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| 90 | </head> |
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| 91 | |
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| 92 | <body lang=EN-US> |
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| 93 | |
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| 94 | <div class=WordSection1> |
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| 95 | |
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| 96 | <p class=MsoNormal><span style='font-size:16.0pt;line-height:115%;font-family: |
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| 97 | "Times New Roman","serif"'>Smear Computation </span></p> |
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| 98 | <p class=MsoNormal> </p> |
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| 99 | |
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| 100 | <ul style='margin-top:0in' type=disc> |
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| 101 | <li class=MsoNormal style='line-height:115%'><a href="#Slit Smear"><b>Slit Smear</b></a> |
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| 102 | </li> |
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| 103 | <li class=MsoNormal style='line-height:115%'><a href="#Pinhole Smear"><b>Pinhole Smear</b></a> |
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| 104 | </li> |
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| 105 | <li class=MsoNormal style='line-height:115%'><a href="#2D Smear"><b>2D Smear</b></a> |
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| 106 | </li> |
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| 107 | </ul> |
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| 108 | <p class=MsoNormal> </p> |
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| 109 | <p class=MsoNormal> </p> |
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| 110 | <p class=MsoListParagraph><span style='font-size:14.0pt;line-height:115%; |
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| 111 | font-family:"Times New Roman","serif"'><a name="Slit Smear">Slit Smear</a></span></p> |
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| 112 | |
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| 113 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The sit |
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| 114 | smeared scattering intensity for SANS is defined by</span></p> |
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| 115 | |
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| 116 | <p class=MsoNormal><img width=349 height=49 |
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| 117 | src="sm_image002.gif" align=left hspace=12></p> |
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| 118 | |
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| 119 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'> |
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| 120 | 1)</span><br clear=all> |
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| 121 | <span style='font-family:"Times New Roman","serif"'>where Norm = <span |
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| 122 | style='position:relative;top:15.0pt'><img width=137 height=49 |
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| 123 | src="sm_image003.gif"></span>.</span></p> |
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| 124 | |
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| 125 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The |
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| 126 | functions <span style='position:relative;top:6.0pt'><img width=43 height=25 |
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| 127 | src="sm_image004.gif"></span>and <span style='position: |
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| 128 | relative;top:6.0pt'><img width=43 height=25 |
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| 129 | src="sm_image005.gif"></span>refer to the slit width weighting |
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| 130 | function and the slit height weighting determined at the q point, respectively. |
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| 131 | Here, we assumes that the weighting function is described by a rectangular |
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| 132 | function, i.e.,</span></p> |
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| 133 | |
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| 134 | <p class=MsoNormal><span style='position:relative;top:7.0pt'><img width=134 |
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| 135 | height=26 src="sm_image006.gif"> |
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| 136 | </span><span style='font-family:"Times New Roman","serif";position:relative; |
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| 137 | top:7.0pt'>2)</span></p> |
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| 138 | |
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| 139 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>and </span></p> |
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| 140 | |
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| 141 | <p class=MsoNormal><span style='position:relative;top:7.0pt'><img width=136 |
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| 142 | height=26 src="sm_image007.gif"></span>, |
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| 143 | <span style='font-family:"Times New Roman","serif"'>3)</span></p> |
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| 144 | |
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| 145 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>so that </span><span |
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| 146 | style='position:relative;top:6.0pt'><img width=58 height=23 |
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| 147 | src="sm_image008.gif"></span> <span style='position:relative; |
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| 148 | top:16.0pt'><img width=76 height=51 src="sm_image009.gif"></span> <span |
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| 149 | style='font-family:"Times New Roman","serif"'>for</span> <span |
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| 150 | style='position:relative;top:3.0pt'><img width=40 height=15 |
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| 151 | src="sm_image010.gif"></span> <span style='font-family: |
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| 152 | "Times New Roman","serif"'>and <i>u</i>. The </span><span style='position:relative; |
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| 153 | top:6.0pt'><img width=28 height=24 src="sm_image011.gif"></span> <span |
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| 154 | style='font-family:"Times New Roman","serif"'>and </span><span |
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| 155 | style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 156 | src="sm_image012.gif"> </span><span style='font-family: |
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| 157 | "Times New Roman","serif"'>stand for the slit height (FWHM/2) and the slit |
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| 158 | width (FWHM/2) in the q space. Now the integral of Eq. (1) is simplified to</span></p> |
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| 159 | |
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| 160 | <p class=MsoNormal><img width=283 height=52 |
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| 161 | src="sm_image013.gif" align=left hspace=12><span |
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| 162 | style='font-family:"Times New Roman","serif"'> |
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| 163 | 4)</span></p> |
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| 164 | |
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| 165 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"; |
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| 166 | position:relative;top:20.0pt'> </span></p> |
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| 167 | |
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| 168 | <p class=MsoListParagraphCxSpFirst style='margin-left:0in'><b><span |
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| 169 | style='font-family:"Times New Roman","serif"'>Numerical Implementation of Eq. |
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| 170 | (4) </span></b></p> |
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| 171 | |
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| 172 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><span |
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| 173 | style='font-family:"Times New Roman","serif"'>1)<span style='font:7.0pt "Times New Roman"'> |
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| 174 | </span></span><span style='font-family:"Times New Roman","serif"'>For </span><span |
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| 175 | style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 176 | src="sm_image014.gif"></span>= 0 <span style='font-family: |
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| 177 | "Times New Roman","serif"'>and </span><span style='position:relative; |
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| 178 | top:6.0pt'><img width=28 height=24 src="sm_image015.gif"></span> = |
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| 179 | <span style='font-family:"Times New Roman","serif"'>constant:</span></p> |
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| 180 | |
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| 181 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> |
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| 182 | <img |
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| 183 | src="sm_image016.gif"></p> |
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| 184 | |
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| 185 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 186 | style='font-family:"Times New Roman","serif"'>For discrete q values, at the q |
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| 187 | values from the data points and at the q values extended up to q<sub>N</sub>= |
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| 188 | q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img width=28 |
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| 189 | height=24 src="sm_image011.gif"></span><span |
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| 190 | style='font-family:"Times New Roman","serif"'>, the smeared intensity can be |
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| 191 | calculated approximately,</span></p> |
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| 192 | |
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| 193 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><img |
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| 194 | src="sm_image017.gif">. <span |
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| 195 | style='font-family:"Times New Roman","serif"'>5)</span></p> |
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| 196 | |
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| 197 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 198 | style='position:relative;top:7.0pt'><img width=23 height=25 |
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| 199 | src="sm_image018.gif"></span> <span style='font-family: |
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| 200 | "Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span |
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| 201 | style='font-family:"Times New Roman","serif"'>j < i</span></i><span |
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| 202 | style='font-family:"Times New Roman","serif"'> or<i> j>N-1</i>.</span></p> |
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| 203 | |
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| 204 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 205 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 206 | |
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| 207 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><span |
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| 208 | style='font-family:"Times New Roman","serif"'>2)<span style='font:7.0pt "Times New Roman"'> |
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| 209 | </span></span><span style='font-family:"Times New Roman","serif"'>For </span><span |
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| 210 | style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 211 | src="sm_image014.gif"></span>= <span style='font-family: |
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| 212 | "Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and |
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| 213 | </span><span style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 214 | src="sm_image015.gif"></span> = <span style='font-family: |
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| 215 | "Times New Roman","serif"'>0:</span></p> |
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| 216 | |
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| 217 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 218 | style='font-family:"Times New Roman","serif"'>Similarly to 1), we get</span></p> |
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| 219 | |
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| 220 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> |
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| 221 | <img |
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| 222 | src="sm_image019.gif"> |
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| 223 | <span |
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| 224 | style='font-family:"Times New Roman","serif"'>6)</span></p> |
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| 225 | |
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| 226 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 227 | style='font-family:"Times New Roman","serif"'>for q<sub>p</sub> = q<sub>i</sub> |
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| 228 | - </span><span style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 229 | src="sm_image012.gif"></span><span style='font-family: |
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| 230 | "Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub> |
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| 231 | = q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img |
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| 232 | width=28 height=24 src="sm_image012.gif"></span>. <span |
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| 233 | style='position:relative;top:7.0pt'><img width=23 height=25 |
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| 234 | src="sm_image018.gif"></span> <span style='font-family: |
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| 235 | "Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span |
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| 236 | style='font-family:"Times New Roman","serif"'>j < p</span></i><span |
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| 237 | style='font-family:"Times New Roman","serif"'> or<i> j>N-1</i>.</span></p> |
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| 238 | |
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| 239 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> </p> |
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| 240 | |
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| 241 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in;text-indent:-.25in'><span |
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| 242 | style='font-family:"Times New Roman","serif"'>3)<span style='font:7.0pt "Times New Roman"'> |
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| 243 | </span></span><span style='font-family:"Times New Roman","serif"'>For </span><span |
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| 244 | style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 245 | src="sm_image014.gif"></span>= <span style='font-family: |
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| 246 | "Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and |
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| 247 | </span><span style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 248 | src="sm_image015.gif"></span> = <span style='font-family: |
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| 249 | "Times New Roman","serif"'>constant:</span></p> |
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| 250 | |
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| 251 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 252 | style='font-family:"Times New Roman","serif"'>This case, the best way is to |
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| 253 | perform the integration, Eq. (1), numerically for both slit height and width. |
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| 254 | However, the numerical integration is not correct enough unless given a large |
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| 255 | number of iteration, say at least 10000 by 10000 for each element of the matrix |
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| 256 | W, which will take minutes and minutes to finish the calculation for a set of |
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| 257 | typical SANS data. An alternative way which is correct for slit width << |
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| 258 | slit hight, is used in the SANSView: This method is a mixed method that |
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| 259 | combines the method 1) with the numerical integration for the slit width.</span></p> |
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| 260 | |
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| 261 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> |
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| 262 | </p> |
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| 263 | |
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| 264 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> |
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| 265 | <img |
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| 266 | src="sm_image020.gif"> <span style='font-family: |
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| 267 | "Times New Roman","serif"'>(7)</span></p> |
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| 268 | |
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| 269 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 270 | style='font-family:"Times New Roman","serif"'>for q<sub>p</sub> = q<sub>i</sub> |
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| 271 | - </span><span style='position:relative;top:6.0pt'><img width=28 height=24 |
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| 272 | src="sm_image012.gif"></span><span style='font-family: |
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| 273 | "Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub> |
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| 274 | = q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img |
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| 275 | width=28 height=24 src="sm_image012.gif"></span>. <span |
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| 276 | style='position:relative;top:7.0pt'><img width=23 height=25 |
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| 277 | src="sm_image018.gif"></span> <span style='font-family: |
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| 278 | "Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span |
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| 279 | style='font-family:"Times New Roman","serif"'>j < p</span></i><span |
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| 280 | style='font-family:"Times New Roman","serif"'> or<i> j>N-1</i>. </span></p> |
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| 281 | |
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| 282 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 283 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 284 | |
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| 285 | <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span |
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| 286 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 287 | |
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| 288 | <p class=MsoListParagraphCxSpLast><span style='font-size:14.0pt;line-height: |
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| 289 | 115%;font-family:"Times New Roman","serif"'><a name="Pinhole Smear">Pinhole Smear</a></span></p> |
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| 290 | |
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| 291 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The |
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| 292 | pinhole smearing computation is done similar to the Case 2) above except that |
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| 293 | the weight function used was the Gaussian function, so that the Eq. 6) for this |
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| 294 | case becomes</span></p> |
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| 295 | |
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| 296 | <p class=MsoNormal><img |
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| 297 | src="sm_image021.gif"><span |
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| 298 | style='font-family:"Times New Roman","serif"'> (8)</span></p> |
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| 299 | |
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| 300 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>For all |
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| 301 | the cases above, the weighting matrix <i>W</i> is calculated when the smearing |
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| 302 | is called at the first time, and it includes the ~ 60 q values (finely binned |
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| 303 | evenly) below (>0) and above the q range of data in order to cover all data |
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| 304 | points of the smearing computation for a given model and for a given slit size. |
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| 305 | The <i>Norm</i> factor is found numerically with the weighting matrix, and |
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| 306 | considered on <i>I<sub>s</sub></i> computation.</span></p> |
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| 307 | |
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| 308 | <p class=MsoListParagraphCxSpFirst style='margin-left:.25in'><span |
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| 309 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 310 | |
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| 311 | <p class=MsoListParagraphCxSpLast><span style='font-size:14.0pt;line-height: |
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| 312 | 115%;font-family:"Times New Roman","serif"'><a name="2D Smear">2D Smear</a></span></p> |
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| 313 | |
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| 314 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The |
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| 315 | 2D smearing computation is done similar to the 1D pinhole smearing above |
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| 316 | except that the weight function used was the 2D elliptical Gaussian function</span></p> |
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| 317 | |
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| 318 | <p class=MsoNormal><img |
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| 319 | src="sm_image022.gif"><span |
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| 320 | style='font-family:"Times New Roman","serif"'> (9)</span></p> |
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| 321 | |
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| 322 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>In Eq |
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| 323 | (9), x<sub>0</sub> = qcos</span><span style='font-family:Symbol'>q</span><span |
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| 324 | style='font-family:"Times New Roman","serif"'> and y<sub>0</sub>=qsin</span><span |
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| 325 | style='font-family:Symbol'>q</span><span style='font-family:"Times New Roman","serif"'> |
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| 326 | , and the primed axes are in the coordinate rotated by an angle </span><span |
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| 327 | style='font-family:Symbol'>q</span><span style='font-family:"Times New Roman","serif"'> |
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| 328 | around z-axis (below) so that x<sub>0</sub> = x<sub>0</sub>cos</span><span |
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| 329 | style='font-family:Symbol'>q + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> |
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| 330 | sin</span><span style='font-family:Symbol'>q </span><span style='font-family: |
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| 331 | "Times New Roman","serif"'>and y<sub>0</sub> = -x<sub>0</sub>sin</span><span |
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| 332 | style='font-family:Symbol'>q + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> |
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| 333 | cos</span><span style='font-family:Symbol'>q .</span><span style='font-family: |
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| 334 | "Times New Roman","serif"'> Note that the rotation angle is zero for x-y |
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| 335 | symmetric elliptical Gaussian distribution</span><span style='font-family:Symbol'>. |
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| 336 | </span><span style='font-family:"Times New Roman","serif"'>The A is a |
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| 337 | normalization factor.</span></p> |
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| 338 | |
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| 339 | <p class=MsoNormal align=center style='text-align:center'><span |
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| 340 | style='font-family:"Times New Roman","serif"'><img width=439 height=376 |
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| 341 | id="Object 1" src="sm_image023.gif"></span></p> |
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| 342 | |
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| 343 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 344 | |
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| 345 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Now we |
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| 346 | consider a numerical integration where each bins in </span><span |
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| 347 | style='font-family:Symbol'>Q</span><span style='font-family:"Times New Roman","serif"'> |
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| 348 | and R are <b>evenly </b>(this is to simplify the equation below) distributed by |
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| 349 | </span><span style='font-family:Symbol'>DQ </span><span style='font-family: |
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| 350 | "Times New Roman","serif"'>and </span><span style='font-family:Symbol'>D</span><span |
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| 351 | style='font-family:"Times New Roman","serif"'>R, respectively, and it is |
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| 352 | assumed that I(x, y) is constant within the bins which in turn becomes</span></p> |
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| 353 | |
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| 354 | <p class=MsoNormal><img |
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| 355 | src="sm_image024.gif"></p> |
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| 356 | |
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| 357 | <p class=MsoNormal> <span |
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| 358 | style='font-family:"Times New Roman","serif"'>(10)</span></p> |
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| 359 | |
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| 360 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Since we |
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| 361 | have found the weighting factor on each bin points, it is convenient to |
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| 362 | transform x-y back to x-y coordinate (rotating it by -</span><span |
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| 363 | style='font-family:Symbol'>q</span><span style='font-family:"Times New Roman","serif"'> |
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| 364 | around z axis). Then, for the polar symmetric smear,</span></p> |
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| 365 | |
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| 366 | <p class=MsoNormal><img |
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| 367 | src="sm_image025.gif"><span |
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| 368 | style='position:relative;top:35.0pt'> </span>(11)</p> |
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| 369 | |
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| 370 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p> |
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| 371 | |
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| 372 | <p class=MsoNormal><img |
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| 373 | src="sm_image026.gif"></p> |
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| 374 | |
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| 375 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>while |
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| 376 | for the x-y symmetric smear,</span></p> |
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| 377 | |
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| 378 | <p class=MsoNormal><img |
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| 379 | src="sm_image027.gif"><span |
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| 380 | style='font-family:"Times New Roman","serif"'> (12)</span></p> |
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| 381 | |
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| 382 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p> |
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| 383 | |
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| 384 | <p class=MsoNormal><img |
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| 385 | src="sm_image028.gif"></p> |
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| 386 | |
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| 387 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Here, the |
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| 388 | current version of the SANSVIEW uses the Eq. (11) for 2D smearing assuming that |
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| 389 | all the Gaussian weighting functions are aligned in the polar coordinate. </span></p> |
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| 390 | |
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| 391 | </div> |
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| 392 | |
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| 393 | </body> |
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| 394 | |
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| 395 | </html> |
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