source: sasview/invariantview/media/invariant_help.html @ e75eff1

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Last change on this file since e75eff1 was f29a433, checked in by Gervaise Alina <gervyh@…>, 15 years ago

add documentation for invariant
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22<title>The scattering invariant (Q*) is a model-independent quantity that can
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421<body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'>
422
423<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
437that can be easily calculated from scattering data. For two phase systems, the
438scattering invariant, Q*, is defined as the integral of the square of the wave
439transfer (q) multiplied by the scattering cross section over the full range of
440q. </p>
441
442<p class=section1>Q* is given by the following equation<span class=GramE>, </span><sub><!--[if gte vml 1]><v:shapetype
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447  <v:f eqn="if lineDrawn pixelLineWidth 0"/>
448  <v:f eqn="sum @0 1 0"/>
449  <v:f eqn="sum 0 0 @1"/>
450  <v:f eqn="prod @2 1 2"/>
451  <v:f eqn="prod @3 21600 pixelWidth"/>
452  <v:f eqn="prod @3 21600 pixelHeight"/>
453  <v:f eqn="sum @0 0 1"/>
454  <v:f eqn="prod @6 1 2"/>
455  <v:f eqn="prod @7 21600 pixelWidth"/>
456  <v:f eqn="sum @8 21600 0"/>
457  <v:f eqn="prod @7 21600 pixelHeight"/>
458  <v:f eqn="sum @10 21600 0"/>
459 </v:formulas>
460 <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
461 <o:lock v:ext="edit" aspectratio="t"/>
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
466src="../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:
471TimesNewRoman'><span style='mso-spacerun:yes'> </span></span>. This model
472independent quantity (Q*) is calculated from the scattering data can be used to
473determine the volume fraction and the specific area of the sample under
474consideration. &nbsp;These quantities&nbsp;are useful in their own right and
475can be used in&nbsp;further analysis. With this scattering invariant module
476users will also be able to determine the consistency of those properties
477between data.&nbsp; There is no real data defined from zero to infinity, there
478usually have limited range. Q* is not really computed from zero to infinity.
479Methods to deal with this limitation will be addressed.</p>
480
481<p class=MsoNormal style='margin-top:0in;margin-right:0in;margin-bottom:0in;
482margin-left:.25in;margin-bottom:.0001pt;line-height:normal'><span
483style='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:
48622.5pt;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
489style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>The
490scattering 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;
493tab-stops:list .5in'><![if !supportLists]><span style='font-size:12.0pt;
494line-height:115%;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
495Symbol'><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
497style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>I(q)</span></i><span
498style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> =<span
499style='mso-spacerun:yes'>  </span><i style='mso-bidi-font-style:normal'>I(q)</i>-
500background for data without background<o:p></o:p></span></p>
501
502<p class=ListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l8 level1 lfo11;
503tab-stops:list .5in'><![if !supportLists]><span style='font-size:12.0pt;
504line-height:115%;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
505Symbol'><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
507style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>I(q) </span></i><span
508style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>= <i
509style='mso-bidi-font-style:normal'>I(q)* scaling factor </i>, </span><span
510lang=IT style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman";
511mso-ansi-language:IT'><sub><!--[if gte vml 1]><v:shape id="_x0000_i1026"
512 type="#_x0000_t75" style='width:11.25pt;height:14.25pt' o:ole="">
513 <v:imagedata src="../media/image003.wmz" o:title=""/>
514</v:shape><![endif]--><![if !vml]><img width=15 height=19
515src="../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
520style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>I(q) = </span></i><span
521style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'><span
522style='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
526src="../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;
533tab-stops:list .5in'><![if !supportLists]><span style='font-size:12.0pt;
534line-height:115%;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
535Symbol'><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:
537115%;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
541src="../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%;
546font-family:"Times New Roman"'><span style='mso-spacerun:yes'> </span><span
547class=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
552src="../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
557the 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
560style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family:
561Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
562style='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%;
564font-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
568src="../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
573of the data points, <span class=SpellE>q<sub>i</sub></span><sub> </sub>= wave transfer,
574I<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
579src="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
584style='mso-spacerun:yes'> </span>= (q<sub>i+1</sub> – q<sub>i-1</sub>)/2 (;
585note 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
590src="../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
595style='mso-spacerun:yes'>  </span>and <sub><span
596style='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
601src="../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
606style='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
610src="../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
617style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family:
618Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
619style='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%;
621font-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
626src="../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
631there 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
635src="../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
640class=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
643style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family:
644Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
645style='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%;
647font-family:"Times New Roman"'>Higher q-region (&gt;= <span class=SpellE>qmax</span>
648of data): <o:p></o:p></span></p>
649
650<p class=ListParagraphCxSpMiddle style='margin-left:63.9pt;mso-add-space:auto'><span
651class=SpellE><span style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Porod</span></span><span
652style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> (w/o
653background term) function = C/q<sup>4</sup> will be used where the constant <span
654class=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
658src="../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
663fitting part of data with the range of <span class=SpellE>q<sub>N</sub></span><sub>-m</sub>
664to <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
667style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>The
668contribution to Q<sup>*</sup> from this region is described in an explicit
669form, and thus numerical integration is not required (w/ no background is
670assumed): <o:p></o:p></span></p>
671
672<p class=ListParagraphCxSpMiddle style='margin-left:99.9pt;mso-add-space:auto;
673text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span
674style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family:
675Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span
676style='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%;
678font-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
682src="../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;
689text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span
690style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family:
691Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span
692style='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%;
694font-family:"Times New Roman"'>(Also numerical integration can be considered
695for this case).<o:p></o:p></span></p>
696
697<p class=ListParagraphCxSpMiddle style='margin-left:99.9pt;mso-add-space:auto;
698text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span
699style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family:
700Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span
701style='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%;
703font-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
708src="../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
713style='font-size:12.0pt;line-height:115%;font-family:Symbol'>D</span><span
714class=SpellE><span style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>q<sub>v</sub></span></span><span
715style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> is
716given in data, <span class=SpellE>q<sub>N</sub></span> = maximum q (<span
717class=SpellE>q<sub>max</sub></span>) in <span class=GramE>data<span
718style='mso-spacerun:yes'>  </span>and</span> C is obtained by the fitting data
719in 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;
722text-indent:-.25in;mso-list:l7 level1 lfo4'><![if !supportLists]><span
723style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family:
724Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span
725style='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%;
727font-family:"Times New Roman"'>Note that the number of points for fitting is to
728be given by users and should be larger than 2 from the point of <span
729class=SpellE>qmax</span>.<o:p></o:p></span></p>
730
731<p class=ListParagraphCxSpMiddle style='margin-left:.95in;mso-add-space:auto;
732text-indent:-.25in;mso-list:l12 level1 lfo5'><![if !supportLists]><span
733style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family:
734Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
735style='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%;
737font-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
740class=SpellE><span style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Guinier</span></span><span
741style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>
742function = <span class=GramE><i style='mso-bidi-font-style:normal'>I<sub>0</sub>exp(</i></span><i
743style='mso-bidi-font-style:normal'>-R<sub>g</sub><sup>2</sup>q<sup>2</sup>/3)</i>
744where I<sub>0</sub> and <span class=SpellE>R<sub>g</sub></span> are obtained by
745fitting, 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;
748text-indent:-.25in;mso-list:l14 level1 lfo6'><![if !supportLists]><span
749style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family:
750Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span
751style='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%;
753font-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
758src="../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
765style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Numerically<span
766style='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
770src="../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
775class=SpellE>q<sub>N</sub></span> &lt;= <span class=SpellE>q<sub>min</sub></span>
776in data points, q<sub>0</sub> will be set to 1e-04 (1/A) and N =1000(<span
777class=GramE>?,</span> linear binning?).<o:p></o:p></span></p>
778
779<p class=ListParagraphCxSpMiddle style='margin-left:1.45in;mso-add-space:auto;
780text-indent:-.25in;mso-list:l14 level1 lfo6'><![if !supportLists]><span
781style='font-size:12.0pt;line-height:115%;font-family:Wingdings;mso-fareast-font-family:
782Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span
783style='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%;
785font-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
790src="../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
797style='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
801src="../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%;
806font-family:"Times New Roman"'><span style='mso-spacerun:yes'>  </span><span
807class=GramE>where</span> <span class=SpellE>q<sub>min</sub></span> is the
808smallest 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
812numerical 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
816style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height:
817115%;font-family:"Times New Roman"'>Volume <span class=GramE>Fraction<span
818style='font-weight:normal'> </span></span></span></b><span style='font-size:
81912.0pt;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
824src="../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
831style='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
835src="../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%;
840font-family:"Times New Roman"'><span style='mso-spacerun:yes'>  </span><span
841class=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
845src="../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
852style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Thus,<span
853style='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
857src="../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
865style='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
868style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height:
869115%;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
873src="../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
878style='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
881style='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
885src="../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%;
890font-family:"Times New Roman"'><span style='mso-spacerun:yes'> </span><span
891class=GramE>where</span> A and Q<sup>*</sup> are obtained from 1) and 2), and
892the <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;
895mso-outline-level:1'><span style='font-size:12.0pt;font-family:"Times New Roman";
896mso-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:
899normal;mso-list:l3 level1 lfo8;tab-stops:list 45.9pt'><![if !supportLists]><span
900style='font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
901mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
902style='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;
904font-family:"Times New Roman"'>Q&nbsp;:</span></span><span style='font-size:
90512.0pt;font-family:"Times New Roman"'> the magnitude of neutron (or X-ray) momentum
906transfer vector.<o:p></o:p></span></p>
907
908<p class=MsoNormal style='mso-margin-bottom-alt:auto;margin-left:45.9pt;
909text-indent:-.25in;line-height:normal;mso-list:l3 level1 lfo8;tab-stops:list 45.9pt'><![if !supportLists]><span
910style='font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
911mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
912style='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;
914font-family:"Times New Roman"'>I(</span></span><span style='font-size:12.0pt;
915font-family:"Times New Roman"'>Q): the scattering intensity as a function of
916the 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
920style='font-size:12.0pt;line-height:115%;font-family:Symbol;mso-fareast-font-family:
921Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span
922style='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;
924line-height:115%;font-family:"Times New Roman"'>the</span></span><span
925style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>
926invariant total is the sum of the<span style='mso-spacerun:yes'> 
927</span>invariant from data’s q range, invariant resulting from extrapolation at
928low 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:
931normal'>How to Compute Invariant?<o:p></o:p></b></a>
932
933<p class=ListParagraphCxSpFirst style='margin-left:0in;mso-add-space:auto'><span
934style='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
937style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height:
938115%;font-family:"Times New Roman"'>Step1:</span></b><span style='font-size:
93912.0pt;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
942style='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
945style='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
948style='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
952src="../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
955style='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
958style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>Step2:</span></b><span
959style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'> Select
960loaded data from a plot panel by highlighting that it until its color turns
961yellow. Then right click on that the data and selects the option Compute
962Invariant. The application automatically computes the invariant value if the
963data 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
975src="../media/image066.jpg" v:shapes="_x0000_i1051"><![endif]></p>
976
977<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
978style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman"'>More
979options</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
984for 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
988values 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
992available 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
996is also available to explain how the total value of the invariant is computed.</span></p>
997
998</div>
999
1000</body>
1001
1002</html>
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