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1.. pd_help.rst
2
3.. This is a port of the original SasView html help file to ReSTructured text
4.. by S King, ISIS, during SasView CodeCamp-III in Feb 2015.
5
6.. |beta| unicode:: U+03B2
7.. |gamma| unicode:: U+03B3
8.. |mu| unicode:: U+03BC
9.. |sigma| unicode:: U+03C3
10.. |phi| unicode:: U+03C6
11.. |theta| unicode:: U+03B8
12.. |chi| unicode:: U+03C7
13.. |Ang| unicode:: U+212B
14
15.. |inlineimage004| image:: sm_image004.gif
16.. |inlineimage005| image:: sm_image005.gif
17.. |inlineimage008| image:: sm_image008.gif
18.. |inlineimage009| image:: sm_image009.gif
19.. |inlineimage010| image:: sm_image010.gif
20.. |inlineimage011| image:: sm_image011.gif
21.. |inlineimage012| image:: sm_image012.gif
22.. |inlineimage018| image:: sm_image018.gif
23.. |inlineimage019| image:: sm_image019.gif
24
25
26.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
27
28Polydispersity Distributions
29----------------------------
30
31With some models SasView can calculate the average form factor for a population
32of particles that exhibit size and/or orientational polydispersity. The resultant
33form factor is normalized by the average particle volume such that
34
35*P(q) = scale* * \ <F*\F> / *V + bkg*
36
37where F is the scattering amplitude and the \<\> denote an average over the size
38distribution.
39
40Users should note that this computation is very intensive. Applying polydispersion
41to multiple parameters at the same time, or increasing the number of *Npts* values
42in the fit, will require patience! However, the calculations are generally more
43robust with more data points or more angles.
44
45SasView uses the term *PD* for a size distribution (and not to be confused with a
46molecular weight distributions in polymer science) and the term *Sigma* for an
47angular distribution.
48
49The following five distribution functions are provided:
50
51*  *Rectangular Distribution*
52*  *Gaussian Distribution*
53*  *Lognormal Distribution*
54*  *Schulz Distribution*
55*  *Array Distribution*
56
57These are all implemented in SasView as *number-average* distributions.
58
59.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
60
61Rectangular Distribution
62^^^^^^^^^^^^^^^^^^^^^^^^
63
64The Rectangular Distribution is defined as
65
66.. image:: pd_image001.png
67
68where *xmean* is the mean of the distribution, *w* is the half-width, and *Norm* is a
69normalization factor which is determined during the numerical calculation.
70
71Note that the standard deviation and the half width *w* are different!
72
73The standard deviation is
74
75.. image:: pd_image002.png
76
77whilst the polydispersity is
78
79.. image:: pd_image003.png
80
81.. image:: pd_image004.jpg
82
83.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
84
85Gaussian Distribution
86^^^^^^^^^^^^^^^^^^^^^
87
88The Gaussian Distribution is defined as
89
90.. image:: pd_image005.png
91
92where *xmean* is the mean of the distribution and *Norm* is a normalization factor
93which is determined during the numerical calculation.
94
95The polydispersity is
96
97.. image:: pd_image003.png
98
99
100.. image:: pd_image006.jpg
101
102.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
103
104Lognormal Distribution
105^^^^^^^^^^^^^^^^^^^^^^
106
107The Lognormal Distribution is defined as
108
109.. image:: pd_image007.png
110
111where |mu|\ =ln(*xmed*), *xmed* is the median value of the distribution, and
112*Norm* is a normalization factor which will be determined during the numerical
113calculation.
114
115The median value for the distribution will be the value given for the respective
116size parameter in the *Fitting Perspective*, for example, radius = 60.
117
118The polydispersity is given by |sigma|
119
120.. image:: pd_image008.png
121
122For the angular distribution
123
124.. image:: pd_image009.png
125
126The mean value is given by *xmean*\ =exp(|mu|\ +p\ :sup:`2`\ /2). The peak value
127is given by *xpeak*\ =exp(|mu|-p\ :sup:`2`\ ).
128
129.. image:: pd_image010.jpg
130
131This distribution function spreads more, and the peak shifts to the left, as *p*
132increases, requiring higher values of Nsigmas and Npts.
133
134.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
135
136Schulz Distribution
137^^^^^^^^^^^^^^^^^^^
138
139The Schulz distribution is defined as
140
141.. image:: pd_image011.png
142
143where *xmean* is the mean of the distribution and *Norm* is a normalization factor
144which is determined during the numerical calculation, and *z* is a measure of the
145width of the distribution such that
146
147z = (1-p\ :sup:`2`\ ) / p\ :sup:`2`
148
149The polydispersity is
150
151.. image:: pd_image012.png
152
153Note that larger values of PD might need larger values of Npts and Nsigmas.
154For example, at PD=0.7 and radius=60 |Ang|, Npts>=160 and Nsigmas>=15 at least.
155
156.. image:: pd_image013.jpg
157
158For further information on the Schulz distribution see:
159M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461.
160
161.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
162
163Array Distribution
164^^^^^^^^^^^^^^^^^^
165
166This user-definable distribution should be given as as a simple ASCII text file
167where the array is defined by two columns of numbers: *x* and *f(x)*. The *f(x)*
168will be normalized by SasView during the computation.
169
170Example of what an array distribution file should look like:
171
172====  =====
173 30    0.1
174 32    0.3
175 35    0.4
176 36    0.5
177 37    0.6
178 39    0.7
179 41    0.9
180====  =====
181
182SasView only uses these array values during the computation, therefore any mean
183value of the parameter represented by *x* present in the *Fitting Perspective*
184will be ignored.
185
186.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
187
188Note about DLS polydispersity
189^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
190
191Many commercial Dynamic Light Scattering (DLS) instruments produce a size
192polydispersity parameter, sometimes even given the symbol *p*! This parameter is
193defined as the relative standard deviation coefficient of variation of the size
194distribution and is NOT the same as the polydispersity parameters in the Lognormal
195and Schulz distributions above (though they all related) except when the DLS
196polydispersity parameter is <0.13.
197
198For more information see:
199S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143
200
201.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
202
203.. note::  This help document was last changed by Steve King, 01May2015
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