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| 101 | </head> |
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| 102 | |
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| 103 | <body lang=EN-US link=blue vlink=purple> |
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| 104 | |
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| 105 | <div class=Section1> |
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| 106 | |
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| 107 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;line-height:normal'><span |
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| 108 | style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'> |
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| 109 | </span></span><b><span style='font-family:"Times New Roman","serif"'><a |
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| 110 | href="#invariant">Scattering Invariant</a></span></b></p> |
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| 111 | |
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| 112 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;line-height:normal'><span |
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| 113 | style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'> |
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| 114 | </span></span><a href="#volume"><b><span style='font-family:"Times New Roman","serif"'>Volume |
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| 115 | Fraction</span></b></a></p> |
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| 116 | |
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| 117 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;line-height:normal'><span |
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| 118 | style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'> |
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| 119 | </span></span><a href="#surface"><b><span style='font-family:"Times New Roman","serif"'>Specific |
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| 120 | Surface Area</span></b></a></p> |
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| 121 | |
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| 122 | <p class=MsoListParagraphCxSpLast style='text-indent:-.25in;line-height:normal'><span |
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| 123 | style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'> |
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| 124 | </span></span><a href="#howto"><b><span style='font-family:"Times New Roman","serif"'>How |
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| 125 | to Compute</span></b></a></p> |
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| 126 | |
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| 127 | <p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>The |
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| 128 | scattering <a name=invariant>invariant</a> (Q*)</span></b><span |
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| 129 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 130 | |
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| 131 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 132 | |
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| 133 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>The |
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| 134 | scattering invariant (Q*) is a model-independent quantity that can be easily |
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| 135 | calculated from </span></p> |
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| 136 | |
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| 137 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>scattering |
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| 138 | data. For two phase systems, the scattering invariant, Q*, is defined as the |
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| 139 | integral of</span></p> |
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| 140 | |
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| 141 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> the |
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| 142 | square of the wave transfer (q) multiplied by the scattering cross section over |
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| 143 | the full range of q. </span></p> |
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| 144 | |
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| 145 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Q* |
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| 146 | is given by the following equation, </span></p> |
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| 147 | |
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| 148 | <p class=MsoNormal style='line-height:normal'> |
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| 149 | |
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| 150 | <table cellpadding=0 cellspacing=0 align=left> |
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| 151 | <tr> |
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| 152 | <td width=2 height=4></td> |
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| 153 | </tr> |
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| 154 | <tr> |
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| 155 | <td></td> |
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| 156 | <td><img width=127 height=35 src="image001.gif"></td> |
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| 157 | </tr> |
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| 158 | </table> |
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| 159 | |
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| 160 | <span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 161 | |
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| 162 | <p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 163 | |
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| 164 | <br clear=ALL> |
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| 165 | |
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| 166 | <p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>This |
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| 167 | model independent quantity (Q*) is calculated from the scattering data that can |
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| 168 | be used to determine the volume fraction and the specific area of the sample |
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| 169 | under consideration. These quantities are useful in their own right |
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| 170 | and can be used in further analysis. With this scattering invariant module |
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| 171 | users will also be able to determine the consistency of those properties |
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| 172 | between data. There is no real data defined from zero to infinity, there |
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| 173 | usually have limited range. Q* is not really computed from zero to infinity. |
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| 174 | Our maximum q range is 1e-5 ~ 10 (1/A). The lower and/or higher q range than |
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| 175 | data given can be extrapolated by fitting some data nearby.</span></p> |
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| 176 | |
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| 177 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>The |
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| 178 | scattering invariant is computed as follow:</span></p> |
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| 179 | |
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| 180 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span |
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| 181 | style='font-family:"Arial","sans-serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 182 | </span></span><i><span style='font-family:"Times New Roman","serif"'>I(q)</span></i><span |
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| 183 | style='font-family:"Times New Roman","serif"'> = <i>I(q)</i> w/o background : |
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| 184 | If the data includes a background, user sets the value to subtract the background |
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| 185 | for the Q* computation.</span></p> |
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| 186 | |
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| 187 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span |
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| 188 | style='font-family:"Arial","sans-serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 189 | </span></span><span style='font-family:"Times New Roman","serif"'>Reset<i> |
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| 190 | I(q) </i></span><span style='font-family:"Times New Roman","serif"'>= <i>I(q)* |
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| 191 | scaling factor </i>, delta<i>I(q) = </i>delta<i>I(q)*scaling factor</i> : If |
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| 192 | non-zero scaling factor is given, it will be considered.</span></p> |
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| 193 | |
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| 194 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span |
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| 195 | style='font-family:"Arial","sans-serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 196 | </span></span><span style='font-family:"Times New Roman","serif"'> Invariant:</span></p> |
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| 197 | |
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| 198 | <p class=MsoNormal style='margin-left:.25in;line-height:normal'> |
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| 199 | |
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| 200 | <table cellpadding=0 cellspacing=0 align=left> |
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| 201 | <tr> |
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| 202 | <td width=45 height=12></td> |
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| 203 | </tr> |
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| 204 | <tr> |
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| 205 | <td></td> |
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| 206 | <td><img width=127 height=35 src="image001.gif"></td> |
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| 207 | </tr> |
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| 208 | </table> |
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| 209 | |
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| 210 | <span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 211 | |
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| 212 | <br clear=ALL> |
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| 213 | |
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| 214 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 215 | |
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| 216 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 217 | style='font-family:"Times New Roman","serif"'> where, <i>g =q</i> |
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| 218 | for the pinhole geometry and <i>g =<sub> </sub>q<sub>v</sub></i>(the slit |
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| 219 | height) for the slit geometry which can be given in data or as a value.</span></p> |
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| 220 | |
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| 221 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span |
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| 222 | style='font-family:"Arial","sans-serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 223 | </span></span><span style='font-family:"Times New Roman","serif"'> Higher |
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| 224 | q-region (>= qmax in data): </span></p> |
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| 225 | |
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| 226 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> Power |
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| 227 | law (w/o background term) function = C/q<sup>4</sup> will be used </span></p> |
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| 228 | |
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| 229 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 230 | style='font-family:"Times New Roman","serif"'> where the constant |
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| 231 | C(=2pi(delta(rho))S<sub>v</sub>) is to be found by fitting part of data with |
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| 232 | the range of q<sub>N-m</sub> to q<sub>N</sub> (m<N).</span></p> |
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| 233 | |
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| 234 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span |
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| 235 | style='font-family:"Arial","sans-serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 236 | </span></span><span style='font-family:"Times New Roman","serif"'> Lower |
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| 237 | q-region (<= qmin in data): </span></p> |
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| 238 | |
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| 239 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> Guinier |
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| 240 | function = <i>I<sub>0</sub>exp(-R<sub>g</sub><sup>2</sup>q<sup>2</sup>/3)</i> |
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| 241 | where I<sub>0</sub> and R<sub>g</sub> are obtained by fitting, </span></p> |
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| 242 | |
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| 243 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> similarly |
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| 244 | to the high q region above.</span></p> |
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| 245 | |
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| 246 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> Power |
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| 247 | law can also be used.</span></p> |
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| 248 | |
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| 249 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'> </span><b>The |
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| 250 | <a name=volume>volume </a>fraction</b>: </p> |
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| 251 | |
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| 252 | <p class=MsoNormal style='line-height:normal'><span style='position:absolute; |
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| 253 | z-index:3;margin-left:1px;margin-top:9px;width:161px;height:47px'><img |
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| 254 | width=161 height=47 src="image002.gif"></span><span |
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| 255 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 256 | |
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| 257 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 258 | |
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| 259 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>where |
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| 260 | delta(rho) is the SLD contrast of which value is given by users.</span></p> |
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| 261 | |
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| 262 | <p class=MsoNormal style='line-height:normal'><span style='position:absolute; |
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| 263 | z-index:4;margin-left:1px;margin-top:29px;width:101px;height:45px'><img |
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| 264 | width=101 height=45 src="image003.gif"></span><span |
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| 265 | style='font-family:"Times New Roman","serif"'>Thus, </span></p> |
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| 266 | |
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| 267 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 268 | |
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| 269 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 270 | |
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| 271 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> where |
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| 272 | 0 =< <i>A</i> =<1/4 in order for these values to be physically valid.</span></p> |
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| 273 | |
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| 274 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 275 | |
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| 276 | <p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>The |
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| 277 | <a name=surface>specific surface area </a>:</span></b></p> |
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| 278 | |
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| 279 | <p class=MsoNormal style='line-height:normal'> |
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| 280 | |
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| 281 | <table cellpadding=0 cellspacing=0 align=left> |
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| 282 | <tr> |
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| 283 | <td width=1 height=3></td> |
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| 284 | </tr> |
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| 285 | <tr> |
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| 286 | <td></td> |
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| 287 | <td><img width=184 height=47 src="image004.gif"></td> |
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| 288 | </tr> |
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| 289 | </table> |
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| 290 | |
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| 291 | <b><span style='font-family:"Times New Roman","serif"'> </span></b></p> |
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| 292 | |
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| 293 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 294 | |
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| 295 | <br clear=ALL> |
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| 296 | |
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| 297 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> where |
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| 298 | <i>A</i> and <i>Q<sup>*</sup></i> are obtained from previous sections, and the |
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| 299 | Porod constant <i>C<sub>p</sub></i> is given by users. </span></p> |
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| 300 | |
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| 301 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 302 | |
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| 303 | <p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>Definitions:</span></b><span |
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| 304 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 305 | |
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| 306 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Q : |
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| 307 | the magnitude of neutron (or X-ray) momentum transfer vector.</span></p> |
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| 308 | |
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| 309 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>I(Q): |
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| 310 | the scattering intensity as a function of the momentum transfer Q.</span></p> |
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| 311 | |
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| 312 | <p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>Invariant |
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| 313 | total is the sum of the invariant calculated from datas q range and the |
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| 314 | invariant resulting from extrapolation at low q range and at high q range if |
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| 315 | considered.</span></p> |
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| 316 | |
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| 317 | <p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'> </span></p> |
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| 318 | |
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| 319 | <p class=MsoNormal style='line-height:normal'><a name=reference><b><span |
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| 320 | style='font-family:"Times New Roman","serif"'>Reference</span></b></a><b><span |
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| 321 | style='font-family:"Times New Roman","serif"'>: </span></b></p> |
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| 322 | |
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| 323 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Porod's |
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| 324 | chapter (2) from O. Glatter and O. Kratky, "Small Angle X-Ray |
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| 325 | Scattering", Academic Press, New York, 1982</span></p> |
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| 326 | |
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| 327 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Otto's |
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| 328 | web page: </span><a href="http://physchem.kfunigraz.ac.at/sm/"><span |
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| 329 | style='font-family:"Times New Roman","serif"'>http://physchem.kfunigraz.ac.at/sm/</span></a><span |
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| 330 | style='font-family:"Times New Roman","serif"'> </span></p> |
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| 331 | |
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| 332 | <p class=MsoNormal style='line-height:normal'><a name=howto><b><span |
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| 333 | style='font-family:"Times New Roman","serif"'>HowTo</span></b></a><b><span |
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| 334 | style='font-family:"Times New Roman","serif"'>:</span></b></p> |
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| 335 | |
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| 336 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 337 | style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 338 | </span>i)<span style='font:7.0pt "Times New Roman"'> |
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| 339 | </span></span><span style='font-family:"Times New Roman","serif"'>Loading data |
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| 340 | to the panel: Open the data file from File in the menu bar. Select loaded data |
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| 341 | from a plot panel by highlighting that it until its color turns yellow. Then |
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| 342 | right click on that the data and selects the option Compute Invariant. The |
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| 343 | application automatically computes the invariant value if the data loaded is |
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| 344 | valid.</span></p> |
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| 345 | |
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| 346 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 347 | style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 348 | </span>ii)<span style='font:7.0pt "Times New Roman"'> |
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| 349 | </span></span><span style='font-family:"Times New Roman","serif"'>To subtract a |
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| 350 | background or/and to rescale the data, type the values in Customized Input |
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| 351 | box. </span></p> |
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| 352 | |
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| 353 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 354 | style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 355 | </span>iii)<span style='font:7.0pt "Times New Roman"'> |
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| 356 | </span></span><span style='font-family:"Times New Roman","serif"'>If you want |
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| 357 | to calculate the volume fraction and the specific surface area, type the |
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| 358 | optional inputs in the customized input box, and then press Compute button.</span></p> |
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| 359 | |
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| 360 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 361 | style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 362 | </span>iv)<span style='font:7.0pt "Times New Roman"'> |
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| 363 | </span></span><span style='font-family:"Times New Roman","serif"'>The invariant |
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| 364 | can also be calculated including the outside of the data Q range: To include |
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| 365 | the lower Q and/or the higher Q range, check in the enable extrapolation check |
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| 366 | box in Extrapolation box. If the power low is chosen, the power (exponent) |
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| 367 | can be either held or fitted by checking the corresponding radio button. The |
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| 368 | Npts that is being used for the extrapolation can be specified.</span></p> |
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| 369 | |
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| 370 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 371 | style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 372 | </span>v)<span style='font:7.0pt "Times New Roman"'> |
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| 373 | </span></span><span style='font-family:"Times New Roman","serif"'>If the |
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| 374 | invariant calculated from the extrapolated region is too large, it will be warn |
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| 375 | in red at the top of the panel, which means that your data is not proper to |
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| 376 | calculate the invariant.</span></p> |
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| 377 | |
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| 378 | <p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span |
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| 379 | style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'> |
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| 380 | </span>vi)<span style='font:7.0pt "Times New Roman"'> |
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| 381 | </span></span><span style='font-family:"Times New Roman","serif"'>The details |
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| 382 | of the calculation is available by clicking the Details button in the middle |
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| 383 | of the panel. </span></p> |
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| 384 | |
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| 385 | <p class=MsoNormal align=center style='text-align:center;line-height:normal'><span |
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| 386 | style='font-family:"Times New Roman","serif"'><img border=0 width=572 |
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| 387 | height=449 id="Object 4" src="image005.gif"></span></p> |
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| 388 | |
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| 389 | <p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </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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