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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
Last change on this file since c25f1fa was b9958b3, checked in by Jae Cho <jhjcho@…>, 13 years ago

additional doc for shulz
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48<body lang=EN-US>
49
50<div class=WordSection1>
51
52<p class=MsoNormal><h3><span style='font-family:"Times New Roman","serif"'>Polydisperisty
53and Angular Distributions</span></h3></p>
54
55<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Calculates
56the form factor for a polydisperse and/or angular population of particles with
57uniform scattering length density. The resultant form factor is normalized by
58the average particle volume such that P(q) = scale*&lt;F*F&gt;/Vol + bkg, where
59F is the scattering amplitude and the &lt; &gt; denote an average over the size
60distribution.  Users should use PD (polydispersity: this definition is different from the typical definition in polymer science)
61for a size distribution and Sigma for an
62angular distribution (see below).</span></p>
63<p> Note that this computation is very time intensive thus applying polydispersion/angular distrubtion for
64more than one paramters or increasing Npts values might need extensive patience to complete the computation. Also
65note that even though it is time consuming, it is safer to have larger values of Npts and Nsigmas.</p>
66
67<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
68style='font-family:"Times New Roman","serif"'>The following five distribution
69functions are provided;</span></p>
70<p>&nbsp;</p>
71<ul>
72<li><a href="#Rectangular">Rectangular distribution</a></li>
73<li><a href="#Array">Array distribution</a></li>
74<li><a href="#Gaussian">Gaussian distribution</a></li>
75<li><a href="#Lognormal">Lognormal distribution</a></li>
76<li><a href="#Schulz">Schulz distribution</a></li>
77</ul>
78<p>&nbsp;</p>
79<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
80style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
81
82<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
83style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
84
85<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
86style='font-family:"Times New Roman","serif"'><a name="Rectangular"><h4>Rectangular distribution</a></h4></span></p>
87
88<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
89style='font-family:"Times New Roman","serif";position:relative;top:22.0pt'><img
90width=248 height=67 src="pd_image001.png"></span></p>
91
92<p>&nbsp;</p>
93<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
94style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean
95of the distribution, w is the half-width, and Norm is a normalization factor
96which is determined during the numerical calculation. Note that the Sigma and
97the half width <i>w</i> are different.</span></p>
98
99<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
100style='font-family:"Times New Roman","serif"'>The standard deviation is </span></p>
101
102<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
103style='font-family:"Times New Roman","serif";position:relative;top:4.0pt'><img
104width=72 height=24 src="pd_image002.png"></span><span
105style='font-family:"Times New Roman","serif"'>. </span></p>
106
107<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
108style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
109
110<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
111style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p>
112
113<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
114style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
115width=93 height=24 src="pd_image003.png"></span><span
116style='font-family:"Times New Roman","serif"'>.</span></p>
117
118<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
119style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
120
121<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
122style='font-family:"Times New Roman","serif"'><img width=511 height=270
123id="Picture 1" src="pd_image004.jpg" alt=flat.gif></span></p>
124
125<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
126style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
127
128<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
129style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
130
131<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
132style='font-family:"Times New Roman","serif"'><a name="Array"><h4>Array distribution</h4></a></span></p>
133
134<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
135style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
136
137
138<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
139style='font-family:"Times New Roman","serif"'>This distribution is to be given
140by users as a txt file where the array should be defined by two columns in the
141order of x and f(x) values. The f(x) will be normalized by SansView during the
142computation.</span></p>
143
144<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
145style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
146
147<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
148style='font-family:"Times New Roman","serif"'>Example of an array in the file;</span></p>
149
150<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
151style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
152
153<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
154style='font-family:"Times New Roman","serif"'>30        0.1</span></p>
155
156<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
157style='font-family:"Times New Roman","serif"'>32        0.3</span></p>
158
159<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
160style='font-family:"Times New Roman","serif"'>35        0.4</span></p>
161
162<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
163style='font-family:"Times New Roman","serif"'>36        0.5</span></p>
164
165<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
166style='font-family:"Times New Roman","serif"'>37        0.6</span></p>
167
168<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
169style='font-family:"Times New Roman","serif"'>39        0.7</span></p>
170
171<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
172style='font-family:"Times New Roman","serif"'>41        0.9</span></p>
173
174<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
175style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
176
177<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
178style='font-family:"Times New Roman","serif"'>We use only these array values in
179the computation, therefore the mean value given in the control panel, for
180example ‘radius = 60’, will be ignored.</span></p>
181
182<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
183style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
184
185<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
186style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
187
188<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
189style='font-family:"Times New Roman","serif"'><a name="Gaussian"><h4>Gaussian distribution</h4></a></span></p>
190
191<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
192style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
193
194<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
195style='font-family:"Times New Roman","serif";position:relative;top:12.0pt'><img
196width=212 height=44 src="pd_image005.png"></span></p>
197
198<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
199style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
200
201<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
202style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean
203of the distribution and Norm is a normalization factor which is determined
204during the numerical calculation.</span></p>
205
206<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
207style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
208
209<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
210style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p>
211
212<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
213style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
214width=93 height=24 src="pd_image003.png"></span><span
215style='font-family:"Times New Roman","serif"'>.</span></p>
216
217<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
218style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
219
220<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
221style='font-family:"Times New Roman","serif"'><img width=518 height=275
222id="Picture 2" src="pd_image006.jpg" alt=gauss.gif></span></p>
223
224<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
225style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
226
227<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
228style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
229
230<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
231style='font-family:"Times New Roman","serif"'><a name="Lognormal"><h4>Lognormal distribution</h4></a></span></p>
232
233<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
234style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
235
236<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
237style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
238
239<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
240style='font-family:"Times New Roman","serif";position:relative;top:14.0pt'><img
241width=236 height=47 src="pd_image007.png"></span></p>
242
243<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
244style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
245
246<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
247style='font-family:"Times New Roman","serif"'>The mu = ln(x<sub>med</sub>),  x<sub>med</sub>
248is the median value of the distribution, and Norm is a normalization factor
249which will be determined during the numerical calculation. The median value is
250the value given in the size parameter in the control panel, for example,
251“radius = 60”.</span></p>
252
253<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
254style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
255
256<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
257style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is given
258by sigma,</span></p>
259
260<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
261style='font-family:"Times New Roman","serif";position:relative;top:5.0pt'><img
262width=55 height=21 src="pd_image008.png"></span><span
263style='font-family:"Times New Roman","serif"'>.</span></p>
264
265<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
266style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
267
268<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
269style='font-family:"Times New Roman","serif"'>For the angular distribution,</span></p>
270
271<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
272style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
273width=76 height=24 src="pd_image009.png"></span></p>
274
275<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
276style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
277
278<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
279style='font-family:"Times New Roman","serif"'>The mean value is given by x<sub>mean</sub>
280=exp(mu+p^2/2).</span></p>
281
282<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
283style='font-family:"Times New Roman","serif"'>The peak value is given by x<sub>peak</sub>=exp(mu-p^2).</span></p>
284
285<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
286style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
287
288<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
289style='font-family:"Times New Roman","serif"'><img width=450 height=239
290id="Picture 7" src="pd_image010.jpg" alt=lognormal.gif></span></p>
291
292<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
293style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
294
295<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
296style='font-family:"Times New Roman","serif"'>This distribution function
297spreads more and the peak shifts to the left as the p increases, requiring
298higher values of Nsigmas and Npts.</span></p>
299
300<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
301style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
302
303<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
304style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
305
306<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
307style='font-family:"Times New Roman","serif"'><a name="Schulz"><h4>Schulz distribution</h4></a></span></p>
308
309<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
310style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
311
312<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
313style='font-family:"Times New Roman","serif";position:relative;top:15.0pt'><img
314width=347 height=45 src="pd_image011.png"></span></p>
315
316<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
317style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
318
319<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
320style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean
321of the distribution and Norm is a normalization factor which is determined
322during the numerical calculation. </span></p>
323
324<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
325style='font-family:"Times New Roman","serif"'>The z = 1/p^2 – 1.</span></p>
326
327<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
328style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
329
330<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
331style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p>
332
333<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
334style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
335width=80 height=24 src="pd_image012.png"></span><span
336style='font-family:"Times New Roman","serif"'>.</span></p>
337<p/>
338<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
339style='font-family:"Times New Roman","serif"'>Note that the higher PD (polydispersity)
340 might need higher values of Npts and Nsigmas. For example, at PD = 0.7 and  radisus = 60 A,
341 Npts >= 160, and Nsigmas >= 15 at least.</span></p>
342 <p/>
343<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
344style='font-family:"Times New Roman","serif"'><img width=438 height=232
345id="Picture 4" src="pd_image013.jpg" alt=schulz.gif></span></p>
346
347</div>
348
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350
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