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 | |
---|
28 | Smearing Functions |
---|
29 | ================== |
---|
30 | |
---|
31 | Sometimes it will be necessary to correct reduced experimental data for the |
---|
32 | physical effects of the instrumental geometry in use. This process is called |
---|
33 | *desmearing*. However, calculated/simulated data - which by definition will be |
---|
34 | perfect/exact - can be *smeared* to make it more representative of what might |
---|
35 | actually be measured experimentally. |
---|
36 | |
---|
37 | SasView provides the following three smearing algorithms: |
---|
38 | |
---|
39 | * *Slit Smearing* |
---|
40 | * *Pinhole Smearing* |
---|
41 | * *2D Smearing* |
---|
42 | |
---|
43 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
---|
44 | |
---|
45 | Slit 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 | |
---|
51 | The slit-smeared scattering intensity is defined by |
---|
52 | |
---|
53 | .. image:: sm_image002.gif |
---|
54 | |
---|
55 | where *Norm* is given by |
---|
56 | |
---|
57 | .. image:: sm_image003.gif |
---|
58 | |
---|
59 | **[Equation 1]** |
---|
60 | |
---|
61 | The functions |inlineimage004| and |inlineimage005| |
---|
62 | refer to the slit width weighting function and the slit height weighting |
---|
63 | determined at the given *q* point, respectively. It is assumed that the weighting |
---|
64 | function is described by a rectangular function, such that |
---|
65 | |
---|
66 | .. image:: sm_image006.gif |
---|
67 | |
---|
68 | **[Equation 2]** |
---|
69 | |
---|
70 | and |
---|
71 | |
---|
72 | .. image:: sm_image007.gif |
---|
73 | |
---|
74 | **[Equation 3]** |
---|
75 | |
---|
76 | so that |inlineimage008| |inlineimage009| for |inlineimage010| and *u*\ . |
---|
77 | |
---|
78 | Here |inlineimage011| and |inlineimage012| stand for |
---|
79 | the slit height (FWHM/2) and the slit width (FWHM/2) in *q* space. |
---|
80 | |
---|
81 | This simplifies the integral in Equation 1 to |
---|
82 | |
---|
83 | .. image:: sm_image013.gif |
---|
84 | |
---|
85 | **[Equation 4]** |
---|
86 | |
---|
87 | which may be solved numerically, depending on the nature of |inlineimage011| and |inlineimage012| . |
---|
88 | |
---|
89 | Solution 1 |
---|
90 | ^^^^^^^^^^ |
---|
91 | |
---|
92 | **For** |inlineimage012| **= 0 and** |inlineimage011| **= constant.** |
---|
93 | |
---|
94 | .. image:: sm_image016.gif |
---|
95 | |
---|
96 | For discrete *q* values, at the *q* values of the data points and at the *q* |
---|
97 | values extended up to *q*\ :sub:`N`\ = *q*\ :sub:`i` + |inlineimage011| the smeared |
---|
98 | intensity can be approximately calculated as |
---|
99 | |
---|
100 | .. image:: sm_image017.gif |
---|
101 | |
---|
102 | **[Equation 5]** |
---|
103 | |
---|
104 | where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *i* or *j* > *N-1*. |
---|
105 | |
---|
106 | Solution 2 |
---|
107 | ^^^^^^^^^^ |
---|
108 | |
---|
109 | **For** |inlineimage012| **= constant and** |inlineimage011| **= 0.** |
---|
110 | |
---|
111 | Similar 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 | |
---|
117 | where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*. |
---|
118 | |
---|
119 | Solution 3 |
---|
120 | ^^^^^^^^^^ |
---|
121 | |
---|
122 | **For** |inlineimage011| **= constant and** |inlineimage011| **= constant.** |
---|
123 | |
---|
124 | In this case, the best way is to perform the integration of Equation 1 |
---|
125 | numerically for both slit height and slit width. However, the numerical |
---|
126 | integration is imperfect unless a large number of iterations, say, at |
---|
127 | least 10000 by 10000 for each element of the matrix *W*, is performed. |
---|
128 | This is usually too slow for routine use. |
---|
129 | |
---|
130 | An alternative approach is used in SasView which assumes |
---|
131 | slit width << slit height. This method combines Solution 1 with the |
---|
132 | numerical integration for the slit width. Then |
---|
133 | |
---|
134 | .. image:: sm_image020.gif |
---|
135 | |
---|
136 | **[Equation 7]** |
---|
137 | |
---|
138 | for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012| |
---|
139 | |
---|
140 | where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*. |
---|
141 | |
---|
142 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
---|
143 | |
---|
144 | Pinhole Smearing |
---|
145 | ---------------- |
---|
146 | |
---|
147 | **This is the type of smearing normally encountered with data from synchrotron** |
---|
148 | **SAXS cameras and SANS instruments.** |
---|
149 | |
---|
150 | The pinhole smearing computation is performed in a similar fashion to the slit- |
---|
151 | smeared case above except that the weight function used is a Gaussian. Thus |
---|
152 | Equation 6 becomes |
---|
153 | |
---|
154 | .. image:: sm_image021.gif |
---|
155 | |
---|
156 | **[Equation 8]** |
---|
157 | |
---|
158 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
---|
159 | |
---|
160 | 2D Smearing |
---|
161 | ----------- |
---|
162 | |
---|
163 | The 2D smearing computation is performed in a similar fashion to the 1D pinhole |
---|
164 | smearing above except that the weight function used is a 2D elliptical Gaussian. |
---|
165 | Thus |
---|
166 | |
---|
167 | .. image:: sm_image022.gif |
---|
168 | |
---|
169 | **[Equation 9]** |
---|
170 | |
---|
171 | In Equation 9, *x*\ :sub:`0` = *q* cos(|theta|), *y*\ :sub:`0` = *q* sin(|theta|), and |
---|
172 | the primed axes, are all in the coordinate rotated by an angle |theta| about |
---|
173 | the 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 |
---|
176 | elliptical Gaussian distribution. The *A* is a normalization factor. |
---|
177 | |
---|
178 | .. image:: sm_image023.gif |
---|
179 | |
---|
180 | Now 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| |
---|
182 | and |bigdelta|\ R, respectively, and it is further assumed that *I(x',y')* is constant |
---|
183 | within the bins. Then |
---|
184 | |
---|
185 | .. image:: sm_image024.gif |
---|
186 | |
---|
187 | **[Equation 10]** |
---|
188 | |
---|
189 | Since the weighting factor on each of the bins is known, it is convenient to |
---|
190 | transform *x'-y'* back to *x-y* coordinates (by rotating it by -|theta| around the |
---|
191 | *z* axis). |
---|
192 | |
---|
193 | Then, for a polar symmetric smear |
---|
194 | |
---|
195 | .. image:: sm_image025.gif |
---|
196 | |
---|
197 | **[Equation 11]** |
---|
198 | |
---|
199 | where |
---|
200 | |
---|
201 | .. image:: sm_image026.gif |
---|
202 | |
---|
203 | while for a *x-y* symmetric smear |
---|
204 | |
---|
205 | .. image:: sm_image027.gif |
---|
206 | |
---|
207 | **[Equation 12]** |
---|
208 | |
---|
209 | where |
---|
210 | |
---|
211 | .. image:: sm_image028.gif |
---|
212 | |
---|
213 | The current version of the SasView uses Equation 11 for 2D smearing, assuming |
---|
214 | that all the Gaussian weighting functions are aligned in the polar coordinate. |
---|
215 | |
---|
216 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
---|
217 | |
---|
218 | Weighting & Normalization |
---|
219 | ------------------------- |
---|
220 | |
---|
221 | In all the cases above, the weighting matrix *W* is calculated on the first call |
---|
222 | to a smearing function, and includes ~60 *q* values (finely and evenly binned) |
---|
223 | below (>0) and above the *q* range of data in order to smear all data points for |
---|
224 | a given model and slit/pinhole size. The *Norm* factor is found numerically with the |
---|
225 | weighting 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 |
---|