+id="Picture 17" src="../images/html/image016.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><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'>2.4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellModel">CoreShell(Sphere)Model</a></span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a
+monodisperse 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>2.4.1.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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=right style='text-align:right'><sub><img width=643
+height=55 src="../images/html/image017.gif"></sub> (11)</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>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>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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>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;</p>
+
+
+<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.2.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 width=573 height=315 id="Picture 19"
+src="../images/html/image018.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=MsoBodyText>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellCylinderModel">CoreShellCylinderModel</a></span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This model provides the form factor for a monodisperse,
+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>2.5.1.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 align=center style='text-align:center'><sub><img width=184
+height=45 src="../images/html/image019.gif"></sub> (12)</p>
+
+<p class=MsoNormal align=center style='text-align:center'><sub><img width=476
+height=93 src="../images/html/image020.gif"></sub> (13)</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>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.</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=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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 equation 6.</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 style='margin-left:.85in;text-indent:-.35in'><b>2.5.2.<span
+style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>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 equation 7. 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.  Figure 9 shows the result of such a cross-check.</p>
+
+<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
+id="Picture 22" src="../images/html/image021.jpg"
+alt="core_shell_cylinder_1D_validation"></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 style='page-break-after:avoid'><img width=573 height=315
+id="Picture 23" src="../images/html/image022.jpg"
+alt="core_shell_cylinder_2D_average"></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 style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>2.6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="EllipticalCylinderModel">EllipticalCylinderModel</a></span></b></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>This function calculates the scattering from an oriented
+elliptical cylinder. 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 (or all) of the orientation angles, and also for the minor
+radius and the ratio of the ellipse radii.</p>
+
+<p class=MsoNormal>The function calculated is:</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='position:relative;top:16.0pt'><img width=377 height=46
+src="../images/html/image023.gif"></span> </p>
+
+<p class=MsoNormal>with the functions:</p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='position:relative;top:39.0pt'><img width=297 height=114
+src="../images/html/image024.gif"></span>          (13)</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>The returned value is scaled to units of [cm<sup>-1</sup>]
+and the parameters are the following:</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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>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 style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:14.0pt'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="Shape-Independent">Shape-Independent</a></span></b></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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <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><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt'><sub><img width=188 height=76
+src="../images/html/image025.gif"></sub></span></p>
+
+<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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 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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <a name="Lorentz">(Ornstein-Zernicke)
+Lorentz (Model)</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'><sub><img width=212 height=24
+src="../images/html/image026.gif"></sub></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><span style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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><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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <a name="DAB_Model">DAB(Debye-Anderson-Brumberger)_
+Model)</a></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'><sub><img width=221 height=28
+src="../images/html/image027.gif"></sub></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 align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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 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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <a name="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'><sub><img width=161 height=24
+src="../images/html/image028.gif"></sub></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=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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><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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><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'><sub><img width=213 height=45
+src="../images/html/image029.gif"></sub></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>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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>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 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 style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>3.6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="Number Density Fractal"> Number Density Fractal (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 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 align=center style='text-align:center'><span
+style='font-size:14.0pt'><sub><img width=380 height=188
+src="../images/html/image030.gif"></sub></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 align=center style='text-align:center'><span
+style='font-size:14.0pt'>&nbsp;</span></p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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 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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <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'><sub><img width=365 height=125
+src="../images/html/image031.gif"></sub></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 bck is the incoherent background.</p>
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><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 align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt'><sub><img width=183 height=28
+src="../images/html/image032.gif"></sub></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=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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:.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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <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'><sub><img width=368 height=25
+src="../images/html/image033.gif"></sub></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  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'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in'>The background term is added for
+data analysis.</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=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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><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:14.0pt'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="Customized Models">Customized Models</a></span></b></p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoBodyText>These models are defined by users (See SansView tutorial for details). </p>
+
+<p class=MsoBodyText>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>4.1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <a name="A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a></span></b></p>
+
+<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
+
+<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>This function, as an example function, calculates intensity = A + Bcos(2Q) + Csin(2Q). </p>
+
+
+<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
+<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><b><span
+style='font-size:14.0pt'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="Structure Factors">Structure Factors</a></span></b></p>
+
+<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>&nbsp;</p>
+
+<p class=MsoListParagraphCxSpLast style='margin-left:.25in'>The information in
+this section is originated from NIST SANS IgorPro package.</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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'> <a name="HardSphereStructure">HardSphere Structure</a></span></b></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 align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt;position:relative;top:15.0pt'><img width=122 height=48
+src="../images/html/image034.gif"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the
+distance from the center of the sphere of a radius R.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='border-collapse:collapse;border:none'>
+ <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>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>
+</table>
+
+</div>
+
+<p class=MsoNormal><img width=579 height=273 id="Picture 43"
+src="../images/html/image035.gif"></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='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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="SquareWellStructure"> SquareWell Structure</a></span></b></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 align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt;position:relative;top:22.0pt'><img width=174 height=67
+src="../images/html/image036.gif"></span></p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the
+distance from the center of the sphere of a radius R.</span></p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='margin-left:-1.5pt;border-collapse:collapse;border:none'>
+ <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>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>0.2</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>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><img width=546 height=296 id="Picture 53"
+src="../images/html/image037.gif"></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='margin-left:.5in;text-autospace:none'>Sharma, R. V.;
+Sharma, K. C. Physica, 89A, 213. (1977).</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 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="HayterMSAStructure"> HayterMSA Structure</a></span></b></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.  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.  If the
+charge is set to zero the routine will self destruct.  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. 
+At present there is no provision for entering the ionic strength directly nor
+for use of any multivalent salts.  The counterions are also assumed to be monovalent.</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='margin-left:-1.5pt;border-collapse:collapse;border:none'>
+ <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>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><img width=548 height=354 id="Picture 56"
+src="../images/html/image038.gif"></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'>JP Hansen and
+JB Hayter, Molecular Physics 46, 651-656 (1982).</p>
+
+<p class=MsoNormal style='text-autospace:none'>            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=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
+style='font-size:14.0pt'>5.4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+</span></span></b><b><span style='font-size:14.0pt'><a name="StickyHSStructure"> StickyHS Structure</a></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 equation 21 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 align=center style='text-align:center'><span
+style='font-size:14.0pt;position:relative;top:22.0pt'><img width=135 height=67
+src="../images/html/image039.gif"></span></p>
+
+<p class=MsoNormal align=center style='text-align:center'><span
+style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p>
+
+<p class=MsoNormal>where the interaction potential is</p>
+
+<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
+style='font-size:14.0pt;position:relative;top:23.0pt'><img width=186 height=69
+src="../images/html/image040.gif"></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>
+
+<div align=center>
+
+<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
+ style='margin-left:-1.5pt;border-collapse:collapse;border:none'>
+ <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>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>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><img width=524 height=359 id="Picture 111"
+src="../images/html/image041.gif"></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><b><span style='font-size:14.0pt'><a name="References">References</a></span></b></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>&nbsp;</p>
+
+<p class=MsoNormal>&nbsp;</p>
+
+</div>
+
+</body>
+
+</html>
