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1.. sm_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.. |bigdelta| unicode:: U+0394
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
28Smearing Functions
29==================
30
31Sometimes it will be necessary to correct reduced experimental data for the
32physical effects of the instrumental geometry in use. This process is called
33*desmearing*. However, calculated/simulated data - which by definition will be
34perfect/exact - can be *smeared* to make it more representative of what might
35actually be measured experimentally.
36
37SasView provides the following three smearing algorithms:
38
39*  *Slit Smearing*
40*  *Pinhole Smearing*
41*  *2D Smearing*
42
43.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
44
45Slit Smearing
46-------------
47
48**This type of smearing is normally only encountered with data from X-ray Kratky**
49**cameras or X-ray/neutron Bonse-Hart USAXS/USANS instruments.**
50
51The slit-smeared scattering intensity is defined by
52
53.. image:: sm_image002.gif
54
55where *Norm* is given by
56
57.. image:: sm_image003.gif
58
59**[Equation 1]**
60
61The functions |inlineimage004| and |inlineimage005|
62refer to the slit width weighting function and the slit height weighting
63determined at the given *q* point, respectively. It is assumed that the weighting
64function is described by a rectangular function, such that
65
66.. image:: sm_image006.gif
67
68**[Equation 2]**
69
70and
71
72.. image:: sm_image007.gif
73
74**[Equation 3]**
75
76so that |inlineimage008| |inlineimage009| for |inlineimage010| and *u*\ .
77
78Here |inlineimage011| and |inlineimage012| stand for
79the slit height (FWHM/2) and the slit width (FWHM/2) in *q* space.
80
81This simplifies the integral in Equation 1 to
82
83.. image:: sm_image013.gif
84
85**[Equation 4]**
86
87which may be solved numerically, depending on the nature of |inlineimage011| and |inlineimage012| .
88
89Solution 1
90^^^^^^^^^^
91
92**For** |inlineimage012| **= 0 and** |inlineimage011| **= constant.**
93
94.. image:: sm_image016.gif
95
96For discrete *q* values, at the *q* values of the data points and at the *q*
97values extended up to *q*\ :sub:`N`\ = *q*\ :sub:`i` + |inlineimage011| the smeared
98intensity can be approximately calculated as
99
100.. image:: sm_image017.gif
101
102**[Equation 5]**
103
104where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *i* or *j* > *N-1*.
105
106Solution 2
107^^^^^^^^^^
108
109**For** |inlineimage012| **= constant and** |inlineimage011| **= 0.**
110
111Similar to Case 1
112
113|inlineimage019| for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
114
115**[Equation 6]**
116
117where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.
118
119Solution 3
120^^^^^^^^^^
121
122**For** |inlineimage011| **= constant and** |inlineimage011| **= constant.**
123
124In this case, the best way is to perform the integration of Equation 1
125numerically for both slit height and slit width. However, the numerical
126integration is imperfect unless a large number of iterations, say, at
127least 10000 by 10000 for each element of the matrix *W*, is performed.
128This is usually too slow for routine use.
129
130An alternative approach is used in SasView which assumes
131slit width << slit height. This method combines Solution 1 with the
132numerical integration for the slit width. Then
133
134.. image:: sm_image020.gif
135
136**[Equation 7]**
137
138for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
139
140where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.
141
142.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
143
144Pinhole Smearing
145----------------
146
147**This is the type of smearing normally encountered with data from synchrotron**
148**SAXS cameras and SANS instruments.**
149
150The pinhole smearing computation is performed in a similar fashion to the slit-
151smeared case above except that the weight function used is a Gaussian. Thus
152Equation 6 becomes
153
154.. image:: sm_image021.gif
155
156**[Equation 8]**
157
158.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
159
1602D Smearing
161-----------
162
163The 2D smearing computation is performed in a similar fashion to the 1D pinhole
164smearing above except that the weight function used is a 2D elliptical Gaussian.
165Thus
166
167.. image:: sm_image022.gif
168
169**[Equation 9]**
170
171In Equation 9, *x*\ :sub:`0` = *q* cos(|theta|), *y*\ :sub:`0` = *q* sin(|theta|), and
172the primed axes, are all in the coordinate rotated by an angle |theta| about
173the z-axis (see the figure below) so that *x'*\ :sub:`0` = *x*\ :sub:`0` cos(|theta|) +
174*y*\ :sub:`0` sin(|theta|) and *y'*\ :sub:`0` = -*x*\ :sub:`0` sin(|theta|) +
175*y*\ :sub:`0` cos(|theta|). Note that the rotation angle is zero for a x-y symmetric
176elliptical Gaussian distribution. The *A* is a normalization factor.
177
178.. image:: sm_image023.gif
179
180Now we consider a numerical integration where each of the bins in |theta| and *R* are
181*evenly* (this is to simplify the equation below) distributed by |bigdelta|\ |theta|
182and |bigdelta|\ R, respectively, and it is further assumed that *I(x',y')* is constant
183within the bins. Then
184
185.. image:: sm_image024.gif
186
187**[Equation 10]**
188
189Since the weighting factor on each of the bins is known, it is convenient to
190transform *x'-y'* back to *x-y* coordinates (by rotating it by -|theta| around the
191*z* axis).
192
193Then, for a polar symmetric smear
194
195.. image:: sm_image025.gif
196
197**[Equation 11]**
198
199where
200
201.. image:: sm_image026.gif
202
203while for a *x-y* symmetric smear
204
205.. image:: sm_image027.gif
206
207**[Equation 12]**
208
209where
210
211.. image:: sm_image028.gif
212
213The current version of the SasView uses Equation 11 for 2D smearing, assuming
214that all the Gaussian weighting functions are aligned in the polar coordinate.
215
216.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
217
218Weighting & Normalization
219-------------------------
220
221In all the cases above, the weighting matrix *W* is calculated on the first call
222to a smearing function, and includes ~60 *q* values (finely and evenly binned)
223below (>0) and above the *q* range of data in order to smear all data points for
224a given model and slit/pinhole size. The *Norm*  factor is found numerically with the
225weighting matrix and applied on the computation of *I*\ :sub:`s`.
226
227.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
228
229.. note::  This help document was last changed by Steve King, 01May2015
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