source: Tutorials/SeqParametric/ParametricFitting.htm

Last change on this file was 4457, checked in by toby, 18 months ago

add data link to tutorials

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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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1235        mso-tstyle-colband-size:0;
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1238        mso-style-parent:"";
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1240        mso-para-margin:0in;
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1242        mso-pagination:widow-orphan;
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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
1271<P><BL>
1272<LI><B>A video version of this tutorial is available at
1273<A href="https://anl.box.com/v/ParametricFitting" target="_blank">
1274https://anl.box.com/v/ParametricFitting</A></B>
1275<LI>Exercise files are created <A
1276  href="../SeqRefine/SequentialTutorial.htm" target="_blank">in this
1277  tutorial</a>
1278</BL></P><P></P>
1279 
1280
1281<p class=MsoNormal>Once a sequential fit has been performed, any function of
1282the fit parameters from the fit and <i style='mso-bidi-font-style:normal'>their
1283estimated uncertainty</i> (<span class=SpellE>s.u</span>.) can be computed by
1284defining one or more &quot;pseudo-variables&quot;. Note that the covariance
1285matrix from the fit is used to provide appropriate treatment of the correlation
1286in parameters with respect to uncertainty estimation.<span
1287style='mso-spacerun:yes'>  </span>In addition, parametric equations in terms of
1288sample parameters such as temperature and pressure, etc. can be fit to either
1289the parameters from the fit or the pseudo-variables. </p>
1290
1291<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1292
1293<p class=MsoNormal>This tutorial assumes that the steps in the <b
1294style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1295mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1296"Times New Roman"'>Sequential refinement of multiple datasets</span></b><span
1297style='mso-bidi-font-family:"Times New Roman"'> tutorial have already been
1298performed. </span>If you have not done so already, start GSAS-II. </p>
1299
1300<h1>Step 1. Open the previous Sequential Fit</h1>
1301
1302<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1303
1304<p class=MsoListParagraphCxSpFirst style='margin-bottom:6.0pt;mso-add-space:
1305auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
1306style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1307style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1308</span></span></span><![endif]>Select the<b style='mso-bidi-font-weight:normal'>
1309</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1310mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>File/Open
1311Project</span></b> menu item to load file named <span class=SpellE><b
1312style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1313mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>
1314you created in the previous tutorial.</p>
1315
1316<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1317auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span
1318style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1319style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1320</span></span></span><![endif]>Select the <b style='mso-bidi-font-weight:normal'><span
1321style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1322mso-hansi-theme-font:major-latin'>Sequential results</span></b> item in the
1323data tree. The sequential refinement results data table is displayed, as below.
1324</p>
1325
1326<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1327auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype
1328 id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
1329 path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
1330 <v:stroke joinstyle="miter"/>
1331 <v:formulas>
1332  <v:f eqn="if lineDrawn pixelLineWidth 0"/>
1333  <v:f eqn="sum @0 1 0"/>
1334  <v:f eqn="sum 0 0 @1"/>
1335  <v:f eqn="prod @2 1 2"/>
1336  <v:f eqn="prod @3 21600 pixelWidth"/>
1337  <v:f eqn="prod @3 21600 pixelHeight"/>
1338  <v:f eqn="sum @0 0 1"/>
1339  <v:f eqn="prod @6 1 2"/>
1340  <v:f eqn="prod @7 21600 pixelWidth"/>
1341  <v:f eqn="sum @8 21600 0"/>
1342  <v:f eqn="prod @7 21600 pixelHeight"/>
1343  <v:f eqn="sum @10 21600 0"/>
1344 </v:formulas>
1345 <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
1346 <o:lock v:ext="edit" aspectratio="t"/>
1347</v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1054" type="#_x0000_t75"
1348 style='width:6in;height:186pt;visibility:visible;mso-wrap-style:square'>
1349 <v:imagedata src="ParametricFitting_files/image071.png" o:title=""/>
1350</v:shape><![endif]--><![if !vml]><img width=576 height=248
1351src="ParametricFitting_files/image021.png" v:shapes="Picture_x0020_1"><![endif]></span></p>
1352
1353<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1354auto'>If you look through the table, you will see how each variable changed
1355through the temperature series. In particular the CuCrO<sub>4</sub> b &amp; c
1356lattice parameters become nearly identical above ~120K (PWDR OH_24.fxye Bank
13571). Select these two columns (<b style='mso-bidi-font-weight:normal'><span
1358style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1359mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0::b</span></b>
1360&amp; <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1361mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1362major-latin'>0::c</span></b>) and do <b style='mso-bidi-font-weight:normal'><span
1363style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1364mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Columns/Plot
1365selected</span></b></p>
1366
1367<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1368auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1369 id="Picture_x0020_34" o:spid="_x0000_i1053" type="#_x0000_t75" style='width:6in;
1370 height:370.5pt;visibility:visible;mso-wrap-style:square'>
1371 <v:imagedata src="ParametricFitting_files/image002.png" o:title=""/>
1372</v:shape><![endif]--><![if !vml]><img width=576 height=494
1373src="ParametricFitting_files/image023.png" v:shapes="Picture_x0020_34"><![endif]></span></p>
1374
1375<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1376auto'>Select the plot &amp; press <span class=GramE>the ‘<b style='mso-bidi-font-weight:
1377normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1378major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>s</span></b>’</span>
1379key; the popup will show</p>
1380
1381<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1382auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1383 id="Picture_x0020_3" o:spid="_x0000_i1052" type="#_x0000_t75" style='width:240pt;
1384 height:232.5pt;visibility:visible;mso-wrap-style:square'>
1385 <v:imagedata src="ParametricFitting_files/image004.png" o:title=""/>
1386</v:shape><![endif]--><![if !vml]><img width=320 height=310
1387src="ParametricFitting_files/image004.png" v:shapes="Picture_x0020_3"><![endif]></span></p>
1388
1389<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1390auto'>Select <b style='mso-bidi-font-weight:normal'><span style='font-family:
1391"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1392major-latin;mso-bidi-theme-font:major-latin'>Temperature</span></b> and press <b
1393style='mso-bidi-font-weight:normal'><span style='font-family:"Cambria",serif;
1394mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'>OK</span></b>;
1395do <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1396mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1397major-latin'>Columns/Plot selected</span></b> again. The x-axis will be
1398temperature</p>
1399
1400<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
1401auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1402 id="_x0000_i1051" type="#_x0000_t75" style='width:6in;height:370.5pt;
1403 visibility:visible;mso-wrap-style:square'>
1404 <v:imagedata src="ParametricFitting_files/image005.png" o:title=""/>
1405</v:shape><![endif]--><![if !vml]><img width=576 height=494
1406src="ParametricFitting_files/image025.png" v:shapes="_x0000_i1051"><![endif]></span></p>
1407
1408<p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space:
1409auto'>It would seem that there is a phase transition where the orthorhombic low
1410temperature phase is transformed to a miss-indexed tetragonal phase where b=c. This
1411phase transition has been previously explored by <span class=SpellE>Suchomel</span>,
1412Shoemaker &amp; Ribaud (Phys. Rev. B86, 054406, 2012) and the data used in that
1413study is used here in this tutorial. We will be investigating this transition
1414with the parametric fitting tools in GSAS-II </p>
1415
1416<h1>Step 2. Create a Pseudo Variable</h1>
1417
1418<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1419
1420<p class=MsoNormal>While one can plot <i style='mso-bidi-font-style:normal'>a</i>,
1421<i style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1422normal'>c</i> axis lengths by clicking on the appropriate column of the table;
1423it will be more useful to have some mathematical relationship among them. We
1424can define pseudo-variables that allow computation of an arbitrarily defined
1425function of combinations of parameters that are displayed in the sequential
1426refinement table. Any valid Python language expression can be used to define them.
1427For this phase transition it is most useful to view the ratio of <i
1428style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
1429normal'>c</i>.</p>
1430
1431<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1432
1433<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1434style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1435style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1436</span></span></span><![endif]>Click on the <b style='mso-bidi-font-weight:
1437normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1438major-latin;mso-hansi-theme-font:major-latin'>Pseudo <span class=SpellE>Vars</span>/Add
1439Formula</span></b> menu item to create a new pseudo-variable. A window opens to
1440define a formula</p>
1441
1442<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1443
1444<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1445mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1050" type="#_x0000_t75"
1446 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
1447 <v:imagedata src="ParametricFitting_files/image073.png" o:title=""/>
1448</v:shape><![endif]--><![if !vml]><img width=576 height=316
1449src="ParametricFitting_files/image027.png" v:shapes="_x0000_i1050"><![endif]></span></p>
1450
1451<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1452
1453<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
1454 o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
1455 text-align:left;margin-left:178.3pt;margin-top:846.2pt;width:229.5pt;height:86.5pt;
1456 z-index:251658240;visibility:visible;mso-wrap-style:square;
1457 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1458 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1459 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1460 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1461 mso-position-vertical-relative:margin;mso-width-percent:0;
1462 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1463 <v:imagedata src="ParametricFitting_files/image007.png" o:title=""/>
1464 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1465</v:shape><![endif]--><![if !vml]><img width=306 height=115
1466src="ParametricFitting_files/image031.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In
1467this window, type a Python expression<b style='mso-bidi-font-weight:normal'>, </b><span
1468style='mso-bidi-font-family:"Times New Roman"'>&quot;</span><b
1469style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1470mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
1471"Times New Roman"'>b/c</span></b><span style='mso-bidi-font-family:"Times New Roman"'>&quot;,</span>
1472the variables <b style='mso-bidi-font-weight:normal'><span style='font-family:
1473"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1474major-latin'>b</span></b> and <b style='mso-bidi-font-weight:normal'><span
1475style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1476mso-hansi-theme-font:major-latin'>c</span></b> are arbitrary labels. After
1477entering the formula, press the <b style='mso-bidi-font-weight:normal'><span
1478style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1479mso-hansi-theme-font:major-latin'>Validate</span></b> button or wait a few
1480seconds and the validation will be tested automatically. Note that if you pause
1481too long, while typing, an error message may be displayed as the incomplete
1482formula is found to be invalid. If so, make corrections or keep typing until
1483the entire expression is entered.<span style='mso-spacerun:yes'>  </span></p>
1484
1485<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1486
1487<p class=MsoListParagraphCxSpMiddle>When the expression is complete, a new
1488table appears where the arbitrary <b style='mso-bidi-font-weight:normal'><span
1489style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1490mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
1491normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1492major-latin;mso-hansi-theme-font:major-latin'>c</span></b> labels will be
1493assigned.</p>
1494
1495<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1496
1497<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1498mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1049"
1499 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1500 mso-wrap-style:square'>
1501 <v:imagedata src="ParametricFitting_files/image076.png" o:title=""/>
1502</v:shape><![endif]--><![if !vml]><img width=576 height=316
1503src="ParametricFitting_files/image033.png" v:shapes="Picture_x0020_6"><![endif]></span></p>
1504
1505<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1506
1507<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
1508style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1509style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1510</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_6"
1511 o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
1512 text-align:left;margin-left:37.3pt;margin-top:1143.2pt;width:88.5pt;height:65.5pt;
1513 z-index:251659264;visibility:visible;mso-wrap-style:square;
1514 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1515 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1516 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1517 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1518 mso-position-vertical-relative:margin;mso-width-percent:0;
1519 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1520 <v:imagedata src="ParametricFitting_files/image011.png" o:title=""/>
1521 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1522</v:shape><![endif]--><![if !vml]><img width=118 height=87
1523src="ParametricFitting_files/image035.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign
1524the label for the numerator by clicking on the menu button next to label <b
1525style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1526mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>;
1527this will bring up a list of variable types, from this select <b
1528style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1529mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Phase</span></b>
1530to bring up a list of phase variables, use ‘<b style='mso-bidi-font-weight:
1531normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1532major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>lattice’</span></b>
1533as a filter and from that list select variable <b style='mso-bidi-font-weight:
1534normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1535major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>, which is the <i
1536style='mso-bidi-font-style:normal'>b</i> lattice parameter for phase 0 (CuCr<sub>2</sub>O<sub>4</sub>).
1537Press <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1538mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1539major-latin'>OK</span></b>. Repeat this process to select assign to label <b
1540style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1541mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>c</span></b>
1542variable <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1543mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>.</p>
1544
1545<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1546
1547<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1548mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1048"
1549 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
1550 mso-wrap-style:square'>
1551 <v:imagedata src="ParametricFitting_files/image079.png" o:title=""/>
1552</v:shape><![endif]--><![if !vml]><img width=576 height=230
1553src="ParametricFitting_files/image037.png" v:shapes="Picture_x0020_7"><![endif]></span></p>
1554
1555<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1556
1557<p class=MsoListParagraphCxSpMiddle>As each variable is set, the window shows
1558the value for each variable from the first refinement and when the expression can
1559be evaluated, the value for the expression, evaluated for that first
1560refinement. Only when the expression can be evaluated can the <b
1561style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1562mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1563button be pressed. </p>
1564
1565<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1566
1567<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1568mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1047"
1569 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1570 mso-wrap-style:square'>
1571 <v:imagedata src="ParametricFitting_files/image081.png" o:title=""/>
1572</v:shape><![endif]--><![if !vml]><img width=576 height=316
1573src="ParametricFitting_files/image039.png" v:shapes="Picture_x0020_5"><![endif]></span></p>
1574
1575<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1576
1577<p class=MsoListParagraphCxSpMiddle>Press<b style='mso-bidi-font-weight:normal'>
1578</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1579mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
1580to close the window and add the pseudo-variable. Note that a new column (‘<b
1581style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1582mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1583major-latin'>b/c</span></b>’) appears at the end of the sequential refinement
1584data table. </p>
1585
1586<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1587
1588<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1589mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75"
1590 style='width:6in;height:192pt;visibility:visible;mso-wrap-style:square'>
1591 <v:imagedata src="ParametricFitting_files/image015.png" o:title=""/>
1592</v:shape><![endif]--><![if !vml]><img width=576 height=256
1593src="ParametricFitting_files/image041.png" v:shapes="_x0000_i1046"><![endif]></span></p>
1594
1595<p class=MsoListParagraphCxSpMiddle>Also note that positioning the mouse above
1596any cell in the table and waiting for a few seconds causes the <span
1597class=SpellE>s.u</span>. to be shown as a tooltip. </p>
1598
1599<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1600
1601<p class=MsoListParagraphCxSpMiddle>Double clicking on the <b style='mso-bidi-font-weight:
1602normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1603major-latin;mso-hansi-theme-font:major-latin'>b/c</span></b> column will cause
1604the pseudo-variable to be plotted:</p>
1605
1606<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1607
1608<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1609mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75"
1610 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
1611 <v:imagedata src="ParametricFitting_files/image017.png" o:title=""/>
1612</v:shape><![endif]--><![if !vml]><img width=576 height=494
1613src="ParametricFitting_files/image045.png" v:shapes="_x0000_i1045"><![endif]></span></p>
1614
1615<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1616
1617<p class=MsoListParagraphCxSpMiddle>The main feature is the b/c ratio falls to
1618~1 at ~120K and is then essentially constant at higher temperatures. It looks
1619like a second order phase transition since the lattice parameter change isn’t
1620confined to a single temperature. Notice that the b/c ratio shows some
1621instability above the transition. This is because the refinement is somewhat
1622ill conditioned after the phase transition because of parameter degeneracies. This
1623can be improved by modifying the Singular Value Decomposition (SVD) <b
1624style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1625mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
1626major-latin'>zero tolerance</span></b>. Go to the <b style='mso-bidi-font-weight:
1627normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1628major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Controls</span></b>
1629main tree item and change this parameter to <b style='mso-bidi-font-weight:
1630normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1631major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0.0001</span></b>.</p>
1632
1633<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1634mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1044"
1635 type="#_x0000_t75" style='width:6in;height:171.75pt;visibility:visible;
1636 mso-wrap-style:square'>
1637 <v:imagedata src="ParametricFitting_files/image047.png" o:title=""/>
1638</v:shape><![endif]--><![if !vml]><img width=576 height=229
1639src="ParametricFitting_files/image049.png" v:shapes="Picture_x0020_35"><![endif]></span></p>
1640
1641<p class=MsoListParagraphCxSpMiddle>Then repeat the sequential refinement (perhaps
1642twice); the new b/c plot is now much cleaner after the phase transition.</p>
1643
1644<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1645
1646<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1647mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_36" o:spid="_x0000_i1043"
1648 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
1649 mso-wrap-style:square'>
1650 <v:imagedata src="ParametricFitting_files/image072.png" o:title=""/>
1651</v:shape><![endif]--><![if !vml]><img width=576 height=494
1652src="ParametricFitting_files/image074.png" v:shapes="Picture_x0020_36"><![endif]></span></p>
1653
1654<h1>Step 3: Fitting a simple parametric equation to the unit <span class=GramE>cell</span>
1655volume</h1>
1656
1657<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1658
1659<p class=MsoNormal>In the next few steps we will fit increasingly sophisticated
1660equations of state to different parameters. As a simple example, we can
1661double-click on<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1662normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1663major-latin;mso-hansi-theme-font:major-latin'>0::Vol</span></b>, to plot the
1664unit volume for the first phase vs. T, and fit this to a straight line (even
1665though it obviously isn’t ‘straight’)<span style='mso-spacerun:yes'>  </span></p>
1666
1667<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1668
1669<p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US;
1670mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75"
1671 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
1672 <v:imagedata src="ParametricFitting_files/image019.png" o:title=""/>
1673</v:shape><![endif]--><![if !vml]><img width=576 height=494
1674src="ParametricFitting_files/image075.png" v:shapes="_x0000_i1042"><![endif]></span></p>
1675
1676<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1677style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1678style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1679</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16"
1680 o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
1681 text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt;
1682 z-index:251661312;visibility:visible;mso-wrap-style:square;
1683 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1684 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1685 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1686 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1687 mso-position-vertical-relative:margin;mso-width-percent:0;
1688 mso-height-percent:0;mso-width-relative:margin;mso-height-relative:margin'>
1689 <v:imagedata src="ParametricFitting_files/image029.png" o:title=""/>
1690 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1691</v:shape><![endif]--><![if !vml]><img width=97 height=71
1692src="ParametricFitting_files/image077.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
1693on the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1694mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Parametric
1695Fit/Add</span></b><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1696major-latin;mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:
1697normal'>equation</b></span> menu item, which opens a window similar to what was
1698used before to create a pseudo-variable, except that here an equation is
1699defined.<span style='mso-spacerun:yes'>  </span></p>
1700
1701<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1702
1703<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1704mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1041"
1705 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1706 mso-wrap-style:square'>
1707 <v:imagedata src="ParametricFitting_files/image024.png" o:title=""/>
1708</v:shape><![endif]--><![if !vml]><img width=576 height=316
1709src="ParametricFitting_files/image078.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
1710
1711<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
1712
1713<p class=MsoNormal>We will want to fit the unit cell volume to an equation of
1714form <i style='mso-bidi-font-style:normal'>y = <span class=SpellE>mx+b</span></i>,
1715where y will be the volume and x temperature and m and b will be best fit. </p>
1716
1717<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><o:p>&nbsp;</o:p></b></p>
1718
1719<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><b
1720style='mso-bidi-font-weight:normal'><span style='mso-fareast-font-family:"Times New Roman";
1721mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span
1722style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]>Click
1723the<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1724mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span></b><span
1725style='mso-bidi-font-family:"Times New Roman"'>empty</span><b style='mso-bidi-font-weight:
1726normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1727major-latin;mso-hansi-theme-font:major-latin'> </span></b>button in the upper
1728left corner to select a dependent variable for &quot;y&quot;. From the phase
1729list (use ‘<span class=SpellE>vol</span>’ as a filter), select the unit cell
1730volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b style='mso-bidi-font-weight:
1731normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1732"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1733major-latin'>0::Vol</span></b> and then press<b style='mso-bidi-font-weight:
1734normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1735"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1736major-latin'>OK</span>.<o:p></o:p></b></p>
1737
1738<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1739
1740<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1741mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1040"
1742 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
1743 mso-wrap-style:square'>
1744 <v:imagedata src="ParametricFitting_files/image028.png" o:title=""/>
1745</v:shape><![endif]--><![if !vml]><img width=576 height=230
1746src="ParametricFitting_files/image080.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
1747
1748<p class=MsoListParagraphCxSpMiddle>The Expression Editor will be updated to
1749show the new y parameter</p>
1750
1751<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1752mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1039"
1753 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1754 mso-wrap-style:square'>
1755 <v:imagedata src="ParametricFitting_files/image032.png" o:title=""/>
1756</v:shape><![endif]--><![if !vml]><img width=576 height=316
1757src="ParametricFitting_files/image082.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
1758
1759<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1760
1761<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1762style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1763style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1764</span></span></span><![endif]>Now we enter the equation using Python syntax,
1765which is simply &quot;<b style='mso-bidi-font-weight:normal'><span
1766style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1767mso-hansi-theme-font:major-latin'>m*x + b</span></b>&quot;, note that the use
1768of spaces and the choice of labels is arbitrary. If one were to accidentally
1769leave out the &quot;<span style='font-family:Courier'>*</span>&quot; and typed
1770&quot;<span style='font-family:Courier'>mx</span>&quot; then that would be
1771considered a label, if &quot;<span style='font-family:Courier'>m x</span>&quot;
1772were typed, an error message would be displayed. </p>
1773
1774<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1775
1776<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1777mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75"
1778 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
1779 <v:imagedata src="ParametricFitting_files/image036.png" o:title=""/>
1780</v:shape><![endif]--><![if !vml]><img width=576 height=316
1781src="ParametricFitting_files/image083.png" v:shapes="_x0000_i1038"><![endif]></span></p>
1782
1783<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1784
1785<p class=MsoListParagraphCxSpMiddle>Then press <b style='mso-bidi-font-weight:
1786normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1787major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> or wait a few
1788seconds and a window changes, as above, allowing assignment of the three new
1789labels (<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1790mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>,
1791<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1792mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1793and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1794mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>).
1795</p>
1796
1797<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1798
1799<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1800style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1801style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1802</span></span></span><![endif]>Assign <b style='mso-bidi-font-weight:normal'><span
1803style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1804mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
1805normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1806major-latin;mso-hansi-theme-font:major-latin'>m</span></b> as refinable “free”
1807parameters: Click on the button next to b and select<b style='mso-bidi-font-weight:
1808normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1809"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1810major-latin'>Free</span></b>. Then repeat this <span class=SpellE>for<b
1811style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
1812style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1813mso-hansi-theme-font:major-latin'>m</span></b></span>. </p>
1814
1815<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1816mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_16" o:spid="_x0000_i1037"
1817 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1818 mso-wrap-style:square'>
1819 <v:imagedata src="ParametricFitting_files/image040.png" o:title=""/>
1820</v:shape><![endif]--><![if !vml]><img width=576 height=316
1821src="ParametricFitting_files/image084.png" v:shapes="Picture_x0020_16"><![endif]></span></p>
1822
1823<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1824
1825<p class=MsoListParagraphCxSpMiddle>It is not necessary to change the default
1826variable names, the initial values, or the refine flag. </p>
1827
1828<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1829
1830<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1831style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1832style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1833</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_24"
1834 o:spid="_x0000_s1026" type="#_x0000_t75" style='position:absolute;left:0;
1835 text-align:left;margin-left:24.3pt;margin-top:837.2pt;width:75.5pt;height:26pt;
1836 z-index:251663360;visibility:visible;mso-wrap-style:square;
1837 mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
1838 mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
1839 mso-wrap-distance-bottom:0;mso-position-horizontal:right;
1840 mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
1841 mso-position-vertical-relative:margin;mso-width-percent:0;
1842 mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
1843 <v:imagedata src="ParametricFitting_files/image043.png" o:title=""/>
1844 <w:wrap type="square" anchorx="margin" anchory="margin"/>
1845</v:shape><![endif]--><![if !vml]><img width=101 height=35
1846src="ParametricFitting_files/image085.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
1847set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1848mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>
1849as the temperature, click on the<span style='font-family:"Calibri",sans-serif;
1850mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span>button
1851next to <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1852mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
1853select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1854normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1855major-latin;mso-hansi-theme-font:major-latin'>Global</span>,</b> and then
1856select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1857normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1858major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> from the
1859list and click on<b style='mso-bidi-font-weight:normal'> </b><b
1860style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1861mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
1862
1863<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1864
1865<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
1866style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
1867 id="Picture_x0020_17" o:spid="_x0000_i1036" type="#_x0000_t75" style='width:6in;
1868 height:172.5pt;visibility:visible;mso-wrap-style:square'>
1869 <v:imagedata src="ParametricFitting_files/image046.png" o:title=""/>
1870</v:shape><![endif]--><![if !vml]><img width=576 height=230
1871src="ParametricFitting_files/image086.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
1872
1873<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1874
1875<p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values
1876of the dependent variable and the expression for the first refinement, as
1877below.<span style='mso-spacerun:yes'>  </span>Optionally, the values for <b
1878style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1879mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1880and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1881mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1882could be changed manually and the displayed value will be updated when <b
1883style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1884mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b>
1885is pressed or after a short delay.</p>
1886
1887<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1888
1889<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
1890mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_46" o:spid="_x0000_i1035"
1891 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
1892 mso-wrap-style:square'>
1893 <v:imagedata src="ParametricFitting_files/image050.png" o:title=""/>
1894</v:shape><![endif]--><![if !vml]><img width=576 height=316
1895src="ParametricFitting_files/image087.png" v:shapes="Picture_x0020_46"><![endif]></span></p>
1896
1897<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1898
1899<p class=MsoListParagraph style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
1900style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1901style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1902</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
1903style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1904mso-hansi-theme-font:major-latin'>OK</span></b> button to save this parametric
1905fit. The <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1906mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
1907and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
1908mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
1909values are then refined and the console shows the results:</p>
1910
1911<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
1912
1913<p class=MsoNormal style='text-indent:.5in'><span style='mso-fareast-language:
1914EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19"
1915 o:spid="_x0000_i1034" type="#_x0000_t75" style='width:270.75pt;height:115.5pt;
1916 visibility:visible;mso-wrap-style:square'>
1917 <v:imagedata src="ParametricFitting_files/image054.png" o:title=""/>
1918</v:shape><![endif]--><![if !vml]><img width=361 height=154
1919src="ParametricFitting_files/image054.png" v:shapes="Picture_x0020_19"><![endif]></span><span
1920style='font-family:Courier'><o:p></o:p></span></p>
1921
1922<p class=MsoListParagraphCxSpFirst><span style='font-family:"Cambria",serif;
1923mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-fareast-language:
1924EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20"
1925 o:spid="_x0000_i1033" type="#_x0000_t75" style='width:6in;height:370.5pt;
1926 visibility:visible;mso-wrap-style:square'>
1927 <v:imagedata src="ParametricFitting_files/image056.png" o:title=""/>
1928</v:shape><![endif]--><![if !vml]><img width=576 height=494
1929src="ParametricFitting_files/image088.png" v:shapes="Picture_x0020_20"><![endif]></span><span
1930style='font-family:"Cambria",serif;mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:
1931minor-latin'><o:p></o:p></span></p>
1932
1933<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1934
1935<p class=MsoListParagraphCxSpMiddle>Also, the plot shows lines for the data
1936values and the fit. Note that while the values are close to a straight line,
1937the deviations are huge compared to the estimated uncertainty values for the
1938unit cell volume (shown as very small error bars, unless the plot is
1939magnified).<span style='mso-spacerun:yes'>  </span>NB: If a linear relationship
1940had been defined as a constraint (as can be done with other software packages) <i>the
1941unit cell parameters would have been forced onto this line with no obvious
1942indication that this was not a correct result.</i> </p>
1943
1944<p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p>
1945
1946<h1>Step 4: Fitting to a <span class=SpellE>Pseudovariable</span></h1>
1947
1948<p class=MsoNormal><o:p>&nbsp;</o:p></p>
1949
1950<p class=MsoNormal>Here we will look at how more sophisticated Python
1951capabilities can be used to fit more complex behavior and use a <span
1952class=SpellE>pseudovariable</span><span style='mso-spacerun:yes'>  </span></p>
1953
1954<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1955style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1956style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1957</span></span></span><![endif]>Double-click on the<b style='mso-bidi-font-weight:
1958normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1959"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1960major-latin'>b/c</span> </b>column. We will be trying to extract the critical
1961exponent for this 2<sup>nd</sup> order phase transition from this data.</p>
1962
1963<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
1964mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_37" o:spid="_x0000_i1032"
1965 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
1966 mso-wrap-style:square'>
1967 <v:imagedata src="ParametricFitting_files/image089.png" o:title=""/>
1968</v:shape><![endif]--><![if !vml]><img width=576 height=494
1969src="ParametricFitting_files/image090.png" v:shapes="Picture_x0020_37"><![endif]></span></p>
1970
1971<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1972
1973<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1974style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1975style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1976</span></span></span><![endif]>As a simple (but incorrect) example, we will now
1977fit <i style='mso-bidi-font-style:normal'>b/c </i>as a function of form 1<span
1978class=GramE>/(</span>1 + <span class=SpellE>exp</span>[k(t-t<sub>0</sub>)]).
1979Select the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
1980normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
1981major-latin;mso-hansi-theme-font:major-latin'>Parametric Fit/Add</span></b><span
1982style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
1983mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:normal'>equation</b></span>
1984menu item (as before). For the dependent variable, click on<b style='mso-bidi-font-weight:
1985normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1986"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1987major-latin'>Global</span></b> and then select<b style='mso-bidi-font-weight:
1988normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
1989"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
1990major-latin'>b/c</span></b>.<span style='mso-spacerun:yes'>  </span></p>
1991
1992<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
1993
1994<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
1995style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
1996style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
1997</span></span></span><![endif]>We then enter expression</p>
1998
1999<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2000
2001<p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><b
2002style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2003mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a + b / (1+<span
2004class=GramE>np.exp(</span>k*(t-t0)))<o:p></o:p></span></b></p>
2005
2006<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
2007mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
2008
2009<p class=MsoListParagraphCxSpLast>Note that use of spaces here is arbitrary, but
2010all parentheses are required. Also note that the exponential function [<span
2011class=SpellE>exp</span>(x) or e<sup>x</sup>] is supplied in Python in the <span
2012class=SpellE>numpy</span> modules as <span class=SpellE>np.exp</span>().
2013Further, there are also a small number of common functions that are defined
2014without the prefix for convenience [these include pi, <span class=SpellE><span
2015class=GramE>exp</span></span><span class=GramE>(</span>), <span class=SpellE>sqrt</span>(),
2016<span class=SpellE>sind</span>(), <span class=SpellE>cosd</span>(), <span
2017class=SpellE>tand</span>() where d indicates an argument in degrees.]</p>
2018
2019<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2020
2021<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2022style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2023style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2024</span></span></span><![endif]>Assign free variables to<b style='mso-bidi-font-weight:
2025normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
2026"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2027major-latin'>a</span>, </b><b style='mso-bidi-font-weight:normal'><span
2028style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2029mso-hansi-theme-font:major-latin'>b</span>, </b><b style='mso-bidi-font-weight:
2030normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2031major-latin;mso-hansi-theme-font:major-latin'>k</span> </b>and<b
2032style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
2033style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2034mso-hansi-theme-font:major-latin'>t0</span></b>, similar as to what was done
2035before. Note that the<b style='mso-bidi-font-weight:normal'> </b>variable name
2036for the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2037mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
2038label was set automatically to <b style='mso-bidi-font-weight:normal'><span
2039style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2040mso-hansi-theme-font:major-latin'>b_1</span></b> because the symbol ‘<b
2041style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2042mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2043major-latin'>b</span></b>’ was used in the previous equation for fitting the
2044volume. If the same variable name (<b style='mso-bidi-font-weight:normal'><span
2045style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2046mso-hansi-theme-font:major-latin'>b</span></b>) was used again here, the same
2047value would be used to fit both equations, which is certainly not what we want
2048here.<span style='mso-spacerun:yes'>  </span>Finally, since this equation is
2049too complex to fit without some reasonable starting values, specify an
2050approximate value, <b style='mso-bidi-font-weight:normal'><span
2051style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2052mso-hansi-theme-font:major-latin'>120</span></b>, for the vicinity of where the
2053<i style='mso-bidi-font-style:normal'>c</i> lattice parameter starts to change
2054(<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2055mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>)
2056and non-zero values for variables <b>c</b> and <b>k</b>.</p>
2057
2058<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2059
2060<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2061mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_38" o:spid="_x0000_i1031"
2062 type="#_x0000_t75" style='width:6in;height:267.75pt;visibility:visible;
2063 mso-wrap-style:square'>
2064 <v:imagedata src="ParametricFitting_files/image091.png" o:title=""/>
2065</v:shape><![endif]--><![if !vml]><img width=576 height=357
2066src="ParametricFitting_files/image092.png" v:shapes="Picture_x0020_38"><![endif]></span></p>
2067
2068<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2069
2070<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2071style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2072style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2073</span></span></span><![endif]>Optionally, press the <b style='mso-bidi-font-weight:
2074normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2075major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b> button to see if
2076the equation can be refined with starting values of zero for the other three parameters.<span
2077style='mso-spacerun:yes'>  </span>A fit is obtained and is plotted; the fit is
2078quite poor.</p>
2079
2080<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2081mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_39" o:spid="_x0000_i1030"
2082 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
2083 mso-wrap-style:square'>
2084 <v:imagedata src="ParametricFitting_files/image093.png" o:title=""/>
2085</v:shape><![endif]--><![if !vml]><img width=576 height=494
2086src="ParametricFitting_files/image094.png" v:shapes="Picture_x0020_39"><![endif]></span><span
2087style='mso-spacerun:yes'> </span></p>
2088
2089<p class=MsoListParagraphCxSpLast>To give it more reasonable starting values,
2090set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2091mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2092major-latin'>a</span></b>=<b style='mso-bidi-font-weight:normal'><span
2093style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2094mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>1.0, b</span></b>=<b
2095style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2096mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2097major-latin'>0.1</span></b><span class=GramE>,<b style='mso-bidi-font-weight:
2098normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2099major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>k</span></b></span>=<b
2100style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2101mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2102major-latin'>0.1</span></b> and <b style='mso-bidi-font-weight:normal'><span
2103style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2104mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>t0</span></b>=<b
2105style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2106mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
2107major-latin'>120.</span></b> Press <b style='mso-bidi-font-weight:normal'><span
2108style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2109mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Fit</span></b>
2110again; the plot shows a better (but not right) fit.</p>
2111
2112<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2113 id="Picture_x0020_40" o:spid="_x0000_i1029" type="#_x0000_t75" style='width:6in;
2114 height:370.5pt;visibility:visible;mso-wrap-style:square'>
2115 <v:imagedata src="ParametricFitting_files/image095.png" o:title=""/>
2116</v:shape><![endif]--><![if !vml]><img width=576 height=494
2117src="ParametricFitting_files/image096.png" v:shapes="Picture_x0020_40"><![endif]></span></p>
2118
2119<p class=MsoListParagraphCxSpFirst><o:p>&nbsp;</o:p></p>
2120
2121<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
2122style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2123style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2124</span></span></span><![endif]>Press <b style='mso-bidi-font-weight:normal'><span
2125style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2126mso-hansi-theme-font:major-latin'>OK</span></b> to add this equation to the
2127parametric fits and both equations are refit. The fit is ok but this is not the
2128correct expression for critical phenomena.</p>
2129
2130<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2131
2132<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
2133
2134<p class=MsoNormal>If we look at the above plot, it appears there are two
2135regions. We can treat the lower range as a power-law decay, <span class=SpellE>T<sup>k</sup></span>,
2136and the upper region as linear (m*T + b). </p>
2137
2138<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2139
2140<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2141style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2142style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2143</span></span></span><![endif]>As before, add an equation<b style='mso-bidi-font-weight:
2144normal'> </b>(or edit an existing one). For the dependent variable, select<b
2145style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
2146style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2147mso-hansi-theme-font:major-latin'>Global</span></b> to bring up that selection
2148of variables and select<b style='mso-bidi-font-weight:normal'> </b><b
2149style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2150mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b/c</span> </b>and
2151press<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
2152normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2153major-latin;mso-hansi-theme-font:major-latin'>OK</span>. </b></p>
2154
2155<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2156
2157<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2158style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2159style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2160</span></span></span><![endif]>Then enter the equation
2161(you can copy and paste the equation above into the text box) as:
2162</p>
2163
2164<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2165
2166<p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
2167auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2168mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
2169"Lucida Grande";color:black'>a<span class=GramE>+(</span>(T0-T)/T0)**k if T
2170&lt; T0 else <span class=SpellE>d+e</span>*T</span></b><b style='mso-bidi-font-weight:
2171normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2172major-latin;mso-hansi-theme-font:major-latin'><o:p></o:p></span></b></p>
2173
2174<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
2175mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p>&nbsp;</o:p></span></p>
2176
2177<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2178style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2179style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2180</span></span></span><![endif]>Assign the free and fixed variables, as shown
2181below.
2182Note: if no option under Phase for ‘Temperature’, then select variable
2183type <B>Global</B> and select <B>Global Temperature</B>.
2184Note that the exponent (<b style='mso-bidi-font-weight:normal'><span
2185style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2186mso-hansi-theme-font:major-latin'>k</span></b>) and scaling factor (<b
2187style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2188mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a</span></b>),
2189cannot start as zero; <b style='mso-bidi-font-weight:normal'><span
2190style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2191mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>T0</span></b>
2192should be set to <b style='mso-bidi-font-weight:normal'><span style='font-family:
2193"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2194major-latin;mso-bidi-theme-font:major-latin'>120</span></b><span class=GramE>.</span><b style='mso-bidi-font-weight:
2195normal'> </b>
2196</p>
2197
2198<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2199
2200<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
2201mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_41" o:spid="_x0000_i1028"
2202 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
2203 mso-wrap-style:square'>
2204 <v:imagedata src="ParametricFitting_files/image097.png" o:title=""/>
2205</v:shape><![endif]--><![if !vml]><img width=576 height=365
2206src="ParametricFitting_files/image098.png" v:shapes="Picture_x0020_41"><![endif]></span></p>
2207
2208<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2209
2210<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span
2211style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
2212style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2213</span></span></span><![endif]>Press the <b style='mso-bidi-font-weight:normal'><span
2214style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2215mso-hansi-theme-font:major-latin'>Fit</span></b> button and note that all the
2216free variables are adjusted </p>
2217
2218<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2219
2220<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2221mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1027"
2222 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
2223 mso-wrap-style:square'>
2224 <v:imagedata src="ParametricFitting_files/image099.png" o:title=""/>
2225</v:shape><![endif]--><![if !vml]><img width=576 height=365
2226src="ParametricFitting_files/image100.png" v:shapes="Picture_x0020_42"><![endif]></span><span
2227style='mso-spacerun:yes'> </span></p>
2228
2229<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2230
2231<p class=MsoListParagraphCxSpMiddle>Also, that the plotted result is close to
2232the values from the refinement.</p>
2233
2234<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
2235
2236<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
2237mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_43" o:spid="_x0000_i1026"
2238 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
2239 mso-wrap-style:square'>
2240 <v:imagedata src="ParametricFitting_files/image101.png" o:title=""/>
2241</v:shape><![endif]--><![if !vml]><img width=576 height=494
2242src="ParametricFitting_files/image102.png" v:shapes="Picture_x0020_43"><![endif]></span></p>
2243
2244<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
2245
2246<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
2247
2248<p class=MsoNormal>This completes this tutorial on use of Pseudo Variables and
2249parametric fitting. </p>
2250
2251<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2252
2253<h1>Advanced Use: Externally-Defined Functions</h1>
2254
2255<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2256
2257<p class=MsoNormal>The parametric refinement section of GSAS-II is intended to
2258be quite flexible, but for very complex fitting, a user may wish to define a
2259custom Python function. Such a function can be defined in a separate module
2260that will be accessed with a parametric equation (or pseudo variable).</p>
2261
2262<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2263
2264<p class=MsoNormal>As an example of that, if we create a file called <b
2265style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
2266mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>fittest.py</span></b>
2267and locate that file in any directory in the Python path (most simply, in the
2268directory where the project file, <span class=SpellE><b style='mso-bidi-font-weight:
2269normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2270major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>, is
2271located) and place a routine <span class=SpellE><b style='mso-bidi-font-weight:
2272normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2273major-latin;mso-hansi-theme-font:major-latin'>fitfxn</span></b></span> in that
2274module:</p>
2275
2276<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'>
2277
2278<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre></blockquote>
2279
2280</blockquote>
2281
2282<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2283normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2284mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2285EN-US'>#file fittest.py<span style='mso-spacerun:yes'>                </span><o:p></o:p></span></b></p>
2286
2287<p class=MsoNormal style='margin-left:1.0in'><span class=SpellE><span
2288class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;
2289font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2290major-latin;mso-fareast-language:EN-US'>def</span></b></span></span><b
2291style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:
2292"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2293major-latin;mso-fareast-language:EN-US'> <span class=SpellE>fitfxn</span>(T,T0,a,k,d,e):<span
2294style='mso-spacerun:yes'>       </span><o:p></o:p></span></b></p>
2295
2296<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2297normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2298mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2299EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>if</span> T
2300&lt; T0:<span style='mso-spacerun:yes'>                  </span><o:p></o:p></span></b></p>
2301
2302<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2303normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2304mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2305EN-US'><span style='mso-spacerun:yes'>        </span><span class=GramE>return</span>
2306a+((T0-T)/T0)**k <o:p></o:p></span></b></p>
2307
2308<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2309normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2310mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2311EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>else</span>:<span
2312style='mso-spacerun:yes'>                       </span><o:p></o:p></span></b></p>
2313
2314<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
2315normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
2316mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
2317EN-US'><span style='mso-spacerun:yes'>         </span><span class=GramE>return</span>
2318<span class=SpellE>d+e</span>*T<o:p></o:p></span></b></p>
2319
2320<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
2321style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2322major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><o:p>&nbsp;</o:p></span></b></p>
2323
2324<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
2325style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
2326major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><span
2327style='mso-spacerun:yes'>           </span></span></b>The function <span
2328class=SpellE><b style='mso-bidi-font-weight:normal'><span style='font-family:
2329"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
2330major-latin'>fittest.fitfxn</span></b></span> can then be used to duplicate the
2331previous example: </p>
2332
2333<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2334
2335<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
2336 id="Picture_x0020_44" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:6in;
2337 height:289.5pt;visibility:visible;mso-wrap-style:square'>
2338 <v:imagedata src="ParametricFitting_files/image103.png" o:title=""/>
2339</v:shape><![endif]--><![if !vml]><img width=576 height=386
2340src="ParametricFitting_files/image104.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
2341
2342<p class=MsoNormal><o:p>&nbsp;</o:p></p>
2343
2344<p class=MsoNormal>Note that module <b style='mso-bidi-font-weight:normal'><span
2345style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
2346mso-hansi-theme-font:major-latin'>fittest</span></b> is automatically located
2347and loaded, provided it is found in the path. No changes to the standard
2348distributed GSAS-II code are needed. Care to avoid use of a module name already
2349found in GSAS-II or Python is wise.</p>
2350
2351</div>
2352
2353</body>
2354
2355</html>
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