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| 6 | |
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| 17 | o\:* {behavior:url(#default#VML);} |
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| 18 | w\:* {behavior:url(#default#VML);} |
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| 19 | .shape {behavior:url(#default#VML);} |
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| 22 | <o:DocumentProperties> |
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| 23 | <o:Author>B Toby</o:Author> |
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| 24 | <o:LastAuthor>Von Dreele</o:LastAuthor> |
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| 25 | <o:Revision>28</o:Revision> |
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| 26 | <o:TotalTime>953</o:TotalTime> |
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| 27 | <o:Created>2014-05-09T19:20:00Z</o:Created> |
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| 28 | <o:LastSaved>2014-07-22T18:01:00Z</o:LastSaved> |
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| 29 | <o:Pages>8</o:Pages> |
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| 30 | <o:Words>1630</o:Words> |
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| 31 | <o:Characters>9293</o:Characters> |
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| 32 | <o:Company>Argonne National Laboratory</o:Company> |
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| 34 | <o:Paragraphs>21</o:Paragraphs> |
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| 36 | <o:Version>14.00</o:Version> |
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| 37 | </o:DocumentProperties> |
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| 40 | </o:OfficeDocumentSettings> |
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| 307 | <w:LsdException Locked="false" Priority="61" SemiHidden="false" |
|---|
| 308 | UnhideWhenUsed="false" Name="Light List Accent 6"/> |
|---|
| 309 | <w:LsdException Locked="false" Priority="62" SemiHidden="false" |
|---|
| 310 | UnhideWhenUsed="false" Name="Light Grid Accent 6"/> |
|---|
| 311 | <w:LsdException Locked="false" Priority="63" SemiHidden="false" |
|---|
| 312 | UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/> |
|---|
| 313 | <w:LsdException Locked="false" Priority="64" SemiHidden="false" |
|---|
| 314 | UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/> |
|---|
| 315 | <w:LsdException Locked="false" Priority="65" SemiHidden="false" |
|---|
| 316 | UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/> |
|---|
| 317 | <w:LsdException Locked="false" Priority="66" SemiHidden="false" |
|---|
| 318 | UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/> |
|---|
| 319 | <w:LsdException Locked="false" Priority="67" SemiHidden="false" |
|---|
| 320 | UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> |
|---|
| 321 | <w:LsdException Locked="false" Priority="68" SemiHidden="false" |
|---|
| 322 | UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> |
|---|
| 323 | <w:LsdException Locked="false" Priority="69" SemiHidden="false" |
|---|
| 324 | UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> |
|---|
| 325 | <w:LsdException Locked="false" Priority="70" SemiHidden="false" |
|---|
| 326 | UnhideWhenUsed="false" Name="Dark List Accent 6"/> |
|---|
| 327 | <w:LsdException Locked="false" Priority="71" SemiHidden="false" |
|---|
| 328 | UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> |
|---|
| 329 | <w:LsdException Locked="false" Priority="72" SemiHidden="false" |
|---|
| 330 | UnhideWhenUsed="false" Name="Colorful List Accent 6"/> |
|---|
| 331 | <w:LsdException Locked="false" Priority="73" SemiHidden="false" |
|---|
| 332 | UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> |
|---|
| 333 | <w:LsdException Locked="false" Priority="19" SemiHidden="false" |
|---|
| 334 | UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> |
|---|
| 335 | <w:LsdException Locked="false" Priority="21" SemiHidden="false" |
|---|
| 336 | UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> |
|---|
| 337 | <w:LsdException Locked="false" Priority="31" SemiHidden="false" |
|---|
| 338 | UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> |
|---|
| 339 | <w:LsdException Locked="false" Priority="32" SemiHidden="false" |
|---|
| 340 | UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> |
|---|
| 341 | <w:LsdException Locked="false" Priority="33" SemiHidden="false" |
|---|
| 342 | UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> |
|---|
| 343 | <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> |
|---|
| 344 | <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> |
|---|
| 345 | </w:LatentStyles> |
|---|
| 346 | </xml><![endif]--> |
|---|
| 347 | <style> |
|---|
| 348 | <!-- |
|---|
| 349 | /* Font Definitions */ |
|---|
| 350 | @font-face |
|---|
| 351 | {font-family:Courier; |
|---|
| 352 | panose-1:2 7 4 9 2 2 5 2 4 4; |
|---|
| 353 | mso-font-alt:"Courier New"; |
|---|
| 354 | mso-font-charset:0; |
|---|
| 355 | mso-generic-font-family:modern; |
|---|
| 356 | mso-font-format:other; |
|---|
| 357 | mso-font-pitch:fixed; |
|---|
| 358 | mso-font-signature:3 0 0 0 1 0;} |
|---|
| 359 | @font-face |
|---|
| 360 | {font-family:Courier; |
|---|
| 361 | panose-1:2 7 4 9 2 2 5 2 4 4; |
|---|
| 362 | mso-font-alt:"Courier New"; |
|---|
| 363 | mso-font-charset:0; |
|---|
| 364 | mso-generic-font-family:modern; |
|---|
| 365 | mso-font-format:other; |
|---|
| 366 | mso-font-pitch:fixed; |
|---|
| 367 | mso-font-signature:3 0 0 0 1 0;} |
|---|
| 368 | @font-face |
|---|
| 369 | {font-family:Cambria; |
|---|
| 370 | panose-1:2 4 5 3 5 4 6 3 2 4; |
|---|
| 371 | mso-font-charset:0; |
|---|
| 372 | mso-generic-font-family:roman; |
|---|
| 373 | mso-font-pitch:variable; |
|---|
| 374 | mso-font-signature:-536870145 1073743103 0 0 415 0;} |
|---|
| 375 | @font-face |
|---|
| 376 | {font-family:Calibri; |
|---|
| 377 | panose-1:2 15 5 2 2 2 4 3 2 4; |
|---|
| 378 | mso-font-charset:0; |
|---|
| 379 | mso-generic-font-family:swiss; |
|---|
| 380 | mso-font-pitch:variable; |
|---|
| 381 | mso-font-signature:-536870145 1073786111 1 0 415 0;} |
|---|
| 382 | @font-face |
|---|
| 383 | {font-family:"Lucida Grande"; |
|---|
| 384 | mso-font-alt:"Times New Roman"; |
|---|
| 385 | mso-font-charset:0; |
|---|
| 386 | mso-generic-font-family:auto; |
|---|
| 387 | mso-font-pitch:variable; |
|---|
| 388 | mso-font-signature:0 1342218751 0 0 447 0;} |
|---|
| 389 | @font-face |
|---|
| 390 | {font-family:Consolas; |
|---|
| 391 | panose-1:2 11 6 9 2 2 4 3 2 4; |
|---|
| 392 | mso-font-charset:0; |
|---|
| 393 | mso-generic-font-family:modern; |
|---|
| 394 | mso-font-pitch:fixed; |
|---|
| 395 | mso-font-signature:-520092929 1073806591 9 0 415 0;} |
|---|
| 396 | /* Style Definitions */ |
|---|
| 397 | p.MsoNormal, li.MsoNormal, div.MsoNormal |
|---|
| 398 | {mso-style-unhide:no; |
|---|
| 399 | mso-style-qformat:yes; |
|---|
| 400 | mso-style-parent:""; |
|---|
| 401 | margin:0in; |
|---|
| 402 | margin-bottom:.0001pt; |
|---|
| 403 | mso-pagination:widow-orphan; |
|---|
| 404 | font-size:12.0pt; |
|---|
| 405 | mso-bidi-font-size:10.0pt; |
|---|
| 406 | font-family:"Times New Roman","serif"; |
|---|
| 407 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 408 | mso-fareast-theme-font:minor-fareast; |
|---|
| 409 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 410 | mso-bidi-theme-font:minor-bidi; |
|---|
| 411 | mso-fareast-language:JA;} |
|---|
| 412 | h1 |
|---|
| 413 | {mso-style-priority:9; |
|---|
| 414 | mso-style-unhide:no; |
|---|
| 415 | mso-style-qformat:yes; |
|---|
| 416 | mso-style-link:"Heading 1 Char"; |
|---|
| 417 | mso-style-next:Normal; |
|---|
| 418 | margin-top:24.0pt; |
|---|
| 419 | margin-right:0in; |
|---|
| 420 | margin-bottom:0in; |
|---|
| 421 | margin-left:0in; |
|---|
| 422 | margin-bottom:.0001pt; |
|---|
| 423 | mso-pagination:widow-orphan lines-together; |
|---|
| 424 | page-break-after:avoid; |
|---|
| 425 | mso-outline-level:1; |
|---|
| 426 | font-size:18.0pt; |
|---|
| 427 | mso-bidi-font-size:16.0pt; |
|---|
| 428 | font-family:"Times New Roman","serif"; |
|---|
| 429 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 430 | mso-fareast-theme-font:major-fareast; |
|---|
| 431 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 432 | mso-bidi-theme-font:major-bidi; |
|---|
| 433 | mso-font-kerning:0pt; |
|---|
| 434 | mso-fareast-language:JA;} |
|---|
| 435 | h2 |
|---|
| 436 | {mso-style-noshow:yes; |
|---|
| 437 | mso-style-priority:9; |
|---|
| 438 | mso-style-qformat:yes; |
|---|
| 439 | mso-style-link:"Heading 2 Char"; |
|---|
| 440 | mso-style-next:Normal; |
|---|
| 441 | margin-top:10.0pt; |
|---|
| 442 | margin-right:0in; |
|---|
| 443 | margin-bottom:0in; |
|---|
| 444 | margin-left:0in; |
|---|
| 445 | margin-bottom:.0001pt; |
|---|
| 446 | mso-pagination:widow-orphan lines-together; |
|---|
| 447 | page-break-after:avoid; |
|---|
| 448 | mso-outline-level:2; |
|---|
| 449 | font-size:13.0pt; |
|---|
| 450 | font-family:"Calibri","sans-serif"; |
|---|
| 451 | mso-ascii-font-family:Calibri; |
|---|
| 452 | mso-ascii-theme-font:major-latin; |
|---|
| 453 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 454 | mso-fareast-theme-font:major-fareast; |
|---|
| 455 | mso-hansi-font-family:Calibri; |
|---|
| 456 | mso-hansi-theme-font:major-latin; |
|---|
| 457 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 458 | mso-bidi-theme-font:major-bidi; |
|---|
| 459 | color:#4F81BD; |
|---|
| 460 | mso-themecolor:accent1; |
|---|
| 461 | mso-fareast-language:JA;} |
|---|
| 462 | a:link, span.MsoHyperlink |
|---|
| 463 | {mso-style-noshow:yes; |
|---|
| 464 | mso-style-priority:99; |
|---|
| 465 | color:blue; |
|---|
| 466 | mso-themecolor:hyperlink; |
|---|
| 467 | text-decoration:underline; |
|---|
| 468 | text-underline:single;} |
|---|
| 469 | a:visited, span.MsoHyperlinkFollowed |
|---|
| 470 | {mso-style-noshow:yes; |
|---|
| 471 | mso-style-priority:99; |
|---|
| 472 | color:purple; |
|---|
| 473 | mso-themecolor:followedhyperlink; |
|---|
| 474 | text-decoration:underline; |
|---|
| 475 | text-underline:single;} |
|---|
| 476 | pre |
|---|
| 477 | {mso-style-noshow:yes; |
|---|
| 478 | mso-style-priority:99; |
|---|
| 479 | mso-style-link:"HTML Preformatted Char"; |
|---|
| 480 | margin:0in; |
|---|
| 481 | margin-bottom:.0001pt; |
|---|
| 482 | mso-pagination:widow-orphan; |
|---|
| 483 | font-size:10.0pt; |
|---|
| 484 | font-family:"Courier New"; |
|---|
| 485 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 486 | mso-fareast-theme-font:minor-fareast;} |
|---|
| 487 | p.MsoAcetate, li.MsoAcetate, div.MsoAcetate |
|---|
| 488 | {mso-style-noshow:yes; |
|---|
| 489 | mso-style-priority:99; |
|---|
| 490 | mso-style-link:"Balloon Text Char"; |
|---|
| 491 | margin:0in; |
|---|
| 492 | margin-bottom:.0001pt; |
|---|
| 493 | mso-pagination:widow-orphan; |
|---|
| 494 | font-size:9.0pt; |
|---|
| 495 | font-family:"Lucida Grande"; |
|---|
| 496 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 497 | mso-fareast-theme-font:minor-fareast; |
|---|
| 498 | mso-fareast-language:JA;} |
|---|
| 499 | p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph |
|---|
| 500 | {mso-style-priority:34; |
|---|
| 501 | mso-style-unhide:no; |
|---|
| 502 | mso-style-qformat:yes; |
|---|
| 503 | margin-top:0in; |
|---|
| 504 | margin-right:0in; |
|---|
| 505 | margin-bottom:0in; |
|---|
| 506 | margin-left:.5in; |
|---|
| 507 | margin-bottom:.0001pt; |
|---|
| 508 | mso-add-space:auto; |
|---|
| 509 | mso-pagination:widow-orphan; |
|---|
| 510 | font-size:12.0pt; |
|---|
| 511 | mso-bidi-font-size:10.0pt; |
|---|
| 512 | font-family:"Times New Roman","serif"; |
|---|
| 513 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 514 | mso-fareast-theme-font:minor-fareast; |
|---|
| 515 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 516 | mso-bidi-theme-font:minor-bidi; |
|---|
| 517 | mso-fareast-language:JA;} |
|---|
| 518 | p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst |
|---|
| 519 | {mso-style-priority:34; |
|---|
| 520 | mso-style-unhide:no; |
|---|
| 521 | mso-style-qformat:yes; |
|---|
| 522 | mso-style-type:export-only; |
|---|
| 523 | margin-top:0in; |
|---|
| 524 | margin-right:0in; |
|---|
| 525 | margin-bottom:0in; |
|---|
| 526 | margin-left:.5in; |
|---|
| 527 | margin-bottom:.0001pt; |
|---|
| 528 | mso-add-space:auto; |
|---|
| 529 | mso-pagination:widow-orphan; |
|---|
| 530 | font-size:12.0pt; |
|---|
| 531 | mso-bidi-font-size:10.0pt; |
|---|
| 532 | font-family:"Times New Roman","serif"; |
|---|
| 533 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 534 | mso-fareast-theme-font:minor-fareast; |
|---|
| 535 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 536 | mso-bidi-theme-font:minor-bidi; |
|---|
| 537 | mso-fareast-language:JA;} |
|---|
| 538 | p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle |
|---|
| 539 | {mso-style-priority:34; |
|---|
| 540 | mso-style-unhide:no; |
|---|
| 541 | mso-style-qformat:yes; |
|---|
| 542 | mso-style-type:export-only; |
|---|
| 543 | margin-top:0in; |
|---|
| 544 | margin-right:0in; |
|---|
| 545 | margin-bottom:0in; |
|---|
| 546 | margin-left:.5in; |
|---|
| 547 | margin-bottom:.0001pt; |
|---|
| 548 | mso-add-space:auto; |
|---|
| 549 | mso-pagination:widow-orphan; |
|---|
| 550 | font-size:12.0pt; |
|---|
| 551 | mso-bidi-font-size:10.0pt; |
|---|
| 552 | font-family:"Times New Roman","serif"; |
|---|
| 553 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 554 | mso-fareast-theme-font:minor-fareast; |
|---|
| 555 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 556 | mso-bidi-theme-font:minor-bidi; |
|---|
| 557 | mso-fareast-language:JA;} |
|---|
| 558 | p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast |
|---|
| 559 | {mso-style-priority:34; |
|---|
| 560 | mso-style-unhide:no; |
|---|
| 561 | mso-style-qformat:yes; |
|---|
| 562 | mso-style-type:export-only; |
|---|
| 563 | margin-top:0in; |
|---|
| 564 | margin-right:0in; |
|---|
| 565 | margin-bottom:0in; |
|---|
| 566 | margin-left:.5in; |
|---|
| 567 | margin-bottom:.0001pt; |
|---|
| 568 | mso-add-space:auto; |
|---|
| 569 | mso-pagination:widow-orphan; |
|---|
| 570 | font-size:12.0pt; |
|---|
| 571 | mso-bidi-font-size:10.0pt; |
|---|
| 572 | font-family:"Times New Roman","serif"; |
|---|
| 573 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 574 | mso-fareast-theme-font:minor-fareast; |
|---|
| 575 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 576 | mso-bidi-theme-font:minor-bidi; |
|---|
| 577 | mso-fareast-language:JA;} |
|---|
| 578 | span.Heading1Char |
|---|
| 579 | {mso-style-name:"Heading 1 Char"; |
|---|
| 580 | mso-style-priority:9; |
|---|
| 581 | mso-style-unhide:no; |
|---|
| 582 | mso-style-locked:yes; |
|---|
| 583 | mso-style-link:"Heading 1"; |
|---|
| 584 | mso-ansi-font-size:18.0pt; |
|---|
| 585 | mso-bidi-font-size:16.0pt; |
|---|
| 586 | font-family:"Times New Roman","serif"; |
|---|
| 587 | mso-ascii-font-family:"Times New Roman"; |
|---|
| 588 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 589 | mso-fareast-theme-font:major-fareast; |
|---|
| 590 | mso-hansi-font-family:"Times New Roman"; |
|---|
| 591 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 592 | mso-bidi-theme-font:major-bidi; |
|---|
| 593 | font-weight:bold;} |
|---|
| 594 | span.Heading2Char |
|---|
| 595 | {mso-style-name:"Heading 2 Char"; |
|---|
| 596 | mso-style-noshow:yes; |
|---|
| 597 | mso-style-priority:9; |
|---|
| 598 | mso-style-unhide:no; |
|---|
| 599 | mso-style-locked:yes; |
|---|
| 600 | mso-style-link:"Heading 2"; |
|---|
| 601 | mso-ansi-font-size:13.0pt; |
|---|
| 602 | mso-bidi-font-size:13.0pt; |
|---|
| 603 | font-family:"Calibri","sans-serif"; |
|---|
| 604 | mso-ascii-font-family:Calibri; |
|---|
| 605 | mso-ascii-theme-font:major-latin; |
|---|
| 606 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 607 | mso-fareast-theme-font:major-fareast; |
|---|
| 608 | mso-hansi-font-family:Calibri; |
|---|
| 609 | mso-hansi-theme-font:major-latin; |
|---|
| 610 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 611 | mso-bidi-theme-font:major-bidi; |
|---|
| 612 | color:#4F81BD; |
|---|
| 613 | mso-themecolor:accent1; |
|---|
| 614 | font-weight:bold;} |
|---|
| 615 | span.BalloonTextChar |
|---|
| 616 | {mso-style-name:"Balloon Text Char"; |
|---|
| 617 | mso-style-noshow:yes; |
|---|
| 618 | mso-style-priority:99; |
|---|
| 619 | mso-style-unhide:no; |
|---|
| 620 | mso-style-locked:yes; |
|---|
| 621 | mso-style-link:"Balloon Text"; |
|---|
| 622 | mso-ansi-font-size:9.0pt; |
|---|
| 623 | mso-bidi-font-size:9.0pt; |
|---|
| 624 | font-family:"Lucida Grande"; |
|---|
| 625 | mso-ascii-font-family:"Lucida Grande"; |
|---|
| 626 | mso-hansi-font-family:"Lucida Grande"; |
|---|
| 627 | mso-bidi-font-family:"Lucida Grande";} |
|---|
| 628 | span.HTMLPreformattedChar |
|---|
| 629 | {mso-style-name:"HTML Preformatted Char"; |
|---|
| 630 | mso-style-noshow:yes; |
|---|
| 631 | mso-style-priority:99; |
|---|
| 632 | mso-style-unhide:no; |
|---|
| 633 | mso-style-locked:yes; |
|---|
| 634 | mso-style-link:"HTML Preformatted"; |
|---|
| 635 | font-family:Consolas; |
|---|
| 636 | mso-ascii-font-family:Consolas; |
|---|
| 637 | mso-hansi-font-family:Consolas;} |
|---|
| 638 | span.SpellE |
|---|
| 639 | {mso-style-name:""; |
|---|
| 640 | mso-spl-e:yes;} |
|---|
| 641 | span.GramE |
|---|
| 642 | {mso-style-name:""; |
|---|
| 643 | mso-gram-e:yes;} |
|---|
| 644 | .MsoChpDefault |
|---|
| 645 | {mso-style-type:export-only; |
|---|
| 646 | mso-default-props:yes; |
|---|
| 647 | font-size:10.0pt; |
|---|
| 648 | mso-ansi-font-size:10.0pt; |
|---|
| 649 | mso-bidi-font-size:10.0pt; |
|---|
| 650 | font-family:"Cambria","serif"; |
|---|
| 651 | mso-ascii-font-family:Cambria; |
|---|
| 652 | mso-ascii-theme-font:minor-latin; |
|---|
| 653 | mso-fareast-font-family:"Times New Roman"; |
|---|
| 654 | mso-fareast-theme-font:minor-fareast; |
|---|
| 655 | mso-hansi-font-family:Cambria; |
|---|
| 656 | mso-hansi-theme-font:minor-latin; |
|---|
| 657 | mso-bidi-font-family:"Times New Roman"; |
|---|
| 658 | mso-bidi-theme-font:minor-bidi; |
|---|
| 659 | mso-fareast-language:JA;} |
|---|
| 660 | @page WordSection1 |
|---|
| 661 | {size:8.5in 11.0in; |
|---|
| 662 | margin:1.0in 1.25in 1.0in 1.25in; |
|---|
| 663 | mso-header-margin:.5in; |
|---|
| 664 | mso-footer-margin:.5in; |
|---|
| 665 | mso-paper-source:0;} |
|---|
| 666 | div.WordSection1 |
|---|
| 667 | {page:WordSection1;} |
|---|
| 668 | /* List Definitions */ |
|---|
| 669 | @list l0 |
|---|
| 670 | {mso-list-id:109671306; |
|---|
| 671 | mso-list-type:hybrid; |
|---|
| 672 | mso-list-template-ids:129770836 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;} |
|---|
| 673 | @list l0:level1 |
|---|
| 674 | {mso-level-tab-stop:none; |
|---|
| 675 | mso-level-number-position:left; |
|---|
| 676 | text-indent:-.25in;} |
|---|
| 677 | @list l0:level2 |
|---|
| 678 | {mso-level-number-format:alpha-lower; |
|---|
| 679 | mso-level-tab-stop:none; |
|---|
| 680 | mso-level-number-position:left; |
|---|
| 681 | text-indent:-.25in;} |
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| 937 | |
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| 938 | <body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'> |
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| 939 | |
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| 940 | <div class=WordSection1> |
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| 941 | |
|---|
| 942 | <h1>Parametric Fitting and Pseudo Variables for Sequential Fits</h1> |
|---|
| 943 | |
|---|
| 944 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 945 | |
|---|
| 946 | <p class=MsoNormal>Once a sequential fit has been performed, any function of |
|---|
| 947 | the fit parameters from the fit and <i style='mso-bidi-font-style:normal'>their |
|---|
| 948 | estimated uncertainty</i> (<span class=SpellE>s.u</span>.) can be computed by |
|---|
| 949 | defining one or more "pseudo-variables" (Pseudo <span class=SpellE>Vars</span>) |
|---|
| 950 | Note that the covariance matrix from the fit is used to provide appropriate |
|---|
| 951 | treatment of the correlation in parameters with respect to uncertainty |
|---|
| 952 | estimation.<span style='mso-spacerun:yes'> </span>In addition, parametric |
|---|
| 953 | equations in terms of sample parameters such as temperature and pressure, etc. |
|---|
| 954 | can be fit to either the parameters from the fit or the Pseudo Vars. </p> |
|---|
| 955 | |
|---|
| 956 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 957 | |
|---|
| 958 | <p class=MsoNormal>This tutorial assumes that the steps in the <span |
|---|
| 959 | style='mso-bidi-font-family:"Times New Roman"'><a href="SequentialTutorial.htm">Sequential |
|---|
| 960 | refinement of multiple datasets</a> tutorial have already been performed. </span>If |
|---|
| 961 | you have not done so already, <a href="Starting%20GSAS.htm">start GSAS-II</a>. |
|---|
| 962 | Some of the pictures in this tutorial are from a Mac OSX machine; these |
|---|
| 963 | illustrate the differences in menu location from Windows/Linux machines.<o:p></o:p></p> |
|---|
| 964 | |
|---|
| 965 | <h1><span class=GramE>Step 1.</span> Open the previous Sequential Fit</h1> |
|---|
| 966 | |
|---|
| 967 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 968 | |
|---|
| 969 | <p class=MsoListParagraphCxSpFirst style='margin-bottom:6.0pt;mso-add-space: |
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| 970 | auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span |
|---|
| 971 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 972 | style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 973 | </span></span></span><![endif]>Select the<b style='mso-bidi-font-weight:normal'> |
|---|
| 974 | </b><b style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif"; |
|---|
| 975 | mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>File/Open |
|---|
| 976 | Project</span></b> menu item to load file named <span class=SpellE><b |
|---|
| 977 | style='mso-bidi-font-weight:normal'><span style='font-family:"Calibri","sans-serif"; |
|---|
| 978 | mso-ascii-theme-font:major-latin;mso-hansi-theme-font:major-latin'>SeqTut.gpx</span></b></span>. |
|---|
| 979 | </p> |
|---|
| 980 | |
|---|
| 981 | <p class=MsoListParagraphCxSpMiddle style='margin-bottom:6.0pt;mso-add-space: |
|---|
| 982 | auto;text-indent:-.25in;mso-list:l4 level1 lfo2'><![if !supportLists]><span |
|---|
| 983 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 984 | style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 985 | </span></span></span><![endif]>Select the <b style='mso-bidi-font-weight:normal'><span |
|---|
| 986 | style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:major-latin; |
|---|
| 987 | mso-hansi-theme-font:major-latin'>Sequential results</span></b> item in the |
|---|
| 988 | data tree. The sequential refinement results data table is displayed, as below. |
|---|
| 989 | </p> |
|---|
| 990 | |
|---|
| 991 | <p class=MsoListParagraphCxSpLast style='margin-bottom:6.0pt;mso-add-space: |
|---|
| 992 | auto'><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype |
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| 1009 | </v:formulas> |
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| 1016 | src="ParametricFitting_files/image069.png" v:shapes="Picture_x0020_36"><![endif]></span></p> |
|---|
| 1017 | |
|---|
| 1018 | <h1>Step 2. Create a Pseudo <span class=SpellE>Var</span></h1> |
|---|
| 1019 | |
|---|
| 1020 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1021 | |
|---|
| 1022 | <p class=MsoNormal>Pseudo <span class=SpellE>Vars</span> allow computation of |
|---|
| 1023 | an arbitrarily defined function of combinations of the variables that are |
|---|
| 1024 | displayed in the sequential refinement table. Any valid Python language |
|---|
| 1025 | expression can be used to define this. While one can plot <i style='mso-bidi-font-style: |
|---|
| 1026 | normal'>a</i>, <i style='mso-bidi-font-style:normal'>b</i> and <i |
|---|
| 1027 | style='mso-bidi-font-style:normal'>c</i> axis lengths by clicking on the |
|---|
| 1028 | appropriate column of the table, to compare them one might want to scale them. |
|---|
| 1029 | Alternately, as will be done here, we will plot the ratio of <i |
|---|
| 1030 | style='mso-bidi-font-style:normal'>b</i> and <i style='mso-bidi-font-style: |
|---|
| 1031 | normal'>c</i>.</p> |
|---|
| 1032 | |
|---|
| 1033 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1034 | |
|---|
| 1035 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span |
|---|
| 1036 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1037 | style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1038 | </span></span></span><![endif]><b style='mso-bidi-font-weight:normal'>Click on |
|---|
| 1039 | the Pseudo <span class=SpellE>Vars</span>/Add</b> menu item to create a new |
|---|
| 1040 | Pseudo Var.</p> |
|---|
| 1041 | |
|---|
| 1042 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1043 | |
|---|
| 1044 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1045 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1052" |
|---|
| 1046 | type="#_x0000_t75" style='width:159pt;height:45.75pt;visibility:visible; |
|---|
| 1047 | mso-wrap-style:square'> |
|---|
| 1048 | <v:imagedata src="ParametricFitting_files/image003.png" o:title=""/> |
|---|
| 1049 | </v:shape><![endif]--><![if !vml]><img border=0 width=212 height=61 |
|---|
| 1050 | src="ParametricFitting_files/image072.jpg" v:shapes="Picture_x0020_2"><![endif]></span></p> |
|---|
| 1051 | |
|---|
| 1052 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1053 | |
|---|
| 1054 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span |
|---|
| 1055 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1056 | style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1057 | </span></span></span><![endif]>A window opens to define a formula</p> |
|---|
| 1058 | |
|---|
| 1059 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1060 | |
|---|
| 1061 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1062 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1051" |
|---|
| 1063 | type="#_x0000_t75" style='width:309.75pt;height:152.25pt;visibility:visible; |
|---|
| 1064 | mso-wrap-style:square'> |
|---|
| 1065 | <v:imagedata src="ParametricFitting_files/image005.png" o:title="" croptop="4214f" |
|---|
| 1066 | cropbottom="8982f" cropleft="3852f" cropright="3654f"/> |
|---|
| 1067 | </v:shape><![endif]--><![if !vml]><img border=0 width=413 height=203 |
|---|
| 1068 | src="ParametricFitting_files/image073.png" v:shapes="Picture_x0020_3"><![endif]></span></p> |
|---|
| 1069 | |
|---|
| 1070 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1071 | |
|---|
| 1072 | <p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_4" |
|---|
| 1073 | o:spid="_x0000_s1031" type="#_x0000_t75" style='position:absolute;left:0; |
|---|
| 1074 | text-align:left;margin-left:178.3pt;margin-top:846.2pt;width:229.5pt;height:86.5pt; |
|---|
| 1075 | z-index:251658240;visibility:visible;mso-wrap-style:square; |
|---|
| 1076 | mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt; |
|---|
| 1077 | mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; |
|---|
| 1078 | mso-wrap-distance-bottom:0;mso-position-horizontal:right; |
|---|
| 1079 | mso-position-horizontal-relative:margin;mso-position-vertical:absolute; |
|---|
| 1080 | mso-position-vertical-relative:margin;mso-width-percent:0; |
|---|
| 1081 | mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'> |
|---|
| 1082 | <v:imagedata src="ParametricFitting_files/image007.png" o:title=""/> |
|---|
| 1083 | <w:wrap type="square" anchorx="margin" anchory="margin"/> |
|---|
| 1084 | </v:shape><![endif]--><![if !vml]><img width=306 height=115 |
|---|
| 1085 | src="ParametricFitting_files/image074.jpg" align=right hspace=12 v:shapes="Picture_x0020_4"><![endif]>In |
|---|
| 1086 | this window, <b style='mso-bidi-font-weight:normal'>type a Python expression, |
|---|
| 1087 | "b/c",</b> but note that the variables <b style='mso-bidi-font-weight: |
|---|
| 1088 | normal'>c</b> and <b style='mso-bidi-font-weight:normal'>b</b> here are |
|---|
| 1089 | arbitrary labels. After entering the formula, <b style='mso-bidi-font-weight: |
|---|
| 1090 | normal'>press the Validate button</b> or wait a few seconds and the validation |
|---|
| 1091 | will be tested automatically. Note that if you pause too long, while typing, an |
|---|
| 1092 | error message may be displayed as the incomplete formula is found to be |
|---|
| 1093 | invalid. If so, make corrections or keep typing until the entire expression is |
|---|
| 1094 | entered.<span style='mso-spacerun:yes'> </span>The error message will |
|---|
| 1095 | sometimes show where error is encountered, though not in this case. </p> |
|---|
| 1096 | |
|---|
| 1097 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1098 | |
|---|
| 1099 | <p class=MsoListParagraphCxSpMiddle>When the expression is complete, a new |
|---|
| 1100 | table appears where the arbitrary <b style='mso-bidi-font-weight:normal'>c</b> |
|---|
| 1101 | and <b style='mso-bidi-font-weight:normal'>b</b> labels will be assigned.</p> |
|---|
| 1102 | |
|---|
| 1103 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1104 | |
|---|
| 1105 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1106 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1050" |
|---|
| 1107 | type="#_x0000_t75" style='width:309.75pt;height:152.25pt;visibility:visible; |
|---|
| 1108 | mso-wrap-style:square'> |
|---|
| 1109 | <v:imagedata src="ParametricFitting_files/image009.png" o:title="" croptop="4435f" |
|---|
| 1110 | cropbottom="8537f" cropleft="3852f" cropright="3650f"/> |
|---|
| 1111 | </v:shape><![endif]--><![if !vml]><img border=0 width=413 height=203 |
|---|
| 1112 | src="ParametricFitting_files/image075.png" v:shapes="Picture_x0020_5"><![endif]></span></p> |
|---|
| 1113 | |
|---|
| 1114 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1115 | |
|---|
| 1116 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l2 level1 lfo4'><![if !supportLists]><span |
|---|
| 1117 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1118 | style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1119 | </span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_6" |
|---|
| 1120 | o:spid="_x0000_s1030" type="#_x0000_t75" style='position:absolute;left:0; |
|---|
| 1121 | text-align:left;margin-left:37.3pt;margin-top:1143.2pt;width:88.5pt;height:65.5pt; |
|---|
| 1122 | z-index:251659264;visibility:visible;mso-wrap-style:square; |
|---|
| 1123 | mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt; |
|---|
| 1124 | mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; |
|---|
| 1125 | mso-wrap-distance-bottom:0;mso-position-horizontal:right; |
|---|
| 1126 | mso-position-horizontal-relative:margin;mso-position-vertical:absolute; |
|---|
| 1127 | mso-position-vertical-relative:margin;mso-width-percent:0; |
|---|
| 1128 | mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'> |
|---|
| 1129 | <v:imagedata src="ParametricFitting_files/image011.png" o:title=""/> |
|---|
| 1130 | <w:wrap type="square" anchorx="margin" anchory="margin"/> |
|---|
| 1131 | </v:shape><![endif]--><![if !vml]><img width=118 height=87 |
|---|
| 1132 | src="ParametricFitting_files/image076.jpg" align=right hspace=12 v:shapes="Picture_x0020_6"><![endif]>Assign |
|---|
| 1133 | the label for the numerator by <b style='mso-bidi-font-weight:normal'>clicking |
|---|
| 1134 | on the menu button next to label b</b>; this will bring up a list of variable |
|---|
| 1135 | types, from this <b style='mso-bidi-font-weight:normal'>select Phase</b> to bring |
|---|
| 1136 | up a list of phase variables, from that list <b style='mso-bidi-font-weight: |
|---|
| 1137 | normal'>select variable </b><b style='mso-bidi-font-weight:normal'><span |
|---|
| 1138 | style='font-family:Courier'>0::b</span>,</b> which is the <i style='mso-bidi-font-style: |
|---|
| 1139 | normal'>b</i> lattice parameter for phase 0 (CuCr<sub>2</sub>O<sub>4</sub>). |
|---|
| 1140 | Repeat this process to select <b style='mso-bidi-font-weight:normal'>assign to |
|---|
| 1141 | label c variable </b><b style='mso-bidi-font-weight:normal'><span |
|---|
| 1142 | style='font-family:Courier'>0::b</span></b>.</p> |
|---|
| 1143 | |
|---|
| 1144 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1145 | |
|---|
| 1146 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1147 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_8" o:spid="_x0000_i1049" |
|---|
| 1148 | type="#_x0000_t75" style='width:315.75pt;height:129pt;visibility:visible; |
|---|
| 1149 | mso-wrap-style:square'> |
|---|
| 1150 | <v:imagedata src="ParametricFitting_files/image013.png" o:title="" croptop="4597f" |
|---|
| 1151 | cropbottom="9873f" cropleft="3348f" cropright="3506f"/> |
|---|
| 1152 | </v:shape><![endif]--><![if !vml]><img border=0 width=421 height=172 |
|---|
| 1153 | src="ParametricFitting_files/image077.png" v:shapes="Picture_x0020_8"><![endif]></span></p> |
|---|
| 1154 | |
|---|
| 1155 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1156 | |
|---|
| 1157 | <p class=MsoListParagraphCxSpMiddle>As each variable is set, the window shows |
|---|
| 1158 | the value for each variable from the first refinement and when the expression |
|---|
| 1159 | can be evaluated, the value for the expression, evaluated for that first |
|---|
| 1160 | refinement. Only when the expression can be evaluated can the OK button be |
|---|
| 1161 | pressed. </p> |
|---|
| 1162 | |
|---|
| 1163 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1164 | |
|---|
| 1165 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1166 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_9" o:spid="_x0000_i1048" |
|---|
| 1167 | type="#_x0000_t75" style='width:314.25pt;height:153.75pt;visibility:visible; |
|---|
| 1168 | mso-wrap-style:square'> |
|---|
| 1169 | <v:imagedata src="ParametricFitting_files/image015.png" o:title="" croptop="3770f" |
|---|
| 1170 | cropbottom="8549f" cropleft="3371f" cropright="3424f"/> |
|---|
| 1171 | </v:shape><![endif]--><![if !vml]><img border=0 width=419 height=205 |
|---|
| 1172 | src="ParametricFitting_files/image078.png" v:shapes="Picture_x0020_9"><![endif]></span></p> |
|---|
| 1173 | |
|---|
| 1174 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1175 | |
|---|
| 1176 | <p class=MsoListParagraphCxSpMiddle><b style='mso-bidi-font-weight:normal'>Press |
|---|
| 1177 | OK</b> to close the window and add the Pseudo Var. Note that a new column |
|---|
| 1178 | appears at the end of the sequential refinement data table. </p> |
|---|
| 1179 | |
|---|
| 1180 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1181 | |
|---|
| 1182 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1183 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_10" o:spid="_x0000_i1047" |
|---|
| 1184 | type="#_x0000_t75" style='width:354pt;height:180.75pt;visibility:visible; |
|---|
| 1185 | mso-wrap-style:square'> |
|---|
| 1186 | <v:imagedata src="ParametricFitting_files/image017.png" o:title="" croptop="2743f" |
|---|
| 1187 | cropbottom="7533f" cropleft="2917f" cropright="3097f"/> |
|---|
| 1188 | </v:shape><![endif]--><![if !vml]><img border=0 width=472 height=241 |
|---|
| 1189 | src="ParametricFitting_files/image079.png" v:shapes="Picture_x0020_10"><![endif]></span></p> |
|---|
| 1190 | |
|---|
| 1191 | <p class=MsoListParagraphCxSpMiddle><!--[if gte vml 1]><v:shape id="Picture_x0020_11" |
|---|
| 1192 | o:spid="_x0000_s1029" type="#_x0000_t75" style='position:absolute;left:0; |
|---|
| 1193 | text-align:left;margin-left:42.3pt;margin-top:666.2pt;width:93.5pt;height:39pt; |
|---|
| 1194 | z-index:251660288;visibility:visible;mso-wrap-style:square; |
|---|
| 1195 | mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt; |
|---|
| 1196 | mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; |
|---|
| 1197 | mso-wrap-distance-bottom:0;mso-position-horizontal:right; |
|---|
| 1198 | mso-position-horizontal-relative:margin;mso-position-vertical:absolute; |
|---|
| 1199 | mso-position-vertical-relative:margin;mso-width-percent:0; |
|---|
| 1200 | mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'> |
|---|
| 1201 | <v:imagedata src="ParametricFitting_files/image019.png" o:title=""/> |
|---|
| 1202 | <w:wrap type="square" anchorx="margin" anchory="margin"/> |
|---|
| 1203 | </v:shape><![endif]--><![if !vml]><img width=125 height=52 |
|---|
| 1204 | src="ParametricFitting_files/image080.jpg" align=right hspace=12 v:shapes="Picture_x0020_11"><![endif]>Also |
|---|
| 1205 | note that positioning the mouse above any cell in the table and waiting for a |
|---|
| 1206 | few seconds causes the <span class=SpellE>s.u</span>. to be shown as a tooltip. |
|---|
| 1207 | </p> |
|---|
| 1208 | |
|---|
| 1209 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1210 | |
|---|
| 1211 | <p class=MsoListParagraphCxSpMiddle><b style='mso-bidi-font-weight:normal'>Double |
|---|
| 1212 | clicking on the column</b> will cause the pseudo <span class=SpellE>var</span> |
|---|
| 1213 | to be plotted:</p> |
|---|
| 1214 | |
|---|
| 1215 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1216 | |
|---|
| 1217 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1218 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_12" o:spid="_x0000_i1046" |
|---|
| 1219 | type="#_x0000_t75" style='width:351.75pt;height:303pt;visibility:visible; |
|---|
| 1220 | mso-wrap-style:square'> |
|---|
| 1221 | <v:imagedata src="ParametricFitting_files/image021.png" o:title="" croptop="2107f" |
|---|
| 1222 | cropbottom="4939f" cropleft="3133f" cropright="3309f"/> |
|---|
| 1223 | </v:shape><![endif]--><![if !vml]><img border=0 width=469 height=404 |
|---|
| 1224 | src="ParametricFitting_files/image081.png" v:shapes="Picture_x0020_12"><![endif]></span></p> |
|---|
| 1225 | |
|---|
| 1226 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1227 | |
|---|
| 1228 | <p class=MsoListParagraphCxSpMiddle><b style='mso-bidi-font-weight:normal'>Press |
|---|
| 1229 | the "s" key</b> inside the plot to <b style='mso-bidi-font-weight: |
|---|
| 1230 | normal'>select Temperature</b> as the x-axis. </p> |
|---|
| 1231 | |
|---|
| 1232 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1233 | |
|---|
| 1234 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1235 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1045" |
|---|
| 1236 | type="#_x0000_t75" style='width:162.75pt;height:156pt;visibility:visible; |
|---|
| 1237 | mso-wrap-style:square'> |
|---|
| 1238 | <v:imagedata src="ParametricFitting_files/image023.png" o:title="" croptop="4105f" |
|---|
| 1239 | cropbottom="8920f" cropleft="6109f" cropright="5911f"/> |
|---|
| 1240 | </v:shape><![endif]--><![if !vml]><img border=0 width=217 height=208 |
|---|
| 1241 | src="ParametricFitting_files/image082.png" v:shapes="Picture_x0020_13"><![endif]></span></p> |
|---|
| 1242 | |
|---|
| 1243 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1244 | |
|---|
| 1245 | <p class=MsoListParagraphCxSpMiddle>The plot will then appear as below:</p> |
|---|
| 1246 | |
|---|
| 1247 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1248 | |
|---|
| 1249 | <p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US; |
|---|
| 1250 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1044" |
|---|
| 1251 | type="#_x0000_t75" style='width:353.25pt;height:303.75pt;visibility:visible; |
|---|
| 1252 | mso-wrap-style:square'> |
|---|
| 1253 | <v:imagedata src="ParametricFitting_files/image025.png" o:title="" croptop="1859f" |
|---|
| 1254 | cropbottom="5066f" cropleft="3024f" cropright="3203f"/> |
|---|
| 1255 | </v:shape><![endif]--><![if !vml]><img border=0 width=471 height=405 |
|---|
| 1256 | src="ParametricFitting_files/image083.png" v:shapes="Picture_x0020_14"><![endif]></span></p> |
|---|
| 1257 | |
|---|
| 1258 | <h2>Step 3: Fitting a simple parametric equation to the unit cell volume</h2> |
|---|
| 1259 | |
|---|
| 1260 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1261 | |
|---|
| 1262 | <p class=MsoNormal>In the next few steps we will fit increasingly sophisticated |
|---|
| 1263 | equations of state to different parameters. As a simple example, we can <b |
|---|
| 1264 | style='mso-bidi-font-weight:normal'>double-click on </b><b style='mso-bidi-font-weight: |
|---|
| 1265 | normal'><span style='font-family:Courier'>0::<span class=SpellE>Vol</span></span></b>, |
|---|
| 1266 | to plot the unit volume for the first phase vs. T, and fit this to a straight |
|---|
| 1267 | line.<span style='mso-spacerun:yes'> </span></p> |
|---|
| 1268 | |
|---|
| 1269 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1270 | |
|---|
| 1271 | <p class=MsoListParagraphCxSpFirst><span style='mso-fareast-language:EN-US; |
|---|
| 1272 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_15" o:spid="_x0000_i1043" |
|---|
| 1273 | type="#_x0000_t75" style='width:356.25pt;height:305.25pt;visibility:visible; |
|---|
| 1274 | mso-wrap-style:square'> |
|---|
| 1275 | <v:imagedata src="ParametricFitting_files/image027.png" o:title="" croptop="2107f" |
|---|
| 1276 | cropbottom="4567f" cropleft="3024f" cropright="2769f"/> |
|---|
| 1277 | </v:shape><![endif]--><![if !vml]><img border=0 width=475 height=407 |
|---|
| 1278 | src="ParametricFitting_files/image084.png" v:shapes="Picture_x0020_15"><![endif]></span></p> |
|---|
| 1279 | |
|---|
| 1280 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span |
|---|
| 1281 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1282 | style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1283 | </span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_16" |
|---|
| 1284 | o:spid="_x0000_s1028" type="#_x0000_t75" style='position:absolute;left:0; |
|---|
| 1285 | text-align:left;margin-left:21.8pt;margin-top:891.2pt;width:73pt;height:53pt; |
|---|
| 1286 | z-index:251661312;visibility:visible;mso-wrap-style:square; |
|---|
| 1287 | mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt; |
|---|
| 1288 | mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; |
|---|
| 1289 | mso-wrap-distance-bottom:0;mso-position-horizontal:right; |
|---|
| 1290 | mso-position-horizontal-relative:margin;mso-position-vertical:absolute; |
|---|
| 1291 | mso-position-vertical-relative:margin;mso-width-percent:0; |
|---|
| 1292 | mso-height-percent:0;mso-width-relative:margin;mso-height-relative:margin'> |
|---|
| 1293 | <v:imagedata src="ParametricFitting_files/image029.png" o:title=""/> |
|---|
| 1294 | <w:wrap type="square" anchorx="margin" anchory="margin"/> |
|---|
| 1295 | </v:shape><![endif]--><![if !vml]><img width=97 height=71 |
|---|
| 1296 | src="ParametricFitting_files/image085.jpg" align=right hspace=12 v:shapes="Picture_x0020_16"><![endif]><b |
|---|
| 1297 | style='mso-bidi-font-weight:normal'>Click on the Parametric Fit/Add</b> <b |
|---|
| 1298 | style='mso-bidi-font-weight:normal'>equation</b> menu item, which opens a |
|---|
| 1299 | window similar to what was used before to create a Pseudo <span class=SpellE>var</span>, |
|---|
| 1300 | except that here an equation is defined.<span style='mso-spacerun:yes'> |
|---|
| 1301 | </span></p> |
|---|
| 1302 | |
|---|
| 1303 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1304 | |
|---|
| 1305 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1306 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_17" o:spid="_x0000_i1042" |
|---|
| 1307 | type="#_x0000_t75" style='width:311.25pt;height:152.25pt;visibility:visible; |
|---|
| 1308 | mso-wrap-style:square'> |
|---|
| 1309 | <v:imagedata src="ParametricFitting_files/image031.png" o:title="" croptop="4213f" |
|---|
| 1310 | cropbottom="8982f" cropleft="3490f" cropright="3653f"/> |
|---|
| 1311 | </v:shape><![endif]--><![if !vml]><img border=0 width=415 height=203 |
|---|
| 1312 | src="ParametricFitting_files/image086.png" v:shapes="Picture_x0020_17"><![endif]></span></p> |
|---|
| 1313 | |
|---|
| 1314 | <p class=MsoListParagraphCxSpLast><o:p> </o:p></p> |
|---|
| 1315 | |
|---|
| 1316 | <p class=MsoNormal>We will want to fit the unit cell volume to an equation of |
|---|
| 1317 | form <i style='mso-bidi-font-style:normal'>y = <span class=SpellE>mx+b</span></i>, |
|---|
| 1318 | where y will be the volume and x temperature and m and b will be best fit. </p> |
|---|
| 1319 | |
|---|
| 1320 | <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><o:p> </o:p></b></p> |
|---|
| 1321 | |
|---|
| 1322 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><b |
|---|
| 1323 | style='mso-bidi-font-weight:normal'><span style='mso-fareast-font-family:"Times New Roman"; |
|---|
| 1324 | mso-bidi-font-family:"Times New Roman"'><span style='mso-list:Ignore'>2.<span |
|---|
| 1325 | style='font:7.0pt "Times New Roman"'> </span></span></span></b><![endif]><b |
|---|
| 1326 | style='mso-bidi-font-weight:normal'>Click the menu button</b> to select a |
|---|
| 1327 | dependent variable for "y". From the phase list, select the unit cell |
|---|
| 1328 | volume for the CuCr<sub>2</sub>O<sub>4</sub> phase, <b style='mso-bidi-font-weight: |
|---|
| 1329 | normal'>click on </b><b style='mso-bidi-font-weight:normal'><span |
|---|
| 1330 | style='font-family:Courier'>0::<span class=SpellE>Vol</span></span></b> and |
|---|
| 1331 | then <b style='mso-bidi-font-weight:normal'>press OK.<o:p></o:p></b></p> |
|---|
| 1332 | |
|---|
| 1333 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1334 | |
|---|
| 1335 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1336 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19" o:spid="_x0000_i1041" |
|---|
| 1337 | type="#_x0000_t75" style='width:314.25pt;height:132pt;visibility:visible; |
|---|
| 1338 | mso-wrap-style:square'> |
|---|
| 1339 | <v:imagedata src="ParametricFitting_files/image033.png" o:title="" croptop="3830f" |
|---|
| 1340 | cropbottom="9359f" cropleft="3705f" cropright="3502f"/> |
|---|
| 1341 | </v:shape><![endif]--><![if !vml]><img border=0 width=419 height=176 |
|---|
| 1342 | src="ParametricFitting_files/image087.png" v:shapes="Picture_x0020_19"><![endif]></span></p> |
|---|
| 1343 | |
|---|
| 1344 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1345 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_20" o:spid="_x0000_i1040" |
|---|
| 1346 | type="#_x0000_t75" style='width:311.25pt;height:155.25pt;visibility:visible; |
|---|
| 1347 | mso-wrap-style:square'> |
|---|
| 1348 | <v:imagedata src="ParametricFitting_files/image035.png" o:title="" croptop="3548f" |
|---|
| 1349 | cropbottom="8549f" cropleft="3611f" cropright="3665f"/> |
|---|
| 1350 | </v:shape><![endif]--><![if !vml]><img border=0 width=415 height=207 |
|---|
| 1351 | src="ParametricFitting_files/image088.png" v:shapes="Picture_x0020_20"><![endif]></span></p> |
|---|
| 1352 | |
|---|
| 1353 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1354 | |
|---|
| 1355 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span |
|---|
| 1356 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1357 | style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1358 | </span></span></span><![endif]>Now we <b style='mso-bidi-font-weight:normal'>enter |
|---|
| 1359 | the equation</b> using Python syntax, which is simply "<b |
|---|
| 1360 | style='mso-bidi-font-weight:normal'><span style='font-family:Courier'>m*x + b</span></b>", |
|---|
| 1361 | note that the use of spaces and the choice of labels is arbitrary. If one were |
|---|
| 1362 | to accidentally leave out the "<span style='font-family:Courier'>*</span>" |
|---|
| 1363 | and typed "<span style='font-family:Courier'>mx</span>" then that |
|---|
| 1364 | would be considered a label, if "<span style='font-family:Courier'>m x</span>" |
|---|
| 1365 | were typed, an error message would be displayed. </p> |
|---|
| 1366 | |
|---|
| 1367 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1368 | |
|---|
| 1369 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1370 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_21" o:spid="_x0000_i1039" |
|---|
| 1371 | type="#_x0000_t75" style='width:315.75pt;height:155.25pt;visibility:visible; |
|---|
| 1372 | mso-wrap-style:square'> |
|---|
| 1373 | <v:imagedata src="ParametricFitting_files/image037.png" o:title="" croptop="3326f" |
|---|
| 1374 | cropbottom="8766f" cropleft="3010f" cropright="3419f"/> |
|---|
| 1375 | </v:shape><![endif]--><![if !vml]><img border=0 width=421 height=207 |
|---|
| 1376 | src="ParametricFitting_files/image089.png" v:shapes="Picture_x0020_21"><![endif]></span></p> |
|---|
| 1377 | |
|---|
| 1378 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1379 | |
|---|
| 1380 | <p class=MsoListParagraphCxSpMiddle>Then press Validate or wait a few seconds |
|---|
| 1381 | and a window changes, as above, allowing assignment of the three new labels (m, |
|---|
| 1382 | x, and b). </p> |
|---|
| 1383 | |
|---|
| 1384 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1385 | |
|---|
| 1386 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span |
|---|
| 1387 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1388 | style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1389 | </span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_22" |
|---|
| 1390 | o:spid="_x0000_s1027" type="#_x0000_t75" style='position:absolute;left:0; |
|---|
| 1391 | text-align:left;margin-left:43.8pt;margin-top:603.2pt;width:95pt;height:35pt; |
|---|
| 1392 | z-index:251662336;visibility:visible;mso-wrap-style:square; |
|---|
| 1393 | mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt; |
|---|
| 1394 | mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; |
|---|
| 1395 | mso-wrap-distance-bottom:0;mso-position-horizontal:right; |
|---|
| 1396 | mso-position-horizontal-relative:margin;mso-position-vertical:absolute; |
|---|
| 1397 | mso-position-vertical-relative:margin;mso-width-percent:0; |
|---|
| 1398 | mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'> |
|---|
| 1399 | <v:imagedata src="ParametricFitting_files/image039.png" o:title=""/> |
|---|
| 1400 | <w:wrap type="square" anchorx="margin" anchory="margin"/> |
|---|
| 1401 | </v:shape><![endif]--><![if !vml]><img width=127 height=47 |
|---|
| 1402 | src="ParametricFitting_files/image090.jpg" align=right hspace=12 v:shapes="Picture_x0020_22"><![endif]>Assign |
|---|
| 1403 | b and m as refinable parameters: <b style='mso-bidi-font-weight:normal'>Click |
|---|
| 1404 | on the menu button</b> next to b and <b style='mso-bidi-font-weight:normal'>select |
|---|
| 1405 | Free</b>. Then <b style='mso-bidi-font-weight:normal'>repeat this for m</b>. </p> |
|---|
| 1406 | |
|---|
| 1407 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1408 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_23" o:spid="_x0000_i1038" |
|---|
| 1409 | type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible; |
|---|
| 1410 | mso-wrap-style:square'> |
|---|
| 1411 | <v:imagedata src="ParametricFitting_files/image041.png" o:title="" croptop="3992f" |
|---|
| 1412 | cropbottom="8541f" cropleft="3250f" cropright="3415f"/> |
|---|
| 1413 | </v:shape><![endif]--><![if !vml]><img border=0 width=419 height=204 |
|---|
| 1414 | src="ParametricFitting_files/image091.png" v:shapes="Picture_x0020_23"><![endif]></span></p> |
|---|
| 1415 | |
|---|
| 1416 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1417 | |
|---|
| 1418 | <p class=MsoListParagraphCxSpMiddle>It is not necessary to change the default |
|---|
| 1419 | variable names, the initial values, or the refine flag. </p> |
|---|
| 1420 | |
|---|
| 1421 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1422 | |
|---|
| 1423 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span |
|---|
| 1424 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1425 | style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1426 | </span></span></span><![endif]><!--[if gte vml 1]><v:shape id="Picture_x0020_24" |
|---|
| 1427 | o:spid="_x0000_s1026" type="#_x0000_t75" style='position:absolute;left:0; |
|---|
| 1428 | text-align:left;margin-left:24.3pt;margin-top:837.2pt;width:75.5pt;height:26pt; |
|---|
| 1429 | z-index:251663360;visibility:visible;mso-wrap-style:square; |
|---|
| 1430 | mso-width-percent:0;mso-height-percent:0;mso-wrap-distance-left:9pt; |
|---|
| 1431 | mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; |
|---|
| 1432 | mso-wrap-distance-bottom:0;mso-position-horizontal:right; |
|---|
| 1433 | mso-position-horizontal-relative:margin;mso-position-vertical:absolute; |
|---|
| 1434 | mso-position-vertical-relative:margin;mso-width-percent:0; |
|---|
| 1435 | mso-height-percent:0;mso-width-relative:page;mso-height-relative:page'> |
|---|
| 1436 | <v:imagedata src="ParametricFitting_files/image043.png" o:title=""/> |
|---|
| 1437 | <w:wrap type="square" anchorx="margin" anchory="margin"/> |
|---|
| 1438 | </v:shape><![endif]--><![if !vml]><img width=101 height=35 |
|---|
| 1439 | src="ParametricFitting_files/image092.jpg" align=right hspace=12 v:shapes="Picture_x0020_24"><![endif]>To |
|---|
| 1440 | set x as the temperature, <b style='mso-bidi-font-weight:normal'>click on the |
|---|
| 1441 | menu button</b> next to x, <b style='mso-bidi-font-weight:normal'>select |
|---|
| 1442 | Global,</b> and then <b style='mso-bidi-font-weight:normal'>select Temperature</b> |
|---|
| 1443 | from the list and <b style='mso-bidi-font-weight:normal'>click on OK</b>.</p> |
|---|
| 1444 | |
|---|
| 1445 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1446 | |
|---|
| 1447 | <p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span |
|---|
| 1448 | style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape |
|---|
| 1449 | id="Picture_x0020_25" o:spid="_x0000_i1037" type="#_x0000_t75" style='width:245.25pt; |
|---|
| 1450 | height:108.75pt;visibility:visible;mso-wrap-style:square'> |
|---|
| 1451 | <v:imagedata src="ParametricFitting_files/image045.png" o:title="" croptop="5483f" |
|---|
| 1452 | cropbottom="10984f" cropleft="4191f" cropright="4159f"/> |
|---|
| 1453 | </v:shape><![endif]--><![if !vml]><img border=0 width=327 height=145 |
|---|
| 1454 | src="ParametricFitting_files/image093.png" v:shapes="Picture_x0020_25"><![endif]></span></p> |
|---|
| 1455 | |
|---|
| 1456 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1457 | |
|---|
| 1458 | <p class=MsoListParagraphCxSpMiddle>The window then changes, showing the values |
|---|
| 1459 | of the dependent variable and the expression for the first refinement, as |
|---|
| 1460 | below.<span style='mso-spacerun:yes'> </span>Optionally, the values for m and |
|---|
| 1461 | b could be changed manually and the displayed value will be updated when |
|---|
| 1462 | Validate is pressed or after a short delay.</p> |
|---|
| 1463 | |
|---|
| 1464 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1465 | |
|---|
| 1466 | <p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US; |
|---|
| 1467 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_26" o:spid="_x0000_i1036" |
|---|
| 1468 | type="#_x0000_t75" style='width:314.25pt;height:153pt;visibility:visible; |
|---|
| 1469 | mso-wrap-style:square'> |
|---|
| 1470 | <v:imagedata src="ParametricFitting_files/image047.png" o:title="" croptop="3992f" |
|---|
| 1471 | cropbottom="8541f" cropleft="3490f" cropright="3294f"/> |
|---|
| 1472 | </v:shape><![endif]--><![if !vml]><img border=0 width=419 height=204 |
|---|
| 1473 | src="ParametricFitting_files/image094.png" v:shapes="Picture_x0020_26"><![endif]></span></p> |
|---|
| 1474 | |
|---|
| 1475 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1476 | |
|---|
| 1477 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo6'><![if !supportLists]><span |
|---|
| 1478 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1479 | style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1480 | </span></span></span><![endif]><b style='mso-bidi-font-weight:normal'>Press the |
|---|
| 1481 | OK button</b> to save this parametric fit. The b and m values are then refined |
|---|
| 1482 | and the console shows the results:</p> |
|---|
| 1483 | |
|---|
| 1484 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space: |
|---|
| 1485 | auto'><span style='font-family:Courier'>Fit Results<o:p></o:p></span></p> |
|---|
| 1486 | |
|---|
| 1487 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space: |
|---|
| 1488 | auto'><span style='font-family:Courier'><span style='mso-spacerun:yes'> |
|---|
| 1489 | </span>:<span class=GramE>:m</span><span style='mso-spacerun:yes'> </span>= |
|---|
| 1490 | -0.4476(5)<o:p></o:p></span></p> |
|---|
| 1491 | |
|---|
| 1492 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space: |
|---|
| 1493 | auto'><span style='font-family:Courier'><span style='mso-spacerun:yes'> |
|---|
| 1494 | </span>:<span class=GramE>:b</span><span style='mso-spacerun:yes'> </span>= |
|---|
| 1495 | 577.73(8)<o:p></o:p></span></p> |
|---|
| 1496 | |
|---|
| 1497 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space: |
|---|
| 1498 | auto'><span style='font-family:Courier'><o:p> </o:p></span></p> |
|---|
| 1499 | |
|---|
| 1500 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1501 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_27" o:spid="_x0000_i1035" |
|---|
| 1502 | type="#_x0000_t75" style='width:350.25pt;height:302.25pt;visibility:visible; |
|---|
| 1503 | mso-wrap-style:square'> |
|---|
| 1504 | <v:imagedata src="ParametricFitting_files/image049.png" o:title="" croptop="2231f" |
|---|
| 1505 | cropbottom="5061f" cropleft="3133f" cropright="3524f"/> |
|---|
| 1506 | </v:shape><![endif]--><![if !vml]><img border=0 width=467 height=403 |
|---|
| 1507 | src="ParametricFitting_files/image095.png" v:shapes="Picture_x0020_27"><![endif]></span><span |
|---|
| 1508 | style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin; |
|---|
| 1509 | mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p> |
|---|
| 1510 | |
|---|
| 1511 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1512 | |
|---|
| 1513 | <p class=MsoListParagraphCxSpMiddle>Also, a plot shows the actual values and |
|---|
| 1514 | the fit. Note that while the values are close to a straight line, the |
|---|
| 1515 | deviations are huge compared to the estimated uncertainty values for the unit |
|---|
| 1516 | cell volume (shown as very small error bars, unless the plot is |
|---|
| 1517 | magnified).<span style='mso-spacerun:yes'> </span>If a linear relationship had |
|---|
| 1518 | been defined as a constraint (as can be done with other software packages) <i>the |
|---|
| 1519 | unit cell parameters would have been forced onto this line with no obvious |
|---|
| 1520 | indication that this was not a correct result.</i> </p> |
|---|
| 1521 | |
|---|
| 1522 | <p class=MsoListParagraphCxSpLast><span style='mso-spacerun:yes'> </span></p> |
|---|
| 1523 | |
|---|
| 1524 | <h2>Step 4: Examples of more complex fitting equations</h2> |
|---|
| 1525 | |
|---|
| 1526 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1527 | |
|---|
| 1528 | <p class=MsoNormal>Here we will look at how more sophisticated Python |
|---|
| 1529 | capabilities can be used to fit more complex behavior.<span |
|---|
| 1530 | style='mso-spacerun:yes'> </span></p> |
|---|
| 1531 | |
|---|
| 1532 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1533 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1534 | style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1535 | </span></span></span><![endif]>Optionally <b style='mso-bidi-font-weight:normal'>Double-click |
|---|
| 1536 | on the </b><b style='mso-bidi-font-weight:normal'><span style='font-family: |
|---|
| 1537 | Courier'>0::c</span> column</b>. Note the changes in the <i style='mso-bidi-font-style: |
|---|
| 1538 | normal'>c</i> unit cell parameter as a function of temperature, as shown below. |
|---|
| 1539 | </p> |
|---|
| 1540 | |
|---|
| 1541 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1542 | |
|---|
| 1543 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1544 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_28" o:spid="_x0000_i1034" |
|---|
| 1545 | type="#_x0000_t75" style='width:350.25pt;height:303pt;visibility:visible; |
|---|
| 1546 | mso-wrap-style:square'> |
|---|
| 1547 | <v:imagedata src="ParametricFitting_files/image051.png" o:title="" croptop="658f" |
|---|
| 1548 | cropbottom="2861f" cropleft="1709f" cropright="1878f"/> |
|---|
| 1549 | </v:shape><![endif]--><![if !vml]><img border=0 width=467 height=404 |
|---|
| 1550 | src="ParametricFitting_files/image096.png" v:shapes="Picture_x0020_28"><![endif]></span></p> |
|---|
| 1551 | |
|---|
| 1552 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1553 | |
|---|
| 1554 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1555 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1556 | style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1557 | </span></span></span><![endif]>We will now fit <i style='mso-bidi-font-style: |
|---|
| 1558 | normal'>c</i> as a function of form 1<span class=GramE>/(</span>1 + <span |
|---|
| 1559 | class=SpellE>exp</span>[k(t-t<sub>0</sub>)]). <b style='mso-bidi-font-weight: |
|---|
| 1560 | normal'>Click on the Parametric Fit/Add</b> <b style='mso-bidi-font-weight: |
|---|
| 1561 | normal'>equation</b> menu item (as before). For the dependent variable, <b |
|---|
| 1562 | style='mso-bidi-font-weight:normal'>click on Phase</b> and then <b |
|---|
| 1563 | style='mso-bidi-font-weight:normal'>select </b><b style='mso-bidi-font-weight: |
|---|
| 1564 | normal'><span style='font-family:Courier'>0::c</span></b>.<span |
|---|
| 1565 | style='mso-spacerun:yes'> </span></p> |
|---|
| 1566 | |
|---|
| 1567 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1568 | |
|---|
| 1569 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1570 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1571 | style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1572 | </span></span></span><![endif]>We then <b style='mso-bidi-font-weight:normal'>enter |
|---|
| 1573 | expression</b></p> |
|---|
| 1574 | |
|---|
| 1575 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1576 | |
|---|
| 1577 | <p class=MsoListParagraphCxSpMiddle style='text-indent:.5in'><span |
|---|
| 1578 | style='font-family:Courier'>b + c / (1+<span class=GramE>math.exp(</span>k*(t-t0)))<o:p></o:p></span></p> |
|---|
| 1579 | |
|---|
| 1580 | <p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria","serif"; |
|---|
| 1581 | mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p> </o:p></span></p> |
|---|
| 1582 | |
|---|
| 1583 | <p class=MsoListParagraphCxSpLast>Note that use of spaces here are arbitrary, |
|---|
| 1584 | but all parentheses are required. Also note that the exponential function [<span |
|---|
| 1585 | class=SpellE>exp</span>(x) or e<sup>x</sup>] is supplied in Python in the math |
|---|
| 1586 | and <span class=SpellE>numpy</span> modules as <span class=SpellE><span |
|---|
| 1587 | class=GramE>math.exp</span></span><span class=GramE>(</span>), <span |
|---|
| 1588 | class=SpellE>numpy.exp</span>(). Both of these can be used as well as the |
|---|
| 1589 | convenience abbreviation of <span class=SpellE><span class=GramE>np.exp</span></span><span |
|---|
| 1590 | class=GramE>(</span>). Further, there are also a small number of common |
|---|
| 1591 | functions that are defined without the prefix for convenience [these include |
|---|
| 1592 | pi, <span class=SpellE><span class=GramE>exp</span></span><span class=GramE>(</span>), |
|---|
| 1593 | <span class=SpellE>sqrt</span>(), <span class=SpellE>sind</span>(), <span |
|---|
| 1594 | class=SpellE>cosd</span>(), <span class=SpellE>tand</span>() where d indicates |
|---|
| 1595 | an argument in degrees.]</p> |
|---|
| 1596 | |
|---|
| 1597 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1598 | |
|---|
| 1599 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1600 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1601 | style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1602 | </span></span></span><![endif]><b style='mso-bidi-font-weight:normal'>Assign |
|---|
| 1603 | free variables to b, c, k and t0</b>, similar as to what was done before. Note |
|---|
| 1604 | that the<b style='mso-bidi-font-weight:normal'> </b>variable name for the <span |
|---|
| 1605 | style='font-family:Courier'>b</span> label was set automatically to <span |
|---|
| 1606 | style='font-family:Courier'>b_1</span>. If the same variable name (<span |
|---|
| 1607 | style='font-family:Courier'>b</span>) was used again here, the same value would |
|---|
| 1608 | be used to fit both equations, which is certainly not what we want here.<span |
|---|
| 1609 | style='mso-spacerun:yes'> </span>Finally, since this equation is too complex |
|---|
| 1610 | to fit without some reasonable starting values, specify an approximate value |
|---|
| 1611 | for the vicinity of where <i style='mso-bidi-font-style:normal'>c</i> starts to |
|---|
| 1612 | change, <b style='mso-bidi-font-weight:normal'>enter 120 for </b><b |
|---|
| 1613 | style='mso-bidi-font-weight:normal'><span style='font-family:Courier'>t0</span>.</b></p> |
|---|
| 1614 | |
|---|
| 1615 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1616 | |
|---|
| 1617 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1618 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_40" o:spid="_x0000_i1033" |
|---|
| 1619 | type="#_x0000_t75" style='width:317.25pt;height:171pt;visibility:visible; |
|---|
| 1620 | mso-wrap-style:square'> |
|---|
| 1621 | <v:imagedata src="ParametricFitting_files/image053.png" o:title="" croptop="3272f" |
|---|
| 1622 | cropbottom="7856f" cropleft="3218f" cropright="3436f"/> |
|---|
| 1623 | </v:shape><![endif]--><![if !vml]><img border=0 width=423 height=228 |
|---|
| 1624 | src="ParametricFitting_files/image097.png" v:shapes="Picture_x0020_40"><![endif]></span></p> |
|---|
| 1625 | |
|---|
| 1626 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1627 | |
|---|
| 1628 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1629 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1630 | style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1631 | </span></span></span><![endif]>Optionally, <b style='mso-bidi-font-weight:normal'>press |
|---|
| 1632 | the Fit button</b> to see if the equation can be refined with starting values |
|---|
| 1633 | of zero for the other three parameters.<span style='mso-spacerun:yes'> |
|---|
| 1634 | </span>A fit is obtained and is plotted. </p> |
|---|
| 1635 | |
|---|
| 1636 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1637 | |
|---|
| 1638 | <p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span><span |
|---|
| 1639 | style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape |
|---|
| 1640 | id="Picture_x0020_41" o:spid="_x0000_i1032" type="#_x0000_t75" style='width:317.25pt; |
|---|
| 1641 | height:168.75pt;visibility:visible;mso-wrap-style:square'> |
|---|
| 1642 | <v:imagedata src="ParametricFitting_files/image055.png" o:title="" croptop="3477f" |
|---|
| 1643 | cropbottom="8265f" cropleft="3218f" cropright="3436f"/> |
|---|
| 1644 | </v:shape><![endif]--><![if !vml]><img border=0 width=423 height=225 |
|---|
| 1645 | src="ParametricFitting_files/image098.png" v:shapes="Picture_x0020_41"><![endif]></span></p> |
|---|
| 1646 | |
|---|
| 1647 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1648 | |
|---|
| 1649 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1650 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1651 | style='mso-list:Ignore'>6.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1652 | </span></span></span><![endif]><b style='mso-bidi-font-weight:normal'>Press OK</b> |
|---|
| 1653 | to add this equation to the parametric fits and both equations are refit (and |
|---|
| 1654 | if the label for b had not been changed, applying the constraint that b must be |
|---|
| 1655 | the same for both equations.) </p> |
|---|
| 1656 | |
|---|
| 1657 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1658 | |
|---|
| 1659 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1660 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_32" o:spid="_x0000_i1031" |
|---|
| 1661 | type="#_x0000_t75" style='width:353.25pt;height:300.75pt;visibility:visible; |
|---|
| 1662 | mso-wrap-style:square'> |
|---|
| 1663 | <v:imagedata src="ParametricFitting_files/image057.png" o:title="" croptop="2602f" |
|---|
| 1664 | cropbottom="4943f" cropleft="3024f" cropright="3097f"/> |
|---|
| 1665 | </v:shape><![endif]--><![if !vml]><img border=0 width=471 height=401 |
|---|
| 1666 | src="ParametricFitting_files/image099.png" v:shapes="Picture_x0020_32"><![endif]></span></p> |
|---|
| 1667 | |
|---|
| 1668 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1669 | |
|---|
| 1670 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l1 level1 lfo8'><![if !supportLists]><span |
|---|
| 1671 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1672 | style='mso-list:Ignore'>7.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1673 | </span></span></span><![endif]>We will now try a more complex equation for the |
|---|
| 1674 | same data, where we set the value to <span style='font-family:Courier'>b</span> |
|---|
| 1675 | for temperatures less than <span style='font-family:Courier'>t0</span> and a |
|---|
| 1676 | critical phenomenon formulism [(t-t<sub>0</sub>)<sup>-a</sup>] for temperatures |
|---|
| 1677 | above <span style='font-family:Courier'>t0</span>. To do this we use the Python |
|---|
| 1678 | syntax "<span style='font-family:Courier'>x1 if <span class=SpellE>testexp</span> |
|---|
| 1679 | else x2</span>" which is <span style='font-family:Courier'>x1</span> where |
|---|
| 1680 | the test expression (<span class=SpellE><span style='font-family:Courier'>testexp</span></span> |
|---|
| 1681 | here) is <span style='font-family:Courier'>True</span> and <span |
|---|
| 1682 | style='font-family:Courier'>x2</span> elsewhere. Thus we can use an expression |
|---|
| 1683 | such as:</p> |
|---|
| 1684 | |
|---|
| 1685 | <p class=MsoListParagraphCxSpMiddle><span style='mso-spacerun:yes'> </span></p> |
|---|
| 1686 | |
|---|
| 1687 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: |
|---|
| 1688 | auto'><span class=GramE><span style='font-family:Courier'>b</span></span><span |
|---|
| 1689 | style='font-family:Courier'> if t <= t0 else b + c * (1-(t-t0)**d)<o:p></o:p></span></p> |
|---|
| 1690 | |
|---|
| 1691 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: |
|---|
| 1692 | auto'><span style='font-family:Courier'><o:p> </o:p></span></p> |
|---|
| 1693 | |
|---|
| 1694 | <p class=MsoListParagraphCxSpMiddle>This may be easier to read with the |
|---|
| 1695 | inclusion of extra parentheses (that have no effect on the execution):<span |
|---|
| 1696 | style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin; |
|---|
| 1697 | mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p> |
|---|
| 1698 | |
|---|
| 1699 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1700 | |
|---|
| 1701 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: |
|---|
| 1702 | auto'><span class=GramE><span style='font-family:Courier'>b</span></span><span |
|---|
| 1703 | style='font-family:Courier'> if (t <= t0) else (b + c * (1-(t-t0)**d))<o:p></o:p></span></p> |
|---|
| 1704 | |
|---|
| 1705 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: |
|---|
| 1706 | auto'><span style='font-family:Courier'><o:p> </o:p></span></p> |
|---|
| 1707 | |
|---|
| 1708 | <p class=MsoListParagraphCxSpMiddle>As before, <b style='mso-bidi-font-weight: |
|---|
| 1709 | normal'>add an equation </b>(or edit an existing one), <b style='mso-bidi-font-weight: |
|---|
| 1710 | normal'>set the dependent variable</b> to <span style='font-family:Courier'>0::c</span>, |
|---|
| 1711 | <b style='mso-bidi-font-weight:normal'>enter the expression</b> and <b |
|---|
| 1712 | style='mso-bidi-font-weight:normal'>assign the free and fixed variables</b>, as |
|---|
| 1713 | before. <b style='mso-bidi-font-weight:normal'>Set the exponent and the t0</b> |
|---|
| 1714 | values to initial values of -0.025 and 120, respectively, and turn off their |
|---|
| 1715 | refinement flags. Then <b style='mso-bidi-font-weight:normal'>press Fit.</b><span |
|---|
| 1716 | style='font-family:"Cambria","serif";mso-ascii-theme-font:minor-latin; |
|---|
| 1717 | mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p> |
|---|
| 1718 | |
|---|
| 1719 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1720 | mso-no-proof:yes'><o:p> </o:p></span></p> |
|---|
| 1721 | |
|---|
| 1722 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1723 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_42" o:spid="_x0000_i1030" |
|---|
| 1724 | type="#_x0000_t75" style='width:315.75pt;height:168.75pt;visibility:visible; |
|---|
| 1725 | mso-wrap-style:square'> |
|---|
| 1726 | <v:imagedata src="ParametricFitting_files/image059.png" o:title="" croptop="3691f" |
|---|
| 1727 | cropbottom="7919f" cropleft="2889f" cropright="3536f"/> |
|---|
| 1728 | </v:shape><![endif]--><![if !vml]><img border=0 width=421 height=225 |
|---|
| 1729 | src="ParametricFitting_files/image100.png" v:shapes="Picture_x0020_42"><![endif]></span></p> |
|---|
| 1730 | |
|---|
| 1731 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1732 | |
|---|
| 1733 | <p class=MsoListParagraphCxSpMiddle>Then <b style='mso-bidi-font-weight:normal'>turn |
|---|
| 1734 | refinement of t0 back on</b> and <b style='mso-bidi-font-weight:normal'>press |
|---|
| 1735 | OK</b>. The resulting fit shows the trends quite well: </p> |
|---|
| 1736 | |
|---|
| 1737 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1738 | |
|---|
| 1739 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1740 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_35" o:spid="_x0000_i1029" |
|---|
| 1741 | type="#_x0000_t75" style='width:351pt;height:302.25pt;visibility:visible; |
|---|
| 1742 | mso-wrap-style:square'> |
|---|
| 1743 | <v:imagedata src="ParametricFitting_files/image061.png" o:title="" croptop="2354f" |
|---|
| 1744 | cropbottom="4937f" cropleft="3349f" cropright="3198f"/> |
|---|
| 1745 | </v:shape><![endif]--><![if !vml]><img border=0 width=468 height=403 |
|---|
| 1746 | src="ParametricFitting_files/image101.png" v:shapes="Picture_x0020_35"><![endif]></span></p> |
|---|
| 1747 | |
|---|
| 1748 | <p class=MsoListParagraphCxSpLast><o:p> </o:p></p> |
|---|
| 1749 | |
|---|
| 1750 | <h2>Step 5: Fitting to a Pseudo <span class=SpellE>Var</span></h2> |
|---|
| 1751 | |
|---|
| 1752 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1753 | |
|---|
| 1754 | <p class=MsoNormal>One can fit a function to a pseudo <span class=SpellE>var</span>, |
|---|
| 1755 | just as to a direct variable. If we look at the plot shown in Step 2, it |
|---|
| 1756 | appears there are two regions. We can treat the lower range as a power-law |
|---|
| 1757 | growth, <span class=SpellE>T<sup>k</sup></span>, and the upper region as linear |
|---|
| 1758 | (m*T + b). </p> |
|---|
| 1759 | |
|---|
| 1760 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1761 | |
|---|
| 1762 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span |
|---|
| 1763 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1764 | style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1765 | </span></span></span><![endif]>As before, <b style='mso-bidi-font-weight:normal'>add |
|---|
| 1766 | an equation </b>(or edit an existing one). For the dependent variable, <b |
|---|
| 1767 | style='mso-bidi-font-weight:normal'>select Global</b> to bring up that |
|---|
| 1768 | selection of variables and <b style='mso-bidi-font-weight:normal'>select b/c </b>and |
|---|
| 1769 | <b style='mso-bidi-font-weight:normal'>press OK. </b></p> |
|---|
| 1770 | |
|---|
| 1771 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1772 | |
|---|
| 1773 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span |
|---|
| 1774 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
|---|
| 1775 | style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'> |
|---|
| 1776 | </span></span></span><![endif]>Then <b style='mso-bidi-font-weight:normal'>enter |
|---|
| 1777 | the equation</b> as </p> |
|---|
| 1778 | |
|---|
| 1779 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
|---|
| 1780 | |
|---|
| 1781 | <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: |
|---|
| 1782 | auto'><span style='font-family:Courier;mso-bidi-font-family:"Lucida Grande"; |
|---|
| 1783 | color:black'>(o1 + (a1*T<span class=GramE>)*</span>*k) if T < 120 else (o2 + |
|---|
| 1784 | a2*T</span><span style='font-family:Courier'>)<o:p></o:p></span></p> |
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| 1785 | |
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| 1786 | <p class=MsoListParagraphCxSpMiddle><span style='font-family:"Cambria","serif"; |
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| 1787 | mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p> </o:p></span></p> |
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| 1788 | |
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| 1789 | <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span |
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| 1790 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
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| 1791 | style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'> |
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| 1792 | </span></span></span><![endif]><b style='mso-bidi-font-weight:normal'>Assign |
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| 1793 | the free and fixed variables</b>, as shown below. Note that the exponent (k) |
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| 1794 | and scaling factor (a1<span class=GramE>),</span> cannot start as zero<b |
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| 1795 | style='mso-bidi-font-weight:normal'>. </b></p> |
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| 1796 | |
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| 1797 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
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| 1798 | |
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| 1799 | <p class=MsoListParagraphCxSpLast><span style='mso-fareast-language:EN-US; |
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| 1800 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_44" o:spid="_x0000_i1028" |
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| 1801 | type="#_x0000_t75" style='width:332.25pt;height:186pt;visibility:visible; |
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| 1802 | mso-wrap-style:square'> |
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| 1803 | <v:imagedata src="ParametricFitting_files/image063.png" o:title="" croptop="3190f" |
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| 1804 | cropbottom="8105f" cropleft="3416f" cropright="3246f"/> |
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| 1805 | </v:shape><![endif]--><![if !vml]><img border=0 width=443 height=248 |
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| 1806 | src="ParametricFitting_files/image102.png" v:shapes="Picture_x0020_44"><![endif]></span></p> |
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| 1807 | |
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| 1808 | <p class=MsoNormal><o:p> </o:p></p> |
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| 1809 | |
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| 1810 | <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l3 level1 lfo10'><![if !supportLists]><span |
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| 1811 | style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Times New Roman"'><span |
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| 1812 | style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'> |
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| 1813 | </span></span></span><![endif]><b style='mso-bidi-font-weight:normal'>Press the |
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| 1814 | Fit button</b> and note that all the free variables are adjusted </p> |
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| 1815 | |
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| 1816 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
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| 1817 | |
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| 1818 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
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| 1819 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_45" o:spid="_x0000_i1027" |
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| 1820 | type="#_x0000_t75" style='width:333.75pt;height:189pt;visibility:visible; |
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| 1821 | mso-wrap-style:square'> |
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| 1822 | <v:imagedata src="ParametricFitting_files/image065.png" o:title="" croptop="2815f" |
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| 1823 | cropbottom="7542f" cropleft="3074f" cropright="3244f"/> |
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| 1824 | </v:shape><![endif]--><![if !vml]><img border=0 width=445 height=252 |
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| 1825 | src="ParametricFitting_files/image103.png" v:shapes="Picture_x0020_45"><![endif]></span><span |
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| 1826 | style='mso-spacerun:yes'> </span></p> |
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| 1827 | |
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| 1828 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
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| 1829 | |
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| 1830 | <p class=MsoListParagraphCxSpMiddle>Also, that the plotted result is close to |
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| 1831 | the values from the refinement.</p> |
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| 1832 | |
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| 1833 | <p class=MsoListParagraphCxSpMiddle><o:p> </o:p></p> |
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| 1834 | |
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| 1835 | <p class=MsoListParagraphCxSpMiddle><span style='mso-fareast-language:EN-US; |
|---|
| 1836 | mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_48" o:spid="_x0000_i1026" |
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| 1837 | type="#_x0000_t75" style='width:348.75pt;height:302.25pt;visibility:visible; |
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| 1838 | mso-wrap-style:square'> |
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| 1839 | <v:imagedata src="ParametricFitting_files/image067.png" o:title="" croptop="2231f" |
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| 1840 | cropbottom="5061f" cropleft="3350f" cropright="3415f"/> |
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| 1841 | </v:shape><![endif]--><![if !vml]><img border=0 width=465 height=403 |
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| 1842 | src="ParametricFitting_files/image104.png" v:shapes="Picture_x0020_48"><![endif]></span></p> |
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| 1843 | |
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| 1844 | <p class=MsoListParagraphCxSpLast><o:p> </o:p></p> |
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| 1845 | |
|---|
| 1846 | <p class=MsoNormal style='margin-left:.25in'><o:p> </o:p></p> |
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| 1847 | |
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| 1848 | <p class=MsoNormal>This completes this tutorial on use of Pseudo <span |
|---|
| 1849 | class=SpellE>Vars</span> and parametric fitting. </p> |
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| 1850 | |
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| 1851 | <p class=MsoNormal><o:p> </o:p></p> |
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| 1852 | |
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| 1853 | <h2>Advanced Use: Externally-Defined Functions</h2> |
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| 1854 | |
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| 1855 | <p class=MsoNormal><o:p> </o:p></p> |
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| 1856 | |
|---|
| 1857 | <p class=MsoNormal>The parametric refinement section of GSAS-II is intended to |
|---|
| 1858 | be quite flexible, but for very complex fitting, a user may wish to define a |
|---|
| 1859 | custom Python function. Such a function can be defined in a separate module |
|---|
| 1860 | that will be accessed with a parametric equation (or pseudo var).</p> |
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| 1861 | |
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| 1862 | <p class=MsoNormal><o:p> </o:p></p> |
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| 1863 | |
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| 1864 | <p class=MsoNormal>As an example of that, if we create a file called <span |
|---|
| 1865 | style='font-family:Courier'>fittest.py</span> and locate that file in any |
|---|
| 1866 | directory in the Python path (most simply, in the directory where the project |
|---|
| 1867 | file, <span class=SpellE><span style='font-family:Courier'>SeqTut.gpx</span></span>, |
|---|
| 1868 | is located) and place a routine <span class=SpellE><span style='font-family: |
|---|
| 1869 | Courier'>fitfxn</span></span> in that module:</p> |
|---|
| 1870 | |
|---|
| 1871 | <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'> |
|---|
| 1872 | |
|---|
| 1873 | <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><pre><o:p> </o:p></pre><pre># file fittest.py</pre><pre>def fitfxn(T,o1,o2,a1,a2,k):</pre><pre><span style='mso-spacerun:yes'> </span>if T < 120:</pre><pre><span style='mso-spacerun:yes'> </span>return o1 + (a1*T)**k</pre><pre><span style='mso-spacerun:yes'> </span>else:</pre><pre><span style='mso-spacerun:yes'> </span>return o2 + a2*T</pre></blockquote> |
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| 1874 | |
|---|
| 1875 | </blockquote> |
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| 1876 | |
|---|
| 1877 | <p class=MsoNormal>The function <span class=SpellE><span style='font-family: |
|---|
| 1878 | Courier'>fittest.fitfxn</span></span> can then be used to duplicate the |
|---|
| 1879 | previous example: </p> |
|---|
| 1880 | |
|---|
| 1881 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1882 | |
|---|
| 1883 | <p class=MsoNormal><span style='mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape |
|---|
| 1884 | id="Picture_x0020_47" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:315pt; |
|---|
| 1885 | height:185.25pt;visibility:visible;mso-wrap-style:square'> |
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| 1886 | <v:imagedata src="ParametricFitting_files/image070.png" o:title="" croptop="3393f" |
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| 1887 | cropbottom="7847f" cropleft="3471f" cropright="3290f"/> |
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| 1888 | </v:shape><![endif]--><![if !vml]><img border=0 width=420 height=247 |
|---|
| 1889 | src="ParametricFitting_files/image105.png" v:shapes="Picture_x0020_47"><![endif]></span></p> |
|---|
| 1890 | |
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| 1891 | <p class=MsoNormal><o:p> </o:p></p> |
|---|
| 1892 | |
|---|
| 1893 | <p class=MsoNormal>Note that module <span style='font-family:Courier'>fittest</span> |
|---|
| 1894 | is automatically located and loaded, provided it is found in the path. No |
|---|
| 1895 | changes to the standard distributed GSAS-II code are needed. Care to avoid use |
|---|
| 1896 | of a module name already found in GSAS-II or Python is wise.</p> |
|---|
| 1897 | |
|---|
| 1898 | </div> |
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| 1899 | |
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| 1900 | </body> |
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| 1901 | |
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| 1902 | </html> |
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