source: trunk/help/ParametricFitting.htm @ 1434

Last change on this file since 1434 was 1434, checked in by vondreele, 8 years ago

fix properties

  • Property svn:mime-type set to text/html
File size: 105.1 KB
Line 
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6
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23  <o:Author>B Toby</o:Author>
24  <o:LastAuthor>Von Dreele</o:LastAuthor>
25  <o:Revision>2</o:Revision>
26  <o:TotalTime>984</o:TotalTime>
27  <o:Created>2014-07-22T21:21:00Z</o:Created>
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32  <o:Company>Argonne National Laboratory</o:Company>
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382        mso-font-alt:"Times New Roman";
383        mso-font-charset:0;
384        mso-generic-font-family:auto;
385        mso-font-pitch:variable;
386        mso-font-signature:0 1342218751 0 0 447 0;}
387@font-face
388        {font-family:Consolas;
389        panose-1:2 11 6 9 2 2 4 3 2 4;
390        mso-font-charset:0;
391        mso-generic-font-family:modern;
392        mso-font-pitch:fixed;
393        mso-font-signature:-520092929 1073806591 9 0 415 0;}
394 /* Style Definitions */
395 p.MsoNormal, li.MsoNormal, div.MsoNormal
396        {mso-style-unhide:no;
397        mso-style-qformat:yes;
398        mso-style-parent:"";
399        margin:0in;
400        margin-bottom:.0001pt;
401        mso-pagination:widow-orphan;
402        font-size:12.0pt;
403        mso-bidi-font-size:10.0pt;
404        font-family:"Times New Roman","serif";
405        mso-fareast-font-family:"Times New Roman";
406        mso-fareast-theme-font:minor-fareast;
407        mso-bidi-font-family:"Times New Roman";
408        mso-bidi-theme-font:minor-bidi;
409        mso-fareast-language:JA;}
410h1
411        {mso-style-priority:9;
412        mso-style-unhide:no;
413        mso-style-qformat:yes;
414        mso-style-link:"Heading 1 Char";
415        mso-style-next:Normal;
416        margin-top:24.0pt;
417        margin-right:0in;
418        margin-bottom:0in;
419        margin-left:0in;
420        margin-bottom:.0001pt;
421        mso-pagination:widow-orphan lines-together;
422        page-break-after:avoid;
423        mso-outline-level:1;
424        font-size:18.0pt;
425        mso-bidi-font-size:16.0pt;
426        font-family:"Times New Roman","serif";
427        mso-fareast-font-family:"Times New Roman";
428        mso-fareast-theme-font:major-fareast;
429        mso-bidi-font-family:"Times New Roman";
430        mso-bidi-theme-font:major-bidi;
431        mso-font-kerning:0pt;
432        mso-fareast-language:JA;}
433h2
434        {mso-style-noshow:yes;
435        mso-style-priority:9;
436        mso-style-qformat:yes;
437        mso-style-link:"Heading 2 Char";
438        mso-style-next:Normal;
439        margin-top:10.0pt;
440        margin-right:0in;
441        margin-bottom:0in;
442        margin-left:0in;
443        margin-bottom:.0001pt;
444        mso-pagination:widow-orphan lines-together;
445        page-break-after:avoid;
446        mso-outline-level:2;
447        font-size:13.0pt;
448        font-family:"Calibri","sans-serif";
449        mso-ascii-font-family:Calibri;
450        mso-ascii-theme-font:major-latin;
451        mso-fareast-font-family:"Times New Roman";
452        mso-fareast-theme-font:major-fareast;
453        mso-hansi-font-family:Calibri;
454        mso-hansi-theme-font:major-latin;
455        mso-bidi-font-family:"Times New Roman";
456        mso-bidi-theme-font:major-bidi;
457        color:#4F81BD;
458        mso-themecolor:accent1;
459        mso-fareast-language:JA;}
460a:link, span.MsoHyperlink
461        {mso-style-noshow:yes;
462        mso-style-priority:99;
463        color:blue;
464        mso-themecolor:hyperlink;
465        text-decoration:underline;
466        text-underline:single;}
467a:visited, span.MsoHyperlinkFollowed
468        {mso-style-noshow:yes;
469        mso-style-priority:99;
470        color:purple;
471        mso-themecolor:followedhyperlink;
472        text-decoration:underline;
473        text-underline:single;}
474pre
475        {mso-style-noshow:yes;
476        mso-style-priority:99;
477        mso-style-link:"HTML Preformatted Char";
478        margin:0in;
479        margin-bottom:.0001pt;
480        mso-pagination:widow-orphan;
481        font-size:10.0pt;
482        font-family:"Courier New";
483        mso-fareast-font-family:"Times New Roman";
484        mso-fareast-theme-font:minor-fareast;}
485p.MsoAcetate, li.MsoAcetate, div.MsoAcetate
486        {mso-style-noshow:yes;
487        mso-style-priority:99;
488        mso-style-link:"Balloon Text Char";
489        margin:0in;
490        margin-bottom:.0001pt;
491        mso-pagination:widow-orphan;
492        font-size:9.0pt;
493        font-family:"Lucida Grande";
494        mso-fareast-font-family:"Times New Roman";
495        mso-fareast-theme-font:minor-fareast;
496        mso-fareast-language:JA;}
497p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph
498        {mso-style-priority:34;
499        mso-style-unhide:no;
500        mso-style-qformat:yes;
501        margin-top:0in;
502        margin-right:0in;
503        margin-bottom:0in;
504        margin-left:.5in;
505        margin-bottom:.0001pt;
506        mso-add-space:auto;
507        mso-pagination:widow-orphan;
508        font-size:12.0pt;
509        mso-bidi-font-size:10.0pt;
510        font-family:"Times New Roman","serif";
511        mso-fareast-font-family:"Times New Roman";
512        mso-fareast-theme-font:minor-fareast;
513        mso-bidi-font-family:"Times New Roman";
514        mso-bidi-theme-font:minor-bidi;
515        mso-fareast-language:JA;}
516p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst
517        {mso-style-priority:34;
518        mso-style-unhide:no;
519        mso-style-qformat:yes;
520        mso-style-type:export-only;
521        margin-top:0in;
522        margin-right:0in;
523        margin-bottom:0in;
524        margin-left:.5in;
525        margin-bottom:.0001pt;
526        mso-add-space:auto;
527        mso-pagination:widow-orphan;
528        font-size:12.0pt;
529        mso-bidi-font-size:10.0pt;
530        font-family:"Times New Roman","serif";
531        mso-fareast-font-family:"Times New Roman";
532        mso-fareast-theme-font:minor-fareast;
533        mso-bidi-font-family:"Times New Roman";
534        mso-bidi-theme-font:minor-bidi;
535        mso-fareast-language:JA;}
536p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle
537        {mso-style-priority:34;
538        mso-style-unhide:no;
539        mso-style-qformat:yes;
540        mso-style-type:export-only;
541        margin-top:0in;
542        margin-right:0in;
543        margin-bottom:0in;
544        margin-left:.5in;
545        margin-bottom:.0001pt;
546        mso-add-space:auto;
547        mso-pagination:widow-orphan;
548        font-size:12.0pt;
549        mso-bidi-font-size:10.0pt;
550        font-family:"Times New Roman","serif";
551        mso-fareast-font-family:"Times New Roman";
552        mso-fareast-theme-font:minor-fareast;
553        mso-bidi-font-family:"Times New Roman";
554        mso-bidi-theme-font:minor-bidi;
555        mso-fareast-language:JA;}
556p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast
557        {mso-style-priority:34;
558        mso-style-unhide:no;
559        mso-style-qformat:yes;
560        mso-style-type:export-only;
561        margin-top:0in;
562        margin-right:0in;
563        margin-bottom:0in;
564        margin-left:.5in;
565        margin-bottom:.0001pt;
566        mso-add-space:auto;
567        mso-pagination:widow-orphan;
568        font-size:12.0pt;
569        mso-bidi-font-size:10.0pt;
570        font-family:"Times New Roman","serif";
571        mso-fareast-font-family:"Times New Roman";
572        mso-fareast-theme-font:minor-fareast;
573        mso-bidi-font-family:"Times New Roman";
574        mso-bidi-theme-font:minor-bidi;
575        mso-fareast-language:JA;}
576span.Heading1Char
577        {mso-style-name:"Heading 1 Char";
578        mso-style-priority:9;
579        mso-style-unhide:no;
580        mso-style-locked:yes;
581        mso-style-link:"Heading 1";
582        mso-ansi-font-size:18.0pt;
583        mso-bidi-font-size:16.0pt;
584        font-family:"Times New Roman","serif";
585        mso-ascii-font-family:"Times New Roman";
586        mso-fareast-font-family:"Times New Roman";
587        mso-fareast-theme-font:major-fareast;
588        mso-hansi-font-family:"Times New Roman";
589        mso-bidi-font-family:"Times New Roman";
590        mso-bidi-theme-font:major-bidi;
591        font-weight:bold;}
592span.Heading2Char
593        {mso-style-name:"Heading 2 Char";
594        mso-style-noshow:yes;
595        mso-style-priority:9;
596        mso-style-unhide:no;
597        mso-style-locked:yes;
598        mso-style-link:"Heading 2";
599        mso-ansi-font-size:13.0pt;
600        mso-bidi-font-size:13.0pt;
601        font-family:"Calibri","sans-serif";
602        mso-ascii-font-family:Calibri;
603        mso-ascii-theme-font:major-latin;
604        mso-fareast-font-family:"Times New Roman";
605        mso-fareast-theme-font:major-fareast;
606        mso-hansi-font-family:Calibri;
607        mso-hansi-theme-font:major-latin;
608        mso-bidi-font-family:"Times New Roman";
609        mso-bidi-theme-font:major-bidi;
610        color:#4F81BD;
611        mso-themecolor:accent1;
612        font-weight:bold;}
613span.HTMLPreformattedChar
614        {mso-style-name:"HTML Preformatted Char";
615        mso-style-noshow:yes;
616        mso-style-priority:99;
617        mso-style-unhide:no;
618        mso-style-locked:yes;
619        mso-style-link:"HTML Preformatted";
620        font-family:Consolas;
621        mso-ascii-font-family:Consolas;
622        mso-hansi-font-family:Consolas;
623        mso-bidi-font-family:Consolas;}
624span.BalloonTextChar
625        {mso-style-name:"Balloon Text Char";
626        mso-style-noshow:yes;
627        mso-style-priority:99;
628        mso-style-unhide:no;
629        mso-style-locked:yes;
630        mso-style-link:"Balloon Text";
631        mso-ansi-font-size:9.0pt;
632        mso-bidi-font-size:9.0pt;
633        font-family:"Lucida Grande";
634        mso-ascii-font-family:"Lucida Grande";
635        mso-hansi-font-family:"Lucida Grande";
636        mso-bidi-font-family:"Lucida Grande";}
637.MsoChpDefault
638        {mso-style-type:export-only;
639        mso-default-props:yes;
640        font-size:10.0pt;
641        mso-ansi-font-size:10.0pt;
642        mso-bidi-font-size:10.0pt;
643        font-family:"Cambria","serif";
644        mso-ascii-font-family:Cambria;
645        mso-ascii-theme-font:minor-latin;
646        mso-fareast-font-family:"Times New Roman";
647        mso-fareast-theme-font:minor-fareast;
648        mso-hansi-font-family:Cambria;
649        mso-hansi-theme-font:minor-latin;
650        mso-bidi-font-family:"Times New Roman";
651        mso-bidi-theme-font:minor-bidi;
652        mso-fareast-language:JA;}
653@page WordSection1
654        {size:8.5in 11.0in;
655        margin:1.0in 1.25in 1.0in 1.25in;
656        mso-header-margin:.5in;
657        mso-footer-margin:.5in;
658        mso-paper-source:0;}
659div.WordSection1
660        {page:WordSection1;}
661 /* List Definitions */
662 @list l0
663        {mso-list-id:109671306;
664        mso-list-type:hybrid;
665        mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
666@list l0:level1
667        {mso-level-tab-stop:none;
668        mso-level-number-position:left;
669        text-indent:-.25in;}
670@list l0:level2
671        {mso-level-number-format:alpha-lower;
672        mso-level-tab-stop:none;
673        mso-level-number-position:left;
674        text-indent:-.25in;}
675@list l0:level3
676        {mso-level-number-format:roman-lower;
677        mso-level-tab-stop:none;
678        mso-level-number-position:right;
679        text-indent:-9.0pt;}
680@list l0:level4
681        {mso-level-tab-stop:none;
682        mso-level-number-position:left;
683        text-indent:-.25in;}
684@list l0:level5
685        {mso-level-number-format:alpha-lower;
686        mso-level-tab-stop:none;
687        mso-level-number-position:left;
688        text-indent:-.25in;}
689@list l0:level6
690        {mso-level-number-format:roman-lower;
691        mso-level-tab-stop:none;
692        mso-level-number-position:right;
693        text-indent:-9.0pt;}
694@list l0:level7
695        {mso-level-tab-stop:none;
696        mso-level-number-position:left;
697        text-indent:-.25in;}
698@list l0:level8
699        {mso-level-number-format:alpha-lower;
700        mso-level-tab-stop:none;
701        mso-level-number-position:left;
702        text-indent:-.25in;}
703@list l0:level9
704        {mso-level-number-format:roman-lower;
705        mso-level-tab-stop:none;
706        mso-level-number-position:right;
707        text-indent:-9.0pt;}
708@list l1
709        {mso-list-id:235746779;
710        mso-list-type:hybrid;
711        mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
712@list l1:level1
713        {mso-level-tab-stop:none;
714        mso-level-number-position:left;
715        text-indent:-.25in;}
716@list l1:level2
717        {mso-level-number-format:alpha-lower;
718        mso-level-tab-stop:none;
719        mso-level-number-position:left;
720        text-indent:-.25in;}
721@list l1:level3
722        {mso-level-number-format:roman-lower;
723        mso-level-tab-stop:none;
724        mso-level-number-position:right;
725        text-indent:-9.0pt;}
726@list l1:level4
727        {mso-level-tab-stop:none;
728        mso-level-number-position:left;
729        text-indent:-.25in;}
730@list l1:level5
731        {mso-level-number-format:alpha-lower;
732        mso-level-tab-stop:none;
733        mso-level-number-position:left;
734        text-indent:-.25in;}
735@list l1:level6
736        {mso-level-number-format:roman-lower;
737        mso-level-tab-stop:none;
738        mso-level-number-position:right;
739        text-indent:-9.0pt;}
740@list l1:level7
741        {mso-level-tab-stop:none;
742        mso-level-number-position:left;
743        text-indent:-.25in;}
744@list l1:level8
745        {mso-level-number-format:alpha-lower;
746        mso-level-tab-stop:none;
747        mso-level-number-position:left;
748        text-indent:-.25in;}
749@list l1:level9
750        {mso-level-number-format:roman-lower;
751        mso-level-tab-stop:none;
752        mso-level-number-position:right;
753        text-indent:-9.0pt;}
754@list l2
755        {mso-list-id:584074573;
756        mso-list-type:hybrid;
757        mso-list-template-ids:-926633506 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
758@list l2:level1
759        {mso-level-tab-stop:none;
760        mso-level-number-position:left;
761        text-indent:-.25in;}
762@list l2:level2
763        {mso-level-number-format:alpha-lower;
764        mso-level-tab-stop:none;
765        mso-level-number-position:left;
766        text-indent:-.25in;}
767@list l2:level3
768        {mso-level-number-format:roman-lower;
769        mso-level-tab-stop:none;
770        mso-level-number-position:right;
771        text-indent:-9.0pt;}
772@list l2:level4
773        {mso-level-tab-stop:none;
774        mso-level-number-position:left;
775        text-indent:-.25in;}
776@list l2:level5
777        {mso-level-number-format:alpha-lower;
778        mso-level-tab-stop:none;
779        mso-level-number-position:left;
780        text-indent:-.25in;}
781@list l2:level6
782        {mso-level-number-format:roman-lower;
783        mso-level-tab-stop:none;
784        mso-level-number-position:right;
785        text-indent:-9.0pt;}
786@list l2:level7
787        {mso-level-tab-stop:none;
788        mso-level-number-position:left;
789        text-indent:-.25in;}
790@list l2:level8
791        {mso-level-number-format:alpha-lower;
792        mso-level-tab-stop:none;
793        mso-level-number-position:left;
794        text-indent:-.25in;}
795@list l2:level9
796        {mso-level-number-format:roman-lower;
797        mso-level-tab-stop:none;
798        mso-level-number-position:right;
799        text-indent:-9.0pt;}
800@list l3
801        {mso-list-id:845174183;
802        mso-list-type:hybrid;
803        mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
804@list l3:level1
805        {mso-level-tab-stop:none;
806        mso-level-number-position:left;
807        text-indent:-.25in;}
808@list l3:level2
809        {mso-level-number-format:alpha-lower;
810        mso-level-tab-stop:none;
811        mso-level-number-position:left;
812        text-indent:-.25in;}
813@list l3:level3
814        {mso-level-number-format:roman-lower;
815        mso-level-tab-stop:none;
816        mso-level-number-position:right;
817        text-indent:-9.0pt;}
818@list l3:level4
819        {mso-level-tab-stop:none;
820        mso-level-number-position:left;
821        text-indent:-.25in;}
822@list l3:level5
823        {mso-level-number-format:alpha-lower;
824        mso-level-tab-stop:none;
825        mso-level-number-position:left;
826        text-indent:-.25in;}
827@list l3:level6
828        {mso-level-number-format:roman-lower;
829        mso-level-tab-stop:none;
830        mso-level-number-position:right;
831        text-indent:-9.0pt;}
832@list l3:level7
833        {mso-level-tab-stop:none;
834        mso-level-number-position:left;
835        text-indent:-.25in;}
836@list l3:level8
837        {mso-level-number-format:alpha-lower;
838        mso-level-tab-stop:none;
839        mso-level-number-position:left;
840        text-indent:-.25in;}
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920<![endif]-->
921<meta name=Title content="">
922<meta name=Keywords content="">
923<!--[if gte mso 9]><xml>
924 <o:shapedefaults v:ext="edit" spidmax="1032"/>
925</xml><![endif]--><!--[if gte mso 9]><xml>
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929</head>
930
931<body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'>
932
933<div class=WordSection1>
934
935<h1>Parametric Fitting and Pseudo Variables for Sequential Fits</h1>
936
937<p class=MsoNormal><o:p>&nbsp;</o:p></p>
938
939<p class=MsoNormal>Once a sequential fit has been performed, any function of
940the fit parameters from the fit and <i style='mso-bidi-font-style:normal'>their
941estimated uncertainty</i> (s.u.) can be computed by defining one or more
942&quot;pseudo-variables&quot;. Note that the covariance matrix from the fit is
943used to provide appropriate treatment of the correlation in parameters with
944respect to uncertainty estimation.<span style='mso-spacerun:yes'>  </span>In
945addition, parametric equations in terms of sample parameters such as
946temperature and pressure, etc. can be fit to either the parameters from the fit
947or the pseudo-variables. </p>
948
949<p class=MsoNormal><o:p>&nbsp;</o:p></p>
950
951<p class=MsoNormal>This tutorial assumes that the steps in the <span
952style='mso-bidi-font-family:"Times New Roman"'><a href="SequentialTutorial.htm">Sequential
953refinement of multiple datasets</a> tutorial have already been performed. </span>If
954you have not done so already, <a href="Starting%20GSAS.htm">start GSAS-II</a>.
955Some of the pictures in this tutorial are from a Mac OSX machine; these
956illustrate the differences in menu location from Windows/Linux machines.</p>
957
958<h1>Step 1. Open the previous Sequential Fit</h1>
959
960<p class=MsoNormal><o:p>&nbsp;</o:p></p>
961
962<p class=MsoListParagraphCxSpFirst style='margin-bottom:6.0pt;mso-add-space:
963auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
964style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
965style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
966</span></span></span><![endif]>Select the<b style='mso-bidi-font-weight:normal'>
967</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
968mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>File/Open
969Project</span></b> menu item to load file named <b style='mso-bidi-font-weight:
970normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
971major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b>. </p>
972
973<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
974auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
975style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
976style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
977</span></span></span><![endif]>Select the <b style='mso-bidi-font-weight:normal'><span
978style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
979mso-hansi-theme-font:major-latin'>Sequential results</span></b> item in the
980data tree. The sequential refinement results data table is displayed, as below.
981</p>
982
983<p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space:
984auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype
985 id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
986 path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
987 <v:stroke joinstyle="miter"/>
988 <v:formulas>
989  <v:f eqn="if lineDrawn pixelLineWidth 0"/>
990  <v:f eqn="sum @0 1 0"/>
991  <v:f eqn="sum 0 0 @1"/>
992  <v:f eqn="prod @2 1 2"/>
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994  <v:f eqn="prod @3 21600 pixelHeight"/>
995  <v:f eqn="sum @0 0 1"/>
996  <v:f eqn="prod @6 1 2"/>
997  <v:f eqn="prod @7 21600 pixelWidth"/>
998  <v:f eqn="sum @8 21600 0"/>
999  <v:f eqn="prod @7 21600 pixelHeight"/>
1000  <v:f eqn="sum @10 21600 0"/>
1001 </v:formulas>
1002 <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
1003 <o:lock v:ext="edit" aspectratio="t"/>
1004</v:shapetype><v:shape id="Picture_x0020_36" o:spid="_x0000_i1053" type="#_x0000_t75"
1005 style='width:525pt;height:262.5pt;visibility:visible;mso-wrap-style:square'>
1006 <v:imagedata src="ParametricFitting_files/image001.png" o:title=""/>
1007</v:shape><![endif]--><![if !vml]><img border=0 width=700 height=350
1008src="ParametricFitting_files/image001.png" v:shapes="Picture_x0020_36"><![endif]></span></p>
1009
1010<h1>Step 2. Create a Pseudo Variable</h1>
1011
1012<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1013
1014<p class=MsoNormal>Pseudo-variables allow computation of an arbitrarily defined
1015function of combinations of the variables that are displayed in the sequential
1016refinement table. Any valid Python language expression can be used to define
1017this. While one can plot <i style='mso-bidi-font-style:normal'>a</i>, <i
1018style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1019normal'>c</i> axis lengths by clicking on the appropriate column of the table,
1020to compare them one might want to scale them. Alternately, as will be done
1021here, we will plot the ratio of <i style='mso-bidi-font-style:normal'>b</i> and
1022<i style='mso-bidi-font-style:normal'>c</i>.</p>
1023
1024<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1025
1026<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1027style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1028style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1029</span></span></span><![endif]>Click on the <b style='mso-bidi-font-weight:
1030normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1031major-latin;mso-hansi-theme-font:major-latin'>Pseudo Vars/Add</span></b> menu
1032item to create a new pseudo-variable.</p>
1033
1034<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1035
1036<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1037mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1052"
1038 type="#_x0000_t75" style='width:159pt;height:45.75pt;visibility:visible;
1039 mso-wrap-style:square'>
1040 <v:imagedata src="ParametricFitting_files/image002.png" o:title=""/>
1041</v:shape><![endif]--><![if !vml]><img border=0 width=212 height=61
1042src="ParametricFitting_files/image003.jpg" v:shapes="Picture_x0020_2"><![endif]></span></p>
1043
1044<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1045
1046<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1047style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1048style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1049</span></span></span><![endif]>A window opens to define a formula</p>
1050
1051<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1052
1053<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1054mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1051"
1055 type="#_x0000_t75" style='width:309.75pt;height:152.25pt;visibility:visible;
1056 mso-wrap-style:square'>
1057 <v:imagedata src="ParametricFitting_files/image004.png" o:title="" croptop="4214f"
1058  cropbottom="8982f" cropleft="3852f" cropright="3654f"/>
1059</v:shape><![endif]--><![if !vml]><img border=0 width=413 height=203
1060src="ParametricFitting_files/image005.png" v:shapes="Picture_x0020_3"><![endif]></span></p>
1061
1062<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1063
1064<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
1065 o:spid="_x0000_s1031" type="#_x0000_t75" style='position:absolute;left:0;
1066 text-align:left;margin-left:178.3pt;margin-top:846.2pt;width:229.5pt;height:86.5pt;
1067 z-index:251658240;visibility:visible;mso-wrap-style:square;
1068 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1069 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1070 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1071 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1072 mso-position-vertical-relative:margin;mso-width-percent:0;
1073 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1074 <v:imagedata src="ParametricFitting_files/image006.png" o:title=""/>
1075 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1076</v:shape><![endif]--><![if !vml]><img width=306 height=115
1077src="ParametricFitting_files/image007.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In
1078this window, type a Python expression<b style='mso-bidi-font-weight:normal'>, </b><span
1079style='mso-bidi-font-family:"Times New Roman"'>&quot;</span><b
1080style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1081mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1082"Times New Roman"'>b/c</span></b><span style='mso-bidi-font-family:"Times New Roman"'>&quot;,</span>
1083but note that the variables <b style='mso-bidi-font-weight:normal'><span
1084style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1085mso-hansi-theme-font:major-latin'>c</span></b> and <b style='mso-bidi-font-weight:
1086normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1087major-latin;mso-hansi-theme-font:major-latin'>b</span></b> here are arbitrary
1088labels. After entering the formula, press the <b style='mso-bidi-font-weight:
1089normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1090major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> button or
1091wait a few seconds and the validation will be tested automatically. Note that if
1092you pause too long, while typing, an error message may be displayed as the
1093incomplete formula is found to be invalid. If so, make corrections or keep
1094typing until the entire expression is entered.<span style='mso-spacerun:yes'> 
1095</span>The error message will sometimes show where error is encountered, though
1096not in this case. </p>
1097
1098<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1099
1100<p class=MsoListParagraphCxSpMiddle>When the expression is complete, a new
1101table appears where the arbitrary <b style='mso-bidi-font-weight:normal'><span
1102style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1103mso-hansi-theme-font:major-latin'>c</span></b> and <b style='mso-bidi-font-weight:
1104normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1105major-latin;mso-hansi-theme-font:major-latin'>b</span></b> labels will be
1106assigned.</p>
1107
1108<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1109
1110<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1111mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1050"
1112 type="#_x0000_t75" style='width:309.75pt;height:152.25pt;visibility:visible;
1113 mso-wrap-style:square'>
1114 <v:imagedata src="ParametricFitting_files/image008.png" o:title="" croptop="4435f"
1115  cropbottom="8537f" cropleft="3852f" cropright="3650f"/>
1116</v:shape><![endif]--><![if !vml]><img border=0 width=413 height=203
1117src="ParametricFitting_files/image009.png" v:shapes="Picture_x0020_5"><![endif]></span></p>
1118
1119<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1120
1121<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1122style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1123style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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1125 o:spid="_x0000_s1030" type="#_x0000_t75" style='position:absolute;left:0;
1126 text-align:left;margin-left:37.3pt;margin-top:1143.2pt;width:88.5pt;height:65.5pt;
1127 z-index:251659264;visibility:visible;mso-wrap-style:square;
1128 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1129 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1130 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1131 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1132 mso-position-vertical-relative:margin;mso-width-percent:0;
1133 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1134 <v:imagedata src="ParametricFitting_files/image010.png" o:title=""/>
1135 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1136</v:shape><![endif]--><![if !vml]><img width=118 height=87
1137src="ParametricFitting_files/image011.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign
1138the label for the numerator by clicking on the menu button next to label <b
1139style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1140mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>;
1141this will bring up a list of variable types, from this select <b
1142style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1143mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Phase</span></b>
1144to bring up a list of phase variables, from that list select variable <b
1145style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1146mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>,
1147which is the <i style='mso-bidi-font-style:normal'>b</i> lattice parameter for
1148phase 0 (CuCr<sub>2</sub>O<sub>4</sub>). Repeat this process to select assign
1149to label <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1150mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>c</span></b>
1151variable <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1152mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>.</p>
1153
1154<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1155
1156<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
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1158 type="#_x0000_t75" style='width:315.75pt;height:129pt;visibility:visible;
1159 mso-wrap-style:square'>
1160 <v:imagedata src="ParametricFitting_files/image012.png" o:title="" croptop="4597f"
1161  cropbottom="9873f" cropleft="3348f" cropright="3506f"/>
1162</v:shape><![endif]--><![if !vml]><img border=0 width=421 height=172
1163src="ParametricFitting_files/image013.png" v:shapes="Picture_x0020_8"><![endif]></span></p>
1164
1165<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1166
1167<p class=MsoListParagraphCxSpMiddle>As each variable is set, the window shows
1168the value for each variable from the first refinement and when the expression
1169can be evaluated, the value for the expression, evaluated for that first
1170refinement. Only when the expression can be evaluated can the <b
1171style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1172mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1173button be pressed. </p>
1174
1175<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1176
1177<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1178mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_9" o:spid="_x0000_i1048"
1179 type="#_x0000_t75" style='width:314.25pt;height:153.75pt;visibility:visible;
1180 mso-wrap-style:square'>
1181 <v:imagedata src="ParametricFitting_files/image014.png" o:title="" croptop="3770f"
1182  cropbottom="8549f" cropleft="3371f" cropright="3424f"/>
1183</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=205
1184src="ParametricFitting_files/image015.png" v:shapes="Picture_x0020_9"><![endif]></span></p>
1185
1186<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1187
1188<p class=MsoListParagraphCxSpMiddle>Press<b style='mso-bidi-font-weight:normal'>
1189</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1190mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1191to close the window and add the pseudo-variable. Note that a new column appears
1192at the end of the sequential refinement data table. </p>
1193
1194<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1195
1196<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1197mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_10" o:spid="_x0000_i1047"
1198 type="#_x0000_t75" style='width:354pt;height:180.75pt;visibility:visible;
1199 mso-wrap-style:square'>
1200 <v:imagedata src="ParametricFitting_files/image016.png" o:title="" croptop="2743f"
1201  cropbottom="7533f" cropleft="2917f" cropright="3097f"/>
1202</v:shape><![endif]--><![if !vml]><img border=0 width=472 height=241
1203src="ParametricFitting_files/image017.png" v:shapes="Picture_x0020_10"><![endif]></span></p>
1204
1205<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_11"
1206 o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
1207 text-align:left;margin-left:42.3pt;margin-top:666.2pt;width:93.5pt;height:39pt;
1208 z-index:251660288;visibility:visible;mso-wrap-style:square;
1209 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1210 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1211 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1212 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1213 mso-position-vertical-relative:margin;mso-width-percent:0;
1214 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1215 <v:imagedata src="ParametricFitting_files/image018.png" o:title=""/>
1216 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1217</v:shape><![endif]--><![if !vml]><img width=125 height=52
1218src="ParametricFitting_files/image019.jpg" align=right hspace=12 v:shapes="Picture_x0020_11"><![endif]>Also
1219note that positioning the mouse above any cell in the table and waiting for a
1220few seconds causes the s.u. to be shown as a tooltip. </p>
1221
1222<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1223
1224<p class=MsoListParagraphCxSpMiddle>Double clicking on the <b style='mso-bidi-font-weight:
1225normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1226major-latin;mso-hansi-theme-font:major-latin'>b/c</span></b> column will cause
1227the pseudo-variable to be plotted:</p>
1228
1229<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1230
1231<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1232mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1046"
1233 type="#_x0000_t75" style='width:351.75pt;height:303pt;visibility:visible;
1234 mso-wrap-style:square'>
1235 <v:imagedata src="ParametricFitting_files/image020.png" o:title="" croptop="2107f"
1236  cropbottom="4939f" cropleft="3133f" cropright="3309f"/>
1237</v:shape><![endif]--><![if !vml]><img border=0 width=469 height=404
1238src="ParametricFitting_files/image021.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
1239
1240<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1241
1242<p class=MsoListParagraphCxSpMiddle>Press the &quot;<b style='mso-bidi-font-weight:
1243normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1244major-latin;mso-hansi-theme-font:major-latin'>s</span></b>&quot; key inside the
1245plot to select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1246normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1247major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> as the
1248x-axis. </p>
1249
1250<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1251
1252<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1253mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1045"
1254 type="#_x0000_t75" style='width:162.75pt;height:156pt;visibility:visible;
1255 mso-wrap-style:square'>
1256 <v:imagedata src="ParametricFitting_files/image022.png" o:title="" croptop="4105f"
1257  cropbottom="8920f" cropleft="6109f" cropright="5911f"/>
1258</v:shape><![endif]--><![if !vml]><img border=0 width=217 height=208
1259src="ParametricFitting_files/image023.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
1260
1261<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1262
1263<p class=MsoListParagraphCxSpMiddle>The plot will then appear as below:</p>
1264
1265<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1266
1267<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1268mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1044"
1269 type="#_x0000_t75" style='width:353.25pt;height:303.75pt;visibility:visible;
1270 mso-wrap-style:square'>
1271 <v:imagedata src="ParametricFitting_files/image024.png" o:title="" croptop="1859f"
1272  cropbottom="5066f" cropleft="3024f" cropright="3203f"/>
1273</v:shape><![endif]--><![if !vml]><img border=0 width=471 height=405
1274src="ParametricFitting_files/image025.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
1275
1276<h1>Step 3: Fitting a simple parametric equation to the unit cell volume</h1>
1277
1278<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1279
1280<p class=MsoNormal>In the next few steps we will fit increasingly sophisticated
1281equations of state to different parameters. As a simple example, we can
1282double-click on<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1283normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1284major-latin;mso-hansi-theme-font:major-latin'>0::Vol</span></b>, to plot the
1285unit volume for the first phase vs. T, and fit this to a straight line.<span
1286style='mso-spacerun:yes'>  </span></p>
1287
1288<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1289
1290<p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US;
1291mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_15" o:spid="_x0000_i1043"
1292 type="#_x0000_t75" style='width:356.25pt;height:305.25pt;visibility:visible;
1293 mso-wrap-style:square'>
1294 <v:imagedata src="ParametricFitting_files/image026.png" o:title="" croptop="2107f"
1295  cropbottom="4567f" cropleft="3024f" cropright="2769f"/>
1296</v:shape><![endif]--><![if !vml]><img border=0 width=475 height=407
1297src="ParametricFitting_files/image027.png" v:shapes="Picture_x0020_15"><![endif]></span></p>
1298
1299<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1300style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1301style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1302</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16"
1303 o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
1304 text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt;
1305 z-index:251661312;visibility:visible;mso-wrap-style:square;
1306 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1307 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1308 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1309 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1310 mso-position-vertical-relative:margin;mso-width-percent:0;
1311 mso-height-percent:0;mso-width-relative:margin;mso-height-relative:margin'>
1312 <v:imagedata src="ParametricFitting_files/image028.png" o:title=""/>
1313 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1314</v:shape><![endif]--><![if !vml]><img width=97 height=71
1315src="ParametricFitting_files/image029.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
1316on the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1317mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Parametric
1318Fit/Add</span></b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1319major-latin;mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:
1320normal'>equation</b></span> menu item, which opens a window similar to what was
1321used before to create a pseudo-variable, except that here an equation is
1322defined.<span style='mso-spacerun:yes'>  </span></p>
1323
1324<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1325
1326<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1327mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_17" o:spid="_x0000_i1042"
1328 type="#_x0000_t75" style='width:311.25pt;height:152.25pt;visibility:visible;
1329 mso-wrap-style:square'>
1330 <v:imagedata src="ParametricFitting_files/image030.png" o:title="" croptop="4213f"
1331  cropbottom="8982f" cropleft="3490f" cropright="3653f"/>
1332</v:shape><![endif]--><![if !vml]><img border=0 width=415 height=203
1333src="ParametricFitting_files/image031.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
1334
1335<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1336
1337<p class=MsoNormal>We will want to fit the unit cell volume to an equation of
1338form <i style='mso-bidi-font-style:normal'>y = mx+b</i>, where y will be the
1339volume and x temperature and m and b will be best fit. </p>
1340
1341<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><o:p>&nbsp;</o:p></b></p>
1342
1343<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><b
1344style='mso-bidi-font-weight:normal'><span style='mso-fareast-font-family:"Times New Roman";
1345mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span
1346style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]>Click
1347the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1348normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1349major-latin;mso-hansi-theme-font:major-latin'>&#8597; </span></b>button to select
1350a dependent variable for &quot;y&quot;. From the phase list, select the unit
1351cell volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b
1352style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1353style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1354mso-hansi-theme-font:major-latin'>0::Vol</span></b> and then press<b
1355style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1356style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1357mso-hansi-theme-font:major-latin'>OK</span>.<o:p></o:p></b></p>
1358
1359<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1360
1361<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1362mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19" o:spid="_x0000_i1041"
1363 type="#_x0000_t75" style='width:314.25pt;height:132pt;visibility:visible;
1364 mso-wrap-style:square'>
1365 <v:imagedata src="ParametricFitting_files/image032.png" o:title="" croptop="3830f"
1366  cropbottom="9359f" cropleft="3705f" cropright="3502f"/>
1367</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=176
1368src="ParametricFitting_files/image033.png" v:shapes="Picture_x0020_19"><![endif]></span></p>
1369
1370<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1371mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20" o:spid="_x0000_i1040"
1372 type="#_x0000_t75" style='width:311.25pt;height:155.25pt;visibility:visible;
1373 mso-wrap-style:square'>
1374 <v:imagedata src="ParametricFitting_files/image034.png" o:title="" croptop="3548f"
1375  cropbottom="8549f" cropleft="3611f" cropright="3665f"/>
1376</v:shape><![endif]--><![if !vml]><img border=0 width=415 height=207
1377src="ParametricFitting_files/image035.png" v:shapes="Picture_x0020_20"><![endif]></span></p>
1378
1379<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1380
1381<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1382style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1383style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1384</span></span></span><![endif]>Now we enter the equation using Python syntax,
1385which is simply &quot;<b style='mso-bidi-font-weight:normal'><span
1386style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1387mso-hansi-theme-font:major-latin'>m*x + b</span></b>&quot;, note that the use
1388of spaces and the choice of labels is arbitrary. If one were to accidentally
1389leave out the &quot;<span style='font-family:Courier'>*</span>&quot; and typed
1390&quot;<span style='font-family:Courier'>mx</span>&quot; then that would be considered
1391a label, if &quot;<span style='font-family:Courier'>m x</span>&quot; were
1392typed, an error message would be displayed. </p>
1393
1394<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1395
1396<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1397mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_21" o:spid="_x0000_i1039"
1398 type="#_x0000_t75" style='width:315.75pt;height:155.25pt;visibility:visible;
1399 mso-wrap-style:square'>
1400 <v:imagedata src="ParametricFitting_files/image036.png" o:title="" croptop="3326f"
1401  cropbottom="8766f" cropleft="3010f" cropright="3419f"/>
1402</v:shape><![endif]--><![if !vml]><img border=0 width=421 height=207
1403src="ParametricFitting_files/image037.png" v:shapes="Picture_x0020_21"><![endif]></span></p>
1404
1405<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1406
1407<p class=MsoListParagraphCxSpMiddle>Then press <b style='mso-bidi-font-weight:
1408normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1409major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> or wait a few
1410seconds and a window changes, as above, allowing assignment of the three new
1411labels (<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1412mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>,
1413<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1414mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1415and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1416mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>).
1417</p>
1418
1419<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1420
1421<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1422style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1423style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1424</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_22"
1425 o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
1426 text-align:left;margin-left:43.8pt;margin-top:603.2pt;width:95pt;height:35pt;
1427 z-index:251662336;visibility:visible;mso-wrap-style:square;
1428 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1429 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1430 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1431 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1432 mso-position-vertical-relative:margin;mso-width-percent:0;
1433 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1434 <v:imagedata src="ParametricFitting_files/image038.png" o:title=""/>
1435 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1436</v:shape><![endif]--><![if !vml]><img width=127 height=47
1437src="ParametricFitting_files/image039.jpg" align=right hspace=12 v:shapes="Picture_x0020_22"><![endif]>Assign
1438<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1439mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1440and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1441mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1442as refinable “free” parameters: Click on the <b style='mso-bidi-font-weight:
1443normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1444major-latin;mso-hansi-theme-font:major-latin'>&#8597;</span></b><span
1445style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1446mso-hansi-theme-font:major-latin'> </span>button next to b and select<b
1447style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1448style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1449mso-hansi-theme-font:major-latin'>Free</span></b>. Then repeat this for<b
1450style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1451style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1452mso-hansi-theme-font:major-latin'>m</span></b>. </p>
1453
1454<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1455mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_23" o:spid="_x0000_i1038"
1456 type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible;
1457 mso-wrap-style:square'>
1458 <v:imagedata src="ParametricFitting_files/image040.png" o:title="" croptop="3992f"
1459  cropbottom="8541f" cropleft="3250f" cropright="3415f"/>
1460</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=204
1461src="ParametricFitting_files/image041.png" v:shapes="Picture_x0020_23"><![endif]></span></p>
1462
1463<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1464
1465<p class=MsoListParagraphCxSpMiddle>It is not necessary to change the default
1466variable names, the initial values, or the refine flag. </p>
1467
1468<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1469
1470<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1471style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1472style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1473</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_24"
1474 o:spid="_x0000_s1026" type="#_x0000_t75" style='position:absolute;left:0;
1475 text-align:left;margin-left:24.3pt;margin-top:837.2pt;width:75.5pt;height:26pt;
1476 z-index:251663360;visibility:visible;mso-wrap-style:square;
1477 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1478 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1479 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1480 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1481 mso-position-vertical-relative:margin;mso-width-percent:0;
1482 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1483 <v:imagedata src="ParametricFitting_files/image042.png" o:title=""/>
1484 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1485</v:shape><![endif]--><![if !vml]><img width=101 height=35
1486src="ParametricFitting_files/image043.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
1487set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1488mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>
1489as the temperature, click on the <b style='mso-bidi-font-weight:normal'><span
1490style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1491mso-hansi-theme-font:major-latin'>&#8597;</span></b><span style='font-family:
1492"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1493major-latin'> </span>button next to <b style='mso-bidi-font-weight:normal'><span
1494style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1495mso-hansi-theme-font:major-latin'>x</span></b>, select<b style='mso-bidi-font-weight:
1496normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1497"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1498major-latin'>Global</span>,</b> and then select<b style='mso-bidi-font-weight:
1499normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1500"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1501major-latin'>Temperature</span></b> from the list and click on<b
1502style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1503style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1504mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
1505
1506<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1507
1508<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1509style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1510 id="Picture_x0020_25" o:spid="_x0000_i1037" type="#_x0000_t75" style='width:245.25pt;
1511 height:108.75pt;visibility:visible;mso-wrap-style:square'>
1512 <v:imagedata src="ParametricFitting_files/image044.png" o:title="" croptop="5483f"
1513  cropbottom="10984f" cropleft="4191f" cropright="4159f"/>
1514</v:shape><![endif]--><![if !vml]><img border=0 width=327 height=145
1515src="ParametricFitting_files/image045.png" v:shapes="Picture_x0020_25"><![endif]></span></p>
1516
1517<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1518
1519<p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values
1520of the dependent variable and the expression for the first refinement, as
1521below. <span style='mso-spacerun:yes'> </span>Optionally, the values for <b
1522style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1523mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1524and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1525mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1526could be changed manually and the displayed value will be updated when <b
1527style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1528mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b>
1529is pressed or after a short delay.</p>
1530
1531<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1532
1533<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1534mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_26" o:spid="_x0000_i1036"
1535 type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible;
1536 mso-wrap-style:square'>
1537 <v:imagedata src="ParametricFitting_files/image046.png" o:title="" croptop="3992f"
1538  cropbottom="8541f" cropleft="3490f" cropright="3294f"/>
1539</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=204
1540src="ParametricFitting_files/image047.png" v:shapes="Picture_x0020_26"><![endif]></span></p>
1541
1542<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1543
1544<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1545style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1546style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1547</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1548style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1549mso-hansi-theme-font:major-latin'>OK</span></b> button to save this parametric
1550fit. The <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1551mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1552and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1553mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1554values are then refined and the console shows the results:</p>
1555
1556<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1557style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1558style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1559</span></span></span><![endif]><o:p>&nbsp;</o:p></p>
1560
1561<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1562auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'>Fit
1563Results<o:p></o:p></span></b></p>
1564
1565<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1566auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
1567style='mso-spacerun:yes'>   </span>::m<span style='mso-spacerun:yes'>  </span>=
1568-0.4476(5)<o:p></o:p></span></b></p>
1569
1570<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1571auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
1572style='mso-spacerun:yes'>   </span>::b<span style='mso-spacerun:yes'>  </span>=
1573577.73(8)<o:p></o:p></span></b></p>
1574
1575<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1576auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
1577
1578<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1579mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_27" o:spid="_x0000_i1035"
1580 type="#_x0000_t75" style='width:350.25pt;height:302.25pt;visibility:visible;
1581 mso-wrap-style:square'>
1582 <v:imagedata src="ParametricFitting_files/image048.png" o:title="" croptop="2231f"
1583  cropbottom="5061f" cropleft="3133f" cropright="3524f"/>
1584</v:shape><![endif]--><![if !vml]><img border=0 width=467 height=403
1585src="ParametricFitting_files/image049.png" v:shapes="Picture_x0020_27"><![endif]></span><span
1586style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin;
1587mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p>
1588
1589<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1590
1591<p class=MsoListParagraphCxSpMiddle>Also, the plot shows the actual values and
1592the fit. Note that while the values are close to a straight line, the
1593deviations are huge compared to the estimated uncertainty values for the unit
1594cell volume (shown as very small error bars, unless the plot is
1595magnified).<span style='mso-spacerun:yes'>  </span>If a linear relationship had
1596been defined as a constraint (as can be done with other software packages) <i>the
1597unit cell parameters would have been forced onto this line with no obvious
1598indication that this was not a correct result.</i> </p>
1599
1600<p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p>
1601
1602<h1>Step 4: Examples of more complex fitt<span class=Heading1Char><span
1603style='font-weight:normal'>i</span></span>ng equations</h1>
1604
1605<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1606
1607<p class=MsoNormal>Here we will look at how more sophisticated Python
1608capabilities can be used to fit more complex behavior.<span
1609style='mso-spacerun:yes'>  </span></p>
1610
1611<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1612style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1613style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1614</span></span></span><![endif]>Optionally Double-click on the<b
1615style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1616style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1617mso-hansi-theme-font:major-latin'>0::c</span> </b>column. Note the changes in
1618the <i style='mso-bidi-font-style:normal'>c</i> unit cell parameter as a
1619function of temperature, as shown below. </p>
1620
1621<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1622
1623<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1624mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_28" o:spid="_x0000_i1034"
1625 type="#_x0000_t75" style='width:350.25pt;height:303pt;visibility:visible;
1626 mso-wrap-style:square'>
1627 <v:imagedata src="ParametricFitting_files/image050.png" o:title="" croptop="658f"
1628  cropbottom="2861f" cropleft="1709f" cropright="1878f"/>
1629</v:shape><![endif]--><![if !vml]><img border=0 width=467 height=404
1630src="ParametricFitting_files/image051.png" v:shapes="Picture_x0020_28"><![endif]></span></p>
1631
1632<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1633
1634<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1635style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1636style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1637</span></span></span><![endif]>We will now fit <i style='mso-bidi-font-style:
1638normal'>c</i> as a function of form 1/(1 + exp[k(t-t<sub>0</sub>)]). Select the<b
1639style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1640style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1641mso-hansi-theme-font:major-latin'>Parametric Fit/Add</span></b><span
1642style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1643mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:normal'>equation</b></span>
1644menu item (as before). For the dependent variable, click on<b style='mso-bidi-font-weight:
1645normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1646"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1647major-latin'>Phase</span></b> and then select<b style='mso-bidi-font-weight:
1648normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1649"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1650major-latin'>0::c</span></b>.<span style='mso-spacerun:yes'>  </span></p>
1651
1652<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1653
1654<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1655style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1656style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1657</span></span></span><![endif]>We then enter expression</p>
1658
1659<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1660
1661<p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><b
1662style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1663mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b + c /
1664(1+math.exp(k*(t-t0)))<o:p></o:p></span></b></p>
1665
1666<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria","serif";
1667mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
1668
1669<p class=MsoListParagraphCxSpLast>Note that use of spaces here is arbitrary,
1670but all parentheses are required. Also note that the exponential function
1671[exp(x) or e<sup>x</sup>] is supplied in Python in the math and numpy modules
1672as math.exp(), numpy.exp(). Either of these can be used as well as the
1673convenience abbreviation of np.exp(). Further, there are also a small number of
1674common functions that are defined without the prefix for convenience [these
1675include pi, exp(), sqrt(), sind(), cosd(), tand() where d indicates an argument
1676in degrees.]</p>
1677
1678<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1679
1680<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1681style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1682style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1683</span></span></span><![endif]>Assign free variables to<b style='mso-bidi-font-weight:
1684normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1685"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1686major-latin'>b</span>, </b><b style='mso-bidi-font-weight:normal'><span
1687style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1688mso-hansi-theme-font:major-latin'>c</span>, </b><b style='mso-bidi-font-weight:
1689normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1690major-latin;mso-hansi-theme-font:major-latin'>k</span> </b>and<b
1691style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1692style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1693mso-hansi-theme-font:major-latin'>t0</span></b>, similar as to what was done
1694before. Note that the<b style='mso-bidi-font-weight:normal'> </b>variable name
1695for the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1696mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1697label was set automatically to <b style='mso-bidi-font-weight:normal'><span
1698style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1699mso-hansi-theme-font:major-latin'>b_1</span></b>. If the same variable name (<b
1700style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1701mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>)
1702was used again here, the same value would be used to fit both equations, which
1703is certainly not what we want here.<span style='mso-spacerun:yes'>  </span>Finally,
1704since this equation is too complex to fit without some reasonable starting
1705values, specify an approximate value for the vicinity of where <i
1706style='mso-bidi-font-style:normal'>c</i> starts to change, enter<b
1707style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1708style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1709mso-hansi-theme-font:major-latin'>120</span> </b>for <b style='mso-bidi-font-weight:
1710normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1711major-latin;mso-hansi-theme-font:major-latin'>t0</span>.</b></p>
1712
1713<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1714
1715<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1716mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_40" o:spid="_x0000_i1033"
1717 type="#_x0000_t75" style='width:317.25pt;height:171pt;visibility:visible;
1718 mso-wrap-style:square'>
1719 <v:imagedata src="ParametricFitting_files/image052.png" o:title="" croptop="3272f"
1720  cropbottom="7856f" cropleft="3218f" cropright="3436f"/>
1721</v:shape><![endif]--><![if !vml]><img border=0 width=423 height=228
1722src="ParametricFitting_files/image053.png" v:shapes="Picture_x0020_40"><![endif]></span></p>
1723
1724<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1725
1726<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1727style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1728style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1729</span></span></span><![endif]>Optionally, press the <b style='mso-bidi-font-weight:
1730normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1731major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b> button to see if
1732the equation can be refined with starting values of zero for the other three
1733parameters.<span style='mso-spacerun:yes'>  </span>A fit is obtained and is
1734plotted. </p>
1735
1736<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1737
1738<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1739style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1740 id="Picture_x0020_41" o:spid="_x0000_i1032" type="#_x0000_t75" style='width:317.25pt;
1741 height:168.75pt;visibility:visible;mso-wrap-style:square'>
1742 <v:imagedata src="ParametricFitting_files/image054.png" o:title="" croptop="3477f"
1743  cropbottom="8265f" cropleft="3218f" cropright="3436f"/>
1744</v:shape><![endif]--><![if !vml]><img border=0 width=423 height=225
1745src="ParametricFitting_files/image055.png" v:shapes="Picture_x0020_41"><![endif]></span></p>
1746
1747<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1748
1749<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1750style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1751style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1752</span></span></span><![endif]>Press <b style='mso-bidi-font-weight:normal'><span
1753style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1754mso-hansi-theme-font:major-latin'>OK</span></b> to add this equation to the
1755parametric fits and both equations are refit (and if the label for <b
1756style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1757mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1758had not been changed, applying the constraint that <b style='mso-bidi-font-weight:
1759normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1760major-latin;mso-hansi-theme-font:major-latin'>b</span></b> must be the same for
1761both equations.) </p>
1762
1763<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1764
1765<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1766mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_32" o:spid="_x0000_i1031"
1767 type="#_x0000_t75" style='width:353.25pt;height:300.75pt;visibility:visible;
1768 mso-wrap-style:square'>
1769 <v:imagedata src="ParametricFitting_files/image056.png" o:title="" croptop="2602f"
1770  cropbottom="4943f" cropleft="3024f" cropright="3097f"/>
1771</v:shape><![endif]--><![if !vml]><img border=0 width=471 height=401
1772src="ParametricFitting_files/image057.png" v:shapes="Picture_x0020_32"><![endif]></span></p>
1773
1774<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1775
1776<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1777style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1778style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1779</span></span></span><![endif]>We will now try a more complex equation for the
1780same data, where we set the value to <b style='mso-bidi-font-weight:normal'><span
1781style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1782mso-hansi-theme-font:major-latin'>b</span></b> for temperatures less than <b
1783style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1784mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
1785and a critical phenomenon formulism [(t-t<sub>0</sub>)<sup>-a</sup>] for
1786temperatures above <b style='mso-bidi-font-weight:normal'><span
1787style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1788mso-hansi-theme-font:major-latin'>t0</span></b>. To do this we use the Python
1789syntax &quot;<b style='mso-bidi-font-weight:normal'><span style='font-family:
1790"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1791major-latin'>x1 if testexp else x2</span></b>&quot; which is <b
1792style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1793mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x1</span></b>
1794where the test expression (<b style='mso-bidi-font-weight:normal'><span
1795style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1796mso-hansi-theme-font:major-latin'>testexp</span></b> here) is <b
1797style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1798mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>True</span></b>
1799and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1800mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x2</span></b>
1801otherwise. Thus we can use an expression such as:</p>
1802
1803<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span></p>
1804
1805<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1806auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1807mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b if t &lt;=
1808t0 else b + c * (1-(t-t0)**d)<o:p></o:p></span></b></p>
1809
1810<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1811auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
1812
1813<p class=MsoListParagraphCxSpMiddle>This may be easier to read with the
1814inclusion of extra parentheses (that have no effect on the execution):<span
1815style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin;
1816mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p>
1817
1818<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1819
1820<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1821auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1822mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b if (t
1823&lt;= t0) else (b + c * (1-(t-t0)**d))<o:p></o:p></span></b></p>
1824
1825<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1826auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
1827
1828<p class=MsoListParagraphCxSpMiddle>As before, add an equation (or edit an
1829existing one), set the dependent variable to <b style='mso-bidi-font-weight:
1830normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1831major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>, enter the
1832expression and assign the free and fixed variables, as before. Set the exponent
1833and the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1834mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
1835values to initial values of <b style='mso-bidi-font-weight:normal'><span
1836style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1837mso-hansi-theme-font:major-latin'>-0.025</span></b> and <b style='mso-bidi-font-weight:
1838normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1839major-latin;mso-hansi-theme-font:major-latin'>120</span></b>, respectively, and
1840turn off their refinement flags. Then press <b style='mso-bidi-font-weight:
1841normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1842major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b>.<span
1843style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin;
1844mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p>
1845
1846<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1847mso-no-proof:yes'><o:p>&nbsp;</o:p></span></p>
1848
1849<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1850mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1030"
1851 type="#_x0000_t75" style='width:315.75pt;height:168.75pt;visibility:visible;
1852 mso-wrap-style:square'>
1853 <v:imagedata src="ParametricFitting_files/image058.png" o:title="" croptop="3691f"
1854  cropbottom="7919f" cropleft="2889f" cropright="3536f"/>
1855</v:shape><![endif]--><![if !vml]><img border=0 width=421 height=225
1856src="ParametricFitting_files/image059.png" v:shapes="Picture_x0020_42"><![endif]></span></p>
1857
1858<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1859
1860<p class=MsoListParagraphCxSpMiddle>Then turn refinement of <b
1861style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1862mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
1863back on and press <b style='mso-bidi-font-weight:normal'><span
1864style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1865mso-hansi-theme-font:major-latin'>OK</span></b>. The resulting fit shows the
1866trends quite well: </p>
1867
1868<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1869
1870<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1871mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1029"
1872 type="#_x0000_t75" style='width:351pt;height:302.25pt;visibility:visible;
1873 mso-wrap-style:square'>
1874 <v:imagedata src="ParametricFitting_files/image060.png" o:title="" croptop="2354f"
1875  cropbottom="4937f" cropleft="3349f" cropright="3198f"/>
1876</v:shape><![endif]--><![if !vml]><img border=0 width=468 height=403
1877src="ParametricFitting_files/image061.png" v:shapes="Picture_x0020_35"><![endif]></span></p>
1878
1879<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1880
1881<h1>Step 5: Fitting to a Pseudo Variable</h1>
1882
1883<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1884
1885<p class=MsoNormal>One can fit a function to a pseudo variable, just as to a
1886direct variable. If we look at the plot shown in Step 2, it appears there are
1887two regions. We can treat the lower range as a power-law growth, T<sup>k</sup>,
1888and the upper region as linear (m*T + b). </p>
1889
1890<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1891
1892<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
1893style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1894style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1895</span></span></span><![endif]>As before, add an equation<b style='mso-bidi-font-weight:
1896normal'> </b>(or edit an existing one). For the dependent variable, select<b
1897style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1898style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1899mso-hansi-theme-font:major-latin'>Global</span></b> to bring up that selection
1900of variables and select<b style='mso-bidi-font-weight:normal'> </b><b
1901style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1902mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b/c</span> </b>and
1903press<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1904normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1905major-latin;mso-hansi-theme-font:major-latin'>OK</span>. </b></p>
1906
1907<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1908
1909<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
1910style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1911style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1912</span></span></span><![endif]>Then enter the equation as </p>
1913
1914<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1915
1916<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1917auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1918mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1919"Lucida Grande";color:black'>(o1 + (a1*T)**k) if T &lt; 120 else (o2 + a2*T</span></b><b
1920style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1921mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>)<o:p></o:p></span></b></p>
1922
1923<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria","serif";
1924mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
1925
1926<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
1927style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1928style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1929</span></span></span><![endif]>Assign the free and fixed variables, as shown
1930below. Note that the exponent (<b style='mso-bidi-font-weight:normal'><span
1931style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1932mso-hansi-theme-font:major-latin'>k</span></b>) and scaling factor (<b
1933style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1934mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a1</span></b>),
1935cannot start as zero<b style='mso-bidi-font-weight:normal'>. </b></p>
1936
1937<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1938
1939<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1940mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_44" o:spid="_x0000_i1028"
1941 type="#_x0000_t75" style='width:332.25pt;height:186pt;visibility:visible;
1942 mso-wrap-style:square'>
1943 <v:imagedata src="ParametricFitting_files/image062.png" o:title="" croptop="3190f"
1944  cropbottom="8105f" cropleft="3416f" cropright="3246f"/>
1945</v:shape><![endif]--><![if !vml]><img border=0 width=443 height=248
1946src="ParametricFitting_files/image063.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
1947
1948<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1949
1950<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
1951style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1952style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1953</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1954style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1955mso-hansi-theme-font:major-latin'>Fit</span></b> button and note that all the
1956free variables are adjusted </p>
1957
1958<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1959
1960<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1961mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_45" o:spid="_x0000_i1027"
1962 type="#_x0000_t75" style='width:333.75pt;height:189pt;visibility:visible;
1963 mso-wrap-style:square'>
1964 <v:imagedata src="ParametricFitting_files/image064.png" o:title="" croptop="2815f"
1965  cropbottom="7542f" cropleft="3074f" cropright="3244f"/>
1966</v:shape><![endif]--><![if !vml]><img border=0 width=445 height=252
1967src="ParametricFitting_files/image065.png" v:shapes="Picture_x0020_45"><![endif]></span><span
1968style='mso-spacerun:yes'> </span></p>
1969
1970<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1971
1972<p class=MsoListParagraphCxSpMiddle>Also, that the plotted result is close to
1973the values from the refinement.</p>
1974
1975<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1976
1977<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1978mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_48" o:spid="_x0000_i1026"
1979 type="#_x0000_t75" style='width:348.75pt;height:302.25pt;visibility:visible;
1980 mso-wrap-style:square'>
1981 <v:imagedata src="ParametricFitting_files/image066.png" o:title="" croptop="2231f"
1982  cropbottom="5061f" cropleft="3350f" cropright="3415f"/>
1983</v:shape><![endif]--><![if !vml]><img border=0 width=465 height=403
1984src="ParametricFitting_files/image067.png" v:shapes="Picture_x0020_48"><![endif]></span></p>
1985
1986<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1987
1988<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
1989
1990<p class=MsoNormal>This completes this tutorial on use of Pseudo Variables and
1991parametric fitting. </p>
1992
1993<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1994
1995<h1>Advanced Use: Externally-Defined Functions</h1>
1996
1997<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1998
1999<p class=MsoNormal>The parametric refinement section of GSAS-II is intended to
2000be quite flexible, but for very complex fitting, a user may wish to define a custom
2001Python function. Such a function can be defined in a separate module that will
2002be accessed with a parametric equation (or pseudo variable).</p>
2003
2004<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2005
2006<p class=MsoNormal>As an example of that, if we create a file called <b
2007style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2008mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fittest.py</span></b>
2009and locate that file in any directory in the Python path (most simply, in the
2010directory where the project file, <b style='mso-bidi-font-weight:normal'><span
2011style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
2012mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b>, is located) and place
2013a routine <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2014mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fitfxn</span></b>
2015in that module:</p>
2016
2017<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'>
2018
2019<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre><pre><b
2020style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2021mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'># file fittest.py<o:p></o:p></span></b></pre><pre><b
2022style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2023mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>def fitfxn(T,o1,o2,a1,a2,k):<o:p></o:p></span></b></pre><pre><b
2024style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2025mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>    </span>if T &lt; 120:<o:p></o:p></span></b></pre><pre><b
2026style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2027mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>        </span>return o1 + (a1*T)**k<o:p></o:p></span></b></pre><pre><b
2028style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2029mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>    </span>else:<o:p></o:p></span></b></pre><pre><b
2030style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2031mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>        </span>return o2 + a2*T<o:p></o:p></span></b></pre></blockquote>
2032
2033</blockquote>
2034
2035<p class=MsoNormal>The function <b style='mso-bidi-font-weight:normal'><span
2036style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
2037mso-hansi-theme-font:major-latin'>fittest.fitfxn</span></b> can then be used to
2038duplicate the previous example: </p>
2039
2040<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2041
2042<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2043 id="Picture_x0020_47" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:315pt;
2044 height:185.25pt;visibility:visible;mso-wrap-style:square'>
2045 <v:imagedata src="ParametricFitting_files/image068.png" o:title="" croptop="3393f"
2046  cropbottom="7847f" cropleft="3471f" cropright="3290f"/>
2047</v:shape><![endif]--><![if !vml]><img border=0 width=420 height=247
2048src="ParametricFitting_files/image069.png" v:shapes="Picture_x0020_47"><![endif]></span></p>
2049
2050<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2051
2052<p class=MsoNormal>Note that module <b style='mso-bidi-font-weight:normal'><span
2053style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
2054mso-hansi-theme-font:major-latin'>fittest</span></b> is automatically located
2055and loaded, provided it is found in the path. No changes to the standard
2056distributed GSAS-II code are needed. Care to avoid use of a module name already
2057found in GSAS-II or Python is wise.</p>
2058
2059</div>
2060
2061</body>
2062
2063</html>
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