source: sasview/sansview/doc/model_functions.html @ 588f84f

ESS_GUIESS_GUI_DocsESS_GUI_batch_fittingESS_GUI_bumps_abstractionESS_GUI_iss1116ESS_GUI_iss879ESS_GUI_iss959ESS_GUI_openclESS_GUI_orderingESS_GUI_sync_sascalccostrafo411magnetic_scattrelease-4.1.1release-4.1.2release-4.2.2release_4.0.1ticket-1009ticket-1094-headlessticket-1242-2d-resolutionticket-1243ticket-1249ticket885unittest-saveload
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118</head>
119
120<body lang=EN-US vlink=purple>
121
122<div class=Section1>
123
124<p class=MsoNormal><b><span style='font-size:16.0pt'>Model Functions</span></b></p>
125
126<p class=MsoNormal>&nbsp;</p>
127
128<ul>
129<li><a href="#Introduction"><b>Introduction</b></a></li>
130<li><a href="#Shapes"><b>Shapes</b></a>
131<a href="#SphereModel">SphereModel</a>,
132<a href="#CylinderModel">CylinderModel</a>,
133<a href="#EllipsoidModel">EllipsoidModel</a>,
134<a href="#CoreShellModel">CoreShellModel</a>,
135<a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>,
136<a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>.
137</li>
138<li><a href="#Shape-Independent"><b>Shape-Independent</b></a>:
139<a href="#Debye">Debye</a>,
140<a href="#Lorentz">Lorentz</a>,
141<a href="#DAB_Model">DAB_Model</a>,
142<a href="#Power_Law">Power_Law</a>,
143<a href="#Teubner Strey">Teubner Strey</a>,
144<a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>,
145<a href="#Number Density Fractal">Number Density Fractal</a>,
146<a href="#Guinier">Guinier</a>,
147<a href="#PorodModel">PorodModel</a>. </li>
148<li><a href="#Customized Models"><b>Customized Models</b></a>
149<a href="#A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a>. </li>
150<li><a href="#Structure Factors"><b>Structure Factors</b></a>
151<a href="#HardSphereStructure">HardSphereStructure</a>,
152<a href="#SquareWellStructure">SquareWellStructure</a>,
153<a href="#HayterMSAStructure">HayterMSAStructure</a>,
154<a href="#StickyHSStructure">StickyHSStructure</a>.</li> 
155<li><a href="#References"><b>References</b></a></li>
156</ul>
157
158<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
159style='font-size:14.0pt'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
160</span></span></b><b><span style='font-size:14.0pt'><a name="Introduction">Introduction</a></span></b></p>
161
162<p class=MsoNormal>The present text documents the modules made available by the
163DANSE software for SANS. </p>
164
165<p class=MsoNormal>&nbsp;</p>
166
167<p class=MsoNormal>Readers are also referred to the SANS/DANSE wiki page:</p>
168
169<p class=MsoNormal>&nbsp;</p>
170
171<p class=MsoNormal style='text-indent:.25in'><i>http://danse.us/trac/sans</i></p>
172
173<p class=MsoNormal>&nbsp;</p>
174
175<p class=MsoNormal>Users can report a bug at the following URL:</p>
176
177<p class=MsoNormal>&nbsp;</p>
178
179<p class=MsoNormal style='text-indent:.25in'><i>http://danse.us/trac/sans/newticket</i></p>
180
181<p class=MsoNormal><b>&nbsp;</b></p>
182
183
184<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
185style='font-size:14.0pt'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
186</span></span></b><b><span style='font-size:14.0pt'><a name="Shapes">Shapes (Scattering intensity models)</a></span></b></p>
187
188<p class=MsoNormal>&nbsp;</p>
189
190<p class=MsoNormal>This software provides form factors for various particle
191shapes. After giving a mathematical definition of each model, we draw the list
192of parameters available to the user. Validation plots for each model are also
193presented. Instructions on how to use the software is available with the source
194code, available from SVN:</p>
195
196<p class=MsoNormal>&nbsp;</p>
197
198<p class=MsoNormal style='text-indent:.25in'><i>svn co svn://danse.us/sans/releases/sansmodels-x.x</i></p>
199
200<p class=MsoNormal>&nbsp;</p>
201
202<p class=MsoNormal>To easily compare to the scattering intensity measured in
203experiments, we normalize the form factors by the volume of the particle:</p>
204
205<p class=MsoNormal align=center style='text-align:center'><sub><img width=192
206height=43 src="../images/html/image001.gif"></sub>            (1)</p>
207
208<p class=MsoNormal>with</p>
209
210<p class=MsoNormal align=center style='text-align:center'><sub><img width=159
211height=29 src="../images/html/image002.gif"></sub>        (2)</p>
212
213<p class=MsoNormal>where <i>P</i><sub>0</sub><i>(<b>q</b>)</i> is the
214un-normalized form factor, <i>&#961;(<b>r</b>)</i> is the scattering length
215density at a given point in space and the integration is done over the volume <i>V</i>
216of the scatterer. </p>
217
218<p class=MsoNormal>&nbsp;</p>
219
220<p class=MsoNormal>For systems without inter-particle interference, the form
221factors we provide can be related to the scattering intensity by the particle
222volume fraction:<sub><img width=92 height=21
223src="../images/html/image003.gif"></sub>.</p>
224
225<p class=MsoNormal>&nbsp;</p>
226
227<p class=MsoNormal>Our so-called 1D scattering intensity functions provide <i>P(q)
228</i>for the case where the scatterer is randomly oriented. In that case, the
229scattering intensity only depends on the length of q. The intensity measured on
230the plane of the SANS detector will have an azimuthal symmetry around <i>q</i>=0.</p>
231
232<p class=MsoNormal>&nbsp;</p>
233
234<p class=MsoNormal>Our so-called 2D scattering intensity functions provide <i>P(q,
235</i><i><span style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> for an
236oriented system as a function of a q-vector in the plane of the detector. We
237define the angle <span style='font-family:"Arial","sans-serif"'>&#966;</span>
238as the angle between the q vector and the horizontal (x) axis of the plane of
239the detector.</p>
240
241<p class=MsoNormal>&nbsp;</p>
242
243
244<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
245style='font-size:14.0pt'>2.1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
246</span></span></b><b><span style='font-size:14.0pt'><a name="SphereModel">SphereModel</a></span></b></p>
247
248<p class=MsoNormal>&nbsp;</p>
249
250<p class=MsoNormal>This model provides the form factor, P(q), for a
251monodisperse spherical particle with uniform scattering length density. The
252form factor is normalized by the particle volume as described below.</p>
253
254<p class=MsoNormal>&nbsp;</p>
255
256<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.1.<span
257style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p>
258
259<p class=MsoNormal>&nbsp;</p>
260
261<p class=MsoNormal>The 1D scattering intensity is calculated in the following way
262(Guinier, 1955):</p>
263
264<p class=MsoNormal align=center style='text-align:center'><sub><img width=337
265height=53 src="../images/html/image004.gif"></sub>            (3)</p>
266
267<p class=MsoNormal>where <i>scale</i> is a scale factor, V is the volume of the
268scatterer, <i>r</i> is the radius of the sphere, <i>bkg</i> is the background
269level and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i>
270(contrast)  is the scattering length density difference between the scatterer
271and the solvent it is in.</p>
272
273<p class=MsoNormal>&nbsp;</p>
274
275<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above,
276regardless of the orientation of the q vector.</p>
277
278<p class=MsoNormal>&nbsp;</p>
279
280<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
281and the parameters of the sphere model are the following:</p>
282
283<div align=center>
284
285<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
286 style='border-collapse:collapse;border:none'>
287 <tr style='height:13.5pt'>
288  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
289  padding:0in 5.4pt 0in 5.4pt;height:13.5pt'>
290  <p class=MsoBodyText>Parameter name</p>
291  </td>
292  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
293  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:13.5pt'>
294  <p class=MsoBodyText>Units</p>
295  </td>
296  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
297  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:13.5pt'>
298  <p class=MsoBodyText>Default value</p>
299  </td>
300 </tr>
301 <tr style='height:12.8pt'>
302  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
303  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
304  <p class=MsoBodyText>Scale</p>
305  </td>
306  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
307  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
308  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
309  <p class=MsoBodyText>None</p>
310  </td>
311  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
312  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
313  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
314  <p class=MsoBodyText>1.0e-6</p>
315  </td>
316 </tr>
317 <tr style='height:12.8pt'>
318  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
319  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
320  <p class=MsoBodyText>Radius</p>
321  </td>
322  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
323  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
324  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
325  <p class=MsoBodyText>Å</p>
326  </td>
327  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
328  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
329  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
330  <p class=MsoBodyText>60</p>
331  </td>
332 </tr>
333 <tr style='height:12.8pt'>
334  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
335  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
336  <p class=MsoBodyText>Contrast</p>
337  </td>
338  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
339  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
340  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
341  <p class=MsoBodyText>Å<sup> -2</sup></p>
342  </td>
343  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
344  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
345  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
346  <p class=MsoBodyText>1</p>
347  </td>
348 </tr>
349 <tr style='height:12.8pt'>
350  <td width=135 valign=top style='width:101.0pt;border:solid windowtext 1.0pt;
351  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
352  <p class=MsoBodyText>Background</p>
353  </td>
354  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
355  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
356  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
357  <p class=MsoBodyText>cm<sup>-1</sup></p>
358  </td>
359  <td width=135 valign=top style='width:101.0pt;border-top:none;border-left:
360  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
361  padding:0in 5.4pt 0in 5.4pt;height:12.8pt'>
362  <p class=MsoBodyText>0</p>
363  </td>
364 </tr>
365</table>
366
367</div>
368
369<p class=MsoNormal>&nbsp;</p>
370
371<p class=MsoNormal>Our model uses the form factor calculations implemented in a
372c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
373
374<p class=MsoNormal>&nbsp;</p>
375
376
377<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.2.<span
378style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation
379of the sphere model</b></p>
380
381<p class=MsoNormal>&nbsp;</p>
382
383<p class=MsoNormal>Validation of our code was done by comparing the output of
384the 1D model to the output of the software provided by the NIST (Kline, 2006).
385Figure 1 shows a comparison of the output of our model and the output of the
386NIST software.</p>
387
388<p class=MsoNormal>&nbsp;</p>
389
390<p class=MsoBodyText align=center style='text-align:center;page-break-after:
391avoid'><img width=573 height=315 id="Picture 5"
392src="../images/html/image005.jpg" alt="sphere_1D_validation"></p>
393
394<p class=MsoCaption>Figure 1: Comparison of the DANSE scattering intensity for
395a sphere with the output of the NIST SANS analysis software. The parameters
396were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>, and
397Background=0.01 cm<sup> -1</sup>.</p>
398
399<p class=MsoNormal>&nbsp;</p>
400
401
402<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
403style='font-size:14.0pt'>2.2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
404</span></span></b><b><span style='font-size:14.0pt'><a name="CylinderModel">CylinderModel</a></span></b></p>
405
406<p class=MsoNormal>&nbsp;</p>
407
408<p class=MsoNormal>This model provides the form factor for a monodisperse right
409circular cylinder with uniform scattering length density. The form factor is
410normalized by the particle volume.</p>
411
412<p class=MsoNormal>&nbsp;</p>
413
414<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.2.1.<span
415style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p>
416
417<p class=MsoNormal>&nbsp;</p>
418
419<p class=MsoNormal>The output of the 2D scattering intensity function for
420oriented cylinders is given by (Guinier, 1955):</p>
421
422<p class=MsoNormal align=center style='text-align:center'><sub><img width=184
423height=41 src="../images/html/image006.gif"></sub>  (4)</p>
424
425<p class=MsoNormal align=center style='text-align:center'><sub><img width=373
426height=45 src="../images/html/image007.gif"></sub>   (5)</p>
427
428<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span>
429is the angle between the axis of the cylinder and the q-vector, V is the volume
430of the cylinder, L is the length of the cylinder, r is the radius of the
431cylinder, and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i>
432(contrast) is the scattering length density difference between the scatterer
433and the solvent. J<sub>1</sub> is the first order Bessel function.</p>
434
435<p class=MsoNormal>&nbsp;</p>
436
437<p class=MsoNormal>To provide easy access to the orientation of the cylinder, we
438define the axis of the cylinder using two angles &#952; and <span
439style='font-family:"Arial","sans-serif"'>&#966;</span>. Those angles are
440defined on Figure 2.</p>
441
442<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
443width=300 height=281 id="Picture 8" src="../images/html/image008.jpg"
444alt=angles></p>
445
446<p class=MsoCaption align=center style='text-align:center'><a
447name="_Ref173306040"></a><a name="_Ref173213915">Figure </a>2: Definition of the angles for oriented cylinders.</p>
448
449<p class=MsoNormal>&nbsp;</p>
450
451
452<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
453and the parameters of the cylinder model are the following:</p>
454
455<p class=MsoNormal>&nbsp;</p>
456
457<div align=center>
458
459<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
460 style='border-collapse:collapse;border:none'>
461 <tr style='height:19.85pt'>
462  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
463  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
464  <p class=MsoBodyText>Parameter name</p>
465  </td>
466  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
467  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
468  <p class=MsoBodyText>Units</p>
469  </td>
470  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
471  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
472  <p class=MsoBodyText>Default value</p>
473  </td>
474 </tr>
475 <tr style='height:19.85pt'>
476  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
477  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
478  <p class=MsoBodyText>Scale</p>
479  </td>
480  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
481  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
482  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
483  <p class=MsoBodyText>None</p>
484  </td>
485  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
486  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
487  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
488  <p class=MsoBodyText>1.0</p>
489  </td>
490 </tr>
491 <tr style='height:19.85pt'>
492  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
493  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
494  <p class=MsoBodyText>Radius</p>
495  </td>
496  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
497  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
498  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
499  <p class=MsoBodyText>Å</p>
500  </td>
501  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
502  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
503  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
504  <p class=MsoBodyText>20.0</p>
505  </td>
506 </tr>
507 <tr style='height:19.85pt'>
508  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
509  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
510  <p class=MsoBodyText>Length</p>
511  </td>
512  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
513  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
514  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
515  <p class=MsoBodyText>Å</p>
516  </td>
517  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
518  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
519  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
520  <p class=MsoBodyText>400.0</p>
521  </td>
522 </tr>
523 <tr style='height:19.85pt'>
524  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
525  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
526  <p class=MsoBodyText>Contrast</p>
527  </td>
528  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
529  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
530  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
531  <p class=MsoBodyText>Å<sup> -2</sup></p>
532  </td>
533  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
534  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
535  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
536  <p class=MsoBodyText>3.0e-6</p>
537  </td>
538 </tr>
539 <tr style='height:19.85pt'>
540  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
541  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
542  <p class=MsoBodyText>Background</p>
543  </td>
544  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
545  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
546  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
547  <p class=MsoBodyText>cm<sup>-1</sup></p>
548  </td>
549  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
550  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
551  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
552  <p class=MsoBodyText>0.0</p>
553  </td>
554 </tr>
555 <tr style='height:19.85pt'>
556  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
557  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
558  <p class=MsoBodyText>Cyl_theta</p>
559  </td>
560  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
561  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
562  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
563  <p class=MsoBodyText>Radian</p>
564  </td>
565  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
566  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
567  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
568  <p class=MsoBodyText>1.0</p>
569  </td>
570 </tr>
571 <tr style='height:20.6pt'>
572  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
573  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
574  <p class=MsoBodyText>Cyl_phi</p>
575  </td>
576  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
577  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
578  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
579  <p class=MsoBodyText>Radian</p>
580  </td>
581  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
582  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
583  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
584  <p class=MsoBodyText>1.0</p>
585  </td>
586 </tr>
587</table>
588
589</div>
590
591<p class=MsoNormal>&nbsp;</p>
592
593<p class=MsoNormal>The output of the 1D scattering intensity function for
594randomly oriented cylinders is then given by:</p>
595
596<p class=MsoNormal align=center style='text-align:center'><a
597name="_Ref173306479"></a><a name="_Ref173306528"><sub><img width=253 height=56
598src="../images/html/image009.gif"></sub>         (</a>6)</p>
599
600<p class=MsoNormal>&nbsp;</p>
601
602<p class=MsoNormal>The <i>cyl_theta</i> and <i>cyl_phi</i> parameter are not
603used for the 1D output. Our implementation of the scattering kernel and the 1D
604scattering intensity use the c-library from NIST.</p>
605
606<p class=MsoNormal>&nbsp;</p>
607
608
609<p class=MsoNormal>&nbsp;</p>
610
611<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.2.2.<span
612style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation
613of the cylinder model</b></p>
614
615<p class=MsoNormal>&nbsp;</p>
616
617<p class=MsoNormal>Validation of our code was done by comparing the output of
618the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure
6193 shows a comparison of the 1D output of our model and the output of the NIST
620software.</p>
621
622<p class=MsoNormal>&nbsp;</p>
623
624<p class=MsoNormal>In general, averaging over a distribution of orientations is
625done by evaluating the following:</p>
626
627<p class=MsoNormal align=center style='text-align:center'><sub><img width=244
628height=51 src="../images/html/image010.gif"></sub>           (7)</p>
629
630<p class=MsoNormal>where <i>p(&#952;,</i><i><span style='font-family:"Arial","sans-serif"'>&#966;</span>)</i>
631is the probability distribution for the orientation and <i>P<sub>0</sub>(q,</i><i><span
632style='font-family:"Arial","sans-serif"'>&#945;</span>)</i> is the scattering
633intensity for the fully oriented system. Since we have no other software to
634compare the implementation of the intensity for fully oriented cylinders, we
635can compare the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span
636style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 4
637shows the result of such a cross-check.</p>
638
639<p class=MsoNormal>&nbsp;</p>
640
641<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
642id="Picture 11" src="../images/html/image011.jpg"
643alt="cylinder_1D_validation"></p>
644
645<p class=MsoCaption><a name="_Ref173211049"></a><a name="_Ref173211066">Figure </a>3: Comparison of the DANSE scattering intensity for a cylinder with the output of the
646NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 Å,
647Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p>
648
649<p class=MsoCaption>&nbsp;</p>
650
651<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
652id="Picture 12" src="../images/html/image012.jpg"
653alt="cylinder_2D_average"></p>
654
655<p class=MsoCaption><a name="_Ref173213305">Figure </a>4: Comparison of the
656intensity for uniformly distributed cylinders calculated from our 2D model and
657the intensity from the NIST SANS analysis software. The parameters used were:
658Scale=1.0, Radius=20 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and
659Background=0.0 cm<sup> -1</sup>.</p>
660
661<p class=MsoCaption>&nbsp;</p>
662
663<p class=MsoNormal>&nbsp;</p>
664
665
666<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
667style='font-size:14.0pt'>2.3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
668</span></span></b><b><span style='font-size:14.0pt'><a name="EllipsoidModel">EllipsoidModel</a></span></b></p>
669
670<p class=MsoNormal>&nbsp;</p>
671
672<p class=MsoNormal>This model provides the form factor for a monodisperse
673ellipsoid (ellipsoid of revolution) with uniform scattering length density. The
674form factor is normalized by the particle volume.</p>
675
676<p class=MsoNormal>&nbsp;</p>
677
678<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.3.1.<span
679style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p>
680
681<p class=MsoNormal>&nbsp;</p>
682
683<p class=MsoNormal>The output of the 2D scattering intensity function for
684oriented ellipsoids is given by (Feigin, 1987):</p>
685
686<p class=MsoCaption align=center style='text-align:center'><sub><img width=184
687height=41 src="../images/html/image006.gif"></sub>   <span
688style='font-size:12.0pt;font-weight:normal'>(8)</span></p>
689
690<p class=MsoCaption align=center style='text-align:center'><sub><img width=371
691height=47 src="../images/html/image013.gif"></sub>    <span
692style='font-size:12.0pt;font-weight:normal'>(9)</span></p>
693
694<p class=MsoCaption align=center style='text-align:center'><sub><img width=272
695height=29 src="../images/html/image014.gif"></sub>     <span
696style='font-size:12.0pt;font-weight:normal'>(10)</span></p>
697
698<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span>
699is the angle between the axis of the ellipsoid and the q-vector, V is the
700volume of the ellipsoid, R<sub>a</sub> is the radius along the rotation axis of
701the ellipsoid, R<sub>b</sub> is the radius perpendicular to the rotation axis of
702the ellipsoid and <i><span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;</i>
703(contrast) is the scattering length density difference between the scatterer
704and the solvent. </p>
705
706<p class=MsoNormal>&nbsp;</p>
707
708<p class=MsoNormal>To provide easy access to the orientation of the ellipsoid,
709we define the rotation axis of the ellipsoid using two angles &#952; and <span
710style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
711of the cylinder, those angles are defined on Figure 2. For the ellipsoid,
712&#952; is the angle between the rotation axis and the z-axis. </p>
713
714<p class=MsoNormal>&nbsp;</p>
715
716<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
717and the parameters of the ellipsoid model are the following:</p>
718
719<p class=MsoNormal>&nbsp;</p>
720
721<div align=center>
722
723<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
724 style='border-collapse:collapse;border:none'>
725 <tr style='height:19.85pt'>
726  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
727  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
728  <p class=MsoBodyText>Parameter name</p>
729  </td>
730  <td width=105 valign=top style='width:78.9pt;border:solid windowtext 1.0pt;
731  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
732  <p class=MsoBodyText>Units</p>
733  </td>
734  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
735  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
736  <p class=MsoBodyText>Default value</p>
737  </td>
738 </tr>
739 <tr style='height:19.85pt'>
740  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
741  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
742  <p class=MsoBodyText>Scale</p>
743  </td>
744  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
745  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
746  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
747  <p class=MsoBodyText>None</p>
748  </td>
749  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
750  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
751  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
752  <p class=MsoBodyText>1.0</p>
753  </td>
754 </tr>
755 <tr style='height:19.85pt'>
756  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
757  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
758  <p class=MsoBodyText>Radius_a (parallel)</p>
759  </td>
760  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
761  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
762  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
763  <p class=MsoBodyText>Å</p>
764  </td>
765  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
766  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
767  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
768  <p class=MsoBodyText>20.0</p>
769  </td>
770 </tr>
771 <tr style='height:19.85pt'>
772  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
773  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
774  <p class=MsoBodyText>Radius_b (perpendicular)</p>
775  </td>
776  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
777  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
778  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
779  <p class=MsoBodyText>Å</p>
780  </td>
781  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
782  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
783  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
784  <p class=MsoBodyText>400.0</p>
785  </td>
786 </tr>
787 <tr style='height:19.85pt'>
788  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
789  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
790  <p class=MsoBodyText>Contrast</p>
791  </td>
792  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
793  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
794  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
795  <p class=MsoBodyText>Å<sup> -2</sup></p>
796  </td>
797  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
798  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
799  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
800  <p class=MsoBodyText>3.0e-6</p>
801  </td>
802 </tr>
803 <tr style='height:19.85pt'>
804  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
805  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
806  <p class=MsoBodyText>Background</p>
807  </td>
808  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
809  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
810  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
811  <p class=MsoBodyText>cm<sup>-1</sup></p>
812  </td>
813  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
814  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
815  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
816  <p class=MsoBodyText>0.0</p>
817  </td>
818 </tr>
819 <tr style='height:19.85pt'>
820  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
821  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
822  <p class=MsoBodyText>Axis_theta</p>
823  </td>
824  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
825  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
826  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
827  <p class=MsoBodyText>Radian</p>
828  </td>
829  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
830  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
831  padding:0in 5.4pt 0in 5.4pt;height:19.85pt'>
832  <p class=MsoBodyText>1.57</p>
833  </td>
834 </tr>
835 <tr style='height:20.6pt'>
836  <td width=180 valign=top style='width:135.1pt;border:solid windowtext 1.0pt;
837  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
838  <p class=MsoBodyText>Axis_phi</p>
839  </td>
840  <td width=105 valign=top style='width:78.9pt;border-top:none;border-left:
841  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
842  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
843  <p class=MsoBodyText>Radian</p>
844  </td>
845  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
846  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
847  padding:0in 5.4pt 0in 5.4pt;height:20.6pt'>
848  <p class=MsoBodyText>0.0</p>
849  </td>
850 </tr>
851</table>
852
853</div>
854
855<p class=MsoNormal>&nbsp;</p>
856
857<p class=MsoNormal>The output of the 1D scattering intensity function for
858randomly oriented ellipsoids is then given by equation 6.</p>
859
860<p class=MsoNormal>&nbsp;</p>
861
862<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not
863used for the 1D output. Our implementation of the scattering kernel and the 1D
864scattering intensity use the c-library from NIST.</p>
865
866<p class=MsoNormal>&nbsp;</p>
867
868
869<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.3.2.<span
870style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation
871of the ellipsoid model</b></p>
872
873<p class=MsoNormal>&nbsp;</p>
874
875<p class=MsoNormal>Validation of our code was done by comparing the output of
876the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure
8775 shows a comparison of the 1D output of our model and the output of the NIST
878software.</p>
879
880<p class=MsoNormal>&nbsp;</p>
881
882<p class=MsoNormal>Averaging over a distribution of orientation is done by
883evaluating equation 7. Since we have no other software to compare the
884implementation of the intensity for fully oriented ellipsoids, we can compare
885the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span
886style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 6 shows the result of such a cross-check.</p>
887
888<p class=MsoNormal align=center style='text-align:center'><b><i><span
889style='color:#3366FF'>&nbsp;</span></i></b></p>
890
891<p class=MsoNormal>The discrepancy above q=0.3 Å<sup> -1</sup> is due to the
892way the form factors are calculated in the c-library provided by NIST. A numerical
893integration has to be performed to obtain P(q) for randomly oriented particles
894(equation 6). The NIST software performs that integration with a 76-point Gaussian
895quadrature rule, which will become imprecise at high q where the amplitude
896varies quickly as a function of q. The DANSE result shown has been obtained by
897summing over 501 equidistant points in <span style='font-family:"Arial","sans-serif"'>&#945;</span>.
898Our result was found to be stable over the range of q shown for a number of
899points higher than 500.</p>
900
901<p class=MsoNormal style='page-break-after:avoid'><b><img width=573 height=315
902id="Picture 16" src="../images/html/image015.jpg"
903alt="ellipsoid_1D_validation"></b></p>
904
905<p class=MsoCaption><a name="_Ref173222904">Figure </a>5: Comparison of the
906DANSE scattering intensity for an ellipsoid with the output of the NIST SANS
907analysis software. The parameters were set to: Scale=1.0, Radius_a=20 Å, Radius_b=400
908Å, </p>
909
910<p class=MsoCaption>Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup>
911-1</sup>.</p>
912
913<p class=MsoNormal><b>&nbsp;</b></p>
914
915
916<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
917id="Picture 17" src="../images/html/image016.jpg"
918alt="ellipsoid_2D_average"></p>
919
920<p class=MsoCaption><a name="_Ref173223004">Figure </a>6: Comparison of the
921intensity for uniformly distributed ellipsoids calculated from our 2D model and
922the intensity from the NIST SANS analysis software. The parameters used were:
923Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and
924Background=0.0 cm<sup> -1</sup>.</p>
925
926<p class=MsoCaption>&nbsp;</p>
927
928<p class=MsoNormal><b>&nbsp;</b></p>
929
930<p class=MsoNormal><b>&nbsp;</b></p>
931
932<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
933style='font-size:14.0pt'>2.4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
934</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellModel">CoreShell(Sphere)Model</a></span></b></p>
935
936<p class=MsoNormal>&nbsp;</p>
937
938<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a
939monodisperse spherical particle with a core-shell structure. The form factor is
940normalized by the particle volume.</p>
941
942<p class=MsoNormal>&nbsp;</p>
943
944<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.1.<span
945style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p>
946
947<p class=MsoNormal>&nbsp;</p>
948
949<p class=MsoNormal>The 1D scattering intensity is calculated in the following
950way (Guinier, 1955):</p>
951
952<p class=MsoNormal align=right style='text-align:right'><sub><img width=643
953height=55 src="../images/html/image017.gif"></sub> (11)</p>
954
955<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>s</sub></i>
956is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the
957core, <i>r<sub>s</sub></i> is the radius of the shell, <i>r<sub>c</sub></i> is
958the radius of the core, <i>&#961;<sub>c</sub></i> is the scattering length
959density of the core, <i>&#961;<sub>s</sub></i> is the scattering length density
960of the shell, &#961;<sub>solv</sub> is the scattering length density of the
961solvent, and <i>bkg</i> is the background level.</p>
962
963<p class=MsoNormal>&nbsp;</p>
964
965<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above,
966regardless of the orientation of the q vector.</p>
967
968<p class=MsoNormal>&nbsp;</p>
969
970<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
971and the parameters of the core-shell sphere model are the following:</p>
972
973<p class=MsoNormal>&nbsp;</p>
974
975<div align=center>
976
977<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
978 style='border-collapse:collapse;border:none'>
979 <tr style='height:18.8pt'>
980  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
981  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
982  <p class=MsoBodyText>Parameter name</p>
983  </td>
984  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
985  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
986  <p class=MsoBodyText>Units</p>
987  </td>
988  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
989  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
990  <p class=MsoBodyText>Default value</p>
991  </td>
992 </tr>
993 <tr style='height:18.8pt'>
994  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
995  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
996  <p class=MsoBodyText>Scale</p>
997  </td>
998  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
999  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1000  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1001  <p class=MsoBodyText>None</p>
1002  </td>
1003  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1004  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1005  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1006  <p class=MsoBodyText>1.0</p>
1007  </td>
1008 </tr>
1009 <tr style='height:18.8pt'>
1010  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1011  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1012  <p class=MsoBodyText>Radius</p>
1013  </td>
1014  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1015  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1016  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1017  <p class=MsoBodyText>Å</p>
1018  </td>
1019  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1020  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1021  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1022  <p class=MsoBodyText>60</p>
1023  </td>
1024 </tr>
1025 <tr style='height:18.8pt'>
1026  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1027  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1028  <p class=MsoBodyText>Thickness</p>
1029  </td>
1030  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1031  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1032  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1033  <p class=MsoBodyText>Å</p>
1034  </td>
1035  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1036  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1037  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1038  <p class=MsoBodyText>10</p>
1039  </td>
1040 </tr>
1041 <tr style='height:18.8pt'>
1042  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1043  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1044  <p class=MsoBodyText>Core_sld</p>
1045  </td>
1046  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1047  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1048  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1049  <p class=MsoBodyText>Å<sup> -2</sup></p>
1050  </td>
1051  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1052  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1053  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1054  <p class=MsoBodyText>1e-6</p>
1055  </td>
1056 </tr>
1057 <tr style='height:18.8pt'>
1058  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1059  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1060  <p class=MsoBodyText>Shell_sld</p>
1061  </td>
1062  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1063  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1064  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1065  <p class=MsoBodyText>Å<sup> -2</sup></p>
1066  </td>
1067  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1068  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1069  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1070  <p class=MsoBodyText>2e-6</p>
1071  </td>
1072 </tr>
1073 <tr style='height:18.8pt'>
1074  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1075  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1076  <p class=MsoBodyText>Solvent_sld</p>
1077  </td>
1078  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1079  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1080  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1081  <p class=MsoBodyText>Å<sup> -2</sup></p>
1082  </td>
1083  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1084  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1085  padding:0in 5.4pt 0in 5.4pt;height:18.8pt'>
1086  <p class=MsoBodyText>3e-6</p>
1087  </td>
1088 </tr>
1089 <tr style='height:19.5pt'>
1090  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1091  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
1092  <p class=MsoBodyText>Background</p>
1093  </td>
1094  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1095  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1096  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
1097  <p class=MsoBodyText>cm<sup>-1</sup></p>
1098  </td>
1099  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1100  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1101  padding:0in 5.4pt 0in 5.4pt;height:19.5pt'>
1102  <p class=MsoBodyText>0.001</p>
1103  </td>
1104 </tr>
1105</table>
1106
1107</div>
1108
1109<p class=MsoNormal>&nbsp;</p>
1110
1111<p class=MsoNormal>Our model uses the form factor calculations implemented in a
1112c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
1113
1114<p class=MsoNormal>&nbsp;</p>
1115
1116
1117<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.2.<span
1118style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation
1119of the core-shell sphere model</b></p>
1120
1121<p class=MsoNormal>&nbsp;</p>
1122
1123<p class=MsoNormal>Validation of our code was done by comparing the output of
1124the 1D model to the output of the software provided by the NIST (Kline, 2006).
1125Figure 1 shows a comparison of the output of our model and the output of the
1126NIST software.</p>
1127
1128<p class=MsoNormal>&nbsp;</p>
1129
1130<p class=MsoBodyText align=center style='text-align:center;page-break-after:
1131avoid'><img width=573 height=315 id="Picture 19"
1132src="../images/html/image018.jpg" alt="core_shell_sphere_1D_validation"></p>
1133
1134<p class=MsoCaption>Figure 7: Comparison of the DANSE scattering intensity for
1135a core-shell sphere with the output of the NIST SANS analysis software. The
1136parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>,
1137and Background=0.001 cm<sup> -1</sup>.</p>
1138
1139<p class=MsoBodyText>&nbsp;</p>
1140
1141<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1142style='font-size:14.0pt'>2.5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1143</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellCylinderModel">CoreShellCylinderModel</a></span></b></p>
1144
1145<p class=MsoNormal>&nbsp;</p>
1146
1147<p class=MsoNormal>This model provides the form factor for a monodisperse,
1148circular cylinder with a core-shell scattering length density profile. The form
1149factor is normalized by the particle volume.</p>
1150
1151<p class=MsoNormal>&nbsp;</p>
1152
1153<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.5.1.<span
1154style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Definition</b></p>
1155
1156<p class=MsoNormal>&nbsp;</p>
1157
1158<p class=MsoNormal>The output of the 2D scattering intensity function for
1159oriented core-shell cylinders is given by (Kline, 2006):</p>
1160
1161<p class=MsoNormal align=center style='text-align:center'><sub><img width=184
1162height=45 src="../images/html/image019.gif"></sub> (12)</p>
1163
1164<p class=MsoNormal align=center style='text-align:center'><sub><img width=476
1165height=93 src="../images/html/image020.gif"></sub> (13)</p>
1166
1167<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>&#945;</span>
1168is the angle between the axis of the cylinder and the q-vector, <i>V<sub>s</sub></i>
1169is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the
1170core, <i>L</i> is the length of the core, <i>r</i> is the radius of the core, <i>t</i>
1171is the thickness of the shell, <i>&#961;<sub>c</sub></i> is the scattering
1172length density of the core, <i>&#961;<sub>s</sub></i> is the scattering length
1173density of the shell, &#961;<sub>solv</sub> is the scattering length density of
1174the solvent, and <i>bkg</i> is the background level. The outer radius of the
1175shell is given by <i>r+t</i> and the total length of the outer shell is given
1176by <i>L+2t</i>. J<sub>1</sub> is the first order Bessel function.</p>
1177
1178<p class=MsoNormal>&nbsp;</p>
1179
1180<p class=MsoNormal>To provide easy access to the orientation of the core-shell cylinder,
1181we define the axis of the cylinder using two angles &#952; and <span
1182style='font-family:"Arial","sans-serif"'>&#966;</span>. Similarly to the case
1183of the cylinder, those angles are defined on Figure 2.</p>
1184
1185<p class=MsoNormal>&nbsp;</p>
1186
1187<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
1188and the parameters of the core-shell cylinder model are the following:</p>
1189
1190<p class=MsoNormal>&nbsp;</p>
1191
1192<div align=center>
1193
1194<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
1195 style='border-collapse:collapse;border:none'>
1196 <tr style='height:19.25pt'>
1197  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1198  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1199  <p class=MsoBodyText>Parameter name</p>
1200  </td>
1201  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1202  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1203  <p class=MsoBodyText>Units</p>
1204  </td>
1205  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1206  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1207  <p class=MsoBodyText>Default value</p>
1208  </td>
1209 </tr>
1210 <tr style='height:19.25pt'>
1211  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1212  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1213  <p class=MsoBodyText>Scale</p>
1214  </td>
1215  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1216  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1217  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1218  <p class=MsoBodyText>None</p>
1219  </td>
1220  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1221  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1222  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1223  <p class=MsoBodyText>1.0</p>
1224  </td>
1225 </tr>
1226 <tr style='height:19.25pt'>
1227  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1228  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1229  <p class=MsoBodyText>Radius</p>
1230  </td>
1231  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1232  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1233  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1234  <p class=MsoBodyText>Å</p>
1235  </td>
1236  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1237  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1238  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1239  <p class=MsoBodyText>20.0</p>
1240  </td>
1241 </tr>
1242 <tr style='height:19.25pt'>
1243  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1244  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1245  <p class=MsoBodyText>Thickness</p>
1246  </td>
1247  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1248  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1249  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1250  <p class=MsoBodyText>Å</p>
1251  </td>
1252  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1253  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1254  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1255  <p class=MsoBodyText>10.0</p>
1256  </td>
1257 </tr>
1258 <tr style='height:19.25pt'>
1259  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1260  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1261  <p class=MsoBodyText>Length</p>
1262  </td>
1263  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1264  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1265  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1266  <p class=MsoBodyText>Å</p>
1267  </td>
1268  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1269  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1270  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1271  <p class=MsoBodyText>400.0</p>
1272  </td>
1273 </tr>
1274 <tr style='height:19.25pt'>
1275  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1276  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1277  <p class=MsoBodyText>Core_sld</p>
1278  </td>
1279  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1280  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1281  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1282  <p class=MsoBodyText>Å<sup> -2</sup></p>
1283  </td>
1284  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1285  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1286  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1287  <p class=MsoBodyText>1e-6</p>
1288  </td>
1289 </tr>
1290 <tr style='height:19.25pt'>
1291  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1292  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1293  <p class=MsoBodyText>Shell_sld</p>
1294  </td>
1295  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1296  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1297  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1298  <p class=MsoBodyText>Å<sup> -2</sup></p>
1299  </td>
1300  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1301  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1302  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1303  <p class=MsoBodyText>4e-6</p>
1304  </td>
1305 </tr>
1306 <tr style='height:19.25pt'>
1307  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1308  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1309  <p class=MsoBodyText>Solvent_sld</p>
1310  </td>
1311  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1312  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1313  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1314  <p class=MsoBodyText>Å<sup> -2</sup></p>
1315  </td>
1316  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1317  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1318  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1319  <p class=MsoBodyText>1e-6</p>
1320  </td>
1321 </tr>
1322 <tr style='height:19.25pt'>
1323  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1324  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1325  <p class=MsoBodyText>Background</p>
1326  </td>
1327  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1328  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1329  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1330  <p class=MsoBodyText>cm<sup>-1</sup></p>
1331  </td>
1332  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1333  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1334  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1335  <p class=MsoBodyText>0.0</p>
1336  </td>
1337 </tr>
1338 <tr style='height:19.25pt'>
1339  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1340  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1341  <p class=MsoBodyText>Axis_theta</p>
1342  </td>
1343  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1344  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1345  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1346  <p class=MsoBodyText>Radian</p>
1347  </td>
1348  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1349  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1350  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1351  <p class=MsoBodyText>1.57</p>
1352  </td>
1353 </tr>
1354 <tr style='height:20.0pt'>
1355  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1356  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:20.0pt'>
1357  <p class=MsoBodyText>Axis_phi</p>
1358  </td>
1359  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1360  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1361  padding:0in 5.4pt 0in 5.4pt;height:20.0pt'>
1362  <p class=MsoBodyText>Radian</p>
1363  </td>
1364  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1365  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1366  padding:0in 5.4pt 0in 5.4pt;height:20.0pt'>
1367  <p class=MsoBodyText>0.0</p>
1368  </td>
1369 </tr>
1370</table>
1371
1372</div>
1373
1374<p class=MsoNormal>&nbsp;</p>
1375
1376<p class=MsoNormal>The output of the 1D scattering intensity function for
1377randomly oriented cylinders is then given by equation 6.</p>
1378
1379<p class=MsoNormal>&nbsp;</p>
1380
1381<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not
1382used for the 1D output. Our implementation of the scattering kernel and the 1D
1383scattering intensity use the c-library from NIST.</p>
1384
1385<p class=MsoNormal>&nbsp;</p>
1386
1387<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.5.2.<span
1388style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b>Validation
1389of the core-shell cylinder model</b></p>
1390
1391<p class=MsoNormal>&nbsp;</p>
1392
1393<p class=MsoNormal>Validation of our code was done by comparing the output of
1394the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure
13958 shows a comparison of the 1D output of our model and the output of the NIST
1396software.</p>
1397
1398<p class=MsoNormal>&nbsp;</p>
1399
1400<p class=MsoNormal>Averaging over a distribution of orientation is done by
1401evaluating equation 7. Since we have no other software to compare the
1402implementation of the intensity for fully oriented core-shell cylinders, we can
1403compare the result of averaging our 2D output using a uniform distribution <i>p(&#952;,</i><i><span
1404style='font-family:"Arial","sans-serif"'>&#966;</span>)</i> = 1.0.  Figure 9 shows the result of such a cross-check.</p>
1405
1406<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
1407id="Picture 22" src="../images/html/image021.jpg"
1408alt="core_shell_cylinder_1D_validation"></p>
1409
1410<p class=MsoCaption><a name="_Ref173307149">Figure </a>8: Comparison of the
1411DANSE scattering intensity for a core-shell cylinder with the output of the
1412NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 Å,
1413Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup>
1414-2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p>
1415
1416<p class=MsoCaption>&nbsp;</p>
1417
1418<p class=MsoNormal>&nbsp;</p>
1419
1420<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
1421id="Picture 23" src="../images/html/image022.jpg"
1422alt="core_shell_cylinder_2D_average"></p>
1423
1424<p class=MsoCaption><a name="_Ref173307204">Figure </a>9: Comparison of the
1425intensity for uniformly distributed core-shell cylinders calculated from our 2D
1426model and the intensity from the NIST SANS analysis software. The parameters
1427used were: Scale=1.0, Radius=20 Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup>
1428-2</sup>, Shell_sld=4e-6 Å<sup> -2</sup>, Solvent_sld=1e-6 Å<sup> -2</sup>, and
1429Background=0.0 cm<sup> -1</sup>.</p>
1430
1431<p class=MsoNormal>&nbsp;</p>
1432<p class=MsoNormal>&nbsp;</p>
1433
1434<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1435style='font-size:14.0pt'>2.6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1436</span></span></b><b><span style='font-size:14.0pt'><a name="EllipticalCylinderModel">EllipticalCylinderModel</a></span></b></p>
1437
1438<p class=MsoNormal>&nbsp;</p>
1439
1440<p class=MsoNormal>This function calculates the scattering from an oriented
1441elliptical cylinder. The angles theta and phi define the orientation of the
1442axis of the cylinder. The  angle psi is defined as the orientation of the major
1443axis of the ellipse with respect to the vector Q. A gaussian poydispersity can
1444be added to any (or all) of the orientation angles, and also for the minor
1445radius and the ratio of the ellipse radii.</p>
1446
1447<p class=MsoNormal>The function calculated is:</p>
1448
1449<p class=MsoNormal align=center style='text-align:center'><span
1450style='position:relative;top:16.0pt'><img width=377 height=46
1451src="../images/html/image023.gif"></span> </p>
1452
1453<p class=MsoNormal>with the functions:</p>
1454
1455<p class=MsoNormal align=center style='text-align:center'><span
1456style='position:relative;top:39.0pt'><img width=297 height=114
1457src="../images/html/image024.gif"></span>          (13)</p>
1458
1459<p class=MsoNormal>and the angle psi is defined as the orientation of the major
1460axis of the ellipse with respect to the vector Q.</p>
1461
1462<p class=MsoNormal>&nbsp;</p>
1463
1464<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
1465and the parameters are the following:</p>
1466
1467<p class=MsoNormal>&nbsp;</p>
1468
1469<div align=center>
1470
1471<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
1472 style='border-collapse:collapse;border:none'>
1473 <tr style='height:19.25pt'>
1474  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1475  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1476  <p class=MsoBodyText>Parameter name</p>
1477  </td>
1478  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1479  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1480  <p class=MsoBodyText>Units</p>
1481  </td>
1482  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1483  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1484  <p class=MsoBodyText>Default value</p>
1485  </td>
1486 </tr>
1487 <tr style='height:19.25pt'>
1488  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1489  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1490  <p class=MsoBodyText>scale</p>
1491  </td>
1492  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1493  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1494  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1495  <p class=MsoBodyText>None</p>
1496  </td>
1497  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1498  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1499  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1500  <p class=MsoBodyText>1.0</p>
1501  </td>
1502 </tr>
1503 <tr style='height:19.25pt'>
1504  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1505  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1506  <p class=MsoBodyText>r_minor</p>
1507  </td>
1508  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1509  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1510  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1511  <p class=MsoBodyText>Å</p>
1512  </td>
1513  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1514  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1515  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1516  <p class=MsoBodyText>20.0</p>
1517  </td>
1518 </tr>
1519 <tr style='height:19.25pt'>
1520  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1521  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1522  <p class=MsoBodyText>r_ratio</p>
1523  </td>
1524  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1525  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1526  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1527  <p class=MsoBodyText>Å</p>
1528  </td>
1529  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1530  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1531  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1532  <p class=MsoBodyText>1.5</p>
1533  </td>
1534 </tr>
1535 <tr style='height:19.25pt'>
1536  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1537  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1538  <p class=MsoBodyText>length</p>
1539  </td>
1540  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1541  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1542  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1543  <p class=MsoBodyText>Å</p>
1544  </td>
1545  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1546  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1547  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1548  <p class=MsoBodyText>400.0</p>
1549  </td>
1550 </tr>
1551 <tr style='height:19.25pt'>
1552  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1553  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1554  <p class=MsoBodyText>contrast</p>
1555  </td>
1556  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1557  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1558  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1559  <p class=MsoBodyText>Å<sup> -2</sup></p>
1560  </td>
1561  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1562  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1563  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1564  <p class=MsoBodyText>3e-6</p>
1565  </td>
1566 </tr>
1567 <tr style='height:19.25pt'>
1568  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1569  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1570  <p class=MsoBodyText>cyl_phi</p>
1571  </td>
1572  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1573  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1574  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1575  <p class=MsoBodyText>Å<sup> -2</sup></p>
1576  </td>
1577  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1578  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1579  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1580  <p class=MsoBodyText>0</p>
1581  </td>
1582 </tr>
1583 <tr style='height:19.25pt'>
1584  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1585  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1586  <p class=MsoBodyText>cyl_psi</p>
1587  </td>
1588  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1589  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1590  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1591  <p class=MsoBodyText>Å<sup> -2</sup></p>
1592  </td>
1593  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1594  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1595  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1596  <p class=MsoBodyText>0</p>
1597  </td>
1598 </tr>
1599 <tr style='height:19.25pt'>
1600  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1601  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1602  <p class=MsoBodyText>cyl_theta</p>
1603  </td>
1604  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1605  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1606  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1607  <p class=MsoBodyText>radian</p>
1608  </td>
1609  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1610  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1611  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1612  <p class=MsoBodyText>1.57</p>
1613  </td>
1614 </tr>
1615 <tr style='height:19.25pt'>
1616  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1617  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1618  <p class=MsoBodyText>background</p>
1619  </td>
1620  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1621  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1622  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1623  <p class=MsoBodyText>&nbsp;</p>
1624  </td>
1625  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1626  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1627  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1628  <p class=MsoBodyText>0</p>
1629  </td>
1630 </tr>
1631</table>
1632
1633</div>
1634
1635<p class=MsoNormal>&nbsp;</p>
1636
1637<p class=MsoNormal>REFERENCE</p>
1638
1639<p class=MsoNormal>&nbsp;</p>
1640
1641<p class=MsoNormal style='text-indent:.25in'>L. A. Feigin and D. I. Svergun
1642“Structure Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New
1643York, (1987).</p>
1644
1645<p class=MsoNormal>&nbsp;</p>
1646
1647<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
1648style='font-size:14.0pt'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
1649</span></span></b><b><span style='font-size:14.0pt'><a name="Shape-Independent">Shape-Independent</a></span></b></p>
1650
1651<p class=MsoBodyText>&nbsp;</p>
1652
1653<p class=MsoBodyText>The following are models used for shape-independent SANS
1654analysis.</p>
1655
1656<p class=MsoBodyText>&nbsp;</p>
1657
1658<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1659style='font-size:14.0pt'>3.1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1660</span></span></b><b><span style='font-size:14.0pt'> <a name="Debye">Debye (Model)</a></span></b></p>
1661
1662<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
1663
1664<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The Debye
1665model is a form factor for a linear polymer chain. In addition to the radius of
1666gyration, Rg, a scale factor &quot;scale&quot;, and a constant background term
1667are included in the calculation. </p>
1668
1669<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
1670
1671<p class=MsoNormal align=center style='text-align:center'><span
1672style='font-size:14.0pt'><sub><img width=188 height=76
1673src="../images/html/image025.gif"></sub></span></p>
1674
1675<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
1676
1677<div align=center>
1678
1679<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
1680 style='border-collapse:collapse;border:none'>
1681 <tr style='height:19.25pt'>
1682  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1683  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1684  <p class=MsoBodyText>Parameter name</p>
1685  </td>
1686  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1687  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1688  <p class=MsoBodyText>Units</p>
1689  </td>
1690  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1691  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1692  <p class=MsoBodyText>Default value</p>
1693  </td>
1694 </tr>
1695 <tr style='height:19.25pt'>
1696  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1697  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1698  <p class=MsoBodyText>Scale</p>
1699  </td>
1700  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1701  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1702  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1703  <p class=MsoBodyText>None</p>
1704  </td>
1705  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1706  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1707  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1708  <p class=MsoBodyText>1.0</p>
1709  </td>
1710 </tr>
1711 <tr style='height:19.25pt'>
1712  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1713  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1714  <p class=MsoBodyText>Rg</p>
1715  </td>
1716  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1717  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1718  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1719  <p class=MsoBodyText>Å</p>
1720  </td>
1721  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1722  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1723  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1724  <p class=MsoBodyText>50.0</p>
1725  </td>
1726 </tr>
1727 <tr style='height:19.25pt'>
1728  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1729  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1730  <p class=MsoBodyText>Background</p>
1731  </td>
1732  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1733  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1734  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1735  <p class=MsoBodyText>cm<sup>-1</sup></p>
1736  </td>
1737  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1738  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1739  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1740  <p class=MsoBodyText>0.0</p>
1741  </td>
1742 </tr>
1743</table>
1744
1745</div>
1746
1747<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
1748
1749<p class=MsoNormal style='margin-left:.25in'>Reference: Roe, R.-J.,
1750&quot;Methods of X-Ray and Neutron Scattering in Polymer Science&quot;, Oxford University Press, New York (2000).</p>
1751
1752<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1753
1754<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1755style='font-size:14.0pt'>3.2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1756</span></span></b><b><span style='font-size:14.0pt'> <a name="Lorentz">(Ornstein-Zernicke)
1757Lorentz (Model)</span></b></p>
1758
1759<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
1760
1761<p class=MsoNormal style='text-indent:.25in'>The Ornstein-Zernicke model is
1762defined by:</p>
1763
1764<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
1765
1766<p class=MsoNormal align=center style='text-align:center'><span
1767style='font-size:14.0pt'><sub><img width=212 height=24
1768src="../images/html/image026.gif"></sub></span></p>
1769
1770<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
1771
1772<p class=MsoNormal style='text-indent:.25in'>The parameter L is referred to as
1773the screening length.</p>
1774
1775<p class=MsoNormal><span style='font-size:14.0pt'>&nbsp;</span></p>
1776
1777<div align=center>
1778
1779<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
1780 style='border-collapse:collapse;border:none'>
1781 <tr style='height:19.25pt'>
1782  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1783  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1784  <p class=MsoBodyText>Parameter name</p>
1785  </td>
1786  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1787  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1788  <p class=MsoBodyText>Units</p>
1789  </td>
1790  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1791  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1792  <p class=MsoBodyText>Default value</p>
1793  </td>
1794 </tr>
1795 <tr style='height:19.25pt'>
1796  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1797  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1798  <p class=MsoBodyText>Scale</p>
1799  </td>
1800  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1801  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1802  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1803  <p class=MsoBodyText>None</p>
1804  </td>
1805  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1806  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1807  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1808  <p class=MsoBodyText>1.0</p>
1809  </td>
1810 </tr>
1811 <tr style='height:19.25pt'>
1812  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1813  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1814  <p class=MsoBodyText>Length</p>
1815  </td>
1816  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1817  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1818  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1819  <p class=MsoBodyText>Å</p>
1820  </td>
1821  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1822  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1823  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1824  <p class=MsoBodyText>50.0</p>
1825  </td>
1826 </tr>
1827 <tr style='height:19.25pt'>
1828  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1829  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1830  <p class=MsoBodyText>Background</p>
1831  </td>
1832  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1833  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1834  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1835  <p class=MsoBodyText>cm<sup>-1</sup></p>
1836  </td>
1837  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1838  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1839  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1840  <p class=MsoBodyText>0.0</p>
1841  </td>
1842 </tr>
1843</table>
1844
1845</div>
1846
1847<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1848
1849<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1850
1851
1852<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1853style='font-size:14.0pt'>3.3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1854</span></span></b><b><span style='font-size:14.0pt'> <a name="DAB_Model">DAB(Debye-Anderson-Brumberger)_
1855Model)</a></span></b></p>
1856
1857<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Calculates the
1858scattering from a randomly distributed, two-phase system based on the
1859Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system is
1860characterized by a single length scale, the correlation length, which is a
1861measure of the average spacing between regions of phase 1 and phase 2. The
1862model also assumes smooth interfaces between the phases and hence exhibits
1863Porod behavior (I ~ Q<sup>-4</sup>) at large Q (Q*correlation length &gt;&gt;
18641). </p>
1865
1866<p class=MsoNormal style='text-indent:.25in'>&nbsp;</p>
1867
1868<p class=MsoNormal align=center style='text-align:center'><span
1869style='font-size:14.0pt'><sub><img width=221 height=28
1870src="../images/html/image027.gif"></sub></span></p>
1871
1872<p class=MsoNormal align=center style='text-align:center'><span
1873style='font-size:14.0pt'>&nbsp;</span></p>
1874
1875<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The parameter
1876L is referred to as the correlation length.</p>
1877
1878<p class=MsoNormal align=center style='text-align:center'><span
1879style='font-size:14.0pt'>&nbsp;</span></p>
1880
1881<div align=center>
1882
1883<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
1884 style='border-collapse:collapse;border:none'>
1885 <tr style='height:19.25pt'>
1886  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1887  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1888  <p class=MsoBodyText>Parameter name</p>
1889  </td>
1890  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1891  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1892  <p class=MsoBodyText>Units</p>
1893  </td>
1894  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1895  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1896  <p class=MsoBodyText>Default value</p>
1897  </td>
1898 </tr>
1899 <tr style='height:19.25pt'>
1900  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1901  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1902  <p class=MsoBodyText>Scale</p>
1903  </td>
1904  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1905  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1906  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1907  <p class=MsoBodyText>None</p>
1908  </td>
1909  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1910  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1911  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1912  <p class=MsoBodyText>1.0</p>
1913  </td>
1914 </tr>
1915 <tr style='height:19.25pt'>
1916  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1917  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1918  <p class=MsoBodyText>Length</p>
1919  </td>
1920  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1921  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1922  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1923  <p class=MsoBodyText>Å</p>
1924  </td>
1925  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1926  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1927  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1928  <p class=MsoBodyText>50.0</p>
1929  </td>
1930 </tr>
1931 <tr style='height:19.25pt'>
1932  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1933  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1934  <p class=MsoBodyText>Background</p>
1935  </td>
1936  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1937  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1938  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1939  <p class=MsoBodyText>cm<sup>-1</sup></p>
1940  </td>
1941  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
1942  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
1943  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1944  <p class=MsoBodyText>0.0</p>
1945  </td>
1946 </tr>
1947</table>
1948
1949</div>
1950
1951<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1952
1953<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
1954
1955<p class=MsoNormal style='margin-left:.5in'>Debye, Anderson, Brumberger,
1956&quot;Scattering by an Inhomogeneous Solid. II. The Correlation Function and
1957its Application&quot;, J. Appl. Phys. 28 (6), 679 (1957).</p>
1958
1959<p class=MsoNormal style='margin-left:.5in'>&nbsp;</p>
1960
1961<p class=MsoNormal style='margin-left:.5in'>Debye, Bueche, &quot;Scattering by
1962an Inhomogeneous Solid&quot;, J. Appl. Phys. 20, 518 (1949).</p>
1963
1964<p class=MsoNormal>&nbsp;</p>
1965
1966<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1967
1968<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1969style='font-size:14.0pt'>3.4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1970</span></span></b><b><span style='font-size:14.0pt'> <a name="Power_Law">Power_Law</a></span></b></p>
1971
1972<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1973
1974<p class=MsoNormal style='margin-left:.25in'>This model describes a power law
1975with background.</p>
1976
1977<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
1978
1979<p class=MsoNormal align=center style='text-align:center'><span
1980style='font-size:14.0pt'><sub><img width=161 height=24
1981src="../images/html/image028.gif"></sub></span></p>
1982
1983<p class=MsoNormal align=center style='text-align:center'><span
1984style='font-size:14.0pt'>&nbsp;</span></p>
1985
1986<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Note the minus
1987sign in front of the exponent.</p>
1988
1989<p class=MsoNormal align=center style='text-align:center'><span
1990style='font-size:14.0pt'>&nbsp;</span></p>
1991
1992<div align=center>
1993
1994<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
1995 style='border-collapse:collapse;border:none'>
1996 <tr style='height:19.25pt'>
1997  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
1998  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
1999  <p class=MsoBodyText>Parameter name</p>
2000  </td>
2001  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2002  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2003  <p class=MsoBodyText>Units</p>
2004  </td>
2005  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2006  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2007  <p class=MsoBodyText>Default value</p>
2008  </td>
2009 </tr>
2010 <tr style='height:19.25pt'>
2011  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2012  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2013  <p class=MsoBodyText>Scale</p>
2014  </td>
2015  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2016  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2017  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2018  <p class=MsoBodyText>None</p>
2019  </td>
2020  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2021  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2022  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2023  <p class=MsoBodyText>1.0</p>
2024  </td>
2025 </tr>
2026 <tr style='height:19.25pt'>
2027  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2028  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2029  <p class=MsoBodyText>m</p>
2030  </td>
2031  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2032  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2033  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2034  <p class=MsoBodyText>None</p>
2035  </td>
2036  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2037  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2038  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2039  <p class=MsoBodyText>4</p>
2040  </td>
2041 </tr>
2042 <tr style='height:19.25pt'>
2043  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2044  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2045  <p class=MsoBodyText>Background</p>
2046  </td>
2047  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2048  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2049  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2050  <p class=MsoBodyText>cm<sup>-1</sup></p>
2051  </td>
2052  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2053  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2054  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2055  <p class=MsoBodyText>0.0</p>
2056  </td>
2057 </tr>
2058</table>
2059
2060</div>
2061
2062<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2063
2064<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2065
2066<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2067style='font-size:14.0pt'>3.5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2068</span></span></b><b><span style='font-size:14.0pt'><a name="Teubner Strey"> Teubner Strey (Model) </a></span></b></p>
2069
2070<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2071
2072<p class=MsoNormal style='margin-left:.25in'>This function calculates the
2073scattered intensity of a two-component system using the Teubner-Strey model.</p>
2074
2075<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2076
2077<p class=MsoNormal align=center style='text-align:center'><span
2078style='font-size:14.0pt'><sub><img width=213 height=45
2079src="../images/html/image029.gif"></sub></span></p>
2080
2081<p class=MsoNormal align=center style='text-align:center'><span
2082style='font-size:14.0pt'>&nbsp;</span></p>
2083
2084<p class=MsoNormal align=center style='text-align:center'><span
2085style='font-size:14.0pt'>&nbsp;</span></p>
2086
2087<div align=center>
2088
2089<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2090 style='border-collapse:collapse;border:none'>
2091 <tr style='height:19.25pt'>
2092  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2093  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2094  <p class=MsoBodyText>Parameter name</p>
2095  </td>
2096  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2097  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2098  <p class=MsoBodyText>Units</p>
2099  </td>
2100  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2101  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2102  <p class=MsoBodyText>Default value</p>
2103  </td>
2104 </tr>
2105 <tr style='height:19.25pt'>
2106  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2107  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2108  <p class=MsoBodyText>A</p>
2109  </td>
2110  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2111  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2112  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2113  <p class=MsoBodyText>None</p>
2114  </td>
2115  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2116  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2117  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2118  <p class=MsoBodyText>0.1</p>
2119  </td>
2120 </tr>
2121 <tr style='height:19.25pt'>
2122  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2123  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2124  <p class=MsoBodyText>C1</p>
2125  </td>
2126  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2127  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2128  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2129  <p class=MsoBodyText>None</p>
2130  </td>
2131  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2132  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2133  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2134  <p class=MsoBodyText>-30.0</p>
2135  </td>
2136 </tr>
2137 <tr style='height:19.25pt'>
2138  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2139  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2140  <p class=MsoBodyText>C2</p>
2141  </td>
2142  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2143  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2144  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2145  <p class=MsoBodyText>None</p>
2146  </td>
2147  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2148  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2149  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2150  <p class=MsoBodyText>5000.0</p>
2151  </td>
2152 </tr>
2153 <tr style='height:19.25pt'>
2154  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2155  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2156  <p class=MsoBodyText>Background</p>
2157  </td>
2158  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2159  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2160  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2161  <p class=MsoBodyText>cm<sup>-1</sup></p>
2162  </td>
2163  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2164  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2165  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2166  <p class=MsoBodyText>0.0</p>
2167  </td>
2168 </tr>
2169</table>
2170
2171</div>
2172
2173<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2174
2175<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
2176
2177<p class=MsoNormal style='margin-left:.5in'>Teubner, M; Strey, R. J. Chem.
2178Phys., 87, 3195 (1987).</p>
2179
2180<p class=MsoNormal style='margin-left:.5in'>&nbsp;</p>
2181
2182<p class=MsoNormal style='margin-left:.5in'>Schubert, K-V., Strey, R., Kline,
2183S. R. and E. W. Kaler, J. Chem. Phys., 101, 5343 (1994).</p>
2184
2185<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2186
2187<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2188
2189
2190<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2191style='font-size:14.0pt'>3.6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2192</span></span></b><b><span style='font-size:14.0pt'><a name="Number Density Fractal"> Number Density Fractal (Model)</a></span></b></p>
2193
2194<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2195
2196<p class=MsoNormal style='margin-left:.25in'>Calculates the scattering from
2197fractal-like aggregates built from spherical building blocks following the
2198Texiera reference. The value returned is in cm<sup>-1</sup>.</p>
2199
2200<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
2201
2202
2203<p class=MsoNormal align=center style='text-align:center'><span
2204style='font-size:14.0pt'><sub><img width=380 height=188
2205src="../images/html/image030.gif"></sub></span></p>
2206
2207<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2208
2209<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The scale
2210parameter is the volume fraction of the building blocks, R<sub>0</sub> is the
2211radius of the building block, D<sub>f</sub> is the fractal dimension, &#958; is
2212the correlation length, <i>&#961;<sub>solvent</sub></i> is the scattering
2213length density of the solvent, and <i>&#961;<sub>block</sub></i> is the
2214scattering length density of the building blocks. </p>
2215
2216<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
2217
2218<p class=MsoNormal align=center style='text-align:center'><span
2219style='font-size:14.0pt'>&nbsp;</span></p>
2220
2221<div align=center>
2222
2223<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2224 style='border-collapse:collapse;border:none'>
2225 <tr style='height:19.25pt'>
2226  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2227  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2228  <p class=MsoBodyText>Parameter name</p>
2229  </td>
2230  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2231  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2232  <p class=MsoBodyText>Units</p>
2233  </td>
2234  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2235  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2236  <p class=MsoBodyText>Default value</p>
2237  </td>
2238 </tr>
2239 <tr style='height:19.25pt'>
2240  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2241  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2242  <p class=MsoBodyText>Scale</p>
2243  </td>
2244  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2245  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2246  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2247  <p class=MsoBodyText>None</p>
2248  </td>
2249  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2250  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2251  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2252  <p class=MsoBodyText>0.05</p>
2253  </td>
2254 </tr>
2255 <tr style='height:19.25pt'>
2256  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2257  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2258  <p class=MsoBodyText>Radius</p>
2259  </td>
2260  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2261  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2262  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2263  <p class=MsoBodyText>Å</p>
2264  </td>
2265  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2266  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2267  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2268  <p class=MsoBodyText>5.0</p>
2269  </td>
2270 </tr>
2271 <tr style='height:19.25pt'>
2272  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2273  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2274  <p class=MsoBodyText>Fractal_dim</p>
2275  </td>
2276  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2277  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2278  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2279  <p class=MsoBodyText>None</p>
2280  </td>
2281  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2282  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2283  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2284  <p class=MsoBodyText>2</p>
2285  </td>
2286 </tr>
2287 <tr style='height:19.25pt'>
2288  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2289  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2290  <p class=MsoBodyText>Corr_length</p>
2291  </td>
2292  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2293  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2294  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2295  <p class=MsoBodyText>Å</p>
2296  </td>
2297  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2298  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2299  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2300  <p class=MsoBodyText>100.0</p>
2301  </td>
2302 </tr>
2303 <tr style='height:19.25pt'>
2304  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2305  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2306  <p class=MsoBodyText>Block_sld</p>
2307  </td>
2308  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2309  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2310  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2311  <p class=MsoBodyText>Å<sup>-2</sup></p>
2312  </td>
2313  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2314  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2315  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2316  <p class=MsoBodyText>2e-6</p>
2317  </td>
2318 </tr>
2319 <tr style='height:19.25pt'>
2320  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2321  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2322  <p class=MsoBodyText>Solvent_sld</p>
2323  </td>
2324  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2325  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2326  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2327  <p class=MsoBodyText>Å<sup>-2</sup></p>
2328  </td>
2329  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2330  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2331  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2332  <p class=MsoBodyText>6e-6</p>
2333  </td>
2334 </tr>
2335 <tr style='height:19.25pt'>
2336  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2337  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2338  <p class=MsoBodyText>Background</p>
2339  </td>
2340  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2341  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2342  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2343  <p class=MsoBodyText>cm<sup>-1</sup></p>
2344  </td>
2345  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2346  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2347  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2348  <p class=MsoBodyText>0.0</p>
2349  </td>
2350 </tr>
2351</table>
2352
2353</div>
2354
2355
2356<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
2357
2358<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
2359
2360<p class=MsoNormal style='margin-left:.25in;text-indent:.25in;text-autospace:
2361none'>J. Teixeira, (1988) J. Appl. Cryst., vol. 21, p781-785</p>
2362
2363<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2364
2365<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2366
2367<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2368style='font-size:14.0pt'>3.7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2369</span></span></b><b><span style='font-size:14.0pt'> <a name="BEPolyelectrolyte">BEPolyelectrolyte Model</a></span></b></p>
2370
2371<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2372
2373<p class=MsoNormal style='margin-left:.25in'>Calculates the structure factor of
2374a polyelectrolyte solution with the RPA expression derived by Borue and
2375Erukhimovich. The value returned is in cm<sup>-1</sup>.</p>
2376
2377<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
2378
2379<p class=MsoNormal align=center style='text-align:center'><span
2380style='font-size:14.0pt'><sub><img width=365 height=125
2381src="../images/html/image031.gif"></sub></span></p>
2382
2383<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
2384
2385<p class=MsoNormal style='margin-left:.25in'>K is a contrast factor of the
2386polymer, L<sub>b</sub> is the Bjerrum length, h is the virial parameter, b is
2387the monomer length, C<sub>s</sub> is the concentration of monovalent salt,
2388&#945; is the ionization degree, C<sub>a</sub> is the polymer molar
2389concentration, and bck is the incoherent background.</p>
2390
2391<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2392
2393<div align=center>
2394
2395<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2396 style='border-collapse:collapse;border:none'>
2397 <tr style='height:19.25pt'>
2398  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2399  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2400  <p class=MsoBodyText>Parameter name</p>
2401  </td>
2402  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2403  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2404  <p class=MsoBodyText>Units</p>
2405  </td>
2406  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2407  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2408  <p class=MsoBodyText>Default value</p>
2409  </td>
2410 </tr>
2411 <tr style='height:19.25pt'>
2412  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2413  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2414  <p class=MsoBodyText>K</p>
2415  </td>
2416  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2417  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2418  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2419  <p class=MsoBodyText>Barns = 10<sup>-24</sup> cm<sup>2</sup></p>
2420  </td>
2421  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2422  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2423  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2424  <p class=MsoBodyText>10</p>
2425  </td>
2426 </tr>
2427 <tr style='height:19.25pt'>
2428  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2429  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2430  <p class=MsoBodyText>L<sub>b</sub></p>
2431  </td>
2432  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2433  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2434  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2435  <p class=MsoBodyText>Å</p>
2436  </td>
2437  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2438  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2439  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2440  <p class=MsoBodyText>7.1</p>
2441  </td>
2442 </tr>
2443 <tr style='height:19.25pt'>
2444  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2445  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2446  <p class=MsoBodyText>h</p>
2447  </td>
2448  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2449  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2450  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2451  <p class=MsoBodyText>Å<sup>-3</sup></p>
2452  </td>
2453  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2454  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2455  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2456  <p class=MsoBodyText>12</p>
2457  </td>
2458 </tr>
2459 <tr style='height:19.25pt'>
2460  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2461  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2462  <p class=MsoBodyText>b</p>
2463  </td>
2464  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2465  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2466  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2467  <p class=MsoBodyText>Å</p>
2468  </td>
2469  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2470  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2471  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2472  <p class=MsoBodyText>10</p>
2473  </td>
2474 </tr>
2475 <tr style='height:19.25pt'>
2476  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2477  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2478  <p class=MsoBodyText>Cs</p>
2479  </td>
2480  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2481  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2482  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2483  <p class=MsoBodyText>Mol/L</p>
2484  </td>
2485  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2486  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2487  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2488  <p class=MsoBodyText>0</p>
2489  </td>
2490 </tr>
2491 <tr style='height:19.25pt'>
2492  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2493  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2494  <p class=MsoBodyText>Alpha</p>
2495  </td>
2496  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2497  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2498  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2499  <p class=MsoBodyText>None</p>
2500  </td>
2501  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2502  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2503  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2504  <p class=MsoBodyText>0.05</p>
2505  </td>
2506 </tr>
2507 <tr style='height:19.25pt'>
2508  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2509  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2510  <p class=MsoBodyText>Ca</p>
2511  </td>
2512  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2513  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2514  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2515  <p class=MsoBodyText>Mol/L</p>
2516  </td>
2517  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2518  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2519  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2520  <p class=MsoBodyText>0.7</p>
2521  </td>
2522 </tr>
2523 <tr style='height:19.25pt'>
2524  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2525  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2526  <p class=MsoBodyText>Background</p>
2527  </td>
2528  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2529  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2530  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2531  <p class=MsoBodyText>cm<sup>-1</sup></p>
2532  </td>
2533  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2534  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2535  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2536  <p class=MsoBodyText>0.0</p>
2537  </td>
2538 </tr>
2539</table>
2540
2541</div>
2542
2543<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2544
2545<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>References: </p>
2546
2547<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Borue, V. Y.,
2548Erukhimovich, I. Y. Macromolecules 21, 3240 (1988).</p>
2549
2550<p class=MsoNormal style='margin-left:.5in;text-autospace:none'><span lang=FR>Joanny,
2551J.-F., Leibler, L. Journal de Physique 51, 545 (1990).</span></p>
2552
2553<p class=MsoNormal style='margin-left:.5in;text-autospace:none'><span lang=FR>Moussaid,
2554A., Schosseler, F., Munch, J.-P., Candau, S. J. Journal de Physique II France </span></p>
2555
2556<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>3, 573 (1993).</p>
2557
2558<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Raphaël, E.,
2559Joanny, J.-F., Europhysics Letters 11, 179 (1990).</p>
2560
2561<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>&nbsp;</p>
2562
2563<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2564
2565<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2566style='font-size:14.0pt'>3.8.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2567</span></span></b><b><span style='font-size:14.0pt'><a name="Guinier"> Guinier (Model)</a></span></b></p>
2568
2569<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2570
2571<p class=MsoNormal style='margin-left:.25in'>A Guinier analysis is done by linearizing
2572the data at low q by plotting it as log(I) versus Q<sup>2</sup>. The Guinier
2573radius R<sub>g</sub> can be obtained by fitting the following model:</p>
2574
2575<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
2576
2577<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2578style='font-size:14.0pt'><sub><img width=183 height=28
2579src="../images/html/image032.gif"></sub></span></p>
2580
2581<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
2582style='font-size:14.0pt'>&nbsp;</span></b></p>
2583
2584<div align=center>
2585
2586<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2587 style='border-collapse:collapse;border:none'>
2588 <tr style='height:19.25pt'>
2589  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2590  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2591  <p class=MsoBodyText>Parameter name</p>
2592  </td>
2593  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2594  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2595  <p class=MsoBodyText>Units</p>
2596  </td>
2597  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2598  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2599  <p class=MsoBodyText>Default value</p>
2600  </td>
2601 </tr>
2602 <tr style='height:19.25pt'>
2603  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2604  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2605  <p class=MsoBodyText>scale</p>
2606  </td>
2607  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2608  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2609  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2610  <p class=MsoBodyText>cm<sup>-1</sup></p>
2611  </td>
2612  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2613  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2614  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2615  <p class=MsoBodyText>1.0</p>
2616  </td>
2617 </tr>
2618 <tr style='height:19.25pt'>
2619  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2620  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2621  <p class=MsoBodyText>Rg</p>
2622  </td>
2623  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2624  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2625  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2626  <p class=MsoBodyText>Å</p>
2627  </td>
2628  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2629  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2630  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2631  <p class=MsoBodyText>60</p>
2632  </td>
2633 </tr>
2634</table>
2635
2636</div>
2637
2638<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2639<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2640<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2641
2642<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2643style='font-size:14.0pt'>3.9.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2644</span></span></b><b><span style='font-size:14.0pt'> <a name="PorodModel">PorodModel</a></span></b></p>
2645
2646<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2647
2648<p class=MsoNormal style='margin-left:.25in'>A Porod analysis is done by
2649linearizing the data at high q by plotting it as log(I) versus log(Q). In the
2650high q region we can fit the following model: </p>
2651
2652<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
2653
2654<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2655style='font-size:14.0pt'><sub><img width=368 height=25
2656src="../images/html/image033.gif"></sub></span></p>
2657
2658<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2659style='font-size:14.0pt'>&nbsp;</span></p>
2660
2661<p class=MsoNormal style='margin-left:.25in'>C is the scale factor and  Sv is
2662the specific surface area of the sample and <span style='font-family:"Arial","sans-serif"'>&#916;</span>&#961;
2663is the contrast factor. </p>
2664
2665<p class=MsoNormal style='margin-left:.25in'>&nbsp;</p>
2666
2667<p class=MsoNormal style='margin-left:.25in'>The background term is added for
2668data analysis.</p>
2669
2670<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
2671style='font-size:14.0pt'>&nbsp;</span></b></p>
2672
2673<div align=center>
2674
2675<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2676 style='border-collapse:collapse;border:none'>
2677 <tr style='height:19.25pt'>
2678  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2679  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2680  <p class=MsoBodyText>Parameter name</p>
2681  </td>
2682  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2683  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2684  <p class=MsoBodyText>Units</p>
2685  </td>
2686  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2687  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2688  <p class=MsoBodyText>Default value</p>
2689  </td>
2690 </tr>
2691 <tr style='height:19.25pt'>
2692  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2693  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2694  <p class=MsoBodyText>Scale</p>
2695  </td>
2696  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2697  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2698  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2699  <p class=MsoBodyText>Å<sup>-4</sup></p>
2700  </td>
2701  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2702  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2703  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2704  <p class=MsoBodyText>0.1</p>
2705  </td>
2706 </tr>
2707 <tr style='height:19.25pt'>
2708  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2709  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2710  <p class=MsoBodyText>Background</p>
2711  </td>
2712  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2713  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2714  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2715  <p class=MsoBodyText>cm<sup>-1</sup></p>
2716  </td>
2717  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2718  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2719  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2720  <p class=MsoBodyText>0</p>
2721  </td>
2722 </tr>
2723</table>
2724
2725</div>
2726
2727<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2728
2729<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2730
2731<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2732
2733
2734
2735
2736<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
2737style='font-size:14.0pt'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
2738</span></span></b><b><span style='font-size:14.0pt'><a name="Customized Models">Customized Models</a></span></b></p>
2739
2740<p class=MsoBodyText>&nbsp;</p>
2741
2742<p class=MsoBodyText>These models are defined by users (See SansView tutorial for details). </p>
2743
2744<p class=MsoBodyText>&nbsp;</p>
2745
2746<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2747style='font-size:14.0pt'>4.1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2748</span></span></b><b><span style='font-size:14.0pt'> <a name="A+Bcos(2x)+Csin(2x)">A+Bcos(2x)+Csin(2x)</a></span></b></p>
2749
2750<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
2751
2752<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>This function, as an example function, calculates intensity = A + Bcos(2Q) + Csin(2Q). </p>
2753
2754
2755<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
2756<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><b><span
2757style='font-size:14.0pt'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
2758</span></span></b><b><span style='font-size:14.0pt'><a name="Structure Factors">Structure Factors</a></span></b></p>
2759
2760<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>&nbsp;</p>
2761
2762<p class=MsoListParagraphCxSpLast style='margin-left:.25in'>The information in
2763this section is originated from NIST SANS IgorPro package.</p>
2764
2765<p class=MsoBodyText>&nbsp;</p>
2766
2767<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
2768style='font-size:14.0pt'>5.1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2769</span></span></b><b><span style='font-size:14.0pt'> <a name="HardSphereStructure">HardSphere Structure</a></span></b></p>
2770
2771<p class=MsoNormal>&nbsp;</p>
2772
2773<p class=MsoNormal>This calculates the interparticle structure factor for
2774monodisperse spherical particles interacting through hard sphere (excluded
2775volume) interactions. The calculation uses the Percus-Yevick closure where the
2776interparticle potential is: </p>
2777
2778<p class=MsoNormal>&nbsp;</p>
2779
2780<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2781style='font-size:14.0pt;position:relative;top:15.0pt'><img width=122 height=48
2782src="../images/html/image034.gif"></span></p>
2783
2784<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2785style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p>
2786
2787<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the
2788distance from the center of the sphere of a radius R.</span></p>
2789
2790<p class=MsoNormal>&nbsp;</p>
2791
2792<p class=MsoNormal>&nbsp;</p>
2793
2794<div align=center>
2795
2796<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2797 style='border-collapse:collapse;border:none'>
2798 <tr style='height:19.25pt'>
2799  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2800  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2801  <p class=MsoBodyText>Parameter name</p>
2802  </td>
2803  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2804  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2805  <p class=MsoBodyText>Units</p>
2806  </td>
2807  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2808  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2809  <p class=MsoBodyText>Default value</p>
2810  </td>
2811 </tr>
2812 <tr style='height:19.25pt'>
2813  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2814  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2815  <p class=MsoBodyText>radius</p>
2816  </td>
2817  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2818  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2819  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2820  <p class=MsoBodyText>Å</p>
2821  </td>
2822  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2823  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2824  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2825  <p class=MsoBodyText>50.0</p>
2826  </td>
2827 </tr>
2828 <tr style='height:19.25pt'>
2829  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2830  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2831  <p class=MsoBodyText>volfraction</p>
2832  </td>
2833  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2834  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2835  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2836  <p class=MsoBodyText>&nbsp;</p>
2837  </td>
2838  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2839  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2840  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2841  <p class=MsoBodyText>0.2</p>
2842  </td>
2843 </tr>
2844</table>
2845
2846</div>
2847
2848<p class=MsoNormal><img width=579 height=273 id="Picture 43"
2849src="../images/html/image035.gif"></p>
2850
2851<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
2852
2853<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
2854
2855<p class=MsoNormal style='text-autospace:none'>References: </p>
2856
2857<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Percus, J. K.;
2858Yevick, J. Phys. Rev. 110, 1. (1958).</p>
2859
2860<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
2861
2862<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
2863
2864<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
2865
2866<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
2867style='font-size:14.0pt'>5.2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2868</span></span></b><b><span style='font-size:14.0pt'><a name="SquareWellStructure"> SquareWell Structure</a></span></b></p>
2869
2870<p class=MsoNormal>&nbsp;</p>
2871
2872<p class=MsoNormal>This calculates the interparticle structure factor for a
2873squar well fluid spherical particles The mean spherical approximation (MSA)
2874closure was used for this calculation, and is not the most appropriate closure
2875for an attractive interparticle potential. This solution has been compared to
2876Monte Carlo simulations for a square well fluid, showing this calculation to be
2877limited in applicability to well depths e &lt; 1.5 kT and volume fractions f
2878&lt; 0.08.</p>
2879
2880<p class=MsoNormal>&nbsp;</p>
2881
2882<p class=MsoNormal>Positive well depths correspond to an attractive potential
2883well. Negative well depths correspond to a potential &quot;shoulder&quot;,
2884which may or may not be physically reasonable.</p>
2885
2886<p class=MsoNormal>&nbsp;</p>
2887
2888<p class=MsoNormal>The well width (l) is defined as multiples of the particle
2889diameter (2*R)</p>
2890
2891<p class=MsoNormal>&nbsp;</p>
2892
2893<p class=MsoNormal>The interaction potential is:</p>
2894
2895<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2896style='font-size:14.0pt;position:relative;top:22.0pt'><img width=174 height=67
2897src="../images/html/image036.gif"></span></p>
2898
2899<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2900style='font-size:14.0pt;position:relative;top:7.0pt'>&nbsp;</span></p>
2901
2902<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the
2903distance from the center of the sphere of a radius R.</span></p>
2904
2905<p class=MsoNormal>&nbsp;</p>
2906
2907<p class=MsoNormal>&nbsp;</p>
2908
2909<div align=center>
2910
2911<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
2912 style='margin-left:-1.5pt;border-collapse:collapse;border:none'>
2913 <tr style='height:19.25pt'>
2914  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
2915  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2916  <p class=MsoBodyText>Parameter name</p>
2917  </td>
2918  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2919  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2920  <p class=MsoBodyText>Units</p>
2921  </td>
2922  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
2923  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2924  <p class=MsoBodyText>Default value</p>
2925  </td>
2926 </tr>
2927 <tr style='height:19.25pt'>
2928  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
2929  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2930  <p class=MsoBodyText>radius</p>
2931  </td>
2932  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2933  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2934  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2935  <p class=MsoBodyText>Å</p>
2936  </td>
2937  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2938  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2939  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2940  <p class=MsoBodyText>50.0</p>
2941  </td>
2942 </tr>
2943 <tr style='height:19.25pt'>
2944  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
2945  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2946  <p class=MsoBodyText>volfraction</p>
2947  </td>
2948  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2949  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2950  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2951  <p class=MsoBodyText>&nbsp;</p>
2952  </td>
2953  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2954  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2955  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2956  <p class=MsoBodyText>0.2</p>
2957  </td>
2958 </tr>
2959 <tr style='height:19.25pt'>
2960  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
2961  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2962  <p class=MsoBodyText>welldepth</p>
2963  </td>
2964  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2965  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2966  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2967  <p class=MsoBodyText>kT</p>
2968  </td>
2969  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2970  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2971  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2972  <p class=MsoBodyText>1.5</p>
2973  </td>
2974 </tr>
2975 <tr style='height:19.25pt'>
2976  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
2977  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2978  <p class=MsoBodyText>wellwidth</p>
2979  </td>
2980  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2981  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2982  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2983  <p class=MsoBodyText>diameters</p>
2984  </td>
2985  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
2986  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
2987  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
2988  <p class=MsoBodyText>1.2</p>
2989  </td>
2990 </tr>
2991</table>
2992
2993</div>
2994
2995<p class=MsoNormal>&nbsp;</p>
2996
2997<p class=MsoNormal><img width=546 height=296 id="Picture 53"
2998src="../images/html/image037.gif"></p>
2999
3000<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3001
3002<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3003
3004<p class=MsoNormal style='text-autospace:none'>References: </p>
3005
3006<p class=MsoNormal style='margin-left:.5in;text-autospace:none'>Sharma, R. V.;
3007Sharma, K. C. Physica, 89A, 213. (1977).</p>
3008
3009<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3010
3011<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
3012style='font-size:14.0pt'>5.3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
3013</span></span></b><b><span style='font-size:14.0pt'><a name="HayterMSAStructure"> HayterMSA Structure</a></span></b></p>
3014
3015<p class=MsoNormal>&nbsp;</p>
3016
3017<p class=MsoNormal>This calculates the Structure factor (the Fourier transform
3018of the pair correlation function g(r)) for a system of charged, spheroidal
3019objects in a dielectric medium.  When combined with an appropriate form factor
3020(such as sphere, core+shell, ellipsoid etc…), this allows for inclusion of the
3021interparticle interference effects due to screened coulomb repulsion between
3022charged particles. This routine only works for charged particles.  If the
3023charge is set to zero the routine will self destruct.  For non-charged
3024particles use a hard sphere potential.</p>
3025
3026<p class=MsoNormal>&nbsp;</p>
3027
3028<p class=MsoNormal>The salt concentration is used to compute the ionic strength
3029of the solution which in turn is used to compute the Debye screening length. 
3030At present there is no provision for entering the ionic strength directly nor
3031for use of any multivalent salts.  The counterions are also assumed to be monovalent.</p>
3032
3033<p class=MsoNormal>&nbsp;</p>
3034
3035<p class=MsoNormal>&nbsp;</p>
3036
3037<div align=center>
3038
3039<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
3040 style='margin-left:-1.5pt;border-collapse:collapse;border:none'>
3041 <tr style='height:19.25pt'>
3042  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3043  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3044  <p class=MsoBodyText>Parameter name</p>
3045  </td>
3046  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
3047  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3048  <p class=MsoBodyText>Units</p>
3049  </td>
3050  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
3051  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3052  <p class=MsoBodyText>Default value</p>
3053  </td>
3054 </tr>
3055 <tr style='height:19.25pt'>
3056  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3057  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3058  <p class=MsoBodyText>radius</p>
3059  </td>
3060  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3061  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3062  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3063  <p class=MsoBodyText>Å</p>
3064  </td>
3065  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3066  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3067  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3068  <p class=MsoBodyText>20.8</p>
3069  </td>
3070 </tr>
3071 <tr style='height:19.25pt'>
3072  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3073  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3074  <p class=MsoBodyText>charge</p>
3075  </td>
3076  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3077  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3078  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3079  <p class=MsoBodyText>&nbsp;</p>
3080  </td>
3081  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3082  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3083  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3084  <p class=MsoBodyText>19</p>
3085  </td>
3086 </tr>
3087 <tr style='height:19.25pt'>
3088  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3089  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3090  <p class=MsoBodyText>volfraction</p>
3091  </td>
3092  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3093  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3094  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3095  <p class=MsoBodyText>&nbsp;</p>
3096  </td>
3097  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3098  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3099  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3100  <p class=MsoBodyText>0.2</p>
3101  </td>
3102 </tr>
3103 <tr style='height:19.25pt'>
3104  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3105  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3106  <p class=MsoBodyText>temperature</p>
3107  </td>
3108  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3109  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3110  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3111  <p class=MsoBodyText>K</p>
3112  </td>
3113  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3114  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3115  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3116  <p class=MsoBodyText>318</p>
3117  </td>
3118 </tr>
3119 <tr style='height:19.25pt'>
3120  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3121  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3122  <p class=MsoBodyText>salt conc</p>
3123  </td>
3124  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3125  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3126  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3127  <p class=MsoBodyText>M</p>
3128  </td>
3129  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3130  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3131  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3132  <p class=MsoBodyText>0</p>
3133  </td>
3134 </tr>
3135 <tr style='height:19.25pt'>
3136  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3137  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3138  <p class=MsoBodyText>dielectconst</p>
3139  </td>
3140  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3141  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3142  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3143  <p class=MsoBodyText>&nbsp;</p>
3144  </td>
3145  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3146  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3147  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3148  <p class=MsoBodyText>71.1</p>
3149  </td>
3150 </tr>
3151</table>
3152
3153</div>
3154
3155<p class=MsoNormal>&nbsp;</p>
3156
3157<p class=MsoNormal><img width=548 height=354 id="Picture 56"
3158src="../images/html/image038.gif"></p>
3159
3160<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3161
3162<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3163
3164<p class=MsoNormal style='text-autospace:none'>References: </p>
3165
3166<p class=MsoNormal style='text-indent:.5in;text-autospace:none'>JP Hansen and
3167JB Hayter, Molecular Physics 46, 651-656 (1982).</p>
3168
3169<p class=MsoNormal style='text-autospace:none'>            JB Hayter and J
3170Penfold, Molecular Physics 42, 109-118 (1981).</p>
3171
3172<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3173
3174<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3175
3176<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
3177style='font-size:14.0pt'>5.4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
3178</span></span></b><b><span style='font-size:14.0pt'><a name="StickyHSStructure"> StickyHS Structure</a></span></b></p>
3179
3180<p class=MsoNormal>&nbsp;</p>
3181
3182<p class=MsoNormal>This calculates the interparticle structure factor for a
3183hard sphere fluid with a narrow attractive well. A perturbative solution of the
3184Percus-Yevick closure is used. The strength of the attractive well is described
3185in terms of &quot;stickiness&quot; as defined below. The returned value is a
3186dimensionless structure factor, S(q).</p>
3187
3188<p class=MsoNormal>&nbsp;</p>
3189
3190<p class=MsoNormal>The perturb (perturbation parameter), epsilon, should be
3191held between 0.01 and 0.1. It is best to hold the perturbation parameter fixed
3192and let the &quot;stickiness&quot; vary to adjust the interaction strength. The
3193stickiness, tau, is defined in equation 21 and is a function of both the
3194perturbation parameter and the interaction strength. Tau and epsilon are
3195defined in terms of the hard sphere diameter (sigma = 2R), the width of the
3196square well, delta (same units as R), and the depth of the well, u<sub>o</sub>,
3197in units of kT. From the definition, it is clear that smaller tau mean stronger
3198attraction.</p>
3199
3200<p class=MsoNormal align=center style='text-align:center'><span
3201style='font-size:14.0pt;position:relative;top:22.0pt'><img width=135 height=67
3202src="../images/html/image039.gif"></span></p>
3203
3204<p class=MsoNormal align=center style='text-align:center'><span
3205style='font-size:14.0pt;position:relative;top:27.0pt'>&nbsp;</span></p>
3206
3207<p class=MsoNormal>where the interaction potential is</p>
3208
3209<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
3210style='font-size:14.0pt;position:relative;top:23.0pt'><img width=186 height=69
3211src="../images/html/image040.gif"></span></p>
3212
3213<p class=MsoNormal>The Percus-Yevick (PY) closure was used for this
3214calculation, and is an adequate closure for an attractive interparticle
3215potential. This solution has been compared to Monte Carlo simulations for a
3216square well fluid, with good agreement.</p>
3217
3218<p class=MsoNormal>&nbsp;</p>
3219
3220<p class=MsoNormal>The true particle volume fraction, f, is not equal to h,
3221which appears in most of the reference. The two are related in equation (24) of
3222the reference. The reference also describes the relationship between this
3223perturbation solution and the original sticky hard sphere (or adhesive sphere)
3224model by Baxter.</p>
3225
3226<p class=MsoNormal>&nbsp;</p>
3227
3228<p class=MsoNormal>NOTES: The calculation can go haywire for certain
3229combinations of the input parameters, producing unphysical solutions - in this
3230case errors are reported to the command window and the S(q) is set to -1 (it
3231will disappear on a log-log plot). Use tight bounds to keep the parameters to
3232values that you know are physical (test them) and keep nudging them until the
3233optimization does not hit the constraints.</p>
3234
3235<p class=MsoNormal>&nbsp;</p>
3236
3237<div align=center>
3238
3239<table class=MsoTableGrid border=1 cellspacing=0 cellpadding=0
3240 style='margin-left:-1.5pt;border-collapse:collapse;border:none'>
3241 <tr style='height:19.25pt'>
3242  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3243  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3244  <p class=MsoBodyText>Parameter name</p>
3245  </td>
3246  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
3247  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3248  <p class=MsoBodyText>Units</p>
3249  </td>
3250  <td width=143 valign=top style='width:107.0pt;border:solid windowtext 1.0pt;
3251  border-left:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3252  <p class=MsoBodyText>Default value</p>
3253  </td>
3254 </tr>
3255 <tr style='height:19.25pt'>
3256  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3257  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3258  <p class=MsoBodyText>radius</p>
3259  </td>
3260  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3261  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3262  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3263  <p class=MsoBodyText>Å</p>
3264  </td>
3265  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3266  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3267  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3268  <p class=MsoBodyText>20.8</p>
3269  </td>
3270 </tr>
3271 <tr style='height:19.25pt'>
3272  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3273  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3274  <p class=MsoBodyText>perturb </p>
3275  </td>
3276  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3277  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3278  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3279  <p class=MsoBodyText>&nbsp;</p>
3280  </td>
3281  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3282  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3283  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3284  <p class=MsoBodyText>0.05</p>
3285  </td>
3286 </tr>
3287 <tr style='height:19.25pt'>
3288  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3289  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3290  <p class=MsoBodyText>volfraction</p>
3291  </td>
3292  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3293  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3294  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3295  <p class=MsoBodyText>&nbsp;</p>
3296  </td>
3297  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3298  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3299  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3300  <p class=MsoBodyText>0.1</p>
3301  </td>
3302 </tr>
3303 <tr style='height:19.25pt'>
3304  <td width=145 valign=top style='width:108.5pt;border:solid windowtext 1.0pt;
3305  border-top:none;padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3306  <p class=MsoBodyText>stickiness</p>
3307  </td>
3308  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3309  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3310  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3311  <p class=MsoBodyText>K</p>
3312  </td>
3313  <td width=143 valign=top style='width:107.0pt;border-top:none;border-left:
3314  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
3315  padding:0in 5.4pt 0in 5.4pt;height:19.25pt'>
3316  <p class=MsoBodyText>0.2</p>
3317  </td>
3318 </tr>
3319</table>
3320
3321</div>
3322
3323<p class=MsoNormal>&nbsp;</p>
3324
3325<p class=MsoNormal><img width=524 height=359 id="Picture 111"
3326src="../images/html/image041.gif"></p>
3327
3328<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3329
3330<p class=MsoNormal style='text-autospace:none'>&nbsp;</p>
3331
3332<p class=MsoNormal style='text-autospace:none'>References: </p>
3333
3334<p class=MsoNormal style='text-indent:.5in;text-autospace:none'>Menon, S. V.
3335G., Manohar, C. and K. Srinivas Rao J. Chem. Phys., 95(12), 9186-9190 (1991).</p>
3336
3337
3338
3339<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
3340
3341<p class=MsoNormal><b><span style='font-size:14.0pt'>&nbsp;</span></b></p>
3342
3343
3344<p class=MsoBodyText><b><span style='font-size:14.0pt'><a name="References">References</a></span></b></p>
3345
3346<p class=MsoBodyText>Feigin, L. A, and D. I. Svergun (1987) &quot;Structure
3347Analysis by Small-Angle X-Ray and Neutron Scattering&quot;, Plenum Press, New York.</p>
3348
3349<p class=MsoBodyText>Guinier, A. and G. Fournet (1955) &quot;Small-Angle
3350Scattering of X-Rays&quot;, John Wiley and Sons, New York.</p>
3351
3352<p class=MsoBodyText>Kline, S. R. (2006) <i>J Appl. Cryst.</i> <b>39</b>(6),
3353895.</p>
3354
3355<p class=MsoBodyText>Hansen, S., (1990)<i> J. Appl. Cryst. </i>23, 344-346.</p>
3356
3357<p class=MsoBodyText>Henderson, S.J. (1996) <i>Biophys. J. </i>70, 1618-1627</p>
3358
3359<p class=MsoBodyText>Stöckel, P., May, R., Strell, I., Cejka, Z., Hoppe, W.,
3360Heumann, H., Zillig, W. and Crespi, H. (1980) <i>Eur. J. Biochem. </i>112,
3361411-417.</p>
3362
3363<p class=MsoBodyText>McAlister, B.C. and Grady, B.P., (1998) J. Appl. Cryst. 31,
3364594-599.</p>
3365
3366<p class=MsoBodyText>Porod, G. (1982) in Small Angle X-ray Scattering, editors
3367Glatter, O. and Kratky, O., Academic Press.</p>
3368
3369<p class=MsoBodyText>&nbsp;</p>
3370
3371<p class=MsoNormal>&nbsp;</p>
3372
3373</div>
3374
3375</body>
3376
3377</html>
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