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Notebook[{
Cell[TextData[
"Approximate angular scattering functions for N2"], "Subsubtitle",
  Evaluatable->False,
  CellHorizontalScrolling->False,
  TextAlignment->Center],

Cell[CellGroupData[{

Cell["Setup notebook  enviroment ", "SmallText",
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Cell["ClearAll[\"Global`*\"]; ", "Input"],

Cell[CellGroupData[{

Cell["Remove[\"Global`*\"]; ", "Input"],

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}, Open  ]],

Cell["startclock = SessionTime[];", "Input"],

Cell["Off[General::spell]", "Input"],

Cell["Off[General::spell1]", "Input"],

Cell["The following does not work", "SmallText"],

Cell["Off[Solve::ifun]", "Input"],

Cell["<< Graphics`Graphics`", "Input"],

Cell[BoxData[
    \(<< Statistics`DataManipulation`\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(now\  = \ 
      StringForm["\<``/``/`` ``:``:``\>", \(Date[]\)[\([2]\)], \
\(Date[]\)[\([3]\)], \(Date[]\)[\([1]\)], \(Date[]\)[\([4]\)], \
\(Date[]\)[\([5]\)], \(Date[]\)[\([6]\)]]\)], "Input"],

Cell[BoxData[
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\\!\\(21\\):\\!\\(13\\):\\!\\(54\\)\"\>",
      StringForm[ "``/``/`` ``:``:``", 9, 17, 2000, 21, 13, 54],
      Editable->False]], "Output"]
}, Open  ]],

Cell[TextData[
"We wish to generate a set of approximate angular distributions for use in \
Monte Carlo calculations.  For that purpose we have modified the screened \
coulomb formula as described below."], "Text",
  Evaluatable->False],

Cell[TextData[
"Modified Screened Coulomb scattering notes\n\nThe reason for the \
modification is to allow for the fitting to data that is predominantly \
backward\nscattering and yet go over to the conventional form in the limit of \
predominantly forward scattering."], "SmallText",
  Evaluatable->False],

Cell[TextData[
"The modified differential scattering cross section i(theta,beta) is assumed \
to be"], "SmallText",
  Evaluatable->False],

Cell[BoxData[
    \(i := a\/\((1 - \((1 - 2\ beta)\)\ Cos[theta])\)\^2\)], "Input",
  PageWidth->Infinity],

Cell[TextData[
"where theta is the scattering angle and beta is the \"modified screening \
parameter\".  In the limit of \nsmall theta this has the same form as for \
conventional screened Coulomb scattering."], "SmallText",
  Evaluatable->False],

Cell["Series[i,{theta,0,2}]", "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[OutputFormData["\<\
SeriesData[theta, 0, {a/(4*beta^2), 0, 
   -(a*(1 - 2*beta))/(8*beta^3)}, 0, 3, 1]










































\
\>", "\<\
                              2
   a      a (1 - 2 beta) theta            3
------- - --------------------- + O[theta]
      2                3
4 beta           8 beta\
\>"], "SmallText",
  Evaluatable->False],

Cell[TextData[
"In the limit of small beta we recover Coulomb scattering"], "SmallText",
  Evaluatable->False],

Cell[TextData["Series[i,{beta,0,2}]"], "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[TextData[
"SeriesData[beta, 0, {a/(1 - Cos[theta])^2, \n   (-4*a*Cos[theta])/(1 - \
Cos[theta])^3, \n   (12*a*Cos[theta]^2)/(1 - Cos[theta])^4}, 0, 3, 1]"], 
  "SmallText",
  Evaluatable->False],

Cell[TextData["Integration yields the total cross section qt"], "SmallText",
  Evaluatable->False],

Cell[TextData[
"qt = Together[2*Pi* Integrate[Sin[theta]*\n              i,{theta,0,Pi}]]"], 
  "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[TextData["qt = -((a*Pi)/((-1 + beta)*beta));"], "SmallText",
  Evaluatable->False],

Cell[TextData["so that"], "SmallText",
  Evaluatable->False],

Cell[BoxData[
    \(\(a := \(qt\ \((1 - beta)\)\ beta\)\/\[Pi];\)\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[TextData["and the momentum transfer cross section qm"], "SmallText",
  Evaluatable->False],

Cell[TextData[
"qm = Together[2*Pi* Integrate[Sin[theta]*\n                  \
(1-Cos[theta])*i,{theta,0,Pi}]]"], "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[TextData[
"qm = (2*Pi*(a - 2*a*beta - a*Log[2 - 2*beta] + \n      a*beta*Log[2 - \
2*beta] + a*Log[2*beta] -\n      a*beta*Log[2*beta])\n    )/((-1 + beta)*(-1 \
+ 2*beta)^2)"], "SmallText",
  Evaluatable->False],

Cell[TextData["so that the ratio of cross sectons is"], "SmallText",
  Evaluatable->False],

Cell[TextData["ratio = Cancel[qm/qt]"], "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[BoxData[
    \(\(ratio = \(2\ beta\ \((\(-1\) + 2\ beta + Log[2 - 2\ beta] - beta\ \
Log[2 - 2\ beta] - Log[2\ beta] + beta\ Log[2\ beta])\)\)\/\((\(-1\) + 2\ \
beta)\)\^2;\)\)], "Input",
  PageWidth->Infinity],

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Cell[TextData["ratio /. beta -> 0.1"], "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[BoxData[
    \(0.3679694123758117\)], "Output",
  PageWidth->Infinity,
  Evaluatable->False]
}, Open  ]],

Cell[TextData[
"The differential scattering cross sections normalized to the total cross \
section is"], "SmallText",
  Evaluatable->False],

Cell[BoxData[
    \(normi := i[theta, eta]\/qt[beta]\)], "Input",
  PageWidth->Infinity],

Cell["\<\
so that probability of scattering through an angle less than theta0 times qt \
is\
\>", "SmallText",
  Evaluatable->False],

Cell[TextData[
"qtxprob[theta0_] = Together[Integrate[2*Pi*Sin[theta]*\n                  \
i,{theta,0,theta0}]]"], "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[BoxData[
    \(\(qtxprob = \(a\ \[Pi] - a\ \[Pi]\ Cos[theta0]\)\/\(beta\ \((1 - \
Cos[theta0] + 2\ beta\ Cos[theta0])\)\);\)\)], "Input",
  PageWidth->Infinity],

Cell["\<\
This allows one to calculate the cumulative probability, qtxprob/qt, needed \
for Monte Carlo calculatioms.\
\>", "SmallText",
  Evaluatable->False],

Cell["\<\
For theta0 = Pi this is qt as expected.  Solve for the scattering angle \
theta0 in terms of the random number, randomnum.\
\>", "SmallText",
  Evaluatable->False],

Cell["rule1 = Solve[qtxprob == randomnum*qt,theta0]", "SmallText",
  Evaluatable->False],

Cell[TextData["gives"], "SmallText",
  Evaluatable->False],

Cell[TextData[
"{{theta0 -> ArcCos[(1 - beta - randomnum)/\n      (1 - beta - randomnum + \
2*beta*randomnum)]}}"], "SmallText",
  Evaluatable->False],

Cell["Resolution of i[theta] into spherical harmonics.", "Subsubsection"],

Cell[BoxData[
    \(\(n = 5;\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qn\  = \ 
      2*Pi*Integrate[
          i*LegendreP[n, \ Cos[theta]]*Sin[theta], {theta, 0, Pi}]\)], "Input"],

Cell[BoxData[
    \(\(\(1\/\(\((\(-1\) + beta)\)\ \((\(-1\) + 2\ beta)\)\^6\)\)\((\((1 - 
            beta)\)\ qt\ \((\(-1\) - 95\ beta + 65\ beta\^2 + 710\ beta\^3 - 
            1130\ beta\^4 + 452\ beta\^5 + 
            30\ beta\ Log[\(-2\)\ \((\(-1\) + beta)\)] + 
            150\ beta\^2\ Log[\(-2\)\ \((\(-1\) + beta)\)] - 
            300\ beta\^3\ Log[\(-2\)\ \((\(-1\) + beta)\)] + 
            180\ beta\^5\ Log[\(-2\)\ \((\(-1\) + beta)\)] - 
            60\ beta\^6\ Log[\(-2\)\ \((\(-1\) + beta)\)] - 
            30\ beta\ Log[2\ beta] - 150\ beta\^2\ Log[2\ beta] + 
            300\ beta\^3\ Log[2\ beta] - 180\ beta\^5\ Log[2\ beta] + 
            60\ beta\^6\ Log[2\ beta])\))\)\)\)], "Output"]
}, Open  ]],

Cell[TextData["Elastic scattering of electons by N2"], "Subsection",
  Evaluatable->False],

Cell["\<\
From Phelps and Pitchford, Phys. Rev. A 31, 2932 (1985), Table I, we obtain \
empirical expressions for q1(en)/q0(en) and qm(en)/q0(en).  These are set \
equal to the expression for \"ratio\" and used to find values of beta for a \
wide range of energies.  We then fit an empirical expresion to the beta(en) \
values by trial and error.  The graphs below show the degree of fit.\
\>", "SmallText",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{en, beta, q1o10, qmoq0, ratiorule, betalist, \n\t\tenergies, 
        empiricalbeta, ratiolist, betavalues, ratiovalues, 
        realbetaplot, \n\t\tempbetaplot, qmoq0a, qt1, beta1}, \n\t
      q1oq0 = \(-\(\(0.2\ en\^0.5\)\/\(0.025 + 
                  en\^0.5\)\)\) + \(1.2\ \((16\ en\^0.5 + en)\)\)\/\(100 + 16\
\ en\^0.5 + en\); \n\tqmoq0 := 1 - q1oq0; \n\t
      qmoq0a := PowerExpand[
          N[qmoq0 /. en \[Rule] 1\/100\ \((10\^\(1/4\))\)\^j]]; \n\t
      ratiorule = 
        Table[FindRoot[N[qmoq0a] == ratio, {beta, 0.1}, 
            AccuracyGoal \[Rule] 5], {j, 0, 20}]; \n\t
      betalist = Re[beta /. ratiorule]; \n\t
      energies = Table[N[1\/10\ \((10\^\(1/4\))\)\^j], {j, 0, 20}]; \n\t
      empiricalbeta :=  .6\/\((1 + \((en\/500)\)\^0.5 + \
\((en\/200)\)\^1.01)\)\^0.99; \n\t
      betatab = empiricalbeta /. en \[Rule] energies; \n\t
      ratiolist = N[ratio /. beta \[Rule] betalist]; \n\t
      betavalues = Transpose[{energies, betalist}]; \n\t
      ratiovalues = Transpose[{energies, ratiolist}]; \n\t
      realbetaplot = 
        LogLogListPlot[betavalues, 
          PlotStyle \[Rule] {PointSize[0.03]}, \n\t\t\tPlotRange \[Rule] All, 
          DisplayFunction \[Rule] Identity]; \n\t
      empbetaplot = 
        LogLogListPlot[Transpose[{energies, betatab}], 
          PlotJoined \[Rule] True, \n\t\t\tPlotRange \[Rule] All, 
          DisplayFunction \[Rule] Identity]; \n\t
      Show[realbetaplot, empbetaplot, 
        DisplayFunction \[Rule] $DisplayFunction, \n\t\tAxesLabel \[Rule] \
{"\<energy (eV)\>", "\<beta\>"}]; \n\t
      LogLogListPlot[ratiovalues, 
        PlotStyle \[Rule] 
          PointSize[
            0.03], \n\t\tAxesLabel \[Rule] {"\<energy (eV)\>", "\<qm/qt\>"}]; \
\n\ta = 1; \n\tbeta1 := N[empiricalbeta /. en \[Rule] 1000]; \n\t
      Plot[i /. beta \[Rule] beta1, {theta, 0, \[Pi]}, 
        AxesLabel \[Rule] {"\<theta (rd)\>", "\<i(theta)\>"}];]\)], "Input",
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Cell[TextData["Lower triplet states- A, B, and W."], "Subsection",
  Evaluatable->False],

Cell["\<\
From Phelps and Pitchford, Phys. Rev. A 31, 2932 (1985), Table I, we obtain \
empirical expressions for q1(en)/q0(en) and qm(en)/q0(en).  These are set \
equal to the expression for \"ratio\" and used to find values of beta for a \
wide range of energies.  We then fit an empirical expresion to the beta(en) \
values by trial and error.  The graphs below show the degree of fit.\
\>", "SmallText",
  Evaluatable->False],

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Cell[BoxData[
    \(Block[{en, beta, q1o10, qmoq0, ratiorule, betalist, energies, 
        empiricalbeta, ratiolist, betavalues, ratiovalues, realbetaplot, 
        empbetaplot, qmoq0a}, q1oq0 = \(-\(en\^2\/\(1500 + en\^2\)\)\); 
      qmoq0 := 1 - q1oq0; 
      qmoq0a := PowerExpand[
          N[qmoq0 /. en \[Rule] 1\/100\ \((10\^\(1/4\))\)\^j]]; 
      ratiorule = 
        Table[FindRoot[N[qmoq0a] == ratio, {beta, 0.1}, 
            AccuracyGoal \[Rule] 5], {j, 0, 20}]; 
      betalist = Re[beta /. ratiorule]; 
      energies = Table[N[1\/10\ \((10\^\(1/4\))\)\^j], {j, 0, 20}]; 
      empiricalbeta := \( .5\ \((1 + \((en\/200)\)\^2)\)\)\/\(1 + \((en\/282)\
\)\^2\); betatab = empiricalbeta /. en \[Rule] energies; 
      ratiolist = N[ratio /. beta \[Rule] betalist]; 
      betavalues = Transpose[{energies, betalist}]; 
      ratiovalues = Transpose[{energies, ratiolist}]; 
      realbetaplot = 
        LogLinearListPlot[betavalues, PlotStyle \[Rule] {PointSize[0.03]}, 
          PlotRange \[Rule] {0, 2}, DisplayFunction \[Rule] Identity]; 
      empbetaplot = 
        LogLinearListPlot[Transpose[{energies, betatab}], 
          PlotJoined \[Rule] True, PlotRange \[Rule] {0, 2}, 
          DisplayFunction \[Rule] Identity]; 
      Show[realbetaplot, empbetaplot, 
        DisplayFunction \[Rule] $DisplayFunction, 
        AxesLabel \[Rule] {"\<energy (eV)\>", "\<beta\>"}]; 
      LogLinearListPlot[ratiovalues, PlotStyle \[Rule] PointSize[0.03], 
        AxesLabel \[Rule] {"\<energy (eV)\>", "\<qm/qt\>"}]; a = 1; 
      beta1 = N[empiricalbeta /. en \[Rule] 15]; 
      Plot[i /. {beta \[Rule] beta1, qt \[Rule] qt1}, {theta, 0, \[Pi]}, 
        PlotRange \[Rule] {0, Automatic}, 
        AxesLabel \[Rule] {"\<theta (rd)\>", "\<i(theta)\>"}];]\)], "Input",
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}, Open  ]],

Cell[TextData["Upper triplet state - C."], "Subsection",
  Evaluatable->False],

Cell["\<\
From Phelps and Pitchford, Phys. Rev. A 31, 2932 (1985), Table I, we obtain \
empirical expressions for q1(en)/q0(en) and qm(en)/q0(en).  These are set \
equal to the expression for \"ratio\" and used to find values of beta for a \
wide range of energies.  We then fit an empirical expresion to the beta(en) \
values by trial and error.  The graphs below show the degree of fit.\
\>", "SmallText",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{en, beta, q1o10, qmoq0, ratiorule, energies, empiricalbeta, 
        ratiolist, betavalues, ratiovalues, \n\t\trealbetaplot, empbetaplot, 
        qmoq0a, qt1, beta1}, \n\tq1oq0 = \(-0.2\); \n\t
      qmoq0 := 1 - q1oq0; \n\t
      qmoq0a := PowerExpand[
          N[qmoq0 /. en \[Rule] 1\/100\ \((10\^\(1/4\))\)\^j]]; \n\t
      ratiorule = 
        Table[FindRoot[N[qmoq0a] == ratio, {beta, 0.1}, 
            AccuracyGoal \[Rule] 5], {j, 0, 20}]; \n\t
      betalist = Re[beta /. ratiorule]; 
      energies = Table[N[1\/10\ \((10\^\(1/4\))\)\^j], {j, 0, 20}]; \n\t
      empiricalbeta :=  .647431\ en\^0.000001; \n\t
      betatab = empiricalbeta /. en \[Rule] energies; \n\t
      ratiolist = N[ratio /. beta \[Rule] betalist]; \n\t
      betavalues = Transpose[{energies, betalist}]; \n\t
      ratiovalues = Transpose[{energies, ratiolist}]; \n\t
      realbetaplot = 
        LogLinearListPlot[betavalues, PlotStyle \[Rule] {PointSize[0.03]}, 
          PlotRange \[Rule] {0, 1}, \n\t\t\tDisplayFunction \[Rule] 
            Identity]; \n\t
      empbetaplot = 
        LogLinearListPlot[Transpose[{energies, betatab}], 
          PlotJoined \[Rule] True, \n\t\t\tPlotRange \[Rule] {0, 1}, 
          DisplayFunction \[Rule] Identity]; \n\t
      Show[realbetaplot, empbetaplot, 
        DisplayFunction \[Rule] $DisplayFunction, \n\t\tAxesLabel \[Rule] \
{"\<energy (eV)\>", "\<beta\>"}]; \n\t
      LogLinearListPlot[ratiovalues, PlotStyle \[Rule] PointSize[0.03], 
        AxesLabel \[Rule] {"\<energy (eV)\>", "\<qm/qt\>"}]; \n\ta = 1; \n\t
      beta1 := N[empiricalbeta /. en \[Rule] 25]; \n\t
      Plot[i /. {beta \[Rule] beta1, qt \[Rule] qt1}, {theta, 0, \[Pi]}, 
        PlotRange \[Rule] {0, 
            Automatic}, \n\t\tAxesLabel \[Rule] {"\<theta (rd)\>", \
"\<i(theta)\>"}];]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

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\>"],
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}, Open  ]],

Cell[TextData["Lower singlet state - a1Pi."], "Subsection",
  Evaluatable->False],

Cell["\<\
From Phelps and Pitchford, Phys. Rev. A 31, 2932 (1985), Table I, we obtain \
empirical expressions for q1(en)/q0(en) and qm(en)/q0(en).  These are set \
equal to the expression for \"ratio\" and used to find values of beta for a \
wide range of energies.  We then fit an empirical expresion to the beta(en) \
values by trial and error.  The graphs below show the degree of fit.\
\>", "SmallText",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{en, beta, q1o10, qmoq0, ratiorule, betalist, energies, 
        empiricalbeta, ratiolist, betavalues, \n\t\tratiovalues, 
        realbetaplot, empbetaplot, qmoq0a, qt1, beta1}, \n\t
      q1oq0 = \(200000 + 160\ en\^2 + en\^4\)\/\(200000 + 2600\ en\^2 + en\^4\
\); \n\tqmoq0 := 1 - q1oq0; \n\t
      qmoq0a := PowerExpand[
          N[qmoq0 /. en \[Rule] 1\/100\ \((10\^\(1/4\))\)\^j]]; \n\t
      ratiorule = 
        Table[FindRoot[N[qmoq0a] == ratio, {beta, 0.1}, 
            AccuracyGoal \[Rule] 5], {j, 0, 20}]; \n\t
      betalist = Re[beta /. ratiorule]; 
      energies = Table[N[1\/10\ \((10\^\(1/4\))\)\^j], {j, 0, 20}]; \n\t
      empiricalbeta := \(0.4\ \((en\/150)\)\^2\)\/\(1 + \((en\/300)\)\^2 + \
\((en\/220)\)\^4\); \n\tbetatab = empiricalbeta /. en \[Rule] energies; \n\t
      ratiolist = N[ratio /. beta \[Rule] betalist]; \n\t
      betavalues = Transpose[{energies, betalist}]; \n\t
      ratiovalues = Transpose[{energies, ratiolist}]; \n\t
      realbetaplot = 
        LogLogListPlot[betavalues, PlotStyle \[Rule] {PointSize[0.03]}, 
          PlotRange \[Rule] {0.01, 1}, \n\t\t\tDisplayFunction \[Rule] 
            Identity]; \n\t
      empbetaplot = 
        LogLogListPlot[Transpose[{energies, betatab}], 
          PlotJoined \[Rule] True, \n\t\t\tPlotRange \[Rule] {0.01, 1}, 
          DisplayFunction \[Rule] Identity]; \n\t
      Show[realbetaplot, empbetaplot, 
        DisplayFunction \[Rule] $DisplayFunction, \n\t\tAxesLabel \[Rule] \
{"\<energy (eV)\>", "\<beta\>"}]; \n\t
      LogLinearListPlot[ratiovalues, PlotStyle \[Rule] PointSize[0.03], 
        AxesLabel \[Rule] {"\<energy (eV)\>", "\<qm/qt\>"}]; \n\ta = 1; \n\t
      beta1 := N[empiricalbeta /. en \[Rule] 500]; \n\t
      Plot[i /. {beta \[Rule] beta1, qt \[Rule] qt1}, {theta, 0, \[Pi]}, 
        AxesLabel \[Rule] {"\<theta (rd)\>", "\<i(theta)\>"}];]\)], "Input",
  PageWidth->Infinity,
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  FontSize->12,
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  FontColor->GrayLevel[0],
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\>"],
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}, Open  ]],

Cell["Sum of singlets.", "Subsection",
  Evaluatable->False],

Cell["\<\
From Phelps and Pitchford, Phys. Rev. A 31, 2932 (1985), Table I, we obtain \
empirical expressions for q1(en)/q0(en) and qm(en)/q0(en).  These are set \
equal to the expression for \"ratio\" and used to find values of beta for a \
wide range of energies.  We then fit an empirical expresion to the beta(en) \
values by trial and error.  The graphs below show the degree of fit.\
\>", "SmallText",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{en, beta, q1o10, qmoq0, ratiorule, betalist, energies, 
        empiricalbeta, ratiolist, \n\t\tbetavalues, ratiovalues, 
        realbetaplot, empbetaplot, qmoq0a, qt1, beta1}, \n\t
      q1oq0 = en\^2\/\(2500 + en\^2\); \n\tqmoq0 := 1 - q1oq0; \n\t
      qmoq0a := PowerExpand[
          N[qmoq0 /. en \[Rule] 1\/100\ \((10\^\(1/4\))\)\^j]]; \n\t
      ratiorule = 
        Table[FindRoot[N[qmoq0a] == ratio, {beta, 0.1}, 
            AccuracyGoal \[Rule] 5], {j, 0, 20}]; \n\t
      betalist = Re[beta /. ratiorule]; 
      energies = Table[N[1\/10\ \((10\^\(1/4\))\)\^j], {j, 0, 20}]; \n\t
      empiricalbeta :=  .5\/\(1 + \((en\/370)\)\^2.5\); \n\t
      betatab = empiricalbeta /. en \[Rule] energies; \n\t
      ratiolist = N[ratio /. beta \[Rule] betalist]; \n\t
      betavalues = Transpose[{energies, betalist}]; \n\t
      ratiovalues = Transpose[{energies, ratiolist}]; \n\t
      realbetaplot = 
        LogLogListPlot[betavalues, 
          PlotStyle \[Rule] {PointSize[
                0.03]}, \n\t\t\tPlotRange \[Rule] {0.001, 1}, 
          DisplayFunction \[Rule] Identity]; \n\t
      empbetaplot = 
        LogLogListPlot[Transpose[{energies, betatab}], 
          PlotJoined \[Rule] True, \n\t\t\tPlotRange \[Rule] {0.001, 1}, 
          DisplayFunction \[Rule] Identity]; \n\t
      Show[realbetaplot, empbetaplot, 
        DisplayFunction \[Rule] $DisplayFunction, \n\t\tAxesLabel \[Rule] \
{"\<energy (eV)\>", "\<beta\>"}]; \n\t
      LogLinearListPlot[ratiovalues, PlotStyle \[Rule] PointSize[0.03], 
        AxesLabel \[Rule] {"\<energy (eV)\>", "\<qm/qt\>"}]; \n\ta = 1; \n\t
      beta1 := N[empiricalbeta /. en \[Rule] 500]; \n\t
      Plot[i /. {beta \[Rule] beta1, qt \[Rule] qt1}, {theta, 0, \[Pi]}, 
        AxesLabel \[Rule] {"\<theta (rd)\>", "\<i(theta)\>"}];]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

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