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8<body lang=EN-US>
9
10<div class=WordSection1>
11
12<p class=MsoNormal><span style='font-size:16.0pt;line-height:115%;font-family:
13"Times New Roman","serif"'><h4>Smear Computation </h4></span></p>
14
15
16<ul style='margin-top:0in' type=disc>
17 <li class=MsoNormal style='line-height:115%'><a href="#Slit Smear"><b>Slit Smear</b></a>
18     </li>
19 <li class=MsoNormal style='line-height:115%'><a href="#Pinhole Smear"><b>Pinhole Smear</b></a>
20        </li>
21 <li class=MsoNormal style='line-height:115%'><a href="#2D Smear"><b>2D Smear</b></a> 
22        </li>
23</ul>
24
25<p class=MsoListParagraph><span style='font-size:14.0pt;line-height:115%;
26font-family:"Times New Roman","serif"'><h5><a name="Slit Smear">Slit Smear</a></h5></span></p>
27
28<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The sit
29smeared scattering intensity for SAS is defined by</span></p>
30
31<p class=MsoNormal><img 
32src="img/sm_image002.gif" align=left hspace=12></p>
33
34<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>           
35                                 ---- 1)</span><br clear=all>
36<span style='font-family:"Times New Roman","serif"'>where Norm = <span
37style='position:relative;top:15.0pt'><img 
38src="img/sm_image003.gif"></span>.</span></p>
39<br>
40<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The
41functions <span style='position:relative;top:6.0pt'><img 
42src="img/sm_image004.gif"></span> and <span style='position:
43relative;top:6.0pt'><img 
44src="img/sm_image005.gif"></span> refer to the slit width weighting
45function and the slit height weighting determined at the q point, respectively.
46 Here, we assumes that the weighting function is described by a rectangular
47function, i.e.,</span></p>
48
49<p class=MsoNormal><span style='position:relative;top:7.0pt'><img 
50 src="img/sm_image006.gif">                                                                                                       
51  </span><span style='font-family:"Times New Roman","serif";position:relative;
52top:7.0pt'> ---- 2)</span></p>
53
54<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>and </span></p>
55
56<p class=MsoNormal><span style='position:relative;top:7.0pt'><img 
57 src="img/sm_image007.gif"></span>,                                                                             
58                         <span style='font-family:"Times New Roman","serif"'> ---- 3)</span></p>
59
60<p>so that <img 
61src="img/sm_image008.gif"> <img src="img/sm_image009.gif">  for <img 
62src="img/sm_image010.gif"> and <i>u</i>. The <img src="img/sm_image011.gif"> 
63 and <img src="img/sm_image012.gif">  stand for the slit height (FWHM/2) and the slit
64width (FWHM/2) in the q space. Now the integral of Eq. (1) is simplified to</span></p>
65
66<p class=MsoNormal><img 
67src="img/sm_image013.gif" align=left hspace=12><span
68style='font-family:"Times New Roman","serif"'>                                                 
69          ---- 4)</span></p>
70
71<p class=MsoNormal><span style='font-family:"Times New Roman","serif";
72position:relative;top:20.0pt'>&nbsp;</span></p>
73
74<p class=MsoListParagraphCxSpFirst style='margin-left:0in'><b><span
75style='font-family:"Times New Roman","serif"'>Numerical Implementation of Eq.
76(4) </span></b></p>
77
78<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><span
79style='font-family:"Times New Roman","serif"'>1)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
80</span></span><span style='font-family:"Times New Roman","serif"'>For </span><span
81style='position:relative;top:6.0pt'><img 
82src="img/sm_image012.gif"></span>= 0  <span style='font-family:
83"Times New Roman","serif"'>and </span><span style='position:relative;
84top:6.0pt'><img src="img/sm_image011.gif"></span> =
85<span style='font-family:"Times New Roman","serif"'>constant:</span></p>
86
87<p>
88<img src="img/sm_image016.gif"></p>
89
90<p> For discrete q values, at the q
91values from the data points and at the q values extended up to  q<sub>N</sub>=
92q<sub>i</sub> +  <img src="img/sm_image011.gif"> , the smeared intensity can be
93calculated approximately, </p>
94
95<p><img 
96src="img/sm_image017.gif">.                                                           
97 ---- 5)</p>
98
99<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
100style='position:relative;top:7.0pt'><img 
101src="img/sm_image018.gif"></span> <span style='font-family:
102"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span
103style='font-family:"Times New Roman","serif"'>j &lt; i</span></i><span
104style='font-family:"Times New Roman","serif"'> or<i> j&gt;N-1</i>.</span></p>
105
106<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
107style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
108
109<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><span
110style='font-family:"Times New Roman","serif"'>2)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
111</span></span><span style='font-family:"Times New Roman","serif"'>For  </span><span
112style='position:relative;top:6.0pt'><img 
113src="img/sm_image012.gif"></span>= <span style='font-family:
114"Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and
115</span><span style='position:relative;top:6.0pt'><img 
116src="img/sm_image011.gif"></span> = <span style='font-family:
117"Times New Roman","serif"'>0:</span></p>
118
119<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
120style='font-family:"Times New Roman","serif"'>Similarly to 1), we get</span></p>
121
122<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>
123<img src="img/sm_image019.gif">                                                                                       
124<span style='font-family:"Times New Roman","serif"'> ---- 6)</span></p>
125
126<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
127style='font-family:"Times New Roman","serif"'>for  q<sub>p</sub> = q<sub>i</sub>
128- </span><span style='position:relative;top:6.0pt'><img 
129src="img/sm_image012.gif"></span><span style='font-family:
130"Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub>
131= q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img
132 src="img/sm_image012.gif"></span>.  <span
133style='position:relative;top:7.0pt'><img 
134src="img/sm_image018.gif"></span> <span style='font-family:
135"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span
136style='font-family:"Times New Roman","serif"'>j &lt; p</span></i><span
137style='font-family:"Times New Roman","serif"'> or<i> j&gt;N-1</i>.</span></p>
138
139<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>&nbsp;</p>
140
141<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><span
142style='font-family:"Times New Roman","serif"'>3)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
143</span></span><span style='font-family:"Times New Roman","serif"'>For  </span><span
144style='position:relative;top:6.0pt'><img 
145src="img/sm_image011.gif"></span>= <span style='font-family:
146"Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and
147</span><span style='position:relative;top:6.0pt'><img 
148src="img/sm_image011.gif"></span> = <span style='font-family:
149"Times New Roman","serif"'>constant:</span></p>
150
151<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
152style='font-family:"Times New Roman","serif"'>This case, the best way is to
153perform the integration, Eq. (1), numerically for both slit height and width.
154However, the numerical integration is not correct enough unless given a large
155number of iteration, say at least 10000 by 10000 for each element of the matrix
156W, which will take minutes and minutes to finish the calculation for a set of
157typical SAS data. An alternative way which is correct for slit width &lt;&lt;
158slit hight, is used in the SASView:  This method is a mixed method that
159combines the method 1) with the numerical integration for the slit width.</span></p>
160
161<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>
162</p>
163
164<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'>
165<img src="img/sm_image020.gif">    <span style='font-family:
166"Times New Roman","serif"'> ---- (7)</span></p>
167
168<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
169style='font-family:"Times New Roman","serif"'>for  q<sub>p</sub> = q<sub>i</sub>
170- </span><span style='position:relative;top:6.0pt'><img 
171src="img/sm_image012.gif"></span><span style='font-family:
172"Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub>
173= q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img
174 src="img/sm_image012.gif"></span>.  <span
175style='position:relative;top:7.0pt'><img 
176src="img/sm_image018.gif"></span> <span style='font-family:
177"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span
178style='font-family:"Times New Roman","serif"'>j &lt; p</span></i><span
179style='font-family:"Times New Roman","serif"'> or<i> j&gt;N-1</i>. </span></p>
180
181<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
182style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
183
184<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span
185style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
186
187<p class=MsoListParagraphCxSpLast><span style='font-size:14.0pt;line-height:
188115%;font-family:"Times New Roman","serif"'><h5><a name="Pinhole Smear">Pinhole Smear</a></h5></span></p>
189
190<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The
191pinhole smearing computation is done similar to the Case 2) above except that
192the weight function used was the Gaussian function, so that the Eq. 6) for this
193case becomes</span></p>
194
195<p class=MsoNormal><img src="img/sm_image021.gif"><span
196style='font-family:"Times New Roman","serif"'>   ---- (8)</span></p>
197
198<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>For all
199the cases above, the weighting matrix <i>W</i> is calculated when the smearing
200is called at the first time, and it includes the ~ 60 q values (finely binned
201evenly) below (&gt;0) and above the q range of data in order to cover all data
202points of the smearing computation for a given model and for a given slit size.
203 The <i>Norm</i> factor is found numerically with the weighting matrix, and
204considered on <i>I<sub>s</sub></i> computation.</span></p>
205
206<p class=MsoListParagraphCxSpFirst style='margin-left:.25in'><span
207style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
208
209<p class=MsoListParagraphCxSpLast><span style='font-size:14.0pt;line-height:
210115%;font-family:"Times New Roman","serif"'><h5><a name="2D Smear">2D Smear</a></h5></span></p>
211
212<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The
2132D smearing computation is done similar to the 1D pinhole smearing above
214except that the weight function used was the 2D elliptical Gaussian function</span></p>
215
216<p class=MsoNormal><img src="img/sm_image022.gif"><span
217style='font-family:"Times New Roman","serif"'>     ---- (9)</span></p>
218
219<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>In Eq
220(9), x<sub>0</sub> = qcos&#952;</span><span
221style='font-family:"Times New Roman","serif"'> and y<sub>0</sub> = qsin&#952;</span><span style='font-family:"Times New Roman","serif"'>
222, and the primed axes are in the coordinate rotated by an angle &#952;</span><span style='font-family:"Times New Roman","serif"'>
223around z-axis (below) so that x’<sub>0</sub> =  x<sub>0</sub>cos&#952; + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub>
224sin&#952;  </span><span style='font-family:
225"Times New Roman","serif"'>and y’<sub>0</sub> =  -x<sub>0</sub>sin&#952; + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub>
226cos&#952;.</span><span style='font-family:
227"Times New Roman","serif"'> Note that the rotation angle is zero for x-y
228symmetric elliptical Gaussian distribution</span>.
229<span style='font-family:"Times New Roman","serif"'>The  A is a
230normalization factor.</span></p>
231
232<p class=MsoNormal align=center style='text-align:center'><span
233style='font-family:"Times New Roman","serif"'><img 
234id="Object 1" src="img/sm_image023.gif"></span></p>
235
236<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p>
237
238<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Now we
239consider a numerical integration where each bins in </span> &#920; </span><span style='font-family:"Times New Roman","serif"'>
240and R are <b>evenly </b>(this is to simplify the equation below) distributed by
241</span>&#916;&#920; </span><span style='font-family:
242"Times New Roman","serif"'>and </span> &#916;</span><span
243style='font-family:"Times New Roman","serif"'>R, respectively, and it is
244assumed that I(x’, y’) is constant within the bins which in turn becomes</span></p>
245
246<p class=MsoNormal><img src="img/sm_image024.gif"></p>
247
248<p class=MsoNormal>                                                                                                                                                                                <span
249style='font-family:"Times New Roman","serif"'> ---- (10)</span></p>
250
251<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Since we
252have found the weighting factor on each bin points, it is convenient to
253transform x’-y’ back to x-y coordinate (rotating it by -&#952;</span><span style='font-family:"Times New Roman","serif"'>
254around z axis).  Then, for the polar symmetric smear,</span></p>
255
256<p class=MsoNormal><img src="img/sm_image025.gif"><span
257style='position:relative;top:35.0pt'>      </span> ---- (11)</p>
258
259<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p>
260
261<p class=MsoNormal><img src="img/sm_image026.gif">,</p>
262
263<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>while
264for the x-y symmetric smear,</span></p>
265
266<p class=MsoNormal><img src="img/sm_image027.gif"><span
267style='font-family:"Times New Roman","serif"'>  ---- (12)</span></p>
268
269<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p>
270
271<p class=MsoNormal><img src="img/sm_image028.gif"></p>
272
273<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Here, the
274current version of the SASVIEW uses the Eq. (11) for 2D smearing assuming that
275all the Gaussian weighting functions are aligned in the polar coordinate. </span></p>
276<p> In the control panel, the higher accuracy indicates more and finer binnng points
277so that it costs more in time. </p>
278
279
280</div>
281
282</body>
283
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