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100
101</head>
102
103<body lang=EN-US link=blue vlink=purple>
104
105<div class=Section1>
106
107<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;line-height:normal'><span
108style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
109</span></span><b><span style='font-family:"Times New Roman","serif"'><a
110href="#invariant">Scattering Invariant</a></span></b></p>
111
112<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;line-height:normal'><span
113style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
114</span></span><a href="#volume"><b><span style='font-family:"Times New Roman","serif"'>Volume
115Fraction</span></b></a></p>
116
117<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;line-height:normal'><span
118style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
119</span></span><a href="#surface"><b><span style='font-family:"Times New Roman","serif"'>Specific
120Surface Area</span></b></a></p>
121
122<p class=MsoListParagraphCxSpLast style='text-indent:-.25in;line-height:normal'><span
123style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
124</span></span><a href="#howto"><b><span style='font-family:"Times New Roman","serif"'>How
125to Compute</span></b></a></p>
126
127<p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>The
128scattering <a name=invariant>invariant</a> (Q*)</span></b><span
129style='font-family:"Times New Roman","serif"'> </span></p>
130
131<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
132
133<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>The
134scattering invariant (Q*) is a model-independent quantity that can be easily
135calculated from </span></p>
136
137<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>scattering
138data. For two phase systems, the scattering invariant, Q*, is defined as the
139integral of</span></p>
140
141<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> the
142square of the wave transfer (q) multiplied by the scattering cross section over
143the full range of q. </span></p>
144
145<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Q*
146is given by the following equation, </span></p>
147
148<p class=MsoNormal style='line-height:normal'>
149
150<table cellpadding=0 cellspacing=0 align=left>
151 <tr>
152  <td width=2 height=4></td>
153 </tr>
154 <tr>
155  <td></td>
156  <td><img width=127 height=35 src="image001.gif"></td>
157 </tr>
158</table>
159
160<span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
161
162<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
163
164<br clear=ALL>
165
166<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>This
167model independent quantity (Q*) is calculated from the scattering data that can
168be used to determine the volume fraction and the specific area of the sample
169under consideration. &nbsp;These quantities&nbsp;are useful in their own right
170and can be used in&nbsp;further analysis. With this scattering invariant module
171users will also be able to determine the consistency of those properties
172between data.&nbsp; There is no real data defined from zero to infinity, there
173usually have limited range. Q* is not really computed from zero to infinity.
174Our maximum q range is 1e-5 ~ 10 (1/A). The lower and/or higher q range than
175data given can be extrapolated by fitting some data nearby.</span></p>
176
177<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>The
178scattering invariant is computed as follow:</span></p>
179
180<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span
181style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
182</span></span><i><span style='font-family:"Times New Roman","serif"'>I(q)</span></i><span
183style='font-family:"Times New Roman","serif"'> =  <i>I(q)</i> w/o background :
184If the data includes a background, user sets the value to subtract the background
185for the Q* computation.</span></p>
186
187<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span
188style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
189</span></span><span style='font-family:"Times New Roman","serif"'>Reset<i>
190I(q)  </i></span><span style='font-family:"Times New Roman","serif"'>= <i>I(q)*
191scaling factor </i>, delta<i>I(q) =  </i>delta<i>I(q)*scaling factor</i> : If
192non-zero scaling factor is given, it will be considered.</span></p>
193
194<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span
195style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
196</span></span><span style='font-family:"Times New Roman","serif"'> Invariant:</span></p>
197
198<p class=MsoNormal style='margin-left:.25in;line-height:normal'>
199
200<table cellpadding=0 cellspacing=0 align=left>
201 <tr>
202  <td width=45 height=12></td>
203 </tr>
204 <tr>
205  <td></td>
206  <td><img width=127 height=35 src="image001.gif"></td>
207 </tr>
208</table>
209
210<span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
211
212<br clear=ALL>
213
214<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p>
215
216<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
217style='font-family:"Times New Roman","serif"'>            where, <i>g =q</i>
218for the pinhole geometry and <i>g =<sub> </sub>q<sub>v</sub></i>(the slit
219height) for the slit geometry which can be given in data or as a value.</span></p>
220
221<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span
222style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
223</span></span><span style='font-family:"Times New Roman","serif"'>&nbsp;Higher
224q-region (&gt;= qmax in data): </span></p>
225
226<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            Power
227law (w/o background term) function = C/q<sup>4</sup> will be used </span></p>
228
229<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
230style='font-family:"Times New Roman","serif"'>            where the constant
231C(=2pi(delta(rho))S<sub>v</sub>) is to be found by fitting part of data with
232the range of q<sub>N-m</sub>  to q<sub>N</sub> (m&lt;N).</span></p>
233
234<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span
235style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
236</span></span><span style='font-family:"Times New Roman","serif"'> Lower
237q-region (&lt;= qmin in data): </span></p>
238
239<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            Guinier
240function = <i>I<sub>0</sub>exp(-R<sub>g</sub><sup>2</sup>q<sup>2</sup>/3)</i>
241where I<sub>0</sub> and R<sub>g</sub> are obtained by fitting, </span></p>
242
243<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            similarly
244to the high q region above.</span></p>
245
246<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            Power
247law can also be used.</span></p>
248
249<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>&nbsp;</span><b>The
250<a name=volume>volume </a>fraction</b>: </p>
251
252<p class=MsoNormal style='line-height:normal'><span style='position:absolute;
253z-index:3;margin-left:1px;margin-top:9px;width:161px;height:47px'><img
254width=161 height=47 src="image002.gif"></span><span
255style='font-family:"Times New Roman","serif"'> </span></p>
256
257<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p>
258
259<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>where
260delta(rho) is the SLD contrast of which value is given by users.</span></p>
261
262<p class=MsoNormal style='line-height:normal'><span style='position:absolute;
263z-index:4;margin-left:1px;margin-top:29px;width:101px;height:45px'><img
264width=101 height=45 src="image003.gif"></span><span
265style='font-family:"Times New Roman","serif"'>Thus,  </span></p>
266
267<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
268
269<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
270
271<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> where
2720 =&lt; <i>A</i> =&lt;1/4 in order for these values to be physically valid.</span></p>
273
274<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
275
276<p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>The
277<a name=surface>specific surface area </a>:</span></b></p>
278
279<p class=MsoNormal style='line-height:normal'>
280
281<table cellpadding=0 cellspacing=0 align=left>
282 <tr>
283  <td width=1 height=3></td>
284 </tr>
285 <tr>
286  <td></td>
287  <td><img width=184 height=47 src="image004.gif"></td>
288 </tr>
289</table>
290
291<b><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></b></p>
292
293<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
294
295<br clear=ALL>
296
297<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> where
298<i>A</i> and <i>Q<sup>*</sup></i> are obtained from previous sections, and the
299Porod constant <i>C<sub>p</sub></i> is given by users. </span></p>
300
301<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
302
303<p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>Definitions:</span></b><span
304style='font-family:"Times New Roman","serif"'> </span></p>
305
306<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Q&nbsp;:
307the magnitude of neutron (or X-ray) momentum transfer vector.</span></p>
308
309<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>I(Q):
310the scattering intensity as a function of the momentum transfer Q.</span></p>
311
312<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>Invariant
313total is the sum of the invariant calculated from data’s q range and the
314invariant resulting from extrapolation at low q range and at high q range if
315considered.</span></p>
316
317<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
318
319<p class=MsoNormal style='line-height:normal'><a name=reference><b><span
320style='font-family:"Times New Roman","serif"'>Reference</span></b></a><b><span
321style='font-family:"Times New Roman","serif"'>: </span></b></p>
322
323<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Porod's
324chapter (2) from  O. Glatter and O. Kratky, &quot;Small Angle X-Ray
325Scattering&quot;, Academic Press, New York, 1982</span></p>
326
327<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Otto's
328web page: </span><a href="http://physchem.kfunigraz.ac.at/sm/"><span
329style='font-family:"Times New Roman","serif"'>http://physchem.kfunigraz.ac.at/sm/</span></a><span
330style='font-family:"Times New Roman","serif"'> </span></p>
331
332<p class=MsoNormal style='line-height:normal'><a name=howto><b><span
333style='font-family:"Times New Roman","serif"'>HowTo</span></b></a><b><span
334style='font-family:"Times New Roman","serif"'>:</span></b></p>
335
336<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
337style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
338</span>i)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
339</span></span><span style='font-family:"Times New Roman","serif"'>Loading data
340to the panel: Open the data file from File in the menu bar. Select loaded data
341from a plot panel by highlighting that it until its color turns yellow. Then
342right click on that the data and selects the option Compute Invariant. The
343application automatically computes the invariant value if the data loaded is
344valid.</span></p>
345
346<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
347style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
348</span>ii)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
349</span></span><span style='font-family:"Times New Roman","serif"'>To subtract a
350background or/and to rescale the data, type the values in Customized Input
351box.  </span></p>
352
353<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
354style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;
355</span>iii)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
356</span></span><span style='font-family:"Times New Roman","serif"'>If you want
357to calculate the volume fraction and the specific surface area, type the
358optional inputs in the customized input box, and then press ‘Compute’ button.</span></p>
359
360<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
361style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;
362</span>iv)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
363</span></span><span style='font-family:"Times New Roman","serif"'>The invariant
364can also be calculated including the outside of the data Q range:  To include
365the lower Q and/or the higher Q range, check in the enable extrapolation check
366box in ‘Extrapolation ‘box. If the power low is chosen, the power (exponent)
367can be either held or fitted by checking the corresponding radio button.  The
368Npts that is being used for the extrapolation can be specified.</span></p>
369
370<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
371style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
372</span>v)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
373</span></span><span style='font-family:"Times New Roman","serif"'>If the
374invariant calculated from the extrapolated region is too large, it will be warn
375in red at the top of the panel, which means that your data is not proper to
376calculate the invariant.</span></p>
377
378<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span
379style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;
380</span>vi)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
381</span></span><span style='font-family:"Times New Roman","serif"'>The details
382of the calculation is available by clicking the ‘Details’ button in the middle
383of the panel. </span></p>
384
385<p class=MsoNormal align=center style='text-align:center;line-height:normal'><span
386style='font-family:"Times New Roman","serif"'><img border=0 width=572
387height=449 id="Object 4" src="image005.gif"></span></p>
388
389<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
390
391</div>
392
393</body>
394
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