source: Tutorials/SeqParametric/ParametricFitting.htm @ 4271

Last change on this file since 4271 was 3553, checked in by toby, 3 years ago

finish web tutorials for now

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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";
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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><B>A video version of this tutorial is available at
1271<A href="https://anl.box.com/v/ParametricFitting" target="_blank">
1272https://anl.box.com/v/ParametricFitting</A></B></P>
1273
1274<p class=MsoNormal>Once a sequential fit has been performed, any function of
1275the fit parameters from the fit and <i style='mso-bidi-font-style:normal'>their
1276estimated uncertainty</i> (<span class=SpellE>s.u</span>.) can be computed by
1277defining one or more &quot;pseudo-variables&quot;. Note that the covariance
1278matrix from the fit is used to provide appropriate treatment of the correlation
1279in parameters with respect to uncertainty estimation.<span
1280style='mso-spacerun:yes'>  </span>In addition, parametric equations in terms of
1281sample parameters such as temperature and pressure, etc. can be fit to either
1282the parameters from the fit or the pseudo-variables. </p>
1283
1284<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1285
1286<p class=MsoNormal>This tutorial assumes that the steps in the <b
1287style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1288mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1289"Times New Roman"'>Sequential refinement of multiple datasets</span></b><span
1290style='mso-bidi-font-family:"Times New Roman"'> tutorial have already been
1291performed. </span>If you have not done so already, start GSAS-II. </p>
1292
1293<h1>Step 1. Open the previous Sequential Fit</h1>
1294
1295<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1296
1297<p class=MsoListParagraphCxSpFirst style='margin-bottom:6.0pt;mso-add-space:
1298auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
1299style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1300style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1301</span></span></span><![endif]>Select the<b style='mso-bidi-font-weight:normal'>
1302</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1303mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>File/Open
1304Project</span></b> menu item to load file named <span class=SpellE><b
1305style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1306mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>
1307you created in the previous tutorial.</p>
1308
1309<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1310auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
1311style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1312style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1313</span></span></span><![endif]>Select the <b style='mso-bidi-font-weight:normal'><span
1314style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1315mso-hansi-theme-font:major-latin'>Sequential results</span></b> item in the
1316data tree. The sequential refinement results data table is displayed, as below.
1317</p>
1318
1319<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1320auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype
1321 id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
1322 path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
1323 <v:stroke joinstyle="miter"/>
1324 <v:formulas>
1325  <v:f eqn="if lineDrawn pixelLineWidth 0"/>
1326  <v:f eqn="sum @0 1 0"/>
1327  <v:f eqn="sum 0 0 @1"/>
1328  <v:f eqn="prod @2 1 2"/>
1329  <v:f eqn="prod @3 21600 pixelWidth"/>
1330  <v:f eqn="prod @3 21600 pixelHeight"/>
1331  <v:f eqn="sum @0 0 1"/>
1332  <v:f eqn="prod @6 1 2"/>
1333  <v:f eqn="prod @7 21600 pixelWidth"/>
1334  <v:f eqn="sum @8 21600 0"/>
1335  <v:f eqn="prod @7 21600 pixelHeight"/>
1336  <v:f eqn="sum @10 21600 0"/>
1337 </v:formulas>
1338 <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
1339 <o:lock v:ext="edit" aspectratio="t"/>
1340</v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1054" type="#_x0000_t75"
1341 style='width:6in;height:186pt;visibility:visible;mso-wrap-style:square'>
1342 <v:imagedata src="ParametricFitting_files/image071.png" o:title=""/>
1343</v:shape><![endif]--><![if !vml]><img width=576 height=248
1344src="ParametricFitting_files/image021.png" v:shapes="Picture_x0020_1"><![endif]></span></p>
1345
1346<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1347auto'>If you look through the table, you will see how each variable changed
1348through the temperature series. In particular the CuCrO<sub>4</sub> b &amp; c
1349lattice parameters become nearly identical above ~120K (PWDR OH_24.fxye Bank
13501). Select these two columns (<b style='mso-bidi-font-weight:normal'><span
1351style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1352mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0::b</span></b>
1353&amp; <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1354mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1355major-latin'>0::c</span></b>) and do <b style='mso-bidi-font-weight:normal'><span
1356style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1357mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Columns/Plot
1358selected</span></b></p>
1359
1360<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1361auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1362 id="Picture_x0020_34" o:spid="_x0000_i1053" type="#_x0000_t75" style='width:6in;
1363 height:370.5pt;visibility:visible;mso-wrap-style:square'>
1364 <v:imagedata src="ParametricFitting_files/image002.png" o:title=""/>
1365</v:shape><![endif]--><![if !vml]><img width=576 height=494
1366src="ParametricFitting_files/image023.png" v:shapes="Picture_x0020_34"><![endif]></span></p>
1367
1368<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1369auto'>Select the plot &amp; press <span class=GramE>the ‘<b style='mso-bidi-font-weight:
1370normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1371major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>s</span></b>’</span>
1372key; the popup will show</p>
1373
1374<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1375auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1376 id="Picture_x0020_3" o:spid="_x0000_i1052" type="#_x0000_t75" style='width:240pt;
1377 height:232.5pt;visibility:visible;mso-wrap-style:square'>
1378 <v:imagedata src="ParametricFitting_files/image004.png" o:title=""/>
1379</v:shape><![endif]--><![if !vml]><img width=320 height=310
1380src="ParametricFitting_files/image004.png" v:shapes="Picture_x0020_3"><![endif]></span></p>
1381
1382<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1383auto'>Select <b style='mso-bidi-font-weight:normal'><span style='font-family:
1384"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1385major-latin;mso-bidi-theme-font:major-latin'>Temperature</span></b> and press <b
1386style='mso-bidi-font-weight:normal'><span style='font-family:"Cambria",serif;
1387mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'>OK</span></b>;
1388do <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1389mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1390major-latin'>Columns/Plot selected</span></b> again. The x-axis will be
1391temperature</p>
1392
1393<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1394auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1395 id="_x0000_i1051" type="#_x0000_t75" style='width:6in;height:370.5pt;
1396 visibility:visible;mso-wrap-style:square'>
1397 <v:imagedata src="ParametricFitting_files/image005.png" o:title=""/>
1398</v:shape><![endif]--><![if !vml]><img width=576 height=494
1399src="ParametricFitting_files/image025.png" v:shapes="_x0000_i1051"><![endif]></span></p>
1400
1401<p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space:
1402auto'>It would seem that there is a phase transition where the orthorhombic low
1403temperature phase is transformed to a miss-indexed tetragonal phase where b=c. This
1404phase transition has been previously explored by <span class=SpellE>Suchomel</span>,
1405Shoemaker &amp; Ribaud (Phys. Rev. B86, 054406, 2012) and the data used in that
1406study is used here in this tutorial. We will be investigating this transition
1407with the parametric fitting tools in GSAS-II </p>
1408
1409<h1>Step 2. Create a Pseudo Variable</h1>
1410
1411<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1412
1413<p class=MsoNormal>While one can plot <i style='mso-bidi-font-style:normal'>a</i>,
1414<i style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1415normal'>c</i> axis lengths by clicking on the appropriate column of the table;
1416it will be more useful to have some mathematical relationship among them. We
1417can define pseudo-variables that allow computation of an arbitrarily defined
1418function of combinations of parameters that are displayed in the sequential
1419refinement table. Any valid Python language expression can be used to define them.
1420For this phase transition it is most useful to view the ratio of <i
1421style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1422normal'>c</i>.</p>
1423
1424<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1425
1426<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1427style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1428style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1429</span></span></span><![endif]>Click on the <b style='mso-bidi-font-weight:
1430normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1431major-latin;mso-hansi-theme-font:major-latin'>Pseudo <span class=SpellE>Vars</span>/Add
1432Formula</span></b> menu item to create a new pseudo-variable. A window opens to
1433define a formula</p>
1434
1435<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1436
1437<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1438mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1050" type="#_x0000_t75"
1439 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
1440 <v:imagedata src="ParametricFitting_files/image073.png" o:title=""/>
1441</v:shape><![endif]--><![if !vml]><img width=576 height=316
1442src="ParametricFitting_files/image027.png" v:shapes="_x0000_i1050"><![endif]></span></p>
1443
1444<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1445
1446<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
1447 o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
1448 text-align:left;margin-left:178.3pt;margin-top:846.2pt;width:229.5pt;height:86.5pt;
1449 z-index:251658240;visibility:visible;mso-wrap-style:square;
1450 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1451 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1452 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1453 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1454 mso-position-vertical-relative:margin;mso-width-percent:0;
1455 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1456 <v:imagedata src="ParametricFitting_files/image007.png" o:title=""/>
1457 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1458</v:shape><![endif]--><![if !vml]><img width=306 height=115
1459src="ParametricFitting_files/image031.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In
1460this window, type a Python expression<b style='mso-bidi-font-weight:normal'>, </b><span
1461style='mso-bidi-font-family:"Times New Roman"'>&quot;</span><b
1462style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1463mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1464"Times New Roman"'>b/c</span></b><span style='mso-bidi-font-family:"Times New Roman"'>&quot;,</span>
1465the variables <b style='mso-bidi-font-weight:normal'><span style='font-family:
1466"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1467major-latin'>b</span></b> and <b style='mso-bidi-font-weight:normal'><span
1468style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1469mso-hansi-theme-font:major-latin'>c</span></b> are arbitrary labels. After
1470entering the formula, press the <b style='mso-bidi-font-weight:normal'><span
1471style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1472mso-hansi-theme-font:major-latin'>Validate</span></b> button or wait a few
1473seconds and the validation will be tested automatically. Note that if you pause
1474too long, while typing, an error message may be displayed as the incomplete
1475formula is found to be invalid. If so, make corrections or keep typing until
1476the entire expression is entered.<span style='mso-spacerun:yes'>  </span></p>
1477
1478<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1479
1480<p class=MsoListParagraphCxSpMiddle>When the expression is complete, a new
1481table appears where the arbitrary <b style='mso-bidi-font-weight:normal'><span
1482style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1483mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
1484normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1485major-latin;mso-hansi-theme-font:major-latin'>c</span></b> labels will be
1486assigned.</p>
1487
1488<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1489
1490<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1491mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1049"
1492 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1493 mso-wrap-style:square'>
1494 <v:imagedata src="ParametricFitting_files/image076.png" o:title=""/>
1495</v:shape><![endif]--><![if !vml]><img width=576 height=316
1496src="ParametricFitting_files/image033.png" v:shapes="Picture_x0020_6"><![endif]></span></p>
1497
1498<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1499
1500<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1501style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1502style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1503</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_6"
1504 o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
1505 text-align:left;margin-left:37.3pt;margin-top:1143.2pt;width:88.5pt;height:65.5pt;
1506 z-index:251659264;visibility:visible;mso-wrap-style:square;
1507 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1508 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1509 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1510 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1511 mso-position-vertical-relative:margin;mso-width-percent:0;
1512 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1513 <v:imagedata src="ParametricFitting_files/image011.png" o:title=""/>
1514 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1515</v:shape><![endif]--><![if !vml]><img width=118 height=87
1516src="ParametricFitting_files/image035.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign
1517the label for the numerator by clicking on the menu button next to label <b
1518style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1519mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>;
1520this will bring up a list of variable types, from this select <b
1521style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1522mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Phase</span></b>
1523to bring up a list of phase variables, use ‘<b style='mso-bidi-font-weight:
1524normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1525major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>lattice’</span></b>
1526as a filter and from that list select variable <b style='mso-bidi-font-weight:
1527normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1528major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>, which is the <i
1529style='mso-bidi-font-style:normal'>b</i> lattice parameter for phase 0 (CuCr<sub>2</sub>O<sub>4</sub>).
1530Press <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1531mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1532major-latin'>OK</span></b>. Repeat this process to select assign to label <b
1533style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1534mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>c</span></b>
1535variable <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1536mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>.</p>
1537
1538<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1539
1540<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1541mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1048"
1542 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
1543 mso-wrap-style:square'>
1544 <v:imagedata src="ParametricFitting_files/image079.png" o:title=""/>
1545</v:shape><![endif]--><![if !vml]><img width=576 height=230
1546src="ParametricFitting_files/image037.png" v:shapes="Picture_x0020_7"><![endif]></span></p>
1547
1548<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1549
1550<p class=MsoListParagraphCxSpMiddle>As each variable is set, the window shows
1551the value for each variable from the first refinement and when the expression can
1552be evaluated, the value for the expression, evaluated for that first
1553refinement. Only when the expression can be evaluated can the <b
1554style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1555mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1556button be pressed. </p>
1557
1558<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1559
1560<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1561mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1047"
1562 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1563 mso-wrap-style:square'>
1564 <v:imagedata src="ParametricFitting_files/image081.png" o:title=""/>
1565</v:shape><![endif]--><![if !vml]><img width=576 height=316
1566src="ParametricFitting_files/image039.png" v:shapes="Picture_x0020_5"><![endif]></span></p>
1567
1568<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1569
1570<p class=MsoListParagraphCxSpMiddle>Press<b style='mso-bidi-font-weight:normal'>
1571</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1572mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1573to close the window and add the pseudo-variable. Note that a new column (‘<b
1574style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1575mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1576major-latin'>b/c</span></b>’) appears at the end of the sequential refinement
1577data table. </p>
1578
1579<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1580
1581<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1582mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75"
1583 style='width:6in;height:192pt;visibility:visible;mso-wrap-style:square'>
1584 <v:imagedata src="ParametricFitting_files/image015.png" o:title=""/>
1585</v:shape><![endif]--><![if !vml]><img width=576 height=256
1586src="ParametricFitting_files/image041.png" v:shapes="_x0000_i1046"><![endif]></span></p>
1587
1588<p class=MsoListParagraphCxSpMiddle>Also note that positioning the mouse above
1589any cell in the table and waiting for a few seconds causes the <span
1590class=SpellE>s.u</span>. to be shown as a tooltip. </p>
1591
1592<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1593
1594<p class=MsoListParagraphCxSpMiddle>Double clicking on the <b style='mso-bidi-font-weight:
1595normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1596major-latin;mso-hansi-theme-font:major-latin'>b/c</span></b> column will cause
1597the pseudo-variable to be plotted:</p>
1598
1599<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1600
1601<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1602mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75"
1603 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
1604 <v:imagedata src="ParametricFitting_files/image017.png" o:title=""/>
1605</v:shape><![endif]--><![if !vml]><img width=576 height=494
1606src="ParametricFitting_files/image045.png" v:shapes="_x0000_i1045"><![endif]></span></p>
1607
1608<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1609
1610<p class=MsoListParagraphCxSpMiddle>The main feature is the b/c ratio falls to
1611~1 at ~120K and is then essentially constant at higher temperatures. It looks
1612like a second order phase transition since the lattice parameter change isn’t
1613confined to a single temperature. Notice that the b/c ratio shows some
1614instability above the transition. This is because the refinement is somewhat
1615ill conditioned after the phase transition because of parameter degeneracies. This
1616can be improved by modifying the Singular Value Decomposition (SVD) <b
1617style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1618mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1619major-latin'>zero tolerance</span></b>. Go to the <b style='mso-bidi-font-weight:
1620normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1621major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Controls</span></b>
1622main tree item and change this parameter to <b style='mso-bidi-font-weight:
1623normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1624major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0.0001</span></b>.</p>
1625
1626<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1627mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1044"
1628 type="#_x0000_t75" style='width:6in;height:171.75pt;visibility:visible;
1629 mso-wrap-style:square'>
1630 <v:imagedata src="ParametricFitting_files/image047.png" o:title=""/>
1631</v:shape><![endif]--><![if !vml]><img width=576 height=229
1632src="ParametricFitting_files/image049.png" v:shapes="Picture_x0020_35"><![endif]></span></p>
1633
1634<p class=MsoListParagraphCxSpMiddle>Then repeat the sequential refinement (perhaps
1635twice); the new b/c plot is now much cleaner after the phase transition.</p>
1636
1637<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1638
1639<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1640mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_36" o:spid="_x0000_i1043"
1641 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
1642 mso-wrap-style:square'>
1643 <v:imagedata src="ParametricFitting_files/image072.png" o:title=""/>
1644</v:shape><![endif]--><![if !vml]><img width=576 height=494
1645src="ParametricFitting_files/image074.png" v:shapes="Picture_x0020_36"><![endif]></span></p>
1646
1647<h1>Step 3: Fitting a simple parametric equation to the unit <span class=GramE>cell</span>
1648volume</h1>
1649
1650<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1651
1652<p class=MsoNormal>In the next few steps we will fit increasingly sophisticated
1653equations of state to different parameters. As a simple example, we can
1654double-click on<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1655normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1656major-latin;mso-hansi-theme-font:major-latin'>0::Vol</span></b>, to plot the
1657unit volume for the first phase vs. T, and fit this to a straight line (even
1658though it obviously isn’t ‘straight’)<span style='mso-spacerun:yes'>  </span></p>
1659
1660<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1661
1662<p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US;
1663mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75"
1664 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
1665 <v:imagedata src="ParametricFitting_files/image019.png" o:title=""/>
1666</v:shape><![endif]--><![if !vml]><img width=576 height=494
1667src="ParametricFitting_files/image075.png" v:shapes="_x0000_i1042"><![endif]></span></p>
1668
1669<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1670style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1671style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1672</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16"
1673 o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
1674 text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt;
1675 z-index:251661312;visibility:visible;mso-wrap-style:square;
1676 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1677 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1678 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1679 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1680 mso-position-vertical-relative:margin;mso-width-percent:0;
1681 mso-height-percent:0;mso-width-relative:margin;mso-height-relative:margin'>
1682 <v:imagedata src="ParametricFitting_files/image029.png" o:title=""/>
1683 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1684</v:shape><![endif]--><![if !vml]><img width=97 height=71
1685src="ParametricFitting_files/image077.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
1686on the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1687mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Parametric
1688Fit/Add</span></b><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1689major-latin;mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:
1690normal'>equation</b></span> menu item, which opens a window similar to what was
1691used before to create a pseudo-variable, except that here an equation is
1692defined.<span style='mso-spacerun:yes'>  </span></p>
1693
1694<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1695
1696<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1697mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1041"
1698 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1699 mso-wrap-style:square'>
1700 <v:imagedata src="ParametricFitting_files/image024.png" o:title=""/>
1701</v:shape><![endif]--><![if !vml]><img width=576 height=316
1702src="ParametricFitting_files/image078.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
1703
1704<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1705
1706<p class=MsoNormal>We will want to fit the unit cell volume to an equation of
1707form <i style='mso-bidi-font-style:normal'>y = <span class=SpellE>mx+b</span></i>,
1708where y will be the volume and x temperature and m and b will be best fit. </p>
1709
1710<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><o:p>&nbsp;</o:p></b></p>
1711
1712<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><b
1713style='mso-bidi-font-weight:normal'><span style='mso-fareast-font-family:"Times New Roman";
1714mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span
1715style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]>Click
1716the<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1717mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span></b><span
1718style='mso-bidi-font-family:"Times New Roman"'>empty</span><b style='mso-bidi-font-weight:
1719normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1720major-latin;mso-hansi-theme-font:major-latin'> </span></b>button in the upper
1721left corner to select a dependent variable for &quot;y&quot;. From the phase
1722list (use ‘<span class=SpellE>vol</span>’ as a filter), select the unit cell
1723volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b style='mso-bidi-font-weight:
1724normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1725"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1726major-latin'>0::Vol</span></b> and then press<b style='mso-bidi-font-weight:
1727normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1728"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1729major-latin'>OK</span>.<o:p></o:p></b></p>
1730
1731<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1732
1733<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1734mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1040"
1735 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
1736 mso-wrap-style:square'>
1737 <v:imagedata src="ParametricFitting_files/image028.png" o:title=""/>
1738</v:shape><![endif]--><![if !vml]><img width=576 height=230
1739src="ParametricFitting_files/image080.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
1740
1741<p class=MsoListParagraphCxSpMiddle>The Expression Editor will be updated to
1742show the new y parameter</p>
1743
1744<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1745mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1039"
1746 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1747 mso-wrap-style:square'>
1748 <v:imagedata src="ParametricFitting_files/image032.png" o:title=""/>
1749</v:shape><![endif]--><![if !vml]><img width=576 height=316
1750src="ParametricFitting_files/image082.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
1751
1752<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1753
1754<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1755style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1756style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1757</span></span></span><![endif]>Now we enter the equation using Python syntax,
1758which is simply &quot;<b style='mso-bidi-font-weight:normal'><span
1759style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1760mso-hansi-theme-font:major-latin'>m*x + b</span></b>&quot;, note that the use
1761of spaces and the choice of labels is arbitrary. If one were to accidentally
1762leave out the &quot;<span style='font-family:Courier'>*</span>&quot; and typed
1763&quot;<span style='font-family:Courier'>mx</span>&quot; then that would be
1764considered a label, if &quot;<span style='font-family:Courier'>m x</span>&quot;
1765were typed, an error message would be displayed. </p>
1766
1767<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1768
1769<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1770mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75"
1771 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
1772 <v:imagedata src="ParametricFitting_files/image036.png" o:title=""/>
1773</v:shape><![endif]--><![if !vml]><img width=576 height=316
1774src="ParametricFitting_files/image083.png" v:shapes="_x0000_i1038"><![endif]></span></p>
1775
1776<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1777
1778<p class=MsoListParagraphCxSpMiddle>Then press <b style='mso-bidi-font-weight:
1779normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1780major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> or wait a few
1781seconds and a window changes, as above, allowing assignment of the three new
1782labels (<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'>m</span></b>,
1784<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1785mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1786and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1787mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>).
1788</p>
1789
1790<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1791
1792<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1793style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1794style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1795</span></span></span><![endif]>Assign <b style='mso-bidi-font-weight:normal'><span
1796style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1797mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
1798normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1799major-latin;mso-hansi-theme-font:major-latin'>m</span></b> as refinable “free”
1800parameters: Click on the button next to b and select<b style='mso-bidi-font-weight:
1801normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1802"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1803major-latin'>Free</span></b>. Then repeat this <span class=SpellE>for<b
1804style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1805style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1806mso-hansi-theme-font:major-latin'>m</span></b></span>. </p>
1807
1808<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1809mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_16" o:spid="_x0000_i1037"
1810 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1811 mso-wrap-style:square'>
1812 <v:imagedata src="ParametricFitting_files/image040.png" o:title=""/>
1813</v:shape><![endif]--><![if !vml]><img width=576 height=316
1814src="ParametricFitting_files/image084.png" v:shapes="Picture_x0020_16"><![endif]></span></p>
1815
1816<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1817
1818<p class=MsoListParagraphCxSpMiddle>It is not necessary to change the default
1819variable names, the initial values, or the refine flag. </p>
1820
1821<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1822
1823<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1824style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1825style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1826</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_24"
1827 o:spid="_x0000_s1026" type="#_x0000_t75" style='position:absolute;left:0;
1828 text-align:left;margin-left:24.3pt;margin-top:837.2pt;width:75.5pt;height:26pt;
1829 z-index:251663360;visibility:visible;mso-wrap-style:square;
1830 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1831 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1832 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1833 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1834 mso-position-vertical-relative:margin;mso-width-percent:0;
1835 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1836 <v:imagedata src="ParametricFitting_files/image043.png" o:title=""/>
1837 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1838</v:shape><![endif]--><![if !vml]><img width=101 height=35
1839src="ParametricFitting_files/image085.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
1840set <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>
1842as the temperature, click on the<span style='font-family:"Calibri",sans-serif;
1843mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span>button
1844next to <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1845mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1846select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1847normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1848major-latin;mso-hansi-theme-font:major-latin'>Global</span>,</b> and then
1849select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1850normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1851major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> from the
1852list and click on<b style='mso-bidi-font-weight:normal'> </b><b
1853style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1854mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
1855
1856<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1857
1858<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1859style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1860 id="Picture_x0020_17" o:spid="_x0000_i1036" type="#_x0000_t75" style='width:6in;
1861 height:172.5pt;visibility:visible;mso-wrap-style:square'>
1862 <v:imagedata src="ParametricFitting_files/image046.png" o:title=""/>
1863</v:shape><![endif]--><![if !vml]><img width=576 height=230
1864src="ParametricFitting_files/image086.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
1865
1866<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1867
1868<p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values
1869of the dependent variable and the expression for the first refinement, as
1870below.<span style='mso-spacerun:yes'>  </span>Optionally, the values for <b
1871style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1872mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1873and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1874mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1875could be changed manually and the displayed value will be updated when <b
1876style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1877mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b>
1878is pressed or after a short delay.</p>
1879
1880<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1881
1882<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1883mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_46" o:spid="_x0000_i1035"
1884 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1885 mso-wrap-style:square'>
1886 <v:imagedata src="ParametricFitting_files/image050.png" o:title=""/>
1887</v:shape><![endif]--><![if !vml]><img width=576 height=316
1888src="ParametricFitting_files/image087.png" v:shapes="Picture_x0020_46"><![endif]></span></p>
1889
1890<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1891
1892<p class=MsoListParagraph style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1893style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1894style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1895</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1896style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1897mso-hansi-theme-font:major-latin'>OK</span></b> button to save this parametric
1898fit. The <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1899mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1900and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1901mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1902values are then refined and the console shows the results:</p>
1903
1904<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
1905
1906<p class=MsoNormal style='text-indent:.5in'><span style='mso-fareast-language:
1907EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19"
1908 o:spid="_x0000_i1034" type="#_x0000_t75" style='width:270.75pt;height:115.5pt;
1909 visibility:visible;mso-wrap-style:square'>
1910 <v:imagedata src="ParametricFitting_files/image054.png" o:title=""/>
1911</v:shape><![endif]--><![if !vml]><img width=361 height=154
1912src="ParametricFitting_files/image054.png" v:shapes="Picture_x0020_19"><![endif]></span><span
1913style='font-family:Courier'><o:p></o:p></span></p>
1914
1915<p class=MsoListParagraphCxSpFirst><span style='font-family:"Cambria",serif;
1916mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-fareast-language:
1917EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20"
1918 o:spid="_x0000_i1033" type="#_x0000_t75" style='width:6in;height:370.5pt;
1919 visibility:visible;mso-wrap-style:square'>
1920 <v:imagedata src="ParametricFitting_files/image056.png" o:title=""/>
1921</v:shape><![endif]--><![if !vml]><img width=576 height=494
1922src="ParametricFitting_files/image088.png" v:shapes="Picture_x0020_20"><![endif]></span><span
1923style='font-family:"Cambria",serif;mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:
1924minor-latin'><o:p></o:p></span></p>
1925
1926<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1927
1928<p class=MsoListParagraphCxSpMiddle>Also, the plot shows lines for the data
1929values and the fit. Note that while the values are close to a straight line,
1930the deviations are huge compared to the estimated uncertainty values for the
1931unit cell volume (shown as very small error bars, unless the plot is
1932magnified).<span style='mso-spacerun:yes'>  </span>NB: If a linear relationship
1933had been defined as a constraint (as can be done with other software packages) <i>the
1934unit cell parameters would have been forced onto this line with no obvious
1935indication that this was not a correct result.</i> </p>
1936
1937<p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p>
1938
1939<h1>Step 4: Fitting to a <span class=SpellE>Pseudovariable</span></h1>
1940
1941<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1942
1943<p class=MsoNormal>Here we will look at how more sophisticated Python
1944capabilities can be used to fit more complex behavior and use a <span
1945class=SpellE>pseudovariable</span><span style='mso-spacerun:yes'>  </span></p>
1946
1947<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1948style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1949style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1950</span></span></span><![endif]>Double-click on the<b style='mso-bidi-font-weight:
1951normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1952"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1953major-latin'>b/c</span> </b>column. We will be trying to extract the critical
1954exponent for this 2<sup>nd</sup> order phase transition from this data.</p>
1955
1956<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1957mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_37" o:spid="_x0000_i1032"
1958 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
1959 mso-wrap-style:square'>
1960 <v:imagedata src="ParametricFitting_files/image089.png" o:title=""/>
1961</v:shape><![endif]--><![if !vml]><img width=576 height=494
1962src="ParametricFitting_files/image090.png" v:shapes="Picture_x0020_37"><![endif]></span></p>
1963
1964<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1965
1966<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1967style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1968style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1969</span></span></span><![endif]>As a simple (but incorrect) example, we will now
1970fit <i style='mso-bidi-font-style:normal'>b/c </i>as a function of form 1<span
1971class=GramE>/(</span>1 + <span class=SpellE>exp</span>[k(t-t<sub>0</sub>)]).
1972Select the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1973normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1974major-latin;mso-hansi-theme-font:major-latin'>Parametric Fit/Add</span></b><span
1975style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1976mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:normal'>equation</b></span>
1977menu item (as before). For the dependent variable, click on<b style='mso-bidi-font-weight:
1978normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1979"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1980major-latin'>Global</span></b> and then select<b style='mso-bidi-font-weight:
1981normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1982"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1983major-latin'>b/c</span></b>.<span style='mso-spacerun:yes'>  </span></p>
1984
1985<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1986
1987<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1988style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1989style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1990</span></span></span><![endif]>We then enter expression</p>
1991
1992<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1993
1994<p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><b
1995style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1996mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a + b / (1+<span
1997class=GramE>np.exp(</span>k*(t-t0)))<o:p></o:p></span></b></p>
1998
1999<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
2000mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
2001
2002<p class=MsoListParagraphCxSpLast>Note that use of spaces here is arbitrary, but
2003all parentheses are required. Also note that the exponential function [<span
2004class=SpellE>exp</span>(x) or e<sup>x</sup>] is supplied in Python in the <span
2005class=SpellE>numpy</span> modules as <span class=SpellE>np.exp</span>().
2006Further, there are also a small number of common functions that are defined
2007without the prefix for convenience [these include pi, <span class=SpellE><span
2008class=GramE>exp</span></span><span class=GramE>(</span>), <span class=SpellE>sqrt</span>(),
2009<span class=SpellE>sind</span>(), <span class=SpellE>cosd</span>(), <span
2010class=SpellE>tand</span>() where d indicates an argument in degrees.]</p>
2011
2012<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2013
2014<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2015style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2016style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2017</span></span></span><![endif]>Assign free variables to<b style='mso-bidi-font-weight:
2018normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
2019"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2020major-latin'>a</span>, </b><b style='mso-bidi-font-weight:normal'><span
2021style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2022mso-hansi-theme-font:major-latin'>b</span>, </b><b style='mso-bidi-font-weight:
2023normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2024major-latin;mso-hansi-theme-font:major-latin'>k</span> </b>and<b
2025style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
2026style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2027mso-hansi-theme-font:major-latin'>t0</span></b>, similar as to what was done
2028before. Note that the<b style='mso-bidi-font-weight:normal'> </b>variable name
2029for the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2030mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
2031label was set automatically to <b style='mso-bidi-font-weight:normal'><span
2032style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2033mso-hansi-theme-font:major-latin'>b_1</span></b> because the symbol ‘<b
2034style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2035mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2036major-latin'>b</span></b>’ was used in the previous equation for fitting the
2037volume. If the same variable name (<b style='mso-bidi-font-weight:normal'><span
2038style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2039mso-hansi-theme-font:major-latin'>b</span></b>) was used again here, the same
2040value would be used to fit both equations, which is certainly not what we want
2041here.<span style='mso-spacerun:yes'>  </span>Finally, since this equation is
2042too complex to fit without some reasonable starting values, specify an
2043approximate value, <b style='mso-bidi-font-weight:normal'><span
2044style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2045mso-hansi-theme-font:major-latin'>120</span></b>, for the vicinity of where the
2046<i style='mso-bidi-font-style:normal'>c</i> lattice parameter starts to change
2047(<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2048mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>)
2049and non-zero values for variables <b>c</b> and <b>k</b>.</p>
2050
2051<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2052
2053<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2054mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_38" o:spid="_x0000_i1031"
2055 type="#_x0000_t75" style='width:6in;height:267.75pt;visibility:visible;
2056 mso-wrap-style:square'>
2057 <v:imagedata src="ParametricFitting_files/image091.png" o:title=""/>
2058</v:shape><![endif]--><![if !vml]><img width=576 height=357
2059src="ParametricFitting_files/image092.png" v:shapes="Picture_x0020_38"><![endif]></span></p>
2060
2061<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2062
2063<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2064style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2065style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2066</span></span></span><![endif]>Optionally, press the <b style='mso-bidi-font-weight:
2067normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2068major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b> button to see if
2069the equation can be refined with starting values of zero for the other three parameters.<span
2070style='mso-spacerun:yes'>  </span>A fit is obtained and is plotted; the fit is
2071quite poor.</p>
2072
2073<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2074mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_39" o:spid="_x0000_i1030"
2075 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
2076 mso-wrap-style:square'>
2077 <v:imagedata src="ParametricFitting_files/image093.png" o:title=""/>
2078</v:shape><![endif]--><![if !vml]><img width=576 height=494
2079src="ParametricFitting_files/image094.png" v:shapes="Picture_x0020_39"><![endif]></span><span
2080style='mso-spacerun:yes'> </span></p>
2081
2082<p class=MsoListParagraphCxSpLast>To give it more reasonable starting values,
2083set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2084mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2085major-latin'>a</span></b>=<b style='mso-bidi-font-weight:normal'><span
2086style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2087mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>1.0, b</span></b>=<b
2088style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2089mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2090major-latin'>0.1</span></b><span class=GramE>,<b style='mso-bidi-font-weight:
2091normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2092major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>k</span></b></span>=<b
2093style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2094mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2095major-latin'>0.1</span></b> and <b style='mso-bidi-font-weight:normal'><span
2096style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2097mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>t0</span></b>=<b
2098style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2099mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2100major-latin'>120.</span></b> Press <b style='mso-bidi-font-weight:normal'><span
2101style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2102mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Fit</span></b>
2103again; the plot shows a better (but not right) fit.</p>
2104
2105<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2106 id="Picture_x0020_40" o:spid="_x0000_i1029" type="#_x0000_t75" style='width:6in;
2107 height:370.5pt;visibility:visible;mso-wrap-style:square'>
2108 <v:imagedata src="ParametricFitting_files/image095.png" o:title=""/>
2109</v:shape><![endif]--><![if !vml]><img width=576 height=494
2110src="ParametricFitting_files/image096.png" v:shapes="Picture_x0020_40"><![endif]></span></p>
2111
2112<p class=MsoListParagraphCxSpFirst><o:p>&nbsp;</o:p></p>
2113
2114<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2115style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2116style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2117</span></span></span><![endif]>Press <b style='mso-bidi-font-weight:normal'><span
2118style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2119mso-hansi-theme-font:major-latin'>OK</span></b> to add this equation to the
2120parametric fits and both equations are refit. The fit is ok but this is not the
2121correct expression for critical phenomena.</p>
2122
2123<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2124
2125<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
2126
2127<p class=MsoNormal>If we look at the above plot, it appears there are two
2128regions. We can treat the lower range as a power-law decay, <span class=SpellE>T<sup>k</sup></span>,
2129and the upper region as linear (m*T + b). </p>
2130
2131<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2132
2133<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2134style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2135style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2136</span></span></span><![endif]>As before, add an equation<b style='mso-bidi-font-weight:
2137normal'> </b>(or edit an existing one). For the dependent variable, select<b
2138style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
2139style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2140mso-hansi-theme-font:major-latin'>Global</span></b> to bring up that selection
2141of variables and select<b style='mso-bidi-font-weight:normal'> </b><b
2142style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2143mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b/c</span> </b>and
2144press<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
2145normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2146major-latin;mso-hansi-theme-font:major-latin'>OK</span>. </b></p>
2147
2148<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2149
2150<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2151style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2152style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2153</span></span></span><![endif]>Then enter the equation
2154(you can copy and paste the equation above into the text box) as:
2155</p>
2156
2157<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2158
2159<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
2160auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2161mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
2162"Lucida Grande";color:black'>a<span class=GramE>+(</span>(T0-T)/T0)**k if T
2163&lt; T0 else <span class=SpellE>d+e</span>*T</span></b><b style='mso-bidi-font-weight:
2164normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2165major-latin;mso-hansi-theme-font:major-latin'><o:p></o:p></span></b></p>
2166
2167<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
2168mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
2169
2170<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2171style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2172style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2173</span></span></span><![endif]>Assign the free and fixed variables, as shown
2174below.
2175Note: if no option under Phase for ‘Temperature’, then select variable
2176type <B>Global</B> and select <B>Global Temperature</B>.
2177Note that the exponent (<b style='mso-bidi-font-weight:normal'><span
2178style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2179mso-hansi-theme-font:major-latin'>k</span></b>) and scaling factor (<b
2180style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2181mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a</span></b>),
2182cannot start as zero; <b style='mso-bidi-font-weight:normal'><span
2183style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2184mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>T0</span></b>
2185should be set to <b style='mso-bidi-font-weight:normal'><span style='font-family:
2186"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2187major-latin;mso-bidi-theme-font:major-latin'>120</span></b><span class=GramE>.</span><b style='mso-bidi-font-weight:
2188normal'> </b>
2189</p>
2190
2191<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2192
2193<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
2194mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_41" o:spid="_x0000_i1028"
2195 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
2196 mso-wrap-style:square'>
2197 <v:imagedata src="ParametricFitting_files/image097.png" o:title=""/>
2198</v:shape><![endif]--><![if !vml]><img width=576 height=365
2199src="ParametricFitting_files/image098.png" v:shapes="Picture_x0020_41"><![endif]></span></p>
2200
2201<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2202
2203<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2204style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2205style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2206</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
2207style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2208mso-hansi-theme-font:major-latin'>Fit</span></b> button and note that all the
2209free variables are adjusted </p>
2210
2211<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2212
2213<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2214mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1027"
2215 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
2216 mso-wrap-style:square'>
2217 <v:imagedata src="ParametricFitting_files/image099.png" o:title=""/>
2218</v:shape><![endif]--><![if !vml]><img width=576 height=365
2219src="ParametricFitting_files/image100.png" v:shapes="Picture_x0020_42"><![endif]></span><span
2220style='mso-spacerun:yes'> </span></p>
2221
2222<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2223
2224<p class=MsoListParagraphCxSpMiddle>Also, that the plotted result is close to
2225the values from the refinement.</p>
2226
2227<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2228
2229<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2230mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_43" o:spid="_x0000_i1026"
2231 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
2232 mso-wrap-style:square'>
2233 <v:imagedata src="ParametricFitting_files/image101.png" o:title=""/>
2234</v:shape><![endif]--><![if !vml]><img width=576 height=494
2235src="ParametricFitting_files/image102.png" v:shapes="Picture_x0020_43"><![endif]></span></p>
2236
2237<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
2238
2239<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
2240
2241<p class=MsoNormal>This completes this tutorial on use of Pseudo Variables and
2242parametric fitting. </p>
2243
2244<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2245
2246<h1>Advanced Use: Externally-Defined Functions</h1>
2247
2248<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2249
2250<p class=MsoNormal>The parametric refinement section of GSAS-II is intended to
2251be quite flexible, but for very complex fitting, a user may wish to define a
2252custom Python function. Such a function can be defined in a separate module
2253that will be accessed with a parametric equation (or pseudo variable).</p>
2254
2255<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2256
2257<p class=MsoNormal>As an example of that, if we create a file called <b
2258style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2259mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fittest.py</span></b>
2260and locate that file in any directory in the Python path (most simply, in the
2261directory where the project file, <span class=SpellE><b style='mso-bidi-font-weight:
2262normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2263major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>, is
2264located) and place a routine <span class=SpellE><b style='mso-bidi-font-weight:
2265normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2266major-latin;mso-hansi-theme-font:major-latin'>fitfxn</span></b></span> in that
2267module:</p>
2268
2269<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'>
2270
2271<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre></blockquote>
2272
2273</blockquote>
2274
2275<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2276normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2277mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2278EN-US'>#file fittest.py<span style='mso-spacerun:yes'>                </span><o:p></o:p></span></b></p>
2279
2280<p class=MsoNormal style='margin-left:1.0in'><span class=SpellE><span
2281class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;
2282font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2283major-latin;mso-fareast-language:EN-US'>def</span></b></span></span><b
2284style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:
2285"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2286major-latin;mso-fareast-language:EN-US'> <span class=SpellE>fitfxn</span>(T,T0,a,k,d,e):<span
2287style='mso-spacerun:yes'>       </span><o:p></o:p></span></b></p>
2288
2289<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2290normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2291mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2292EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>if</span> T
2293&lt; T0:<span style='mso-spacerun:yes'>                  </span><o:p></o:p></span></b></p>
2294
2295<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2296normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2297mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2298EN-US'><span style='mso-spacerun:yes'>        </span><span class=GramE>return</span>
2299a+((T0-T)/T0)**k <o:p></o:p></span></b></p>
2300
2301<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2302normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2303mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2304EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>else</span>:<span
2305style='mso-spacerun:yes'>                       </span><o:p></o:p></span></b></p>
2306
2307<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2308normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2309mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2310EN-US'><span style='mso-spacerun:yes'>         </span><span class=GramE>return</span>
2311<span class=SpellE>d+e</span>*T<o:p></o:p></span></b></p>
2312
2313<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
2314style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2315major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><o:p>&nbsp;</o:p></span></b></p>
2316
2317<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
2318style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2319major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><span
2320style='mso-spacerun:yes'>           </span></span></b>The function <span
2321class=SpellE><b style='mso-bidi-font-weight:normal'><span style='font-family:
2322"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2323major-latin'>fittest.fitfxn</span></b></span> can then be used to duplicate the
2324previous example: </p>
2325
2326<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2327
2328<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2329 id="Picture_x0020_44" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:6in;
2330 height:289.5pt;visibility:visible;mso-wrap-style:square'>
2331 <v:imagedata src="ParametricFitting_files/image103.png" o:title=""/>
2332</v:shape><![endif]--><![if !vml]><img width=576 height=386
2333src="ParametricFitting_files/image104.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
2334
2335<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2336
2337<p class=MsoNormal>Note that module <b style='mso-bidi-font-weight:normal'><span
2338style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2339mso-hansi-theme-font:major-latin'>fittest</span></b> is automatically located
2340and loaded, provided it is found in the path. No changes to the standard
2341distributed GSAS-II code are needed. Care to avoid use of a module name already
2342found in GSAS-II or Python is wise.</p>
2343
2344</div>
2345
2346</body>
2347
2348</html>
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