Ignore:
Timestamp:
Apr 21, 2010 2:35:06 PM (15 years ago)
Author:
Jae Cho <jhjcho@…>
Branches:
master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, costrafo411, magnetic_scatt, release-4.1.1, release-4.1.2, release-4.2.2, release_4.0.1, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
Children:
4834cba
Parents:
d10c217
Message:

new help docs

File:
1 edited

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  • invariantview/media/invariant_help.html

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     94 ol 
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    39596ul 
    … …  
    39798--> 
    39899</style> 
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    402  table.MsoNormalTable 
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     100 
    419101</head> 
    420102 
    421 <body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'> 
     103<body lang=EN-US link=blue vlink=purple> 
    422104 
    423105<div class=Section1> 
    424 <ul> 
    425 <li><a class=section1 style='mso-outline-level:1' href="#Scattering Invariant">Scattering Invariant</a></li> 
    426 <li><a class=section1 style='mso-outline-level:1' href="#Volume Fraction">Volume Fraction</a></li> 
    427 <li><a class=section1 style='mso-outline-level:1' href="#Surface Area">Specific Surface Area</a></li> 
    428 <li><a class=section1 style='mso-outline-level:1'href="#How to Compute Invariant?">How to Compute Invariant?</a></li> 
    429 </ul> 
    430 <p class=section1 style='margin-left:26.1pt'><o:p>&nbsp;</o:p></p> 
    431  
    432 <a name="Scattering Invariant" class=section1><b style='mso-bidi-font-weight:normal'>The Scattering Invariant<o:p></o:p></b></a> 
    433  
    434 <p class=section1><o:p>&nbsp;</o:p></p> 
    435  
    436 <p class=section1>The scattering invariant (Q*) is a model-independent quantity 
    437 that can be easily calculated from scattering data. For two phase systems, the 
    438 scattering invariant, Q*, is defined as the integral of the square of the wave 
    439 transfer (q) multiplied by the scattering cross section over the full range of 
    440 q. </p> 
    441  
    442 <p class=section1>Q* is given by the following equation<span class=GramE>, </span><sub><!--[if gte vml 1]><v:shapetype 
    443  id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" 
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    456   <v:f eqn="sum @8 21600 0"/> 
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    462 </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:96pt; 
    463  height:26.25pt' o:ole=""> 
    464  <v:imagedata src="../media/image001.wmz" o:title=""/> 
    465 </v:shape><![endif]--><![if !vml]><img width=128 height=35 
    466 src="../media/image047.gif" v:shapes="_x0000_i1025"><![endif]><!--[if gte mso 9]><xml> 
    467  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1025" 
    468   DrawAspect="Content" ObjectID="_1333134530"> 
    469  </o:OLEObject> 
    470 </xml><![endif]--></sub><span style='font-family:TimesNewRoman;mso-bidi-font-family: 
    471 TimesNewRoman'><span style='mso-spacerun:yes'> </span></span>. This model 
    472 independent quantity (Q*) is calculated from the scattering data can be used to 
    473 determine the volume fraction and the specific area of the sample under 
    474 consideration. &nbsp;These quantities&nbsp;are useful in their own right and 
    475 can be used in&nbsp;further analysis. With this scattering invariant module 
     106 
     107<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;line-height:normal'><span 
     108style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     109</span></span><b><span style='font-family:"Times New Roman","serif"'><a 
     110href="#invariant">Scattering Invariant</a></span></b></p> 
     111 
     112<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;line-height:normal'><span 
     113style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     114</span></span><a href="#volume"><b><span style='font-family:"Times New Roman","serif"'>Volume 
     115Fraction</span></b></a></p> 
     116 
     117<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;line-height:normal'><span 
     118style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     119</span></span><a href="#surface"><b><span style='font-family:"Times New Roman","serif"'>Specific 
     120Surface Area</span></b></a></p> 
     121 
     122<p class=MsoListParagraphCxSpLast style='text-indent:-.25in;line-height:normal'><span 
     123style='font-family:Symbol'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     124</span></span><a href="#howto"><b><span style='font-family:"Times New Roman","serif"'>How 
     125to Compute</span></b></a></p> 
     126 
     127<p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>The 
     128scattering <a name=invariant>invariant</a> (Q*)</span></b><span 
     129style='font-family:"Times New Roman","serif"'> </span></p> 
     130 
     131<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     132 
     133<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>The 
     134scattering invariant (Q*) is a model-independent quantity that can be easily 
     135calculated from </span></p> 
     136 
     137<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>scattering 
     138data. For two phase systems, the scattering invariant, Q*, is defined as the 
     139integral of</span></p> 
     140 
     141<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> the 
     142square of the wave transfer (q) multiplied by the scattering cross section over 
     143the full range of q. </span></p> 
     144 
     145<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Q* 
     146is given by the following equation, </span></p> 
     147 
     148<p class=MsoNormal style='line-height:normal'> 
     149 
     150<table cellpadding=0 cellspacing=0 align=left> 
     151 <tr> 
     152  <td width=2 height=4></td> 
     153 </tr> 
     154 <tr> 
     155  <td></td> 
     156  <td><img width=127 height=35 src="image001.gif"></td> 
     157 </tr> 
     158</table> 
     159 
     160<span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     161 
     162<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     163 
     164<br clear=ALL> 
     165 
     166<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>This 
     167model independent quantity (Q*) is calculated from the scattering data that can 
     168be used to determine the volume fraction and the specific area of the sample 
     169under consideration. &nbsp;These quantities&nbsp;are useful in their own right 
     170and can be used in&nbsp;further analysis. With this scattering invariant module 
    476171users will also be able to determine the consistency of those properties 
    477172between data.&nbsp; There is no real data defined from zero to infinity, there 
    478173usually have limited range. Q* is not really computed from zero to infinity. 
    479 Methods to deal with this limitation will be addressed.</p> 
    480  
    481 <p class=MsoNormal style='margin-top:0in;margin-right:0in;margin-bottom:0in; 
    482 margin-left:.25in;margin-bottom:.0001pt;line-height:normal'><span 
    483 style='font-size:12.0pt;font-family:"Times New Roman"'>&nbsp;<o:p></o:p></span></p> 
    484  
    485 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;text-indent: 
    486 22.5pt;line-height:normal'><span style='font-size:12.0pt;font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> 
    487  
    488 <p class=ListParagraphCxSpFirst style='margin-left:0in;mso-add-space:auto'><span 
    489 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>The 
    490 scattering invariant is derived as follow:<o:p></o:p></span></p> 
    491  
    492 <p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l13 level1 lfo10; 
    493 tab-stops:list .5in'><![if !supportLists]><span style='font-size:12.0pt; 
    494 line-height:115%;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: 
    495 Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    496 </span></span></span><![endif]><i style='mso-bidi-font-style:normal'><span 
    497 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>I(q)</span></i><span 
    498 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> =<span 
    499 style='mso-spacerun:yes'>  </span><i style='mso-bidi-font-style:normal'>I(q)</i>- 
    500 background for data without background<o:p></o:p></span></p> 
    501  
    502 <p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l8 level1 lfo11; 
    503 tab-stops:list .5in'><![if !supportLists]><span style='font-size:12.0pt; 
    504 line-height:115%;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: 
    505 Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    506 </span></span></span><![endif]><i style='mso-bidi-font-style:normal'><span 
    507 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>I(q) </span></i><span 
    508 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>= <i 
    509 style='mso-bidi-font-style:normal'>I(q)* scaling factor </i>, </span><span 
    510 lang=IT style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"; 
    511 mso-ansi-language:IT'><sub><!--[if gte vml 1]><v:shape id="_x0000_i1026" 
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    513  <v:imagedata src="../media/image003.wmz" o:title=""/> 
    514 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    515 src="../media/image048.gif" v:shapes="_x0000_i1026"><![endif]></sub><!--[if gte mso 9]><xml> 
    516  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1026" 
    517   DrawAspect="Content" ObjectID="_1333134531"> 
    518  </o:OLEObject> 
    519 </xml><![endif]--></span><i style='mso-bidi-font-style:normal'><span 
    520 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>I(q) = </span></i><span 
    521 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><span 
    522 style='mso-spacerun:yes'> </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1027" 
    523  type="#_x0000_t75" style='width:11.25pt;height:14.25pt' o:ole=""> 
    524  <v:imagedata src="../media/image005.wmz" o:title=""/> 
    525 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    526 src="../media/image048.gif" v:shapes="_x0000_i1027"><![endif]></sub><!--[if gte mso 9]><xml> 
    527  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1027" 
    528   DrawAspect="Content" ObjectID="_1333134532"> 
    529  </o:OLEObject> 
    530 </xml><![endif]--><i style='mso-bidi-font-style:normal'>I(q)*scaling factor</i><o:p></o:p></span></p> 
    531  
    532 <p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l8 level1 lfo11; 
    533 tab-stops:list .5in'><![if !supportLists]><span style='font-size:12.0pt; 
    534 line-height:115%;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: 
    535 Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    536 </span></span></span><![endif]><sub><span style='font-size:12.0pt;line-height: 
    537 115%;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape id="_x0000_i1028" 
    538  type="#_x0000_t75" style='width:96pt;height:26.25pt' o:ole=""> 
    539  <v:imagedata src="../media/image001.wmz" o:title=""/> 
    540 </v:shape><![endif]--><![if !vml]><img width=128 height=35 
    541 src="../media/image047.gif" v:shapes="_x0000_i1028"><![endif]><!--[if gte mso 9]><xml> 
    542  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1028" 
    543   DrawAspect="Content" ObjectID="_1333134533"> 
    544  </o:OLEObject> 
    545 </xml><![endif]--></span></sub><span style='font-size:12.0pt;line-height:115%; 
    546 font-family:"Times New Roman"'><span style='mso-spacerun:yes'> </span><span 
    547 class=GramE>where</span> g =q for the pinhole geometry and g =<sub><!--[if gte vml 1]><v:shape 
    548  id="_x0000_i1029" type="#_x0000_t75" style='width:11.25pt;height:12.75pt' 
    549  o:ole=""> 
    550  <v:imagedata src="../media/image006.wmz" o:title=""/> 
    551 </v:shape><![endif]--><![if !vml]><img width=15 height=17 
    552 src="../media/image049.gif" v:shapes="_x0000_i1029"><![endif]></sub><!--[if gte mso 9]><xml> 
    553  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1029" 
    554   DrawAspect="Content" ObjectID="_1333134534"> 
    555  </o:OLEObject> 
    556 </xml><![endif]--><span class=SpellE>q<sub>v</sub></span>(the slit height) for 
    557 the slit geometry which can be given in data or as a value.<o:p></o:p></span></p> 
    558  
    559 <p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l6 level1 lfo3'><![if !supportLists]><span 
    560 style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family: 
    561 Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    562 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    563 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    564 font-family:"Times New Roman"'>Numerical approach: <sub><!--[if gte vml 1]><v:shape 
    565  id="_x0000_i1030" type="#_x0000_t75" style='width:87pt;height:33.75pt' o:ole=""> 
    566  <v:imagedata src="../media/image008.wmz" o:title=""/> 
    567 </v:shape><![endif]--><![if !vml]><img width=116 height=45 
    568 src="../media/image050.gif" v:shapes="_x0000_i1030"><![endif]><!--[if gte mso 9]><xml> 
    569  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1030" 
    570   DrawAspect="Content" ObjectID="_1333134535"> 
    571  </o:OLEObject> 
    572 </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>where N = number 
    573 of the data points, <span class=SpellE>q<sub>i</sub></span><sub> </sub>= wave transfer, 
    574 I<sub>i</sub> = Scattering intensity at <span class=SpellE>q<sub>i</sub></span> 
    575 , and <sub><!--[if gte vml 1]><v:shape id="_x0000_i1031" type="#_x0000_t75" 
    576  style='width:11.25pt;height:14.25pt' o:ole=""> 
    577  <v:imagedata src="../media/image010.wmz" o:title=""/> 
    578 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    579 src="invariant_help_files/image048.gif" v:shapes="_x0000_i1031"><![endif]></sub><!--[if gte mso 9]><xml> 
    580  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1031" 
    581   DrawAspect="Content" ObjectID="_1333134536"> 
    582  </o:OLEObject> 
    583 </xml><![endif]--><span class=SpellE>q<sub>i</sub></span><sub> </sub><span 
    584 style='mso-spacerun:yes'> </span>= (q<sub>i+1</sub> – q<sub>i-1</sub>)/2 (; 
    585 note that<span style='mso-spacerun:yes'>  </span><sub><!--[if gte vml 1]><v:shape 
    586  id="_x0000_i1032" type="#_x0000_t75" style='width:11.25pt;height:14.25pt' 
    587  o:ole=""> 
    588  <v:imagedata src="../media/image011.wmz" o:title=""/> 
    589 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    590 src="../media/image048.gif" v:shapes="_x0000_i1032"><![endif]></sub><!--[if gte mso 9]><xml> 
    591  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1032" 
    592   DrawAspect="Content" ObjectID="_1333134537"> 
    593  </o:OLEObject> 
    594 </xml><![endif]-->q<sub>0 </sub>= (q<sub>1</sub>- q<sub>0</sub>)/2<span 
    595 style='mso-spacerun:yes'>  </span>and <sub><span 
    596 style='mso-spacerun:yes'> </span></sub><span style='mso-spacerun:yes'> </span><sub><!--[if gte vml 1]><v:shape 
    597  id="_x0000_i1033" type="#_x0000_t75" style='width:11.25pt;height:14.25pt' 
    598  o:ole=""> 
    599  <v:imagedata src="../media/image012.wmz" o:title=""/> 
    600 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    601 src="../media/image048.gif" v:shapes="_x0000_i1033"><![endif]></sub><!--[if gte mso 9]><xml> 
    602  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1033" 
    603   DrawAspect="Content" ObjectID="_1333134538"> 
    604  </o:OLEObject> 
    605 </xml><![endif]--><span class=SpellE>q<sub>N</sub></span><sub> </sub><span 
    606 style='mso-spacerun:yes'> </span>=<sub><!--[if gte vml 1]><v:shape id="_x0000_i1034" 
    607  type="#_x0000_t75" style='width:11.25pt;height:14.25pt' o:ole=""> 
    608  <v:imagedata src="../media/image013.wmz" o:title=""/> 
    609 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    610 src="../media/image048.gif" v:shapes="_x0000_i1034"><![endif]></sub><!--[if gte mso 9]><xml> 
    611  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1034" 
    612   DrawAspect="Content" ObjectID="_1333134539"> 
    613  </o:OLEObject> 
    614 </xml><![endif]-->q<sub>N-1</sub> /2).<o:p></o:p></span></p> 
    615  
    616 <p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l6 level1 lfo3'><![if !supportLists]><span 
    617 style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family: 
    618 Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    619 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    620 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    621 font-family:"Times New Roman"'>Uncertainty:<span style='mso-spacerun:yes'>  
    622 </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1035" type="#_x0000_t75" 
    623  style='width:137.25pt;height:38.25pt' o:ole=""> 
    624  <v:imagedata src="../media/image014.wmz" o:title=""/> 
    625 </v:shape><![endif]--><![if !vml]><img width=183 height=51 
    626 src="../media/image051.gif" v:shapes="_x0000_i1035"><![endif]><!--[if gte mso 9]><xml> 
    627  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1035" 
    628   DrawAspect="Content" ObjectID="_1333134540"> 
    629  </o:OLEObject> 
    630 </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>assuming that 
    631 there is no correlation between each <sub><!--[if gte vml 1]><v:shape id="_x0000_i1036" 
    632  type="#_x0000_t75" style='width:11.25pt;height:14.25pt' o:ole=""> 
    633  <v:imagedata src="../media/image016.wmz" o:title=""/> 
    634 </v:shape><![endif]--><![if !vml]><img width=15 height=19 
    635 src="../media/image048.gif" v:shapes="_x0000_i1036"><![endif]></sub><!--[if gte mso 9]><xml> 
    636  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1036" 
    637   DrawAspect="Content" ObjectID="_1333134541"> 
    638  </o:OLEObject> 
    639 </xml><![endif]-->I<sub>i</sub> (uncertainty of I, say <span class=SpellE><span 
    640 class=GramE>dy</span></span>).<o:p></o:p></span></p> 
    641  
    642 <p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l6 level1 lfo3'><![if !supportLists]><span 
    643 style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family: 
    644 Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    645 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    646 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    647 font-family:"Times New Roman"'>Higher q-region (&gt;= <span class=SpellE>qmax</span> 
    648 of data): <o:p></o:p></span></p> 
    649  
    650 <p class=ListParagraphCxSpMiddle style='margin-left:63.9pt;mso-add-space:auto'><span 
    651 class=SpellE><span style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Porod</span></span><span 
    652 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> (w/o 
    653 background term) function = C/q<sup>4</sup> will be used where the constant <span 
    654 class=GramE>C(</span>=2<sub><!--[if gte vml 1]><v:shape id="_x0000_i1037" 
    655  type="#_x0000_t75" style='width:24pt;height:15.75pt' o:ole=""> 
    656  <v:imagedata src="../media/image017.wmz" o:title=""/> 
    657 </v:shape><![endif]--><![if !vml]><img width=32 height=21 
    658 src="../media/image052.gif" v:shapes="_x0000_i1037"><![endif]></sub><!--[if gte mso 9]><xml> 
    659  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1037" 
    660   DrawAspect="Content" ObjectID="_1333134542"> 
    661  </o:OLEObject> 
    662 </xml><![endif]--><span class=SpellE>S<sub>v</sub></span>) is to be found by 
    663 fitting part of data with the range of <span class=SpellE>q<sub>N</sub></span><sub>-m</sub> 
    664 to <span class=SpellE>q<sub>N</sub></span> (m&lt;N).<o:p></o:p></span></p> 
    665  
    666 <p class=ListParagraphCxSpMiddle style='margin-left:0in;mso-add-space:auto'><span 
    667 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>The 
    668 contribution to Q<sup>*</sup> from this region is described in an explicit 
    669 form, and thus numerical integration is not required (w/ no background is 
    670 assumed): <o:p></o:p></span></p> 
    671  
    672 <p class=ListParagraphCxSpMiddle style='margin-left:99.9pt;mso-add-space:auto; 
    673 text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span 
    674 style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family: 
    675 Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span 
    676 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    677 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    678 font-family:"Times New Roman"'>For pinhole: <sub><!--[if gte vml 1]><v:shape 
    679  id="_x0000_i1038" type="#_x0000_t75" style='width:180pt;height:27.75pt' o:ole=""> 
    680  <v:imagedata src="../media/image019.wmz" o:title=""/> 
    681 </v:shape><![endif]--><![if !vml]><img width=240 height=37 
    682 src="../media/image053.gif" v:shapes="_x0000_i1038"><![endif]><!--[if gte mso 9]><xml> 
    683  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1038" 
    684   DrawAspect="Content" ObjectID="_1333134543"> 
    685  </o:OLEObject> 
    686 </xml><![endif]--></sub>,<o:p></o:p></span></p> 
    687  
    688 <p class=ListParagraphCxSpMiddle style='margin-left:99.9pt;mso-add-space:auto; 
    689 text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span 
    690 style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family: 
    691 Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span 
    692 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    693 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    694 font-family:"Times New Roman"'>(Also numerical integration can be considered 
    695 for this case).<o:p></o:p></span></p> 
    696  
    697 <p class=ListParagraphCxSpMiddle style='margin-left:99.9pt;mso-add-space:auto; 
    698 text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span 
    699 style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family: 
    700 Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span 
    701 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    702 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    703 font-family:"Times New Roman"'>For slit smear: <sub><!--[if gte vml 1]><v:shape 
    704  id="_x0000_i1039" type="#_x0000_t75" style='width:216.75pt;height:38.25pt' 
    705  o:ole=""> 
    706  <v:imagedata src="../media/image021.wmz" o:title=""/> 
    707 </v:shape><![endif]--><![if !vml]><img width=289 height=51 
    708 src="../media/image054.gif" v:shapes="_x0000_i1039"><![endif]><!--[if gte mso 9]><xml> 
    709  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1039" 
    710   DrawAspect="Content" ObjectID="_1333134545"> 
    711  </o:OLEObject> 
    712 </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>where </span><span 
    713 style='font-size:12.0pt;line-height:115%;font-family:Symbol'>D</span><span 
    714 class=SpellE><span style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>q<sub>v</sub></span></span><span 
    715 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> is 
    716 given in data, <span class=SpellE>q<sub>N</sub></span> = maximum q (<span 
    717 class=SpellE>q<sub>max</sub></span>) in <span class=GramE>data<span 
    718 style='mso-spacerun:yes'>  </span>and</span> C is obtained by the fitting data 
    719 in larger q region as described above.<o:p></o:p></span></p> 
    720  
    721 <p class=ListParagraphCxSpMiddle style='margin-left:99.9pt;mso-add-space:auto; 
    722 text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span 
    723 style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family: 
    724 Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span 
    725 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    726 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    727 font-family:"Times New Roman"'>Note that the number of points for fitting is to 
    728 be given by users and should be larger than 2 from the point of <span 
    729 class=SpellE>qmax</span>.<o:p></o:p></span></p> 
    730  
    731 <p class=ListParagraphCxSpMiddle style='margin-left:.95in;mso-add-space:auto; 
    732 text-indent:-.25in;mso-list:l12 level1 lfo5'><![if !supportLists]><span 
    733 style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family: 
    734 Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    735 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    736 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    737 font-family:"Times New Roman"'>Lower q-region:<o:p></o:p></span></p> 
    738  
    739 <p class=ListParagraphCxSpMiddle style='margin-left:.95in;mso-add-space:auto'><span 
    740 class=SpellE><span style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Guinier</span></span><span 
    741 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> 
    742 function = <span class=GramE><i style='mso-bidi-font-style:normal'>I<sub>0</sub>exp(</i></span><i 
    743 style='mso-bidi-font-style:normal'>-R<sub>g</sub><sup>2</sup>q<sup>2</sup>/3)</i> 
    744 where I<sub>0</sub> and <span class=SpellE>R<sub>g</sub></span> are obtained by 
    745 fitting, similarly to the high q region above.<o:p></o:p></span></p> 
    746  
    747 <p class=ListParagraphCxSpMiddle style='margin-left:1.45in;mso-add-space:auto; 
    748 text-indent:-.25in;mso-list:l14 level1 lfo6'><![if !supportLists]><span 
    749 style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family: 
    750 Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span 
    751 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    752 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    753 font-family:"Times New Roman"'>For pinhole: <sub><!--[if gte vml 1]><v:shape 
    754  id="_x0000_i1040" type="#_x0000_t75" style='width:158.25pt;height:27.75pt' 
    755  o:ole=""> 
    756  <v:imagedata src="../media/image023.wmz" o:title=""/> 
    757 </v:shape><![endif]--><![if !vml]><img width=211 height=37 
    758 src="../media/image055.gif" v:shapes="_x0000_i1040"><![endif]><!--[if gte mso 9]><xml> 
    759  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1040" 
    760   DrawAspect="Content" ObjectID="_1333134546"> 
    761  </o:OLEObject> 
    762 </xml><![endif]--></sub><o:p></o:p></span></p> 
    763  
    764 <p class=ListParagraphCxSpMiddle style='margin-left:1.45in;mso-add-space:auto'><span 
    765 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Numerically<span 
    766 style='mso-spacerun:yes'>  </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1041" 
    767  type="#_x0000_t75" style='width:158.25pt;height:33.75pt' o:ole=""> 
    768  <v:imagedata src="../media/image025.wmz" o:title=""/> 
    769 </v:shape><![endif]--><![if !vml]><img width=211 height=45 
    770 src="../media/image056.gif" v:shapes="_x0000_i1041"><![endif]><!--[if gte mso 9]><xml> 
    771  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1041" 
    772   DrawAspect="Content" ObjectID="_1333134547"> 
    773  </o:OLEObject> 
    774 </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>where <span 
    775 class=SpellE>q<sub>N</sub></span> &lt;= <span class=SpellE>q<sub>min</sub></span> 
    776 in data points, q<sub>0</sub> will be set to 1e-04 (1/A) and N =1000(<span 
    777 class=GramE>?,</span> linear binning?).<o:p></o:p></span></p> 
    778  
    779 <p class=ListParagraphCxSpMiddle style='margin-left:1.45in;mso-add-space:auto; 
    780 text-indent:-.25in;mso-list:l14 level1 lfo6'><![if !supportLists]><span 
    781 style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family: 
    782 Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span 
    783 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    784 </span></span></span><![endif]><span style='font-size:12.0pt;line-height:115%; 
    785 font-family:"Times New Roman"'>For slit smear: <sub><!--[if gte vml 1]><v:shape 
    786  id="_x0000_i1042" type="#_x0000_t75" style='width:161.25pt;height:27.75pt' 
    787  o:ole=""> 
    788  <v:imagedata src="../media/image027.wmz" o:title=""/> 
    789 </v:shape><![endif]--><![if !vml]><img width=215 height=37 
    790 src="../media/image057.gif" v:shapes="_x0000_i1042"><![endif]><!--[if gte mso 9]><xml> 
    791  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1042" 
    792   DrawAspect="Content" ObjectID="_1333134548"> 
    793  </o:OLEObject> 
    794 </xml><![endif]--></sub><o:p></o:p></span></p> 
    795  
    796 <p class=ListParagraphCxSpLast style='margin-left:207.9pt;mso-add-space:auto'><sub><span 
    797 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape 
    798  id="_x0000_i1043" type="#_x0000_t75" style='width:117pt;height:36.75pt' o:ole=""> 
    799  <v:imagedata src="../media/image029.wmz" o:title=""/> 
    800 </v:shape><![endif]--><![if !vml]><img width=156 height=49 
    801 src="../media/image058.gif" v:shapes="_x0000_i1043"><![endif]><!--[if gte mso 9]><xml> 
    802  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1043" 
    803   DrawAspect="Content" ObjectID="_1333134549"> 
    804  </o:OLEObject> 
    805 </xml><![endif]--></span></sub><span style='font-size:12.0pt;line-height:115%; 
    806 font-family:"Times New Roman"'><span style='mso-spacerun:yes'>  </span><span 
    807 class=GramE>where</span> <span class=SpellE>q<sub>min</sub></span> is the 
    808 smallest q value in data points.<o:p></o:p></span></p> 
    809  
    810 <p class=MsoNormal><span style='font-size:12.0pt;line-height:115%;font-family: 
    811 "Times New Roman"'><span style='mso-tab-count:3'>                                    </span>(Also 
    812 numerical integration can be considered for this case)<o:p></o:p></span></p> 
    813  
    814 <p class=section1><o:p>&nbsp;</o:p></p> 
    815 <a name="Volume Fraction" class=ListParagraphCxSpFirst style='margin-left:0in;mso-add-space:auto'><b 
    816 style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height: 
    817 115%;font-family:"Times New Roman"'>Volume <span class=GramE>Fraction<span 
    818 style='font-weight:normal'> </span></span></span></b><span style='font-size: 
    819 12.0pt;line-height:115%;font-family:"Times New Roman"'><sub><!--[if gte vml 1]><v:shape 
    820  id="_x0000_i1044" type="#_x0000_t75" style='width:9.75pt;height:15.75pt' 
    821  o:ole=""> 
    822  <v:imagedata src="../media/image031.wmz" o:title=""/> 
    823 </v:shape><![endif]--><![if !vml]><img width=13 height=21 
    824 src="../media/image059.gif" v:shapes="_x0000_i1044"><![endif]></sub><!--[if gte mso 9]><xml> 
    825  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1044" 
    826   DrawAspect="Content" ObjectID="_1333134550"> 
    827  </o:OLEObject> 
    828 </xml><![endif]-->: <o:p></o:p></span></a> 
    829  
    830 <p class=ListParagraphCxSpMiddle style='margin-left:27.9pt;mso-add-space:auto'><sub><span 
    831 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape 
    832  id="_x0000_i1045" type="#_x0000_t75" style='width:125.25pt;height:36pt' o:ole=""> 
    833  <v:imagedata src="../media/image033.wmz" o:title=""/> 
    834 </v:shape><![endif]--><![if !vml]><img width=167 height=48 
    835 src="../media/image060.gif" v:shapes="_x0000_i1045"><![endif]><!--[if gte mso 9]><xml> 
    836  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1045" 
    837   DrawAspect="Content" ObjectID="_1333134551"> 
    838  </o:OLEObject> 
    839 </xml><![endif]--></span></sub><span style='font-size:12.0pt;line-height:115%; 
    840 font-family:"Times New Roman"'><span style='mso-spacerun:yes'>  </span><span 
    841 class=GramE>where</span> <sub><!--[if gte vml 1]><v:shape id="_x0000_i1046" 
    842  type="#_x0000_t75" style='width:18.75pt;height:15.75pt' o:ole=""> 
    843  <v:imagedata src="../media/image035.wmz" o:title=""/> 
    844 </v:shape><![endif]--><![if !vml]><img width=25 height=21 
    845 src="../media/image061.gif" v:shapes="_x0000_i1046"><![endif]></sub><!--[if gte mso 9]><xml> 
    846  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1046" 
    847   DrawAspect="Content" ObjectID="_1333134552"> 
    848  </o:OLEObject> 
    849 </xml><![endif]-->is the SLD contrast of which value is given by users.<o:p></o:p></span></p> 
    850  
    851 <p class=ListParagraphCxSpMiddle style='margin-left:27.9pt;mso-add-space:auto'><span 
    852 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Thus,<span 
    853 style='mso-spacerun:yes'>  </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1047" 
    854  type="#_x0000_t75" style='width:78pt;height:33.75pt' o:ole=""> 
    855  <v:imagedata src="../media/image037.wmz" o:title=""/> 
    856 </v:shape><![endif]--><![if !vml]><img width=104 height=45 
    857 src="../media/image062.gif" v:shapes="_x0000_i1047"><![endif]><!--[if gte mso 9]><xml> 
    858  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1047" 
    859   DrawAspect="Content" ObjectID="_1333134553"> 
    860  </o:OLEObject> 
    861 </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>where 0 =&lt; A 
    862 =&lt;1/4 in order for these values to be physically valid.<o:p></o:p></span></p> 
    863  
    864 <p class=ListParagraphCxSpMiddle style='margin-left:27.9pt;mso-add-space:auto'><span 
    865 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> 
    866  
    867 <a name="Surface Area" class=ListParagraphCxSpMiddle style='margin-left:0in;mso-add-space:auto'><b 
    868 style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height: 
    869 115%;font-family:"Times New Roman"'>Specific Surface Area</span> <sub><!--[if gte vml 1]><v:shape 
    870  id="_x0000_i1048" type="#_x0000_t75" style='width:14.25pt;height:18pt' o:ole=""> 
    871  <v:imagedata src="../media/image039.wmz" o:title=""/> 
    872 </v:shape><![endif]--><![if !vml]><img width=19 height=24 
    873 src="../media/image063.gif" v:shapes="_x0000_i1048"><![endif]></sub><!--[if gte mso 9]><xml> 
    874  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1048" 
    875   DrawAspect="Content" ObjectID="_1333134554"> 
    876  </o:OLEObject> 
    877 </xml><![endif]--></b><b style='mso-bidi-font-weight:normal'><span 
    878 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><o:p></o:p></span></b></a> 
    879  
    880 <p class=ListParagraphCxSpLast style='margin-left:27.9pt;mso-add-space:auto'><sub><span 
    881 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape 
    882  id="_x0000_i1049" type="#_x0000_t75" style='width:143.25pt;height:36pt' o:ole=""> 
    883  <v:imagedata src="../media/image041.wmz" o:title=""/> 
    884 </v:shape><![endif]--><![if !vml]><img width=191 height=48 
    885 src="../media/image064.gif" v:shapes="_x0000_i1049"><![endif]><!--[if gte mso 9]><xml> 
    886  <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1049" 
    887   DrawAspect="Content" ObjectID="_1333134555"> 
    888  </o:OLEObject> 
    889 </xml><![endif]--></span></sub><span style='font-size:12.0pt;line-height:115%; 
    890 font-family:"Times New Roman"'><span style='mso-spacerun:yes'> </span><span 
    891 class=GramE>where</span> A and Q<sup>*</sup> are obtained from 1) and 2), and 
    892 the <span class=SpellE>Porod</span> constant C<sub>p</sub> is given by users. <o:p></o:p></span></p> 
    893  
    894 <p class=MsoNormal style='mso-margin-top-alt:auto;line-height:normal; 
    895 mso-outline-level:1'><span style='font-size:12.0pt;font-family:"Times New Roman"; 
    896 mso-bidi-font-weight:bold'>Definitions<o:p></o:p></span></p> 
    897  
    898 <p class=MsoNormal style='margin-left:45.9pt;text-indent:-.25in;line-height: 
    899 normal;mso-list:l3 level1 lfo8;tab-stops:list 45.9pt'><![if !supportLists]><span 
    900 style='font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; 
    901 mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    902 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    903 </span></span></span><![endif]><span class=GramE><span style='font-size:12.0pt; 
    904 font-family:"Times New Roman"'>Q&nbsp;:</span></span><span style='font-size: 
    905 12.0pt;font-family:"Times New Roman"'> the magnitude of neutron (or X-ray) momentum 
    906 transfer vector.<o:p></o:p></span></p> 
    907  
    908 <p class=MsoNormal style='mso-margin-bottom-alt:auto;margin-left:45.9pt; 
    909 text-indent:-.25in;line-height:normal;mso-list:l3 level1 lfo8;tab-stops:list 45.9pt'><![if !supportLists]><span 
    910 style='font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; 
    911 mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    912 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    913 </span></span></span><![endif]><span class=GramE><span style='font-size:12.0pt; 
    914 font-family:"Times New Roman"'>I(</span></span><span style='font-size:12.0pt; 
    915 font-family:"Times New Roman"'>Q): the scattering intensity as a function of 
    916 the momentum transfer Q.<o:p></o:p></span></p> 
    917  
    918 <p class=ListParagraph style='margin-left:45.9pt;mso-add-space:auto;text-indent: 
    919 -.25in;mso-list:l3 level1 lfo8;tab-stops:list 45.9pt'><![if !supportLists]><span 
    920 style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family: 
    921 Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span 
    922 style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    923 </span></span></span><![endif]><span class=GramE><span style='font-size:12.0pt; 
    924 line-height:115%;font-family:"Times New Roman"'>the</span></span><span 
    925 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> 
    926 invariant total is the sum of the<span style='mso-spacerun:yes'>  
    927 </span>invariant from data’s q range, invariant resulting from extrapolation at 
    928 low q range and high q range.<o:p></o:p></span></p> 
    929  
    930 <a name="How to Compute Invariant?" class=section1 style='mso-outline-level:1'><b style='mso-bidi-font-weight: 
    931 normal'>How to Compute Invariant?<o:p></o:p></b></a> 
    932  
    933 <p class=ListParagraphCxSpFirst style='margin-left:0in;mso-add-space:auto'><span 
    934 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> 
    935  
    936 <p class=ListParagraphCxSpMiddle style='margin-left:0in;mso-add-space:auto'><b 
    937 style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height: 
    938 115%;font-family:"Times New Roman"'>Step1:</span></b><span style='font-size: 
    939 12.0pt;line-height:115%;font-family:"Times New Roman"'> Load data file.<o:p></o:p></span></p> 
    940  
    941 <p class=ListParagraphCxSpMiddle style='margin-left:0in;mso-add-space:auto'><span 
    942 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> 
    943  
    944 <p class=ListParagraphCxSpMiddle style='margin-left:0in;mso-add-space:auto'><span 
    945 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> 
    946  
    947 <p class=ListParagraphCxSpMiddle style='margin-left:0in;mso-add-space:auto'><span 
    948 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape 
    949  id="_x0000_i1050" type="#_x0000_t75" style='width:395.25pt;height:180pt'> 
    950  <v:imagedata src="../media/image043.png" o:title=""/> 
    951 </v:shape><![endif]--><![if !vml]><img width=527 height=240 
    952 src="../media/image065.jpg" v:shapes="_x0000_i1050"><![endif]><o:p></o:p></span></p> 
    953  
    954 <p class=ListParagraphCxSpLast style='margin-left:0in;mso-add-space:auto'><span 
    955 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> 
    956  
    957 <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span 
    958 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Step2:</span></b><span 
    959 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> Select 
    960 loaded data from a plot panel by highlighting that it until its color turns 
    961 yellow. Then right click on that the data and selects the option Compute 
    962 Invariant. The application automatically computes the invariant value if the 
    963 data loaded is valid.</span></p> 
    964  
    965 <p class=MsoNormal><o:p>&nbsp;</o:p></p> 
    966  
    967 <p class=MsoNormal><o:p>&nbsp;</o:p></p> 
    968  
    969 <p class=MsoNormal><o:p>&nbsp;</o:p></p> 
    970  
    971 <p class=MsoNormal><!--[if gte vml 1]><v:shape id="_x0000_i1051" type="#_x0000_t75" 
    972  style='width:431.25pt;height:262.5pt'> 
    973  <v:imagedata src="../media/image045.png" o:title=""/> 
    974 </v:shape><![endif]--><![if !vml]><img width=575 height=350 
    975 src="../media/image066.jpg" v:shapes="_x0000_i1051"><![endif]></p> 
    976  
    977 <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span 
    978 style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>More 
    979 options</span></b><span style='font-size:12.0pt;line-height:115%;font-family: 
    980 "Times New Roman"'>:<o:p></o:p></span></p> 
    981  
    982 <p class=MsoNormal><span style='font-size:12.0pt;line-height:115%;font-family: 
    983 "Times New Roman"'>To compute the volume, the user must enter a numerical value 
    984 for the contrast.<o:p></o:p></span></p> 
    985  
    986 <p class=MsoNormal><span style='font-size:12.0pt;line-height:115%;font-family: 
    987 "Times New Roman"'>To compute the surface area, the user must enter numerical 
    988 values for contrast and <span class=SpellE>Porod</span> constant.<o:p></o:p></span></p> 
    989  
    990 <p class=MsoNormal><span style='font-size:12.0pt;line-height:115%;font-family: 
    991 "Times New Roman"'>The user can enable extrapolation with check buttons 
    992 available and change options necessary for computations.<o:p></o:p></span></p> 
    993  
    994 <p class=MsoNormal><span style='font-size:12.0pt;line-height:115%;font-family: 
    995 "Times New Roman"'>A <b style='mso-bidi-font-weight:normal'>detail</b> button 
    996 is also available to explain how the total value of the invariant is computed.</span></p> 
     174Our maximum q range is 1e-5 ~ 10 (1/A). The lower and/or higher q range than 
     175data given can be extrapolated by fitting some data nearby.</span></p> 
     176 
     177<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>The 
     178scattering invariant is computed as follow:</span></p> 
     179 
     180<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span 
     181style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     182</span></span><i><span style='font-family:"Times New Roman","serif"'>I(q)</span></i><span 
     183style='font-family:"Times New Roman","serif"'> =  <i>I(q)</i> w/o background : 
     184If the data includes a background, user sets the value to subtract the background 
     185for the Q* computation.</span></p> 
     186 
     187<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span 
     188style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     189</span></span><span style='font-family:"Times New Roman","serif"'>Reset<i> 
     190I(q)  </i></span><span style='font-family:"Times New Roman","serif"'>= <i>I(q)* 
     191scaling factor </i>, delta<i>I(q) =  </i>delta<i>I(q)*scaling factor</i> : If 
     192non-zero scaling factor is given, it will be considered.</span></p> 
     193 
     194<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span 
     195style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     196</span></span><span style='font-family:"Times New Roman","serif"'> Invariant:</span></p> 
     197 
     198<p class=MsoNormal style='margin-left:.25in;line-height:normal'> 
     199 
     200<table cellpadding=0 cellspacing=0 align=left> 
     201 <tr> 
     202  <td width=45 height=12></td> 
     203 </tr> 
     204 <tr> 
     205  <td></td> 
     206  <td><img width=127 height=35 src="image001.gif"></td> 
     207 </tr> 
     208</table> 
     209 
     210<span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     211 
     212<br clear=ALL> 
     213 
     214<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> 
     215 
     216<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     217style='font-family:"Times New Roman","serif"'>            where, <i>g =q</i> 
     218for the pinhole geometry and <i>g =<sub> </sub>q<sub>v</sub></i>(the slit 
     219height) for the slit geometry which can be given in data or as a value.</span></p> 
     220 
     221<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span 
     222style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     223</span></span><span style='font-family:"Times New Roman","serif"'>&nbsp;Higher 
     224q-region (&gt;= qmax in data): </span></p> 
     225 
     226<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            Power 
     227law (w/o background term) function = C/q<sup>4</sup> will be used </span></p> 
     228 
     229<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     230style='font-family:"Times New Roman","serif"'>            where the constant 
     231C(=2pi(delta(rho))S<sub>v</sub>) is to be found by fitting part of data with 
     232the range of q<sub>N-m</sub>  to q<sub>N</sub> (m&lt;N).</span></p> 
     233 
     234<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:normal'><span 
     235style='font-family:"Arial","sans-serif"'>•<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     236</span></span><span style='font-family:"Times New Roman","serif"'> Lower 
     237q-region (&lt;= qmin in data): </span></p> 
     238 
     239<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            Guinier 
     240function = <i>I<sub>0</sub>exp(-R<sub>g</sub><sup>2</sup>q<sup>2</sup>/3)</i> 
     241where I<sub>0</sub> and R<sub>g</sub> are obtained by fitting, </span></p> 
     242 
     243<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            similarly 
     244to the high q region above.</span></p> 
     245 
     246<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>            Power 
     247law can also be used.</span></p> 
     248 
     249<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>&nbsp;</span><b>The 
     250<a name=volume>volume </a>fraction</b>: </p> 
     251 
     252<p class=MsoNormal style='line-height:normal'><span style='position:absolute; 
     253z-index:3;margin-left:1px;margin-top:9px;width:161px;height:47px'><img 
     254width=161 height=47 src="image002.gif"></span><span 
     255style='font-family:"Times New Roman","serif"'> </span></p> 
     256 
     257<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> </span></p> 
     258 
     259<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>where 
     260delta(rho) is the SLD contrast of which value is given by users.</span></p> 
     261 
     262<p class=MsoNormal style='line-height:normal'><span style='position:absolute; 
     263z-index:4;margin-left:1px;margin-top:29px;width:101px;height:45px'><img 
     264width=101 height=45 src="image003.gif"></span><span 
     265style='font-family:"Times New Roman","serif"'>Thus,  </span></p> 
     266 
     267<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     268 
     269<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     270 
     271<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> where 
     2720 =&lt; <i>A</i> =&lt;1/4 in order for these values to be physically valid.</span></p> 
     273 
     274<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     275 
     276<p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>The 
     277<a name=surface>specific surface area </a>:</span></b></p> 
     278 
     279<p class=MsoNormal style='line-height:normal'> 
     280 
     281<table cellpadding=0 cellspacing=0 align=left> 
     282 <tr> 
     283  <td width=1 height=3></td> 
     284 </tr> 
     285 <tr> 
     286  <td></td> 
     287  <td><img width=184 height=47 src="image004.gif"></td> 
     288 </tr> 
     289</table> 
     290 
     291<b><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></b></p> 
     292 
     293<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     294 
     295<br clear=ALL> 
     296 
     297<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'> where 
     298<i>A</i> and <i>Q<sup>*</sup></i> are obtained from previous sections, and the 
     299Porod constant <i>C<sub>p</sub></i> is given by users. </span></p> 
     300 
     301<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     302 
     303<p class=MsoNormal style='line-height:normal'><b><span style='font-family:"Times New Roman","serif"'>Definitions:</span></b><span 
     304style='font-family:"Times New Roman","serif"'> </span></p> 
     305 
     306<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Q&nbsp;: 
     307the magnitude of neutron (or X-ray) momentum transfer vector.</span></p> 
     308 
     309<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>I(Q): 
     310the scattering intensity as a function of the momentum transfer Q.</span></p> 
     311 
     312<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>Invariant 
     313total is the sum of the invariant calculated from data’s q range and the 
     314invariant resulting from extrapolation at low q range and at high q range if 
     315considered.</span></p> 
     316 
     317<p class=MsoNormal style='line-height:150%'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     318 
     319<p class=MsoNormal style='line-height:normal'><a name=reference><b><span 
     320style='font-family:"Times New Roman","serif"'>Reference</span></b></a><b><span 
     321style='font-family:"Times New Roman","serif"'>: </span></b></p> 
     322 
     323<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Porod's 
     324chapter (2) from  O. Glatter and O. Kratky, &quot;Small Angle X-Ray 
     325Scattering&quot;, Academic Press, New York, 1982</span></p> 
     326 
     327<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>Otto's 
     328web page: </span><a href="http://physchem.kfunigraz.ac.at/sm/"><span 
     329style='font-family:"Times New Roman","serif"'>http://physchem.kfunigraz.ac.at/sm/</span></a><span 
     330style='font-family:"Times New Roman","serif"'> </span></p> 
     331 
     332<p class=MsoNormal style='line-height:normal'><a name=howto><b><span 
     333style='font-family:"Times New Roman","serif"'>HowTo</span></b></a><b><span 
     334style='font-family:"Times New Roman","serif"'>:</span></b></p> 
     335 
     336<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     337style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     338</span>i)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     339</span></span><span style='font-family:"Times New Roman","serif"'>Loading data 
     340to the panel: Open the data file from File in the menu bar. Select loaded data 
     341from a plot panel by highlighting that it until its color turns yellow. Then 
     342right click on that the data and selects the option Compute Invariant. The 
     343application automatically computes the invariant value if the data loaded is 
     344valid.</span></p> 
     345 
     346<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     347style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     348</span>ii)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     349</span></span><span style='font-family:"Times New Roman","serif"'>To subtract a 
     350background or/and to rescale the data, type the values in Customized Input 
     351box.  </span></p> 
     352 
     353<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     354style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     355</span>iii)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     356</span></span><span style='font-family:"Times New Roman","serif"'>If you want 
     357to calculate the volume fraction and the specific surface area, type the 
     358optional inputs in the customized input box, and then press ‘Compute’ button.</span></p> 
     359 
     360<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     361style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     362</span>iv)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     363</span></span><span style='font-family:"Times New Roman","serif"'>The invariant 
     364can also be calculated including the outside of the data Q range:  To include 
     365the lower Q and/or the higher Q range, check in the enable extrapolation check 
     366box in ‘Extrapolation ‘box. If the power low is chosen, the power (exponent) 
     367can be either held or fitted by checking the corresponding radio button.  The 
     368Npts that is being used for the extrapolation can be specified.</span></p> 
     369 
     370<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     371style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     372</span>v)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     373</span></span><span style='font-family:"Times New Roman","serif"'>If the 
     374invariant calculated from the extrapolated region is too large, it will be warn 
     375in red at the top of the panel, which means that your data is not proper to 
     376calculate the invariant.</span></p> 
     377 
     378<p class=MsoNormal style='margin-left:.5in;text-indent:-.5in;line-height:normal'><span 
     379style='font-family:"Times New Roman","serif"'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; 
     380</span>vi)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
     381</span></span><span style='font-family:"Times New Roman","serif"'>The details 
     382of the calculation is available by clicking the ‘Details’ button in the middle 
     383of the panel. </span></p> 
     384 
     385<p class=MsoNormal align=center style='text-align:center;line-height:normal'><span 
     386style='font-family:"Times New Roman","serif"'><img border=0 width=572 
     387height=449 id="Object 4" src="image005.gif"></span></p> 
     388 
     389<p class=MsoNormal style='line-height:normal'><span style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    997390 
    998391</div> 
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