Changeset fb071900 in sasview


Ignore:
Timestamp:
Apr 16, 2010 1:53:25 PM (15 years ago)
Author:
Jae Cho <jhjcho@…>
Branches:
master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, costrafo411, magnetic_scatt, release-4.1.1, release-4.1.2, release-4.2.2, release_4.0.1, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
Children:
f536cbf
Parents:
a5775b1
Message:

new model function help html

File:
1 edited

Legend:

Unmodified
Added
Removed

  • sansview/media/model_functions.html

    r3e0d308 rfb071900  
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    756286<p class=MsoNormal>&nbsp;</p> 
    757287 
    758 <ul type=disc> 
     288<ul style='margin-top:0in' type=disc> 
    759289 <li class=MsoNormal style='line-height:115%'><a href="#Introduction"><b>Introduction</b></a> 
    760290     </li> 
    761291 <li class=MsoNormal style='line-height:115%'><a href="#Shapes"><b>Shapes</b></a>: 
    762      <a href="#SphereModel">SphereModel</a>, 
    763      <a href="#CoreShellModel">CoreShellModel</a>, 
    764      <a href="#VesiclelModel">VesicleModel</a>, 
    765      <a href="#MultiShellModel">MultiShellModel</a>, 
    766      <a href="#BinaryHSModel">BinaryHSModel</a>, 
    767      <a href="#CylinderModel">CylinderModel</a>, 
    768      <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, 
    769      <a href="#HollowCylinderModel">HollowCylinderModel</a>, 
    770      <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, 
    771      <a href="#StackedDisksModel">StackedDisksModel</a>, 
    772      <a href="#ParallelepipedModel">ParallelepipedModel</a>, 
    773      <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, 
    774      <a href="#EllipsoidModel">EllipsoidModel</a>, 
    775      <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, 
    776      <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, 
    777      <a href="#LamellarModel">LamellarModel</a>, 
    778      <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, 
    779      <a href="#LamellarPSModel">LamellarPSModel</a>, 
    780      <a href="#LamellarPSHGModel">LamellarPSHGModel</a>.</li> 
     292     <a href="#SphereModel">SphereModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, 
     293     <a href="#CoreShellModel">CoreShellModel</a>, <a href="#CoreFourShellModel">CoreFourShellModel</a>, 
     294     <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, 
     295     <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#CylinderModel">CylinderModel</a>, 
     296     <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a 
     297     href="#HollowCylinderModel">HollowCylinderModel</a>, <a 
     298     href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a 
     299     href="#FlexCylEllipXModel">FlexCylEllipXModel</a>, <a 
     300     href="#StackedDisksModel">StackedDisksModel</a>, <a 
     301     href="#ParallelepipedModel">ParallelepipedModel</a>, <a 
     302     href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a 
     303     href="#EllipsoidModel">EllipsoidModel</a>, <a 
     304     href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a 
     305     href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a 
     306     href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, 
     307     <a href="#LamellarPSModel">LamellarPSModel</a>, <a 
     308     href="#LamellarPSHGModel">LamellarPSHGModel</a>.</li> 
    781309 <li class=MsoNormal style='line-height:115%'><a href="#Shape-Independent"><b>Shape-Independent</b></a>: 
    782      <a href="#Debye">Debye</a>, 
    783      <a href="#Lorentz">Lorentz</a>, 
    784      <a href="#DAB_Model">DAB_Model</a>, 
    785      <a href="#Power_Law">Power_Law</a>, 
    786      <a href="#Teubner Strey">Teubner Strey</a>,  
    787         <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, 
    788      <a href="#Number Density Fractal">Number Density Fractal</a>,  
    789      <a href="#Guinier">Guinier</a>, 
    790      <a href="#PorodModel">PorodModel</a>, 
    791      <a href="#Peak Gauss Model">Peak Gauss Model</a>,  
    792 <a href="#Peak Lorentz Model">Peak Lorentz Model</a>,  
    793 <a href="#LineModel">LineModel</a>.</li> 
    794  <li class=MsoNormal style='line-height:115%'> 
    795 <a href="#Customized Models"><b>Customized 
    796      Models</b></a>: <a href="#A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a>, 
    797      <a href="#sinpoly_poly">sin(poly)/poly</a>.</li> 
    798  <li class=MsoNormal style='line-height:115%'> 
    799 <a href="#Structure Factors"><b>Structure 
    800      Factors</b></a>:  
    801      <a href="#HardsphereStructure">HardSphereStructure</a>, 
    802      <a href="#SquareWellStructure">SquareWellStructure</a>, 
    803      <a href="#HayterMSAStructure">HayterMSAStructure</a>, 
    804      <a href="#StickyHSStructure">StickyHSStructure</a>. 
    805      </li> 
    806  <li class=MsoNormal style='line-height:115%'> 
    807 <a href="#References"><b>References</b></a> 
     310     <a href="#Debye">Debye</a>, <a href="#Lorentz">Lorentz</a>, <a 
     311     href="#DAB_Model">DAB_Model</a>, <a href="#Absolute Power_Law">Power_Law</a>, <a 
     312     href="#Teubner Strey">Teubner Strey</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, 
     313     <a href="#FractalModel">FractalModel</a>, <a href="#Guinier">Guinier</a>, 
     314     <a href="#PorodModel">PorodModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, 
     315     <a href="#Peak Gauss Model">Peak Gauss Model</a>, <a 
     316     href="#Peak Lorentz Model">Peak Lorentz Model</a>, <a href="#LineModel">LineModel</a>.</li> 
     317 <li class=MsoNormal style='line-height:115%'><a href="#Customized_Models"><b>Customized 
     318     Models</b></a>: <a href="#A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a>, <a 
     319     href="#sinpoly_poly">sin(poly)/poly</a>.</li> 
     320 <li class=MsoNormal style='line-height:115%'><a href="#Structure_Factors"><b>Structure 
     321     Factors</b></a>: <a href="#HardsphereStructure">HardSphereStructure</a>, <a 
     322     href="#SquareWellStructure">SquareWellStructure</a>, <a 
     323     href="#HayterMSAStructure">HayterMSAStructure</a>, <a 
     324     href="#StickyHSStructure">StickyHSStructure</a>. </li> 
     325 <li class=MsoNormal style='line-height:115%'><a href="#References"><b>References</b></a> 
    808326     </li> 
    809327</ul> 
    … …  
    812330 
    813331<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    814 style='font-size:16.0pt'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
    815 </span></span></b><b><span style='font-size:16.0pt'><a name="Introduction"> Introduction </a></span></b></p> 
     332style='font-size:16.0pt'>1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp; 
     333</span></b><a name=Introduction><b><span style='font-size:16.0pt'>Introduction </span></b></a></p> 
    816334 
    817335<p class=MsoNormal>The present text documents the modules made available by the 
    … …  
    836354<p class=MsoNormal><b>&nbsp;</b></p> 
    837355 
    838 <p class=MsoNormal>Note that our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research and thus that some contents and figures in this documentation are originated from or shared with the NIST Igor SANS analysis package by permission (S. Kline, NIST, 2006). </p> 
    839  
    840 <p class=MsoNormal><b>&nbsp;</b></p> 
     356<p class=MsoNormal><b>&nbsp;Note: Our model uses the form factor calculations 
     357implemented in a c-library provided by the NIST Center for Neutron Research and 
     358thus some content and figures in this document are originated from or shared 
     359with the NIST Igor analysis package by permission (S. Kline, NIST, 2006).</b></p> 
     360 
     361<p class=MsoNormal>&nbsp;</p> 
    841362 
    842363<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    843 style='font-size:16.0pt'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
    844 </span></span></b><b><span style='font-size:16.0pt'><a name="Shapes">Shapes (Scattering Intensity 
    845 Models)</a></span></b></p> 
     364style='font-size:16.0pt'>2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp; 
     365</span></b><a name=Shapes><b><span style='font-size:16.0pt'>Shapes (Scattering 
     366Intensity Models)</span></b></a></p> 
    846367 
    847368<p class=MsoNormal>&nbsp;</p> 
    … …  
    864385<p class=MsoNormal>&nbsp;</p> 
    865386 
    866 <p class=MsoNormal align=center style='text-align:center'><span 
    867 style='position:relative;top:12.0pt'><img border=0 width=192 height=43 
    868 src="../images/html/image001.png"></span>            (1)</p> 
     387<p class=MsoNormal align=center style='text-align:center'><img border=0 
     388width=192 height=43 src="../images/html/image001.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     389</p> 
    869390 
    870391<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    872393<p class=MsoNormal>with</p> 
    873394 
    874 <p class=MsoNormal align=center style='text-align:center'><span 
    875 style='position:relative;top:8.0pt'><img border=0 width=159 height=29 
    876 src="../images/html/image002.png"></span>        (2)</p> 
    877  
    878 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     395<p class=MsoNormal align=center style='text-align:center'><img border=0 
     396width=159 height=29 src="../images/html/image002.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
    879397 
    880398<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    889407<p class=MsoNormal>For systems without inter-particle interference, the form 
    890408factors we provide can be related to the scattering intensity by the particle 
    891 volume fraction:<span style='position:relative;top:5.0pt'><img border=0 
    892 width=92 height=21 src="../images/html/image003.png"></span>.</p> 
     409volume fraction:<img border=0 width=66 height=15 
     410src="../images/html/image003.png">.</p> 
    893411 
    894412<p class=MsoNormal>&nbsp;</p> 
    … …  
    913431 
    914432<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    915 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; 
    916 </span></span></b><b><span style='font-size:14.0pt'><a name="SphereModel">Sphere Model</a></span></b></p> 
     433style='font-size:14.0pt'>2.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     434</span></b><a name=SphereModel><b><span style='font-size:14.0pt'>Sphere Model</span></b></a></p> 
    917435 
    918436<p class=MsoNormal>&nbsp;</p> 
    … …  
    924442<p class=MsoNormal>&nbsp;</p> 
    925443 
    926 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
    927 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     444<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     445style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    928446 
    929447<p class=MsoNormal>&nbsp;</p> 
    … …  
    934452<p class=MsoNormal>&nbsp;</p> 
    935453 
    936 <p class=MsoNormal align=center style='text-align:center'><span 
    937 style='position:relative;top:16.0pt'><img border=0 width=337 height=53 
    938 src="../images/html/image004.png"></span>            (3)</p> 
    939  
    940 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    941  
    942 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    943  
    944 <p class=MsoNormal>where <i>scale</i> is a scale factor, V is the volume of the 
    945 scatterer, <i>r</i> is the radius of the sphere, <i>bkg</i> is the background 
    946 level and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
    947 (contrast)  is the scattering length density difference between the scatterer 
    948 and the solvent it is in.</p> 
     454<p class=MsoNormal align=center style='text-align:center'><img border=0 
     455width=337 height=53 src="../images/html/image004.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     456</p> 
     457 
     458<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     459 
     460<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     461 
     462<p class=MsoNormal>where <i>scale</i> is a scale factor* volume fraction, V is 
     463the volume of the scatterer, <i>r</i> is the radius of the sphere, <i>bkg</i> 
     464is the background level and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
     465(contrast)&nbsp; is the scattering length density (SLD) &nbsp;difference 
     466between the scatterer and the solvent it is in.</p> 
    949467 
    950468<p class=MsoNormal>&nbsp;</p> 
    … …  
    962480<div align=center> 
    963481 
    964 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    965  style='border-collapse:collapse;border:none'> 
     482<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     483 style='border-collapse:collapse'> 
    966484 <tr style='height:13.5pt'> 
    967485  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt; 
    … …  
    981499  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt; 
    982500  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
    983   <p class=MsoBodyText>Scale</p> 
     501  <p class=MsoBodyText>scale</p> 
    984502  </td> 
    985503  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left: 
    … …  
    991509  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    992510  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
    993   <p class=MsoBodyText>1.0e-6</p> 
     511  <p class=MsoBodyText>1</p> 
    994512  </td> 
    995513 </tr> 
    … …  
    997515  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt; 
    998516  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
    999   <p class=MsoBodyText>Radius</p> 
     517  <p class=MsoBodyText>radius</p> 
    1000518  </td> 
    1001519  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left: 
    … …  
    1013531  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt; 
    1014532  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
    1015   <p class=MsoBodyText>Contrast</p> 
     533  <p class=MsoBodyText>sldSph</p> 
    1016534  </td> 
    1017535  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left: 
    … …  
    1023541  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1024542  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
    1025   <p class=MsoBodyText>1</p> 
     543  <p class=MsoBodyText>2.0e-6</p> 
    1026544  </td> 
    1027545 </tr> 
    … …  
    1029547  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt; 
    1030548  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
    1031   <p class=MsoBodyText>Background</p> 
     549  <p class=MsoBodyText>sldSolv</p> 
     550  </td> 
     551  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left: 
     552  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     553  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
     554  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     555  </td> 
     556  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left: 
     557  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     558  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
     559  <p class=MsoBodyText>1.0e-6</p> 
     560  </td> 
     561 </tr> 
     562 <tr style='height:12.8pt'> 
     563  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt; 
     564  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'> 
     565  <p class=MsoBodyText>background</p> 
    1032566  </td> 
    1033567  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left: 
    … …  
    1057591<p class=MsoNormal>&nbsp;</p> 
    1058592 
    1059 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
    1060 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    1061 of the sphere model</b></p> 
     593<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span 
     594style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the 
     595sphere model</b></p> 
    1062596 
    1063597<p class=MsoNormal>&nbsp;</p> 
    … …  
    1071605 
    1072606<p class=MsoBodyText align=center style='text-align:center;page-break-after: 
    1073 avoid'><img border=0 width=521 height=287 id="Picture 5" 
     607avoid'><img border=0 width=439 height=287 
    1074608src="../images/html/image005.jpg" alt="sphere_1D_validation"></p> 
    1075609 
    … …  
    1086620<p class=MsoNormal><b>&nbsp;</b></p> 
    1087621 
    1088 <p class=MsoNormal><b>&nbsp;</b></p> 
    1089  
    1090 <p class=MsoNormal><b>&nbsp;</b></p> 
    1091  
    1092  
    1093  
    1094622<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    1095 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; 
    1096 </span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellModel">Core Shell (Sphere) Model</a></span></b></p> 
    1097  
    1098 <p class=MsoNormal>&nbsp;</p> 
    1099  
    1100 <p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
    1101 spherical particle with a core-shell structure. The form factor is normalized 
    1102 by the particle volume.</p> 
    1103  
    1104 <p class=MsoNormal>&nbsp;</p> 
    1105  
    1106 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
    1107 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     623style='font-size:14.0pt'>2.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     624</span></b><a name=StickyHSStructure></a><a name=FuzzySphereModel></a><b><span 
     625style='font-size:14.0pt'>FuzzySphereModel</span></b></p> 
     626 
     627<p class=MsoNormal>&nbsp;</p> 
     628 
     629<p class=MsoNormal><b>&nbsp;</b>This model is to calculate the scattering from 
     630spherical particles with a &quot;fuzzy&quot; interface. </p> 
     631 
     632<p class=MsoNormal>&nbsp;</p> 
     633 
     634<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     635style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    1108636 
    1109637<p class=MsoNormal>&nbsp;</p> 
    … …  
    1112640way (Guinier, 1955):</p> 
    1113641 
     642<p class=MsoNormal>The returned value is scaled to units of [cm-1 sr-1], absolute 
     643scale.</p> 
     644 
     645<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     646 
     647<p class=MsoNormal>The scattering intensity I(q) is calculated as:</p> 
     648 
     649<p class=MsoNormal>&nbsp;</p> 
     650 
    1114651<p class=MsoNormal align=center style='text-align:center'><span 
    1115 style='position:relative;top:16.0pt'><img border=0 width=642 height=55 
    1116 src="../images/html/image006.png"></span> (11)</p> 
     652style='position:relative;top:5pt'><img border=0 width=12 height=23 
     653src="../images/html/image101.gif"><img border=0 width=233 height=41 
     654src="../images/html/image102.gif"></p> 
     655 
     656<p class=MsoNormal>&nbsp;</p> 
     657 
     658<p class=MsoNormal>where the amplitude A(q) is given as the typical sphere 
     659scattering convoluted with a Gaussian to get a gradual drop-off in the 
     660scattering length density:</p> 
     661 
     662<p class=MsoNormal>&nbsp;</p> 
     663 
     664<p class=MsoNormal>&nbsp;</p> 
     665 
     666<p class=MsoNormal align=center style='text-align:center'><span 
     667style='position:relative;top:18pt'><img border=0 width=315 height=56 
     668src="../images/html/image103.gif"></p> 
     669 
     670<p class=MsoNormal>&nbsp;</p> 
     671 
     672<p class=MsoNormal>&nbsp;</p> 
     673 
     674<p class=MsoNormal>Here A<sup>2</sup>(q) is the form factor, P(q). The ‘scale’ 
     675is equivalent to the volume fraction of spheres, each of volume, V. Contrast (<i><span 
     676style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> ) is the 
     677difference of scattering length densities of the sphere and the surrounding 
     678solvent.</p> 
     679 
     680<p class=MsoNormal>&nbsp;</p> 
     681 
     682<p class=MsoNormal>The poly-dispersion in radius and in fuzziness is provided.</p> 
     683 
     684<p class=MsoNormal>&nbsp;</p> 
     685 
     686<p class=MsoNormal>(direct from the reference)</p> 
     687 
     688<p class=MsoNormal>The &quot;fuzziness&quot; of the interface is defined by the 
     689parameter (sigma)<sub>fuzzy</sub>. The particle radius R represents the radius 
     690of the particle where the scattering length density profile decreased to 1/2 of 
     691the core density. The (sigma)<sub>fuzzy</sub>&nbsp; is the width of the smeared 
     692particle surface: i.e., the standard deviation from the average height of the 
     693fuzzy interface. The inner regions [] that display a higher density are 
     694described by the radial box profile extending to a radius of approximately Rbox 
     695~ R - 2(sigma). the profile approaches zero as Rsans ~ R + 2(sigma).</p> 
     696 
     697<p class=MsoNormal>For 2D data: The 2D scattering intensity is calculated in 
     698the same way as 1D, where the <i>q</i> vector is defined as<span 
     699style='font-size:14.0pt'><img border=0 width=82 height=26 
     700src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
     701 
     702<p class=MsoNormal>&nbsp;</p> 
     703 
     704<p class=MsoNormal>REFERENCE</p> 
     705 
     706<p class=MsoNormal>M. Stieger, J. S. Pedersen, P. Lindner, W. Richtering, 
     707Langmuir 20 (2004) 7283-7292.</p> 
     708 
     709<p class=MsoNormal>&nbsp;</p> 
     710 
     711<p class=MsoNormal>TEST DATASET</p> 
     712 
     713<p class=MsoNormal>This example dataset is produced by running the 
     714FuzzySphereModel, using 200 data points, qmin = 0.001 Å<sup>-1</sup>,&nbsp; 
     715qmax = 0.7 A<sup>-1</sup> and the default coef_fuzz values:</p> 
     716 
     717<p class=MsoNormal>&nbsp;</p> 
     718 
     719<div align=center> 
     720 
     721<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     722 style='border-collapse:collapse'> 
     723 <tr style='height:18.8pt'> 
     724  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     725  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     726  <p class=MsoBodyText>Parameter name</p> 
     727  </td> 
     728  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     729  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     730  <p class=MsoBodyText>Units</p> 
     731  </td> 
     732  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     733  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     734  <p class=MsoBodyText>Default value</p> 
     735  </td> 
     736 </tr> 
     737 <tr style='height:18.8pt'> 
     738  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     739  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     740  <p class=MsoBodyText>scale</p> 
     741  </td> 
     742  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     743  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     744  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     745  <p class=MsoBodyText>None</p> 
     746  </td> 
     747  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     748  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     749  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     750  <p class=MsoBodyText>1.0</p> 
     751  </td> 
     752 </tr> 
     753 <tr style='height:18.8pt'> 
     754  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     755  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     756  <p class=MsoBodyText>radius</p> 
     757  </td> 
     758  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     759  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     760  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     761  <p class=MsoBodyText>Å</p> 
     762  </td> 
     763  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     764  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     765  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     766  <p class=MsoBodyText>60</p> 
     767  </td> 
     768 </tr> 
     769 <tr style='height:18.8pt'> 
     770  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     771  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     772  <p class=MsoBodyText>fuzziness</p> 
     773  </td> 
     774  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     775  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     776  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     777  <p class=MsoBodyText>Å</p> 
     778  </td> 
     779  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     780  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     781  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     782  <p class=MsoBodyText>10</p> 
     783  </td> 
     784 </tr> 
     785 <tr style='height:18.8pt'> 
     786  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     787  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     788  <p class=MsoBodyText>sldSolv</p> 
     789  </td> 
     790  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     791  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     792  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     793  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     794  </td> 
     795  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     796  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     797  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     798  <p class=MsoBodyText>3e-6</p> 
     799  </td> 
     800 </tr> 
     801 <tr style='height:18.8pt'> 
     802  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     803  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     804  <p class=MsoBodyText>sldSph</p> 
     805  </td> 
     806  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     807  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     808  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     809  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     810  </td> 
     811  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     812  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     813  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     814  <p class=MsoBodyText>1e-6</p> 
     815  </td> 
     816 </tr> 
     817 <tr style='height:18.8pt'> 
     818  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     819  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     820  <p class=MsoBodyText>background</p> 
     821  </td> 
     822  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     823  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     824  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     825  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     826  </td> 
     827  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     828  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     829  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     830  <p class=MsoBodyText>0.001</p> 
     831  </td> 
     832 </tr> 
     833</table> 
     834 
     835</div> 
     836 
     837<p class=MsoNormal>&nbsp;</p> 
     838 
     839<p class=MsoNormal align=center style='text-align:center'><img border=0 
     840width=442 height=275 src="../images/html/image104.jpg"></p> 
     841 
     842<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     8431D plot using the default values (w/200 data point).</b></p> 
     844 
     845<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     846 
     847<p class=MsoNormal>&nbsp;</p> 
     848 
     849<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     850style='font-size:14.0pt'>2.3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     851</span></b><a name=CoreShellModel><b><span style='font-size:14.0pt'>Core Shell 
     852(Sphere) Model</span></b></a></p> 
     853 
     854<p class=MsoNormal>&nbsp;</p> 
     855 
     856<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a spherical 
     857particle with a core-shell structure. The form factor is normalized by the 
     858particle volume.</p> 
     859 
     860<p class=MsoNormal>&nbsp;</p> 
     861 
     862<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     863style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
     864 
     865<p class=MsoNormal>&nbsp;</p> 
     866 
     867<p class=MsoNormal>The 1D scattering intensity is calculated in the following 
     868way (Guinier, 1955):</p> 
     869 
     870<p class=MsoNormal align=center style='text-align:center'><img border=0 
     871width=387 height=55 src="../images/html/image006.png">&nbsp;</p> 
    1117872 
    1118873<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    1150905<div align=center> 
    1151906 
    1152 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    1153  style='border-collapse:collapse;border:none'> 
     907<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     908 style='border-collapse:collapse'> 
    1154909 <tr style='height:18.8pt'> 
    1155910  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    1185940  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1186941  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1187   <p class=MsoBodyText>radius</p> 
     942  <p class=MsoBodyText>(core) radius</p> 
    1188943  </td> 
    1189944  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    12871042c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    12881043 
    1289 <p class=MsoNormal>            </p> 
     1044<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1045</p> 
    12901046 
    12911047<p class=MsoNormal>REFERENCE</p> 
    … …  
    12981054<p class=MsoNormal>&nbsp;</p> 
    12991055 
    1300 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
    1301 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    1302 of the core-shell sphere model</b></p> 
     1056<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span 
     1057style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the 
     1058core-shell sphere model</b></p> 
    13031059 
    13041060<p class=MsoNormal>&nbsp;</p> 
    … …  
    13121068 
    13131069<p class=MsoBodyText align=center style='text-align:center;page-break-after: 
    1314 avoid'><img border=0 width=526 height=289 id="Picture 19" 
     1070avoid'><img border=0 width=466 height=289 
    13151071src="../images/html/image007.jpg" alt="core_shell_sphere_1D_validation"></p> 
    13161072 
    … …  
    13241080<p class=MsoNormal>&nbsp;</p> 
    13251081 
     1082<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     1083style='font-size:14.0pt'>2.4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1084</span></b><a name=CoreFourShellModel><b><span style='font-size:14.0pt'>CoreFourShell(Sphere)Model</span></b></a></p> 
     1085 
     1086<p class=MsoNormal>&nbsp;</p> 
     1087 
     1088<p class=MsoNormal>This model provides the scattering from monodisperse core 4 
     1089shell structures. It &nbsp;has a core of a specified radius, with four shells. 
     1090The SLDs of the core and each shell are individually specified.&nbsp;</p> 
     1091 
     1092<p class=MsoNormal>&nbsp;</p> 
     1093 
     1094<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     1095style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
     1096 
     1097<p class=MsoNormal>&nbsp;</p> 
     1098 
     1099<p class=MsoNormal>The returned value is scaled to units of [cm-1sr-1], 
     1100absolute scale.</p> 
     1101 
     1102<p class=MsoNormal>&nbsp;</p> 
     1103 
     1104<p class=MsoNormal>This model is a trivial extension of the CoreShell function 
     1105to a larger number of shells. See the CoreShell function for a diagram and 
     1106documentation.</p> 
     1107 
     1108<p class=MsoNormal>&nbsp;</p> 
     1109 
     1110<p class=MsoNormal>Be careful that the SLDs and scale can be highly correlated. 
     1111Hold as many of these fixed as possible.</p> 
     1112 
     1113<p class=MsoNormal>&nbsp;</p> 
     1114 
     1115<p class=MsoNormal>The 2D scattering intensity is the same as P(q) of 1D, 
     1116regardless of the orientation of the q vector.</p> 
     1117 
     1118<p class=MsoNormal>&nbsp;</p> 
     1119 
     1120<p class=MsoNormal>For P*S: The outer most radius (= radius + 4 thicknesses) is 
     1121used as the effective radius toward S(Q) if P(Q)*S(Q) is applied. </p> 
     1122 
     1123<p class=MsoNormal>&nbsp;</p> 
     1124 
     1125<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
     1126and the parameters of the CoreFourshell sphere model are the following:</p> 
     1127 
     1128<p class=MsoNormal>Here, rad_core = the radius of the core, &nbsp;thick_shelli 
     1129= the thickness of the shell i and sld_shelli = the SLD of the shell i.</p> 
     1130 
     1131<p class=MsoNormal>And the sld_core and the sld_solv are the SLD of the core 
     1132and the solvent, respectively.</p> 
     1133 
     1134<p class=MsoNormal>&nbsp;</p> 
     1135 
     1136<div align=center> 
     1137 
     1138<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     1139 style='border-collapse:collapse'> 
     1140 <tr style='height:18.8pt'> 
     1141  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1142  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1143  <p class=MsoBodyText>Parameter name</p> 
     1144  </td> 
     1145  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1146  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1147  <p class=MsoBodyText>Units</p> 
     1148  </td> 
     1149  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1150  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1151  <p class=MsoBodyText>Default value</p> 
     1152  </td> 
     1153 </tr> 
     1154 <tr style='height:18.8pt'> 
     1155  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1156  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1157  <p class=MsoBodyText>scale</p> 
     1158  </td> 
     1159  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1160  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1161  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1162  <p class=MsoBodyText>None</p> 
     1163  </td> 
     1164  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1165  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1166  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1167  <p class=MsoBodyText>1.0</p> 
     1168  </td> 
     1169 </tr> 
     1170 <tr style='height:18.8pt'> 
     1171  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1172  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1173  <p class=MsoBodyText>rad_core</p> 
     1174  </td> 
     1175  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1176  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1177  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1178  <p class=MsoBodyText>Å</p> 
     1179  </td> 
     1180  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1181  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1182  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1183  <p class=MsoBodyText>60</p> 
     1184  </td> 
     1185 </tr> 
     1186 <tr style='height:18.8pt'> 
     1187  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1188  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1189  <p class=MsoBodyText>sld_core</p> 
     1190  </td> 
     1191  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1192  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1193  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1194  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1195  </td> 
     1196  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1197  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1198  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1199  <p class=MsoBodyText>6.4e-6</p> 
     1200  </td> 
     1201 </tr> 
     1202 <tr style='height:18.8pt'> 
     1203  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1204  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1205  <p class=MsoBodyText>sld_shell1</p> 
     1206  </td> 
     1207  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1208  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1209  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1210  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1211  </td> 
     1212  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1213  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1214  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1215  <p class=MsoBodyText>1e-6</p> 
     1216  </td> 
     1217 </tr> 
     1218 <tr style='height:18.8pt'> 
     1219  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1220  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1221  <p class=MsoBodyText>sld_shell2</p> 
     1222  </td> 
     1223  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1224  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1225  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1226  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1227  </td> 
     1228  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1229  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1230  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1231  <p class=MsoBodyText>2e-6</p> 
     1232  </td> 
     1233 </tr> 
     1234 <tr style='height:18.8pt'> 
     1235  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1236  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1237  <p class=MsoBodyText>sld_shell3</p> 
     1238  </td> 
     1239  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1240  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1241  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1242  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1243  </td> 
     1244  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1245  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1246  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1247  <p class=MsoBodyText>3e-6</p> 
     1248  </td> 
     1249 </tr> 
     1250 <tr style='height:18.8pt'> 
     1251  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1252  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1253  <p class=MsoBodyText>sld_shell4</p> 
     1254  </td> 
     1255  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1256  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1257  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1258  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1259  </td> 
     1260  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1261  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1262  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1263  <p class=MsoBodyText>4e-6</p> 
     1264  </td> 
     1265 </tr> 
     1266 <tr style='height:18.8pt'> 
     1267  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1268  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1269  <p class=MsoBodyText>sld_solv</p> 
     1270  </td> 
     1271  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1272  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1273  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1274  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1275  </td> 
     1276  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1277  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1278  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1279  <p class=MsoBodyText>6.4e-6</p> 
     1280  </td> 
     1281 </tr> 
     1282 <tr style='height:18.8pt'> 
     1283  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1284  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1285  <p class=MsoBodyText>thick_shell1</p> 
     1286  </td> 
     1287  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1288  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1289  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1290  <p class=MsoBodyText>Å</p> 
     1291  </td> 
     1292  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1293  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1294  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1295  <p class=MsoBodyText>10</p> 
     1296  </td> 
     1297 </tr> 
     1298 <tr style='height:18.8pt'> 
     1299  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1300  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1301  <p class=MsoBodyText>thick_shell2</p> 
     1302  </td> 
     1303  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1304  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1305  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1306  <p class=MsoBodyText>Å</p> 
     1307  </td> 
     1308  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1309  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1310  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1311  <p class=MsoBodyText>10</p> 
     1312  </td> 
     1313 </tr> 
     1314 <tr style='height:18.8pt'> 
     1315  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1316  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1317  <p class=MsoBodyText>thick_shell3</p> 
     1318  </td> 
     1319  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1320  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1321  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1322  <p class=MsoBodyText>Å</p> 
     1323  </td> 
     1324  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1325  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1326  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1327  <p class=MsoBodyText>10</p> 
     1328  </td> 
     1329 </tr> 
     1330 <tr style='height:18.8pt'> 
     1331  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1332  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1333  <p class=MsoBodyText>thick_shell4</p> 
     1334  </td> 
     1335  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1336  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1337  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1338  <p class=MsoBodyText>Å</p> 
     1339  </td> 
     1340  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1341  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1342  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1343  <p class=MsoBodyText>10</p> 
     1344  </td> 
     1345 </tr> 
     1346 <tr style='height:19.5pt'> 
     1347  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1348  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
     1349  <p class=MsoBodyText>background</p> 
     1350  </td> 
     1351  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1352  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1353  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
     1354  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     1355  </td> 
     1356  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1357  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1358  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
     1359  <p class=MsoBodyText>0.001</p> 
     1360  </td> 
     1361 </tr> 
     1362</table> 
     1363 
     1364</div> 
     1365 
     1366<p class=MsoNormal>&nbsp;</p> 
     1367 
     1368<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     1369c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     1370 
     1371<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1372</p> 
     1373 
     1374<p class=MsoNormal>REFERENCE</p> 
     1375 
     1376<p class=MsoNormal>&nbsp;</p> 
     1377 
     1378<p class=MsoNormal>See the CoreShell documentation.&nbsp;</p> 
     1379 
     1380<p class=MsoNormal>&nbsp;</p> 
     1381 
     1382<p class=MsoNormal>TEST DATASET</p> 
     1383 
     1384<p class=MsoNormal>&nbsp;</p> 
     1385 
     1386<p class=MsoNormal>This example dataset is produced by running the 
     1387CoreFourShellModel using 200 data points, qmin = 0.001 Å-1,&nbsp; qmax = 0.7 
     1388Å-1 and the above default values.</p> 
     1389 
     1390<p class=MsoNormal>&nbsp;</p> 
     1391 
     1392<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1393width=412 height=287 src="../images/html/image105.jpg"></p> 
     1394 
     1395<p class=MsoNormal>&nbsp;</p> 
     1396 
     1397<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     13981D plot using the default values (w/200 data point).</b></p> 
     1399 
     1400<p class=MsoNormal>&nbsp;</p> 
     1401 
     1402<p class=MsoNormal>&nbsp;</p> 
     1403 
     1404<p class=MsoNormal>&nbsp;</p> 
     1405 
    13261406<p class=MsoNormal><b>&nbsp;</b></p> 
    13271407 
    13281408<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    1329 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; 
    1330 </span></span></b><b><span style='font-size:14.0pt'><a name="VesicleModel">VesicleModel</a></span></b></p> 
     1409style='font-size:14.0pt'>2.5.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1410</span></b><a name=VesicleModel><b><span style='font-size:14.0pt'>VesicleModel</span></b></a></p> 
    13311411 
    13321412<p class=MsoNormal>&nbsp;</p> 
    … …  
    13401420<p class=MsoNormal>&nbsp;</p> 
    13411421 
    1342 <p class=MsoNormal align=center style='text-align:center'><span 
    1343 style='position:relative;top:16.0pt'><img border=0 width=448 height=54 
    1344 src="../images/html/image008.png"></span> (11)</p> 
     1422<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1423width=377 height=54 src="../images/html/image008.png">&nbsp;</p> 
    13451424 
    13461425<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    13601439except that the scattering is normalized by the volume that is contributing to 
    13611440the scattering, namely the volume of the shell alone. Also, the vesicle is best 
    1362 defined in terms of a core radius (= R1) and a shell thickness, t.  </p> 
     1441defined in terms of a core radius (= R1) and a shell thickness, t. &nbsp;</p> 
    13631442 
    13641443<p class=MsoNormal>&nbsp;</p> 
    … …  
    13671446 
    13681447<p class=MsoNormal align=center style='text-align:center'><img border=0 
    1369 width=149 height=143 id="Picture 31" src="../images/html/image009.jpg"></p> 
     1448width=149 height=143 src="../images/html/image009.jpg"></p> 
    13701449 
    13711450<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    13741453 
    13751454<p class=MsoNormal>The 2D scattering intensity is the same as <i>P</i>(<i>q</i>) 
    1376 above, regardless of the orientation of the <i>q</i> vector  which is defined 
    1377 as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 
    1378 width=103 height=33 src="../images/html/image010.png"></span><span 
    1379 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     1455above, regardless of the orientation of the <i>q</i> vector&nbsp; which is 
     1456defined as<span style='font-size:14.0pt'><img border=0 width=69 height=22 
     1457src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    13801458 
    13811459<p class=MsoNormal>For P*S: The outer most radius (= radius + thickness) is 
    … …  
    13951473<div align=center> 
    13961474 
    1397 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    1398  style='border-collapse:collapse;border:none'> 
     1475<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     1476 style='border-collapse:collapse'> 
    13991477 <tr style='height:18.8pt'> 
    14001478  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    15161594 
    15171595<p class=MsoNormal align=center style='text-align:center'><img border=0 
    1518 width=454 height=356 id="Picture 158" src="../images/html/image011.png"></p> 
     1596width=454 height=275 src="../images/html/image011.png"></p> 
    15191597 
    15201598<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    15471625<p class=MsoNormal><b>&nbsp;</b></p> 
    15481626 
    1549  
    15501627<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    1551 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; 
    1552 </span></span></b><b><span style='font-size:14.0pt'><a name="MultiShellModel">MultiShellModel</a></span></b></p> 
     1628style='font-size:14.0pt'>2.6.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1629</span></b><a name=MultiShellModel><b><span style='font-size:14.0pt'>MultiShellModel</span></b></a></p> 
    15531630 
    15541631<p class=MsoNormal>&nbsp;</p> 
    15551632 
    15561633<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
    1557 multi-lamellar vesicle with N shells where the core is filled with solvent and 
    1558 the shells are interleaved with layers of solvent. For N = 1, this return to 
    1559 the vesicle model (above).</p> 
     1634multi-lamellar vesicle with N shells where the core is filled with solvent and the 
     1635shells are interleaved with layers of solvent. For N = 1, this return to the 
     1636vesicle model (above).</p> 
    15601637 
    15611638<p class=MsoNormal>&nbsp;</p> 
    … …  
    15641641 
    15651642<p class=MsoNormal align=center style='text-align:center'><img border=0 
    1566 width=255 height=224 id="Picture 32" src="../images/html/image012.jpg"></p> 
     1643width=255 height=224 src="../images/html/image012.jpg"></p> 
    15671644 
    15681645<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    15721649<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of 
    15731650the orientation of the <i>q</i> vector which is defined as<span 
    1574 style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
    1575 height=33 src="../images/html/image010.png"></span><span 
    1576 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
    1577  
    1578 <p class=MsoNormal>For P*S: The outer most radius (= core_radius + n_pairs  * 
    1579 s_thickness  + (n_pairs -1) * w_thickness) is used as the effective radius 
    1580 toward S(Q) when P(Q)*S(Q) is applied. </p> 
     1651style='font-size:14.0pt'><img border=0 width=59 height=19 
     1652src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
     1653 
     1654<p class=MsoNormal>For P*S: The outer most radius (= core_radius + 
     1655n_pairs&nbsp; * s_thickness&nbsp; + (n_pairs -1) * w_thickness) is used as the 
     1656effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
    15811657 
    15821658<p class=MsoNormal>&nbsp;</p> 
    … …  
    15951671<div align=center> 
    15961672 
    1597 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    1598  style='border-collapse:collapse;border:none'> 
     1673<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     1674 style='border-collapse:collapse'> 
    15991675 <tr style='height:18.8pt'> 
    16001676  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    17481824 
    17491825<p class=MsoNormal align=center style='text-align:center'><img border=0 
    1750 width=447 height=340 id="Picture 173" src="../images/html/image013.jpg"></p> 
     1826width=447 height=289 src="../images/html/image013.jpg"></p> 
    17511827 
    17521828<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    17771853 
    17781854<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    1779 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; 
    1780 </span></span></b><b><span style='font-size:14.0pt'><a name="BinaryHSModel">BinaryHSModel</a></span></b></p> 
    1781  
    1782 <p class=MsoNormal>&nbsp;</p> 
    1783  
    1784 <p class=MsoNormal>This model (binary hard sphere model) provides the scattering 
    1785 intensity, for binary mixture of spheres including hard sphere interaction 
    1786 between those particles. Using Percus-Yevick closure, the calculation is an 
    1787 exact multi-component solution:</p> 
    1788  
    1789 <p class=MsoNormal>&nbsp;</p> 
    1790  
    1791 <p class=MsoNormal align=center style='text-align:center'><span 
    1792 style='position:relative;top:5.0pt'><img border=0 width=479 height=25 
    1793 src="../images/html/image014.png"></span></p> 
    1794  
    1795 <p class=MsoNormal align=center style='text-align:center'><span 
    1796 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    1797  
    1798 <p class=MsoNormal align=center style='text-align:center'><span 
    1799 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    1800  
    1801 <p class=MsoNormal><span style='position:relative;top:16.0pt'>where S<sub>ij</sub> 
    1802 are the partial structure factors and f<sub>i</sub> are the scattering 
    1803 amplitudes of the particles. And the subscript 1 is for the smaller particle 
    1804 and 2 is for the larger. The number fraction of the larger particle, (<i>x</i> 
    1805 =  n<sub>2</sub>/(n<sub>1</sub>+n<sub>2</sub>), n = the number density) is 
    1806 internally calculated based on:</span></p> 
    1807  
    1808 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    1809  
    1810 <p class=MsoNormal>&nbsp;</p> 
    1811  
    1812 <p class=MsoNormal align=center style='text-align:center'><span 
    1813 style='position:relative;top:24.0pt'><img border=0 width=248 height=98 
    1814 src="../images/html/image015.png"></span><span style='position:relative; 
    1815 top:16.0pt'>.</span></p> 
    1816  
    1817 <p class=MsoNormal align=center style='text-align:center'><span 
    1818 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    1819  
    1820 <p class=MsoNormal align=center style='text-align:center'><span 
    1821 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     1855style='font-size:14.0pt'>2.7.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1856</span></b><a name=BinaryHSModel><b><span style='font-size:14.0pt'>BinaryHSModel</span></b></a></p> 
     1857 
     1858<p class=MsoNormal>&nbsp;</p> 
     1859 
     1860<p class=MsoNormal>This model (binary hard sphere model) provides the 
     1861scattering intensity, for binary mixture of spheres including hard sphere 
     1862interaction between those particles. Using Percus-Yevick closure, the 
     1863calculation is an exact multi-component solution:</p> 
     1864 
     1865<p class=MsoNormal>&nbsp;</p> 
     1866 
     1867<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1868width=349 height=25 src="../images/html/image014.png"></p> 
     1869 
     1870<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1871 
     1872<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1873 
     1874<p class=MsoNormal>where S<sub>ij</sub> are the partial structure factors and f<sub>i</sub> 
     1875are the scattering amplitudes of the particles. And the subscript 1 is for the 
     1876smaller particle and 2 is for the larger. The number fraction of the larger 
     1877particle, (<i>x</i> = &nbsp;n<sub>2</sub>/(n<sub>1</sub>+n<sub>2</sub>), n = 
     1878the number density) is internally calculated based on:</p> 
     1879 
     1880<p class=MsoNormal>&nbsp;</p> 
     1881 
     1882<p class=MsoNormal>&nbsp;</p> 
     1883 
     1884<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1885width=248 height=98 src="../images/html/image015.png">.</p> 
     1886 
     1887<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1888 
     1889<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    18221890 
    18231891<p class=MsoNormal>&nbsp;</p> 
    … …  
    18251893<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of 
    18261894the orientation of the <i>q</i> vector which is defined as<span 
    1827 style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
    1828 height=33 src="../images/html/image010.png"></span><span 
    1829 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     1895style='font-size:14.0pt'><img border=0 width=69 height=22 
     1896src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    18301897 
    18311898<p class=MsoNormal>&nbsp;</p> 
    … …  
    18411908<div align=center> 
    18421909 
    1843 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    1844  style='border-collapse:collapse;border:none'> 
     1910<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     1911 style='border-collapse:collapse'> 
    18451912 <tr style='height:18.8pt'> 
    18461913  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    19942061 
    19952062<p class=MsoNormal align=center style='text-align:center'><img border=0 
    1996 width=484 height=361 id="Picture 197" src="../images/html/image016.png"></p> 
     2063width=447 height=313 src="../images/html/image016.png"></p> 
    19972064 
    19982065<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    20262093 
    20272094<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    2028 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; 
    2029 </span></span></b><b><span style='font-size:14.0pt'><a name="CylinderModel">Cylinder Model</a></span></b></p> 
     2095style='font-size:14.0pt'>2.8.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2096</span></b><a name=CylinderModel><b><span style='font-size:14.0pt'>Cylinder 
     2097Model</span></b></a></p> 
    20302098 
    20312099<p class=MsoNormal>&nbsp;</p> 
    … …  
    20372105<p class=MsoNormal>&nbsp;</p> 
    20382106 
    2039 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
    2040 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     2107<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     2108style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    20412109 
    20422110<p class=MsoNormal>&nbsp;</p> 
    … …  
    20492117<p class=MsoNormal>&nbsp;</p> 
    20502118 
    2051 <p class=MsoNormal align=center style='text-align:center'><span 
    2052 style='position:relative;top:12.0pt'><img border=0 width=184 height=41 
    2053 src="../images/html/image017.png"></span>  (4)</p> 
    2054  
    2055 <p class=MsoNormal align=center style='text-align:center'><span 
    2056 style='position:relative;top:14.0pt'><img border=0 width=373 height=45 
    2057 src="../images/html/image018.png"></span>   (5)</p> 
     2119<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2120width=184 height=41 src="../images/html/image017.png">&nbsp; (4)</p> 
     2121 
     2122<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2123width=359 height=45 src="../images/html/image018.png">&nbsp;&nbsp; </p> 
    20582124 
    20592125<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    20632129<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span> 
    20642130is the angle between the axis of the cylinder and the q-vector, V is the volume 
    2065 of the cylinder, L is the length of the cylinder, r is the radius of the 
    2066 cylinder, and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
     2131of the cylinder, L is the length of the cylinder, r is the radius of the cylinder, 
     2132and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
    20672133(contrast) is the scattering length density difference between the scatterer 
    20682134and the solvent. J<sub>1</sub> is the first order Bessel function.</p> 
    … …  
    20772143 
    20782144<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    2079 border=0 width=478 height=258 id="Picture 14" 
    2080 src="../images/html/image019.jpg" alt=cylinderangles.gif></p> 
     2145border=0 width=478 height=258 src="../images/html/image019.jpg" 
     2146alt=cylinderangles.gif></p> 
    20812147 
    20822148<p class=MsoCaption align=center style='text-align:center'><a 
    2083 name="_Ref173306040"></a><a name="_Ref173213915">Figure </a>2a. Definition of the angles for oriented cylinders.</p> 
     2149name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure 2a. Definition of 
     2150the angles for oriented cylinders.</p> 
    20842151 
    20852152<p class=MsoNormal>&nbsp;</p> 
    … …  
    20912158alt=cylinderangles2.gif></p> 
    20922159 
    2093 <p class=MsoCaption align=center style='text-align:center'>Figure 2b. Examples of the angles for oriented cylinders against the detector plane.</p> 
     2160<p class=MsoCaption align=center style='text-align:center'>Figure 2b. Examples 
     2161of the angles for oriented cylinders against the detector plane.</p> 
    20942162 
    20952163<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    21122180<div align=center> 
    21132181 
    2114 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    2115  style='border-collapse:collapse;border:none'> 
     2182<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     2183 style='border-collapse:collapse'> 
    21162184 <tr style='height:19.85pt'> 
    21172185  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    22562324 
    22572325<p class=MsoNormal align=center style='text-align:center'><a 
    2258 name="_Ref173306479"></a><a name="_Ref173306528"><span style='position:relative; 
    2259 top:16.0pt'><img border=0 width=253 height=56 
    2260 src="../images/html/image021.png"></span>         (</a>6)</p> 
     2326name="_Ref173306528"></a><a name="_Ref173306479"></a><img border=0 width=253 
     2327height=56 src="../images/html/image021.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2328</p> 
    22612329 
    22622330<p class=MsoNormal>&nbsp;</p> 
    … …  
    22742342<p class=MsoNormal>&nbsp;</p> 
    22752343 
    2276 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
    2277 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    2278 of the cylinder model</b></p> 
     2344<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span 
     2345style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the 
     2346cylinder model</b></p> 
    22792347 
    22802348<p class=MsoNormal>&nbsp;</p> 
    22812349 
    22822350<p class=MsoNormal>Validation of our code was done by comparing the output of 
    2283 the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 
    2284 3 shows a comparison of the 1D output of our model and the output of the NIST 
    2285 software.</p> 
     2351the 1D model to the output of the software provided by the NIST (Kline, 2006). 
     2352Figure 3 shows a comparison of the 1D output of our model and the output of the 
     2353NIST software.</p> 
    22862354 
    22872355<p class=MsoNormal>&nbsp;</p> 
    … …  
    22922360<p class=MsoNormal>&nbsp;</p> 
    22932361 
    2294 <p class=MsoNormal align=center style='text-align:center'><span 
    2295 style='position:relative;top:16.0pt'><img border=0 width=244 height=51 
    2296 src="../images/html/image022.png"></span>           (7)</p> 
     2362<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2363width=244 height=51 src="../images/html/image022.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2364</p> 
    22972365 
    22982366<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    23062374compare the implementation of the intensity for fully oriented cylinders, we 
    23072375can compare the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span 
    2308 style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 4 
    2309 shows the result of such a cross-check.</p> 
     2376style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.&nbsp; Figure 
     23774 shows the result of such a cross-check.</p> 
    23102378 
    23112379<p class=MsoNormal>&nbsp;</p> 
    … …  
    23142382 
    23152383<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    2316 border=0 width=512 height=282 id="Picture 11" 
    2317 src="../images/html/image023.jpg" alt="cylinder_1D_validation"></p> 
     2384border=0 width=512 height=282 src="../images/html/image023.jpg" 
     2385alt="cylinder_1D_validation"></p> 
    23182386 
    23192387<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    23202388 
    2321 <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 
    2322 NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 
    2323 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p> 
     2389<p class=MsoCaption><a name="_Ref173211066"></a><a name="_Ref173211049"></a>Figure 
     23903: Comparison of the DANSE scattering intensity for a cylinder with the output 
     2391of the NIST SANS analysis software. The parameters were set to: Scale=1.0, 
     2392Radius=20 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 
     2393cm<sup> -1</sup>.</p> 
    23242394 
    23252395<p class=MsoNormal>&nbsp;</p> 
    … …  
    23282398 
    23292399<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    2330 border=0 width=521 height=287 id="Picture 12" 
    2331 src="../images/html/image024.jpg" alt="cylinder_2D_average"></p> 
     2400border=0 width=467 height=287 src="../images/html/image024.jpg" 
     2401alt="cylinder_2D_average"></p> 
    23322402 
    23332403<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    23342404 
    2335 <p class=MsoCaption><a name="_Ref173213305">Figure </a>4: Comparison of the 
    2336 intensity for uniformly distributed cylinders calculated from our 2D model and 
    2337 the intensity from the NIST SANS analysis software. The parameters used were: 
     2405<p class=MsoCaption><a name="_Ref173213305">Figure </a>4: Comparison of the intensity 
     2406for uniformly distributed cylinders calculated from our 2D model and the 
     2407intensity from the NIST SANS analysis software. The parameters used were: 
    23382408Scale=1.0, Radius=20 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and 
    23392409Background=0.0 cm<sup> -1</sup>.</p> 
    … …  
    23462416 
    23472417<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    2348 style='font-size:14.0pt'>2.7.<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; 
    2349 </span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellCylinderModel">Core-Shell Cylinder Model</a></span></b></p> 
     2418style='font-size:14.0pt'>2.9.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2419</span></b><a name=CoreShellCylinderModel><b><span style='font-size:14.0pt'>Core-Shell 
     2420Cylinder Model</span></b></a></p> 
    23502421 
    23512422<p class=MsoNormal>&nbsp;</p> 
    … …  
    23572428<p class=MsoNormal>&nbsp;</p> 
    23582429 
    2359 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
    2360 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     2430<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     2431style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    23612432 
    23622433<p class=MsoNormal>&nbsp;</p> 
    … …  
    23692440<p class=MsoNormal>&nbsp;</p> 
    23702441 
    2371 <p class=MsoNormal align=center style='text-align:center'><span 
    2372 style='position:relative;top:15.0pt'><img border=0 width=184 height=45 
    2373 src="../images/html/image025.png"></span> (12)</p> 
    2374  
    2375 <p class=MsoNormal align=center style='text-align:center'><span 
    2376 style='position:relative;top:32.0pt'><img border=0 width=476 height=93 
    2377 src="../images/html/image026.png"></span> (13)</p> 
    2378  
    2379 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2442<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2443width=184 height=45 src="../images/html/image025.png">&nbsp;</p> 
     2444 
     2445<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2446width=357 height=93 src="../images/html/image026.png">&nbsp;</p> 
    23802447 
    23812448<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    23972464 
    23982465<p class=MsoNormal align=center style='text-align:center'><img border=0 
    2399 width=384 height=196 id="Picture 34" src="../images/html/image027.jpg"></p> 
     2466width=384 height=196 src="../images/html/image027.jpg"></p> 
    24002467 
    24012468<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    24062473cylinder, we define the axis of the cylinder using two angles &#952; and <span 
    24072474style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
    2408 of the cylinder, those angles are defined on Figure 2.</p> 
     2475of the cylinder, those angles are defined on Figure 2 in Cylinder Model.</p> 
    24092476 
    24102477<p class=MsoNormal>&nbsp;</p> 
    … …  
    24262493<div align=center> 
    24272494 
    2428 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    2429  style='border-collapse:collapse;border:none'> 
     2495<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     2496 style='border-collapse:collapse'> 
    24302497 <tr style='height:19.25pt'> 
    24312498  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    26082675<p class=MsoNormal>&nbsp;</p> 
    26092676 
    2610 <p class=MsoNormal>The output of the 1D scattering intensity function for 
    2611 randomly oriented cylinders is then given by equation 6.</p> 
     2677<p class=MsoNormal>The output of the 1D scattering intensity function for randomly 
     2678oriented cylinders is then given by the equation above.</p> 
    26122679 
    26132680<p class=MsoNormal>&nbsp;</p> 
    … …  
    26212688<p class=MsoNormal>&nbsp;</p> 
    26222689 
    2623 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
    2624 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    2625 of the core-shell cylinder model</b></p> 
     2690<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span 
     2691style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the 
     2692core-shell cylinder model</b></p> 
    26262693 
    26272694<p class=MsoNormal>&nbsp;</p> 
    26282695 
    26292696<p class=MsoNormal>Validation of our code was done by comparing the output of 
    2630 the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 
    2631 8 shows a comparison of the 1D output of our model and the output of the NIST 
    2632 software.</p> 
     2697the 1D model to the output of the software provided by the NIST (Kline, 2006). 
     2698Figure 8 shows a comparison of the 1D output of our model and the output of the 
     2699NIST software.</p> 
    26332700 
    26342701<p class=MsoNormal>&nbsp;</p> 
    26352702 
    26362703<p class=MsoNormal>Averaging over a distribution of orientation is done by 
    2637 evaluating equation 7. Since we have no other software to compare the 
     2704evaluating the equation above. Since we have no other software to compare the 
    26382705implementation of the intensity for fully oriented core-shell cylinders, we can 
    26392706compare the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span 
    2640 style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 9 shows the result of such a cross-check.</p> 
     2707style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.&nbsp; Figure 
     27089 shows the result of such a cross-check.</p> 
    26412709 
    26422710<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    2643 border=0 width=509 height=280 id="Picture 22" 
    2644 src="../images/html/image028.jpg" alt="core_shell_cylinder_1D_validation"></p> 
     2711border=0 width=454 height=272 src="../images/html/image028.jpg" 
     2712alt="core_shell_cylinder_1D_validation"></p> 
    26452713 
    26462714<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    … …  
    26492717DANSE scattering intensity for a core-shell cylinder with the output of the 
    26502718NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 
    2651 Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup> 
    2652 -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p> 
     2719Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6 
     2720Å<sup> -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> 
     2721-1</sup>.</p> 
    26532722 
    26542723<p class=MsoCaption>&nbsp;</p> 
    … …  
    26592728 
    26602729<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    2661 border=0 width=518 height=285 id="Picture 23" 
    2662 src="../images/html/image029.jpg" alt="core_shell_cylinder_2D_average"></p> 
     2730border=0 width=458 height=285 src="../images/html/image029.jpg" 
     2731alt="core_shell_cylinder_2D_average"></p> 
    26632732 
    26642733<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    … …  
    26672736intensity for uniformly distributed core-shell cylinders calculated from our 2D 
    26682737model and the intensity from the NIST SANS analysis software. The parameters 
    2669 used were: Scale=1.0, Radius=20 Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> 
    2670 -2</sup>, Shell_sld=4e-6 Å<sup> -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and 
    2671 Background=0.0 cm<sup> -1</sup>.</p> 
     2738used were: Scale=1.0, Radius=20 Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 
     2739Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup> -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, 
     2740and Background=0.0 cm<sup> -1</sup>.</p> 
    26722741 
    26732742<p class=MsoNormal>&nbsp;</p> 
    … …  
    26792748<p class=MsoNormal><b>&nbsp;</b></p> 
    26802749 
    2681  
    26822750<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    2683 style='font-size:14.0pt'>2.8.<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; 
    2684 </span></span></b><b><span style='font-size:14.0pt'><a name="HollowCylinderModel">HollowCylinderModel</span></b></p> 
     2751style='font-size:14.0pt'>2.10.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2752</span></b><a name=HollowCylinderModel><b><span style='font-size:14.0pt'>HollowCylinderModel</span></b></a></p> 
    26852753 
    26862754<p class=MsoNormal>&nbsp;</p> 
    … …  
    26912759 
    26922760<p class=MsoNormal>P(q) = scale*&lt;f^2&gt;/V<sub>shell</sub>+background where 
    2693 the averaging &lt; &gt; id applied only for the 1D calculation.  The inside and 
    2694 outside of the hollow cylinder have the same SLD.</p> 
     2761the averaging &lt; &gt; id applied only for the 1D calculation. &nbsp;The 
     2762inside and outside of the hollow cylinder have the same SLD.</p> 
    26952763 
    26962764<p class=MsoNormal>The 1D scattering intensity is calculated in the following 
    … …  
    26992767<p class=MsoNormal>&nbsp;</p> 
    27002768 
    2701 <p class=MsoNormal align=center style='text-align:center'><span 
    2702 style='position:relative;top:44.0pt'><img border=0 width=514 height=176 
    2703 src="../images/html/image030.png"></span> (11)</p> 
    2704  
    2705 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2769<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2770width=406 height=176 src="../images/html/image030.png">&nbsp;&nbsp;</p> 
    27062771 
    27072772<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    27162781 
    27172782<p class=MsoNormal align=center style='text-align:center'><img border=0 
    2718 width=393 height=217 id="Picture 33" src="../images/html/image031.jpg"></p> 
     2783width=393 height=217 src="../images/html/image031.jpg"></p> 
    27192784 
    27202785<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    27292794cylinder, we define the axis of the cylinder using two angles &#952; and <span 
    27302795style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
    2731 of the cylinder, those angles are defined on Figure 2 of CylinderModel.</p> 
     2796of the cylinder, those angles are defined on Figure 2 in Cylinder Model.</p> 
    27322797 
    27332798<p class=MsoNormal>&nbsp;</p> 
    … …  
    27482813<div align=center> 
    27492814 
    2750 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    2751  style='border-collapse:collapse;border:none'> 
     2815<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     2816 style='border-collapse:collapse'> 
    27522817 <tr style='height:18.8pt'> 
    27532818  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    28312896  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    28322897  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    2833   <p class=MsoBodyText>contrast</p> 
     2898  <p class=MsoBodyText>sldCyl</p> 
    28342899  </td> 
    28352900  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    28412906  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    28422907  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2908  <p class=MsoBodyText>6.3e-6</p> 
     2909  </td> 
     2910 </tr> 
     2911 <tr style='height:18.8pt'> 
     2912  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2913  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2914  <p class=MsoBodyText>sldSolv</p> 
     2915  </td> 
     2916  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2917  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2918  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2919  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     2920  </td> 
     2921  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2922  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2923  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    28432924  <p class=MsoBodyText>5e-06</p> 
    28442925  </td> 
    … …  
    28712952 
    28722953<p class=MsoNormal align=center style='text-align:center'><img border=0 
    2873 width=468 height=344 id="Picture 220" src="../images/html/image032.png"></p> 
     2954width=443 height=259 src="../images/html/image106.jpg"></p> 
    28742955 
    28752956<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    29022983 
    29032984<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    2904 style='font-size:14.0pt'>2.9.<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; 
    2905 </span></span></b><b><span style='font-size:14.0pt'><a name="FlexibleCylinderModel">FlexibleCylinderModel</a></span></b></p> 
     2985style='font-size:14.0pt'>2.11.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2986</span></b><a name=FlexibleCylinderModel></a><b><span style='font-size:14.0pt'>FlexibleCylinderModel</span></b></p> 
    29062987 
    29072988<p class=MsoNormal>&nbsp;</p> 
    … …  
    29102991flexible cylinder where the form factor is normalized by the volume of the 
    29112992cylinder: Inter-cylinder interactions are NOT included. P(q) = 
    2912 scale*&lt;f^2&gt;/V+background where the averaging &lt; &gt;  is applied over 
    2913 all orientation for 1D.  The 2D scattering intensity is the same as 1D, 
    2914 regardless of the orientation of the <i>q</i> vector which is defined as<span 
    2915 style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
    2916 height=33 src="../images/html/image010.png"></span><span 
    2917 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     2993scale*&lt;f^2&gt;/V+background where the averaging &lt; &gt;&nbsp; is applied 
     2994over all orientation for 1D. &nbsp;The 2D scattering intensity is the same as 
     29951D, regardless of the orientation of the <i>q</i> vector which is defined as<span 
     2996style='font-size:14.0pt'><img border=0 width=71 height=23 
     2997src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    29182998 
    29192999<p class=MsoNormal>&nbsp;</p> 
    29203000 
    29213001<p class=MsoNormal align=center style='text-align:center'><img border=0 
    2922 width=329 height=150 id="Picture 35" src="../images/html/image033.jpg"></p> 
     3002width=329 height=150 src="../images/html/image033.jpg"></p> 
    29233003 
    29243004<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    29313011cylinder can be considered a rigid rod. The Kuhn length (b = 2*lp) is also used 
    29323012to describe the stiffness of a chain. The returned value is in units of [cm-1], 
    2933 on absolute scale. In the parameters, the ‘contrast’ represents SLD (chain) - 
    2934 SLD (solvent).</p> 
     3013on absolute scale. In the parameters, the sldCyl and sldSolv&nbsp; represent 
     3014SLD (chain/cylinder) &nbsp;and SLD (solvent) respectively.</p> 
    29353015 
    29363016<p class=MsoNormal><sub>&nbsp;</sub></p> 
    … …  
    29403020<div align=center> 
    29413021 
    2942 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    2943  style='border-collapse:collapse;border:none'> 
     3022<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     3023 style='border-collapse:collapse'> 
    29443024 <tr style='height:18.8pt'> 
    29453025  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    30073087  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    30083088  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    3009   <p class=MsoBodyText>contrast</p> 
     3089  <p class=MsoBodyText>sldCyl</p> 
    30103090  </td> 
    30113091  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    30173097  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    30183098  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    3019   <p class=MsoBodyText>5e-06</p> 
     3099  <p class=MsoBodyText>1e-06</p> 
     3100  </td> 
     3101 </tr> 
     3102 <tr style='height:18.8pt'> 
     3103  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3104  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3105  <p class=MsoBodyText>sldSolv</p> 
     3106  </td> 
     3107  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3108  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3109  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3110  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3111  </td> 
     3112  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3113  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3114  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3115  <p class=MsoBodyText>6.3e-06</p> 
    30203116  </td> 
    30213117 </tr> 
    … …  
    30613157 
    30623158<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3063 width=434 height=322 id="Picture 228" src="../images/html/image034.jpg"></p> 
     3159width=434 height=291 src="../images/html/image034.jpg"></p> 
    30643160 
    30653161<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    30783174Volume&quot; is used. The model is a parametrization of simulations of a 
    30793175discrete representation of the worm-like chain model of Kratky and Porod 
    3080 applied in the pseudocontinuous limit.  See equations (13,26-27) in the 
    3081 original reference for the details.</p> 
     3176applied in the pseudocontinuous limit.&nbsp; See equations (13,26-27) in the original 
     3177reference for the details.</p> 
    30823178 
    30833179<p class=MsoNormal>&nbsp;</p> 
    … …  
    30993195<p class=MsoNormal>&nbsp;</p> 
    31003196 
     3197<p class=MsoNormal>&nbsp;</p> 
     3198 
     3199<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     3200style='font-size:14.0pt'>2.12.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3201</span></b><a name=FlexCylEllipXModel><b><span style='font-size:14.0pt'>FlexCylEllipXModel</span></b></a></p> 
     3202 
     3203<p class=MsoNormal>&nbsp;</p> 
     3204 
     3205<p class=MsoNormal><b>Flexible Cylinder with Elliptical Cross-Section: &nbsp;</b>Calculates 
     3206the form factor for a flexible cylinder with an elliptical cross section and a 
     3207uniform scattering length density. The non-negligible diameter of the cylinder 
     3208is included by accounting for excluded volume interactions within the walk of a 
     3209single cylinder. The form factor is normalized by the particle volume such that 
     3210P(q) = scale*&lt;f^2&gt;/Vol + bkg, where &lt; &gt; is an average over all 
     3211possible orientations of the flexible cylinder. </p> 
     3212 
     3213<p class=MsoNormal>&nbsp;</p> 
     3214 
     3215<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     3216style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
     3217 
     3218<p class=MsoNormal>&nbsp;</p> 
     3219 
     3220<p class=MsoNormal>The function calculated is from the reference given below. 
     3221From that paper, &quot;Method 3 With Excluded Volume&quot; is used. The model 
     3222is a parameterization of simulations of a discrete representation of the 
     3223worm-like chain model of Kratky and Porod applied in the pseudo-continuous limit.&nbsp; 
     3224See equations (13, 26-27) in the original reference for the details.</p> 
     3225 
     3226<p class=MsoNormal>&nbsp;</p> 
     3227 
     3228<p class=MsoNormal>NOTE: there are several typos in the original reference that 
     3229have been corrected by WRC. Details of the corrections are in the reference 
     3230below.</p> 
     3231 
     3232<p class=MsoNormal>&nbsp;- Equation (13): the term (1-w(QR)) should swap 
     3233position with w(QR)</p> 
     3234 
     3235<p class=MsoNormal>&nbsp;- Equations (23) and (24) are incorrect. WRC has 
     3236entered these into Mathematica and solved analytically. The results were 
     3237converted to code.</p> 
     3238 
     3239<p class=MsoNormal>&nbsp;- Equation (27) should be q0 = max(a3/sqrt(RgSquare),3) 
     3240instead of max(a3*b/sqrt(RgSquare),3)</p> 
     3241 
     3242<p class=MsoNormal>&nbsp;- The scattering function is negative for a range of 
     3243parameter values and q-values that are experimentally accessible. A correction 
     3244function has been added to give the proper behavior.</p> 
     3245 
     3246<p class=MsoNormal>&nbsp;</p> 
     3247 
     3248<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3249width=329 height=150 src="../images/html/image033.jpg"></p> 
     3250 
     3251<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3252 
     3253<p class=MsoNormal>The chain of contour length, L, (the total length) can be 
     3254described a chain of some number of locally stiff segments of length lp. The 
     3255persistence length, lp, is the length along the cylinder over which the 
     3256flexible cylinder can be considered a rigid rod. The Kuhn length (b) used in 
     3257the model is also used to describe the stiffness of a chain, and is simply b = 
     32582*lp.</p> 
     3259 
     3260<p class=MsoNormal>&nbsp;</p> 
     3261 
     3262<p class=MsoNormal>The cross section of the cylinder is elliptical, with minor 
     3263radius a. The major radius is larger, so of course, the 'axis_ratio'  
     3264must be greater than one. Simple constraints should be applied during curve 
     3265fitting to maintain this inequality.</p> 
     3266 
     3267<p class=MsoNormal>&nbsp;</p> 
     3268 
     3269<p class=MsoNormal>The returned value is in units of [cm-1], on absolute scale.</p> 
     3270 
     3271<p class=MsoNormal>&nbsp;</p> 
     3272 
     3273<p class=MsoNormal>The sldCyl&nbsp; = SLD (chain), sldSolv = SLD (solvent). The 
     3274‘scale’, and the contrast are both multiplicative factors in the model and are 
     3275perfectly correlated. One or both of these parameters must be held fixed during 
     3276model fitting.</p> 
     3277 
     3278<p class=MsoNormal>&nbsp;</p> 
     3279 
     3280<p class=MsoNormal>If the ‘scale’ is set equal to the particle volume fraction, 
     3281<span style='font-family:Symbol'>f</span>, the returned value is the scattered 
     3282intensity per unit volume, I(q) = <span style='font-family:Symbol'>f</span>*P(q). 
     3283However, no inter-particle interference effects are included in this 
     3284calculation.</p> 
     3285 
     3286<p class=MsoNormal>For 2D data: The 2D scattering intensity is calculated in 
     3287the same way as 1D, where the <i>q</i> vector is defined as<span 
     3288style='font-size:14.0pt'><img border=0 width=82 height=26 
     3289src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
     3290 
     3291<p class=MsoNormal>&nbsp;</p> 
     3292 
     3293<p class=MsoNormal>REFERENCE</p> 
     3294 
     3295<p class=MsoNormal>Pedersen, J. S. and P. Schurtenberger (1996). “Scattering 
     3296functions of semiflexible polymers with and without excluded volume effects.” 
     3297Macromolecules 29: 7602-7612. </p> 
     3298 
     3299<p class=MsoNormal>&nbsp;</p> 
     3300 
     3301<p class=MsoNormal>Corrections are in:</p> 
     3302 
     3303<p class=MsoNormal>&nbsp;</p> 
     3304 
     3305<p class=MsoNormal>Wei-Ren Chen, Paul D. Butler, and Linda J. Magid, 
     3306&quot;Incorporating Intermicellar Interactions in the Fitting of SANS Data from 
     3307Cationic Wormlike Micelles&quot; Langmuir, August 2006.</p> 
     3308 
     3309<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3310</p> 
     3311 
     3312<p class=MsoNormal>TEST DATASET</p> 
     3313 
     3314<p class=MsoNormal>This example dataset is produced by running the Macro FlexCylEllipXModel, 
     3315using 200 data points, qmin = 0.001 Å-1,&nbsp; qmax = 0.7 Å-1 and the default 
     3316values below.</p> 
     3317 
     3318<p class=MsoNormal>&nbsp;</p> 
     3319 
     3320<div align=center> 
     3321 
     3322<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     3323 style='border-collapse:collapse'> 
     3324 <tr style='height:18.8pt'> 
     3325  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3326  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3327  <p class=MsoBodyText>Parameter name</p> 
     3328  </td> 
     3329  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3330  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3331  <p class=MsoBodyText>Units</p> 
     3332  </td> 
     3333  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3334  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3335  <p class=MsoBodyText>Default value</p> 
     3336  </td> 
     3337 </tr> 
     3338 <tr style='height:18.8pt'> 
     3339  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3340  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3341  <p class=MsoBodyText>axis_ratio</p> 
     3342  </td> 
     3343  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3344  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3345  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'></td> 
     3346  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3347  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3348  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3349  <p class=MsoBodyText>1.5</p> 
     3350  </td> 
     3351 </tr> 
     3352 <tr style='height:18.8pt'> 
     3353  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3354  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3355  <p class=MsoBodyText>background</p> 
     3356  </td> 
     3357  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3358  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3359  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3360  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     3361  </td> 
     3362  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3363  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3364  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3365  <p class=MsoBodyText>0.0001</p> 
     3366  </td> 
     3367 </tr> 
     3368 <tr style='height:18.8pt'> 
     3369  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3370  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3371  <p class=MsoBodyText>Kuhn_length</p> 
     3372  </td> 
     3373  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3374  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3375  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3376  <p class=MsoBodyText>Å</p> 
     3377  </td> 
     3378  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3379  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3380  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3381  <p class=MsoBodyText>100</p> 
     3382  </td> 
     3383 </tr> 
     3384 <tr style='height:18.8pt'> 
     3385  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3386  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3387  <p class=MsoBodyText>(Contour) length</p> 
     3388  </td> 
     3389  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3390  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3391  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3392  <p class=MsoBodyText>Å</p> 
     3393  </td> 
     3394  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3395  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3396  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3397  <p class=MsoBodyText>1e+3</p> 
     3398  </td> 
     3399 </tr> 
     3400 <tr style='height:18.8pt'> 
     3401  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3402  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3403  <p class=MsoBodyText>radius</p> 
     3404  </td> 
     3405  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3406  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3407  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3408  <p class=MsoBodyText>Å</p> 
     3409  </td> 
     3410  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3411  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3412  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3413  <p class=MsoBodyText>20.0</p> 
     3414  </td> 
     3415 </tr> 
     3416 <tr style='height:18.8pt'> 
     3417  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3418  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3419  <p class=MsoBodyText>scale</p> 
     3420  </td> 
     3421  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3422  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3423  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'></td> 
     3424  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3425  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3426  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3427  <p class=MsoBodyText>1.0</p> 
     3428  </td> 
     3429 </tr> 
     3430 <tr style='height:18.8pt'> 
     3431  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3432  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3433  <p class=MsoBodyText>sldCyl</p> 
     3434  </td> 
     3435  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3436  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3437  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3438  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3439  </td> 
     3440  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3441  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3442  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3443  <p class=MsoBodyText>1e-6</p> 
     3444  </td> 
     3445 </tr> 
     3446 <tr style='height:18.8pt'> 
     3447  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3448  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3449  <p class=MsoBodyText>sldSolv</p> 
     3450  </td> 
     3451  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3452  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3453  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3454  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3455  </td> 
     3456  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3457  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3458  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3459  <p class=MsoBodyText>6.3e-6</p> 
     3460  </td> 
     3461 </tr> 
     3462</table> 
     3463 
     3464</div> 
     3465 
     3466<p class=MsoNormal>&nbsp;</p> 
     3467 
     3468<p class=MsoNormal>&nbsp;</p> 
     3469 
     3470<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3471width=440 height=300 src="../images/html/image107.jpg"></p> 
     3472 
     3473<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     34741D plot using the default values (w/200 data point).</b></p> 
     3475 
     3476<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3477 
     3478<p class=MsoNormal>&nbsp;</p> 
     3479 
    31013480<p class=MsoNormal><b>&nbsp;</b></p> 
    31023481 
    31033482<p class=MsoNormal><b>&nbsp;</b></p> 
    31043483 
     3484<p class=MsoNormal><b>&nbsp;</b></p> 
    31053485 
    31063486<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    3107 style='font-size:14.0pt'>2.10.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    3108 </span></span></b><b><span style='font-size:14.0pt'><a name="StackedDisksModel"> StackedDisksModel </a></span></b></p> 
    3109  
    3110 <p class=MsoNormal>&nbsp;</p> 
    3111  
    3112 <p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for stacked 
    3113 discs (tactoids) with a core/layer structure where the form factor is 
    3114 normalized by the volume of the cylinder.  Assuming the next neighbor distance 
    3115 (d-spacing) in a stack of parallel discs obeys a Gaussian distribution, a 
    3116 structure factor S(q) proposed by Kratky and Porod in 1949 is used in this 
    3117 function. Note that the resolution smearing calculation uses 76 Gauss 
    3118 quadrature points to properly smear the model since the function is HIGHLY 
    3119 oscillatory, especially around the q-values that correspond to the repeat 
    3120 distance of the layers.</p> 
     3487style='font-size:14.0pt'>2.13.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3488</span></b><a name=StackedDisksModel><b><span style='font-size:14.0pt'>StackedDisksModel 
     3489</span></b></a></p> 
     3490 
     3491<p class=MsoNormal>&nbsp;</p> 
     3492 
     3493<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for 
     3494stacked discs (tactoids) with a core/layer structure where the form factor is 
     3495normalized by the volume of the cylinder. &nbsp;Assuming the next neighbor 
     3496distance (d-spacing) in a stack of parallel discs obeys a Gaussian 
     3497distribution, a structure factor S(q) proposed by Kratky and Porod in 1949 is 
     3498used in this function. Note that the resolution smearing calculation uses 76 
     3499Gauss quadrature points to properly smear the model since the function is 
     3500HIGHLY oscillatory, especially around the q-values that correspond to the 
     3501repeat distance of the layers.</p> 
    31213502 
    31223503<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of 
    31233504the orientation of the <i>q</i> vector which is defined as<span 
    3124 style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
    3125 height=33 src="../images/html/image010.png"></span><span 
    3126 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
    3127  
    3128 <p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
    3129  
    3130 <p class=MsoNormal>&nbsp;</p> 
    3131  
    3132 <p class=MsoNormal align=center style='text-align:center'>         <img 
    3133 border=0 width=214 height=93 id="Picture 36" 
    3134 src="../images/html/image035.png"></p> 
     3505style='font-size:14.0pt'><img border=0 width=68 height=22 
     3506src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
     3507 
     3508<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
     3509 
     3510<p class=MsoNormal>&nbsp;</p> 
     3511 
     3512<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3513<img border=0 width=214 height=93 src="../images/html/image035.png"></p> 
    31353514 
    31363515<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    31373516 
    31383517<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3139 width=173 height=136 id="Picture 37" src="../images/html/image036.png"></p> 
     3518width=173 height=136 src="../images/html/image036.png"></p> 
    31403519 
    31413520<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    31523531<p class=MsoNormal>&nbsp;</p> 
    31533532 
    3154 <p class=MsoNormal align=center style='text-align:center'><span 
    3155 style='position:relative;top:9.0pt'><img border=0 width=514 height=38 
    3156 src="../images/html/image037.png"></span></p> 
    3157  
    3158 <p class=MsoNormal align=center style='text-align:center'><span 
    3159 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3533<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3534width=325 height=38 src="../images/html/image037.png"></p> 
     3535 
     3536<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    31603537 
    31613538<p class=MsoNormal>where the contrast,</p> 
    31623539 
    3163 <p class=MsoNormal align=center style='text-align:center'><span 
    3164 style='position:relative;top:6.0pt'><img border=0 width=115 height=24 
    3165 src="../images/html/image038.png"></span></p> 
    3166  
    3167 <p class=MsoNormal align=center style='text-align:center'><span 
    3168 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3169  
    3170 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3171  
    3172 <p class=MsoNormal><span style='position:relative;top:16.0pt'>N is the number 
    3173 of discs per unit volume, </span><span style='font-family:Symbol;position:relative; 
    3174 top:16.0pt'>a</span><span style='position:relative;top:16.0pt'> is the angle 
    3175 between the axis of the disc and q, and Vt and Vc are the total volume and the 
    3176 core volume of a single disc, respectively.</span></p> 
    3177  
    3178 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3179  
    3180 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3181  
    3182 <p class=MsoNormal align=center style='text-align:center'><span 
    3183 style='position:relative;top:39.0pt'><img border=0 width=424 height=113 
    3184 src="../images/html/image039.png"></span></p> 
    3185  
    3186 <p class=MsoNormal align=center style='text-align:center'><span 
    3187 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3188  
    3189 <p class=MsoNormal align=center style='text-align:center'><span 
    3190 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3540<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3541width=115 height=24 src="../images/html/image038.png"></p> 
     3542 
     3543<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3544 
     3545<p class=MsoNormal>&nbsp;</p> 
     3546 
     3547<p class=MsoNormal>N is the number of discs per unit volume, <span 
     3548style='font-family:Symbol'>a</span>&nbsp;is the angle between the axis of the 
     3549disc and q, and Vt and Vc are the total volume and the core volume of a single 
     3550disc, respectively.</p> 
     3551 
     3552<p class=MsoNormal>&nbsp;</p> 
     3553 
     3554<p class=MsoNormal>&nbsp;</p> 
     3555 
     3556<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3557width=374 height=113 src="../images/html/image039.png"></p> 
     3558 
     3559<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3560 
     3561<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    31913562 
    31923563<p class=MsoNormal>where d = thickness of the layer (layer_thick), 2h= core 
    3193 thickness (core_thick),  and R = radius of the disc (radius).</p> 
    3194  
    3195 <p class=MsoNormal>&nbsp;</p> 
    3196  
    3197 <p class=MsoNormal>&nbsp;</p> 
    3198  
    3199 <p class=MsoNormal align=center style='text-align:center'><span 
    3200 style='position:relative;top:14.0pt'><img border=0 width=399 height=46 
    3201 src="../images/html/image040.png"></span></p> 
    3202  
    3203 <p class=MsoNormal align=center style='text-align:center'><span 
    3204 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3205  
    3206 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3207  
    3208 <p class=MsoNormal><span style='position:relative;top:16.0pt'>where n = the 
    3209 total number of the disc stacked (n_stacking), D=the next neighbor center to 
    3210 cent distance (d-spacing), and </span><span style='font-family:Symbol; 
    3211 position:relative;top:16.0pt'>s</span><sub><span style='position:relative; 
    3212 top:16.0pt'>D</span></sub><span style='position:relative;top:16.0pt'> = the 
    3213 Gaussian standard deviation of the d-spacing (sigma_d).</span></p> 
    3214  
    3215 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3216  
    3217 <p class=MsoNormal>To provide easy access to the orientation of the stackeddisks, 
    3218 we define the axis of the cylinder using two angles &#952; and <span 
     3564thickness (core_thick),&nbsp; and R = radius of the disc (radius).</p> 
     3565 
     3566<p class=MsoNormal>&nbsp;</p> 
     3567 
     3568<p class=MsoNormal>&nbsp;</p> 
     3569 
     3570<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3571width=348 height=46 src="../images/html/image040.png"></p> 
     3572 
     3573<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3574 
     3575<p class=MsoNormal>&nbsp;</p> 
     3576 
     3577<p class=MsoNormal>where n = the total number of the disc stacked (n_stacking), 
     3578D=the next neighbor center to cent distance (d-spacing), and <span 
     3579style='font-family:Symbol'>s</span><sub>D</sub>&nbsp;= the Gaussian standard 
     3580deviation of the d-spacing (sigma_d).</p> 
     3581 
     3582<p class=MsoNormal>&nbsp;</p> 
     3583 
     3584<p class=MsoNormal>To provide easy access to the orientation of the 
     3585stackeddisks, we define the axis of the cylinder using two angles &#952; and <span 
    32193586style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
    32203587of the cylinder, those angles are defined on Figure 2 of CylinderModel.</p> 
    32213588 
    3222 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3223  
    3224 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3225  
    3226 <p class=MsoNormal><span style='position:relative;top:16.0pt'> </span>For P*S: 
    3227 The 2<sup>nd</sup> virial coefficient of the solid cylinder is calculate based 
    3228 on the (radius) and length = n_stacking*(core_thick +2*layer_thick) values, and 
    3229 used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     3589<p class=MsoNormal>&nbsp;</p> 
     3590 
     3591<p class=MsoNormal>&nbsp;</p> 
     3592 
     3593<p class=MsoNormal>&nbsp;For P*S: The 2<sup>nd</sup> virial coefficient of the 
     3594solid cylinder is calculate based on the (radius) and length = 
     3595n_stacking*(core_thick +2*layer_thick) values, and used as the effective radius 
     3596toward S(Q) when P(Q)*S(Q) is applied. </p> 
    32303597 
    32313598<p class=MsoNormal>&nbsp;</p> 
    … …  
    32373604<div align=center> 
    32383605 
    3239 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    3240  style='border-collapse:collapse;border:none'> 
     3606<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     3607 style='border-collapse:collapse'> 
    32413608 <tr style='height:18.8pt'> 
    32423609  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    34223789 
    34233790<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3424 width=451 height=334 id="Picture 233" src="../images/html/image041.png"></p> 
     3791width=451 height=297 src="../images/html/image041.png"></p> 
    34253792 
    34263793<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    34323799 
    34333800<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3434 width=377 height=215 id="Picture 251" src="../images/html/image042.jpg" 
     3801width=377 height=215 src="../images/html/image042.jpg" 
    34353802alt=stackdiskangles.gif></p> 
    34363803 
    … …  
    34723839 
    34733840<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    3474 style='font-size:14.0pt'>2.11.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    3475 </span></span></b><b><span style='font-size:14.0pt'><a name="ParallelepipedModel">ParallelepipedModel</a></span></b></p> 
     3841style='font-size:14.0pt'>2.14.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3842</span></b><a name=ParallelepipedModel><b><span style='font-size:14.0pt'>ParallelepipedModel</span></b></a></p> 
    34763843 
    34773844<p class=MsoNormal>&nbsp;</p> 
    … …  
    34803847cylinder (below) where the form factor is normalized by the volume of the 
    34813848cylinder. P(q) = scale*&lt;f^2&gt;/V+background where the volume V= ABC and the 
    3482 averaging &lt; &gt;  is applied over all orientation for 1D.  </p> 
    3483  
    3484 <p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     3849averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;</p> 
     3850 
     3851<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    34853852 
    34863853<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3487 width=326 height=247 id="Picture 38" src="../images/html/image043.jpg"></p> 
     3854width=239 height=181 src="../images/html/image043.jpg"></p> 
    34883855 
    34893856<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    35013868<p class=MsoNormal>&nbsp;</p> 
    35023869 
    3503 <p class=MsoNormal align=center style='text-align:center'><span 
    3504 style='position:relative;top:48.0pt'><img border=0 width=360 height=145 
    3505 src="../images/html/image044.png"></span></p> 
    3506  
    3507 <p class=MsoNormal align=center style='text-align:center'><span 
    3508 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3870<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3871width=347 height=145 src="../images/html/image044.png"></p> 
     3872 
     3873<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    35093874 
    35103875<p class=MsoNormal>The contrast is defined as</p> 
    … …  
    35123877<p class=MsoNormal>&nbsp;</p> 
    35133878 
    3514 <p class=MsoNormal align=center style='text-align:center'><span 
    3515 style='position:relative;top:6.0pt'><img border=0 width=117 height=24 
    3516 src="../images/html/image045.png"></span></p> 
    3517  
    3518 <p class=MsoNormal align=center style='text-align:center'><span 
    3519 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3520  
    3521 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    3522  
    3523 <p class=MsoNormal><span style='position:relative;top:16.0pt'>The scattering 
    3524 intensity per unit volume is returned in the unit of [cm<sup>-1</sup>];  I(q) = 
    3525 </span><span style='font-family:Symbol;position:relative;top:16.0pt'>f</span><span 
    3526 style='position:relative;top:16.0pt'>P(q).</span></p> 
    3527  
    3528 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3879<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3880width=117 height=24 src="../images/html/image045.png"></p> 
     3881 
     3882<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3883 
     3884<p class=MsoNormal>&nbsp;</p> 
     3885 
     3886<p class=MsoNormal>The scattering intensity per unit volume is returned in the 
     3887unit of [cm<sup>-1</sup>]; &nbsp;I(q) = <span style='font-family:Symbol'>f</span>P(q).</p> 
     3888 
     3889<p class=MsoNormal>&nbsp;</p> 
    35293890 
    35303891<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
    … …  
    35373898<p class=MsoNormal>&nbsp;</p> 
    35383899 
    3539 <p class=MsoNormal>To provide easy access to the orientation of the parallelepiped, 
    3540 we define the axis of the cylinder using two angles &#952; , <span 
     3900<p class=MsoNormal>To provide easy access to the orientation of the 
     3901parallelepiped, we define the axis of the cylinder using two angles &#952; , <span 
    35413902style='font-family:"Arial","sans-serif"'>&#966; </span>and<span 
    35423903style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span 
    35433904style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the 
    3544 cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     3905cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
    35453906are defined on Figure 2 of CylinderModel. The angle <span style='font-family: 
    35463907Symbol'>Y </span>is the rotational angle around its own long_c axis against the 
    3547 q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the short_b 
    3548 axis is parallel to the x-axis of the detector.</p> 
     3908q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the 
     3909short_b axis is parallel to the x-axis of the detector.</p> 
    35493910 
    35503911<p class=MsoNormal>&nbsp;</p> 
    35513912 
    35523913<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3553 width=352 height=264 id="Picture 243" src="../images/html/image046.jpg" 
     3914width=352 height=264 src="../images/html/image046.jpg" 
    35543915alt=pprangles.gif></p> 
    35553916 
    … …  
    35763937<div align=center> 
    35773938 
    3578 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    3579  style='border-collapse:collapse;border:none'> 
     3939<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     3940 style='border-collapse:collapse'> 
    35803941 <tr style='height:18.8pt'> 
    35813942  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    36974058 
    36984059<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3699 width=431 height=332 id="Picture 492" src="../images/html/image048.jpg"></p> 
     4060width=431 height=298 src="../images/html/image048.jpg"></p> 
    37004061 
    37014062<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    37094070 
    37104071<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span 
    3711 style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    3712 </span></span><b>Validation of the parallelepiped 2D model</b></p> 
     4072style='font-family:Symbol'>·</span><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4073</span><b>Validation of the parallelepiped 2D model</b></p> 
    37134074 
    37144075<p class=MsoNormal>&nbsp;</p> 
    … …  
    37174078the 1D calculation to the angular average of the output of 2 D calculation over 
    37184079all possible angles. The Figure below shows the comparison where the solid dot 
    3719 refers to averaged 2D while the line represents the result of 1D calculation (for 
    3720 the averaging, 76, 180, 76 points are taken over the angles of theta, phi, and 
    3721 psi respectively).</p> 
     4080refers to averaged 2D while the line represents the result of 1D calculation 
     4081(for the averaging, 76, 180, 76 points are taken over the angles of theta, phi, 
     4082and psi respectively).</p> 
    37224083 
    37234084<p class=MsoNormal>&nbsp;</p> 
    … …  
    37264087 
    37274088<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3728 width=481 height=299 id="Picture 693" src="../images/html/image049.png"></p> 
     4089width=441 height=299 src="../images/html/image049.png"></p> 
    37294090 
    37304091<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    37604121 
    37614122<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    3762 style='font-size:14.0pt'>2.12.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    3763 </span></span></b><b><span style='font-size:14.0pt'><a name="EllipticalCylinderModel">Elliptical Cylinder Model</a></span></b></p> 
     4123style='font-size:14.0pt'>2.15.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4124</span></b><a name=EllipticalCylinderModel><b><span style='font-size:14.0pt'>Elliptical 
     4125Cylinder Model</span></b></a></p> 
    37644126 
    37654127<p class=MsoNormal>&nbsp;</p> 
    … …  
    37834145 
    37844146<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3785 width=352 height=172 id="Picture 40" src="../images/html/image050.png"></p> 
     4147width=297 height=145 src="../images/html/image050.png"></p> 
    37864148 
    37874149<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    37884150 
    37894151<p class=MsoNormal align=center style='text-align:center'><b>Figure. a= r_minor 
    3790 and </b><b><span style='font-family:Symbol'>n</span>= r_ratio (i.e., r_major/r_minor).</b></p> 
     4152and </b><b><span style='font-family:Symbol'>n</span>= r_ratio (i.e., 
     4153r_major/r_minor).</b></p> 
    37914154 
    37924155<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    37964159<p class=MsoNormal>&nbsp;</p> 
    37974160 
    3798 <p class=MsoNormal align=center style='text-align:center'><span 
    3799 style='position:relative;top:16.0pt'><img border=0 width=377 height=46 
    3800 src="../images/html/image051.png"></span> </p> 
     4161<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4162width=346 height=46 src="../images/html/image051.png">&nbsp;</p> 
    38014163 
    38024164<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    38084170<p class=MsoNormal>&nbsp;</p> 
    38094171 
    3810 <p class=MsoNormal align=center style='text-align:center'><span 
    3811 style='position:relative;top:39.0pt'><img border=0 width=297 height=114 
    3812 src="../images/html/image052.png"></span>          (13)</p> 
     4172<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4173width=297 height=114 src="../images/html/image052.png">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4174(13)</p> 
    38134175 
    38144176<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    38414203style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span 
    38424204style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the 
    3843 cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     4205cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
    38444206are defined on Figure 2 of CylinderModel. The angle <span style='font-family: 
    38454207Symbol'>Y </span>is the rotational angle around its own long_c axis against the 
    3846 q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the r_minor 
    3847 axis is parallel to the x-axis of the detector.</p> 
    3848  
    3849 <p class=MsoNormal>&nbsp;</p> 
    3850  
    3851 <p class=MsoNormal>All angle parameters are valid and given only for 2D calculation 
    3852 (Oriented system).</p> 
     4208q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the 
     4209r_minor axis is parallel to the x-axis of the detector.</p> 
     4210 
     4211<p class=MsoNormal>&nbsp;</p> 
     4212 
     4213<p class=MsoNormal>All angle parameters are valid and given only for 2D 
     4214calculation (Oriented system).</p> 
    38534215 
    38544216<p class=MsoNormal>&nbsp;</p> 
    … …  
    38574219 
    38584220<p class=MsoNormal align=center style='text-align:center'><img border=0 
    3859 width=370 height=277 id="Picture 239" src="../images/html/image053.jpg" 
     4221width=370 height=277 src="../images/html/image053.jpg" 
    38604222alt=ellcylinderangles.gif></p> 
    38614223 
    … …  
    38694231alt=cylinderangles2.gif></p> 
    38704232 
    3871 <p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
    3872 the angles for oriented elliptical cylinders against the detector plane.</p> 
     4233<p class=MsoCaption align=center style='text-align:center'><span 
     4234style='font-size:12.0pt'>Figure. Examples of the angles for oriented elliptical 
     4235cylinders </span></p> 
     4236 
     4237<p class=MsoCaption align=center style='text-align:center'><span 
     4238style='font-size:12.0pt'>against the detector plane.</span></p> 
    38734239 
    38744240<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    38784244<p class=MsoNormal><b>For P*S</b>: The 2<sup>nd</sup> virial coefficient of the 
    38794245solid cylinder is calculate based on the averaged radius 
    3880 (=sqrt(r_minor^2*r_ratio))  and length values, and used as the effective radius 
    3881 toward S(Q) when P(Q)*S(Q) is applied. </p> 
     4246(=sqrt(r_minor^2*r_ratio))&nbsp; and length values, and used as the effective 
     4247radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
    38824248 
    38834249<p class=MsoNormal>&nbsp;</p> 
    … …  
    38874253<div align=center> 
    38884254 
    3889 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    3890  style='border-collapse:collapse;border:none'> 
     4255<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     4256 style='border-collapse:collapse'> 
    38914257 <tr style='height:19.25pt'> 
    38924258  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    40564422 
    40574423<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4058 width=443 height=328 id="Picture 503" src="../images/html/image054.jpg"></p> 
     4424width=443 height=293 src="../images/html/image054.jpg"></p> 
    40594425 
    40604426<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    40664432 
    40674433<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span 
    4068 style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    4069 </span></span><b>Validation of the elliptical cylinder 2D model</b></p> 
     4434style='font-family:Symbol'>·</span><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4435</span><b>Validation of the elliptical cylinder 2D model</b></p> 
    40704436 
    40714437<p class=MsoNormal>&nbsp;</p> 
    … …  
    40744440the 1D calculation to the angular average of the output of 2 D calculation over 
    40754441all possible angles. The Figure below shows the comparison where the solid dot 
    4076 refers to averaged 2D while the line represents the result of 1D calculation (for 
    4077 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and 
    4078 psi respectively).</p> 
     4442refers to averaged 2D while the line represents the result of 1D calculation 
     4443(for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, 
     4444and psi respectively).</p> 
    40794445 
    40804446<p class=MsoNormal>&nbsp;</p> 
    … …  
    40834449 
    40844450<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4085 width=448 height=278 id="Picture 692" src="../images/html/image055.png"></p> 
     4451width=448 height=278 src="../images/html/image055.png"></p> 
    40864452 
    40874453<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    40914457 
    40924458<p class=MsoNormal style='text-autospace:none'>In the 2D average, more binning 
    4093 in the angle phi is necessary to get the proper result. The following figure 
    4094 shows the results of the averaging by varying the number of bin over angles.</p> 
     4459in the angle phi is necessary to get the proper result. The following figure shows 
     4460the results of the averaging by varying the number of bin over angles.</p> 
    40954461 
    40964462<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    40974463 
    40984464<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><img 
    4099 border=0 width=409 height=303 id="Picture 690" 
    4100 src="../images/html/image056.png"></p> 
     4465border=0 width=409 height=303 src="../images/html/image056.png"></p> 
    41014466 
    41024467<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    4103 The intensities averaged from 2D over different number of points of binning of 
    4104 angles.</b></p> 
     4468The intensities averaged from 2D over different number </b></p> 
     4469 
     4470<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>of 
     4471points of binning of angles.</b></p> 
    41054472 
    41064473<p class=MsoNormal align=center style='text-align:center;text-autospace:none'>&nbsp;</p> 
    … …  
    41124479<p class=MsoNormal>&nbsp;</p> 
    41134480 
    4114 <p class=MsoNormal style='text-indent:.25in'>L. A. Feigin and D. I. Svergun 
    4115 “Structure Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New 
    4116 York, (1987).</p> 
     4481<p class=MsoNormal style='text-indent:.25in'>L. A. Feigin and D. I. Svergun “Structure 
     4482Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New York, 
     4483(1987).</p> 
    41174484 
    41184485<p class=MsoNormal>&nbsp;</p> 
    … …  
    41214488 
    41224489<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    4123 style='font-size:14.0pt'>2.13.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    4124 </span></span></b><b><span style='font-size:14.0pt'><a name="EllipsoidModel">Ellipsoid Model</a></span></b></p> 
     4490style='font-size:14.0pt'>2.16.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4491</span></b><a name=EllipsoidModel><b><span style='font-size:14.0pt'>Ellipsoid 
     4492Model</span></b></a></p> 
    41254493 
    41264494<p class=MsoNormal>&nbsp;</p> 
    … …  
    41324500<p class=MsoNormal>&nbsp;</p> 
    41334501 
    4134 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
    4135 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     4502<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.</b><b><span 
     4503style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    41364504 
    41374505<p class=MsoNormal>&nbsp;</p> 
    … …  
    41444512<p class=MsoNormal>&nbsp;</p> 
    41454513 
    4146 <p class=MsoCaption align=center style='text-align:center'><span 
    4147 style='position:relative;top:12.0pt'><img border=0 width=184 height=41 
    4148 src="../images/html/image017.png"></span>   <span style='font-size:12.0pt; 
    4149 font-weight:normal'>(8)</span></p> 
    4150  
    4151 <p class=MsoNormal>&nbsp;</p> 
    4152  
    4153 <p class=MsoCaption align=center style='text-align:center'><span 
    4154 style='position:relative;top:15.0pt'><img border=0 width=371 height=47 
    4155 src="../images/html/image057.png"></span>    <span style='font-size: 
    4156 12.0pt;font-weight:normal'>(9)</span></p> 
    4157  
    4158 <p class=MsoNormal>&nbsp;</p> 
    4159  
    4160 <p class=MsoCaption align=center style='text-align:center'><span 
    4161 style='position:relative;top:6.0pt'><img border=0 width=272 height=29 
    4162 src="../images/html/image058.png"></span>     <span style='font-size: 
    4163 12.0pt;font-weight:normal'>(10)</span></p> 
     4514<p class=MsoCaption align=center style='text-align:center'><img border=0 
     4515width=184 height=41 src="../images/html/image017.png">&nbsp;&nbsp; </p> 
     4516 
     4517<p class=MsoNormal>&nbsp;</p> 
     4518 
     4519<p class=MsoCaption align=center style='text-align:center'><img border=0 
     4520width=340 height=47 src="../images/html/image057.png">&nbsp;&nbsp;&nbsp; 
     4521</p> 
     4522 
     4523<p class=MsoNormal>&nbsp;</p> 
     4524 
     4525<p class=MsoCaption align=center style='text-align:center'><img border=0 
     4526width=272 height=29 src="../images/html/image058.png">&nbsp;&nbsp;&nbsp;&nbsp; 
     4527</p> 
    41644528 
    41654529<p class=MsoNormal>&nbsp;</p> 
    … …  
    42024566<div align=center> 
    42034567 
    4204 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    4205  style='border-collapse:collapse;border:none'> 
     4568<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     4569 style='border-collapse:collapse'> 
    42064570 <tr style='height:19.85pt'> 
    42074571  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    … …  
    42694633  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    42704634  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    4271   <p class=MsoBodyText>contrast</p> 
     4635  <p class=MsoBodyText>sldEll</p> 
    42724636  </td> 
    42734637  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    … …  
    42794643  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    42804644  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    4281   <p class=MsoBodyText>3.0e-6</p> 
     4645  <p class=MsoBodyText>4.0e-6</p> 
     4646  </td> 
     4647 </tr> 
     4648 <tr style='height:19.85pt'> 
     4649  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4650  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4651  <p class=MsoBodyText>sldSolv</p> 
     4652  </td> 
     4653  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4654  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4655  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4656  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     4657  </td> 
     4658  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4659  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4660  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4661  <p class=MsoBodyText>1.0e-6</p> 
    42824662  </td> 
    42834663 </tr> 
    … …  
    42954675  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    42964676  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    4297   <p class=MsoBodyText>0.0</p> 
     4677  <p class=MsoBodyText>0.0 </p> 
    42984678  </td> 
    42994679 </tr> 
    … …  
    43374717 
    43384718<p class=MsoNormal>The output of the 1D scattering intensity function for 
    4339 randomly oriented ellipsoids is then given by equation 6.</p> 
     4719randomly oriented ellipsoids is then given by the equation above.</p> 
    43404720 
    43414721<p class=MsoNormal>&nbsp;</p> 
    … …  
    43514731alt=ellipsoangles.gif></p> 
    43524732 
    4353 <p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
    4354 the angles for oriented ellipsoid against the detector plane.</p> 
    4355  
    4356 <p class=MsoNormal>&nbsp;</p> 
    4357  
    4358 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    4359  
    4360 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
    4361 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    4362 of the ellipsoid model</b></p> 
     4733<p class=MsoCaption align=center style='text-align:center'><span 
     4734style='font-size:12.0pt'>Figure. Examples of the angles for oriented ellipsoid </span></p> 
     4735 
     4736<p class=MsoCaption align=center style='text-align:center'><span 
     4737style='font-size:12.0pt'>against the detector plane</span>.</p> 
     4738 
     4739<p class=MsoNormal>&nbsp;</p> 
     4740 
     4741<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4742 
     4743<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.</b><b><span 
     4744style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the 
     4745ellipsoid model</b></p> 
    43634746 
    43644747<p class=MsoNormal>&nbsp;</p> 
    43654748 
    43664749<p class=MsoNormal>Validation of our code was done by comparing the output of 
    4367 the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 
    4368 5 shows a comparison of the 1D output of our model and the output of the NIST 
    4369 software.</p> 
     4750the 1D model to the output of the software provided by the NIST (Kline, 2006). 
     4751Figure 5 shows a comparison of the 1D output of our model and the output of the 
     4752NIST software.</p> 
    43704753 
    43714754<p class=MsoNormal>&nbsp;</p> 
    43724755 
    43734756<p class=MsoNormal>Averaging over a distribution of orientation is done by 
    4374 evaluating equation 7. Since we have no other software to compare the 
     4757evaluating the equation above. Since we have no other software to compare the 
    43754758implementation of the intensity for fully oriented ellipsoids, we can compare 
    43764759the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span 
    4377 style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 6 shows the result of such a cross-check.</p> 
     4760style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.&nbsp; Figure 
     47616 shows the result of such a cross-check.</p> 
    43784762 
    43794763<p class=MsoNormal align=center style='text-align:center'><b><i><span 
    … …  
    43834767way the form factors are calculated in the c-library provided by NIST. A 
    43844768numerical integration has to be performed to obtain P(q) for randomly oriented 
    4385 particles (equation 6). The NIST software performs that integration with a 
    4386 76-point Gaussian quadrature rule, which will become imprecise at high q where 
    4387 the amplitude varies quickly as a function of q. The DANSE result shown has been 
    4388 obtained by summing over 501 equidistant points in <span style='font-family: 
    4389 "Arial","sans-serif"'>&#945;</span>. Our result was found to be stable over the 
    4390 range of q shown for a number of points higher than 500.</p> 
     4769particles. The NIST software performs that integration with a 76-point Gaussian 
     4770quadrature rule, which will become imprecise at high q where the amplitude 
     4771varies quickly as a function of q. The DANSE result shown has been obtained by 
     4772summing over 501 equidistant points in <span style='font-family:"Arial","sans-serif"'>&#945;</span>. 
     4773Our result was found to be stable over the range of q shown for a number of 
     4774points higher than 500.</p> 
    43914775 
    43924776<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><b><img 
    4393 border=0 width=469 height=258 id="Picture 16" 
    4394 src="../images/html/image060.jpg" alt="ellipsoid_1D_validation"></b></p> 
     4777border=0 width=449 height=258 src="../images/html/image060.jpg" 
     4778alt="ellipsoid_1D_validation"></b></p> 
    43954779 
    43964780<p class=MsoCaption><a name="_Ref173222904">Figure </a>5: Comparison of the 
    … …  
    44074791 
    44084792<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    4409 border=0 width=450 height=247 id="Picture 17" 
    4410 src="../images/html/image061.jpg" alt="ellipsoid_2D_average"></p> 
     4793border=0 width=426 height=247 src="../images/html/image061.jpg" 
     4794alt="ellipsoid_2D_average"></p> 
    44114795 
    44124796<p class=MsoCaption><a name="_Ref173223004">Figure </a>6: Comparison of the 
    … …  
    44284812<p class=MsoNormal><b>&nbsp;</b></p> 
    44294813 
    4430  
    4431  
    44324814<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    4433 style='font-size:14.0pt'>2.14.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    4434 </span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellEllipsoidModel"> CoreShellEllipsoidModel </a></span></b></p> 
     4815style='font-size:14.0pt'>2.17.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4816</span></b><a name=CoreShellEllipsoidModel><b><span style='font-size:14.0pt'>CoreShellEllipsoidModel 
     4817</span></b></a></p> 
    44354818 
    44364819<p class=MsoNormal>&nbsp;</p> 
    … …  
    44394822shell ellipsoid (below) where the form factor is normalized by the volume of 
    44404823the cylinder. P(q) = scale*&lt;f^2&gt;/V+background where the volume V= 4pi/3*r<sub>maj</sub>*r<sub>min</sub><sup>2</sup> 
    4441 and the averaging &lt; &gt;  is applied over all orientation for 1D.  </p> 
    4442  
    4443 <p class=MsoNormal>&nbsp;</p> 
    4444  
    4445 <p class=MsoNormal>&nbsp;</p> 
    4446  
    4447 <p class=MsoNormal align=center style='text-align:center'>   <img border=0 
    4448 width=246 height=131 id="Picture 41" src="../images/html/image062.png"></p> 
     4824and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. 
     4825&nbsp;</p> 
     4826 
     4827<p class=MsoNormal>&nbsp;</p> 
     4828 
     4829<p class=MsoNormal>&nbsp;</p> 
     4830 
     4831<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp; <img 
     4832border=0 width=246 height=131 src="../images/html/image062.png"></p> 
    44494833 
    44504834<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    … …  
    44614845<p class=MsoNormal>&nbsp;</p> 
    44624846 
    4463 <p class=MsoNormal align=center style='text-align:center'><span 
    4464 style='position:relative;top:48.0pt'><img border=0 width=334 height=137 
    4465 src="../images/html/image063.png"></span></p> 
    4466  
    4467 <p class=MsoNormal align=center style='text-align:center'><span 
    4468 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4847<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4848width=334 height=137 src="../images/html/image063.png"></p> 
     4849 
     4850<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    44694851 
    44704852<p class=MsoNormal>&nbsp;</p> 
    … …  
    44754857ellipsoid, we define the axis of the solid ellipsoid using two angles &#952; , <span 
    44764858style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
    4477 of the cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     4859of the cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
    44784860are defined on Figure 2 of CylinderModel. </p> 
    44794861 
    … …  
    44864868radius of the shell).</p> 
    44874869 
    4488 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4870<p class=MsoNormal>&nbsp;</p> 
    44894871 
    44904872<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
    4491 ellipsoid is calculate based on the radius_a (= polar_shell) and radius_b (= equat_shell) 
    4492 values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. 
    4493 </p> 
     4873ellipsoid is calculate based on the radius_a (= polar_shell) and radius_b (= 
     4874equat_shell) values, and used as the effective radius toward S(Q) when 
     4875P(Q)*S(Q) is applied. </p> 
    44944876 
    44954877<p class=MsoNormal><sub>&nbsp;</sub></p> 
    … …  
    44974879<div align=center> 
    44984880 
    4499 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    4500  style='border-collapse:collapse;border:none'> 
     4881<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     4882 style='border-collapse:collapse'> 
    45014883 <tr style='height:18.8pt'> 
    45024884  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    46265008 </tr> 
    46275009 <tr style='height:18.8pt'> 
    4628   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    4629   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    4630   <p class=MsoBodyText>contrast</p> 
    4631   </td> 
    4632   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    4633   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5010  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5011  solid windowtext 1.0pt;border-bottom:none;border-right:solid windowtext 1.0pt; 
     5012  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5013  <p class=MsoBodyText>sld_core</p> 
     5014  </td> 
     5015  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt; 
    46345016  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    46355017  <p class=MsoBodyText>Å<sup> -2</sup></p> 
    46365018  </td> 
     5019  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt; 
     5020  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5021  <p class=MsoBodyText>2e-006</p> 
     5022  </td> 
     5023 </tr> 
     5024 <tr style='height:18.8pt'> 
     5025  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5026  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5027  <p class=MsoBodyText>sld_shell</p> 
     5028  </td> 
     5029  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5030  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5031  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5032  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5033  </td> 
    46375034  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    46385035  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    … …  
    46505047 
    46515048<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4652 width=426 height=333 id="Picture 526" src="../images/html/image064.jpg"></p> 
     5049width=424 height=243 src="../images/html/image108.jpg"></p> 
    46535050 
    46545051<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    46605057 
    46615058<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4662 width=396 height=297 id="Picture 245" src="../images/html/image059.jpg" 
     5059width=396 height=297 src="../images/html/image059.jpg" 
    46635060alt=ellipsoangles.gif></p> 
    46645061 
    … …  
    46785075<p class=MsoNormal>Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.</p> 
    46795076 
    4680 <p class=MsoNormal>Berr, S.  J. Phys. Chem., 1987, 91, 4760.</p> 
     5077<p class=MsoNormal>Berr, S.&nbsp; J. Phys. Chem., 1987, 91, 4760.</p> 
    46815078 
    46825079<p class=MsoNormal>&nbsp;</p> 
    … …  
    46915088 
    46925089<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    4693 style='font-size:14.0pt'>2.15.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    4694 </span></span></b><b><span style='font-size:14.0pt'><a name="TriaxialEllipsoidModel">TriaxialEllipsoidModel</a></span></b></p> 
     5090style='font-size:14.0pt'>2.18.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5091</span></b><a name=TriaxialEllipsoidModel><b><span style='font-size:14.0pt'>TriaxialEllipsoidModel</span></b></a></p> 
    46955092 
    46965093<p class=MsoNormal>&nbsp;</p> 
    46975094 
    46985095<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for an 
    4699 ellipsoid (below) where all three axes are of different lengths, i.e.,  R<sub>a</sub> 
     5096ellipsoid (below) where all three axes are of different lengths, i.e.,&nbsp; R<sub>a</sub> 
    47005097=&lt; R<sub>b</sub> =&lt; R<sub>c</sub> (Note that users should maintains this 
    4701 inequality for the all calculations).  P(q) = scale*&lt;f^2&gt;/V+background 
    4702 where the volume V= 4pi/3*R<sub>a</sub>*R<sub>b</sub>*R<sub>c</sub>, and the 
    4703 averaging &lt; &gt;  is applied over all orientation for 1D.  </p> 
    4704  
    4705 <p class=MsoNormal>&nbsp;</p> 
    4706  
    4707 <p class=MsoNormal align=center style='text-align:center'>   <img border=0 
    4708 width=261 height=157 id="Picture 42" src="../images/html/image065.jpg"></p> 
    4709  
    4710 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    4711  
    4712 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    4713  
    4714 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5098inequality for the all calculations).&nbsp; P(q) = 
     5099scale*&lt;f^2&gt;/V+background where the volume V= 4pi/3*R<sub>a</sub>*R<sub>b</sub>*R<sub>c</sub>, 
     5100and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. 
     5101&nbsp;</p> 
     5102 
     5103<p class=MsoNormal>&nbsp;</p> 
     5104 
     5105<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp; <img 
     5106border=0 width=261 height=157 src="../images/html/image065.jpg"></p> 
     5107 
     5108<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5109 
     5110<p class=MsoNormal align=center style='text-align:center'>&nbsp;&nbsp;</p> 
    47155111 
    47165112<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
    … …  
    47215117<p class=MsoNormal>&nbsp;</p> 
    47225118 
    4723 <p class=MsoNormal align=center style='text-align:center'><span 
    4724 style='position:relative;top:33.0pt'><img border=0 width=472 height=97 
    4725 src="../images/html/image066.png"></span></p> 
    4726  
    4727 <p class=MsoNormal align=center style='text-align:center'><span 
    4728 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    4729  
    4730 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    4731  
    4732 <p class=MsoNormal>&nbsp;</p> 
    4733  
    4734 <p class=MsoNormal>&nbsp;</p> 
    4735  
    4736 <p class=MsoNormal>To provide easy access to the orientation of the triaxial 
    4737 ellipsoid, we define the axis of the cylinder using the angles &#952; , <span 
     5119<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5120width=396 height=97 src="../images/html/image066.png"></p> 
     5121 
     5122<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5123 
     5124<p class=MsoNormal>&nbsp;</p> 
     5125 
     5126<p class=MsoNormal>&nbsp;</p> 
     5127 
     5128<p class=MsoNormal>&nbsp;</p> 
     5129 
     5130<p class=MsoNormal>To provide easy access to the orientation of the triaxial ellipsoid, 
     5131we define the axis of the cylinder using the angles &#952; , <span 
    47385132style='font-family:"Arial","sans-serif"'>&#966; </span>and<span 
    47395133style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span 
    47405134style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the 
    4741 cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     5135cylinder, those angles, &#952; &nbsp;and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
    47425136are defined on Figure 2 of CylinderModel. The angle <span style='font-family: 
    47435137Symbol'>Y </span>is the rotational angle around its own semi_axisC axis against 
    4744 the q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the 
    4745 semi_axisA axis is parallel to the x-axis of the detector.</p> 
     5138the q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when 
     5139the semi_axisA axis is parallel to the x-axis of the detector.</p> 
    47465140 
    47475141<p class=MsoNormal>&nbsp;</p> 
    47485142 
    47495143<p class=MsoNormal>The radius of gyration for this system is R<sub>g</sub><sup>2</sup> 
    4750 = (R<sub>a</sub><sup>2</sup>*R<sub>b</sub><sup>2</sup>*R<sub>c</sub><sup>2</sup>)/5.  
    4751 The contrast is defined as SLD(ellipsoid) – SLD(solvent). In the parameters, semi_axisA 
    4752 = R<sub>a</sub> (or minor equatorial radius), semi_axisB = R<sub>b</sub> (or 
    4753 major equatorial radius), and semi_axisC = R<sub>c</sub> (or polar radius of 
    4754 the ellipsoid).</p> 
    4755  
    4756 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5144= (R<sub>a</sub><sup>2</sup>*R<sub>b</sub><sup>2</sup>*R<sub>c</sub><sup>2</sup>)/5.&nbsp; 
     5145The contrast is defined as SLD(ellipsoid) – SLD(solvent). In the parameters, 
     5146semi_axisA = R<sub>a</sub> (or minor equatorial radius), semi_axisB = R<sub>b</sub> 
     5147(or major equatorial radius), and semi_axisC = R<sub>c</sub> (or polar radius 
     5148of the ellipsoid).</p> 
     5149 
     5150<p class=MsoNormal>&nbsp;</p> 
    47575151 
    47585152<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
    4759 ellipsoid is calculate based on the radius_a (=semi_axisC) and radius_b (=sqrt(semi_axisA* 
    4760 semi_axisB))  values, and used as the effective radius toward S(Q) when 
    4761 P(Q)*S(Q) is applied. </p> 
     5153ellipsoid is calculate based on the radius_a (=semi_axisC) and radius_b 
     5154(=sqrt(semi_axisA* semi_axisB)) &nbsp;values, and used as the effective radius 
     5155toward S(Q) when P(Q)*S(Q) is applied. </p> 
    47625156 
    47635157<p class=MsoNormal><sub>&nbsp;</sub></p> 
    … …  
    47675161<div align=center> 
    47685162 
    4769 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    4770  style='border-collapse:collapse;border:none'> 
     5163<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     5164 style='border-collapse:collapse'> 
    47715165 <tr style='height:18.8pt'> 
    47725166  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    48645258 </tr> 
    48655259 <tr style='height:18.8pt'> 
    4866   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    4867   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    4868   <p class=MsoBodyText>contrast</p> 
    4869   </td> 
    4870   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    4871   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5260  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5261  solid windowtext 1.0pt;border-bottom:none;border-right:solid windowtext 1.0pt; 
     5262  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5263  <p class=MsoBodyText>sldEll</p> 
     5264  </td> 
     5265  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt; 
    48725266  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    48735267  <p class=MsoBodyText>Å<sup> -2</sup></p> 
    48745268  </td> 
    4875   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    4876   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    4877   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    4878   <p class=MsoBodyText>5e-006</p> 
     5269  <td width=143 valign=top style='width:107.0pt;border:none;border-right:solid windowtext 1.0pt; 
     5270  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5271  <p class=MsoBodyText>1.0e-006</p> 
     5272  </td> 
     5273 </tr> 
     5274 <tr style='height:18.8pt'> 
     5275  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5276  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5277  <p class=MsoBodyText>sldSolv</p> 
     5278  </td> 
     5279  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5280  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5281  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5282  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5283  </td> 
     5284  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5285  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5286  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5287  <p class=MsoBodyText>6.3e-006</p> 
    48795288  </td> 
    48805289 </tr> 
    … …  
    48885297 
    48895298<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4890 width=439 height=341 id="Picture 545" src="../images/html/image067.png"></p> 
     5299width=441 height=282 src="../images/html/image109.jpg"></p> 
    48915300 
    48925301<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    48985307 
    48995308<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span 
    4900 style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    4901 </span></span><b>Validation of the triaxialellipsoid 2D model</b></p> 
    4902  
    4903 <p class=MsoNormal>&nbsp;</p> 
    4904  
    4905 <p class=MsoNormal>Validation of our code was done by comparing the output of 
    4906 the 1D calculation to the angular average of the output of 2 D calculation over 
    4907 all possible angles. The Figure below shows the comparison where the solid dot 
    4908 refers to averaged 2D while the line represents the result of 1D calculation (for 
    4909 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and 
    4910 psi respectively).</p> 
     5309style='font-family:Symbol'>·</span><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5310</span><b>Validation of the triaxialellipsoid 2D model</b></p> 
     5311 
     5312<p class=MsoNormal>&nbsp;</p> 
     5313 
     5314<p class=MsoNormal>Validation of our code was done by comparing the output of the 
     53151D calculation to the angular average of the output of 2 D calculation over all 
     5316possible angles. The Figure below shows the comparison where the solid dot 
     5317refers to averaged 2D while the line represents the result of 1D calculation 
     5318(for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, 
     5319and psi respectively).</p> 
    49115320 
    49125321<p class=MsoNormal>&nbsp;</p> 
    … …  
    49155324 
    49165325<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4917 width=438 height=272 id="Picture 691" src="../images/html/image068.png"></p> 
     5326width=438 height=272 src="../images/html/image068.png"></p> 
    49185327 
    49195328<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    49255334 
    49265335<p class=MsoNormal align=center style='text-align:center'><img border=0 
    4927 width=396 height=297 id="Picture 248" src="../images/html/image069.jpg" 
     5336width=396 height=297 src="../images/html/image069.jpg" 
    49285337alt=triellrangles.gif></p> 
    49295338 
    … …  
    49525361 
    49535362<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    4954 style='font-size:14.0pt'>2.16.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    4955 </span></span></b><b><span style='font-size:14.0pt'><a name="LamellarModel">LamellarModel</a></span></b></p> 
     5363style='font-size:14.0pt'>2.19.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5364</span></b><a name=LamellarModel><b><span style='font-size:14.0pt'>LamellarModel</span></b></a></p> 
    49565365 
    49575366<p class=MsoNormal>&nbsp;</p> 
    … …  
    49595368<p class=MsoNormal>This model provides the scattering intensity, I(<i>q</i>), 
    49605369for a lyotropic lamellar phase where a uniform SLD and random distribution in 
    4961 solution are assumed.  The ploydispersion in the bilayer thickness can be 
     5370solution are assumed. &nbsp;The ploydispersion in the bilayer thickness can be 
    49625371applied from the GUI.</p> 
    49635372 
    … …  
    49685377<p class=MsoNormal>&nbsp;</p> 
    49695378 
    4970 <p class=MsoNormal align=center style='text-align:center'><span 
    4971 style='position:relative;top:15.0pt'><img border=0 width=103 height=46 
    4972 src="../images/html/image070.png"></span></p> 
     5379<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5380width=103 height=46 src="../images/html/image070.png"></p> 
    49735381 
    49745382<p class=MsoNormal>The form factor is,</p> 
    … …  
    49765384<p class=MsoNormal>&nbsp;</p> 
    49775385 
    4978 <p class=MsoNormal align=center style='text-align:center'><span 
    4979 style='position:relative;top:15.0pt'><img border=0 width=174 height=48 
    4980 src="../images/html/image071.png"></span></p> 
    4981  
    4982 <p class=MsoNormal align=center style='text-align:center'><span 
    4983 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5386<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5387width=174 height=48 src="../images/html/image071.png"></p> 
     5388 
     5389<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    49845390 
    49855391<p class=MsoNormal>where <span style='font-family:Symbol'>d</span> = bilayer 
    … …  
    49875393 
    49885394<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
    4989 1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
    4990 position:relative;top:8.0pt'><img border=0 width=103 height=33 
    4991 src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
    4992 position:relative;top:6.0pt'>.</span></p> 
    4993  
    4994 <p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     53951D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img 
     5396border=0 width=69 height=22 src="../images/html/image010.png"></span><span 
     5397style='font-size:14.0pt'>.</span></p> 
     5398 
     5399<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    49955400 
    49965401<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
    … …  
    50045409<div align=center> 
    50055410 
    5006 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    5007  style='border-collapse:collapse;border:none'> 
     5411<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     5412 style='border-collapse:collapse'> 
    50085413 <tr style='height:18.8pt'> 
    50095414  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    51095514 
    51105515<p class=MsoNormal align=center style='text-align:center'><img border=0 
    5111 width=476 height=351 id="Picture 571" src="../images/html/image072.png"></p> 
     5516width=457 height=275 src="../images/html/image072.png"></p> 
    51125517 
    51135518<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    51285533487-502.</p> 
    51295534 
    5130 <p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
    5131 11081-11088.</p> 
     5535<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5536also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 
    51325537 
    51335538<p class=MsoNormal>&nbsp;</p> 
    … …  
    51395544<p class=MsoNormal><b>&nbsp;</b></p> 
    51405545 
    5141  
    5142  
    51435546<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    5144 style='font-size:14.0pt'>2.17.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    5145 </span></span></b><b><span style='font-size:14.0pt'><a name="LamellarFFHGModel">LamellarFFHGModel</a></span></b></p> 
     5547style='font-size:14.0pt'>2.20.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5548</span></b><a name=LamellarFFHGModel><b><span style='font-size:14.0pt'>LamellarFFHGModel</span></b></a></p> 
    51465549 
    51475550<p class=MsoNormal>&nbsp;</p> 
    … …  
    51495552<p class=MsoNormal>This model provides the scattering intensity, I(<i>q</i>), 
    51505553for a lyotropic lamellar phase where a random distribution in solution are 
    5151 assumed.  The SLD of the head region is taken to be different from the SLD of 
    5152 the tail region.</p> 
     5554assumed. &nbsp;The SLD of the head region is taken to be different from the SLD 
     5555of the tail region.</p> 
    51535556 
    51545557<p class=MsoNormal>&nbsp;</p> 
    … …  
    51605563<p class=MsoNormal>&nbsp;</p> 
    51615564 
    5162 <p class=MsoNormal align=center style='text-align:center'><span 
    5163 style='position:relative;top:15.0pt'><img border=0 width=151 height=46 
    5164 src="../images/html/image073.png"></span></p> 
    5165  
    5166  
     5565<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5566width=151 height=46 src="../images/html/image073.png"></p> 
     5567 
     5568<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    51675569 
    51685570<p class=MsoNormal>The form factor is,</p> 
    … …  
    51705572<p class=MsoNormal>&nbsp;</p> 
    51715573 
    5172 <p class=MsoNormal align=center style='text-align:center'><span 
    5173 style='position:relative;top:15.0pt'><img border=0 width=398 height=46 
    5174 src="../images/html/image078.png"></span></p> 
    5175  
    5176 <p class=MsoNormal align=center style='text-align:center'><span 
    5177 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5178  
    5179 <p class=MsoNormal>where delta<sub>T</sub> 
    5180 = tail length (or t_length), delta<sub>H</sub> 
    5181 = head thickness (or h_thickness) , <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i><sub>H</sub> 
    5182 = SLD (headgroup) - SLD(solvent), and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i><sub>T</sub> 
     5574<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5575width=398 height=46 src="../images/html/image078.png"></p> 
     5576 
     5577<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5578 
     5579<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5580 
     5581<p class=MsoNormal>where <span style='font-family:Symbol'>d<sub>T</sub></span> 
     5582= tail length (or t_length), <span style='font-family:Symbol'>d<sub>H</sub></span> 
     5583= heasd thickness (or h_thickness) , <span style='font-family:Symbol'>Dr</span><sub>H</sub> 
     5584= SLD (headgroup) - SLD(solvent), and <span style='font-family:Symbol'>Dr</span><sub>T</sub> 
    51835585= SLD (tail) - SLD(headgroup).</p> 
    51845586 
     5587<p class=MsoNormal>&nbsp;</p> 
     5588 
    51855589<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
    5186 1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
    5187 position:relative;top:8.0pt'><img border=0 width=103 height=33 
    5188 src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
    5189 position:relative;top:6.0pt'>.</span></p> 
    5190  
    5191 <p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     55901D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img 
     5591border=0 width=71 height=23 src="../images/html/image010.png"></span><span 
     5592style='font-size:14.0pt'>.</span></p> 
     5593 
     5594<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    51925595 
    51935596<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
    … …  
    52015604<div align=center> 
    52025605 
    5203 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    5204  style='border-collapse:collapse;border:none'> 
     5606<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     5607 style='border-collapse:collapse'> 
    52055608 <tr style='height:18.8pt'> 
    52065609  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    53385741 
    53395742<p class=MsoNormal align=center style='text-align:center'><img border=0 
    5340 width=444 height=343 src="../images/html/image079.png"></p> 
     5743width=444 height=279 src="../images/html/image079.png"></p> 
    53415744 
    53425745<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    53575760487-502.</p> 
    53585761 
    5359 <p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
    5360 11081-11088.</p> 
    5361  
    5362 <p class=MsoNormal>&nbsp;</p> 
    5363  
    5364 <p class=MsoNormal>&nbsp;</p> 
     5762<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5763also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 
     5764 
     5765<p class=MsoNormal>&nbsp;</p> 
     5766 
     5767<p class=MsoNormal>&nbsp;</p> 
     5768 
    53655769<p class=MsoNormal>&nbsp;</p> 
    53665770 
    … …  
    53695773<p class=MsoNormal><b>&nbsp;</b></p> 
    53705774 
    5371  
    53725775<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    5373 style='font-size:14.0pt'>2.18.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    5374 </span></span></b><b><span style='font-size:14.0pt'><a name="LamellarPSModel">LamellarPSModel</a></span></b></p> 
     5776style='font-size:14.0pt'>2.21.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5777</span></b><a name=LamellarPSModel><b><span style='font-size:14.0pt'>LamellarPSModel</span></b></a></p> 
    53755778 
    53765779<p class=MsoNormal>&nbsp;</p> 
    53775780 
    53785781<p class=MsoNormal>This model provides the scattering intensity (<b>form factor</b> 
    5379 <b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase where 
    5380 a random distribution in solution are assumed.  </p> 
     5782<b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase 
     5783where a random distribution in solution are assumed.</p> 
    53815784 
    53825785<p class=MsoNormal>&nbsp;</p> 
    … …  
    53865789<p class=MsoNormal>&nbsp;</p> 
    53875790 
    5388 <p class=MsoNormal align=center style='text-align:center'><span 
    5389 style='position:relative;top:15.0pt'><img border=0 width=133 height=46 
    5390 src="../images/html/image077.png"></span></p> 
    5391  
    5392 <p class=MsoNormal align=center style='text-align:center'><span 
    5393 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5394  
    5395  
     5791<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5792width=133 height=46 src="../images/html/image077.png"></p> 
     5793 
     5794<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    53965795 
    53975796<p class=MsoNormal>The form factor is</p> 
    … …  
    54015800<p class=MsoNormal>&nbsp;</p> 
    54025801 
    5403 <p class=MsoNormal align=center style='text-align:center'><span 
    5404 style='position:relative;top:15.0pt'><img border=0 width=174 height=48 
    5405 src="../images/html/image071.png"></span></p> 
    5406  
    5407 <p class=MsoNormal><span style='position:relative;top:16.0pt'>And the structure 
    5408 is </span></p> 
    5409  
    5410 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5411  
    5412 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5413  
    5414 <p class=MsoNormal align=center style='text-align:center'><span 
    5415 style='position:relative;top:16.0pt'><img border=0 width=334 height=51 
    5416 src="../images/html/image074.png"></span></p> 
    5417  
    5418 <p class=MsoNormal align=center style='text-align:center'><span 
    5419 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5420  
    5421 <p class=MsoNormal><span style='position:relative;top:16.0pt'>where </span></p> 
    5422  
    5423 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5424  
    5425 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5426  
    5427 <p class=MsoNormal align=center style='text-align:center'><span 
    5428 style='position:relative;top:50.0pt'><img border=0 width=252 height=133 
    5429 src="../images/html/image075.png"></span></p> 
    5430  
    5431 <p class=MsoNormal align=center style='text-align:center'><span 
    5432 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5433  
    5434 <p class=MsoNormal align=center style='text-align:center'><span 
    5435 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5436  
    5437 <p class=MsoNormal>Here d= (repeat) spacing, delta 
    5438 = bilayer thickness, the contrast <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> = 
     5802<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5803width=174 height=48 src="../images/html/image071.png"></p> 
     5804 
     5805<p class=MsoNormal>and the structure is </p> 
     5806 
     5807<p class=MsoNormal>&nbsp;</p> 
     5808 
     5809<p class=MsoNormal>&nbsp;</p> 
     5810 
     5811<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5812width=308 height=51 src="../images/html/image074.png"></p> 
     5813 
     5814<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5815 
     5816<p class=MsoNormal>where </p> 
     5817 
     5818<p class=MsoNormal>&nbsp;</p> 
     5819 
     5820<p class=MsoNormal>&nbsp;</p> 
     5821 
     5822<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5823width=252 height=133 src="../images/html/image075.png"></p> 
     5824 
     5825<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5826 
     5827<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5828 
     5829<p class=MsoNormal>Here d= (repeat) spacing, <span style='font-family:Symbol'>d</span> 
     5830= bilayer thickness, the contrast <span style='font-family:Symbol'>Dr</span> = 
    54395831SLD (headgroup) - SLD(solvent), K=smectic bending elasticity, B=compression 
    54405832modulus, and N = number of lamellar plates (n_plates).</p> 
    … …  
    54475839 
    54485840<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
    5449 1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
    5450 position:relative;top:8.0pt'><img border=0 width=103 height=33 
    5451 src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
    5452 position:relative;top:6.0pt'>.</span></p> 
     58411D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img 
     5842border=0 width=82 height=26 src="../images/html/image010.png"></span><span 
     5843style='font-size:14.0pt'>.</span></p> 
    54535844 
    54545845<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
    … …  
    54615852<div align=center> 
    54625853 
    5463 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    5464  style='border-collapse:collapse;border:none'> 
     5854<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     5855 style='border-collapse:collapse'> 
    54655856 <tr style='height:18.8pt'> 
    54665857  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    55985989 
    55995990<p class=MsoNormal align=center style='text-align:center'><img border=0 
    5600 width=439 height=348 id="Picture 659" src="../images/html/image076.jpg"></p> 
     5991width=439 height=287 src="../images/html/image076.jpg"></p> 
    56015992 
    56025993<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    56176008487-502.</p> 
    56186009 
    5619 <p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
    5620 11081-11088.</p> 
    5621  
    5622 <p class=MsoNormal>&nbsp;</p> 
    5623  
    5624 <p class=MsoNormal>&nbsp;</p> 
    5625  
    5626  
     6010<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6011also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 
     6012 
     6013<p class=MsoNormal>&nbsp;</p> 
     6014 
     6015<p class=MsoNormal>&nbsp;</p> 
    56276016 
    56286017<p class=MsoNormal>&nbsp;</p> 
    … …  
    56326021<p class=MsoNormal><b>&nbsp;</b></p> 
    56336022 
    5634  
    56356023<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    5636 style='font-size:14.0pt'>2.19.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    5637 </span></span></b><b><span style='font-size:14.0pt'><a name="LamellarPSHGModel">LamellarPSHGModel</a></span></b></p> 
     6024style='font-size:14.0pt'>2.22.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6025</span></b><a name=LamellarPSHGModel><b><span style='font-size:14.0pt'>LamellarPSHGModel</span></b></a></p> 
    56386026 
    56396027<p class=MsoNormal>&nbsp;</p> 
    56406028 
    56416029<p class=MsoNormal>This model provides the scattering intensity (<b>form factor</b> 
    5642 <b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase where 
    5643 a random distribution in solution are assumed.  The SLD of the head region is 
    5644 taken to be different from the SLD of the tail region.</p> 
     6030<b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase 
     6031where a random distribution in solution are assumed. &nbsp;The SLD of the head 
     6032region is taken to be different from the SLD of the tail region.</p> 
    56456033 
    56466034<p class=MsoNormal>&nbsp;</p> 
    … …  
    56506038<p class=MsoNormal>&nbsp;</p> 
    56516039 
    5652 <p class=MsoNormal align=center style='text-align:center'><span 
    5653 style='position:relative;top:15.0pt'><img border=0 width=133 height=46 
    5654 src="../images/html/image077.png"></span></p> 
    5655  
    5656 <p class=MsoNormal align=center style='text-align:center'><span 
    5657 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     6040<p class=MsoNormal align=center style='text-align:center'><img border=0 
     6041width=133 height=46 src="../images/html/image077.png"></p> 
     6042 
     6043<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    56586044 
    56596045<p class=MsoNormal>The form factor is,</p> 
    … …  
    56616047<p class=MsoNormal>&nbsp;</p> 
    56626048 
    5663 <p class=MsoNormal align=center style='text-align:center'><span 
    5664 style='position:relative;top:15.0pt'><img border=0 width=398 height=46 
    5665 src="../images/html/image078.png"></span></p> 
    5666  
    5667 <p class=MsoNormal align=center style='text-align:center'><span 
    5668 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     6049<p class=MsoNormal align=center style='text-align:center'><img border=0 
     6050width=343 height=46 src="../images/html/image078.png"></p> 
     6051 
     6052<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    56696053 
    56706054<p class=MsoNormal>The structure factor is</p> 
    … …  
    56726056<p class=MsoNormal>&nbsp;</p> 
    56736057 
    5674 <p class=MsoNormal align=center style='text-align:center'><span 
    5675 style='position:relative;top:16.0pt'><img border=0 width=334 height=51 
    5676 src="../images/html/image074.png"></span></p> 
    5677  
    5678 <p class=MsoNormal align=center style='text-align:center'><span 
    5679 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5680  
    5681 <p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    5682  
    5683 <p class=MsoNormal><span style='position:relative;top:16.0pt'>where </span></p> 
    5684  
    5685 <p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5686  
    5687 <p class=MsoNormal align=center style='text-align:center'><span 
    5688 style='position:relative;top:50.0pt'><img border=0 width=252 height=133 
    5689 src="../images/html/image075.png"></span></p> 
    5690  
    5691 <p class=MsoNormal align=center style='text-align:center'><span 
    5692 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5693  
    5694 <p class=MsoNormal align=center style='text-align:center'><span 
    5695 style='position:relative;top:16.0pt'>&nbsp;</span></p> 
    5696  
    5697 <p class=MsoNormal>&nbsp;</p> 
    5698  
    5699 <p class=MsoNormal>where delta<sub>T</sub> 
    5700 = tail length (or t_length), delta<sub>H</sub> 
    5701 = head thickness (or h_thickness) , <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i><sub>H</sub> 
    5702 = SLD (headgroup) - SLD(solvent), and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i><sub>T</sub> 
     6058<p class=MsoNormal align=center style='text-align:center'><img border=0 
     6059width=334 height=51 src="../images/html/image074.png"></p> 
     6060 
     6061<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     6062 
     6063<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     6064 
     6065<p class=MsoNormal>where </p> 
     6066 
     6067<p class=MsoNormal>&nbsp;</p> 
     6068 
     6069<p class=MsoNormal align=center style='text-align:center'><img border=0 
     6070width=252 height=133 src="../images/html/image075.png"></p> 
     6071 
     6072<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     6073 
     6074<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     6075 
     6076<p class=MsoNormal>&nbsp;</p> 
     6077 
     6078<p class=MsoNormal>where <span style='font-family:Symbol'>d<sub>T</sub></span> 
     6079= tail length (or t_length), <span style='font-family:Symbol'>d<sub>H</sub></span> 
     6080= heasd thickness (or h_thickness) , <span style='font-family:Symbol'>Dr</span><sub>H</sub> 
     6081= SLD (headgroup) - SLD(solvent), and <span style='font-family:Symbol'>Dr</span><sub>T</sub> 
    57036082= SLD (tail) - SLD(headgroup). Here d= (repeat) spacing, K=smectic bending 
    57046083elasticity, B=compression modulus, and N = number of lamellar plates 
    … …  
    57146093 
    57156094<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
    5716 1D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
    5717 position:relative;top:8.0pt'><img border=0 width=103 height=33 
    5718 src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
    5719 position:relative;top:6.0pt'>.</span></p> 
    5720  
    5721 <p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     60951D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt'><img 
     6096border=0 width=70 height=22 src="../images/html/image010.png"></span><span 
     6097style='font-size:14.0pt'>.</span></p> 
     6098 
     6099<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    57226100 
    57236101<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
    … …  
    57316109<div align=center> 
    57326110 
    5733 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    5734  style='border-collapse:collapse;border:none'> 
     6111<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     6112 style='border-collapse:collapse'> 
    57356113 <tr style='height:18.8pt'> 
    57366114  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    59166294 
    59176295<p class=MsoNormal align=center style='text-align:center'><img border=0 
    5918 width=463 height=360 id="Picture 687" src="../images/html/image080.png"></p> 
     6296width=453 height=298 src="../images/html/image080.png"></p> 
    59196297 
    59206298<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    59356313487-502.</p> 
    59366314 
    5937 <p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
    5938 11081-11088.</p> 
     6315<p class=MsoNormal>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6316also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 
    59396317 
    59406318<p class=MsoNormal>&nbsp;</p> 
    … …  
    59536331 
    59546332<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    5955 style='font-size:16.0pt'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
    5956 </span></span></b><b><span style='font-size:16.0pt'><a name="Shape-Independent">Shape-Independent Models </a></span></b></p> 
     6333style='font-size:16.0pt'>3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp; 
     6334</span></b><a name=Shape-Independent><b><span style='font-size:16.0pt'>Shape-Independent 
     6335Models </span></b></a></p> 
    59576336 
    59586337<p class=MsoBodyText>&nbsp;</p> 
    … …  
    59646343 
    59656344<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    5966 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; 
    5967 </span></span></b><b><span style='font-size:14.0pt'> <a name="Debye">Debye (Model)</a></span></b></p> 
     6345style='font-size:14.0pt'>3.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6346</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=Debye>Debye (Model)</a></span></b></p> 
    59686347 
    59696348<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    59796358 
    59806359<p class=MsoNormal align=center style='text-align:center'><span 
    5981 style='font-size:14.0pt;position:relative;top:27.0pt'><img border=0 width=188 
    5982 height=76 src="../images/html/image081.png"></span></p> 
    5983  
    5984 <p class=MsoNormal align=center style='text-align:center'><span 
    5985 style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p> 
     6360style='font-size:14.0pt'><img border=0 width=188 height=76 
     6361src="../images/html/image081.png"></span></p> 
    59866362 
    59876363<p class=MsoNormal align=center style='text-align:center'><span 
    59886364style='font-size:14.0pt'>&nbsp;</span></p> 
    59896365 
    5990 <p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    5991 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    5992 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    5993 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6366<p class=MsoNormal align=center style='text-align:center'><span 
     6367style='font-size:14.0pt'>&nbsp;</span></p> 
     6368 
     6369<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is defined 
     6370as <span style='font-size:14.0pt'><img border=0 width=73 height=23 
     6371src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    59946372 
    59956373<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    … …  
    59976375<div align=center> 
    59986376 
    5999 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6000  style='border-collapse:collapse;border:none'> 
     6377<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     6378 style='border-collapse:collapse'> 
    60016379 <tr style='height:19.25pt'> 
    60026380  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    60166394  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    60176395  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6018   <p class=MsoBodyText>Scale</p> 
     6396  <p class=MsoBodyText>scale</p> 
    60196397  </td> 
    60206398  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    60326410  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    60336411  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6034   <p class=MsoBodyText>Rg</p> 
     6412  <p class=MsoBodyText>rg</p> 
    60356413  </td> 
    60366414  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    60486426  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    60496427  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6050   <p class=MsoBodyText>Background</p> 
     6428  <p class=MsoBodyText>background</p> 
    60516429  </td> 
    60526430  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    60616439  </td> 
    60626440 </tr> 
    6063  <tr style='height:19.25pt'> 
    6064   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    6065   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6066   <p class=MsoBodyText>&nbsp;</p> 
    6067   </td> 
    6068   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    6069   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    6070   padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6071   <p class=MsoBodyText>&nbsp;</p> 
    6072   </td> 
    6073   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    6074   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    6075   padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6076   <p class=MsoBodyText>&nbsp;</p> 
    6077   </td> 
    6078  </tr> 
    60796441</table> 
    60806442 
    … …  
    60836445<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    60846446 
     6447<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><img 
     6448border=0 width=423 height=273 src="../images/html/image110.jpg"></p> 
     6449 
     6450<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     64511D plot using the default values (w/200 data point).</b></p> 
     6452 
     6453<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'>&nbsp;</p> 
     6454 
     6455<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     6456 
    60856457<p class=MsoNormal style='margin-left:.25in'>Reference: Roe, R.-J., 
    6086 &quot;Methods of X-Ray and Neutron Scattering in Polymer Science&quot;, Oxford University Press, New York (2000).</p> 
     6458&quot;Methods of X-Ray and Neutron Scattering in Polymer Science&quot;, Oxford 
     6459University Press, New York (2000).</p> 
    60876460 
    60886461<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    60896462 
    60906463<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6091 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; 
    6092 </span></span></b><b><span style='font-size:14.0pt'> <a name="Lorentz">(Ornstein-Zernicke) Lorentz 
    6093 (Model)</a></span></b></p> 
     6464style='font-size:14.0pt'>3.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6465</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=Lorentz>(Ornstein-Zernicke) 
     6466Lorentz (Model)</a></span></b></p> 
    60946467 
    60956468<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    61016474 
    61026475<p class=MsoNormal align=center style='text-align:center'><span 
    6103 style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=212 
    6104 height=24 src="../images/html/image082.png"></span></p> 
     6476style='font-size:14.0pt'><img border=0 width=212 height=24 
     6477src="../images/html/image082.png"></span></p> 
    61056478 
    61066479<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    61076480 
    6108 <p class=MsoNormal style='text-indent:.25in'>The parameter L is referred to as 
    6109 the screening length.</p> 
     6481<p class=MsoNormal style='text-indent:.25in'>The parameter L is referred to as the 
     6482screening length.</p> 
    61106483 
    61116484<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p> 
    61126485 
    61136486<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    6114 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    6115 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    6116 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6487defined as <span style='font-size:14.0pt'><img border=0 width=103 height=33 
     6488src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    61176489 
    61186490<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p> 
    … …  
    61226494<div align=center> 
    61236495 
    6124 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6125  style='border-collapse:collapse;border:none'> 
     6496<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     6497 style='border-collapse:collapse'> 
    61266498 <tr style='height:19.25pt'> 
    61276499  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    61416513  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    61426514  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6143   <p class=MsoBodyText>Scale</p> 
     6515  <p class=MsoBodyText>scale</p> 
    61446516  </td> 
    61456517  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    61576529  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    61586530  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6159   <p class=MsoBodyText>Length</p> 
     6531  <p class=MsoBodyText>length</p> 
    61606532  </td> 
    61616533  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    61736545  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    61746546  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6175   <p class=MsoBodyText>Background</p> 
     6547  <p class=MsoBodyText>background</p> 
    61766548  </td> 
    61776549  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    61926564<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    61936565 
     6566<p class=MsoNormal align=center style='text-align:center'><b><span 
     6567style='font-size:14.0pt'><img border=0 width=422 height=273 
     6568src="../images/html/image111.jpg"></span></b></p> 
     6569 
     6570<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b><span 
     6571style='font-size:14.0pt'>&nbsp;</span>Figure. 1D plot using the default values 
     6572(w/200 data point).</b></p> 
     6573 
     6574<p class=MsoNormal>&nbsp;</p> 
     6575 
    61946576<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    61956577 
    61966578<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    61976579 
    6198 <p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    6199  
    62006580<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6201 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; 
    6202 </span></span></b><b><span style='font-size:14.0pt'> <a name="DAB_Model">DAB (Debye-Anderson-Brumberger)_Model</a></span></b></p> 
     6581style='font-size:14.0pt'>3.3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6582</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="DAB_Model">DAB 
     6583(Debye-Anderson-Brumberger)_Model</a></span></b></p> 
    62036584 
    62046585<p class=MsoNormal style='margin-left:.55in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    62056586 
    62066587<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Calculates the 
    6207 scattering from a randomly distributed, two-phase system based on the 
    6208 Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system is 
    6209 characterized by a single length scale, the correlation length, which is a 
    6210 measure of the average spacing between regions of phase 1 and phase 2. The 
    6211 model also assumes smooth interfaces between the phases and hence exhibits 
    6212 Porod behavior (I ~ Q<sup>-4</sup>) at large Q (Q*correlation length &gt;&gt; 
    6213 1). </p> 
     6588scattering from a randomly distributed, two-phase system based on the Debye-Anderson-Brumberger 
     6589(DAB) model for such systems. The two-phase system is characterized by a single 
     6590length scale, the correlation length, which is a measure of the average spacing 
     6591between regions of phase 1 and phase 2. The model also assumes smooth interfaces 
     6592between the phases and hence exhibits Porod behavior (I ~ Q<sup>-4</sup>) at 
     6593large Q (Q*correlation length &gt;&gt; 1). </p> 
    62146594 
    62156595<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p> 
    62166596 
    62176597<p class=MsoNormal align=center style='text-align:center'><span 
    6218 style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=221 
    6219 height=28 src="../images/html/image083.png"></span></p> 
     6598style='font-size:14.0pt'><img border=0 width=221 height=28 
     6599src="../images/html/image083.png"></span></p> 
    62206600 
    62216601<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    62316611 
    62326612<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    6233 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    6234 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    6235 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6613defined as <span style='font-size:14.0pt'><img border=0 width=75 height=24 
     6614src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    62366615 
    62376616<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    … …  
    62426621<div align=center> 
    62436622 
    6244 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6245  style='border-collapse:collapse;border:none'> 
     6623<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     6624 style='border-collapse:collapse'> 
    62466625 <tr style='height:19.25pt'> 
    62476626  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    62616640  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    62626641  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6263   <p class=MsoBodyText>Scale</p> 
     6642  <p class=MsoBodyText>scale</p> 
    62646643  </td> 
    62656644  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    62776656  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    62786657  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6279   <p class=MsoBodyText>Length</p> 
     6658  <p class=MsoBodyText>length</p> 
    62806659  </td> 
    62816660  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    62936672  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    62946673  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6295   <p class=MsoBodyText>Background</p> 
     6674  <p class=MsoBodyText>background</p> 
    62966675  </td> 
    62976676  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    63116690 
    63126691<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     6692 
     6693<p class=MsoNormal align=center style='text-align:center'><b><span 
     6694style='font-size:14.0pt'><img border=0 width=450 height=285 
     6695src="../images/html/image112.jpg"></span></b></p> 
     6696 
     6697<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b><span 
     6698style='font-size:14.0pt'>&nbsp;</span>Figure. 1D plot using the default values 
     6699(w/200 data point).</b></p> 
     6700 
     6701<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     6702 
     6703<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    63136704 
    63146705<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p> 
    … …  
    63286719 
    63296720<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6330 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; 
    6331 </span></span></b><b><span style='font-size:14.0pt'> <a name="Power_Law"> Power_Law </a></span></b></p> 
     6721style='font-size:14.0pt'>3.4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6722</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="Absolute Power_Law"> 
     6723Power_Law </a></span></b></p> 
    63326724 
    63336725<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    63396731 
    63406732<p class=MsoNormal align=center style='text-align:center'><span 
    6341 style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=149 
    6342 height=24 src="../images/html/image084.png"></span></p> 
     6733style='font-size:14.0pt'><img border=0 width=149 height=24 
     6734src="../images/html/image084.png"></span></p> 
    63436735 
    63446736<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    63566748<div align=center> 
    63576749 
    6358 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6359  style='border-collapse:collapse;border:none'> 
     6750<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     6751 style='border-collapse:collapse'> 
    63606752 <tr style='height:19.25pt'> 
    63616753  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    64266818<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    64276819 
     6820<p class=MsoNormal align=center style='text-align:center'><img border=0 
     6821width=416 height=259 src="../images/html/image113.jpg"></p> 
     6822 
     6823<p class=MsoNormal align=center style='text-align:center'><b>Figure. 1D plot 
     6824using the default values (w/200 data point).</b></p> 
     6825 
    64286826<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    64296827 
    … …  
    64316829 
    64326830<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6433 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; 
    6434 </span></span></b><b><span style='font-size:14.0pt'> <a name="Teubner Strey">Teubner Strey (Model)</a></span></b></p> 
     6831style='font-size:14.0pt'>3.5.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6832</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="Teubner Strey">Teubner 
     6833Strey (Model)</a></span></b></p> 
    64356834 
    64366835<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    64426841 
    64436842<p class=MsoNormal align=center style='text-align:center'><span 
    6444 style='font-size:14.0pt;position:relative;top:15.0pt'><img border=0 width=213 
    6445 height=45 src="../images/html/image085.png"></span></p> 
     6843style='font-size:14.0pt'><img border=0 width=213 height=45 
     6844src="../images/html/image085.png"></span></p> 
    64466845 
    64476846<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    64526851 
    64536852<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    6454 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    6455 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    6456 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6853defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23 
     6854src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    64576855 
    64586856<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    … …  
    64606858<div align=center> 
    64616859 
    6462 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6463  style='border-collapse:collapse;border:none'> 
     6860<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     6861 style='border-collapse:collapse'> 
    64646862 <tr style='height:19.25pt'> 
    64656863  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    64796877  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    64806878  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6481   <p class=MsoBodyText>A</p> 
     6879  <p class=MsoBodyText>scale</p> 
    64826880  </td> 
    64836881  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    64956893  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    64966894  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6497   <p class=MsoBodyText>C1</p> 
     6895  <p class=MsoBodyText>c1</p> 
    64986896  </td> 
    64996897  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    65116909  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    65126910  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6513   <p class=MsoBodyText>C2</p> 
     6911  <p class=MsoBodyText>c2</p> 
    65146912  </td> 
    65156913  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    65276925  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    65286926  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6529   <p class=MsoBodyText>Background</p> 
     6927  <p class=MsoBodyText>background</p> 
    65306928  </td> 
    65316929  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    65456943 
    65466944<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     6945 
     6946<p class=MsoNormal align=center style='text-align:center'><img border=0 
     6947width=426 height=275 src="../images/html/image114.jpg"></p> 
     6948 
     6949<p class=MsoNormal align=center style='text-align:center'><b>Figure. 1D plot 
     6950using the default values (w/200 data point).</b></p> 
     6951 
     6952<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    65476953 
    65486954<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p> 
    … …  
    65636969 
    65646970<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6565 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; 
    6566 </span></span></b><b><span style='font-size:14.0pt'> <a name="Number Density Fractal">Number Density Fractal (Model)</a></span></b></p> 
     6971style='font-size:14.0pt'>3.6.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6972</span></b><b><span style='font-size:14.0pt'>&nbsp;<a 
     6973name="Number_Density_Fractal"> </a><a name=FractalModel>FractalModel</a></span></b></p> 
    65676974 
    65686975<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    65776984 
    65786985<p class=MsoNormal align=center style='text-align:center'><span 
    6579 style='font-size:14.0pt;position:relative;top:85.0pt'><img border=0 width=332 
    6580 height=188 src="../images/html/image086.png"></span></p> 
     6986style='font-size:14.0pt'><img border=0 width=332 height=188 
     6987src="../images/html/image086.png"></span></p> 
    65816988 
    65826989<p class=MsoNormal align=center style='text-align:center'><span 
    6583 style='font-size:14.0pt;position:relative;top:85.0pt'>&nbsp;</span></p> 
     6990style='font-size:14.0pt'>&nbsp;</span></p> 
    65846991 
    65856992<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    65977004<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    65987005 
     7006<p class=MsoNormal style='margin-left:.25in;text-autospace:none'><b>The 
     7007polydispersion in radius is provided.</b></p> 
     7008 
    65997009<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    6600 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    6601 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    6602 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7010defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23 
     7011src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    66037012 
    66047013<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    6605  
    6606 <p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    66077014 
    66087015<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    66117018<div align=center> 
    66127019 
    6613 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6614  style='border-collapse:collapse;border:none'> 
     7020<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7021 style='border-collapse:collapse'> 
    66157022 <tr style='height:19.25pt'> 
    66167023  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    66307037  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    66317038  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    6632   <p class=MsoBodyText>Scale</p> 
     7039  <p class=MsoBodyText>scale</p> 
    66337040  </td> 
    66347041  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    67457152<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    67467153 
    6747 <p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     7154<p class=MsoNormal align=center style='margin-left:.25in;text-align:center; 
     7155text-autospace:none'><img border=0 width=445 height=280 
     7156src="../images/html/image115.jpg"></p> 
     7157 
     7158<p class=MsoNormal align=center style='text-align:center'><b>Figure. 1D plot 
     7159using the default values (w/200 data point).</b></p> 
     7160 
     7161<p class=MsoNormal align=center style='margin-left:.25in;text-align:center; 
     7162text-autospace:none'>&nbsp;</p> 
     7163 
     7164<p class=MsoNormal align=center style='margin-left:.25in;text-align:center; 
     7165text-autospace:none'>&nbsp;</p> 
    67487166 
    67497167<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p> 
    … …  
    67577175 
    67587176<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6759 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; 
    6760 </span></span></b><b><span style='font-size:14.0pt'> <a name="BEPolyelectrolyte"> BEPolyelectrolyte Model</a></span></b></p> 
     7177style='font-size:14.0pt'>3.7.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7178</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=BEPolyelectrolyte> 
     7179BEPolyelectrolyte Model</a></span></b></p> 
    67617180 
    67627181<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    67697188 
    67707189<p class=MsoNormal align=center style='text-align:center'><span 
    6771 style='font-size:14.0pt;position:relative;top:43.0pt'><img border=0 width=420 
    6772 height=125 src="../images/html/image087.png"></span></p> 
    6773  
    6774 <p class=MsoNormal align=center style='text-align:center'><span 
    6775 style='font-size:14.0pt;position:relative;top:43.0pt'>&nbsp;</span></p> 
     7190style='font-size:14.0pt'><img border=0 width=346 height=125 
     7191src="../images/html/image087.png"></span></p> 
    67767192 
    67777193<p class=MsoNormal align=center style='text-align:center'><span 
    67787194style='font-size:14.0pt'>&nbsp;</span></p> 
    67797195 
     7196<p class=MsoNormal align=center style='text-align:center'><span 
     7197style='font-size:14.0pt'>&nbsp;</span></p> 
     7198 
    67807199<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    67817200 
    6782 <p class=MsoNormal style='margin-left:.25in'>K is a contrast factor of the 
    6783 polymer, L<sub>b</sub> is the Bjerrum length, h is the virial parameter, b is 
    6784 the monomer length, C<sub>s</sub> is the concentration of monovalent salt, 
    6785 &#945; is the ionization degree, C<sub>a</sub> is the polymer molar 
    6786 concentration, and background is the incoherent background.</p> 
     7201<p class=MsoNormal style='margin-left:.25in'>K is a contrast factor of the polymer, 
     7202L<sub>b</sub> is the Bjerrum length, h is the virial parameter, b is the 
     7203monomer length, C<sub>s</sub> is the concentration of monovalent salt, &#945; 
     7204is the ionization degree, C<sub>a</sub> is the polymer molar concentration, and 
     7205background is the incoherent background.</p> 
    67877206 
    67887207<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    67897208 
    67907209<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    6791 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    6792 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    6793 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7210defined as <span style='font-size:14.0pt'><img border=0 width=75 height=24 
     7211src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    67947212 
    67957213<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    … …  
    67997217<div align=center> 
    68007218 
    6801 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    6802  style='border-collapse:collapse;border:none'> 
     7219<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7220 style='border-collapse:collapse'> 
    68037221 <tr style='height:19.25pt'> 
    68047222  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    69517369<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p> 
    69527370 
    6953 <p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Borue, V. Y., 
    6954 Erukhimovich, I. Y. Macromolecules 21, 3240 (1988).</p> 
     7371<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Borue, V. Y., Erukhimovich, 
     7372I. Y. Macromolecules 21, 3240 (1988).</p> 
    69557373 
    69567374<p class=MsoNormal style='margin-left:.5in;text-autospace:none'><span lang=FR>Joanny, 
    … …  
    69707388 
    69717389<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    6972 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; 
    6973 </span></span></b><b><span style='font-size:14.0pt'> <a name="Guinier">Guinier (Model)</a></span></b></p> 
     7390style='font-size:14.0pt'>3.8.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7391</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=Guinier>Guinier 
     7392(Model)</a></span></b></p> 
    69747393 
    69757394<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    69857404 
    69867405<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    6987 style='font-size:14.0pt;position:relative;top:7.0pt'><img border=0 width=183 
    6988 height=28 src="../images/html/image088.png"></span></p> 
     7406style='font-size:14.0pt'><img border=0 width=183 height=28 
     7407src="../images/html/image088.png"></span></p> 
    69897408 
    69907409<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    … …  
    69957414 
    69967415<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    6997 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    6998 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    6999 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7416defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23 
     7417src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    70007418 
    70017419<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
    … …  
    70047422<div align=center> 
    70057423 
    7006 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7007  style='border-collapse:collapse;border:none'> 
     7424<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7425 style='border-collapse:collapse'> 
    70087426 <tr style='height:19.25pt'> 
    70097427  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    70617479 
    70627480<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    7063 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; 
    7064 </span></span></b><b><span style='font-size:14.0pt'> <a name="PorodModel">PorodModel</a></span></b></p> 
     7481style='font-size:14.0pt'>3.9.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7482</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=PorodModel>PorodModel</a></span></b></p> 
    70657483 
    70667484<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    70677485 
    7068 <p class=MsoNormal style='margin-left:.25in'>A Porod analysis is done by 
    7069 linearizing the data at high q by plotting it as log(I) versus log(Q). In the 
    7070 high q region we can fit the following model: </p> 
     7486<p class=MsoNormal style='margin-left:.25in'>A Porod analysis is done by linearizing 
     7487the data at high q by plotting it as log(I) versus log(Q). In the high q region 
     7488we can fit the following model: </p> 
    70717489 
    70727490<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    70737491 
    70747492<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7075 style='font-size:14.0pt;position:relative;top:6.0pt'><img border=0 width=368 
    7076 height=25 src="../images/html/image089.png"></span></p> 
     7493style='font-size:14.0pt'><img border=0 width=319 height=25 
     7494src="../images/html/image089.png"></span></p> 
    70777495 
    70787496<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    … …  
    70827500style='font-size:14.0pt'>&nbsp;</span></p> 
    70837501 
    7084 <p class=MsoNormal style='margin-left:.25in'>C is the scale factor and  Sv is 
    7085 the specific surface area of the sample and <span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961; 
     7502<p class=MsoNormal style='margin-left:.25in'>C is the scale factor and &nbsp;Sv 
     7503is the specific surface area of the sample and <span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961; 
    70867504is the contrast factor. </p> 
    70877505 
    … …  
    70927510 
    70937511<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    7094 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    7095 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    7096 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7512defined as <span style='font-size:14.0pt'><img border=0 width=65 height=21 
     7513src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    70977514 
    70987515<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    … …  
    71037520<div align=center> 
    71047521 
    7105 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7106  style='border-collapse:collapse;border:none'> 
     7522<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7523 style='border-collapse:collapse'> 
    71077524 <tr style='height:19.25pt'> 
    71087525  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    71577574<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    71587575 
    7159 <p class=MsoNormal style='margin-left:.5in;text-autospace:none'>&nbsp;</p> 
     7576<p class=MsoNormal>&nbsp;</p> 
     7577 
     7578<p class=MsoBodyText>&nbsp;</p> 
     7579 
     7580<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     7581style='font-size:14.0pt'>3.10&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <a 
     7582name="Poly_GaussCoil">Poly_GaussCoil</a> (Model)</span></b></p> 
     7583 
     7584<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     7585 
     7586<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Polydisperse Gaussian 
     7587Coil: Calculate an empirical functional form for scattering from a polydisperse 
     7588polymer chain ina good solvent. The polymer is polydisperse with a Schulz-Zimm 
     7589polydispersity of the molecular weight distribution.&nbsp;</p> 
     7590 
     7591<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The returned 
     7592value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     7593 
     7594<p class=MsoNormal align=center style='margin-left:.25in;text-align:center; 
     7595text-autospace:none'><span style='position:relative;top:15pt'><img border=0 
     7596width=261 height=48 src="../images/html/image116.gif"></p> 
    71607597 
    71617598<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    71627599 
     7600<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>where the 
     7601dimensionless chain dimension is:</p> 
     7602 
     7603<p class=MsoNormal align=center style='margin-left:.25in;text-align:center; 
     7604text-autospace:none'><span style='position:relative;top:12pt'><img border=0 
     7605width=73 height=47 src="../images/html/image117.gif"></p> 
     7606 
     7607<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     7608 
     7609<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>and the 
     7610polydispersion is</p> 
     7611 
     7612<p class=MsoNormal align=center style='margin-left:.25in;text-align:center; 
     7613text-autospace:none'><span style='position:relative;top:15pt'><img border=0 
     7614width=81 height=47 src="../images/html/image118.gif">.</p> 
     7615 
     7616<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The scattering 
     7617intensity I(q) is calculated as:</p> 
     7618 
     7619<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     7620 
     7621<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The 
     7622polydispersion in ‘rg’ is provided.</p> 
     7623 
     7624<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
     7625 
     7626<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7627defined as <span style='font-size:14.0pt'><img border=0 width=73 height=23 
     7628src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
     7629 
     7630<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7631 
     7632<p class=MsoNormal style='margin-left:.25in'>TEST DATASET</p> 
     7633 
     7634<p class=MsoNormal style='margin-left:.25in'>&nbsp;This example dataset is 
     7635produced by running the Poly_GaussCoil, using 200 data points, qmin = 0.001 
     7636Å-1,&nbsp; qmax = 0.7 Å-1&nbsp;&nbsp; and the default values below.</p> 
     7637 
     7638<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7639 
     7640<div align=center> 
     7641 
     7642<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7643 style='border-collapse:collapse'> 
     7644 <tr style='height:19.25pt'> 
     7645  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7646  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7647  <p class=MsoBodyText>Parameter name</p> 
     7648  </td> 
     7649  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7650  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7651  <p class=MsoBodyText>Units</p> 
     7652  </td> 
     7653  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7654  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7655  <p class=MsoBodyText>Default value</p> 
     7656  </td> 
     7657 </tr> 
     7658 <tr style='height:19.25pt'> 
     7659  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7660  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7661  <p class=MsoBodyText>Scale</p> 
     7662  </td> 
     7663  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7664  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7665  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7666  <p class=MsoBodyText>None</p> 
     7667  </td> 
     7668  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7669  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7670  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7671  <p class=MsoBodyText>1.0</p> 
     7672  </td> 
     7673 </tr> 
     7674 <tr style='height:19.25pt'> 
     7675  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7676  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7677  <p class=MsoBodyText>rg</p> 
     7678  </td> 
     7679  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7680  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7681  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7682  <p class=MsoBodyText>Å</p> 
     7683  </td> 
     7684  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7685  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7686  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7687  <p class=MsoBodyText>60.0</p> 
     7688  </td> 
     7689 </tr> 
     7690 <tr style='height:19.25pt'> 
     7691  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7692  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7693  <p class=MsoBodyText>poly_m</p> 
     7694  </td> 
     7695  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7696  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7697  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7698  <p class=MsoBodyText>Mw/Mn</p> 
     7699  </td> 
     7700  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7701  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7702  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7703  <p class=MsoBodyText>2</p> 
     7704  </td> 
     7705 </tr> 
     7706 <tr style='height:19.25pt'> 
     7707  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7708  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7709  <p class=MsoBodyText>background</p> 
     7710  </td> 
     7711  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7712  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7713  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7714  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     7715  </td> 
     7716  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7717  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7718  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7719  <p class=MsoBodyText>0.001</p> 
     7720  </td> 
     7721 </tr> 
     7722</table> 
     7723 
     7724</div> 
     7725 
     7726<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'>&nbsp;</p> 
     7727 
     7728<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><img 
     7729border=0 width=431 height=268 src="../images/html/image119.jpg"></p> 
     7730 
     7731<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     77321D plot using the default values (w/200 data point).</b></p> 
     7733 
     7734<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'>&nbsp;</p> 
     7735 
     7736<p class=MsoNormal style='margin-left:.25in'>Reference: </p> 
     7737 
     7738<p class=MsoNormal style='margin-left:.25in'>Glatter &amp; Kratky - pg.404.</p> 
     7739 
     7740<p class=MsoNormal style='margin-left:.25in'>J.S. Higgins, and H.C. Benoit, “Polymers 
     7741and Neutron Scattering”, Oxford Science</p> 
     7742 
     7743<p class=MsoNormal style='margin-left:.25in'>Publications (1996).</p> 
     7744 
     7745<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     7746 
     7747<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     7748 
    71637749<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    71647750 
    7165  
    7166  
    71677751<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    7168 style='font-size:14.0pt'>3.10.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    7169 </span></span></b><b><span style='font-size:14.0pt'><a name="Peak Gauss Model"> Peak Gauss Model</a></span></b></p> 
     7752style='font-size:14.0pt'>3.11.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7753</span></b><a name="Peak Gauss Model"><b><span style='font-size:14.0pt'>&nbsp;Peak 
     7754Gauss Model</span></b></a></p> 
    71707755 
    71717756<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    71777762 
    71787763<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7179 style='font-size:14.0pt;position:relative;top:6.0pt'><img border=0 width=334 
    7180 height=25 src="../images/html/image090.png"></span></p> 
     7764style='font-size:14.0pt'><img border=0 width=334 height=25 
     7765src="../images/html/image090.png"></span></p> 
    71817766 
    71827767<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7183 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7768style='font-size:14.0pt'>&nbsp;</span></p> 
    71847769 
    71857770<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7186 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
    7187  
    7188 <p class=MsoNormal style='margin-left:.25in'><span style='position:relative; 
    7189 top:7.0pt'>where ‘scale’ is the peak height centered at q<sub>0</sub>, and B 
    7190 refers to the standard deviation of the function (equivalently, the FWHM is 
    7191 2.54*B).</span></p> 
     7771style='font-size:14.0pt'>&nbsp;</span></p> 
     7772 
     7773<p class=MsoNormal style='margin-left:.25in'>where ‘scale’ is the peak height centered 
     7774at q<sub>0</sub>, and B refers to the standard deviation of the function 
     7775(equivalently, the FWHM is 2.54*B).</p> 
    71927776 
    71937777<p class=MsoNormal style='margin-left:.25in'>The background term is added for 
    … …  
    71977781 
    71987782<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    7199 defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    7200 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    7201 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7783defined as <span style='font-size:14.0pt'><img border=0 width=71 height=23 
     7784src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    72027785 
    72037786<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    … …  
    72107793<div align=center> 
    72117794 
    7212 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7213  style='border-collapse:collapse;border:none'> 
     7795<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7796 style='border-collapse:collapse'> 
    72147797 <tr style='height:19.25pt'> 
    72157798  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    73037886 
    73047887<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    7305 style='font-size:14.0pt'>3.11.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    7306 </span></span></b><b><span style='font-size:14.0pt'> <a name="Peak Lorentz Model">Peak Lorentz Model</a></span></b></p> 
     7888style='font-size:14.0pt'>3.12.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7889</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name="Peak Lorentz Model">Peak 
     7890Lorentz Model</a></span></b></p> 
    73077891 
    73087892<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    73147898 
    73157899<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7316 style='font-size:14.0pt;position:relative;top:34.0pt'><img border=0 width=241 
    7317 height=70 src="../images/html/image091.png"></span></p> 
    7318  
    7319 <p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7320 style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
    7321  
    7322 <p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7323 style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     7900style='font-size:14.0pt'><img border=0 width=241 height=70 
     7901src="../images/html/image091.png"></span></p> 
    73247902 
    73257903<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    73267904style='font-size:14.0pt'>&nbsp;</span></p> 
    73277905 
    7328 <p class=MsoNormal style='margin-left:.25in'><span style='position:relative; 
    7329 top:7.0pt'>where ‘scale’ is the peak height centered at q<sub>0</sub>, and B 
    7330 refers to the standard deviation of the function.</span></p> 
     7906<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7907style='font-size:14.0pt'>&nbsp;</span></p> 
     7908 
     7909<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7910style='font-size:14.0pt'>&nbsp;</span></p> 
     7911 
     7912<p class=MsoNormal style='margin-left:.25in'>where ‘scale’ is the peak height 
     7913centered at q<sub>0</sub>, and B refers to the standard deviation of the 
     7914function.</p> 
    73317915 
    73327916<p class=MsoNormal style='margin-left:.25in'>The background term is added for 
    … …  
    73367920 
    73377921<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    7338 defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    7339 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    7340 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7922defined as<span style='font-size:14.0pt'><img border=0 width=77 height=25 
     7923src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    73417924 
    73427925<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
    … …  
    73457928<div align=center> 
    73467929 
    7347 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7348  style='border-collapse:collapse;border:none'> 
     7930<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     7931 style='border-collapse:collapse'> 
    73497932 <tr style='height:19.25pt'> 
    73507933  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    74368019 
    74378020<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    7438 style='font-size:14.0pt'>3.12.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    7439 </span></span></b><b><span style='font-size:14.0pt'> <a name="LineModel"> LineModel</a></span></b></p> 
     8021style='font-size:14.0pt'>3.13.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8022</span></b><b><span style='font-size:14.0pt'>&nbsp;<a name=LineModel> LineModel</a></span></b></p> 
    74408023 
    74418024<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    74478030 
    74488031<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7449 style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=92 
    7450 height=21 src="../images/html/image092.png"></span></p> 
    7451  
    7452 <p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7453 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8032style='font-size:14.0pt'><img border=0 width=92 height=21 
     8033src="../images/html/image092.png"></span></p> 
    74548034 
    74558035<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    74568036style='font-size:14.0pt'>&nbsp;</span></p> 
    74578037 
     8038<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     8039style='font-size:14.0pt'>&nbsp;</span></p> 
     8040 
    74588041<p class=MsoNormal style='margin-left:.25in'>where A and B are the coefficients 
    74598042of the first and second order terms.</p> 
    74608043 
    7461 <p class=MsoNormal style='margin-left:.25in'>Note: For 2D plot, I(q) = I(q<sub>x</sub>)*I(q<sub>y</sub>)  
    7462 which is defined differently from other shape independent models.</p> 
    7463  
    74648044<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    74658045 
     8046<p class=MsoNormal style='margin-left:.25in'><b>Note:</b> For 2D plot, I(q) = 
     8047I(q<sub>x</sub>)*I(q<sub>y</sub>)&nbsp; which is defined differently from other 
     8048shape independent models.</p> 
     8049 
     8050<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     8051 
    74668052<div align=center> 
    74678053 
    7468 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7469  style='border-collapse:collapse;border:none'> 
     8054<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     8055 style='border-collapse:collapse'> 
    74708056 <tr style='height:19.25pt'> 
    74718057  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    75258111 
    75268112<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    7527 style='font-size:16.0pt'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
    7528 </span></span></b><b><span style='font-size:16.0pt'><a name="Customized Models">Customized Models </a></span></b></p> 
     8113style='font-size:16.0pt'>4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp; 
     8114</span></b><a name="Customized_Models"><b><span style='font-size:16.0pt'>Customized 
     8115Models </span></b></a></p> 
    75298116 
    75308117<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    75318118 
    7532 <p class=MsoBodyText style='margin-left:.25in'>These model functions can be redefined 
    7533 by users (See SansView tutorial for details). </p> 
     8119<p class=MsoBodyText style='margin-left:.25in'>These model functions can be 
     8120redefined by users (See SansView tutorial for details). </p> 
    75348121 
    75358122<p class=MsoBodyText style='margin-left:.25in'>&nbsp;</p> 
    75368123 
    7537 <p class=MsoListParagraphCxSpFirst style='margin-left:.55in;text-indent:-.3in'><b><span 
    7538 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; 
    7539 </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> 
    7540  
    7541 <p class=MsoListParagraphCxSpLast style='margin-left:.55in'><b><span 
    7542 style='font-size:14.0pt'>&nbsp;</span></b></p> 
    7543  
    7544 <p class=MsoNormal>This function, as a sample function, calculates the intensity 
    7545 = A + Bcos(2q) + Csin(2q).</p> 
    7546  
    7547 <p class=MsoNormal>&nbsp;</p> 
    7548  
    7549 <p class=MsoNormal>&nbsp;</p> 
    7550  
    7551  
    7552 <p class=MsoListParagraphCxSpLast style='margin-left:.55in;text-indent:-.3in'><b><span 
    7553 style='font-size:14.0pt'>4.2.<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; 
    7554 </span></span></b><b><span style='font-size:14.0pt'><a name ="sinpoly_poly"> sin(poly)/poly </a></span></b></p> 
     8124<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
     8125style='font-size:14.0pt'>4.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8126</span></b><a name="A+Bcos(2x)+Csin(2x)"><b><span style='font-size:14.0pt'>A+Bcos(2x)+Csin(2x)</span></b></a></p> 
     8127 
     8128<p class=MsoListParagraph style='margin-left:.55in'><b><span style='font-size: 
     812914.0pt'>&nbsp;</span></b></p> 
     8130 
     8131<p class=MsoNormal>This function, as a sample function, calculates the 
     8132intensity = A + Bcos(2q) + Csin(2q).</p> 
     8133 
     8134<p class=MsoNormal>&nbsp;</p> 
     8135 
     8136<p class=MsoNormal>&nbsp;</p> 
     8137 
     8138<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
     8139style='font-size:14.0pt'>4.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8140</span></b><a name="sinpoly_poly"><b><span style='font-size:14.0pt'>sin(poly)/poly 
     8141</span></b></a></p> 
    75558142 
    75568143<p class=MsoNormal><b>&nbsp;</b></p> 
    75578144 
    7558 <p class=MsoNormal>This function calculates the intensity =  scale * sin(f)/f, 
    7559 where f = A + Bq + Cq<sup>2</sup> + Dq<sup>3 </sup>+ Eq<sup>4</sup> + Fq<sup>5</sup>.</p> 
    7560  
    7561 <p class=MsoNormal>&nbsp;</p> 
    7562  
    7563 <p class=MsoNormal>&nbsp;</p> 
    7564  
    7565 <p class=MsoNormal>&nbsp;</p> 
    7566  
    7567 <p class=MsoListParagraphCxSpFirst style='margin-left:.25in;text-indent:-.25in'><b><span 
    7568 style='font-size:16.0pt'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
    7569 </span></span></b><b><span style='font-size:16.0pt'><a name="Structure Factors"> Structure Factors</a></span></b></p> 
    7570  
    7571 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>&nbsp;</p> 
    7572  
    7573 <p class=MsoListParagraphCxSpLast style='margin-left:.25in'>The information in 
    7574 this section is originated from NIST SANS IgorPro package.</p> 
     8145<p class=MsoNormal>This function calculates the intensity =&nbsp; scale * 
     8146sin(f)/f, where f = A + Bq + Cq<sup>2</sup> + Dq<sup>3 </sup>+ Eq<sup>4</sup> + 
     8147Fq<sup>5</sup>.</p> 
     8148 
     8149<p class=MsoNormal>&nbsp;</p> 
     8150 
     8151<p class=MsoNormal>&nbsp;</p> 
     8152 
     8153<p class=MsoNormal>&nbsp;</p> 
     8154 
     8155<p class=MsoListParagraph style='margin-left:.25in;text-indent:-.25in'><b><span 
     8156style='font-size:16.0pt'>5.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp; 
     8157</span></b><a name="Structure_Factors"><b><span style='font-size:16.0pt'>Structure 
     8158Factors</span></b></a></p> 
     8159 
     8160<p class=MsoListParagraph style='margin-left:.25in'>&nbsp;</p> 
     8161 
     8162<p class=MsoListParagraph style='margin-left:.25in'>The information in this 
     8163section is originated from NIST SANS IgorPro package.</p> 
    75758164 
    75768165<p class=MsoBodyText>&nbsp;</p> 
    … …  
    75788167<p class=MsoBodyText>&nbsp;</p> 
    75798168 
    7580 <p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b> 
    7581 <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; 
    7582 </span></span></b><b><span style='font-size:14.0pt'><a name="HardsphereStructure"> HardSphere Structure </a></span></b></p> 
    7583  
    7584 <p class=MsoNormal>&nbsp;</p> 
    7585  
    7586 <p class=MsoNormal>This calculates the interparticle structure factor for 
    7587 monodisperse spherical particles interacting through hard sphere (excluded 
    7588 volume) interactions. The calculation uses the Percus-Yevick closure where the 
     8169<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
     8170style='font-size:14.0pt'>5.1.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8171</span></b><a name=HardsphereStructure><b><span style='font-size:14.0pt'>HardSphere 
     8172Structure </span></b></a></p> 
     8173 
     8174<p class=MsoNormal>&nbsp;</p> 
     8175 
     8176<p class=MsoNormal>This calculates the interparticle structure factor for monodisperse 
     8177spherical particles interacting through hard sphere (excluded volume) 
     8178interactions. The calculation uses the Percus-Yevick closure where the 
    75898179interparticle potential is: </p> 
    75908180 
    … …  
    75948184 
    75958185<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7596 style='font-size:14.0pt;position:relative;top:15.0pt'><img border=0 width=122 
    7597 height=48 src="../images/html/image093.png"></span></p> 
     8186style='font-size:14.0pt'><img border=0 width=122 height=48 
     8187src="../images/html/image093.png"></span></p> 
    75988188 
    75998189<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7600 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8190style='font-size:14.0pt'>&nbsp;</span></p> 
    76018191 
    76028192<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7603 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
    7604  
    7605 <p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the 
    7606 distance from the center of the sphere of a radius R.</span></p> 
    7607  
    7608 <p class=MsoNormal><span style='position:relative;top:7.0pt'>&nbsp;</span></p> 
     8193style='font-size:14.0pt'>&nbsp;</span></p> 
     8194 
     8195<p class=MsoNormal>where r is the distance from the center of the sphere of a 
     8196radius R.</p> 
     8197 
     8198<p class=MsoNormal>&nbsp;</p> 
    76098199 
    76108200<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    7611 defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    7612 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    7613 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
    7614  
    7615 <p class=MsoNormal><span style='position:relative;top:7.0pt'>&nbsp;</span></p> 
     8201defined as<span style='font-size:14.0pt'><img border=0 width=72 height=23 
     8202src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
     8203 
     8204<p class=MsoNormal>&nbsp;</p> 
    76168205 
    76178206<p class=MsoNormal>&nbsp;</p> 
    … …  
    76218210<div align=center> 
    76228211 
    7623 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7624  style='border-collapse:collapse;border:none'> 
     8212<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     8213 style='border-collapse:collapse'> 
    76258214 <tr style='height:19.25pt'> 
    76268215  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    … …  
    76928281 
    76938282<p class=MsoNormal align=center style='text-align:center'><img border=0 
    7694 width=484 height=331 id="Picture 111" src="../images/html/image094.png"></p> 
     8283width=454 height=295 src="../images/html/image094.png"></p> 
    76958284 
    76968285<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    77138302 
    77148303<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    7715 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; 
    7716 </span></span></b><b><span style='font-size:14.0pt'><a name="SquareWellStructure"> SquareWell Structure </a></span></b></p> 
     8304style='font-size:14.0pt'>5.2.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8305</span></b><a name=SquareWellStructure><b><span style='font-size:14.0pt'>&nbsp;SquareWell 
     8306Structure </span></b></a></p> 
    77178307 
    77188308<p class=MsoNormal>&nbsp;</p> 
    … …  
    77468336 
    77478337<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7748 style='font-size:14.0pt;position:relative;top:22.0pt'><img border=0 width=174 
    7749 height=67 src="../images/html/image095.png"></span></p> 
     8338style='font-size:14.0pt'><img border=0 width=174 height=67 
     8339src="../images/html/image095.png"></span></p> 
    77508340 
    77518341<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7752 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8342style='font-size:14.0pt'>&nbsp;</span></p> 
    77538343 
    77548344<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7755 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8345style='font-size:14.0pt'>&nbsp;</span></p> 
    77568346 
    77578347<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    7758 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
    7759  
    7760 <p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the 
    7761 distance from the center of the sphere of a radius R.</span></p> 
    7762  
    7763 <p class=MsoNormal><span style='position:relative;top:7.0pt'>&nbsp;</span></p> 
     8348style='font-size:14.0pt'>&nbsp;</span></p> 
     8349 
     8350<p class=MsoNormal>where r is the distance from the center of the sphere of a 
     8351radius R.</p> 
     8352 
     8353<p class=MsoNormal>&nbsp;</p> 
    77648354 
    77658355<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    7766 defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    7767 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    7768 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     8356defined as<span style='font-size:14.0pt'><img border=0 width=71 height=23 
     8357src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    77698358 
    77708359<p class=MsoNormal>&nbsp;</p> 
    … …  
    77748363<div align=center> 
    77758364 
    7776 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7777  style='margin-left:-1.5pt;border-collapse:collapse;border:none'> 
     8365<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     8366 style='margin-left:-1.5pt;border-collapse:collapse'> 
    77788367 <tr style='height:19.25pt'> 
    77798368  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt; 
    … …  
    78618450 
    78628451<p class=MsoNormal align=center style='text-align:center'><img border=0 
    7863 width=510 height=346 id="Picture 110" src="../images/html/image096.png"></p> 
     8452width=453 height=286 src="../images/html/image096.png"></p> 
    78648453 
    78658454<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    78808469 
    78818470<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    7882 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; 
    7883 </span></span></b><b><span style='font-size:14.0pt'><a name="HayterMSAStructure"> HayterMSA Structure </a></span></b></p> 
     8471style='font-size:14.0pt'>5.3.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8472</span></b><a name=HayterMSAStructure><b><span style='font-size:14.0pt'>&nbsp;HayterMSA 
     8473Structure </span></b></a></p> 
    78848474 
    78858475<p class=MsoNormal>&nbsp;</p> 
    … …  
    78878477<p class=MsoNormal>This calculates the Structure factor (the Fourier transform 
    78888478of the pair correlation function g(r)) for a system of charged, spheroidal 
    7889 objects in a dielectric medium.  When combined with an appropriate form factor 
    7890 (such as sphere, core+shell, ellipsoid etc 
    7891 ), this allows for inclusion of the 
    7892 interparticle interference effects due to screened coulomb repulsion between 
    7893 charged particles. This routine only works for charged particles.  If the 
    7894 charge is set to zero the routine will self destruct.  For non-charged 
    7895 particles use a hard sphere potential.</p> 
     8479objects in a dielectric medium.&nbsp; When combined with an appropriate form 
     8480factor (such as sphere, core+shell, ellipsoid etc 
     8481), this allows for inclusion 
     8482of the interparticle interference effects due to screened coulomb repulsion 
     8483between charged particles. This routine only works for charged particles.&nbsp; 
     8484If the charge is set to zero the routine will self destruct.&nbsp; For 
     8485non-charged particles use a hard sphere potential.</p> 
    78968486 
    78978487<p class=MsoNormal>&nbsp;</p> 
    78988488 
    78998489<p class=MsoNormal>The salt concentration is used to compute the ionic strength 
    7900 of the solution which in turn is used to compute the Debye screening length.  
    7901 At present there is no provision for entering the ionic strength directly nor 
    7902 for use of any multivalent salts.  The counterions are also assumed to be 
    7903 monovalent.</p> 
     8490of the solution which in turn is used to compute the Debye screening 
     8491length.&nbsp; At present there is no provision for entering the ionic strength 
     8492directly nor for use of any multivalent salts.&nbsp; The counterions are also 
     8493assumed to be monovalent.</p> 
    79048494 
    79058495<p class=MsoNormal>&nbsp;</p> 
    79068496 
    79078497<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    7908 defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    7909 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    7910 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     8498defined as<span style='font-size:14.0pt'><img border=0 width=71 height=23 
     8499src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    79118500 
    79128501<p class=MsoNormal>&nbsp;</p> 
    … …  
    79168505<div align=center> 
    79178506 
    7918 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    7919  style='margin-left:-1.5pt;border-collapse:collapse;border:none'> 
     8507<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     8508 style='margin-left:-1.5pt;border-collapse:collapse'> 
    79208509 <tr style='height:19.25pt'> 
    79218510  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt; 
    … …  
    80358624 
    80368625<p class=MsoNormal align=center style='text-align:center'><img border=0 
    8037 width=498 height=333 id="Picture 112" src="../images/html/image097.png"></p> 
     8626width=451 height=273 src="../images/html/image097.png"></p> 
    80388627 
    80398628<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    80498638JB Hayter, Molecular Physics 46, 651-656 (1982).</p> 
    80508639 
    8051 <p class=MsoNormal style='text-autospace:none'>            JB Hayter and J 
    8052 Penfold, Molecular Physics 42, 109-118 (1981).</p> 
     8640<p class=MsoNormal style='text-autospace:none'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8641JB Hayter and J Penfold, Molecular Physics 42, 109-118 (1981).</p> 
    80538642 
    80548643<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    80598648 
    80608649<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    8061 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; 
    8062 </span></span></b><b><span style='font-size:14.0pt'> <a name="StickyHSStructure">StickyHS Structure </a></span></b></p> 
     8650style='font-size:14.0pt'>5.4.</span></b><b><span style='font-size:7.0pt'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     8651</span></b><b><span style='font-size:14.0pt'>&nbsp;StickyHS Structure </span></b></p> 
    80638652 
    80648653<p class=MsoNormal>&nbsp;</p> 
    … …  
    80758664held between 0.01 and 0.1. It is best to hold the perturbation parameter fixed 
    80768665and let the &quot;stickiness&quot; vary to adjust the interaction strength. The 
    8077 stickiness, tau, is defined in equation 21 and is a function of both the 
     8666stickiness, tau, is defined in the equation below and is a function of both the 
    80788667perturbation parameter and the interaction strength. Tau and epsilon are 
    80798668defined in terms of the hard sphere diameter (sigma = 2R), the width of the 
    … …  
    80878676 
    80888677<p class=MsoNormal align=center style='text-align:center'><span 
    8089 style='font-size:14.0pt;position:relative;top:22.0pt'><img border=0 width=135 
    8090 height=67 src="../images/html/image098.png"></span></p> 
     8678style='font-size:14.0pt'><img border=0 width=135 height=67 
     8679src="../images/html/image098.png"></span></p> 
    80918680 
    80928681<p class=MsoNormal align=center style='text-align:center'><span 
    8093 style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p> 
     8682style='font-size:14.0pt'>&nbsp;</span></p> 
    80948683 
    80958684<p class=MsoNormal align=center style='text-align:center'><span 
    8096 style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p> 
     8685style='font-size:14.0pt'>&nbsp;</span></p> 
    80978686 
    80988687<p class=MsoNormal>where the interaction potential is</p> 
    … …  
    81038692 
    81048693<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    8105 style='font-size:14.0pt;position:relative;top:23.0pt'><img border=0 width=186 
    8106 height=69 src="../images/html/image099.png"></span></p> 
     8694style='font-size:14.0pt'><img border=0 width=186 height=69 
     8695src="../images/html/image099.png"></span></p> 
    81078696 
    81088697<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    8109 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8698style='font-size:14.0pt'>&nbsp;</span></p> 
    81108699 
    81118700<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    8112 style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8701style='font-size:14.0pt'>&nbsp;</span></p> 
    81138702 
    81148703<p class=MsoNormal>The Percus-Yevick (PY) closure was used for this 
    … …  
    81218710<p class=MsoNormal>The true particle volume fraction, f, is not equal to h, 
    81228711which appears in most of the reference. The two are related in equation (24) of 
    8123 the reference. The reference also describes the relationship between this 
    8124 perturbation solution and the original sticky hard sphere (or adhesive sphere) 
    8125 model by Baxter.</p> 
     8712the reference. The reference also describes the relationship between this perturbation 
     8713solution and the original sticky hard sphere (or adhesive sphere) model by 
     8714Baxter.</p> 
    81268715 
    81278716<p class=MsoNormal>&nbsp;</p> 
    … …  
    81378726 
    81388727<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
    8139 defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
    8140 border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
    8141 style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     8728defined as<span style='font-size:14.0pt'><img border=0 width=73 height=23 
     8729src="../images/html/image010.png"></span><span style='font-size:14.0pt'>.</span></p> 
    81428730 
    81438731<p class=MsoNormal>&nbsp;</p> 
    … …  
    81478735<div align=center> 
    81488736 
    8149 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    8150  style='margin-left:-1.5pt;border-collapse:collapse;border:none'> 
     8737<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 
     8738 style='margin-left:-1.5pt;border-collapse:collapse'> 
    81518739 <tr style='height:19.25pt'> 
    81528740  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt; 
    … …  
    82348822 
    82358823<p class=MsoNormal align=center style='text-align:center'><img border=0 
    8236 width=502 height=328 id="Picture 113" src="../images/html/image100.png"></p> 
     8824width=463 height=273 src="../images/html/image100.png"></p> 
    82378825 
    82388826<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
    … …  
    82528840<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    82538841 
    8254 <p class=MsoBodyText><b><span style='font-size:14.0pt'><a name="References">References</a></span></b></p> 
     8842<p class=MsoBodyText><a name=References><b><span style='font-size:14.0pt'>References</span></b></a></p> 
    82558843 
    82568844<p class=MsoBodyText>Feigin, L. A, and D. I. Svergun (1987) &quot;Structure 
    … …  
    82728860411-417.</p> 
    82738861 
    8274 <p class=MsoBodyText>McAlister, B.C. and Grady, B.P., (1998) J. Appl. Cryst. 31, 
    8275 594-599.</p> 
     8862<p class=MsoBodyText>McAlister, B.C. and Grady, B.P., (1998) J. Appl. Cryst. 
     886331, 594-599.</p> 
    82768864 
    82778865<p class=MsoBodyText>Porod, G. (1982) in Small Angle X-ray Scattering, editors 
    82788866Glatter, O. and Kratky, O., Academic Press.</p> 
    82798867 
     8868<p class=MsoBodyText>*Also, see the references in the each model function 
     8869descriptions.</p> 
     8870 
    82808871<p class=MsoBodyText>&nbsp;</p> 
    82818872 
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