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Notebook[{
Cell["\<\
Approximate angular scattering functions for He-He collisions\
\>", "Subtitle",
  Evaluatable->False,
  CellHorizontalScrolling->False,
  TextAlignment->Center],

Cell["\<\
We wish to generate a set of approximate angular distributions for use in \
Monte Carlo calculations.  For that purpose we have used the modified the \
screened Coulomb (or Rutherford) formula as described below.  This notebook \
is based on our previous work on electrons in N2 and Ar-Ar elastic collisions \
and we will not repeat the derivations given in the Ar-Ar notebook.\
\>", "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell["Setup notebook environment", "Text"],

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

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

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

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

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

Cell["Off[NumberForm::sigz]", "Input"],

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

Cell[CellGroupData[{

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

Cell[BoxData[
    InterpretationBox["\<\"\\!\\(12\\)/\\!\\(9\\)/\\!\\(2000\\) \
\\!\\(21\\):\\!\\(20\\):\\!\\(24\\)\"\>",
      StringForm[ "``/``/`` ``:``:``", 12, 9, 2000, 21, 20, 24],
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Differential scattering cross section", "Subsection",
  Evaluatable->False],

Cell["Data from Newman, Smith, and Stebbings (1985).", "Subsubsection",
  Evaluatable->False],

Cell["\<\
From their tabulation of the He-He differential cross sections we have\
\>", "SmallText",
  Evaluatable->False],

Cell["!! c:\\mathbook\\helium\\hehecollisions\\newman85.dat", "Input",
  PageWidth->Infinity,
  Evaluatable->False],

Cell["\<\
Note that entries of 1E-30 are used instead of a blank entry or zero so that \
our plotting program ProPlot will work for log plots.\
\>", "SmallText",
  Evaluatable->False],

Cell["\<\
Read in the data from this file, transform to relative angle, and prepare \
plots\
\>", "Text"],

Cell["\<\
            inputfile := \
\"c:\\\\mathbook\\\\helium\\\\hehecollisions\\\\newman85.dat\";
            stream = OpenRead[inputfile];
            Find[stream,\"%\"];
            Find[stream,\"%\"];
            data1 = ReadList[stream, Number, RecordLists ->True];
            Close[stream];
            angle1 = Part[Transpose[data1],1];
            relangle1 = angle1*2;
            q1diff500 = Part[Transpose[data1],2]*10^16;
            q1diff1500 =  Part[Transpose[data1],3]*10^16;
            q1diff5000 = Part[Transpose[data1],4]*10^16;
            i1expcm500 = \
Table[Part[q1diff500,j]/4/Cos[Part[angle1*Pi/180,j]],{j,1,Length[angle1]}];
            i1expcm1500 = \
Table[Part[q1diff1500,j]/4/Cos[Part[angle1*Pi/180,j]],{j,1,Length[angle1]}];
            i1expcm5000 = \
Table[Part[q1diff5000,j]/4/Cos[Part[angle1*Pi/180,j]],{j,1,Length[angle1]}];
            data1qdiff500 = Transpose[{relangle1,i1expcm500}];
            data1qdiff1500 = Transpose[{relangle1,i1expcm1500}];
            data1qdiff5000 = Transpose[{relangle1,i1expcm5000}];
            data1q500plot = LogLogListPlot[data1qdiff500,
                         PlotStyle ->{PointSize[0.025],Hue[0.55]},
                         PlotRange -> {{0.1,20},{1,10000}},
                         (*PlotRange -> {0.1,1000},*)
                         DisplayFunction ->Identity];
            data1q1500plot = LogLogListPlot[data1qdiff1500,             
                         PlotStyle ->{PointSize[0.025],Hue[1]},
                         PlotRange ->  {{0.1,20},{1,10000}},
                         DisplayFunction -> Identity];
            data1q5000plot = LogLogListPlot[data1qdiff5000,             
                         PlotStyle ->{PointSize[0.025],Hue[0.4]},
                         PlotRange ->  {{0.1,20},{1,10000}},
                         DisplayFunction -> Identity];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Show[data1q500plot,data1q1500plot,data1q5000plot,
            DisplayFunction -> $DisplayFunction, GridLines -> Automatic,
             AxesLabel -> {\"angle (deg)\",\"I(angle) (10^-20 m^2/str)\"}];\
\>", "Input",
  PageWidth->Infinity,
  Evaluatable->False,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["!! c:\\mathbook\\helium\\hehecollisions\\nitz87a.dat", "Input",
  PageWidth->Infinity,
  Evaluatable->False],

Cell["\<\
            inputfile := \
\"c:\\\\mathbook\\\\helium\\\\hehecollisions\\\\nitz87a.dat\";
            stream = OpenRead[inputfile];
            Find[stream,\"%\"];
            Find[stream,\"%\"];
            data2 = ReadList[stream, Number, RecordLists ->True];
            Close[stream];
            angle2 = Part[Transpose[data2],1];
            relangle2 = angle2*2;
            q2diff500 = Part[Transpose[data2],2];
            q2diff1500 =  Part[Transpose[data2],3];
            q2diff5000 = Part[Transpose[data2],4];
            i2expcm500 = \
Table[Part[q2diff500,j]/4/Cos[Part[angle2*Pi/180,j]],{j,1,Length[angle2]}];
            i2expcm1500 = \
Table[Part[q2diff1500,j]/4/Cos[Part[angle2*Pi/180,j]],{j,1,Length[angle2]}];
            i2expcm5000 = \
Table[Part[q2diff5000,j]/4/Cos[Part[angle2*Pi/180,j]],{j,1,Length[angle2]}];
            data2qdiff500 = Transpose[{relangle2,i2expcm500}];
            data2qdiff1500 = Transpose[{relangle2,i2expcm1500}];
            data2qdiff5000 = Transpose[{relangle2,i2expcm5000}];
          data2q500plot = LogLogListPlot[data2qdiff500,
                         PlotStyle ->{PointSize[0.025],Hue[0.55]},
                         PlotRange -> {{0.1,10},{1,100000}},
                         DisplayFunction ->Identity];
            data2q1500plot = LogLogListPlot[data2qdiff1500,             
                         PlotStyle ->{PointSize[0.025],Hue[1]},
                         PlotRange ->  {{0.1,10},{1,100000}},
                         DisplayFunction -> Identity];
            data2q5000plot = LogLogListPlot[data2qdiff5000,             
                         PlotStyle ->{PointSize[0.025],Hue[0.4]},
                         PlotRange ->  {{0.1,10},{1,100000}},
                         DisplayFunction -> Identity];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Show[data2q500plot,data2q1500plot,data2q5000plot,data1q500plot,data1q1500plot,\
data1q5000plot,
            DisplayFunction -> $DisplayFunction, GridLines -> Automatic,
             AxesLabel -> {\"rel. angle (deg)\",\"I(angle) (10^-20 \
m^2/str)\"}];\
\>", "Input",
  PageWidth->Infinity,
  Evaluatable->False,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]]
}, Open  ]],

Cell["Analysis of 500 eV data", "Subsection"],

Cell["We will compare with screened-Coulomb scattering", "Subsubsection"],

Cell["\<\
In center-of-mass the screened-Coulomb differential scattering formula is\
\>", "SmallText"],

Cell[BoxData[
    \(\(\(\ \)\(icoulombcm\  := \ 
      a \((1 - \ Cos[thetacm]\  + \ 2*beta)\)^\(-2\)\)\)\)], "Input"],

Cell["\<\
where thetacm is the scattering angle in center-of-mass.  The \
total,diffusion, and viscosity cross sections are\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qtcoulomb\  = \ 
      2*Pi*Integrate[icoulombcm*Sin[thetacm], {thetacm, 0, Pi}]\  // \ 
        Simplify\)], "Input"],

Cell[BoxData[
    \(\(a\ \[Pi]\)\/\(beta + beta\^2\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qdcoulomb\  = \ 
      2*Pi*Integrate[
            icoulombcm*\((1 - Cos[thetacm])\)*Sin[thetacm], {thetacm, 0, 
              Pi}]\  // \ Simplify\)], "Input"],

Cell[BoxData[
    \(\(-\(\(2\ a\ \[Pi]\ \((1 + \((1 + beta)\)\ Log[\(-2\)\ beta] - \((1 + 
                      beta)\)\ Log[\(-2\)\ \((1 + beta)\)])\)\)\/\(1 + 
            beta\)\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qvcoulomb\  = \ 
      2*Pi*Integrate[
            icoulombcm*\((Sin[thetacm])\)^3, {thetacm, 0, Pi}]\  // \ 
        Simplify\)], "Input"],

Cell[BoxData[
    \(\(-4\)\ a\ \[Pi]\ \((2 + \((1 + 2\ beta)\)\ Log[\(-2\)\ beta] - \((1 + 
                2\ beta)\)\ Log[\(-2\)\ \((1 + beta)\)])\)\)], "Output"]
}, Open  ]],

Cell["\<\
 We need the differential scattering expression transformed to the laboratory \
frame.  From McDaniel (1964) we need the solution to\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(coulombrule\  = \ 
      Solve[{\((icoulombcm\ *\ Sin[thetacm]\  /. \ thetacm\  -> \ 2*theta)\)*
              dthetacm\  == \ 
            icoulomblab*Sin[theta]*
              dtheta, \ \n\t\t\ \ \ \ \ \ \ \ \ \ dthetacm\  == \ 2*dtheta}, 
        icoulomblab]\)], "Input"],

Cell[BoxData[
    \({{icoulomblab \[Rule] \(4\ a\ Cos[theta]\)\/\((\(-1\) - 2\ beta + Cos[2\
\ theta])\)\^2}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(icoulomblab\  = \ Part[icoulomblab\  /. \ coulombrule, 1]\)], "Input"],

Cell[BoxData[
    \(\(4\ a\ Cos[theta]\)\/\((\(-1\) - 2\ beta + Cos[2\ theta])\)\^2\)], \
"Output"]
}, Open  ]],

Cell["\<\
From Zhkarofsky et al. Eq. (7-11) when we replace the relative energy by the \
laboratory energy m*v^2/2 = mr*v^2 = en \
\>", "SmallText"],

Cell[BoxData[
    \(a\  := \ \(\(10^20\)\(*\)\(\((z^2*0.529*^-10*13.6/
                en)\)^2\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\( (*\ 
          10^\(-20\) m^2/str*) \)\)\)], "Input"],

Cell["\<\
We seem to find the best fit with z = 2 and beta = 0.004, so that for en = \
500 eV we have\
\>", "SmallText"],

Cell[BoxData[
    \(\(\(\t\)\(coulomb500plot\  = \ 
        LogLogPlot[\((icoulombcm\  //. {z\  -> 1.8, en\  -> 500, 
                beta\  -> \ 0.002, 
                thetacm\  -> \ 
                  relangle*
                    Pi/180})\), \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {relangle, 0.1, 20}, \ 
          PlotRange\  -> \ {{0.1, 20}, {1, 100000}}, \ 
          DisplayFunction\  -> \ Identity];\)\)\)], "Input"],

Cell["The Rice University experimental data  for 500 eV", "Subsubsection"],

Cell[BoxData[
    \(\(list500\  = \ 
        Union[Drop[Drop[data1qdiff500, {6}], {10}], 
          data2qdiff500];\)\)], "Input"],

Cell[BoxData[
    \(\(dataqdiff500\  = \ 
        DeleteCases[
          Table[\ If[Part[list500, j, 2]\  >= \ 10^\(-8\), 
              Part[list500, j]], {j, 1, Length[list500]}], 
          Null];\)\)], "Input"],

Cell[BoxData[
    \(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \)\(dataq500plot\  = \ 
        LogLogListPlot[
          dataqdiff500, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotStyle\  -> {PointSize[0.025], 
              Hue[0.55]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotRange\  -> \ {{0.1, 20}, {1, 
                100000}}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
DisplayFunction\  -> Identity];\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Show[dataq500plot, 
        coulomb500plot,  (*\(AxesLabel -> {"\<rel. angle (deg)\>", "\<i (rel. \
angle)\>"}\)\(,\)*) \n\t\tDisplayFunction -> $DisplayFunction];\)\)], "Input"],

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Cell[CellGroupData[{

Cell[BoxData[
    \(qdiff500int\  = \ Interpolation[dataqdiff500]\)], "Input"],

Cell[BoxData[
    TagBox[\(InterpolatingFunction[{{0.036`, 10.16`}}, "<>"]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(norm\  = \ 
      2*Pi*NIntegrate[Sin[angle*3.1416/180], \ {angle, 0, 180}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
    \(12.566370614189621`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qt500\  = \ 
      2*Pi*NIntegrate[
          qdiff500int[relangle]*Sin[relangle*3.1416/180], \ {relangle, 0.036, 
            10.16}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.7082968749999999`\)."\)], "Message"],

Cell[BoxData[
    \(13.806061997804553`\)], "Output"]
}, Open  ]],

Cell["\<\
Presumably the following is closer to the true total cross section.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qt500\  = \ 
      2*Pi*NIntegrate[
          qdiff500int[relangle]*Sin[relangle*3.1416/180], \ {relangle, 0.036, 
            10.16}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.7082968749999999`\)."\)], "Message"],

Cell[BoxData[
    \(13.806061997804553`\)], "Output"]
}, Open  ]],

Cell["\<\
This value is about 15% larger than that given by Newman et al. (1985), but \
is only 7% larger than the experimental value of Savola (1973) as obtained \
using the detector angle of 0.056 deg.  This difference may be the result of \
the improved differential scattering data available at low angles from the \
experiments of Nitz as used in our calculation but not in Newman's \
calculation.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qm500\  = \ 
      2*Pi*NIntegrate[
          qdiff500int[relangle]*\((1 - Cos[relangle*3.1416/180])\)*
            Sin[relangle*3.1416/
                  180], \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {relangle, 0.036, 10.16}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.8664843749999999`\)."\)], "Message"],

Cell[BoxData[
    \(0.011621917815295224`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qv500\  = \ 
      2*Pi*NIntegrate[
          qdiff500int[
              relangle]*\((Sin[
                  relangle*3.1416/
                      180])\)^3, \ \n\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t{relangle, 0.036, 10.16}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.8664843749999999`\)."\)], "Message"],

Cell[BoxData[
    \(0.023160590632078017`\)], "Output"]
}, Open  ]],

Cell["\<\
These qm and qv values are very small and suggest we need to look very \
carefully at the large angle scattering, e.g., how does experiment compare \
with screened Coulomb scattering.\
\>", "SmallText"],

Cell[BoxData[
    \(\(angleqdiff500\  = \ Part[Transpose[dataqdiff500], 1];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[%]\)], "Input"],

Cell[BoxData[
    \(83\)], "Output"]
}, Open  ]],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(Length[%]\)], "Input"],

Cell[BoxData[
    \(83\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(integrandqv500\  = \ 
        Table[\((Part[listqdiff500, 
                j]*\((Sin[
                    Part[angleqdiff500, j]*
                      Pi/180])\)^3)\), \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {j, 1, 
            Length[angleqdiff500]}];\)\)], "Input"],

Cell[BoxData[
    \(\(tableintegrandqv500\  = \ 
        Transpose[{angleqdiff500, integrandqv500}];\)\)], "Input"],

Cell[BoxData[
    \(\(\ \ \ \ \ \ \ \ \ \ \ 
    \(integrand500plot\  = \ 
      LogLogListPlot[tableintegrandqv500, 
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*
          log\ log\ plot*) \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotStyle\  -> {PointSize[0.025], Hue[0.55]}, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotRange\  -> \ {{0.1, 20}, {1*^-5, 0.1}}, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        DisplayFunction\  -> Identity]; \)\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(integrand500plot\  = \ 
        ListPlot[
          tableintegrandqv500, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*linear\ plot*) \n\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotStyle\  -> {PointSize[0.025], 
              Hue[0.55]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotRange\  -> \ {{0, 90}, {0, 
                0.04}}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
DisplayFunction\  -> Identity];\)\(\n\)\)\)], "Input"],

Cell["\<\
The integrand for the viscosity integral and screened Coulomb scattering  \
is\
\>", "SmallText"],

Cell[BoxData[
    \(\(\tcoulombint500plot\  = 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*
        log\ log\ plot*) 
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ 
      \((LogLogPlot[
          \((\((icoulombcm*\((Sin[thetacm])\)^3)\)\  //. {z\  -> 1.8, 
                en\  -> 500, beta\  -> \ 0.002, 
                thetacm\  -> \ relangle*Pi/180})\), \n
          \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ {relangle, 0.1, 20}, \ \ 
          AxesLabel\  -> \ {"\<rel. angle (deg)\>", \ "\<i\>"}, \n
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t\
\t\tPlotRange\  -> \ {{0.1, 10}, {1*^-5, 0.1}}, \ 
          DisplayFunction\  -> \ Identity]; \)\)\)], "Input",
  Evaluatable->False],

Cell["\<\
coulombint500plot =                                (*linear plot*)
   Plot[icoulombcm*Sin[thetacm]^3 //. 
     {z -> 1.8, en ->500, beta -> 0.002, thetacm -> relangle*Pi/180}, \
{relangle, 0, 90}, 
    AxesLabel -> {\"rel. angle (deg)\", \"i\"}, 
    PlotRange -> {{0, 90}, {0, 0.04}}, DisplayFunction -> Identity]; \
\>", "Input"],

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(delqt500\  = \ 
      2*Pi*NIntegrate[\((icoulombcm\  //. {z\  -> 1.8, en\  -> 500, 
                  beta\  -> \ 0.002, 
                  thetacm\  -> \ 
                    relangle*
                      Pi/180})\)\ \ *\ \ \n\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sin[
              relangle*3.1416/180], {relangle, 10.16, 180}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["\<\
delqm500 = 2*Pi*NIntegrate[(icoulombcm //.  {z -> 1.8, en -> 500, beta -> \
0.002, thetacm -> relangle*Pi/180})*
                (1 - Cos[relangle*Pi/180])* Sin[relangle*Pi/180], {relangle, \
10.16, 180}]* Pi/180\
\>", "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(delqv500\  = \ 
      2*Pi*NIntegrate[\((icoulombcm\  //. {z\  -> 1.8, en\  -> 500, 
                  beta\  -> \ 0.002, 
                  thetacm\  -> \ 
                    relangle*
                      Pi/180})\)*\n\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \((Sin[relangle*3.1416/180])\)^3, \ {relangle, 
            10.16, 180}]*3.1416/180\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(coulombqv500\  = \ 
      2*Pi*NIntegrate[\((icoulombcm\  //. {z\  -> 1.8, en\  -> 500, 
                  beta\  -> \ 0.002, 
                  thetacm -> \ 
                    relangle*
                      Pi/180})\)*\n\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((Sin[
                  relangle*3.1416/180])\)^3, \ {relangle, 0, 180}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell["\<\
Something may be wrong with these three integrals - they seem too large.\
\>", "Subsubtitle"],

Cell["Analysis of 1500 eV data", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

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Cell[BoxData[
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PlotStyle\  -> {PointSize[0.025], 
              Hue[0.55]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotRange\  -> \ {{0.1, 20}, {1, 
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DisplayFunction\  -> Identity];\)\)\)], "Input"],

Cell[BoxData[
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        LogLogPlot[\((icoulombcm\  //. {z\  -> 1.8, en\  -> \ 1500, 
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Cell[CellGroupData[{

Cell[BoxData[
    \(qdiff1500int\  = \ Interpolation[dataqdiff1500]\)], "Input"],

Cell[BoxData[
    TagBox[\(InterpolatingFunction[{{0.036`, 9.04`}}, "<>"]\),
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      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(norm\  = \ 
      2*Pi*NIntegrate[Sin[relangle*3.1416/180], \ {relangle, 0, 180}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
    \(12.566370614189621`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qt1500\  = \ 
      2*Pi*NIntegrate[
          qdiff1500int[relangle]*Sin[relangle*3.1416/180], \ {relangle, 
            0.036, 9.04}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.2881485954920154`\)."\)], "Message"],

Cell[BoxData[
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}, Open  ]],

Cell["\<\
This value is about 15% larger than that given by Newman et al. (1985), but \
is only 7% larger than the experimental value of Savola (1973) as obtained \
using the detector angle of 0.056 deg.  This difference may be the result of \
the improved differential scattering data available at low angles from the \
experiments of Nitz as used in our calculation but not in Newman's \
calculation.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qm1500\  = \ 
      2*Pi*NIntegrate[
          qdiff1500int[relangle]*\((1 - Cos[relangle*3.1416/180])\)*
            Sin[relangle*3.1416/
                  180], \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {relangle, 0.036, 9.04}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(2.3925156249999993`\)."\)], "Message"],

Cell[BoxData[
    \(0.0031648735707903827`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qv1500\  = \ 
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          qdiff1500int[
              relangle]*\((Sin[
                  relangle*3.1416/
                      180])\)^3, \ \n\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t{relangle, 0.036, 9.04}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(2.3925156249999993`\)."\)], "Message"],

Cell[BoxData[
    \(0.00631847953384979`\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\t\(coulombint1500plot\  = \ 
      LogLogPlot[
        \((\((icoulombcm*\((Sin[thetacm])\)^3)\)\  //. {z\  -> 1.8, 
              en\  -> \ 1500, beta\  -> \ 0.001, 
              thetacm\  -> \ relangle*Pi/180})\), {relangle, 0.1, 20}, \ \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ AxesLabel\  -> \ {"\<angle (deg)\>", \ "\<i(angle)\>"}, 
        PlotRange\  -> \ {{0.1, 10}, {1*^-5, 0.01}}, \n
        \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ DisplayFunction\  -> \ Identity]; \)\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \  (*linear\ plot*) \n\t
        Plot[\((\((icoulombcm*\((Sin[thetacm])\)^3)\)\  //. {z\  -> 1.8, 
                en\  -> \ 1500, beta\  -> \ 0.001, 
                thetacm\  -> \ relangle*Pi/180})\), {relangle, 0, 
            90}, \ \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesLabel\  -> \ {"\<rel. angle (deg)\>", \ \
"\<i\>"}, 
          PlotRange\  -> \ {{0, 90}, {0, 
                0.01}}, \n\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction\  -> \ Identity];\)\)\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
    \(\(integrandqv1500\  = \ 
        Table[\((Part[listqdiff1500, 
                j]*\((Sin[
                    Part[angleqdiff1500, j]*
                      Pi/180])\)^3)\), \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {j, 1, 
            Length[angleqdiff1500]}];\)\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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    \(integrand1500plot\  = \ 
      LogLogListPlot[tableintegrandqv1500, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotStyle\  -> {PointSize[0.025], Hue[0.55]}, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotRange\  -> \ {{0.1, 20}, {1*^-5, 0.01}}, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        DisplayFunction\  -> Identity]; \)\n\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(integrand1500plot\  = \ 
        ListPlot[
          tableintegrandqv1500, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \
(*linear\ plot*) \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotStyle\  -> {PointSize[0.025], 
              Hue[0.55]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotRange\  -> \ {{0, 90}, {0, 
                0.01}}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
DisplayFunction\  -> Identity];\)\(\n\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

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Cell["Analysis of 5000 eV data", "Subsection"],

Cell[CellGroupData[{

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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qdiff5000int\  = \ Interpolation[dataqdiff5000]\)], "Input"],

Cell[BoxData[
    TagBox[\(InterpolatingFunction[{{0.036`, 7.56`}}, "<>"]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(norm\  = \ 
      2*Pi*NIntegrate[Sin[relangle*3.1416/180], \ {relangle, 0, 90}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
    \(6.283208386607923`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qt5000\  = \ 
      2*Pi*NIntegrate[
          qdiff5000int[relangle]*Sin[relangle*3.1416/180], \ {relangle, 
            0.036, 7.56}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.5356406249999999`\)."\)], "Message"],

Cell[BoxData[
    \(7.727604727922762`\)], "Output"]
}, Open  ]],

Cell["\<\
This value is about 15% larger than that given by Newman et al. (1985), but \
is only 7% larger than the experimental value of Savola (1973) as obtained \
using the detector angle of 0.056 deg.  This difference may be the result of \
the improved differential scattering data available at low angles from the \
experiments of Nitz as used in our calculation but not in Newman's \
calculation.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qm5000\  = \ 
      2*Pi*NIntegrate[
          qdiff5000int[relangle]*\((1 - Cos[relangle*3.1416/180])\)*
            Sin[relangle*3.1416/
                  180], \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {relangle, 0.036, 7.56}]*3.1416/180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.8883281249999999`\)."\)], "Message"],

Cell[BoxData[
    \(0.0004927448116516897`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qv5000\  = \ 
      2*Pi*NIntegrate[
          qdiff5000int[
              relangle]*\((Sin[
                  relangle*3.1416/
                      180])\)^3, \ \n\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t{relangle, 0.036, 7.56}]*3.1416/
          180\)], "Input"],

Cell[BoxData[
    \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) 
      "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \
recursive bisections in \!\(relangle\) near \!\(relangle\) = \
\!\(0.8883281249999999`\)."\)], "Message"],

Cell[BoxData[
    \(0.000984317960030413`\)], "Output"]
}, Open  ]],

Cell["\<\
These qm and qv values are very small and suggests we need to look very \
carefully at the large angle scattering, e.g., how does experiment compare \
with screened Coulomb scattering.\
\>", "SmallText"],

Cell[BoxData[
    \(\(\t\(coulomb5000plot\  = \ 
      LogLogPlot[
        \((\((icoulombcm*\((Sin[thetacm])\)^3)\)\  //. {z\  -> 2, 
              en\  -> \ 5000, beta\  -> \ 0.0004, 
              thetacm\  -> \ relangle*Pi/180})\), {relangle, 0.1, 20}, \ \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ AxesLabel\  -> \ {"\<angle (deg)\>", \ "\<i(angle)\>"}, 
        PlotRange\  -> \ {{0.1, 10}, {1*^-5, 0.001}}, \n
        \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ DisplayFunction\  -> \ Identity]; \)\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(\(\t\)\(coulomb5000plot\  = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*linear\ plot*) \ \
\ Plot[\((\((icoulombcm*\((Sin[thetacm])\)^3)\)\  //. {z\  -> 2, 
                en\  -> \ 5000, beta\  -> \ 0.0004, 
                thetacm\  -> \ relangle*Pi/180})\), {relangle, 0, 
            90}, \ \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesLabel\  -> \ {"\<angle (deg)\>", \ \
"\<i(angle)\>"}, 
          PlotRange\  -> \ {{\(-10\), 90}, {0, 
                0.002}}, \n\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction\  -> \ 
            Identity];\)\)\)], "Input"],

Cell[BoxData[
    \(\(angleqdiff5000\  = \ Part[Transpose[dataqdiff5000], 1];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[%]\)], "Input"],

Cell[BoxData[
    \(61\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(listqdiff5000\  = \ \ Part[Transpose[dataqdiff5000], 2];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[%]\)], "Input"],

Cell[BoxData[
    \(61\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(integrandqv5000\  = \ 
        Table[\((Part[listqdiff5000, 
                j]*\((Sin[
                    Part[angleqdiff5000, j]*
                      Pi/180])\)^3)\), \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {j, 1, 
            Length[angleqdiff5000]}];\)\)], "Input"],

Cell[BoxData[
    \(\(tableintegrandqv5000\  = \ 
        Transpose[{angleqdiff5000, integrandqv5000}];\)\)], "Input"],

Cell[BoxData[
    \(\(\ \ \ \ \ \ \ \ \ \ \ 
    \(integrand5000plot\  = \ 
      LogLogListPlot[tableintegrandqv5000, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotStyle\  -> {PointSize[0.025], Hue[0.55]}, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotRange\  -> \ {{0.1, 10}, {1*^-5, 0.001}}, \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        DisplayFunction\  -> Identity]; \)\n\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(integrand5000plot\  = \ 
        ListPlot[
          tableintegrandqv5000, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ PlotStyle\  -> {PointSize[0.025], 
              Hue[0.55]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
PlotRange\  -> \ {{\(-10\), 90}, {0, 
                0.002}}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
DisplayFunction\  -> Identity];\)\(\n\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Show[integrand5000plot, coulomb5000plot, \ 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
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Cell[BoxData[
    \(Quit[]\)], "Input"],

Cell["Not converted to He-He beyond here", "Subtitle",
  Background->RGBColor[1, 0, 0]],

Cell["\<\
Examples of relative differential scattering cross sections for the \
parameters of our fit.\
\>", "Subsubsection",
  Evaluatable->False],

Cell["f[x_] :=  a*(1 -(1 - 2*beta)*Cos[x])^-2", "Input",
  PageWidth->Infinity,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Plot[{(f[theta]/f[0] /. beta -> empiricalbeta /. enrel  -> 0.01 (*eV*)), 
        (f[theta]/f[0] /. beta -> empiricalbeta /. enrel  -> 1 (*eV*)),
        (f[theta]/f[0] /. beta -> empiricalbeta /. enrel  -> 10 (*eV*)),
        (f[theta]/f[0] /. beta -> empiricalbeta /. enrel  -> 100 (*eV*)),
        (f[theta]/f[0] /. beta -> empiricalbeta /. enrel  -> 1000 (*eV*))},
              {theta,0,Pi}, 
               PlotRange -> {0,1.2},
               PlotStyle -> {Dashing[{ }],  (*0.01 eV - solid*)
                                     Dashing[{0.003,0.003}], (*1 eV - dots*)
                                     Dashing[{0.01,0.01}], (*10 eV - short \
dashes*)
                                     Dashing[{0.05,0.01}], (*100 eV - long \
dashes*)
                                     Dashing[{0.05,0.01,0.01,0.01}]}, (*1000 \
eV - long & short dashes*)
               AxesLabel  -> {\"theta (rd)\",\"i(theta)/i(0)\"}];\
\>", "Input",
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  FontSize->12,
  FontWeight->"Bold",
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Cell["Export data to file.", "Section",
  Evaluatable->False],

Cell["Parameters and and resuktant cross sections. ", "Subsubsection"],

Cell[BoxData[
    \(\(fittedqdqvqt\  = \ 
      Transpose[{energies, empiricalbetalist, empiricalalist, empiricalblist, 
          \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ qvfitlist, qdfitlist, 
          qtfitlist}]; \)\)], "Input"],

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    \(\(fittedresult\  = \ 
      Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm]&, 
        fittedqdqvqt, {2}]; \)\)], "Input"],

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fitteddataout1 = TableForm[fittedresult,TableHeadings -> {None,
                   {\"%rel. en.\",\"beta\",\"a\",\"b\",\"Qv\", \
\"Qd\",\"Qt\"}},
                   TableSpacing -> {0,1}]\
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  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

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Block[
{stream1,theoryfile,today},
SetOptions[$Output,PageWidth->120];
outputfile := \"c:\\\\proplot\\\\arctprop\\\\ararxsec.dat\";
(*DeleteFile[outputfile];*)
stream1 = OpenWrite[outputfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Diffusion, viscosity, and total cross \
sections\"],
                    \"\\n\",
                    ToString[fitteddataout1]] ;
Close[stream1];
SetOptions[$Output,PageWidth->53];
]\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

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  PageWidth->Infinity],

Cell["Diffferential cross sections ", "Subsubsection"],

Cell[BoxData[
    \(\(ithetadatafunction\  = \ Interpolation[ithetadataout]; \)\)], "Input"],

Cell[BoxData[
    \(\(ithetadatatable\  = \ Table[ithetadatafunction[2*j], {j, 2, 44/2}]; 
    \)\)], "Input"],

Cell[BoxData[
    \(\(thetatable\  = \ Table[2*j, {j, 2, 44/2}]; \)\)], "Input"],

Cell[BoxData[
    \(\(ithetafittable\  = \ 
      \(i\ \  //. 
          \ \ {\ beta\  -> \ empiricalbeta, \ a\  -> \ empiricala, \ 
            b\  -> \ empiricalb\ }\)\  //. \ \ \n
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {
          enrel\ \  -> \ 0.03, \ theta\  -> \ thetatable*Pi/90}; \)\)], 
  "Input"],

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    \(\(fittedithetaresult\  = \ 
      Transpose[{thetatable, ithetadatatable, ithetafittable}]; \)\)], "Input"],

Cell["\<\
fittedithetadataout = TableForm[fittedithetaresult,TableHeadings -> {None,
                   {\"%theta\",\"i expt.\",\"i fit\"}},
                   TableSpacing -> {0,1}]\
\>", "Input",
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Block[
{stream1,theoryfile,today},
SetOptions[$Output,PageWidth->120];
outputfile := \"c:\\\\proplot\\\\arctprop\\\\tparson.dat\";
(*DeleteFile[outputfile];*)
stream1 = OpenWrite[outputfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Diffferential scattering I(theta)\"],
                    \"\\n\",
                     StringForm[\"Parson,Sioska, and Lee, J. Chem. Phys. 56, \
1511 (1972)\"],
                    \"\\n\",
                    ToString[fittedithetadataout]];
Close[stream1];
SetOptions[$Output,PageWidth->53];
]\
\>", "Input",
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  Evaluatable->False,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
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  Evaluatable->False],

Cell["Discussion", "Subsubsection",
  Evaluatable->False],

Cell["\<\
1. The objective of this notebook is to use measured and/or calculated cross \
sections to infer the effectve width of the forward peaked differential cross \
section for the elastic scattering of He by He.  
2. At high energies the width is small and we have shown that the momentum \
tramsfer and viscosity cross sections weight the differential cross section \
the same way, i.e., theta^2, except that the ratio of the cross sections at \
high energies approaches 2:1.  Therefore we learn very little from their \
ratio.
3. We are trying to fit three observables, i.e., Qv(enrel), Qm(enrel), and \
Qt(enrel) with a formula having three parameters a(enrel), b(enrel), and \
beta(enrel).  
4. At low energies one one cannot asssume that the differential cross section \
approaches isotropic and beta approaches 0.5.  In fact our analysis give a \
higher degree of anisotropy, i.e., larger beta, than at high energies.
5. We have emphasized the fit to the experimental viscosity coefficient data \
at low energies and to the theoretical viscosity cross section at high \
energies because we plan to use the results for the modeling of the transport \
of He atoms through He where the viscosity angular weighting is the important \
consideration.
6. We get a reasonable fit between the measured differential cross sections \
from Rice and the formula for Couomb scattering when we use the Newman et al \
(1985) values of the effective z, e.g., z = 2 at 5000 eV.\
\>", "SmallText",
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  Evaluatable->False,
  FontFamily->"Helv",
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
We have determined total, diffusion, and viscosity cross sections from \
differential scattering cross sections for He-He at 500, 1500, and 5000 eV.\
\>", "SmallText",
  Evaluatable->False],

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(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)