source: Tutorials/SeqParametric/ParametricFitting.htm @ 1705

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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=""/>
1201</v:shape><![endif]--><![if !vml]><img border=0 width=472 height=241
1202src="ParametricFitting_files/image016.png" v:shapes="Picture_x0020_10"><![endif]></span></p>
1203
1204<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_11"
1205 o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
1206 text-align:left;margin-left:42.3pt;margin-top:666.2pt;width:93.5pt;height:39pt;
1207 z-index:251660288;visibility:visible;mso-wrap-style:square;
1208 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1209 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1210 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1211 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1212 mso-position-vertical-relative:margin;mso-width-percent:0;
1213 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1214 <v:imagedata src="ParametricFitting_files/image018.png" o:title=""/>
1215 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1216</v:shape><![endif]--><![if !vml]><img width=125 height=52
1217src="ParametricFitting_files/image019.jpg" align=right hspace=12 v:shapes="Picture_x0020_11"><![endif]>Also
1218note that positioning the mouse above any cell in the table and waiting for a
1219few seconds causes the s.u. to be shown as a tooltip. </p>
1220
1221<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1222
1223<p class=MsoListParagraphCxSpMiddle>Double clicking on the <b style='mso-bidi-font-weight:
1224normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1225major-latin;mso-hansi-theme-font:major-latin'>b/c</span></b> column will cause
1226the pseudo-variable to be plotted:</p>
1227
1228<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1229
1230<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1231mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1046"
1232 type="#_x0000_t75" style='width:351.75pt;height:303pt;visibility:visible;
1233 mso-wrap-style:square'>
1234 <v:imagedata src="ParametricFitting_files/image020.png" o:title=""/>
1235</v:shape><![endif]--><![if !vml]><img border=0 width=469 height=404
1236src="ParametricFitting_files/image020.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
1237
1238<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1239
1240<p class=MsoListParagraphCxSpMiddle>Press the &quot;<b style='mso-bidi-font-weight:
1241normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1242major-latin;mso-hansi-theme-font:major-latin'>s</span></b>&quot; key inside the
1243plot to select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1244normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1245major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> as the
1246x-axis. </p>
1247
1248<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1249
1250<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1251mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1045"
1252 type="#_x0000_t75" style='width:162.75pt;height:156pt;visibility:visible;
1253 mso-wrap-style:square'>
1254 <v:imagedata src="ParametricFitting_files/image022.png" o:title="" croptop="4105f"
1255  cropbottom="8920f" cropleft="6109f" cropright="5911f"/>
1256</v:shape><![endif]--><![if !vml]><img border=0 width=217 height=208
1257src="ParametricFitting_files/image023.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
1258
1259<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1260
1261<p class=MsoListParagraphCxSpMiddle>The plot will then appear as below:</p>
1262
1263<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1264
1265<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1266mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1044"
1267 type="#_x0000_t75" style='width:353.25pt;height:303.75pt;visibility:visible;
1268 mso-wrap-style:square'>
1269 <v:imagedata src="ParametricFitting_files/image024.png" o:title=""/>
1270</v:shape><![endif]--><![if !vml]><img border=0 width=471 height=405
1271src="ParametricFitting_files/image024.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
1272
1273<h1>Step 3: Fitting a simple parametric equation to the unit cell volume</h1>
1274
1275<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1276
1277<p class=MsoNormal>In the next few steps we will fit increasingly sophisticated
1278equations of state to different parameters. As a simple example, we can
1279double-click on<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1280normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1281major-latin;mso-hansi-theme-font:major-latin'>0::Vol</span></b>, to plot the
1282unit volume for the first phase vs. T, and fit this to a straight line.<span
1283style='mso-spacerun:yes'>  </span></p>
1284
1285<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1286
1287<p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US;
1288mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_15" o:spid="_x0000_i1043"
1289 type="#_x0000_t75" style='width:356.25pt;height:305.25pt;visibility:visible;
1290 mso-wrap-style:square'>
1291 <v:imagedata src="ParametricFitting_files/image026.png" o:title=""/>
1292</v:shape><![endif]--><![if !vml]><img border=0 width=475 height=407
1293src="ParametricFitting_files/image026.png" v:shapes="Picture_x0020_15"><![endif]></span></p>
1294
1295<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1296style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1297style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1298</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16"
1299 o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
1300 text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt;
1301 z-index:251661312;visibility:visible;mso-wrap-style:square;
1302 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1303 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1304 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1305 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1306 mso-position-vertical-relative:margin;mso-width-percent:0;
1307 mso-height-percent:0;mso-width-relative:margin;mso-height-relative:margin'>
1308 <v:imagedata src="ParametricFitting_files/image028.png" o:title=""/>
1309 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1310</v:shape><![endif]--><![if !vml]><img width=97 height=71
1311src="ParametricFitting_files/image029.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
1312on the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1313mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Parametric
1314Fit/Add</span></b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1315major-latin;mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:
1316normal'>equation</b></span> menu item, which opens a window similar to what was
1317used before to create a pseudo-variable, except that here an equation is
1318defined.<span style='mso-spacerun:yes'>  </span></p>
1319
1320<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1321
1322<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1323mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_17" o:spid="_x0000_i1042"
1324 type="#_x0000_t75" style='width:311.25pt;height:152.25pt;visibility:visible;
1325 mso-wrap-style:square'>
1326 <v:imagedata src="ParametricFitting_files/image030.png" o:title="" croptop="4213f"
1327  cropbottom="8982f" cropleft="3490f" cropright="3653f"/>
1328</v:shape><![endif]--><![if !vml]><img border=0 width=415 height=203
1329src="ParametricFitting_files/image031.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
1330
1331<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1332
1333<p class=MsoNormal>We will want to fit the unit cell volume to an equation of
1334form <i style='mso-bidi-font-style:normal'>y = mx+b</i>, where y will be the
1335volume and x temperature and m and b will be best fit. </p>
1336
1337<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><o:p>&nbsp;</o:p></b></p>
1338
1339<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><b
1340style='mso-bidi-font-weight:normal'><span style='mso-fareast-font-family:"Times New Roman";
1341mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span
1342style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]>Click
1343the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1344normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1345major-latin;mso-hansi-theme-font:major-latin'>&#8597; </span></b>button to select
1346a dependent variable for &quot;y&quot;. From the phase list, select the unit
1347cell volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b
1348style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1349style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1350mso-hansi-theme-font:major-latin'>0::Vol</span></b> and then press<b
1351style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1352style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1353mso-hansi-theme-font:major-latin'>OK</span>.<o:p></o:p></b></p>
1354
1355<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1356
1357<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1358mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19" o:spid="_x0000_i1041"
1359 type="#_x0000_t75" style='width:314.25pt;height:132pt;visibility:visible;
1360 mso-wrap-style:square'>
1361 <v:imagedata src="ParametricFitting_files/image032.png" o:title="" croptop="3830f"
1362  cropbottom="9359f" cropleft="3705f" cropright="3502f"/>
1363</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=176
1364src="ParametricFitting_files/image033.png" v:shapes="Picture_x0020_19"><![endif]></span></p>
1365
1366<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1367mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20" o:spid="_x0000_i1040"
1368 type="#_x0000_t75" style='width:311.25pt;height:155.25pt;visibility:visible;
1369 mso-wrap-style:square'>
1370 <v:imagedata src="ParametricFitting_files/image034.png" o:title="" croptop="3548f"
1371  cropbottom="8549f" cropleft="3611f" cropright="3665f"/>
1372</v:shape><![endif]--><![if !vml]><img border=0 width=415 height=207
1373src="ParametricFitting_files/image035.png" v:shapes="Picture_x0020_20"><![endif]></span></p>
1374
1375<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1376
1377<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1378style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1379style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1380</span></span></span><![endif]>Now we enter the equation using Python syntax,
1381which is simply &quot;<b style='mso-bidi-font-weight:normal'><span
1382style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1383mso-hansi-theme-font:major-latin'>m*x + b</span></b>&quot;, note that the use
1384of spaces and the choice of labels is arbitrary. If one were to accidentally
1385leave out the &quot;<span style='font-family:Courier'>*</span>&quot; and typed
1386&quot;<span style='font-family:Courier'>mx</span>&quot; then that would be considered
1387a label, if &quot;<span style='font-family:Courier'>m x</span>&quot; were
1388typed, an error message would be displayed. </p>
1389
1390<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1391
1392<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1393mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_21" o:spid="_x0000_i1039"
1394 type="#_x0000_t75" style='width:315.75pt;height:155.25pt;visibility:visible;
1395 mso-wrap-style:square'>
1396 <v:imagedata src="ParametricFitting_files/image036.png" o:title="" croptop="3326f"
1397  cropbottom="8766f" cropleft="3010f" cropright="3419f"/>
1398</v:shape><![endif]--><![if !vml]><img border=0 width=421 height=207
1399src="ParametricFitting_files/image037.png" v:shapes="Picture_x0020_21"><![endif]></span></p>
1400
1401<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1402
1403<p class=MsoListParagraphCxSpMiddle>Then press <b style='mso-bidi-font-weight:
1404normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1405major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> or wait a few
1406seconds and a window changes, as above, allowing assignment of the three new
1407labels (<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1408mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>,
1409<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1410mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1411and <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'>b</span></b>).
1413</p>
1414
1415<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1416
1417<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1418style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1419style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1420</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_22"
1421 o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
1422 text-align:left;margin-left:43.8pt;margin-top:603.2pt;width:95pt;height:35pt;
1423 z-index:251662336;visibility:visible;mso-wrap-style:square;
1424 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1425 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1426 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1427 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1428 mso-position-vertical-relative:margin;mso-width-percent:0;
1429 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1430 <v:imagedata src="ParametricFitting_files/image038.png" o:title=""/>
1431 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1432</v:shape><![endif]--><![if !vml]><img width=127 height=47
1433src="ParametricFitting_files/image039.jpg" align=right hspace=12 v:shapes="Picture_x0020_22"><![endif]>Assign
1434<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1435mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1436and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1437mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1438as refinable “free” parameters: Click on the <b style='mso-bidi-font-weight:
1439normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1440major-latin;mso-hansi-theme-font:major-latin'>&#8597;</span></b><span
1441style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1442mso-hansi-theme-font:major-latin'> </span>button next to b and select<b
1443style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1444style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1445mso-hansi-theme-font:major-latin'>Free</span></b>. Then repeat this for<b
1446style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1447style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1448mso-hansi-theme-font:major-latin'>m</span></b>. </p>
1449
1450<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1451mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_23" o:spid="_x0000_i1038"
1452 type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible;
1453 mso-wrap-style:square'>
1454 <v:imagedata src="ParametricFitting_files/image040.png" o:title="" croptop="3992f"
1455  cropbottom="8541f" cropleft="3250f" cropright="3415f"/>
1456</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=204
1457src="ParametricFitting_files/image041.png" v:shapes="Picture_x0020_23"><![endif]></span></p>
1458
1459<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1460
1461<p class=MsoListParagraphCxSpMiddle>It is not necessary to change the default
1462variable names, the initial values, or the refine flag. </p>
1463
1464<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1465
1466<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1467style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1468style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1469</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_24"
1470 o:spid="_x0000_s1026" type="#_x0000_t75" style='position:absolute;left:0;
1471 text-align:left;margin-left:24.3pt;margin-top:837.2pt;width:75.5pt;height:26pt;
1472 z-index:251663360;visibility:visible;mso-wrap-style:square;
1473 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1474 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1475 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1476 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1477 mso-position-vertical-relative:margin;mso-width-percent:0;
1478 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1479 <v:imagedata src="ParametricFitting_files/image042.png" o:title=""/>
1480 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1481</v:shape><![endif]--><![if !vml]><img width=101 height=35
1482src="ParametricFitting_files/image043.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
1483set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1484mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>
1485as the temperature, click on the <b style='mso-bidi-font-weight:normal'><span
1486style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1487mso-hansi-theme-font:major-latin'>&#8597;</span></b><span style='font-family:
1488"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1489major-latin'> </span>button next to <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'>x</span></b>, select<b style='mso-bidi-font-weight:
1492normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1493"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1494major-latin'>Global</span>,</b> and then select<b style='mso-bidi-font-weight:
1495normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1496"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1497major-latin'>Temperature</span></b> from the list and click on<b
1498style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1499style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1500mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
1501
1502<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1503
1504<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1505style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1506 id="Picture_x0020_25" o:spid="_x0000_i1037" type="#_x0000_t75" style='width:245.25pt;
1507 height:108.75pt;visibility:visible;mso-wrap-style:square'>
1508 <v:imagedata src="ParametricFitting_files/image044.png" o:title="" croptop="5483f"
1509  cropbottom="10984f" cropleft="4191f" cropright="4159f"/>
1510</v:shape><![endif]--><![if !vml]><img border=0 width=327 height=145
1511src="ParametricFitting_files/image045.png" v:shapes="Picture_x0020_25"><![endif]></span></p>
1512
1513<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1514
1515<p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values
1516of the dependent variable and the expression for the first refinement, as
1517below. <span style='mso-spacerun:yes'> </span>Optionally, the values for <b
1518style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1519mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1520and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1521mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1522could be changed manually and the displayed value will be updated when <b
1523style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1524mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b>
1525is pressed or after a short delay.</p>
1526
1527<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1528
1529<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1530mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_26" o:spid="_x0000_i1036"
1531 type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible;
1532 mso-wrap-style:square'>
1533 <v:imagedata src="ParametricFitting_files/image046.png" o:title="" croptop="3992f"
1534  cropbottom="8541f" cropleft="3490f" cropright="3294f"/>
1535</v:shape><![endif]--><![if !vml]><img border=0 width=419 height=204
1536src="ParametricFitting_files/image047.png" v:shapes="Picture_x0020_26"><![endif]></span></p>
1537
1538<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1539
1540<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1541style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1542style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1543</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1544style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1545mso-hansi-theme-font:major-latin'>OK</span></b> button to save this parametric
1546fit. The <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1547mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1548and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1549mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1550values are then refined and the console shows the results:</p>
1551
1552<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1553style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1554style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1555</span></span></span><![endif]><o:p>&nbsp;</o:p></p>
1556
1557<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1558auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'>Fit
1559Results<o:p></o:p></span></b></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'><span
1563style='mso-spacerun:yes'>  </span>0::Vol = m*x +b<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>x<span style='mso-spacerun:yes'>  </span>=
1568Temperature<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>=
1573561.7722(26)<o:p></o:p></span></b></p>
1574
1575<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1576auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
1577style='mso-spacerun:yes'>   </span>m<span style='mso-spacerun:yes'>  </span>=
15780.014833(14)<o:p></o:p></span></b></p>
1579
1580
1581<p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
1582auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
1583
1584<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1585mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_27" o:spid="_x0000_i1035"
1586 type="#_x0000_t75" style='width:350.25pt;height:302.25pt;visibility:visible;
1587 mso-wrap-style:square'>
1588 <v:imagedata src="ParametricFitting_files/image048.png" o:title=""/>
1589</v:shape><![endif]--><![if !vml]><img border=0 width=467 height=403
1590src="ParametricFitting_files/image048.png" v:shapes="Picture_x0020_27"><![endif]></span><span
1591style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin;
1592mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p>
1593
1594<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1595
1596<p class=MsoListParagraphCxSpMiddle>Also, the plot shows the actual values and
1597the fit. Note that while the values are close to a straight line, the
1598deviations are huge compared to the estimated uncertainty values for the unit
1599cell volume (shown as very small error bars, unless the plot is
1600magnified).<span style='mso-spacerun:yes'>  </span>If a linear relationship had
1601been defined as a constraint (as can be done with other software packages) <i>the
1602unit cell parameters would have been forced onto this line with no obvious
1603indication that this was not a correct result.</i> </p>
1604
1605<p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p>
1606
1607<h1>Step 4: Examples of more complex fitt<span class=Heading1Char><span
1608style='font-weight:normal'>i</span></span>ng equations</h1>
1609
1610<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1611
1612<p class=MsoNormal>Here we will look at how more sophisticated Python
1613capabilities can be used to fit more complex behavior.<span
1614style='mso-spacerun:yes'>  </span></p>
1615
1616<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1617style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1618style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1619</span></span></span><![endif]>Optionally Double-click on the<b
1620style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1621style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1622mso-hansi-theme-font:major-latin'>0::c</span> </b>column. Note the changes in
1623the <i style='mso-bidi-font-style:normal'>c</i> unit cell parameter as a
1624function of temperature, as shown below. </p>
1625
1626<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1627
1628<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1629mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_28" o:spid="_x0000_i1034"
1630 type="#_x0000_t75" style='width:350.25pt;height:303pt;visibility:visible;
1631 mso-wrap-style:square'>
1632 <v:imagedata src="ParametricFitting_files/image050.png" o:title=""/>
1633</v:shape><![endif]--><![if !vml]><img border=0 width=467 height=404
1634src="ParametricFitting_files/image050.png" v:shapes="Picture_x0020_28"><![endif]></span></p>
1635
1636<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1637
1638<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1639style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1640style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1641</span></span></span><![endif]>We will now fit <i style='mso-bidi-font-style:
1642normal'>c</i> as a function of form 1/(1 + exp[k(t-t<sub>0</sub>)]). Select the<b
1643style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1644style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1645mso-hansi-theme-font:major-latin'>Parametric Fit/Add</span></b><span
1646style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1647mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:normal'>equation</b></span>
1648menu item (as before). For the dependent variable, click on<b style='mso-bidi-font-weight:
1649normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1650"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1651major-latin'>Phase</span></b> and then select<b style='mso-bidi-font-weight:
1652normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1653"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1654major-latin'>0::c</span></b>.<span style='mso-spacerun:yes'>  </span></p>
1655
1656<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1657
1658<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1659style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1660style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1661</span></span></span><![endif]>We then enter expression</p>
1662
1663<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1664
1665<p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><b
1666style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1667mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b + c /
1668(1+math.exp(k*(t-t0)))<o:p></o:p></span></b></p>
1669
1670<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria","serif";
1671mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
1672
1673<p class=MsoListParagraphCxSpLast>Note that use of spaces here is arbitrary,
1674but all parentheses are required. Also note that the exponential function
1675[exp(x) or e<sup>x</sup>] is supplied in Python in the math and numpy modules
1676as math.exp(), numpy.exp(). Either of these can be used as well as the
1677convenience abbreviation of np.exp(). Further, there are also a small number of
1678common functions that are defined without the prefix for convenience [these
1679include pi, exp(), sqrt(), sind(), cosd(), tand() where d indicates an argument
1680in degrees.]</p>
1681
1682<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1683
1684<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1685style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1686style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1687</span></span></span><![endif]>Assign free variables to<b style='mso-bidi-font-weight:
1688normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1689"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1690major-latin'>b</span>, </b><b style='mso-bidi-font-weight:normal'><span
1691style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1692mso-hansi-theme-font:major-latin'>c</span>, </b><b style='mso-bidi-font-weight:
1693normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1694major-latin;mso-hansi-theme-font:major-latin'>k</span> </b>and<b
1695style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1696style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1697mso-hansi-theme-font:major-latin'>t0</span></b>, similar as to what was done
1698before. Note that the<b style='mso-bidi-font-weight:normal'> </b>variable name
1699for the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1700mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1701label was set automatically to <b style='mso-bidi-font-weight:normal'><span
1702style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1703mso-hansi-theme-font:major-latin'>b_1</span></b>. If the same variable name (<b
1704style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1705mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>)
1706was used again here, the same value would be used to fit both equations, which
1707is certainly not what we want here.<span style='mso-spacerun:yes'>  </span>Finally,
1708since this equation is too complex to fit without some reasonable starting
1709values, specify an approximate value, <b style='mso-bidi-font-weight:normal'><span
1710style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1711mso-hansi-theme-font:major-latin'>120</span></b>, for the vicinity of where the <i
1712style='mso-bidi-font-style:normal'>c</i> lattice parameter starts to
1713change (<b style='mso-bidi-font-weight:
1714normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1715major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>) and
1716non-zero values for variables <B>c</B> and <B>k</B>.</p>
1717
1718<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1719
1720<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1721mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_40" o:spid="_x0000_i1033"
1722 type="#_x0000_t75" style='width:317.25pt;height:171pt;visibility:visible;
1723 mso-wrap-style:square'>
1724 <v:imagedata src="ParametricFitting_files/image052.png" o:title=""/>
1725</v:shape><![endif]--><![if !vml]><img border=0 width=423 height=228
1726src="ParametricFitting_files/image052.png" v:shapes="Picture_x0020_40"><![endif]></span></p>
1727
1728<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1729
1730<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1731style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1732style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1733</span></span></span><![endif]>Optionally, press the <b style='mso-bidi-font-weight:
1734normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1735major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b> button to see if
1736the equation can be refined with starting values of zero for the other three
1737parameters.<span style='mso-spacerun:yes'>  </span>A fit is obtained and is
1738plotted. </p>
1739
1740<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1741
1742<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1743style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1744 id="Picture_x0020_41" o:spid="_x0000_i1032" type="#_x0000_t75" style='width:317.25pt;
1745 height:168.75pt;visibility:visible;mso-wrap-style:square'>
1746 <v:imagedata src="ParametricFitting_files/image054.png" o:title=""/>
1747</v:shape><![endif]--><![if !vml]><img border=0 width=423 height=225
1748src="ParametricFitting_files/image054.png" v:shapes="Picture_x0020_41"><![endif]></span></p>
1749
1750<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1751
1752<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1753style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1754style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1755</span></span></span><![endif]>Press <b style='mso-bidi-font-weight:normal'><span
1756style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1757mso-hansi-theme-font:major-latin'>OK</span></b> to add this equation to the
1758parametric fits and both equations are refit (and if the label for <b
1759style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1760mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1761had not been changed, applying the constraint that <b style='mso-bidi-font-weight:
1762normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1763major-latin;mso-hansi-theme-font:major-latin'>b</span></b> must be the same for
1764both equations.) </p>
1765
1766<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1767
1768<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1769mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_32" o:spid="_x0000_i1031"
1770 type="#_x0000_t75" style='width:353.25pt;height:300.75pt;visibility:visible;
1771 mso-wrap-style:square'>
1772 <v:imagedata src="ParametricFitting_files/image056.png" o:title=""/>
1773</v:shape><![endif]--><![if !vml]><img border=0 width=471 height=401
1774src="ParametricFitting_files/image056.png" v:shapes="Picture_x0020_32"><![endif]></span></p>
1775
1776<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1777
1778<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1779style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1780style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1781</span></span></span><![endif]>We will now try a more complex equation for the
1782same data, where we set the value to <b style='mso-bidi-font-weight:normal'><span
1783style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1784mso-hansi-theme-font:major-latin'>b</span></b> for temperatures less than <b
1785style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1786mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
1787and a critical phenomenon formulism [(t-t<sub>0</sub>)<sup>-a</sup>] for
1788temperatures above <b style='mso-bidi-font-weight:normal'><span
1789style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1790mso-hansi-theme-font:major-latin'>t0</span></b>. To do this we use the Python
1791syntax &quot;<b style='mso-bidi-font-weight:normal'><span style='font-family:
1792"Calibri","sans-serif";mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1793major-latin'>x1 if testexp else x2</span></b>&quot; which is <b
1794style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1795mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x1</span></b>
1796where the test expression (<b style='mso-bidi-font-weight:normal'><span
1797style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1798mso-hansi-theme-font:major-latin'>testexp</span></b> here) is <b
1799style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1800mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>True</span></b>
1801and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1802mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x2</span></b>
1803otherwise. Thus we can use an expression such as:</p>
1804
1805<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span></p>
1806
1807<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1808auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1809mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b if t &lt;=
1810t0 else b + c * (1-(t-t0)**d)<o:p></o:p></span></b></p>
1811
1812<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1813auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
1814
1815<p class=MsoListParagraphCxSpMiddle>This may be easier to read with the
1816inclusion of extra parentheses (that have no effect on the execution):<span
1817style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin;
1818mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p>
1819
1820<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1821
1822<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1823auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1824mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b if (t
1825&lt;= t0) else (b + c * (1-(t-t0)**d))<o:p></o:p></span></b></p>
1826
1827<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
1828auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
1829
1830<p class=MsoListParagraphCxSpMiddle>As before, add an equation (or edit an
1831existing one), set the dependent variable to <b style='mso-bidi-font-weight:
1832normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1833major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>, enter the
1834expression and assign the free and fixed variables, as before. Set the exponent
1835and the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1836mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
1837values to initial values of <b style='mso-bidi-font-weight:normal'><span
1838style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1839mso-hansi-theme-font:major-latin'>-0.025</span></b> and <b style='mso-bidi-font-weight:
1840normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1841major-latin;mso-hansi-theme-font:major-latin'>120</span></b>, respectively, and
1842turn off their refinement flags. Then press <b style='mso-bidi-font-weight:
1843normal'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:
1844major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b>.<span
1845style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin;
1846mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p>
1847
1848<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1849mso-no-proof:yes'><o:p>&nbsp;</o:p></span></p>
1850
1851<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1852mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1030"
1853 type="#_x0000_t75" style='width:315.75pt;height:168.75pt;visibility:visible;
1854 mso-wrap-style:square'>
1855 <v:imagedata src="ParametricFitting_files/image058.png" o:title=""/>
1856</v:shape><![endif]--><![if !vml]><img border=0 width=421 height=225
1857src="ParametricFitting_files/image058.png" v:shapes="Picture_x0020_42"><![endif]></span></p>
1858
1859<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1860
1861<p class=MsoListParagraphCxSpMiddle>Then turn refinement of <b
1862style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
1863mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
1864back on and press <b style='mso-bidi-font-weight:normal'><span
1865style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1866mso-hansi-theme-font:major-latin'>OK</span></b>. The resulting fit shows the
1867trends quite well: </p>
1868
1869<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1870
1871<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1872mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1029"
1873 type="#_x0000_t75" style='width:351pt;height:302.25pt;visibility:visible;
1874 mso-wrap-style:square'>
1875 <v:imagedata src="ParametricFitting_files/image060.png" o:title=""/>
1876</v:shape><![endif]--><![if !vml]><img border=0 width=468 height=403
1877src="ParametricFitting_files/image060.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=""/>
1944</v:shape><![endif]--><![if !vml]><img border=0 width=443 height=248
1945src="ParametricFitting_files/image062.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
1946
1947<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1948
1949<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
1950style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1951style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1952</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1953style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
1954mso-hansi-theme-font:major-latin'>Fit</span></b> button and note that all the
1955free variables are adjusted </p>
1956
1957<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1958
1959<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1960mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_45" o:spid="_x0000_i1027"
1961 type="#_x0000_t75" style='width:333.75pt;height:189pt;visibility:visible;
1962 mso-wrap-style:square'>
1963 <v:imagedata src="ParametricFitting_files/image064.png" o:title=""/>
1964</v:shape><![endif]--><![if !vml]><img border=0 width=445 height=252
1965src="ParametricFitting_files/image064.png" v:shapes="Picture_x0020_45"><![endif]></span><span
1966style='mso-spacerun:yes'> </span></p>
1967
1968<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1969
1970<p class=MsoListParagraphCxSpMiddle>Also, that the plotted result is close to
1971the values from the refinement.</p>
1972
1973<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1974
1975<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1976mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_48" o:spid="_x0000_i1026"
1977 type="#_x0000_t75" style='width:348.75pt;height:302.25pt;visibility:visible;
1978 mso-wrap-style:square'>
1979 <v:imagedata src="ParametricFitting_files/image066.png" o:title=""/>
1980</v:shape><![endif]--><![if !vml]><img border=0 width=465 height=403
1981src="ParametricFitting_files/image066.png" v:shapes="Picture_x0020_48"><![endif]></span></p>
1982
1983<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1984
1985<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
1986
1987<p class=MsoNormal>This completes this tutorial on use of Pseudo Variables and
1988parametric fitting. </p>
1989
1990<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1991
1992<h1>Advanced Use: Externally-Defined Functions</h1>
1993
1994<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1995
1996<p class=MsoNormal>The parametric refinement section of GSAS-II is intended to
1997be quite flexible, but for very complex fitting, a user may wish to define a custom
1998Python function. Such a function can be defined in a separate module that will
1999be accessed with a parametric equation (or pseudo variable).</p>
2000
2001<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2002
2003<p class=MsoNormal>As an example of that, if we create a file called <b
2004style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2005mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fittest.py</span></b>
2006and locate that file in any directory in the Python path (most simply, in the
2007directory where the project file, <b style='mso-bidi-font-weight:normal'><span
2008style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
2009mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b>, is located) and place
2010a routine <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2011mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fitfxn</span></b>
2012in that module:</p>
2013
2014<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'>
2015
2016<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre><pre><b
2017style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2018mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'># file fittest.py<o:p></o:p></span></b></pre><pre><b
2019style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2020mso-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
2021style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2022mso-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
2023style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2024mso-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
2025style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2026mso-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
2027style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif";
2028mso-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>
2029
2030</blockquote>
2031
2032<p class=MsoNormal>The function <b style='mso-bidi-font-weight:normal'><span
2033style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
2034mso-hansi-theme-font:major-latin'>fittest.fitfxn</span></b> can then be used to
2035duplicate the previous example: </p>
2036
2037<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2038
2039<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2040 id="Picture_x0020_47" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:315pt;
2041 height:185.25pt;visibility:visible;mso-wrap-style:square'>
2042 <v:imagedata src="ParametricFitting_files/image068.png" o:title="" croptop="3393f"
2043  cropbottom="7847f" cropleft="3471f" cropright="3290f"/>
2044</v:shape><![endif]--><![if !vml]><img border=0 width=420 height=247
2045src="ParametricFitting_files/image069.png" v:shapes="Picture_x0020_47"><![endif]></span></p>
2046
2047<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2048
2049<p class=MsoNormal>Note that module <b style='mso-bidi-font-weight:normal'><span
2050style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin;
2051mso-hansi-theme-font:major-latin'>fittest</span></b> is automatically located
2052and loaded, provided it is found in the path. No changes to the standard
2053distributed GSAS-II code are needed. Care to avoid use of a module name already
2054found in GSAS-II or Python is wise.</p>
2055
2056</div>
2057
2058</body>
2059
2060</html>
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