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Notebook[{
Cell["Thermal Transport Cross Sections for Ar -Ar Collisions", "Subtitle"],

Cell[CellGroupData[{

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

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Cell["ClearAll[vo,temp,r,visctemp,s]", "Input"]
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        \(Date[]\)[\([5]\)], \(Date[]\)[\([6]\)]]\)], "Input"],

Cell[BoxData[
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\\!\\(10\\):\\!\\(12\\):\\!\\(27\\)\"\>",
      StringForm[ "``/``/`` ``:``:``", 11, 6, 1998, 10, 12, 27],
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Cell["\<\
We wish to determine the viscosity and momentum transfer cross sections for \
Ar in Ar from calculated thermal viscosity and diffusion data, e.g., Dymond, \
J. Phys. B 4, 621 (1971). We use this reference primarily because it is \
convienient, rather than because we have done a review of the experimental \
literature. We consider diffusion first because it is more familiar to us \
than viscosity.\
\>", "SmallText"],

Cell["Momentum transfer or diffusion cross section", "Section"],

Cell["\<\
It should be kept in mind that the theory of self diffusion is messy.  \
However, my reading of Chapman and Cowling, Sect. 14.5 is that in the end one \
uses the theory for mixed gases with equal mass.\
\>", "SmallText"],

Cell["Input data", "Subsection"],

Cell["\<\
The results can be expressed in terms of self-diffusion coefficients versus \
temperature.  The data from Table 4 of Dymond in the form \
{temperature,diffusion coefficient} are:\
\>", "SmallText"],

Cell["\<\
normdifftable = {{1000, \
1.53},{2000,5.06},{3000,10.3},{4000,17.0},{5000,25.2}};\
\>", "Input"],

Cell["\<\
The units are degrees Kelvin and cm^2/s at 1 atm pressure.  We begin by \
normalizing the diffusion coefficients to the Ar density and converting to \
MKS units.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["\<\
normdifflist = Table[10^-21*100*Part[normdifftable,j,2]*
                            2.69*10^19*273/
                            Part[normdifftable,j,1],
                     {j,1,5}]\
\>", "Input"],

Cell[BoxData[
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      3.12107250000000036`, 3.70122479999999987`}\)], "Output"]
}, Open  ]],

Cell["\<\
The 2.69E19*273/T converts the diffusion coefficients at 1atm to density \
normalized values in cm^-1s^-1.  The factor of 100 to converts the diffusion \
coefficients from cm^-1s^-1 to m^-1s^-1.  The 1E-21 is a scale factor to give \
smaller numbers and will be removed later.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["templist = Table[Part[normdifftable,j,1],{j,1,5}]", "Input"],

Cell[BoxData[
    \({1000, 2000, 3000, 4000, 5000}\)], "Output"]
}, Open  ]],

Cell["\<\
normdifflistplot = LogLogListPlot[Transpose[{templist,normdifflist}],
                                      PlotRange -> {{100,10000},{0.1,10}},
                                      PlotStyle -> PointSize[0.02],
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell["\<\
From Chapman and Cowling, p. 267 the self-diffusion coefficient at NTP is \
0.157 cm^2/s so that the normalized diffusion coefficient is 0.422 *10^21 \
m^-1s^-1.\
\>", "SmallText"],

Cell["\<\
ccnormdifflistplot = LogLogListPlot[{{273,0.422}},
                                      PlotRange -> {{100,10000},{0.1,10}},
                                      PlotStyle -> PointSize[0.02],
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell["\<\
In the following we get a better fit with r = 0.74, but as a compromise with \
the viscosity data we use r = 0.73.\
\>", "SmallText"],

Cell[BoxData[
    \(\(r\  = \ 0.73; \)\)], "Input"],

Cell["empnormdiff = 1.125*(temp/1000)^0.73;  (*10^19 m^-1s^-1*)", "Input"],

Cell["\<\
empnormdiffplot = LogLogPlot[empnormdiff, {temp,100,10000},
                                      PlotRange -> {0.1,10} ,
                                      DisplayFunction -> Identity];\
\>", "Input"],

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Cell["\<\
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\>", "Input"],

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Cell["\<\
Momentum transfer cross section relation to diffusion for isotope diffusion\
\>", "Subsection"],

Cell[CellGroupData[{

Cell["From Massey and Burhop,  p. 367", "Subsubsection"],

Cell["\<\
normtheordiff := 3/16*Sqrt[Pi]*vo^7/p12r                               (*Eq. \
3.1 (7)*)\
\>", "Input"],

Cell["where", "SmallText"],

Cell["vo = Sqrt[2*k*temp/mr];", "Input"],

Cell["\<\
We used to believe the expression given by Massey and Burhop should be \
reduced by a factor of 2, but now use their relation\
\>", "SmallText"],

Cell["\<\
p12 := 2*Integrate[v^5*qd*Exp[-mr*v^2/(2*k*temp)],{v,0,Infinity},
           Assumptions->{mr/k/temp>0}]                               (*Eq.3.1 \
(8)*)\
\>", "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{p12}, p12]\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(16\ k\^3\ qd\ temp\^3\)\/mr\^3\)], "Output"]
}, Open  ]],

Cell["Apparently Mathematica assumed qd was a constant", "SmallText"],

Cell[BoxData[
    \(\(qd\  := \ 
      2*Pi*Integrate[
          \((1 - cos[theta])\) I[theta, v] sin[theta], {theta, 0, Pi}]\  (*
      Eq . \ 3.1\ \((6)\)*) \)\)], "Input",
  Evaluatable->False],

Cell["\<\
For a cross sectiion that can be approximated by qd = qdr*(v/vr)^(2*rm), \
where qdr is the cross section at vo.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["\<\
p12r = qdr/vr^(2*rm)*Integrate[v^5*v^(2*rm)*Exp[-v^2/vo^2],{v,0,Infinity},
                           Assumptions -> {mr/(k*temp)> 0,rm > -2,vo^2 >0}]  \
//PowerExpand\
\>", "Input"],

Cell[BoxData[
    \(2\^\(2 + rm\)\ k\^\(3 + rm\)\ mr\^\(\(-3\) - rm\)\ qdr\ 
      temp\^\(3 + rm\)\ vr\^\(\(-2\)\ rm\)\ Gamma[3 + rm]\)], "Output"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(normtheordiffr\  = \ 
      normtheordiff\ \  /. \ mr\  -> m/2\  // PowerExpand\n\)\)], "Input"],

Cell[BoxData[
    \(\(3\ 2\^\(\(-2\) - 2\ rm\)\ k\^\(1\/2 - rm\)\ 
        m\^\(\(-\(1\/2\)\) + rm\)\ \@\[Pi]\ temp\^\(1\/2 - rm\)\ 
        vr\^\(2\ rm\)\)\/\(qdr\ Gamma[3 + rm]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Block[{r,vo},
normtheordiff0 = normtheordiffr /. {rm -> 0,vo -> Sqrt[4*k*temp/m],mr->m/2} \
// PowerExpand;
normtheordiff1 = normtheordiff0 //. temp -> Pi*m/16/k*vbar^2 // PowerExpand
]\
\>", "Input"],

Cell[BoxData[
    \(\(3\ \[Pi]\ vbar\)\/\(32\ qdr\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell["This agrees approximately the classical (1/3)(vbar/qdr).", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(normtheordiff0\)], "Input"],

Cell[BoxData[
    \(\(3\ \@k\ \@\[Pi]\ \@temp\)\/\(8\ \@m\ qdr\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["From Curtis, Hirshfelder and Bird,  p. 527", "Subsubsection"],

Cell["\<\

theordiffpure := 3/8*Sqrt[Pi*m*k*temp]/Pi/sigma^2/omega11star/n/m  (*Eq. \
8.2-9*);\
\>", "Input"],

Cell["where sigma is the diameter of a reference hard sphere,", "SmallText"],

Cell["\<\
omega11star  :=  omega11*Sqrt[Pi*m/k/temp]/Pi/sigma^2    (*Eq.8.2-8 for l=1 \
and s=1*);\
\>", "Input"],

Cell["\<\
omega11 := Sqrt[k*temp/Pi/m]*Integrate[y^5*omega1*Exp[-y^2],{y,0,Infinity},
                   Assumptions->{rc>-2}];                                   \
(*Eq. 8.2-3*)\
\>", "Input"],

Cell["where y^2 = m*v^2/4/k/temp and", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(omega1\  = \ 
      2*Pi*Integrate[\((1 - cos[theta])\) I[theta] sin[theta], {theta, 0, Pi}]
          \  (*Eq . \ 8.2 - 2*) \)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(2\ \[Pi]\ 
      \(\[Integral]\_0\%\[Pi]\( I[theta]\ \((1 - cos[theta])\)\ sin[theta]\) 
          \[DifferentialD]theta\)\)], "Output"]
}, Open  ]],

Cell["\<\
In the last equation we used the relation that b db = I[theta] sin[theta] \
dtheta from McDaniel,Mitchell, and Rudd, p. 8, Eq. 3-7-1.  We note that 
omega1 = qd from above and from Masey and Burhop, p. 367.\
\>", "SmallText",
  PageWidth->WindowWidth],

Cell[CellGroupData[{

Cell["Evaluation of omega11 without specification of  omega1:", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(theordiffpure\  // \ PowerExpand\)], "Input"],

Cell[BoxData[
    \(\(3\ \@k\ \@\[Pi]\ \@temp\)\/\(8\ \@m\ n\ omega1\)\)], "Output"]
}, Open  ]],

Cell["\<\
Mathematica assumed omega1 was a constant.   The result agrees with that \
obtained with the formulas originally given on p. 367 of Massey and Burhop.\
\>", "SmallText"]
}, Open  ]],

Cell["\<\
For a cross sectiion that can be approximated by omega1 = qd = \
qdr*(v/vr)^(2*rc) = qdr*y^(2*rc)*(vo/vr)^(2*rc), where qdr is the cross \
section at vo.\
\>", "SmallText"],

Cell[BoxData[
    \(\(omega1\  = \ qdr*y^\((2*rc)\)*\((vo/vr)\)^\((2*rc)\)\ ; \)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(omega11\ \  // PowerExpand\)], "Input"],

Cell[BoxData[
    \(\(2\^\(\(-1\) + rc\)\ k\^\(1\/2 + rc\)\ mr\^\(-rc\)\ qdr\ 
        temp\^\(1\/2 + rc\)\ vr\^\(\(-2\)\ rc\)\ Gamma[3 + rc]\)\/\(\@m\ 
        \@\[Pi]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(theornormdiffpure\  = \ 
      theordiffpure\ *n\ \  /. \ mr\  -> m/2\  // \ PowerExpand\)], "Input"],

Cell[BoxData[
    \(\(3\ 2\^\(\(-2\) - 2\ rc\)\ k\^\(1\/2 - rc\)\ 
        m\^\(\(-\(1\/2\)\) + rc\)\ \@\[Pi]\ temp\^\(1\/2 - rc\)\ 
        vr\^\(2\ rc\)\)\/\(qdr\ Gamma[3 + rc]\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell["Determination of empirical isotope diffusion cross section", 
  "Subsection"],

Cell[CellGroupData[{

Cell["\<\
empnormdiff = empnormdiff /. temp -> mr*vo^2/2/k //PowerExpand\
\>", "Input"],

Cell[BoxData[
    RowBox[{"0.00726361007663988011`", " ", 
      SuperscriptBox["temp", 
        StyleBox["0.729999999999999982`",
          StyleBoxAutoDelete->True,
          PrintPrecision->2]]}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(coefemp\  = \ Coefficient[empnormdiff, temp^0.73]\)], "Input"],

Cell[BoxData[
    \(0.00726361007663988011`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(theornormdiffpure\)], "Input"],

Cell[BoxData[
    \(\(3\ 2\^\(\(-2\) - 2\ rc\)\ k\^\(1\/2 - rc\)\ 
        m\^\(\(-\(1\/2\)\) + rc\)\ \@\[Pi]\ temp\^\(1\/2 - rc\)\ 
        vr\^\(2\ rc\)\)\/\(qdr\ Gamma[3 + rc]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(coeftheor\  = \ 
      Coefficient[theornormdiffpure, temp^\((1/2 - \ \ rc)\)]\)], "Input"],

Cell[BoxData[
    \(\(3\ 2\^\(\(-2\) - 2\ rc\)\ k\^\(1\/2 - rc\)\ 
        m\^\(\(-\(1\/2\)\) + rc\)\ \@\[Pi]\ vr\^\(2\ rc\)\)\/\(qdr\ 
        Gamma[3 + rc]\)\)], "Output"]
}, Open  ]],

Cell["\<\
In the present case the experimental data is consistent with 0.5 - rc = 0.73 \
or\
\>", "SmallText"],

Cell["\<\
rc = 0.5 - r;
k = 1.381*10^(-23);
mr = 20*1.66*10^(-27);
qe = 1.602*10^(-19);
m=mr*2;
vr = (2*qe*1/mr)^0.5;    (*for vr at a relative energy of 1 eV*)\
\>", "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Gamma[2.77]\)], "Input"],

Cell[BoxData[
    \(1.63506073608500157`\)], "Output"]
}, Open  ]],

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Cell[BoxData[
    \(rc\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(coeftheor\)], "Input"],

Cell[BoxData[
    \(1.36234555185962302`\/qdr\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(vr\)], "Input"],

Cell[BoxData[
    \(3106.54187315068375`\)], "Output"]
}, Open  ]],

Cell["\<\
Here we converted vo to an equivalent center of mass or relative energy u in \
eV.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["soldiff = Solve[coeftheor == 10^21*coefemp, {qdr}]", "Input"],

Cell[BoxData[
    \({{qdr \[Rule] 1.87557638348593691`*^-19}}\)], "Output"]
}, Open  ]],

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Cell["qd1 = (qdr /. soldiff )*u^rc", "Input"],

Cell[BoxData[
    \({1.87557638348593691`*^-19\/u\^0.229999999999999982`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["qd1 /. u -> 0.01  (*laboratory energy = 0.02 eV*)", "Input"],

Cell[BoxData[
    \({5.40922137649370693`*^-19}\)], "Output"]
}, Open  ]],

Cell["Viscosity cross section", "Section"],

Cell["\<\

We determine the viscosity cross section the same way as we did the momentum \
transfer cross section\
\>", "SmallText"],

Cell["Input data", "Subsection"],

Cell["\<\
The results can be expressed in terms of viscosity coefficients versus \
temperature.  Representative calculated values from Dymond (1971), Table 2 in \

the form {temperature,viscosity coefficient} are:\
\>", "SmallText"],

Cell["\<\

viscinput = {{1500, 720},{2100,910},{2700,1087},{3300,1254},{4000,1441},
                      {4500,1574},{5500,1827},{7000,2189}};\
\>", "Input"],

Cell["\<\
The units are degrees Kelvin and 1E-6 gm.cm^-1.s^-1 at 1 atm pressure.  First \
we change the units to MKS by\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["\<\

visclist = Table[10^4*100./1000.*10^-6*Part[viscinput,j,2], {j,1,8}]\
\>", "Input"],

Cell[BoxData[
    \({0.719999999999999928`, 0.909999999999999964`, 1.08699999999999995`, 
      1.254`, 1.44100000000000001`, 1.57400000000000002`, 
      1.82699999999999995`, 2.18900000000000005`}\)], "Output"]
}, Open  ]],

Cell["\<\
The factor of 100/1000 converts the viscosity from gm.cm^-1s^-1 to \
kg.m^-1s^-1.  The 1E4 is a scale factor to give larger numbers and will be 
removed later.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["visctemplist = Table[Part[viscinput,j,1],{j,1,8}]", "Input"],

Cell[BoxData[
    \({1500, 2100, 2700, 3300, 4000, 4500, 5500, 7000}\)], "Output"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \({1500, 2100, 2700, 3300, 4000, 4500, 5500, 7000}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Transpose[{visctemplist, visclist}]\)], "Input"],

Cell[BoxData[
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        4500, 1.57400000000000002`}, {5500, 1.82699999999999995`}, {7000, 
        2.18900000000000005`}}\)], "Output"]
}, Open  ]],

Cell["\<\

visclistplot = LogLogListPlot[Transpose[{visctemplist,visclist}],
                                      PlotRange -> {{100,10000},{0.1,10}},
                                      PlotStyle -> PointSize[0.02],
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell["\<\
The data given by Chapman and Cowling, p. 228 is visc = 2120*10^-7 gm/cm/sec \
= 0.212 *10^-4 kg/m/s at STP\
\>", "SmallText"],

Cell["\<\
ccvisclistplot = LogLogListPlot[{{273,0.212}},
                                      PlotRange -> {{100,10000},{0.1,10}},
                                      PlotStyle -> PointSize[0.02],
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell["\<\
The data given by Serikov and Nanbu (1996) is visc = 2.2275*10^-5 kg/m/s at \
STP\
\>", "SmallText"],

Cell["\<\
snvisclistplot = LogLogListPlot[{{273,0.222}},
                                      PlotRange -> {{100,10000},{0.1,10}},
                                      PlotStyle -> {Hue[1],PointSize[0.02]},
                                      (*PlotStyle -> PointSize[0.1],*)
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell["\<\
In the following we get a better fit to Dymond's results with s = 0.72, but \
we compromise with the diffusion data and use s = 0.73.\
\>", "SmallText"],

Cell[BoxData[
    \(\(s = 0.73; \)\)], "Input"],

Cell["empvisc = 1.21*(visctemp/3000)^0.73;  (*10^-4 kg.m^-1s^-1*)", "Input"],

Cell["\<\
empviscplot = LogLogPlot[empvisc, {visctemp,100,10000},
                                      PlotRange -> {0.1,10} ,
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell["\<\
Serikov and Nanbu (1996) give the viscosity as 2.23*^{-5} (temp/273)^{0.811} \
Ns/m^2 or kgm/m/s.  This is shown below by the dashed curve.\
\>", "SmallText"],

Cell[BoxData[
    \(\(serikovvisc\  := \ 10^4*2.23*10^{\(-5\)}*\((visctemp/273)\)^0.811\ ; 
    \)\)], "Input"],

Cell["\<\
serikovviscplot = LogLogPlot[Evaluate[serikovvisc], {visctemp,100,10000},
                                      PlotRange -> {0.1,10},
                                      PlotStyle -> Dashing[{0.05,0.05}],
                                      DisplayFunction -> Identity];\
\>", "Input"],

Cell[CellGroupData[{

Cell["\<\
Show[empviscplot,visclistplot,snvisclistplot,ccvisclistplot,serikovviscplot,
               DisplayFunction -> $DisplayFunction];
\
\>", "Input"],

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Cell["ClearAll[k,mr,vo,s,vr,m,qe]", "Input"],

Cell["Viscosity cross section relation to viscosity coefficient", "Subsection"],

Cell[CellGroupData[{

Cell["From Massey and Burhop,  p. 368", "Subsubsection"],

Cell["\<\
theorvisc := 10*k^3*visctemp^3/m^2*Sqrt[4*Pi*k*visctemp/m]^3/Pi/r11 // \
PowerExpand      (*Eq. 3.1 (10)*)\
\>", "Input"],

Cell["In general", "SmallText"],

Cell["\<\
r11 := 1/2*Integrate[v^7*qv*Exp[-m*v^2/(4*k*visctemp)],{v,0,Infinity},
                                     Assumptions ->{m/k/visctemp>0}]      \
(*Eq. 3.1 (11)*)\
\>", "Input",
  Evaluatable->False],

Cell["where", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(qv = \ 
      2*Pi*Integrate[I[theta] \((sin[theta])\)^3, {theta, 0, Pi}]
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*Eq . \ 3.1\ \((11)\)*) 
      \)\)], "Input",
  Evaluatable->False],

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Cell["Block[{theorvisc}, N[theorvisc  // PowerExpand]   ]", "Input"],

Cell[BoxData[
    \(\(80\ k\^\(9/2\)\ \@\[Pi]\ visctemp\^\(9/2\)\)\/\(m\^\(7/2\)\ r11\)\)], 
  "Output"]
}, Open  ]],

Cell["Here Mathematica assumed qv was a constant.", "SmallText"]
}, Open  ]],

Cell["\<\
For a cross section that can be approximated by qvs*(v/vr)^(2*sm), where qvs \
is the viscosity cross section at vr.\
\>", "SmallText"],

Cell["\<\
r11s := qvs/2/vr^(2*sm)*Integrate[v^7*v^(2*sm)*Exp[-m*v^2/(4*k*visctemp)],{v,\
0,Infinity},
                   Assumptions->{sm>-2,m/k/visctemp>0}] // PowerExpand\
\>", "Input"],

Cell[CellGroupData[{

Cell["theorviscs = theorvisc //.  r11->r11s ", "Input"],

Cell[BoxData[
    \(\(5\ 2\^\(\(-2\) - 2\ sm\)\ k\^\(1\/2 - sm\)\ m\^\(1\/2 + sm\)\ 
        \@\[Pi]\ visctemp\^\(1\/2 - sm\)\ vr\^\(2\ sm\)\)\/\(qvs\ 
        Gamma[4 + sm]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["theorvisc0 = theorviscs /. sm -> 1", "Input"],

Cell[BoxData[
    \(\(5\ m\^\(3/2\)\ \@\[Pi]\ vr\^2\)\/\(384\ \@k\ qvs\ \@visctemp\)\)], 
  "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["From Curtis, Hirshfelder and Bird,  p. 527", "Subsubsection"],

Cell["\<\
theorviscpure := 5/16*Sqrt[Pi*m*k*visctempc]/(Pi*sigma^2*omega22star)  (*Eq. \
8.2-10*);\
\>", "Input"],

Cell["\<\
where sigma is the radius of a reference hard sphere.  Using mr = m/2\
\>", "SmallText"],

Cell["\<\
omega22star  :=  omega22*Sqrt[2*Pi*mr/(k*visctempc)]/(Pi*sigma^2*3*(1-(1/3))) \
;(*Eq.8.2-8 for l=2 and s=2*)\
\>", "Input"],

Cell["\<\

omega22 := \
Sqrt[k*visctempc/(2*Pi*mr)]*Integrate[y^7*omega2*Exp[-y^2],{y,0,Infinity}];
                            (*Eq. 8.2-3*)\
\>", "Input"],

Cell["and", "SmallText"],

Cell[BoxData[
    \(\(omega2\  = \ 
      2*Pi*Integrate[\((1 - \((cos[theta])\)^2)\) I[theta] sin[theta], \n
          \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {
            theta, 0, Infinity}]
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*Eq . \ 8.2 - 2*) 
      \)\)], "Input",
  Evaluatable->False],

Cell["\<\
In the last equation we used the relation that b db = I[theta] sin[theta] \
dtheta from McDaniel,Mitchell, and Rudd, p. 8, Eq. 3-7-1.  We note that 
omega2 = qv from above and from Masey and Burhop, p. 368.\
\>", "SmallText"],

Cell["\<\
For a cross sectiion that can be approximated by qv = qvs*(v/vr)^(2*sc) = \
qvs*y^(2*sc)*(vos/vr)^(2*sc), where qvs is the viscosity cross section at vr.\
\
\>", "SmallText"],

Cell[BoxData[
    \(omega2\  = \ qvs*\((y*vos/vr)\)^\((2*sc)\); 
    vos\  = \ Sqrt[2*k*visctempc/mr]; mr = m/2; \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(theorviscpuresc\  = \ N[theorviscpure\  // \ PowerExpand]\)], "Input"],

Cell[BoxData[
    \(\(8.86226925452758074`\ 2.`\^\(\(-2.`\) - 2.`\ sc\)\ 
        k\^\(\(0.5`\[InvisibleSpace]\) - 1.`\ sc\)\ 
        m\^\(\(0.5`\[InvisibleSpace]\) + sc\)\ 
        visctempc\^\(\(0.5`\[InvisibleSpace]\) - 1.`\ sc\)\ 
        vr\^\(2.`\ sc\)\)\/\(qvs\ Gamma[\(4.`\[InvisibleSpace]\) + sc]\)\)], 
  "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(theorviscpure1\  = \ theorviscpuresc\  /. \ sc\  -> \ 1\)], "Input"],

Cell[BoxData[
    \(\(0.0230788261836655728`\ m\^1.5`\ vr\^2.`\)\/\(k\^0.5`\ qvs\ 
        visctempc\^0.5`\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(solqvs\  = \ Solve[theorviscpuresc\  == \ exptvisc, {qvs}]\)], "Input"],

Cell[BoxData[
    \({{qvs \[Rule] 
          \(2.21556731363189518`\ 2.`\^\(\(-2\)\ sc\)\ k\^\(1\/2 - sc\)\ 
              m\^\(1\/2 + sc\)\ visctempc\^\(1\/2 - sc\)\ 
              vr\^\(2\ sc\)\)\/\(exptvisc\ Gamma[4 + sc]\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qvsc1\  = \ 
      \((\((qvs\ \  /. solqvs)\)\  /. \ vr\  -> \ Sqrt[2*qe*1/mr])\)*\ 
          u^sc\  // \ PowerExpand\)], "Input"],

Cell[BoxData[
    \({\(2.21556731363189518`\ 1.`\^\(2\ sc\)\ k\^\(1\/2 - sc\)\ \@m\ qe\^sc\ 
          u\^sc\ visctempc\^\(1\/2 - sc\)\)\/\(exptvisc\ Gamma[4 + sc]\)}\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qvsc1\  /. \ sc\  -> 0\)], "Input"],

Cell[BoxData[
    \({\(0.369261218938649227`\ \@k\ \@m\ \@visctempc\)\/exptvisc}\)], 
  "Output"]
}, Open  ]],

Cell["From Bird (1981),  p. 250", "Subsubsection"],

Cell["\<\
For a cross sectiion that can be approximated by  qv = qvr*(c/cr)^(2*b) , \
where qdr is the cross section at cr.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["\<\
viscbird = \
5/8*Sqrt[Pi*m*k*temp]*(m/(4*k*temp))^-4/Integrate[c^7*qvr*(c/cr)^(2*b)*Exp[-m*\
c^2/(4*k*temp)], 
       {c, 0, Infinity}, Assumptions -> m/k/temp > 0] // PowerExpand\
\>", "Input"],

Cell[BoxData[
    \(\(5\ 2\^\(\(-2\) - 2\ b\)\ cr\^\(2\ b\)\ k\^\(1\/2 - b\)\ 
        m\^\(1\/2 + b\)\ \@\[Pi]\ temp\^\(1\/2 - b\)\)\/\(qvr\ 
        Gamma[4 + b]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(viscbird1\  = \ viscbird\  /. \ b\  -> \ 0\)], "Input"],

Cell[BoxData[
    \(\(5\ \@k\ \@m\ \@\[Pi]\ \@temp\)\/\(24\ qvr\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(viscbirdb\  = \ 
      viscbird\  /. \ cr\  -> \ Sqrt[2*qe*ur/\((m/2)\)]\  // \ PowerExpand
        \)], "Input"],

Cell[BoxData[
    \(\(5\ k\^\(1\/2 - b\)\ \@m\ \@\[Pi]\ qe\^b\ temp\^\(1\/2 - b\)\ 
        ur\^b\)\/\(4\ qvr\ Gamma[4 + b]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[viscbirdb\  == \ viscexpt, qvr]\)], "Input"],

Cell[BoxData[
    \({{qvr \[Rule] 
          \(5\ k\^\(1\/2 - b\)\ \@m\ \@\[Pi]\ qe\^b\ temp\^\(1\/2 - b\)\ 
              ur\^b\)\/\(4\ viscexpt\ Gamma[4 + b]\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qvr1\  = \ viscbirdb*qvr/viscexpt\)], "Input"],

Cell[BoxData[
    \(\(5\ k\^\(1\/2 - b\)\ \@m\ \@\[Pi]\ qe\^b\ temp\^\(1\/2 - b\)\ 
        ur\^b\)\/\(4\ viscexpt\ Gamma[4 + b]\)\)], "Output"]
}, Open  ]],

Cell["Determination of empirical viscosity cross section", "Subsection"],

Cell[CellGroupData[{

Cell["empvisc1 = 10^-4*empvisc  //PowerExpand", "Input"],

Cell[BoxData[
    RowBox[{"3.50337683794548571`*^-7", " ", 
      SuperscriptBox["visctemp", 
        StyleBox["0.729999999999999982`",
          StyleBoxAutoDelete->True,
          PrintPrecision->2]]}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(coefempv\  = \ Coefficient[empvisc1, visctemp^0.73]\)], "Input"],

Cell[BoxData[
    \(3.50337683794548571`*^-7\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(theorviscpuresc\ \)\)], "Input"],

Cell[BoxData[
    \(\(8.86226925452758074`\ 2.`\^\(\(-2.`\) - 2.`\ sc\)\ 
        k\^\(\(0.5`\[InvisibleSpace]\) - 1.`\ sc\)\ 
        m\^\(\(0.5`\[InvisibleSpace]\) + sc\)\ 
        visctempc\^\(\(0.5`\[InvisibleSpace]\) - 1.`\ sc\)\ 
        vr\^\(2.`\ sc\)\)\/\(qvs\ Gamma[\(4.`\[InvisibleSpace]\) + sc]\)\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(coeftheor\  = \ Coefficient[viscbirdb, temp^\((1/2 - b)\)]\)], "Input"],

Cell[BoxData[
    \(\(5\ k\^\(1\/2 - b\)\ \@m\ \@\[Pi]\ qe\^b\ ur\^b\)\/\(4\ qvr\ 
        Gamma[4 + b]\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(coeftheorc\  = \ 
      Coefficient[theorviscpuresc, visctempc^\((0.5 - 1. \ sc)\)]\)], "Input"],

Cell[BoxData[
    \(\(2.21556731363189518`\ 2.`\^\(\(-2.`\)\ sc\)\ 
        k\^\(\(0.5`\[InvisibleSpace]\) - 1.`\ sc\)\ 
        m\^\(\(0.5`\[InvisibleSpace]\) + sc\)\ vr\^\(2.`\ sc\)\)\/\(qvs\ 
        Gamma[\(4.`\[InvisibleSpace]\) + sc]\)\)], "Output"]
}, Open  ]],

Cell["\<\
In the present case the experimental data is consistent with 0.5 - sc = 0.73 \
or\
\>", "SmallText"],

Cell["\<\
s = 0.73;
sc = 0.5 - s;
b = sc;
k = 1.381*10^(-23);
mr = 20*1.66*10^(-27);
qe = 1.602*10^(-19);
m=mr*2;
vr = (2*qe*1/mr)^0.5;    (*for vr at a relative energy of 1 eV*)
ur = 1;\
\>", "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(coeftheor\)], "Input"],

Cell[BoxData[
    \(5.44282458745360919`*^-26\/qvr\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["solviscb = Solve[coeftheor == coefempv, {qvr}]", "Input"],

Cell[BoxData[
    \({{qvr \[Rule] 1.55359381511624361`*^-19}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["qv1 = (qvr /. solviscb )*u^b", "Input"],

Cell[BoxData[
    \({1.55359381511624361`*^-19\/u\^0.229999999999999982`}\)], "Output"]
}, Open  ]],

Cell["\<\
Next we convert to an equivalent center of mass or relative energy u in eV.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["qv1 /. u -> 0.01  (*laboratory energy = 0.02 eV*)", "Input"],

Cell[BoxData[
    \({4.48061350585789863`*^-19}\)], "Output"]
}, Open  ]],

Cell["An alternate procedure is:", "SmallText"],

Cell[CellGroupData[{

Cell["solvisc = Solve[coeftheorc == coefempv, {qvs}]", "Input"],

Cell[BoxData[
    \({{qvs \[Rule] 1.55359381511624299`*^-19}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["qv1 = (qvs /. solvisc )*u^sc", "Input"],

Cell[BoxData[
    \({1.55359381511624299`*^-19\/u\^0.229999999999999982`}\)], "Output"]
}, Open  ]],

Cell["\<\
Next we convert to an equivalent center of mass or relative energy u in eV.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["qv1 /. u -> 0.01  (*laboratory energy = 0.02 eV*)", "Input"],

Cell[BoxData[
    \({4.48061350585789597`*^-19}\)], "Output"]
}, Open  ]],

Cell["Comparison of viscosity and isotopic diffusion results", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(crossratio\  = \ qv1/qd1\)], "Input"],

Cell[BoxData[
    \({0.828328736059654247`}\)], "Output"]
}, Open  ]],

Cell["\<\
The viscosity cross sections are 80% of the diffusion cross sections, \
compared to the ratio of about 2/3 expected for an enegy independent cross \
section. See Hirshfelder, Curtis, and Bird, p. 526.\
\>", "SmallText"],

Cell["\<\
 We should check to see whether the viscosity and diffusion coefficients have \
the expected relationship from Chapman and Cowling, Eq. (14.5,1), i.e., 
normtheordiff*m/theorvisc = 6/5;\
\>", "SmallText"],

Cell["We find from our empirical fits", "SmallText"],

Cell[CellGroupData[{

Cell["\<\
Block[{mr},
mr = 20*1.66*10^(-27);
expratio = 10^21*empnormdiff/(10^-4*empvisc)*(2*mr) /. temp -> visctemp
]\
\>", "Input"],

Cell[BoxData[
    \(1.37668235933114546`\)], "Output"]
}, Open  ]],

Cell["\<\
This value is somewhat higher than the value of 1.33 given on p. 267 by \
Chapman and Cowling.  Unfortunately, Dymond does not discuss this ratio.\
\>", "SmallText"],

Cell["\<\
From the Massey and Burhop formulas with the power law temperature dependence\
\
\>", "SmallText"],

Cell[CellGroupData[{

Cell["\<\
Block[{mr,vo,qvs,qdr},
mr = m/2;
qvs = 2*qdr/3;
sm = rm;
temp = visctemp;
tranratio= m*normtheordiffr/theorviscs // PowerExpand
]\
\>", "Input"],

Cell[BoxData[
    \(\(2.65599999999999969`*^-26\ 
        6.64000000000000056`*^-26\^\(\(-\(1\/2\)\) - rm\)\ 
        1.50602409638554224`*^25\^\(1\/2 - rm\)\ Gamma[4 + rm]\)\/Gamma[
        3 + rm]\)], "Output"],

Cell[CellGroupData[{

Cell[BoxData[
    \(tranratio\  //. \ rm -> 0\)], "Input"],

Cell[BoxData[
    \(1.20000000000000017`\)], "Output"]
}, Open  ]],

Cell["This gives the expected theoretical value of 1.2.", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(tranratio\  //. \ rm\  -> 0.5\)], "Input"],

Cell[BoxData[
    \(1.40000000000000017`\)], "Output"]
}, Open  ]],

Cell["\<\
This is to be compared with the expected theoretical value of 1.55 for \
Maxwellian molecules.  See Chapman and Cowling p. 265.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(tranratio\  //. \ rm\  -> 1\)], "Input"],

Cell[BoxData[
    \(1.60000000000000008`\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell["Comparison with published cross sections", "Subsection"],

Cell["\<\
Comparison with Serikov and Nanbu (1996) where these authors use \
\>", "SmallText"],

Cell[BoxData[
    \(viscref\  = \ 2.2275*10^\(-5\); \  (*N\ s/m^2*) \ntref\  = \ 273; \n
    omegasn\  = \ 0.811; \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(crossrefsn\  = \ 
      Pi*15 \((m*k*tref/Pi)\)^0.5/
          \((2*\((5 - 2  omegasn)\)*\((7 - 2  omegasn)\) viscref)\)\)], 
  "Input"],

Cell[BoxData[
    \(5.19756879547501782`*^-19\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(crosssn\  = \ 
      crossrefsn*
        \((k*tref/\((qe*u)\))\)^\((omegasn - 0.5)\)/Gamma[5/2 - omegasn]\ 
    \)\)], "Input"],

Cell[BoxData[
    \(1.78641007516216054`*^-19\ \((1\/u)\)\^0.311000000000000031`\)], 
  "Output"]
}, Open  ]],

Cell["The cross section formula Nanbu sent me by email is:", "SmallText"],

Cell[BoxData[
    \(\(qvhs\  := \ 60.3*\((0.0235/enrel)\)^0.311\ \ \  (*10^\(-20\)\ m^2*) 
      \)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qvhs\  /. \ enrel\  -> \ 1\)], "Input"],

Cell[BoxData[
    \(18.7810304712315012`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(qvhs\  /. \ enrel\  -> \ 0.01\)], "Input"],

Cell[BoxData[
    \(78.6537470763529888`\)], "Output"]
}, Open  ]],

Cell["\<\
Thus, the two results from Nanbu et al. agree.  However, this value is about \
1.2 times that we obtain from our empirical fit to viscosity data.  It also \
has a much larger exponent.\
\>", "SmallText"],

Cell["\<\
Comparison with Kersh, Morokoff, and Werner (1994) where these authors use \
\>", "SmallText"],

Cell[BoxData[
    \(\(crossrefkmw\  := \ 
      Pi*15 \((m*k*trefkmw/Pi)\)^0.5/
          \((2*\((5 - 2  omegakmw)\)*\((7 - 2  omegakmw)\) viscrefkmw)\); 
    \)\)], "Input"],

Cell["\<\
Check of Gamma relations.  This did not work in general, so we use several \
specific numbers.\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\t\t
    N[Gamma[5/2 - omegakmw]\  - \ 
          Gamma[9/2 - omegakmw]/
            \((\((7 - 2*omegakmw)\)*\((5 - 2*omegakmw)\)/4)\)\  /. \ \ 
        omegakmw\  -> {1, 0, 2}]\)\)], "Input"],

Cell[BoxData[
    \({0, 0, 0}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(crosskmw\  := \ 
      crossrefkmw*
          \((k*trefkmw/\((qe*u)\))\)^\((omegakmw - 0.5)\)/
            Gamma[5/2 - omegakmw]\  /. \n\t\t
        Gamma[5/2 - omegakmw]\  -> 
          Gamma[9/2 - omegakmw]/
            \((\((7 - 2*omegakmw)\)*\((5 - 2*omegakmw)\)/4)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{trefkmw, omegakmw, viscrefkmw}, \n
      viscrefkmw\  = \ 2.162*10^\(-5\); \  (*\(kg/m\)/s\  = \ N\ s/m^2*) \n
      trefkmw\  = \ 273; \nomegakmw\  = \ 0.6785; \ncrosskmw1\  = \ crosskmw\n
      \t]\)], "Input"],

Cell[BoxData[
    \(2.58557720646675193`*^-19\ \((1\/u)\)\^0.178499999999999996`\)], 
  "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(ClearAll[k, m, mr]; \)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Block[{omegakmw}, \nomegakmw\  = \ 0.5; \ncrosskmw2\  = \ crosskmw\n\t]
      \)], "Input"],

Cell[BoxData[
    FractionBox[
      RowBox[{"0.55389182840797364`", " ", 
        SuperscriptBox[\((k\ m\ trefkmw)\), 
          StyleBox["0.5`",
            StyleBoxAutoDelete->True,
            PrintPrecision->1]]}], "viscrefkmw"]], "Output"]
}, Open  ]],

Cell["\<\
Kersh et al. give a normalized diffusion coefficient of 2.057 m^2/s at 5 \
mTorr and 273 K which corresponds to\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(ndiff\  = \ 2.057*5*^-3*2.69*^25/760\ \ \  (*m^\(-1\) s^\(-1\)*) 
      \)\)], "Input"],

Cell[BoxData[
    \(3.64034868421052504`*^20\)], "Output"]
}, Open  ]],

Cell["\<\
This is to be compared with Chapman and Cowling's value of 0.422 *10^21 \
m^-1s^-1.\
\>", "SmallText"],

Cell["\<\
            qd1plot = LogLogPlot[1*^20*Part[qd1,1],  {u,0.01,1000},           \

                        PlotStyle -> {{Dashing[{0.03,0.03}],Thickness[.01]}},
                        (* PlotStyle -> Hue[0.55],*)
                         PlotRange -> {{0.01,1000}, {0.1,100}},
                         DisplayFunction -> Identity];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
            qv1plot = LogLogPlot[1*^20*Part[qv1,1],  {u,0.01,1000},           \

                        (*PlotStyle -> Dashing[{0.03,0.03}],*)
                         PlotStyle -> {{Hue[0.55],Thickness[.01]}},
                         PlotRange ->  {{0.01,1000}, {0.1,100}},
                         DisplayFunction -> Identity];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
            qvsnplot = LogLogPlot[1*^20*crosssn,  {u,0.01,1000},           
                        (*PlotStyle -> Dashing[{0.03,0.03}],*)
                         PlotStyle -> {{Hue[0.3],Thickness[.01]}},
                         PlotRange ->  {{0.01,1000},{0.1,100}},
                         DisplayFunction -> Identity];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
            qvkmwplot = LogLogPlot[1*^20*crosskmw1,  {u,0.01,1000},           \

                        (*PlotStyle -> Dashing[{0.03,0.03}],*)
                         PlotStyle -> {{Hue[1],Thickness[.01]}},
                         PlotRange ->  {{0.01,1000}, {0.1,100}},
                         DisplayFunction -> Identity];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[CellGroupData[{

Cell["\<\
Show[qd1plot,qv1plot,qvsnplot,qvkmwplot,
            DisplayFunction -> $DisplayFunction, GridLines -> Automatic,
             AxesLabel -> {\"rel. energy (eV)\",\"q (10^-20 m^2)\"}];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[GraphicsData["PostScript", "\<\
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