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1<head>
2<meta http-equiv=Content-Type content="text/html; charset=windows-1252">
3<meta name=Generator content="Microsoft Word 12 (filtered)">
4
5</head>
6
7<body lang=EN-US>
8
9<div class=WordSection1>
10
11<p class=MsoNormal><h3><span style='font-family:"Times New Roman","serif"'>Polydisperisty
12and Angular Distributions</span></h3></p>
13
14<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Calculates
15the form factor for a polydisperse and/or angular population of particles with
16uniform scattering length density. The resultant form factor is normalized by
17the average particle volume such that P(q) = scale*&lt;F*F&gt;/Vol + bkg, where
18F is the scattering amplitude and the &lt; &gt; denote an average over the size
19distribution.  Users should use PD (polydispersity: this definition is different from the typical definition in polymer science)
20for a size distribution and Sigma for an
21angular distribution (see below).</span></p>
22<p> Note that this computation is very time intensive thus applying polydispersion/angular distrubtion for
23more than one paramters or increasing Npts values might need extensive patience to complete the computation. Also
24note that even though it is time consuming, it is safer to have larger values of Npts and Nsigmas.</p>
25
26<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
27style='font-family:"Times New Roman","serif"'>The following five distribution
28functions are provided;</span></p>
29<ul>
30<li><a href="#Rectangular">Rectangular distribution</a></li>
31<li><a href="#Array">Array distribution</a></li>
32<li><a href="#Gaussian">Gaussian distribution</a></li>
33<li><a href="#Lognormal">Lognormal distribution</a></li>
34<li><a href="#Schulz">Schulz distribution</a></li>
35</ul>
36<p>&nbsp;</p>
37<p><a name="Rectangular"><h4>Rectangular distribution</a></h4></p>
38<p><img src="img/pd_image001.png"></p>
39<p>&nbsp;</p>
40<p>The x<sub>mean</sub> is the mean
41of the distribution, w is the half-width, and Norm is a normalization factor
42which is determined during the numerical calculation. Note that the Sigma and
43the half width <i>w</i> are different.</p>
44<p>The standard deviation is </p>
45<p><img src="img/pd_image002.png"></p>
46<p>&nbsp;</p>
47<p>The PD (polydispersity) is </p>
48<p><img src="img/pd_image003.png"></p>
49<p>&nbsp;</p>
50<p><img width=511 height=270
51id="Picture 1" src="img/pd_image004.jpg" alt=flat.gif></p>
52<p>&nbsp;</p>
53<p>&nbsp;</p>
54<p><a name="Array"><h4>Array distribution</h4></a></p>
55
56<p>This distribution is to be given
57by users as a txt file where the array should be defined by two columns in the
58order of x and f(x) values. The f(x) will be normalized by SasView during the
59computation.</p>
60
61<p>Example of an array in the file;</p>
62
63<p>30        0.1</p>
64
65<p>32        0.3</p>
66
67<p>35        0.4</p>
68
69<p>36        0.5</p>
70
71<p>37        0.6</p>
72
73<p>39        0.7</p>
74
75<p>41        0.9</p>
76
77<p'>&nbsp;</p>
78
79<p>We use only these array values in
80the computation, therefore the mean value given in the control panel, for
81example ‘radius = 60’, will be ignored.</p>
82<p>&nbsp;</p>
83
84
85<p><a name="Gaussian"><h4>Gaussian distribution</h4></a></p>
86<p>&nbsp;</p>
87
88<p><img src="img/pd_image005.png"></p>
89
90<p>&nbsp;</p>
91
92<p>The x<sub>mean</sub> is the mean
93of the distribution and Norm is a normalization factor which is determined
94during the numerical calculation.</p>
95
96<p>&nbsp;</p>
97
98<p>The PD (polydispersity) is </p>
99
100<p><img src="img/pd_image003.png"></p>
101
102<p>&nbsp;</p>
103
104<p><img width=518 height=275
105id="Picture 2" src="img/pd_image006.jpg" alt=gauss.gif></p>
106
107<p>&nbsp;</p>
108
109<p><a name="Lognormal"><h4>Lognormal distribution</h4></a></p>
110
111<p>&nbsp;</p>
112
113<p><img src="img/pd_image007.png"></p>
114
115<p>&nbsp;</p>
116
117<p>The &#956; = ln(x<sub>med</sub>),  x<sub>med</sub>
118is the median value of the distribution, and Norm is a normalization factor
119which will be determined during the numerical calculation. The median value is
120the value given in the size parameter in the control panel, for example,
121“radius = 60”.</p>
122
123<p >&nbsp;</p>
124
125<p>The PD (polydispersity) is given
126by &#963;,</p>
127
128<p><img src="img/pd_image008.png"></p>
129
130<p>&nbsp;</p>
131
132<p>For the angular distribution,</p>
133
134<p><img src="img/pd_image009.png"></p>
135
136<p>&nbsp;</p>
137
138<p>The mean value is given by x<sub>mean</sub>
139=exp(&#956;+p<sup>2</sup>/2).</p>
140
141<p>The peak value is given by x<sub>peak</sub>=exp(&#956;-p<sup>2</sup>).</span></p>
142
143<p>&nbsp;</p>
144
145<p><img width=450 height=239
146id="Picture 7" src="img/pd_image010.jpg" alt=lognormal.gif></p>
147
148<p>&nbsp;</p>
149
150<p>This distribution function
151spreads more and the peak shifts to the left as the p increases, requiring
152higher values of Nsigmas and Npts.</p>
153
154<p>&nbsp;</p>
155
156<p><a name="Schulz"><h4>Schulz distribution</h4></a></p>
157
158<p>&nbsp;</p>
159
160<p><img src="img/pd_image011.png"></p>
161
162<p>&nbsp;</p>
163
164<p>The x<sub>mean</sub> is the mean
165of the distribution and Norm is a normalization factor which is determined
166during the numerical calculation. </p>
167
168<p>The z = 1/p<sup>2</sup> – 1.</p>
169<p>&nbsp;</p>
170
171<p>The PD (polydispersity) is </p>
172<p'><img src="img/pd_image012.png"></p>
173<p>Note that the higher PD (polydispersity)
174 might need higher values of Npts and Nsigmas. For example, at PD = 0.7 and  radisus = 60 A,
175 Npts >= 160, and Nsigmas >= 15 at least.</p>
176 <p/>
177<p><img width=438 height=232
178id="Picture 4" src="img/pd_image013.jpg" alt=schulz.gif></p>
179
180</div>
181<br>
182</body>
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