model_functions.html @ 164c7d0

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> </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
159	style='font-size:14.0pt'>1.<span style='font:7.0pt "Times New Roman"'>
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
163	DANSE software for SANS. </p>
164
165	<p class=MsoNormal> </p>
166
167	<p class=MsoNormal>Readers are also referred to the SANS/DANSE wiki page:</p>
168
169	<p class=MsoNormal> </p>
170
171	<p class=MsoNormal style='text-indent:.25in'><i>http://danse.us/trac/sans</i></p>
172
173	<p class=MsoNormal> </p>
174
175	<p class=MsoNormal>Users can report a bug at the following URL:</p>
176
177	<p class=MsoNormal> </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> </b></p>
182
183
184	<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
185	style='font-size:14.0pt'>2.<span style='font:7.0pt "Times New Roman"'>
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> </p>
189
190	<p class=MsoNormal>This software provides form factors for various particle
191	shapes. After giving a mathematical definition of each model, we draw the list
192	of parameters available to the user. Validation plots for each model are also
193	presented. Instructions on how to use the software is available with the source
194	code, available from SVN:</p>
195
196	<p class=MsoNormal> </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> </p>
201
202	<p class=MsoNormal>To easily compare to the scattering intensity measured in
203	experiments, 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
206	height=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
211	height=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
214	un-normalized form factor, <i>ρ(<b>r</b>)</i> is the scattering length
215	density at a given point in space and the integration is done over the volume <i>V</i>
216	of the scatterer. </p>
217
218	<p class=MsoNormal> </p>
219
220	<p class=MsoNormal>For systems without inter-particle interference, the form
221	factors we provide can be related to the scattering intensity by the particle
222	volume fraction:<sub><img width=92 height=21
223	src="../images/html/image003.gif"></sub>.</p>
224
225	<p class=MsoNormal> </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
229	scattering intensity only depends on the length of q. The intensity measured on
230	the plane of the SANS detector will have an azimuthal symmetry around <i>q</i>=0.</p>
231
232	<p class=MsoNormal> </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"'>φ</span>)</i> for an
236	oriented system as a function of a q-vector in the plane of the detector. We
237	define the angle <span style='font-family:"Arial","sans-serif"'>φ</span>
238	as the angle between the q vector and the horizontal (x) axis of the plane of
239	the detector.</p>
240
241	<p class=MsoNormal> </p>
242
243
244	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
245	style='font-size:14.0pt'>2.1.<span style='font:7.0pt "Times New Roman"'>
246	</span></span></b><b><span style='font-size:14.0pt'><a name="SphereModel">SphereModel</a></span></b></p>
247
248	<p class=MsoNormal> </p>
249
250	<p class=MsoNormal>This model provides the form factor, P(q), for a
251	monodisperse spherical particle with uniform scattering length density. The
252	form factor is normalized by the particle volume as described below.</p>
253
254	<p class=MsoNormal> </p>
255
256	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.1.<span
257	style='font:7.0pt "Times New Roman"'>      </span></b><b>Definition</b></p>
258
259	<p class=MsoNormal> </p>
260
261	<p class=MsoNormal>The 1D scattering intensity is calculated in the following way
262	(Guinier, 1955):</p>
263
264	<p class=MsoNormal align=center style='text-align:center'><sub><img width=337
265	height=53 src="../images/html/image004.gif"></sub> (3)</p>
266
267	<p class=MsoNormal>where <i>scale</i> is a scale factor, V is the volume of the
268	scatterer, <i>r</i> is the radius of the sphere, <i>bkg</i> is the background
269	level and <i><span style='font-family:"Arial","sans-serif"'>Δ</span>ρ</i>
270	(contrast) is the scattering length density difference between the scatterer
271	and the solvent it is in.</p>
272
273	<p class=MsoNormal> </p>
274
275	<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above,
276	regardless of the orientation of the q vector.</p>
277
278	<p class=MsoNormal> </p>
279
280	<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
281	and 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> </p>
370
371	<p class=MsoNormal>Our model uses the form factor calculations implemented in a
372	c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
373
374	<p class=MsoNormal> </p>
375
376
377	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.1.2.<span
378	style='font:7.0pt "Times New Roman"'>      </span></b><b>Validation
379	of the sphere model</b></p>
380
381	<p class=MsoNormal> </p>
382
383	<p class=MsoNormal>Validation of our code was done by comparing the output of
384	the 1D model to the output of the software provided by the NIST (Kline, 2006).
385	Figure 1 shows a comparison of the output of our model and the output of the
386	NIST software.</p>
387
388	<p class=MsoNormal> </p>
389
390	<p class=MsoBodyText align=center style='text-align:center;page-break-after:
391	avoid'><img width=573 height=315 id="Picture 5"
392	src="../images/html/image005.jpg" alt="sphere_1D_validation"></p>
393
394	<p class=MsoCaption>Figure 1: Comparison of the DANSE scattering intensity for
395	a sphere with the output of the NIST SANS analysis software. The parameters
396	were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>, and
397	Background=0.01 cm<sup> -1</sup>.</p>
398
399	<p class=MsoNormal> </p>
400
401
402	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
403	style='font-size:14.0pt'>2.2.<span style='font:7.0pt "Times New Roman"'>
404	</span></span></b><b><span style='font-size:14.0pt'><a name="CylinderModel">CylinderModel</a></span></b></p>
405
406	<p class=MsoNormal> </p>
407
408	<p class=MsoNormal>This model provides the form factor for a monodisperse right
409	circular cylinder with uniform scattering length density. The form factor is
410	normalized by the particle volume.</p>
411
412	<p class=MsoNormal> </p>
413
414	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.2.1.<span
415	style='font:7.0pt "Times New Roman"'>      </span></b><b>Definition</b></p>
416
417	<p class=MsoNormal> </p>
418
419	<p class=MsoNormal>The output of the 2D scattering intensity function for
420	oriented cylinders is given by (Guinier, 1955):</p>
421
422	<p class=MsoNormal align=center style='text-align:center'><sub><img width=184
423	height=41 src="../images/html/image006.gif"></sub> (4)</p>
424
425	<p class=MsoNormal align=center style='text-align:center'><sub><img width=373
426	height=45 src="../images/html/image007.gif"></sub> (5)</p>
427
428	<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>α</span>
429	is the angle between the axis of the cylinder and the q-vector, V is the volume
430	of the cylinder, L is the length of the cylinder, r is the radius of the
431	cylinder, and <i><span style='font-family:"Arial","sans-serif"'>Δ</span>ρ</i>
432	(contrast) is the scattering length density difference between the scatterer
433	and the solvent. J<sub>1</sub> is the first order Bessel function.</p>
434
435	<p class=MsoNormal> </p>
436
437	<p class=MsoNormal>To provide easy access to the orientation of the cylinder, we
438	define the axis of the cylinder using two angles θ and <span
439	style='font-family:"Arial","sans-serif"'>φ</span>. Those angles are
440	defined on Figure 2.</p>
441
442	<p class=MsoNormal align=center style='text-align:center;page-break-after:avoid'><img
443	width=300 height=281 id="Picture 8" src="../images/html/image008.jpg"
444	alt=angles></p>
445
446	<p class=MsoCaption align=center style='text-align:center'><a
447	name="_Ref173306040"></a><a name="_Ref173213915">Figure </a>2: Definition of the angles for oriented cylinders.</p>
448
449	<p class=MsoNormal> </p>
450
451
452	<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
453	and the parameters of the cylinder model are the following:</p>
454
455	<p class=MsoNormal> </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> </p>
592
593	<p class=MsoNormal>The output of the 1D scattering intensity function for
594	randomly oriented cylinders is then given by:</p>
595
596	<p class=MsoNormal align=center style='text-align:center'><a
597	name="_Ref173306479"></a><a name="_Ref173306528"><sub><img width=253 height=56
598	src="../images/html/image009.gif"></sub> (</a>6)</p>
599
600	<p class=MsoNormal> </p>
601
602	<p class=MsoNormal>The <i>cyl_theta</i> and <i>cyl_phi</i> parameter are not
603	used for the 1D output. Our implementation of the scattering kernel and the 1D
604	scattering intensity use the c-library from NIST.</p>
605
606	<p class=MsoNormal> </p>
607
608
609	<p class=MsoNormal> </p>
610
611	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.2.2.<span
612	style='font:7.0pt "Times New Roman"'>      </span></b><b>Validation
613	of the cylinder model</b></p>
614
615	<p class=MsoNormal> </p>
616
617	<p class=MsoNormal>Validation of our code was done by comparing the output of
618	the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure
619	3 shows a comparison of the 1D output of our model and the output of the NIST
620	software.</p>
621
622	<p class=MsoNormal> </p>
623
624	<p class=MsoNormal>In general, averaging over a distribution of orientations is
625	done by evaluating the following:</p>
626
627	<p class=MsoNormal align=center style='text-align:center'><sub><img width=244
628	height=51 src="../images/html/image010.gif"></sub> (7)</p>
629
630	<p class=MsoNormal>where <i>p(θ,</i><i><span style='font-family:"Arial","sans-serif"'>φ</span>)</i>
631	is the probability distribution for the orientation and <i>P<sub>0</sub>(q,</i><i><span
632	style='font-family:"Arial","sans-serif"'>α</span>)</i> is the scattering
633	intensity for the fully oriented system. Since we have no other software to
634	compare the implementation of the intensity for fully oriented cylinders, we
635	can compare the result of averaging our 2D output using a uniform distribution <i>p(θ,</i><i><span
636	style='font-family:"Arial","sans-serif"'>φ</span>)</i> = 1.0. Figure 4
637	shows the result of such a cross-check.</p>
638
639	<p class=MsoNormal> </p>
640
641	<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
642	id="Picture 11" src="../images/html/image011.jpg"
643	alt="cylinder_1D_validation"></p>
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
646	NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 Å,
647	Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup> -1</sup>.</p>
648
649	<p class=MsoCaption> </p>
650
651	<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
652	id="Picture 12" src="../images/html/image012.jpg"
653	alt="cylinder_2D_average"></p>
654
655	<p class=MsoCaption><a name="_Ref173213305">Figure </a>4: Comparison of the
656	intensity for uniformly distributed cylinders calculated from our 2D model and
657	the intensity from the NIST SANS analysis software. The parameters used were:
658	Scale=1.0, Radius=20 Å, Length=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and
659	Background=0.0 cm<sup> -1</sup>.</p>
660
661	<p class=MsoCaption> </p>
662
663	<p class=MsoNormal> </p>
664
665
666	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
667	style='font-size:14.0pt'>2.3.<span style='font:7.0pt "Times New Roman"'>
668	</span></span></b><b><span style='font-size:14.0pt'><a name="EllipsoidModel">EllipsoidModel</a></span></b></p>
669
670	<p class=MsoNormal> </p>
671
672	<p class=MsoNormal>This model provides the form factor for a monodisperse
673	ellipsoid (ellipsoid of revolution) with uniform scattering length density. The
674	form factor is normalized by the particle volume.</p>
675
676	<p class=MsoNormal> </p>
677
678	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.3.1.<span
679	style='font:7.0pt "Times New Roman"'>      </span></b><b>Definition</b></p>
680
681	<p class=MsoNormal> </p>
682
683	<p class=MsoNormal>The output of the 2D scattering intensity function for
684	oriented ellipsoids is given by (Feigin, 1987):</p>
685
686	<p class=MsoCaption align=center style='text-align:center'><sub><img width=184
687	height=41 src="../images/html/image006.gif"></sub> <span
688	style='font-size:12.0pt;font-weight:normal'>(8)</span></p>
689
690	<p class=MsoCaption align=center style='text-align:center'><sub><img width=371
691	height=47 src="../images/html/image013.gif"></sub> <span
692	style='font-size:12.0pt;font-weight:normal'>(9)</span></p>
693
694	<p class=MsoCaption align=center style='text-align:center'><sub><img width=272
695	height=29 src="../images/html/image014.gif"></sub> <span
696	style='font-size:12.0pt;font-weight:normal'>(10)</span></p>
697
698	<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>α</span>
699	is the angle between the axis of the ellipsoid and the q-vector, V is the
700	volume of the ellipsoid, R<sub>a</sub> is the radius along the rotation axis of
701	the ellipsoid, R<sub>b</sub> is the radius perpendicular to the rotation axis of
702	the ellipsoid and <i><span style='font-family:"Arial","sans-serif"'>Δ</span>ρ</i>
703	(contrast) is the scattering length density difference between the scatterer
704	and the solvent. </p>
705
706	<p class=MsoNormal> </p>
707
708	<p class=MsoNormal>To provide easy access to the orientation of the ellipsoid,
709	we define the rotation axis of the ellipsoid using two angles θ and <span
710	style='font-family:"Arial","sans-serif"'>φ</span>. Similarly to the case
711	of the cylinder, those angles are defined on Figure 2. For the ellipsoid,
712	θ is the angle between the rotation axis and the z-axis. </p>
713
714	<p class=MsoNormal> </p>
715
716	<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
717	and the parameters of the ellipsoid model are the following:</p>
718
719	<p class=MsoNormal> </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> </p>
856
857	<p class=MsoNormal>The output of the 1D scattering intensity function for
858	randomly oriented ellipsoids is then given by equation 6.</p>
859
860	<p class=MsoNormal> </p>
861
862	<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not
863	used for the 1D output. Our implementation of the scattering kernel and the 1D
864	scattering intensity use the c-library from NIST.</p>
865
866	<p class=MsoNormal> </p>
867
868
869	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.3.2.<span
870	style='font:7.0pt "Times New Roman"'>      </span></b><b>Validation
871	of the ellipsoid model</b></p>
872
873	<p class=MsoNormal> </p>
874
875	<p class=MsoNormal>Validation of our code was done by comparing the output of
876	the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure
877	5 shows a comparison of the 1D output of our model and the output of the NIST
878	software.</p>
879
880	<p class=MsoNormal> </p>
881
882	<p class=MsoNormal>Averaging over a distribution of orientation is done by
883	evaluating equation 7. Since we have no other software to compare the
884	implementation of the intensity for fully oriented ellipsoids, we can compare
885	the result of averaging our 2D output using a uniform distribution <i>p(θ,</i><i><span
886	style='font-family:"Arial","sans-serif"'>φ</span>)</i> = 1.0. Figure 6 shows the result of such a cross-check.</p>
887
888	<p class=MsoNormal align=center style='text-align:center'><b><i><span
889	style='color:#3366FF'> </span></i></b></p>
890
891	<p class=MsoNormal>The discrepancy above q=0.3 Å<sup> -1</sup> is due to the
892	way the form factors are calculated in the c-library provided by NIST. A numerical
893	integration has to be performed to obtain P(q) for randomly oriented particles
894	(equation 6). The NIST software performs that integration with a 76-point Gaussian
895	quadrature rule, which will become imprecise at high q where the amplitude
896	varies quickly as a function of q. The DANSE result shown has been obtained by
897	summing over 501 equidistant points in <span style='font-family:"Arial","sans-serif"'>α</span>.
898	Our result was found to be stable over the range of q shown for a number of
899	points higher than 500.</p>
900
901	<p class=MsoNormal style='page-break-after:avoid'><b><img width=573 height=315
902	id="Picture 16" src="../images/html/image015.jpg"
903	alt="ellipsoid_1D_validation"></b></p>
904
905	<p class=MsoCaption><a name="_Ref173222904">Figure </a>5: Comparison of the
906	DANSE scattering intensity for an ellipsoid with the output of the NIST SANS
907	analysis software. The parameters were set to: Scale=1.0, Radius_a=20 Å, Radius_b=400
908	Å, </p>
909
910	<p class=MsoCaption>Contrast=3e-6 Å<sup> -2</sup>, and Background=0.01 cm<sup>
911	-1</sup>.</p>
912
913	<p class=MsoNormal><b> </b></p>
914
915
916	<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
917	id="Picture 17" src="../images/html/image016.jpg"
918	alt="ellipsoid_2D_average"></p>
919
920	<p class=MsoCaption><a name="_Ref173223004">Figure </a>6: Comparison of the
921	intensity for uniformly distributed ellipsoids calculated from our 2D model and
922	the intensity from the NIST SANS analysis software. The parameters used were:
923	Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å<sup> -2</sup>, and
924	Background=0.0 cm<sup> -1</sup>.</p>
925
926	<p class=MsoCaption> </p>
927
928	<p class=MsoNormal><b> </b></p>
929
930	<p class=MsoNormal><b> </b></p>
931
932	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
933	style='font-size:14.0pt'>2.4.<span style='font:7.0pt "Times New Roman"'>
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> </p>
937
938	<p class=MsoNormal>This model provides the form factor, P(<i>q</i>), for a
939	monodisperse spherical particle with a core-shell structure. The form factor is
940	normalized by the particle volume.</p>
941
942	<p class=MsoNormal> </p>
943
944	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.1.<span
945	style='font:7.0pt "Times New Roman"'>      </span></b><b>Definition</b></p>
946
947	<p class=MsoNormal> </p>
948
949	<p class=MsoNormal>The 1D scattering intensity is calculated in the following
950	way (Guinier, 1955):</p>
951
952	<p class=MsoNormal align=right style='text-align:right'><sub><img width=643
953	height=55 src="../images/html/image017.gif"></sub> (11)</p>
954
955	<p class=MsoNormal>where <i>scale</i> is a scale factor, <i>V<sub>s</sub></i>
956	is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the
957	core, <i>r<sub>s</sub></i> is the radius of the shell, <i>r<sub>c</sub></i> is
958	the radius of the core, <i>ρ<sub>c</sub></i> is the scattering length
959	density of the core, <i>ρ<sub>s</sub></i> is the scattering length density
960	of the shell, ρ<sub>solv</sub> is the scattering length density of the
961	solvent, and <i>bkg</i> is the background level.</p>
962
963	<p class=MsoNormal> </p>
964
965	<p class=MsoNormal>The 2D scattering intensity is the same as P(q) above,
966	regardless of the orientation of the q vector.</p>
967
968	<p class=MsoNormal> </p>
969
970	<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
971	and the parameters of the core-shell sphere model are the following:</p>
972
973	<p class=MsoNormal> </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> </p>
1110
1111	<p class=MsoNormal>Our model uses the form factor calculations implemented in a
1112	c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p>
1113
1114	<p class=MsoNormal> </p>
1115
1116
1117	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.4.2.<span
1118	style='font:7.0pt "Times New Roman"'>      </span></b><b>Validation
1119	of the core-shell sphere model</b></p>
1120
1121	<p class=MsoNormal> </p>
1122
1123	<p class=MsoNormal>Validation of our code was done by comparing the output of
1124	the 1D model to the output of the software provided by the NIST (Kline, 2006).
1125	Figure 1 shows a comparison of the output of our model and the output of the
1126	NIST software.</p>
1127
1128	<p class=MsoNormal> </p>
1129
1130	<p class=MsoBodyText align=center style='text-align:center;page-break-after:
1131	avoid'><img width=573 height=315 id="Picture 19"
1132	src="../images/html/image018.jpg" alt="core_shell_sphere_1D_validation"></p>
1133
1134	<p class=MsoCaption>Figure 7: Comparison of the DANSE scattering intensity for
1135	a core-shell sphere with the output of the NIST SANS analysis software. The
1136	parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å<sup> -2</sup>,
1137	and Background=0.001 cm<sup> -1</sup>.</p>
1138
1139	<p class=MsoBodyText> </p>
1140
1141	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1142	style='font-size:14.0pt'>2.5.<span style='font:7.0pt "Times New Roman"'>
1143	</span></span></b><b><span style='font-size:14.0pt'><a name="CoreShellCylinderModel">CoreShellCylinderModel</a></span></b></p>
1144
1145	<p class=MsoNormal> </p>
1146
1147	<p class=MsoNormal>This model provides the form factor for a monodisperse,
1148	circular cylinder with a core-shell scattering length density profile. The form
1149	factor is normalized by the particle volume.</p>
1150
1151	<p class=MsoNormal> </p>
1152
1153	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.5.1.<span
1154	style='font:7.0pt "Times New Roman"'>      </span></b><b>Definition</b></p>
1155
1156	<p class=MsoNormal> </p>
1157
1158	<p class=MsoNormal>The output of the 2D scattering intensity function for
1159	oriented core-shell cylinders is given by (Kline, 2006):</p>
1160
1161	<p class=MsoNormal align=center style='text-align:center'><sub><img width=184
1162	height=45 src="../images/html/image019.gif"></sub> (12)</p>
1163
1164	<p class=MsoNormal align=center style='text-align:center'><sub><img width=476
1165	height=93 src="../images/html/image020.gif"></sub> (13)</p>
1166
1167	<p class=MsoNormal>where <span style='font-family:"Arial","sans-serif"'>α</span>
1168	is the angle between the axis of the cylinder and the q-vector, <i>V<sub>s</sub></i>
1169	is the volume of the outer shell, <i>V<sub>c</sub></i> is the volume of the
1170	core, <i>L</i> is the length of the core, <i>r</i> is the radius of the core, <i>t</i>
1171	is the thickness of the shell, <i>ρ<sub>c</sub></i> is the scattering
1172	length density of the core, <i>ρ<sub>s</sub></i> is the scattering length
1173	density of the shell, ρ<sub>solv</sub> is the scattering length density of
1174	the solvent, and <i>bkg</i> is the background level. The outer radius of the
1175	shell is given by <i>r+t</i> and the total length of the outer shell is given
1176	by <i>L+2t</i>. J<sub>1</sub> is the first order Bessel function.</p>
1177
1178	<p class=MsoNormal> </p>
1179
1180	<p class=MsoNormal>To provide easy access to the orientation of the core-shell cylinder,
1181	we define the axis of the cylinder using two angles θ and <span
1182	style='font-family:"Arial","sans-serif"'>φ</span>. Similarly to the case
1183	of the cylinder, those angles are defined on Figure 2.</p>
1184
1185	<p class=MsoNormal> </p>
1186
1187	<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
1188	and the parameters of the core-shell cylinder model are the following:</p>
1189
1190	<p class=MsoNormal> </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> </p>
1375
1376	<p class=MsoNormal>The output of the 1D scattering intensity function for
1377	randomly oriented cylinders is then given by equation 6.</p>
1378
1379	<p class=MsoNormal> </p>
1380
1381	<p class=MsoNormal>The <i>axis_theta</i> and axis<i>_phi</i> parameters are not
1382	used for the 1D output. Our implementation of the scattering kernel and the 1D
1383	scattering intensity use the c-library from NIST.</p>
1384
1385	<p class=MsoNormal> </p>
1386
1387	<p class=MsoNormal style='margin-left:.85in;text-indent:-.35in'><b>2.5.2.<span
1388	style='font:7.0pt "Times New Roman"'>      </span></b><b>Validation
1389	of the core-shell cylinder model</b></p>
1390
1391	<p class=MsoNormal> </p>
1392
1393	<p class=MsoNormal>Validation of our code was done by comparing the output of
1394	the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure
1395	8 shows a comparison of the 1D output of our model and the output of the NIST
1396	software.</p>
1397
1398	<p class=MsoNormal> </p>
1399
1400	<p class=MsoNormal>Averaging over a distribution of orientation is done by
1401	evaluating equation 7. Since we have no other software to compare the
1402	implementation of the intensity for fully oriented core-shell cylinders, we can
1403	compare the result of averaging our 2D output using a uniform distribution <i>p(θ,</i><i><span
1404	style='font-family:"Arial","sans-serif"'>φ</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
1407	id="Picture 22" src="../images/html/image021.jpg"
1408	alt="core_shell_cylinder_1D_validation"></p>
1409
1410	<p class=MsoCaption><a name="_Ref173307149">Figure </a>8: Comparison of the
1411	DANSE scattering intensity for a core-shell cylinder with the output of the
1412	NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 Å,
1413	Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å<sup> -2</sup>, Shell_sld=4e-6 Å<sup>
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> </p>
1417
1418	<p class=MsoNormal> </p>
1419
1420	<p class=MsoNormal style='page-break-after:avoid'><img width=573 height=315
1421	id="Picture 23" src="../images/html/image022.jpg"
1422	alt="core_shell_cylinder_2D_average"></p>
1423
1424	<p class=MsoCaption><a name="_Ref173307204">Figure </a>9: Comparison of the
1425	intensity for uniformly distributed core-shell cylinders calculated from our 2D
1426	model and the intensity from the NIST SANS analysis software. The parameters
1427	used 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
1429	Background=0.0 cm<sup> -1</sup>.</p>
1430
1431	<p class=MsoNormal> </p>
1432	<p class=MsoNormal> </p>
1433
1434	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1435	style='font-size:14.0pt'>2.6.<span style='font:7.0pt "Times New Roman"'>
1436	</span></span></b><b><span style='font-size:14.0pt'><a name="EllipticalCylinderModel">EllipticalCylinderModel</a></span></b></p>
1437
1438	<p class=MsoNormal> </p>
1439
1440	<p class=MsoNormal>This function calculates the scattering from an oriented
1441	elliptical cylinder. The angles theta and phi define the orientation of the
1442	axis of the cylinder. The angle psi is defined as the orientation of the major
1443	axis of the ellipse with respect to the vector Q. A gaussian poydispersity can
1444	be added to any (or all) of the orientation angles, and also for the minor
1445	radius and the ratio of the ellipse radii.</p>
1446
1447	<p class=MsoNormal>The function calculated is:</p>
1448
1449	<p class=MsoNormal align=center style='text-align:center'><span
1450	style='position:relative;top:16.0pt'><img width=377 height=46
1451	src="../images/html/image023.gif"></span> </p>
1452
1453	<p class=MsoNormal>with the functions:</p>
1454
1455	<p class=MsoNormal align=center style='text-align:center'><span
1456	style='position:relative;top:39.0pt'><img width=297 height=114
1457	src="../images/html/image024.gif"></span> (13)</p>
1458
1459	<p class=MsoNormal>and the angle psi is defined as the orientation of the major
1460	axis of the ellipse with respect to the vector Q.</p>
1461
1462	<p class=MsoNormal> </p>
1463
1464	<p class=MsoNormal>The returned value is scaled to units of [cm<sup>-1</sup>]
1465	and the parameters are the following:</p>
1466
1467	<p class=MsoNormal> </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> </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> </p>
1636
1637	<p class=MsoNormal>REFERENCE</p>
1638
1639	<p class=MsoNormal> </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
1643	York, (1987).</p>
1644
1645	<p class=MsoNormal> </p>
1646
1647	<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
1648	style='font-size:14.0pt'>3.<span style='font:7.0pt "Times New Roman"'>
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> </p>
1652
1653	<p class=MsoBodyText>The following are models used for shape-independent SANS
1654	analysis.</p>
1655
1656	<p class=MsoBodyText> </p>
1657
1658	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1659	style='font-size:14.0pt'>3.1.<span style='font:7.0pt "Times New Roman"'>
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'> </p>
1663
1664	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The Debye
1665	model is a form factor for a linear polymer chain. In addition to the radius of
1666	gyration, Rg, a scale factor "scale", and a constant background term
1667	are included in the calculation. </p>
1668
1669	<p class=MsoNormal><span style='font-size:14.0pt'> </span></p>
1670
1671	<p class=MsoNormal align=center style='text-align:center'><span
1672	style='font-size:14.0pt'><sub><img width=188 height=76
1673	src="../images/html/image025.gif"></sub></span></p>
1674
1675	<p class=MsoNormal><span style='font-size:14.0pt'> </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'> </p>
1748
1749	<p class=MsoNormal style='margin-left:.25in'>Reference: Roe, R.-J.,
1750	"Methods of X-Ray and Neutron Scattering in Polymer Science", Oxford University Press, New York (2000).</p>
1751
1752	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
1753
1754	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1755	style='font-size:14.0pt'>3.2.<span style='font:7.0pt "Times New Roman"'>
1756	</span></span></b><b><span style='font-size:14.0pt'> <a name="Lorentz">(Ornstein-Zernicke)
1757	Lorentz (Model)</span></b></p>
1758
1759	<p class=MsoNormal style='text-autospace:none'> </p>
1760
1761	<p class=MsoNormal style='text-indent:.25in'>The Ornstein-Zernicke model is
1762	defined by:</p>
1763
1764	<p class=MsoNormal><span style='font-size:14.0pt'> </span></p>
1765
1766	<p class=MsoNormal align=center style='text-align:center'><span
1767	style='font-size:14.0pt'><sub><img width=212 height=24
1768	src="../images/html/image026.gif"></sub></span></p>
1769
1770	<p class=MsoNormal><span style='font-size:14.0pt'> </span></p>
1771
1772	<p class=MsoNormal style='text-indent:.25in'>The parameter L is referred to as
1773	the screening length.</p>
1774
1775	<p class=MsoNormal><span style='font-size:14.0pt'> </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'> </span></b></p>
1848
1849	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
1850
1851
1852	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1853	style='font-size:14.0pt'>3.3.<span style='font:7.0pt "Times New Roman"'>
1854	</span></span></b><b><span style='font-size:14.0pt'> <a name="DAB_Model">DAB(Debye-Anderson-Brumberger)_
1855	Model)</a></span></b></p>
1856
1857	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Calculates the
1858	scattering from a randomly distributed, two-phase system based on the
1859	Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system is
1860	characterized by a single length scale, the correlation length, which is a
1861	measure of the average spacing between regions of phase 1 and phase 2. The
1862	model also assumes smooth interfaces between the phases and hence exhibits
1863	Porod behavior (I ~ Q<sup>-4</sup>) at large Q (Q*correlation length >>
1864	1). </p>
1865
1866	<p class=MsoNormal style='text-indent:.25in'> </p>
1867
1868	<p class=MsoNormal align=center style='text-align:center'><span
1869	style='font-size:14.0pt'><sub><img width=221 height=28
1870	src="../images/html/image027.gif"></sub></span></p>
1871
1872	<p class=MsoNormal align=center style='text-align:center'><span
1873	style='font-size:14.0pt'> </span></p>
1874
1875	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The parameter
1876	L is referred to as the correlation length.</p>
1877
1878	<p class=MsoNormal align=center style='text-align:center'><span
1879	style='font-size:14.0pt'> </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'> </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	"Scattering by an Inhomogeneous Solid. II. The Correlation Function and
1957	its Application", J. Appl. Phys. 28 (6), 679 (1957).</p>
1958
1959	<p class=MsoNormal style='margin-left:.5in'> </p>
1960
1961	<p class=MsoNormal style='margin-left:.5in'>Debye, Bueche, "Scattering by
1962	an Inhomogeneous Solid", J. Appl. Phys. 20, 518 (1949).</p>
1963
1964	<p class=MsoNormal> </p>
1965
1966	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
1967
1968	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
1969	style='font-size:14.0pt'>3.4.<span style='font:7.0pt "Times New Roman"'>
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'> </span></b></p>
1973
1974	<p class=MsoNormal style='margin-left:.25in'>This model describes a power law
1975	with background.</p>
1976
1977	<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'> </span></b></p>
1978
1979	<p class=MsoNormal align=center style='text-align:center'><span
1980	style='font-size:14.0pt'><sub><img width=161 height=24
1981	src="../images/html/image028.gif"></sub></span></p>
1982
1983	<p class=MsoNormal align=center style='text-align:center'><span
1984	style='font-size:14.0pt'> </span></p>
1985
1986	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>Note the minus
1987	sign in front of the exponent.</p>
1988
1989	<p class=MsoNormal align=center style='text-align:center'><span
1990	style='font-size:14.0pt'> </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'> </span></b></p>
2063
2064	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2065
2066	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2067	style='font-size:14.0pt'>3.5.<span style='font:7.0pt "Times New Roman"'>
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'> </span></b></p>
2071
2072	<p class=MsoNormal style='margin-left:.25in'>This function calculates the
2073	scattered 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'> </span></b></p>
2076
2077	<p class=MsoNormal align=center style='text-align:center'><span
2078	style='font-size:14.0pt'><sub><img width=213 height=45
2079	src="../images/html/image029.gif"></sub></span></p>
2080
2081	<p class=MsoNormal align=center style='text-align:center'><span
2082	style='font-size:14.0pt'> </span></p>
2083
2084	<p class=MsoNormal align=center style='text-align:center'><span
2085	style='font-size:14.0pt'> </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'> </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.
2178	Phys., 87, 3195 (1987).</p>
2179
2180	<p class=MsoNormal style='margin-left:.5in'> </p>
2181
2182	<p class=MsoNormal style='margin-left:.5in'>Schubert, K-V., Strey, R., Kline,
2183	S. R. and E. W. Kaler, J. Chem. Phys., 101, 5343 (1994).</p>
2184
2185	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2186
2187	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2188
2189
2190	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2191	style='font-size:14.0pt'>3.6.<span style='font:7.0pt "Times New Roman"'>
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'> </span></b></p>
2195
2196	<p class=MsoNormal style='margin-left:.25in'>Calculates the scattering from
2197	fractal-like aggregates built from spherical building blocks following the
2198	Texiera reference. The value returned is in cm<sup>-1</sup>.</p>
2199
2200	<p class=MsoNormal style='margin-left:.25in'> </p>
2201
2202
2203	<p class=MsoNormal align=center style='text-align:center'><span
2204	style='font-size:14.0pt'><sub><img width=380 height=188
2205	src="../images/html/image030.gif"></sub></span></p>
2206
2207	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2208
2209	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'>The scale
2210	parameter is the volume fraction of the building blocks, R<sub>0</sub> is the
2211	radius of the building block, D<sub>f</sub> is the fractal dimension, ξ is
2212	the correlation length, <i>ρ<sub>solvent</sub></i> is the scattering
2213	length density of the solvent, and <i>ρ<sub>block</sub></i> is the
2214	scattering length density of the building blocks. </p>
2215
2216	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'> </p>
2217
2218	<p class=MsoNormal align=center style='text-align:center'><span
2219	style='font-size:14.0pt'> </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'> </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:
2361	none'>J. Teixeira, (1988) J. Appl. Cryst., vol. 21, p781-785</p>
2362
2363	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2364
2365	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2366
2367	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2368	style='font-size:14.0pt'>3.7.<span style='font:7.0pt "Times New Roman"'>
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'> </span></b></p>
2372
2373	<p class=MsoNormal style='margin-left:.25in'>Calculates the structure factor of
2374	a polyelectrolyte solution with the RPA expression derived by Borue and
2375	Erukhimovich. The value returned is in cm<sup>-1</sup>.</p>
2376
2377	<p class=MsoNormal style='margin-left:.25in'> </p>
2378
2379	<p class=MsoNormal align=center style='text-align:center'><span
2380	style='font-size:14.0pt'><sub><img width=365 height=125
2381	src="../images/html/image031.gif"></sub></span></p>
2382
2383	<p class=MsoNormal style='margin-left:.25in'> </p>
2384
2385	<p class=MsoNormal style='margin-left:.25in'>K is a contrast factor of the
2386	polymer, L<sub>b</sub> is the Bjerrum length, h is the virial parameter, b is
2387	the monomer length, C<sub>s</sub> is the concentration of monovalent salt,
2388	α is the ionization degree, C<sub>a</sub> is the polymer molar
2389	concentration, and bck is the incoherent background.</p>
2390
2391	<p class=MsoNormal><b><span style='font-size:14.0pt'> </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'> </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.,
2548	Erukhimovich, I. Y. Macromolecules 21, 3240 (1988).</p>
2549
2550	<p class=MsoNormal style='margin-left:.5in;text-autospace:none'><span lang=FR>Joanny,
2551	J.-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,
2554	A., 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.,
2559	Joanny, J.-F., Europhysics Letters 11, 179 (1990).</p>
2560
2561	<p class=MsoNormal style='margin-left:.25in;text-autospace:none'> </p>
2562
2563	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2564
2565	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2566	style='font-size:14.0pt'>3.8.<span style='font:7.0pt "Times New Roman"'>
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'> </span></b></p>
2570
2571	<p class=MsoNormal style='margin-left:.25in'>A Guinier analysis is done by linearizing
2572	the data at low q by plotting it as log(I) versus Q<sup>2</sup>. The Guinier
2573	radius R<sub>g</sub> can be obtained by fitting the following model:</p>
2574
2575	<p class=MsoNormal style='margin-left:.25in'> </p>
2576
2577	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2578	style='font-size:14.0pt'><sub><img width=183 height=28
2579	src="../images/html/image032.gif"></sub></span></p>
2580
2581	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
2582	style='font-size:14.0pt'> </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'> </span></b></p>
2639	<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'> </span></b></p>
2640	<p class=MsoNormal style='margin-left:.25in'><b><span style='font-size:14.0pt'> </span></b></p>
2641
2642	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2643	style='font-size:14.0pt'>3.9.<span style='font:7.0pt "Times New Roman"'>
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'> </span></b></p>
2647
2648	<p class=MsoNormal style='margin-left:.25in'>A Porod analysis is done by
2649	linearizing the data at high q by plotting it as log(I) versus log(Q). In the
2650	high q region we can fit the following model: </p>
2651
2652	<p class=MsoNormal style='margin-left:.25in'> </p>
2653
2654	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2655	style='font-size:14.0pt'><sub><img width=368 height=25
2656	src="../images/html/image033.gif"></sub></span></p>
2657
2658	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2659	style='font-size:14.0pt'> </span></p>
2660
2661	<p class=MsoNormal style='margin-left:.25in'>C is the scale factor and Sv is
2662	the specific surface area of the sample and <span style='font-family:"Arial","sans-serif"'>Δ</span>ρ
2663	is the contrast factor. </p>
2664
2665	<p class=MsoNormal style='margin-left:.25in'> </p>
2666
2667	<p class=MsoNormal style='margin-left:.25in'>The background term is added for
2668	data analysis.</p>
2669
2670	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><b><span
2671	style='font-size:14.0pt'> </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'> </span></b></p>
2728
2729	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2730
2731	<p class=MsoNormal><b><span style='font-size:14.0pt'> </span></b></p>
2732
2733
2734
2735
2736	<p class=MsoNormal style='margin-left:.25in;text-indent:-.25in'><b><span
2737	style='font-size:14.0pt'>4.<span style='font:7.0pt "Times New Roman"'>
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> </p>
2741
2742	<p class=MsoBodyText>These models are defined by users (See SansView tutorial for details). </p>
2743
2744	<p class=MsoBodyText> </p>
2745
2746	<p class=MsoNormal style='margin-left:.55in;text-indent:-.3in'><b><span
2747	style='font-size:14.0pt'>4.1.<span style='font:7.0pt "Times New Roman"'>
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'> </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'> </span></b></p>
2756	<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><b><span
2757	style='font-size:14.0pt'>5.<span style='font:7.0pt "Times New Roman"'>
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'> </p>
2761
2762	<p class=MsoListParagraphCxSpLast style='margin-left:.25in'>The information in
2763	this section is originated from NIST SANS IgorPro package.</p>
2764
2765	<p class=MsoBodyText> </p>
2766
2767	<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
2768	style='font-size:14.0pt'>5.1.<span style='font:7.0pt "Times New Roman"'>
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> </p>
2772
2773	<p class=MsoNormal>This calculates the interparticle structure factor for
2774	monodisperse spherical particles interacting through hard sphere (excluded
2775	volume) interactions. The calculation uses the Percus-Yevick closure where the
2776	interparticle potential is: </p>
2777
2778	<p class=MsoNormal> </p>
2779
2780	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2781	style='font-size:14.0pt;position:relative;top:15.0pt'><img width=122 height=48
2782	src="../images/html/image034.gif"></span></p>
2783
2784	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2785	style='font-size:14.0pt;position:relative;top:7.0pt'> </span></p>
2786
2787	<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the
2788	distance from the center of the sphere of a radius R.</span></p>
2789
2790	<p class=MsoNormal> </p>
2791
2792	<p class=MsoNormal> </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> </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"
2849	src="../images/html/image035.gif"></p>
2850
2851	<p class=MsoNormal style='text-autospace:none'> </p>
2852
2853	<p class=MsoNormal style='text-autospace:none'> </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.;
2858	Yevick, J. Phys. Rev. 110, 1. (1958).</p>
2859
2860	<p class=MsoNormal style='text-autospace:none'> </p>
2861
2862	<p class=MsoNormal style='text-autospace:none'> </p>
2863
2864	<p class=MsoNormal style='text-autospace:none'> </p>
2865
2866	<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
2867	style='font-size:14.0pt'>5.2.<span style='font:7.0pt "Times New Roman"'>
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> </p>
2871
2872	<p class=MsoNormal>This calculates the interparticle structure factor for a
2873	squar well fluid spherical particles The mean spherical approximation (MSA)
2874	closure was used for this calculation, and is not the most appropriate closure
2875	for an attractive interparticle potential. This solution has been compared to
2876	Monte Carlo simulations for a square well fluid, showing this calculation to be
2877	limited in applicability to well depths e < 1.5 kT and volume fractions f
2878	< 0.08.</p>
2879
2880	<p class=MsoNormal> </p>
2881
2882	<p class=MsoNormal>Positive well depths correspond to an attractive potential
2883	well. Negative well depths correspond to a potential "shoulder",
2884	which may or may not be physically reasonable.</p>
2885
2886	<p class=MsoNormal> </p>
2887
2888	<p class=MsoNormal>The well width (l) is defined as multiples of the particle
2889	diameter (2*R)</p>
2890
2891	<p class=MsoNormal> </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
2896	style='font-size:14.0pt;position:relative;top:22.0pt'><img width=174 height=67
2897	src="../images/html/image036.gif"></span></p>
2898
2899	<p class=MsoNormal align=center style='margin-left:.25in;text-align:center'><span
2900	style='font-size:14.0pt;position:relative;top:7.0pt'> </span></p>
2901
2902	<p class=MsoNormal><span style='position:relative;top:7.0pt'>where r is the
2903	distance from the center of the sphere of a radius R.</span></p>
2904
2905	<p class=MsoNormal> </p>
2906
2907	<p class=MsoNormal> </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> </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> </p>
2996
2997	<p class=MsoNormal><img width=546 height=296 id="Picture 53"
2998	src="../images/html/image037.gif"></p>
2999
3000	<p class=MsoNormal style='text-autospace:none'> </p>
3001
3002	<p class=MsoNormal style='text-autospace:none'> </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.;
3007	Sharma, K. C. Physica, 89A, 213. (1977).</p>
3008
3009	<p class=MsoNormal style='text-autospace:none'> </p>
3010
3011	<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
3012	style='font-size:14.0pt'>5.3.<span style='font:7.0pt "Times New Roman"'>
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> </p>
3016
3017	<p class=MsoNormal>This calculates the Structure factor (the Fourier transform
3018	of the pair correlation function g(r)) for a system of charged, spheroidal
3019	objects 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
3021	interparticle interference effects due to screened coulomb repulsion between
3022	charged particles. This routine only works for charged particles. If the
3023	charge is set to zero the routine will self destruct. For non-charged
3024	particles use a hard sphere potential.</p>
3025
3026	<p class=MsoNormal> </p>
3027
3028	<p class=MsoNormal>The salt concentration is used to compute the ionic strength
3029	of the solution which in turn is used to compute the Debye screening length.
3030	At present there is no provision for entering the ionic strength directly nor
3031	for use of any multivalent salts. The counterions are also assumed to be monovalent.</p>
3032
3033	<p class=MsoNormal> </p>
3034
3035	<p class=MsoNormal> </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> </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> </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> </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> </p>
3156
3157	<p class=MsoNormal><img width=548 height=354 id="Picture 56"
3158	src="../images/html/image038.gif"></p>
3159
3160	<p class=MsoNormal style='text-autospace:none'> </p>
3161
3162	<p class=MsoNormal style='text-autospace:none'> </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
3167	JB Hayter, Molecular Physics 46, 651-656 (1982).</p>
3168
3169	<p class=MsoNormal style='text-autospace:none'> JB Hayter and J
3170	Penfold, Molecular Physics 42, 109-118 (1981).</p>
3171
3172	<p class=MsoNormal style='text-autospace:none'> </p>
3173
3174	<p class=MsoNormal style='text-autospace:none'> </p>
3175
3176	<p class=MsoListParagraph style='margin-left:.55in;text-indent:-.3in'><b><span
3177	style='font-size:14.0pt'>5.4.<span style='font:7.0pt "Times New Roman"'>
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> </p>
3181
3182	<p class=MsoNormal>This calculates the interparticle structure factor for a
3183	hard sphere fluid with a narrow attractive well. A perturbative solution of the
3184	Percus-Yevick closure is used. The strength of the attractive well is described
3185	in terms of "stickiness" as defined below. The returned value is a
3186	dimensionless structure factor, S(q).</p>
3187
3188	<p class=MsoNormal> </p>
3189
3190	<p class=MsoNormal>The perturb (perturbation parameter), epsilon, should be
3191	held between 0.01 and 0.1. It is best to hold the perturbation parameter fixed
3192	and let the "stickiness" vary to adjust the interaction strength. The
3193	stickiness, tau, is defined in equation 21 and is a function of both the
3194	perturbation parameter and the interaction strength. Tau and epsilon are
3195	defined in terms of the hard sphere diameter (sigma = 2R), the width of the
3196	square well, delta (same units as R), and the depth of the well, u<sub>o</sub>,
3197	in units of kT. From the definition, it is clear that smaller tau mean stronger
3198	attraction.</p>
3199
3200	<p class=MsoNormal align=center style='text-align:center'><span
3201	style='font-size:14.0pt;position:relative;top:22.0pt'><img width=135 height=67
3202	src="../images/html/image039.gif"></span></p>
3203
3204	<p class=MsoNormal align=center style='text-align:center'><span
3205	style='font-size:14.0pt;position:relative;top:27.0pt'> </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
3210	style='font-size:14.0pt;position:relative;top:23.0pt'><img width=186 height=69
3211	src="../images/html/image040.gif"></span></p>
3212
3213	<p class=MsoNormal>The Percus-Yevick (PY) closure was used for this
3214	calculation, and is an adequate closure for an attractive interparticle
3215	potential. This solution has been compared to Monte Carlo simulations for a
3216	square well fluid, with good agreement.</p>
3217
3218	<p class=MsoNormal> </p>
3219
3220	<p class=MsoNormal>The true particle volume fraction, f, is not equal to h,
3221	which appears in most of the reference. The two are related in equation (24) of
3222	the reference. The reference also describes the relationship between this
3223	perturbation solution and the original sticky hard sphere (or adhesive sphere)
3224	model by Baxter.</p>
3225
3226	<p class=MsoNormal> </p>
3227
3228	<p class=MsoNormal>NOTES: The calculation can go haywire for certain
3229	combinations of the input parameters, producing unphysical solutions - in this
3230	case errors are reported to the command window and the S(q) is set to -1 (it
3231	will disappear on a log-log plot). Use tight bounds to keep the parameters to
3232	values that you know are physical (test them) and keep nudging them until the
3233	optimization does not hit the constraints.</p>
3234
3235	<p class=MsoNormal> </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> </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> </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> </p>
3324
3325	<p class=MsoNormal><img width=524 height=359 id="Picture 111"
3326	src="../images/html/image041.gif"></p>
3327
3328	<p class=MsoNormal style='text-autospace:none'> </p>
3329
3330	<p class=MsoNormal style='text-autospace:none'> </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.
3335	G., 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'> </span></b></p>
3340
3341	<p class=MsoNormal><b><span style='font-size:14.0pt'> </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) "Structure
3347	Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York.</p>
3348
3349	<p class=MsoBodyText>Guinier, A. and G. Fournet (1955) "Small-Angle
3350	Scattering of X-Rays", 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),
3353	895.</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.,
3360	Heumann, H., Zillig, W. and Crespi, H. (1980) <i>Eur. J. Biochem. </i>112,
3361	411-417.</p>
3362
3363	<p class=MsoBodyText>McAlister, B.C. and Grady, B.P., (1998) J. Appl. Cryst. 31,
3364	594-599.</p>
3365
3366	<p class=MsoBodyText>Porod, G. (1982) in Small Angle X-ray Scattering, editors
3367	Glatter, O. and Kratky, O., Academic Press.</p>
3368
3369	<p class=MsoBodyText> </p>
3370
3371	<p class=MsoNormal> </p>
3372
3373	</div>
3374
3375	</body>
3376
3377	</html>

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