Changeset e0405592 in sasview for sansmodels


Ignore:
Timestamp:
Jul 2, 2012 5:01:29 PM (12 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:
d560a37
Parents:
732c548
Message:

added Raspberry model help/doc

Location:
sansmodels/src/sans/models/media
Files:
2 added
1 edited

Legend:

Unmodified
Added
Removed
  • sansmodels/src/sans/models/media/model_functions.html

    r940aca7 re0405592  
    444444{margin-bottom:0in;} 
    445445--> 
    446 </style><title></title></head> 
     446</style><title></title> 
     447 
     448</head> 
    447449<body lang="EN-US" link="blue" vlink="purple"> 
    448450<div class="WordSection1"> 
     
    453455<li class="MsoNormal" style="line-height: 115%;"><a href="#Introduction"><b>Introduction</b></a> 
    454456</li> 
    455 <li class="MsoNormal" style="line-height: 115%;"><a href="#Shapes"><b>Shapes</b></a>: <a href="#SphereModel">SphereModel</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#CoreShellModel">CoreShellModel</a>,&nbsp;<a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, &nbsp;<a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, 
     457<li class="MsoNormal" style="line-height: 115%;"><a href="#Shapes"><b>Shapes</b></a>: <a href="#SphereModel">SphereModel</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel</a>,&nbsp;<a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, &nbsp;<a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, 
    456458<a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, 
    457459<a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, 
     
    531533particle:</p> 
    532534<p class="MsoNormal">&nbsp;</p> 
    533 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image001.PNG" ></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     535<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image001.PNG"></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    534536</p> 
    535537<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    536538<p class="MsoNormal">with</p> 
    537 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 8pt;"><img src="image002.PNG" ></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
     539<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 8pt;"><img src="image002.PNG"></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
    538540<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    539541<p class="MsoNormal">where <i>P</i><sub>0</sub><i>(<b>q</b>)</i> 
     
    550552particle 
    551553volume fraction:<span style="position: relative; top: 5pt;"> 
    552 <img src="image003.PNG" ></span>.</p> 
     554<img src="image003.PNG"></span>.</p> 
    553555<p class="MsoNormal">&nbsp;</p> 
    554556<p class="MsoNormal">Our so-called 1D scattering intensity 
     
    588590way (Guinier, 1955):</p> 
    589591<p class="MsoNormal">&nbsp;</p> 
    590 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image004.PNG" ></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     592<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image004.PNG"></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    591593</p> 
    592594<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    726728calculation is an exact multi-component solution:</p> 
    727729<p class="MsoNormal">&nbsp;</p> 
    728 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 5pt;"><img src="image006.PNG" ></span></p> 
     730<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 5pt;"><img src="image006.PNG"></span></p> 
    729731<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    730732<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    740742<p class="MsoNormal">&nbsp;</p> 
    741743<p class="MsoNormal">&nbsp;</p> 
    742 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 24pt;"><img src="image007.PNG" ></span>.</p> 
     744<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 24pt;"><img src="image007.PNG"></span>.</p> 
    743745<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    744746<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    746748<p class="MsoNormal">The 2D scattering intensity is the 
    747749same as 1D, regardless of 
    748 the orientation of the <i>q</i> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" ></span><span style="font-size: 14pt;">.</span></p> 
     750the orientation of the <i>q</i> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
    749751<p class="MsoNormal">&nbsp;</p> 
    750752<p class="MsoNormal">&nbsp;</p> 
     
    904906calculated as:</p> 
    905907<p class="MsoNormal">&nbsp;</p> 
    906 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image010.PNG" ></span></p> 
     908<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image010.PNG"></span></p> 
    907909<p class="MsoNormal">&nbsp;</p> 
    908910<p class="MsoNormal">where the amplitude A(q) is given as 
     
    912914<p class="MsoNormal">&nbsp;</p> 
    913915<p class="MsoNormal">&nbsp;</p> 
    914 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 18pt;"><img src="image011.PNG" ></span></p> 
     916<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 18pt;"><img src="image011.PNG"></span></p> 
    915917<p class="MsoNormal">&nbsp;</p> 
    916918<p class="MsoNormal">&nbsp;</p> 
     
    947949<p class="MsoNormal">For 2D data: The 2D scattering 
    948950intensity is calculated in 
    949 the same way as 1D, where the <i>q</i> vector is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" ></span><span style="font-size: 14pt;">.</span></p> 
     951the same way as 1D, where the <i>q</i> vector is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
    950952<p class="MsoNormal">&nbsp;</p> 
    951953<p class="MsoNormal">REFERENCE</p> 
     
    10491051<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    10501052<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     1053<p class="MsoNormal"> 
     1054</p> 
    10511055<p class="MsoNormal">&nbsp;</p> 
    10521056<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.4.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1057</span></b><a name="RaspBerryModel"></a><b><span style="font-size: 14pt;">RaspBerryModel</span></b></p> 
     1058&nbsp;<b> <br> 
     1059</b>Calculates the form factor, P(q), for a "Raspberry-like" 
     1060structure where there are smaller spheres at the surface of a larger 
     1061sphere, such as the structure of a Pickering emulsion.&nbsp; 
     1062<p class="MsoNormal">&nbsp;</p> 
     1063<p class="MsoNormal" style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; 
     1064</span>Definition</b></p> 
     1065<p class="MsoNormal">&nbsp;</p> 
     1066The structure is:<br> 
     1067<img src="raspberry_pic.jpg"> 
     1068<p class="MsoNormal"><br> 
     1069Ro = the radius of the&nbsp;large sphere<br> 
     1070Rp = the radius of the smaller sphere on the surface<br> 
     1071delta = the fractional penetration depth<br> 
     1072surface coverage = fractional coverage of the large sphere surface (0.9 
     1073max)<br> 
     1074<br> 
     1075<br> 
     1076The large and small spheres have their own SLD, as well as the solvent. 
     1077The surface coverage term is a fractional coverage (maximum of 
     1078approximately 0.9 for hexagonally packed spheres on a surface). Since 
     1079not all of the small spheres are necessarily attached to the surface, 
     1080the excess free (small) spheres scattering is also included in the 
     1081calculation. The function calculated follows equations (8)-(12) of the 
     1082reference below, and the equations are not reproduced here.<br> 
     1083<br> 
     1084The returned value is scaled to units of [cm-1]. No interparticle 
     1085scattering is included in this model.<br> 
     1086<br> 
     1087</p> 
     1088<p class="MsoNormal">For 2D data: The 2D scattering 
     1089intensity is calculated in 
     1090the same way as 1D, where the <i>q</i> vector is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
     1091<p class="MsoNormal">&nbsp;</p> 
     1092&nbsp;&nbsp; REFERENCE<br> 
     1093Kjersta Larson-Smith, Andrew Jackson, and Danilo C 
     1094Pozzo, "Small angle scattering model for Pickering emulsions and 
     1095raspberry particles." Journal of Colloid and Interface Science (2010) 
     1096vol. 343 (1) pp. 36-41.<br> 
     1097<p class="MsoNormal">&nbsp;</p> 
     1098<p class="MsoNormal">TEST DATASET</p> 
     1099<p class="MsoNormal">This example dataset is produced by 
     1100running the RaspBerryModel, using 2000 data points, qmin = 0.0001 Å<sup>-1</sup>,&nbsp; 
     1101qmax = 0.2 A<sup>-1</sup> and the default values, where 
     1102Ssph/Lsph stands for Smaller/Large sphere and&nbsp;surfrac_Ssph for 
     1103the surface fraction of the smaller spheres.</p> 
     1104<p class="MsoNormal">&nbsp;</p> 
     1105<div align="center"> 
     1106<table class="MsoNormalTable" style="border-collapse: collapse;" border="2" cellpadding="0" cellspacing="0"> 
     1107<tbody> 
     1108<tr style="height: 18.8pt;"> 
     1109<td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1110<p class="MsoBodyText">Parameter name</p> 
     1111</td> 
     1112<td style="border-style: solid solid solid none; border-color: windowtext windowtext windowtext -moz-use-text-color; border-width: 1pt 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1113<p class="MsoBodyText">Units</p> 
     1114</td> 
     1115<td style="border-style: solid solid solid none; border-color: windowtext windowtext windowtext -moz-use-text-color; border-width: 1pt 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1116<p class="MsoBodyText">Default value</p> 
     1117</td> 
     1118</tr> 
     1119<tr> 
     1120<td>&nbsp; delta_Ssph</td> 
     1121<td></td> 
     1122<td>&nbsp; 0</td> 
     1123</tr> 
     1124<tr> 
     1125<td>&nbsp; radius_Lsph</td> 
     1126<td>&nbsp; Å</td> 
     1127<td>&nbsp; 5000</td> 
     1128</tr> 
     1129<tr> 
     1130<td>&nbsp; radius_Ssph</td> 
     1131<td>&nbsp; Å</td> 
     1132<td>&nbsp; 100</td> 
     1133</tr> 
     1134<tr> 
     1135<td>&nbsp; sld_Lsph</td> 
     1136<td>&nbsp;&nbsp;Å<sup> -2</sup></td> 
     1137<td>&nbsp; -4e-07</td> 
     1138</tr> 
     1139<tr style="height: 18.8pt;"> 
     1140<td style="border-style: none solid solid; border-color: -moz-use-text-color windowtext windowtext; border-width: medium 1pt 1pt; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1141<p class="MsoBodyText">sld_Ssph</p> 
     1142</td> 
     1143<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; vertical-align: top; width: 107pt; height: 18.8pt;"> 
     1144<p class="MsoBodyText">Å<sup> -2</sup></p> 
     1145</td> 
     1146<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1147<p class="MsoBodyText">3.5e-6</p> 
     1148</td> 
     1149</tr> 
     1150<tr style="height: 18.8pt;"> 
     1151<td style="border-style: none solid solid; border-color: -moz-use-text-color windowtext windowtext; border-width: medium 1pt 1pt; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1152<p class="MsoBodyText">sld_solv</p> 
     1153</td> 
     1154<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1155<p class="MsoBodyText">Å<sup> -2</sup></p> 
     1156</td> 
     1157<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1158<p class="MsoBodyText">6.3e-6</p> 
     1159</td> 
     1160</tr> 
     1161<tr style="height: 18.8pt;"> 
     1162<td style="border-style: none solid solid; border-color: -moz-use-text-color windowtext windowtext; border-width: medium 1pt 1pt; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1163<p class="MsoBodyText">surfrac_Ssph</p> 
     1164</td> 
     1165<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1166<p class="MsoBodyText"><br> 
     1167</p> 
     1168</td> 
     1169<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1170<p class="MsoBodyText">0.4</p> 
     1171</td> 
     1172</tr> 
     1173<tr style="height: 18.8pt;"> 
     1174<td style="border-style: none solid solid; border-color: -moz-use-text-color windowtext windowtext; border-width: medium 1pt 1pt; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1175<p class="MsoBodyText">volf_Lsph</p> 
     1176</td> 
     1177<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1178<p class="MsoBodyText">&nbsp;</p> 
     1179</td> 
     1180<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1181<p class="MsoBodyText">0.05</p> 
     1182</td> 
     1183</tr> 
     1184<tr style="height: 18.8pt;"> 
     1185<td style="border-style: none solid solid; border-color: -moz-use-text-color windowtext windowtext; border-width: medium 1pt 1pt; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1186<p class="MsoBodyText">volf_Lsph</p> 
     1187</td> 
     1188<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1189<p class="MsoBodyText"><br> 
     1190</p> 
     1191</td> 
     1192<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1193<p class="MsoBodyText">0.005</p> 
     1194</td> 
     1195</tr> 
     1196<tr style="height: 18.8pt;"> 
     1197<td style="border-style: none solid solid; border-color: -moz-use-text-color windowtext windowtext; border-width: medium 1pt 1pt; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1198<p class="MsoBodyText">background</p> 
     1199</td> 
     1200<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1201<p class="MsoBodyText">cm<sup>-1</sup></p> 
     1202</td> 
     1203<td style="border-style: none solid solid none; border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 
     1204<p class="MsoBodyText">0</p> 
     1205</td> 
     1206</tr> 
     1207</tbody> 
     1208</table> 
     1209</div> 
     1210<p class="MsoNormal">&nbsp;</p> 
     1211<p class="MsoNormal" style="text-align: center;" align="center"><img src="raspberry_plot.jpg"></p> 
     1212<p class="MsoNormal" style="text-align: center;" align="center"><b>Figure. 
     12131D plot using the values of /2000 data points.</b></p> 
     1214<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     1215<p class="MsoNormal" style="text-align: center;" align="center"></p> 
     1216<p class="MsoNormal">&nbsp;</p> 
     1217<p class="MsoNormal"></p> 
     1218<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.5.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    10531219</span></b><a name="CoreShellModel"><b><span style="font-size: 14pt;">Core Shell 
    10541220(Sphere) Model</span></b></a></p> 
     
    10671233way (Guinier, 1955):</p> 
    10681234<p class="MsoNormal">&nbsp;</p> 
    1069 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image013.PNG" ></span>&nbsp;</p> 
     1235<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image013.PNG"></span>&nbsp;</p> 
    10701236<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    10711237<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    12291395<p class="MsoNormal">&nbsp;</p> 
    12301396<p class="MsoNormal">&nbsp;</p> 
    1231 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.5.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1397<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.6.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    12321398</span></b><a name="Core2ndMomentModel"><b><span style="font-size: 14pt;">Core2ndMomentModel</span></b></a></p> 
    12331399<p class="MsoNormal">&nbsp;</p> 
     
    13991565<p class="MsoNormal">&nbsp;</p> 
    14001566<p class="MsoNormal">&nbsp;</p> 
    1401 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.6.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1567<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.7.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    14021568</span></b><a name="CoreMultiShellModel"><b><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model</span></b></a></p> 
    14031569<p class="MsoNormal">&nbsp;</p> 
     
    16381804<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    16391805<p class="MsoNormal"><b>&nbsp;</b></p> 
    1640 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.7.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1806<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.8.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    16411807</span></b><a name="VesicleModel"><b><span style="font-size: 14pt;">VesicleModel</span></b></a></p> 
    16421808<p class="MsoNormal">&nbsp;</p> 
     
    16831849above, regardless of the orientation of the <i>q</i> 
    16841850vector&nbsp; which is 
    1685 defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" ></span><span style="font-size: 14pt;">.</span></p> 
     1851defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
    16861852<p class="MsoNormal">For P*S: The outer most radius (= 
    16871853radius + thickness) is 
     
    18021968<p class="MsoNormal"><b>&nbsp;</b></p> 
    18031969<p class="MsoNormal"><b>&nbsp;</b></p> 
    1804 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.8.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     1970<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.9.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    18051971</span></b><a name="MultiShellModel"><b><span style="font-size: 14pt;">MultiShellModel</span></b></a></p> 
    18061972<p class="MsoNormal">&nbsp;</p> 
     
    18191985<p class="MsoNormal">The 2D scattering intensity is the 
    18201986same as 1D, regardless of 
    1821 the orientation of the <i>q</i> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" ></span><span style="font-size: 14pt;">.</span></p> 
     1987the orientation of the <i>q</i> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
    18221988<p class="MsoNormal">For P*S: The outer most radius (= 
    18231989core_radius + 
     
    19632129<p class="MsoNormal">&nbsp;</p> 
    19642130<p class="MsoNormal"><b>&nbsp;</b></p> 
    1965 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.9.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2131<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.10.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    19662132</span></b><a name="OnionExpShellModel"><b><span style="font-size: 14pt;">OnionExpShellModel</span></b></a></p> 
    19672133<p class="MsoNormal">&nbsp;&nbsp;</p> 
     
    22222388<p class="MsoNormal">&nbsp;</p> 
    22232389<p class="MsoNormal"><b>&nbsp;</b></p> 
    2224 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.10.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2390<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.11.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    22252391</span></b><a name="SphericalSLDModel"><b><span style="font-size: 14pt;">SphericalSLDModel</span></b></a></p> 
    22262392<p class="MsoNormal">&nbsp;&nbsp;</p> 
     
    22652431<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image024.gif" height="41" width="200"></span></p> 
    22662432<p class="MsoNormal">so that</p> 
    2267 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image044.gif" ></span></p> 
    2268 <p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    2269 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image045.gif" ></span></p> 
    2270 <p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    2271 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image046.gif" ></span></p> 
    2272 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image047.gif" ></span></p> 
    2273 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image048.gif" ></span></p> 
    2274 <p class="MsoNormal">&nbsp;</p> 
    2275 <p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image027.gif" ></span></p> 
     2433<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image044.gif"></span></p> 
     2434<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     2435<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image045.gif"></span></p> 
     2436<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     2437<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image046.gif"></span></p> 
     2438<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image047.gif"></span></p> 
     2439<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image048.gif"></span></p> 
     2440<p class="MsoNormal">&nbsp;</p> 
     2441<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image027.gif"></span></p> 
    22762442<p class="MsoNormal">&nbsp;</p> 
    22772443<p class="MsoNormal">&nbsp;</p> 
     
    22852451<p class="MsoListParagraph" style="text-indent: -0.25in;">1)<span style="font-family: &quot;Times New Roman&quot;; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    22862452</span>Exp;&nbsp;&nbsp;&nbsp; </p> 
    2287 <p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image049.gif" ></span></p> 
     2453<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image049.gif"></span></p> 
    22882454<p class="MsoNormal">&nbsp;</p> 
    22892455<p class="MsoListParagraph" style="text-indent: -0.25in;">2)<span style="font-family: &quot;Times New Roman&quot;; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     
    22912457<p class="MsoNormal" style="margin-left: 0.25in;">&nbsp;</p> 
    22922458<p class="MsoListParagraph">&nbsp;</p> 
    2293 <p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image050.gif" ></span></p> 
     2459<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image050.gif"></span></p> 
    22942460<p class="MsoNormal">&nbsp;</p> 
    22952461<p class="MsoNormal">&nbsp;</p> 
     
    22972463</span>Erf;</p> 
    22982464<p class="MsoNormal" style="margin-left: 0.25in;">&nbsp;</p> 
    2299 <p class="MsoListParagraph"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image051.gif" ></span></p> 
     2465<p class="MsoListParagraph"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image051.gif"></span></p> 
    23002466<p class="MsoListParagraph">&nbsp;</p> 
    23012467<p class="MsoListParagraph">&nbsp;</p> 
     
    23212487<p class="MsoNormal">&nbsp;</p> 
    23222488<p class="MsoNormal">In the equation, </p> 
    2323 <p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image055.gif" ></span></p> 
     2489<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image055.gif"></span></p> 
    23242490<p class="MsoNormal">&nbsp;</p> 
    23252491<p class="MsoNormal">&nbsp;</p> 
     
    23272493calculated by</p> 
    23282494<p class="MsoNormal">&nbsp;</p> 
    2329 <p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image037.gif" ></span></p> 
     2495<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image037.gif"></span></p> 
    23302496<p class="MsoNormal">where </p> 
    23312497<p class="MsoNormal">&nbsp;</p> 
    2332 <p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image038.gif" ></span></p> 
     2498<p class="MsoNormal"><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;"><img src="image038.gif"></span></p> 
    23332499<p class="MsoNormal"><span style="font-family: Symbol;">&nbsp;</span></p> 
    23342500<p class="MsoNormal"><span style="font-family: Symbol;">&nbsp;</span></p> 
    2335 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-family: Symbol;"><img id="Object 20" src="image056.gif" ></span></p> 
     2501<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-family: Symbol;"><img id="Object 20" src="image056.gif"></span></p> 
    23362502<p class="MsoNormal" style="text-align: center;" align="center"><span style="font-family: Symbol;">&nbsp;</span></p> 
    23372503<p class="MsoNormal">&nbsp;</p> 
     
    23402506above, regardless of the orientation of the <i>q</i> 
    23412507vector&nbsp; which is 
    2342 defined as<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;; position: relative; top: 4.5pt;"><img src="image040.gif" ></span><span style="font-size: 14pt;">.</span></p> 
     2508defined as<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;; position: relative; top: 4.5pt;"><img src="image040.gif"></span><span style="font-size: 14pt;">.</span></p> 
    23432509<p class="MsoNormal">&nbsp;</p> 
    23442510<p class="MsoNormal">For P*S: The outer most radius is 
     
    25352701<p class="MsoNormal">&nbsp;</p> 
    25362702<p class="MsoNormal">&nbsp;</p> 
    2537 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.11.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2703<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.12.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    25382704</span></b><b><a name="LinearPearlsModel"><span style="font-size: 14pt;">LinearPearlsModel</span></a></b></p> 
    25392705<p class="MsoNormal"><a name="LinearPearlsModel">&nbsp;</a></p> 
     
    26762842<p class="MsoNormal"><a name="LinearPearlsModel">&nbsp;</a></p> 
    26772843<p class="MsoNormal"><a name="LinearPearlsModel">&nbsp;</a></p> 
    2678 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><a name="LinearPearlsModel"><b><span style="font-size: 14pt;">2.12.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     2844<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><a name="LinearPearlsModel"><b><span style="font-size: 14pt;">2.13.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    26792845</span></b></a><b><a name="PearlNecklaceModel"><span style="font-size: 14pt;">PearlNecklaceModel</span></a></b></p> 
    26802846<p class="MsoNormal"><a name="PearlNecklaceModel">&nbsp;</a></p> 
     
    28513017<p class="MsoNormal"><a name="PearlNecklaceModel">&nbsp;</a></p> 
    28523018<p class="MsoNormal"><a name="PearlNecklaceModel">&nbsp;</a></p> 
    2853 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"><b><span style="font-size: 14pt;">2.13.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3019<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"><b><span style="font-size: 14pt;">2.14.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    28543020</span></b></a><a name="CylinderModel"><b><span style="font-size: 14pt;">Cylinder 
    28553021Model</span></b></a></p> 
     
    28693035<p class="MsoNormal">&nbsp;</p> 
    28703036<p class="MsoNormal">&nbsp;</p> 
    2871 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG" ></span>&nbsp; 
     3037<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG"></span>&nbsp; 
    28723038</p> 
    2873 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image060.PNG" ></span>&nbsp;&nbsp; 
     3039<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image060.PNG"></span>&nbsp;&nbsp; 
    28743040</p> 
    28753041<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    30133179randomly oriented cylinders is then given by:</p> 
    30143180<p class="MsoNormal">&nbsp;</p> 
    3015 <p class="MsoNormal" style="text-align: center;" align="center"><a name="_Ref173306528"></a><a name="_Ref173306479"></a><span style="position: relative; top: 16pt;"><img src="image063.PNG" ></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3181<p class="MsoNormal" style="text-align: center;" align="center"><a name="_Ref173306528"></a><a name="_Ref173306479"></a><span style="position: relative; top: 16pt;"><img src="image063.PNG"></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    30163182</p> 
    30173183<p class="MsoNormal">&nbsp;</p> 
     
    30413207done by evaluating the following:</p> 
    30423208<p class="MsoNormal">&nbsp;</p> 
    3043 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image064.PNG" ></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3209<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image064.PNG"></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    30443210</p> 
    30453211<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    30843250<p class="MsoNormal"><b>&nbsp;</b></p> 
    30853251<p class="MsoNormal"><b>&nbsp;</b></p> 
    3086 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.14.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3252<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.15.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    30873253</span></b><a name="CoreShellCylinderModel"><b><span style="font-size: 14pt;">Core-Shell 
    30883254Cylinder Model</span></b></a></p> 
     
    31003266oriented core-shell cylinders is given by (Kline, 2006):</p> 
    31013267<p class="MsoNormal">&nbsp;</p> 
    3102 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image067.PNG" ></span>&nbsp;</p> 
    3103 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 32pt;"><img src="image068.PNG" ></span>&nbsp;</p> 
     3268<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image067.PNG"></span>&nbsp;</p> 
     3269<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 32pt;"><img src="image068.PNG"></span>&nbsp;</p> 
    31043270<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    31053271<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    33383504<p class="MsoNormal">&nbsp;</p> 
    33393505<p class="MsoNormal">&nbsp;</p> 
    3340 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.15.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3506<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.16.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    33413507</span></b><a name="CoreShellBicelleModel"><b><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></b></a></p> 
    33423508<p class="MsoNormal">&nbsp;</p> 
     
    35153681<p class="MsoNormal"><b>&nbsp;</b></p> 
    35163682<p class="MsoNormal"><b>&nbsp;</b></p> 
    3517 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.16.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3683<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.17.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    35183684</span></b><a name="HollowCylinderModel"><b><span style="font-size: 14pt;">HollowCylinderModel</span></b></a></p> 
    35193685<p class="MsoNormal">&nbsp;</p> 
     
    35323698way (Guinier, 1955):</p> 
    35333699<p class="MsoNormal">&nbsp;</p> 
    3534 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image072.PNG" ></p> 
     3700<p class="MsoNormal" style="text-align: center;" align="center"><img src="image072.PNG"></p> 
    35353701<br> 
    35363702<nr> 
     3703</nr> 
    35373704<p class="MsoNormal">where <i>scale</i> is a 
    35383705scale factor, <i>J<sub>1</sub></i> 
     
    36823849<p class="MsoNormal"><b>&nbsp;</b></p> 
    36833850<p class="MsoNormal"><b>&nbsp;</b></p> 
    3684 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.17.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     3851<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.18.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    36853852</span></b><a name="FlexibleCylinderModel"></a><b><span style="font-size: 14pt;">FlexibleCylinderModel</span></b></p> 
    36863853<p class="MsoNormal">&nbsp;</p> 
     
    38454012<p class="MsoNormal">&nbsp;</p> 
    38464013<p class="MsoNormal">&nbsp;</p> 
    3847 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.18.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4014<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.19.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    38484015</span></b><b><span style="font-size: 14pt;">FlexCylEllipXMo<a name="FlexCylEllipXModel"></a>del</span></b></p> 
    38494016<p class="MsoNormal">&nbsp;</p> 
     
    39394106<p class="MsoNormal">For 2D data: The 2D scattering 
    39404107intensity is calculated in 
    3941 the same way as 1D, where the <i>q</i> vector is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" ></span><span style="font-size: 14pt;">.</span></p> 
     4108the same way as 1D, where the <i>q</i> vector is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
    39424109<p class="MsoNormal">&nbsp;</p> 
    39434110<p class="MsoNormal">REFERENCE</p> 
     
    40724239<p class="MsoNormal">&nbsp;</p> 
    40734240<p class="MsoNormal"><b>&nbsp;</b></p> 
    4074 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.19.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4241<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.20.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    40754242</span></b><a name="StackedDisksModel"><b><span style="font-size: 14pt;">StackedDisksModel 
    40764243</span></b></a></p> 
     
    40944261<p class="MsoNormal">The 2D scattering intensity is the 
    40954262same as 1D, regardless of 
    4096 the orientation of the <i>q</i> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" ></span><span style="font-size: 14pt;">.</span></p> 
     4263the orientation of the <i>q</i> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 
    40974264<p class="MsoNormal"><span style="font-size: 14pt;">&nbsp;</span></p> 
    40984265<p class="MsoNormal">&nbsp;</p> 
     
    41084275<p class="MsoNormal">The scattering intensity I(q) is:</p> 
    41094276<p class="MsoNormal">&nbsp;</p> 
    4110 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 9pt;"><img src="image081.PNG" ></span></p> 
     4277<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 9pt;"><img src="image081.PNG"></span></p> 
    41114278<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    41124279<p class="MsoNormal">where the contrast,</p> 
    4113 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image082.PNG" ></span></p> 
     4280<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image082.PNG"></span></p> 
    41144281<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    41154282<p class="MsoNormal">&nbsp;</p> 
     
    41214288disc, respectively.</p> 
    41224289<p class="MsoNormal">&nbsp;</p> 
    4123 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image083.PNG" ></p> 
    4124 <br><br> 
    4125 <p class="MsoNormal"></p><p class="MsoNormal">where d = thickness of the layer 
     4290<p class="MsoNormal" style="text-align: center;" align="center"><img src="image083.PNG"></p> 
     4291<br> 
     4292<br> 
     4293<p class="MsoNormal"></p> 
     4294<p class="MsoNormal">where d = thickness of the layer 
    41264295(layer_thick), 2h= core 
    41274296thickness (core_thick),&nbsp; and R = radius of the disc (radius).</p> 
    4128  
    4129 <p class="MsoNormal">&nbsp;</p> 
    4130 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image084.PNG" ></span></p> 
     4297<p class="MsoNormal">&nbsp;</p> 
     4298<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image084.PNG"></span></p> 
    41314299<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    41324300<p class="MsoNormal">&nbsp;</p> 
     
    43144482<p class="MsoNormal"><b>&nbsp;</b></p> 
    43154483<p class="MsoNormal"><b>&nbsp;</b></p> 
    4316 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.20.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4484<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.21.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    43174485</span></b><a name="ParallelepipedModel"><b><span style="font-size: 14pt;">ParallelepipedModel</span></b></a></p> 
    43184486<p class="MsoNormal">&nbsp;</p> 
     
    43394507c=C/B&gt;1, the form factor,</p> 
    43404508<p class="MsoNormal">&nbsp;</p> 
    4341 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 48pt;"><img src="image088.PNG" ></span></p> 
     4509<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 48pt;"><img src="image088.PNG"></span></p> 
    43424510<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    43434511<p class="MsoNormal">The contrast is defined as</p> 
    43444512<p class="MsoNormal">&nbsp;</p> 
    4345 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image089.PNG" ></span></p> 
     4513<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image089.PNG"></span></p> 
    43464514<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    43474515<p class="MsoNormal">&nbsp;</p> 
     
    45094677<p class="MsoNormal"><b>&nbsp;</b></p> 
    45104678<p class="MsoNormal"><b>&nbsp;</b></p> 
    4511 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.21.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4679<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.22.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    45124680</span></b><b><span style="font-size: 14pt;"><a name="CSParallelepipedModel">CSParallelepipedModel</a></span></b></p> 
    45134681<p class="MsoNormal">&nbsp;</p> 
     
    45424710<p class="MsoNormal">The volume of the solid is:</p> 
    45434711<p class="MsoNormal">&nbsp;</p> 
    4544 <p class="MsoNormal"><span style="position: relative; top: 5pt;"><img src="image095.PNG" ></span></p> 
     4712<p class="MsoNormal"><span style="position: relative; top: 5pt;"><img src="image095.PNG"></span></p> 
    45454713<p class="MsoNormal">meaning that there are "gaps" at the 
    45464714corners of 
     
    48184986<p class="MsoNormal">&nbsp;</p> 
    48194987<p class="MsoNormal">&nbsp;</p> 
    4820 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.22.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     4988<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.23.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    48214989</span></b><a name="EllipticalCylinderModel"><b><span style="font-size: 14pt;">Elliptical 
    48224990Cylinder Model</span></b></a></p> 
     
    48475015<p class="MsoNormal">The function calculated is:</p> 
    48485016<p class="MsoNormal">&nbsp;</p> 
    4849 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image099.PNG" ></span>&nbsp;</p> 
     5017<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image099.PNG"></span>&nbsp;</p> 
    48505018<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    48515019<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    48525020<p class="MsoNormal">with the functions:</p> 
    48535021<br> 
    4854 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image100.PNG" > 
    4855 </p><br> 
    4856  
     5022<p class="MsoNormal" style="text-align: center;" align="center"><img src="image100.PNG"> 
     5023</p> 
     5024<br> 
    48575025<p class="MsoNormal">and the angle psi is defined as the 
    48585026orientation of the major 
     
    50505218<p class="MsoNormal">&nbsp;</p> 
    50515219<p class="MsoNormal">&nbsp;</p> 
    5052 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.23.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5220<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.24.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    50535221</span></b><a name="BarBellModel"><b><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></b></a></p> 
    50545222<p class="MsoNormal">&nbsp;</p> 
     
    50905258calculated as:</p> 
    50915259<p class="MsoNormal">&nbsp;</p> 
    5092 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image106.PNG" ></span></p> 
     5260<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image106.PNG"></span></p> 
    50935261<p class="MsoNormal">&nbsp;</p> 
    50945262<p class="MsoNormal">where the amplitude A(q) is given as:</p> 
    50955263<br> 
    5096 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image107.PNG" ></p> 
     5264<p class="MsoNormal" style="text-align: center;" align="center"><img src="image107.PNG"></p> 
    50975265<br> 
    50985266<p class="MsoNormal">The &lt; &gt; brackets denote 
     
    52545422<p class="MsoNormal">&nbsp;</p> 
    52555423<p class="MsoNormal">&nbsp;</p> 
    5256 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.24.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5424<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.25.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    52575425</span></b><a name="CappedCylinderModel"><b><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></b></a></p> 
    52585426<p class="MsoNormal">&nbsp;</p> 
     
    54575625<p class="MsoNormal">&nbsp;</p> 
    54585626<p class="MsoNormal">&nbsp;</p> 
    5459 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.25.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5627<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.26.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    54605628</span></b><a name="EllipsoidModel"><b><span style="font-size: 14pt;">Ellipsoid 
    54615629Model</span></b></a></p> 
     
    54755643<p class="MsoNormal">&nbsp;</p> 
    54765644<p class="MsoNormal">&nbsp;</p> 
    5477 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG" ></span>&nbsp;&nbsp; 
     5645<p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG"></span>&nbsp;&nbsp; 
    54785646</p> 
    54795647<p class="MsoNormal">&nbsp;</p> 
    5480 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image119.PNG" ></span>&nbsp;&nbsp;&nbsp; 
     5648<p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image119.PNG"></span>&nbsp;&nbsp;&nbsp; 
    54815649</p> 
    54825650<p class="MsoNormal">&nbsp;</p> 
    5483 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image120.PNG" ></span>&nbsp;&nbsp;&nbsp;&nbsp; 
     5651<p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image120.PNG"></span>&nbsp;&nbsp;&nbsp;&nbsp; 
    54845652</p> 
    54855653<p class="MsoNormal">&nbsp;</p> 
     
    57135881<p class="MsoNormal"><b>&nbsp;</b></p> 
    57145882<p class="MsoNormal"><b>&nbsp;</b></p> 
    5715 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.26.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     5883<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.27.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    57165884</span></b><a name="CoreShellEllipsoidModel"><b><span style="font-size: 14pt;">CoreShellEllipsoidModel 
    57175885</span></b></a></p> 
     
    57375905<p class="MsoNormal">The form factor calculated is:</p> 
    57385906<br> 
    5739 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image126.PNG" ></p> 
     5907<p class="MsoNormal" style="text-align: center;" align="center"><img src="image126.PNG"></p> 
    57405908<br> 
    57415909<p class="MsoNormal">To provide easy access to the 
     
    57495917<p class="MsoNormal">The contrast is defined as SLD(core) 
    57505918– SLD(shell) or 
    5751 SLD(shell – solvent). In the parameters, equat_core = equatorial core radius,  
    5752 polar_core = polar core radius, equat_shell = r<sub>min</sub> 
     5919SLD(shell – solvent). In the parameters, equat_core = equatorial core 
     5920radius, polar_core = polar core radius, equat_shell = r<sub>min</sub> 
    57535921(or equatorial outer radius), and polar_shell = = r<sub>maj</sub> 
    57545922(or polar outer radius).</p> 
     
    59056073<p class="MsoNormal"><b>&nbsp;</b></p> 
    59066074<p class="MsoNormal"><b>&nbsp;</b></p> 
    5907 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.27.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6075<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.28.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    59086076</span></b><a name="TriaxialEllipsoidModel"><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></a></p> 
    59096077<p class="MsoNormal">&nbsp;</p> 
     
    59286096<p class="MsoNormal">The form factor calculated is:</p> 
    59296097<p class="MsoNormal">&nbsp;</p> 
    5930 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 33pt;"><img src="image129.PNG" ></span></p> 
     6098<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 33pt;"><img src="image129.PNG"></span></p> 
    59316099<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    59326100<p class="MsoNormal">&nbsp;</p> 
     
    61026270<p class="MsoNormal"><b>&nbsp;</b></p> 
    61036271<p class="MsoNormal"><b>&nbsp;</b></p> 
    6104 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.28.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6272<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.29.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    61056273</span></b><a name="LamellarModel"><b><span style="font-size: 14pt;">LamellarModel</span></b></a></p> 
    61066274<p class="MsoNormal">&nbsp;</p> 
     
    61146282<p class="MsoNormal">&nbsp;</p> 
    61156283<p class="MsoNormal">The scattering intensity I(q) is:</p> 
    6116 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image133.PNG" ></span></p> 
     6284<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image133.PNG"></span></p> 
    61176285<p class="MsoNormal">The form factor is,</p> 
    61186286<p class="MsoNormal">&nbsp;</p> 
    6119 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG" ></span></p> 
     6287<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG"></span></p> 
    61206288<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    61216289<p class="MsoNormal">where <span style="font-family: Symbol;">d</span> = bilayer 
     
    62266394<p class="MsoNormal"><b>&nbsp;</b></p> 
    62276395<p class="MsoNormal"><b>&nbsp;</b></p> 
    6228 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.29.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6396<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.30.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    62296397</span></b><a name="LamellarFFHGModel"><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></a></p> 
    62306398<p class="MsoNormal">&nbsp;</p> 
     
    62386406<p class="MsoNormal">&nbsp;</p> 
    62396407<p class="MsoNormal">The scattering intensity I(q) is:</p> 
    6240 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image136.PNG" ></span></p> 
     6408<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image136.PNG"></span></p> 
    62416409<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    62426410<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
     
    63796547<p class="MsoNormal"><b>&nbsp;</b></p> 
    63806548<p class="MsoNormal"><b>&nbsp;</b></p> 
    6381 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.30.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6549<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.31.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    63826550</span></b><a name="LamellarPSModel"><b><span style="font-size: 14pt;">LamellarPSModel</span></b></a></p> 
    63836551<p class="MsoNormal">&nbsp;</p> 
     
    63896557<p class="MsoNormal">&nbsp;</p> 
    63906558<p class="MsoNormal">The scattering intensity I(q) is:</p> 
    6391 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image139.PNG" ></span></p> 
     6559<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image139.PNG"></span></p> 
    63926560<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    63936561<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    63946562<p class="MsoNormal">The form factor is</p> 
    6395 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG" ></span></p> 
     6563<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG"></span></p> 
    63966564<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    63976565<p class="MsoNormal">and the structure is </p> 
    6398 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image140.PNG" ></span></p> 
     6566<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image140.PNG"></span></p> 
    63996567<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    64006568<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    64016569<p class="MsoNormal">where </p> 
    64026570<br> 
    6403 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG" ></p> 
     6571<p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG"></p> 
    64046572<br> 
    64056573<p class="MsoNormal">Here d= (repeat) spacing, <span style="font-family: Symbol;">d</span> 
     
    65436711<p class="MsoNormal"><b>&nbsp;</b></p> 
    65446712<p class="MsoNormal"><b>&nbsp;</b></p> 
    6545 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.31.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6713<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.32.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    65466714</span></b><a name="LamellarPSHGModel"><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></a></p> 
    65476715<p class="MsoNormal">&nbsp;</p> 
     
    65556723<p class="MsoNormal">&nbsp;</p> 
    65566724<p class="MsoNormal">The scattering intensity I(q) is:</p> 
    6557 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image139.PNG" ></span></p> 
     6725<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image139.PNG"></span></p> 
    65586726<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    65596727<p class="MsoNormal">The form factor is,</p> 
    6560 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image143.PNG" ></span></p> 
     6728<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image143.PNG"></span></p> 
    65616729<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    65626730<p class="MsoNormal">The structure factor is</p> 
    6563 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image140.PNG" ></span></p> 
     6731<p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image140.PNG"></span></p> 
    65646732<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    65656733<p class="MsoNormal" style="text-align: center;" align="center">&nbsp;</p> 
    65666734<p class="MsoNormal">where </p> 
    65676735<br> 
    6568 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG" ></p> 
     6736<p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG"></p> 
    65696737<br> 
    65706738<p class="MsoNormal">where <span style="font-family: Symbol;">d<sub>T</sub></span> 
     
    67466914<p class="MsoNormal"><b>&nbsp;</b></p> 
    67476915<p class="MsoNormal"><b>&nbsp;</b></p> 
    6748 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.32.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     6916<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.33.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    67496917</span></b><a name="LamellarPCrystalModel"><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></a></p> 
    67506918<p class="MsoNormal">&nbsp;</p> 
     
    69197087<p class="MsoNormal"><b>&nbsp;</b></p> 
    69207088<p class="MsoNormal"><b>&nbsp;</b></p> 
    6921 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.33.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7089<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.34.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    69227090</span></b><a name="SCCrystalModel"><b><span style="font-size: 14pt;">SC(Simple 
    69237091Cubic Para-)CrystalModel</span></b></a></p> 
     
    71377305<p class="MsoNormal">&nbsp;</p> 
    71387306<p class="MsoNormal"><b>&nbsp;</b></p> 
    7139 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.34.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7307<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.35.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    71407308</span></b><a name="FCCrystalModel"><b><span style="font-size: 14pt;">FC(Face 
    71417309Centered Cubic Para-)CrystalModel</span></b></a></p> 
     
    73617529<p class="MsoNormal">&nbsp;</p> 
    73627530<p class="MsoNormal"><b>&nbsp;</b></p> 
    7363 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.35.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     7531<p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.36.</span></b><b><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    73647532</span></b><a name="BCCrystalModel"><b><span style="font-size: 14pt;">BC(Body 
    73657533Centered Cubic Para-)CrystalModel</span></b></a></p> 
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