Changeset 3094


Ignore:
Timestamp:
Sep 21, 2017 2:36:00 PM (8 years ago)
Author:
vondreele
Message:

revise & update to 2 frame

Location:
Tutorials/SeqParametric
Files:
42 added
46 deleted
22 edited

Legend:

Unmodified
Added
Removed

  • TabularUnified Tutorials/SeqParametric/ParametricFitting.htm

    r2324 r3094  
    2222 <o:DocumentProperties>
    2323  <o:Author>B Toby</o:Author>
    24   <o:LastAuthor>Von Dreele</o:LastAuthor>
    25   <o:Revision>3</o:Revision>
    26   <o:TotalTime>986</o:TotalTime>
     24  <o:LastAuthor>vondreele</o:LastAuthor>
     25  <o:Revision>6</o:Revision>
     26  <o:TotalTime>1765</o:TotalTime>
    2727  <o:Created>2014-07-22T21:21:00Z</o:Created>
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    3232  <o:Company>Argonne National Laboratory</o:Company>
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    12731285"Times New Roman"'>Sequential refinement of multiple datasets</span></b><span
    12741286style='mso-bidi-font-family:"Times New Roman"'> tutorial have already been
    1275 performed. </span>If you have not done so already, start GSAS-II. Some of the
    1276 pictures in this tutorial are from a Mac OSX machine; these illustrate the
    1277 differences in menu location from Windows/Linux machines.</p>
     1287performed. </span>If you have not done so already, start GSAS-II. </p>
    12781288
    12791289<h1>Step 1. Open the previous Sequential Fit</h1>
    … …  
    13031313</p>
    13041314
    1305 <p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space:
     1315<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
    13061316auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype
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    … …  
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    1328  <v:imagedata src="ParametricFitting_files/image001.png" o:title=""/>
    1329 </v:shape><![endif]--><![if !vml]><img width=700 height=350
    1330 src="ParametricFitting_files/image002.png" v:shapes="Picture_x0020_36"><![endif]></span></p>
     1336</v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1054" type="#_x0000_t75"
     1337 style='width:6in;height:186pt;visibility:visible;mso-wrap-style:square'>
     1338 <v:imagedata src="ParametricFitting_files/image071.png" o:title=""/>
     1339</v:shape><![endif]--><![if !vml]><img width=576 height=248
     1340src="ParametricFitting_files/image021.png" v:shapes="Picture_x0020_1"><![endif]></span></p>
     1341
     1342<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
     1343auto'>If you look through the table, you will see how each variable changed
     1344through the temperature series. In particular the CuCrO<sub>4</sub> b &amp; c
     1345lattice parameters become nearly identical above ~120K (PWDR OH_24.fxye Bank
     13461). Select these two columns (<b style='mso-bidi-font-weight:normal'><span
     1347style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     1348mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0::b</span></b>
     1349&amp; <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1350mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     1351major-latin'>0::c</span></b>) and do <b style='mso-bidi-font-weight:normal'><span
     1352style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     1353mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Columns/Plot
     1354selected</span></b></p>
     1355
     1356<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
     1357auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
     1358 id="Picture_x0020_34" o:spid="_x0000_i1053" type="#_x0000_t75" style='width:6in;
     1359 height:370.5pt;visibility:visible;mso-wrap-style:square'>
     1360 <v:imagedata src="ParametricFitting_files/image002.png" o:title=""/>
     1361</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1362src="ParametricFitting_files/image023.png" v:shapes="Picture_x0020_34"><![endif]></span></p>
     1363
     1364<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
     1365auto'>Select the plot &amp; press <span class=GramE>the ‘<b style='mso-bidi-font-weight:
     1366normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1367major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>s</span></b>’</span>
     1368key; the popup will show</p>
     1369
     1370<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
     1371auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
     1372 id="Picture_x0020_3" o:spid="_x0000_i1052" type="#_x0000_t75" style='width:240pt;
     1373 height:232.5pt;visibility:visible;mso-wrap-style:square'>
     1374 <v:imagedata src="ParametricFitting_files/image004.png" o:title=""/>
     1375</v:shape><![endif]--><![if !vml]><img width=320 height=310
     1376src="ParametricFitting_files/image004.png" v:shapes="Picture_x0020_3"><![endif]></span></p>
     1377
     1378<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
     1379auto'>Select <b style='mso-bidi-font-weight:normal'><span style='font-family:
     1380"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1381major-latin;mso-bidi-theme-font:major-latin'>Temperature</span></b> and press <b
     1382style='mso-bidi-font-weight:normal'><span style='font-family:"Cambria",serif;
     1383mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'>OK</span></b>;
     1384do <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1385mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     1386major-latin'>Columns/Plot selected</span></b> again. The x-axis will be
     1387temperature</p>
     1388
     1389<p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space:
     1390auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
     1391 id="_x0000_i1051" type="#_x0000_t75" style='width:6in;height:370.5pt;
     1392 visibility:visible;mso-wrap-style:square'>
     1393 <v:imagedata src="ParametricFitting_files/image005.png" o:title=""/>
     1394</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1395src="ParametricFitting_files/image025.png" v:shapes="_x0000_i1051"><![endif]></span></p>
     1396
     1397<p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space:
     1398auto'>It would seem that there is a phase transition where the orthorhombic low
     1399temperature phase is transformed to a miss-indexed tetragonal phase where b=c. This
     1400phase transition has been previously explored by <span class=SpellE>Suchomel</span>,
     1401Shoemaker &amp; Ribaud (Phys. Rev. B86, 054406, 2012) and the data used in that
     1402study is used here in this tutorial. We will be investigating this transition
     1403with the parametric fitting tools in GSAS-II </p>
    13311404
    13321405<h1>Step 2. Create a Pseudo Variable</h1>
    … …  
    13341407<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    13351408
    1336 <p class=MsoNormal>Pseudo-variables allow computation of an arbitrarily defined
    1337 function of combinations of the variables that are displayed in the sequential
    1338 refinement table. Any valid Python language expression can be used to define
    1339 this. While one can plot <i style='mso-bidi-font-style:normal'>a</i>, <i
     1409<p class=MsoNormal>While one can plot <i style='mso-bidi-font-style:normal'>a</i>,
     1410<i style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
     1411normal'>c</i> axis lengths by clicking on the appropriate column of the table;
     1412it will be more useful to have some mathematical relationship among them. We
     1413can define pseudo-variables that allow computation of an arbitrarily defined
     1414function of combinations of parameters that are displayed in the sequential
     1415refinement table. Any valid Python language expression can be used to define them.
     1416For this phase transition it is most useful to view the ratio of <i
    13401417style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style:
    1341 normal'>c</i> axis lengths by clicking on the appropriate column of the table,
    1342 to compare them one might want to scale them. Alternately, as will be done
    1343 here, we will plot the ratio of <i style='mso-bidi-font-style:normal'>b</i> and
    1344 <i style='mso-bidi-font-style:normal'>c</i>.</p>
     1418normal'>c</i>.</p>
    13451419
    13461420<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    … …  
    13511425</span></span></span><![endif]>Click on the <b style='mso-bidi-font-weight:
    13521426normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1353 major-latin;mso-hansi-theme-font:major-latin'>Pseudo <span
    1354 class=SpellE>Vars</span>/Add Formula</span></b>
    1355 menu item to create a new pseudo-variable.</p>
     1427major-latin;mso-hansi-theme-font:major-latin'>Pseudo <span class=SpellE>Vars</span>/Add
     1428Formula</span></b> menu item to create a new pseudo-variable. A window opens to
     1429define a formula</p>
    13561430
    13571431<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    13581432
    13591433<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1360 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1052"
    1361  type="#_x0000_t75" style='width:159pt;height:45.75pt;visibility:visible;
    1362  mso-wrap-style:square'>
    1363  <v:imagedata src="ParametricFitting_files/image003.png" o:title=""/>
    1364 </v:shape><![endif]--><![if !vml]><img width=212 height=61
    1365 src="ParametricFitting_files/image004.jpg" v:shapes="Picture_x0020_2"><![endif]></span></p>
    1366 
    1367 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    1368 
    1369 <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
    1370 style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    1371 style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    1372 </span></span></span><![endif]>A window opens to define a formula</p>
    1373 
    1374 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    1375 
    1376 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
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    1379  mso-wrap-style:square'>
    1380  <v:imagedata src="ParametricFitting_files/image005.png" o:title="" croptop="4214f"
    1381   cropbottom="8982f" cropleft="3852f" cropright="3654f"/>
    1382 </v:shape><![endif]--><![if !vml]><img width=413 height=203
    1383 src="ParametricFitting_files/image006.png" v:shapes="Picture_x0020_3"><![endif]></span></p>
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     1437</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1438src="ParametricFitting_files/image027.png" v:shapes="_x0000_i1050"><![endif]></span></p>
    13841439
    13851440<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    13861441
    13871442<p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
    1388  o:spid="_x0000_s1031" type="#_x0000_t75" style='position:absolute;left:0;
     1443 o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
    13891444 text-align:left;margin-left:178.3pt;margin-top:846.2pt;width:229.5pt;height:86.5pt;
    13901445 z-index:251658240;visibility:visible;mso-wrap-style:square;
    … …  
    13981453 <w:wrap type="square" anchorx="margin" anchory="margin"/>
    13991454</v:shape><![endif]--><![if !vml]><img width=306 height=115
    1400 src="ParametricFitting_files/image008.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In
     1455src="ParametricFitting_files/image031.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In
    14011456this window, type a Python expression<b style='mso-bidi-font-weight:normal'>, </b><span
    14021457style='mso-bidi-font-family:"Times New Roman"'>&quot;</span><b
    … …  
    14041459mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
    14051460"Times New Roman"'>b/c</span></b><span style='mso-bidi-font-family:"Times New Roman"'>&quot;,</span>
    1406 but note that the variables <b style='mso-bidi-font-weight:normal'><span
    1407 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1408 mso-hansi-theme-font:major-latin'>c</span></b> and <b style='mso-bidi-font-weight:
    1409 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1410 major-latin;mso-hansi-theme-font:major-latin'>b</span></b> here are arbitrary
    1411 labels. After entering the formula, press the <b style='mso-bidi-font-weight:
    1412 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1413 major-latin;mso-hansi-theme-font:major-latin'>Validate</span></b> button or
    1414 wait a few seconds and the validation will be tested automatically. Note that
    1415 if you pause too long, while typing, an error message may be displayed as the
    1416 incomplete formula is found to be invalid. If so, make corrections or keep
    1417 typing until the entire expression is entered.<span style='mso-spacerun:yes'> 
    1418 </span>The error message will sometimes show where error is encountered, though
    1419 not in this case. </p>
     1461the variables <b style='mso-bidi-font-weight:normal'><span style='font-family:
     1462"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1463major-latin'>b</span></b> and <b style='mso-bidi-font-weight:normal'><span
     1464style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     1465mso-hansi-theme-font:major-latin'>c</span></b> are arbitrary labels. After
     1466entering the formula, press the <b style='mso-bidi-font-weight:normal'><span
     1467style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     1468mso-hansi-theme-font:major-latin'>Validate</span></b> button or wait a few
     1469seconds and the validation will be tested automatically. Note that if you pause
     1470too long, while typing, an error message may be displayed as the incomplete
     1471formula is found to be invalid. If so, make corrections or keep typing until
     1472the entire expression is entered.<span style='mso-spacerun:yes'>  </span></p>
    14201473
    14211474<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    14241477table appears where the arbitrary <b style='mso-bidi-font-weight:normal'><span
    14251478style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1426 mso-hansi-theme-font:major-latin'>c</span></b> and <b style='mso-bidi-font-weight:
     1479mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
    14271480normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1428 major-latin;mso-hansi-theme-font:major-latin'>b</span></b> labels will be
     1481major-latin;mso-hansi-theme-font:major-latin'>c</span></b> labels will be
    14291482assigned.</p>
    14301483
    … …  
    14321485
    14331486<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1434 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1050"
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    1437  <v:imagedata src="ParametricFitting_files/image009.png" o:title="" croptop="4435f"
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    1440 src="ParametricFitting_files/image010.png" v:shapes="Picture_x0020_5"><![endif]></span></p>
     1490 <v:imagedata src="ParametricFitting_files/image076.png" o:title=""/>
     1491</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1492src="ParametricFitting_files/image033.png" v:shapes="Picture_x0020_6"><![endif]></span></p>
    14411493
    14421494<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    14441496<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span
    14451497style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    1446 style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
     1498style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    14471499</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_6"
    1448  o:spid="_x0000_s1030" type="#_x0000_t75" style='position:absolute;left:0;
     1500 o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
    14491501 text-align:left;margin-left:37.3pt;margin-top:1143.2pt;width:88.5pt;height:65.5pt;
    14501502 z-index:251659264;visibility:visible;mso-wrap-style:square;
    … …  
    14581510 <w:wrap type="square" anchorx="margin" anchory="margin"/>
    14591511</v:shape><![endif]--><![if !vml]><img width=118 height=87
    1460 src="ParametricFitting_files/image012.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign
     1512src="ParametricFitting_files/image035.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign
    14611513the label for the numerator by clicking on the menu button next to label <b
    14621514style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    … …  
    14651517style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    14661518mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Phase</span></b>
    1467 to bring up a list of phase variables, from that list select variable <b
     1519to bring up a list of phase variables, use ‘<b style='mso-bidi-font-weight:
     1520normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1521major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>lattice’</span></b>
     1522as a filter and from that list select variable <b style='mso-bidi-font-weight:
     1523normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1524major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>, which is the <i
     1525style='mso-bidi-font-style:normal'>b</i> lattice parameter for phase 0 (CuCr<sub>2</sub>O<sub>4</sub>).
     1526Press <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1527mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     1528major-latin'>OK</span></b>. Repeat this process to select assign to label <b
    14681529style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1469 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>,
    1470 which is the <i style='mso-bidi-font-style:normal'>b</i> lattice parameter for
    1471 phase 0 (CuCr<sub>2</sub>O<sub>4</sub>). Repeat this process to select assign
    1472 to label <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    14731530mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>c</span></b>
    14741531variable <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1475 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::b</span></b>.</p>
     1532mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>.</p>
    14761533
    14771534<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    14781535
    14791536<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1480 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_8" o:spid="_x0000_i1049"
    1481  type="#_x0000_t75" style='width:315.75pt;height:129pt;visibility:visible;
     1537mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1048"
     1538 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
    14821539 mso-wrap-style:square'>
    1483  <v:imagedata src="ParametricFitting_files/image013.png" o:title="" croptop="4597f"
    1484   cropbottom="9873f" cropleft="3348f" cropright="3506f"/>
    1485 </v:shape><![endif]--><![if !vml]><img width=421 height=172
    1486 src="ParametricFitting_files/image014.png" v:shapes="Picture_x0020_8"><![endif]></span></p>
     1540 <v:imagedata src="ParametricFitting_files/image079.png" o:title=""/>
     1541</v:shape><![endif]--><![if !vml]><img width=576 height=230
     1542src="ParametricFitting_files/image037.png" v:shapes="Picture_x0020_7"><![endif]></span></p>
    14871543
    14881544<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    14891545
    14901546<p class=MsoListParagraphCxSpMiddle>As each variable is set, the window shows
    1491 the value for each variable from the first refinement and when the expression
    1492 can be evaluated, the value for the expression, evaluated for that first
     1547the value for each variable from the first refinement and when the expression can
     1548be evaluated, the value for the expression, evaluated for that first
    14931549refinement. Only when the expression can be evaluated can the <b
    14941550style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    … …  
    14991555
    15001556<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1501 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_9" o:spid="_x0000_i1048"
    1502  type="#_x0000_t75" style='width:314.25pt;height:153.75pt;visibility:visible;
     1557mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1047"
     1558 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
    15031559 mso-wrap-style:square'>
    1504  <v:imagedata src="ParametricFitting_files/image015.png" o:title="" croptop="3770f"
    1505   cropbottom="8549f" cropleft="3371f" cropright="3424f"/>
    1506 </v:shape><![endif]--><![if !vml]><img width=419 height=205
    1507 src="ParametricFitting_files/image016.png" v:shapes="Picture_x0020_9"><![endif]></span></p>
     1560 <v:imagedata src="ParametricFitting_files/image081.png" o:title=""/>
     1561</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1562src="ParametricFitting_files/image039.png" v:shapes="Picture_x0020_5"><![endif]></span></p>
    15081563
    15091564<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    15121567</b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    15131568mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>
    1514 to close the window and add the pseudo-variable. Note that a new column appears
    1515 at the end of the sequential refinement data table. </p>
     1569to close the window and add the pseudo-variable. Note that a new column (‘<b
     1570style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1571mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     1572major-latin'>b/c</span></b>’) appears at the end of the sequential refinement
     1573data table. </p>
    15161574
    15171575<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    15181576
    15191577<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1520 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_10" o:spid="_x0000_i1047"
    1521  type="#_x0000_t75" style='width:354pt;height:180.75pt;visibility:visible;
    1522  mso-wrap-style:square'>
    1523  <v:imagedata src="ParametricFitting_files/image017.png" o:title=""/>
    1524 </v:shape><![endif]--><![if !vml]><img width=472 height=241
    1525 src="ParametricFitting_files/image018.jpg" v:shapes="Picture_x0020_10"><![endif]></span></p>
    1526 
    1527 <p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_11"
    1528  o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0;
    1529  text-align:left;margin-left:42.3pt;margin-top:666.2pt;width:93.5pt;height:39pt;
    1530  z-index:251660288;visibility:visible;mso-wrap-style:square;
    1531  mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
    1532  mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
    1533  mso-wrap-distance-bottom:0;mso-position-horizontal:right;
    1534  mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
    1535  mso-position-vertical-relative:margin;mso-width-percent:0;
    1536  mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
    1537  <v:imagedata src="ParametricFitting_files/image019.png" o:title=""/>
    1538  <w:wrap type="square" anchorx="margin" anchory="margin"/>
    1539 </v:shape><![endif]--><![if !vml]><img width=125 height=52
    1540 src="ParametricFitting_files/image020.jpg" align=right hspace=12 v:shapes="Picture_x0020_11"><![endif]>Also
    1541 note that positioning the mouse above any cell in the table and waiting for a
    1542 few seconds causes the <span class=SpellE>s.u</span>. to be shown as a tooltip.
    1543 </p>
     1578mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75"
     1579 style='width:6in;height:192pt;visibility:visible;mso-wrap-style:square'>
     1580 <v:imagedata src="ParametricFitting_files/image015.png" o:title=""/>
     1581</v:shape><![endif]--><![if !vml]><img width=576 height=256
     1582src="ParametricFitting_files/image041.png" v:shapes="_x0000_i1046"><![endif]></span></p>
     1583
     1584<p class=MsoListParagraphCxSpMiddle>Also note that positioning the mouse above
     1585any cell in the table and waiting for a few seconds causes the <span
     1586class=SpellE>s.u</span>. to be shown as a tooltip. </p>
    15441587
    15451588<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    15531596
    15541597<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1555 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1046"
    1556  type="#_x0000_t75" style='width:351.75pt;height:303pt;visibility:visible;
     1598mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75"
     1599 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
     1600 <v:imagedata src="ParametricFitting_files/image017.png" o:title=""/>
     1601</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1602src="ParametricFitting_files/image045.png" v:shapes="_x0000_i1045"><![endif]></span></p>
     1603
     1604<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
     1605
     1606<p class=MsoListParagraphCxSpMiddle>The main feature is the b/c ratio falls to
     1607~1 at ~120K and is then essentially constant at higher temperatures. It looks
     1608like a second order phase transition since the lattice parameter change isn’t
     1609confined to a single temperature. Notice that the b/c ratio shows some
     1610instability above the transition. This is because the refinement is somewhat
     1611ill conditioned after the phase transition because of parameter degeneracies. This
     1612can be improved by modifying the Singular Value Decomposition (SVD) <b
     1613style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1614mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     1615major-latin'>zero tolerance</span></b>. Go to the <b style='mso-bidi-font-weight:
     1616normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1617major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Controls</span></b>
     1618main tree item and change this parameter to <b style='mso-bidi-font-weight:
     1619normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1620major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>0.0001</span></b>.</p>
     1621
     1622<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
     1623mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1044"
     1624 type="#_x0000_t75" style='width:6in;height:171.75pt;visibility:visible;
    15571625 mso-wrap-style:square'>
    1558  <v:imagedata src="ParametricFitting_files/image021.png" o:title=""/>
    1559 </v:shape><![endif]--><![if !vml]><img width=469 height=404
    1560 src="ParametricFitting_files/image022.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
    1561 
    1562 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    1563 
    1564 <p class=MsoListParagraphCxSpMiddle>Press the &quot;<b style='mso-bidi-font-weight:
    1565 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1566 major-latin;mso-hansi-theme-font:major-latin'>s</span></b>&quot; key inside the
    1567 plot to select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
    1568 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1569 major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> as the
    1570 x-axis. </p>
    1571 
    1572 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    1573 
    1574 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1575 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1045"
    1576  type="#_x0000_t75" style='width:162.75pt;height:156pt;visibility:visible;
     1626 <v:imagedata src="ParametricFitting_files/image047.png" o:title=""/>
     1627</v:shape><![endif]--><![if !vml]><img width=576 height=229
     1628src="ParametricFitting_files/image049.png" v:shapes="Picture_x0020_35"><![endif]></span></p>
     1629
     1630<p class=MsoListParagraphCxSpMiddle>Then repeat the sequential refinement (perhaps
     1631twice); the new b/c plot is now much cleaner after the phase transition.</p>
     1632
     1633<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
     1634
     1635<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
     1636mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_36" o:spid="_x0000_i1043"
     1637 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
    15771638 mso-wrap-style:square'>
    1578  <v:imagedata src="ParametricFitting_files/image023.png" o:title="" croptop="4105f"
    1579   cropbottom="8920f" cropleft="6109f" cropright="5911f"/>
    1580 </v:shape><![endif]--><![if !vml]><img width=217 height=208
    1581 src="ParametricFitting_files/image024.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
    1582 
    1583 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    1584 
    1585 <p class=MsoListParagraphCxSpMiddle>The plot will then appear as below:</p>
    1586 
    1587 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    1588 
    1589 <p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
    1590 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1044"
    1591  type="#_x0000_t75" style='width:353.25pt;height:303.75pt;visibility:visible;
    1592  mso-wrap-style:square'>
    1593  <v:imagedata src="ParametricFitting_files/image025.png" o:title=""/>
    1594 </v:shape><![endif]--><![if !vml]><img width=471 height=405
    1595 src="ParametricFitting_files/image026.jpg" v:shapes="Picture_x0020_14"><![endif]></span></p>
    1596 
    1597 <h1>Step 3: Fitting a simple parametric equation to the unit cell volume</h1>
     1639 <v:imagedata src="ParametricFitting_files/image072.png" o:title=""/>
     1640</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1641src="ParametricFitting_files/image074.png" v:shapes="Picture_x0020_36"><![endif]></span></p>
     1642
     1643<h1>Step 3: Fitting a simple parametric equation to the unit <span class=GramE>cell</span>
     1644volume</h1>
    15981645
    15991646<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    … …  
    16041651normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    16051652major-latin;mso-hansi-theme-font:major-latin'>0::Vol</span></b>, to plot the
    1606 unit volume for the first phase vs. T, and fit this to a straight line.<span
    1607 style='mso-spacerun:yes'>  </span></p>
     1653unit volume for the first phase vs. T, and fit this to a straight line (even
     1654though it obviously isn’t ‘straight’)<span style='mso-spacerun:yes'>  </span></p>
    16081655
    16091656<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    16101657
    16111658<p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US;
    1612 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_15" o:spid="_x0000_i1043"
    1613  type="#_x0000_t75" style='width:356.25pt;height:305.25pt;visibility:visible;
    1614  mso-wrap-style:square'>
    1615  <v:imagedata src="ParametricFitting_files/image027.png" o:title=""/>
    1616 </v:shape><![endif]--><![if !vml]><img width=475 height=407
    1617 src="ParametricFitting_files/image028.jpg" v:shapes="Picture_x0020_15"><![endif]></span></p>
     1659mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75"
     1660 style='width:6in;height:370.5pt;visibility:visible;mso-wrap-style:square'>
     1661 <v:imagedata src="ParametricFitting_files/image019.png" o:title=""/>
     1662</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1663src="ParametricFitting_files/image075.png" v:shapes="_x0000_i1042"><![endif]></span></p>
    16181664
    16191665<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
    … …  
    16211667style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    16221668</span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16"
    1623  o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0;
     1669 o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
    16241670 text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt;
    16251671 z-index:251661312;visibility:visible;mso-wrap-style:square;
    … …  
    16331679 <w:wrap type="square" anchorx="margin" anchory="margin"/>
    16341680</v:shape><![endif]--><![if !vml]><img width=97 height=71
    1635 src="ParametricFitting_files/image030.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
     1681src="ParametricFitting_files/image077.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]>Click
    16361682on the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    16371683mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Parametric
    … …  
    16451691
    16461692<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1647 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_17" o:spid="_x0000_i1042"
    1648  type="#_x0000_t75" style='width:311.25pt;height:152.25pt;visibility:visible;
     1693mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1041"
     1694 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
    16491695 mso-wrap-style:square'>
    1650  <v:imagedata src="ParametricFitting_files/image031.png" o:title="" croptop="4213f"
    1651   cropbottom="8982f" cropleft="3490f" cropright="3653f"/>
    1652 </v:shape><![endif]--><![if !vml]><img width=415 height=203
    1653 src="ParametricFitting_files/image032.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
     1696 <v:imagedata src="ParametricFitting_files/image024.png" o:title=""/>
     1697</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1698src="ParametricFitting_files/image078.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
    16541699
    16551700<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
    … …  
    16651710mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span
    16661711style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span></b><![endif]>Click
    1667 the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
     1712the<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1713mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span></b><span
     1714style='mso-bidi-font-family:"Times New Roman"'>empty</span><b style='mso-bidi-font-weight:
    16681715normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1669 major-latin;mso-hansi-theme-font:major-latin'>&#8597; </span></b>button to
    1670 select a dependent variable for &quot;y&quot;. From the phase list, select the
    1671 unit cell volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b
    1672 style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
    1673 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1674 mso-hansi-theme-font:major-latin'>0::Vol</span></b> and then press<b
    1675 style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
    1676 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1677 mso-hansi-theme-font:major-latin'>OK</span>.<o:p></o:p></b></p>
     1716major-latin;mso-hansi-theme-font:major-latin'> </span></b>button in the upper
     1717left corner to select a dependent variable for &quot;y&quot;. From the phase
     1718list (use ‘<span class=SpellE>vol</span>’ as a filter), select the unit cell
     1719volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, select<b style='mso-bidi-font-weight:
     1720normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
     1721"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1722major-latin'>0::Vol</span></b> and then press<b style='mso-bidi-font-weight:
     1723normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
     1724"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1725major-latin'>OK</span>.<o:p></o:p></b></p>
    16781726
    16791727<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    16801728
    16811729<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1682 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19" o:spid="_x0000_i1041"
    1683  type="#_x0000_t75" style='width:314.25pt;height:132pt;visibility:visible;
     1730mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1040"
     1731 type="#_x0000_t75" style='width:6in;height:172.5pt;visibility:visible;
    16841732 mso-wrap-style:square'>
    1685  <v:imagedata src="ParametricFitting_files/image033.png" o:title="" croptop="3830f"
    1686   cropbottom="9359f" cropleft="3705f" cropright="3502f"/>
    1687 </v:shape><![endif]--><![if !vml]><img width=419 height=176
    1688 src="ParametricFitting_files/image034.png" v:shapes="Picture_x0020_19"><![endif]></span></p>
     1733 <v:imagedata src="ParametricFitting_files/image028.png" o:title=""/>
     1734</v:shape><![endif]--><![if !vml]><img width=576 height=230
     1735src="ParametricFitting_files/image080.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
     1736
     1737<p class=MsoListParagraphCxSpMiddle>The Expression Editor will be updated to
     1738show the new y parameter</p>
    16891739
    16901740<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1691 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20" o:spid="_x0000_i1040"
    1692  type="#_x0000_t75" style='width:311.25pt;height:155.25pt;visibility:visible;
     1741mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1039"
     1742 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
    16931743 mso-wrap-style:square'>
    1694  <v:imagedata src="ParametricFitting_files/image035.png" o:title="" croptop="3548f"
    1695   cropbottom="8549f" cropleft="3611f" cropright="3665f"/>
    1696 </v:shape><![endif]--><![if !vml]><img width=415 height=207
    1697 src="ParametricFitting_files/image036.png" v:shapes="Picture_x0020_20"><![endif]></span></p>
     1744 <v:imagedata src="ParametricFitting_files/image032.png" o:title=""/>
     1745</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1746src="ParametricFitting_files/image082.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
    16981747
    16991748<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    17151764
    17161765<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1717 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_21" o:spid="_x0000_i1039"
    1718  type="#_x0000_t75" style='width:315.75pt;height:155.25pt;visibility:visible;
    1719  mso-wrap-style:square'>
    1720  <v:imagedata src="ParametricFitting_files/image037.png" o:title="" croptop="3326f"
    1721   cropbottom="8766f" cropleft="3010f" cropright="3419f"/>
    1722 </v:shape><![endif]--><![if !vml]><img width=421 height=207
    1723 src="ParametricFitting_files/image038.png" v:shapes="Picture_x0020_21"><![endif]></span></p>
     1766mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75"
     1767 style='width:6in;height:237pt;visibility:visible;mso-wrap-style:square'>
     1768 <v:imagedata src="ParametricFitting_files/image036.png" o:title=""/>
     1769</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1770src="ParametricFitting_files/image083.png" v:shapes="_x0000_i1038"><![endif]></span></p>
    17241771
    17251772<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    17421789style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    17431790style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    1744 </span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_22"
    1745  o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0;
    1746  text-align:left;margin-left:43.8pt;margin-top:603.2pt;width:95pt;height:35pt;
    1747  z-index:251662336;visibility:visible;mso-wrap-style:square;
    1748  mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt;
    1749  mso-wrap-distance-top:0;mso-wrap-distance-right:9pt;
    1750  mso-wrap-distance-bottom:0;mso-position-horizontal:right;
    1751  mso-position-horizontal-relative:margin;mso-position-vertical:absolute;
    1752  mso-position-vertical-relative:margin;mso-width-percent:0;
    1753  mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'>
    1754  <v:imagedata src="ParametricFitting_files/image039.png" o:title=""/>
    1755  <w:wrap type="square" anchorx="margin" anchory="margin"/>
    1756 </v:shape><![endif]--><![if !vml]><img width=127 height=47
    1757 src="ParametricFitting_files/image040.jpg" align=right hspace=12 v:shapes="Picture_x0020_22"><![endif]>Assign
    1758 <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1759 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
    1760 and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1761 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
    1762 as refinable “free” parameters: Click on the <b style='mso-bidi-font-weight:
     1791</span></span></span><![endif]>Assign <b style='mso-bidi-font-weight:normal'><span
     1792style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     1793mso-hansi-theme-font:major-latin'>b</span></b> and <b style='mso-bidi-font-weight:
    17631794normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1764 major-latin;mso-hansi-theme-font:major-latin'>&#8597;</span></b><span
    1765 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1766 mso-hansi-theme-font:major-latin'> </span>button next to b and select<b
     1795major-latin;mso-hansi-theme-font:major-latin'>m</span></b> as refinable “free”
     1796parameters: Click on the button next to b and select<b style='mso-bidi-font-weight:
     1797normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
     1798"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1799major-latin'>Free</span></b>. Then repeat this <span class=SpellE>for<b
    17671800style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
    17681801style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1769 mso-hansi-theme-font:major-latin'>Free</span></b>. Then repeat this <span
    1770 class=SpellE>for<b style='mso-bidi-font-weight:normal'> </b><b
    1771 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1772 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b></span>.
    1773 </p>
     1802mso-hansi-theme-font:major-latin'>m</span></b></span>. </p>
    17741803
    17751804<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1776 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_23" o:spid="_x0000_i1038"
    1777  type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible;
     1805mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_16" o:spid="_x0000_i1037"
     1806 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
    17781807 mso-wrap-style:square'>
    1779  <v:imagedata src="ParametricFitting_files/image041.png" o:title="" croptop="3992f"
    1780   cropbottom="8541f" cropleft="3250f" cropright="3415f"/>
    1781 </v:shape><![endif]--><![if !vml]><img width=419 height=204
    1782 src="ParametricFitting_files/image042.png" v:shapes="Picture_x0020_23"><![endif]></span></p>
     1808 <v:imagedata src="ParametricFitting_files/image040.png" o:title=""/>
     1809</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1810src="ParametricFitting_files/image084.png" v:shapes="Picture_x0020_16"><![endif]></span></p>
    17831811
    17841812<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    18051833 <w:wrap type="square" anchorx="margin" anchory="margin"/>
    18061834</v:shape><![endif]--><![if !vml]><img width=101 height=35
    1807 src="ParametricFitting_files/image044.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
     1835src="ParametricFitting_files/image085.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To
    18081836set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    18091837mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>
    1810 as the temperature, click on the <b style='mso-bidi-font-weight:normal'><span
    1811 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1812 mso-hansi-theme-font:major-latin'>&#8597;</span></b><span style='font-family:
    1813 "Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
    1814 major-latin'> </span>button next to <b style='mso-bidi-font-weight:normal'><span
    1815 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1816 mso-hansi-theme-font:major-latin'>x</span></b>, select<b style='mso-bidi-font-weight:
    1817 normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
    1818 "Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
    1819 major-latin'>Global</span>,</b> and then select<b style='mso-bidi-font-weight:
    1820 normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
    1821 "Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
    1822 major-latin'>Temperature</span></b> from the list and click on<b
    1823 style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
    1824 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1825 mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
     1838as the temperature, click on the<span style='font-family:"Calibri",sans-serif;
     1839mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> </span>button
     1840next to <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1841mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x</span></b>,
     1842select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
     1843normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1844major-latin;mso-hansi-theme-font:major-latin'>Global</span>,</b> and then
     1845select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
     1846normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1847major-latin;mso-hansi-theme-font:major-latin'>Temperature</span></b> from the
     1848list and click on<b style='mso-bidi-font-weight:normal'> </b><b
     1849style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     1850mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>OK</span></b>.</p>
    18261851
    18271852<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    18291854<p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
    18301855style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
    1831  id="Picture_x0020_25" o:spid="_x0000_i1037" type="#_x0000_t75" style='width:245.25pt;
    1832  height:108.75pt;visibility:visible;mso-wrap-style:square'>
    1833  <v:imagedata src="ParametricFitting_files/image045.png" o:title="" croptop="5483f"
    1834   cropbottom="10984f" cropleft="4191f" cropright="4159f"/>
    1835 </v:shape><![endif]--><![if !vml]><img width=327 height=145
    1836 src="ParametricFitting_files/image046.png" v:shapes="Picture_x0020_25"><![endif]></span></p>
     1856 id="Picture_x0020_17" o:spid="_x0000_i1036" type="#_x0000_t75" style='width:6in;
     1857 height:172.5pt;visibility:visible;mso-wrap-style:square'>
     1858 <v:imagedata src="ParametricFitting_files/image046.png" o:title=""/>
     1859</v:shape><![endif]--><![if !vml]><img width=576 height=230
     1860src="ParametricFitting_files/image086.png" v:shapes="Picture_x0020_17"><![endif]></span></p>
    18371861
    18381862<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    18401864<p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values
    18411865of the dependent variable and the expression for the first refinement, as
    1842 below. <span style='mso-spacerun:yes'> </span>Optionally, the values for <b
     1866below.<span style='mso-spacerun:yes'>  </span>Optionally, the values for <b
    18431867style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    18441868mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>m</span></b>
    … …  
    18531877
    18541878<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
    1855 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_26" o:spid="_x0000_i1036"
    1856  type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible;
     1879mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_46" o:spid="_x0000_i1035"
     1880 type="#_x0000_t75" style='width:6in;height:237pt;visibility:visible;
    18571881 mso-wrap-style:square'>
    1858  <v:imagedata src="ParametricFitting_files/image047.png" o:title="" croptop="3992f"
    1859   cropbottom="8541f" cropleft="3490f" cropright="3294f"/>
    1860 </v:shape><![endif]--><![if !vml]><img width=419 height=204
    1861 src="ParametricFitting_files/image048.png" v:shapes="Picture_x0020_26"><![endif]></span></p>
     1882 <v:imagedata src="ParametricFitting_files/image050.png" o:title=""/>
     1883</v:shape><![endif]--><![if !vml]><img width=576 height=316
     1884src="ParametricFitting_files/image087.png" v:shapes="Picture_x0020_46"><![endif]></span></p>
    18621885
    18631886<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    18641887
    1865 <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
     1888<p class=MsoListParagraph style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
    18661889style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    18671890style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    … …  
    18751898values are then refined and the console shows the results:</p>
    18761899
    1877 <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span
    1878 style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    1879 style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    1880 </span></span></span><![endif]><o:p>&nbsp;</o:p></p>
    1881 
    1882 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
    1883 auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'>Fit
    1884 Results<o:p></o:p></span></b></p>
    1885 
    1886 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
    1887 auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
    1888 style='mso-spacerun:yes'>  </span>0::Vol = m*x +b<o:p></o:p></span></b></p>
    1889 
    1890 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
    1891 auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
    1892 style='mso-spacerun:yes'>   </span><span class=GramE>x<span
    1893 style='mso-spacerun:yes'>  </span>=</span> Temperature<o:p></o:p></span></b></p>
    1894 
    1895 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
    1896 auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
    1897 style='mso-spacerun:yes'>   </span><span class=GramE>b<span
    1898 style='mso-spacerun:yes'>  </span>=</span> 561.7722(26)<o:p></o:p></span></b></p>
    1899 
    1900 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
    1901 auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:Courier'><span
    1902 style='mso-spacerun:yes'>   </span><span class=GramE>m<span
    1903 style='mso-spacerun:yes'>  </span>=</span> 0.014833(14)<o:p></o:p></span></b></p>
    1904 
    1905 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space:
    1906 auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
    1907 
    1908 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1909 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_27" o:spid="_x0000_i1035"
    1910  type="#_x0000_t75" style='width:350.25pt;height:302.25pt;visibility:visible;
    1911  mso-wrap-style:square'>
    1912  <v:imagedata src="ParametricFitting_files/image049.png" o:title=""/>
    1913 </v:shape><![endif]--><![if !vml]><img width=467 height=403
    1914 src="ParametricFitting_files/image050.jpg" v:shapes="Picture_x0020_27"><![endif]></span><span
     1900<p class=MsoNormal style='margin-left:.25in'><o:p>&nbsp;</o:p></p>
     1901
     1902<p class=MsoNormal style='text-indent:.5in'><span style='mso-fareast-language:
     1903EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19"
     1904 o:spid="_x0000_i1034" type="#_x0000_t75" style='width:270.75pt;height:115.5pt;
     1905 visibility:visible;mso-wrap-style:square'>
     1906 <v:imagedata src="ParametricFitting_files/image054.png" o:title=""/>
     1907</v:shape><![endif]--><![if !vml]><img width=361 height=154
     1908src="ParametricFitting_files/image054.png" v:shapes="Picture_x0020_19"><![endif]></span><span
     1909style='font-family:Courier'><o:p></o:p></span></p>
     1910
     1911<p class=MsoListParagraphCxSpFirst><span style='font-family:"Cambria",serif;
     1912mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-fareast-language:
     1913EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20"
     1914 o:spid="_x0000_i1033" type="#_x0000_t75" style='width:6in;height:370.5pt;
     1915 visibility:visible;mso-wrap-style:square'>
     1916 <v:imagedata src="ParametricFitting_files/image056.png" o:title=""/>
     1917</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1918src="ParametricFitting_files/image088.png" v:shapes="Picture_x0020_20"><![endif]></span><span
    19151919style='font-family:"Cambria",serif;mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:
    19161920minor-latin'><o:p></o:p></span></p>
    … …  
    19181922<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    19191923
    1920 <p class=MsoListParagraphCxSpMiddle>Also, the plot shows the actual values and
    1921 the fit. Note that while the values are close to a straight line, the
    1922 deviations are huge compared to the estimated uncertainty values for the unit
    1923 cell volume (shown as very small error bars, unless the plot is
    1924 magnified).<span style='mso-spacerun:yes'>  </span>If a linear relationship had
    1925 been defined as a constraint (as can be done with other software packages) <i>the
     1924<p class=MsoListParagraphCxSpMiddle>Also, the plot shows lines for the data
     1925values and the fit. Note that while the values are close to a straight line,
     1926the deviations are huge compared to the estimated uncertainty values for the
     1927unit cell volume (shown as very small error bars, unless the plot is
     1928magnified).<span style='mso-spacerun:yes'>  </span>NB: If a linear relationship
     1929had been defined as a constraint (as can be done with other software packages) <i>the
    19261930unit cell parameters would have been forced onto this line with no obvious
    19271931indication that this was not a correct result.</i> </p>
    … …  
    19291933<p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p>
    19301934
    1931 <h1>Step 4: Examples of more complex fitt<span class=Heading1Char><span
    1932 style='font-weight:normal'>i</span></span>ng equations</h1>
     1935<h1>Step 4: Fitting to a <span class=SpellE>Pseudovariable</span></h1>
    19331936
    19341937<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    19351938
    19361939<p class=MsoNormal>Here we will look at how more sophisticated Python
    1937 capabilities can be used to fit more complex behavior.<span
    1938 style='mso-spacerun:yes'>  </span></p>
     1940capabilities can be used to fit more complex behavior and use a <span
     1941class=SpellE>pseudovariable</span><span style='mso-spacerun:yes'>  </span></p>
    19391942
    19401943<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
    19411944style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    19421945style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    1943 </span></span></span><![endif]>Optionally Double-click on the<b
    1944 style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:normal'><span
    1945 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    1946 mso-hansi-theme-font:major-latin'>0::c</span> </b>column. Note the changes in
    1947 the <i style='mso-bidi-font-style:normal'>c</i> unit cell parameter as a
    1948 function of temperature, as shown below. </p>
    1949 
    1950 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
     1946</span></span></span><![endif]>Double-click on the<b style='mso-bidi-font-weight:
     1947normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
     1948"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1949major-latin'>b/c</span> </b>column. We will be trying to extract the critical
     1950exponent for this 2<sup>nd</sup> order phase transition from this data.</p>
    19511951
    19521952<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    1953 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_28" o:spid="_x0000_i1034"
    1954  type="#_x0000_t75" style='width:350.25pt;height:303pt;visibility:visible;
     1953mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_37" o:spid="_x0000_i1032"
     1954 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
    19551955 mso-wrap-style:square'>
    1956  <v:imagedata src="ParametricFitting_files/image051.png" o:title=""/>
    1957 </v:shape><![endif]--><![if !vml]><img width=467 height=404
    1958 src="ParametricFitting_files/image052.jpg" v:shapes="Picture_x0020_28"><![endif]></span></p>
     1956 <v:imagedata src="ParametricFitting_files/image089.png" o:title=""/>
     1957</v:shape><![endif]--><![if !vml]><img width=576 height=494
     1958src="ParametricFitting_files/image090.png" v:shapes="Picture_x0020_37"><![endif]></span></p>
    19591959
    19601960<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    19631963style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    19641964style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    1965 </span></span></span><![endif]>We will now fit <i style='mso-bidi-font-style:
    1966 normal'>c</i> as a function of form 1<span class=GramE>/(</span>1 + <span
    1967 class=SpellE>exp</span>[k(t-t<sub>0</sub>)]). Select the<b style='mso-bidi-font-weight:
     1965</span></span></span><![endif]>As a simple (but incorrect) example, we will now
     1966fit <i style='mso-bidi-font-style:normal'>b/c </i>as a function of form 1<span
     1967class=GramE>/(</span>1 + <span class=SpellE>exp</span>[k(t-t<sub>0</sub>)]).
     1968Select the<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
     1969normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     1970major-latin;mso-hansi-theme-font:major-latin'>Parametric Fit/Add</span></b><span
     1971style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     1972mso-hansi-theme-font:major-latin'> <b style='mso-bidi-font-weight:normal'>equation</b></span>
     1973menu item (as before). For the dependent variable, click on<b style='mso-bidi-font-weight:
    19681974normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
    19691975"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
    1970 major-latin'>Parametric Fit/Add</span></b><span style='font-family:"Calibri",sans-serif;
    1971 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> <b
    1972 style='mso-bidi-font-weight:normal'>equation</b></span> menu item (as before).
    1973 For the dependent variable, click on<b style='mso-bidi-font-weight:normal'> </b><b
    1974 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1975 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>Phase</span></b>
    1976 and then select<b style='mso-bidi-font-weight:normal'> </b><b style='mso-bidi-font-weight:
    1977 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    1978 major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>.<span
    1979 style='mso-spacerun:yes'>  </span></p>
     1976major-latin'>Global</span></b> and then select<b style='mso-bidi-font-weight:
     1977normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
     1978"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     1979major-latin'>b/c</span></b>.<span style='mso-spacerun:yes'>  </span></p>
    19801980
    19811981<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    19901990<p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><b
    19911991style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    1992 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b + c / (1+<span
    1993 class=GramE>math.exp(</span>k*(t-t0)))<o:p></o:p></span></b></p>
     1992mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a + b / (1+<span
     1993class=GramE>np.exp(</span>k*(t-t0)))<o:p></o:p></span></b></p>
    19941994
    19951995<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
    … …  
    19981998<p class=MsoListParagraphCxSpLast>Note that use of spaces here is arbitrary, but
    19991999all parentheses are required. Also note that the exponential function [<span
    2000 class=SpellE>exp</span>(x) or e<sup>x</sup>] is supplied in Python in the math
    2001 and <span class=SpellE>numpy</span> modules as <span class=SpellE><span
    2002 class=GramE>math.exp</span></span><span class=GramE>(</span>), <span
    2003 class=SpellE>numpy.exp</span>(). Either of these can be used as well as the
    2004 convenience abbreviation of <span class=SpellE><span class=GramE>np.exp</span></span><span
    2005 class=GramE>(</span>). Further, there are also a small number of common
    2006 functions that are defined without the prefix for convenience [these include
    2007 pi, <span class=SpellE><span class=GramE>exp</span></span><span class=GramE>(</span>),
    2008 <span class=SpellE>sqrt</span>(), <span class=SpellE>sind</span>(), <span
    2009 class=SpellE>cosd</span>(), <span class=SpellE>tand</span>() where d indicates
    2010 an argument in degrees.]</p>
     2000class=SpellE>exp</span>(x) or e<sup>x</sup>] is supplied in Python in the <span
     2001class=SpellE>numpy</span> modules as <span class=SpellE>np.exp</span>().
     2002Further, there are also a small number of common functions that are defined
     2003without the prefix for convenience [these include pi, <span class=SpellE><span
     2004class=GramE>exp</span></span><span class=GramE>(</span>), <span class=SpellE>sqrt</span>(),
     2005<span class=SpellE>sind</span>(), <span class=SpellE>cosd</span>(), <span
     2006class=SpellE>tand</span>() where d indicates an argument in degrees.]</p>
    20112007
    20122008<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    … …  
    20182014normal'> </b><b style='mso-bidi-font-weight:normal'><span style='font-family:
    20192015"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
    2020 major-latin'>b</span>, </b><b style='mso-bidi-font-weight:normal'><span
    2021 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2022 mso-hansi-theme-font:major-latin'>c</span>, </b><b style='mso-bidi-font-weight:
     2016major-latin'>a</span>, </b><b style='mso-bidi-font-weight:normal'><span
     2017style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2018mso-hansi-theme-font:major-latin'>b</span>, </b><b style='mso-bidi-font-weight:
    20232019normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    20242020major-latin;mso-hansi-theme-font:major-latin'>k</span> </b>and<b
    … …  
    20312027label was set automatically to <b style='mso-bidi-font-weight:normal'><span
    20322028style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2033 mso-hansi-theme-font:major-latin'>b_1</span></b>. If the same variable name (<b
     2029mso-hansi-theme-font:major-latin'>b_1</span></b> because the symbol ‘<b
    20342030style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2035 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>)
    2036 was used again here, the same value would be used to fit both equations, which
    2037 is certainly not what we want here.<span style='mso-spacerun:yes'> 
    2038 </span>Finally, since this equation is too complex to fit without some
    2039 reasonable starting values, specify an approximate value, <b style='mso-bidi-font-weight:
    2040 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2041 major-latin;mso-hansi-theme-font:major-latin'>120</span></b>, for the vicinity
    2042 of where the <i style='mso-bidi-font-style:normal'>c</i> lattice parameter
    2043 starts to change (<b style='mso-bidi-font-weight:normal'><span
    2044 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2045 mso-hansi-theme-font:major-latin'>t0</span></b>) and non-zero values for
    2046 variables <b>c</b> and <b>k</b>.</p>
     2031mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     2032major-latin'>b</span></b>’ was used in the previous equation for fitting the
     2033volume. If the same variable name (<b style='mso-bidi-font-weight:normal'><span
     2034style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2035mso-hansi-theme-font:major-latin'>b</span></b>) was used again here, the same
     2036value would be used to fit both equations, which is certainly not what we want
     2037here.<span style='mso-spacerun:yes'>  </span>Finally, since this equation is
     2038too complex to fit without some reasonable starting values, specify an
     2039approximate value, <b style='mso-bidi-font-weight:normal'><span
     2040style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2041mso-hansi-theme-font:major-latin'>120</span></b>, for the vicinity of where the
     2042<i style='mso-bidi-font-style:normal'>c</i> lattice parameter starts to change
     2043(<b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     2044mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>)
     2045and non-zero values for variables <b>c</b> and <b>k</b>.</p>
    20472046
    20482047<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    20492048
    20502049<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2051 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_40" o:spid="_x0000_i1033"
    2052  type="#_x0000_t75" style='width:317.25pt;height:171pt;visibility:visible;
     2050mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_38" o:spid="_x0000_i1031"
     2051 type="#_x0000_t75" style='width:6in;height:267.75pt;visibility:visible;
    20532052 mso-wrap-style:square'>
    2054  <v:imagedata src="ParametricFitting_files/image053.png" o:title=""/>
    2055 </v:shape><![endif]--><![if !vml]><img width=423 height=228
    2056 src="ParametricFitting_files/image054.jpg" v:shapes="Picture_x0020_40"><![endif]></span></p>
     2053 <v:imagedata src="ParametricFitting_files/image091.png" o:title=""/>
     2054</v:shape><![endif]--><![if !vml]><img width=576 height=357
     2055src="ParametricFitting_files/image092.png" v:shapes="Picture_x0020_38"><![endif]></span></p>
    20572056
    20582057<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    … …  
    20642063normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    20652064major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b> button to see if
    2066 the equation can be refined with starting values of zero for the other three
    2067 parameters.<span style='mso-spacerun:yes'>  </span>A fit is obtained and is
    2068 plotted. </p>
    2069 
    2070 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    2071 
    2072 <p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span
    2073 style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
    2074  id="Picture_x0020_41" o:spid="_x0000_i1032" type="#_x0000_t75" style='width:317.25pt;
    2075  height:168.75pt;visibility:visible;mso-wrap-style:square'>
    2076  <v:imagedata src="ParametricFitting_files/image055.png" o:title=""/>
    2077 </v:shape><![endif]--><![if !vml]><img width=423 height=225
    2078 src="ParametricFitting_files/image056.jpg" v:shapes="Picture_x0020_41"><![endif]></span></p>
    2079 
    2080 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
     2065the equation can be refined with starting values of zero for the other three parameters.<span
     2066style='mso-spacerun:yes'>  </span>A fit is obtained and is plotted; the fit is
     2067quite poor.</p>
     2068
     2069<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
     2070mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_39" o:spid="_x0000_i1030"
     2071 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
     2072 mso-wrap-style:square'>
     2073 <v:imagedata src="ParametricFitting_files/image093.png" o:title=""/>
     2074</v:shape><![endif]--><![if !vml]><img width=576 height=494
     2075src="ParametricFitting_files/image094.png" v:shapes="Picture_x0020_39"><![endif]></span><span
     2076style='mso-spacerun:yes'> </span></p>
     2077
     2078<p class=MsoListParagraphCxSpLast>To give it more reasonable starting values,
     2079set <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     2080mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     2081major-latin'>a</span></b>=<b style='mso-bidi-font-weight:normal'><span
     2082style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2083mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>1.0, b</span></b>=<b
     2084style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     2085mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     2086major-latin'>0.1</span></b><span class=GramE>,<b style='mso-bidi-font-weight:
     2087normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     2088major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>k</span></b></span>=<b
     2089style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     2090mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     2091major-latin'>0.1</span></b> and <b style='mso-bidi-font-weight:normal'><span
     2092style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2093mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>t0</span></b>=<b
     2094style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
     2095mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-theme-font:
     2096major-latin'>120.</span></b> Press <b style='mso-bidi-font-weight:normal'><span
     2097style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2098mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>Fit</span></b>
     2099again; the plot shows a better (but not right) fit.</p>
     2100
     2101<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
     2102 id="Picture_x0020_40" o:spid="_x0000_i1029" type="#_x0000_t75" style='width:6in;
     2103 height:370.5pt;visibility:visible;mso-wrap-style:square'>
     2104 <v:imagedata src="ParametricFitting_files/image095.png" o:title=""/>
     2105</v:shape><![endif]--><![if !vml]><img width=576 height=494
     2106src="ParametricFitting_files/image096.png" v:shapes="Picture_x0020_40"><![endif]></span></p>
     2107
     2108<p class=MsoListParagraphCxSpFirst><o:p>&nbsp;</o:p></p>
    20812109
    20822110<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
    … …  
    20862114style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    20872115mso-hansi-theme-font:major-latin'>OK</span></b> to add this equation to the
    2088 parametric fits and both equations are refit (and if the label for <b
    2089 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2090 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>b</span></b>
    2091 had not been changed, applying the constraint that <b style='mso-bidi-font-weight:
    2092 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2093 major-latin;mso-hansi-theme-font:major-latin'>b</span></b> must be the same for
    2094 both equations.) </p>
    2095 
    2096 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    2097 
    2098 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2099 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_32" o:spid="_x0000_i1031"
    2100  type="#_x0000_t75" style='width:353.25pt;height:300.75pt;visibility:visible;
    2101  mso-wrap-style:square'>
    2102  <v:imagedata src="ParametricFitting_files/image057.png" o:title=""/>
    2103 </v:shape><![endif]--><![if !vml]><img width=471 height=401
    2104 src="ParametricFitting_files/image058.jpg" v:shapes="Picture_x0020_32"><![endif]></span></p>
    2105 
    2106 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    2107 
    2108 <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span
    2109 style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span
    2110 style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    2111 </span></span></span><![endif]>We will now try a more complex equation for the
    2112 same data, where we set the value to <b style='mso-bidi-font-weight:normal'><span
    2113 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2114 mso-hansi-theme-font:major-latin'>b</span></b> for temperatures less than <b
    2115 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2116 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
    2117 and a critical phenomenon formulism [(t-t<sub>0</sub>)<sup>-a</sup>] for
    2118 temperatures above <b style='mso-bidi-font-weight:normal'><span
    2119 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2120 mso-hansi-theme-font:major-latin'>t0</span></b>. To do this we use the Python
    2121 syntax &quot;<b style='mso-bidi-font-weight:normal'><span style='font-family:
    2122 "Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
    2123 major-latin'>x1 if <span class=SpellE>testexp</span> else x2</span></b>&quot;
    2124 which is <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2125 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x1</span></b>
    2126 where the test expression (<span class=SpellE><b style='mso-bidi-font-weight:
    2127 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2128 major-latin;mso-hansi-theme-font:major-latin'>testexp</span></b></span> here)
    2129 is <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2130 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>True</span></b>
    2131 and <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2132 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>x2</span></b>
    2133 otherwise. Thus we can use an expression such as:</p>
    2134 
    2135 <p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span></p>
    2136 
    2137 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
    2138 auto'><span class=GramE><b style='mso-bidi-font-weight:normal'><span
    2139 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2140 mso-hansi-theme-font:major-latin'>b</span></b></span><b style='mso-bidi-font-weight:
    2141 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2142 major-latin;mso-hansi-theme-font:major-latin'> if t &lt;= t0 else b + c *
    2143 (1-(t-t0)**d)<o:p></o:p></span></b></p>
    2144 
    2145 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
    2146 auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
    2147 
    2148 <p class=MsoListParagraphCxSpMiddle>This may be easier to read with the inclusion
    2149 of extra parentheses (that have no effect on the execution):<span
    2150 style='font-family:"Cambria",serif;mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:
    2151 minor-latin'><o:p></o:p></span></p>
    2152 
    2153 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    2154 
    2155 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
    2156 auto'><span class=GramE><b style='mso-bidi-font-weight:normal'><span
    2157 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2158 mso-hansi-theme-font:major-latin'>b</span></b></span><b style='mso-bidi-font-weight:
    2159 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2160 major-latin;mso-hansi-theme-font:major-latin'> if (t &lt;= t0) else (b + c *
    2161 (1-(t-t0)**d))<o:p></o:p></span></b></p>
    2162 
    2163 <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space:
    2164 auto'><span style='font-family:Courier'><o:p>&nbsp;</o:p></span></p>
    2165 
    2166 <p class=MsoListParagraphCxSpMiddle>As before, add an equation (or edit an
    2167 existing one), set the dependent variable to <b style='mso-bidi-font-weight:
    2168 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2169 major-latin;mso-hansi-theme-font:major-latin'>0::c</span></b>, enter the
    2170 expression and assign the free and fixed variables, as before. Set the exponent
    2171 and the <b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2172 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
    2173 values to initial values of <b style='mso-bidi-font-weight:normal'><span
    2174 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2175 mso-hansi-theme-font:major-latin'>-0.025</span></b> and <b style='mso-bidi-font-weight:
    2176 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2177 major-latin;mso-hansi-theme-font:major-latin'>120</span></b>, respectively, and
    2178 turn off their refinement flags. Then press <b style='mso-bidi-font-weight:
    2179 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2180 major-latin;mso-hansi-theme-font:major-latin'>Fit</span></b>.<span
    2181 style='font-family:"Cambria",serif;mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:
    2182 minor-latin'><o:p></o:p></span></p>
    2183 
    2184 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2185 mso-no-proof:yes'><o:p>&nbsp;</o:p></span></p>
    2186 
    2187 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2188 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1030"
    2189  type="#_x0000_t75" style='width:315.75pt;height:168.75pt;visibility:visible;
    2190  mso-wrap-style:square'>
    2191  <v:imagedata src="ParametricFitting_files/image059.png" o:title=""/>
    2192 </v:shape><![endif]--><![if !vml]><img width=421 height=225
    2193 src="ParametricFitting_files/image060.jpg" v:shapes="Picture_x0020_42"><![endif]></span></p>
    2194 
    2195 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    2196 
    2197 <p class=MsoListParagraphCxSpMiddle>Then turn refinement of <b
    2198 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2199 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>t0</span></b>
    2200 back on and press <b style='mso-bidi-font-weight:normal'><span
    2201 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2202 mso-hansi-theme-font:major-latin'>OK</span></b>. The resulting fit shows the
    2203 trends quite well: </p>
    2204 
    2205 <p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    2206 
    2207 <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2208 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1029"
    2209  type="#_x0000_t75" style='width:351pt;height:302.25pt;visibility:visible;
    2210  mso-wrap-style:square'>
    2211  <v:imagedata src="ParametricFitting_files/image061.png" o:title=""/>
    2212 </v:shape><![endif]--><![if !vml]><img width=468 height=403
    2213 src="ParametricFitting_files/image062.jpg" v:shapes="Picture_x0020_35"><![endif]></span></p>
     2116parametric fits and both equations are refit. The fit is ok but this is not the
     2117correct expression for critical phenomena.</p>
     2118
     2119<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    22142120
    22152121<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
    22162122
    2217 <h1>Step 5: Fitting to a Pseudo Variable</h1>
    2218 
    2219 <p class=MsoNormal><o:p>&nbsp;</o:p></p>
    2220 
    2221 <p class=MsoNormal>One can fit a function to a pseudo variable, just as to a
    2222 direct variable. If we look at the plot shown in Step 2, it appears there are
    2223 two regions. We can treat the lower range as a power-law growth, <span
    2224 class=SpellE>T<sup>k</sup></span>, and the upper region as linear (m*T + b). </p>
     2123<p class=MsoNormal>If we look at the above plot, it appears there are two
     2124regions. We can treat the lower range as a power-law decay, <span class=SpellE>T<sup>k</sup></span>,
     2125and the upper region as linear (m*T + b). </p>
    22252126
    22262127<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    … …  
    22532154auto'><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    22542155mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-bidi-font-family:
    2255 "Lucida Grande";color:black'>(o1 + (a1*T<span class=GramE>)*</span>*k) if T
    2256 &lt; 120 else (o2 + a2*T</span></b><b style='mso-bidi-font-weight:normal'><span
    2257 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2258 mso-hansi-theme-font:major-latin'>)<o:p></o:p></span></b></p>
     2156"Lucida Grande";color:black'>a<span class=GramE>+(</span>(T0-T)/T0)**k if T
     2157&lt; T0 else <span class=SpellE>d+e</span>*T</span></b><b style='mso-bidi-font-weight:
     2158normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     2159major-latin;mso-hansi-theme-font:major-latin'><o:p></o:p></span></b></p>
    22592160
    22602161<p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria",serif;
    … …  
    22692170mso-hansi-theme-font:major-latin'>k</span></b>) and scaling factor (<b
    22702171style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2271 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a1</span></b>),
    2272 cannot start as zero<b style='mso-bidi-font-weight:normal'>. </b></p>
     2172mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>a</span></b>),
     2173cannot start as zero; <b style='mso-bidi-font-weight:normal'><span
     2174style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
     2175mso-hansi-theme-font:major-latin;mso-bidi-theme-font:major-latin'>T0</span></b>
     2176should be set to <b style='mso-bidi-font-weight:normal'><span style='font-family:
     2177"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     2178major-latin;mso-bidi-theme-font:major-latin'>120</span></b><span class=GramE>.<b
     2179style='mso-bidi-font-weight:normal'>.</b></span><b style='mso-bidi-font-weight:
     2180normal'> </b></p>
    22732181
    22742182<p class=MsoListParagraphCxSpMiddle><o:p>&nbsp;</o:p></p>
    22752183
    22762184<p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US;
    2277 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_44" o:spid="_x0000_i1028"
    2278  type="#_x0000_t75" style='width:332.25pt;height:186pt;visibility:visible;
     2185mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_41" o:spid="_x0000_i1028"
     2186 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
    22792187 mso-wrap-style:square'>
    2280  <v:imagedata src="ParametricFitting_files/image063.png" o:title=""/>
    2281 </v:shape><![endif]--><![if !vml]><img width=443 height=248
    2282 src="ParametricFitting_files/image064.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
     2188 <v:imagedata src="ParametricFitting_files/image097.png" o:title=""/>
     2189</v:shape><![endif]--><![if !vml]><img width=576 height=365
     2190src="ParametricFitting_files/image098.png" v:shapes="Picture_x0020_41"><![endif]></span></p>
    22832191
    22842192<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    … …  
    22952203
    22962204<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2297 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_45" o:spid="_x0000_i1027"
    2298  type="#_x0000_t75" style='width:333.75pt;height:189pt;visibility:visible;
     2205mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1027"
     2206 type="#_x0000_t75" style='width:6in;height:273.75pt;visibility:visible;
    22992207 mso-wrap-style:square'>
    2300  <v:imagedata src="ParametricFitting_files/image065.png" o:title=""/>
    2301 </v:shape><![endif]--><![if !vml]><img width=445 height=252
    2302 src="ParametricFitting_files/image066.png" v:shapes="Picture_x0020_45"><![endif]></span><span
     2208 <v:imagedata src="ParametricFitting_files/image099.png" o:title=""/>
     2209</v:shape><![endif]--><![if !vml]><img width=576 height=365
     2210src="ParametricFitting_files/image100.png" v:shapes="Picture_x0020_42"><![endif]></span><span
    23032211style='mso-spacerun:yes'> </span></p>
    23042212
    … …  
    23112219
    23122220<p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US;
    2313 mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_48" o:spid="_x0000_i1026"
    2314  type="#_x0000_t75" style='width:348.75pt;height:302.25pt;visibility:visible;
     2221mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_43" o:spid="_x0000_i1026"
     2222 type="#_x0000_t75" style='width:6in;height:370.5pt;visibility:visible;
    23152223 mso-wrap-style:square'>
    2316  <v:imagedata src="ParametricFitting_files/image067.png" o:title=""/>
    2317 </v:shape><![endif]--><![if !vml]><img width=465 height=403
    2318 src="ParametricFitting_files/image068.png" v:shapes="Picture_x0020_48"><![endif]></span></p>
     2224 <v:imagedata src="ParametricFitting_files/image101.png" o:title=""/>
     2225</v:shape><![endif]--><![if !vml]><img width=576 height=494
     2226src="ParametricFitting_files/image102.png" v:shapes="Picture_x0020_43"><![endif]></span></p>
    23192227
    23202228<p class=MsoListParagraphCxSpLast><o:p>&nbsp;</o:p></p>
    … …  
    23522260<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'>
    23532261
    2354 <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre><pre><b
    2355 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2356 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'># <span
    2357 class=GramE>file</span> fittest.py<o:p></o:p></span></b></pre><pre><span
    2358 class=SpellE><span class=GramE><b style='mso-bidi-font-weight:normal'><span
    2359 style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;
    2360 mso-hansi-theme-font:major-latin'>def</span></b></span></span><b
    2361 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2362 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'> <span
    2363 class=SpellE>fitfxn</span>(T,o1,o2,a1,a2,k):<o:p></o:p></span></b></pre><pre><b
    2364 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2365 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>    </span><span
    2366 class=GramE>if</span> T &lt; 120:<o:p></o:p></span></b></pre><pre><b
    2367 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2368 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>        </span><span
    2369 class=GramE>return</span> o1 + (a1*T)**k<o:p></o:p></span></b></pre><pre><b
    2370 style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri",sans-serif;
    2371 mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>    </span><span
    2372 class=GramE>else</span>:<o:p></o:p></span></b></pre><pre><b style='mso-bidi-font-weight:
    2373 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2374 major-latin;mso-hansi-theme-font:major-latin'><span style='mso-spacerun:yes'>        </span><span
    2375 class=GramE>return</span> o2 + a2*T<o:p></o:p></span></b></pre></blockquote>
     2262<blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p>&nbsp;</o:p></pre></blockquote>
    23762263
    23772264</blockquote>
    23782265
    2379 <p class=MsoNormal>The function <span class=SpellE><b style='mso-bidi-font-weight:
    2380 normal'><span style='font-family:"Calibri",sans-serif;mso-ascii-theme-font:
    2381 major-latin;mso-hansi-theme-font:major-latin'>fittest.fitfxn</span></b></span>
    2382 can then be used to duplicate the previous example: </p>
     2266<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
     2267normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
     2268mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
     2269EN-US'>#file fittest.py<span style='mso-spacerun:yes'>                </span><o:p></o:p></span></b></p>
     2270
     2271<p class=MsoNormal style='margin-left:1.0in'><span class=SpellE><span
     2272class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;
     2273font-family:"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     2274major-latin;mso-fareast-language:EN-US'>def</span></b></span></span><b
     2275style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:
     2276"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     2277major-latin;mso-fareast-language:EN-US'> <span class=SpellE>fitfxn</span>(T,T0,a,k,d,e):<span
     2278style='mso-spacerun:yes'>       </span><o:p></o:p></span></b></p>
     2279
     2280<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
     2281normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
     2282mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
     2283EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>if</span> T
     2284&lt; T0:<span style='mso-spacerun:yes'>                  </span><o:p></o:p></span></b></p>
     2285
     2286<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
     2287normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
     2288mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
     2289EN-US'><span style='mso-spacerun:yes'>        </span><span class=GramE>return</span>
     2290a+((T0-T)/T0)**k <o:p></o:p></span></b></p>
     2291
     2292<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
     2293normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
     2294mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
     2295EN-US'><span style='mso-spacerun:yes'>    </span><span class=GramE>else</span>:<span
     2296style='mso-spacerun:yes'>                       </span><o:p></o:p></span></b></p>
     2297
     2298<p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight:
     2299normal'><span style='font-size:10.0pt;font-family:"Calibri",sans-serif;
     2300mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:
     2301EN-US'><span style='mso-spacerun:yes'>         </span><span class=GramE>return</span>
     2302<span class=SpellE>d+e</span>*T<o:p></o:p></span></b></p>
     2303
     2304<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
     2305style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     2306major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><o:p>&nbsp;</o:p></span></b></p>
     2307
     2308<p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span
     2309style='font-size:10.0pt;font-family:"Calibri",sans-serif;mso-ascii-theme-font:
     2310major-latin;mso-hansi-theme-font:major-latin;mso-fareast-language:EN-US'><span
     2311style='mso-spacerun:yes'>           </span></span></b>The function <span
     2312class=SpellE><b style='mso-bidi-font-weight:normal'><span style='font-family:
     2313"Calibri",sans-serif;mso-ascii-theme-font:major-latin;mso-hansi-theme-font:
     2314major-latin'>fittest.fitfxn</span></b></span> can then be used to duplicate the
     2315previous example: </p>
    23832316
    23842317<p class=MsoNormal><o:p>&nbsp;</o:p></p>
    23852318
    23862319<p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape
    2387  id="Picture_x0020_47" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:315pt;
    2388  height:185.25pt;visibility:visible;mso-wrap-style:square'>
    2389  <v:imagedata src="ParametricFitting_files/image069.png" o:title="" croptop="3393f"
    2390   cropbottom="7847f" cropleft="3471f" cropright="3290f"/>
    2391 </v:shape><![endif]--><![if !vml]><img width=420 height=247
    2392 src="ParametricFitting_files/image070.png" v:shapes="Picture_x0020_47"><![endif]></span></p>
     2320 id="Picture_x0020_44" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:6in;
     2321 height:289.5pt;visibility:visible;mso-wrap-style:square'>
     2322 <v:imagedata src="ParametricFitting_files/image103.png" o:title=""/>
     2323</v:shape><![endif]--><![if !vml]><img width=576 height=386
     2324src="ParametricFitting_files/image104.png" v:shapes="Picture_x0020_44"><![endif]></span></p>
    23932325
    23942326<p class=MsoNormal><o:p>&nbsp;</o:p></p>

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