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