source: Tutorials/SeqParametric/ParametricFitting.htm @ 3094

Last change on this file since 3094 was 3094, checked in by vondreele, 4 years ago

revise & update to 2 frame

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415  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
416  <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
417  <w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
418  <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
419  <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
420  <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
421  <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
422  <w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
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429  <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
430  <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
431  <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
432  <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
433  <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
434  <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
435  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
436  <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
437  <w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
438  <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
439  <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
440  <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
441  <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
442  <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
443  <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
444  <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
445  <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
446  <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
447  <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
448  <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
449  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
450  <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
451  <w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
452  <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
453  <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
454  <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
455  <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
456  <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
457  <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
458  <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
459  <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
460  <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
461  <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
462  <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
463  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
464  <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
465  <w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
466  <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
467  <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
468  <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
469  <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
470  <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
471  <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
472  <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
473  <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
474  <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
475  <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
476  <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
477  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
478  <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
479  <w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
480  <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
481  <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
482  <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
483  <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
484  <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
485  <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
486  <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
487  <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
488  <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
489  <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
490  <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
491  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
492  <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
493  <w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
494  <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
495  <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
496  <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
497  <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
498  <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
499  <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
500  <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
501  <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
502  <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
503  <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
504  <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
505  <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
506  <w:LsdException Locked="false" Priority="19" QFormat="true"
507   Name="Subtle Emphasis"/>
508  <w:LsdException Locked="false" Priority="21" QFormat="true"
509   Name="Intense Emphasis"/>
510  <w:LsdException Locked="false" Priority="31" QFormat="true"
511   Name="Subtle Reference"/>
512  <w:LsdException Locked="false" Priority="32" QFormat="true"
513   Name="Intense Reference"/>
514  <w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
515  <w:LsdException Locked="false" Priority="37" SemiHidden="true"
516   UnhideWhenUsed="true" Name="Bibliography"/>
517  <w:LsdException Locked="false" Priority="39" SemiHidden="true"
518   UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
519  <w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
520  <w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
521  <w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
522  <w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
523  <w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
524  <w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
525  <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
526  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
527  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
528  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
529  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
530  <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
531  <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
532  <w:LsdException Locked="false" Priority="46"
533   Name="Grid Table 1 Light Accent 1"/>
534  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
535  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
536  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
537  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
538  <w:LsdException Locked="false" Priority="51"
539   Name="Grid Table 6 Colorful Accent 1"/>
540  <w:LsdException Locked="false" Priority="52"
541   Name="Grid Table 7 Colorful Accent 1"/>
542  <w:LsdException Locked="false" Priority="46"
543   Name="Grid Table 1 Light Accent 2"/>
544  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
545  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
546  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
547  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
548  <w:LsdException Locked="false" Priority="51"
549   Name="Grid Table 6 Colorful Accent 2"/>
550  <w:LsdException Locked="false" Priority="52"
551   Name="Grid Table 7 Colorful Accent 2"/>
552  <w:LsdException Locked="false" Priority="46"
553   Name="Grid Table 1 Light Accent 3"/>
554  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
555  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
556  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
557  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
558  <w:LsdException Locked="false" Priority="51"
559   Name="Grid Table 6 Colorful Accent 3"/>
560  <w:LsdException Locked="false" Priority="52"
561   Name="Grid Table 7 Colorful Accent 3"/>
562  <w:LsdException Locked="false" Priority="46"
563   Name="Grid Table 1 Light Accent 4"/>
564  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
565  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
566  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
567  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
568  <w:LsdException Locked="false" Priority="51"
569   Name="Grid Table 6 Colorful Accent 4"/>
570  <w:LsdException Locked="false" Priority="52"
571   Name="Grid Table 7 Colorful Accent 4"/>
572  <w:LsdException Locked="false" Priority="46"
573   Name="Grid Table 1 Light Accent 5"/>
574  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
575  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
576  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
577  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
578  <w:LsdException Locked="false" Priority="51"
579   Name="Grid Table 6 Colorful Accent 5"/>
580  <w:LsdException Locked="false" Priority="52"
581   Name="Grid Table 7 Colorful Accent 5"/>
582  <w:LsdException Locked="false" Priority="46"
583   Name="Grid Table 1 Light Accent 6"/>
584  <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
585  <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
586  <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
587  <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
588  <w:LsdException Locked="false" Priority="51"
589   Name="Grid Table 6 Colorful Accent 6"/>
590  <w:LsdException Locked="false" Priority="52"
591   Name="Grid Table 7 Colorful Accent 6"/>
592  <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
593  <w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
594  <w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
595  <w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
596  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
597  <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
598  <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
599  <w:LsdException Locked="false" Priority="46"
600   Name="List Table 1 Light Accent 1"/>
601  <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
602  <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
603  <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
604  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
605  <w:LsdException Locked="false" Priority="51"
606   Name="List Table 6 Colorful Accent 1"/>
607  <w:LsdException Locked="false" Priority="52"
608   Name="List Table 7 Colorful Accent 1"/>
609  <w:LsdException Locked="false" Priority="46"
610   Name="List Table 1 Light Accent 2"/>
611  <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
612  <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
613  <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/>
614  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/>
615  <w:LsdException Locked="false" Priority="51"
616   Name="List Table 6 Colorful Accent 2"/>
617  <w:LsdException Locked="false" Priority="52"
618   Name="List Table 7 Colorful Accent 2"/>
619  <w:LsdException Locked="false" Priority="46"
620   Name="List Table 1 Light Accent 3"/>
621  <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 3"/>
622  <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 3"/>
623  <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 3"/>
624  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 3"/>
625  <w:LsdException Locked="false" Priority="51"
626   Name="List Table 6 Colorful Accent 3"/>
627  <w:LsdException Locked="false" Priority="52"
628   Name="List Table 7 Colorful Accent 3"/>
629  <w:LsdException Locked="false" Priority="46"
630   Name="List Table 1 Light Accent 4"/>
631  <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 4"/>
632  <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 4"/>
633  <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 4"/>
634  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 4"/>
635  <w:LsdException Locked="false" Priority="51"
636   Name="List Table 6 Colorful Accent 4"/>
637  <w:LsdException Locked="false" Priority="52"
638   Name="List Table 7 Colorful Accent 4"/>
639  <w:LsdException Locked="false" Priority="46"
640   Name="List Table 1 Light Accent 5"/>
641  <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 5"/>
642  <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/>
643  <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 5"/>
644  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 5"/>
645  <w:LsdException Locked="false" Priority="51"
646   Name="List Table 6 Colorful Accent 5"/>
647  <w:LsdException Locked="false" Priority="52"
648   Name="List Table 7 Colorful Accent 5"/>
649  <w:LsdException Locked="false" Priority="46"
650   Name="List Table 1 Light Accent 6"/>
651  <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 6"/>
652  <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 6"/>
653  <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 6"/>
654  <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 6"/>
655  <w:LsdException Locked="false" Priority="51"
656   Name="List Table 6 Colorful Accent 6"/>
657  <w:LsdException Locked="false" Priority="52"
658   Name="List Table 7 Colorful Accent 6"/>
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782        mso-style-priority:99;
783        color:purple;
784        mso-themecolor:followedhyperlink;
785        text-decoration:underline;
786        text-underline:single;}
787pre
788        {mso-style-noshow:yes;
789        mso-style-priority:99;
790        mso-style-link:"HTML Preformatted Char";
791        margin:0in;
792        margin-bottom:.0001pt;
793        mso-pagination:widow-orphan;
794        font-size:10.0pt;
795        font-family:"Courier New";
796        mso-fareast-font-family:"Times New Roman";
797        mso-fareast-theme-font:minor-fareast;}
798p.MsoAcetate, li.MsoAcetate, div.MsoAcetate
799        {mso-style-noshow:yes;
800        mso-style-priority:99;
801        mso-style-link:"Balloon Text Char";
802        margin:0in;
803        margin-bottom:.0001pt;
804        mso-pagination:widow-orphan;
805        font-size:9.0pt;
806        font-family:"Lucida Grande";
807        mso-fareast-font-family:"Times New Roman";
808        mso-fareast-theme-font:minor-fareast;
809        mso-fareast-language:JA;}
810p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph
811        {mso-style-priority:34;
812        mso-style-unhide:no;
813        mso-style-qformat:yes;
814        margin-top:0in;
815        margin-right:0in;
816        margin-bottom:0in;
817        margin-left:.5in;
818        margin-bottom:.0001pt;
819        mso-add-space:auto;
820        mso-pagination:widow-orphan;
821        font-size:12.0pt;
822        mso-bidi-font-size:10.0pt;
823        font-family:"Times New Roman",serif;
824        mso-fareast-font-family:"Times New Roman";
825        mso-fareast-theme-font:minor-fareast;
826        mso-bidi-font-family:"Times New Roman";
827        mso-bidi-theme-font:minor-bidi;
828        mso-fareast-language:JA;}
829p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst
830        {mso-style-priority:34;
831        mso-style-unhide:no;
832        mso-style-qformat:yes;
833        mso-style-type:export-only;
834        margin-top:0in;
835        margin-right:0in;
836        margin-bottom:0in;
837        margin-left:.5in;
838        margin-bottom:.0001pt;
839        mso-add-space:auto;
840        mso-pagination:widow-orphan;
841        font-size:12.0pt;
842        mso-bidi-font-size:10.0pt;
843        font-family:"Times New Roman",serif;
844        mso-fareast-font-family:"Times New Roman";
845        mso-fareast-theme-font:minor-fareast;
846        mso-bidi-font-family:"Times New Roman";
847        mso-bidi-theme-font:minor-bidi;
848        mso-fareast-language:JA;}
849p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle
850        {mso-style-priority:34;
851        mso-style-unhide:no;
852        mso-style-qformat:yes;
853        mso-style-type:export-only;
854        margin-top:0in;
855        margin-right:0in;
856        margin-bottom:0in;
857        margin-left:.5in;
858        margin-bottom:.0001pt;
859        mso-add-space:auto;
860        mso-pagination:widow-orphan;
861        font-size:12.0pt;
862        mso-bidi-font-size:10.0pt;
863        font-family:"Times New Roman",serif;
864        mso-fareast-font-family:"Times New Roman";
865        mso-fareast-theme-font:minor-fareast;
866        mso-bidi-font-family:"Times New Roman";
867        mso-bidi-theme-font:minor-bidi;
868        mso-fareast-language:JA;}
869p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast
870        {mso-style-priority:34;
871        mso-style-unhide:no;
872        mso-style-qformat:yes;
873        mso-style-type:export-only;
874        margin-top:0in;
875        margin-right:0in;
876        margin-bottom:0in;
877        margin-left:.5in;
878        margin-bottom:.0001pt;
879        mso-add-space:auto;
880        mso-pagination:widow-orphan;
881        font-size:12.0pt;
882        mso-bidi-font-size:10.0pt;
883        font-family:"Times New Roman",serif;
884        mso-fareast-font-family:"Times New Roman";
885        mso-fareast-theme-font:minor-fareast;
886        mso-bidi-font-family:"Times New Roman";
887        mso-bidi-theme-font:minor-bidi;
888        mso-fareast-language:JA;}
889span.Heading1Char
890        {mso-style-name:"Heading 1 Char";
891        mso-style-priority:9;
892        mso-style-unhide:no;
893        mso-style-locked:yes;
894        mso-style-link:"Heading 1";
895        mso-ansi-font-size:18.0pt;
896        mso-bidi-font-size:16.0pt;
897        font-family:"Times New Roman",serif;
898        mso-ascii-font-family:"Times New Roman";
899        mso-fareast-font-family:"Times New Roman";
900        mso-fareast-theme-font:major-fareast;
901        mso-hansi-font-family:"Times New Roman";
902        mso-bidi-font-family:"Times New Roman";
903        mso-bidi-theme-font:major-bidi;
904        font-weight:bold;}
905span.Heading2Char
906        {mso-style-name:"Heading 2 Char";
907        mso-style-noshow:yes;
908        mso-style-priority:9;
909        mso-style-unhide:no;
910        mso-style-locked:yes;
911        mso-style-link:"Heading 2";
912        mso-ansi-font-size:13.0pt;
913        mso-bidi-font-size:13.0pt;
914        font-family:"Calibri",sans-serif;
915        mso-ascii-font-family:Calibri;
916        mso-ascii-theme-font:major-latin;
917        mso-fareast-font-family:"Times New Roman";
918        mso-fareast-theme-font:major-fareast;
919        mso-hansi-font-family:Calibri;
920        mso-hansi-theme-font:major-latin;
921        mso-bidi-font-family:"Times New Roman";
922        mso-bidi-theme-font:major-bidi;
923        color:#4F81BD;
924        mso-themecolor:accent1;
925        font-weight:bold;}
926span.HTMLPreformattedChar
927        {mso-style-name:"HTML Preformatted Char";
928        mso-style-noshow:yes;
929        mso-style-priority:99;
930        mso-style-unhide:no;
931        mso-style-locked:yes;
932        mso-style-link:"HTML Preformatted";
933        font-family:Consolas;
934        mso-ascii-font-family:Consolas;
935        mso-hansi-font-family:Consolas;
936        mso-bidi-font-family:Consolas;}
937p.msonormal0, li.msonormal0, div.msonormal0
938        {mso-style-name:msonormal;
939        mso-style-unhide:no;
940        mso-margin-top-alt:auto;
941        margin-right:0in;
942        mso-margin-bottom-alt:auto;
943        margin-left:0in;
944        mso-pagination:widow-orphan;
945        font-size:12.0pt;
946        font-family:"Times New Roman",serif;
947        mso-fareast-font-family:"Times New Roman";
948        mso-fareast-theme-font:minor-fareast;}
949span.BalloonTextChar
950        {mso-style-name:"Balloon Text Char";
951        mso-style-noshow:yes;
952        mso-style-priority:99;
953        mso-style-unhide:no;
954        mso-style-locked:yes;
955        mso-style-link:"Balloon Text";
956        mso-ansi-font-size:9.0pt;
957        mso-bidi-font-size:9.0pt;
958        font-family:"Lucida Grande";
959        mso-ascii-font-family:"Lucida Grande";
960        mso-hansi-font-family:"Lucida Grande";
961        mso-bidi-font-family:"Lucida Grande";}
962span.SpellE
963        {mso-style-name:"";
964        mso-spl-e:yes;}
965span.GramE
966        {mso-style-name:"";
967        mso-gram-e:yes;}
968.MsoChpDefault
969        {mso-style-type:export-only;
970        mso-default-props:yes;
971        font-size:10.0pt;
972        mso-ansi-font-size:10.0pt;
973        mso-bidi-font-size:10.0pt;
974        font-family:"Cambria",serif;
975        mso-ascii-font-family:Cambria;
976        mso-ascii-theme-font:minor-latin;
977        mso-fareast-font-family:"Times New Roman";
978        mso-fareast-theme-font:minor-fareast;
979        mso-hansi-font-family:Cambria;
980        mso-hansi-theme-font:minor-latin;
981        mso-bidi-font-family:"Times New Roman";
982        mso-bidi-theme-font:minor-bidi;
983        mso-fareast-language:JA;}
984@page WordSection1
985        {size:8.5in 11.0in;
986        margin:1.0in 1.25in 1.0in 1.25in;
987        mso-header-margin:.5in;
988        mso-footer-margin:.5in;
989        mso-paper-source:0;}
990div.WordSection1
991        {page:WordSection1;}
992 /* List Definitions */
993 @list l0
994        {mso-list-id:109671306;
995        mso-list-type:hybrid;
996        mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
997@list l0:level1
998        {mso-level-tab-stop:none;
999        mso-level-number-position:left;
1000        text-indent:-.25in;}
1001@list l0:level2
1002        {mso-level-number-format:alpha-lower;
1003        mso-level-tab-stop:none;
1004        mso-level-number-position:left;
1005        text-indent:-.25in;}
1006@list l0:level3
1007        {mso-level-number-format:roman-lower;
1008        mso-level-tab-stop:none;
1009        mso-level-number-position:right;
1010        text-indent:-9.0pt;}
1011@list l0:level4
1012        {mso-level-tab-stop:none;
1013        mso-level-number-position:left;
1014        text-indent:-.25in;}
1015@list l0:level5
1016        {mso-level-number-format:alpha-lower;
1017        mso-level-tab-stop:none;
1018        mso-level-number-position:left;
1019        text-indent:-.25in;}
1020@list l0:level6
1021        {mso-level-number-format:roman-lower;
1022        mso-level-tab-stop:none;
1023        mso-level-number-position:right;
1024        text-indent:-9.0pt;}
1025@list l0:level7
1026        {mso-level-tab-stop:none;
1027        mso-level-number-position:left;
1028        text-indent:-.25in;}
1029@list l0:level8
1030        {mso-level-number-format:alpha-lower;
1031        mso-level-tab-stop:none;
1032        mso-level-number-position:left;
1033        text-indent:-.25in;}
1034@list l0:level9
1035        {mso-level-number-format:roman-lower;
1036        mso-level-tab-stop:none;
1037        mso-level-number-position:right;
1038        text-indent:-9.0pt;}
1039@list l1
1040        {mso-list-id:235746779;
1041        mso-list-type:hybrid;
1042        mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
1043@list l1:level1
1044        {mso-level-tab-stop:none;
1045        mso-level-number-position:left;
1046        text-indent:-.25in;}
1047@list l1:level2
1048        {mso-level-number-format:alpha-lower;
1049        mso-level-tab-stop:none;
1050        mso-level-number-position:left;
1051        text-indent:-.25in;}
1052@list l1:level3
1053        {mso-level-number-format:roman-lower;
1054        mso-level-tab-stop:none;
1055        mso-level-number-position:right;
1056        text-indent:-9.0pt;}
1057@list l1:level4
1058        {mso-level-tab-stop:none;
1059        mso-level-number-position:left;
1060        text-indent:-.25in;}
1061@list l1:level5
1062        {mso-level-number-format:alpha-lower;
1063        mso-level-tab-stop:none;
1064        mso-level-number-position:left;
1065        text-indent:-.25in;}
1066@list l1:level6
1067        {mso-level-number-format:roman-lower;
1068        mso-level-tab-stop:none;
1069        mso-level-number-position:right;
1070        text-indent:-9.0pt;}
1071@list l1:level7
1072        {mso-level-tab-stop:none;
1073        mso-level-number-position:left;
1074        text-indent:-.25in;}
1075@list l1:level8
1076        {mso-level-number-format:alpha-lower;
1077        mso-level-tab-stop:none;
1078        mso-level-number-position:left;
1079        text-indent:-.25in;}
1080@list l1:level9
1081        {mso-level-number-format:roman-lower;
1082        mso-level-tab-stop:none;
1083        mso-level-number-position:right;
1084        text-indent:-9.0pt;}
1085@list l2
1086        {mso-list-id:584074573;
1087        mso-list-type:hybrid;
1088        mso-list-template-ids:-926633506 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
1089@list l2:level1
1090        {mso-level-tab-stop:none;
1091        mso-level-number-position:left;
1092        text-indent:-.25in;}
1093@list l2:level2
1094        {mso-level-number-format:alpha-lower;
1095        mso-level-tab-stop:none;
1096        mso-level-number-position:left;
1097        text-indent:-.25in;}
1098@list l2:level3
1099        {mso-level-number-format:roman-lower;
1100        mso-level-tab-stop:none;
1101        mso-level-number-position:right;
1102        text-indent:-9.0pt;}
1103@list l2:level4
1104        {mso-level-tab-stop:none;
1105        mso-level-number-position:left;
1106        text-indent:-.25in;}
1107@list l2:level5
1108        {mso-level-number-format:alpha-lower;
1109        mso-level-tab-stop:none;
1110        mso-level-number-position:left;
1111        text-indent:-.25in;}
1112@list l2:level6
1113        {mso-level-number-format:roman-lower;
1114        mso-level-tab-stop:none;
1115        mso-level-number-position:right;
1116        text-indent:-9.0pt;}
1117@list l2:level7
1118        {mso-level-tab-stop:none;
1119        mso-level-number-position:left;
1120        text-indent:-.25in;}
1121@list l2:level8
1122        {mso-level-number-format:alpha-lower;
1123        mso-level-tab-stop:none;
1124        mso-level-number-position:left;
1125        text-indent:-.25in;}
1126@list l2:level9
1127        {mso-level-number-format:roman-lower;
1128        mso-level-tab-stop:none;
1129        mso-level-number-position:right;
1130        text-indent:-9.0pt;}
1131@list l3
1132        {mso-list-id:845174183;
1133        mso-list-type:hybrid;
1134        mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
1135@list l3:level1
1136        {mso-level-tab-stop:none;
1137        mso-level-number-position:left;
1138        text-indent:-.25in;}
1139@list l3:level2
1140        {mso-level-number-format:alpha-lower;
1141        mso-level-tab-stop:none;
1142        mso-level-number-position:left;
1143        text-indent:-.25in;}
1144@list l3:level3
1145        {mso-level-number-format:roman-lower;
1146        mso-level-tab-stop:none;
1147        mso-level-number-position:right;
1148        text-indent:-9.0pt;}
1149@list l3:level4
1150        {mso-level-tab-stop:none;
1151        mso-level-number-position:left;
1152        text-indent:-.25in;}
1153@list l3:level5
1154        {mso-level-number-format:alpha-lower;
1155        mso-level-tab-stop:none;
1156        mso-level-number-position:left;
1157        text-indent:-.25in;}
1158@list l3:level6
1159        {mso-level-number-format:roman-lower;
1160        mso-level-tab-stop:none;
1161        mso-level-number-position:right;
1162        text-indent:-9.0pt;}
1163@list l3:level7
1164        {mso-level-tab-stop:none;
1165        mso-level-number-position:left;
1166        text-indent:-.25in;}
1167@list l3:level8
1168        {mso-level-number-format:alpha-lower;
1169        mso-level-tab-stop:none;
1170        mso-level-number-position:left;
1171        text-indent:-.25in;}
1172@list l3:level9
1173        {mso-level-number-format:roman-lower;
1174        mso-level-tab-stop:none;
1175        mso-level-number-position:right;
1176        text-indent:-9.0pt;}
1177@list l4
1178        {mso-list-id:1447432924;
1179        mso-list-type:hybrid;
1180        mso-list-template-ids:2060980778 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;}
1181@list l4:level1
1182        {mso-level-tab-stop:none;
1183        mso-level-number-position:left;
1184        text-indent:-.25in;}
1185@list l4:level2
1186        {mso-level-number-format:alpha-lower;
1187        mso-level-tab-stop:none;
1188        mso-level-number-position:left;
1189        text-indent:-.25in;}
1190@list l4:level3
1191        {mso-level-number-format:roman-lower;
1192        mso-level-tab-stop:none;
1193        mso-level-number-position:right;
1194        text-indent:-9.0pt;}
1195@list l4:level4
1196        {mso-level-tab-stop:none;
1197        mso-level-number-position:left;
1198        text-indent:-.25in;}
1199@list l4:level5
1200        {mso-level-number-format:alpha-lower;
1201        mso-level-tab-stop:none;
1202        mso-level-number-position:left;
1203        text-indent:-.25in;}
1204@list l4:level6
1205        {mso-level-number-format:roman-lower;
1206        mso-level-tab-stop:none;
1207        mso-level-number-position:right;
1208        text-indent:-9.0pt;}
1209@list l4:level7
1210        {mso-level-tab-stop:none;
1211        mso-level-number-position:left;
1212        text-indent:-.25in;}
1213@list l4:level8
1214        {mso-level-number-format:alpha-lower;
1215        mso-level-tab-stop:none;
1216        mso-level-number-position:left;
1217        text-indent:-.25in;}
1218@list l4:level9
1219        {mso-level-number-format:roman-lower;
1220        mso-level-tab-stop:none;
1221        mso-level-number-position:right;
1222        text-indent:-9.0pt;}
1223ol
1224        {margin-bottom:0in;}
1225ul
1226        {margin-bottom:0in;}
1227-->
1228</style>
1229<!--[if gte mso 10]>
1230<style>
1231 /* Style Definitions */
1232 table.MsoNormalTable
1233        {mso-style-name:"Table Normal";
1234        mso-tstyle-rowband-size:0;
1235        mso-tstyle-colband-size:0;
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1239        mso-padding-alt:0in 5.4pt 0in 5.4pt;
1240        mso-para-margin:0in;
1241        mso-para-margin-bottom:.0001pt;
1242        mso-pagination:widow-orphan;
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1248        mso-hansi-theme-font:minor-latin;
1249        mso-fareast-language:JA;}
1250</style>
1251<![endif]-->
1252<meta name=Title content="">
1253<meta name=Keywords content="">
1254<!--[if gte mso 9]><xml>
1255 <o:shapedefaults v:ext="edit" spidmax="1030"/>
1256</xml><![endif]--><!--[if gte mso 9]><xml>
1257 <o:shapelayout v:ext="edit">
1258  <o:idmap v:ext="edit" data="1"/>
1259 </o:shapelayout></xml><![endif]-->
1260</head>
1261
1262<body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'>
1263
1264<div class=WordSection1>
1265
1266<h1>Parametric Fitting and Pseudo Variables for Sequential Fits</h1>
1267
1268<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1269
1270<p class=MsoNormal>Once a sequential fit has been performed, any function of
1271the fit parameters from the fit and <i style='mso-bidi-font-style:normal'>their
1272estimated uncertainty</i> (<span class=SpellE>s.u</span>.) can be computed by
1273defining one or more &quot;pseudo-variables&quot;. Note that the covariance
1274matrix from the fit is used to provide appropriate treatment of the correlation
1275in parameters with respect to uncertainty estimation.<span
1276style='mso-spacerun:yes'>  </span>In addition, parametric equations in terms of
1277sample parameters such as temperature and pressure, etc. can be fit to either
1278the parameters from the fit or the pseudo-variables. </p>
1279
1280<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1281
1282<p class=MsoNormal>This tutorial assumes that the steps in the <b
1283style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1284mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1285"Times New Roman"'>Sequential refinement of multiple datasets</span></b><span
1286style='mso-bidi-font-family:"Times New Roman"'> tutorial have already been
1287performed. </span>If you have not done so already, start GSAS-II. </p>
1288
1289<h1>Step 1. Open the previous Sequential Fit</h1>
1290
1291<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1292
1293<p class=MsoListParagraphCxSpFirst style='margin-bottom:6.0pt;mso-add-space:
1294auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
1295style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1296style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1297</span></span></span><![endif]>Select the<b style='mso-bidi-font-weight:normal'>
1298</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1299mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>File/Open
1300Project</span></b> menu item to load file named <span class=SpellE><b
1301style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1302mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>
1303you created in the previous tutorial.</p>
1304
1305<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1306auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
1307style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1308style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1309</span></span></span><![endif]>Select the <b style='mso-bidi-font-weight:normal'><span
1310style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1311mso-hansi-theme-font:major-latin'>Sequential results</span></b> item in the
1312data tree. The sequential refinement results data table is displayed, as below.
1313</p>
1314
1315<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1316auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype
1317 id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
1318 path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
1319 <v:stroke joinstyle="miter"/>
1320 <v:formulas>
1321  <v:f eqn="if lineDrawn pixelLineWidth 0"/>
1322  <v:f eqn="sum @0 1 0"/>
1323  <v:f eqn="sum 0 0 @1"/>
1324  <v:f eqn="prod @2 1 2"/>
1325  <v:f eqn="prod @3 21600 pixelWidth"/>
1326  <v:f eqn="prod @3 21600 pixelHeight"/>
1327  <v:f eqn="sum @0 0 1"/>
1328  <v:f eqn="prod @6 1 2"/>
1329  <v:f eqn="prod @7 21600 pixelWidth"/>
1330  <v:f eqn="sum @8 21600 0"/>
1331  <v:f eqn="prod @7 21600 pixelHeight"/>
1332  <v:f eqn="sum @10 21600 0"/>
1333 </v:formulas>
1334 <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
1335 <o:lock v:ext="edit" aspectratio="t"/>
1336</v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1054" type="#_x0000_t75"
1337 style='width:6in;height:186pt;visibility:visible;mso-wrap-style:square'>
1338 <v:imagedata src="ParametricFitting_files/image071.png" o:title=""/>
1339</v:shape><![endif]--><![if !vml]><img width=576 height=248
1340src="ParametricFitting_files/image021.png" v:shapes="Picture_x0020_1"><![endif]></span></p>
1341
1342<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1343auto'>If you look through the table, you will see how each variable changed
1344through the temperature series. In particular the CuCrO<sub>4</sub> b &amp; c
1345lattice parameters become nearly identical above ~120K (PWDR OH_24.fxye Bank
13461). Select these two columns (<b style='mso-bidi-font-weight:normal'><span
1347style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1348mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0::b</span></b>
1349&amp; <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1350mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1351major-latin'>0::c</span></b>) and do <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;mso-bidi-theme-font:major-latin'>Columns/Plot
1354selected</span></b></p>
1355
1356<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1357auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1358 id="Picture_x0020_34" o:spid="_x0000_i1053" type="#_x0000_t75" style='width:6in;
1359 height:370.5pt;visibility:visible;mso-wrap-style:square'>
1360 <v:imagedata src="ParametricFitting_files/image002.png" o:title=""/>
1361</v:shape><![endif]--><![if !vml]><img width=576 height=494
1362src="ParametricFitting_files/image023.png" v:shapes="Picture_x0020_34"><![endif]></span></p>
1363
1364<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1365auto'>Select the plot &amp; press <span class=GramE>the ‘<b style='mso-bidi-font-weight:
1366normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1367major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>s</span></b>’</span>
1368key; the popup will show</p>
1369
1370<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1371auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1372 id="Picture_x0020_3" o:spid="_x0000_i1052" type="#_x0000_t75" style='width:240pt;
1373 height:232.5pt;visibility:visible;mso-wrap-style:square'>
1374 <v:imagedata src="ParametricFitting_files/image004.png" o:title=""/>
1375</v:shape><![endif]--><![if !vml]><img width=320 height=310
1376src="ParametricFitting_files/image004.png" v:shapes="Picture_x0020_3"><![endif]></span></p>
1377
1378<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1379auto'>Select <b style='mso-bidi-font-weight:normal'><span style='font-family:
1380"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1381major-latin;mso-bidi-theme-font:major-latin'>Temperature</span></b> and press <b
1382style='mso-bidi-font-weight:normal'><span style='font-family:"Cambria",serif;
1383mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'>OK</span></b>;
1384do <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1385mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1386major-latin'>Columns/Plot selected</span></b> again. The x-axis will be
1387temperature</p>
1388
1389<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1390auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1391 id="_x0000_i1051" type="#_x0000_t75" style='width:6in;height:370.5pt;
1392 visibility:visible;mso-wrap-style:square'>
1393 <v:imagedata src="ParametricFitting_files/image005.png" o:title=""/>
1394</v:shape><![endif]--><![if !vml]><img width=576 height=494
1395src="ParametricFitting_files/image025.png" v:shapes="_x0000_i1051"><![endif]></span></p>
1396
1397<p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space:
1398auto'>It would seem that there is a phase transition where the orthorhombic low
1399temperature phase is transformed to a miss-indexed tetragonal phase where b=c. This
1400phase transition has been previously explored by <span class=SpellE>Suchomel</span>,
1401Shoemaker &amp; Ribaud (Phys. Rev. B86, 054406, 2012) and the data used in that
1402study is used here in this tutorial. We will be investigating this transition
1403with the parametric fitting tools in GSAS-II </p>
1404
1405<h1>Step 2. Create a Pseudo Variable</h1>
1406
1407<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1408
1409<p class=MsoNormal>While one can plot <i style='mso-bidi-font-style:normal'>a</i>,
1410<i style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1411normal'>c</i> axis lengths by clicking on the appropriate column of the table;
1412it will be more useful to have some mathematical relationship among them. We
1413can define pseudo-variables that allow computation of an arbitrarily defined
1414function of combinations of parameters that are displayed in the sequential
1415refinement table. Any valid Python language expression can be used to define them.
1416For this phase transition it is most useful to view the ratio of <i
1417style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1418normal'>c</i>.</p>
1419
1420<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1421
1422<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1423style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1424style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1425</span></span></span><![endif]>Click on the <b style='mso-bidi-font-weight:
1426normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1427major-latin;mso-hansi-theme-font:major-latin'>Pseudo <span class=SpellE>Vars</span>/Add
1428Formula</span></b> menu item to create a new pseudo-variable. A window opens to
1429define a formula</p>
1430
1431<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1432
1433<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1434mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1050" type="#_x0000_t75"
1435 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
1436 <v:imagedata src="ParametricFitting_files/image073.png" o:title=""/>
1437</v:shape><![endif]--><![if !vml]><img width=576 height=316
1438src="ParametricFitting_files/image027.png" v:shapes="_x0000_i1050"><![endif]></span></p>
1439
1440<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1441
1442<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
1443 o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
1444 text-align:left;margin-left:178.3pt;margin-top:846.2pt;width:229.5pt;height:86.5pt;
1445 z-index:251658240;visibility:visible;mso-wrap-style:square;
1446 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1447 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1448 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1449 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1450 mso-position-vertical-relative:margin;mso-width-percent:0;
1451 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1452 <v:imagedata src="ParametricFitting_files/image007.png" o:title=""/>
1453 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1454</v:shape><![endif]--><![if !vml]><img width=306 height=115
1455src="ParametricFitting_files/image031.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In
1456this window, type a Python expression<b style='mso-bidi-font-weight:normal'>, </b><span
1457style='mso-bidi-font-family:"Times New Roman"'>&quot;</span><b
1458style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1459mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1460"Times New Roman"'>b/c</span></b><span style='mso-bidi-font-family:"Times New Roman"'>&quot;,</span>
1461the variables <b style='mso-bidi-font-weight:normal'><span style='font-family:
1462"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1463major-latin'>b</span></b> and <b style='mso-bidi-font-weight:normal'><span
1464style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1465mso-hansi-theme-font:major-latin'>c</span></b> are arbitrary labels. After
1466entering the formula, press the <b style='mso-bidi-font-weight:normal'><span
1467style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1468mso-hansi-theme-font:major-latin'>Validate</span></b> button or wait a few
1469seconds and the validation will be tested automatically. Note that if you pause
1470too long, while typing, an error message may be displayed as the incomplete
1471formula is found to be invalid. If so, make corrections or keep typing until
1472the entire expression is entered.<span style='mso-spacerun:yes'>  </span></p>
1473
1474<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1475
1476<p class=MsoListParagraphCxSpMiddle>When the expression is complete, a new
1477table appears where the arbitrary <b style='mso-bidi-font-weight:normal'><span
1478style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1479mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
1480normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1481major-latin;mso-hansi-theme-font:major-latin'>c</span></b> labels will be
1482assigned.</p>
1483
1484<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1485
1486<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1487mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1049"
1488 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1489 mso-wrap-style:square'>
1490 <v:imagedata src="ParametricFitting_files/image076.png" o:title=""/>
1491</v:shape><![endif]--><![if !vml]><img width=576 height=316
1492src="ParametricFitting_files/image033.png" v:shapes="Picture_x0020_6"><![endif]></span></p>
1493
1494<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1495
1496<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1497style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1498style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1499</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_6"
1500 o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
1501 text-align:left;margin-left:37.3pt;margin-top:1143.2pt;width:88.5pt;height:65.5pt;
1502 z-index:251659264;visibility:visible;mso-wrap-style:square;
1503 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1504 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1505 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1506 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1507 mso-position-vertical-relative:margin;mso-width-percent:0;
1508 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1509 <v:imagedata src="ParametricFitting_files/image011.png" o:title=""/>
1510 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1511</v:shape><![endif]--><![if !vml]><img width=118 height=87
1512src="ParametricFitting_files/image035.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign
1513the label for the numerator by clicking on the menu button next to label <b
1514style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1515mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>;
1516this will bring up a list of variable types, from this select <b
1517style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1518mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Phase</span></b>
1519to bring up a list of phase variables, use ‘<b style='mso-bidi-font-weight:
1520normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1521major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>lattice’</span></b>
1522as a filter and from that list select variable <b style='mso-bidi-font-weight:
1523normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1524major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>, which is the <i
1525style='mso-bidi-font-style:normal'>b</i> lattice parameter for phase 0 (CuCr<sub>2</sub>O<sub>4</sub>).
1526Press <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1527mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1528major-latin'>OK</span></b>. Repeat this process to select assign to label <b
1529style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1530mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>c</span></b>
1531variable <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1532mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>.</p>
1533
1534<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1535
1536<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1537mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1048"
1538 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
1539 mso-wrap-style:square'>
1540 <v:imagedata src="ParametricFitting_files/image079.png" o:title=""/>
1541</v:shape><![endif]--><![if !vml]><img width=576 height=230
1542src="ParametricFitting_files/image037.png" v:shapes="Picture_x0020_7"><![endif]></span></p>
1543
1544<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1545
1546<p class=MsoListParagraphCxSpMiddle>As each variable is set, the window shows
1547the value for each variable from the first refinement and when the expression can
1548be evaluated, the value for the expression, evaluated for that first
1549refinement. Only when the expression can be evaluated can the <b
1550style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1551mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1552button be pressed. </p>
1553
1554<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1555
1556<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1557mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1047"
1558 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1559 mso-wrap-style:square'>
1560 <v:imagedata src="ParametricFitting_files/image081.png" o:title=""/>
1561</v:shape><![endif]--><![if !vml]><img width=576 height=316
1562src="ParametricFitting_files/image039.png" v:shapes="Picture_x0020_5"><![endif]></span></p>
1563
1564<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1565
1566<p class=MsoListParagraphCxSpMiddle>Press<b style='mso-bidi-font-weight:normal'>
1567</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1568mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1569to close the window and add the pseudo-variable. Note that a new column (‘<b
1570style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1571mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1572major-latin'>b/c</span></b>’) appears at the end of the sequential refinement
1573data table. </p>
1574
1575<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1576
1577<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1578mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75"
1579 style='width:6in;height:192pt;visibility:visible;mso-wrap-style:square'>
1580 <v:imagedata src="ParametricFitting_files/image015.png" o:title=""/>
1581</v:shape><![endif]--><![if !vml]><img width=576 height=256
1582src="ParametricFitting_files/image041.png" v:shapes="_x0000_i1046"><![endif]></span></p>
1583
1584<p class=MsoListParagraphCxSpMiddle>Also note that positioning the mouse above
1585any cell in the table and waiting for a few seconds causes the <span
1586class=SpellE>s.u</span>. to be shown as a tooltip. </p>
1587
1588<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1589
1590<p class=MsoListParagraphCxSpMiddle>Double clicking on the <b style='mso-bidi-font-weight:
1591normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1592major-latin;mso-hansi-theme-font:major-latin'>b/c</span></b> column will cause
1593the pseudo-variable to be plotted:</p>
1594
1595<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1596
1597<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1598mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75"
1599 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
1600 <v:imagedata src="ParametricFitting_files/image017.png" o:title=""/>
1601</v:shape><![endif]--><![if !vml]><img width=576 height=494
1602src="ParametricFitting_files/image045.png" v:shapes="_x0000_i1045"><![endif]></span></p>
1603
1604<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1605
1606<p class=MsoListParagraphCxSpMiddle>The main feature is the b/c ratio falls to
1607~1 at ~120K and is then essentially constant at higher temperatures. It looks
1608like a second order phase transition since the lattice parameter change isn’t
1609confined to a single temperature. Notice that the b/c ratio shows some
1610instability above the transition. This is because the refinement is somewhat
1611ill conditioned after the phase transition because of parameter degeneracies. This
1612can be improved by modifying the Singular Value Decomposition (SVD) <b
1613style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1614mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1615major-latin'>zero tolerance</span></b>. Go to the <b style='mso-bidi-font-weight:
1616normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1617major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Controls</span></b>
1618main tree item and change this parameter to <b style='mso-bidi-font-weight:
1619normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1620major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0.0001</span></b>.</p>
1621
1622<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1623mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1044"
1624 type="#_x0000_t75" style='width:6in;height:171.75pt;visibility:visible;
1625 mso-wrap-style:square'>
1626 <v:imagedata src="ParametricFitting_files/image047.png" o:title=""/>
1627</v:shape><![endif]--><![if !vml]><img width=576 height=229
1628src="ParametricFitting_files/image049.png" v:shapes="Picture_x0020_35"><![endif]></span></p>
1629
1630<p class=MsoListParagraphCxSpMiddle>Then repeat the sequential refinement (perhaps
1631twice); the new b/c plot is now much cleaner after the phase transition.</p>
1632
1633<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1634
1635<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1636mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_36" o:spid="_x0000_i1043"
1637 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
1638 mso-wrap-style:square'>
1639 <v:imagedata src="ParametricFitting_files/image072.png" o:title=""/>
1640</v:shape><![endif]--><![if !vml]><img width=576 height=494
1641src="ParametricFitting_files/image074.png" v:shapes="Picture_x0020_36"><![endif]></span></p>
1642
1643<h1>Step 3: Fitting a simple parametric equation to the unit <span class=GramE>cell</span>
1644volume</h1>
1645
1646<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1647
1648<p class=MsoNormal>In the next few steps we will fit increasingly sophisticated
1649equations of state to different parameters. As a simple example, we can
1650double-click on<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1651normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1652major-latin;mso-hansi-theme-font:major-latin'>0::Vol</span></b>, to plot the
1653unit volume for the first phase vs. T, and fit this to a straight line (even
1654though it obviously isn’t ‘straight’)<span style='mso-spacerun:yes'>  </span></p>
1655
1656<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1657
1658<p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US;
1659mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75"
1660 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
1661 <v:imagedata src="ParametricFitting_files/image019.png" o:title=""/>
1662</v:shape><![endif]--><![if !vml]><img width=576 height=494
1663src="ParametricFitting_files/image075.png" v:shapes="_x0000_i1042"><![endif]></span></p>
1664
1665<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1666style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1667style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1668</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16"
1669 o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
1670 text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt;
1671 z-index:251661312;visibility:visible;mso-wrap-style:square;
1672 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1673 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1674 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1675 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1676 mso-position-vertical-relative:margin;mso-width-percent:0;
1677 mso-height-percent:0;mso-width-relative:margin;mso-height-relative:margin'>
1678 <v:imagedata src="ParametricFitting_files/image029.png" o:title=""/>
1679 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1680</v:shape><![endif]--><![if !vml]><img width=97 height=71
1681src="ParametricFitting_files/image077.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
1682on the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1683mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Parametric
1684Fit/Add</span></b><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1685major-latin;mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:
1686normal'>equation</b></span> menu item, which opens a window similar to what was
1687used before to create a pseudo-variable, except that here an equation is
1688defined.<span style='mso-spacerun:yes'>  </span></p>
1689
1690<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1691
1692<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1693mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1041"
1694 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1695 mso-wrap-style:square'>
1696 <v:imagedata src="ParametricFitting_files/image024.png" o:title=""/>
1697</v:shape><![endif]--><![if !vml]><img width=576 height=316
1698src="ParametricFitting_files/image078.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
1699
1700<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1701
1702<p class=MsoNormal>We will want to fit the unit cell volume to an equation of
1703form <i style='mso-bidi-font-style:normal'>y = <span class=SpellE>mx+b</span></i>,
1704where y will be the volume and x temperature and m and b will be best fit. </p>
1705
1706<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><o:p>&nbsp;</o:p></b></p>
1707
1708<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><b
1709style='mso-bidi-font-weight:normal'><span style='mso-fareast-font-family:"Times New Roman";
1710mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span
1711style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]>Click
1712the<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1713mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span></b><span
1714style='mso-bidi-font-family:"Times New Roman"'>empty</span><b style='mso-bidi-font-weight:
1715normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1716major-latin;mso-hansi-theme-font:major-latin'> </span></b>button in the upper
1717left corner to select a dependent variable for &quot;y&quot;. From the phase
1718list (use ‘<span class=SpellE>vol</span>’ as a filter), select the unit cell
1719volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b style='mso-bidi-font-weight:
1720normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1721"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1722major-latin'>0::Vol</span></b> and then press<b style='mso-bidi-font-weight:
1723normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1724"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1725major-latin'>OK</span>.<o:p></o:p></b></p>
1726
1727<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1728
1729<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1730mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1040"
1731 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
1732 mso-wrap-style:square'>
1733 <v:imagedata src="ParametricFitting_files/image028.png" o:title=""/>
1734</v:shape><![endif]--><![if !vml]><img width=576 height=230
1735src="ParametricFitting_files/image080.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
1736
1737<p class=MsoListParagraphCxSpMiddle>The Expression Editor will be updated to
1738show the new y parameter</p>
1739
1740<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1741mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1039"
1742 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1743 mso-wrap-style:square'>
1744 <v:imagedata src="ParametricFitting_files/image032.png" o:title=""/>
1745</v:shape><![endif]--><![if !vml]><img width=576 height=316
1746src="ParametricFitting_files/image082.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
1747
1748<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1749
1750<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1751style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1752style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1753</span></span></span><![endif]>Now we enter the equation using Python syntax,
1754which is simply &quot;<b style='mso-bidi-font-weight:normal'><span
1755style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1756mso-hansi-theme-font:major-latin'>m*x + b</span></b>&quot;, note that the use
1757of spaces and the choice of labels is arbitrary. If one were to accidentally
1758leave out the &quot;<span style='font-family:Courier'>*</span>&quot; and typed
1759&quot;<span style='font-family:Courier'>mx</span>&quot; then that would be
1760considered a label, if &quot;<span style='font-family:Courier'>m x</span>&quot;
1761were typed, an error message would be displayed. </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="_x0000_i1038" type="#_x0000_t75"
1767 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
1768 <v:imagedata src="ParametricFitting_files/image036.png" o:title=""/>
1769</v:shape><![endif]--><![if !vml]><img width=576 height=316
1770src="ParametricFitting_files/image083.png" v:shapes="_x0000_i1038"><![endif]></span></p>
1771
1772<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1773
1774<p class=MsoListParagraphCxSpMiddle>Then press <b style='mso-bidi-font-weight:
1775normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1776major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> or wait a few
1777seconds and a window changes, as above, allowing assignment of the three new
1778labels (<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1779mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>,
1780<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1781mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1782and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1783mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>).
1784</p>
1785
1786<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1787
1788<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1789style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1790style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1791</span></span></span><![endif]>Assign <b style='mso-bidi-font-weight:normal'><span
1792style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1793mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
1794normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1795major-latin;mso-hansi-theme-font:major-latin'>m</span></b> as refinable “free”
1796parameters: Click on the button next to b and select<b style='mso-bidi-font-weight:
1797normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1798"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1799major-latin'>Free</span></b>. Then repeat this <span class=SpellE>for<b
1800style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1801style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1802mso-hansi-theme-font:major-latin'>m</span></b></span>. </p>
1803
1804<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1805mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_16" o:spid="_x0000_i1037"
1806 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1807 mso-wrap-style:square'>
1808 <v:imagedata src="ParametricFitting_files/image040.png" o:title=""/>
1809</v:shape><![endif]--><![if !vml]><img width=576 height=316
1810src="ParametricFitting_files/image084.png" v:shapes="Picture_x0020_16"><![endif]></span></p>
1811
1812<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1813
1814<p class=MsoListParagraphCxSpMiddle>It is not necessary to change the default
1815variable names, the initial values, or the refine flag. </p>
1816
1817<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1818
1819<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1820style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1821style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1822</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_24"
1823 o:spid="_x0000_s1026" type="#_x0000_t75" style='position:absolute;left:0;
1824 text-align:left;margin-left:24.3pt;margin-top:837.2pt;width:75.5pt;height:26pt;
1825 z-index:251663360;visibility:visible;mso-wrap-style:square;
1826 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1827 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1828 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1829 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1830 mso-position-vertical-relative:margin;mso-width-percent:0;
1831 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1832 <v:imagedata src="ParametricFitting_files/image043.png" o:title=""/>
1833 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1834</v:shape><![endif]--><![if !vml]><img width=101 height=35
1835src="ParametricFitting_files/image085.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
1836set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1837mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>
1838as the temperature, click on the<span style='font-family:"Calibri",sans-serif;
1839mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span>button
1840next to <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1841mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1842select<b style='mso-bidi-font-weight:normal'> </b><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'>Global</span>,</b> and then
1845select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1846normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1847major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> from the
1848list and click on<b style='mso-bidi-font-weight:normal'> </b><b
1849style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1850mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
1851
1852<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1853
1854<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1855style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1856 id="Picture_x0020_17" o:spid="_x0000_i1036" type="#_x0000_t75" style='width:6in;
1857 height:172.5pt;visibility:visible;mso-wrap-style:square'>
1858 <v:imagedata src="ParametricFitting_files/image046.png" o:title=""/>
1859</v:shape><![endif]--><![if !vml]><img width=576 height=230
1860src="ParametricFitting_files/image086.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
1861
1862<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1863
1864<p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values
1865of the dependent variable and the expression for the first refinement, as
1866below.<span style='mso-spacerun:yes'>  </span>Optionally, the values for <b
1867style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1868mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1869and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1870mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1871could be changed manually and the displayed value will be updated when <b
1872style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1873mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b>
1874is pressed or after a short delay.</p>
1875
1876<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1877
1878<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1879mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_46" o:spid="_x0000_i1035"
1880 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1881 mso-wrap-style:square'>
1882 <v:imagedata src="ParametricFitting_files/image050.png" o:title=""/>
1883</v:shape><![endif]--><![if !vml]><img width=576 height=316
1884src="ParametricFitting_files/image087.png" v:shapes="Picture_x0020_46"><![endif]></span></p>
1885
1886<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1887
1888<p class=MsoListParagraph style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1889style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1890style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1891</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1892style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1893mso-hansi-theme-font:major-latin'>OK</span></b> button to save this parametric
1894fit. The <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1895mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1896and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1897mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1898values are then refined and the console shows the results:</p>
1899
1900<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
1901
1902<p class=MsoNormal style='text-indent:.5in'><span style='mso-fareast-language:
1903EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19"
1904 o:spid="_x0000_i1034" type="#_x0000_t75" style='width:270.75pt;height:115.5pt;
1905 visibility:visible;mso-wrap-style:square'>
1906 <v:imagedata src="ParametricFitting_files/image054.png" o:title=""/>
1907</v:shape><![endif]--><![if !vml]><img width=361 height=154
1908src="ParametricFitting_files/image054.png" v:shapes="Picture_x0020_19"><![endif]></span><span
1909style='font-family:Courier'><o:p></o:p></span></p>
1910
1911<p class=MsoListParagraphCxSpFirst><span style='font-family:"Cambria",serif;
1912mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-fareast-language:
1913EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20"
1914 o:spid="_x0000_i1033" type="#_x0000_t75" style='width:6in;height:370.5pt;
1915 visibility:visible;mso-wrap-style:square'>
1916 <v:imagedata src="ParametricFitting_files/image056.png" o:title=""/>
1917</v:shape><![endif]--><![if !vml]><img width=576 height=494
1918src="ParametricFitting_files/image088.png" v:shapes="Picture_x0020_20"><![endif]></span><span
1919style='font-family:"Cambria",serif;mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:
1920minor-latin'><o:p></o:p></span></p>
1921
1922<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1923
1924<p class=MsoListParagraphCxSpMiddle>Also, the plot shows lines for the data
1925values and the fit. Note that while the values are close to a straight line,
1926the deviations are huge compared to the estimated uncertainty values for the
1927unit cell volume (shown as very small error bars, unless the plot is
1928magnified).<span style='mso-spacerun:yes'>  </span>NB: If a linear relationship
1929had been defined as a constraint (as can be done with other software packages) <i>the
1930unit cell parameters would have been forced onto this line with no obvious
1931indication that this was not a correct result.</i> </p>
1932
1933<p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p>
1934
1935<h1>Step 4: Fitting to a <span class=SpellE>Pseudovariable</span></h1>
1936
1937<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1938
1939<p class=MsoNormal>Here we will look at how more sophisticated Python
1940capabilities can be used to fit more complex behavior and use a <span
1941class=SpellE>pseudovariable</span><span style='mso-spacerun:yes'>  </span></p>
1942
1943<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1944style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1945style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1946</span></span></span><![endif]>Double-click on the<b style='mso-bidi-font-weight:
1947normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1948"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1949major-latin'>b/c</span> </b>column. We will be trying to extract the critical
1950exponent for this 2<sup>nd</sup> order phase transition from this data.</p>
1951
1952<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1953mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_37" o:spid="_x0000_i1032"
1954 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
1955 mso-wrap-style:square'>
1956 <v:imagedata src="ParametricFitting_files/image089.png" o:title=""/>
1957</v:shape><![endif]--><![if !vml]><img width=576 height=494
1958src="ParametricFitting_files/image090.png" v:shapes="Picture_x0020_37"><![endif]></span></p>
1959
1960<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1961
1962<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1963style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1964style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1965</span></span></span><![endif]>As a simple (but incorrect) example, we will now
1966fit <i style='mso-bidi-font-style:normal'>b/c </i>as a function of form 1<span
1967class=GramE>/(</span>1 + <span class=SpellE>exp</span>[k(t-t<sub>0</sub>)]).
1968Select the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1969normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1970major-latin;mso-hansi-theme-font:major-latin'>Parametric Fit/Add</span></b><span
1971style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1972mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:normal'>equation</b></span>
1973menu item (as before). For the dependent variable, click on<b style='mso-bidi-font-weight:
1974normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1975"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1976major-latin'>Global</span></b> and then select<b style='mso-bidi-font-weight:
1977normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1978"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1979major-latin'>b/c</span></b>.<span style='mso-spacerun:yes'>  </span></p>
1980
1981<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1982
1983<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1984style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1985style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1986</span></span></span><![endif]>We then enter expression</p>
1987
1988<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1989
1990<p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><b
1991style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1992mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a + b / (1+<span
1993class=GramE>np.exp(</span>k*(t-t0)))<o:p></o:p></span></b></p>
1994
1995<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
1996mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
1997
1998<p class=MsoListParagraphCxSpLast>Note that use of spaces here is arbitrary, but
1999all parentheses are required. Also note that the exponential function [<span
2000class=SpellE>exp</span>(x) or e<sup>x</sup>] is supplied in Python in the <span
2001class=SpellE>numpy</span> modules as <span class=SpellE>np.exp</span>().
2002Further, there are also a small number of common functions that are defined
2003without the prefix for convenience [these include pi, <span class=SpellE><span
2004class=GramE>exp</span></span><span class=GramE>(</span>), <span class=SpellE>sqrt</span>(),
2005<span class=SpellE>sind</span>(), <span class=SpellE>cosd</span>(), <span
2006class=SpellE>tand</span>() where d indicates an argument in degrees.]</p>
2007
2008<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2009
2010<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2011style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2012style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2013</span></span></span><![endif]>Assign free variables to<b style='mso-bidi-font-weight:
2014normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
2015"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2016major-latin'>a</span>, </b><b style='mso-bidi-font-weight:normal'><span
2017style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2018mso-hansi-theme-font:major-latin'>b</span>, </b><b style='mso-bidi-font-weight:
2019normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2020major-latin;mso-hansi-theme-font:major-latin'>k</span> </b>and<b
2021style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
2022style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2023mso-hansi-theme-font:major-latin'>t0</span></b>, similar as to what was done
2024before. Note that the<b style='mso-bidi-font-weight:normal'> </b>variable name
2025for the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2026mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
2027label was set automatically to <b style='mso-bidi-font-weight:normal'><span
2028style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2029mso-hansi-theme-font:major-latin'>b_1</span></b> because the symbol ‘<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;mso-bidi-theme-font:
2032major-latin'>b</span></b>’ was used in the previous equation for fitting the
2033volume. If the same variable name (<b style='mso-bidi-font-weight:normal'><span
2034style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2035mso-hansi-theme-font:major-latin'>b</span></b>) was used again here, the same
2036value would be used to fit both equations, which is certainly not what we want
2037here.<span style='mso-spacerun:yes'>  </span>Finally, since this equation is
2038too complex to fit without some reasonable starting values, specify an
2039approximate value, <b style='mso-bidi-font-weight:normal'><span
2040style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2041mso-hansi-theme-font:major-latin'>120</span></b>, for the vicinity of where the
2042<i style='mso-bidi-font-style:normal'>c</i> lattice parameter starts to change
2043(<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2044mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>)
2045and non-zero values for variables <b>c</b> and <b>k</b>.</p>
2046
2047<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2048
2049<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2050mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_38" o:spid="_x0000_i1031"
2051 type="#_x0000_t75" style='width:6in;height:267.75pt;visibility:visible;
2052 mso-wrap-style:square'>
2053 <v:imagedata src="ParametricFitting_files/image091.png" o:title=""/>
2054</v:shape><![endif]--><![if !vml]><img width=576 height=357
2055src="ParametricFitting_files/image092.png" v:shapes="Picture_x0020_38"><![endif]></span></p>
2056
2057<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2058
2059<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2060style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2061style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2062</span></span></span><![endif]>Optionally, press the <b style='mso-bidi-font-weight:
2063normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2064major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b> button to see if
2065the equation can be refined with starting values of zero for the other three parameters.<span
2066style='mso-spacerun:yes'>  </span>A fit is obtained and is plotted; the fit is
2067quite poor.</p>
2068
2069<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2070mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_39" o:spid="_x0000_i1030"
2071 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
2072 mso-wrap-style:square'>
2073 <v:imagedata src="ParametricFitting_files/image093.png" o:title=""/>
2074</v:shape><![endif]--><![if !vml]><img width=576 height=494
2075src="ParametricFitting_files/image094.png" v:shapes="Picture_x0020_39"><![endif]></span><span
2076style='mso-spacerun:yes'> </span></p>
2077
2078<p class=MsoListParagraphCxSpLast>To give it more reasonable starting values,
2079set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2080mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2081major-latin'>a</span></b>=<b style='mso-bidi-font-weight:normal'><span
2082style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2083mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>1.0, b</span></b>=<b
2084style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2085mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2086major-latin'>0.1</span></b><span class=GramE>,<b style='mso-bidi-font-weight:
2087normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2088major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>k</span></b></span>=<b
2089style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2090mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2091major-latin'>0.1</span></b> and <b style='mso-bidi-font-weight:normal'><span
2092style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2093mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>t0</span></b>=<b
2094style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2095mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2096major-latin'>120.</span></b> Press <b style='mso-bidi-font-weight:normal'><span
2097style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2098mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Fit</span></b>
2099again; the plot shows a better (but not right) fit.</p>
2100
2101<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2102 id="Picture_x0020_40" o:spid="_x0000_i1029" type="#_x0000_t75" style='width:6in;
2103 height:370.5pt;visibility:visible;mso-wrap-style:square'>
2104 <v:imagedata src="ParametricFitting_files/image095.png" o:title=""/>
2105</v:shape><![endif]--><![if !vml]><img width=576 height=494
2106src="ParametricFitting_files/image096.png" v:shapes="Picture_x0020_40"><![endif]></span></p>
2107
2108<p class=MsoListParagraphCxSpFirst><o:p>&nbsp;</o:p></p>
2109
2110<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2111style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2112style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2113</span></span></span><![endif]>Press <b style='mso-bidi-font-weight:normal'><span
2114style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2115mso-hansi-theme-font:major-latin'>OK</span></b> to add this equation to the
2116parametric fits and both equations are refit. The fit is ok but this is not the
2117correct expression for critical phenomena.</p>
2118
2119<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2120
2121<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
2122
2123<p class=MsoNormal>If we look at the above plot, it appears there are two
2124regions. We can treat the lower range as a power-law decay, <span class=SpellE>T<sup>k</sup></span>,
2125and the upper region as linear (m*T + b). </p>
2126
2127<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2128
2129<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2130style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2131style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2132</span></span></span><![endif]>As before, add an equation<b style='mso-bidi-font-weight:
2133normal'> </b>(or edit an existing one). For the dependent variable, select<b
2134style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
2135style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2136mso-hansi-theme-font:major-latin'>Global</span></b> to bring up that selection
2137of variables and select<b style='mso-bidi-font-weight:normal'> </b><b
2138style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2139mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b/c</span> </b>and
2140press<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
2141normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2142major-latin;mso-hansi-theme-font:major-latin'>OK</span>. </b></p>
2143
2144<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2145
2146<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2147style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2148style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2149</span></span></span><![endif]>Then enter the equation as </p>
2150
2151<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2152
2153<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
2154auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2155mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
2156"Lucida Grande";color:black'>a<span class=GramE>+(</span>(T0-T)/T0)**k if T
2157&lt; T0 else <span class=SpellE>d+e</span>*T</span></b><b style='mso-bidi-font-weight:
2158normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2159major-latin;mso-hansi-theme-font:major-latin'><o:p></o:p></span></b></p>
2160
2161<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
2162mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
2163
2164<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2165style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2166style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2167</span></span></span><![endif]>Assign the free and fixed variables, as shown
2168below. Note that the exponent (<b style='mso-bidi-font-weight:normal'><span
2169style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2170mso-hansi-theme-font:major-latin'>k</span></b>) and scaling factor (<b
2171style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2172mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a</span></b>),
2173cannot start as zero; <b style='mso-bidi-font-weight:normal'><span
2174style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2175mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>T0</span></b>
2176should be set to <b style='mso-bidi-font-weight:normal'><span style='font-family:
2177"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2178major-latin;mso-bidi-theme-font:major-latin'>120</span></b><span class=GramE>.<b
2179style='mso-bidi-font-weight:normal'>.</b></span><b style='mso-bidi-font-weight:
2180normal'> </b></p>
2181
2182<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2183
2184<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
2185mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_41" o:spid="_x0000_i1028"
2186 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
2187 mso-wrap-style:square'>
2188 <v:imagedata src="ParametricFitting_files/image097.png" o:title=""/>
2189</v:shape><![endif]--><![if !vml]><img width=576 height=365
2190src="ParametricFitting_files/image098.png" v:shapes="Picture_x0020_41"><![endif]></span></p>
2191
2192<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2193
2194<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2195style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2196style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2197</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
2198style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2199mso-hansi-theme-font:major-latin'>Fit</span></b> button and note that all the
2200free variables are adjusted </p>
2201
2202<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2203
2204<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2205mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1027"
2206 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
2207 mso-wrap-style:square'>
2208 <v:imagedata src="ParametricFitting_files/image099.png" o:title=""/>
2209</v:shape><![endif]--><![if !vml]><img width=576 height=365
2210src="ParametricFitting_files/image100.png" v:shapes="Picture_x0020_42"><![endif]></span><span
2211style='mso-spacerun:yes'> </span></p>
2212
2213<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2214
2215<p class=MsoListParagraphCxSpMiddle>Also, that the plotted result is close to
2216the values from the refinement.</p>
2217
2218<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2219
2220<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2221mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_43" o:spid="_x0000_i1026"
2222 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
2223 mso-wrap-style:square'>
2224 <v:imagedata src="ParametricFitting_files/image101.png" o:title=""/>
2225</v:shape><![endif]--><![if !vml]><img width=576 height=494
2226src="ParametricFitting_files/image102.png" v:shapes="Picture_x0020_43"><![endif]></span></p>
2227
2228<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
2229
2230<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
2231
2232<p class=MsoNormal>This completes this tutorial on use of Pseudo Variables and
2233parametric fitting. </p>
2234
2235<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2236
2237<h1>Advanced Use: Externally-Defined Functions</h1>
2238
2239<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2240
2241<p class=MsoNormal>The parametric refinement section of GSAS-II is intended to
2242be quite flexible, but for very complex fitting, a user may wish to define a
2243custom Python function. Such a function can be defined in a separate module
2244that will be accessed with a parametric equation (or pseudo variable).</p>
2245
2246<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2247
2248<p class=MsoNormal>As an example of that, if we create a file called <b
2249style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2250mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fittest.py</span></b>
2251and locate that file in any directory in the Python path (most simply, in the
2252directory where the project file, <span class=SpellE><b style='mso-bidi-font-weight:
2253normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2254major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>, is
2255located) and place a routine <span class=SpellE><b style='mso-bidi-font-weight:
2256normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2257major-latin;mso-hansi-theme-font:major-latin'>fitfxn</span></b></span> in that
2258module:</p>
2259
2260<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'>
2261
2262<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre></blockquote>
2263
2264</blockquote>
2265
2266<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2267normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2268mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2269EN-US'>#file fittest.py<span style='mso-spacerun:yes'>                </span><o:p></o:p></span></b></p>
2270
2271<p class=MsoNormal style='margin-left:1.0in'><span class=SpellE><span
2272class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;
2273font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2274major-latin;mso-fareast-language:EN-US'>def</span></b></span></span><b
2275style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:
2276"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2277major-latin;mso-fareast-language:EN-US'> <span class=SpellE>fitfxn</span>(T,T0,a,k,d,e):<span
2278style='mso-spacerun:yes'>       </span><o:p></o:p></span></b></p>
2279
2280<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2281normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2282mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2283EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>if</span> T
2284&lt; T0:<span style='mso-spacerun:yes'>                  </span><o:p></o:p></span></b></p>
2285
2286<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2287normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2288mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2289EN-US'><span style='mso-spacerun:yes'>        </span><span class=GramE>return</span>
2290a+((T0-T)/T0)**k <o:p></o:p></span></b></p>
2291
2292<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2293normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2294mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2295EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>else</span>:<span
2296style='mso-spacerun:yes'>                       </span><o:p></o:p></span></b></p>
2297
2298<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2299normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2300mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2301EN-US'><span style='mso-spacerun:yes'>         </span><span class=GramE>return</span>
2302<span class=SpellE>d+e</span>*T<o:p></o:p></span></b></p>
2303
2304<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
2305style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2306major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><o:p>&nbsp;</o:p></span></b></p>
2307
2308<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
2309style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2310major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><span
2311style='mso-spacerun:yes'>           </span></span></b>The function <span
2312class=SpellE><b style='mso-bidi-font-weight:normal'><span style='font-family:
2313"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2314major-latin'>fittest.fitfxn</span></b></span> can then be used to duplicate the
2315previous example: </p>
2316
2317<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2318
2319<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2320 id="Picture_x0020_44" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:6in;
2321 height:289.5pt;visibility:visible;mso-wrap-style:square'>
2322 <v:imagedata src="ParametricFitting_files/image103.png" o:title=""/>
2323</v:shape><![endif]--><![if !vml]><img width=576 height=386
2324src="ParametricFitting_files/image104.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
2325
2326<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2327
2328<p class=MsoNormal>Note that module <b style='mso-bidi-font-weight:normal'><span
2329style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2330mso-hansi-theme-font:major-latin'>fittest</span></b> is automatically located
2331and loaded, provided it is found in the path. No changes to the standard
2332distributed GSAS-II code are needed. Care to avoid use of a module name already
2333found in GSAS-II or Python is wise.</p>
2334
2335</div>
2336
2337</body>
2338
2339</html>
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