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Kelly, 1996-2002" }], "Text", TextAlignment->Center, TextJustification->0, FontSize->14, FontSlant->"Italic"], Cell["\<\ The statistical mechanics of ideal Fermi-Dirac systems is developed in terms \ of special functions defined by integrating the mean occupation number \ against powers of the single-particle energy. The analytical and numerical \ properties of these fermi functions are studied in considerable detail. Here \ we emphasize nonrelativistic systems in three dimensions, but other systems \ can be explored easily with appropriate modifications of this notebook. The \ important special case of nearly degenerate Fermi gases is also developed in \ some detail, with applications made to the specific heat of metals and to the \ properties of atomic nuclei.\ \>", "1ColumnBox"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "\tAccording to the Fermi-Dirac distribution function, which is derived \ within the notebook ", StyleBox[ButtonBox["occupy.nb", ButtonData:>{"occupy.nb", None}, ButtonStyle->"Hyperlink"], FontSlant->"Italic"], ", the mean occupancy for a single-particle orbital with energy ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\)]], " is" }], "Text"], Cell[BoxData[ \(n\&_[\[CurlyEpsilon]]\ = \ 1\/\(Exp[\(\[CurlyEpsilon] - \[Mu]\)\/\(\(k\_B\) T\)]\ + \ 1\)\)], \ "DisplayFormula", Evaluatable->False, TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "where the chemical potential ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], " is a function of density and temperature. At ", Cell[BoxData[ \(TraditionalForm\`T = 0\)]], ", the argument of the exponential is ", Cell[BoxData[ \(TraditionalForm\`\(-\[Infinity]\)\)]], " when ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon] < \[Mu]\)]], " or ", Cell[BoxData[ \(TraditionalForm\`\(+\[Infinity]\)\)]], " when ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon] > \[Mu]\)]], "; hence, the occupancy is unity for all states with ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\)]], " below ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], " and is zero for all states with ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\)]], " above ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], ". Therefore, at absolute zero a Fermi gas is described as ", StyleBox["completely degenerate", FontSlant->"Italic"], " and is characterized by a frozen distribution in which all orbitals with \ ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon] < \[CurlyEpsilon]\_F\)]], " are occupied and all orbitals with ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon] > \[CurlyEpsilon]\_F\)]], " are vacant, where the Fermi energy and Fermi temperature are defined in \ terms of the chemical potential as" }], "Text"], Cell[BoxData[ \(T = \(0\ \ \ \ \[DoubleLongRightArrow]\ \ \ \ \[Mu] = \(\[CurlyEpsilon]\ \_F\ = \ \(k\_B\) T\_F\)\)\)], "DisplayFormula", Evaluatable->False, TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "and are functions of the density of the system. Thus, it is useful to \ express the thermodynamic properties of Fermi systems at finite temperature, \ which are no longer completely degenerate, in terms of reduced energy and \ temperature variables scaled to ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_F\)]], " or ", Cell[BoxData[ \(TraditionalForm\`T\_F\)]], ", respectively. " }], "Text"], Cell["\<\ \tThe dependence of the chemical potential upon density and temperature is \ determined by integrating the occupation numbers to obtain the total particle \ number\ \>", "Text"], Cell[BoxData[ \(N\ = \ \[Integral]\_0\%\[Infinity] n\&_[\[CurlyEpsilon]]\ \ \[ScriptCapitalD][\[CurlyEpsilon]] \[DifferentialD]\[CurlyEpsilon]\)], \ "DisplayFormula", Evaluatable->False, TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "where for large systems the density of states, ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD][\[CurlyEpsilon]]\)]], ", is approximated by a continuous function of energy. For an ordinary \ nonrelativistic gas in three dimensions, the single-particle energy is simply \ ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\ = \ \(\(\[HBar]\^2\) k\^2\)\/\(2\ \ m\)\)]], " and the density of momentum states is ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD][k\&\[RightVector]]\ \(d\^3\) k = \ g\ V\ \ \(\(d\^3\) k\)\/\((2 \[Pi])\)\^3\)]], " where ", Cell[BoxData[ \(TraditionalForm\`g = 2 s + 1\)]], " is the intrinsic degeneracy for a particle with spin ", Cell[BoxData[ \(TraditionalForm\`s\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\)]], " is the volume of the system. Generalizations to one- or two-dimensional \ systems or to relativistic kinematics are developed in the problems at the \ end of the notebook. The chemical potential enters the above integral \ through ", Cell[BoxData[ \(TraditionalForm\`n\&_[\[CurlyEpsilon]]\)]], " in a rather nontrivial fashion that requires numerical evaluation. \ Similarly, other thermodynamic properties can be expressed in terms of \ integrals of this type. Therefore, it is useful to define the ", StyleBox["fermi function", FontSlant->"Italic"], " of order ", Cell[BoxData[ \(TraditionalForm\`\[Nu]\)]], " as" }], "Text"], Cell[BoxData[ \(fermi[\[Nu], z]\ = \ \(1\/\[CapitalGamma][\[Nu]]\) \ \(\[Integral]\_0\%\[Infinity]\( x\^\(\[Nu] - 1\)\/\(\(z\^\(-1\)\) Exp[x] + 1\)\) \[DifferentialD]x\)\)], "DisplayFormula", Evaluatable->False, TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`z\ = \ Exp[\[Mu]\/\(\(k\_B\) T\)]\)]], " is known as the ", StyleBox["fugacity", FontSlant->"Italic"], " and where ", Cell[BoxData[ \(TraditionalForm\`x = \[CurlyEpsilon]/\[CurlyEpsilon]\_F\)]], " is the energy in units of the Fermi energy. The gamma function is \ included to ensure a convenient normalization, namely ", Cell[BoxData[ \(TraditionalForm\`fermi[\[Nu], z] \[Rule] z\)]], " for small ", Cell[BoxData[ \(TraditionalForm\`z\)]], ". Using these fermi functions, we will show below that some of the common \ thermodynamic functions for an ideal nonrelativistic Fermi gas in three \ dimensions become" }], "Text"], Cell[BoxData[ \(n\_Q = \ \(\(N\ \[Lambda]\^3\)\/\(g\ V\) = fermi[3\/2, z]\)\)], "DisplayFormula", Evaluatable->False, FontFamily->"Times New Roman"], Cell[BoxData[ \(\[ScriptCapitalG] = \(\(-p\)\ V\ = \ \(-N\)\ \(k\_B\) T\ fermi[5\/2, z]\/fermi[3\/2, z]\)\)], "DisplayFormula", Evaluatable->False, FontFamily->"Times New Roman"], Cell[BoxData[ \(U\ = \ \(3\/2\) N\ \(k\_B\) T fermi[5\/2, z]\/fermi[3\/2, z]\)], "DisplayFormula", Evaluatable->False, FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ SubscriptBox["n", "Q"], TraditionalForm]]], " is the quantum concentration and ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", SqrtBox[ FractionBox[ RowBox[{"2", " ", "\[Pi]", " ", SuperscriptBox["\[HBar]", "2"]}], RowBox[{"m", " ", SubscriptBox["k", "B"], " ", "T"}]]]}], TraditionalForm]]], " is the thermal wavelength. The properties of fermi functions are \ investigated below in considerable detail." }], "Text"], Cell[CellGroupData[{ Cell["Glossary", "Subsection"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalN]\)]], "\t= number of particles\n\t", Cell[BoxData[ \(TraditionalForm\`V\)]], "\t= volume\n\t", Cell[BoxData[ \(TraditionalForm\`T\)]], "\t= temperature\n\t", Cell[BoxData[ \(TraditionalForm\`k\_B\)]], "\t= Boltzmann constant\n\t", Cell[BoxData[ \(TraditionalForm\`g\)]], "\t= energy-level degeneracy\n\t", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\)]], "\t= single-particle energy\n\t", Cell[BoxData[ \(TraditionalForm\`k\)]], "\t= single-particle wave number\n\t", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD][\[CurlyEpsilon]]\)]], "\t= density of states wrt energy\n\t", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD][k]\)]], "\t= density of states wrt momentum\n\t", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)]], "\t= thermal wavelength\n\t", Cell[BoxData[ \(TraditionalForm\`n\_Q\)]], "\t= quantum concentration\n\t", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], "\t= chemical potential\n\t", Cell[BoxData[ \(TraditionalForm\`z\)]], "\t= fugacity\n\t", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalZ]\_1\%FD\)]], "\t= single-particle grand-partition function\n\t", Cell[BoxData[ \(TraditionalForm\`n\&_\)]], "\t= mean occupation number\n\t", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_F\)]], "\t= Fermi energy\n\t", Cell[BoxData[ \(TraditionalForm\`k\_F\)]], "\t= Fermi momentum\n\t", Cell[BoxData[ \(TraditionalForm\`T\_F\)]], "\t= Fermi temperature (", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_F/k\_B\)]], ")\n\t", Cell[BoxData[ \(TraditionalForm\`\[Tau]\)]], "\t= reduced temperature (", Cell[BoxData[ \(TraditionalForm\`T/T\_F\)]], ")\n\t", Cell[BoxData[ FormBox[ StyleBox["\[ScriptCapitalG]", FontFamily->"Brush Script MT", FontSlant->"Italic"], TraditionalForm]]], "\t= grand potential\n\t", Cell[BoxData[ \(TraditionalForm\`U\)]], "\t= internal energy\n\t", Cell[BoxData[ \(TraditionalForm\`p\)]], "\t= pressure\n\t", Cell[BoxData[ \(TraditionalForm\`S\)]], "\t= entropy\n\t", Cell[BoxData[ \(TraditionalForm\`F\)]], "\t= free energy\n\t", Cell[BoxData[ \(TraditionalForm\`H\)]], "\t= enthalpy\n\t", Cell[BoxData[ \(TraditionalForm\`C\_V\)]], "\t= isochoric heat capacity\n\t", Cell[BoxData[ \(TraditionalForm\`C\_p\)]], "\t= isochoric heat capacity\n\t", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\ = \ C\_p\/C\_V\)]], "\t= ratio of principal heat capacities\n\t", Cell[BoxData[ \(TraditionalForm\`fermi[n, z]\)]], "\t= Fermi-Dirac function of order ", Cell[BoxData[ \(TraditionalForm\`n\)]], "\n\t", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\ = \ 1\/V\ \((\[PartialD]V\/\[PartialD]T)\)\_p\)]], " \t= isobaric expansivity\n\t", Cell[BoxData[ \(TraditionalForm\`\[Kappa]\_T = \ \ \(-\(1\/V\)\)\ \((\[PartialD]V\/\ \[PartialD]p)\)\_T\)]], "\t= isothermal compressibility\n\t", Cell[BoxData[ \(TraditionalForm\`\[Kappa]\_S = \ \ \(-\(1\/V\)\)\ \((\[PartialD]V\/\ \[PartialD]p)\)\_S\)]], "\t= adiabatic compressibility\n" }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[CellGroupData[{ Cell["Defaults, packages, and symbols", "Subsection"], Cell[BoxData[{ \(\(ClearAll["\"];\)\), "\[IndentingNewLine]", \(\(Off[General::spell, General::spell1];\)\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[{ \(\($DefaultFont = {"\", 12};\)\), "\n", \(\($TextStyle = {FontFamily \[Rule] "\", FontSize \[Rule] 12, FontSlant \[Rule] "\"};\)\)}], "Input", CellLabel->"In[3]:="], Cell["\<\ Needs[\"Utilities`Notation`\"]; 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Here we repeat the derivation of the mean occupation number, briefly, \ but then investigate its properties for nearly degenerate Fermi systems in \ more detail. The goal in this section is to motivate the definition of the \ Fermi functions and to explain how expansions with respect to reduced \ temperature can be developed for various thermodynamic functions." }], "Text"], Cell[CellGroupData[{ Cell["Mean occupation number", "Subsection"], Cell["\<\ The mean occupation number, as a function of energy and chemical potential, \ is determined from the grand partition function for a single orbital. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalZ]\_1\%FD = \[Sum]\+\(n = 0\)\%1 Exp[\(-n\)\ \[Beta]\ \((\[CurlyEpsilon] - \ \[Mu])\)]\)], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \(1 + \[ExponentialE]\^\(\(-\[Beta]\)\ \((\[CurlyEpsilon] - \ \[Mu])\)\)\)], "Output", CellLabel->"Out[32]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(n\_FD\)\&_ = k\_B\ T\ \[PartialD]\_\(\(\ \ \)\(\[Mu]\)\)\((Log[\[ScriptCapitalZ]\_1\%FD] /. {\[Beta] \[Rule] 1\/\(k\_B\ T\)})\)\ // FullSimplify\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \(1\/\(1 + \[ExponentialE]\^\(\(\[CurlyEpsilon] - \[Mu]\)\/\(k\_B\ \ T\)\)\)\)], "Output", CellLabel->"Out[33]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(n\_FD\)\&_ /. muToZ\ // FullSimplify\)], "Input", CellLabel->"In[34]:="], Cell[BoxData[ \(z\/\(\[ExponentialE]\^\(\[CurlyEpsilon]\/\(k\_B\ T\)\) + z\)\)], \ "Output", CellLabel->"Out[34]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Evaluate Fermi energy and momentum", "Subsection"], Cell[TextData[{ "The Fermi momentum, ", Cell[BoxData[ FormBox[ SubscriptBox["k", "F"], TraditionalForm]]], ", is defined to be the highest occupied momentum when all lower states \ are fully occupied. The corresponding energy is the Fermi energy, ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyEpsilon]", "F"], TraditionalForm]]], "." }], "Text"], Cell[BoxData[ \(\(n\_Q = \(\[ScriptCapitalN]\ \[Lambda]\^3\)\/\(g\ V\);\)\)], "Input", CellLabel->"In[35]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(kFrule\)\(=\)\(\(Solve[\[Integral]\_0\%\(k\_F\)\[ScriptCapitalD][ k] \[DifferentialD]k \[Equal] \[ScriptCapitalN], k\_F]\)\[LeftDoubleBracket]3\[RightDoubleBracket]\)\(\ \)\)\)], \ "Input", CellLabel->"In[36]:="], Cell[BoxData[ \({k\_F \[Rule] \(6\^\(1/3\)\ \[Pi]\^\(2/3\)\ \ \[ScriptCapitalN]\^\(1/3\)\)\/\(g\^\(1/3\)\ V\^\(1/3\)\)}\)], "Output", CellLabel->"Out[36]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(eFrule\ = \ {\[CurlyEpsilon]\_F \[Rule] \((\[HBar]\ k\_F)\)\^2\/\(2\ m\ \)\ } /. kFrule\)], "Input", CellLabel->"In[37]:="], Cell[BoxData[ \({\[CurlyEpsilon]\_F \[Rule] \(3\^\(2/3\)\ \[Pi]\^\(4/3\)\ \ \[ScriptCapitalN]\^\(2/3\)\ \[HBar]\^2\)\/\(2\^\(1/3\)\ g\^\(2/3\)\ m\ \ V\^\(2/3\)\)}\)], "Output", CellLabel->"Out[37]="] }, Closed]], Cell[TextData[{ "It is useful to recognize that the density of states, for either momentum \ or energy, can be expressed in terms of simple functions of the Fermi \ momentum or energy which display the scaling relations in a transparent \ manner. Thus, if we define ", Cell[BoxData[ FormBox[ RowBox[{"\[Kappa]", " ", "=", " ", RowBox[{"k", "/", SubscriptBox["k", "F"]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", "=", RowBox[{"\[CurlyEpsilon]", "/", SubscriptBox["\[CurlyEpsilon]", "F"]}]}], TraditionalForm]]], " we can produce the simpler density of state functions below. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalD][\[Kappa]] = \((\(\[ScriptCapitalD][k]\ \ k\_F\^3\ /. kFrule // PowerExpand\) // Simplify)\) /. k \[Rule] \[Kappa]\)], "Input", CellLabel->"In[38]:="], Cell[BoxData[ \(3\ \[ScriptCapitalN]\ \[Kappa]\^2\)], "Output", CellLabel->"Out[38]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalD][\[Epsilon]] = \((\(\[ScriptCapitalD][\[CurlyEpsilon]]\ \ \ \[CurlyEpsilon]\_F\^\(3/2\)\ /. eFrule // PowerExpand\) // Simplify)\) /. \[CurlyEpsilon] \[Rule] \[Epsilon]\)], "Input", CellLabel->"In[39]:="], Cell[BoxData[ \(\(3\ \[ScriptCapitalN]\ \@\[Epsilon]\)\/2\)], "Output", CellLabel->"Out[39]="] }, Closed]], Cell[TextData[{ "Thus, we can evaluate the mean energy or momentum, in units of the ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_F\)]], " or ", Cell[BoxData[ \(TraditionalForm\`k\_F\)]], ", for a completely degenerate nonrelativistic Fermi gas in three \ dimensions using simple dimensional arguments." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Epsilon]\&_\ = \ \(\[Integral]\_0\%1 \[Epsilon]\ \[ScriptCapitalD][\ \[Epsilon]] \[DifferentialD]\[Epsilon]\)\/\(\[Integral]\_0\%1\ \ \[ScriptCapitalD][\[Epsilon]] \[DifferentialD]\[Epsilon]\)\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[ \(3\/5\)], "Output", CellLabel->"Out[40]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Kappa]\&_\ = \ \(\[Integral]\_0\%1 \[Kappa]\ \[ScriptCapitalD][\ \[Kappa]] \[DifferentialD]\[Kappa]\)\/\(\[Integral]\_0\%1\ \[ScriptCapitalD][\ \[Kappa]] \[DifferentialD]\[Kappa]\)\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(3\/4\)], "Output", CellLabel->"Out[41]="] }, Closed]], Cell[TextData[{ "Therefore, we conclude that the internal energy for a completely \ degenerate Fermi gas is simply ", Cell[BoxData[ \(TraditionalForm\`T = \(0\ \ \[DoubleLongRightArrow]\ \ U = \ \(3\/5\) \ \[ScriptCapitalN]\ \[CurlyEpsilon]\_F\)\)]], ", which will provide an important consistency check on the more detailed \ numerical evaluations to follow for finite temperature. Furthermore, \ recognizing that the pressure is simply two-thirds of the energy density for \ an ideal nonrelativistic gas, regardless of its permutation symmetry, we \ conclude that ", Cell[BoxData[ \(TraditionalForm\`T = \(0\ \ \[DoubleLongRightArrow]\ \ p\ V = \ \ \(2\/5\) \[ScriptCapitalN]\ \[CurlyEpsilon]\_F\)\)]], " for a Fermi gas. The failure of the energy density and pressure to \ approach zero as the temperature approaches zero, as expected for a classical \ gas, represents the most dramatic consequence of the Pauli exclusion \ principle for fermions. For dense systems the degeneracy energy and \ pressure, namely their values at zero temperature, can be extremely large." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Occupation numbers for low temperature", "Subsection"], Cell["\<\ Permutation symmetry has its greatest effect on the thermodynamics of Fermi \ systems when the reduced temperature is small. The mean occupation number \ then approaches a step function of temperature, near unity below the Fermi \ energy or near zero above the Fermi energy. Particles occupying states far \ below the Fermi level cannot change orbitals without absorbing a relatively \ large amount of energy and hence do not participate in thermodynamic \ processes. Therefore, many properties of the system are governed primarily \ by the states that are in the immediate vicinity of the Fermi level. We \ illustrate by comparing the low temperature occupation number distribution \ with that for a completely degenerate system. 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Most of the action occurs near the \ Fermi surface. From this plot it is clear that the chemical potential is \ equal to the single-particle energy for which the mean occupation number is ", Cell[BoxData[ FormBox[ FractionBox["1", "2"], TraditionalForm]]], "." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Thermodynamic properties of nearly degenerate systems", "Section"], Cell["\<\ In this section we express various thermodynamic functions as expansions with \ respect to reduced temperature, \[Tau], using the properties of the \ occupation number distribution. This procedure provides considerable insight \ into the behavior of the important class of systems that are nearly \ degenerate. Some of these results will be rederived later as low-temperature \ limits of more general expressions based upon Fermi-Dirac functions.\ \>", "Text"], Cell[CellGroupData[{ Cell["Expansion with respect to temperature", "Subsection"], Cell["\<\ Generally a thermodynamic function is constructed by integrating some \ function of the single-particle energy against the density of states and the \ mean occupation number. Deviations with respect to the properties of a \ completely degenerate system arise at finite temperature because the mean \ occupation number spreads out a little near the Fermi level. The effect of \ this change in occupation distributions can be studied using generic \ integrals of the form\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(A[\[Tau]]\ = \[Integral]\_0\%\[Infinity] n[\[Epsilon], \[Tau]]\ \[PartialD]\_\[Epsilon] f[\[Epsilon]] \[DifferentialD]\[Epsilon]\)], "Input", CellLabel->"In[45]:="], Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], RowBox[{ RowBox[{\(n[\[Epsilon], \[Tau]]\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "\[Epsilon]", "]"}]}], \(\[DifferentialD]\[Epsilon]\)}]}]], "Output", CellLabel->"Out[45]="] }, Closed]], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\[Epsilon] f[\[Epsilon]]\)]], " includes both the density of states and the single-particle function of \ interest. Integration by parts is accomplished using the following rule." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(partsRule\ = \ \[Integral]\_a_\%b_ g_\ \[PartialD]\_\[Epsilon] f_[\[Epsilon]] \[DifferentialD]\[Epsilon]\ \[RuleDelayed] \ \((g\ f[\[Epsilon]] /. \[Epsilon] \[Rule] b)\)\ - \ \((g\ f[\[Epsilon]] /. \[Epsilon] \[Rule] a)\) - \[Integral]\_a\%b f[\[Epsilon]]\ \[PartialD]\_\[Epsilon] g\ \[DifferentialD]\[Epsilon]\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ RowBox[{ RowBox[{\(\[Integral]\_a_\%b_\), RowBox[{ RowBox[{"g_", " ", RowBox[{ SuperscriptBox["f_", "\[Prime]", MultilineFunction->None], "[", "\[Epsilon]", "]"}]}], \(\[DifferentialD]\[Epsilon]\)}]}], "\[RuleDelayed]", \(\((g\ f[\[Epsilon]] /. \[InvisibleSpace]\[Epsilon] \ \[Rule] b)\) - \((g\ f[\[Epsilon]] /. \[InvisibleSpace]\[Epsilon] \[Rule] a)\) - \[Integral]\_a\%b\( f[\[Epsilon]]\ \[PartialD]\_\[Epsilon] g\) \[DifferentialD]\[Epsilon]\)}]], "Output", CellLabel->"Out[46]="] }, Closed]], Cell[TextData[{ "Hence, assuming that ", Cell[BoxData[ FormBox[ RowBox[{"f", "[", "\[Epsilon]", "]"}], TraditionalForm]]], " includes the density of states and vanishes at ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", "=", "0"}], TraditionalForm]]], " while the mean occupation number vanishes at ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", "=", "\[Infinity]"}], TraditionalForm]]], ", we can discard the surface terms and obtain" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{\(A[\[Tau]] /. partsRule\), " ", "/.", RowBox[{"{", RowBox[{\(f[0] \[Rule] 0\), ",", " ", RowBox[{ RowBox[{"n", "[", RowBox[{ InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], ",", "\[Tau]"}], "]"}], "\[Rule]", "0"}]}], "}"}]}]}]], "Input", CellLabel->"In[47]:="], Cell[BoxData[ RowBox[{"-", RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], RowBox[{ RowBox[{\(f[\[Epsilon]]\), " ", RowBox[{ SuperscriptBox["n", TagBox[\((1, 0)\), Derivative], 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Furthermore, we can extend the range of integration to ", Cell[BoxData[ \(TraditionalForm\`\((\(-\[Infinity]\), \(+\[Infinity]\))\)\)]], " and use a change of variables to simplify the integrand, recognizing that \ dnfermi is symmetric about its peak." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"integrand", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{\(-f[\[Epsilon]]\), " ", RowBox[{ SuperscriptBox["n", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(\[Epsilon], \[Tau]\), "]"}]}], "/.", RowBox[{"{", RowBox[{\(f[\[Epsilon]] \[Rule] Normal[Series[f[\[Epsilon]], {\[Epsilon], 1, 4}]]\), ",", RowBox[{ RowBox[{ SuperscriptBox["n", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(\[Epsilon], \[Tau]\), "]"}], "\[Rule]", \(dnfermi[\[Epsilon], \[Tau]]\)}]}], "}"}]}], "/.", \({\[Epsilon] \[Rule] x\ \[Tau]\ + \ 1}\)}], "//", "Expand"}]}]], "Input", CellLabel->"In[50]:="], Cell[BoxData[ RowBox[{\(\(\[ExponentialE]\^x\ f[ 1]\)\/\(\((1 + \[ExponentialE]\^x)\)\^2\ \[Tau]\)\), "+", FractionBox[ RowBox[{\(\[ExponentialE]\^x\), " ", "x", " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "1", "]"}]}], \(\((1 + \[ExponentialE]\^x)\)\^2\)], "+", FractionBox[ RowBox[{\(\[ExponentialE]\^x\), " ", \(x\^2\), " ", "\[Tau]", " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "1", "]"}]}], \(2\ \((1 + \[ExponentialE]\^x)\)\^2\)], "+", FractionBox[ RowBox[{\(\[ExponentialE]\^x\), " ", \(x\^3\), " ", \(\[Tau]\^2\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "1", "]"}]}], \(6\ \((1 + \[ExponentialE]\^x)\)\^2\)], "+", FractionBox[ RowBox[{\(\[ExponentialE]\^x\), " ", \(x\^4\), " ", \(\[Tau]\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "1", "]"}]}], \(24\ \((1 + \[ExponentialE]\^x)\)\^2\)]}]], "Output", CellLabel->"Out[50]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(expansion\ = \((\[Tau] \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ integrand \[DifferentialD]x\)\ )\) // Simplify\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ RowBox[{\(f[1]\), "+", RowBox[{\(1\/6\), " ", \(\[Pi]\^2\), " ", \(\[Tau]\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "1", "]"}]}], "+", RowBox[{\(7\/360\), " ", \(\[Pi]\^4\), " ", \(\[Tau]\^4\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "1", "]"}]}]}]], "Output", CellLabel->"Out[51]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Chemical potential", "Subsection"], Cell[TextData[{ "We can apply this expansion to the relationship between density and \ chemical potential by identifying ", Cell[BoxData[ FormBox[ FractionBox[\(\[PartialD]f\), \(\[PartialD]\[Epsilon]\), MultilineFunction->None], TraditionalForm]]], " with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD][\[Epsilon]]\)]], " and making the necessary substitutions into the expansion formula." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f\_N\ = \[Integral]\[ScriptCapitalD][\[Epsilon]] \[DifferentialD]\ \[Epsilon]\)], "Input", CellLabel->"In[52]:="], Cell[BoxData[ \(\[ScriptCapitalN]\ \[Epsilon]\^\(3/2\)\)], "Output", CellLabel->"Out[52]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Nexpansion\ = \((\ \(expansion\ /. f[x_] \[Rule] f\_N\)\ /. \(\(Derivative[n_]\)[f]\)[1] \[Rule] D[f\_N, {\[Epsilon], n}])\) /. {\[Epsilon] \[Rule] \[Mu], \[Tau] \[Rule] \ \(k\_B\) T}\)], "Input", CellLabel->"In[53]:="], Cell[BoxData[ \(\(7\ k\_B\%4\ \[Pi]\^4\ T\^4\ \[ScriptCapitalN]\)\/\(640\ \ \[Mu]\^\(5/2\)\) + \(k\_B\%2\ \[Pi]\^2\ T\^2\ \[ScriptCapitalN]\)\/\(8\ \@\ \[Mu]\) + \[ScriptCapitalN]\ \[Mu]\^\(3/2\)\)], "Output", CellLabel->"Out[53]="] }, Closed]], Cell[TextData[{ "Realizing that ", Cell[BoxData[ FormBox["\[Mu]", TraditionalForm]]], " is near ", Cell[BoxData[ FormBox[ SubscriptBox["\[CurlyEpsilon]", "F"], TraditionalForm]]], " for low temperatures, we use the substitution ", Cell[BoxData[ FormBox[ RowBox[{"\[Mu]", "\[Rule]", RowBox[{ SubscriptBox["\[CurlyEpsilon]", "F"], "+", "\[Delta]"}]}], TraditionalForm]]], " to develop an equation for the correction term valid to second order in ", Cell[BoxData[ FormBox["\[Delta]", TraditionalForm]]], ". After solving this equation, we expand with respect to temperature, \ being careful to limit that expansion to the same order used for Nexpansion, \ and select the physically meaningful solution." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(stuff\ = \ \(\(\(Nexpansion\)\(\ \)\)\/\(\[ScriptCapitalN]\ \ \[CurlyEpsilon]\_F\^\(3/2\)\)\ /. \[Mu] \[Rule] \[CurlyEpsilon]\_F + \ \[Delta] // Series[#, {\[Delta], 0, 2}] &\)\ // Normal\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \(\(\[Delta]\^2\ \((\(49\ k\_B\%4\ \[Pi]\^4\ T\^4\ \ \[ScriptCapitalN]\)\/\(1024\ \[CurlyEpsilon]\_F\%\(9/2\)\) + \(3\ k\_B\%2\ \ \[Pi]\^2\ T\^2\ \[ScriptCapitalN]\)\/\(64\ \[CurlyEpsilon]\_F\%\(5/2\)\) + \ \(3\ \[ScriptCapitalN]\)\/\(8\ \@\[CurlyEpsilon]\_F\))\)\)\/\(\ \[ScriptCapitalN]\ \[CurlyEpsilon]\_F\%\(3/2\)\) + \(\[Delta]\ \((\(-\(\(7\ k\ \_B\%4\ \[Pi]\^4\ T\^4\ \[ScriptCapitalN]\)\/\(256\ \[CurlyEpsilon]\_F\%\(7/2\ \)\)\)\) - \(k\_B\%2\ \[Pi]\^2\ T\^2\ \[ScriptCapitalN]\)\/\(16\ \ \[CurlyEpsilon]\_F\%\(3/2\)\) + \(3\ \[ScriptCapitalN]\ \ \@\[CurlyEpsilon]\_F\)\/2)\)\)\/\(\[ScriptCapitalN]\ \ \[CurlyEpsilon]\_F\%\(3/2\)\) + \(\(7\ k\_B\%4\ \[Pi]\^4\ T\^4\ \ \[ScriptCapitalN]\)\/\(640\ \[CurlyEpsilon]\_F\%\(5/2\)\) + \(k\_B\%2\ \ \[Pi]\^2\ T\^2\ \[ScriptCapitalN]\)\/\(8\ \@\[CurlyEpsilon]\_F\) + \ \[ScriptCapitalN]\ \[CurlyEpsilon]\_F\%\(3/2\)\)\/\(\[ScriptCapitalN]\ \ \[CurlyEpsilon]\_F\%\(3/2\)\)\)], "Output", CellLabel->"Out[54]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Solve[ stuff \[Equal] 1, \[Delta]]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input",\ CellLabel->"In[55]:="], Cell[BoxData[ \({\[Delta] \[Rule] \((2\ \((35\ k\_B\%4\ \[Pi]\^4\ T\^4\ \[CurlyEpsilon]\ \_F + 80\ k\_B\%2\ \[Pi]\^2\ T\^2\ \[CurlyEpsilon]\_F\%3 - 1920\ \[CurlyEpsilon]\_F\%5 - \@5\ \@\(\(-441\)\ k\_B\%8\ \ \[Pi]\^8\ T\^8\ \[CurlyEpsilon]\_F\%2 - 7392\ k\_B\%6\ \[Pi]\^6\ T\^6\ \ \[CurlyEpsilon]\_F\%4 - 38656\ k\_B\%4\ \[Pi]\^4\ T\^4\ \[CurlyEpsilon]\_F\%6 \ - 122880\ k\_B\%2\ \[Pi]\^2\ T\^2\ \[CurlyEpsilon]\_F\%8 + 737280\ \ \[CurlyEpsilon]\_F\%10\))\))\)/\((5\ \((49\ k\_B\%4\ \[Pi]\^4\ T\^4 + 48\ k\_B\%2\ \[Pi]\^2\ T\^2\ \[CurlyEpsilon]\_F\%2 + 384\ \[CurlyEpsilon]\_F\%4)\))\)}\)], "Output", CellLabel->"Out[55]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Delta]sol = \(\((\(\[Delta] /. Solve[stuff \[Equal] 1, \[Delta]]\ // Series[#, {T, 0, 4}] &\)\ // Normal)\) // PowerExpand\) // Simplify\)], "Input", CellLabel->"In[56]:="], Cell[BoxData[ \({\(41\ k\_B\%4\ \[Pi]\^4\ T\^4\)\/\(80\ \[CurlyEpsilon]\_F\%3\) + \(3\ \ k\_B\%2\ \[Pi]\^2\ T\^2\)\/\(4\ \[CurlyEpsilon]\_F\) - 4\ \[CurlyEpsilon]\_F, \(-\(\(3\ k\_B\%4\ \[Pi]\^4\ T\^4 + 20\ k\_B\%2\ \[Pi]\^2\ T\^2\ \[CurlyEpsilon]\_F\%2\)\/\(240\ \ \[CurlyEpsilon]\_F\%3\)\)\)}\)], "Output", CellLabel->"Out[56]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Mu]Expansion\ = \ \[CurlyEpsilon]\_F + \[Delta]sol\ \[LeftDoubleBracket]2\[RightDoubleBracket]\ /. {T \[Rule] \[Tau]\ \ \[CurlyEpsilon]\_F\ /\ k\_B} // Collect[#, {\[CurlyEpsilon]\_F}] &\)], "Input",\ CellLabel->"In[57]:="], Cell[BoxData[ \(\[CurlyEpsilon]\_F\ \((1 - \(\[Pi]\^2\ \[Tau]\^2\)\/12 - \(\[Pi]\^4\ \ \[Tau]\^4\)\/80)\)\)], "Output", CellLabel->"Out[57]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Internal energy", "Subsection"], Cell[TextData[{ "To evaluate the temperature dependence of the total energy, in units of ", Cell[BoxData[ FormBox["\[Mu]", TraditionalForm]]], ", we identify ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[PartialD]", "f"}], RowBox[{"\[PartialD]", "\[Epsilon]"}], MultilineFunction->None], TraditionalForm]]], " with ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", " ", RowBox[{"\[Rho]", "[", "\[Epsilon]", "]"}]}], TraditionalForm]]], " and make the necessary substitutions into the expansion formula." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f\_\[Epsilon]\ = \[Integral]\[Epsilon]\ \[ScriptCapitalD][\[Epsilon]] \ \(\(\[DifferentialD]\[Epsilon]\)\(\ \)\)\)], "Input", CellLabel->"In[58]:="], Cell[BoxData[ \(3\/5\ \[ScriptCapitalN]\ \[Epsilon]\^\(5/2\)\)], "Output", CellLabel->"Out[58]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Ustuff\ = \((\ \(\(\[CurlyEpsilon]\_F\^\(\(-3\)/2\)\) expansion\ /. f[x_] \[Rule] f\_\[Epsilon]\)\ /. \(\(Derivative[n_]\)[f]\)[ 1] \[Rule] D[f\_\[Epsilon], {\[Epsilon], n}])\) /. {\[Epsilon] \[Rule] \[Mu], \[Tau] \[Rule] \ \(k\_B\) T}\)], "Input", CellLabel->"In[59]:="], Cell[BoxData[ \(\(\(-\(\(7\ k\_B\%4\ \[Pi]\^4\ T\^4\ \[ScriptCapitalN]\)\/\(640\ \ \[Mu]\^\(3/2\)\)\)\) + 3\/8\ k\_B\%2\ \[Pi]\^2\ T\^2\ \[ScriptCapitalN]\ \@\ \[Mu] + 3\/5\ \[ScriptCapitalN]\ \ \[Mu]\^\(5/2\)\)\/\[CurlyEpsilon]\_F\%\(3/2\)\)], "Output", CellLabel->"Out[59]="] }, Closed]], Cell[TextData[{ "Finally, we use our expansions for the temperature dependence of ", Cell[BoxData[ FormBox["\[Mu]", TraditionalForm]]], " to obtain a simple formula for the temperature dependence of the internal \ energy." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Uexpansion\ = \ \(\(\((\(Ustuff\ /. \[Mu] \[Rule] \[Mu]Expansion\)\ /. \ {T \[Rule] \[Tau]\ \[CurlyEpsilon]\_F\ /\ k\_B})\) // Series[#, {\[Tau], 0, 4}] &\) // Normal\)\ // Collect[#, {\[ScriptCapitalN], \[CurlyEpsilon]\_F}] &\)], "Input", CellLabel->"In[60]:="], Cell[BoxData[ \(\[ScriptCapitalN]\ \[CurlyEpsilon]\_F\ \((3\/5 + \(\[Pi]\^2\ \ \[Tau]\^2\)\/4 - \(3\ \[Pi]\^4\ \[Tau]\^4\)\/80)\)\)], "Output", CellLabel->"Out[60]="] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Fermi-Dirac functions", "Section"], Cell["\<\ \tAlthough the expansions developed in the preceding section are very useful \ at low temperatures, a more complete analysis over a broad range of \ temperatures requires that we develop more general techniques. Thus, it is \ useful to define Fermi functions in terms of integrals of powers of the \ energy against the occupation number such that\ \>", "Text"], Cell[BoxData[ \(fermi[\[Nu], z]\ = \ \(1\/\[CapitalGamma][\[Nu]]\) \ \(\[Integral]\_0\%\[Infinity]\( x\^\(\[Nu] - 1\)\/\(\(z\^\(-1\)\) Exp[x] + 1\)\) \[DifferentialD]x\)\)], "DisplayFormula", Evaluatable->False, TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "where the ", StyleBox["fugacity", FontSlant->"Italic"], " is defined as ", Cell[BoxData[ \(TraditionalForm\`\(\(z = Exp[\ \[Mu]\/\(\(k\_B\) T\)]\)\(\ \)\)\)]], ". The gamma function is included to ensure a convenient normalization, \ namely ", Cell[BoxData[ \(TraditionalForm\`fermi[\[Nu], z] \[Rule] z\)]], " for small ", Cell[BoxData[ \(TraditionalForm\`z\)]], ". Since the chemical potential for a dilute system is large and negative, \ the classical limit corresponds to small ", Cell[BoxData[ \(TraditionalForm\`z\)]], ". Conversely, at low temperatures the chemical potential approaches the \ Fermi energy so that the fugacity for a Fermi system is large. In later \ sections we will show that the thermodynamic properties of Fermi systems can \ be expressed in terms of these Fermi functions for arbitrary temperature and \ density." }], "Text"], Cell[TextData[{ "\tIn this section we define the Fermi function and develop rules to \ exploit some of its analytic properties. We then display several fermi \ functions graphically. Finally, we develop expansions that apply to both the \ classical and degenerate regimes. 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" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Small z expansion", "Subsection"], Cell[TextData[{ "Inspection of the Taylor series around ", Cell[BoxData[ \(TraditionalForm\`z \[Rule] 0\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[fermi[\[Nu], z], {z, 0, 3}]\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ InterpretationBox[ RowBox[{ "z", "-", \(2\^\(-\[Nu]\)\ z\^2\), "+", \(3\^\(-\[Nu]\)\ z\^3\), "+", InterpretationBox[\(O[z]\^4\), SeriesData[ z, 0, {}, 1, 4, 1]]}], SeriesData[ z, 0, {1, Times[ -1, Power[ 2, Times[ -1, \[Nu]]]], Power[ 3, Times[ -1, \[Nu]]]}, 1, 4, 1]]], "Output", CellLabel->"Out[63]="] }, Closed]], Cell["suggests that the general power series takes the form", "Text"], Cell[BoxData[ \(fermiZexpansion[\[Nu]_, z_, \(k\_max\) : _] := \(-\(\[Sum]\+\(k = 1\)\%\(k\_max\)\((\(-z\))\)\ \^k\/k\^\[Nu]\)\)\)], "Input", CellLabel->"In[64]:="], Cell["\<\ Below we verify that the infinite series does reproduce the desired function.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fermi[\[Nu], z] \[Equal] fermiZexpansion[\[Nu], z, \[Infinity]]\)], "Input", CellLabel->"In[65]:="], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[65]="] }, Closed]], Cell[TextData[{ "It will be useful to define a rule for small ", StyleBox["z", FontSlant->"Italic"], " as follows." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zSmallRule = \(-fermi[\[Nu]_, z]\) \[Rule] \(-fermiZexpansion[\[Nu], z, 3]\)\)], "Input", CellLabel->"In[66]:="], Cell[BoxData[ \(PolyLog[\[Nu]_, \(-z\)] \[Rule] \(-z\) + 2\^\(-\[Nu]\)\ z\^2 - 3\^\(-\[Nu]\)\ z\^3\)], "Output", CellLabel->"Out[66]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Large z expansion", "Subsection"], Cell[TextData[{ "The degenerate Fermi gas is an important special case characterized by \ large ", Cell[BoxData[ \(TraditionalForm\`z\)]], ". At low temperatures the occupation probability is nearly a square \ function, unity below and zero above the chemical potential, with deviations \ from the square function confined to a narrow region, of width approximately \ equal to ", Cell[BoxData[ \(TraditionalForm\`k\_B\ T\)]], ", surrounding ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], ". It is useful to separate the deviations from the completely degenerate \ behavior into two regions, one below and the other above ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], ". The correction below ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], " is given by integrand1 and above ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], " by integrand2. Since the integrands become very small for energies more \ than a few ", Cell[BoxData[ \(TraditionalForm\`k\_B\ T\)]], " away from ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]], ", we can extend the range of integration to -\[Infinity] for integrand1 or \ to \[Infinity] for integrand2. The exponentially small error we make in \ extending the energy range to-\[Infinity] can be treated in more detail, if \ desired. With appropriate changes of variables, these two terms can be \ combined into a single integral over the range ", Cell[BoxData[ \(TraditionalForm\`{x, 0, \[Infinity]}\)]], ". A series expansion of the integrand is then made so that the integral \ can be performed. " }], "Text"], Cell[BoxData[ \(Clear[integrand1, integrand2, integrand]\)], "Input", CellLabel->"In[67]:="], Cell[BoxData[ \(integrand1[x_] := x\^\(\[Nu] - 1\)\ \((1 - 1\/\(Exp[x - y] + 1\))\) /. {x \[Rule] y - x}\)], "Input", CellLabel->"In[68]:="], Cell[BoxData[ \(integrand2[x_] := x\^\(\[Nu] - 1\)\/\(Exp[x - y] + 1\) /. {x \[Rule] y + x}\)], "Input", CellLabel->"In[69]:="], Cell[CellGroupData[{ Cell[BoxData[ \(temp = integrand2[x] - integrand1[x]\ // Simplify\)], "Input", CellLabel->"In[70]:="], Cell[BoxData[ \(\(\(-\((\(-x\) + y)\)\^\(\(-1\) + \[Nu]\)\) + \((x + y)\)\^\(\(-1\) + \ \[Nu]\)\)\/\(1 + \[ExponentialE]\^x\)\)], "Output", CellLabel->"Out[70]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(integrand = Normal[\((Series[#1, {a, 0, 3}] &)\)[ temp /. {\((x + y)\)\^\[Nu]_ \[Rule] y\^\[Nu]\ \((\(a\ x\)\/y + 1)\)\^\[Nu], \((\(-x\) + y)\)\^\ \[Nu]_ \[Rule] y\^\[Nu]\ \((\(-\(\(a\ x\)\/y\)\) + 1)\)\^\[Nu]}]] /. a \[Rule] 1\)], "Input", CellLabel->"In[71]:="], Cell[BoxData[ \(\(2\ x\ y\^\(\(-2\) + \[Nu]\)\ \((\(-1\) + \[Nu])\)\)\/\(1 + \ \[ExponentialE]\^x\) + \(x\^3\ y\^\(\(-4\) + \[Nu]\)\ \((\(-3\) + \[Nu])\)\ \ \((\(-2\) + \[Nu])\)\ \((\(-1\) + \[Nu])\)\)\/\(3\ \((1 + \[ExponentialE]\^x)\ \)\)\)], "Output", CellLabel->"Out[71]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(correction = \[Integral]\_0\%\[Infinity] integrand \[DifferentialD]x\)], "Input", CellLabel->"In[72]:="], Cell[BoxData[ \(1\/360\ \[Pi]\^2\ y\^\(\(-4\) + \[Nu]\)\ \((\(-1\) + \[Nu])\)\ \((60\ y\ \^2 + 7\ \[Pi]\^2\ \((6 - 5\ \[Nu] + \[Nu]\^2)\))\)\)], "Output", CellLabel->"Out[72]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(zLargeRule = \(\(-fermi[\[Nu]_, z]\) \[Rule] \(-\((\(y\^\[Nu]\/\[Nu] + correction\)\/Gamma[\ \[Nu]])\)\) /. {Gamma[\[Nu]] \[Rule] Gamma[\[Nu] + 1]\/\[Nu]}\) /. y \[Rule] Log[z]\ // Simplify\)], "Input", CellLabel->"In[73]:="], Cell[BoxData[ \(PolyLog[\[Nu]_, \(-z\)] \[Rule] \(-\(\(\[Nu]\ \((Log[z]\^\[Nu]\/\[Nu] + 1\/360\ \[Pi]\^2\ \((\(-1\) + \[Nu])\)\ Log[z]\^\(\(-4\) + \ \[Nu]\)\ \((7\ \[Pi]\^2\ \((6 - 5\ \[Nu] + \[Nu]\^2)\) + 60\ Log[z]\^2)\))\)\)\/Gamma[ 1 + \[Nu]]\)\)\)], "Output", CellLabel->"Out[73]="] }, Closed]], Cell["\<\ Therefore, we have obtained an asymptotic expansion of the Fermi function for \ large fugacity, known as Sommerfeld's lemma, that is very useful in \ developing low-temperature approximations. With sufficient patience we could \ carry this expansion to high order, if desired.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Recursion relations for symbolic derivatives", "Subsection"], Cell["\<\ It is also useful to note the following downward recursion relation for \ derivatives.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_z fermi[\[Nu], z]\)], "Input", CellLabel->"In[74]:="], Cell[BoxData[ \(\(-\(PolyLog[\(-1\) + \[Nu], \(-z\)]\/z\)\)\)], "Output", CellLabel->"Out[74]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(z \[PartialD]\_z fermi[\[Nu], z] \[Equal] fermi[\[Nu] - 1, z]\)], "Input", CellLabel->"In[75]:="], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[75]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Rules for manipulation of Fermi-Dirac integrals", "Subsection"], Cell[TextData[{ "Unfortunately, the ", StyleBox["Mathematica", FontSlant->"Italic"], " Integrate function still fails to recognize several simple variants of \ the Fermi-Dirac integral. Furthermore, we would like to employ symbolic \ integrals without needing to specify the parameters. Therefore, in order to \ guide ", StyleBox["Mathematica", FontSlant->"Italic"], " toward proper evaluation of these integrals, we need to temporarily \ unprotect the Integrate function so that we can install our own rules for the \ relevant types of integrals. 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Although this equation usually must be solved numerically, it will be \ useful to develop symbolic relationship valid for either low or high \ temperature limits. 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Thus, the chemical potential for a Fermi gas is positive for ", Cell[BoxData[ \(TraditionalForm\`T < T\_F\)]], " and is negative for ", Cell[BoxData[ \(TraditionalForm\`T > \ T\_F\)]], ". A Fermi gas is described as completely degenerate for ", Cell[BoxData[ \(TraditionalForm\`T \[LessLess] T\_F\)]], ". For very large temperatures, ", Cell[BoxData[ \(TraditionalForm\`T \[GreaterGreater] T\_F\ \[DoubleLongRightArrow]\[Mu] \[LessLess] \(-k\_B\) T\)]], ", the chemical potential becomes large and negative such that the \ nondegenerate Fermi gas approaches classsical behavior. For intermediate \ temperatures, ", Cell[BoxData[ \(TraditionalForm\`T \[Tilde] T\_F\ \[DoubleLongRightArrow]\ \[Mu]\ \[Tilde] \ 0\)]], ", the partially degenerate Fermi gas can be expected to display \ thermodynamic properties intermediate between the degenerate and classical \ situations." }], "Text"], Cell["\<\ Next we compare the interpolation formula for fugacity with the limiting \ behaviors derived in the preceding section.\ \>", "Text"], Cell[BoxData[ \(fugacity[\[Tau]_ /; \[Tau] > 0] := Exp[chemicalPotential[\[Tau]]\/\[Tau]]\)], "Input", CellLabel->"In[96]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\(DisplayTogether[ Plot[fugacity[\[Tau]], {\[Tau], 0.1, 5}, PlotStyle \[Rule] RGBColor[1, 0, 0]], Plot[zlow, {\[Tau], 0.01, 0.45}, PlotStyle \[Rule] GrayLevel[0]], Plot[zhigh, {\[Tau], 0.5, 3}], PlotLabel \[Rule] "\", PlotRange \[Rule] {Automatic, {0, 10}}, GridLines \[Rule] Automatic, Frame \[Rule] True, FrameLabel \[Rule] {"\<\[Tau]\>", "\"}];\)\)], "Input", CellLabel->"In[97]:="], 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However, the divergence of the fugacity as ", Cell[BoxData[ \(TraditionalForm\`\[Tau] \[Rule] 0\)]], " requires special handling of the low-temperature limits for many \ thermodynamic functions. We will employ the large ", StyleBox["z", FontSlant->"Italic"], " expansion to provide well-behaved limiting formulas as needed. We also \ find that small inaccuracies in the interpolation formula can sometimes \ produce small imaginary parts in some of the thermodynamic functions. 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.61249 .57866 .60419 Metetra .722 .804 .915 r .57866 .60419 .58856 .61249 .60775 .60374 .59796 .5952 Metetra .72 .8 .914 r .59796 .5952 .60775 .60374 .62704 .59503 .61735 .58625 Metetra .719 .796 .913 r .61735 .58625 .62704 .59503 .64643 .58636 .63685 .57734 Metetra .717 .793 .911 r .63685 .57734 .64643 .58636 .66593 .57773 .65645 .56848 Metetra .715 .789 .91 r .65645 .56848 .66593 .57773 .68553 .56915 .67616 .55967 Metetra .713 .786 .909 r .67616 .55967 .68553 .56915 .70525 .56062 .69598 .55091 Metetra .711 .783 .908 r .69598 .55091 .70525 .56062 .72507 .55213 .7159 .54221 Metetra .709 .779 .907 r .7159 .54221 .72507 .55213 .74502 .54369 .73595 .53356 Metetra .707 .776 .906 r .73595 .53356 .74502 .54369 .76507 .53531 .75611 .52497 Metetra .705 .773 .905 r .75611 .52497 .76507 .53531 .78525 .52697 .77639 .51644 Metetra .703 .771 .904 r .77639 .51644 .78525 .52697 .80555 .51868 .7968 .50796 Metetra .702 .768 .904 r .7968 .50796 .80555 .51868 .82598 .51044 .81733 .49953 Metetra .7 .766 .903 r .81733 .49953 .82598 .51044 .84654 .50225 .83799 .49116 Metetra .698 .763 .902 r .83799 .49116 .84654 .50225 .86723 .4941 .85878 .48284 Metetra .696 .761 .902 r .85878 .48284 .86723 .4941 .88805 .486 .87971 .47457 Metetra .694 .759 .901 r .87971 .47457 .88805 .486 .90901 .47794 .90078 .46635 Metetra .693 .757 .901 r .90078 .46635 .90901 .47794 .9301 .46991 .92198 .45817 Metetra .691 .755 .901 r .92198 .45817 .9301 .46991 .95134 .46192 .94333 .45003 Metetra .69 .754 .9 r .94333 .45003 .95134 .46192 .97273 .45396 .96483 .44193 Metetra .733 .843 .932 r .37957 .68892 .39074 .69478 .40913 .68578 .39808 .67971 Metetra .733 .84 .93 r .39808 .67971 .40913 .68578 .42762 .67675 .41668 .67045 Metetra .733 .836 .928 r .41668 .67045 .42762 .67675 .44619 .66769 .43537 .66115 Metetra .733 .833 .927 r .43537 .66115 .44619 .66769 .46484 .65861 .45414 .65183 Metetra .732 .829 .925 r .45414 .65183 .46484 .65861 .48359 .64952 .47299 .64249 Metetra .731 .826 .923 r .47299 .64249 .48359 .64952 .50242 .64043 .49194 .63315 Metetra .731 .822 .921 r .49194 .63315 .50242 .64043 .52134 .63134 .51097 .6238 Metetra .729 .818 .92 r .51097 .6238 .52134 .63134 .54035 .62227 .5301 .61447 Metetra .728 .814 .918 r .5301 .61447 .54035 .62227 .55946 .61322 .54931 .60516 Metetra .727 .81 .917 r .54931 .60516 .55946 .61322 .57866 .60419 .56862 .59587 Metetra .725 .806 .915 r .56862 .59587 .57866 .60419 .59796 .5952 .58803 .58663 Metetra .723 .802 .913 r .58803 .58663 .59796 .5952 .61735 .58625 .60753 .57742 Metetra .721 .798 .912 r .60753 .57742 .61735 .58625 .63685 .57734 .62713 .56826 Metetra .72 .795 .911 r .62713 .56826 .63685 .57734 .65645 .56848 .64684 .55915 Metetra .718 .791 .909 r .64684 .55915 .65645 .56848 .67616 .55967 .66665 .5501 Metetra .716 .787 .908 r .66665 .5501 .67616 .55967 .69598 .55091 .68658 .5411 Metetra .713 .784 .907 r .68658 .5411 .69598 .55091 .7159 .54221 .70661 .53217 Metetra .711 .78 .906 r .70661 .53217 .7159 .54221 .73595 .53356 .72676 .5233 Metetra .709 .777 .905 r .72676 .5233 .73595 .53356 .75611 .52497 .74702 .51449 Metetra .707 .774 .904 r .74702 .51449 .75611 .52497 .77639 .51644 .76741 .50575 Metetra .705 .771 .903 r .76741 .50575 .77639 .51644 .7968 .50796 .78792 .49707 Metetra .703 .768 .903 r .78792 .49707 .7968 .50796 .81733 .49953 .80855 .48845 Metetra .701 .766 .902 r .80855 .48845 .81733 .49953 .83799 .49116 .82932 .47989 Metetra .699 .763 .901 r .82932 .47989 .83799 .49116 .85878 .48284 .85022 .4714 Metetra .697 .761 .901 r .85022 .4714 .85878 .48284 .87971 .47457 .87126 .46295 Metetra .696 .759 .9 r .87126 .46295 .87971 .47457 .90078 .46635 .89243 .45457 Metetra .694 .757 .9 r .89243 .45457 .90078 .46635 .92198 .45817 .91375 .44623 Metetra .692 .755 .9 r .91375 .44623 .92198 .45817 .94333 .45003 .93521 .43794 Metetra .691 .753 .9 r .93521 .43794 .94333 .45003 .96483 .44193 .95682 .42969 Metetra .735 .847 .933 r .36823 .68314 .37957 .68892 .39808 .67971 .38687 .67371 Metetra .736 .844 .931 r .38687 .67371 .39808 .67971 .41668 .67045 .40558 .66423 Metetra .736 .84 .929 r .40558 .66423 .41668 .67045 .43537 .66115 .42438 .65469 Metetra .736 .837 .927 r .42438 .65469 .43537 .66115 .45414 .65183 .44327 .64512 Metetra .735 .833 .925 r .44327 .64512 .45414 .65183 .47299 .64249 .46225 .63552 Metetra .735 .829 .923 r .46225 .63552 .47299 .64249 .49194 .63315 .48131 .6259 Metetra .734 .825 .921 r .48131 .6259 .49194 .63315 .51097 .6238 .50045 .61629 Metetra .733 .821 .92 r .50045 .61629 .51097 .6238 .5301 .61447 .51969 .60669 Metetra .731 .817 .918 r .51969 .60669 .5301 .61447 .54931 .60516 .53902 .5971 Metetra .73 .813 .916 r .53902 .5971 .54931 .60516 .56862 .59587 .55844 .58754 Metetra .728 .809 .915 r .55844 .58754 .56862 .59587 .58803 .58663 .57796 .57802 Metetra .726 .805 .913 r .57796 .57802 .58803 .58663 .60753 .57742 .59757 .56854 Metetra .725 .8 .911 r .59757 .56854 .60753 .57742 .62713 .56826 .61728 .55911 Metetra .723 .796 .91 r .61728 .55911 .62713 .56826 .64684 .55915 .6371 .54974 Metetra .72 .792 .909 r .6371 .54974 .64684 .55915 .66665 .5501 .65702 .54043 Metetra .718 .789 .907 r .65702 .54043 .66665 .5501 .68658 .5411 .67705 .53118 Metetra .716 .785 .906 r .67705 .53118 .68658 .5411 .70661 .53217 .69719 .522 Metetra .714 .781 .905 r .69719 .522 .70661 .53217 .72676 .5233 .71744 .5129 Metetra .712 .778 .904 r .71744 .5129 .72676 .5233 .74702 .51449 .73781 .50386 Metetra .709 .775 .903 r .73781 .50386 .74702 .51449 .76741 .50575 .7583 .4949 Metetra .707 .771 .902 r .7583 .4949 .76741 .50575 .78792 .49707 .77892 .48601 Metetra .705 .769 .902 r .77892 .48601 .78792 .49707 .80855 .48845 .79966 .47719 Metetra .703 .766 .901 r .79966 .47719 .80855 .48845 .82932 .47989 .82053 .46844 Metetra .701 .763 .9 r .82053 .46844 .82932 .47989 .85022 .4714 .84154 .45976 Metetra .699 .761 .9 r .84154 .45976 .85022 .4714 .87126 .46295 .86268 .45114 Metetra .697 .758 .899 r .86268 .45114 .87126 .46295 .89243 .45457 .88396 .44258 Metetra .695 .756 .899 r .88396 .44258 .89243 .45457 .91375 .44623 .90539 .43408 Metetra .693 .754 .899 r .90539 .43408 .91375 .44623 .93521 .43794 .92697 .42563 Metetra .691 .753 .899 r .92697 .42563 .93521 .43794 .95682 .42969 .94869 .41723 Metetra .738 .851 .933 r .35672 .67746 .36823 .68314 .38687 .67371 .37547 .66781 Metetra .739 .848 .931 r .37547 .66781 .38687 .67371 .40558 .66423 .39431 .65809 Metetra .739 .844 .929 r .39431 .65809 .40558 .66423 .42438 .65469 .41323 .6483 Metetra .739 .841 .927 r .41323 .6483 .42438 .65469 .44327 .64512 .43224 .63847 Metetra .739 .837 .925 r .43224 .63847 .44327 .64512 .46225 .63552 .45133 .6286 Metetra .738 .833 .923 r .45133 .6286 .46225 .63552 .48131 .6259 .47051 .61871 Metetra .737 .829 .921 r .47051 .61871 .48131 .6259 .50045 .61629 .48978 .60882 Metetra .736 .825 .919 r .48978 .60882 .50045 .61629 .51969 .60669 .50914 .59892 Metetra .735 .82 .918 r .50914 .59892 .51969 .60669 .53902 .5971 .52858 .58904 Metetra .734 .816 .916 r .52858 .58904 .53902 .5971 .55844 .58754 .54812 .57919 Metetra .732 .812 .914 r .54812 .57919 .55844 .58754 .57796 .57802 .56775 .56938 Metetra .73 .807 .912 r .56775 .56938 .57796 .57802 .59757 .56854 .58747 .55961 Metetra .728 .803 .911 r .58747 .55961 .59757 .56854 .61728 .55911 .6073 .54989 Metetra .726 .798 .909 r .6073 .54989 .61728 .55911 .6371 .54974 .62722 .54024 Metetra .724 .794 .908 r .62722 .54024 .6371 .54974 .65702 .54043 .64725 .53066 Metetra .721 .79 .906 r .64725 .53066 .65702 .54043 .67705 .53118 .66739 .52115 Metetra .719 .786 .905 r .66739 .52115 .67705 .53118 .69719 .522 .68763 .51171 Metetra .716 .782 .904 r .68763 .51171 .69719 .522 .71744 .5129 .70799 .50236 Metetra .714 .779 .903 r .70799 .50236 .71744 .5129 .73781 .50386 .72847 .49308 Metetra .712 .775 .902 r .72847 .49308 .73781 .50386 .7583 .4949 .74907 .48389 Metetra .709 .772 .901 r .74907 .48389 .7583 .4949 .77892 .48601 .76979 .47477 Metetra .707 .769 .9 r .76979 .47477 .77892 .48601 .79966 .47719 .79064 .46574 Metetra .705 .766 .9 r .79064 .46574 .79966 .47719 .82053 .46844 .81162 .45679 Metetra .702 .763 .899 r .81162 .45679 .82053 .46844 .84154 .45976 .83273 .44791 Metetra .7 .76 .899 r .83273 .44791 .84154 .45976 .86268 .45114 .85399 .4391 Metetra .698 .758 .898 r .85399 .4391 .86268 .45114 .88396 .44258 .87538 .43037 Metetra .696 .756 .898 r .87538 .43037 .88396 .44258 .90539 .43408 .89692 .4217 Metetra .694 .754 .898 r .89692 .4217 .90539 .43408 .92697 .42563 .9186 .41309 Metetra .692 .752 .897 r .9186 .41309 .92697 .42563 .94869 .41723 .94044 .40454 Metetra .74 .855 .934 r .34502 .67187 .35672 .67746 .37547 .66781 .36389 .66201 Metetra .741 .852 .932 r .36389 .66201 .37547 .66781 .39431 .65809 .38285 .65205 Metetra .742 .849 .93 r .38285 .65205 .39431 .65809 .41323 .6483 .4019 .64201 Metetra .742 .845 .927 r .4019 .64201 .41323 .6483 .43224 .63847 .42104 .63191 Metetra .742 .841 .925 r .42104 .63191 .43224 .63847 .45133 .6286 .44025 .62176 Metetra .742 .837 .923 r .44025 .62176 .45133 .6286 .47051 .61871 .45956 .61158 Metetra .741 .833 .921 r .45956 .61158 .47051 .61871 .48978 .60882 .47895 .60139 Metetra .74 .828 .919 r .47895 .60139 .48978 .60882 .50914 .59892 .49842 .59119 Metetra .739 .824 .917 r .49842 .59119 .50914 .59892 .52858 .58904 .51799 .581 Metetra .737 .819 .915 r .51799 .581 .52858 .58904 .54812 .57919 .53764 .57084 Metetra .736 .814 .913 r .53764 .57084 .54812 .57919 .56775 .56938 .55739 .56071 Metetra .734 .81 .912 r .55739 .56071 .56775 .56938 .58747 .55961 .57723 .55064 Metetra .731 .805 .91 r .57723 .55064 .58747 .55961 .6073 .54989 .59717 .54062 Metetra .729 .8 .908 r .59717 .54062 .6073 .54989 .62722 .54024 .61721 .53067 Metetra .727 .796 .907 r .61721 .53067 .62722 .54024 .64725 .53066 .63735 .52079 Metetra .724 .791 .905 r .63735 .52079 .64725 .53066 .66739 .52115 .65759 .511 Metetra .722 .787 .904 r .65759 .511 .66739 .52115 .68763 .51171 .67795 .50128 Metetra .719 .783 .903 r .67795 .50128 .68763 .51171 .70799 .50236 .69841 .49166 Metetra .717 .779 .902 r .69841 .49166 .70799 .50236 .72847 .49308 .719 .48213 Metetra .714 .775 .901 r .719 .48213 .72847 .49308 .74907 .48389 .7397 .47269 Metetra .711 .772 .9 r .7397 .47269 .74907 .48389 .76979 .47477 .76053 .46335 Metetra .709 .769 .899 r .76053 .46335 .76979 .47477 .79064 .46574 .78149 .45409 Metetra .706 .765 .898 r .78149 .45409 .79064 .46574 .81162 .45679 .80258 .44493 Metetra .704 .763 .898 r .80258 .44493 .81162 .45679 .83273 .44791 .8238 .43585 Metetra .702 .76 .897 r .8238 .43585 .83273 .44791 .85399 .4391 .84516 .42685 Metetra .699 .757 .897 r .84516 .42685 .85399 .4391 .87538 .43037 .86667 .41793 Metetra .697 .755 .897 r .86667 .41793 .87538 .43037 .89692 .4217 .88832 .40909 Metetra .695 .753 .896 r .88832 .40909 .89692 .4217 .9186 .41309 .91012 .40032 Metetra .693 .751 .896 r .91012 .40032 .9186 .41309 .94044 .40454 .93207 .39162 Metetra .742 .859 .934 r .33312 .66639 .34502 .67187 .36389 .66201 .35213 .65631 Metetra .744 .856 .932 r .35213 .65631 .36389 .66201 .38285 .65205 .37122 .64611 Metetra .745 .853 .93 r .37122 .64611 .38285 .65205 .4019 .64201 .39039 .63582 Metetra .745 .85 .928 r .39039 .63582 .4019 .64201 .42104 .63191 .40965 .62545 Metetra .746 .846 .925 r .40965 .62545 .42104 .63191 .44025 .62176 .429 .61501 Metetra .745 .842 .923 r .429 .61501 .44025 .62176 .45956 .61158 .44843 .60453 Metetra .745 .837 .921 r .44843 .60453 .45956 .61158 .47895 .60139 .46795 .59402 Metetra .744 .832 .919 r .46795 .59402 .47895 .60139 .49842 .59119 .48755 .5835 Metetra .743 .828 .917 r .48755 .5835 .49842 .59119 .51799 .581 .50724 .57298 Metetra .741 .823 .915 r .50724 .57298 .51799 .581 .53764 .57084 .52702 .56249 Metetra .74 .818 .913 r .52702 .56249 .53764 .57084 .55739 .56071 .54688 .55203 Metetra .738 .813 .911 r .54688 .55203 .55739 .56071 .57723 .55064 .56684 .54163 Metetra .735 .808 .909 r .56684 .54163 .57723 .55064 .59717 .54062 .5869 .53128 Metetra .733 .803 .907 r .5869 .53128 .59717 .54062 .61721 .53067 .60705 .52101 Metetra .73 .798 .905 r .60705 .52101 .61721 .53067 .63735 .52079 .6273 .51082 Metetra .728 .793 .904 r .6273 .51082 .63735 .52079 .65759 .511 .64766 .50072 Metetra .725 .788 .903 r .64766 .50072 .65759 .511 .67795 .50128 .66812 .49072 Metetra .722 .784 .901 r .66812 .49072 .67795 .50128 .69841 .49166 .6887 .48081 Metetra .719 .78 .9 r .6887 .48081 .69841 .49166 .719 .48213 .70939 .47101 Metetra .716 .776 .899 r .70939 .47101 .719 .48213 .7397 .47269 .73021 .46131 Metetra .714 .772 .898 r .73021 .46131 .7397 .47269 .76053 .46335 .75115 .45172 Metetra .711 .769 .897 r .75115 .45172 .76053 .46335 .78149 .45409 .77221 .44223 Metetra .708 .765 .897 r .77221 .44223 .78149 .45409 .80258 .44493 .79341 .43284 Metetra .706 .762 .896 r .79341 .43284 .80258 .44493 .8238 .43585 .81474 .42356 Metetra .703 .759 .896 r .81474 .42356 .8238 .43585 .84516 .42685 .83622 .41436 Metetra .701 .757 .895 r .83622 .41436 .84516 .42685 .86667 .41793 .85783 .40526 Metetra .698 .754 .895 r .85783 .40526 .86667 .41793 .88832 .40909 .8796 .39624 Metetra .696 .752 .895 r .8796 .39624 .88832 .40909 .91012 .40032 .90151 .3873 Metetra .694 .75 .895 r .90151 .3873 .91012 .40032 .93207 .39162 .92358 .37844 Metetra .744 .863 .935 r .32103 .66102 .33312 .66639 .35213 .65631 .34016 .65073 Metetra .746 .861 .933 r .34016 .65073 .35213 .65631 .37122 .64611 .35939 .6403 Metetra .748 .858 .93 r .35939 .6403 .37122 .64611 .39039 .63582 .37869 .62975 Metetra .749 .854 .928 r .37869 .62975 .39039 .63582 .40965 .62545 .39809 .6191 Metetra .749 .85 .926 r .39809 .6191 .40965 .62545 .429 .61501 .41757 .60836 Metetra .749 .846 .923 r .41757 .60836 .429 .61501 .44843 .60453 .43713 .59757 Metetra .749 .842 .921 r .43713 .59757 .44843 .60453 .46795 .59402 .45678 .58673 Metetra .748 .837 .919 r .45678 .58673 .46795 .59402 .48755 .5835 .47651 .57587 Metetra .747 .832 .916 r .47651 .57587 .48755 .5835 .50724 .57298 .49633 .56501 Metetra .746 .826 .914 r .49633 .56501 .50724 .57298 .52702 .56249 .51623 .55416 Metetra .744 .821 .912 r .51623 .55416 .52702 .56249 .54688 .55203 .53622 .54335 Metetra .742 .816 .91 r .53622 .54335 .54688 .55203 .56684 .54163 .5563 .53259 Metetra .74 .81 .908 r .5563 .53259 .56684 .54163 .5869 .53128 .57647 .5219 Metetra .737 .805 .906 r .57647 .5219 .5869 .53128 .60705 .52101 .59674 .51128 Metetra .734 .8 .904 r .59674 .51128 .60705 .52101 .6273 .51082 .61711 .50075 Metetra .731 .795 .903 r .61711 .50075 .6273 .51082 .64766 .50072 .63758 .49033 Metetra .728 .79 .901 r .63758 .49033 .64766 .50072 .66812 .49072 .65816 .48001 Metetra .725 .785 .9 r .65816 .48001 .66812 .49072 .6887 .48081 .67885 .4698 Metetra .722 .78 .899 r .67885 .4698 .6887 .48081 .70939 .47101 .69965 .45971 Metetra .719 .776 .897 r .69965 .45971 .70939 .47101 .73021 .46131 .72058 .44974 Metetra .716 .772 .897 r .72058 .44974 .73021 .46131 .75115 .45172 .74162 .43988 Metetra .713 .768 .896 r .74162 .43988 .75115 .45172 .77221 .44223 .7628 .43015 Metetra .71 .765 .895 r .7628 .43015 .77221 .44223 .79341 .43284 .78411 .42053 Metetra .707 .762 .895 r .78411 .42053 .79341 .43284 .81474 .42356 .80555 .41102 Metetra .704 .759 .894 r .80555 .41102 .81474 .42356 .83622 .41436 .82713 .40162 Metetra .702 .756 .894 r .82713 .40162 .83622 .41436 .85783 .40526 .84886 .39233 Metetra .699 .753 .894 r .84886 .39233 .85783 .40526 .8796 .39624 .87074 .38313 Metetra .697 .751 .894 r .87074 .38313 .8796 .39624 .90151 .3873 .89277 .37402 Metetra .694 .749 .894 r .89277 .37402 .90151 .3873 .92358 .37844 .91496 .365 Metetra .746 .867 .936 r .30873 .65576 .32103 .66102 .34016 .65073 .328 .64527 Metetra .749 .865 .933 r .328 .64527 .34016 .65073 .35939 .6403 .34736 .63461 Metetra .75 .862 .931 r .34736 .63461 .35939 .6403 .37869 .62975 .3668 .6238 Metetra .752 .859 .928 r .3668 .6238 .37869 .62975 .39809 .6191 .38633 .61287 Metetra .753 .855 .926 r .38633 .61287 .39809 .6191 .41757 .60836 .40595 .60183 Metetra .753 .851 .923 r .40595 .60183 .41757 .60836 .43713 .59757 .42565 .59071 Metetra .753 .846 .921 r .42565 .59071 .43713 .59757 .45678 .58673 .44543 .57953 Metetra .753 .841 .918 r .44543 .57953 .45678 .58673 .47651 .57587 .4653 .56831 Metetra .752 .836 .916 r .4653 .56831 .47651 .57587 .49633 .56501 .48524 .55708 Metetra .75 .83 .913 r .48524 .55708 .49633 .56501 .51623 .55416 .50528 .54586 Metetra .749 .825 .911 r .50528 .54586 .51623 .55416 .53622 .54335 .52539 .53466 Metetra .746 .819 .909 r .52539 .53466 .53622 .54335 .5563 .53259 .5456 .52352 Metetra .744 .813 .906 r .5456 .52352 .5563 .53259 .57647 .5219 .5659 .51245 Metetra .741 .807 .904 r .5659 .51245 .57647 .5219 .59674 .51128 .58629 .50147 Metetra .738 .802 .903 r .58629 .50147 .59674 .51128 .61711 .50075 .60677 .49058 Metetra .735 .796 .901 r .60677 .49058 .61711 .50075 .63758 .49033 .62736 .4798 Metetra .732 .791 .899 r .62736 .4798 .63758 .49033 .65816 .48001 .64805 .46914 Metetra .729 .786 .898 r .64805 .46914 .65816 .48001 .67885 .4698 .66886 .45861 Metetra .725 .781 .897 r .66886 .45861 .67885 .4698 .69965 .45971 .68977 .44821 Metetra .722 .776 .896 r .68977 .44821 .69965 .45971 .72058 .44974 .71081 .43795 Metetra .719 .772 .895 r .71081 .43795 .72058 .44974 .74162 .43988 .73196 .42782 Metetra .715 .768 .894 r .73196 .42782 .74162 .43988 .7628 .43015 .75325 .41783 Metetra .712 .764 .893 r .75325 .41783 .7628 .43015 .78411 .42053 .77467 .40796 Metetra .709 .761 .893 r .77467 .40796 .78411 .42053 .80555 .41102 .79622 .39823 Metetra .706 .758 .892 r .79622 .39823 .80555 .41102 .82713 .40162 .81792 .38862 Metetra .703 .755 .892 r .81792 .38862 .82713 .40162 .84886 .39233 .83976 .37913 Metetra .7 .752 .892 r .83976 .37913 .84886 .39233 .87074 .38313 .86176 .36975 Metetra .698 .75 .892 r .86176 .36975 .87074 .38313 .89277 .37402 .8839 .36047 Metetra .695 .748 .892 r .8839 .36047 .89277 .37402 .91496 .365 .90621 .35129 Metetra .748 .871 .937 r .29622 .6506 .30873 .65576 .328 .64527 .31562 .63993 Metetra .751 .869 .934 r .31562 .63993 .328 .64527 .34736 .63461 .33512 .62905 Metetra .753 .867 .931 r .33512 .62905 .34736 .63461 .3668 .6238 .3547 .61799 Metetra .755 .864 .929 r .3547 .61799 .3668 .6238 .38633 .61287 .37438 .60678 Metetra .756 .86 .926 r .37438 .60678 .38633 .61287 .40595 .60183 .39413 .59543 Metetra .757 .856 .923 r .39413 .59543 .40595 .60183 .42565 .59071 .41397 .58397 Metetra .758 .851 .92 r .41397 .58397 .42565 .59071 .44543 .57953 .4339 .57243 Metetra .757 .846 .918 r .4339 .57243 .44543 .57953 .4653 .56831 .4539 .56083 Metetra .757 .84 .915 r .4539 .56083 .4653 .56831 .48524 .55708 .47399 .54921 Metetra .755 .834 .912 r .47399 .54921 .48524 .55708 .50528 .54586 .49415 .53759 Metetra .754 .828 .91 r .49415 .53759 .50528 .54586 .52539 .53466 .51441 .52599 Metetra .751 .822 .907 r .51441 .52599 .52539 .53466 .5456 .52352 .53474 .51444 Metetra .749 .816 .905 r .53474 .51444 .5456 .52352 .5659 .51245 .55516 .50296 Metetra .746 .81 .903 r .55516 .50296 .5659 .51245 .58629 .50147 .57568 .49157 Metetra .743 .804 .901 r .57568 .49157 .58629 .50147 .60677 .49058 .59629 .48029 Metetra .739 .798 .899 r .59629 .48029 .60677 .49058 .62736 .4798 .61699 .46913 Metetra .736 .792 .897 r .61699 .46913 .62736 .4798 .64805 .46914 .6378 .45811 Metetra .732 .786 .896 r .6378 .45811 .64805 .46914 .66886 .45861 .65872 .44723 Metetra .728 .781 .895 r .65872 .44723 .66886 .45861 .68977 .44821 .67975 .4365 Metetra .725 .776 .893 r .67975 .4365 .68977 .44821 .71081 .43795 .70089 .42593 Metetra .721 .772 .893 r .70089 .42593 .71081 .43795 .73196 .42782 .72216 .41551 Metetra .718 .767 .892 r .72216 .41551 .73196 .42782 .75325 .41783 .74356 .40524 Metetra .714 .763 .891 r .74356 .40524 .75325 .41783 .77467 .40796 .76509 .39512 Metetra .711 .76 .891 r .76509 .39512 .77467 .40796 .79622 .39823 .78676 .38516 Metetra .707 .756 .89 r .78676 .38516 .79622 .39823 .81792 .38862 .80857 .37533 Metetra .704 .753 .89 r .80857 .37533 .81792 .38862 .83976 .37913 .83053 .36563 Metetra .701 .751 .89 r .83053 .36563 .83976 .37913 .86176 .36975 .85263 .35607 Metetra .698 .748 .89 r .85263 .35607 .86176 .36975 .8839 .36047 .8749 .34662 Metetra .696 .746 .891 r .8749 .34662 .8839 .36047 .90621 .35129 .89733 .33728 Metetra .749 .875 .937 r .28349 .64554 .29622 .6506 .31562 .63993 .30303 .6347 Metetra .752 .873 .935 r .30303 .6347 .31562 .63993 .33512 .62905 .32267 .62363 Metetra .755 .871 .932 r .32267 .62363 .33512 .62905 .3547 .61799 .3424 .61233 Metetra .758 .868 .929 r .3424 .61233 .3547 .61799 .37438 .60678 .36221 .60084 Metetra .76 .865 .926 r .36221 .60084 .37438 .60678 .39413 .59543 .38212 .58917 Metetra .761 .861 .923 r .38212 .58917 .39413 .59543 .41397 .58397 .4021 .57737 Metetra .762 .856 .92 r .4021 .57737 .41397 .58397 .4339 .57243 .42217 .56546 Metetra .762 .851 .917 r .42217 .56546 .4339 .57243 .4539 .56083 .44232 .55346 Metetra .762 .845 .914 r .44232 .55346 .4539 .56083 .47399 .54921 .46255 .54142 Metetra .761 .839 .911 r .46255 .54142 .47399 .54921 .49415 .53759 .48286 .52937 Metetra .759 .832 .908 r .48286 .52937 .49415 .53759 .51441 .52599 .50325 .51733 Metetra .757 .826 .906 r .50325 .51733 .51441 .52599 .53474 .51444 .52372 .50534 Metetra .754 .819 .903 r .52372 .50534 .53474 .51444 .55516 .50296 .54427 .49341 Metetra .751 .812 .901 r .54427 .49341 .55516 .50296 .57568 .49157 .56491 .48159 Metetra .747 .806 .899 r .56491 .48159 .57568 .49157 .59629 .48029 .58564 .46989 Metetra .744 .799 .897 r .58564 .46989 .59629 .48029 .61699 .46913 .60647 .45832 Metetra .74 .793 .895 r .60647 .45832 .61699 .46913 .6378 .45811 .6274 .4469 Metetra .736 .787 .894 r .6274 .4469 .6378 .45811 .65872 .44723 .64843 .43565 Metetra .732 .781 .892 r .64843 .43565 .65872 .44723 .67975 .4365 .66958 .42456 Metetra .728 .776 .891 r .66958 .42456 .67975 .4365 .70089 .42593 .69084 .41366 Metetra .724 .771 .89 r .69084 .41366 .70089 .42593 .72216 .41551 .71222 .40293 Metetra .72 .766 .889 r .71222 .40293 .72216 .41551 .74356 .40524 .73373 .39237 Metetra .716 .762 .889 r .73373 .39237 .74356 .40524 .76509 .39512 .75537 .38199 Metetra .712 .758 .888 r .75537 .38199 .76509 .39512 .78676 .38516 .77715 .37178 Metetra .709 .755 .888 r .77715 .37178 .78676 .38516 .80857 .37533 .79907 .36173 Metetra .705 .752 .888 r .79907 .36173 .80857 .37533 .83053 .36563 .82115 .35183 Metetra .702 .749 .888 r .82115 .35183 .83053 .36563 .85263 .35607 .84337 .34208 Metetra .699 .747 .888 r .84337 .34208 .85263 .35607 .8749 .34662 .86576 .33246 Metetra .696 .745 .889 r .86576 .33246 .8749 .34662 .89733 .33728 .8883 .32296 Metetra .749 .878 .939 r .27054 .64057 .28349 .64554 .30303 .6347 .29022 .6296 Metetra .753 .877 .936 r .29022 .6296 .30303 .6347 .32267 .62363 .31 .61834 Metetra .757 .876 .933 r .31 .61834 .32267 .62363 .3424 .61233 .32987 .60682 Metetra .761 .873 .929 r .32987 .60682 .3424 .61233 .36221 .60084 .34984 .59505 Metetra .763 .87 .926 r .34984 .59505 .36221 .60084 .38212 .58917 .36989 .58308 Metetra .765 .866 .923 r .36989 .58308 .38212 .58917 .4021 .57737 .39003 .57093 Metetra .767 .861 .92 r .39003 .57093 .4021 .57737 .42217 .56546 .41025 .55863 Metetra .767 .856 .916 r .41025 .55863 .42217 .56546 .44232 .55346 .43055 .54622 Metetra .767 .85 .913 r .43055 .54622 .44232 .55346 .46255 .54142 .45093 .53373 Metetra .766 .843 .91 r .45093 .53373 .46255 .54142 .48286 .52937 .47138 .52122 Metetra .765 .836 .907 r .47138 .52122 .48286 .52937 .50325 .51733 .49191 .5087 Metetra .762 .829 .904 r .49191 .5087 .50325 .51733 .52372 .50534 .51252 .49623 Metetra .76 .822 .901 r .51252 .49623 .52372 .50534 .54427 .49341 .53321 .48383 Metetra .756 .815 .899 r .53321 .48383 .54427 .49341 .56491 .48159 .55398 .47153 Metetra .753 .808 .896 r .55398 .47153 .56491 .48159 .58564 .46989 .57484 .45936 Metetra .749 .801 .894 r .57484 .45936 .58564 .46989 .60647 .45832 .5958 .44734 Metetra .744 .794 .892 r .5958 .44734 .60647 .45832 .6274 .4469 .61685 .4355 Metetra .74 .787 .891 r .61685 .4355 .6274 .4469 .64843 .43565 .638 .42384 Metetra .736 .781 .889 r .638 .42384 .64843 .43565 .66958 .42456 .65926 .41237 Metetra .731 .775 .888 r .65926 .41237 .66958 .42456 .69084 .41366 .68063 .40111 Metetra .727 .77 .887 r .68063 .40111 .69084 .41366 .71222 .40293 .70213 .39005 Metetra .722 .765 .887 r .70213 .39005 .71222 .40293 .73373 .39237 .72375 .3792 Metetra .718 .761 .886 r .72375 .3792 .73373 .39237 .75537 .38199 .7455 .36854 Metetra .714 .757 .886 r .7455 .36854 .75537 .38199 .77715 .37178 .7674 .35808 Metetra .71 .753 .886 r .7674 .35808 .77715 .37178 .79907 .36173 .78944 .3478 Metetra .706 .75 .886 r .78944 .3478 .79907 .36173 .82115 .35183 .81163 .3377 Metetra .703 .747 .886 r .81163 .3377 .82115 .35183 .84337 .34208 .83397 .32776 Metetra .7 .745 .887 r .83397 .32776 .84337 .34208 .86576 .33246 .85647 .31797 Metetra .696 .743 .887 r .85647 .31797 .86576 .33246 .8883 .32296 .87914 .30832 Metetra .749 .881 .94 r .25735 .63567 .27054 .64057 .29022 .6296 .27717 .62459 Metetra .754 .881 .937 r .27717 .62459 .29022 .6296 .31 .61834 .29709 .61318 Metetra .759 .88 .933 r .29709 .61318 .31 .61834 .32987 .60682 .31712 .60146 Metetra .763 .878 .93 r .31712 .60146 .32987 .60682 .34984 .59505 .33724 .58945 Metetra .766 .875 .926 r .33724 .58945 .34984 .59505 .36989 .58308 .35745 .57717 Metetra .769 .872 .923 r .35745 .57717 .36989 .58308 .39003 .57093 .37774 .56467 Metetra .771 .867 .919 r .37774 .56467 .39003 .57093 .41025 .55863 .39812 .55197 Metetra .772 .861 .916 r .39812 .55197 .41025 .55863 .43055 .54622 .41858 .53912 Metetra .773 .855 .912 r .41858 .53912 .43055 .54622 .45093 .53373 .43911 .52617 Metetra .772 .848 .909 r .43911 .52617 .45093 .53373 .47138 .52122 .45971 .51316 Metetra .771 .841 .905 r .45971 .51316 .47138 .52122 .49191 .5087 .48039 .50013 Metetra .768 .833 .902 r .48039 .50013 .49191 .5087 .51252 .49623 .50115 .48713 Metetra .766 .826 .899 r .50115 .48713 .51252 .49623 .53321 .48383 .52198 .4742 Metetra .762 .818 .896 r .52198 .4742 .53321 .48383 .55398 .47153 .54289 .46138 Metetra .758 .81 .894 r .54289 .46138 .55398 .47153 .57484 .45936 .56388 .4487 Metetra .754 .802 .892 r .56388 .4487 .57484 .45936 .5958 .44734 .58496 .4362 Metetra .749 .795 .89 r .58496 .4362 .5958 .44734 .61685 .4355 .60613 .42388 Metetra .744 .788 .888 r .60613 .42388 .61685 .4355 .638 .42384 .62741 .41178 Metetra .739 .781 .886 r .62741 .41178 .638 .42384 .65926 .41237 .64878 .39991 Metetra .734 .775 .885 r .64878 .39991 .65926 .41237 .68063 .40111 .67027 .38826 Metetra .73 .769 .884 r .67027 .38826 .68063 .40111 .70213 .39005 .69188 .37686 Metetra .725 .764 .884 r .69188 .37686 .70213 .39005 .72375 .3792 .71362 .36569 Metetra .72 .759 .883 r .71362 .36569 .72375 .3792 .7455 .36854 .73549 .35474 Metetra .716 .755 .883 r .73549 .35474 .7455 .36854 .7674 .35808 .7575 .34403 Metetra .711 .751 .883 r .7575 .34403 .7674 .35808 .78944 .3478 .77965 .33352 Metetra .707 .748 .884 r .77965 .33352 .78944 .3478 .81163 .3377 .80196 .32321 Metetra .703 .745 .884 r .80196 .32321 .81163 .3377 .83397 .32776 .82442 .31308 Metetra .7 .743 .884 r .82442 .31308 .83397 .32776 .85647 .31797 .84705 .30313 Metetra .697 .74 .885 r .84705 .30313 .85647 .31797 .87914 .30832 .86984 .29334 Metetra .747 .884 .942 r .24393 .6308 .25735 .63567 .27717 .62459 .26389 .61967 Metetra .754 .884 .938 r .26389 .61967 .27717 .62459 .29709 .61318 .28395 .60815 Metetra .759 .884 .935 r .28395 .60815 .29709 .61318 .31712 .60146 .30413 .59627 Metetra .765 .883 .931 r .30413 .59627 .31712 .60146 .33724 .58945 .3244 .58403 Metetra .769 .881 .927 r .3244 .58403 .33724 .58945 .35745 .57717 .34477 .57146 Metetra .773 .877 .923 r .34477 .57146 .35745 .57717 .37774 .56467 .36523 .55861 Metetra .776 .873 .919 r .36523 .55861 .37774 .56467 .39812 .55197 .38577 .54551 Metetra .778 .867 .915 r .38577 .54551 .39812 .55197 .41858 .53912 .40639 .53221 Metetra .778 .861 .911 r .40639 .53221 .41858 .53912 .43911 .52617 .42709 .51876 Metetra .778 .854 .907 r .42709 .51876 .43911 .52617 .45971 .51316 .44785 .50522 Metetra .777 .846 .903 r .44785 .50522 .45971 .51316 .48039 .50013 .46869 .49163 Metetra .775 .838 .9 r .46869 .49163 .48039 .50013 .50115 .48713 .48959 .47806 Metetra .772 .829 .897 r .48959 .47806 .50115 .48713 .52198 .4742 .51057 .46455 Metetra .768 .821 .893 r .51057 .46455 .52198 .4742 .54289 .46138 .53162 .45116 Metetra .764 .812 .891 r .53162 .45116 .54289 .46138 .56388 .4487 .55275 .43792 Metetra .759 .804 .888 r .55275 .43792 .56388 .4487 .58496 .4362 .57396 .42487 Metetra .754 .795 .886 r .57396 .42487 .58496 .4362 .60613 .42388 .59526 .41204 Metetra .749 .788 .884 r .59526 .41204 .60613 .42388 .62741 .41178 .61666 .39946 Metetra .743 .78 .883 r .61666 .39946 .62741 .41178 .64878 .39991 .63815 .38714 Metetra .738 .774 .882 r .63815 .38714 .64878 .39991 .67027 .38826 .65976 .37508 Metetra .732 .767 .881 r .65976 .37508 .67027 .38826 .69188 .37686 .68149 .36331 Metetra .727 .762 .88 r .68149 .36331 .69188 .37686 .71362 .36569 .70334 .35181 Metetra .722 .757 .88 r .70334 .35181 .71362 .36569 .73549 .35474 .72532 .34057 Metetra .717 .752 .88 r .72532 .34057 .73549 .35474 .7575 .34403 .74744 .32959 Metetra .712 .749 .88 r .74744 .32959 .7575 .34403 .77965 .33352 .76971 .31885 Metetra .708 .745 .881 r .76971 .31885 .77965 .33352 .80196 .32321 .79214 .30834 Metetra .704 .742 .882 r .79214 .30834 .80196 .32321 .82442 .31308 .81472 .29804 Metetra .7 .74 .882 r .81472 .29804 .82442 .31308 .84705 .30313 .83747 .28793 Metetra .696 .738 .883 r .83747 .28793 .84705 .30313 .86984 .29334 .86039 .278 Metetra .745 .885 .943 r .23027 .62593 .24393 .6308 .26389 .61967 .25036 .6148 Metetra .752 .887 .94 r .25036 .6148 .26389 .61967 .28395 .60815 .27057 .60323 Metetra .759 .888 .936 r .27057 .60323 .28395 .60815 .30413 .59627 .29089 .59122 Metetra .766 .888 .932 r .29089 .59122 .30413 .59627 .3244 .58403 .31133 .57879 Metetra .771 .886 .928 r .31133 .57879 .3244 .58403 .34477 .57146 .33186 .56597 Metetra .776 .883 .923 r .33186 .56597 .34477 .57146 .36523 .55861 .35248 .55278 Metetra .78 .879 .919 r .35248 .55278 .36523 .55861 .38577 .54551 .3732 .53928 Metetra .783 .873 .914 r .3732 .53928 .38577 .54551 .40639 .53221 .39399 .52552 Metetra .785 .867 .91 r .39399 .52552 .40639 .53221 .42709 .51876 .41485 .51154 Metetra .785 .859 .905 r .41485 .51154 .42709 .51876 .44785 .50522 .43579 .49743 Metetra .784 .851 .901 r .43579 .49743 .44785 .50522 .46869 .49163 .45679 .48323 Metetra .782 .842 .897 r .45679 .48323 .46869 .49163 .48959 .47806 .47785 .46903 Metetra .779 .833 .894 r .47785 .46903 .48959 .47806 .51057 .46455 .49898 .45488 Metetra .775 .824 .89 r .49898 .45488 .51057 .46455 .53162 .45116 .52018 .44085 Metetra .771 .814 .887 r .52018 .44085 .53162 .45116 .55275 .43792 .54145 .42698 Metetra .766 .805 .884 r .54145 .42698 .55275 .43792 .57396 .42487 .5628 .41333 Metetra .76 .796 .882 r .5628 .41333 .57396 .42487 .59526 .41204 .58423 .39994 Metetra .754 .787 .88 r .58423 .39994 .59526 .41204 .61666 .39946 .60575 .38682 Metetra .748 .779 .879 r .60575 .38682 .61666 .39946 .63815 .38714 .62737 .37402 Metetra .741 .772 .878 r .62737 .37402 .63815 .38714 .65976 .37508 .64909 .36153 Metetra .735 .765 .877 r .64909 .36153 .65976 .37508 .68149 .36331 .67093 .34936 Metetra .729 .759 .877 r .67093 .34936 .68149 .36331 .70334 .35181 .6929 .33751 Metetra .724 .754 .877 r .6929 .33751 .70334 .35181 .72532 .34057 .715 .32597 Metetra .718 .75 .877 r .715 .32597 .72532 .34057 .74744 .32959 .73724 .31473 Metetra .713 .746 .877 r .73724 .31473 .74744 .32959 .76971 .31885 .75962 .30377 Metetra .708 .742 .878 r .75962 .30377 .76971 .31885 .79214 .30834 .78217 .29306 Metetra .704 .739 .879 r .78217 .29306 .79214 .30834 .81472 .29804 .80487 .2826 Metetra .7 .737 .88 r .80487 .2826 .81472 .29804 .83747 .28793 .82774 .27235 Metetra .696 .735 .881 r .82774 .27235 .83747 .28793 .86039 .278 .85079 .2623 Metetra .741 .886 .946 r .21637 .62099 .23027 .62593 .25036 .6148 .23658 .60993 Metetra .75 .889 .942 r .23658 .60993 .25036 .6148 .27057 .60323 .25693 .59837 Metetra .758 .891 .938 r .25693 .59837 .27057 .60323 .29089 .59122 .27741 .58631 Metetra .766 .892 .933 r .27741 .58631 .29089 .59122 .31133 .57879 .298 .57374 Metetra .773 .891 .929 r .298 .57374 .31133 .57879 .33186 .56597 .3187 .5607 Metetra .779 .889 .924 r .3187 .5607 .33186 .56597 .35248 .55278 .33949 .54721 Metetra .784 .885 .919 r .33949 .54721 .35248 .55278 .3732 .53928 .36038 .53331 Metetra .788 .88 .913 r .36038 .53331 .3732 .53928 .39399 .52552 .38135 .51907 Metetra .791 .874 .908 r .38135 .51907 .39399 .52552 .41485 .51154 .40239 .50455 Metetra .792 .866 .903 r .40239 .50455 .41485 .51154 .43579 .49743 .4235 .48982 Metetra .792 .857 .899 r .4235 .48982 .43579 .49743 .45679 .48323 .44468 .47497 Metetra .79 .847 .894 r .44468 .47497 .45679 .48323 .47785 .46903 .46591 .46007 Metetra .787 .837 .89 r .46591 .46007 .47785 .46903 .49898 .45488 .4872 .4452 Metetra .783 .827 .886 r .4872 .4452 .49898 .45488 .52018 .44085 .50856 .43045 Metetra .778 .816 .883 r .50856 .43045 .52018 .44085 .54145 .42698 .52998 .41589 Metetra .772 .806 .88 r .52998 .41589 .54145 .42698 .5628 .41333 .55146 .40156 Metetra .766 .796 .878 r .55146 .40156 .5628 .41333 .58423 .39994 .57303 .38754 Metetra .759 .787 .876 r .57303 .38754 .58423 .39994 .60575 .38682 .59467 .37385 Metetra .752 .778 .874 r .59467 .37385 .60575 .38682 .62737 .37402 .61641 .36051 Metetra .745 .77 .873 r .61641 .36051 .62737 .37402 .64909 .36153 .63826 .34755 Metetra .738 .763 .873 r .63826 .34755 .64909 .36153 .67093 .34936 .66022 .33497 Metetra .732 .756 .872 r .66022 .33497 .67093 .34936 .6929 .33751 .6823 .32276 Metetra .725 .751 .873 r .6823 .32276 .6929 .33751 .715 .32597 .70451 .31092 Metetra .719 .746 .873 r .70451 .31092 .715 .32597 .73724 .31473 .72687 .29941 Metetra .714 .742 .874 r .72687 .29941 .73724 .31473 .75962 .30377 .74937 .28824 Metetra .708 .739 .875 r .74937 .28824 .75962 .30377 .78217 .29306 .77204 .27735 Metetra .704 .736 .876 r .77204 .27735 .78217 .29306 .80487 .2826 .79486 .26674 Metetra .699 .734 .878 r .79486 .26674 .80487 .2826 .82774 .27235 .81786 .25637 Metetra .695 .732 .879 r .81786 .25637 .82774 .27235 .85079 .2623 .84104 .24622 Metetra .736 .885 .948 r .20222 .61591 .21637 .62099 .23658 .60993 .22256 .60501 Metetra .746 .889 .944 r .22256 .60501 .23658 .60993 .25693 .59837 .24304 .59354 Metetra .755 .893 .94 r .24304 .59354 .25693 .59837 .27741 .58631 .26366 .58149 Metetra .764 .895 .935 r .26366 .58149 .27741 .58631 .298 .57374 .28441 .56886 Metetra .773 .896 .93 r .28441 .56886 .298 .57374 .3187 .5607 .30528 .55566 Metetra .781 .895 .924 r .30528 .55566 .3187 .5607 .33949 .54721 .32625 .5419 Metetra .788 .892 .919 r .32625 .5419 .33949 .54721 .36038 .53331 .34732 .52764 Metetra .793 .887 .913 r .34732 .52764 .36038 .53331 .38135 .51907 .36847 .51293 Metetra .797 .881 .907 r .36847 .51293 .38135 .51907 .40239 .50455 .3897 .49783 Metetra .799 .873 .901 r .3897 .49783 .40239 .50455 .4235 .48982 .411 .48245 Metetra .8 .863 .896 r .411 .48245 .4235 .48982 .44468 .47497 .43236 .46687 Metetra .799 .853 .891 r .43236 .46687 .44468 .47497 .46591 .46007 .45377 .4512 Metetra .796 .842 .886 r .45377 .4512 .46591 .46007 .4872 .4452 .47523 .43553 Metetra .791 .83 .882 r .47523 .43553 .4872 .4452 .50856 .43045 .49675 .41997 Metetra .786 .818 .878 r .49675 .41997 .50856 .43045 .52998 .41589 .51832 .40462 Metetra .779 .807 .875 r .51832 .40462 .52998 .41589 .55146 .40156 .53995 .38954 Metetra .772 .796 .872 r .53995 .38954 .55146 .40156 .57303 .38754 .56165 .37481 Metetra .764 .785 .87 r .56165 .37481 .57303 .38754 .59467 .37385 .58343 .36047 Metetra .756 .776 .869 r .58343 .36047 .59467 .37385 .61641 .36051 .6053 .34656 Metetra .749 .767 .868 r .6053 .34656 .61641 .36051 .63826 .34755 .62726 .33309 Metetra .741 .759 .868 r .62726 .33309 .63826 .34755 .66022 .33497 .64934 .32007 Metetra .733 .753 .868 r .64934 .32007 .66022 .33497 .6823 .32276 .67154 .3075 Metetra .726 .747 .868 r .67154 .3075 .6823 .32276 .70451 .31092 .69387 .29535 Metetra .72 .742 .869 r .69387 .29535 .70451 .31092 .72687 .29941 .71634 .2836 Metetra .714 .738 .87 r .71634 .2836 .72687 .29941 .74937 .28824 .73896 .27222 Metetra .708 .735 .872 r .73896 .27222 .74937 .28824 .77204 .27735 .76175 .26118 Metetra .703 .732 .874 r .76175 .26118 .77204 .27735 .79486 .26674 .7847 .25045 Metetra .698 .731 .876 r .7847 .25045 .79486 .26674 .81786 .25637 .80783 .23999 Metetra .694 .729 .877 r .80783 .23999 .81786 .25637 .84104 .24622 .83114 .22976 Metetra .73 .882 .951 r .18783 .6106 .20222 .61591 .22256 .60501 .20829 .59994 Metetra .74 .888 .947 r .20829 .59994 .22256 .60501 .24304 .59354 .2289 .58866 Metetra .751 .894 .943 r .2289 .58866 .24304 .59354 .26366 .58149 .24966 .57672 Metetra .761 .898 .938 r .24966 .57672 .26366 .58149 .28441 .56886 .27055 .56412 Metetra .772 .9 .932 r .27055 .56412 .28441 .56886 .30528 .55566 .29158 .55083 Metetra .782 .9 .926 r .29158 .55083 .30528 .55566 .32625 .5419 .31273 .53688 Metetra .79 .898 .919 r .31273 .53688 .32625 .5419 .34732 .52764 .33399 .52229 Metetra .798 .894 .912 r .33399 .52229 .34732 .52764 .36847 .51293 .35534 .50713 Metetra .803 .888 .906 r .35534 .50713 .36847 .51293 .3897 .49783 .37676 .49145 Metetra .807 .88 .899 r .37676 .49145 .3897 .49783 .411 .48245 .39826 .47537 Metetra .809 .87 .893 r .39826 .47537 .411 .48245 .43236 .46687 .41981 .459 Metetra .808 .859 .887 r .41981 .459 .43236 .46687 .45377 .4512 .44141 .44246 Metetra .805 .846 .881 r .44141 .44246 .45377 .4512 .47523 .43553 .46306 .42589 Metetra .8 .833 .877 r .46306 .42589 .47523 .43553 .49675 .41997 .48475 .40941 Metetra .794 .82 .872 r .48475 .40941 .49675 .41997 .51832 .40462 .50648 .39315 Metetra .787 .807 .869 r .50648 .39315 .51832 .40462 .53995 .38954 .52827 .37721 Metetra .779 .795 .866 r .52827 .37721 .53995 .38954 .56165 .37481 .55011 .36168 Metetra .77 .783 .864 r .55011 .36168 .56165 .37481 .58343 .36047 .57202 .34662 Metetra .761 .773 .863 r .57202 .34662 .58343 .36047 .6053 .34656 .59401 .33208 Metetra .752 .763 .862 r .59401 .33208 .6053 .34656 .62726 .33309 .6161 .31807 Metetra .743 .755 .862 r .6161 .31807 .62726 .33309 .64934 .32007 .63829 .3046 Metetra .735 .748 .862 r .63829 .3046 .64934 .32007 .67154 .3075 .66061 .29166 Metetra .727 .742 .864 r .66061 .29166 .67154 .3075 .69387 .29535 .68306 .27922 Metetra .72 .737 .865 r .68306 .27922 .69387 .29535 .71634 .2836 .70565 .26724 Metetra .713 .733 .867 r .70565 .26724 .71634 .2836 .73896 .27222 .72839 .2557 Metetra .707 .731 .869 r .72839 .2557 .73896 .27222 .76175 .26118 .7513 .24453 Metetra .702 .728 .871 r .7513 .24453 .76175 .26118 .7847 .25045 .77438 .23371 Metetra .697 .727 .874 r .77438 .23371 .7847 .25045 .80783 .23999 .79764 .22319 Metetra .692 .726 .876 r .79764 .22319 .80783 .23999 .83114 .22976 .82108 .21292 Metetra .721 .877 .953 r .17322 .60491 .18783 .6106 .20829 .59994 .19377 .59458 Metetra .732 .885 .95 r .19377 .59458 .20829 .59994 .2289 .58866 .2145 .5836 Metetra .744 .892 .946 r .2145 .5836 .2289 .58866 .24966 .57672 .23538 .57189 Metetra .756 .898 .941 r .23538 .57189 .24966 .57672 .27055 .56412 .25643 .55943 Metetra .769 .903 .935 r .25643 .55943 .27055 .56412 .29158 .55083 .27762 .54617 Metetra .781 .905 .928 r .27762 .54617 .29158 .55083 .31273 .53688 .29894 .53212 Metetra .792 .904 .92 r .29894 .53212 .31273 .53688 .33399 .52229 .32038 .51728 Metetra .802 .901 .912 r .32038 .51728 .33399 .52229 .35534 .50713 .34193 .50169 Metetra .809 .896 .904 r .34193 .50169 .35534 .50713 .37676 .49145 .36356 .48544 Metetra .815 .888 .896 r .36356 .48544 .37676 .49145 .39826 .47537 .38527 .46863 Metetra .818 .877 .889 r .38527 .46863 .39826 .47537 .41981 .459 .40703 .45139 Metetra .818 .865 .882 r .40703 .45139 .41981 .459 .44141 .44246 .42884 .43388 Metetra .815 .851 .876 r .42884 .43388 .44141 .44246 .46306 .42589 .45068 .41627 Metetra .81 .836 .87 r .45068 .41627 .46306 .42589 .48475 .40941 .47255 .39873 Metetra .804 .821 .865 r .47255 .39873 .48475 .40941 .50648 .39315 .49446 .38143 Metetra .795 .807 .861 r .49446 .38143 .50648 .39315 .52827 .37721 .5164 .36449 Metetra .786 .793 .858 r .5164 .36449 .52827 .37721 .55011 .36168 .53839 .34805 Metetra .776 .78 .856 r .53839 .34805 .55011 .36168 .57202 .34662 .56044 .33219 Metetra .765 .768 .855 r .56044 .33219 .57202 .34662 .59401 .33208 .58256 .31695 Metetra .755 .758 .855 r .58256 .31695 .59401 .33208 .6161 .31807 .60477 .30237 Metetra .745 .749 .855 r .60477 .30237 .6161 .31807 .63829 .3046 .62708 .28845 Metetra .736 .742 .857 r .62708 .28845 .63829 .3046 .66061 .29166 .64951 .27515 Metetra .727 .736 .858 r .64951 .27515 .66061 .29166 .68306 .27922 .67208 .26245 Metetra .719 .731 .86 r .67208 .26245 .68306 .27922 .70565 .26724 .69479 .25029 Metetra .712 .728 .863 r .69479 .25029 .70565 .26724 .72839 .2557 .71765 .23862 Metetra .705 .725 .866 r .71765 .23862 .72839 .2557 .7513 .24453 .74069 .22738 Metetra .699 .724 .869 r .74069 .22738 .7513 .24453 .77438 .23371 .7639 .21653 Metetra .694 .723 .872 r .7639 .21653 .77438 .23371 .79764 .22319 .78729 .20599 Metetra .69 .723 .875 r .78729 .20599 .79764 .22319 .82108 .21292 .81088 .19573 Metetra .711 .869 .956 r .15838 .59871 .17322 .60491 .19377 .59458 .17902 .58881 Metetra .723 .879 .953 r .17902 .58881 .19377 .59458 .2145 .5836 .19985 .57823 Metetra .735 .889 .95 r .19985 .57823 .2145 .5836 .23538 .57189 .22085 .56689 Metetra .749 .897 .945 r .22085 .56689 .23538 .57189 .25643 .55943 .24203 .55471 Metetra .763 .904 .938 r .24203 .55471 .25643 .55943 .27762 .54617 .26336 .54163 Metetra .778 .908 .93 r .26336 .54163 .27762 .54617 .29894 .53212 .28486 .52761 Metetra .792 .91 .922 r .28486 .52761 .29894 .53212 .32038 .51728 .30649 .51262 Metetra .804 .909 .913 r .30649 .51262 .32038 .51728 .34193 .50169 .32823 .4967 Metetra .815 .904 .903 r .32823 .4967 .34193 .50169 .36356 .48544 .35008 .47989 Metetra .823 .896 .894 r .35008 .47989 .36356 .48544 .38527 .46863 .37201 .46232 Metetra .827 .885 .885 r .37201 .46232 .38527 .46863 .40703 .45139 .394 .44414 Metetra .829 .872 .876 r .394 .44414 .40703 .45139 .42884 .43388 .41603 .42553 Metetra .827 .856 .869 r .41603 .42553 .42884 .43388 .45068 .41627 .43808 .40672 Metetra .822 .84 .862 r .43808 .40672 .45068 .41627 .47255 .39873 .46016 .38794 Metetra .814 .823 .857 r .46016 .38794 .47255 .39873 .49446 .38143 .48225 .36941 Metetra .804 .806 .853 r .48225 .36941 .49446 .38143 .5164 .36449 .50436 .35132 Metetra .793 .79 .85 r .50436 .35132 .5164 .36449 .53839 .34805 .5265 .33383 Metetra .781 .775 .848 r .5265 .33383 .53839 .34805 .56044 .33219 .54869 .31706 Metetra .769 .763 .847 r .54869 .31706 .56044 .33219 .58256 .31695 .57094 .30108 Metetra .757 .751 .847 r .57094 .30108 .58256 .31695 .60477 .30237 .59327 .2859 Metetra .746 .742 .848 r .59327 .2859 .60477 .30237 .62708 .28845 .6157 .27152 Metetra .736 .735 .85 r .6157 .27152 .62708 .28845 .64951 .27515 .63825 .25791 Metetra .726 .729 .853 r .63825 .25791 .64951 .27515 .67208 .26245 .66093 .24499 Metetra .717 .725 .856 r .66093 .24499 .67208 .26245 .69479 .25029 .68376 .23271 Metetra .709 .722 .859 r .68376 .23271 .69479 .25029 .71765 .23862 .70675 .22099 Metetra .703 .72 .863 r .70675 .22099 .71765 .23862 .74069 .22738 .72991 .20975 Metetra .697 .719 .867 r .72991 .20975 .74069 .22738 .7639 .21653 .75326 .19892 Metetra .691 .719 .871 r .75326 .19892 .7639 .21653 .78729 .20599 .77679 .18843 Metetra .687 .72 .874 r .77679 .18843 .78729 .20599 .81088 .19573 .80052 .17822 Metetra .7 .858 .958 r .14334 .59185 .15838 .59871 .17902 .58881 .16406 .58247 Metetra .711 .87 .957 r .16406 .58247 .17902 .58881 .19985 .57823 .18496 .57241 Metetra .724 .882 .954 r .18496 .57241 .19985 .57823 .22085 .56689 .20606 .56157 Metetra .739 .893 .949 r .20606 .56157 .22085 .56689 .24203 .55471 .22735 .54985 Metetra .755 .903 .943 r .22735 .54985 .24203 .55471 .26336 .54163 .24882 .53713 Metetra .771 .911 .935 r .24882 .53713 .26336 .54163 .28486 .52761 .27047 .52332 Metetra .788 .916 .925 r .27047 .52332 .28486 .52761 .30649 .51262 .29228 .50836 Metetra .804 .917 .914 r .29228 .50836 .30649 .51262 .32823 .4967 .31423 .49223 Metetra .819 .914 .903 r .31423 .49223 .32823 .4967 .35008 .47989 .3363 .47495 Metetra .83 .907 .891 r .3363 .47495 .35008 .47989 .37201 .46232 .35847 .45663 Metetra .838 .895 .88 r .35847 .45663 .37201 .46232 .394 .44414 .3807 .43742 Metetra .841 .881 .87 r .3807 .43742 .394 .44414 .41603 .42553 .40297 .41757 Metetra .84 .863 .861 r .40297 .41757 .41603 .42553 .43808 .40672 .42526 .39736 Metetra .834 .844 .854 r .42526 .39736 .43808 .40672 .46016 .38794 .44755 .37709 Metetra .825 .824 .847 r .44755 .37709 .46016 .38794 .48225 .36941 .46984 .35709 Metetra .814 .805 .843 r .46984 .35709 .48225 .36941 .50436 .35132 .49213 .33763 Metetra .801 .787 .839 r .49213 .33763 .50436 .35132 .5265 .33383 .51443 .31892 Metetra .787 .77 .838 r .51443 .31892 .5265 .33383 .54869 .31706 .53676 .30112 Metetra .773 .756 .838 r .53676 .30112 .54869 .31706 .57094 .30108 .55914 .28432 Metetra .759 .744 .839 r .55914 .28432 .57094 .30108 .59327 .2859 .5816 .26854 Metetra .746 .734 .841 r .5816 .26854 .59327 .2859 .6157 .27152 .60415 .25375 Metetra .734 .727 .844 r .60415 .25375 .6157 .27152 .63825 .25791 .62681 .23988 Metetra .724 .721 .848 r .62681 .23988 .63825 .25791 .66093 .24499 .64961 .22684 Metetra .714 .718 .852 r .64961 .22684 .66093 .24499 .68376 .23271 .67256 .21453 Metetra .706 .716 .857 r .67256 .21453 .68376 .23271 .70675 .22099 .69568 .20284 Metetra .699 .715 .861 r .69568 .20284 .70675 .22099 .72991 .20975 .71898 .19168 Metetra .693 .715 .866 r .71898 .19168 .72991 .20975 .75326 .19892 .74246 .18095 Metetra .688 .716 .871 r .74246 .18095 .75326 .19892 .77679 .18843 .76614 .17057 Metetra .683 .718 .875 r .76614 .17057 .77679 .18843 .80052 .17822 .79002 .16047 Metetra .687 .844 .958 r .12812 .58414 .14334 .59185 .16406 .58247 .14889 .57534 Metetra .697 .857 .959 r .14889 .57534 .16406 .58247 .18496 .57241 .16986 .5659 Metetra .71 .871 .957 r .16986 .5659 .18496 .57241 .20606 .56157 .19104 .55571 Metetra .725 .885 .954 r .19104 .55571 .20606 .56157 .22735 .54985 .21242 .54463 Metetra .742 .899 .949 r .21242 .54463 .22735 .54985 .24882 .53713 .234 .53248 Metetra .761 .91 .941 r .234 .53248 .24882 .53713 .27047 .52332 .25579 .51914 Metetra .781 .919 .93 r .25579 .51914 .27047 .52332 .29228 .50836 .27776 .50446 Metetra .801 .924 .918 r .27776 .50446 .29228 .50836 .31423 .49223 .2999 .48834 Metetra .82 .924 .904 r .2999 .48834 .31423 .49223 .3363 .47495 .32219 .47074 Metetra .836 .918 .89 r .32219 .47074 .3363 .47495 .35847 .45663 .3446 .45172 Metetra .848 .907 .876 r .3446 .45172 .35847 .45663 .3807 .43742 .3671 .43144 Metetra .854 .891 .863 r .3671 .43144 .3807 .43742 .40297 .41757 .38963 .41018 Metetra .854 .871 .852 r .38963 .41018 .40297 .41757 .42526 .39736 .41218 .3883 Metetra .849 .849 .843 r .41218 .3883 .42526 .39736 .44755 .37709 .43472 .36623 Metetra .838 .826 .836 r .43472 .36623 .44755 .37709 .46984 .35709 .45724 .34443 Metetra .825 .803 .831 r .45724 .34443 .46984 .35709 .49213 .33763 .47972 .32329 Metetra .809 .782 .827 r .47972 .32329 .49213 .33763 .51443 .31892 .5022 .30314 Metetra .792 .763 .826 r .5022 .30314 .51443 .31892 .53676 .30112 .52467 .28418 Metetra .775 .747 .827 r .52467 .28418 .53676 .30112 .55914 .28432 .54719 .26652 Metetra .759 .734 .829 r .54719 .26652 .55914 .28432 .5816 .26854 .56976 .25016 Metetra .744 .724 .833 r .56976 .25016 .5816 .26854 .60415 .25375 .59243 .23503 Metetra .731 .717 .838 r .59243 .23503 .60415 .25375 .62681 .23988 .61521 .22102 Metetra .72 .713 .843 r .61521 .22102 .62681 .23988 .64961 .22684 .63813 .20798 Metetra .71 .71 .849 r .63813 .20798 .64961 .22684 .67256 .21453 .66121 .19577 Metetra .701 .71 .855 r .66121 .19577 .67256 .21453 .69568 .20284 .68446 .18424 Metetra .694 .711 .861 r .68446 .18424 .69568 .20284 .71898 .19168 .70789 .17326 Metetra .688 .712 .867 r .70789 .17326 .71898 .19168 .74246 .18095 .73152 .16271 Metetra .683 .714 .872 r .73152 .16271 .74246 .18095 .76614 .17057 .75535 .15251 Metetra .679 .717 .877 r .75535 .15251 .76614 .17057 .79002 .16047 .77937 .14256 Metetra .674 .826 .957 r .11274 .57542 .12812 .58414 .14889 .57534 .13355 .5672 Metetra .683 .84 .959 r .13355 .5672 .14889 .57534 .16986 .5659 .15457 .55845 Metetra .694 .855 .96 r .15457 .55845 .16986 .5659 .19104 .55571 .1758 .54902 Metetra .708 .872 .959 r .1758 .54902 .19104 .55571 .21242 .54463 .19725 .53874 Metetra .725 .889 .955 r .19725 .53874 .21242 .54463 .234 .53248 .21891 .52742 Metetra .745 .905 .948 r .21891 .52742 .234 .53248 .25579 .51914 .2408 .51483 Metetra .769 .919 .938 r .2408 .51483 .25579 .51914 .27776 .50446 .26291 .50076 Metetra .793 .929 .924 r .26291 .50076 .27776 .50446 .2999 .48834 .28523 .48499 Metetra .818 .934 .907 r .28523 .48499 .2999 .48834 .32219 .47074 .30773 .46736 Metetra .84 .931 .889 r .30773 .46736 .32219 .47074 .3446 .45172 .33039 .44782 Metetra .857 .921 .871 r .33039 .44782 .3446 .45172 .3671 .43144 .35317 .42648 Metetra .868 .904 .855 r .35317 .42648 .3671 .43144 .38963 .41018 .376 .40363 Metetra .87 .882 .841 r .376 .40363 .38963 .41018 .41218 .3883 .39885 .37973 Metetra .865 .855 .83 r .39885 .37973 .41218 .3883 .43472 .36623 .42167 .35541 Metetra .853 .827 .821 r .42167 .35541 .43472 .36623 .45724 .34443 .44444 .33134 Metetra .837 .8 .816 r .44444 .33134 .45724 .34443 .47972 .32329 .46714 .30812 Metetra .817 .775 .813 r .46714 .30812 .47972 .32329 .5022 .30314 .48979 .28623 Metetra .796 .753 .813 r .48979 .28623 .5022 .30314 .52467 .28418 .51242 .26597 Metetra .776 .735 .815 r .51242 .26597 .52467 .28418 .54719 .26652 .53507 .24744 Metetra .757 .722 .819 r .53507 .24744 .54719 .26652 .56976 .25016 .55776 .23059 Metetra .74 .712 .825 r .55776 .23059 .56976 .25016 .59243 .23503 .58054 .21529 Metetra .726 .706 .832 r .58054 .21529 .59243 .23503 .61521 .22102 .60344 .20133 Metetra .714 .704 .839 r .60344 .20133 .61521 .22102 .63813 .20798 .62649 .18848 Metetra .703 .703 .847 r .62649 .18848 .63813 .20798 .66121 .19577 .6497 .17654 Metetra .695 .705 .855 r .6497 .17654 .66121 .19577 .68446 .18424 .67308 .16531 Metetra .688 .707 .862 r .67308 .16531 .68446 .18424 .70789 .17326 .69666 .15463 Metetra .683 .71 .869 r .69666 .15463 .70789 .17326 .73152 .16271 .72043 .14436 Metetra .678 .714 .875 r .72043 .14436 .73152 .16271 .75535 .15251 .74441 .13439 Metetra .675 .717 .881 r .74441 .13439 .75535 .15251 .77937 .14256 .7686 .12463 Metetra .662 .807 .953 r .09722 .56556 .11274 .57542 .13355 .5672 .11808 .55788 Metetra .668 .819 .956 r .11808 .55788 .13355 .5672 .15457 .55845 .13913 .54979 Metetra .677 .834 .959 r .13913 .54979 .15457 .55845 .1758 .54902 .1604 .54115 Metetra .689 .851 .961 r .1604 .54115 .1758 .54902 .19725 .53874 .18188 .53179 Metetra .704 .871 .96 r .18188 .53179 .19725 .53874 .21891 .52742 .2036 .5215 Metetra .724 .892 .956 r .2036 .5215 .21891 .52742 .2408 .51483 .22555 .50999 Metetra .749 .913 .948 r .22555 .50999 .2408 .51483 .26291 .50076 .24776 .49695 Metetra .777 .93 .934 r .24776 .49695 .26291 .50076 .28523 .48499 .27021 .48201 Metetra .808 .942 .914 r .27021 .48201 .28523 .48499 .30773 .46736 .2929 .46482 Metetra .838 .944 .891 r .2929 .46482 .30773 .46736 .33039 .44782 .3158 .44514 Metetra .864 .937 .867 r .3158 .44514 .33039 .44782 .35317 .42648 .33886 .4229 Metetra .882 .92 .846 r .33886 .4229 .35317 .42648 .376 .40363 .36203 .39832 Metetra .889 .894 .827 r .36203 .39832 .376 .40363 .39885 .37973 .38523 .37198 Metetra .884 .863 .813 r .38523 .37198 .39885 .37973 .42167 .35541 .40838 .34476 Metetra .87 .829 .804 r .40838 .34476 .42167 .35541 .44444 .33134 .43144 .3177 Metetra .849 .795 .798 r .43144 .3177 .44444 .33134 .46714 .30812 .45439 .29181 Metetra .824 .765 .796 r .45439 .29181 .46714 .30812 .48979 .28623 .47724 .26782 Metetra .799 .74 .798 r .47724 .26782 .48979 .28623 .51242 .26597 .50003 .24613 Metetra .774 .72 .802 r .50003 .24613 .51242 .26597 .53507 .24744 .5228 .22681 Metetra .752 .707 .809 r .5228 .22681 .53507 .24744 .55776 .23059 .54561 .20972 Metetra .733 .699 .818 r .54561 .20972 .55776 .23059 .58054 .21529 .5685 .19453 Metetra .718 .695 .828 r .5685 .19453 .58054 .21529 .60344 .20133 .59152 .18091 Metetra .705 .695 .838 r .59152 .18091 .60344 .20133 .62649 .18848 .61469 .16852 Metetra .695 .698 .848 r .61469 .16852 .62649 .18848 .6497 .17654 .63803 .15706 Metetra .687 .701 .858 r .63803 .15706 .6497 .17654 .67308 .16531 .66156 .14628 Metetra .681 .706 .867 r .66156 .14628 .67308 .16531 .69666 .15463 .68529 .136 Metetra .677 .711 .874 r .68529 .136 .69666 .15463 .72043 .14436 .70921 .12606 Metetra .673 .716 .881 r .70921 .12606 .72043 .14436 .74441 .13439 .73335 .11635 Metetra .671 .72 .886 r .73335 .11635 .74441 .13439 .7686 .12463 .75769 .10679 Metetra .653 .789 .947 r .08157 .55456 .09722 .56556 .11808 .55788 .10247 .54727 Metetra .656 .798 .95 r .10247 .54727 .11808 .55788 .13913 .54979 .12357 .53971 Metetra .662 .81 .954 r .12357 .53971 .13913 .54979 .1604 .54115 .14486 .53179 Metetra .67 .825 .958 r .14486 .53179 .1604 .54115 .18188 .53179 .16637 .52335 Metetra .682 .845 .961 r .16637 .52335 .18188 .53179 .2036 .5215 .18811 .51418 Metetra .698 .868 .962 r .18811 .51418 .2036 .5215 .22555 .50999 .21009 .504 Metetra .721 .895 .958 r .21009 .504 .22555 .50999 .24776 .49695 .23234 .49242 Metetra .751 .921 .947 r .23234 .49242 .24776 .49695 .27021 .48201 .25486 .47892 Metetra .787 .943 .927 r .25486 .47892 .27021 .48201 .2929 .46482 .27768 .46293 Metetra .827 .956 .899 r .27768 .46293 .2929 .46482 .3158 .44514 .30078 .44383 Metetra .865 .955 .866 r .30078 .44383 .3158 .44514 .33886 .4229 .32412 .42117 Metetra .894 .939 .835 r .32412 .42117 .33886 .4229 .36203 .39832 .34765 .39489 Metetra .909 .91 .81 r .34765 .39489 .36203 .39832 .38523 .37198 .37126 .36556 Metetra .907 .872 .792 r .37126 .36556 .38523 .37198 .40838 .34476 .39483 .33446 Metetra .89 .829 .781 r .39483 .33446 .40838 .34476 .43144 .3177 .41826 .30331 Metetra .863 .788 .776 r .41826 .30331 .43144 .3177 .45439 .29181 .4415 .27385 Metetra .83 .751 .776 r .4415 .27385 .45439 .29181 .47724 .26782 .46456 .2473 Metetra .797 .721 .781 r .46456 .2473 .47724 .26782 .50003 .24613 .4875 .22418 Metetra .767 .701 .789 r .4875 .22418 .50003 .24613 .5228 .22681 .51039 .20441 Metetra .742 .688 .8 r .51039 .20441 .5228 .22681 .54561 .20972 .53331 .18753 Metetra .721 .684 .813 r .53331 .18753 .54561 .20972 .5685 .19453 .55631 .17295 Metetra .705 .684 .827 r .55631 .17295 .5685 .19453 .59152 .18091 .57945 .16008 Metetra .694 .689 .841 r .57945 .16008 .59152 .18091 .61469 .16852 .60275 .14845 Metetra .685 .695 .854 r .60275 .14845 .61469 .16852 .63803 .15706 .62623 .13767 Metetra .678 .702 .865 r .62623 .13767 .63803 .15706 .66156 .14628 .6499 .12746 Metetra .674 .709 .874 r .6499 .12746 .66156 .14628 .68529 .136 .67378 .11763 Metetra .671 .715 .882 r .67378 .11763 .68529 .136 .70921 .12606 .69786 .10802 Metetra .669 .721 .888 r .69786 .10802 .70921 .12606 .73335 .11635 .72215 .09854 Metetra .667 .725 .893 r .72215 .09854 .73335 .11635 .75769 .10679 .74666 .08913 Metetra .647 .774 .941 r .06579 .54256 .08157 .55456 .10247 .54727 .08675 .53547 Metetra .649 .78 .943 r .08675 .53547 .10247 .54727 .12357 .53971 .10789 .52823 Metetra .651 .787 .947 r .10789 .52823 .12357 .53971 .14486 .53179 .12923 .52079 Metetra .655 .798 .951 r .12923 .52079 .14486 .53179 .16637 .52335 .15077 .51307 Metetra .661 .813 .956 r .15077 .51307 .16637 .52335 .18811 .51418 .17252 .50491 Metetra .672 .834 .961 r .17252 .50491 .18811 .51418 .21009 .504 .1945 .49608 Metetra .688 .861 .964 r .1945 .49608 .21009 .504 .23234 .49242 .21674 .48622 Metetra .714 .894 .962 r .21674 .48622 .23234 .49242 .25486 .47892 .23926 .47477 Metetra .75 .93 .948 r .23926 .47477 .25486 .47892 .27768 .46293 .26211 .46093 Metetra .797 .959 .917 r .26211 .46093 .27768 .46293 .30078 .44383 .28532 .44364 Metetra .85 .971 .873 r .28532 .44364 .30078 .44383 .32412 .42117 .30888 .42174 Metetra .898 .961 .825 r .30888 .42174 .32412 .42117 .34765 .39489 .33278 .39432 Metetra .928 .93 .786 r .33278 .39432 .34765 .39489 .37126 .36556 .35688 .36148 Metetra .933 .883 .762 r .35688 .36148 .37126 .36556 .39483 .33446 .38099 .32493 Metetra .913 .83 .75 r .38099 .32493 .39483 .33446 .41826 .30331 .4049 .28776 Metetra .876 .776 .748 r .4049 .28776 .41826 .30331 .4415 .27385 .42851 .25332 Metetra .831 .73 .753 r .42851 .25332 .4415 .27385 .46456 .2473 .4518 .22376 Metetra .788 .696 .762 r .4518 .22376 .46456 .2473 .4875 .22418 .47488 .19961 Metetra .752 .675 .776 r .47488 .19961 .4875 .22418 .51039 .20441 .49787 .1802 Metetra .724 .668 .794 r .49787 .1802 .51039 .20441 .53331 .18753 .52088 .16439 Metetra .704 .67 .814 r .52088 .16439 .53331 .18753 .55631 .17295 .54398 .1511 Metetra .69 .678 .833 r .54398 .1511 .55631 .17295 .57945 .16008 .56724 .13945 Metetra .681 .688 .851 r .56724 .13945 .57945 .16008 .60275 .14845 .59067 .12882 Metetra .674 .699 .865 r .59067 .12882 .60275 .14845 .62623 .13767 .61429 .11882 Metetra .67 .709 .877 r .61429 .11882 .62623 .13767 .6499 .12746 .63811 .10918 Metetra .668 .717 .885 r .63811 .10918 .6499 .12746 .67378 .11763 .66214 .09972 Metetra .666 .723 .892 r .66214 .09972 .67378 .11763 .69786 .10802 .68638 .09035 Metetra .665 .728 .896 r .68638 .09035 .69786 .10802 .72215 .09854 .71082 .08101 Metetra .665 .732 .9 r .71082 .08101 .72215 .09854 .74666 .08913 .73548 .07164 Metetra .645 .766 .936 r .04982 .52987 .06579 .54256 .08675 .53547 .07086 .52278 Metetra .646 .768 .937 r .07086 .52278 .08675 .53547 .10789 .52823 .09208 .51562 Metetra .646 .771 .939 r .09208 .51562 .10789 .52823 .12923 .52079 .11349 .50837 Metetra .647 .776 .942 r .11349 .50837 .12923 .52079 .15077 .51307 .13508 .50099 Metetra .649 .784 .946 r .13508 .50099 .15077 .51307 .17252 .50491 .15687 .49344 Metetra .653 .796 .951 r .15687 .49344 .17252 .50491 .1945 .49608 .17886 .48559 Metetra .66 .816 .958 r .17886 .48559 .1945 .49608 .21674 .48622 .20108 .47724 Metetra .674 .846 .965 r .20108 .47724 .21674 .48622 .23926 .47477 .22356 .46798 Metetra .699 .887 .967 r .22356 .46798 .23926 .47477 .26211 .46093 .24634 .45708 Metetra .742 .937 .95 r .24634 .45708 .26211 .46093 .28532 .44364 .26949 .44324 Metetra .806 .977 .9 r .26949 .44324 .28532 .44364 .30888 .42174 .29311 .4244 Metetra .88 .984 .825 r .29311 .4244 .30888 .42174 .33278 .39432 .31727 .39789 Metetra .939 .954 .756 r .31727 .39789 .33278 .39432 .35688 .36148 .34192 .3617 Metetra .96 .897 .718 r .34192 .3617 .35688 .36148 .38099 .32493 .36678 .3171 Metetra .939 .828 .708 r .36678 .3171 .38099 .32493 .4049 .28776 .39142 .27012 Metetra .887 .758 .714 r .39142 .27012 .4049 .28776 .42851 .25332 .41551 .22838 Metetra .822 .697 .727 r .41551 .22838 .42851 .25332 .4518 .22376 .43902 .19592 Metetra .765 .659 .745 r .43902 .19592 .4518 .22376 .47488 .19961 .46219 .17228 Metetra .724 .645 .77 r .46219 .17228 .47488 .19961 .49787 .1802 .48523 .15493 Metetra .697 .65 .798 r .48523 .15493 .49787 .1802 .52088 .16439 .50831 .14144 Metetra .682 .665 .827 r .50831 .14144 .52088 .16439 .54398 .1511 .53151 .1301 Metetra .673 .683 .851 r .53151 .1301 .54398 .1511 .56724 .13945 .55489 .11989 Metetra .669 .699 .869 r .55489 .11989 .56724 .13945 .59067 .12882 .57845 .11025 Metetra .666 .712 .882 r .57845 .11025 .59067 .12882 .61429 .11882 .60222 .10087 Metetra .665 .722 .891 r .60222 .10087 .61429 .11882 .63811 .10918 .62618 .09158 Metetra .664 .728 .897 r .62618 .09158 .63811 .10918 .66214 .09972 .65036 .0823 Metetra .664 .733 .901 r .65036 .0823 .66214 .09972 .68638 .09035 .67474 .07299 Metetra .664 .736 .903 r .67474 .07299 .68638 .09035 .71082 .08101 .69934 .06363 Metetra .664 .738 .905 r .69934 .06363 .71082 .08101 .73548 .07164 .72416 .0542 Metetra .645 .763 .935 r .03363 .5168 .04982 .52987 .07086 .52278 .05476 .50963 Metetra .645 .764 .935 r .05476 .50963 .07086 .52278 .09208 .51562 .07608 .50239 Metetra .645 .764 .935 r .07608 .50239 .09208 .51562 .11349 .50837 .09758 .49508 Metetra .645 .765 .936 r .09758 .49508 .11349 .50837 .13508 .50099 .11926 .48772 Metetra .645 .767 .937 r .11926 .48772 .13508 .50099 .15687 .49344 .14112 .48028 Metetra .646 .771 .939 r .14112 .48028 .15687 .49344 .17886 .48559 .16318 .47275 Metetra .646 .778 .943 r .16318 .47275 .17886 .48559 .20108 .47724 .18544 .46509 Metetra .649 .791 .95 r .18544 .46509 .20108 .47724 .22356 .46798 .2079 .4572 Metetra .655 .817 .961 r .2079 .4572 .22356 .46798 .24634 .45708 .2306 .44879 Metetra .672 .863 .972 r .2306 .44879 .24634 .45708 .26949 .44324 .25359 .43911 Metetra .715 .934 .96 r .25359 .43911 .26949 .44324 .29311 .4244 .27698 .42627 Metetra .802 .993 .873 r .27698 .42627 .29311 .4244 .31727 .39789 .30103 .40588 Metetra .914 .98 .732 r .30103 .40588 .31727 .39789 .34192 .3617 .32606 .3702 Metetra .98 .911 .65 r .32606 .3702 .34192 .3617 .36678 .3171 .35204 .31368 Metetra .969 .825 .646 r .35204 .31368 .36678 .3171 .39142 .27012 .37797 .24779 Metetra .889 .726 .674 r .37797 .24779 .39142 .27012 .41551 .22838 .40271 .19529 Metetra .786 .642 .703 r .40271 .19529 .41551 .22838 .43902 .19592 .42631 .16309 Metetra .715 .609 .738 r .42631 .16309 .43902 .19592 .46219 .17228 .44942 .14407 Metetra .68 .62 .782 r .44942 .14407 .46219 .17228 .48523 .15493 .47244 .1312 Metetra .668 .652 .825 r .47244 .1312 .48523 .15493 .50831 .14144 .49558 .12077 Metetra .664 .684 .859 r .49558 .12077 .50831 .14144 .53151 .1301 .51889 .11125 Metetra .663 .707 .88 r .51889 .11125 .53151 .1301 .55489 .11989 .54238 .10202 Metetra .663 .722 .893 r .54238 .10202 .55489 .11989 .57845 .11025 .56608 .09285 Metetra .663 .731 .9 r .56608 .09285 .57845 .11025 .60222 .10087 .58998 .08365 Metetra .664 .735 .903 r .58998 .08365 .60222 .10087 .62618 .09158 .61409 .07439 Metetra .664 .738 .905 r .61409 .07439 .62618 .09158 .65036 .0823 .63841 .06506 Metetra .664 .739 .906 r .63841 .06506 .65036 .0823 .67474 .07299 .66294 .05565 Metetra .664 .74 .907 r .66294 .05565 .67474 .07299 .69934 .06363 .68769 .04616 Metetra .664 .74 .907 r .68769 .04616 .69934 .06363 .72416 .0542 .71266 .03659 Metetra .645 .763 .934 r .01719 .50352 .03363 .5168 .05476 .50963 .03842 .49623 Metetra .645 .763 .934 r .03842 .49623 .05476 .50963 .07608 .50239 .05983 .48887 Metetra .645 .763 .934 r .05983 .48887 .07608 .50239 .09758 .49508 .08143 .48146 Metetra .645 .763 .934 r .08143 .48146 .09758 .49508 .11926 .48772 .10321 .47398 Metetra .645 .763 .935 r .10321 .47398 .11926 .48772 .14112 .48028 .12518 .46643 Metetra .645 .763 .935 r .12518 .46643 .14112 .48028 .16318 .47275 .14735 .45882 Metetra .645 .764 .935 r .14735 .45882 .16318 .47275 .18544 .46509 .1697 .45115 Metetra .645 .765 .936 r .1697 .45115 .18544 .46509 .2079 .4572 .19225 .4434 Metetra .645 .769 .939 r .19225 .4434 .2079 .4572 .2306 .44879 .215 .43559 Metetra .645 .78 .946 r .215 .43559 .2306 .44879 .25359 .43911 .23796 .42767 Metetra .645 .811 .963 r .23796 .42767 .25359 .43911 .27698 .42627 .26113 .41948 Metetra .653 .889 .982 r .26113 .41948 .27698 .42627 .30103 .40588 .28462 .40977 Metetra .737 .992 .849 r .28462 .40977 .30103 .40588 .32606 .3702 .30893 .3905 Metetra .958 .92 .568 r .30893 .3905 .32606 .3702 .35204 .31368 .33598 .3262 Metetra .996 .815 .544 r .33598 .3262 .35204 .31368 .37797 .24779 .36517 .20977 Metetra .846 .654 .642 r .36517 .20977 .37797 .24779 .40271 .19529 .39039 .15029 Metetra .673 .538 .692 r .39039 .15029 .40271 .19529 .42631 .16309 .41348 .13177 Metetra .645 .582 .77 r .41348 .13177 .42631 .16309 .44942 .14407 .43638 .12146 Metetra .654 .658 .842 r .43638 .12146 .44942 .14407 .47244 .1312 .45941 .11233 Metetra .66 .706 .881 r .45941 .11233 .47244 .1312 .49558 .12077 .48264 .1033 Metetra .663 .727 .897 r .48264 .1033 .49558 .12077 .51889 .11125 .50606 .09422 Metetra .663 .736 .904 r .50606 .09422 .51889 .11125 .54238 .10202 .52969 .08507 Metetra .664 .739 .906 r .52969 .08507 .54238 .10202 .56608 .09285 .55352 .07583 Metetra .664 .74 .907 r .55352 .07583 .56608 .09285 .58998 .08365 .57756 .06652 Metetra .664 .741 .907 r .57756 .06652 .58998 .08365 .61409 .07439 .60181 .05712 Metetra .664 .741 .907 r .60181 .05712 .61409 .07439 .63841 .06506 .62628 .04765 Metetra .664 .741 .907 r .62628 .04765 .63841 .06506 .66294 .05565 .65095 .03809 Metetra .664 .741 .907 r .65095 .03809 .66294 .05565 .68769 .04616 .67585 .02844 Metetra .664 .741 .907 r .67585 .02844 .68769 .04616 .71266 .03659 .70097 .01871 Metetra .645 .763 .934 r .00049 .49002 .01719 .50352 .03842 .49623 .02182 .48262 Metetra .645 .763 .934 r .02182 .48262 .03842 .49623 .05983 .48887 .04333 .47515 Metetra .645 .763 .934 r .04333 .47515 .05983 .48887 .08143 .48146 .06503 .46761 Metetra .645 .763 .934 r .06503 .46761 .08143 .48146 .10321 .47398 .08691 .46001 Metetra .645 .763 .934 r .08691 .46001 .10321 .47398 .12518 .46643 .10899 .45235 Metetra .645 .763 .934 r .10899 .45235 .12518 .46643 .14735 .45882 .13126 .44461 Metetra .645 .763 .934 r .13126 .44461 .14735 .45882 .1697 .45115 .15372 .43681 Metetra .645 .763 .934 r .15372 .43681 .1697 .45115 .19225 .4434 .17639 .42894 Metetra .645 .763 .934 r .17639 .42894 .19225 .4434 .215 .43559 .19926 .421 Metetra .645 .763 .935 r .19926 .421 .215 .43559 .23796 .42767 .22233 .41299 Metetra .645 .764 .935 r .22233 .41299 .23796 .42767 .26113 .41948 .2456 .40491 Metetra .645 .767 .937 r .2456 .40491 .26113 .41948 .28462 .40977 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