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1329
1330	<body lang=EN-US link="#0563C1" vlink="#954F72" style='tab-interval:.5in'>
1331
1332	<div class=WordSection1>
1333
1334	<p class=MsoTitle><a name="_GoBack"></a>Use of Parameter Limits</p>
1335
1336	<p class=MsoNormal><span style='mso-bidi-font-size:14.0pt;mso-bidi-font-family:
1337	Calibri;mso-bidi-theme-font:minor-latin'>Note: Exercise files are found <a
1338	href="data" target="_blank">here</a><o:p></o:p></span></p>
1339
1340	<h1>Introduction</h1>
1341
1342	<p class=MsoNormal>It is fairly common to see parameters in Rietveld fitting
1343	refine to values that do not make sense, either because the values fall in a
1344	range that is not physically possible or because one has <i>a priori</i> knowledge
1345	about the material being studied. When this occurs, most likely one of the
1346	following cases is true: </p>
1347
1348	<p class=MsoNormal><o:p> </o:p></p>
1349
1350	<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l2 level1 lfo2'><![if !supportLists]><span
1351	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1352	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1353	</span></span></span><![endif]><span dir=LTR></span>There is a significant
1354	problem with the model, for example for an x-ray fit, if an O atom is placed
1355	where a S atom is actually present, the <span class=SpellE>Uiso</span> will
1356	refine artificially small or the occupancy much larger than unity to try to
1357	compensate for the missing electrons.</p>
1358
1359	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo2'><![if !supportLists]><span
1360	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1361	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1362	</span></span></span><![endif]><span dir=LTR></span>The data are simply
1363	insensitive to the parameter or combination of parameters. For example, without
1364	sufficient high-Q data, the effects of occupancy and <span class=SpellE>Uiso</span>
1365	value can have compensating effects, so if both are refined together
1366	meaningless results are obtained. Likewise, with neutron data, vanadium atoms (at
1367	natural isotopic abundance) are nearly invisible due to weak coherent
1368	scattering. No parameters can be fit for a V atom with neutrons.</p>
1369
1370	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo2'><![if !supportLists]><span
1371	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1372	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1373	</span></span></span><![endif]><span dir=LTR></span>The parameter is
1374	non-physical (such as a negative <span class=SpellE>Uiso</span> value) but
1375	within 2<span lang=EL style='mso-ansi-language:EL'>σ</span> (<span lang=EL
1376	style='mso-ansi-language:EL'>σ</span><span lang=EL> </span>= standard
1377	uncertainty, aka <span class=SpellE>e.s.d</span>.) of a reasonable value, in
1378	which case the value is not problematic, as it is experimentally
1379	indistinguishable from an expected value.</p>
1380
1381	<p class=MsoListParagraphCxSpLast style='text-indent:-.25in;mso-list:l2 level1 lfo2'><![if !supportLists]><span
1382	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1383	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1384	</span></span></span><![endif]><span dir=LTR></span><span lang=EL
1385	style='mso-ansi-language:EL'>Τ</span>here is a systematic problem with the data
1386	(experimental error).</p>
1387
1388	<p class=MsoNormal><o:p> </o:p></p>
1389
1390	<p class=MsoNormal>In all these cases, the data and fit need to be reviewed by
1391	a crystallographer to decide how to best determine a structural model for these
1392	data. </p>
1393
1394	<p class=MsoNormal><o:p> </o:p></p>
1395
1396	<p class=MsoNormal>A refinement implementation could be constructed that
1397	constrains parameter values to be within the expected ranges, but this is a bad
1398	idea as this can hide that a systematic problem that is present. This would
1399	result in a wrong model, or at best a poor result being obtained. Restraints
1400	are a better choice, as they make it possible to see that the refinement is
1401	being skewed in a direction that does not fit the data, but care is still
1402	needed to learn this. It should be noted that is also possible in GSAS-II for
1403	users to supply their own functions to restrain values to fall in a specified
1404	range. This can be done by supplying an appropriate expression with the
1405	"General" restraints. As is the case with any restraint, one must
1406	probe the “final result” to understand the impact of the restraint weighting on
1407	the fit. If the diffraction fit improves significantly as the restraint is
1408	down-weighted, it likely indicates that the data are in conflict with the
1409	restraint and something is seriously wrong. <o:p></o:p></p>
1410
1411	<p class=MsoNormal><o:p> </o:p></p>
1412
1413	<p class=MsoNormal>What GSAS-II does now implement is a mechanism where refinement
1414	limits can optionally be specified for parameters. Should a variable parameter
1415	refine to a value outside that range, the value is set to the upper or lower
1416	bounding value and the refine flag for that parameter will be ignored in future
1417	cycles of refinement. A parameter that has refined outside its limits is
1418	labeled as “frozen” and will not be refined unless the “frozen” flag is removed
1419	for that parameter. This prevents a refinement from continuing with unexpected
1420	values. Forcing values to remain within limits may prevent a model from falling
1421	into a false minimum in the early stages of fitting or can alert the user to
1422	problems in the model.</p>
1423
1424	<p class=MsoNormal><o:p> </o:p></p>
1425
1426	<p class=MsoNormal>In this exercise we will add parameter limits to the
1427	refinement used in the <a
1428	href="https://subversion.xray.aps.anl.gov/pyGSAS/Tutorials/SeqRefine/SequentialTutorial.htm">Sequential
1429	Fitting tutorial</a>. </p>
1430
1431	<p class=MsoNormal><o:p> </o:p></p>
1432
1433	<h1>Step 1. Providing parameter ranges</h1>
1434
1435	<p class=MsoNormal><o:p> </o:p></p>
1436
1437	<p class=MsoNormal>For this step, a fairly complete refinement is supplied as
1438	file <span class=SpellE><span style='font-family:Courier'>SingleHistFit.gpx</span></span>.
1439	Review of this shows an excellent fit, with a GOF of 1.09 (<span lang=EL
1440	style='mso-ansi-language:EL'>χ</span><sup>2</sup> = 1.2), but some minor
1441	problems. One is that all <span class=SpellE>Uiso</span> values are all small,
1442	with values of 0.0002(2), 0.0001(2), -0.0042(5) Å<sup>2 </sup>respectively for
1443	Cu, Cr and O. (Standard uncertainties are found in the <span class=SpellE><span
1444	style='font-family:Courier'>SingleHistFit.lst</span></span> file.) The reason
1445	for these overall low values is unclear, but is not sample absorption or
1446	extinction, nor correlation between the profile, background and <span
1447	class=SpellE>Uiso</span> values, since introduction of appropriate parameters
1448	and fixing the <span class=SpellE>Uiso</span> values at somewhat more
1449	reasonable values, say 0.01 Å<sup>2</sup>, produces significantly worse fits.
1450	Noting that Cu has 29 electrons, Cr has 24 electrons and O only 8, we can
1451	expect the refinement to have significantly reduced sensitivity to the O <span
1452	class=SpellE>Uiso</span> value, so the slightly negative <span class=SpellE>Uiso</span>
1453	value for O is a concern but not serious. </p>
1454
1455	<p class=MsoNormal><o:p> </o:p></p>
1456
1457	<p class=MsoNormal>One other issue is that no sample broadening has been
1458	introduced for the <span class=SpellE>CuO</span> impurity. To assume that this
1459	phase has no sample broadening potentially skews the phase fraction.</p>
1460
1461	<p class=MsoNormal align=right style='text-align:right'><span
1462	style='mso-spacerun:yes'> </span><span style='mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype
1463	id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
1464	path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
1465	<v:stroke joinstyle="miter"/>
1466	<v:formulas>
1467	<v:f eqn="if lineDrawn pixelLineWidth 0"/>
1468	<v:f eqn="sum @0 1 0"/>
1469	<v:f eqn="sum 0 0 @1"/>
1470	<v:f eqn="prod @2 1 2"/>
1471	<v:f eqn="prod @3 21600 pixelWidth"/>
1472	<v:f eqn="prod @3 21600 pixelHeight"/>
1473	<v:f eqn="sum @0 0 1"/>
1474	<v:f eqn="prod @6 1 2"/>
1475	<v:f eqn="prod @7 21600 pixelWidth"/>
1476	<v:f eqn="sum @8 21600 0"/>
1477	<v:f eqn="prod @7 21600 pixelHeight"/>
1478	<v:f eqn="sum @10 21600 0"/>
1479	</v:formulas>
1480	<v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
1481	<o:lock v:ext="edit" aspectratio="t"/>
1482	</v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1047" type="#_x0000_t75"
1483	style='width:479pt;height:390pt;visibility:visible;mso-wrap-style:square'>
1484	<v:imagedata src="ParameterLimitsUse.fld/image001.png" o:title=""/>
1485	</v:shape><![endif]--><![if !vml]><img border=0 width=479 height=390
1486	src="ParameterLimitsUse.fld/image001.png" v:shapes="Picture_x0020_1"><![endif]></span></p>
1487
1488	<p class=MsoNormal><o:p> </o:p></p>
1489
1490	<h2><span class=MsoIntenseReference>1.1 Refine broadening for <span
1491	class=SpellE>CoO</span><o:p></o:p></span></h2>
1492
1493	<p class=MsoNormal>Without any prior knowledge, it is unclear if size or <span
1494	class=SpellE>microstrain</span> broadening (or both) might be expected for <span
1495	class=SpellE>CuO</span>, so initially, we might try refining both. </p>
1496
1497	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1498	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1046"
1499	type="#_x0000_t75" style='width:469pt;height:350pt;visibility:visible;
1500	mso-wrap-style:square'>
1501	<v:imagedata src="ParameterLimitsUse.fld/image002.png" o:title=""/>
1502	</v:shape><![endif]--><![if !vml]><img border=0 width=469 height=350
1503	src="ParameterLimitsUse.fld/image002.png" v:shapes="Picture_x0020_2"><![endif]></span></p>
1504
1505	<p class=MsoNormal><o:p> </o:p></p>
1506
1507	<p class=MsoNormal>If this is done, we see that an unreasonable (0.001 <span
1508	lang=EL style='mso-ansi-language:EL'>μ</span>m) value is obtained for the size.
1509	With the peaks so much wider, the phase fraction rises to an unrealistic and
1510	much larger value. Note that 0.001 <span lang=EL style='mso-ansi-language:EL'>μ</span>m
1511	value is the smallest that GSAS-II allows and these “hard limits” can result in
1512	an unstable refinement. <span class=GramE>This results</span> are not correct
1513	and should be reset. </p>
1514
1515	<p class=MsoNormal><o:p> </o:p></p>
1516
1517	<p class=MsoNormal>One could simply reset the size parameter to the default and
1518	turn off refinement, but another option is to use parameter limits to do that for
1519	us. </p>
1520
1521	<p class=MsoNormal><o:p> </o:p></p>
1522
1523	<h2><span class=MsoIntenseReference>1.2 Reset the refinement<o:p></o:p></span></h2>
1524
1525	<p class=MsoNormal>Before setting parameter limits we will put the size/<span
1526	class=SpellE>microstrain</span> and phase fraction back to their previous
1527	values.</p>
1528
1529	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1530	yes'><!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75" style='width:355pt;
1531	height:126pt;visibility:visible;mso-wrap-style:square'>
1532	<v:imagedata src="ParameterLimitsUse.fld/image012.png" o:title=""/>
1533	</v:shape><![endif]--><![if !vml]><img border=0 width=355 height=126
1534	src="ParameterLimitsUse.fld/image012.png" v:shapes="_x0000_i1045"><![endif]></span></p>
1535
1536	<p class=MsoNormal align=right style='text-align:right'><o:p> </o:p></p>
1537
1538	<p class=MsoNormal>Press the <u>Reset</u> button for both the <u>domain size</u>
1539	and the <span class=SpellE><u>Mustrain</u></span> values. Note the size is
1540	reset to the 1.0 <span lang=EL style='mso-ansi-language:EL'>μ</span>m initial
1541	value and the <span class=SpellE>microstrain</span> to 1,000 (note that <span
1542	class=SpellE>microstrain</span> is a fractional broadening measure, expressed
1543	as <span lang=EL style='mso-ansi-language:EL'>Δ</span>Q/Q x 10<sup>6</sup>, so it
1544	has no units.)<span style='mso-spacerun:yes'> </span>Likewise, <u>reset</u>
1545	the phase fraction to its approximate initial value, <u>0.03</u>.</p>
1546
1547	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1548	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1044"
1549	type="#_x0000_t75" style='width:351pt;height:100pt;visibility:visible;
1550	mso-wrap-style:square'>
1551	<v:imagedata src="ParameterLimitsUse.fld/image013.png" o:title=""/>
1552	</v:shape><![endif]--><![if !vml]><img border=0 width=351 height=100
1553	src="ParameterLimitsUse.fld/image013.png" v:shapes="Picture_x0020_14"><![endif]></span></p>
1554
1555	<p class=MsoNormal><o:p> </o:p></p>
1556
1557	<h2><span class=MsoIntenseReference>1.3 Set parameter limits<o:p></o:p></span></h2>
1558
1559	<p class=MsoNormal>Use the Calculate<span class=GramE>/“</span>View LS <span
1560	class=SpellE>parms</span>” menu command (shortcut: control-L or <span
1561	style='font-family:"Cambria Math",serif;mso-bidi-font-family:"Cambria Math"'>⌘-L)
1562	</span>to bring up the Least Squares Parameters window, as seen below. Note
1563	this is accessible from any tree item. </p>
1564
1565	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1566	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_4" o:spid="_x0000_i1043"
1567	type="#_x0000_t75" style='width:426pt;height:241pt;visibility:visible;
1568	mso-wrap-style:square'>
1569	<v:imagedata src="ParameterLimitsUse.fld/image004.png" o:title=""/>
1570	</v:shape><![endif]--><![if !vml]><img border=0 width=426 height=241
1571	src="ParameterLimitsUse.fld/image004.png" v:shapes="Picture_x0020_4"><![endif]></span></p>
1572
1573	<p class=MsoNormal><o:p> </o:p></p>
1574
1575	<p class=MsoNormal>The size and <span class=SpellE>microstrain</span> variables
1576	are found in the Phase/<span class=SpellE>Histo</span> parameters (since there
1577	is a value for every phase in each histogram), so <u>select Phase/<span
1578	class=SpellE>Histo</span></u> and to further simplify choices, the <span
1579	class=SpellE>CuO</span> phase (<u>phase #1</u>) only. </p>
1580
1581	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1582	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1042"
1583	type="#_x0000_t75" style='width:418pt;height:189pt;visibility:visible;
1584	mso-wrap-style:square'>
1585	<v:imagedata src="ParameterLimitsUse.fld/image005.png" o:title=""/>
1586	</v:shape><![endif]--><![if !vml]><img border=0 width=418 height=189
1587	src="ParameterLimitsUse.fld/image005.png" v:shapes="Picture_x0020_5"><![endif]></span><span
1588	style='mso-spacerun:yes'> </span></p>
1589
1590	<p class=MsoNormal><o:p> </o:p></p>
1591
1592	<p class=MsoNormal>Then <u>click</u> anywhere on the <u>row</u> that lists the
1593	size variable for phase 1, <u><span style='font-family:Courier'>1:<span
1594	class=GramE>0:Size</span>;i</span></u>, and an editing window opens, as
1595	below.<span style='mso-spacerun:yes'> </span></p>
1596
1597	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1598	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1041"
1599	type="#_x0000_t75" style='width:318pt;height:156pt;visibility:visible;
1600	mso-wrap-style:square'>
1601	<v:imagedata src="ParameterLimitsUse.fld/image006.png" o:title=""/>
1602	</v:shape><![endif]--><![if !vml]><img border=0 width=318 height=156
1603	src="ParameterLimitsUse.fld/image006.png" v:shapes="Picture_x0020_6"><![endif]></span></p>
1604
1605	<p class=MsoNormal><o:p> </o:p></p>
1606
1607	<p class=MsoNormal>The reasonable range for the size requires some thought, but
1608	with a minor component, even though the resolution of the instrument would
1609	likely allow better, 2 <span lang=EL style='mso-ansi-language:EL'>μ</span>m is
1610	a good upper limit and for a lower limit, 0.1 <span lang=EL style='mso-ansi-language:
1611	EL'>μ</span>m is probably a good number, though for this purpose a larger
1612	number could be used. <u>Press</u> the “<u>Add Lower limit</u>” and “<u>Add
1613	Upper limit</u>” buttons, and <u>add</u> the specified <u>values</u>. </p>
1614
1615	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1616	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1040"
1617	type="#_x0000_t75" style='width:380pt;height:158pt;visibility:visible;
1618	mso-wrap-style:square'>
1619	<v:imagedata src="ParameterLimitsUse.fld/image007.png" o:title=""/>
1620	</v:shape><![endif]--><![if !vml]><img border=0 width=380 height=158
1621	src="ParameterLimitsUse.fld/image007.png" v:shapes="Picture_x0020_7"><![endif]></span></p>
1622
1623	<p class=MsoNormal><o:p> </o:p></p>
1624
1625	<p class=MsoNormal>It will not have any effect in this particular refinement,
1626	but adding limits from 1000 to 20,000 for <span class=SpellE>microstrain</span>
1627	(variable <span style='font-family:Courier'>1:<span class=GramE>0:Mustrain</span>;i</span>),
1628	would also make sense to constrain that variable to a reasonable range. </p>
1629
1630	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1631	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_10" o:spid="_x0000_i1039"
1632	type="#_x0000_t75" style='width:559pt;height:240pt;visibility:visible;
1633	mso-wrap-style:square'>
1634	<v:imagedata src="ParameterLimitsUse.fld/image008.png" o:title=""/>
1635	</v:shape><![endif]--><![if !vml]><img border=0 width=559 height=240
1636	src="ParameterLimitsUse.fld/image008.png" v:shapes="Picture_x0020_10"><![endif]></span></p>
1637
1638	<p class=MsoNormal><o:p> </o:p></p>
1639
1640	<h2><span class=MsoIntenseReference>1.4 Redo the refinement with limits in
1641	place</span><b><span style='font-variant:small-caps;color:#4472C4;mso-themecolor:
1642	accent1;letter-spacing:.25pt'><o:p></o:p></span></b></h2>
1643
1644	<p class=MsoNormal>After the limits are set, <u>start a refinement</u> (Calculate/Refine
1645	menu or shortcut: control-R or <span style='font-family:"Cambria Math",serif;
1646	mso-bidi-font-family:"Cambria Math"'>⌘-</span>R). When this is done with limits
1647	in place, a new message will show up in the refinement results showing that a
1648	variable has been frozen:</p>
1649
1650	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1651	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_11" o:spid="_x0000_i1038"
1652	type="#_x0000_t75" style='width:270pt;height:170pt;visibility:visible;
1653	mso-wrap-style:square'>
1654	<v:imagedata src="ParameterLimitsUse.fld/image009.png" o:title=""/>
1655	</v:shape><![endif]--><![if !vml]><img border=0 width=270 height=170
1656	src="ParameterLimitsUse.fld/image009.png" v:shapes="Picture_x0020_11"><![endif]></span></p>
1657
1658	<p class=MsoNormal><o:p> </o:p></p>
1659
1660	<p class=MsoNormal>Likewise, revisiting the Least Squares Parameters window, as
1661	before, will now show that the size parameter has been set at the minimum value
1662	and this variable is now flagged as frozen with an F in the 3<sup>rd</sup>
1663	column, meaning that it will not be refined. Note that the rows of parameters that
1664	are frozen are also highlighted in yellow. </p>
1665
1666	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1667	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1037"
1668	type="#_x0000_t75" style='width:336pt;height:222pt;visibility:visible;
1669	mso-wrap-style:square'>
1670	<v:imagedata src="ParameterLimitsUse.fld/image010.png" o:title=""/>
1671	</v:shape><![endif]--><![if !vml]><img border=0 width=336 height=222
1672	src="ParameterLimitsUse.fld/image010.png" v:shapes="Picture_x0020_12"><![endif]></span></p>
1673
1674	<p class=MsoNormal><o:p> </o:p></p>
1675
1676	<h2><span class=MsoIntenseReference>1.5 Final considerations</span></h2>
1677
1678	<p class=MsoNormal>This has demonstrated how parameters limits are implemented.
1679	In fact, for this histogram, once the refinement of the phase fraction and the <span
1680	class=SpellE>microstrain</span> have settled into reasonable values, the freeze
1681	on the size broadening can be released and then the parameter will refine to
1682	yield a reasonable value. </p>
1683
1684	<p class=MsoNormal><o:p> </o:p></p>
1685
1686	<p class=MsoNormal>There are two ways to release a frozen parameter. One is by
1687	clicking on the row of the now-frozen parameter the Least Squares Parameters
1688	window, where a new button is now present to remove the variable from the list
1689	of frozen parameters. </p>
1690
1691	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1692	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1036"
1693	type="#_x0000_t75" style='width:326pt;height:135pt;visibility:visible;
1694	mso-wrap-style:square'>
1695	<v:imagedata src="ParameterLimitsUse.fld/image011.png" o:title=""/>
1696	</v:shape><![endif]--><![if !vml]><img border=0 width=326 height=135
1697	src="ParameterLimitsUse.fld/image011.png" v:shapes="Picture_x0020_13"><![endif]></span></p>
1698
1699	<p class=MsoNormal>The other way is that all frozen variables can be released
1700	from the Controls tree item. </p>
1701
1702	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1703	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_15" o:spid="_x0000_i1035"
1704	type="#_x0000_t75" style='width:341pt;height:106pt;visibility:visible;
1705	mso-wrap-style:square'>
1706	<v:imagedata src="ParameterLimitsUse.fld/image014.png" o:title=""/>
1707	</v:shape><![endif]--><![if !vml]><img border=0 width=341 height=106
1708	src="ParameterLimitsUse.fld/image014.png" v:shapes="Picture_x0020_15"><![endif]></span></p>
1709
1710	<h1><o:p><span style='text-decoration:none'> </span></o:p></h1>
1711
1712	<h1>Step 2. Sequential Fitting with Limits</h1>
1713
1714	<p class=MsoNormal align=right style='text-align:right'><o:p> </o:p></p>
1715
1716	<p class=MsoNormal>Parameter limits show an even greater value in sequential fitting,
1717	where many thousands of parameters may be varied (sometimes more), since the
1718	same parameter may be repeated for many histograms. In this exercise we will
1719	see how to use parameter limits in a sequential refinement. </p>
1720
1721	<p class=MsoNormal><o:p> </o:p></p>
1722
1723	<p class=MsoNormal>A starting point for a sequential fit is supplied as file <span
1724	class=SpellE><span style='font-family:Courier'>AllDataStart.gpx</span></span><span
1725	style='font-family:Courier'>,</span> which has the final results from Step 1
1726	(where the frozen size variable has been freed). A large number of variables
1727	have been turned off so that an initial fit can be used to account for the
1728	changes in lattice parameters due to temperature and background, before
1729	allowing any other parameters to vary. These are the steps that were performed
1730	for to create this file:</p>
1731
1732	<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1733	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1734	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1735	</span></span></span><![endif]><span dir=LTR></span>The additional 16 datasets were
1736	imported. </p>
1737
1738	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1739	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1740	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1741	</span></span></span><![endif]><span dir=LTR></span>For CuCr<sub>2</sub>O<sub>4</sub>:
1742	stop refining the cell, start refining the <span class=SpellE>D<sub>jj</sub></span>
1743	(Hydrostatic strain) terms; do not refine any size or <span class=SpellE>microstrain</span>
1744	terms. Turn off XU (position & <span class=SpellE>U<sub>iso</sub></span>) refinement
1745	for atoms</p>
1746
1747	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1748	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1749	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1750	</span></span></span><![endif]><span dir=LTR></span>For <span class=SpellE>CuO</span>:
1751	stop refining the cell, start refining the <span class=SpellE>D<sub>jj</sub></span>
1752	(Hydrostatic strain), but not the D13 term (related to the <span lang=EL
1753	style='mso-ansi-language:EL'>β</span> angle); do not refine phase fraction, size
1754	or <span class=SpellE>microstrain</span> terms.</p>
1755
1756	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1757	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1758	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1759	</span></span></span><![endif]><span dir=LTR></span>For background, do not
1760	refine the background peak’s position or width. </p>
1761
1762	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1763	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1764	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1765	</span></span></span><![endif]><span dir=LTR></span>For sample parameters, do
1766	not refine the sample displacement. </p>
1767
1768	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1769	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1770	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1771	</span></span></span><![endif]><span dir=LTR></span>Copy the parameters: Under
1772	the main histogram tree item use Commands<span class=GramE>/“</span>Copy <span
1773	class=SpellE>params</span>” to copy data limits, background, sample <span
1774	class=SpellE>param</span>. flags, etc. from histogram 0 to the rest.</p>
1775
1776	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1777	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1778	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1779	</span></span></span><![endif]><span dir=LTR></span>For both phases, use “Copy Data”
1780	in the Phases Data tab to duplicate the fit values as the initial values.</p>
1781
1782	<p class=MsoListParagraphCxSpLast style='text-indent:-.25in;mso-list:l1 level1 lfo3'><![if !supportLists]><span
1783	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1784	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1785	</span></span></span><![endif]><span dir=LTR></span>Controls: set maximum
1786	cycles to 10; select all 17 datasets for sequential refinement and set the “Copy
1787	results to next histogram” flag.</p>
1788
1789	<p class=MsoNormal><o:p> </o:p></p>
1790
1791	<h2><span class=MsoIntenseReference>2.1 Set <span class=SpellE>U<sub>iso</sub></span>
1792	parameter limits<o:p></o:p></span></h2>
1793
1794	<p class=MsoNormal>Here we will set limits on the size and <span class=SpellE>microstrain</span>
1795	parameters in all histograms, as well as set a lower limit for the <span
1796	class=SpellE>U<sub>iso</sub></span> values. </p>
1797
1798	<p class=MsoNormal><o:p> </o:p></p>
1799
1800	<p class=MsoNormal>Open the Least Squares Parameters window (Calculate<span
1801	class=GramE>/“</span><u>View LS <span class=SpellE>parms</span></u>” menu
1802	command or control-L/<span style='font-family:"Cambria Math",serif;mso-bidi-font-family:
1803	"Cambria Math"'>⌘-</span>L). Since none of the parameters to get limits are
1804	currently being refined, we will need to set to view all parameters rather than
1805	only refined parameters in the “Refinement status” radio buttons and then select
1806	the specific parameters. </p>
1807
1808	<p class=MsoNormal><o:p> </o:p></p>
1809
1810	<p class=MsoNormal>Let’s start with the <span class=SpellE>Uiso</span> value,
1811	which is a phase parameter and has variables named as <span class=GramE><i>p</i><span
1812	style='font-family:Courier'>::</span></span><span class=SpellE><span
1813	style='font-family:Courier'>AUiso:</span><i>a</i></span>. In <u>Parameter type</u>
1814	select <u>Phase</u>; then select Parameter <span class=SpellE>AUiso</span>; and
1815	finally, the “<u>Refinement status</u>” <u>All</u> radio button. The <span
1816	class=SpellE>Uiso</span> parameters for the five atoms across the two phases are
1817	shown, as below. </p>
1818
1819	<p class=MsoNormal><o:p> </o:p></p>
1820
1821	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1822	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_21" o:spid="_x0000_i1034"
1823	type="#_x0000_t75" style='width:362pt;height:198pt;visibility:visible;
1824	mso-wrap-style:square'>
1825	<v:imagedata src="ParameterLimitsUse.fld/image003.png" o:title=""/>
1826	</v:shape><![endif]--><![if !vml]><img border=0 width=362 height=198
1827	src="ParameterLimitsUse.fld/image003.png" v:shapes="Picture_x0020_21"><![endif]></span><span
1828	style='mso-spacerun:yes'> </span></p>
1829
1830	<p class=MsoNormal><o:p> </o:p></p>
1831
1832	<p class=MsoNormal><u>Click</u> anywhere <u>in the row</u> for any of the atoms
1833	in the CuCr<sub>2</sub>O<sub>4</sub> (<u>phase #0</u>) in the window shown
1834	above and, as in step 1, a window opens for that parameter, as seen below: <o:p></o:p></p>
1835
1836	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1837	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_22" o:spid="_x0000_i1033"
1838	type="#_x0000_t75" style='width:330pt;height:110pt;visibility:visible;
1839	mso-wrap-style:square'>
1840	<v:imagedata src="ParameterLimitsUse.fld/image019.png" o:title=""/>
1841	</v:shape><![endif]--><![if !vml]><img border=0 width=330 height=110
1842	src="ParameterLimitsUse.fld/image019.png" v:shapes="Picture_x0020_22"><![endif]></span></p>
1843
1844	<p class=MsoNormal><o:p> </o:p></p>
1845
1846	<p class=MsoNormal><o:p> </o:p></p>
1847
1848	<p class=MsoNormal>In that window, <u>click</u> on the “<u>Add Lower limit</u>”
1849	button and the window changes, as shown below. In that window, <u>set</u> the <u>limit</u>
1850	to <u>0.0</u>. Also set “<u>Match all atoms</u>” so that the limit is applied
1851	to all atoms in that phase. </p>
1852
1853	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1854	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_23" o:spid="_x0000_i1032"
1855	type="#_x0000_t75" style='width:328pt;height:118pt;visibility:visible;
1856	mso-wrap-style:square'>
1857	<v:imagedata src="ParameterLimitsUse.fld/image020.png" o:title=""/>
1858	</v:shape><![endif]--><![if !vml]><img border=0 width=328 height=118
1859	src="ParameterLimitsUse.fld/image020.png" v:shapes="Picture_x0020_23"><![endif]></span><span
1860	style='mso-spacerun:yes'> </span></p>
1861
1862	<p class=MsoNormal><o:p> </o:p></p>
1863
1864	<p class=MsoNormal>After pressing OK, note that the limits are now seen on the View
1865	Parameters window for the three intended atoms, as below:</p>
1866
1867	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1868	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20" o:spid="_x0000_i1031"
1869	type="#_x0000_t75" style='width:354pt;height:201pt;visibility:visible;
1870	mso-wrap-style:square'>
1871	<v:imagedata src="ParameterLimitsUse.fld/image018.png" o:title=""/>
1872	</v:shape><![endif]--><![if !vml]><img border=0 width=354 height=201
1873	src="ParameterLimitsUse.fld/image018.png" v:shapes="Picture_x0020_20"><![endif]></span></p>
1874
1875	<p class=MsoNormal><o:p> </o:p></p>
1876
1877	<h2><span class=MsoIntenseReference>2.2 Modify Size/<span class=SpellE>microstrain</span>
1878	parameter limits<o:p></o:p></span></h2>
1879
1880	<p class=MsoNormal>The limits set previously for crystallite size and <span
1881	lang=EL style='mso-ansi-language:EL'>μ</span>strain were previously set only
1882	for the first histogram. We want to set them for all histograms. </p>
1883
1884	<p class=MsoNormal>Switch to Parameter type <u>Phase/<span class=SpellE>Histo</span></u>;
1885	as before the “<u>Refinement status</u>” radio button needs to be <u>All</u>; <u>Select</u>
1886	Phase <u>1</u> (<span class=SpellE>CuO</span>) and for Parameter name, select “<span
1887	class=SpellE><span class=GramE><span style='font-family:Courier'>Size;i</span></span></span>”
1888	(<span class=SpellE><span style='font-family:Courier'>i</span></span> for
1889	isotropic). The window should appear as below:</p>
1890
1891	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1892	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_26" o:spid="_x0000_i1030"
1893	type="#_x0000_t75" style='width:309pt;height:203pt;visibility:visible;
1894	mso-wrap-style:square'>
1895	<v:imagedata src="ParameterLimitsUse.fld/image021.png" o:title=""/>
1896	</v:shape><![endif]--><![if !vml]><img border=0 width=309 height=203
1897	src="ParameterLimitsUse.fld/image021.png" v:shapes="Picture_x0020_26"><![endif]></span><span
1898	style='mso-spacerun:yes'> </span></p>
1899
1900	<p class=MsoNormal><o:p> </o:p></p>
1901
1902	<p class=MsoNormal>To extend the limits to all histograms, click anywhere in
1903	the first row, so that the window below opens:</p>
1904
1905	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1906	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_27" o:spid="_x0000_i1029"
1907	type="#_x0000_t75" style='width:312pt;height:128pt;visibility:visible;
1908	mso-wrap-style:square'>
1909	<v:imagedata src="ParameterLimitsUse.fld/image022.png" o:title=""/>
1910	</v:shape><![endif]--><![if !vml]><img border=0 width=312 height=128
1911	src="ParameterLimitsUse.fld/image022.png" v:shapes="Picture_x0020_27"><![endif]></span></p>
1912
1913	<p class=MsoNormal><o:p> </o:p></p>
1914
1915	<p class=MsoNormal>Click on the <u>two</u> “<u>Match all histograms</u>”
1916	buttons and <u>OK</u> and the limits now apply to all histograms:</p>
1917
1918	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
1919	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_28" o:spid="_x0000_i1028"
1920	type="#_x0000_t75" style='width:320pt;height:211pt;visibility:visible;
1921	mso-wrap-style:square'>
1922	<v:imagedata src="ParameterLimitsUse.fld/image023.png" o:title=""/>
1923	</v:shape><![endif]--><![if !vml]><img border=0 width=320 height=211
1924	src="ParameterLimitsUse.fld/image023.png" v:shapes="Picture_x0020_28"><![endif]></span></p>
1925
1926	<p class=MsoNormal align=right style='text-align:right'><o:p> </o:p></p>
1927
1928	<p class=MsoNormal><u>Repeat</u> the above for the <u>“</u><span class=SpellE><span
1929	class=GramE><u><span style='font-family:Courier'>Mustrain;i</span></u></span></span><u>”
1930	parameter</u> extending that to <u>all histograms</u>. </p>
1931
1932	<h2><span class=MsoIntenseReference><o:p> </o:p></span></h2>
1933
1934	<h2><span class=MsoIntenseReference>2.3 Refinements<o:p></o:p></span></h2>
1935
1936	<p class=MsoNormal><span class=MsoIntenseReference>A)</span> For the first
1937	sequential refinement, no additional parameters need to be added to the
1938	refinement. Use <u>Calculate<span class=GramE>/“</span>Sequential Refine”</u>
1939	(or Control-R/<span style='font-family:"Cambria Math",serif;mso-bidi-font-family:
1940	"Cambria Math"'>⌘-</span>L) to start the fitting. </p>
1941
1942	<p class=MsoNormal><o:p> </o:p></p>
1943
1944	<p class=MsoNormal><span class=MsoIntenseReference>B)</span> For the second sequential
1945	refinement add the following parameters:</p>
1946
1947	<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo4'><![if !supportLists]><span
1948	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1949	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1950	</span></span></span><![endif]><span dir=LTR></span>In one histogram, add the <u>Sample
1951	X</u> displacement <u>refine</u> flag; Use Command<span class=GramE>/“</span><u>Copy
1952	Flags</u>” to copy this setting to all histograms.</p>
1953
1954	<p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo4'><![if !supportLists]><span
1955	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1956	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1957	</span></span></span><![endif]><span dir=LTR></span>In the <u>CuCr2O4 phas</u>e
1958	in the <u>Data tab</u>, turn on refinement of the <u>size</u> and the <u>two <span
1959	class=SpellE>microstrain</span></u> flags. Use “Edit Phase”<span class=GramE>/“</span><u>Copy
1960	Flags</u>” to copy this setting to all histograms.</p>
1961
1962	<p class=MsoListParagraphCxSpLast style='text-indent:-.25in;mso-list:l0 level1 lfo4'><![if !supportLists]><span
1963	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1964	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1965	</span></span></span><![endif]><span dir=LTR></span>In the <span class=SpellE><u>CuO</u></span><u>
1966	phas</u>e in the <u>Data tab</u>, turn on refinement of the <u>Phase fraction </u>flag.
1967	Use “Edit Phase”<span class=GramE>/“</span><u>Copy Flags</u>” to copy this
1968	setting to all histograms.</p>
1969
1970	<p class=MsoNormal style='margin-left:.25in'>Use <u>Calculate<span class=GramE>/“</span>Sequential
1971	Refine”</u> (or Control-R/<span style='font-family:"Cambria Math",serif;
1972	mso-bidi-font-family:"Cambria Math"'>⌘-</span>L) to start the fitting again.<o:p></o:p></p>
1973
1974	<p class=MsoNormal><o:p> </o:p></p>
1975
1976	<p class=MsoNormal><span class=MsoIntenseReference>C)</span> For the third sequential
1977	refinement we will add parameters that have limits:<o:p></o:p></p>
1978
1979	<p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo4'><![if !supportLists]><span
1980	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1981	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1982	</span></span></span><![endif]><span dir=LTR></span>In the <u>CuCr2O4 phas</u>e
1983	in the <u>Atoms tab</u>, turn on refinement of the <u>XU</u> flags for all
1984	atoms. </p>
1985
1986	<p class=MsoListParagraphCxSpLast style='text-indent:-.25in;mso-list:l0 level1 lfo4'><![if !supportLists]><span
1987	style='font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:
1988	Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>
1989	</span></span></span><![endif]><span dir=LTR></span>In the <span class=SpellE><u>CuO</u></span><u>
1990	phas</u>e in the <u>Data tab</u>, turn on refinement of the <u>Phase fraction </u>flag.
1991	Use “Edit Phase”<span class=GramE>/“</span><u>Copy Flags</u>” to copy this
1992	setting to all histograms.</p>
1993
1994	<p class=MsoNormal style='margin-left:.25in'>Use <u>Calculate<span class=GramE>/“</span>Sequential
1995	Refine”</u> (or Control-R/<span style='font-family:"Cambria Math",serif;
1996	mso-bidi-font-family:"Cambria Math"'>⌘-</span>L) to start the fitting again.
1997	Note that after this refinement a number of parameters are noted as frozen:<o:p></o:p></p>
1998
1999	<p class=MsoNormal><o:p> </o:p></p>
2000
2001	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
2002	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_29" o:spid="_x0000_i1027"
2003	type="#_x0000_t75" style='width:251pt;height:137pt;visibility:visible;
2004	mso-wrap-style:square'>
2005	<v:imagedata src="ParameterLimitsUse.fld/image015.png" o:title=""/>
2006	</v:shape><![endif]--><![if !vml]><img border=0 width=251 height=137
2007	src="ParameterLimitsUse.fld/image015.png" v:shapes="Picture_x0020_29"><![endif]></span></p>
2008
2009	<p class=MsoNormal><o:p> </o:p></p>
2010
2011	<p class=MsoNormal>Also, a new column appears listing the number of frozen
2012	variables in the sequential results table.</p>
2013
2014	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
2015	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_30" o:spid="_x0000_i1026"
2016	type="#_x0000_t75" style='width:365pt;height:224pt;visibility:visible;
2017	mso-wrap-style:square'>
2018	<v:imagedata src="ParameterLimitsUse.fld/image016.png" o:title=""/>
2019	</v:shape><![endif]--><![if !vml]><img border=0 width=365 height=224
2020	src="ParameterLimitsUse.fld/image016.png" v:shapes="Picture_x0020_30"><![endif]></span></p>
2021
2022	<p class=MsoNormal align=right style='text-align:right'><o:p> </o:p></p>
2023
2024	<p class=MsoNormal>Resting the mouse above a cell in the table where a
2025	parameter is frozen will show that rather than an uncertainty (<span
2026	class=SpellE>e.s.d</span>.). </p>
2027
2028	<p class=MsoNormal align=right style='text-align:right'><span style='mso-no-proof:
2029	yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_31" o:spid="_x0000_i1025"
2030	type="#_x0000_t75" style='width:112pt;height:158pt;visibility:visible;
2031	mso-wrap-style:square'>
2032	<v:imagedata src="ParameterLimitsUse.fld/image017.png" o:title=""/>
2033	</v:shape><![endif]--><![if !vml]><img border=0 width=112 height=158
2034	src="ParameterLimitsUse.fld/image017.png" v:shapes="Picture_x0020_31"><![endif]></span></p>
2035
2036	<p class=MsoNormal><o:p> </o:p></p>
2037
2038	<p class=MsoNormal><o:p> </o:p></p>
2039
2040	<h2><span class=MsoIntenseReference>2.4 Finishing up<o:p></o:p></span></h2>
2041
2042	<p class=MsoNormal>Based on these results, it is clear that size and <span
2043	lang=EL style='mso-ansi-language:EL'>μ</span>strain cannot be refined for the <span
2044	class=SpellE>CuO</span> minor impurity and the fluctuations in this are causing
2045	the <span lang=EL style='mso-ansi-language:EL'>μ</span>strain and phase
2046	fraction values to vary unreliably. Reset any frozen <span lang=EL
2047	style='mso-ansi-language:EL'>μ</span>strain variables in the Least Squares
2048	Parameter viewer. Set reasonable values for size and <span lang=EL
2049	style='mso-ansi-language:EL'>μ</span>strain using the results in the first histogram
2050	and turn off the size refine flag. Using “Copy Selected Data” copy the Scale,
2051	Size and <span class=SpellE>Mustrain</span> to all other histograms and repeat
2052	the refinement. </p>
2053
2054	<p class=MsoNormal><o:p> </o:p></p>
2055
2056	<p class=MsoNormal>The O atoms now have their <span class=SpellE>Uiso</span>
2057	values fixed at 0.0 at lower temperatures, but these values do refine as the
2058	temperature increases. Weight fraction of <span class=SpellE>CuO</span> <span
2059	class=GramE>and <span style='mso-spacerun:yes'> </span><span lang=EL
2060	style='mso-ansi-language:EL'>μ</span></span>strain values now vary reasonably
2061	smoothly (considering uncertainties). A second sequential fit causes the fit to
2062	converge well, as noted by small changes <span class=GramE>in <span
2063	style='mso-spacerun:yes'> </span><span class=SpellE><span lang=EL
2064	style='mso-ansi-language:EL'>Δχ</span></span></span><sup>2</sup>. The final
2065	result is provided as file <span class=SpellE><span style='font-family:Courier'>SeqFit.gpx</span></span>,
2066	though one could contemplate adding a few more parameters to the refinement,
2067	such as the width and position of the background peak and the <span
2068	class=SpellE>CuO</span> D13 variable. The <span class=SpellE><span
2069	style='font-family:Courier'>SeqFit.lst</span></span> file shows the final
2070	goodness-of-fit values (which range from 1.0 to 1.2, corresponding to a maximum
2071	reduced <span lang=EL style='mso-ansi-language:EL'>χ</span><sup>2</sup> value
2072	of 1.4, so there is very little improvement to improve these models). Likewise,
2073	the variables that are frozen are shown in the .<span class=SpellE>lst</span>
2074	file as well. </p>
2075
2076	</div>
2077
2078	</body>
2079
2080	</html>

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