Changeset e0405592 in sasview for sansmodels/src/sans/models/media/model_functions.html
- Timestamp:
- Jul 2, 2012 3:01:29 PM (12 years ago)
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sansmodels/src/sans/models/media/model_functions.html
r940aca7 re0405592 444 444 {margin-bottom:0in;} 445 445 --> 446 </style><title></title></head> 446 </style><title></title> 447 448 </head> 447 449 <body lang="EN-US" link="blue" vlink="purple"> 448 450 <div class="WordSection1"> … … 453 455 <li class="MsoNormal" style="line-height: 115%;"><a href="#Introduction"><b>Introduction</b></a> 454 456 </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>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <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>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <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>, 456 458 <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, 457 459 <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, … … 531 533 particle:</p> 532 534 <p class="MsoNormal"> </p> 533 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image001.PNG" 535 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image001.PNG"></span> 534 536 </p> 535 537 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 536 538 <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" 539 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 8pt;"><img src="image002.PNG"></span> </p> 538 540 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 539 541 <p class="MsoNormal">where <i>P</i><sub>0</sub><i>(<b>q</b>)</i> … … 550 552 particle 551 553 volume fraction:<span style="position: relative; top: 5pt;"> 552 <img src="image003.PNG" 554 <img src="image003.PNG"></span>.</p> 553 555 <p class="MsoNormal"> </p> 554 556 <p class="MsoNormal">Our so-called 1D scattering intensity … … 588 590 way (Guinier, 1955):</p> 589 591 <p class="MsoNormal"> </p> 590 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image004.PNG" 592 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image004.PNG"></span> 591 593 </p> 592 594 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 726 728 calculation is an exact multi-component solution:</p> 727 729 <p class="MsoNormal"> </p> 728 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 5pt;"><img src="image006.PNG" 730 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 5pt;"><img src="image006.PNG"></span></p> 729 731 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 730 732 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 740 742 <p class="MsoNormal"> </p> 741 743 <p class="MsoNormal"> </p> 742 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 24pt;"><img src="image007.PNG" 744 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 24pt;"><img src="image007.PNG"></span>.</p> 743 745 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 744 746 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 746 748 <p class="MsoNormal">The 2D scattering intensity is the 747 749 same 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" 750 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> 749 751 <p class="MsoNormal"> </p> 750 752 <p class="MsoNormal"> </p> … … 904 906 calculated as:</p> 905 907 <p class="MsoNormal"> </p> 906 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image010.PNG" 908 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image010.PNG"></span></p> 907 909 <p class="MsoNormal"> </p> 908 910 <p class="MsoNormal">where the amplitude A(q) is given as … … 912 914 <p class="MsoNormal"> </p> 913 915 <p class="MsoNormal"> </p> 914 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 18pt;"><img src="image011.PNG" 916 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 18pt;"><img src="image011.PNG"></span></p> 915 917 <p class="MsoNormal"> </p> 916 918 <p class="MsoNormal"> </p> … … 947 949 <p class="MsoNormal">For 2D data: The 2D scattering 948 950 intensity 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" 951 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> 950 952 <p class="MsoNormal"> </p> 951 953 <p class="MsoNormal">REFERENCE</p> … … 1049 1051 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 1050 1052 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 1053 <p class="MsoNormal"> 1054 </p> 1051 1055 <p class="MsoNormal"> </p> 1052 1056 <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;"> 1057 </span></b><a name="RaspBerryModel"></a><b><span style="font-size: 14pt;">RaspBerryModel</span></b></p> 1058 <b> <br> 1059 </b>Calculates the form factor, P(q), for a "Raspberry-like" 1060 structure where there are smaller spheres at the surface of a larger 1061 sphere, such as the structure of a Pickering emulsion. 1062 <p class="MsoNormal"> </p> 1063 <p class="MsoNormal" style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> 1064 </span>Definition</b></p> 1065 <p class="MsoNormal"> </p> 1066 The structure is:<br> 1067 <img src="raspberry_pic.jpg"> 1068 <p class="MsoNormal"><br> 1069 Ro = the radius of the large sphere<br> 1070 Rp = the radius of the smaller sphere on the surface<br> 1071 delta = the fractional penetration depth<br> 1072 surface coverage = fractional coverage of the large sphere surface (0.9 1073 max)<br> 1074 <br> 1075 <br> 1076 The large and small spheres have their own SLD, as well as the solvent. 1077 The surface coverage term is a fractional coverage (maximum of 1078 approximately 0.9 for hexagonally packed spheres on a surface). Since 1079 not all of the small spheres are necessarily attached to the surface, 1080 the excess free (small) spheres scattering is also included in the 1081 calculation. The function calculated follows equations (8)-(12) of the 1082 reference below, and the equations are not reproduced here.<br> 1083 <br> 1084 The returned value is scaled to units of [cm-1]. No interparticle 1085 scattering is included in this model.<br> 1086 <br> 1087 </p> 1088 <p class="MsoNormal">For 2D data: The 2D scattering 1089 intensity is calculated in 1090 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> 1091 <p class="MsoNormal"> </p> 1092 REFERENCE<br> 1093 Kjersta Larson-Smith, Andrew Jackson, and Danilo C 1094 Pozzo, "Small angle scattering model for Pickering emulsions and 1095 raspberry particles." Journal of Colloid and Interface Science (2010) 1096 vol. 343 (1) pp. 36-41.<br> 1097 <p class="MsoNormal"> </p> 1098 <p class="MsoNormal">TEST DATASET</p> 1099 <p class="MsoNormal">This example dataset is produced by 1100 running the RaspBerryModel, using 2000 data points, qmin = 0.0001 Å<sup>-1</sup>, 1101 qmax = 0.2 A<sup>-1</sup> and the default values, where 1102 Ssph/Lsph stands for Smaller/Large sphere and surfrac_Ssph for 1103 the surface fraction of the smaller spheres.</p> 1104 <p class="MsoNormal"> </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> delta_Ssph</td> 1121 <td></td> 1122 <td> 0</td> 1123 </tr> 1124 <tr> 1125 <td> radius_Lsph</td> 1126 <td> Å</td> 1127 <td> 5000</td> 1128 </tr> 1129 <tr> 1130 <td> radius_Ssph</td> 1131 <td> Å</td> 1132 <td> 100</td> 1133 </tr> 1134 <tr> 1135 <td> sld_Lsph</td> 1136 <td> Å<sup> -2</sup></td> 1137 <td> -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"> </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"> </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. 1213 1D plot using the values of /2000 data points.</b></p> 1214 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 1215 <p class="MsoNormal" style="text-align: center;" align="center"></p> 1216 <p class="MsoNormal"> </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;"> 1053 1219 </span></b><a name="CoreShellModel"><b><span style="font-size: 14pt;">Core Shell 1054 1220 (Sphere) Model</span></b></a></p> … … 1067 1233 way (Guinier, 1955):</p> 1068 1234 <p class="MsoNormal"> </p> 1069 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image013.PNG" 1235 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image013.PNG"></span> </p> 1070 1236 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 1071 1237 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 1229 1395 <p class="MsoNormal"> </p> 1230 1396 <p class="MsoNormal"> </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;"> 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;"> 1232 1398 </span></b><a name="Core2ndMomentModel"><b><span style="font-size: 14pt;">Core2ndMomentModel</span></b></a></p> 1233 1399 <p class="MsoNormal"> </p> … … 1399 1565 <p class="MsoNormal"> </p> 1400 1566 <p class="MsoNormal"> </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;"> 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;"> 1402 1568 </span></b><a name="CoreMultiShellModel"><b><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model</span></b></a></p> 1403 1569 <p class="MsoNormal"> </p> … … 1638 1804 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 1639 1805 <p class="MsoNormal"><b> </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;"> 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;"> 1641 1807 </span></b><a name="VesicleModel"><b><span style="font-size: 14pt;">VesicleModel</span></b></a></p> 1642 1808 <p class="MsoNormal"> </p> … … 1683 1849 above, regardless of the orientation of the <i>q</i> 1684 1850 vector which is 1685 defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG" 1851 defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="image008.PNG"></span><span style="font-size: 14pt;">.</span></p> 1686 1852 <p class="MsoNormal">For P*S: The outer most radius (= 1687 1853 radius + thickness) is … … 1802 1968 <p class="MsoNormal"><b> </b></p> 1803 1969 <p class="MsoNormal"><b> </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;"> 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;"> 1805 1971 </span></b><a name="MultiShellModel"><b><span style="font-size: 14pt;">MultiShellModel</span></b></a></p> 1806 1972 <p class="MsoNormal"> </p> … … 1819 1985 <p class="MsoNormal">The 2D scattering intensity is the 1820 1986 same 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" 1987 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> 1822 1988 <p class="MsoNormal">For P*S: The outer most radius (= 1823 1989 core_radius + … … 1963 2129 <p class="MsoNormal"> </p> 1964 2130 <p class="MsoNormal"><b> </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;"> 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;"> 1966 2132 </span></b><a name="OnionExpShellModel"><b><span style="font-size: 14pt;">OnionExpShellModel</span></b></a></p> 1967 2133 <p class="MsoNormal"> </p> … … 2222 2388 <p class="MsoNormal"> </p> 2223 2389 <p class="MsoNormal"><b> </b></p> 2224 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 0.</span></b><b><span style="font-size: 7pt;"> 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;"> 2225 2391 </span></b><a name="SphericalSLDModel"><b><span style="font-size: 14pt;">SphericalSLDModel</span></b></a></p> 2226 2392 <p class="MsoNormal"> </p> … … 2265 2431 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image024.gif" height="41" width="200"></span></p> 2266 2432 <p class="MsoNormal">so that</p> 2267 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image044.gif" 2268 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 2269 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image045.gif" 2270 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 2271 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image046.gif" 2272 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image047.gif" 2273 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image048.gif" 2274 <p class="MsoNormal"> </p> 2275 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image027.gif" 2433 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image044.gif"></span></p> 2434 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 2435 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image045.gif"></span></p> 2436 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 2437 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image046.gif"></span></p> 2438 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image047.gif"></span></p> 2439 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image048.gif"></span></p> 2440 <p class="MsoNormal"> </p> 2441 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image027.gif"></span></p> 2276 2442 <p class="MsoNormal"> </p> 2277 2443 <p class="MsoNormal"> </p> … … 2285 2451 <p class="MsoListParagraph" style="text-indent: -0.25in;">1)<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> 2286 2452 </span>Exp; </p> 2287 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image049.gif" 2453 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image049.gif"></span></p> 2288 2454 <p class="MsoNormal"> </p> 2289 2455 <p class="MsoListParagraph" style="text-indent: -0.25in;">2)<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> … … 2291 2457 <p class="MsoNormal" style="margin-left: 0.25in;"> </p> 2292 2458 <p class="MsoListParagraph"> </p> 2293 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image050.gif" 2459 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image050.gif"></span></p> 2294 2460 <p class="MsoNormal"> </p> 2295 2461 <p class="MsoNormal"> </p> … … 2297 2463 </span>Erf;</p> 2298 2464 <p class="MsoNormal" style="margin-left: 0.25in;"> </p> 2299 <p class="MsoListParagraph"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image051.gif" 2465 <p class="MsoListParagraph"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image051.gif"></span></p> 2300 2466 <p class="MsoListParagraph"> </p> 2301 2467 <p class="MsoListParagraph"> </p> … … 2321 2487 <p class="MsoNormal"> </p> 2322 2488 <p class="MsoNormal">In the equation, </p> 2323 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image055.gif" 2489 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image055.gif"></span></p> 2324 2490 <p class="MsoNormal"> </p> 2325 2491 <p class="MsoNormal"> </p> … … 2327 2493 calculated by</p> 2328 2494 <p class="MsoNormal"> </p> 2329 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image037.gif" 2495 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image037.gif"></span></p> 2330 2496 <p class="MsoNormal">where </p> 2331 2497 <p class="MsoNormal"> </p> 2332 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image038.gif" 2498 <p class="MsoNormal"><span style="font-size: 12pt; font-family: "Times New Roman","serif";"><img src="image038.gif"></span></p> 2333 2499 <p class="MsoNormal"><span style="font-family: Symbol;"> </span></p> 2334 2500 <p class="MsoNormal"><span style="font-family: Symbol;"> </span></p> 2335 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-family: Symbol;"><img id="Object 20" src="image056.gif" 2501 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-family: Symbol;"><img id="Object 20" src="image056.gif"></span></p> 2336 2502 <p class="MsoNormal" style="text-align: center;" align="center"><span style="font-family: Symbol;"> </span></p> 2337 2503 <p class="MsoNormal"> </p> … … 2340 2506 above, regardless of the orientation of the <i>q</i> 2341 2507 vector which is 2342 defined as<span style="font-size: 12pt; font-family: "Times New Roman","serif"; position: relative; top: 4.5pt;"><img src="image040.gif" 2508 defined as<span style="font-size: 12pt; font-family: "Times New Roman","serif"; position: relative; top: 4.5pt;"><img src="image040.gif"></span><span style="font-size: 14pt;">.</span></p> 2343 2509 <p class="MsoNormal"> </p> 2344 2510 <p class="MsoNormal">For P*S: The outer most radius is … … 2535 2701 <p class="MsoNormal"> </p> 2536 2702 <p class="MsoNormal"> </p> 2537 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 1.</span></b><b><span style="font-size: 7pt;"> 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;"> 2538 2704 </span></b><b><a name="LinearPearlsModel"><span style="font-size: 14pt;">LinearPearlsModel</span></a></b></p> 2539 2705 <p class="MsoNormal"><a name="LinearPearlsModel"> </a></p> … … 2676 2842 <p class="MsoNormal"><a name="LinearPearlsModel"> </a></p> 2677 2843 <p class="MsoNormal"><a name="LinearPearlsModel"> </a></p> 2678 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><a name="LinearPearlsModel"><b><span style="font-size: 14pt;">2.1 2.</span></b><b><span style="font-size: 7pt;"> 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;"> 2679 2845 </span></b></a><b><a name="PearlNecklaceModel"><span style="font-size: 14pt;">PearlNecklaceModel</span></a></b></p> 2680 2846 <p class="MsoNormal"><a name="PearlNecklaceModel"> </a></p> … … 2851 3017 <p class="MsoNormal"><a name="PearlNecklaceModel"> </a></p> 2852 3018 <p class="MsoNormal"><a name="PearlNecklaceModel"> </a></p> 2853 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"><b><span style="font-size: 14pt;">2.1 3.</span></b><b><span style="font-size: 7pt;"> 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;"> 2854 3020 </span></b></a><a name="CylinderModel"><b><span style="font-size: 14pt;">Cylinder 2855 3021 Model</span></b></a></p> … … 2869 3035 <p class="MsoNormal"> </p> 2870 3036 <p class="MsoNormal"> </p> 2871 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG" 3037 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG"></span> 2872 3038 </p> 2873 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image060.PNG" 3039 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image060.PNG"></span> 2874 3040 </p> 2875 3041 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 3013 3179 randomly oriented cylinders is then given by:</p> 3014 3180 <p class="MsoNormal"> </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" 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> 3016 3182 </p> 3017 3183 <p class="MsoNormal"> </p> … … 3041 3207 done by evaluating the following:</p> 3042 3208 <p class="MsoNormal"> </p> 3043 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image064.PNG" 3209 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image064.PNG"></span> 3044 3210 </p> 3045 3211 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 3084 3250 <p class="MsoNormal"><b> </b></p> 3085 3251 <p class="MsoNormal"><b> </b></p> 3086 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 4.</span></b><b><span style="font-size: 7pt;"> 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;"> 3087 3253 </span></b><a name="CoreShellCylinderModel"><b><span style="font-size: 14pt;">Core-Shell 3088 3254 Cylinder Model</span></b></a></p> … … 3100 3266 oriented core-shell cylinders is given by (Kline, 2006):</p> 3101 3267 <p class="MsoNormal"> </p> 3102 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image067.PNG" 3103 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 32pt;"><img src="image068.PNG" 3268 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image067.PNG"></span> </p> 3269 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 32pt;"><img src="image068.PNG"></span> </p> 3104 3270 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 3105 3271 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 3338 3504 <p class="MsoNormal"> </p> 3339 3505 <p class="MsoNormal"> </p> 3340 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 5.</span></b><b><span style="font-size: 7pt;"> 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;"> 3341 3507 </span></b><a name="CoreShellBicelleModel"><b><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></b></a></p> 3342 3508 <p class="MsoNormal"> </p> … … 3515 3681 <p class="MsoNormal"><b> </b></p> 3516 3682 <p class="MsoNormal"><b> </b></p> 3517 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 6.</span></b><b><span style="font-size: 7pt;"> 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;"> 3518 3684 </span></b><a name="HollowCylinderModel"><b><span style="font-size: 14pt;">HollowCylinderModel</span></b></a></p> 3519 3685 <p class="MsoNormal"> </p> … … 3532 3698 way (Guinier, 1955):</p> 3533 3699 <p class="MsoNormal"> </p> 3534 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image072.PNG" 3700 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image072.PNG"></p> 3535 3701 <br> 3536 3702 <nr> 3703 </nr> 3537 3704 <p class="MsoNormal">where <i>scale</i> is a 3538 3705 scale factor, <i>J<sub>1</sub></i> … … 3682 3849 <p class="MsoNormal"><b> </b></p> 3683 3850 <p class="MsoNormal"><b> </b></p> 3684 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 7.</span></b><b><span style="font-size: 7pt;"> 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;"> 3685 3852 </span></b><a name="FlexibleCylinderModel"></a><b><span style="font-size: 14pt;">FlexibleCylinderModel</span></b></p> 3686 3853 <p class="MsoNormal"> </p> … … 3845 4012 <p class="MsoNormal"> </p> 3846 4013 <p class="MsoNormal"> </p> 3847 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1 8.</span></b><b><span style="font-size: 7pt;"> 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;"> 3848 4015 </span></b><b><span style="font-size: 14pt;">FlexCylEllipXMo<a name="FlexCylEllipXModel"></a>del</span></b></p> 3849 4016 <p class="MsoNormal"> </p> … … 3939 4106 <p class="MsoNormal">For 2D data: The 2D scattering 3940 4107 intensity 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" 4108 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> 3942 4109 <p class="MsoNormal"> </p> 3943 4110 <p class="MsoNormal">REFERENCE</p> … … 4072 4239 <p class="MsoNormal"> </p> 4073 4240 <p class="MsoNormal"><b> </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;"> 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;"> 4075 4242 </span></b><a name="StackedDisksModel"><b><span style="font-size: 14pt;">StackedDisksModel 4076 4243 </span></b></a></p> … … 4094 4261 <p class="MsoNormal">The 2D scattering intensity is the 4095 4262 same 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" 4263 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> 4097 4264 <p class="MsoNormal"><span style="font-size: 14pt;"> </span></p> 4098 4265 <p class="MsoNormal"> </p> … … 4108 4275 <p class="MsoNormal">The scattering intensity I(q) is:</p> 4109 4276 <p class="MsoNormal"> </p> 4110 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 9pt;"><img src="image081.PNG" 4277 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 9pt;"><img src="image081.PNG"></span></p> 4111 4278 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4112 4279 <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" 4280 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image082.PNG"></span></p> 4114 4281 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4115 4282 <p class="MsoNormal"> </p> … … 4121 4288 disc, respectively.</p> 4122 4289 <p class="MsoNormal"> </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 4126 4295 (layer_thick), 2h= core 4127 4296 thickness (core_thick), and R = radius of the disc (radius).</p> 4128 4129 <p class="MsoNormal"> </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"> </p> 4298 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 14pt;"><img src="image084.PNG"></span></p> 4131 4299 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4132 4300 <p class="MsoNormal"> </p> … … 4314 4482 <p class="MsoNormal"><b> </b></p> 4315 4483 <p class="MsoNormal"><b> </b></p> 4316 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 0.</span></b><b><span style="font-size: 7pt;"> 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;"> 4317 4485 </span></b><a name="ParallelepipedModel"><b><span style="font-size: 14pt;">ParallelepipedModel</span></b></a></p> 4318 4486 <p class="MsoNormal"> </p> … … 4339 4507 c=C/B>1, the form factor,</p> 4340 4508 <p class="MsoNormal"> </p> 4341 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 48pt;"><img src="image088.PNG" 4509 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 48pt;"><img src="image088.PNG"></span></p> 4342 4510 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4343 4511 <p class="MsoNormal">The contrast is defined as</p> 4344 4512 <p class="MsoNormal"> </p> 4345 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image089.PNG" 4513 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image089.PNG"></span></p> 4346 4514 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4347 4515 <p class="MsoNormal"> </p> … … 4509 4677 <p class="MsoNormal"><b> </b></p> 4510 4678 <p class="MsoNormal"><b> </b></p> 4511 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 1.</span></b><b><span style="font-size: 7pt;"> 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;"> 4512 4680 </span></b><b><span style="font-size: 14pt;"><a name="CSParallelepipedModel">CSParallelepipedModel</a></span></b></p> 4513 4681 <p class="MsoNormal"> </p> … … 4542 4710 <p class="MsoNormal">The volume of the solid is:</p> 4543 4711 <p class="MsoNormal"> </p> 4544 <p class="MsoNormal"><span style="position: relative; top: 5pt;"><img src="image095.PNG" 4712 <p class="MsoNormal"><span style="position: relative; top: 5pt;"><img src="image095.PNG"></span></p> 4545 4713 <p class="MsoNormal">meaning that there are "gaps" at the 4546 4714 corners of … … 4818 4986 <p class="MsoNormal"> </p> 4819 4987 <p class="MsoNormal"> </p> 4820 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 2.</span></b><b><span style="font-size: 7pt;"> 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;"> 4821 4989 </span></b><a name="EllipticalCylinderModel"><b><span style="font-size: 14pt;">Elliptical 4822 4990 Cylinder Model</span></b></a></p> … … 4847 5015 <p class="MsoNormal">The function calculated is:</p> 4848 5016 <p class="MsoNormal"> </p> 4849 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image099.PNG" 5017 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image099.PNG"></span> </p> 4850 5018 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4851 5019 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 4852 5020 <p class="MsoNormal">with the functions:</p> 4853 5021 <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> 4857 5025 <p class="MsoNormal">and the angle psi is defined as the 4858 5026 orientation of the major … … 5050 5218 <p class="MsoNormal"> </p> 5051 5219 <p class="MsoNormal"> </p> 5052 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 3.</span></b><b><span style="font-size: 7pt;"> 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;"> 5053 5221 </span></b><a name="BarBellModel"><b><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></b></a></p> 5054 5222 <p class="MsoNormal"> </p> … … 5090 5258 calculated as:</p> 5091 5259 <p class="MsoNormal"> </p> 5092 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image106.PNG" 5260 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image106.PNG"></span></p> 5093 5261 <p class="MsoNormal"> </p> 5094 5262 <p class="MsoNormal">where the amplitude A(q) is given as:</p> 5095 5263 <br> 5096 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image107.PNG" 5264 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image107.PNG"></p> 5097 5265 <br> 5098 5266 <p class="MsoNormal">The < > brackets denote … … 5254 5422 <p class="MsoNormal"> </p> 5255 5423 <p class="MsoNormal"> </p> 5256 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 4.</span></b><b><span style="font-size: 7pt;"> 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;"> 5257 5425 </span></b><a name="CappedCylinderModel"><b><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></b></a></p> 5258 5426 <p class="MsoNormal"> </p> … … 5457 5625 <p class="MsoNormal"> </p> 5458 5626 <p class="MsoNormal"> </p> 5459 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 5.</span></b><b><span style="font-size: 7pt;"> 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;"> 5460 5628 </span></b><a name="EllipsoidModel"><b><span style="font-size: 14pt;">Ellipsoid 5461 5629 Model</span></b></a></p> … … 5475 5643 <p class="MsoNormal"> </p> 5476 5644 <p class="MsoNormal"> </p> 5477 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG" 5645 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="image059.PNG"></span> 5478 5646 </p> 5479 5647 <p class="MsoNormal"> </p> 5480 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image119.PNG" 5648 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image119.PNG"></span> 5481 5649 </p> 5482 5650 <p class="MsoNormal"> </p> 5483 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image120.PNG" 5651 <p class="MsoCaption" style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="image120.PNG"></span> 5484 5652 </p> 5485 5653 <p class="MsoNormal"> </p> … … 5713 5881 <p class="MsoNormal"><b> </b></p> 5714 5882 <p class="MsoNormal"><b> </b></p> 5715 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 6.</span></b><b><span style="font-size: 7pt;"> 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;"> 5716 5884 </span></b><a name="CoreShellEllipsoidModel"><b><span style="font-size: 14pt;">CoreShellEllipsoidModel 5717 5885 </span></b></a></p> … … 5737 5905 <p class="MsoNormal">The form factor calculated is:</p> 5738 5906 <br> 5739 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image126.PNG" 5907 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image126.PNG"></p> 5740 5908 <br> 5741 5909 <p class="MsoNormal">To provide easy access to the … … 5749 5917 <p class="MsoNormal">The contrast is defined as SLD(core) 5750 5918 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>5919 SLD(shell solvent). In the parameters, equat_core = equatorial core 5920 radius, polar_core = polar core radius, equat_shell = r<sub>min</sub> 5753 5921 (or equatorial outer radius), and polar_shell = = r<sub>maj</sub> 5754 5922 (or polar outer radius).</p> … … 5905 6073 <p class="MsoNormal"><b> </b></p> 5906 6074 <p class="MsoNormal"><b> </b></p> 5907 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 7.</span></b><b><span style="font-size: 7pt;"> 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;"> 5908 6076 </span></b><a name="TriaxialEllipsoidModel"><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></a></p> 5909 6077 <p class="MsoNormal"> </p> … … 5928 6096 <p class="MsoNormal">The form factor calculated is:</p> 5929 6097 <p class="MsoNormal"> </p> 5930 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 33pt;"><img src="image129.PNG" 6098 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 33pt;"><img src="image129.PNG"></span></p> 5931 6099 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 5932 6100 <p class="MsoNormal"> </p> … … 6102 6270 <p class="MsoNormal"><b> </b></p> 6103 6271 <p class="MsoNormal"><b> </b></p> 6104 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 8.</span></b><b><span style="font-size: 7pt;"> 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;"> 6105 6273 </span></b><a name="LamellarModel"><b><span style="font-size: 14pt;">LamellarModel</span></b></a></p> 6106 6274 <p class="MsoNormal"> </p> … … 6114 6282 <p class="MsoNormal"> </p> 6115 6283 <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" 6284 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image133.PNG"></span></p> 6117 6285 <p class="MsoNormal">The form factor is,</p> 6118 6286 <p class="MsoNormal"> </p> 6119 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG" 6287 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG"></span></p> 6120 6288 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6121 6289 <p class="MsoNormal">where <span style="font-family: Symbol;">d</span> = bilayer … … 6226 6394 <p class="MsoNormal"><b> </b></p> 6227 6395 <p class="MsoNormal"><b> </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;"> 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;"> 6229 6397 </span></b><a name="LamellarFFHGModel"><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></a></p> 6230 6398 <p class="MsoNormal"> </p> … … 6238 6406 <p class="MsoNormal"> </p> 6239 6407 <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" 6408 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image136.PNG"></span></p> 6241 6409 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6242 6410 <p class="MsoNormal" style="text-align: center;" align="center"> </p> … … 6379 6547 <p class="MsoNormal"><b> </b></p> 6380 6548 <p class="MsoNormal"><b> </b></p> 6381 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 0.</span></b><b><span style="font-size: 7pt;"> 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;"> 6382 6550 </span></b><a name="LamellarPSModel"><b><span style="font-size: 14pt;">LamellarPSModel</span></b></a></p> 6383 6551 <p class="MsoNormal"> </p> … … 6389 6557 <p class="MsoNormal"> </p> 6390 6558 <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" 6559 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image139.PNG"></span></p> 6392 6560 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6393 6561 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6394 6562 <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" 6563 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image134.PNG"></span></p> 6396 6564 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6397 6565 <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" 6566 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image140.PNG"></span></p> 6399 6567 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6400 6568 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6401 6569 <p class="MsoNormal">where </p> 6402 6570 <br> 6403 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG" 6571 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG"></p> 6404 6572 <br> 6405 6573 <p class="MsoNormal">Here d= (repeat) spacing, <span style="font-family: Symbol;">d</span> … … 6543 6711 <p class="MsoNormal"><b> </b></p> 6544 6712 <p class="MsoNormal"><b> </b></p> 6545 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 1.</span></b><b><span style="font-size: 7pt;"> 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;"> 6546 6714 </span></b><a name="LamellarPSHGModel"><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></a></p> 6547 6715 <p class="MsoNormal"> </p> … … 6555 6723 <p class="MsoNormal"> </p> 6556 6724 <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" 6725 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image139.PNG"></span></p> 6558 6726 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6559 6727 <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" 6728 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="image143.PNG"></span></p> 6561 6729 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6562 6730 <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" 6731 <p class="MsoNormal" style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="image140.PNG"></span></p> 6564 6732 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6565 6733 <p class="MsoNormal" style="text-align: center;" align="center"> </p> 6566 6734 <p class="MsoNormal">where </p> 6567 6735 <br> 6568 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG" 6736 <p class="MsoNormal" style="text-align: center;" align="center"><img src="image141.PNG"></p> 6569 6737 <br> 6570 6738 <p class="MsoNormal">where <span style="font-family: Symbol;">d<sub>T</sub></span> … … 6746 6914 <p class="MsoNormal"><b> </b></p> 6747 6915 <p class="MsoNormal"><b> </b></p> 6748 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 2.</span></b><b><span style="font-size: 7pt;"> 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;"> 6749 6917 </span></b><a name="LamellarPCrystalModel"><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></a></p> 6750 6918 <p class="MsoNormal"> </p> … … 6919 7087 <p class="MsoNormal"><b> </b></p> 6920 7088 <p class="MsoNormal"><b> </b></p> 6921 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 3.</span></b><b><span style="font-size: 7pt;"> 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;"> 6922 7090 </span></b><a name="SCCrystalModel"><b><span style="font-size: 14pt;">SC(Simple 6923 7091 Cubic Para-)CrystalModel</span></b></a></p> … … 7137 7305 <p class="MsoNormal"> </p> 7138 7306 <p class="MsoNormal"><b> </b></p> 7139 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 4.</span></b><b><span style="font-size: 7pt;"> 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;"> 7140 7308 </span></b><a name="FCCrystalModel"><b><span style="font-size: 14pt;">FC(Face 7141 7309 Centered Cubic Para-)CrystalModel</span></b></a></p> … … 7361 7529 <p class="MsoNormal"> </p> 7362 7530 <p class="MsoNormal"><b> </b></p> 7363 <p class="MsoNormal" style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 5.</span></b><b><span style="font-size: 7pt;"> 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;"> 7364 7532 </span></b><a name="BCCrystalModel"><b><span style="font-size: 14pt;">BC(Body 7365 7533 Centered Cubic Para-)CrystalModel</span></b></a></p>
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