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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 149322, 3653]*) (*NotebookOutlinePosition[ 186382, 4962]*) (* CellTagsIndexPosition[ 186338, 4958]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Paschen-Back Effect", "Title", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "from ", StyleBox[ButtonBox["Essential Mathematica for Students of Science", Active->False, ButtonData:>{ URL[ "http://www.physics.umd.edu/courses/CourseWare/EssentialMathematica/"], None}, Active->False, ButtonStyle->"Hyperlink"], Active->False], "\n\[Copyright] James J. Kelly, 1998\nLast revised: January, 2006" }], "Author", Active->True, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "In the absence of magnetic interactions, the ground state of a hydrogen \ atom would display a 4-fold degeneracy arising from the ", Cell[BoxData[ \(TraditionalForm\`2\[Times]2\)]], " combinations its two spin-", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "constituents. The hyperfine interaction between the electron and proton \ spins, represented by an interaction hamiltonian of the form ", Cell[BoxData[ \(TraditionalForm\`H\_hf = \ \(\[CurlyEpsilon]\_0\) S\_e\[CenterDot]S\_p\)]], " alleviates part of this degeneracy. Thus, one finds the singlet state \ (with total spin zero) is approximately ", Cell[BoxData[ \(TraditionalForm\`5.9\ \[Mu]eV\)]], " lower in energy than the triplet state (with total spin 1); nevertheless, \ a 3-fold degeneracy remains among the triplet substates. Now suppose that \ the hydrogen atom is placed in an external magnetic field. The interaction \ between the spins and the external magnetic field can be represented by an \ interaction hamiltonian of the form ", Cell[BoxData[ \(TraditionalForm\`H\_ext = \ \((\(\[Gamma]\_e\) S\_e - \ \(\[Gamma]\_p\) S\_p)\)\[CenterDot]B\)]], " and removes the remaining degeneracy, such that four distinct energy \ eigenvalues emerge. For relatively weak magnetic external fields, the \ hyperfine interaction dominates and it is convenient to express the \ eigenstates in terms of the eigenstates of total spin; the behavior of the \ energy levels under these conditions is described as the ", StyleBox["Zeeman effect", FontSlant->"Italic"], ". For strong external magnetic fields, it is more convenient to employ \ eigenstates of the individual spin components relative to the external field \ and the linear dependence of the energy levels is described as the ", StyleBox["Paschen-Back", FontSlant->"Italic"], " effect. This effect is important because it provides a means for \ measuring the proton gyromagnetic ratio. " }], "Text"], Cell[TextData[{ "Glossary:\n\t", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_0\)]], "\t\t= hyperfine coupling constant\n\t", Cell[BoxData[ \(TraditionalForm\`S\_e\)]], "\t\t= electron spin\n\t", Cell[BoxData[ \(TraditionalForm\`S\_p\)]], "\t\t= proton spin\n\t", Cell[BoxData[ \(TraditionalForm\`B\)]], "\t\t= external magnetic field\n\t", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_e = \(g\_e\) \[Mu]\_B\)]], "\t= magnetic coupling for electron\n\t", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_p = \(g\_p\) \[Mu]\_N\)]], "\t= magnetic coupling for proton\n\t", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_B\)]], "\t\t= Bohr magneton\n\t", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_N\)]], "\t\t= nuclear magneton\n\t", Cell[BoxData[ \(TraditionalForm\`g\_e\)]], "\t\t= electron gyromagnetic ratio\n\t", Cell[BoxData[ \(TraditionalForm\`g\_p\)]], "\t\t= proton gyromagnetic ratio" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[BoxData[{ \(\(ClearAll["\"];\)\), "\n", \(\(Off[General::spell, General::spell1];\)\), "\n", \(\($TextStyle = {FontFamily \[Rule] "\", FontSize \[Rule] 12};\)\)}], "Input"], Cell[BoxData[{ \(\(Needs["\"];\)\), "\[IndentingNewLine]", \(\(Needs["\"];\)\), "\n", \(Needs["\"]\)}], "Input"], Cell[BoxData[ \(\(Off[Symbolize::"\"];\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(\[CurlyEpsilon]\_x_\), NotationBoxTag, Editable->True], "]"}], ";", " ", RowBox[{"Symbolize", "[", TagBox[\(\[Gamma]\_x_\), NotationBoxTag, Editable->True], "]"}], ";", " ", RowBox[{"Symbolize", "[", TagBox[\(\[Delta]\_x_\), NotationBoxTag, Editable->True], "]"}]}]], "Input"], Cell[BoxData[{ \(\(norm[v_] := v . v;\)\), "\n", \(\(normalize[v_] := Module[{vnorm}, vnorm = v . v; vnew = v\/\@vnorm];\)\)}], "Input"], Cell[BoxData[{ \(\(Clear[normalize];\)\), "\[IndentingNewLine]", \(normalize[v_] := v\/\@\(v . v\)\)}], "Input"], Cell["\<\ The following simplification functions use known properties of the physical \ constants to simplify algebraic results as much as possible.\ \>", "Text"], Cell[BoxData[{ \(\(MyAssumptions = {\[Gamma]\_e > 0, \[Gamma]\_p > 0, \[Gamma]\_1 > 0, \[Gamma]\_2 > 0, B > 0, \[CurlyEpsilon]\_0 > 0, ratio > 1, \[Delta]\_e + \[Delta]\_p \[GreaterEqual] \ \[CurlyEpsilon]\_0};\)\), "\[IndentingNewLine]", \(\(MySimplify = Simplify[#, MyAssumptions] &;\)\), "\[IndentingNewLine]", \(\(MyFullSimplify = FullSimplify[#, MyAssumptions] &;\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Establish Hamiltonian", "Section"], Cell[TextData[{ "It is convenient to express the hamiltonian in terms of the eigenstates ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]J\[InvisibleComma] M\[RightAngleBracket]\)]], " of total spin, arranged in the order ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]00\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]10\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]11\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]1\[InvisibleComma]\(-1\)\ \[RightAngleBracket]\)]], ". Thus, evaluating the matrix elements, we obtain the following \ hamiltonian matrix." }], "Text", DelimiterMatching->None], Cell[BoxData[ \(\(hamiltonian = Array[h, {4, 4}];\)\)], "Input"], Cell[BoxData[{ \(h[1, 1] = \(-\(\(\(3\)\(\ \)\)\/4\)\) \[CurlyEpsilon]\_0; \ h[2, 2] = \(1\/4\) \[CurlyEpsilon]\_0;\), "\n", \(\(h[3, 3] = \(1\/4\) \[CurlyEpsilon]\_0 + 1\/2\ \((\[Gamma]\_e - \[Gamma]\_p)\)\ B;\)\), "\n", \(\(h[4, 4] = \(1\/4\) \[CurlyEpsilon]\_0 - \(1\/2\) \((\[Gamma]\_e - \ \[Gamma]\_p)\)\ B;\)\), "\n", \(\(h[1, 2] = \(+\(1\/2\)\) \((\[Gamma]\_e + \[Gamma]\_p)\)\ B;\)\), "\n", \(\(h[1, 3] = \(h[1, 4] = \(h[2, 3] = \(h[2, 4] = \(h[3, 4] = 0\)\)\)\);\)\), "\n", \(h[2, 1] = h[1, 2]; h[3, 1] = h[1, 3]; h[3, 2] = h[2, 3];\), "\n", \(h[4, 1] = h[1, 4]; h[4, 2] = h[2, 4]; h[4, 3] = h[3, 4];\)}], "Input"], Cell[TextData[{ "It is convenient to express the field strengths in Tesla and energies in \ ", Cell[BoxData[ \(10\^\(-6\)\)]], " eV. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(values = \({\[CurlyEpsilon]\_0 \[Rule] 5.87\ microElectronVolt, \[Gamma]\_e \[Rule] 2\ Convert[BohrMagneton, ElectronVolt\/Tesla], \[Gamma]\_p \[Rule] 2\[Times]2.78\ Convert[NuclearMagneton, ElectronVolt\/Tesla]} //. ElectronVolt \[Rule] 10\^6\ microElectronVolt\) //. {microElectronVolt \[Rule] 1, Tesla \[Rule] 1}\)], "Input"], Cell[BoxData[ \({\[CurlyEpsilon]\_0 \[Rule] 5.87`, \[Gamma]\_e \[Rule] 115.76765226106399`, \[Gamma]\_p \[Rule] 0.17527631286606707`}\)], "Output"] }, Closed]], Cell[TextData[{ "It will also be useful to define linear transformations between ", Cell[BoxData[ \(TraditionalForm\`\((\[Gamma]\_e, \[Gamma]\_p)\)\)]], " and their sum and difference ", Cell[BoxData[ \(TraditionalForm\`\((\[Gamma]\_1, \[Gamma]\_2)\)\)]], ", such that" }], "Text"], Cell[BoxData[ \(\(transep12 = {\[Gamma]\_e \[Rule] \(\[Gamma]\_1 + \[Gamma]\_2\)\/2, \ \[Gamma]\_p \[Rule] \(\[Gamma]\_1 - \[Gamma]\_2\)\/2};\)\)], "Input"], Cell[BoxData[ \(\(trans12ep = {\[Gamma]\_1 \[Rule] \[Gamma]\_e + \[Gamma]\_p, \ \[Gamma]\_2 \[Rule] \[Gamma]\_e - \[Gamma]\_p};\)\)], "Input"], Cell[TextData[{ "Note that ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_1 \[TildeTilde] \[Gamma]\_2\)]], "." }], "Text"], 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The substates with ", Cell[BoxData[ \(TraditionalForm\`M = 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J = 0\)]], " or 1, are mixed by the interaction, so that their energies are quadratic \ functions of ", Cell[BoxData[ \(TraditionalForm\`B\)]], ". For small ", Cell[BoxData[ \(TraditionalForm\`B\)]], ", the ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]J, M\[RightAngleBracket]\)]], " states are approximate eigenstates. For large ", Cell[BoxData[ \(TraditionalForm\`B\)]], " the approximate eigenstates are have unique spin-projection along the \ field since the hyperfine interaction becomes irrelevant. Therefore, for \ moderate B the states are ordered with increasing energy as\n\t", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]J, M\[RightAngleBracket]\ = \ \[LeftBracketingBar]0, 0\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]1, \ \(-1\)\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]1, 0\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]1, 1\[RightAngleBracket]\)]], " for small ", Cell[BoxData[ \(TraditionalForm\`B\)]], "\n\t", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]S\_z, I\_z\[RightAngleBracket]\ = \ \[LeftBracketingBar]\(-+\)\ \[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]--\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\(+-\)\[RightAngleBracket]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]++\[RightAngleBracket]\)]], " for moderate ", Cell[BoxData[ \(TraditionalForm\`B\)]], ".\nFor very large ", Cell[BoxData[ \(TraditionalForm\`B\)]], ", the ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\(+-\)\[RightAngleBracket]\)]], " state crosses through the ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]++\[RightAngleBracket]\)]], " and has the higher energy." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Eigenvectors", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(vectors = Eigenvectors[hamiltonian /. transep12]\)], "Input"], Cell[BoxData[ \({{0, 0, 0, 1}, {0, 0, 1, 0}, {\(-\(\(\[CurlyEpsilon]\_0 + \@\(B\^2\ \[Gamma]\_1\%2 + \ \[CurlyEpsilon]\_0\%2\)\)\/\(B\ \[Gamma]\_1\)\)\), 1, 0, 0}, {\(-\(\(\[CurlyEpsilon]\_0 - \@\(B\^2\ \[Gamma]\_1\%2 + \ \[CurlyEpsilon]\_0\%2\)\)\/\(B\ \[Gamma]\_1\)\)\), 1, 0, 0}}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell["Evaluate mixing between M=0 states.", "Subsubsection"], Cell[TextData[{ "We define the tangent of the mixing angle to be the ratio between the \ amplitudes 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"Subsection"], Cell[TextData[{ "The order of operations must be chosen with care to avoid indeterminate \ results in evaluating the ", Cell[BoxData[ \(TraditionalForm\`B \[Rule] 0\)]], " limit." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[normalize /@ MySimplify[Normal[Series[vectors, {B, 0, 2}]]], B \[Rule] 0] // MySimplify\)], "Input"], Cell[BoxData[ \({{0, 0, 0, 1}, {0, 0, 1, 0}, {\(-1\), 0, 0, 0}, {0, 1, 0, 0}}\)], "Output"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Determine ratio of proton to electron magnetic couplings from spin-flip \ transition energies\ \>", "Section"], Cell["\<\ A precise measurement of the proton anomalous magnetic moment can be based \ upon spin-flip transitions in the Paschen-Back (large field) limit. In that \ limit the ratio between proton and electron spin-flip energies is equal to \ the ratio between their magnetic coupling constants.\ \>", "Text"], Cell[TextData[{ "The experiment is done by measuring the absorption of microwave radiation \ tuned to the energy differences between various pairs of levels split by a \ strong magnetic field. The relatively small difference between the lowest \ two energy levels is determined by ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_p\)]], " while the much larger difference between the second and third levels is \ determined by ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_e\)]], ". Hence, to lowest order the ratio between these two frequencies for the \ same magnetic field gives the ratio between gyromagnetic factors, up to small \ corrections. Note that the use of ratios tends to eliminate systematic \ errors. This relationship between the transition energies and gyromagnetic \ factors is derived below, including the corrections due to the hyperfine \ coupling." }], "Text"], Cell[CellGroupData[{ Cell["Determine the transition energies", "Subsection"], Cell[TextData[{ "We must be careful to identify the appropriate eigenvalues, whose order \ may depend upon the version of ", StyleBox["Mathematica", FontSlant->"Italic"], " that is employed." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1 = \({\[Delta]\_e \[Equal] eigenvalues\[LeftDoubleBracket]4\[RightDoubleBracket] - eigenvalues\[LeftDoubleBracket]2\[RightDoubleBracket], \ \[Delta]\_p \[Equal] eigenvalues\[LeftDoubleBracket]2\[RightDoubleBracket] - eigenvalues\[LeftDoubleBracket]3\[RightDoubleBracket]} /. trans12ep\) /. {\[Gamma]\_p \[Rule] \(-ratio\)\ \[Gamma]\_e}\ // MySimplify\)], "Input"], Cell[BoxData[ \({\@\(B\^2\ \((\(-1\) + ratio)\)\^2\ \[Gamma]\_e\%2 + \[CurlyEpsilon]\_0\ \%2\) \[Equal] B\ \((1 + ratio)\)\ \[Gamma]\_e + 2\ \[Delta]\_e + \[CurlyEpsilon]\_0, B\ \((1 + ratio)\)\ \[Gamma]\_e + \[CurlyEpsilon]\_0 + \@\(B\^2\ \ \((\(-1\) + ratio)\)\^2\ \[Gamma]\_e\%2 + \[CurlyEpsilon]\_0\%2\) \[Equal] 2\ \[Delta]\_p}\)], "Output"] }, Closed]], Cell[TextData[{ "Some of the results below depend upon the combination ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_e + \[Delta]\_p\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eigenvalues\[LeftDoubleBracket]2\[RightDoubleBracket] - eigenvalues\[LeftDoubleBracket]3\[RightDoubleBracket] + eigenvalues\[LeftDoubleBracket]4\[RightDoubleBracket] - eigenvalues\[LeftDoubleBracket]2\[RightDoubleBracket] /. trans12ep // MySimplify\)], "Input"], Cell[BoxData[ \(\@\(B\^2\ \((\[Gamma]\_e + \[Gamma]\_p)\)\^2 + \ \[CurlyEpsilon]\_0\%2\)\)], "Output"] }, Closed]], Cell[TextData[{ "which satisfies the inequality ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_e + \[Delta]\_p \[GreaterEqual] \ \[CurlyEpsilon]\_0\)]], ". Therefore, this inequality was included in ", StyleBox["MyAssumptions", "InlineInput", FontWeight->"Bold"], " in order to optimize the simplification functions." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Large field limit", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(eq2 = eq1 /. {\[CurlyEpsilon]\_0 \[Rule] 0}\ // MySimplify\)], "Input"], Cell[BoxData[ \({B\ \[Gamma]\_e + \[Delta]\_e \[Equal] 0, B\ ratio\ \[Gamma]\_e \[Equal] \[Delta]\_p}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[eq2, ratio, B]\)], "Input"], Cell[BoxData[ \({{ratio \[Rule] \(-\(\[Delta]\_p\/\[Delta]\_e\)\)}}\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["General case", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = Solve[eq1, {ratio, B}] // MyFullSimplify\)], "Input"], Cell[BoxData[ \({{ratio \[Rule] \(-\(\(\[Delta]\_e\%2 - \((\[Delta]\_p - \ \[CurlyEpsilon]\_0)\)\ \((\(-\[Delta]\_p\) + \@\(\((\[Delta]\_e + \[Delta]\_p \ - \[CurlyEpsilon]\_0)\)\ \((\[Delta]\_e + \[Delta]\_p + \[CurlyEpsilon]\_0)\)\ \))\) + \[Delta]\_e\ \((\[CurlyEpsilon]\_0 + \@\(\((\[Delta]\_e + \[Delta]\_p \ - 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