(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 962335, 20031]*) (*NotebookOutlinePosition[ 963007, 20054]*) (* CellTagsIndexPosition[ 962963, 20050]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ BCC Alloys in the Bragg-Williams Approximation\ \>", "Title"], Cell["The Model", "Subtitle"], Cell[CellGroupData[{ Cell["Samuel M. Allen\tApril 19, 2006", "Subsection"], Cell[BoxData[{ \(Clear["\"]\), "\[IndentingNewLine]", \(<< Graphics`Legend`\)}], "Input"], Cell["Define values for Agogadro and Boltzmann constants", "Text"], Cell[BoxData[{ \(\(N0 = 6.02214\[Times]10\^23;\)\), "\[IndentingNewLine]", \(\(kB = 1.3806503\[Times]10\^\(-23\);\)\)}], "Input"], Cell[TextData[StyleBox["Bragg-Williams model for b.c.c. alloys with \ first-neighbor pairwise interactions. ", "Subsection"]], "Subsubtitle"], Cell[TextData[StyleBox["Order parameters are the atom fraction of B, XB, and \ the long-range order parameter, \[Eta] (see KoM Fig. 17.4 and Eqs. \ 17.7\[Dash]8 for definitions).", "Subsection", FontSize->16, FontWeight->"Plain", Background->None, FontVariations->{"CompatibilityType"->0}]], "Text"], Cell["\<\ Enter expressions for internal energy,entropy, and free energy as \ in KoM Eqs. 17.13\[Dash]14. In the formuation below \[CapitalDelta]F has \ units of energy/mole\ \>", "Text"], Cell[BoxData[{ \(\(\[CapitalDelta]U[ XB_, \[Eta]_] := \(-4\)\ N0\ W\ \((XB \((1 - XB)\) + \[Eta]\^2)\);\)\), "\[IndentingNewLine]", \(\(\[CapitalDelta]S\ [ XB_, \[Eta]_] := \(-N0\)\ kB \((\((XB + \[Eta])\) Log[XB + \[Eta]] + \((XB - \[Eta])\) Log[XB - \[Eta]] + \((1 - XB + \[Eta])\) Log[1 - XB + \[Eta]] + \((1 - XB - \[Eta])\) Log[1 - XB - \[Eta]])\);\)\), "\[IndentingNewLine]", \(\[CapitalDelta]F[XB_, \[Eta]_, T_] := \[CapitalDelta]U[XB, \[Eta]] - T\ \[CapitalDelta]S[XB, \[Eta]]\)}], "Input"], Cell["\<\ Look first at the possibility for order-disorder transitions by \ calculating the order-disorder transition temperature at XB = 0.5:\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(SecDerivEta = \[PartialD]\_\(\[Eta], \[Eta]\)\[CapitalDelta]F[ XB, \[Eta], T]\), "\[IndentingNewLine]", \(Fetaeta = Limit[SecDerivEta, \[Eta] -> 0]\), "\[IndentingNewLine]", \(Fetaeta /. XB \[Rule] 0.5\)}], "Input"], Cell[BoxData[ \(\(-4.817712`*^24\)\ W + 8.314469397642002`\ T\ \((1\/\(1 - XB - \[Eta]\) + 1\/\(XB - \[Eta]\) + 1\/\(1 - XB + \[Eta]\) + 1\/\(XB + \[Eta]\))\)\)], "Output"], Cell[BoxData[ \(\(-\(\(16.628938795284004`\ T\)\/\((\(-1.`\) + XB)\)\^2\)\) - \ \(4.817712`*^24\ W\)\/\((\(-1.`\) + XB)\)\^2 + \(16.628938795284004`\ T\)\/\(\ \((\(-1.`\) + XB)\)\^2\ XB\) + \(9.635424`*^24\ W\ XB\)\/\((\(-1.`\) + \ XB)\)\^2 - \(4.817712`*^24\ W\ XB\^2\)\/\((\(-1.`\) + XB)\)\^2\)], "Output"], Cell[BoxData[ \(66.51575518113602`\ T - 4.817712000000002`*^24\ W\)], "Output"] }, Open ]], Cell[TextData[{ "We see that there will only be order-disorder transitions (i.e., a \ positive-temperature solution to \n\tLimit", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ RowBox[{"(", FormBox[\(F\_\[Eta]\[Eta]\), "TraditionalForm"], ")"}], \(\[Eta] \[Rule] 0\)], "=", "0"}], TraditionalForm]]], " \nwhen W>0. With this result, we can explore the simpler case of phase \ equilibria with miscibility gaps by considering only W<0 and setting \[Eta]=0 \ in the free energy model. (In this way, we recover the regular solution \ model.)" }], "Text"], Cell["Spinodal Decomposition", "Subtitle"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Calculation of Miscibility Gap and Chemical Spinodal in the \ Bragg-Williams Approximation\ \>", "Subsection"], Cell["\<\ We'll need to make calculations and draw graphs so I begin by \ picking a suitable negative value for W\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(W = \(-1000\)\[Times]kB\)], "Input"], Cell[BoxData[ \(\(-1.3806503000000003`*^-20\)\)], "Output"] }, Open ]], Cell["Begin by evaluating the spinodal temperature at XB = 0.5", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\[PartialD]\_\(XB, XB\)\[CapitalDelta]F[XB, 0, T]\), "\[IndentingNewLine]", \(FXBXB = Limit[\[PartialD]\_\(XB, XB\)\[CapitalDelta]F[XB, 0, T], XB -> 0.5]\)}], "Input"], Cell[BoxData[ \(\(-66515.755181136`\) + 8.314469397642002`\ T\ \((2\/\(1 - XB\) + 2\/XB)\)\)], "Output"], Cell[BoxData[ \(\(-66515.755181136`\) + 66.51575518113602`\ T\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[FXBXB \[Equal] 0, T]\)], "Input"], Cell[BoxData[ \({{T \[Rule] 999.9999999999998`}}\)], "Output"] }, Open ]], Cell["\<\ So the top of the miscibility gap (remember, this free-energy model \ is symmetric about XB = 0.5) is at T = 1000 K.\ \>", "Text"], Cell["\<\ Plot \[CapitalDelta]F vs. XB for T = 800, 900, 1000, 1100 K\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{\[CapitalDelta]F[XB, 0, 800], \[CapitalDelta]F[XB, 0, 900], \[CapitalDelta]F[XB, 0, 1000], \[CapitalDelta]F[XB, 0, 1100]}, {XB, 0.0, 1}, AxesLabel \[Rule] {XB, \[CapitalDelta]F}, PlotStyle \[Rule] {{Hue[ .3]}, {Hue[ .5]}, {Hue[ .7]}, {Hue[ .9]}}, PlotLegend \[Rule] {"\<800 K\>", "\<900 K\>", "\<1000 K\>", "\<1100 \ K\>"}, LegendPosition \[Rule] {1, \(-1\)}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .64023 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 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r .67714 .48899 .68844 .49456 .71312 .48857 .70203 .4827 Metetra .547 .853 .998 r .70203 .4827 .71312 .48857 .7381 .48324 .72719 .47699 Metetra .53 .84 .999 r .72719 .47699 .7381 .48324 .76338 .4787 .75266 .47198 Metetra .508 .823 1 r .75266 .47198 .76338 .4787 .78902 .47514 .77847 .46784 Metetra .481 .8 1 r .77847 .46784 .78902 .47514 .81506 .47282 .80467 .46486 Metetra .444 .77 .997 r .80467 .46486 .81506 .47282 .84159 .47216 .83133 .46341 Metetra .39 .726 .99 r .83133 .46341 .84159 .47216 .86871 .4738 .85856 .46411 Metetra .309 .66 .972 r .85856 .46411 .86871 .4738 .89661 .4788 .88653 .46799 Metetra .169 .547 .928 r .88653 .46799 .89661 .4788 .92565 .48934 .91559 .47716 Metetra 0 .289 .775 r .91559 .47716 .92565 .48934 .95686 .51231 .94672 .49827 Metetra .856 .769 .765 r .33283 .66099 .34694 .67243 .37073 .63816 .35678 .62829 Metetra .823 .799 .829 r .35678 .62829 .37073 .63816 .39393 .61504 .38013 .60627 Metetra .797 .823 .872 r .38013 .60627 .39393 .61504 .41694 .59686 .4033 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.47671 .70203 .4827 .72719 .47699 .71606 .47061 Metetra .542 .846 .999 r .71606 .47061 .72719 .47699 .75266 .47198 .74171 .46511 Metetra .524 .83 1 r .74171 .46511 .75266 .47198 .77847 .46784 .7677 .4604 Metetra .501 .81 1 r .7677 .4604 .77847 .46784 .80467 .46486 .79406 .45673 Metetra .468 .782 .998 r .79406 .45673 .80467 .46486 .83133 .46341 .82086 .45447 Metetra .42 .742 .993 r .82086 .45447 .83133 .46341 .85856 .46411 .84821 .45422 Metetra .346 .682 .979 r .84821 .45422 .85856 .46411 .88653 .46799 .87626 .45698 Metetra .215 .578 .942 r .87626 .45698 .88653 .46799 .91559 .47716 .90534 .46476 Metetra 0 .333 .806 r .90534 .46476 .91559 .47716 .94672 .49827 .93639 .48398 Metetra .85 .77 .773 r .3185 .64937 .33283 .66099 .35678 .62829 .34259 .61825 Metetra .816 .801 .838 r .34259 .61825 .35678 .62829 .38013 .60627 .36608 .59735 Metetra .788 .824 .88 r .36608 .59735 .38013 .60627 .4033 .58896 .3894 .58091 Metetra .761 .842 .913 r .3894 .58091 .4033 .58896 .42644 .57447 .41271 .56713 Metetra .736 .856 .937 r .41271 .56713 .42644 .57447 .44964 .56186 .43607 .5551 Metetra .712 .867 .955 r .43607 .5551 .44964 .56186 .47294 .5506 .45954 .5443 Metetra .689 .874 .968 r .45954 .5443 .47294 .5506 .49637 .54033 .48316 .53439 Metetra .669 .879 .977 r .48316 .53439 .49637 .54033 .51995 .53081 .50694 .52514 Metetra .65 .882 .983 r .50694 .52514 .51995 .53081 .54371 .52187 .53089 .51638 Metetra .634 .883 .987 r .53089 .51638 .54371 .52187 .56766 .51341 .55505 .50802 Metetra .619 .883 .989 r .55505 .50802 .56766 .51341 .59182 .50535 .57941 .49997 Metetra .607 .881 .991 r .57941 .49997 .59182 .50535 .61619 .49765 .60399 .49221 Metetra .596 .878 .992 r .60399 .49221 .61619 .49765 .64078 .49029 .62879 .4847 Metetra .586 .874 .994 r .62879 .4847 .64078 .49029 .66561 .48329 .65383 .47748 Metetra .577 .869 .995 r .65383 .47748 .66561 .48329 .6907 .47671 .67912 .47058 Metetra .567 .861 .996 r .67912 .47058 .6907 .47671 .71606 .47061 .70468 .46408 Metetra .555 .851 .998 r .70468 .46408 .71606 .47061 .74171 .46511 .73053 .4581 Metetra .54 .837 .999 r .73053 .4581 .74171 .46511 .7677 .4604 .7567 .4528 Metetra .52 .819 .999 r .7567 .4528 .7677 .4604 .79406 .45673 .78323 .44843 Metetra .492 .793 .999 r .78323 .44843 .79406 .45673 .82086 .45447 .81018 .44536 Metetra .449 .757 .995 r .81018 .44536 .82086 .45447 .84821 .45422 .83764 .44414 Metetra .381 .703 .984 r .83764 .44414 .84821 .45422 .87626 .45698 .86578 .44575 Metetra .26 .607 .955 r .86578 .44575 .87626 .45698 .90534 .46476 .8949 .45212 Metetra 0 .377 .836 r .8949 .45212 .90534 .46476 .93639 .48398 .92587 .46942 Metetra .845 .772 .781 r .30393 .63757 .3185 .64937 .34259 .61825 .32815 .60803 Metetra .807 .802 .847 r .32815 .60803 .34259 .61825 .36608 .59735 .35178 .58826 Metetra .778 .824 .889 r .35178 .58826 .36608 .59735 .3894 .58091 .37524 .57271 Metetra .75 .842 .92 r .37524 .57271 .3894 .58091 .41271 .56713 .3987 .55965 Metetra .724 .855 .943 r .3987 .55965 .41271 .56713 .43607 .5551 .42223 .5482 Metetra .7 .865 .96 r .42223 .5482 .43607 .5551 .45954 .5443 .44588 .53787 Metetra .678 .873 .972 r .44588 .53787 .45954 .5443 .48316 .53439 .46968 .52833 Metetra .659 .877 .98 r .46968 .52833 .48316 .53439 .50694 .52514 .49365 .51935 Metetra .642 .88 .985 r .49365 .51935 .50694 .52514 .53089 .51638 .5178 .51078 Metetra .628 .882 .988 r .5178 .51078 .53089 .51638 .55505 .50802 .54216 .50251 Metetra .616 .882 .99 r .54216 .50251 .55505 .50802 .57941 .49997 .56673 .49448 Metetra .606 .881 .991 r .56673 .49448 .57941 .49997 .60399 .49221 .59152 .48665 Metetra .597 .879 .992 r .59152 .48665 .60399 .49221 .62879 .4847 .61654 .47899 Metetra .59 .876 .993 r .61654 .47899 .62879 .4847 .65383 .47748 .64179 .47154 Metetra .583 .871 .994 r .64179 .47154 .65383 .47748 .67912 .47058 .66729 .46432 Metetra .576 .864 .996 r .66729 .46432 .67912 .47058 .70468 .46408 .69306 .45742 Metetra .567 .856 .997 r .69306 .45742 .70468 .46408 .73053 .4581 .7191 .45093 Metetra .555 .843 .998 r .7191 .45093 .73053 .4581 .7567 .4528 .74546 .44503 Metetra .539 .827 .999 r .74546 .44503 .7567 .4528 .78323 .44843 .77216 .43996 Metetra .514 .804 .998 r .77216 .43996 .78323 .44843 .81018 .44536 .79927 .43605 Metetra .477 .772 .996 r .79927 .43605 .81018 .44536 .83764 .44414 .82687 .43385 Metetra .416 .722 .988 r .82687 .43385 .83764 .44414 .86578 .44575 .8551 .43429 Metetra .305 .635 .964 r .8551 .43429 .86578 .44575 .8949 .45212 .88425 .43923 Metetra .028 .421 .863 r .88425 .43923 .8949 .45212 .92587 .46942 .91516 .4546 Metetra .838 .774 .79 r .28911 .62556 .30393 .63757 .32815 .60803 .31346 .59763 Metetra .798 .803 .856 r .31346 .59763 .32815 .60803 .35178 .58826 .33721 .579 Metetra .766 .825 .898 r .33721 .579 .35178 .58826 .37524 .57271 .36082 .56435 Metetra .738 .841 .928 r .36082 .56435 .37524 .57271 .3987 .55965 .38442 .55202 Metetra .712 .854 .95 r .38442 .55202 .3987 .55965 .42223 .5482 .40811 .54116 Metetra .688 .864 .965 r .40811 .54116 .42223 .5482 .44588 .53787 .43193 .53131 Metetra .667 .87 .976 r .43193 .53131 .44588 .53787 .46968 .52833 .45591 .52214 Metetra .649 .875 .982 r .45591 .52214 .46968 .52833 .49365 .51935 .48007 .51343 Metetra .634 .878 .987 r .48007 .51343 .49365 .51935 .5178 .51078 .50443 .50505 Metetra .622 .88 .989 r .50443 .50505 .5178 .51078 .54216 .50251 .529 .49688 Metetra .612 .881 .991 r .529 .49688 .54216 .50251 .56673 .49448 .55378 .48887 Metetra .604 .88 .991 r .55378 .48887 .56673 .49448 .59152 .48665 .57878 .48096 Metetra .599 .879 .992 r .57878 .48096 .59152 .48665 .61654 .47899 .60401 .47316 Metetra .594 .877 .993 r .60401 .47316 .61654 .47899 .64179 .47154 .62948 .46547 Metetra .59 .873 .994 r .62948 .46547 .64179 .47154 .66729 .46432 .6552 .45793 Metetra .585 .868 .995 r .6552 .45793 .66729 .46432 .69306 .45742 .68118 .45061 Metetra .579 .86 .996 r .68118 .45061 .69306 .45742 .7191 .45093 .70743 .44361 Metetra .57 .849 .997 r .70743 .44361 .7191 .45093 .74546 .44503 .73398 .4371 Metetra .557 .835 .997 r .73398 .4371 .74546 .44503 .77216 .43996 .76086 .4313 Metetra .536 .814 .997 r .76086 .4313 .77216 .43996 .79927 .43605 .78813 .42654 Metetra .503 .785 .996 r .78813 .42654 .79927 .43605 .82687 .43385 .81587 .42335 Metetra .449 .74 .99 r .81587 .42335 .82687 .43385 .8551 .43429 .8442 .42261 Metetra .348 .661 .972 r .8442 .42261 .8551 .43429 .88425 .43923 .8734 .4261 Metetra .087 .464 .888 r .8734 .4261 .88425 .43923 .91516 .4546 .90425 .43951 Metetra .831 .775 .799 r .27406 .61336 .28911 .62556 .31346 .59763 .29851 .58705 Metetra .788 .804 .866 r .29851 .58705 .31346 .59763 .33721 .579 .32238 .56958 Metetra .754 .825 .907 r .32238 .56958 .33721 .579 .36082 .56435 .34611 .55584 Metetra .724 .84 .936 r .34611 .55584 .36082 .56435 .38442 .55202 .36986 .54424 Metetra .698 .852 .956 r .36986 .54424 .38442 .55202 .40811 .54116 .39371 .53399 Metetra .675 .861 .97 r .39371 .53399 .40811 .54116 .43193 .53131 .4177 .52461 Metetra .655 .868 .979 r .4177 .52461 .43193 .53131 .45591 .52214 .44186 .51582 Metetra .639 .873 .985 r .44186 .51582 .45591 .52214 .48007 .51343 .46621 .5074 Metetra .626 .876 .988 r .46621 .5074 .48007 .51343 .50443 .50505 .49077 .4992 Metetra .616 .878 .99 r .49077 .4992 .50443 .50505 .529 .49688 .51554 .49113 Metetra .608 .88 .991 r .51554 .49113 .529 .49688 .55378 .48887 .54053 .48313 Metetra .603 .88 .992 r .54053 .48313 .55378 .48887 .57878 .48096 .56576 .47516 Metetra .6 .88 .992 r .56576 .47516 .57878 .48096 .60401 .47316 .59121 .46719 Metetra .598 .878 .992 r .59121 .46719 .60401 .47316 .62948 .46547 .6169 .45926 Metetra .596 .875 .993 r .6169 .45926 .62948 .46547 .6552 .45793 .64284 .45139 Metetra .594 .871 .993 r .64284 .45139 .6552 .45793 .68118 .45061 .66903 .44364 Metetra .59 .864 .994 r .66903 .44364 .68118 .45061 .70743 .44361 .6955 .43613 Metetra .584 .855 .995 r .6955 .43613 .70743 .44361 .73398 .4371 .72225 .429 Metetra .574 .842 .995 r .72225 .429 .73398 .4371 .76086 .4313 .74932 .42246 Metetra .557 .823 .995 r .74932 .42246 .76086 .4313 .78813 .42654 .77676 .41684 Metetra .529 .797 .994 r .77676 .41684 .78813 .42654 .81587 .42335 .80464 .41263 Metetra .481 .756 .991 r .80464 .41263 .81587 .42335 .8442 .42261 .83308 .41068 Metetra .389 .686 .978 r .83308 .41068 .8442 .42261 .8734 .4261 .86234 .4127 Metetra .147 .505 .91 r .86234 .4127 .8734 .4261 .90425 .43951 .89314 .42414 Metetra .823 .777 .809 r .25875 .60095 .27406 .61336 .29851 .58705 .28329 .57628 Metetra .776 .805 .876 r .28329 .57628 .29851 .58705 .32238 .56958 .30727 .55997 Metetra .74 .824 .916 r .30727 .55997 .32238 .56958 .34611 .55584 .33112 .54716 Metetra .71 .839 .944 r .33112 .54716 .34611 .55584 .36986 .54424 .35501 .5363 Metetra .683 .85 .963 r .35501 .5363 .36986 .54424 .39371 .53399 .37901 .52666 Metetra .661 .859 .975 r .37901 .52666 .39371 .53399 .4177 .52461 .40317 .51778 Metetra .642 .865 .983 r .40317 .51778 .4177 .52461 .44186 .51582 .42751 .50936 Metetra .628 .87 .988 r .42751 .50936 .44186 .51582 .46621 .5074 .45205 .50122 Metetra .617 .873 .99 r .45205 .50122 .46621 .5074 .49077 .4992 .47681 .49322 Metetra .609 .876 .991 r .47681 .49322 .49077 .4992 .51554 .49113 .50179 .48525 Metetra .604 .878 .992 r .50179 .48525 .51554 .49113 .54053 .48313 .527 .47727 Metetra .602 .88 .992 r .527 .47727 .54053 .48313 .56576 .47516 .55244 .46922 Metetra .601 .88 .992 r .55244 .46922 .56576 .47516 .59121 .46719 .57812 .46109 Metetra .602 .88 .992 r .57812 .46109 .59121 .46719 .6169 .45926 .60404 .45291 Metetra .602 .878 .992 r .60404 .45291 .6169 .45926 .64284 .45139 .6302 .4447 Metetra .603 .874 .992 r .6302 .4447 .64284 .45139 .66903 .44364 .65662 .43652 Metetra .602 .868 .993 r .65662 .43652 .66903 .44364 .6955 .43613 .6833 .42848 Metetra .598 .86 .993 r .6833 .42848 .6955 .43613 .72225 .429 .71026 .42071 Metetra .591 .848 .993 r .71026 .42071 .72225 .429 .74932 .42246 .73752 .41342 Metetra .577 .832 .993 r .73752 .41342 .74932 .42246 .77676 .41684 .76514 .40692 Metetra .553 .808 .993 r .76514 .40692 .77676 .41684 .80464 .41263 .79317 .40168 Metetra .511 .771 .99 r .79317 .40168 .80464 .41263 .83308 .41068 .82173 .39852 Metetra .429 .708 .981 r .82173 .39852 .83308 .41068 .86234 .4127 .85106 .39905 Metetra .206 .546 .929 r .85106 .39905 .86234 .4127 .89314 .42414 .88182 .40847 Metetra .813 .778 .82 r .24318 .58833 .25875 .60095 .28329 .57628 .2678 .56532 Metetra .763 .805 .886 r .2678 .56532 .28329 .57628 .30727 .55997 .29187 .55019 Metetra .725 .823 .926 r .29187 .55019 .30727 .55997 .33112 .54716 .31584 .5383 Metetra .694 .837 .952 r .31584 .5383 .33112 .54716 .35501 .5363 .33986 .52821 Metetra .668 .847 .969 r .33986 .52821 .35501 .5363 .37901 .52666 .36401 .51918 Metetra .646 .855 .98 r .36401 .51918 .37901 .52666 .40317 .51778 .38833 .51079 Metetra .629 .862 .986 r .38833 .51079 .40317 .51778 .42751 .50936 .41284 .50277 Metetra .617 .867 .99 r .41284 .50277 .42751 .50936 .45205 .50122 .43758 .49492 Metetra .608 .871 .992 r .43758 .49492 .45205 .50122 .47681 .49322 .46253 .48711 Metetra .603 .874 .992 r .46253 .48711 .47681 .49322 .50179 .48525 .48772 .47924 Metetra .601 .877 .992 r .48772 .47924 .50179 .48525 .527 .47727 .51315 .47127 Metetra .601 .879 .992 r .51315 .47127 .527 .47727 .55244 .46922 .53882 .46314 Metetra .602 .881 .992 r .53882 .46314 .55244 .46922 .57812 .46109 .56473 .45486 Metetra .605 .881 .991 r .56473 .45486 .57812 .46109 .60404 .45291 .59088 .44642 Metetra .608 .88 .991 r .59088 .44642 .60404 .45291 .6302 .4447 .61727 .43786 Metetra .611 .877 .991 r .61727 .43786 .6302 .4447 .65662 .43652 .64392 .42924 Metetra .612 .872 .991 r .64392 .42924 .65662 .43652 .6833 .42848 .67082 .42066 Metetra .611 .865 .991 r .67082 .42066 .6833 .42848 .71026 .42071 .698 .41224 Metetra .607 .854 .991 r .698 .41224 .71026 .42071 .73752 .41342 .72547 .40419 Metetra .596 .839 .99 r .72547 .40419 .73752 .41342 .76514 .40692 .75327 .39679 Metetra .577 .817 .99 r .75327 .39679 .76514 .40692 .79317 .40168 .78146 .39051 Metetra .54 .785 .988 r .78146 .39051 .79317 .40168 .82173 .39852 .81015 .3861 Metetra .467 .728 .983 r .81015 .3861 .82173 .39852 .85106 .39905 .83955 .38512 Metetra .264 .583 .945 r .83955 .38512 .85106 .39905 .88182 .40847 .87029 .39251 Metetra .803 .779 .831 r .22734 .5755 .24318 .58833 .2678 .56532 .25202 .55416 Metetra .749 .804 .897 r .25202 .55416 .2678 .56532 .29187 .55019 .27618 .54022 Metetra .709 .822 .935 r .27618 .54022 .29187 .55019 .31584 .5383 .30026 .52928 Metetra .677 .834 .959 r .30026 .52928 .31584 .5383 .33986 .52821 .3244 .51995 Metetra .651 .844 .974 r .3244 .51995 .33986 .52821 .36401 .51918 .34869 .51155 Metetra .63 .852 .984 r .34869 .51155 .36401 .51918 .38833 .51079 .37317 .50366 Metetra .615 .858 .989 r .37317 .50366 .38833 .51079 .41284 .50277 .39786 .49603 Metetra .605 .863 .992 r .39786 .49603 .41284 .50277 .43758 .49492 .42278 .48847 Metetra .599 .868 .993 r .42278 .48847 .43758 .49492 .46253 .48711 .44794 .48086 Metetra .596 .872 .993 r .44794 .48086 .46253 .48711 .48772 .47924 .47334 .4731 Metetra .597 .876 .993 r .47334 .4731 .48772 .47924 .51315 .47127 .49899 .46513 Metetra .599 .879 .992 r .49899 .46513 .51315 .47127 .53882 .46314 .52489 .45693 Metetra .604 .881 .991 r .52489 .45693 .53882 .46314 .56473 .45486 .55103 .44847 Metetra .609 .882 .991 r .55103 .44847 .56473 .45486 .59088 .44642 .57741 .43977 Metetra .614 .882 .99 r .57741 .43977 .59088 .44642 .61727 .43786 .60405 .43086 Metetra .619 .88 .99 r .60405 .43086 .61727 .43786 .64392 .42924 .63093 .42179 Metetra .623 .876 .989 r .63093 .42179 .64392 .42924 .67082 .42066 .65806 .41265 Metetra .624 .869 .988 r .65806 .41265 .67082 .42066 .698 .41224 .68546 .40358 Metetra .622 .86 .988 r .68546 .40358 .698 .41224 .72547 .40419 .71314 .39475 Metetra .615 .846 .987 r .71314 .39475 .72547 .40419 .75327 .39679 .74114 .38644 Metetra .599 .827 .987 r .74114 .38644 .75327 .39679 .78146 .39051 .7695 .37909 Metetra .567 .797 .986 r .7695 .37909 .78146 .39051 .81015 .3861 .79832 .37342 Metetra .503 .747 .983 r .79832 .37342 .81015 .3861 .83955 .38512 .82781 .37091 Metetra .32 .619 .956 r .82781 .37091 .83955 .38512 .87029 .39251 .85853 .37625 Metetra .791 .78 .843 r .21123 .56244 .22734 .5755 .25202 .55416 .23596 .54279 Metetra .732 .803 .908 r .23596 .54279 .25202 .55416 .27618 .54022 .26019 .53006 Metetra .691 .819 .944 r .26019 .53006 .27618 .54022 .30026 .52928 .28436 .52007 Metetra .658 .831 .966 r .28436 .52007 .30026 .52928 .3244 .51995 .30862 .51152 Metetra .633 .84 .98 r .30862 .51152 .3244 .51995 .34869 .51155 .33305 .50375 Metetra .614 .847 .988 r .33305 .50375 .34869 .51155 .37317 .50366 .35769 .49638 Metetra .601 .854 .992 r .35769 .49638 .37317 .50366 .39786 .49603 .38255 .48915 Metetra .593 .86 .994 r .38255 .48915 .39786 .49603 .42278 .48847 .40766 .48188 Metetra .59 .865 .994 r .40766 .48188 .42278 .48847 .44794 .48086 .43302 .47447 Metetra .59 .87 .994 r .43302 .47447 .44794 .48086 .47334 .4731 .45863 .46681 Metetra .593 .875 .993 r .45863 .46681 .47334 .4731 .49899 .46513 .4845 .45886 Metetra .598 .878 .992 r .4845 .45886 .49899 .46513 .52489 .45693 .51063 .45057 Metetra .605 .881 .991 r .51063 .45057 .52489 .45693 .55103 .44847 .53701 .44194 Metetra .613 .883 .99 r .53701 .44194 .55103 .44847 .57741 .43977 .56364 .43297 Metetra .62 .884 .989 r .56364 .43297 .57741 .43977 .60405 .43086 .59051 .4237 Metetra .627 .882 .988 r .59051 .4237 .60405 .43086 .63093 .42179 .61764 .41417 Metetra .633 .879 .987 r .61764 .41417 .63093 .42179 .65806 .41265 .64501 .40447 Metetra .637 .874 .986 r .64501 .40447 .65806 .41265 .68546 .40358 .67264 .39471 Metetra .637 .865 .984 r .67264 .39471 .68546 .40358 .71314 .39475 .70054 .38509 Metetra .632 .853 .983 r .70054 .38509 .71314 .39475 .74114 .38644 .72874 .37586 Metetra .62 .835 .983 r .72874 .37586 .74114 .38644 .7695 .37909 .75728 .36742 Metetra .593 .808 .982 r .75728 .36742 .7695 .37909 .79832 .37342 .78625 .36047 Metetra .537 .764 .981 r .78625 .36047 .79832 .37342 .82781 .37091 .81583 .3564 Metetra .374 .651 .965 r .81583 .3564 .82781 .37091 .85853 .37625 .84655 .35967 Metetra .777 .78 .855 r .19484 .54916 .21123 .56244 .23596 .54279 .2196 .53121 Metetra .714 .802 .919 r .2196 .53121 .23596 .54279 .26019 .53006 .24389 .5197 Metetra .67 .816 .953 r .24389 .5197 .26019 .53006 .28436 .52007 .26814 .51068 Metetra .638 .826 .973 r .26814 .51068 .28436 .52007 .30862 .51152 .29252 .50292 Metetra .613 .835 .985 r .29252 .50292 .30862 .51152 .33305 .50375 .31708 .49579 Metetra .596 .843 .991 r .31708 .49579 .33305 .50375 .35769 .49638 .34186 .48893 Metetra .586 .849 .994 r .34186 .48893 .35769 .49638 .38255 .48915 .3669 .48211 Metetra .58 .856 .995 r .3669 .48211 .38255 .48915 .40766 .48188 .39219 .47515 Metetra .58 .862 .995 r .39219 .47515 .40766 .48188 .43302 .47447 .41775 .46793 Metetra .583 .868 .995 r .41775 .46793 .43302 .47447 .45863 .46681 .44358 .46038 Metetra .589 .873 .994 r .44358 .46038 .45863 .46681 .4845 .45886 .46968 .45243 Metetra .597 .878 .992 r .46968 .45243 .4845 .45886 .51063 .45057 .49604 .44406 Metetra .606 .882 .991 r .49604 .44406 .51063 .45057 .53701 .44194 .52266 .43525 Metetra .616 .884 .99 r .52266 .43525 .53701 .44194 .56364 .43297 .54953 .42601 Metetra .626 .885 .988 r .54953 .42601 .56364 .43297 .59051 .4237 .57666 .41637 Metetra .635 .885 .986 r .57666 .41637 .59051 .4237 .61764 .41417 .60403 .40637 Metetra .643 .882 .985 r .60403 .40637 .61764 .41417 .64501 .40447 .63165 .39609 Metetra .649 .877 .983 r .63165 .39609 .64501 .40447 .67264 .39471 .65952 .38565 Metetra .651 .87 .981 r .65952 .38565 .67264 .39471 .70054 .38509 .68765 .37521 Metetra .649 .858 .979 r .68765 .37521 .70054 .38509 .72874 .37586 .71606 .36503 Metetra .639 .842 .978 r .71606 .36503 .72874 .37586 .75728 .36742 .74479 .35549 Metetra .618 .818 .977 r .74479 .35549 .75728 .36742 .78625 .36047 .77392 .34725 Metetra .57 .779 .977 r .77392 .34725 .78625 .36047 .81583 .3564 .8036 .3416 Metetra .426 .681 .97 r .8036 .3416 .81583 .3564 .84655 .35967 .83434 .34277 Metetra .761 .78 .868 r .17817 .53565 .19484 .54916 .2196 .53121 .20293 .51942 Metetra .693 .799 .93 r .20293 .51942 .2196 .53121 .24389 .5197 .22726 .50914 Metetra .648 .811 .962 r .22726 .50914 .24389 .5197 .26814 .51068 .2516 .5011 Metetra .615 .821 .979 r .2516 .5011 .26814 .51068 .29252 .50292 .27607 .49413 Metetra .592 .829 .989 r .27607 .49413 .29252 .50292 .31708 .49579 .30076 .48766 Metetra .578 .837 .994 r .30076 .48766 .31708 .49579 .34186 .48893 .32569 .48133 Metetra .57 .844 .996 r .32569 .48133 .34186 .48893 .3669 .48211 .35089 .47491 Metetra .568 .852 .997 r .35089 .47491 .3669 .48211 .39219 .47515 .37637 .46825 Metetra .57 .859 .996 r .37637 .46825 .39219 .47515 .41775 .46793 .40213 .46124 Metetra .576 .866 .995 r .40213 .46124 .41775 .46793 .44358 .46038 .42817 .45379 Metetra .585 .872 .994 r .42817 .45379 .44358 .46038 .46968 .45243 .4545 .44585 Metetra .596 .878 .993 r .4545 .44585 .46968 .45243 .49604 .44406 .4811 .4374 Metetra .607 .882 .991 r .4811 .4374 .49604 .44406 .52266 .43525 .50796 .42841 Metetra .62 .885 .989 r .50796 .42841 .52266 .43525 .54953 .42601 .53509 .41889 Metetra .632 .887 .987 r .53509 .41889 .54953 .42601 .57666 .41637 .56247 .40886 Metetra .643 .887 .985 r .56247 .40886 .57666 .41637 .60403 .40637 .5901 .39838 Metetra .653 .885 .982 r .5901 .39838 .60403 .40637 .63165 .39609 .61797 .38751 Metetra .66 .881 .98 r .61797 .38751 .63165 .39609 .65952 .38565 .64609 .37637 Metetra .665 .874 .977 r .64609 .37637 .65952 .38565 .68765 .37521 .67445 .36511 Metetra .665 .863 .975 r .67445 .36511 .68765 .37521 .71606 .36503 .70309 .35397 Metetra .658 .849 .973 r .70309 .35397 .71606 .36503 .74479 .35549 .73202 .3433 Metetra .641 .827 .972 r .73202 .3433 .74479 .35549 .77392 .34725 .76131 .33374 Metetra .6 .792 .973 r .76131 .33374 .77392 .34725 .8036 .3416 .79112 .3265 Metetra .474 .708 .971 r .79112 .3265 .8036 .3416 .83434 .34277 .82188 .32553 Metetra .743 .779 .882 r .1612 .5219 .17817 .53565 .20293 .51942 .18595 .5074 Metetra .669 .795 .942 r .18595 .5074 .20293 .51942 .22726 .50914 .21031 .49836 Metetra .623 .805 .97 r .21031 .49836 .22726 .50914 .2516 .5011 .23471 .49131 Metetra .591 .814 .985 r .23471 .49131 .2516 .5011 .27607 .49413 .25928 .48516 Metetra .57 .822 .993 r .25928 .48516 .27607 .49413 .30076 .48766 .28408 .47934 Metetra .558 .83 .997 r .28408 .47934 .30076 .48766 .32569 .48133 .30916 .47355 Metetra .553 .839 .998 r .30916 .47355 .32569 .48133 .35089 .47491 .33452 .46755 Metetra .554 .847 .998 r .33452 .46755 .35089 .47491 .37637 .46825 .36018 .4612 Metetra .56 .855 .997 r .36018 .4612 .37637 .46825 .40213 .46124 .38614 .45439 Metetra .569 .863 .996 r .38614 .45439 .40213 .46124 .42817 .45379 .4124 .44705 Metetra .581 .871 .995 r .4124 .44705 .42817 .45379 .4545 .44585 .43895 .43912 Metetra .594 .877 .993 r .43895 .43912 .4545 .44585 .4811 .4374 .46579 .43057 Metetra .609 .883 .991 r .46579 .43057 .4811 .4374 .50796 .42841 .49291 .42139 Metetra .623 .886 .988 r .49291 .42139 .50796 .42841 .53509 .41889 .5203 .41159 Metetra .638 .889 .986 r .5203 .41159 .53509 .41889 .56247 .40886 .54794 .40117 Metetra .651 .889 .983 r .54794 .40117 .56247 .40886 .5901 .39838 .57583 .3902 Metetra .662 .888 .98 r .57583 .3902 .5901 .39838 .61797 .38751 .60397 .37873 Metetra .672 .884 .977 r .60397 .37873 .61797 .38751 .64609 .37637 .63234 .36687 Metetra .678 .878 .973 r .63234 .36687 .64609 .37637 .67445 .36511 .66096 .35477 Metetra .68 .868 .97 r .66096 .35477 .67445 .36511 .70309 .35397 .68982 .34265 Metetra .676 .854 .967 r .68982 .34265 .70309 .35397 .73202 .3433 .71896 .33084 Metetra .662 .835 .966 r .71896 .33084 .73202 .3433 .76131 .33374 .74844 .31993 Metetra .628 .803 .967 r .74844 .31993 .76131 .33374 .79112 .3265 .77838 .31107 Metetra .52 .731 .97 r .77838 .31107 .79112 .3265 .82188 .32553 .80918 .30795 Metetra .722 .777 .896 r .14392 .5079 .1612 .5219 .18595 .5074 .16865 .49516 Metetra .643 .789 .952 r .16865 .49516 .18595 .5074 .21031 .49836 .19302 .48738 Metetra .595 .798 .977 r .19302 .48738 .21031 .49836 .23471 .49131 .21747 .48133 Metetra .564 .806 .99 r .21747 .48133 .23471 .49131 .25928 .48516 .24213 .47599 Metetra .546 .814 .996 r .24213 .47599 .25928 .48516 .28408 .47934 .26704 .47085 Metetra .537 .823 .998 r .26704 .47085 .28408 .47934 .30916 .47355 .29225 .46559 Metetra .536 .832 .999 r .29225 .46559 .30916 .47355 .33452 .46755 .31777 .46002 Metetra .54 .842 .999 r .31777 .46002 .33452 .46755 .36018 .4612 .34361 .45398 Metetra .549 .852 .998 r .34361 .45398 .36018 .4612 .38614 .45439 .36977 .44738 Metetra .562 .861 .997 r .36977 .44738 .38614 .45439 .4124 .44705 .39625 .44015 Metetra .577 .869 .995 r .39625 .44015 .4124 .44705 .43895 .43912 .42303 .43222 Metetra .593 .877 .993 r .42303 .43222 .43895 .43912 .46579 .43057 .45012 .42358 Metetra .61 .883 .991 r .45012 .42358 .46579 .43057 .49291 .42139 .47749 .41421 Metetra .627 .888 .988 r .47749 .41421 .49291 .42139 .5203 .41159 .50514 .40411 Metetra .643 .89 .985 r .50514 .40411 .5203 .41159 .54794 .40117 .53305 .3933 Metetra .658 .891 .981 r .53305 .3933 .54794 .40117 .57583 .3902 .56122 .38182 Metetra .671 .89 .977 r .56122 .38182 .57583 .3902 .60397 .37873 .58962 .36973 Metetra .682 .887 .973 r .58962 .36973 .60397 .37873 .63234 .36687 .61826 .35714 Metetra .691 .881 .969 r .61826 .35714 .63234 .36687 .66096 .35477 .64714 .34418 Metetra .695 .872 .965 r .64714 .34418 .66096 .35477 .68982 .34265 .67624 .33106 Metetra .693 .859 .962 r .67624 .33106 .68982 .34265 .71896 .33084 .70561 .31809 Metetra .682 .842 .959 r .70561 .31809 .71896 .33084 .74844 .31993 .73528 .30582 Metetra .654 .814 .96 r .73528 .30582 .74844 .31993 .77838 .31107 .76536 .29532 Metetra .562 .752 .966 r .76536 .29532 .77838 .31107 .80918 .30795 .79622 .29002 Metetra .696 .773 .911 r .12634 .49365 .14392 .5079 .16865 .49516 .15101 .48267 Metetra .612 .781 .962 r .15101 .48267 .16865 .49516 .19302 .48738 .17539 .47617 Metetra .564 .788 .983 r .17539 .47617 .19302 .48738 .21747 .48133 .19987 .47114 Metetra .535 .796 .993 r .19987 .47114 .21747 .48133 .24213 .47599 .2246 .46663 Metetra .52 .805 .998 r .2246 .46663 .24213 .47599 .26704 .47085 .24961 .46216 Metetra .515 .815 .999 r .24961 .46216 .26704 .47085 .29225 .46559 .27495 .45746 Metetra .517 .826 1 r .27495 .45746 .29225 .46559 .31777 .46002 .30063 .45231 Metetra .526 .837 .999 r .30063 .45231 .31777 .46002 .34361 .45398 .32665 .44659 Metetra .539 .848 .998 r .32665 .44659 .34361 .45398 .36977 .44738 .35301 .4402 Metetra .555 .858 .997 r .35301 .4402 .36977 .44738 .39625 .44015 .3797 .43307 Metetra .573 .868 .995 r .3797 .43307 .39625 .44015 .42303 .43222 .40672 .42516 Metetra .592 .876 .993 r .40672 .42516 .42303 .43222 .45012 .42358 .43406 .41642 Metetra .611 .883 .99 r .43406 .41642 .45012 .42358 .47749 .41421 .46169 .40685 Metetra .63 .889 .987 r .46169 .40685 .47749 .41421 .50514 .40411 .48961 .39644 Metetra .648 .892 .983 r .48961 .39644 .50514 .40411 .53305 .3933 .5178 .38522 Metetra .665 .893 .979 r .5178 .38522 .53305 .3933 .56122 .38182 .54624 .37323 Metetra .68 .893 .975 r .54624 .37323 .56122 .38182 .58962 .36973 .57493 .36052 Metetra .693 .89 .97 r .57493 .36052 .58962 .36973 .61826 .35714 .60385 .34718 Metetra .702 .884 .965 r .60385 .34718 .61826 .35714 .64714 .34418 .63299 .33334 Metetra .708 .876 .96 r .63299 .33334 .64714 .34418 .67624 .33106 .66235 .3192 Metetra .709 .864 .955 r .66235 .3192 .67624 .33106 .70561 .31809 .69195 .30505 Metetra .701 .847 .952 r .69195 .30505 .70561 .31809 .73528 .30582 .72182 .29139 Metetra .678 .823 .952 r .72182 .29139 .73528 .30582 .76536 .29532 .75207 .27922 Metetra .601 .77 .961 r .75207 .27922 .76536 .29532 .79622 .29002 .783 .27172 Metetra .667 .767 .925 r .10844 .47914 .12634 .49365 .15101 .48267 .13303 .46995 Metetra .578 .771 .971 r .13303 .46995 .15101 .48267 .17539 .47617 .15739 .46473 Metetra .53 .777 .988 r .15739 .46473 .17539 .47617 .19987 .47114 .1819 .46073 Metetra .504 .785 .995 r .1819 .46073 .19987 .47114 .2246 .46663 .20669 .45706 Metetra .492 .795 .999 r .20669 .45706 .2246 .46663 .24961 .46216 .2318 .45328 Metetra .491 .806 1 r .2318 .45328 .24961 .46216 .27495 .45746 .25726 .44913 Metetra .498 .818 1 r .25726 .44913 .27495 .45746 .30063 .45231 .28309 .44442 Metetra .511 .831 1 r .28309 .44442 .30063 .45231 .32665 .44659 .30928 .43902 Metetra .528 .844 .999 r .30928 .43902 .32665 .44659 .35301 .4402 .33584 .43285 Metetra .547 .855 .997 r .33584 .43285 .35301 .4402 .3797 .43307 .36275 .42583 Metetra .568 .866 .996 r .36275 .42583 .3797 .43307 .40672 .42516 .39001 .41791 Metetra .59 .876 .993 r .39001 .41791 .40672 .42516 .43406 .41642 .41759 .40908 Metetra .612 .884 .99 r .41759 .40908 .43406 .41642 .46169 .40685 .44549 .3993 Metetra .634 .89 .986 r .44549 .3993 .46169 .40685 .48961 .39644 .47369 .38859 Metetra .654 .894 .982 r .47369 .38859 .48961 .39644 .5178 .38522 .50216 .37695 Metetra .672 .895 .977 r .50216 .37695 .5178 .38522 .54624 .37323 .53089 .36443 Metetra .689 .895 .972 r .53089 .36443 .54624 .37323 .57493 .36052 .55987 .35107 Metetra .703 .892 .966 r .55987 .35107 .57493 .36052 .60385 .34718 .58907 .33697 Metetra .714 .887 .96 r .58907 .33697 .60385 .34718 .63299 .33334 .61849 .32224 Metetra .721 .879 .954 r .61849 .32224 .63299 .33334 .66235 .3192 .64812 .30706 Metetra .724 .868 .949 r .64812 .30706 .66235 .3192 .69195 .30505 .67797 .29171 Metetra .719 .853 .945 r .67797 .29171 .69195 .30505 .72182 .29139 .70807 .27664 Metetra .701 .83 .944 r .70807 .27664 .72182 .29139 .75207 .27922 .73848 .26278 Metetra .637 .785 .953 r .73848 .26278 .75207 .27922 .783 .27172 .7695 .25305 Metetra .632 .759 .939 r .09021 .46436 .10844 .47914 .13303 .46995 .11469 .45698 Metetra .539 .759 .978 r .11469 .45698 .13303 .46995 .15739 .46473 .13902 .45306 Metetra .493 .763 .991 r .13902 .45306 .15739 .46473 .1819 .46073 .16354 .45009 Metetra .47 .771 .996 r .16354 .45009 .1819 .46073 .20669 .45706 .18838 .44728 Metetra .463 .782 .998 r .18838 .44728 .20669 .45706 .2318 .45328 .21357 .4442 Metetra .467 .796 .999 r .21357 .4442 .2318 .45328 .25726 .44913 .23915 .44061 Metetra .478 .81 1 r .23915 .44061 .25726 .44913 .28309 .44442 .26512 .43634 Metetra .496 .825 .999 r .26512 .43634 .28309 .44442 .30928 .43902 .29149 .43127 Metetra .516 .839 .999 r .29149 .43127 .30928 .43902 .33584 .43285 .31824 .42531 Metetra .54 .853 .998 r .31824 .42531 .33584 .43285 .36275 .42583 .34537 .4184 Metetra .564 .865 .996 r .34537 .4184 .36275 .42583 .39001 .41791 .37287 .41049 Metetra .589 .875 .993 r .37287 .41049 .39001 .41791 .41759 .40908 .40071 .40155 Metetra .614 .884 .99 r .40071 .40155 .41759 .40908 .44549 .3993 .42888 .39156 Metetra .637 .891 .986 r .42888 .39156 .44549 .3993 .47369 .38859 .45736 .38053 Metetra .659 .895 .981 r .45736 .38053 .47369 .38859 .50216 .37695 .48612 .36846 Metetra .679 .897 .975 r .48612 .36846 .50216 .37695 .53089 .36443 .51515 .3554 Metetra .697 .897 .969 r .51515 .3554 .53089 .36443 .55987 .35107 .54443 .34139 Metetra .713 .894 .962 r .54443 .34139 .55987 .35107 .58907 .33697 .57393 .32651 Metetra .725 .889 .955 r .57393 .32651 .58907 .33697 .61849 .32224 .60364 .31087 Metetra .734 .882 .949 r .60364 .31087 .61849 .32224 .64812 .30706 .63356 .29463 Metetra .738 .871 .942 r .63356 .29463 .64812 .30706 .67797 .29171 .66367 .27805 Metetra .736 .857 .937 r .66367 .27805 .67797 .29171 .70807 .27664 .69399 .26156 Metetra .722 .837 .935 r .69399 .26156 .70807 .27664 .73848 .26278 .7246 .24598 Metetra .669 .798 .944 r .7246 .24598 .73848 .26278 .7695 .25305 .75573 .23399 Metetra .591 .747 .952 r .07164 .44932 .09021 .46436 .11469 .45698 .096 .44375 Metetra .495 .742 .983 r .096 .44375 .11469 .45698 .13902 .45306 .12027 .44114 Metetra .452 .746 .992 r .12027 .44114 .13902 .45306 .16354 .45009 .14479 .43923 Metetra .434 .756 .995 r .14479 .43923 .16354 .45009 .18838 .44728 .16966 .43728 Metetra .432 .769 .997 r .16966 .43728 .18838 .44728 .21357 .4442 .19493 .43491 Metetra .441 .784 .998 r .19493 .43491 .21357 .4442 .23915 .44061 .22061 .43189 Metetra .458 .801 .999 r .22061 .43189 .23915 .44061 .26512 .43634 .24673 .42807 Metetra .48 .818 .999 r .24673 .42807 .26512 .43634 .29149 .43127 .27326 .42333 Metetra .505 .834 .999 r .27326 .42333 .29149 .43127 .31824 .42531 .30021 .41759 Metetra .532 .85 .998 r .30021 .41759 .31824 .42531 .34537 .4184 .32756 .41079 Metetra .56 .863 .996 r .32756 .41079 .34537 .4184 .37287 .41049 .35529 .40287 Metetra .588 .875 .994 r .35529 .40287 .37287 .41049 .40071 .40155 .3834 .39382 Metetra .615 .884 .99 r .3834 .39382 .40071 .40155 .42888 .39156 .41184 .38362 Metetra .64 .892 .985 r .41184 .38362 .42888 .39156 .45736 .38053 .44061 .37226 Metetra .664 .896 .979 r .44061 .37226 .45736 .38053 .48612 .36846 .46967 .35976 Metetra .686 .899 .973 r .46967 .35976 .48612 .36846 .51515 .3554 .49901 .34614 Metetra .705 .899 .966 r .49901 .34614 .51515 .3554 .54443 .34139 .5286 .33146 Metetra .722 .896 .958 r .5286 .33146 .54443 .34139 .57393 .32651 .55841 .31578 Metetra .735 .891 .951 r .55841 .31578 .57393 .32651 .60364 .31087 .58843 .29921 Metetra .746 .884 .943 r .58843 .29921 .60364 .31087 .63356 .29463 .61863 .28189 Metetra .751 .874 .936 r .61863 .28189 .63356 .29463 .66367 .27805 .64902 .26407 Metetra .751 .861 .929 r .64902 .26407 .66367 .27805 .69399 .26156 .6796 .24612 Metetra .741 .842 .926 r .6796 .24612 .69399 .26156 .7246 .24598 .71041 .2288 Metetra .699 .809 .933 r .71041 .2288 .7246 .24598 .75573 .23399 .74167 .21453 Metetra .542 .731 .962 r .05272 .43399 .07164 .44932 .096 .44375 .07693 .43025 Metetra .445 .722 .984 r .07693 .43025 .096 .44375 .12027 .44114 .10112 .42898 Metetra .406 .726 .99 r .10112 .42898 .12027 .44114 .14479 .43923 .12562 .42813 Metetra .394 .738 .992 r .12562 .42813 .14479 .43923 .16966 .43728 .15051 .42705 Metetra .399 .754 .994 r .15051 .42705 .16966 .43728 .19493 .43491 .17585 .4254 Metetra .414 .772 .995 r .17585 .4254 .19493 .43491 .22061 .43189 .20163 .42296 Metetra .436 .792 .997 r .20163 .42296 .22061 .43189 .24673 .42807 .22788 .41959 Metetra .463 .811 .998 r .22788 .41959 .24673 .42807 .27326 .42333 .25458 .41519 Metetra .493 .83 .999 r .25458 .41519 .27326 .42333 .30021 .41759 .28172 .40967 Metetra .524 .847 .998 r .28172 .40967 .30021 .41759 .32756 .41079 .30929 .40298 Metetra .556 .862 .997 r .30929 .40298 .32756 .41079 .35529 .40287 .33727 .39506 Metetra .586 .875 .994 r .33727 .39506 .35529 .40287 .3834 .39382 .36563 .3859 Metetra .616 .885 .99 r .36563 .3859 .3834 .39382 .41184 .38362 .39436 .37547 Metetra .644 .893 .984 r .39436 .37547 .41184 .38362 .44061 .37226 .42342 .36378 Metetra .669 .898 .978 r .42342 .36378 .44061 .37226 .46967 .35976 .45279 .35083 Metetra .693 .9 .971 r .45279 .35083 .46967 .35976 .49901 .34614 .48245 .33665 Metetra .713 .9 .963 r .48245 .33665 .49901 .34614 .5286 .33146 .51236 .32128 Metetra .731 .898 .954 r .51236 .32128 .5286 .33146 .55841 .31578 .54249 .30478 Metetra .745 .893 .946 r .54249 .30478 .55841 .31578 .58843 .29921 .57283 .28726 Metetra .757 .886 .937 r .57283 .28726 .58843 .29921 .61863 .28189 .60334 .26884 Metetra .764 .877 .929 r .60334 .26884 .61863 .28189 .64902 .26407 .63402 .24975 Metetra .766 .864 .921 r .63402 .24975 .64902 .26407 .6796 .24612 .66486 .23032 Metetra .759 .847 .917 r .66486 .23032 .6796 .24612 .71041 .2288 .69591 .21124 Metetra .726 .818 .922 r .69591 .21124 .71041 .2288 .74167 .21453 .72731 .19466 Metetra .484 .709 .97 r .03345 .41837 .05272 .43399 .07693 .43025 .05747 .41648 Metetra .389 .697 .982 r .05747 .41648 .07693 .43025 .10112 .42898 .08157 .41655 Metetra .357 .703 .984 r .08157 .41655 .10112 .42898 .12562 .42813 .10603 .41678 Metetra .352 .717 .986 r .10603 .41678 .12562 .42813 .15051 .42705 .13093 .41659 Metetra .364 .737 .989 r .13093 .41659 .15051 .42705 .17585 .4254 .15631 .41566 Metetra .386 .758 .992 r .15631 .41566 .17585 .4254 .20163 .42296 .18219 .41381 Metetra .414 .781 .994 r .18219 .41381 .20163 .42296 .22788 .41959 .20857 .41091 Metetra .446 .803 .997 r .20857 .41091 .22788 .41959 .25458 .41519 .23542 .40685 Metetra .481 .824 .998 r .23542 .40685 .25458 .41519 .28172 .40967 .26276 .40155 Metetra 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{466.188, 0}} -> {-0.081516, 0.0209008, \ 0.00184551, 0.00184551}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] SurfaceGraphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], "'s FindRoot function can be used to map out the miscibility gap. Note that \ the common tangent in this symmetric free energy model will have zero slope. \ FindRoot needs an initial guess and I start from 990 K and work downward \ along one branch of the curve." }], "Text"], Cell["\<\ First, initialize an array for storing pairs of values (T, XB) that \ will form the coexistence curve.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CoexCurvePts = {}\)], "Input"], Cell[BoxData[ \({}\)], "Output"] }, Open ]], Cell["\<\ Here's the general procedure for calculating and extracting the \ value of a point on the coexistence curve:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(T = 990\), "\[IndentingNewLine]", \(XBCoex = FindRoot[\[PartialD]\_XB\ \[CapitalDelta]F[XB, 0, T] \[Equal] 0, {XB, 0.6}]\), "\[IndentingNewLine]", \(XBCoex[\([1, 2]\)]\)}], "Input"], Cell[BoxData[ \(990\)], "Output"], Cell[BoxData[ \({XB \[Rule] 0.5862555349875236`}\)], "Output"], Cell[BoxData[ \(0.5862555349875236`\)], "Output"] }, Open ]], Cell["Now calculate a whole lot 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I collect the two branches of the curve in separate arrays.\ \>", \ "Text"], Cell[BoxData[{ \(\(SpinodalCurvePtsR = {};\)\), "\[IndentingNewLine]", \(\(SpinodalCurvePtsL = {};\)\), "\[IndentingNewLine]", \(Do[\[IndentingNewLine]XBSpin = Flatten[Solve[\[PartialD]\_\(XB, XB\)\[CapitalDelta]F[XB, 0, T] == 0, {XB}]]; \[IndentingNewLine]SpinodalCurvePtsR = Append[SpinodalCurvePtsR, {XBSpin[\([2, 2]\)], T}]; \[IndentingNewLine]SpinodalCurvePtsL = Append[SpinodalCurvePtsL, {XBSpin[\([1, 2]\)], T}], {T, 990, 300, \(-10\)}]\)}], "Input"], Cell["\<\ Now add in the point at the top of the spinodal curve, join the two \ branches of the curve into a single ordered list, and plot the results.\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(SpinodalCurvePts = Prepend[SpinodalCurvePtsR, {0.5, 1000}];\)\), "\[IndentingNewLine]", \(\(SpinodalCurvePts = Join[Reverse[SpinodalCurvePtsL], SpinodalCurvePts];\)\), "\[IndentingNewLine]", \(SpinodalPlot = ListPlot[SpinodalCurvePts, AxesLabel \[Rule] 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Assuming a \ reasonable value for the lattice constant I calculate Kc for the system \ as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(a = 0.29\[Times]10\^\(-9\);\)\), "\[IndentingNewLine]", \(\(VMolar\ = \ a\^3\/2\ N0;\)\), "\[IndentingNewLine]", \(\(VAtomic\ = \ a\^3\/2;\)\), "\[IndentingNewLine]", \(\(\[CapitalDelta]U50 = \[CapitalDelta]U[0.5, 0];\)\), "\[IndentingNewLine]", \(Kc = \(2\/3\) \[CapitalDelta]U50 \(\(\ \)\(\((\@3/2 \ a)\)\^2\)\)\/VMolar\)}], "Input"], Cell[BoxData[ \(4.760863103448276`*^-11\)], "Output"] }, Open ]], Cell["\<\ We will look at the energy of the interface between equilibrium \ phases at T = 900 K. We need the expression for the free energy density \ \[CapitalDelta]fhom(XB) as illustrated in KoM Fig. 18.1. First calculate the \ free energy density of the equilibrium phases. 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