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3<head>
4<meta http-equiv=Content-Type content="text/html; charset=windows-1252">
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9<body lang=EN-US>
10
11<div class=WordSection1>
12
13<p class=MsoNormal><h3><span style='font-family:"Times New Roman","serif"'>Polydisperisty
14and Angular Distributions</span></h3></p>
15
16<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Calculates
17the form factor for a polydisperse and/or angular population of particles with
18uniform scattering length density. The resultant form factor is normalized by
19the average particle volume such that P(q) = scale*&lt;F*F&gt;/Vol + bkg, where
20F is the scattering amplitude and the &lt; &gt; denote an average over the size
21distribution.  Users should use PD (polydispersity: this definition is different from the typical definition in polymer science)
22for a size distribution and Sigma for an
23angular distribution (see below).</span></p>
24<p> Note that this computation is very time intensive thus applying polydispersion/angular distrubtion for
25more than one paramters or increasing Npts values might need extensive patience to complete the computation. Also
26note that even though it is time consuming, it is safer to have larger values of Npts and Nsigmas.</p>
27
28<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
29style='font-family:"Times New Roman","serif"'>The following five distribution
30functions are provided;</span></p>
31<p>&nbsp;</p>
32<ul>
33<li><a href="#Rectangular">Rectangular distribution</a></li>
34<li><a href="#Array">Array distribution</a></li>
35<li><a href="#Gaussian">Gaussian distribution</a></li>
36<li><a href="#Lognormal">Lognormal distribution</a></li>
37<li><a href="#Schulz">Schulz distribution</a></li>
38</ul>
39<p>&nbsp;</p>
40<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
41style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
42
43<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
44style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
45
46<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
47style='font-family:"Times New Roman","serif"'><a name="Rectangular"><h4>Rectangular distribution</a></h4></span></p>
48
49<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
50style='font-family:"Times New Roman","serif";position:relative;top:22.0pt'><img
51width=248 height=67 src="./img/pd_image001.png"></span></p>
52
53<p>&nbsp;</p>
54<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
55style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean
56of the distribution, w is the half-width, and Norm is a normalization factor
57which is determined during the numerical calculation. Note that the Sigma and
58the half width <i>w</i> are different.</span></p>
59
60<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
61style='font-family:"Times New Roman","serif"'>The standard deviation is </span></p>
62
63<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
64style='font-family:"Times New Roman","serif";position:relative;top:4.0pt'><img
65width=72 height=24 src="./img/pd_image002.png"></span><span
66style='font-family:"Times New Roman","serif"'>. </span></p>
67
68<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
69style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
70
71<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
72style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p>
73
74<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
75style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
76width=93 height=24 src="./img/pd_image003.png"></span><span
77style='font-family:"Times New Roman","serif"'>.</span></p>
78
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"'><img width=511 height=270
84id="Picture 1" src="./img/pd_image004.jpg" alt=flat.gif></span></p>
85
86<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
87style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
88
89<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
90style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
91
92<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
93style='font-family:"Times New Roman","serif"'><a name="Array"><h4>Array distribution</h4></a></span></p>
94
95<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
96style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
97
98
99<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
100style='font-family:"Times New Roman","serif"'>This distribution is to be given
101by users as a txt file where the array should be defined by two columns in the
102order of x and f(x) values. The f(x) will be normalized by SansView during the
103computation.</span></p>
104
105<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
106style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
107
108<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
109style='font-family:"Times New Roman","serif"'>Example of an array in the file;</span></p>
110
111<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
112style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
113
114<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
115style='font-family:"Times New Roman","serif"'>30        0.1</span></p>
116
117<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
118style='font-family:"Times New Roman","serif"'>32        0.3</span></p>
119
120<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
121style='font-family:"Times New Roman","serif"'>35        0.4</span></p>
122
123<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
124style='font-family:"Times New Roman","serif"'>36        0.5</span></p>
125
126<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
127style='font-family:"Times New Roman","serif"'>37        0.6</span></p>
128
129<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
130style='font-family:"Times New Roman","serif"'>39        0.7</span></p>
131
132<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
133style='font-family:"Times New Roman","serif"'>41        0.9</span></p>
134
135<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
136style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
137
138<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
139style='font-family:"Times New Roman","serif"'>We use only these array values in
140the computation, therefore the mean value given in the control panel, for
141example ‘radius = 60’, will be ignored.</span></p>
142
143<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
144style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
145
146<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
147style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
148
149<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
150style='font-family:"Times New Roman","serif"'><a name="Gaussian"><h4>Gaussian distribution</h4></a></span></p>
151
152<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
153style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
154
155<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
156style='font-family:"Times New Roman","serif";position:relative;top:12.0pt'><img
157width=212 height=44 src="./img/pd_image005.png"></span></p>
158
159<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
160style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
161
162<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
163style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean
164of the distribution and Norm is a normalization factor which is determined
165during the numerical calculation.</span></p>
166
167<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
168style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
169
170<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
171style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p>
172
173<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
174style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
175width=93 height=24 src="./img/pd_image003.png"></span><span
176style='font-family:"Times New Roman","serif"'>.</span></p>
177
178<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
179style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
180
181<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
182style='font-family:"Times New Roman","serif"'><img width=518 height=275
183id="Picture 2" src="./img/pd_image006.jpg" alt=gauss.gif></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"'>&nbsp;</span></p>
190
191<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
192style='font-family:"Times New Roman","serif"'><a name="Lognormal"><h4>Lognormal distribution</h4></a></span></p>
193
194<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
195style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
196
197<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
198style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
199
200<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
201style='font-family:"Times New Roman","serif";position:relative;top:14.0pt'><img
202width=236 height=47 src="./img/pd_image007.png"></span></p>
203
204<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
205style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
206
207<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
208style='font-family:"Times New Roman","serif"'>The mu = ln(x<sub>med</sub>),  x<sub>med</sub>
209is the median value of the distribution, and Norm is a normalization factor
210which will be determined during the numerical calculation. The median value is
211the value given in the size parameter in the control panel, for example,
212“radius = 60”.</span></p>
213
214<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
215style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
216
217<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
218style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is given
219by sigma,</span></p>
220
221<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
222style='font-family:"Times New Roman","serif";position:relative;top:5.0pt'><img
223width=55 height=21 src="./img/pd_image008.png"></span><span
224style='font-family:"Times New Roman","serif"'>.</span></p>
225
226<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
227style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
228
229<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
230style='font-family:"Times New Roman","serif"'>For the angular distribution,</span></p>
231
232<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
233style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
234width=76 height=24 src="./img/pd_image009.png"></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"'>The mean value is given by x<sub>mean</sub>
241=exp(mu+p^2/2).</span></p>
242
243<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
244style='font-family:"Times New Roman","serif"'>The peak value is given by x<sub>peak</sub>=exp(mu-p^2).</span></p>
245
246<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
247style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
248
249<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
250style='font-family:"Times New Roman","serif"'><img width=450 height=239
251id="Picture 7" src="./img/pd_image010.jpg" alt=lognormal.gif></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"'>This distribution function
258spreads more and the peak shifts to the left as the p increases, requiring
259higher values of Nsigmas and Npts.</span></p>
260
261<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
262style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
263
264<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
265style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
266
267<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
268style='font-family:"Times New Roman","serif"'><a name="Schulz"><h4>Schulz distribution</h4></a></span></p>
269
270<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
271style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
272
273<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
274style='font-family:"Times New Roman","serif";position:relative;top:15.0pt'><img
275width=347 height=45 src="./img/pd_image011.png"></span></p>
276
277<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
278style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
279
280<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
281style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean
282of the distribution and Norm is a normalization factor which is determined
283during the numerical calculation. </span></p>
284
285<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
286style='font-family:"Times New Roman","serif"'>The z = 1/p^2 – 1.</span></p>
287
288<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
289style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
290
291<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
292style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p>
293
294<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
295style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img
296width=80 height=24 src="./img/pd_image012.png"></span><span
297style='font-family:"Times New Roman","serif"'>.</span></p>
298<p/>
299<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
300style='font-family:"Times New Roman","serif"'>Note that the higher PD (polydispersity)
301 might need higher values of Npts and Nsigmas. For example, at PD = 0.7 and  radisus = 60 A,
302 Npts >= 160, and Nsigmas >= 15 at least.</span></p>
303 <p/>
304<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
305style='font-family:"Times New Roman","serif"'><img width=438 height=232
306id="Picture 4" src="./img/pd_image013.jpg" alt=schulz.gif></span></p>
307
308</div>
309
310</body>
311
312</html>
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