Changeset 51b638f in sasview


Ignore:
Timestamp:
Sep 22, 2009 6:00:41 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:
49c5a36
Parents:
a0bc608
Message:

Model function doc and help update

Location:
sansview/doc
Files:
1 added
2 edited

Legend:

Unmodified
Added
Removed

  • sansview/doc/model_functions.html

    rb570686 r51b638f  
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    118748</head> 
    119749 
    120 <body lang=EN-US vlink=purple> 
     750<body lang=EN-US link=blue vlink=purple> 
    121751 
    122752<div class=Section1> 
    … …  
    126756<p class=MsoNormal>&nbsp;</p> 
    127757 
    128 <ul> 
    129 <li><a href="#Introduction"><b>Introduction</b></a></li> 
    130 <li><a href="#Shapes"><b>Shapes</b></a>:   
    131 <a href="#SphereModel">SphereModel</a>,  
    132 <a href="#CylinderModel">CylinderModel</a>,  
    133 <a href="#EllipsoidModel">EllipsoidModel</a>,  
    134 <a href="#CoreShellModel">CoreShellModel</a>,  
    135 <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, 
    136 <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>.  
    137 </li> 
    138 <li><a href="#Shape-Independent"><b>Shape-Independent</b></a>:  
    139 <a href="#Debye">Debye</a>,  
    140 <a href="#Lorentz">Lorentz</a>,  
    141 <a href="#DAB_Model">DAB_Model</a>,  
    142 <a href="#Power_Law">Power_Law</a>, 
    143 <a href="#Teubner Strey">Teubner Strey</a>, 
    144 <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, 
    145 <a href="#Number Density Fractal">Number Density Fractal</a>, 
    146 <a href="#Guinier">Guinier</a>, 
    147 <a href="#PorodModel">PorodModel</a>. </li> 
    148 <li><a href="#Customized Models"><b>Customized Models</b></a>:   
    149 <a href="#A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a>. </li> 
    150 <li><a href="#Structure Factors"><b>Structure Factors</b></a>:   
    151 <a href="#HardSphereStructure">HardSphereStructure</a>,  
    152 <a href="#SquareWellStructure">SquareWellStructure</a>,  
    153 <a href="#HayterMSAStructure">HayterMSAStructure</a>,  
    154 <a href="#StickyHSStructure">StickyHSStructure</a>.</li>  
    155 <li><a href="#References"><b>References</b></a></li> 
     758<ul type=disc> 
     759 <li class=MsoNormal style='line-height:115%'><a href="#Introduction"><b>Introduction</b></a> 
     760     </li> 
     761 <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> 
     781 <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="#sin(poly)/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> 
     808     </li> 
    156809</ul> 
    157810 
     811<p class=MsoNormal>&nbsp;</p> 
     812 
    158813<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    159 style='font-size:14.0pt'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; 
    160 </span></span></b><b><span style='font-size:14.0pt'><a name="Introduction">Introduction</a></span></b></p> 
     814style='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> 
    161816 
    162817<p class=MsoNormal>The present text documents the modules made available by the 
    … …  
    181836<p class=MsoNormal><b>&nbsp;</b></p> 
    182837 
     838<p class=MsoNormal><b>&nbsp;</b></p> 
    183839 
    184840<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    185 style='font-size:14.0pt'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; 
    186 </span></span></b><b><span style='font-size:14.0pt'><a name="Shapes">Shapes (Scattering intensity models)</a></span></b></p> 
     841style='font-size:16.0pt'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     842</span></span></b><b><span style='font-size:16.0pt'><a name="Shapes">Shapes (Scattering Intensity 
     843Models)</a></span></b></p> 
    187844 
    188845<p class=MsoNormal>&nbsp;</p> 
    … …  
    203860experiments, we normalize the form factors by the volume of the particle:</p> 
    204861 
    205 <p class=MsoNormal align=center style='text-align:center'><sub><img width=192 
    206 height=43 src="../images/html/image001.gif"></sub>            (1)</p> 
     862<p class=MsoNormal>&nbsp;</p> 
     863 
     864<p class=MsoNormal align=center style='text-align:center'><span 
     865style='position:relative;top:12.0pt'><img border=0 width=192 height=43 
     866src="../images/html/image001.png"></span>            (1)</p> 
     867 
     868<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    207869 
    208870<p class=MsoNormal>with</p> 
    209871 
    210 <p class=MsoNormal align=center style='text-align:center'><sub><img width=159 
    211 height=29 src="../images/html/image002.gif"></sub>        (2)</p> 
     872<p class=MsoNormal align=center style='text-align:center'><span 
     873style='position:relative;top:8.0pt'><img border=0 width=159 height=29 
     874src="../images/html/image002.png"></span>        (2)</p> 
     875 
     876<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     877 
     878<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    212879 
    213880<p class=MsoNormal>where <i>P</i><sub>0</sub><i>(<b>q</b>)</i> is the 
    … …  
    220887<p class=MsoNormal>For systems without inter-particle interference, the form 
    221888factors we provide can be related to the scattering intensity by the particle 
    222 volume fraction:<sub><img width=92 height=21 
    223 src="../images/html/image003.gif"></sub>.</p> 
     889volume fraction:<span style='position:relative;top:5.0pt'><img border=0 
     890width=92 height=21 src="../images/html/image003.png"></span>.</p> 
    224891 
    225892<p class=MsoNormal>&nbsp;</p> 
    … …  
    241908<p class=MsoNormal>&nbsp;</p> 
    242909 
     910<p class=MsoNormal>&nbsp;</p> 
    243911 
    244912<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    245913style='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; 
    246 </span></span></b><b><span style='font-size:14.0pt'><a name="SphereModel">SphereModel</a></span></b></p> 
     914</span></span></b><b><span style='font-size:14.0pt'><a name="SphereModel">Sphere Model</span></b></p> 
    247915 
    248916<p class=MsoNormal>&nbsp;</p> 
    … …  
    254922<p class=MsoNormal>&nbsp;</p> 
    255923 
    256 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.1.<span 
    257 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
    258  
    259 <p class=MsoNormal>&nbsp;</p> 
    260  
    261 <p class=MsoNormal>The 1D scattering intensity is calculated in the following way 
    262 (Guinier, 1955):</p> 
    263  
    264 <p class=MsoNormal align=center style='text-align:center'><sub><img width=337 
    265 height=53 src="../images/html/image004.gif"></sub>            (3)</p> 
     924<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
     925style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     926 
     927<p class=MsoNormal>&nbsp;</p> 
     928 
     929<p class=MsoNormal>The 1D scattering intensity is calculated in the following 
     930way (Guinier, 1955):</p> 
     931 
     932<p class=MsoNormal>&nbsp;</p> 
     933 
     934<p class=MsoNormal align=center style='text-align:center'><span 
     935style='position:relative;top:16.0pt'><img border=0 width=337 height=53 
     936src="../images/html/image004.png"></span>            (3)</p> 
     937 
     938<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     939 
     940<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    266941 
    267942<p class=MsoNormal>where <i>scale</i> is a scale factor, V is the volume of the 
    268943scatterer, <i>r</i> is the radius of the sphere, <i>bkg</i> is the background 
    269944level and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
    270 (contrast)  is the scattering length density difference between the scatterer 
     945(contrast)  is the scattering length density difference between the scatterer 
    271946and the solvent it is in.</p> 
    272947 
    … …  
    280955<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
    281956and the parameters of the sphere model are the following:</p> 
     957 
     958<p class=MsoNormal>&nbsp;</p> 
    282959 
    283960<div align=center> 
    … …  
    3691046<p class=MsoNormal>&nbsp;</p> 
    3701047 
     1048<p class=MsoNormal>&nbsp;</p> 
     1049 
    3711050<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
    3721051c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    … …  
    3741053<p class=MsoNormal>&nbsp;</p> 
    3751054 
    376  
    377 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.2.<span 
    378 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
     1055<p class=MsoNormal>&nbsp;</p> 
     1056 
     1057<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
     1058style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    3791059of the sphere model</b></p> 
    3801060 
    … …  
    3891069 
    3901070<p class=MsoBodyText align=center style='text-align:center;page-break-after: 
    391 avoid'><img width=573 height=315 id="Picture 5" 
     1071avoid'><img border=0 width=521 height=287 id="Picture 5" 
    3921072src="../images/html/image005.jpg" alt="sphere_1D_validation"></p> 
     1073 
     1074<p class=MsoBodyText align=center style='text-align:center;page-break-after: 
     1075avoid'>&nbsp;</p> 
    3931076 
    3941077<p class=MsoCaption>Figure 1: Comparison of the DANSE scattering intensity for 
    … …  
    3991082<p class=MsoNormal>&nbsp;</p> 
    4001083 
     1084<p class=MsoNormal><b>&nbsp;</b></p> 
     1085 
     1086<p class=MsoNormal><b>&nbsp;</b></p> 
     1087 
     1088<p class=MsoNormal><b>&nbsp;</b></p> 
     1089 
     1090 
    4011091 
    4021092<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    4031093style='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; 
    404 </span></span></b><b><span style='font-size:14.0pt'><a name="CylinderModel">CylinderModel</a></span></b></p> 
    405  
    406 <p class=MsoNormal>&nbsp;</p> 
    407  
    408 <p class=MsoNormal>This model provides the form factor for a monodisperse right 
    409 circular cylinder with uniform scattering length density. The form factor is 
    410 normalized by the particle volume.</p> 
    411  
    412 <p class=MsoNormal>&nbsp;</p> 
    413  
    414 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.2.1.<span 
    415 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     1094</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellModel">Core Shell (Sphere) Model</span></b></p> 
     1095 
     1096<p class=MsoNormal>&nbsp;</p> 
     1097 
     1098<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
     1099spherical particle with a core-shell structure. The form factor is normalized 
     1100by the particle volume.</p> 
     1101 
     1102<p class=MsoNormal>&nbsp;</p> 
     1103 
     1104<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
     1105style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     1106 
     1107<p class=MsoNormal>&nbsp;</p> 
     1108 
     1109<p class=MsoNormal>The 1D scattering intensity is calculated in the following 
     1110way (Guinier, 1955):</p> 
     1111 
     1112<p class=MsoNormal align=center style='text-align:center'><span 
     1113style='position:relative;top:16.0pt'><img border=0 width=642 height=55 
     1114src="../images/html/image006.png"></span> (11)</p> 
     1115 
     1116<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1117 
     1118<p class=MsoNormal align=right style='text-align:right'>&nbsp;</p> 
     1119 
     1120<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>s</sub></i> 
     1121is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the 
     1122core, <i>r<sub>s</sub></i> is the radius of the shell, <i>r<sub>c</sub></i> is 
     1123the radius of the core, <i>&#961;<sub>c</sub></i> is the scattering length 
     1124density of the core, <i>&#961;<sub>s</sub></i> is the scattering length density 
     1125of the shell, &#961;<sub>solv</sub> is the scattering length density of the 
     1126solvent, and <i>bkg</i> is the background level. </p> 
     1127 
     1128<p class=MsoNormal>&nbsp;</p> 
     1129 
     1130<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above, 
     1131regardless of the orientation of the q vector.</p> 
     1132 
     1133<p class=MsoNormal>&nbsp;</p> 
     1134 
     1135<p class=MsoNormal>For P*S: The outer most radius (= radius + thickness) is 
     1136used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     1137 
     1138<p class=MsoNormal>&nbsp;</p> 
     1139 
     1140<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
     1141and the parameters of the core-shell sphere model are the following:</p> 
     1142 
     1143<p class=MsoNormal>Here, radius = the radius of the core and thickness = the 
     1144thickness of the shell.</p> 
     1145 
     1146<p class=MsoNormal>&nbsp;</p> 
     1147 
     1148<div align=center> 
     1149 
     1150<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     1151 style='border-collapse:collapse;border:none'> 
     1152 <tr style='height:18.8pt'> 
     1153  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1154  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1155  <p class=MsoBodyText>Parameter name</p> 
     1156  </td> 
     1157  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1158  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1159  <p class=MsoBodyText>Units</p> 
     1160  </td> 
     1161  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1162  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1163  <p class=MsoBodyText>Default value</p> 
     1164  </td> 
     1165 </tr> 
     1166 <tr style='height:18.8pt'> 
     1167  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1168  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1169  <p class=MsoBodyText>scale</p> 
     1170  </td> 
     1171  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1172  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1173  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1174  <p class=MsoBodyText>None</p> 
     1175  </td> 
     1176  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1177  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1178  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1179  <p class=MsoBodyText>1.0</p> 
     1180  </td> 
     1181 </tr> 
     1182 <tr style='height:18.8pt'> 
     1183  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1184  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1185  <p class=MsoBodyText>radius</p> 
     1186  </td> 
     1187  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1188  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1189  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1190  <p class=MsoBodyText>Å</p> 
     1191  </td> 
     1192  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1193  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1194  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1195  <p class=MsoBodyText>60</p> 
     1196  </td> 
     1197 </tr> 
     1198 <tr style='height:18.8pt'> 
     1199  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1200  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1201  <p class=MsoBodyText>thickness</p> 
     1202  </td> 
     1203  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1204  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1205  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1206  <p class=MsoBodyText>Å</p> 
     1207  </td> 
     1208  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1209  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1210  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1211  <p class=MsoBodyText>10</p> 
     1212  </td> 
     1213 </tr> 
     1214 <tr style='height:18.8pt'> 
     1215  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1216  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1217  <p class=MsoBodyText>core_sld</p> 
     1218  </td> 
     1219  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1220  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1221  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1222  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1223  </td> 
     1224  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1225  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1226  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1227  <p class=MsoBodyText>1e-6</p> 
     1228  </td> 
     1229 </tr> 
     1230 <tr style='height:18.8pt'> 
     1231  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1232  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1233  <p class=MsoBodyText>shell_sld</p> 
     1234  </td> 
     1235  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1236  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1237  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1238  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1239  </td> 
     1240  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1241  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1242  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1243  <p class=MsoBodyText>2e-6</p> 
     1244  </td> 
     1245 </tr> 
     1246 <tr style='height:18.8pt'> 
     1247  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1248  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1249  <p class=MsoBodyText>solvent_sld</p> 
     1250  </td> 
     1251  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1252  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1253  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1254  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1255  </td> 
     1256  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1257  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1258  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1259  <p class=MsoBodyText>3e-6</p> 
     1260  </td> 
     1261 </tr> 
     1262 <tr style='height:19.5pt'> 
     1263  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1264  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
     1265  <p class=MsoBodyText>background</p> 
     1266  </td> 
     1267  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1268  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1269  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
     1270  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     1271  </td> 
     1272  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1273  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1274  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
     1275  <p class=MsoBodyText>0.001</p> 
     1276  </td> 
     1277 </tr> 
     1278</table> 
     1279 
     1280</div> 
     1281 
     1282<p class=MsoNormal>&nbsp;</p> 
     1283 
     1284<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     1285c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     1286 
     1287<p class=MsoNormal>            </p> 
     1288 
     1289<p class=MsoNormal>REFERENCE</p> 
     1290 
     1291<p class=MsoNormal>&nbsp;</p> 
     1292 
     1293<p class=MsoNormal>Guinier, A. and G. Fournet, &quot;Small-Angle Scattering of 
     1294X-Rays&quot;, John Wiley and Sons, New York, (1955).</p> 
     1295 
     1296<p class=MsoNormal>&nbsp;</p> 
     1297 
     1298<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
     1299style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
     1300of the core-shell sphere model</b></p> 
     1301 
     1302<p class=MsoNormal>&nbsp;</p> 
     1303 
     1304<p class=MsoNormal>Validation of our code was done by comparing the output of 
     1305the 1D model to the output of the software provided by the NIST (Kline, 2006). 
     1306Figure 1 shows a comparison of the output of our model and the output of the 
     1307NIST software.</p> 
     1308 
     1309<p class=MsoNormal>&nbsp;</p> 
     1310 
     1311<p class=MsoBodyText align=center style='text-align:center;page-break-after: 
     1312avoid'><img border=0 width=526 height=289 id="Picture 19" 
     1313src="../images/html/image007.jpg" alt="core_shell_sphere_1D_validation"></p> 
     1314 
     1315<p class=MsoCaption>Figure 7: Comparison of the DANSE scattering intensity for 
     1316a core-shell sphere with the output of the NIST SANS analysis software. The 
     1317parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>, 
     1318and Background=0.001 cm<sup> -1</sup>.</p> 
     1319 
     1320<p class=MsoNormal>&nbsp;</p> 
     1321 
     1322<p class=MsoNormal>&nbsp;</p> 
     1323 
     1324<p class=MsoNormal><b>&nbsp;</b></p> 
     1325 
     1326<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     1327style='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; 
     1328</span></span></b><b><span style='font-size:14.0pt'><a name="VesicleModel">VesicleModel</span></b></p> 
     1329 
     1330<p class=MsoNormal>&nbsp;</p> 
     1331 
     1332<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for an 
     1333unilamellar vesicle. The form factor is normalized by the volume of the shell.</p> 
     1334 
     1335<p class=MsoNormal>The 1D scattering intensity is calculated in the following 
     1336way (Guinier, 1955):</p> 
     1337 
     1338<p class=MsoNormal>&nbsp;</p> 
     1339 
     1340<p class=MsoNormal align=center style='text-align:center'><span 
     1341style='position:relative;top:16.0pt'><img border=0 width=448 height=54 
     1342src="../images/html/image008.png"></span> (11)</p> 
     1343 
     1344<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1345 
     1346<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1347 
     1348<p class=MsoNormal align=right style='text-align:right'>&nbsp;</p> 
     1349 
     1350<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>shell</sub></i> 
     1351is the volume of the shell, <i>V<sub>1</sub></i> is the volume of the core, <i>V<sub>2</sub></i> 
     1352is the total volume, <i>R<sub>1</sub></i> is the radius of the core, <i>r<sub>2</sub></i> 
     1353is the outer radius of the shell, <i>&#961;<sub>1</sub></i> is the scattering 
     1354length density of the core and the solvent, <i>&#961;<sub>2</sub></i> is the 
     1355scattering length density of the shell, and <i>bkg</i> is the background level. 
     1356And <i>J<sub>1</sub></i> = (sin<i>x </i>- <i>x</i>cos<i>x</i>)/<i>x</i><sup>2</sup>. 
     1357The functional form is identical to a &quot;typical&quot; core-shell structure, 
     1358except that the scattering is normalized by the volume that is contributing to 
     1359the scattering, namely the volume of the shell alone. Also, the vesicle is best 
     1360defined in terms of a core radius (= R1) and a shell thickness, t.  </p> 
     1361 
     1362<p class=MsoNormal>&nbsp;</p> 
     1363 
     1364<p class=MsoNormal>&nbsp;</p> 
     1365 
     1366<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1367width=149 height=143 id="Picture 31" src="../images/html/image009.jpg"></p> 
     1368 
     1369<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1370 
     1371<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1372 
     1373<p class=MsoNormal>The 2D scattering intensity is the same as <i>P</i>(<i>q</i>) 
     1374above, regardless of the orientation of the <i>q</i> vector  which is defined 
     1375as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 
     1376width=103 height=33 src="../images/html/image010.png"></span><span 
     1377style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     1378 
     1379<p class=MsoNormal>For P*S: The outer most radius (= radius + thickness) is 
     1380used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     1381 
     1382<p class=MsoNormal>&nbsp;</p> 
     1383 
     1384<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
     1385and the parameters of the vesicle model are the following:</p> 
     1386 
     1387<p class=MsoNormal>In the parameters, the ‘radius’ represents the core radius 
     1388(R<sub>1</sub>) and the ‘thickness’ (R<sub>2</sub> – R<sub>1</sub>) is the 
     1389shell thickness.</p> 
     1390 
     1391<p class=MsoNormal>&nbsp;</p> 
     1392 
     1393<div align=center> 
     1394 
     1395<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     1396 style='border-collapse:collapse;border:none'> 
     1397 <tr style='height:18.8pt'> 
     1398  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1399  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1400  <p class=MsoBodyText>Parameter name</p> 
     1401  </td> 
     1402  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1403  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1404  <p class=MsoBodyText>Units</p> 
     1405  </td> 
     1406  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1407  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1408  <p class=MsoBodyText>Default value</p> 
     1409  </td> 
     1410 </tr> 
     1411 <tr style='height:18.8pt'> 
     1412  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1413  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1414  <p class=MsoBodyText>scale</p> 
     1415  </td> 
     1416  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1417  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1418  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1419  <p class=MsoBodyText>None</p> 
     1420  </td> 
     1421  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1422  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1423  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1424  <p class=MsoBodyText>1.0</p> 
     1425  </td> 
     1426 </tr> 
     1427 <tr style='height:18.8pt'> 
     1428  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1429  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1430  <p class=MsoBodyText>radius</p> 
     1431  </td> 
     1432  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1433  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1434  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1435  <p class=MsoBodyText>Å</p> 
     1436  </td> 
     1437  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1438  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1439  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1440  <p class=MsoBodyText>100</p> 
     1441  </td> 
     1442 </tr> 
     1443 <tr style='height:18.8pt'> 
     1444  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1445  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1446  <p class=MsoBodyText>thickness</p> 
     1447  </td> 
     1448  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1449  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1450  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1451  <p class=MsoBodyText>Å</p> 
     1452  </td> 
     1453  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1454  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1455  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1456  <p class=MsoBodyText>30</p> 
     1457  </td> 
     1458 </tr> 
     1459 <tr style='height:18.8pt'> 
     1460  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1461  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1462  <p class=MsoBodyText>core_sld</p> 
     1463  </td> 
     1464  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1465  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1466  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1467  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1468  </td> 
     1469  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1470  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1471  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1472  <p class=MsoBodyText>6.3e-6</p> 
     1473  </td> 
     1474 </tr> 
     1475 <tr style='height:18.8pt'> 
     1476  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1477  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1478  <p class=MsoBodyText>shell_sld</p> 
     1479  </td> 
     1480  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1481  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1482  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1483  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1484  </td> 
     1485  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1486  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1487  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1488  <p class=MsoBodyText>0</p> 
     1489  </td> 
     1490 </tr> 
     1491 <tr style='height:18.8pt'> 
     1492  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1493  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1494  <p class=MsoBodyText>background</p> 
     1495  </td> 
     1496  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1497  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1498  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1499  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     1500  </td> 
     1501  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1502  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1503  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1504  <p class=MsoBodyText>0.0</p> 
     1505  </td> 
     1506 </tr> 
     1507</table> 
     1508 
     1509</div> 
     1510 
     1511<p class=MsoNormal>&nbsp;</p> 
     1512 
     1513<p class=MsoNormal>&nbsp;</p> 
     1514 
     1515<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1516width=454 height=356 id="Picture 158" src="../images/html/image011.png"></p> 
     1517 
     1518<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     15191D plot using the default values (w/200 data point).</b></p> 
     1520 
     1521<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1522 
     1523<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1524 
     1525<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1526 
     1527<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     1528c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     1529 
     1530<p class=MsoNormal>REFERENCE</p> 
     1531 
     1532<p class=MsoNormal>Guinier, A. and G. Fournet, &quot;Small-Angle Scattering of 
     1533X-Rays&quot;, John Wiley and Sons, New York, (1955).</p> 
     1534 
     1535<p class=MsoNormal>&nbsp;</p> 
     1536 
     1537<p class=MsoNormal>&nbsp;</p> 
     1538 
     1539<p class=MsoBodyText>&nbsp;</p> 
     1540 
     1541<p class=MsoNormal>&nbsp;</p> 
     1542 
     1543<p class=MsoNormal><b>&nbsp;</b></p> 
     1544 
     1545<p class=MsoNormal><b>&nbsp;</b></p> 
     1546 
     1547 
     1548<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     1549style='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; 
     1550</span></span></b><b><span style='font-size:14.0pt'><a name="MultiShellModel">MultiShellModel</span></b></p> 
     1551 
     1552<p class=MsoNormal>&nbsp;</p> 
     1553 
     1554<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
     1555multi-lamellar vesicle with N shells where the core is filled with solvent and 
     1556the shells are interleaved with layers of solvent. For N = 1, this return to 
     1557the vesicle model (above).</p> 
     1558 
     1559<p class=MsoNormal>&nbsp;</p> 
     1560 
     1561<p class=MsoNormal>&nbsp;</p> 
     1562 
     1563<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1564width=255 height=224 id="Picture 32" src="../images/html/image012.jpg"></p> 
     1565 
     1566<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1567 
     1568<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1569 
     1570<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of 
     1571the orientation of the <i>q</i> vector which is defined as<span 
     1572style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
     1573height=33 src="../images/html/image010.png"></span><span 
     1574style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     1575 
     1576<p class=MsoNormal>For P*S: The outer most radius (= core_radius + n_pairs  * 
     1577s_thickness  + (n_pairs -1) * w_thickness) is used as the effective radius 
     1578toward S(Q) when P(Q)*S(Q) is applied. </p> 
     1579 
     1580<p class=MsoNormal>&nbsp;</p> 
     1581 
     1582<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
     1583and the parameters of the multi-shell model are the following:</p> 
     1584 
     1585<p class=MsoNormal>In the parameters, the ‘s_thickness’ is the shell thickness 
     1586while the ‘w_thickness’ is the solvent thickness, and the n_pair is the number 
     1587of shells.</p> 
     1588 
     1589<p class=MsoNormal>&nbsp;</p> 
     1590 
     1591<p class=MsoNormal>&nbsp;</p> 
     1592 
     1593<div align=center> 
     1594 
     1595<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     1596 style='border-collapse:collapse;border:none'> 
     1597 <tr style='height:18.8pt'> 
     1598  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1599  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1600  <p class=MsoBodyText>Parameter name</p> 
     1601  </td> 
     1602  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1603  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1604  <p class=MsoBodyText>Units</p> 
     1605  </td> 
     1606  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1607  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1608  <p class=MsoBodyText>Default value</p> 
     1609  </td> 
     1610 </tr> 
     1611 <tr style='height:18.8pt'> 
     1612  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1613  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1614  <p class=MsoBodyText>scale</p> 
     1615  </td> 
     1616  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1617  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1618  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1619  <p class=MsoBodyText>None</p> 
     1620  </td> 
     1621  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1622  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1623  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1624  <p class=MsoBodyText>1.0</p> 
     1625  </td> 
     1626 </tr> 
     1627 <tr style='height:18.8pt'> 
     1628  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1629  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1630  <p class=MsoBodyText>core_radius</p> 
     1631  </td> 
     1632  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1633  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1634  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1635  <p class=MsoBodyText>Å</p> 
     1636  </td> 
     1637  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1638  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1639  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1640  <p class=MsoBodyText>60.0</p> 
     1641  </td> 
     1642 </tr> 
     1643 <tr style='height:18.8pt'> 
     1644  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1645  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1646  <p class=MsoBodyText>n_pairs</p> 
     1647  </td> 
     1648  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1649  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1650  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1651  <p class=MsoBodyText>None</p> 
     1652  </td> 
     1653  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1654  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1655  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1656  <p class=MsoBodyText>2.0</p> 
     1657  </td> 
     1658 </tr> 
     1659 <tr style='height:18.8pt'> 
     1660  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1661  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1662  <p class=MsoBodyText>core_sld</p> 
     1663  </td> 
     1664  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1665  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1666  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1667  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1668  </td> 
     1669  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1670  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1671  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1672  <p class=MsoBodyText>6.3e-6</p> 
     1673  </td> 
     1674 </tr> 
     1675 <tr style='height:18.8pt'> 
     1676  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1677  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1678  <p class=MsoBodyText>shell_sld</p> 
     1679  </td> 
     1680  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1681  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1682  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1683  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1684  </td> 
     1685  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1686  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1687  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1688  <p class=MsoBodyText>0.0</p> 
     1689  </td> 
     1690 </tr> 
     1691 <tr style='height:18.8pt'> 
     1692  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1693  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1694  <p class=MsoBodyText>background</p> 
     1695  </td> 
     1696  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1697  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1698  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1699  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     1700  </td> 
     1701  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1702  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1703  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1704  <p class=MsoBodyText>0.0</p> 
     1705  </td> 
     1706 </tr> 
     1707 <tr style='height:18.8pt'> 
     1708  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1709  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1710  <p class=MsoBodyText>s_thickness</p> 
     1711  </td> 
     1712  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1713  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1714  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1715  <p class=MsoBodyText>Å</p> 
     1716  </td> 
     1717  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1718  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1719  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1720  <p class=MsoBodyText>10</p> 
     1721  </td> 
     1722 </tr> 
     1723 <tr style='height:18.8pt'> 
     1724  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1725  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1726  <p class=MsoBodyText>w_thickness</p> 
     1727  </td> 
     1728  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1729  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1730  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1731  <p class=MsoBodyText>Å</p> 
     1732  </td> 
     1733  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1734  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1735  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1736  <p class=MsoBodyText>10</p> 
     1737  </td> 
     1738 </tr> 
     1739</table> 
     1740 
     1741</div> 
     1742 
     1743<p class=MsoNormal>&nbsp;</p> 
     1744 
     1745<p class=MsoNormal>&nbsp;</p> 
     1746 
     1747<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1748width=447 height=340 id="Picture 173" src="../images/html/image013.jpg"></p> 
     1749 
     1750<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     17511D plot using the default values (w/200 data point).</b></p> 
     1752 
     1753<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1754 
     1755<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     1756 
     1757<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     1758c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     1759 
     1760<p class=MsoNormal>&nbsp;</p> 
     1761 
     1762<p class=MsoNormal>REFERENCE</p> 
     1763 
     1764<p class=MsoNormal>Cabane, B., Small Angle Scattering Methods, Surfactant 
     1765Solutions: New Methods of Investigation, Ch.2, Surfactant Science Series Vol. 
     176622, Ed. R. Zana, M. Dekker, New York, 1987.</p> 
     1767 
     1768<p class=MsoNormal>&nbsp;</p> 
     1769 
     1770<p class=MsoNormal>&nbsp;</p> 
     1771 
     1772<p class=MsoNormal><b>&nbsp;</b></p> 
     1773 
     1774<p class=MsoNormal><b>&nbsp;</b></p> 
     1775 
     1776<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     1777style='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; 
     1778</span></span></b><b><span style='font-size:14.0pt'><a name="BinaryHSModel">BinaryHSModel</span></b></p> 
     1779 
     1780<p class=MsoNormal>&nbsp;</p> 
     1781 
     1782<p class=MsoNormal>This model (binary hard sphere model) provides the scattering 
     1783intensity, for binary mixture of spheres including hard sphere interaction 
     1784between those particles. Using Percus-Yevick closure, the calculation is an 
     1785exact multi-component solution:</p> 
     1786 
     1787<p class=MsoNormal>&nbsp;</p> 
     1788 
     1789<p class=MsoNormal align=center style='text-align:center'><span 
     1790style='position:relative;top:5.0pt'><img border=0 width=479 height=25 
     1791src="../images/html/image014.png"></span></p> 
     1792 
     1793<p class=MsoNormal align=center style='text-align:center'><span 
     1794style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     1795 
     1796<p class=MsoNormal align=center style='text-align:center'><span 
     1797style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     1798 
     1799<p class=MsoNormal><span style='position:relative;top:16.0pt'>where S<sub>ij</sub> 
     1800are the partial structure factors and f<sub>i</sub> are the scattering 
     1801amplitudes of the particles. And the subscript 1 is for the smaller particle 
     1802and 2 is for the larger. The number fraction of the larger particle, (<i>x</i> 
     1803=  n<sub>2</sub>/(n<sub>1</sub>+n<sub>2</sub>), n = the number density) is 
     1804internally calculated based on:</span></p> 
     1805 
     1806<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     1807 
     1808<p class=MsoNormal>&nbsp;</p> 
     1809 
     1810<p class=MsoNormal align=center style='text-align:center'><span 
     1811style='position:relative;top:24.0pt'><img border=0 width=248 height=98 
     1812src="../images/html/image015.png"></span><span style='position:relative; 
     1813top:16.0pt'>.</span></p> 
     1814 
     1815<p class=MsoNormal align=center style='text-align:center'><span 
     1816style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     1817 
     1818<p class=MsoNormal align=center style='text-align:center'><span 
     1819style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     1820 
     1821<p class=MsoNormal>&nbsp;</p> 
     1822 
     1823<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of 
     1824the orientation of the <i>q</i> vector which is defined as<span 
     1825style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
     1826height=33 src="../images/html/image010.png"></span><span 
     1827style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     1828 
     1829<p class=MsoNormal>&nbsp;</p> 
     1830 
     1831<p class=MsoNormal>&nbsp;</p> 
     1832 
     1833<p class=MsoNormal>The parameters of the binary hard sphere are the following 
     1834(in the names, l (or ls) stands for larger spheres while s (or ss) for the 
     1835smaller spheres):</p> 
     1836 
     1837<p class=MsoNormal>&nbsp;</p> 
     1838 
     1839<div align=center> 
     1840 
     1841<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     1842 style='border-collapse:collapse;border:none'> 
     1843 <tr style='height:18.8pt'> 
     1844  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1845  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1846  <p class=MsoBodyText>Parameter name</p> 
     1847  </td> 
     1848  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1849  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1850  <p class=MsoBodyText>Units</p> 
     1851  </td> 
     1852  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1853  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1854  <p class=MsoBodyText>Default value</p> 
     1855  </td> 
     1856 </tr> 
     1857 <tr style='height:18.8pt'> 
     1858  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1859  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1860  <p class=MsoBodyText>background</p> 
     1861  </td> 
     1862  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1863  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1864  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1865  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     1866  </td> 
     1867  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1868  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1869  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1870  <p class=MsoBodyText>0.001</p> 
     1871  </td> 
     1872 </tr> 
     1873 <tr style='height:18.8pt'> 
     1874  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1875  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1876  <p class=MsoBodyText>l_radius</p> 
     1877  </td> 
     1878  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1879  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1880  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1881  <p class=MsoBodyText>Å</p> 
     1882  </td> 
     1883  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1884  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1885  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1886  <p class=MsoBodyText>100.0</p> 
     1887  </td> 
     1888 </tr> 
     1889 <tr style='height:18.8pt'> 
     1890  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1891  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1892  <p class=MsoBodyText>ss_sld</p> 
     1893  </td> 
     1894  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1895  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1896  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1897  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1898  </td> 
     1899  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1900  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1901  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1902  <p class=MsoBodyText>0.0</p> 
     1903  </td> 
     1904 </tr> 
     1905 <tr style='height:18.8pt'> 
     1906  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1907  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1908  <p class=MsoBodyText>ls_sld</p> 
     1909  </td> 
     1910  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1911  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1912  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1913  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1914  </td> 
     1915  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1916  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1917  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1918  <p class=MsoBodyText>3e-6</p> 
     1919  </td> 
     1920 </tr> 
     1921 <tr style='height:18.8pt'> 
     1922  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1923  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1924  <p class=MsoBodyText>solvent_sld</p> 
     1925  </td> 
     1926  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1927  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1928  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1929  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     1930  </td> 
     1931  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1932  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1933  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1934  <p class=MsoBodyText>6e-6</p> 
     1935  </td> 
     1936 </tr> 
     1937 <tr style='height:18.8pt'> 
     1938  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1939  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1940  <p class=MsoBodyText>s_radius</p> 
     1941  </td> 
     1942  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1943  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1944  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1945  <p class=MsoBodyText>Å</p> 
     1946  </td> 
     1947  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1948  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1949  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1950  <p class=MsoBodyText>25.0</p> 
     1951  </td> 
     1952 </tr> 
     1953 <tr style='height:18.8pt'> 
     1954  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1955  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1956  <p class=MsoBodyText>vol_frac_ls</p> 
     1957  </td> 
     1958  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1959  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1960  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1961  <p class=MsoBodyText>Å</p> 
     1962  </td> 
     1963  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1964  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1965  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1966  <p class=MsoBodyText>0.1</p> 
     1967  </td> 
     1968 </tr> 
     1969 <tr style='height:18.8pt'> 
     1970  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     1971  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1972  <p class=MsoBodyText>vol_frac_ss</p> 
     1973  </td> 
     1974  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1975  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1976  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1977  <p class=MsoBodyText>Å</p> 
     1978  </td> 
     1979  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     1980  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     1981  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     1982  <p class=MsoBodyText>0.2</p> 
     1983  </td> 
     1984 </tr> 
     1985</table> 
     1986 
     1987</div> 
     1988 
     1989<p class=MsoNormal>&nbsp;</p> 
     1990 
     1991<p class=MsoNormal>&nbsp;</p> 
     1992 
     1993<p class=MsoNormal align=center style='text-align:center'><img border=0 
     1994width=484 height=361 id="Picture 197" src="../images/html/image016.png"></p> 
     1995 
     1996<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     19971D plot using the default values above (w/200 data point).</b></p> 
     1998 
     1999<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2000 
     2001<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2002 
     2003<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     2004c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     2005 
     2006<p class=MsoNormal>&nbsp;</p> 
     2007 
     2008<p class=MsoNormal>See the reference for details.</p> 
     2009 
     2010<p class=MsoNormal>&nbsp;</p> 
     2011 
     2012<p class=MsoNormal>REFERENCE</p> 
     2013 
     2014<p class=MsoNormal>N. W. Ashcroft and D. C. Langreth, Physical Review, v. 156 
     2015(1967) 685-692.</p> 
     2016 
     2017<p class=MsoNormal>[Errata found in Phys. Rev. 166 (1968) 934.]</p> 
     2018 
     2019<p class=MsoNormal>&nbsp;</p> 
     2020 
     2021<p class=MsoNormal>&nbsp;</p> 
     2022 
     2023<p class=MsoNormal>&nbsp;</p> 
     2024 
     2025<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     2026style='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; 
     2027</span></span></b><b><span style='font-size:14.0pt'><a name="CylinderModel">Cylinder Model</span></b></p> 
     2028 
     2029<p class=MsoNormal>&nbsp;</p> 
     2030 
     2031<p class=MsoNormal>This model provides the form factor for a right circular 
     2032cylinder with uniform scattering length density. The form factor is normalized 
     2033by the particle volume.</p> 
     2034 
     2035<p class=MsoNormal>&nbsp;</p> 
     2036 
     2037<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
     2038style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
    4162039 
    4172040<p class=MsoNormal>&nbsp;</p> 
    … …  
    4202043oriented cylinders is given by (Guinier, 1955):</p> 
    4212044 
    422 <p class=MsoNormal align=center style='text-align:center'><sub><img width=184 
    423 height=41 src="../images/html/image006.gif"></sub>  (4)</p> 
    424  
    425 <p class=MsoNormal align=center style='text-align:center'><sub><img width=373 
    426 height=45 src="../images/html/image007.gif"></sub>   (5)</p> 
     2045<p class=MsoNormal>&nbsp;</p> 
     2046 
     2047<p class=MsoNormal>&nbsp;</p> 
     2048 
     2049<p class=MsoNormal align=center style='text-align:center'><span 
     2050style='position:relative;top:12.0pt'><img border=0 width=184 height=41 
     2051src="../images/html/image017.png"></span>  (4)</p> 
     2052 
     2053<p class=MsoNormal align=center style='text-align:center'><span 
     2054style='position:relative;top:14.0pt'><img border=0 width=373 height=45 
     2055src="../images/html/image018.png"></span>   (5)</p> 
     2056 
     2057<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2058 
     2059<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    4272060 
    4282061<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span> 
    … …  
    4352068<p class=MsoNormal>&nbsp;</p> 
    4362069 
    437 <p class=MsoNormal>To provide easy access to the orientation of the cylinder, we 
    438 define the axis of the cylinder using two angles &#952; and <span 
    439 style='font-family:"Arial","sans-serif"'>&#966;</span>. Those angles are 
    440 defined on Figure 2.</p> 
     2070<p class=MsoNormal>To provide easy access to the orientation of the cylinder, 
     2071we define the axis of the cylinder using two angles theta and phi. Those angles 
     2072are defined on Figure 2.</p> 
     2073 
     2074<p class=MsoNormal>&nbsp;</p> 
    4412075 
    4422076<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
    443 width=300 height=281 id="Picture 8" src="../images/html/image008.jpg" 
    444 alt=angles></p> 
     2077border=0 width=478 height=258 id="Picture 14" 
     2078src="../images/html/image019.jpg" alt=cylinderangles.gif></p> 
    4452079 
    4462080<p class=MsoCaption align=center style='text-align:center'><a 
    447 name="_Ref173306040"></a><a name="_Ref173213915">Figure </a>2: Definition of the angles for oriented cylinders.</p> 
    448  
    449 <p class=MsoNormal>&nbsp;</p> 
    450  
     2081name="_Ref173306040"></a><a name="_Ref173213915">Figure </a>2a. Definition of the angles for oriented cylinders.</p> 
     2082 
     2083<p class=MsoNormal>&nbsp;</p> 
     2084 
     2085<p class=MsoNormal>&nbsp;</p> 
     2086 
     2087<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2088width=409 height=276 src="../images/html/image020.jpg" 
     2089alt=cylinderangles2.gif></p> 
     2090 
     2091<p class=MsoCaption align=center style='text-align:center'>Figure 2b. Examples of the angles for oriented cylinders against the detector plane.</p> 
     2092 
     2093<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2094 
     2095<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2096 
     2097<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the 
     2098cylinder is calculate based on the radius and length values, and used as the 
     2099effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     2100 
     2101<p class=MsoNormal>&nbsp;</p> 
    4512102 
    4522103<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
    4532104and the parameters of the cylinder model are the following:</p> 
     2105 
     2106<p class=MsoNormal>&nbsp;</p> 
    4542107 
    4552108<p class=MsoNormal>&nbsp;</p> 
    … …  
    4762129  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    4772130  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    478   <p class=MsoBodyText>Scale</p> 
     2131  <p class=MsoBodyText>scale</p> 
    4792132  </td> 
    4802133  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    4922145  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    4932146  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    494   <p class=MsoBodyText>Radius</p> 
     2147  <p class=MsoBodyText>radius</p> 
    4952148  </td> 
    4962149  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    5082161  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    5092162  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    510   <p class=MsoBodyText>Length</p> 
     2163  <p class=MsoBodyText>length</p> 
    5112164  </td> 
    5122165  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    5242177  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    5252178  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    526   <p class=MsoBodyText>Contrast</p> 
     2179  <p class=MsoBodyText>contrast</p> 
    5272180  </td> 
    5282181  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    5402193  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    5412194  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    542   <p class=MsoBodyText>Background</p> 
     2195  <p class=MsoBodyText>background</p> 
    5432196  </td> 
    5442197  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    5562209  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    5572210  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    558   <p class=MsoBodyText>Cyl_theta</p> 
     2211  <p class=MsoBodyText>cyl_theta</p> 
    5592212  </td> 
    5602213  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    5722225  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    5732226  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
    574   <p class=MsoBodyText>Cyl_phi</p> 
     2227  <p class=MsoBodyText>cyl_phi</p> 
    5752228  </td> 
    5762229  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    5912244<p class=MsoNormal>&nbsp;</p> 
    5922245 
     2246<p class=MsoNormal>&nbsp;</p> 
     2247 
    5932248<p class=MsoNormal>The output of the 1D scattering intensity function for 
    5942249randomly oriented cylinders is then given by:</p> 
    5952250 
     2251<p class=MsoNormal>&nbsp;</p> 
     2252 
     2253<p class=MsoNormal>&nbsp;</p> 
     2254 
    5962255<p class=MsoNormal align=center style='text-align:center'><a 
    597 name="_Ref173306479"></a><a name="_Ref173306528"><sub><img width=253 height=56 
    598 src="../images/html/image009.gif"></sub>         (</a>6)</p> 
     2256name="_Ref173306479"></a><a name="_Ref173306528"><span style='position:relative; 
     2257top:16.0pt'><img border=0 width=253 height=56 
     2258src="../images/html/image021.png"></span>         (</a>6)</p> 
    5992259 
    6002260<p class=MsoNormal>&nbsp;</p> 
    … …  
    6062266<p class=MsoNormal>&nbsp;</p> 
    6072267 
    608  
    609 <p class=MsoNormal>&nbsp;</p> 
    610  
    611 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.2.2.<span 
    612 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
     2268<p class=MsoNormal>&nbsp;</p> 
     2269 
     2270<p class=MsoNormal>&nbsp;</p> 
     2271 
     2272<p class=MsoNormal>&nbsp;</p> 
     2273 
     2274<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
     2275style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    6132276of the cylinder model</b></p> 
    6142277 
    … …  
    6252288done by evaluating the following:</p> 
    6262289 
    627 <p class=MsoNormal align=center style='text-align:center'><sub><img width=244 
    628 height=51 src="../images/html/image010.gif"></sub>           (7)</p> 
     2290<p class=MsoNormal>&nbsp;</p> 
     2291 
     2292<p class=MsoNormal align=center style='text-align:center'><span 
     2293style='position:relative;top:16.0pt'><img border=0 width=244 height=51 
     2294src="../images/html/image022.png"></span>           (7)</p> 
     2295 
     2296<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2297 
     2298<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    6292299 
    6302300<p class=MsoNormal>where <i>p(&#952;,</i><i><span style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> 
    … …  
    6392309<p class=MsoNormal>&nbsp;</p> 
    6402310 
    641 <p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315 
    642 id="Picture 11" src="../images/html/image011.jpg" 
    643 alt="cylinder_1D_validation"></p> 
     2311<p class=MsoNormal>&nbsp;</p> 
     2312 
     2313<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
     2314border=0 width=512 height=282 id="Picture 11" 
     2315src="../images/html/image023.jpg" alt="cylinder_1D_validation"></p> 
     2316 
     2317<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    6442318 
    6452319<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 
    646 NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 Å, 
    647 Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p> 
     2320NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 
     2321Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p> 
     2322 
     2323<p class=MsoNormal>&nbsp;</p> 
    6482324 
    6492325<p class=MsoCaption>&nbsp;</p> 
    6502326 
    651 <p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315 
    652 id="Picture 12" src="../images/html/image012.jpg" 
    653 alt="cylinder_2D_average"></p> 
     2327<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
     2328border=0 width=521 height=287 id="Picture 12" 
     2329src="../images/html/image024.jpg" alt="cylinder_2D_average"></p> 
     2330 
     2331<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    6542332 
    6552333<p class=MsoCaption><a name="_Ref173213305">Figure </a>4: Comparison of the 
    … …  
    6612339<p class=MsoCaption>&nbsp;</p> 
    6622340 
    663 <p class=MsoNormal>&nbsp;</p> 
    664  
     2341<p class=MsoNormal><b>&nbsp;</b></p> 
     2342 
     2343<p class=MsoNormal><b>&nbsp;</b></p> 
    6652344 
    6662345<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    667 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; 
    668 </span></span></b><b><span style='font-size:14.0pt'><a name="EllipsoidModel">EllipsoidModel</a></span></b></p> 
    669  
    670 <p class=MsoNormal>&nbsp;</p> 
    671  
    672 <p class=MsoNormal>This model provides the form factor for a monodisperse 
    673 ellipsoid (ellipsoid of revolution) with uniform scattering length density. The 
    674 form factor is normalized by the particle volume.</p> 
    675  
    676 <p class=MsoNormal>&nbsp;</p> 
    677  
    678 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.3.1.<span 
    679 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
    680  
    681 <p class=MsoNormal>&nbsp;</p> 
    682  
    683 <p class=MsoNormal>The output of the 2D scattering intensity function for 
    684 oriented ellipsoids is given by (Feigin, 1987):</p> 
    685  
    686 <p class=MsoCaption align=center style='text-align:center'><sub><img width=184 
    687 height=41 src="../images/html/image006.gif"></sub>   <span 
    688 style='font-size:12.0pt;font-weight:normal'>(8)</span></p> 
    689  
    690 <p class=MsoCaption align=center style='text-align:center'><sub><img width=371 
    691 height=47 src="../images/html/image013.gif"></sub>    <span 
    692 style='font-size:12.0pt;font-weight:normal'>(9)</span></p> 
    693  
    694 <p class=MsoCaption align=center style='text-align:center'><sub><img width=272 
    695 height=29 src="../images/html/image014.gif"></sub>     <span 
    696 style='font-size:12.0pt;font-weight:normal'>(10)</span></p> 
    697  
    698 <p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span> 
    699 is the angle between the axis of the ellipsoid and the q-vector, V is the 
    700 volume of the ellipsoid, R<sub>a</sub> is the radius along the rotation axis of 
    701 the ellipsoid, R<sub>b</sub> is the radius perpendicular to the rotation axis of 
    702 the ellipsoid and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
    703 (contrast) is the scattering length density difference between the scatterer 
    704 and the solvent. </p> 
    705  
    706 <p class=MsoNormal>&nbsp;</p> 
    707  
    708 <p class=MsoNormal>To provide easy access to the orientation of the ellipsoid, 
    709 we define the rotation axis of the ellipsoid using two angles &#952; and <span 
    710 style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
    711 of the cylinder, those angles are defined on Figure 2. For the ellipsoid, 
    712 &#952; is the angle between the rotation axis and the z-axis. </p> 
    713  
    714 <p class=MsoNormal>&nbsp;</p> 
    715  
    716 <p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
    717 and the parameters of the ellipsoid model are the following:</p> 
    718  
    719 <p class=MsoNormal>&nbsp;</p> 
    720  
    721 <div align=center> 
    722  
    723 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    724  style='border-collapse:collapse;border:none'> 
    725  <tr style='height:19.85pt'> 
    726   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    727   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    728   <p class=MsoBodyText>Parameter name</p> 
    729   </td> 
    730   <td width=105 valign=top style='width:78.9pt;border:solid windowtext 1.0pt; 
    731   border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    732   <p class=MsoBodyText>Units</p> 
    733   </td> 
    734   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    735   border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    736   <p class=MsoBodyText>Default value</p> 
    737   </td> 
    738  </tr> 
    739  <tr style='height:19.85pt'> 
    740   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    741   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    742   <p class=MsoBodyText>Scale</p> 
    743   </td> 
    744   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    745   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    746   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    747   <p class=MsoBodyText>None</p> 
    748   </td> 
    749   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    750   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    751   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    752   <p class=MsoBodyText>1.0</p> 
    753   </td> 
    754  </tr> 
    755  <tr style='height:19.85pt'> 
    756   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    757   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    758   <p class=MsoBodyText>Radius_a (parallel)</p> 
    759   </td> 
    760   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    761   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    762   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    763   <p class=MsoBodyText>Å</p> 
    764   </td> 
    765   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    766   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    767   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    768   <p class=MsoBodyText>20.0</p> 
    769   </td> 
    770  </tr> 
    771  <tr style='height:19.85pt'> 
    772   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    773   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    774   <p class=MsoBodyText>Radius_b (perpendicular)</p> 
    775   </td> 
    776   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    777   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    778   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    779   <p class=MsoBodyText>Å</p> 
    780   </td> 
    781   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    782   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    783   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    784   <p class=MsoBodyText>400.0</p> 
    785   </td> 
    786  </tr> 
    787  <tr style='height:19.85pt'> 
    788   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    789   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    790   <p class=MsoBodyText>Contrast</p> 
    791   </td> 
    792   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    793   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    794   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    795   <p class=MsoBodyText>Å<sup> -2</sup></p> 
    796   </td> 
    797   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    798   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    799   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    800   <p class=MsoBodyText>3.0e-6</p> 
    801   </td> 
    802  </tr> 
    803  <tr style='height:19.85pt'> 
    804   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    805   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    806   <p class=MsoBodyText>Background</p> 
    807   </td> 
    808   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    809   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    810   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    811   <p class=MsoBodyText>cm<sup>-1</sup></p> 
    812   </td> 
    813   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    814   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    815   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    816   <p class=MsoBodyText>0.0</p> 
    817   </td> 
    818  </tr> 
    819  <tr style='height:19.85pt'> 
    820   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    821   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    822   <p class=MsoBodyText>Axis_theta</p> 
    823   </td> 
    824   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    825   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    826   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    827   <p class=MsoBodyText>Radian</p> 
    828   </td> 
    829   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    830   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    831   padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
    832   <p class=MsoBodyText>1.57</p> 
    833   </td> 
    834  </tr> 
    835  <tr style='height:20.6pt'> 
    836   <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
    837   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
    838   <p class=MsoBodyText>Axis_phi</p> 
    839   </td> 
    840   <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
    841   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    842   padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
    843   <p class=MsoBodyText>Radian</p> 
    844   </td> 
    845   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    846   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    847   padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
    848   <p class=MsoBodyText>0.0</p> 
    849   </td> 
    850  </tr> 
    851 </table> 
    852  
    853 </div> 
    854  
    855 <p class=MsoNormal>&nbsp;</p> 
    856  
    857 <p class=MsoNormal>The output of the 1D scattering intensity function for 
    858 randomly oriented ellipsoids is then given by equation 6.</p> 
    859  
    860 <p class=MsoNormal>&nbsp;</p> 
    861  
    862 <p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not 
    863 used for the 1D output. Our implementation of the scattering kernel and the 1D 
    864 scattering intensity use the c-library from NIST.</p> 
    865  
    866 <p class=MsoNormal>&nbsp;</p> 
    867  
    868  
    869 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.3.2.<span 
    870 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    871 of the ellipsoid model</b></p> 
    872  
    873 <p class=MsoNormal>&nbsp;</p> 
    874  
    875 <p class=MsoNormal>Validation of our code was done by comparing the output of 
    876 the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 
    877 5 shows a comparison of the 1D output of our model and the output of the NIST 
    878 software.</p> 
    879  
    880 <p class=MsoNormal>&nbsp;</p> 
    881  
    882 <p class=MsoNormal>Averaging over a distribution of orientation is done by 
    883 evaluating equation 7. Since we have no other software to compare the 
    884 implementation of the intensity for fully oriented ellipsoids, we can compare 
    885 the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span 
    886 style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 6 shows the result of such a cross-check.</p> 
    887  
    888 <p class=MsoNormal align=center style='text-align:center'><b><i><span 
    889 style='color:#3366FF'>&nbsp;</span></i></b></p> 
    890  
    891 <p class=MsoNormal>The discrepancy above q=0.3 Å<sup> -1</sup> is due to the 
    892 way the form factors are calculated in the c-library provided by NIST. A numerical 
    893 integration has to be performed to obtain P(q) for randomly oriented particles 
    894 (equation 6). The NIST software performs that integration with a 76-point Gaussian 
    895 quadrature rule, which will become imprecise at high q where the amplitude 
    896 varies quickly as a function of q. The DANSE result shown has been obtained by 
    897 summing over 501 equidistant points in <span style='font-family:"Arial","sans-serif"'>&#945;</span>. 
    898 Our result was found to be stable over the range of q shown for a number of 
    899 points higher than 500.</p> 
    900  
    901 <p class=MsoNormal style='page-break-after:avoid'><b><img width=573 height=315 
    902 id="Picture 16" src="../images/html/image015.jpg" 
    903 alt="ellipsoid_1D_validation"></b></p> 
    904  
    905 <p class=MsoCaption><a name="_Ref173222904">Figure </a>5: Comparison of the 
    906 DANSE scattering intensity for an ellipsoid with the output of the NIST SANS 
    907 analysis software. The parameters were set to: Scale=1.0, Radius_a=20 Å, Radius_b=400 
    908 Å, </p> 
    909  
    910 <p class=MsoCaption>Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> 
    911 -1</sup>.</p> 
    912  
    913 <p class=MsoNormal><b>&nbsp;</b></p> 
    914  
    915  
    916 <p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315 
    917 id="Picture 17" src="../images/html/image016.jpg" 
    918 alt="ellipsoid_2D_average"></p> 
    919  
    920 <p class=MsoCaption><a name="_Ref173223004">Figure </a>6: Comparison of the 
    921 intensity for uniformly distributed ellipsoids calculated from our 2D model and 
    922 the intensity from the NIST SANS analysis software. The parameters used were: 
    923 Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and 
    924 Background=0.0 cm<sup> -1</sup>.</p> 
    925  
    926 <p class=MsoCaption>&nbsp;</p> 
    927  
    928 <p class=MsoNormal><b>&nbsp;</b></p> 
    929  
    930 <p class=MsoNormal><b>&nbsp;</b></p> 
    931  
    932 <p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    933 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; 
    934 </span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellModel">CoreShell(Sphere)Model</a></span></b></p> 
    935  
    936 <p class=MsoNormal>&nbsp;</p> 
    937  
    938 <p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
    939 monodisperse spherical particle with a core-shell structure. The form factor is 
     2346style='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; 
     2347</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellCylinderModel">Core-Shell Cylinder Model</span></b></p> 
     2348 
     2349<p class=MsoNormal>&nbsp;</p> 
     2350 
     2351<p class=MsoNormal>This model provides the form factor for a circular cylinder 
     2352with a core-shell scattering length density profile. The form factor is 
    9402353normalized by the particle volume.</p> 
    9412354 
    9422355<p class=MsoNormal>&nbsp;</p> 
    9432356 
    944 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.1.<span 
    945 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
    946  
    947 <p class=MsoNormal>&nbsp;</p> 
    948  
    949 <p class=MsoNormal>The 1D scattering intensity is calculated in the following 
    950 way (Guinier, 1955):</p> 
    951  
    952 <p class=MsoNormal align=right style='text-align:right'><sub><img width=643 
    953 height=55 src="../images/html/image017.gif"></sub> (11)</p> 
    954  
    955 <p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>s</sub></i> 
    956 is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the 
    957 core, <i>r<sub>s</sub></i> is the radius of the shell, <i>r<sub>c</sub></i> is 
    958 the radius of the core, <i>&#961;<sub>c</sub></i> is the scattering length 
    959 density of the core, <i>&#961;<sub>s</sub></i> is the scattering length density 
    960 of the shell, &#961;<sub>solv</sub> is the scattering length density of the 
    961 solvent, and <i>bkg</i> is the background level.</p> 
    962  
    963 <p class=MsoNormal>&nbsp;</p> 
    964  
    965 <p class=MsoNormal>The 2D scattering intensity is the same as P(q) above, 
    966 regardless of the orientation of the q vector.</p> 
    967  
    968 <p class=MsoNormal>&nbsp;</p> 
    969  
    970 <p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
    971 and the parameters of the core-shell sphere model are the following:</p> 
    972  
    973 <p class=MsoNormal>&nbsp;</p> 
    974  
    975 <div align=center> 
    976  
    977 <table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
    978  style='border-collapse:collapse;border:none'> 
    979  <tr style='height:18.8pt'> 
    980   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    981   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    982   <p class=MsoBodyText>Parameter name</p> 
    983   </td> 
    984   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    985   border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    986   <p class=MsoBodyText>Units</p> 
    987   </td> 
    988   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    989   border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    990   <p class=MsoBodyText>Default value</p> 
    991   </td> 
    992  </tr> 
    993  <tr style='height:18.8pt'> 
    994   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    995   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    996   <p class=MsoBodyText>Scale</p> 
    997   </td> 
    998   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    999   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1000   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1001   <p class=MsoBodyText>None</p> 
    1002   </td> 
    1003   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1004   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1005   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1006   <p class=MsoBodyText>1.0</p> 
    1007   </td> 
    1008  </tr> 
    1009  <tr style='height:18.8pt'> 
    1010   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1011   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1012   <p class=MsoBodyText>Radius</p> 
    1013   </td> 
    1014   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1015   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1016   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1017   <p class=MsoBodyText>Å</p> 
    1018   </td> 
    1019   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1020   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1021   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1022   <p class=MsoBodyText>60</p> 
    1023   </td> 
    1024  </tr> 
    1025  <tr style='height:18.8pt'> 
    1026   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1027   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1028   <p class=MsoBodyText>Thickness</p> 
    1029   </td> 
    1030   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1031   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1032   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1033   <p class=MsoBodyText>Å</p> 
    1034   </td> 
    1035   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1036   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1037   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1038   <p class=MsoBodyText>10</p> 
    1039   </td> 
    1040  </tr> 
    1041  <tr style='height:18.8pt'> 
    1042   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1043   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1044   <p class=MsoBodyText>Core_sld</p> 
    1045   </td> 
    1046   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1047   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1048   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1049   <p class=MsoBodyText>Å<sup> -2</sup></p> 
    1050   </td> 
    1051   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1052   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1053   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1054   <p class=MsoBodyText>1e-6</p> 
    1055   </td> 
    1056  </tr> 
    1057  <tr style='height:18.8pt'> 
    1058   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1059   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1060   <p class=MsoBodyText>Shell_sld</p> 
    1061   </td> 
    1062   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1063   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1064   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1065   <p class=MsoBodyText>Å<sup> -2</sup></p> 
    1066   </td> 
    1067   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1068   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1069   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1070   <p class=MsoBodyText>2e-6</p> 
    1071   </td> 
    1072  </tr> 
    1073  <tr style='height:18.8pt'> 
    1074   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1075   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1076   <p class=MsoBodyText>Solvent_sld</p> 
    1077   </td> 
    1078   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1079   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1080   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1081   <p class=MsoBodyText>Å<sup> -2</sup></p> 
    1082   </td> 
    1083   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1084   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1085   padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
    1086   <p class=MsoBodyText>3e-6</p> 
    1087   </td> 
    1088  </tr> 
    1089  <tr style='height:19.5pt'> 
    1090   <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    1091   border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
    1092   <p class=MsoBodyText>Background</p> 
    1093   </td> 
    1094   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1095   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1096   padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
    1097   <p class=MsoBodyText>cm<sup>-1</sup></p> 
    1098   </td> 
    1099   <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    1100   none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    1101   padding:0in 5.4pt 0in 5.4pt;height:19.5pt'> 
    1102   <p class=MsoBodyText>0.001</p> 
    1103   </td> 
    1104  </tr> 
    1105 </table> 
    1106  
    1107 </div> 
    1108  
    1109 <p class=MsoNormal>&nbsp;</p> 
    1110  
    1111 <p class=MsoNormal>Our model uses the form factor calculations implemented in a 
    1112 c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    1113  
    1114 <p class=MsoNormal>&nbsp;</p> 
    1115  
    1116  
    1117 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.2.<span 
    1118 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    1119 of the core-shell sphere model</b></p> 
    1120  
    1121 <p class=MsoNormal>&nbsp;</p> 
    1122  
    1123 <p class=MsoNormal>Validation of our code was done by comparing the output of 
    1124 the 1D model to the output of the software provided by the NIST (Kline, 2006). 
    1125 Figure 1 shows a comparison of the output of our model and the output of the 
    1126 NIST software.</p> 
    1127  
    1128 <p class=MsoNormal>&nbsp;</p> 
    1129  
    1130 <p class=MsoBodyText align=center style='text-align:center;page-break-after: 
    1131 avoid'><img width=573 height=315 id="Picture 19" 
    1132 src="../images/html/image018.jpg" alt="core_shell_sphere_1D_validation"></p> 
    1133  
    1134 <p class=MsoCaption>Figure 7: Comparison of the DANSE scattering intensity for 
    1135 a core-shell sphere with the output of the NIST SANS analysis software. The 
    1136 parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>, 
    1137 and Background=0.001 cm<sup> -1</sup>.</p> 
    1138  
    1139 <p class=MsoBodyText>&nbsp;</p> 
    1140  
    1141 <p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    1142 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; 
    1143 </span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellCylinderModel">CoreShellCylinderModel</a></span></b></p> 
    1144  
    1145 <p class=MsoNormal>&nbsp;</p> 
    1146  
    1147 <p class=MsoNormal>This model provides the form factor for a monodisperse, 
    1148 circular cylinder with a core-shell scattering length density profile. The form 
    1149 factor is normalized by the particle volume.</p> 
    1150  
    1151 <p class=MsoNormal>&nbsp;</p> 
    1152  
    1153 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.5.1.<span 
    1154 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     2357<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
     2358style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
    11552359 
    11562360<p class=MsoNormal>&nbsp;</p> 
    … …  
    11592363oriented core-shell cylinders is given by (Kline, 2006):</p> 
    11602364 
    1161 <p class=MsoNormal align=center style='text-align:center'><sub><img width=184 
    1162 height=45 src="../images/html/image019.gif"></sub> (12)</p> 
    1163  
    1164 <p class=MsoNormal align=center style='text-align:center'><sub><img width=476 
    1165 height=93 src="../images/html/image020.gif"></sub> (13)</p> 
     2365<p class=MsoNormal>&nbsp;</p> 
     2366 
     2367<p class=MsoNormal>&nbsp;</p> 
     2368 
     2369<p class=MsoNormal align=center style='text-align:center'><span 
     2370style='position:relative;top:15.0pt'><img border=0 width=184 height=45 
     2371src="../images/html/image025.png"></span> (12)</p> 
     2372 
     2373<p class=MsoNormal align=center style='text-align:center'><span 
     2374style='position:relative;top:32.0pt'><img border=0 width=476 height=93 
     2375src="../images/html/image026.png"></span> (13)</p> 
     2376 
     2377<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2378 
     2379<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    11662380 
    11672381<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span> 
    … …  
    11782392<p class=MsoNormal>&nbsp;</p> 
    11792393 
    1180 <p class=MsoNormal>To provide easy access to the orientation of the core-shell cylinder, 
    1181 we define the axis of the cylinder using two angles &#952; and <span 
     2394<p class=MsoNormal>&nbsp;</p> 
     2395 
     2396<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2397width=384 height=196 id="Picture 34" src="../images/html/image027.jpg"></p> 
     2398 
     2399<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2400 
     2401<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2402 
     2403<p class=MsoNormal>To provide easy access to the orientation of the core-shell 
     2404cylinder, we define the axis of the cylinder using two angles &#952; and <span 
    11822405style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
    11832406of the cylinder, those angles are defined on Figure 2.</p> 
    … …  
    11852408<p class=MsoNormal>&nbsp;</p> 
    11862409 
     2410<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
     2411cylinder is calculate based on the (radius+thickness) and 2(length +thickness) 
     2412values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. 
     2413</p> 
     2414 
     2415<p class=MsoNormal>&nbsp;</p> 
     2416 
     2417<p class=MsoNormal>&nbsp;</p> 
     2418 
    11872419<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
    11882420and the parameters of the core-shell cylinder model are the following:</p> 
    … …  
    12112443  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    12122444  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1213   <p class=MsoBodyText>Scale</p> 
     2445  <p class=MsoBodyText>scale</p> 
    12142446  </td> 
    12152447  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    12272459  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    12282460  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1229   <p class=MsoBodyText>Radius</p> 
     2461  <p class=MsoBodyText>radius</p> 
    12302462  </td> 
    12312463  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    12432475  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    12442476  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1245   <p class=MsoBodyText>Thickness</p> 
     2477  <p class=MsoBodyText>thickness</p> 
    12462478  </td> 
    12472479  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    12592491  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    12602492  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1261   <p class=MsoBodyText>Length</p> 
     2493  <p class=MsoBodyText>length</p> 
    12622494  </td> 
    12632495  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    12752507  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    12762508  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1277   <p class=MsoBodyText>Core_sld</p> 
     2509  <p class=MsoBodyText>core_sld</p> 
    12782510  </td> 
    12792511  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    12912523  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    12922524  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1293   <p class=MsoBodyText>Shell_sld</p> 
     2525  <p class=MsoBodyText>shell_sld</p> 
    12942526  </td> 
    12952527  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    13072539  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    13082540  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1309   <p class=MsoBodyText>Solvent_sld</p> 
     2541  <p class=MsoBodyText>solvent_sld</p> 
    13102542  </td> 
    13112543  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    13232555  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    13242556  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1325   <p class=MsoBodyText>Background</p> 
     2557  <p class=MsoBodyText>background</p> 
    13262558  </td> 
    13272559  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    13392571  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    13402572  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    1341   <p class=MsoBodyText>Axis_theta</p> 
     2573  <p class=MsoBodyText>axis_theta</p> 
    13422574  </td> 
    13432575  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    13552587  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    13562588  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.0pt'> 
    1357   <p class=MsoBodyText>Axis_phi</p> 
     2589  <p class=MsoBodyText>axis_phi</p> 
    13582590  </td> 
    13592591  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    13852617<p class=MsoNormal>&nbsp;</p> 
    13862618 
    1387 <p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.5.2.<span 
    1388 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
     2619<p class=MsoNormal>&nbsp;</p> 
     2620 
     2621<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
     2622style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
    13892623of the core-shell cylinder model</b></p> 
    13902624 
    … …  
    14042638style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 9 shows the result of such a cross-check.</p> 
    14052639 
    1406 <p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315 
    1407 id="Picture 22" src="../images/html/image021.jpg" 
    1408 alt="core_shell_cylinder_1D_validation"></p> 
     2640<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
     2641border=0 width=509 height=280 id="Picture 22" 
     2642src="../images/html/image028.jpg" alt="core_shell_cylinder_1D_validation"></p> 
     2643 
     2644<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    14092645 
    14102646<p class=MsoCaption><a name="_Ref173307149">Figure </a>8: Comparison of the 
    14112647DANSE scattering intensity for a core-shell cylinder with the output of the 
    1412 NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 Å, 
    1413 Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup> 
     2648NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 
     2649Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup> 
    14142650-2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p> 
    14152651 
    … …  
    14182654<p class=MsoNormal>&nbsp;</p> 
    14192655 
    1420 <p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315 
    1421 id="Picture 23" src="../images/html/image022.jpg" 
    1422 alt="core_shell_cylinder_2D_average"></p> 
     2656<p class=MsoNormal>&nbsp;</p> 
     2657 
     2658<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
     2659border=0 width=518 height=285 id="Picture 23" 
     2660src="../images/html/image029.jpg" alt="core_shell_cylinder_2D_average"></p> 
     2661 
     2662<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'>&nbsp;</p> 
    14232663 
    14242664<p class=MsoCaption><a name="_Ref173307204">Figure </a>9: Comparison of the 
    … …  
    14302670 
    14312671<p class=MsoNormal>&nbsp;</p> 
    1432 <p class=MsoNormal>&nbsp;</p> 
     2672 
     2673<p class=MsoNormal>&nbsp;</p> 
     2674 
     2675<p class=MsoNormal><b>&nbsp;</b></p> 
     2676 
     2677<p class=MsoNormal><b>&nbsp;</b></p> 
     2678 
    14332679 
    14342680<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    1435 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; 
    1436 </span></span></b><b><span style='font-size:14.0pt'><a name="EllipticalCylinderModel">EllipticalCylinderModel</a></span></b></p> 
     2681style='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; 
     2682</span></span></b><b><span style='font-size:14.0pt'><a name="HollowCylinderModel">HollowCylinderModel</span></b></p> 
     2683 
     2684<p class=MsoNormal>&nbsp;</p> 
     2685 
     2686<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
     2687monodisperse hollow right angle circular cylinder (tube) where the form factor 
     2688is normalized by the volume of the tube:</p> 
     2689 
     2690<p class=MsoNormal>P(q) = scale*&lt;f^2&gt;/V<sub>shell</sub>+background where 
     2691the averaging &lt; &gt; id applied only for the 1D calculation.  The inside and 
     2692outside of the hollow cylinder have the same SLD.</p> 
     2693 
     2694<p class=MsoNormal>The 1D scattering intensity is calculated in the following 
     2695way (Guinier, 1955):</p> 
     2696 
     2697<p class=MsoNormal>&nbsp;</p> 
     2698 
     2699<p class=MsoNormal align=center style='text-align:center'><span 
     2700style='position:relative;top:44.0pt'><img border=0 width=514 height=176 
     2701src="../images/html/image030.png"></span> (11)</p> 
     2702 
     2703<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2704 
     2705<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2706 
     2707<p class=MsoNormal align=right style='text-align:right'>&nbsp;</p> 
     2708 
     2709<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>J<sub>1</sub></i> 
     2710is the 1<sup>st</sup> order Bessel function, <i>J<sub>1</sub></i> (x)= (sin<i>x 
     2711</i>- <i>x</i>cos<i>x</i>)/<i>x</i><sup>2</sup>. </p> 
     2712 
     2713<p class=MsoNormal>&nbsp;</p> 
     2714 
     2715<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2716width=393 height=217 id="Picture 33" src="../images/html/image031.jpg"></p> 
     2717 
     2718<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2719 
     2720<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2721 
     2722<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2723 
     2724<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2725 
     2726<p class=MsoNormal>To provide easy access to the orientation of the core-shell 
     2727cylinder, we define the axis of the cylinder using two angles &#952; and <span 
     2728style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
     2729of the cylinder, those angles are defined on Figure 2 of CylinderModel.</p> 
     2730 
     2731<p class=MsoNormal>&nbsp;</p> 
     2732 
     2733<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
     2734cylinder is calculate based on the (radius) and 2(length) values, and used as 
     2735the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     2736 
     2737<p class=MsoNormal>&nbsp;</p> 
     2738 
     2739<p class=MsoNormal>In the parameters, the ‘contrast’ represents SLD (shell) - 
     2740SLD (solvent) and the radius = R<sub>hell</sub> while core_radius = R<sub>core</sub>.</p> 
     2741 
     2742<p class=MsoNormal>&nbsp;</p> 
     2743 
     2744<p class=MsoNormal><sub>&nbsp;</sub></p> 
     2745 
     2746<div align=center> 
     2747 
     2748<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     2749 style='border-collapse:collapse;border:none'> 
     2750 <tr style='height:18.8pt'> 
     2751  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2752  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2753  <p class=MsoBodyText>Parameter name</p> 
     2754  </td> 
     2755  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2756  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2757  <p class=MsoBodyText>Units</p> 
     2758  </td> 
     2759  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2760  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2761  <p class=MsoBodyText>Default value</p> 
     2762  </td> 
     2763 </tr> 
     2764 <tr style='height:18.8pt'> 
     2765  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2766  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2767  <p class=MsoBodyText>scale</p> 
     2768  </td> 
     2769  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2770  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2771  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2772  <p class=MsoBodyText>None</p> 
     2773  </td> 
     2774  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2775  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2776  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2777  <p class=MsoBodyText>1.0</p> 
     2778  </td> 
     2779 </tr> 
     2780 <tr style='height:18.8pt'> 
     2781  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2782  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2783  <p class=MsoBodyText>radius</p> 
     2784  </td> 
     2785  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2786  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2787  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2788  <p class=MsoBodyText>Å</p> 
     2789  </td> 
     2790  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2791  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2792  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2793  <p class=MsoBodyText>30</p> 
     2794  </td> 
     2795 </tr> 
     2796 <tr style='height:18.8pt'> 
     2797  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2798  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2799  <p class=MsoBodyText>length</p> 
     2800  </td> 
     2801  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2802  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2803  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2804  <p class=MsoBodyText>Å</p> 
     2805  </td> 
     2806  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2807  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2808  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2809  <p class=MsoBodyText>400</p> 
     2810  </td> 
     2811 </tr> 
     2812 <tr style='height:18.8pt'> 
     2813  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2814  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2815  <p class=MsoBodyText>core_radius</p> 
     2816  </td> 
     2817  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2818  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2819  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2820  <p class=MsoBodyText>Å</p> 
     2821  </td> 
     2822  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2823  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2824  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2825  <p class=MsoBodyText>20</p> 
     2826  </td> 
     2827 </tr> 
     2828 <tr style='height:18.8pt'> 
     2829  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2830  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2831  <p class=MsoBodyText>contrast</p> 
     2832  </td> 
     2833  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2834  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2835  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2836  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     2837  </td> 
     2838  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2839  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2840  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2841  <p class=MsoBodyText>5e-06</p> 
     2842  </td> 
     2843 </tr> 
     2844 <tr style='height:18.8pt'> 
     2845  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2846  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2847  <p class=MsoBodyText>background</p> 
     2848  </td> 
     2849  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2850  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2851  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2852  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     2853  </td> 
     2854  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2855  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2856  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2857  <p class=MsoBodyText>0.01</p> 
     2858  </td> 
     2859 </tr> 
     2860</table> 
     2861 
     2862</div> 
     2863 
     2864<p class=MsoNormal>&nbsp;</p> 
     2865 
     2866<p class=MsoNormal>&nbsp;</p> 
     2867 
     2868<p class=MsoNormal>&nbsp;</p> 
     2869 
     2870<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2871width=468 height=344 id="Picture 220" src="../images/html/image032.png"></p> 
     2872 
     2873<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     28741D plot using the default values (w/1000 data point).</b></p> 
     2875 
     2876<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2877 
     2878<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2879 
     2880<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     2881c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     2882 
     2883<p class=MsoBodyText>&nbsp;</p> 
     2884 
     2885<p class=MsoBodyText>REFERENCE</p> 
     2886 
     2887<p class=MsoBodyText>Feigin, L. A, and D. I. Svergun, &quot;Structure Analysis 
     2888by Small-Angle X-Ray and Neutron Scattering&quot;, Plenum Press, New York, 
     2889(1987).</p> 
     2890 
     2891<p class=MsoNormal>&nbsp;</p> 
     2892 
     2893<p class=MsoNormal>&nbsp;</p> 
     2894 
     2895<p class=MsoNormal>&nbsp;</p> 
     2896 
     2897<p class=MsoNormal><b>&nbsp;</b></p> 
     2898 
     2899<p class=MsoNormal><b>&nbsp;</b></p> 
     2900 
     2901<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     2902style='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; 
     2903</span></span></b><b><span style='font-size:14.0pt'><a name="FlexibleCylinderModel">FlexibleCylinderModel</span></b></p> 
     2904 
     2905<p class=MsoNormal>&nbsp;</p> 
     2906 
     2907<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a 
     2908flexible cylinder where the form factor is normalized by the volume of the 
     2909cylinder: Inter-cylinder interactions are NOT included. P(q) = 
     2910scale*&lt;f^2&gt;/V+background where the averaging &lt; &gt;  is applied over 
     2911all orientation for 1D.  The 2D scattering intensity is the same as 1D, 
     2912regardless of the orientation of the <i>q</i> vector which is defined as<span 
     2913style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
     2914height=33 src="../images/html/image010.png"></span><span 
     2915style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     2916 
     2917<p class=MsoNormal>&nbsp;</p> 
     2918 
     2919<p class=MsoNormal align=center style='text-align:center'><img border=0 
     2920width=329 height=150 id="Picture 35" src="../images/html/image033.jpg"></p> 
     2921 
     2922<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2923 
     2924<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     2925 
     2926<p class=MsoNormal>The chain of contour length, L, (the total length) can be 
     2927described a chain of some number of locally stiff segments of length lp. The 
     2928persistence length,lp, is the length along the cylinder over which the flexible 
     2929cylinder can be considered a rigid rod. The Kuhn length (b = 2*lp) is also used 
     2930to describe the stiffness of a chain. The returned value is in units of [cm-1], 
     2931on absolute scale. In the parameters, the ‘contrast’ represents SLD (chain) - 
     2932SLD (solvent).</p> 
     2933 
     2934<p class=MsoNormal><sub>&nbsp;</sub></p> 
     2935 
     2936<p class=MsoNormal><sub>&nbsp;</sub></p> 
     2937 
     2938<div align=center> 
     2939 
     2940<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     2941 style='border-collapse:collapse;border:none'> 
     2942 <tr style='height:18.8pt'> 
     2943  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2944  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2945  <p class=MsoBodyText>Parameter name</p> 
     2946  </td> 
     2947  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2948  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2949  <p class=MsoBodyText>Units</p> 
     2950  </td> 
     2951  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2952  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2953  <p class=MsoBodyText>Default value</p> 
     2954  </td> 
     2955 </tr> 
     2956 <tr style='height:18.8pt'> 
     2957  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2958  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2959  <p class=MsoBodyText>scale</p> 
     2960  </td> 
     2961  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2962  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2963  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2964  <p class=MsoBodyText>None</p> 
     2965  </td> 
     2966  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2967  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2968  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2969  <p class=MsoBodyText>1.0</p> 
     2970  </td> 
     2971 </tr> 
     2972 <tr style='height:18.8pt'> 
     2973  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2974  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2975  <p class=MsoBodyText>radius</p> 
     2976  </td> 
     2977  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2978  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2979  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2980  <p class=MsoBodyText>Å</p> 
     2981  </td> 
     2982  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2983  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2984  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2985  <p class=MsoBodyText>20</p> 
     2986  </td> 
     2987 </tr> 
     2988 <tr style='height:18.8pt'> 
     2989  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     2990  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2991  <p class=MsoBodyText>length</p> 
     2992  </td> 
     2993  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2994  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     2995  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     2996  <p class=MsoBodyText>Å</p> 
     2997  </td> 
     2998  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     2999  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3000  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3001  <p class=MsoBodyText>1000</p> 
     3002  </td> 
     3003 </tr> 
     3004 <tr style='height:18.8pt'> 
     3005  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3006  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3007  <p class=MsoBodyText>contrast</p> 
     3008  </td> 
     3009  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3010  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3011  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3012  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3013  </td> 
     3014  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3015  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3016  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3017  <p class=MsoBodyText>5e-06</p> 
     3018  </td> 
     3019 </tr> 
     3020 <tr style='height:18.8pt'> 
     3021  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3022  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3023  <p class=MsoBodyText>background</p> 
     3024  </td> 
     3025  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3026  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3027  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3028  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     3029  </td> 
     3030  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3031  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3032  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3033  <p class=MsoBodyText>0.01</p> 
     3034  </td> 
     3035 </tr> 
     3036 <tr style='height:18.8pt'> 
     3037  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3038  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3039  <p class=MsoBodyText>kuhn_length</p> 
     3040  </td> 
     3041  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3042  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3043  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3044  <p class=MsoBodyText>Å</p> 
     3045  </td> 
     3046  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3047  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3048  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3049  <p class=MsoBodyText>100</p> 
     3050  </td> 
     3051 </tr> 
     3052</table> 
     3053 
     3054</div> 
     3055 
     3056<p class=MsoNormal>&nbsp;</p> 
     3057 
     3058<p class=MsoNormal>&nbsp;</p> 
     3059 
     3060<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3061width=434 height=322 id="Picture 228" src="../images/html/image034.jpg"></p> 
     3062 
     3063<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     30641D plot using the default values (w/1000 data point).</b></p> 
     3065 
     3066<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3067 
     3068<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3069 
     3070<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3071 
     3072<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     3073c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     3074 
     3075<p class=MsoNormal>From the reference, &quot;Method 3 With Excluded 
     3076Volume&quot; is used. The model is a parametrization of simulations of a 
     3077discrete representation of the worm-like chain model of Kratky and Porod 
     3078applied in the pseudocontinuous limit.  See equations (13,26-27) in the 
     3079original reference for the details.</p> 
     3080 
     3081<p class=MsoNormal>&nbsp;</p> 
     3082 
     3083<p class=MsoBodyText>REFERENCE</p> 
     3084 
     3085<p class=MsoNormal>Pedersen, J. S. and P. Schurtenberger (1996). “Scattering 
     3086functions of semiflexible polymers with and without excluded volume effects.” 
     3087Macromolecules 29: 7602-7612. </p> 
     3088 
     3089<p class=MsoNormal>Correction of the formula can be found in:</p> 
     3090 
     3091<p class=MsoNormal>Wei-Ren Chen, Paul D. Butler, and Linda J. Magid, 
     3092&quot;Incorporating Intermicellar Interactions in the Fitting of SANS Data from 
     3093Cationic Wormlike Micelles&quot; Langmuir, August 2006.</p> 
     3094 
     3095<p class=MsoNormal>&nbsp;</p> 
     3096 
     3097<p class=MsoNormal>&nbsp;</p> 
     3098 
     3099<p class=MsoNormal><b>&nbsp;</b></p> 
     3100 
     3101<p class=MsoNormal><b>&nbsp;</b></p> 
     3102 
     3103 
     3104<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     3105style='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; 
     3106</span></span></b><b><span style='font-size:14.0pt'><a name="StackedDisksModel"> StackedDisksModel </a></span></b></p> 
     3107 
     3108<p class=MsoNormal>&nbsp;</p> 
     3109 
     3110<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for stacked 
     3111discs (tactoids) with a core/layer structure where the form factor is 
     3112normalized by the volume of the cylinder.  Assuming the next neighbor distance 
     3113(d-spacing) in a stack of parallel discs obeys a Gaussian distribution, a 
     3114structure factor S(q) proposed by Kratky and Porod in 1949 is used in this 
     3115function. Note that the resolution smearing calculation uses 76 Gauss 
     3116quadrature points to properly smear the model since the function is HIGHLY 
     3117oscillatory, especially around the q-values that correspond to the repeat 
     3118distance of the layers.</p> 
     3119 
     3120<p class=MsoNormal>The 2D scattering intensity is the same as 1D, regardless of 
     3121the orientation of the <i>q</i> vector which is defined as<span 
     3122style='font-size:14.0pt;position:relative;top:8.0pt'><img border=0 width=103 
     3123height=33 src="../images/html/image010.png"></span><span 
     3124style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     3125 
     3126<p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     3127 
     3128<p class=MsoNormal>&nbsp;</p> 
     3129 
     3130<p class=MsoNormal align=center style='text-align:center'>         <img 
     3131border=0 width=214 height=93 id="Picture 36" 
     3132src="../images/html/image035.png"></p> 
     3133 
     3134<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3135 
     3136<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3137width=173 height=136 id="Picture 37" src="../images/html/image036.png"></p> 
     3138 
     3139<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3140 
     3141<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3142 
     3143<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3144 
     3145<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup> sr<sup>-1</sup>], 
     3146on absolute scale. </p> 
     3147 
     3148<p class=MsoNormal>The scattering intensity I(q) is:</p> 
     3149 
     3150<p class=MsoNormal>&nbsp;</p> 
     3151 
     3152<p class=MsoNormal align=center style='text-align:center'><span 
     3153style='position:relative;top:9.0pt'><img border=0 width=514 height=38 
     3154src="../images/html/image037.png"></span></p> 
     3155 
     3156<p class=MsoNormal align=center style='text-align:center'><span 
     3157style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3158 
     3159<p class=MsoNormal>where the contrast,</p> 
     3160 
     3161<p class=MsoNormal align=center style='text-align:center'><span 
     3162style='position:relative;top:6.0pt'><img border=0 width=115 height=24 
     3163src="../images/html/image038.png"></span></p> 
     3164 
     3165<p class=MsoNormal align=center style='text-align:center'><span 
     3166style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3167 
     3168<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3169 
     3170<p class=MsoNormal><span style='position:relative;top:16.0pt'>N is the number 
     3171of discs per unit volume, </span><span style='font-family:Symbol;position:relative; 
     3172top:16.0pt'>a</span><span style='position:relative;top:16.0pt'> is the angle 
     3173between the axis of the disc and q, and Vt and Vc are the total volume and the 
     3174core volume of a single disc, respectively.</span></p> 
     3175 
     3176<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3177 
     3178<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3179 
     3180<p class=MsoNormal align=center style='text-align:center'><span 
     3181style='position:relative;top:39.0pt'><img border=0 width=424 height=113 
     3182src="../images/html/image039.png"></span></p> 
     3183 
     3184<p class=MsoNormal align=center style='text-align:center'><span 
     3185style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3186 
     3187<p class=MsoNormal align=center style='text-align:center'><span 
     3188style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3189 
     3190<p class=MsoNormal>where d = thickness of the layer (layer_thick), 2h= core 
     3191thickness (core_thick),  and R = radius of the disc (radius).</p> 
     3192 
     3193<p class=MsoNormal>&nbsp;</p> 
     3194 
     3195<p class=MsoNormal>&nbsp;</p> 
     3196 
     3197<p class=MsoNormal align=center style='text-align:center'><span 
     3198style='position:relative;top:14.0pt'><img border=0 width=399 height=46 
     3199src="../images/html/image040.png"></span></p> 
     3200 
     3201<p class=MsoNormal align=center style='text-align:center'><span 
     3202style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3203 
     3204<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3205 
     3206<p class=MsoNormal><span style='position:relative;top:16.0pt'>where n = the 
     3207total number of the disc stacked (n_stacking), D=the next neighbor center to 
     3208cent distance (d-spacing), and </span><span style='font-family:Symbol; 
     3209position:relative;top:16.0pt'>s</span><sub><span style='position:relative; 
     3210top:16.0pt'>D</span></sub><span style='position:relative;top:16.0pt'> = the 
     3211Gaussian standard deviation of the d-spacing (sigma_d).</span></p> 
     3212 
     3213<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3214 
     3215<p class=MsoNormal>To provide easy access to the orientation of the stackeddisks, 
     3216we define the axis of the cylinder using two angles &#952; and <span 
     3217style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
     3218of the cylinder, those angles are defined on Figure 2 of CylinderModel.</p> 
     3219 
     3220<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3221 
     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'> </span>For P*S: 
     3225The 2<sup>nd</sup> virial coefficient of the solid cylinder is calculate based 
     3226on the (radius) and length = n_stacking*(core_thick +2*layer_thick) values, and 
     3227used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     3228 
     3229<p class=MsoNormal>&nbsp;</p> 
     3230 
     3231<p class=MsoNormal>&nbsp;</p> 
     3232 
     3233<p class=MsoNormal>&nbsp;</p> 
     3234 
     3235<div align=center> 
     3236 
     3237<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     3238 style='border-collapse:collapse;border:none'> 
     3239 <tr style='height:18.8pt'> 
     3240  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3241  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3242  <p class=MsoBodyText>Parameter name</p> 
     3243  </td> 
     3244  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3245  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3246  <p class=MsoBodyText>Units</p> 
     3247  </td> 
     3248  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3249  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3250  <p class=MsoBodyText>Default value</p> 
     3251  </td> 
     3252 </tr> 
     3253 <tr style='height:18.8pt'> 
     3254  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3255  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3256  <p class=MsoBodyText>background</p> 
     3257  </td> 
     3258  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3259  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3260  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3261  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     3262  </td> 
     3263  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3264  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3265  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3266  <p class=MsoBodyText>0.001</p> 
     3267  </td> 
     3268 </tr> 
     3269 <tr style='height:18.8pt'> 
     3270  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3271  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3272  <p class=MsoBodyText>core_sld</p> 
     3273  </td> 
     3274  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3275  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3276  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3277  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3278  </td> 
     3279  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3280  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3281  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3282  <p class=MsoBodyText>4e-006</p> 
     3283  </td> 
     3284 </tr> 
     3285 <tr style='height:18.8pt'> 
     3286  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3287  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3288  <p class=MsoBodyText>core_thick</p> 
     3289  </td> 
     3290  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3291  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3292  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3293  <p class=MsoBodyText>Å</p> 
     3294  </td> 
     3295  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3296  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3297  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3298  <p class=MsoBodyText>10</p> 
     3299  </td> 
     3300 </tr> 
     3301 <tr style='height:18.8pt'> 
     3302  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3303  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3304  <p class=MsoBodyText>layer_sld</p> 
     3305  </td> 
     3306  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3307  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3308  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3309  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3310  </td> 
     3311  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3312  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3313  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3314  <p class=MsoBodyText>0</p> 
     3315  </td> 
     3316 </tr> 
     3317 <tr style='height:18.8pt'> 
     3318  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3319  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3320  <p class=MsoBodyText>layer_thick</p> 
     3321  </td> 
     3322  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3323  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3324  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3325  <p class=MsoBodyText>Å</p> 
     3326  </td> 
     3327  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3328  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3329  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3330  <p class=MsoBodyText>15</p> 
     3331  </td> 
     3332 </tr> 
     3333 <tr style='height:18.8pt'> 
     3334  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3335  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3336  <p class=MsoBodyText>n_stacking</p> 
     3337  </td> 
     3338  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3339  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3340  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3341  <p class=MsoBodyText>&nbsp;</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'> 
     3346  <p class=MsoBodyText>1</p> 
     3347  </td> 
     3348 </tr> 
     3349 <tr style='height:18.8pt'> 
     3350  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3351  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3352  <p class=MsoBodyText>radius</p> 
     3353  </td> 
     3354  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3355  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3356  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3357  <p class=MsoBodyText>Å</p> 
     3358  </td> 
     3359  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3360  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3361  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3362  <p class=MsoBodyText>3e+003</p> 
     3363  </td> 
     3364 </tr> 
     3365 <tr style='height:18.8pt'> 
     3366  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3367  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3368  <p class=MsoBodyText>scale</p> 
     3369  </td> 
     3370  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3371  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3372  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3373  <p class=MsoBodyText>&nbsp;</p> 
     3374  </td> 
     3375  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3376  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3377  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3378  <p class=MsoBodyText>0.01</p> 
     3379  </td> 
     3380 </tr> 
     3381 <tr style='height:18.8pt'> 
     3382  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3383  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3384  <p class=MsoBodyText>sigma_d</p> 
     3385  </td> 
     3386  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3387  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3388  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3389  <p class=MsoBodyText>&nbsp;</p> 
     3390  </td> 
     3391  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3392  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3393  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3394  <p class=MsoBodyText>0</p> 
     3395  </td> 
     3396 </tr> 
     3397 <tr style='height:18.8pt'> 
     3398  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3399  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3400  <p class=MsoBodyText>solvent_sld</p> 
     3401  </td> 
     3402  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3403  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3404  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3405  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3406  </td> 
     3407  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3408  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3409  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3410  <p class=MsoBodyText>5e-006</p> 
     3411  </td> 
     3412 </tr> 
     3413</table> 
     3414 
     3415</div> 
     3416 
     3417<p class=MsoNormal>&nbsp;</p> 
     3418 
     3419<p class=MsoNormal>&nbsp;</p> 
     3420 
     3421<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3422width=451 height=334 id="Picture 233" src="../images/html/image041.png"></p> 
     3423 
     3424<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     34251D plot using the default values (w/1000 data point).</b></p> 
     3426 
     3427<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3428 
     3429<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3430 
     3431<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3432width=377 height=215 id="Picture 251" src="../images/html/image042.jpg" 
     3433alt=stackdiskangles.gif></p> 
     3434 
     3435<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
     3436the angles for oriented stackeddisks against the detector plane.</p> 
     3437 
     3438<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3439 
     3440<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3441 
     3442<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3443 
     3444<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     3445c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     3446 
     3447<p class=MsoNormal>&nbsp;</p> 
     3448 
     3449<p class=MsoBodyText>REFERENCE</p> 
     3450 
     3451<p class=MsoNormal>Guinier, A. and Fournet, G., &quot;Small-Angle Scattering of 
     3452X-Rays&quot;, John Wiley and Sons, New York, 1955.</p> 
     3453 
     3454<p class=MsoNormal>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949. </p> 
     3455 
     3456<p class=MsoNormal>Higgins, J.S. and Benoit, H.C., &quot;Polymers and Neutron 
     3457Scattering&quot;, Clarendon, Oxford, 1994.</p> 
     3458 
     3459<p class=MsoNormal><b>&nbsp;</b></p> 
     3460 
     3461<p class=MsoNormal><b>&nbsp;</b></p> 
     3462 
     3463<p class=MsoNormal><b>&nbsp;</b></p> 
     3464 
     3465<p class=MsoNormal>&nbsp;</p> 
     3466 
     3467<p class=MsoNormal><b>&nbsp;</b></p> 
     3468 
     3469<p class=MsoNormal><b>&nbsp;</b></p> 
     3470 
     3471<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     3472style='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; 
     3473</span></span></b><b><span style='font-size:14.0pt'><a name="ParallelepipedModel">ParallelepipedModel</span></b></p> 
     3474 
     3475<p class=MsoNormal>&nbsp;</p> 
     3476 
     3477<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a rectangular 
     3478cylinder (below) where the form factor is normalized by the volume of the 
     3479cylinder. P(q) = scale*&lt;f^2&gt;/V+background where the volume V= ABC and the 
     3480averaging &lt; &gt;  is applied over all orientation for 1D.  </p> 
     3481 
     3482<p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     3483 
     3484<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3485width=326 height=247 id="Picture 38" src="../images/html/image043.jpg"></p> 
     3486 
     3487<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3488 
     3489<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3490 
     3491<p class=MsoNormal>The side of the solid must be satisfied the condition of 
     3492A&lt;B&lt;C in order for the calculation to be correct: where A = short_a, 
     3493B=short_b, and C=long_c for the fitting parameters.</p> 
     3494 
     3495<p class=MsoNormal>By this definition, assuming </p> 
     3496 
     3497<p class=MsoNormal>a = A/B&lt;1; b=B/B=1; c=C/B&gt;1, the form factor,</p> 
     3498 
     3499<p class=MsoNormal>&nbsp;</p> 
     3500 
     3501<p class=MsoNormal align=center style='text-align:center'><span 
     3502style='position:relative;top:48.0pt'><img border=0 width=360 height=145 
     3503src="../images/html/image044.png"></span></p> 
     3504 
     3505<p class=MsoNormal align=center style='text-align:center'><span 
     3506style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3507 
     3508<p class=MsoNormal>The contrast is defined as</p> 
     3509 
     3510<p class=MsoNormal>&nbsp;</p> 
     3511 
     3512<p class=MsoNormal align=center style='text-align:center'><span 
     3513style='position:relative;top:6.0pt'><img border=0 width=117 height=24 
     3514src="../images/html/image045.png"></span></p> 
     3515 
     3516<p class=MsoNormal align=center style='text-align:center'><span 
     3517style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3518 
     3519<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3520 
     3521<p class=MsoNormal><span style='position:relative;top:16.0pt'>The scattering 
     3522intensity per unit volume is returned in the unit of [cm<sup>-1</sup>];  I(q) = 
     3523</span><span style='font-family:Symbol;position:relative;top:16.0pt'>f</span><span 
     3524style='position:relative;top:16.0pt'>P(q).</span></p> 
     3525 
     3526<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     3527 
     3528<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
     3529cylinder is calculate based on the averaged radius (= sqrt(short_a*short_b/pi)) 
     3530and length( = long_c) values, and used as the effective radius toward S(Q) when 
     3531P(Q)*S(Q) is applied. </p> 
     3532 
     3533<p class=MsoNormal>&nbsp;</p> 
     3534 
     3535<p class=MsoNormal>&nbsp;</p> 
     3536 
     3537<p class=MsoNormal>To provide easy access to the orientation of the parallelepiped, 
     3538we define the axis of the cylinder using two angles &#952; , <span 
     3539style='font-family:"Arial","sans-serif"'>&#966; </span>and<span 
     3540style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span 
     3541style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the 
     3542cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     3543are defined on Figure 2 of CylinderModel. The angle <span style='font-family: 
     3544Symbol'>Y </span>is the rotational angle around its own long_c axis against the 
     3545q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the short_b 
     3546axis is parallel to the x-axis of the detector.</p> 
     3547 
     3548<p class=MsoNormal>&nbsp;</p> 
     3549 
     3550<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3551width=352 height=264 id="Picture 243" src="../images/html/image046.jpg" 
     3552alt=pprangles.gif></p> 
     3553 
     3554<p class=MsoNormal align=center style='text-align:center'><b>Figure. Definition 
     3555of angels for 2D</b>.</p> 
     3556 
     3557<p class=MsoNormal align=center style='text-align:center'>&nbsp;</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'><img border=0 
     3562width=379 height=256 src="../images/html/image047.jpg" 
     3563alt=cylinderangles2.gif></p> 
     3564 
     3565<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
     3566the angles for oriented elliptical cylinders against the detector plane.</p> 
     3567 
     3568<p class=MsoNormal>&nbsp;</p> 
     3569 
     3570<p class=MsoNormal>&nbsp;</p> 
     3571 
     3572<p class=MsoNormal>&nbsp;</p> 
     3573 
     3574<div align=center> 
     3575 
     3576<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     3577 style='border-collapse:collapse;border:none'> 
     3578 <tr style='height:18.8pt'> 
     3579  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3580  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3581  <p class=MsoBodyText>Parameter name</p> 
     3582  </td> 
     3583  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3584  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3585  <p class=MsoBodyText>Units</p> 
     3586  </td> 
     3587  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3588  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3589  <p class=MsoBodyText>Default value</p> 
     3590  </td> 
     3591 </tr> 
     3592 <tr style='height:18.8pt'> 
     3593  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3594  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3595  <p class=MsoBodyText>background</p> 
     3596  </td> 
     3597  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3598  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3599  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3600  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     3601  </td> 
     3602  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3603  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3604  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3605  <p class=MsoBodyText>0.0</p> 
     3606  </td> 
     3607 </tr> 
     3608 <tr style='height:18.8pt'> 
     3609  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3610  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3611  <p class=MsoBodyText>contrast</p> 
     3612  </td> 
     3613  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3614  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3615  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3616  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3617  </td> 
     3618  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3619  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3620  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3621  <p class=MsoBodyText>5e-006</p> 
     3622  </td> 
     3623 </tr> 
     3624 <tr style='height:18.8pt'> 
     3625  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3626  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3627  <p class=MsoBodyText>long_c</p> 
     3628  </td> 
     3629  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3630  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3631  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3632  <p class=MsoBodyText>Å</p> 
     3633  </td> 
     3634  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3635  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3636  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3637  <p class=MsoBodyText>400</p> 
     3638  </td> 
     3639 </tr> 
     3640 <tr style='height:18.8pt'> 
     3641  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3642  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3643  <p class=MsoBodyText>short_a</p> 
     3644  </td> 
     3645  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3646  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3647  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3648  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     3649  </td> 
     3650  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3651  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3652  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3653  <p class=MsoBodyText>35</p> 
     3654  </td> 
     3655 </tr> 
     3656 <tr style='height:18.8pt'> 
     3657  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3658  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3659  <p class=MsoBodyText>short_b</p> 
     3660  </td> 
     3661  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3662  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3663  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3664  <p class=MsoBodyText>Å</p> 
     3665  </td> 
     3666  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3667  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3668  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3669  <p class=MsoBodyText>75</p> 
     3670  </td> 
     3671 </tr> 
     3672 <tr style='height:18.8pt'> 
     3673  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     3674  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3675  <p class=MsoBodyText>scale</p> 
     3676  </td> 
     3677  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3678  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3679  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3680  <p class=MsoBodyText>&nbsp;</p> 
     3681  </td> 
     3682  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     3683  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     3684  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     3685  <p class=MsoBodyText>1</p> 
     3686  </td> 
     3687 </tr> 
     3688</table> 
     3689 
     3690</div> 
     3691 
     3692<p class=MsoNormal>&nbsp;</p> 
     3693 
     3694<p class=MsoNormal>&nbsp;</p> 
     3695 
     3696<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3697width=431 height=332 id="Picture 492" src="../images/html/image048.jpg"></p> 
     3698 
     3699<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     37001D plot using the default values (w/1000 data point).</b></p> 
     3701 
     3702<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3703 
     3704<p class=MsoNormal>&nbsp;</p> 
     3705 
     3706<p class=MsoNormal>&nbsp;</p> 
     3707 
     3708<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span 
     3709style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3710</span></span><b>Validation of the parallelepiped 2D model</b></p> 
     3711 
     3712<p class=MsoNormal>&nbsp;</p> 
     3713 
     3714<p class=MsoNormal>Validation of our code was done by comparing the output of 
     3715the 1D calculation to the angular average of the output of 2 D calculation over 
     3716all possible angles. The Figure below shows the comparison where the solid dot 
     3717refers to averaged 2D while the line represents the result of 1D calculation (for 
     3718the averaging, 76, 180, 76 points are taken over the angles of theta, phi, and 
     3719psi respectively).</p> 
     3720 
     3721<p class=MsoNormal>&nbsp;</p> 
     3722 
     3723<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3724 
     3725<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3726width=481 height=299 id="Picture 693" src="../images/html/image049.png"></p> 
     3727 
     3728<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     3729Comparison between 1D and averaged 2D.</b></p> 
     3730 
     3731<p class=MsoNormal>&nbsp;</p> 
     3732 
     3733<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3734 
     3735<p class=MsoNormal>&nbsp;</p> 
     3736 
     3737<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3738 
     3739<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     3740c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     3741 
     3742<p class=MsoNormal>&nbsp;</p> 
     3743 
     3744<p class=MsoBodyText>REFERENCE</p> 
     3745 
     3746<p class=MsoNormal>Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 
     3747185-211.</p> 
     3748 
     3749<p class=MsoNormal>Equations (1), (13-14). (in German)</p> 
     3750 
     3751<p class=MsoNormal><b>&nbsp;</b></p> 
     3752 
     3753<p class=MsoNormal>&nbsp;</p> 
     3754 
     3755<p class=MsoNormal>&nbsp;</p> 
     3756 
     3757<p class=MsoNormal>&nbsp;</p> 
     3758 
     3759<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     3760style='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; 
     3761</span></span></b><b><span style='font-size:14.0pt'><a name="EllipticalCylinderModel">Elliptical Cylinder Model</span></b></p> 
    14373762 
    14383763<p class=MsoNormal>&nbsp;</p> 
    14393764 
    14403765<p class=MsoNormal>This function calculates the scattering from an oriented 
    1441 elliptical cylinder. The angles theta and phi define the orientation of the 
    1442 axis of the cylinder. The  angle psi is defined as the orientation of the major 
    1443 axis of the ellipse with respect to the vector Q. A gaussian poydispersity can 
    1444 be added to any (or all) of the orientation angles, and also for the minor 
    1445 radius and the ratio of the ellipse radii.</p> 
     3766elliptical cylinder. </p> 
     3767 
     3768<p class=MsoNormal>&nbsp;</p> 
     3769 
     3770<p class=MsoNormal><b>For 2D (orientated system):</b></p> 
     3771 
     3772<p class=MsoNormal>&nbsp;</p> 
     3773 
     3774<p class=MsoNormal>The angles theta and phi define the orientation of the axis 
     3775of the cylinder. The angle psi is defined as the orientation of the major axis 
     3776of the ellipse with respect to the vector Q. A gaussian poydispersity can be 
     3777added to any of the orientation angles, and also for the minor radius and the 
     3778ratio of the ellipse radii.</p> 
     3779 
     3780<p class=MsoNormal>&nbsp;</p> 
     3781 
     3782<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3783width=352 height=172 id="Picture 40" src="../images/html/image050.png"></p> 
     3784 
     3785<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3786 
     3787<p class=MsoNormal align=center style='text-align:center'><b>Figure. a= r_minor 
     3788and </b><b><span style='font-family:Symbol'>n</span>= r_ratio (i.e., r_major/r_minor).</b></p> 
     3789 
     3790<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    14463791 
    14473792<p class=MsoNormal>The function calculated is:</p> 
    14483793 
    1449 <p class=MsoNormal align=center style='text-align:center'><span 
    1450 style='position:relative;top:16.0pt'><img width=377 height=46 
    1451 src="../images/html/image023.gif"></span> </p> 
     3794<p class=MsoNormal>&nbsp;</p> 
     3795 
     3796<p class=MsoNormal align=center style='text-align:center'><span 
     3797style='position:relative;top:16.0pt'><img border=0 width=377 height=46 
     3798src="../images/html/image051.png"></span> </p> 
     3799 
     3800<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3801 
     3802<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    14523803 
    14533804<p class=MsoNormal>with the functions:</p> 
    14543805 
    1455 <p class=MsoNormal align=center style='text-align:center'><span 
    1456 style='position:relative;top:39.0pt'><img width=297 height=114 
    1457 src="../images/html/image024.gif"></span>          (13)</p> 
     3806<p class=MsoNormal>&nbsp;</p> 
     3807 
     3808<p class=MsoNormal align=center style='text-align:center'><span 
     3809style='position:relative;top:39.0pt'><img border=0 width=297 height=114 
     3810src="../images/html/image052.png"></span>          (13)</p> 
     3811 
     3812<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3813 
     3814<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3815 
     3816<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    14583817 
    14593818<p class=MsoNormal>and the angle psi is defined as the orientation of the major 
    … …  
    14623821<p class=MsoNormal>&nbsp;</p> 
    14633822 
    1464 <p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
    1465 and the parameters are the following:</p> 
     3823<p class=MsoNormal><b>For 1D (no preferred orientation):</b></p> 
     3824 
     3825<p class=MsoNormal>The form factor is averaged over all possible orientation 
     3826before normalized by the particle volume: P(q) = scale*&lt;f^2&gt;/V .</p> 
     3827 
     3828<p class=MsoNormal>&nbsp;</p> 
     3829 
     3830<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]. </p> 
     3831 
     3832<p class=MsoNormal>&nbsp;</p> 
     3833 
     3834<p class=MsoNormal>&nbsp;</p> 
     3835 
     3836<p class=MsoNormal>To provide easy access to the orientation of the elliptical, 
     3837we define the axis of the cylinder using two angles &#952; , <span 
     3838style='font-family:"Arial","sans-serif"'>&#966; </span>and<span 
     3839style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span 
     3840style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the 
     3841cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     3842are defined on Figure 2 of CylinderModel. The angle <span style='font-family: 
     3843Symbol'>Y </span>is the rotational angle around its own long_c axis against the 
     3844q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the r_minor 
     3845axis is parallel to the x-axis of the detector.</p> 
     3846 
     3847<p class=MsoNormal>&nbsp;</p> 
     3848 
     3849<p class=MsoNormal>All angle parameters are valid and given only for 2D calculation 
     3850(Oriented system).</p> 
     3851 
     3852<p class=MsoNormal>&nbsp;</p> 
     3853 
     3854<p class=MsoNormal>&nbsp;</p> 
     3855 
     3856<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3857width=370 height=277 id="Picture 239" src="../images/html/image053.jpg" 
     3858alt=ellcylinderangles.gif></p> 
     3859 
     3860<p class=MsoNormal align=center style='text-align:center'><b>Figure. Definition 
     3861of angels for 2D</b>.</p> 
     3862 
     3863<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3864 
     3865<p class=MsoNormal align=center style='text-align:center'><img border=0 
     3866width=379 height=256 src="../images/html/image047.jpg" 
     3867alt=cylinderangles2.gif></p> 
     3868 
     3869<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
     3870the angles for oriented elliptical cylinders against the detector plane.</p> 
     3871 
     3872<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     3873 
     3874<p class=MsoNormal><b>&nbsp;</b></p> 
     3875 
     3876<p class=MsoNormal><b>For P*S</b>: The 2<sup>nd</sup> virial coefficient of the 
     3877solid cylinder is calculate based on the averaged radius 
     3878(=sqrt(r_minor^2*r_ratio))  and length values, and used as the effective radius 
     3879toward S(Q) when P(Q)*S(Q) is applied. </p> 
     3880 
     3881<p class=MsoNormal>&nbsp;</p> 
    14663882 
    14673883<p class=MsoNormal>&nbsp;</p> 
    … …  
    16344050 
    16354051<p class=MsoNormal>&nbsp;</p> 
     4052 
     4053<p class=MsoNormal>&nbsp;</p> 
     4054 
     4055<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4056width=443 height=328 id="Picture 503" src="../images/html/image054.jpg"></p> 
     4057 
     4058<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     40591D plot using the default values (w/1000 data point).</b></p> 
     4060 
     4061<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4062 
     4063<p class=MsoNormal>&nbsp;</p> 
     4064 
     4065<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span 
     4066style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4067</span></span><b>Validation of the elliptical cylinder 2D model</b></p> 
     4068 
     4069<p class=MsoNormal>&nbsp;</p> 
     4070 
     4071<p class=MsoNormal>Validation of our code was done by comparing the output of 
     4072the 1D calculation to the angular average of the output of 2 D calculation over 
     4073all possible angles. The Figure below shows the comparison where the solid dot 
     4074refers to averaged 2D while the line represents the result of 1D calculation (for 
     40752D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and 
     4076psi respectively).</p> 
     4077 
     4078<p class=MsoNormal>&nbsp;</p> 
     4079 
     4080<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4081 
     4082<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4083width=448 height=278 id="Picture 692" src="../images/html/image055.png"></p> 
     4084 
     4085<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     4086Comparison between 1D and averaged 2D.</b></p> 
     4087 
     4088<p class=MsoNormal style='text-autospace:none'><b>&nbsp;</b></p> 
     4089 
     4090<p class=MsoNormal style='text-autospace:none'>In the 2D average, more binning 
     4091in the angle phi is necessary to get the proper result. The following figure 
     4092shows the results of the averaging by varying the number of bin over angles.</p> 
     4093 
     4094<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     4095 
     4096<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><img 
     4097border=0 width=409 height=303 id="Picture 690" 
     4098src="../images/html/image056.png"></p> 
     4099 
     4100<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     4101The intensities averaged from 2D over different number of points of binning of 
     4102angles.</b></p> 
     4103 
     4104<p class=MsoNormal align=center style='text-align:center;text-autospace:none'>&nbsp;</p> 
     4105 
     4106<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
    16364107 
    16374108<p class=MsoNormal>REFERENCE</p> 
    … …  
    16454116<p class=MsoNormal>&nbsp;</p> 
    16464117 
     4118<p class=MsoNormal>&nbsp;</p> 
     4119 
     4120<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     4121style='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; 
     4122</span></span></b><b><span style='font-size:14.0pt'><a name="EllipsoidModel">Ellipsoid Model</span></b></p> 
     4123 
     4124<p class=MsoNormal>&nbsp;</p> 
     4125 
     4126<p class=MsoNormal>This model provides the form factor for an ellipsoid 
     4127(ellipsoid of revolution) with uniform scattering length density. The form 
     4128factor is normalized by the particle volume.</p> 
     4129 
     4130<p class=MsoNormal>&nbsp;</p> 
     4131 
     4132<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>1.1.<span 
     4133style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p> 
     4134 
     4135<p class=MsoNormal>&nbsp;</p> 
     4136 
     4137<p class=MsoNormal>The output of the 2D scattering intensity function for 
     4138oriented ellipsoids is given by (Feigin, 1987):</p> 
     4139 
     4140<p class=MsoNormal>&nbsp;</p> 
     4141 
     4142<p class=MsoNormal>&nbsp;</p> 
     4143 
     4144<p class=MsoCaption align=center style='text-align:center'><span 
     4145style='position:relative;top:12.0pt'><img border=0 width=184 height=41 
     4146src="../images/html/image017.png"></span>   <span style='font-size:12.0pt; 
     4147font-weight:normal'>(8)</span></p> 
     4148 
     4149<p class=MsoNormal>&nbsp;</p> 
     4150 
     4151<p class=MsoCaption align=center style='text-align:center'><span 
     4152style='position:relative;top:15.0pt'><img border=0 width=371 height=47 
     4153src="../images/html/image057.png"></span>    <span style='font-size: 
     415412.0pt;font-weight:normal'>(9)</span></p> 
     4155 
     4156<p class=MsoNormal>&nbsp;</p> 
     4157 
     4158<p class=MsoCaption align=center style='text-align:center'><span 
     4159style='position:relative;top:6.0pt'><img border=0 width=272 height=29 
     4160src="../images/html/image058.png"></span>     <span style='font-size: 
     416112.0pt;font-weight:normal'>(10)</span></p> 
     4162 
     4163<p class=MsoNormal>&nbsp;</p> 
     4164 
     4165<p class=MsoNormal>&nbsp;</p> 
     4166 
     4167<p class=MsoNormal>&nbsp;</p> 
     4168 
     4169<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span> 
     4170is the angle between the axis of the ellipsoid and the q-vector, V is the 
     4171volume of the ellipsoid, R<sub>a</sub> is the radius along the rotation axis of 
     4172the ellipsoid, R<sub>b</sub> is the radius perpendicular to the rotation axis 
     4173of the ellipsoid and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i> 
     4174(contrast) is the scattering length density difference between the scatterer 
     4175and the solvent. </p> 
     4176 
     4177<p class=MsoNormal>&nbsp;</p> 
     4178 
     4179<p class=MsoNormal>To provide easy access to the orientation of the ellipsoid, 
     4180we define the rotation axis of the ellipsoid using two angles &#952; and <span 
     4181style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
     4182of the cylinder, those angles are defined on Figure 2. For the ellipsoid, 
     4183&#952; is the angle between the rotation axis and the z-axis. </p> 
     4184 
     4185<p class=MsoNormal>&nbsp;</p> 
     4186 
     4187<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
     4188ellipsoid is calculate based on the radius_a and radius_b values, and used as 
     4189the effective radius toward S(Q) when P(Q)*S(Q) is applied. </p> 
     4190 
     4191<p class=MsoNormal>&nbsp;</p> 
     4192 
     4193<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>] 
     4194and the parameters of the ellipsoid model are the following:</p> 
     4195 
     4196<p class=MsoNormal>&nbsp;</p> 
     4197 
     4198<p class=MsoNormal>&nbsp;</p> 
     4199 
     4200<div align=center> 
     4201 
     4202<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     4203 style='border-collapse:collapse;border:none'> 
     4204 <tr style='height:19.85pt'> 
     4205  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4206  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4207  <p class=MsoBodyText>Parameter name</p> 
     4208  </td> 
     4209  <td width=105 valign=top style='width:78.9pt;border:solid windowtext 1.0pt; 
     4210  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4211  <p class=MsoBodyText>Units</p> 
     4212  </td> 
     4213  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4214  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4215  <p class=MsoBodyText>Default value</p> 
     4216  </td> 
     4217 </tr> 
     4218 <tr style='height:19.85pt'> 
     4219  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4220  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4221  <p class=MsoBodyText>scale</p> 
     4222  </td> 
     4223  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4224  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4225  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4226  <p class=MsoBodyText>None</p> 
     4227  </td> 
     4228  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4229  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4230  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4231  <p class=MsoBodyText>1.0</p> 
     4232  </td> 
     4233 </tr> 
     4234 <tr style='height:19.85pt'> 
     4235  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4236  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4237  <p class=MsoBodyText>radius_a (polar)</p> 
     4238  </td> 
     4239  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4240  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4241  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4242  <p class=MsoBodyText>Å</p> 
     4243  </td> 
     4244  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4245  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4246  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4247  <p class=MsoBodyText>20.0</p> 
     4248  </td> 
     4249 </tr> 
     4250 <tr style='height:19.85pt'> 
     4251  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4252  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4253  <p class=MsoBodyText>radius_b (equatorial)</p> 
     4254  </td> 
     4255  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4256  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4257  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4258  <p class=MsoBodyText>Å</p> 
     4259  </td> 
     4260  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4261  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4262  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4263  <p class=MsoBodyText>400.0</p> 
     4264  </td> 
     4265 </tr> 
     4266 <tr style='height:19.85pt'> 
     4267  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4268  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4269  <p class=MsoBodyText>contrast</p> 
     4270  </td> 
     4271  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4272  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4273  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4274  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     4275  </td> 
     4276  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4277  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4278  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4279  <p class=MsoBodyText>3.0e-6</p> 
     4280  </td> 
     4281 </tr> 
     4282 <tr style='height:19.85pt'> 
     4283  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4284  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4285  <p class=MsoBodyText>background</p> 
     4286  </td> 
     4287  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4288  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4289  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4290  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     4291  </td> 
     4292  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4293  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4294  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4295  <p class=MsoBodyText>0.0</p> 
     4296  </td> 
     4297 </tr> 
     4298 <tr style='height:19.85pt'> 
     4299  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4300  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4301  <p class=MsoBodyText>axis_theta</p> 
     4302  </td> 
     4303  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4304  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4305  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4306  <p class=MsoBodyText>Radian</p> 
     4307  </td> 
     4308  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4309  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4310  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'> 
     4311  <p class=MsoBodyText>1.57</p> 
     4312  </td> 
     4313 </tr> 
     4314 <tr style='height:20.6pt'> 
     4315  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt; 
     4316  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
     4317  <p class=MsoBodyText>axis_phi</p> 
     4318  </td> 
     4319  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left: 
     4320  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4321  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
     4322  <p class=MsoBodyText>Radian</p> 
     4323  </td> 
     4324  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4325  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4326  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'> 
     4327  <p class=MsoBodyText>0.0</p> 
     4328  </td> 
     4329 </tr> 
     4330</table> 
     4331 
     4332</div> 
     4333 
     4334<p class=MsoNormal>&nbsp;</p> 
     4335 
     4336<p class=MsoNormal>The output of the 1D scattering intensity function for 
     4337randomly oriented ellipsoids is then given by equation 6.</p> 
     4338 
     4339<p class=MsoNormal>&nbsp;</p> 
     4340 
     4341<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not 
     4342used for the 1D output. Our implementation of the scattering kernel and the 1D 
     4343scattering intensity use the c-library from NIST.</p> 
     4344 
     4345<p class=MsoNormal>&nbsp;</p> 
     4346 
     4347<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4348width=396 height=297 src="../images/html/image059.jpg" 
     4349alt=ellipsoangles.gif></p> 
     4350 
     4351<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
     4352the angles for oriented ellipsoid against the detector plane.</p> 
     4353 
     4354<p class=MsoNormal>&nbsp;</p> 
     4355 
     4356<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4357 
     4358<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.<span 
     4359style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation 
     4360of the ellipsoid model</b></p> 
     4361 
     4362<p class=MsoNormal>&nbsp;</p> 
     4363 
     4364<p class=MsoNormal>Validation of our code was done by comparing the output of 
     4365the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 
     43665 shows a comparison of the 1D output of our model and the output of the NIST 
     4367software.</p> 
     4368 
     4369<p class=MsoNormal>&nbsp;</p> 
     4370 
     4371<p class=MsoNormal>Averaging over a distribution of orientation is done by 
     4372evaluating equation 7. Since we have no other software to compare the 
     4373implementation of the intensity for fully oriented ellipsoids, we can compare 
     4374the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span 
     4375style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 6 shows the result of such a cross-check.</p> 
     4376 
     4377<p class=MsoNormal align=center style='text-align:center'><b><i><span 
     4378style='color:#3366FF'>&nbsp;</span></i></b></p> 
     4379 
     4380<p class=MsoNormal>The discrepancy above q=0.3 Å<sup> -1</sup> is due to the 
     4381way the form factors are calculated in the c-library provided by NIST. A 
     4382numerical integration has to be performed to obtain P(q) for randomly oriented 
     4383particles (equation 6). The NIST software performs that integration with a 
     438476-point Gaussian quadrature rule, which will become imprecise at high q where 
     4385the amplitude varies quickly as a function of q. The DANSE result shown has been 
     4386obtained by summing over 501 equidistant points in <span style='font-family: 
     4387"Arial","sans-serif"'>&#945;</span>. Our result was found to be stable over the 
     4388range of q shown for a number of points higher than 500.</p> 
     4389 
     4390<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><b><img 
     4391border=0 width=469 height=258 id="Picture 16" 
     4392src="../images/html/image060.jpg" alt="ellipsoid_1D_validation"></b></p> 
     4393 
     4394<p class=MsoCaption><a name="_Ref173222904">Figure </a>5: Comparison of the 
     4395DANSE scattering intensity for an ellipsoid with the output of the NIST SANS 
     4396analysis software. The parameters were set to: Scale=1.0, Radius_a=20 Å, 
     4397Radius_b=400 Å, </p> 
     4398 
     4399<p class=MsoCaption>Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> 
     4400-1</sup>.</p> 
     4401 
     4402<p class=MsoNormal><b>&nbsp;</b></p> 
     4403 
     4404<p class=MsoNormal><b>&nbsp;</b></p> 
     4405 
     4406<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img 
     4407border=0 width=450 height=247 id="Picture 17" 
     4408src="../images/html/image061.jpg" alt="ellipsoid_2D_average"></p> 
     4409 
     4410<p class=MsoCaption><a name="_Ref173223004">Figure </a>6: Comparison of the 
     4411intensity for uniformly distributed ellipsoids calculated from our 2D model and 
     4412the intensity from the NIST SANS analysis software. The parameters used were: 
     4413Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and 
     4414Background=0.0 cm<sup> -1</sup>.</p> 
     4415 
     4416<p class=MsoCaption>&nbsp;</p> 
     4417 
     4418<p class=MsoNormal>&nbsp;</p> 
     4419 
     4420<p class=MsoNormal>&nbsp;</p> 
     4421 
     4422<p class=MsoNormal><b>&nbsp;</b></p> 
     4423 
     4424<p class=MsoNormal><b>&nbsp;</b></p> 
     4425 
     4426<p class=MsoNormal><b>&nbsp;</b></p> 
     4427 
     4428 
     4429 
     4430<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     4431style='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; 
     4432</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellEllipsoidModel"> CoreShellEllipsoidModel </a></span></b></p> 
     4433 
     4434<p class=MsoNormal>&nbsp;</p> 
     4435 
     4436<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a core 
     4437shell ellipsoid (below) where the form factor is normalized by the volume of 
     4438the 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> 
     4439and the averaging &lt; &gt;  is applied over all orientation for 1D.  </p> 
     4440 
     4441<p class=MsoNormal>&nbsp;</p> 
     4442 
     4443<p class=MsoNormal>&nbsp;</p> 
     4444 
     4445<p class=MsoNormal align=center style='text-align:center'>   <img border=0 
     4446width=246 height=131 id="Picture 41" src="../images/html/image062.png"></p> 
     4447 
     4448<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4449 
     4450<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4451 
     4452<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4453 
     4454<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
     4455absolute scale. </p> 
     4456 
     4457<p class=MsoNormal>The form factor calculated is:</p> 
     4458 
     4459<p class=MsoNormal>&nbsp;</p> 
     4460 
     4461<p class=MsoNormal align=center style='text-align:center'><span 
     4462style='position:relative;top:48.0pt'><img border=0 width=334 height=137 
     4463src="../images/html/image063.png"></span></p> 
     4464 
     4465<p class=MsoNormal align=center style='text-align:center'><span 
     4466style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4467 
     4468<p class=MsoNormal>&nbsp;</p> 
     4469 
     4470<p class=MsoNormal>&nbsp;</p> 
     4471 
     4472<p class=MsoNormal>To provide easy access to the orientation of the coreshell 
     4473ellipsoid, we define the axis of the solid ellipsoid using two angles &#952; , <span 
     4474style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case 
     4475of the cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     4476are defined on Figure 2 of CylinderModel. </p> 
     4477 
     4478<p class=MsoNormal>&nbsp;</p> 
     4479 
     4480<p class=MsoNormal>The contrast is defined as SLD(core) – SLD(shell) or 
     4481SLD(shell – solvent). In the parameters, equat_core = equatorial radius of the 
     4482core, polar_core = polar radius of the core, equat_shell = r<sub>min</sub> (or 
     4483equatorial radius of the shell), and polar_shell = = r<sub>maj</sub> (or polar 
     4484radius of the shell).</p> 
     4485 
     4486<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4487 
     4488<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
     4489ellipsoid is calculate based on the radius_a (= polar_shell) and radius_b (= equat_shell) 
     4490values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. 
     4491</p> 
     4492 
     4493<p class=MsoNormal><sub>&nbsp;</sub></p> 
     4494 
     4495<div align=center> 
     4496 
     4497<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     4498 style='border-collapse:collapse;border:none'> 
     4499 <tr style='height:18.8pt'> 
     4500  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4501  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4502  <p class=MsoBodyText>Parameter name</p> 
     4503  </td> 
     4504  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4505  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4506  <p class=MsoBodyText>Units</p> 
     4507  </td> 
     4508  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4509  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4510  <p class=MsoBodyText>Default value</p> 
     4511  </td> 
     4512 </tr> 
     4513 <tr style='height:18.8pt'> 
     4514  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4515  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4516  <p class=MsoBodyText>background</p> 
     4517  </td> 
     4518  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4519  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4520  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4521  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     4522  </td> 
     4523  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4524  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4525  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4526  <p class=MsoBodyText>0.001</p> 
     4527  </td> 
     4528 </tr> 
     4529 <tr style='height:18.8pt'> 
     4530  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4531  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4532  <p class=MsoBodyText>equat_core</p> 
     4533  </td> 
     4534  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4535  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4536  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4537  <p class=MsoBodyText>Å</p> 
     4538  </td> 
     4539  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4540  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4541  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4542  <p class=MsoBodyText>200</p> 
     4543  </td> 
     4544 </tr> 
     4545 <tr style='height:18.8pt'> 
     4546  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4547  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4548  <p class=MsoBodyText>equat_shell</p> 
     4549  </td> 
     4550  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4551  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4552  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4553  <p class=MsoBodyText>Å</p> 
     4554  </td> 
     4555  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4556  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4557  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4558  <p class=MsoBodyText>250</p> 
     4559  </td> 
     4560 </tr> 
     4561 <tr style='height:18.8pt'> 
     4562  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4563  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4564  <p class=MsoBodyText>sld_solvent</p> 
     4565  </td> 
     4566  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4567  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4568  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4569  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     4570  </td> 
     4571  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4572  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4573  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4574  <p class=MsoBodyText>6e-006</p> 
     4575  </td> 
     4576 </tr> 
     4577 <tr style='height:18.8pt'> 
     4578  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4579  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4580  <p class=MsoBodyText>ploar_shell</p> 
     4581  </td> 
     4582  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4583  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4584  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4585  <p class=MsoBodyText>Å</p> 
     4586  </td> 
     4587  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4588  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4589  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4590  <p class=MsoBodyText>30</p> 
     4591  </td> 
     4592 </tr> 
     4593 <tr style='height:18.8pt'> 
     4594  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4595  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4596  <p class=MsoBodyText>ploar_core</p> 
     4597  </td> 
     4598  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4599  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4600  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4601  <p class=MsoBodyText>Å</p> 
     4602  </td> 
     4603  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4604  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4605  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4606  <p class=MsoBodyText>20</p> 
     4607  </td> 
     4608 </tr> 
     4609 <tr style='height:18.8pt'> 
     4610  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4611  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4612  <p class=MsoBodyText>scale</p> 
     4613  </td> 
     4614  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4615  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4616  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4617  <p class=MsoBodyText>&nbsp;</p> 
     4618  </td> 
     4619  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4620  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4621  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4622  <p class=MsoBodyText>1</p> 
     4623  </td> 
     4624 </tr> 
     4625 <tr style='height:18.8pt'> 
     4626  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4627  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4628  <p class=MsoBodyText>contrast</p> 
     4629  </td> 
     4630  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4631  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4632  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4633  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     4634  </td> 
     4635  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4636  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4637  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4638  <p class=MsoBodyText>1e-006</p> 
     4639  </td> 
     4640 </tr> 
     4641</table> 
     4642 
     4643</div> 
     4644 
     4645<p class=MsoNormal>&nbsp;</p> 
     4646 
     4647<p class=MsoNormal>&nbsp;</p> 
     4648 
     4649<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4650width=426 height=333 id="Picture 526" src="../images/html/image064.jpg"></p> 
     4651 
     4652<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     46531D plot using the default values (w/1000 data point).</b></p> 
     4654 
     4655<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>&nbsp;</b></p> 
     4656 
     4657<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4658 
     4659<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4660width=396 height=297 id="Picture 245" src="../images/html/image059.jpg" 
     4661alt=ellipsoangles.gif></p> 
     4662 
     4663<p class=MsoNormal align=center style='text-align:center'>Figure. Examples of 
     4664the angles for oriented coreshellellipsoid against the detector plane where a 
     4665=polar axis.</p> 
     4666 
     4667<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4668 
     4669<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     4670c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     4671 
     4672<p class=MsoNormal>&nbsp;</p> 
     4673 
     4674<p class=MsoBodyText>REFERENCE</p> 
     4675 
     4676<p class=MsoNormal>Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.</p> 
     4677 
     4678<p class=MsoNormal>Berr, S.  J. Phys. Chem., 1987, 91, 4760.</p> 
     4679 
     4680<p class=MsoNormal>&nbsp;</p> 
     4681 
     4682<p class=MsoNormal>&nbsp;</p> 
     4683 
     4684<p class=MsoNormal><b>&nbsp;</b></p> 
     4685 
     4686<p class=MsoNormal><b>&nbsp;</b></p> 
     4687 
     4688<p class=MsoNormal><b>&nbsp;</b></p> 
     4689 
     4690<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     4691style='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; 
     4692</span></span></b><b><span style='font-size:14.0pt'><a name="TriaxialEllipsoidModel">TriaxialEllipsoidModel</span></b></p> 
     4693 
     4694<p class=MsoNormal>&nbsp;</p> 
     4695 
     4696<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for an 
     4697ellipsoid (below) where all three axes are of different lengths, i.e.,  R<sub>a</sub> 
     4698=&lt; R<sub>b</sub> =&lt; R<sub>c</sub> (Note that users should maintains this 
     4699inequality for the all calculations).  P(q) = scale*&lt;f^2&gt;/V+background 
     4700where the volume V= 4pi/3*R<sub>a</sub>*R<sub>b</sub>*R<sub>c</sub>, and the 
     4701averaging &lt; &gt;  is applied over all orientation for 1D.  </p> 
     4702 
     4703<p class=MsoNormal>&nbsp;</p> 
     4704 
     4705<p class=MsoNormal align=center style='text-align:center'>   <img border=0 
     4706width=261 height=157 id="Picture 42" src="../images/html/image065.jpg"></p> 
     4707 
     4708<p class=MsoNormal align=center style='text-align:center'>&nbsp;</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>The returned value is in units of [cm<sup>-1</sup>], on 
     4715absolute scale. </p> 
     4716 
     4717<p class=MsoNormal>The form factor calculated is:</p> 
     4718 
     4719<p class=MsoNormal>&nbsp;</p> 
     4720 
     4721<p class=MsoNormal align=center style='text-align:center'><span 
     4722style='position:relative;top:33.0pt'><img border=0 width=472 height=97 
     4723src="../images/html/image066.png"></span></p> 
     4724 
     4725<p class=MsoNormal align=center style='text-align:center'><span 
     4726style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4727 
     4728<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4729 
     4730<p class=MsoNormal>&nbsp;</p> 
     4731 
     4732<p class=MsoNormal>&nbsp;</p> 
     4733 
     4734<p class=MsoNormal>To provide easy access to the orientation of the triaxial 
     4735ellipsoid, we define the axis of the cylinder using the angles &#952; , <span 
     4736style='font-family:"Arial","sans-serif"'>&#966; </span>and<span 
     4737style='font-family:"Arial","sans-serif"'> </span><span style='font-family:Symbol'>Y</span><span 
     4738style='font-family:"Arial","sans-serif"'> </span>. Similarly to the case of the 
     4739cylinder, those angles, &#952;  and <span style='font-family:"Arial","sans-serif"'>&#966;,</span> 
     4740are defined on Figure 2 of CylinderModel. The angle <span style='font-family: 
     4741Symbol'>Y </span>is the rotational angle around its own semi_axisC axis against 
     4742the q plane. For example, <span style='font-family:Symbol'>Y </span>= 0 when the 
     4743semi_axisA axis is parallel to the x-axis of the detector.</p> 
     4744 
     4745<p class=MsoNormal>&nbsp;</p> 
     4746 
     4747<p class=MsoNormal>The radius of gyration for this system is R<sub>g</sub><sup>2</sup> 
     4748= (R<sub>a</sub><sup>2</sup>*R<sub>b</sub><sup>2</sup>*R<sub>c</sub><sup>2</sup>)/5.  
     4749The contrast is defined as SLD(ellipsoid) – SLD(solvent). In the parameters, semi_axisA 
     4750= R<sub>a</sub> (or minor equatorial radius), semi_axisB = R<sub>b</sub> (or 
     4751major equatorial radius), and semi_axisC = R<sub>c</sub> (or polar radius of 
     4752the ellipsoid).</p> 
     4753 
     4754<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4755 
     4756<p class=MsoNormal>For P*S: The 2<sup>nd</sup> virial coefficient of the solid 
     4757ellipsoid is calculate based on the radius_a (=semi_axisC) and radius_b (=sqrt(semi_axisA* 
     4758semi_axisB))  values, and used as the effective radius toward S(Q) when 
     4759P(Q)*S(Q) is applied. </p> 
     4760 
     4761<p class=MsoNormal><sub>&nbsp;</sub></p> 
     4762 
     4763<p class=MsoNormal><sub>&nbsp;</sub></p> 
     4764 
     4765<div align=center> 
     4766 
     4767<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     4768 style='border-collapse:collapse;border:none'> 
     4769 <tr style='height:18.8pt'> 
     4770  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4771  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4772  <p class=MsoBodyText>Parameter name</p> 
     4773  </td> 
     4774  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4775  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4776  <p class=MsoBodyText>Units</p> 
     4777  </td> 
     4778  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4779  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4780  <p class=MsoBodyText>Default value</p> 
     4781  </td> 
     4782 </tr> 
     4783 <tr style='height:18.8pt'> 
     4784  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4785  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4786  <p class=MsoBodyText>background</p> 
     4787  </td> 
     4788  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4789  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4790  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4791  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     4792  </td> 
     4793  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4794  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4795  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4796  <p class=MsoBodyText>0.0</p> 
     4797  </td> 
     4798 </tr> 
     4799 <tr style='height:18.8pt'> 
     4800  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4801  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4802  <p class=MsoBodyText>semi_axisA</p> 
     4803  </td> 
     4804  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4805  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4806  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4807  <p class=MsoBodyText>Å</p> 
     4808  </td> 
     4809  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4810  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4811  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4812  <p class=MsoBodyText>35</p> 
     4813  </td> 
     4814 </tr> 
     4815 <tr style='height:18.8pt'> 
     4816  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4817  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4818  <p class=MsoBodyText>semi_axisB</p> 
     4819  </td> 
     4820  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4821  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4822  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4823  <p class=MsoBodyText>Å</p> 
     4824  </td> 
     4825  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4826  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4827  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4828  <p class=MsoBodyText>100</p> 
     4829  </td> 
     4830 </tr> 
     4831 <tr style='height:18.8pt'> 
     4832  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4833  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4834  <p class=MsoBodyText>semi_axisC</p> 
     4835  </td> 
     4836  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4837  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4838  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4839  <p class=MsoBodyText>Å<sup> </sup></p> 
     4840  </td> 
     4841  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4842  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4843  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4844  <p class=MsoBodyText>400</p> 
     4845  </td> 
     4846 </tr> 
     4847 <tr style='height:18.8pt'> 
     4848  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4849  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4850  <p class=MsoBodyText>scale</p> 
     4851  </td> 
     4852  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4853  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4854  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4855  <p class=MsoBodyText>&nbsp;</p> 
     4856  </td> 
     4857  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4858  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4859  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4860  <p class=MsoBodyText>1</p> 
     4861  </td> 
     4862 </tr> 
     4863 <tr style='height:18.8pt'> 
     4864  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     4865  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4866  <p class=MsoBodyText>contrast</p> 
     4867  </td> 
     4868  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4869  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4870  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4871  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     4872  </td> 
     4873  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     4874  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     4875  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     4876  <p class=MsoBodyText>5e-006</p> 
     4877  </td> 
     4878 </tr> 
     4879</table> 
     4880 
     4881</div> 
     4882 
     4883<p class=MsoNormal>&nbsp;</p> 
     4884 
     4885<p class=MsoNormal>&nbsp;</p> 
     4886 
     4887<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4888width=439 height=341 id="Picture 545" src="../images/html/image067.png"></p> 
     4889 
     4890<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     48911D plot using the default values (w/1000 data point).</b></p> 
     4892 
     4893<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4894 
     4895<p class=MsoNormal>&nbsp;</p> 
     4896 
     4897<p class=MsoListParagraph style='margin-left:1.35in;text-indent:-.25in'><span 
     4898style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4899</span></span><b>Validation of the triaxialellipsoid 2D model</b></p> 
     4900 
     4901<p class=MsoNormal>&nbsp;</p> 
     4902 
     4903<p class=MsoNormal>Validation of our code was done by comparing the output of 
     4904the 1D calculation to the angular average of the output of 2 D calculation over 
     4905all possible angles. The Figure below shows the comparison where the solid dot 
     4906refers to averaged 2D while the line represents the result of 1D calculation (for 
     49072D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and 
     4908psi respectively).</p> 
     4909 
     4910<p class=MsoNormal>&nbsp;</p> 
     4911 
     4912<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4913 
     4914<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4915width=438 height=272 id="Picture 691" src="../images/html/image068.png"></p> 
     4916 
     4917<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     4918Comparison between 1D and averaged 2D.</b></p> 
     4919 
     4920<p class=MsoNormal>&nbsp;</p> 
     4921 
     4922<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     4923 
     4924<p class=MsoNormal align=center style='text-align:center'><img border=0 
     4925width=396 height=297 id="Picture 248" src="../images/html/image069.jpg" 
     4926alt=triellrangles.gif></p> 
     4927 
     4928<p class=MsoCaption align=center style='text-align:center'>Figure. Examples of 
     4929the angles for oriented ellipsoid against the detector plane.</p> 
     4930 
     4931<p class=MsoNormal>&nbsp;</p> 
     4932 
     4933<p class=MsoNormal>&nbsp;</p> 
     4934 
     4935<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     4936c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     4937 
     4938<p class=MsoNormal>&nbsp;</p> 
     4939 
     4940<p class=MsoBodyText>REFERENCE</p> 
     4941 
     4942<p class=MsoNormal>L. A. Feigin and D. I. Svergun “Structure Analysis by 
     4943Small-Angle X-Ray and Neutron Scattering”, Plenum, New York, 1987.</p> 
     4944 
     4945<p class=MsoNormal>&nbsp;</p> 
     4946 
     4947<p class=MsoNormal><b>&nbsp;</b></p> 
     4948 
     4949<p class=MsoNormal><b>&nbsp;</b></p> 
     4950 
     4951<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     4952style='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; 
     4953</span></span></b><b><span style='font-size:14.0pt'><a name="LamellarModel">LamellarModel</span></b></p> 
     4954 
     4955<p class=MsoNormal>&nbsp;</p> 
     4956 
     4957<p class=MsoNormal>This model provides the scattering intensity, I(<i>q</i>), 
     4958for a lyotropic lamellar phase where a uniform SLD and random distribution in 
     4959solution are assumed.  The ploydispersion in the bilayer thickness can be 
     4960applied from the GUI.</p> 
     4961 
     4962<p class=MsoNormal>&nbsp;</p> 
     4963 
     4964<p class=MsoNormal>The scattering intensity I(q) is:</p> 
     4965 
     4966<p class=MsoNormal>&nbsp;</p> 
     4967 
     4968<p class=MsoNormal align=center style='text-align:center'><span 
     4969style='position:relative;top:15.0pt'><img border=0 width=103 height=46 
     4970src="../images/html/image070.png"></span></p> 
     4971 
     4972<p class=MsoNormal>The form factor is,</p> 
     4973 
     4974<p class=MsoNormal>&nbsp;</p> 
     4975 
     4976<p class=MsoNormal align=center style='text-align:center'><span 
     4977style='position:relative;top:15.0pt'><img border=0 width=174 height=48 
     4978src="../images/html/image071.png"></span></p> 
     4979 
     4980<p class=MsoNormal align=center style='text-align:center'><span 
     4981style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     4982 
     4983<p class=MsoNormal>where <span style='font-family:Symbol'>d</span> = bilayer 
     4984thickness.</p> 
     4985 
     4986<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
     49871D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
     4988position:relative;top:8.0pt'><img border=0 width=103 height=33 
     4989src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
     4990position:relative;top:6.0pt'>.</span></p> 
     4991 
     4992<p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     4993 
     4994<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
     4995absolute scale. In the parameters, sld_bi = SLD of the bilayer, sld_sol = SLD 
     4996of the solvent, and bi_thick = the thickness of the bilayer.</p> 
     4997 
     4998<p class=MsoNormal>&nbsp;</p> 
     4999 
     5000<p class=MsoNormal><sub>&nbsp;</sub></p> 
     5001 
     5002<div align=center> 
     5003 
     5004<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     5005 style='border-collapse:collapse;border:none'> 
     5006 <tr style='height:18.8pt'> 
     5007  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5008  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5009  <p class=MsoBodyText>Parameter name</p> 
     5010  </td> 
     5011  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5012  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5013  <p class=MsoBodyText>Units</p> 
     5014  </td> 
     5015  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5016  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5017  <p class=MsoBodyText>Default value</p> 
     5018  </td> 
     5019 </tr> 
     5020 <tr style='height:18.8pt'> 
     5021  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5022  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5023  <p class=MsoBodyText>background</p> 
     5024  </td> 
     5025  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5026  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5027  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5028  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     5029  </td> 
     5030  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5031  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5032  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5033  <p class=MsoBodyText>0.0</p> 
     5034  </td> 
     5035 </tr> 
     5036 <tr style='height:18.8pt'> 
     5037  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5038  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5039  <p class=MsoBodyText>sld_bi</p> 
     5040  </td> 
     5041  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5042  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5043  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5044  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5045  </td> 
     5046  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5047  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5048  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5049  <p class=MsoBodyText>1e-006</p> 
     5050  </td> 
     5051 </tr> 
     5052 <tr style='height:18.8pt'> 
     5053  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5054  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5055  <p class=MsoBodyText>bi_thick</p> 
     5056  </td> 
     5057  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5058  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5059  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5060  <p class=MsoBodyText>Å</p> 
     5061  </td> 
     5062  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5063  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5064  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5065  <p class=MsoBodyText>50</p> 
     5066  </td> 
     5067 </tr> 
     5068 <tr style='height:18.8pt'> 
     5069  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5070  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5071  <p class=MsoBodyText>sld_sol</p> 
     5072  </td> 
     5073  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5074  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5075  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5076  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5077  </td> 
     5078  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5079  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5080  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5081  <p class=MsoBodyText>6e-006</p> 
     5082  </td> 
     5083 </tr> 
     5084 <tr style='height:18.8pt'> 
     5085  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5086  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5087  <p class=MsoBodyText>scale</p> 
     5088  </td> 
     5089  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5090  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5091  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5092  <p class=MsoBodyText>&nbsp;</p> 
     5093  </td> 
     5094  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5095  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5096  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5097  <p class=MsoBodyText>1</p> 
     5098  </td> 
     5099 </tr> 
     5100</table> 
     5101 
     5102</div> 
     5103 
     5104<p class=MsoNormal>&nbsp;</p> 
     5105 
     5106<p class=MsoNormal>&nbsp;</p> 
     5107 
     5108<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5109width=476 height=351 id="Picture 571" src="../images/html/image072.png"></p> 
     5110 
     5111<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     51121D plot using the default values (w/1000 data point).</b></p> 
     5113 
     5114<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5115 
     5116<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5117 
     5118<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     5119c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     5120 
     5121<p class=MsoNormal>&nbsp;</p> 
     5122 
     5123<p class=MsoBodyText>REFERENCE</p> 
     5124 
     5125<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 
     5126487-502.</p> 
     5127 
     5128<p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
     512911081-11088.</p> 
     5130 
     5131<p class=MsoNormal>&nbsp;</p> 
     5132 
     5133<p class=MsoNormal>&nbsp;</p> 
     5134 
     5135<p class=MsoNormal><b>&nbsp;</b></p> 
     5136 
     5137<p class=MsoNormal><b>&nbsp;</b></p> 
     5138 
     5139 
     5140 
     5141<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     5142style='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; 
     5143</span></span></b><b><span style='font-size:14.0pt'><a name="LamellarFFHGModel">LamellarFFHGModel</span></b></p> 
     5144 
     5145<p class=MsoNormal>&nbsp;</p> 
     5146 
     5147<p class=MsoNormal>This model provides the scattering intensity, I(<i>q</i>), 
     5148for a lyotropic lamellar phase where a random distribution in solution are 
     5149assumed.  The SLD of the head region is taken to be different from the SLD of 
     5150the tail region.</p> 
     5151 
     5152<p class=MsoNormal>&nbsp;</p> 
     5153 
     5154<p class=MsoNormal>The scattering intensity I(q) is:</p> 
     5155 
     5156<p class=MsoNormal>&nbsp;</p> 
     5157 
     5158<p class=MsoNormal>&nbsp;</p> 
     5159 
     5160<p class=MsoNormal align=center style='text-align:center'><span 
     5161style='position:relative;top:15.0pt'><img border=0 width=151 height=46 
     5162src="../images/html/image073.png"></span></p> 
     5163 
     5164<p class=MsoNormal>The form factor is</p> 
     5165 
     5166<p class=MsoNormal>&nbsp;</p> 
     5167 
     5168<p class=MsoNormal>&nbsp;</p> 
     5169 
     5170<p class=MsoNormal align=center style='text-align:center'><span 
     5171style='position:relative;top:15.0pt'><img border=0 width=174 height=48 
     5172src="../images/html/image071.png"></span></p> 
     5173 
     5174<p class=MsoNormal><span style='position:relative;top:16.0pt'>And the structure 
     5175is </span></p> 
     5176 
     5177<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5178 
     5179<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5180 
     5181<p class=MsoNormal align=center style='text-align:center'><span 
     5182style='position:relative;top:16.0pt'><img border=0 width=334 height=51 
     5183src="../images/html/image074.png"></span></p> 
     5184 
     5185<p class=MsoNormal align=center style='text-align:center'><span 
     5186style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5187 
     5188<p class=MsoNormal><span style='position:relative;top:16.0pt'>where </span></p> 
     5189 
     5190<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5191 
     5192<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5193 
     5194<p class=MsoNormal align=center style='text-align:center'><span 
     5195style='position:relative;top:50.0pt'><img border=0 width=252 height=133 
     5196src="../images/html/image075.png"></span></p> 
     5197 
     5198<p class=MsoNormal align=center style='text-align:center'><span 
     5199style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5200 
     5201<p class=MsoNormal align=center style='text-align:center'><span 
     5202style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5203 
     5204<p class=MsoNormal>Here d= (repeat) spacing, <span style='font-family:Symbol'>d</span> 
     5205= bilayer thickness, the contrast <span style='font-family:Symbol'>Dr</span> = 
     5206SLD (headgroup) - SLD(solvent), K=smectic bending elasticity, B=compression 
     5207modulus, and N = number of lamellar plates (n_plates).</p> 
     5208 
     5209<p class=MsoNormal>Note: When the Caille parameter is greater than 
     5210approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due 
     5211to the complication of the model function, users are responsible to make sure 
     5212that all the assumptions are handled accurately: see the original reference 
     5213(below) for more details.</p> 
     5214 
     5215<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
     52161D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
     5217position:relative;top:8.0pt'><img border=0 width=103 height=33 
     5218src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
     5219position:relative;top:6.0pt'>.</span></p> 
     5220 
     5221<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
     5222absolute scale. </p> 
     5223 
     5224<p class=MsoNormal>&nbsp;</p> 
     5225 
     5226<p class=MsoNormal><sub>&nbsp;</sub></p> 
     5227 
     5228<div align=center> 
     5229 
     5230<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     5231 style='border-collapse:collapse;border:none'> 
     5232 <tr style='height:18.8pt'> 
     5233  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5234  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5235  <p class=MsoBodyText>Parameter name</p> 
     5236  </td> 
     5237  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5238  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5239  <p class=MsoBodyText>Units</p> 
     5240  </td> 
     5241  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5242  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5243  <p class=MsoBodyText>Default value</p> 
     5244  </td> 
     5245 </tr> 
     5246 <tr style='height:18.8pt'> 
     5247  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5248  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5249  <p class=MsoBodyText>background</p> 
     5250  </td> 
     5251  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5252  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5253  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5254  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     5255  </td> 
     5256  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5257  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5258  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5259  <p class=MsoBodyText>0.0</p> 
     5260  </td> 
     5261 </tr> 
     5262 <tr style='height:18.8pt'> 
     5263  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5264  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5265  <p class=MsoBodyText>contrast</p> 
     5266  </td> 
     5267  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5268  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5269  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5270  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5271  </td> 
     5272  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5273  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5274  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5275  <p class=MsoBodyText>5e-006</p> 
     5276  </td> 
     5277 </tr> 
     5278 <tr style='height:18.8pt'> 
     5279  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5280  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5281  <p class=MsoBodyText>scale</p> 
     5282  </td> 
     5283  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5284  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5285  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5286  <p class=MsoBodyText>&nbsp;</p> 
     5287  </td> 
     5288  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5289  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5290  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5291  <p class=MsoBodyText>1</p> 
     5292  </td> 
     5293 </tr> 
     5294 <tr style='height:18.8pt'> 
     5295  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5296  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5297  <p class=MsoBodyText>delta</p> 
     5298  </td> 
     5299  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5300  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5301  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5302  <p class=MsoBodyText>Å</p> 
     5303  </td> 
     5304  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5305  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5306  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5307  <p class=MsoBodyText>30</p> 
     5308  </td> 
     5309 </tr> 
     5310 <tr style='height:18.8pt'> 
     5311  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5312  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5313  <p class=MsoBodyText>n_plates</p> 
     5314  </td> 
     5315  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5316  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5317  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5318  <p class=MsoBodyText>&nbsp;</p> 
     5319  </td> 
     5320  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5321  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5322  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5323  <p class=MsoBodyText>20</p> 
     5324  </td> 
     5325 </tr> 
     5326 <tr style='height:18.8pt'> 
     5327  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5328  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5329  <p class=MsoBodyText>spacing</p> 
     5330  </td> 
     5331  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5332  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5333  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5334  <p class=MsoBodyText>Å</p> 
     5335  </td> 
     5336  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5337  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5338  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5339  <p class=MsoBodyText>400</p> 
     5340  </td> 
     5341 </tr> 
     5342 <tr style='height:18.8pt'> 
     5343  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5344  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5345  <p class=MsoBodyText>caille</p> 
     5346  </td> 
     5347  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5348  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5349  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5350  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5351  </td> 
     5352  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5353  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5354  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5355  <p class=MsoBodyText>0.1</p> 
     5356  </td> 
     5357 </tr> 
     5358</table> 
     5359 
     5360</div> 
     5361 
     5362<p class=MsoNormal>&nbsp;</p> 
     5363 
     5364<p class=MsoNormal>&nbsp;</p> 
     5365 
     5366<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5367width=439 height=348 id="Picture 659" src="../images/html/image076.jpg"></p> 
     5368 
     5369<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     53701D plot using the default values (w/6000 data point).</b></p> 
     5371 
     5372<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5373 
     5374<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5375 
     5376<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     5377c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     5378 
     5379<p class=MsoNormal>&nbsp;</p> 
     5380 
     5381<p class=MsoBodyText>REFERENCE</p> 
     5382 
     5383<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 
     5384487-502.</p> 
     5385 
     5386<p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
     538711081-11088.</p> 
     5388 
     5389<p class=MsoNormal>&nbsp;</p> 
     5390 
     5391<p class=MsoNormal>&nbsp;</p> 
     5392 
     5393<p class=MsoNormal>&nbsp;</p> 
     5394 
     5395<p class=MsoNormal><b>&nbsp;</b></p> 
     5396 
     5397<p class=MsoNormal><b>&nbsp;</b></p> 
     5398 
     5399 
     5400<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     5401style='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; 
     5402</span></span></b><b><span style='font-size:14.0pt'><a name="LamellarPSModel">LamellarPSModel</span></b></p> 
     5403 
     5404<p class=MsoNormal>&nbsp;</p> 
     5405 
     5406<p class=MsoNormal>This model provides the scattering intensity (<b>form factor</b> 
     5407<b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase where 
     5408a random distribution in solution are assumed.  The SLD of the head region is 
     5409taken to be different from the SLD of the tail region.</p> 
     5410 
     5411<p class=MsoNormal>&nbsp;</p> 
     5412 
     5413<p class=MsoNormal>The scattering intensity I(q) is:</p> 
     5414 
     5415<p class=MsoNormal>&nbsp;</p> 
     5416 
     5417<p class=MsoNormal align=center style='text-align:center'><span 
     5418style='position:relative;top:15.0pt'><img border=0 width=133 height=46 
     5419src="../images/html/image077.png"></span></p> 
     5420 
     5421<p class=MsoNormal align=center style='text-align:center'><span 
     5422style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5423 
     5424<p class=MsoNormal>The form factor is,</p> 
     5425 
     5426<p class=MsoNormal>&nbsp;</p> 
     5427 
     5428<p class=MsoNormal align=center style='text-align:center'><span 
     5429style='position:relative;top:15.0pt'><img border=0 width=398 height=46 
     5430src="../images/html/image078.png"></span></p> 
     5431 
     5432<p class=MsoNormal align=center style='text-align:center'><span 
     5433style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5434 
     5435<p class=MsoNormal>where <span style='font-family:Symbol'>d<sub>T</sub></span> 
     5436= tail length (or t_length), <span style='font-family:Symbol'>d<sub>H</sub></span> 
     5437= heasd thickness (or h_thickness) , <span style='font-family:Symbol'>Dr</span><sub>H</sub> 
     5438= SLD (headgroup) - SLD(solvent), and <span style='font-family:Symbol'>Dr</span><sub>T</sub> 
     5439= SLD (tail) - SLD(headgroup).</p> 
     5440 
     5441<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
     54421D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
     5443position:relative;top:8.0pt'><img border=0 width=103 height=33 
     5444src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
     5445position:relative;top:6.0pt'>.</span></p> 
     5446 
     5447<p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     5448 
     5449<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
     5450absolute scale. In the parameters, sld_tail = SLD of the tail group, and 
     5451sld_head = SLD of the head group.</p> 
     5452 
     5453<p class=MsoNormal>&nbsp;</p> 
     5454 
     5455<p class=MsoNormal><sub>&nbsp;</sub></p> 
     5456 
     5457<div align=center> 
     5458 
     5459<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     5460 style='border-collapse:collapse;border:none'> 
     5461 <tr style='height:18.8pt'> 
     5462  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5463  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5464  <p class=MsoBodyText>Parameter name</p> 
     5465  </td> 
     5466  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5467  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5468  <p class=MsoBodyText>Units</p> 
     5469  </td> 
     5470  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5471  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5472  <p class=MsoBodyText>Default value</p> 
     5473  </td> 
     5474 </tr> 
     5475 <tr style='height:18.8pt'> 
     5476  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5477  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5478  <p class=MsoBodyText>background</p> 
     5479  </td> 
     5480  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5481  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5482  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5483  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     5484  </td> 
     5485  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5486  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5487  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5488  <p class=MsoBodyText>0.0</p> 
     5489  </td> 
     5490 </tr> 
     5491 <tr style='height:18.8pt'> 
     5492  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5493  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5494  <p class=MsoBodyText>sld_head</p> 
     5495  </td> 
     5496  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5497  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5498  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5499  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5500  </td> 
     5501  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5502  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5503  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5504  <p class=MsoBodyText>3e-006</p> 
     5505  </td> 
     5506 </tr> 
     5507 <tr style='height:18.8pt'> 
     5508  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5509  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5510  <p class=MsoBodyText>scale</p> 
     5511  </td> 
     5512  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5513  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5514  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5515  <p class=MsoBodyText>&nbsp;</p> 
     5516  </td> 
     5517  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5518  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5519  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5520  <p class=MsoBodyText>1</p> 
     5521  </td> 
     5522 </tr> 
     5523 <tr style='height:18.8pt'> 
     5524  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5525  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5526  <p class=MsoBodyText>sld_solvent</p> 
     5527  </td> 
     5528  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5529  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5530  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5531  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5532  </td> 
     5533  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5534  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5535  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5536  <p class=MsoBodyText>6e-006</p> 
     5537  </td> 
     5538 </tr> 
     5539 <tr style='height:18.8pt'> 
     5540  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5541  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5542  <p class=MsoBodyText>h_thickness</p> 
     5543  </td> 
     5544  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5545  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5546  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5547  <p class=MsoBodyText>Å</p> 
     5548  </td> 
     5549  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5550  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5551  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5552  <p class=MsoBodyText>10</p> 
     5553  </td> 
     5554 </tr> 
     5555 <tr style='height:18.8pt'> 
     5556  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5557  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5558  <p class=MsoBodyText>t_length</p> 
     5559  </td> 
     5560  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5561  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5562  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5563  <p class=MsoBodyText>Å</p> 
     5564  </td> 
     5565  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5566  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5567  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5568  <p class=MsoBodyText>15</p> 
     5569  </td> 
     5570 </tr> 
     5571 <tr style='height:18.8pt'> 
     5572  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5573  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5574  <p class=MsoBodyText>sld_tail</p> 
     5575  </td> 
     5576  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5577  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5578  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5579  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5580  </td> 
     5581  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5582  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5583  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5584  <p class=MsoBodyText>0</p> 
     5585  </td> 
     5586 </tr> 
     5587</table> 
     5588 
     5589</div> 
     5590 
     5591<p class=MsoNormal>&nbsp;</p> 
     5592 
     5593<p class=MsoNormal>&nbsp;</p> 
     5594 
     5595<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5596width=444 height=343 src="../images/html/image079.png"></p> 
     5597 
     5598<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     55991D plot using the default values (w/1000 data point).</b></p> 
     5600 
     5601<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5602 
     5603<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5604 
     5605<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     5606c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     5607 
     5608<p class=MsoNormal>&nbsp;</p> 
     5609 
     5610<p class=MsoBodyText>REFERENCE</p> 
     5611 
     5612<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 
     5613487-502.</p> 
     5614 
     5615<p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
     561611081-11088.</p> 
     5617 
     5618<p class=MsoNormal>&nbsp;</p> 
     5619 
     5620<p class=MsoNormal>&nbsp;</p> 
     5621 
     5622<p class=MsoNormal>&nbsp;</p> 
     5623 
     5624<p class=MsoNormal><b>&nbsp;</b></p> 
     5625 
     5626<p class=MsoNormal><b>&nbsp;</b></p> 
     5627 
     5628 
     5629<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     5630style='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; 
     5631</span></span></b><b><span style='font-size:14.0pt'><a name="LamellarPSHGModel">LamellarPSHGModel</span></b></p> 
     5632 
     5633<p class=MsoNormal>&nbsp;</p> 
     5634 
     5635<p class=MsoNormal>This model provides the scattering intensity (<b>form factor</b> 
     5636<b>*</b> <b>structure factor</b>), I(<i>q</i>), for a lyotropic lamellar phase where 
     5637a random distribution in solution are assumed.  The SLD of the head region is 
     5638taken to be different from the SLD of the tail region.</p> 
     5639 
     5640<p class=MsoNormal>&nbsp;</p> 
     5641 
     5642<p class=MsoNormal>The scattering intensity I(q) is:</p> 
     5643 
     5644<p class=MsoNormal>&nbsp;</p> 
     5645 
     5646<p class=MsoNormal align=center style='text-align:center'><span 
     5647style='position:relative;top:15.0pt'><img border=0 width=133 height=46 
     5648src="../images/html/image077.png"></span></p> 
     5649 
     5650<p class=MsoNormal align=center style='text-align:center'><span 
     5651style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5652 
     5653<p class=MsoNormal>The form factor is,</p> 
     5654 
     5655<p class=MsoNormal>&nbsp;</p> 
     5656 
     5657<p class=MsoNormal align=center style='text-align:center'><span 
     5658style='position:relative;top:15.0pt'><img border=0 width=398 height=46 
     5659src="../images/html/image078.png"></span></p> 
     5660 
     5661<p class=MsoNormal align=center style='text-align:center'><span 
     5662style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5663 
     5664<p class=MsoNormal>The structure factor is</p> 
     5665 
     5666<p class=MsoNormal>&nbsp;</p> 
     5667 
     5668<p class=MsoNormal align=center style='text-align:center'><span 
     5669style='position:relative;top:16.0pt'><img border=0 width=334 height=51 
     5670src="../images/html/image074.png"></span></p> 
     5671 
     5672<p class=MsoNormal align=center style='text-align:center'><span 
     5673style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5674 
     5675<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5676 
     5677<p class=MsoNormal><span style='position:relative;top:16.0pt'>where </span></p> 
     5678 
     5679<p class=MsoNormal><span style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5680 
     5681<p class=MsoNormal align=center style='text-align:center'><span 
     5682style='position:relative;top:50.0pt'><img border=0 width=252 height=133 
     5683src="../images/html/image075.png"></span></p> 
     5684 
     5685<p class=MsoNormal align=center style='text-align:center'><span 
     5686style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5687 
     5688<p class=MsoNormal align=center style='text-align:center'><span 
     5689style='position:relative;top:16.0pt'>&nbsp;</span></p> 
     5690 
     5691<p class=MsoNormal>&nbsp;</p> 
     5692 
     5693<p class=MsoNormal>where <span style='font-family:Symbol'>d<sub>T</sub></span> 
     5694= tail length (or t_length), <span style='font-family:Symbol'>d<sub>H</sub></span> 
     5695= heasd thickness (or h_thickness) , <span style='font-family:Symbol'>Dr</span><sub>H</sub> 
     5696= SLD (headgroup) - SLD(solvent), and <span style='font-family:Symbol'>Dr</span><sub>T</sub> 
     5697= SLD (tail) - SLD(headgroup). Here d= (repeat) spacing, K=smectic bending 
     5698elasticity, B=compression modulus, and N = number of lamellar plates 
     5699(n_plates).</p> 
     5700 
     5701<p class=MsoNormal>Note: When the Caille parameter is greater than 
     5702approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due 
     5703to the complication of the model function, users are responsible to make sure 
     5704that all the assumptions are handled accurately: see the original reference 
     5705(below) for more details.</p> 
     5706 
     5707<p class=MsoNormal>&nbsp;</p> 
     5708 
     5709<p class=MsoNormal>The 2D scattering intensity is calculated in the same way as 
     57101D, where the <i>q</i> vector is defined as<span style='font-size:14.0pt; 
     5711position:relative;top:8.0pt'><img border=0 width=103 height=33 
     5712src="../images/html/image010.png"></span><span style='font-size:14.0pt; 
     5713position:relative;top:6.0pt'>.</span></p> 
     5714 
     5715<p class=MsoNormal><span style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     5716 
     5717<p class=MsoNormal>The returned value is in units of [cm<sup>-1</sup>], on 
     5718absolute scale. In the parameters, sld_tail = SLD of the tail group, sld_head = 
     5719SLD of the head group, and sld_solvent = SLD of the solvent.</p> 
     5720 
     5721<p class=MsoNormal>&nbsp;</p> 
     5722 
     5723<p class=MsoNormal><sub>&nbsp;</sub></p> 
     5724 
     5725<div align=center> 
     5726 
     5727<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     5728 style='border-collapse:collapse;border:none'> 
     5729 <tr style='height:18.8pt'> 
     5730  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5731  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5732  <p class=MsoBodyText>Parameter name</p> 
     5733  </td> 
     5734  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5735  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5736  <p class=MsoBodyText>Units</p> 
     5737  </td> 
     5738  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5739  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5740  <p class=MsoBodyText>Default value</p> 
     5741  </td> 
     5742 </tr> 
     5743 <tr style='height:18.8pt'> 
     5744  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5745  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5746  <p class=MsoBodyText>background</p> 
     5747  </td> 
     5748  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5749  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5750  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5751  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     5752  </td> 
     5753  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5754  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5755  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5756  <p class=MsoBodyText>0.001</p> 
     5757  </td> 
     5758 </tr> 
     5759 <tr style='height:18.8pt'> 
     5760  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5761  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5762  <p class=MsoBodyText>sld_head</p> 
     5763  </td> 
     5764  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5765  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5766  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5767  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5768  </td> 
     5769  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5770  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5771  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5772  <p class=MsoBodyText>2e-006</p> 
     5773  </td> 
     5774 </tr> 
     5775 <tr style='height:18.8pt'> 
     5776  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5777  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5778  <p class=MsoBodyText>scale</p> 
     5779  </td> 
     5780  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5781  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5782  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5783  <p class=MsoBodyText>&nbsp;</p> 
     5784  </td> 
     5785  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5786  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5787  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5788  <p class=MsoBodyText>1</p> 
     5789  </td> 
     5790 </tr> 
     5791 <tr style='height:18.8pt'> 
     5792  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5793  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5794  <p class=MsoBodyText>sld_solvent</p> 
     5795  </td> 
     5796  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5797  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5798  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5799  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5800  </td> 
     5801  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5802  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5803  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5804  <p class=MsoBodyText>6e-006</p> 
     5805  </td> 
     5806 </tr> 
     5807 <tr style='height:18.8pt'> 
     5808  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5809  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5810  <p class=MsoBodyText>deltaH</p> 
     5811  </td> 
     5812  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5813  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5814  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5815  <p class=MsoBodyText>Å</p> 
     5816  </td> 
     5817  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5818  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5819  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5820  <p class=MsoBodyText>2</p> 
     5821  </td> 
     5822 </tr> 
     5823 <tr style='height:18.8pt'> 
     5824  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5825  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5826  <p class=MsoBodyText>deltaT</p> 
     5827  </td> 
     5828  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5829  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5830  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5831  <p class=MsoBodyText>Å</p> 
     5832  </td> 
     5833  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5834  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5835  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5836  <p class=MsoBodyText>10</p> 
     5837  </td> 
     5838 </tr> 
     5839 <tr style='height:18.8pt'> 
     5840  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5841  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5842  <p class=MsoBodyText>sld_tail</p> 
     5843  </td> 
     5844  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5845  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5846  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5847  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5848  </td> 
     5849  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5850  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5851  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5852  <p class=MsoBodyText>0</p> 
     5853  </td> 
     5854 </tr> 
     5855 <tr style='height:18.8pt'> 
     5856  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5857  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5858  <p class=MsoBodyText>n_plates</p> 
     5859  </td> 
     5860  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5861  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5862  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5863  <p class=MsoBodyText>&nbsp;</p> 
     5864  </td> 
     5865  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5866  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5867  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5868  <p class=MsoBodyText>30</p> 
     5869  </td> 
     5870 </tr> 
     5871 <tr style='height:18.8pt'> 
     5872  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5873  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5874  <p class=MsoBodyText>spacing</p> 
     5875  </td> 
     5876  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5877  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5878  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5879  <p class=MsoBodyText>Å</p> 
     5880  </td> 
     5881  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5882  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5883  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5884  <p class=MsoBodyText>40</p> 
     5885  </td> 
     5886 </tr> 
     5887 <tr style='height:18.8pt'> 
     5888  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     5889  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5890  <p class=MsoBodyText>caille</p> 
     5891  </td> 
     5892  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5893  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5894  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5895  <p class=MsoBodyText>Å<sup> -2</sup></p> 
     5896  </td> 
     5897  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     5898  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     5899  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'> 
     5900  <p class=MsoBodyText>0.001</p> 
     5901  </td> 
     5902 </tr> 
     5903</table> 
     5904 
     5905</div> 
     5906 
     5907<p class=MsoNormal>&nbsp;</p> 
     5908 
     5909<p class=MsoNormal>&nbsp;</p> 
     5910 
     5911<p class=MsoNormal align=center style='text-align:center'><img border=0 
     5912width=463 height=360 id="Picture 687" src="../images/html/image080.png"></p> 
     5913 
     5914<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     59151D plot using the default values (w/6000 data point).</b></p> 
     5916 
     5917<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5918 
     5919<p class=MsoNormal align=center style='text-align:center'>&nbsp;</p> 
     5920 
     5921<p class=MsoNormal>Our model uses the form factor calculations implemented in a 
     5922c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     5923 
     5924<p class=MsoNormal>&nbsp;</p> 
     5925 
     5926<p class=MsoBodyText>REFERENCE</p> 
     5927 
     5928<p class=MsoNormal>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 
     5929487-502.</p> 
     5930 
     5931<p class=MsoNormal>            also in J. Phys. Chem. B, 105, (2001) 
     593211081-11088.</p> 
     5933 
     5934<p class=MsoNormal>&nbsp;</p> 
     5935 
     5936<p class=MsoNormal>&nbsp;</p> 
     5937 
     5938<p class=MsoNormal>&nbsp;</p> 
     5939 
     5940<p class=MsoNormal>&nbsp;</p> 
     5941 
     5942<p class=MsoNormal>&nbsp;</p> 
     5943 
     5944<p class=MsoNormal>&nbsp;</p> 
     5945 
     5946<p class=MsoNormal>&nbsp;</p> 
     5947 
    16475948<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    1648 style='font-size:14.0pt'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; 
    1649 </span></span></b><b><span style='font-size:14.0pt'><a name="Shape-Independent">Shape-Independent</a></span></b></p> 
     5949style='font-size:16.0pt'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     5950</span></span></b><b><span style='font-size:16.0pt'><a name="Shape-Independent">Shape-Independent Models </a></span></b></p> 
    16505951 
    16515952<p class=MsoBodyText>&nbsp;</p> 
    … …  
    16585959<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    16595960style='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; 
    1660 </span></span></b><b><span style='font-size:14.0pt'> <a name="Debye">Debye (Model)</a></span></b></p> 
     5961</span></span></b><b><span style='font-size:14.0pt'> <a name="Debye">Debye (Model)</span></b></p> 
    16615962 
    16625963<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    16675968are included in the calculation. </p> 
    16685969 
     5970<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     5971 
    16695972<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    16705973 
    16715974<p class=MsoNormal align=center style='text-align:center'><span 
    1672 style='font-size:14.0pt'><sub><img width=188 height=76 
    1673 src="../images/html/image025.gif"></sub></span></p> 
     5975style='font-size:14.0pt;position:relative;top:27.0pt'><img border=0 width=188 
     5976height=76 src="../images/html/image081.png"></span></p> 
     5977 
     5978<p class=MsoNormal align=center style='text-align:center'><span 
     5979style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p> 
     5980 
     5981<p class=MsoNormal align=center style='text-align:center'><span 
     5982style='font-size:14.0pt'>&nbsp;</span></p> 
     5983 
     5984<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     5985defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     5986border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     5987style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
    16745988 
    16755989<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    … …  
    17416055  </td> 
    17426056 </tr> 
     6057 <tr style='height:19.25pt'> 
     6058  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     6059  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     6060  <p class=MsoBodyText>&nbsp;</p> 
     6061  </td> 
     6062  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     6063  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     6064  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     6065  <p class=MsoBodyText>&nbsp;</p> 
     6066  </td> 
     6067  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     6068  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     6069  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     6070  <p class=MsoBodyText>&nbsp;</p> 
     6071  </td> 
     6072 </tr> 
    17436073</table> 
    17446074 
    … …  
    17546084<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    17556085style='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; 
    1756 </span></span></b><b><span style='font-size:14.0pt'> <a name="Lorentz">(Ornstein-Zernicke) 
    1757 Lorentz (Model)</span></b></p> 
     6086</span></span></b><b><span style='font-size:14.0pt'> <a name="Lorentz">(Ornstein-Zernicke) Lorentz 
     6087(Model)</span></b></p> 
    17586088 
    17596089<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    17656095 
    17666096<p class=MsoNormal align=center style='text-align:center'><span 
    1767 style='font-size:14.0pt'><sub><img width=212 height=24 
    1768 src="../images/html/image026.gif"></sub></span></p> 
     6097style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=212 
     6098height=24 src="../images/html/image082.png"></span></p> 
    17696099 
    17706100<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    … …  
    17736103the screening length.</p> 
    17746104 
     6105<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p> 
     6106 
     6107<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     6108defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     6109border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     6110style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6111 
     6112<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p> 
     6113 
    17756114<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    17766115 
    … …  
    18496188<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    18506189 
     6190<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     6191 
     6192<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    18516193 
    18526194<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    18536195style='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; 
    1854 </span></span></b><b><span style='font-size:14.0pt'> <a name="DAB_Model">DAB(Debye-Anderson-Brumberger)_ 
    1855 Model)</a></span></b></p> 
     6196</span></span></b><b><span style='font-size:14.0pt'> <a name="DAB_Model">DAB (Debye-Anderson-Brumberger)_Model</span></b></p> 
     6197 
     6198<p class=MsoNormal style='margin-left:.55in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    18566199 
    18576200<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Calculates the 
    … …  
    18676210 
    18686211<p class=MsoNormal align=center style='text-align:center'><span 
    1869 style='font-size:14.0pt'><sub><img width=221 height=28 
    1870 src="../images/html/image027.gif"></sub></span></p> 
     6212style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=221 
     6213height=28 src="../images/html/image083.png"></span></p> 
     6214 
     6215<p class=MsoNormal align=center style='text-align:center'><span 
     6216style='font-size:14.0pt'>&nbsp;</span></p> 
    18716217 
    18726218<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    18756221<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The parameter 
    18766222L is referred to as the correlation length.</p> 
     6223 
     6224<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     6225 
     6226<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     6227defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     6228border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     6229style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6230 
     6231<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    18776232 
    18786233<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    19686323<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    19696324style='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; 
    1970 </span></span></b><b><span style='font-size:14.0pt'> <a name="Power_Law">Power_Law</a></span></b></p> 
     6325</span></span></b><b><span style='font-size:14.0pt'> <a name="Power_Law"> Power_Law </a></span></b></p> 
    19716326 
    19726327<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    19786333 
    19796334<p class=MsoNormal align=center style='text-align:center'><span 
    1980 style='font-size:14.0pt'><sub><img width=161 height=24 
    1981 src="../images/html/image028.gif"></sub></span></p> 
     6335style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=149 
     6336height=24 src="../images/html/image084.png"></span></p> 
     6337 
     6338<p class=MsoNormal align=center style='text-align:center'><span 
     6339style='font-size:14.0pt'>&nbsp;</span></p> 
    19826340 
    19836341<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    20646422<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    20656423 
     6424<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     6425 
    20666426<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    20676427style='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; 
    2068 </span></span></b><b><span style='font-size:14.0pt'><a name="Teubner Strey"> Teubner Strey (Model) </a></span></b></p> 
     6428</span></span></b><b><span style='font-size:14.0pt'> <a name="Teubner Strey">Teubner Strey (Model)</a></span></b></p> 
    20696429 
    20706430<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    20766436 
    20776437<p class=MsoNormal align=center style='text-align:center'><span 
    2078 style='font-size:14.0pt'><sub><img width=213 height=45 
    2079 src="../images/html/image029.gif"></sub></span></p> 
     6438style='font-size:14.0pt;position:relative;top:15.0pt'><img border=0 width=213 
     6439height=45 src="../images/html/image085.png"></span></p> 
    20806440 
    20816441<p class=MsoNormal align=center style='text-align:center'><span 
    … …  
    20846444<p class=MsoNormal align=center style='text-align:center'><span 
    20856445style='font-size:14.0pt'>&nbsp;</span></p> 
     6446 
     6447<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     6448defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     6449border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     6450style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6451 
     6452<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p> 
    20866453 
    20876454<div align=center> 
    … …  
    21876554<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    21886555 
     6556<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    21896557 
    21906558<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    21916559style='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; 
    2192 </span></span></b><b><span style='font-size:14.0pt'><a name="Number Density Fractal"> Number Density Fractal (Model)</a></span></b></p> 
     6560</span></span></b><b><span style='font-size:14.0pt'> <a name="Number Density Fractal">Number Density Fractal (Model)</a></span></b></p> 
    21936561 
    21946562<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    22006568<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    22016569 
    2202  
    2203 <p class=MsoNormal align=center style='text-align:center'><span 
    2204 style='font-size:14.0pt'><sub><img width=380 height=188 
    2205 src="../images/html/image030.gif"></sub></span></p> 
     6570<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     6571 
     6572<p class=MsoNormal align=center style='text-align:center'><span 
     6573style='font-size:14.0pt;position:relative;top:85.0pt'><img border=0 width=332 
     6574height=188 src="../images/html/image086.png"></span></p> 
     6575 
     6576<p class=MsoNormal align=center style='text-align:center'><span 
     6577style='font-size:14.0pt;position:relative;top:85.0pt'>&nbsp;</span></p> 
     6578 
     6579<p class=MsoNormal align=center style='text-align:center'><span 
     6580style='font-size:14.0pt'>&nbsp;</span></p> 
    22066581 
    22076582<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    22166591<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    22176592 
     6593<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     6594defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     6595border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     6596style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6597 
     6598<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     6599 
     6600<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     6601 
    22186602<p class=MsoNormal align=center style='text-align:center'><span 
    22196603style='font-size:14.0pt'>&nbsp;</span></p> 
    … …  
    22566640  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    22576641  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2258   <p class=MsoBodyText>Radius</p> 
     6642  <p class=MsoBodyText>radius</p> 
    22596643  </td> 
    22606644  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    22726656  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    22736657  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2274   <p class=MsoBodyText>Fractal_dim</p> 
     6658  <p class=MsoBodyText>fractal_dim</p> 
    22756659  </td> 
    22766660  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    22886672  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    22896673  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2290   <p class=MsoBodyText>Corr_length</p> 
     6674  <p class=MsoBodyText>corr_length</p> 
    22916675  </td> 
    22926676  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    23046688  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    23056689  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2306   <p class=MsoBodyText>Block_sld</p> 
     6690  <p class=MsoBodyText>block_sld</p> 
    23076691  </td> 
    23086692  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    23206704  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    23216705  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2322   <p class=MsoBodyText>Solvent_sld</p> 
     6706  <p class=MsoBodyText>solvent_sld</p> 
    23236707  </td> 
    23246708  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    23366720  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    23376721  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2338   <p class=MsoBodyText>Background</p> 
     6722  <p class=MsoBodyText>background</p> 
    23396723  </td> 
    23406724  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    23536737</div> 
    23546738 
     6739<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    23556740 
    23566741<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
    … …  
    23676752<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    23686753style='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; 
    2369 </span></span></b><b><span style='font-size:14.0pt'> <a name="BEPolyelectrolyte">BEPolyelectrolyte Model</a></span></b></p> 
     6754</span></span></b><b><span style='font-size:14.0pt'> <a name="BEPolyelectrolyte"> BEPolyelectrolyte Model</span></b></p> 
    23706755 
    23716756<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    23786763 
    23796764<p class=MsoNormal align=center style='text-align:center'><span 
    2380 style='font-size:14.0pt'><sub><img width=365 height=125 
    2381 src="../images/html/image031.gif"></sub></span></p> 
     6765style='font-size:14.0pt;position:relative;top:43.0pt'><img border=0 width=420 
     6766height=125 src="../images/html/image087.png"></span></p> 
     6767 
     6768<p class=MsoNormal align=center style='text-align:center'><span 
     6769style='font-size:14.0pt;position:relative;top:43.0pt'>&nbsp;</span></p> 
     6770 
     6771<p class=MsoNormal align=center style='text-align:center'><span 
     6772style='font-size:14.0pt'>&nbsp;</span></p> 
    23826773 
    23836774<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    … …  
    23876778the monomer length, C<sub>s</sub> is the concentration of monovalent salt, 
    23886779&#945; is the ionization degree, C<sub>a</sub> is the polymer molar 
    2389 concentration, and bck is the incoherent background.</p> 
     6780concentration, and background is the incoherent background.</p> 
     6781 
     6782<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     6783 
     6784<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     6785defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     6786border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     6787style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6788 
     6789<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    23906790 
    23916791<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    24926892  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    24936893  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2494   <p class=MsoBodyText>Alpha</p> 
     6894  <p class=MsoBodyText>alpha</p> 
    24956895  </td> 
    24966896  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    25246924  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    25256925  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2526   <p class=MsoBodyText>Background</p> 
     6926  <p class=MsoBodyText>background</p> 
    25276927  </td> 
    25286928  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    25656965<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    25666966style='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; 
    2567 </span></span></b><b><span style='font-size:14.0pt'><a name="Guinier"> Guinier (Model)</a></span></b></p> 
     6967</span></span></b><b><span style='font-size:14.0pt'> <a name="Guinier">Guinier (Model)</span></b></p> 
    25686968 
    25696969<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    25706970 
    2571 <p class=MsoNormal style='margin-left:.25in'>A Guinier analysis is done by linearizing 
    2572 the data at low q by plotting it as log(I) versus Q<sup>2</sup>. The Guinier 
    2573 radius R<sub>g</sub> can be obtained by fitting the following model:</p> 
     6971<p class=MsoNormal style='margin-left:.25in'>A Guinier analysis is done by 
     6972linearizing the data at low q by plotting it as log(I) versus Q<sup>2</sup>. 
     6973The Guinier radius R<sub>g</sub> can be obtained by fitting the following 
     6974model:</p> 
    25746975 
    25756976<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    25766977 
     6978<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     6979 
    25776980<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    2578 style='font-size:14.0pt'><sub><img width=183 height=28 
    2579 src="../images/html/image032.gif"></sub></span></p> 
     6981style='font-size:14.0pt;position:relative;top:7.0pt'><img border=0 width=183 
     6982height=28 src="../images/html/image088.png"></span></p> 
     6983 
     6984<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     6985style='font-size:14.0pt'>&nbsp;</span></p> 
    25806986 
    25816987<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
    25826988style='font-size:14.0pt'>&nbsp;</span></b></p> 
    25836989 
     6990<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     6991defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     6992border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     6993style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     6994 
     6995<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
     6996style='font-size:14.0pt'>&nbsp;</span></b></p> 
     6997 
    25846998<div align=center> 
    25856999 
    … …  
    26297043  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    26307044  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2631   <p class=MsoBodyText>60</p> 
     7045  <p class=MsoBodyText>0.1</p> 
    26327046  </td> 
    26337047 </tr> 
    … …  
    26377051 
    26387052<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    2639 <p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7053 
    26407054<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    26417055 
    26427056<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
    26437057style='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; 
    2644 </span></span></b><b><span style='font-size:14.0pt'> <a name="PorodModel">PorodModel</a></span></b></p> 
     7058</span></span></b><b><span style='font-size:14.0pt'> <a name="PorodModel">PorodModel</span></b></p> 
    26457059 
    26467060<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    26537067 
    26547068<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    2655 style='font-size:14.0pt'><sub><img width=368 height=25 
    2656 src="../images/html/image033.gif"></sub></span></p> 
     7069style='font-size:14.0pt;position:relative;top:6.0pt'><img border=0 width=368 
     7070height=25 src="../images/html/image089.png"></span></p> 
     7071 
     7072<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7073style='font-size:14.0pt'>&nbsp;</span></p> 
    26577074 
    26587075<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    … …  
    26637080is the contrast factor. </p> 
    26647081 
    2665 <p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
    2666  
    26677082<p class=MsoNormal style='margin-left:.25in'>The background term is added for 
    26687083data analysis.</p> 
    26697084 
     7085<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7086 
     7087<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7088defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     7089border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     7090style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7091 
     7092<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7093 
    26707094<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
    26717095style='font-size:14.0pt'>&nbsp;</span></b></p> 
    … …  
    26927116  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    26937117  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2694   <p class=MsoBodyText>Scale</p> 
     7118  <p class=MsoBodyText>scale</p> 
    26957119  </td> 
    26967120  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    27087132  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    27097133  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2710   <p class=MsoBodyText>Background</p> 
     7134  <p class=MsoBodyText>background</p> 
    27117135  </td> 
    27127136  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    27277151<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    27287152 
     7153<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>&nbsp;</p> 
     7154 
     7155<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     7156 
    27297157<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    27307158 
     7159 
     7160 
     7161<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     7162style='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; 
     7163</span></span></b><b><span style='font-size:14.0pt'><a name="Peak Gauss Model"> Peak Gauss Model</span></b></p> 
     7164 
    27317165<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    27327166 
    2733  
    2734  
     7167<p class=MsoNormal style='margin-left:.25in'>This Gaussian peak model with a 
     7168flat background calculates:</p> 
     7169 
     7170<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7171 
     7172<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7173style='font-size:14.0pt;position:relative;top:6.0pt'><img border=0 width=334 
     7174height=25 src="../images/html/image090.png"></span></p> 
     7175 
     7176<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7177style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7178 
     7179<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7180style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7181 
     7182<p class=MsoNormal style='margin-left:.25in'><span style='position:relative; 
     7183top:7.0pt'>where ‘scale’ is the peak height centered at q<sub>0</sub>, and B 
     7184refers to the standard deviation of the function (equivalently, the FWHM is 
     71852.54*B).</span></p> 
     7186 
     7187<p class=MsoNormal style='margin-left:.25in'>The background term is added for 
     7188data analysis. </p> 
     7189 
     7190<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7191 
     7192<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7193defined as <span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     7194border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     7195style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7196 
     7197<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7198 
     7199<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7200 
     7201<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
     7202style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7203 
     7204<div align=center> 
     7205 
     7206<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     7207 style='border-collapse:collapse;border:none'> 
     7208 <tr style='height:19.25pt'> 
     7209  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7210  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7211  <p class=MsoBodyText>Parameter name</p> 
     7212  </td> 
     7213  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7214  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7215  <p class=MsoBodyText>Units</p> 
     7216  </td> 
     7217  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7218  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7219  <p class=MsoBodyText>Default value</p> 
     7220  </td> 
     7221 </tr> 
     7222 <tr style='height:19.25pt'> 
     7223  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7224  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7225  <p class=MsoBodyText>scale</p> 
     7226  </td> 
     7227  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7228  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7229  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7230  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     7231  </td> 
     7232  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7233  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7234  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7235  <p class=MsoBodyText>100</p> 
     7236  </td> 
     7237 </tr> 
     7238 <tr style='height:19.25pt'> 
     7239  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7240  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7241  <p class=MsoBodyText>B</p> 
     7242  </td> 
     7243  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7244  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7245  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7246  <p class=MsoBodyText>Å<sup>-1</sup></p> 
     7247  </td> 
     7248  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7249  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7250  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7251  <p class=MsoBodyText>0.005</p> 
     7252  </td> 
     7253 </tr> 
     7254 <tr style='height:19.25pt'> 
     7255  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7256  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7257  <p class=MsoBodyText>q<sub>0</sub></p> 
     7258  </td> 
     7259  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7260  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7261  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7262  <p class=MsoBodyText>Å<sup>-1</sup></p> 
     7263  </td> 
     7264  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7265  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7266  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7267  <p class=MsoBodyText>0.05</p> 
     7268  </td> 
     7269 </tr> 
     7270 <tr style='height:19.25pt'> 
     7271  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7272  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7273  <p class=MsoBodyText>background</p> 
     7274  </td> 
     7275  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7276  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7277  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7278  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     7279  </td> 
     7280  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7281  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7282  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7283  <p class=MsoBodyText>1.0</p> 
     7284  </td> 
     7285 </tr> 
     7286</table> 
     7287 
     7288</div> 
     7289 
     7290<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7291 
     7292<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7293 
     7294<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7295 
     7296<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7297 
     7298<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     7299style='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; 
     7300</span></span></b><b><span style='font-size:14.0pt'> <a name="Peak Lorentz Model">Peak Lorentz Model</span></b></p> 
     7301 
     7302<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7303 
     7304<p class=MsoNormal style='margin-left:.25in'>This Lorentzean peak model with a 
     7305flat background calculates:</p> 
     7306 
     7307<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7308 
     7309<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7310style='font-size:14.0pt;position:relative;top:34.0pt'><img border=0 width=241 
     7311height=70 src="../images/html/image091.png"></span></p> 
     7312 
     7313<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7314style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     7315 
     7316<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7317style='font-size:14.0pt;position:relative;top:6.0pt'>&nbsp;</span></p> 
     7318 
     7319<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7320style='font-size:14.0pt'>&nbsp;</span></p> 
     7321 
     7322<p class=MsoNormal style='margin-left:.25in'><span style='position:relative; 
     7323top:7.0pt'>where ‘scale’ is the peak height centered at q<sub>0</sub>, and B 
     7324refers to the standard deviation of the function.</span></p> 
     7325 
     7326<p class=MsoNormal style='margin-left:.25in'>The background term is added for 
     7327data analysis. </p> 
     7328 
     7329<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7330 
     7331<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7332defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     7333border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     7334style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7335 
     7336<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span 
     7337style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7338 
     7339<div align=center> 
     7340 
     7341<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     7342 style='border-collapse:collapse;border:none'> 
     7343 <tr style='height:19.25pt'> 
     7344  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7345  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7346  <p class=MsoBodyText>Parameter name</p> 
     7347  </td> 
     7348  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7349  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7350  <p class=MsoBodyText>Units</p> 
     7351  </td> 
     7352  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7353  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7354  <p class=MsoBodyText>Default value</p> 
     7355  </td> 
     7356 </tr> 
     7357 <tr style='height:19.25pt'> 
     7358  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7359  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7360  <p class=MsoBodyText>scale</p> 
     7361  </td> 
     7362  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7363  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7364  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7365  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     7366  </td> 
     7367  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7368  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7369  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7370  <p class=MsoBodyText>100</p> 
     7371  </td> 
     7372 </tr> 
     7373 <tr style='height:19.25pt'> 
     7374  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7375  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7376  <p class=MsoBodyText>B</p> 
     7377  </td> 
     7378  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7379  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7380  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7381  <p class=MsoBodyText>Å<sup>-1</sup></p> 
     7382  </td> 
     7383  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7384  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7385  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7386  <p class=MsoBodyText>0.005</p> 
     7387  </td> 
     7388 </tr> 
     7389 <tr style='height:19.25pt'> 
     7390  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7391  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7392  <p class=MsoBodyText>q<sub>0</sub></p> 
     7393  </td> 
     7394  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7395  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7396  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7397  <p class=MsoBodyText>Å<sup>-1</sup></p> 
     7398  </td> 
     7399  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7400  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7401  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7402  <p class=MsoBodyText>0.05</p> 
     7403  </td> 
     7404 </tr> 
     7405 <tr style='height:19.25pt'> 
     7406  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7407  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7408  <p class=MsoBodyText>background</p> 
     7409  </td> 
     7410  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7411  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7412  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7413  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     7414  </td> 
     7415  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7416  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7417  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7418  <p class=MsoBodyText>1.0</p> 
     7419  </td> 
     7420 </tr> 
     7421</table> 
     7422 
     7423</div> 
     7424 
     7425<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>&nbsp;</p> 
     7426 
     7427<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p> 
     7428 
     7429<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7430 
     7431<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     7432style='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; 
     7433</span></span></b><b><span style='font-size:14.0pt'> <a name="LineModel"> LineModel</span></b></p> 
     7434 
     7435<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7436 
     7437<p class=MsoNormal style='margin-left:.25in'>This is a linear function that 
     7438calculates:</p> 
     7439 
     7440<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7441 
     7442<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7443style='font-size:14.0pt;position:relative;top:5.0pt'><img border=0 width=92 
     7444height=21 src="../images/html/image092.png"></span></p> 
     7445 
     7446<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7447style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7448 
     7449<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7450style='font-size:14.0pt'>&nbsp;</span></p> 
     7451 
     7452<p class=MsoNormal style='margin-left:.25in'>where A and B are the coefficients 
     7453of the first and second order terms.</p> 
     7454 
     7455<p class=MsoNormal style='margin-left:.25in'>Note: For 2D plot, I(q) = I(q<sub>x</sub>)*I(q<sub>y</sub>)  
     7456which is defined differently from other shape independent models.</p> 
     7457 
     7458<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p> 
     7459 
     7460<div align=center> 
     7461 
     7462<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0 
     7463 style='border-collapse:collapse;border:none'> 
     7464 <tr style='height:19.25pt'> 
     7465  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7466  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7467  <p class=MsoBodyText>Parameter name</p> 
     7468  </td> 
     7469  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7470  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7471  <p class=MsoBodyText>Units</p> 
     7472  </td> 
     7473  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7474  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7475  <p class=MsoBodyText>Default value</p> 
     7476  </td> 
     7477 </tr> 
     7478 <tr style='height:19.25pt'> 
     7479  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7480  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7481  <p class=MsoBodyText>A</p> 
     7482  </td> 
     7483  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7484  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7485  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7486  <p class=MsoBodyText>cm<sup>-1</sup></p> 
     7487  </td> 
     7488  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7489  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7490  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7491  <p class=MsoBodyText>1.0</p> 
     7492  </td> 
     7493 </tr> 
     7494 <tr style='height:19.25pt'> 
     7495  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7496  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7497  <p class=MsoBodyText>B</p> 
     7498  </td> 
     7499  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7500  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7501  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7502  <p class=MsoBodyText>Å</p> 
     7503  </td> 
     7504  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7505  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7506  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7507  <p class=MsoBodyText>1.0</p> 
     7508  </td> 
     7509 </tr> 
     7510</table> 
     7511 
     7512</div> 
     7513 
     7514<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7515 
     7516<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7517 
     7518<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    27357519 
    27367520<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span 
    2737 style='font-size:14.0pt'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; 
    2738 </span></span></b><b><span style='font-size:14.0pt'><a name="Customized Models">Customized Models</a></span></b></p> 
    2739  
    2740 <p class=MsoBodyText>&nbsp;</p> 
    2741  
    2742 <p class=MsoBodyText>These models are defined by users (See SansView tutorial for details). </p> 
    2743  
    2744 <p class=MsoBodyText>&nbsp;</p> 
    2745  
    2746 <p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span 
     7521style='font-size:16.0pt'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     7522</span></span></b><b><span style='font-size:16.0pt'><a name="Customized Models">Customized Models </a></span></b></p> 
     7523 
     7524<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7525 
     7526<p class=MsoBodyText style='margin-left:.25in'>These model functions can be redefined 
     7527by users (See SansView tutorial for details). </p> 
     7528 
     7529<p class=MsoBodyText style='margin-left:.25in'>&nbsp;</p> 
     7530 
     7531<p class=MsoListParagraphCxSpFirst style='margin-left:.55in;text-indent:-.3in'><b><span 
    27477532style='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; 
    2748 </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> 
    2749  
    2750 <p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    2751  
    2752 <p class=MsoNormal style='margin-left:.25in;text-autospace:none'>This function, as an example function, calculates intensity = A + Bcos(2Q) + Csin(2Q). </p> 
    2753  
    2754  
    2755 <p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    2756 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><b><span 
    2757 style='font-size:14.0pt'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; 
    2758 </span></span></b><b><span style='font-size:14.0pt'><a name="Structure Factors">Structure Factors</a></span></b></p> 
     7533</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> 
     7534 
     7535<p class=MsoListParagraphCxSpLast style='margin-left:.55in'><b><span 
     7536style='font-size:14.0pt'>&nbsp;</span></b></p> 
     7537 
     7538<p class=MsoNormal>This function, as a sample function, calculates the intensity 
     7539= A + Bcos(2q) + Csin(2q).</p> 
     7540 
     7541<p class=MsoNormal>&nbsp;</p> 
     7542 
     7543<p class=MsoNormal>&nbsp;</p> 
     7544 
     7545 
     7546<p class=MsoListParagraphCxSpLast style='margin-left:.55in;text-indent:-.3in'><b><span 
     7547style='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; 
     7548</span></span></b><b><span style='font-size:14.0pt'><a name ="sin(poly)/poly"> sin(poly)/poly </a></span></b></p> 
     7549 
     7550<p class=MsoNormal><b>&nbsp;</b></p> 
     7551 
     7552<p class=MsoNormal>This function calculates the intensity =  scale * sin(f)/f, 
     7553where f = A + Bq + Cq<sup>2</sup> + Dq<sup>3 </sup>+ Eq<sup>4</sup> + Fq<sup>5</sup>.</p> 
     7554 
     7555<p class=MsoNormal>&nbsp;</p> 
     7556 
     7557<p class=MsoNormal>&nbsp;</p> 
     7558 
     7559<p class=MsoNormal>&nbsp;</p> 
     7560 
     7561<p class=MsoListParagraphCxSpFirst style='margin-left:.25in;text-indent:-.25in'><b><span 
     7562style='font-size:16.0pt'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     7563</span></span></b><b><span style='font-size:16.0pt'><a name="Structure Factors"> Structure Factors</a></span></b></p> 
    27597564 
    27607565<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>&nbsp;</p> 
    … …  
    27657570<p class=MsoBodyText>&nbsp;</p> 
    27667571 
    2767 <p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    2768 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; 
    2769 </span></span></b><b><span style='font-size:14.0pt'> <a name="HardSphereStructure">HardSphere Structure</a></span></b></p> 
     7572<p class=MsoBodyText>&nbsp;</p> 
     7573 
     7574<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b> 
     7575<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; 
     7576</span></span></b><b><span style='font-size:14.0pt'><a name="HardsphereStructure"> HardSphere Structure </a></span></b></p> 
    27707577 
    27717578<p class=MsoNormal>&nbsp;</p> 
    … …  
    27787585<p class=MsoNormal>&nbsp;</p> 
    27797586 
     7587<p class=MsoNormal>&nbsp;</p> 
     7588 
    27807589<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    2781 style='font-size:14.0pt;position:relative;top:15.0pt'><img width=122 height=48 
    2782 src="../images/html/image034.gif"></span></p> 
     7590style='font-size:14.0pt;position:relative;top:15.0pt'><img border=0 width=122 
     7591height=48 src="../images/html/image093.png"></span></p> 
    27837592 
    27847593<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    27857594style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
    27867595 
     7596<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7597style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7598 
    27877599<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the 
    27887600distance from the center of the sphere of a radius R.</span></p> 
    27897601 
     7602<p class=MsoNormal><span style='position:relative;top:7.0pt'>&nbsp;</span></p> 
     7603 
     7604<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7605defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     7606border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     7607style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7608 
     7609<p class=MsoNormal><span style='position:relative;top:7.0pt'>&nbsp;</span></p> 
     7610 
    27907611<p class=MsoNormal>&nbsp;</p> 
    27917612 
    … …  
    28137634  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
    28147635  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2815   <p class=MsoBodyText>radius</p> 
     7636  <p class=MsoBodyText>effect_radius</p> 
    28167637  </td> 
    28177638  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    28427663  </td> 
    28437664 </tr> 
     7665 <tr style='height:19.25pt'> 
     7666  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt; 
     7667  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7668  <p class=MsoBodyText>&nbsp;</p> 
     7669  </td> 
     7670  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7671  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7672  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7673  <p class=MsoBodyText>&nbsp;</p> 
     7674  </td> 
     7675  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
     7676  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
     7677  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
     7678  <p class=MsoBodyText>&nbsp;</p> 
     7679  </td> 
     7680 </tr> 
    28447681</table> 
    28457682 
    28467683</div> 
    28477684 
    2848 <p class=MsoNormal><img width=579 height=273 id="Picture 43" 
    2849 src="../images/html/image035.gif"></p> 
     7685<p class=MsoNormal>&nbsp;</p> 
     7686 
     7687<p class=MsoNormal align=center style='text-align:center'><img border=0 
     7688width=484 height=331 id="Picture 111" src="../images/html/image094.png"></p> 
    28507689 
    28517690<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     7691 
     7692<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     76931D plot using the default values (in linear scale).</b></p> 
    28527694 
    28537695<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    28667708<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    28677709style='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; 
    2868 </span></span></b><b><span style='font-size:14.0pt'><a name="SquareWellStructure"> SquareWell Structure</a></span></b></p> 
     7710</span></span></b><b><span style='font-size:14.0pt'><a name="SquareWellStructure"> SquareWell Structure </a></span></b></p> 
    28697711 
    28707712<p class=MsoNormal>&nbsp;</p> 
    … …  
    28937735<p class=MsoNormal>The interaction potential is:</p> 
    28947736 
     7737<p class=MsoNormal>&nbsp;</p> 
     7738 
     7739<p class=MsoNormal>&nbsp;</p> 
     7740 
    28957741<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    2896 style='font-size:14.0pt;position:relative;top:22.0pt'><img width=174 height=67 
    2897 src="../images/html/image036.gif"></span></p> 
     7742style='font-size:14.0pt;position:relative;top:22.0pt'><img border=0 width=174 
     7743height=67 src="../images/html/image095.png"></span></p> 
    28987744 
    28997745<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    29007746style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
    29017747 
     7748<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7749style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7750 
     7751<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     7752style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     7753 
    29027754<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the 
    29037755distance from the center of the sphere of a radius R.</span></p> 
    29047756 
     7757<p class=MsoNormal><span style='position:relative;top:7.0pt'>&nbsp;</span></p> 
     7758 
     7759<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7760defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     7761border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     7762style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     7763 
    29057764<p class=MsoNormal>&nbsp;</p> 
    29067765 
    … …  
    29287787  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt; 
    29297788  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2930   <p class=MsoBodyText>radius</p> 
     7789  <p class=MsoBodyText>effect_radius</p> 
    29317790  </td> 
    29327791  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    29547813  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    29557814  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    2956   <p class=MsoBodyText>0.2</p> 
     7815  <p class=MsoBodyText>0.04</p> 
    29577816  </td> 
    29587817 </tr> 
    … …  
    29957854<p class=MsoNormal>&nbsp;</p> 
    29967855 
    2997 <p class=MsoNormal><img width=546 height=296 id="Picture 53" 
    2998 src="../images/html/image037.gif"></p> 
     7856<p class=MsoNormal align=center style='text-align:center'><img border=0 
     7857width=510 height=346 id="Picture 110" src="../images/html/image096.png"></p> 
    29997858 
    30007859<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     7860 
     7861<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     78621D plot using the default values (in linear scale).</b></p> 
    30017863 
    30027864<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    30077869Sharma, K. C. Physica, 89A, 213. (1977).</p> 
    30087870 
     7871<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>&nbsp;</p> 
     7872 
    30097873<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    30107874 
    30117875<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    30127876style='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; 
    3013 </span></span></b><b><span style='font-size:14.0pt'><a name="HayterMSAStructure"> HayterMSA Structure</a></span></b></p> 
     7877</span></span></b><b><span style='font-size:14.0pt'><a name="HayterMSAStructure"> HayterMSA Structure </a></span></b></p> 
    30147878 
    30157879<p class=MsoNormal>&nbsp;</p> 
    … …  
    30307894of the solution which in turn is used to compute the Debye screening length.  
    30317895At present there is no provision for entering the ionic strength directly nor 
    3032 for use of any multivalent salts.  The counterions are also assumed to be monovalent.</p> 
     7896for use of any multivalent salts.  The counterions are also assumed to be 
     7897monovalent.</p> 
     7898 
     7899<p class=MsoNormal>&nbsp;</p> 
     7900 
     7901<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     7902defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     7903border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     7904style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
    30337905 
    30347906<p class=MsoNormal>&nbsp;</p> 
    … …  
    30577929  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt; 
    30587930  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    3059   <p class=MsoBodyText>radius</p> 
     7931  <p class=MsoBodyText>effect_radius</p> 
    30607932  </td> 
    30617933  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    31568028<p class=MsoNormal>&nbsp;</p> 
    31578029 
    3158 <p class=MsoNormal><img width=548 height=354 id="Picture 56" 
    3159 src="../images/html/image038.gif"></p> 
     8030<p class=MsoNormal align=center style='text-align:center'><img border=0 
     8031width=498 height=333 id="Picture 112" src="../images/html/image097.png"></p> 
    31608032 
    31618033<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     8034 
     8035<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     80361D plot using the default values (in linear scale).</b></p> 
    31628037 
    31638038<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    31758050<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    31768051 
     8052<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
     8053 
    31778054<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span 
    31788055style='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; 
    3179 </span></span></b><b><span style='font-size:14.0pt'><a name="StickyHSStructure"> StickyHS Structure</a></span></b></p> 
     8056</span></span></b><b><span style='font-size:14.0pt'> <a name="StickyHSStructure">StickyHS Structure </a></span></b></p> 
    31808057 
    31818058<p class=MsoNormal>&nbsp;</p> 
    … …  
    31998076attraction.</p> 
    32008077 
    3201 <p class=MsoNormal align=center style='text-align:center'><span 
    3202 style='font-size:14.0pt;position:relative;top:22.0pt'><img width=135 height=67 
    3203 src="../images/html/image039.gif"></span></p> 
     8078<p class=MsoNormal>&nbsp;</p> 
     8079 
     8080<p class=MsoNormal>&nbsp;</p> 
     8081 
     8082<p class=MsoNormal align=center style='text-align:center'><span 
     8083style='font-size:14.0pt;position:relative;top:22.0pt'><img border=0 width=135 
     8084height=67 src="../images/html/image098.png"></span></p> 
    32048085 
    32058086<p class=MsoNormal align=center style='text-align:center'><span 
    32068087style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p> 
    32078088 
     8089<p class=MsoNormal align=center style='text-align:center'><span 
     8090style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p> 
     8091 
    32088092<p class=MsoNormal>where the interaction potential is</p> 
    32098093 
     8094<p class=MsoNormal>&nbsp;</p> 
     8095 
     8096<p class=MsoNormal>&nbsp;</p> 
     8097 
    32108098<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
    3211 style='font-size:14.0pt;position:relative;top:23.0pt'><img width=186 height=69 
    3212 src="../images/html/image040.gif"></span></p> 
     8099style='font-size:14.0pt;position:relative;top:23.0pt'><img border=0 width=186 
     8100height=69 src="../images/html/image099.png"></span></p> 
     8101 
     8102<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     8103style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
     8104 
     8105<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span 
     8106style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p> 
    32138107 
    32148108<p class=MsoNormal>The Percus-Yevick (PY) closure was used for this 
    … …  
    32368130<p class=MsoNormal>&nbsp;</p> 
    32378131 
     8132<p class=MsoNormal style='margin-left:.25in'>For 2D plot, the wave transfer is 
     8133defined as<span style='font-size:14.0pt;position:relative;top:8.0pt'><img 
     8134border=0 width=103 height=33 src="../images/html/image010.png"></span><span 
     8135style='font-size:14.0pt;position:relative;top:6.0pt'>.</span></p> 
     8136 
     8137<p class=MsoNormal>&nbsp;</p> 
     8138 
     8139<p class=MsoNormal>&nbsp;</p> 
     8140 
    32388141<div align=center> 
    32398142 
    … …  
    32578160  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt; 
    32588161  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    3259   <p class=MsoBodyText>radius</p> 
     8162  <p class=MsoBodyText>effect_radius</p> 
    32608163  </td> 
    32618164  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left: 
    … …  
    32678170  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; 
    32688171  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'> 
    3269   <p class=MsoBodyText>20.8</p> 
     8172  <p class=MsoBodyText>50</p> 
    32708173  </td> 
    32718174 </tr> 
    … …  
    33248227<p class=MsoNormal>&nbsp;</p> 
    33258228 
    3326 <p class=MsoNormal><img width=524 height=359 id="Picture 111" 
    3327 src="../images/html/image041.gif"></p> 
     8229<p class=MsoNormal align=center style='text-align:center'><img border=0 
     8230width=502 height=328 id="Picture 113" src="../images/html/image100.png"></p> 
     8231 
     8232<p class=MsoNormal align=center style='text-align:center;text-autospace:none'><b>Figure. 
     82331D plot using the default values (in linear scale).</b></p> 
    33288234 
    33298235<p class=MsoNormal style='text-autospace:none'>&nbsp;</p> 
    … …  
    33368242G., Manohar, C. and K. Srinivas Rao J. Chem. Phys., 95(12), 9186-9190 (1991).</p> 
    33378243 
    3338  
    3339  
    33408244<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    33418245 
    33428246<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p> 
    33438247 
    3344  
    3345 <p class=MsoBodyText><b><span style='font-size:14.0pt'><a name="References">References</a></span></b></p> 
     8248<p class=MsoBodyText><b><span style='font-size:14.0pt'>References</span></b></p> 
    33468249 
    33478250<p class=MsoBodyText>Feigin, L. A, and D. I. Svergun (1987) &quot;Structure 
    3348 Analysis by Small-Angle X-Ray and Neutron Scattering&quot;, Plenum Press, New York.</p> 
     8251Analysis by Small-Angle X-Ray and Neutron Scattering&quot;, Plenum Press, New 
     8252York.</p> 
    33498253 
    33508254<p class=MsoBodyText>Guinier, A. and G. Fournet (1955) &quot;Small-Angle 
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