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Notebook[{
Cell["Ionization source term for abnormal cathode fall modeling", "Subtitle"],

Cell["\<\
As of 8/6/00 I decided to export the ion production rather than the \
multiplication used previously.\
\>", "Subsubtitle",
  Background->RGBColor[1, 0, 0]],

Cell["Setup notebook  enviroment ", "SmallText",
  PageWidth->Infinity],

Cell[BoxData[
    \(\(a = 1;\)\)], "Input"],

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

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

Cell["startclock = SessionTime[];", "Input",
  PageWidth->Infinity],

Cell["Off[General::spell]", "Input",
  PageWidth->Infinity],

Cell["Off[General::spell1]", "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(Needs["\<NumericalMath`ListIntegrate`\>"]\)], "Input"],

Cell["Needs[ \"Graphics`Graphics`\"]", "Input",
  PageWidth->Infinity,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
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Cell["Needs[\"Graphics`Graphics3D`\"]", "Input",
  PageWidth->Infinity,
  Evaluatable->False,
  FontFamily->"Times New Roman",
  FontSize->12,
  FontWeight->"Bold",
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Cell[BoxData[
    \(now\  = \ 
      StringForm["\<``/``/`` ``:``:``\>", \(Date[]\)[\([2]\)], \
\(Date[]\)[\([3]\)], \(Date[]\)[\([1]\)], \(Date[]\)[\([4]\)], \
\(Date[]\)[\([5]\)], \(Date[]\)[\([6]\)]]\)], "Input"],

Cell[BoxData[
    \(SetDirectory["\<c:\\Mathbook\\Argon\\CathFallModels\>"]\)], "Input"],

Cell["\<\
Empirical formulas for the ionization source term from Pitchford (11/17/1999) \
\
\>", "Section"],

Cell["\<\
Following Preres et al (1992), assume a linear increasing source term for  \
px<pdc and an exponentially decreasing source term for px>pdc, i.e.,\
\>", "SmallText",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
 snop :=If[px<pdc,smax*px/pdc,smax*Exp[-kp(px-pdc)]]  (*cm^-1 Torr^-1*)\
\>", "Input",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
From Peres et al, J. appl. Phys. 72, 4533 (1992), the source term is defined \
by the relation snop(px)= sn(px)/p = alpha_i(x)/p*F_e(x)/F_i(0),  where sn \
and alpha_i are in units of cm^-1, p is in Torr, and the fluxes F are in \
cm^-2s^-1.  In this paper, the multiplication is defined as M = 1+ \
Integrate[{snop(px)} dpx] and the steady-state condition is M = 1 + \
1/gamma_i.  Evidence for this definition is that when there is a spatially \
uniform electric field (as at breakdown) so that the alpha_i/p is independent \
of position, the relative electron flux grows exponentially with position as \
Exp[(alpha_i/p)*px], the total ion production per electron is \
Integrate[{snop(px)*dpx] = Exp[(alpha_i/p)*pd]-1, and the multiplication is \
Exp[(alpha/p)*pd].

Note that the Monte Carlo calculations by Peres et al of the electron \
behavior were made using an electric field that decreased linearly with \
position such that E/p = 2*V/pdc*(1-px/pdc).  I presume that the same is true \
for the more recent calculations.\
\>", "SmallText"],

Cell["We begin by converting from pressure to density units", "SmallText"],

Cell[BoxData[
    \(snon\  := \ \(\(snop\ /\((3.22*10^16)\)\ \ \  /. \ {px\  -> 
              nx/\((3.22*10^16)\), pdc\  -> ndc/\((3.22*10^16)\), 
            smax\  -> smaxon*3.22*10^16, 
            kp\  -> kon*3.22*10^16}\)\(\ \ \ \ \ \ \ \ \ \)\(//\)\(\ \
\)\(Simplify\)\(\ \ \ \ \ \ \ \)\( (*cm^2*) \)\)\)], "Input"],

Cell["The corresponding ion production is", "SmallText"],

Cell[BoxData[
    \(prodn1\  = \ \(\(Integrate[smaxon*nx/ndc, {nx, 0, ndc}] + 
          Integrate[
            smaxon*Exp[\(-kon\)*\ \((nx - ndc)\)], {nx, ndc, Infinity}, 
            Assumptions\  -> \ {Re[kon]\  > 0\ , 
                ndc\  > 
                  0}]\)\(\ \)\(//\)\(Simplify\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\
\( (*dimensionless*) \)\)\)], "Input"],

Cell["\<\
Here dc is the sheath length, p is the pressure and snop is the normalized \
source term (source term / pressure / electron flux leaving 
the cathode). smax and kp are determined as functions of  V and pdc by \
fitting to Monte Carlo results and using the relations:\
\>", "SmallText",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["The fraction of ion production in the zero field region is", "SmallText"],

Cell[BoxData[
    \(fraction\  := \ \((smaxon/kon)\)/\((\((ndc*smaxon/2)\) + 
              smaxon/kon)\)\  // Cancel\)], "Input"],

Cell["\<\
This fraction is of interest in estimating the effects of radial losses of \
ions from the zero field region, i.e., when the ratio is large the distance \
that should be compared to the tube radius is the length characteristic of \
the zero field region = n/kon.\
\>", "SmallText"],

Cell["\<\
Attempts to use UnitStep[ ] functions instead of If[ ] worked OK for graphs \
but not for the integrals.\
\>", "SmallText"],

Cell["\<\
Pitchford et al express kop and smaxop in terms of V and pdc. \
\>", "SmallText"],

Cell["\<\
kon =((a1*pdc+a2)*v+b1*pdc+b2)/n /. {pdc -> ndc/(3.22*10^16),n->3.22*10^16}\
\>", "Input",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
smaxon =((c1*pdc+c2*v+c3)*v+d1*pdc+d2)/n /. {pdc -> \
ndc/(3.22*10^16),n->3.22*10^16}\
\>", "Input",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
Next we look at the magnitude of the production at very low pdc, where the \
ionization should occur only in the field free region, i.e., after the \
electrons have passed through the sheath.  In this case the electron \
effectively enter the gas at the applied voltage and without collisions so \
that the production should be less than the voltage divided by ~35 eV/ion \
pair.\
\>", "SmallText"],

Cell[BoxData[
    \(lowndcLimit\  = \ Limit[prodn1, \ ndc\  -> 0]\)], "Input"],

Cell["The numerical values of the coefficients are:", "SmallText",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
a1=-2.82*10^-2;
a2=-1.45*10^-2;
b1=39.49;
b2=9.221;
c1=4.99;
c2=-3.4*10^-4;
c3=0.087;
d1=-708.72;
d2=53.99;\
\>", "Input",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
Pitchford says that these formulas are good for pdc > 0.05 Torr cm and V < \
400 V.  Unfortunately, she does not give an upper limit to pdc or a lower \
limit to V.  From Peres et al, (1992), I would guess that these results apply \
for multiplication < 40 and pdc < 0.25 Torr cm.\
\>", "SmallText"],

Cell[BoxData[
    \(mn1\  = \ 1 + \ prodn1\)], "Input"],

Cell[BoxData[
    \(\(multiplicationPlot1 = \ 
        LogLogPlot[{\((mn1\  /. \ {v\  -> \ 150. , 
                  ndc\  -> \ pdc*3.22*10^16})\), \((mn1\  /. \ {v\  -> \ 
                    200. , ndc\  -> \ 
                    pdc*3.22*10^16})\), \((mn1\  /. \ {v\  -> \ 250. , 
                  ndc\  -> \ pdc*3.22*10^16})\), \((mn1\  /. \ {v\  -> \ 
                    300. , ndc\  -> \ 
                    pdc*3.22*10^16})\), \((mn1\  /. \ {v\  -> \ 350. , 
                  ndc\  -> \ pdc*3.22*10^16})\), \((mn1\  /. \ {v\  -> \ 
                    400. , ndc\  -> \ pdc*3.22*10^16})\)}, {pdc, 0.001, 1. }, 
          PlotRange\  -> \ {{0.001, 1. }, {1. , 1000. }}, 
          GridLines\  -> \ Automatic\[IndentingNewLine]];\)\)], "Input"],

Cell[BoxData[
    \(\(fractionPlot1 = \ 
        LogLogPlot[{\((fraction\  /. {\ v\  -> \ 150. , 
                  ndc\  -> \ pdc*3.22*10^16})\), \((fraction\  /. {\ 
                  v\  -> \ 200. , 
                  ndc\  -> \ pdc*3.22*10^16})\), \((fraction\  /. \ {v\  -> \ 
                    250. , ndc\  -> \ 
                    pdc*3.22*10^16})\), \((fraction\  /. \ {v\  -> \ 300. , 
                  ndc\  -> \ pdc*3.22*10^16})\), \((fraction\  /. \ {v\  -> \ 
                    350. , ndc\  -> \ 
                    pdc*3.22*10^16})\), \((fraction\  /. \ {v\  -> \ 400. , 
                  ndc\  -> \ pdc*3.22*10^16})\)}, {pdc, 0.001, 1. }, 
          PlotRange\  -> \ {{0.001, 1. }, {0.01, 1. }}, 
          GridLines\  -> \ Automatic\[IndentingNewLine]];\)\)], "Input"],

Cell["Export Pitchford's fit to data ", "Subsubsection"],

Cell[BoxData[
    \(\(vdList\  = \ {200, 400};\)\)], "Input"],

Cell[BoxData[
    \(\(prodPitchfordTable\  = \ 
        Table[\((\((prodn1\  /. \ {v\  -> \ Part[vdList, j], 
                    ndc\  -> 3.54*10^16*p0dc})\)\ \  //. \ 
              p0dc\  -> \ 0.01*10^\((k/4)\))\), {k, 0, 6}, {j, 1, 
            Length[vdList]}];\)\)], "Input"],

Cell[BoxData[
    \(\(prodPitchfordTable2\  = \ Transpose[prodPitchfordTable];\)\)], "Input"],

Cell[BoxData[
    \(\(p0dcList2 = Table[\ 0.01*10^\((k/4)\), {k, 0, 6}];\)\)], "Input"],

Cell[BoxData[
    \(\(prodPitchfordTable3\  = \ 
        Transpose[Join[{p0dcList2}, prodPitchfordTable2]];\)\)], "Input"],

Cell[BoxData[
    \(\(prodPitchfordTable4\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          prodPitchfordTable3, {2}];\)\)], "Input"],

Cell["\<\
prodPitchfordTable5 = TableForm[prodPitchfordTable4,TableHeadings -> {None,
                   {\"%p0dc\\Vd(V)\",\"200\",\"400\"}},
                   TableSpacing -> {0,1}];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Block[
{stream1,theoryfile,today},
SetOptions[$Output,PageWidth->120];
theoryfile := \"c:\\\\proplot\\\\argon\\\\arabnor\\\\FitToPit.dat\";
stream1 = OpenWrite[theoryfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Pitchford's fit to calculations of ion \
production  \"],now,
                    \"\\n\",
                    ToString[prodPitchfordTable5]];
Close[stream1];
SetOptions[$Output,PageWidth->53];
]\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["!!c:\\proplot\\argon\\arabnor\\FitToPit.dat", "Input",
  PageWidth->Infinity],

Cell["\<\
Next we compare the pdc ->0 limit from Pitchford's formula with the usual 35 \
eV/ion pair.\
\>", "SmallText"],

Cell[BoxData[
    \(\(lowpdcLimitPlot\  = \ 
        Plot[{lowndcLimit, \((1 + v/35)\)}, \ {v, 100, 400}, 
          PlotStyle\  -> \ {{Hue[1]}, {Hue[0.5]}}, 
          AxesLabel -> {"\<Voltage(V)\>", "\<production\>"}, 
          PlotRange\  -> {{0, 400}, {0, 15}}];\)\)], "Input"],

Cell[" Consider the parameters", "SmallText"],

Cell[BoxData[
    \(\(\(v = 400\ ;\)\(\ \)\( (*V*) \)\(\ \)\)\)], "Input",
  Evaluatable->False,
  Background->RGBColor[0, 1, 1]],

Cell[BoxData[
    \(\(\(gammai = 0.025\ ;\)\( (*Torr\ cm*) \)\)\)], "Input"],

Cell["Let us solve for pdc for the desired values of V and gamma_i", \
"SmallText"],

Cell[BoxData[
    \(prodn1a\  = \ 
      prodn1\  /. \ {v\  -> 400. , ndc\  -> \ p0dc*3.54*10^16}\)], "Input"],

Cell[BoxData[
    \(FindRoot[1 + 1/gammai == prodn1a\ , \ {p0dc, 0, 1.0}]\)], "Input"],

Cell[BoxData[
    \(p0dc\  = \ p0dc\  /. \ %\)], "Input"],

Cell[BoxData[
    \(\(p0dc400\  = \ %;\)\)], "Input"],

Cell["\<\
Here p0dc has been adjusted to give the desired multiplication at the given \
voltage.\
\>", "SmallText"],

Cell[BoxData[
    \(\(snop0Plot = \ 
        Plot[snon\ *\((3.54*10^16)\)\  //. \ {v\  -> \ 400. , 
              ndc\  -> \ p0dc*3.54*10^16, nx\  -> p0x*3.54*10^16}, {p0x, 0, 
            1}, PlotRange\  -> \ {{0, 1}, {0, 250. }}];\)\)], "Input"],

Cell["The total ion production is ", "SmallText"],

Cell[BoxData[
    \(prodn1\ \  /. \ {\ v\  -> \ 400. , 
        ndc\  -> \ p0dc*3.54*10^16}\)], "Input"],

Cell[BoxData[
    \(Needs["\<NumericalMath`ListIntegrate`\>"]\)], "Input"],

Cell[BoxData[
    \(empiricalIonProduction\  = \ 
      NIntegrate[
          snon\ \ *\((3.54*10^16)\) //. \ {v\  -> 400. , 
              ndc\  -> \ p0dc*3.54*10^16, nx\  -> p0x*3.54*10^16}, \ {p0x, 0, 
            0.2}] + \[IndentingNewLine]NIntegrate[
          snon\ \ *\((3.54*10^16)\) //. \ {v\  -> 400. , 
              ndc\  -> \ p0dc*3.54*10^16, nx\  -> p0x*3.54*10^16}, \ {p0x, 
            0.2, 1}]\)], "Input"],

Cell["\<\
Pitchford's email included the following statement.  \"This present \
approximation for sn is valid, i.e., the electron multiplication is 
within about 5% of MC, for pdc > 0.05 torr cm and v < 400 V. \"\
\>", "SmallText",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell["\<\
We need to clear pdc in order to get general expressions for the production \
in terms of pdc.\
\>", "SmallText"],

Cell[BoxData[
    \(Clear[p0dc, ndc]\)], "Input"],

Cell["Limiting forms of ionization source term", "Section"],

Cell["\<\
Here we wish to derive the ion production and multiplication in the low and \
high pdc limits.\
\>", "SmallText"],

Cell["Low pd limit", "Subsection"],

Cell["\<\
Note that as pdc approaches zero, e.g., below say 0.01 Torr cm, one has the \
situation of the nonuniform field region is so small the one is effectively \
injecting electrons at the total voltage.  In this limit the secondary \
electrons should be produced only in the zero field region and the only \
ionization is by the primary electrons as they cool off.  In this limit one \
expects the ion production to be less than the applied voltage divided by an \
average energy loss per ion pair, e.g., 400/35 ~ 11 ions as compared to the M \
-1 = ~16 ions shown in Fig. 2 of Peres et al (1992).  Note that as the \
electron energy goes below the threshold for ionization the energy loss per \
ion pair become infinite, not some small number as I was using.  I will try a \
loss of 1E6 so as to keep thing finite.\
\>", "SmallText"],

Cell["\<\
From the notebook ARENERGYLOSSREAD.NB we find that the energy loss per ion \
pair produced when electrons are injected into Ar can be approximated by\
\>", "SmallText"],

Cell[BoxData[
    \(fitLoss\  = \ 
      If[energy > 14.71, 
        26.4 + \((100. /\((energy - 14.71)\))\) + 10*Exp[\(-energy\)/100], 
        1. *10^30]\)], "Input"],

Cell["\<\
Note the essentially infinite value for energies below threshold.\
\>", "Text"],

Cell["\<\
The number of ion pairs produced by an injected electron is then\
\>", "SmallText"],

Cell[BoxData[
    \(\(lowpdIonNumber\  = \ energy/fitLoss;\)\)], "Input"],

Cell["\<\
We need to consider backscattering of primary electrons and the production of \
ions behind the cathode.  For an absorbing cathode, electron absorption by \
the cathode will reduce the ion production at low pdc values by roughly 10% \
from the ion production obtained in energy loss experiments and theory.  See, \
for example, Vasenkov (1998).  Fortunately, the fraction of the ions produced \
within less than 1 mfp (0.003 Torr cm) of the cathode and that have a reduced \
knietic energy compared to that calculated from the E/n at the cathode is \
small.  We use\
\>", "SmallText"],

Cell[BoxData[
    \(\(lowpdproduction\  = \ 0.9*\ lowpdIonNumber;\)\)], "Input"],

Cell["For some purposes we need to cutoff this function", "Text"],

Cell[BoxData[
    \(\(lowpdproduction3\  = \ 
        0.9*\ lowpdIonNumber/\((1 + \((p0dc/1)\)^4)\);\)\)], "Input"],

Cell[BoxData[
    \(\(vdList\  = \ {50, 70, 100, 200, 300, 400, 500, 700, 1000, 1500, 2000, 
          3000};\)\)], "Input"],

Cell[BoxData[
    \(\(lowpdproductionList\  = \ 
        Table[lowpdprouction\  /. \ {energy\  -> \ Part[vdList, j], 
              p0dc -> 0.001}, {j, 1, Length[vdList]}];\)\)], "Input"],

Cell["Export low pdc limit calculation ", "Subsubsection"],

Cell[BoxData[
    \(\(vdList2\  = \ {100, 200, 400, 800, 1600, 3000};\)\)], "Input"],

Cell[BoxData[
    \(\(lowpdproductionList\  = \ 
        Table[lowpdproduction\  /. \ {energy\  -> \ Part[vdList2, j], 
              p0dc -> 0.001}, {j, 1, Length[vdList2]}];\)\)], "Input"],

Cell[BoxData[
    \(\(lowpdproductionTable1\  = \ 
        Transpose[{vdList2, lowpdproductionList}];\)\)], "Input"],

Cell[BoxData[
    \(\(lowpdproductionTable2\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          lowpdproductionTable1, {2}];\)\)], "Input"],

Cell["\<\
lowpdproductionTable3 = TableForm[lowpdproductionTable2,TableHeadings -> \
{None,
                   {\"%Vd(V)\",\"production\"}},
                   TableSpacing -> {0,1}];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Block[
{stream1,theoryfile,today},
SetOptions[$Output,PageWidth->120];
theoryfile := \"c:\\\\proplot\\\\argon\\\\arabnor\\\\lowpdLimit.dat\";
stream1 = OpenWrite[theoryfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Low pd limit of production  \"],now,
                    \"\\n\",
                    ToString[lowpdproductionTable3]];
Close[stream1];
SetOptions[$Output,PageWidth->53];
]\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["!!c:\\proplot\\argon\\arabnor\\lowpdLimit.dat", "Input",
  PageWidth->Infinity],

Cell["\<\
For plotting purposes, we want to express this limit as the voltage required \
to give specified ion pair production determined by 1/gammai.\
\>", "Text"],

Cell[BoxData[
    \(\(gammai = {0.01, 0.02, 0.05, 0.07, 0.1, 0.2};\)\)], "Input"],

Cell[BoxData[
    \(lowpdproduction2\  = \ lowpdproduction\  /. \ p0dc -> 0.001\)], "Input"],

Cell[BoxData[
    \(\(lowpdVcSol = \ 
        Table[FindRoot[
            1/gammai[\([j]\)] == lowpdproduction2, {energy, 100}], {j, 1, 
            6}];\)\)], "Input"],

Cell[BoxData[
    \(lowpdVcTable = 
      Transpose[Join[{gammai, energy\  /. \ lowpdVcSol}]]\)], "Input"],

Cell[BoxData[
    \(Export["\<lowpdVcTable.txt\>", 
      Join[{"\<%Vc versus gammai in low pd limit\>"}, 
        Join[{"\<%gammai   Vc\>"}, lowpdVcTable]], "\<Table\>"]\)], "Input"],

Cell[BoxData[
    \(\(!! lowpdVcTable.txt\)\)], "Input"],

Cell[CellGroupData[{

Cell["Large pd limit - local field model", "Subsection"],

Cell["\<\
Here we wish to find an expression for the ionization produced when pd is \
large enough so that the local field model of electron motion is valid, i.e., \
the ionization coefficient as a function of position decreases as pz \
approaches pd in a manner described by the steady-state, uniform-field  \
ionization coefficient.  In this limit, we expect that the ionization \
produced in the field free region is negligible.\
\>", "SmallText"],

Cell["The Townsend ionization coefficient is approximated by", "SmallText"],

Cell[BoxData[
    \(\(alphaion\  = \ \(+5.5\)*10^\(-21\)*Exp[\(-187\)/eonTd] + 
          3.2*10^\(-20\)*Exp[\(-700\)/eonTd] - 
          1.5*10^\(-20\)*Exp[\(-10000\)/eonTd] + 
          1.1*10^\(-22\)*Exp[\(-72\)/eonTd];\)\)], "Input"],

Cell["The electric field is assumed to be given by", "SmallText"],

Cell[BoxData[
    \(\(eAssumed\  = \ 2. *vd/dc*\((1 - z/dc)\);\)\)], "Input"],

Cell["and the E/n is", "SmallText"],

Cell[BoxData[
    \(\(eonAssumed\  = \ eAssumed/n;\)\)], "Input"],

Cell["so that the number of ion pairs formed is found from", "SmallText"],

Cell[BoxData[
    \(\(dlneFluxodz\  = 
        alphaion*n\  /. \ eonTd\  -> \ eonAssumed*10^21;\)\)], "Input"],

Cell["\<\
or changing variable from z to x = 1/(1-z/dc), where when z = 0, x = 1 and z \
= dc, x = Infinity.  We use dlneFlux/dx = dlneFlux/dz *dz/dx.\
\>", "SmallText"],

Cell[BoxData[
    \(\(dlneFluxodx\  = \ \((dlneFluxodz\  /. \ z\  -> \ dc*\((1 - 1/x)\))\)*
          D[z\  /. \ z\  -> \ dc*\((1 - 1/x)\), x];\)\)], "Input"],

Cell["\<\
We approximate the initial nonlocal effects by using a delay in the onset of \
ionization z0 = dc*(1-1/x0). The escape fraction fesc would be applied along \
with the electron yield gamma_i.  Here both z0 and fesc are assumed functions \
only of the E/n at the cathode E/n(0) = 2*vd/dc/n.\
\>", "SmallText"],

Cell[BoxData[
    \(lneFlux\  = \ 
      Integrate[dlneFluxodx, \ {x, x0, Infinity}, 
          Assumptions\  -> {Re[dc*n/vd] > 0, x0 > 0}\ ]\  // 
        Simplify\)], "Input"]
}, Open  ]],

Cell[BoxData[
    \(\(ionProduction\  = \ Exp[lneFlux] - 1;\)\)], "Input"],

Cell["\<\
 If we neglect the nonequilibrium effects,  x0 =1.  In the following we \
change units such that p0 is in Torr and p0dc is in Torr cm, whereas in the \
above n is in m^-3, alphaion is in m^2, and dc is in m.  \
\>", "SmallText"],

Cell[BoxData[
    \(ionProduction2\  = \ 
      ionProduction\  /. \ {n\  -> 3.54*10^22*p0, x0\  -> 1, \ 
            dc\  -> \ \(p0dc/p0\)/100}\  // Simplify\)], "Input"],

Cell["\<\
To evaluate this expression in QuatroPro we need to replace the Incomplete \
Gamma functions with approximations from Abramowitz and Stegun (1964).  Thus, \
from Eq. 5.145, Gamma(-1,z) = (1/z)E_2(z) and one can use Eq. 5.1.53 for 0 <= \
z <= 1 and Eq. 5.1.54 for 1 <= z < infinity.\
\>", "Text"],

Cell[BoxData[
    \(lowz := \(-Log[z]\) - 0.57721566 + 0.99999193*z - 0.24991055*z^2 + 
        0.05519968*z^3 - 0.00976004*z^4 + 0.00107857*z^5\)], "Input"],

Cell[BoxData[
    \(highz\  := \ 
      Exp[\(-z\)]/
          z*\((0.250621 + 2.334733*z + z^2)\)/\((1.681534 + 3.330657*z + 
              z^2)\)\)], "Input"],

Cell[BoxData[
    \(expInt1 := \ If[z <= 1, lowz, highz]\)], "Input"],

Cell["or", "Text"],

Cell[BoxData[
    \(expInt1 := \ \((1 - UnitStep[z - 1])\)*
          lowz + \(\(UnitStep[z - 1]\)\(*\)\(highz\)\(\ \)\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(\(\ \)\(expInt2 := \ Exp[\(-z\)] - z*expInt1\)\)\)], "Input"],

Cell[BoxData[
    \(expInt2\  //. \ z -> 0.0001\)], "Input"],

Cell[BoxData[
    \(expInt1\  //. \ z -> 8\)], "Input"],

Cell["\<\
This is so complicated that it would be a real problem to apply it four times \
to QuatroPro\
\>", "Text"],

Cell["\<\
We tabulate these large p0d limit results for use in a 3-D plot\
\>", "SmallText"],

Cell[BoxData[
    \(\(largepdLimitTable\  = \ 
        Table[Evaluate[
              Log[10, ionProduction2]]\  /. \ {p0dc\  -> \ 
                0.1*10^\((k/4)\), \ vd\  -> \ Part[vdList, j]}, {j, 1, 
            Length[vdList]}, {k, 1, 8}];\)\)], "Input"],

Cell[BoxData[
    \(\(ListPlot3D[largepdLimitTable, PlotRange -> All, 
        BoxRatios -> {1, 1, 1}];\)\)], "Input"],

Cell[TextData[{
  "Note that negative values of the vertical axes result from \
log[ionProduction2]\n",
  StyleBox["Our problem now is to come up with an analytical fit to all of \
this data.", "Subsubtitle"]
}], "Subsubtitle"],

Cell["Export large p0d limit to production cases ", "Subsubsection"],

Cell[BoxData[
    \(\(vdList3\  = \ {100, 120, 135, 150, 170, 200, 250, 300, 400, 500, 
          1000, 2000, 3000};\)\)], "Input"],

Cell[BoxData[
    \(\(largep0dLimitTable\  = \ 
        Table[ionProduction2\  /. \ {vd\  -> \ Part[vdList3, j], 
              p0dc\  -> 0.1*10^\((k/4)\)}, {k, 0, 12}, {j, 1, 
            Length[vdList3]}];\)\)], "Input"],

Cell[BoxData[
    \(\(largep0dLimitTable2\  = \ Transpose[largep0dLimitTable];\)\)], "Input"],

Cell[BoxData[
    \(\(p0dcList2 = Table[0.1*\ 10^\((k/4)\), {k, 0, 12}];\)\)], "Input"],

Cell[BoxData[
    \(\(largep0dLimitTable3\  = \ 
        Transpose[Join[{p0dcList2}, largep0dLimitTable2]];\)\)], "Input"],

Cell[BoxData[
    \(\(largep0dLimitTable4\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          largep0dLimitTable3, {2}];\)\)], "Input"],

Cell["\<\
largep0dLimitTable5 = TableForm[largep0dLimitTable4,TableHeadings -> {None,
                   {\"%p0dc\\Vd(V)\",\"100\",\"120\",\"135\",\"150\",\"170\",\
\"200\",\"250\",\"300\",\"400\",\"500\",\"1000\",\"2000\",\"3000\"}},
                   TableSpacing -> {0,1}];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Block[
{stream1,theoryfile,today},
SetOptions[$Output,PageWidth->120];
theoryfile := \"c:\\\\proplot\\\\argon\\\\arabnor\\\\HipdLimit.dat\";
stream1 = OpenWrite[theoryfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Ion production in local field limit  \
\"],now,
                    \"\\n\",
                    ToString[largep0dLimitTable5]];
Close[stream1];
SetOptions[$Output,PageWidth->53];
]\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["!!c:\\proplot\\argon\\arabnor\\HipdLimit.dat", "Input",
  PageWidth->Infinity],

Cell["\<\
The following does not work because of an essential singularity in \
Gamma[-1,y].\
\>", "SmallText"],

Cell[BoxData[
    \(Limit[ionProduction2, p0dc -> Infinity]\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(Limit[ionProduction2, p0dc -> 0]\)], "Input"],

Cell["\<\
We would like to get values of p0dc versus vd for various values of gammai by \
solving 1/gammai= ionProduction2.  In the following gammi=0.05.  Hoever, the \
following was not working as of 9/13/02.\
\>", "Text"],

Cell[BoxData[
    \(\(Table[
        FindRoot[
          1/0.05 == \((ionProduction2\  /. \ 
                vd\  -> vdList3[\([j]\)])\), {p0dc,  .1}, 
          DampingFactor -> 0.01, MaxIterations -> 10000], {j, 1, 
          Length[vdList3]}];\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(Table[
        FindRoot[
          1/0.05 == \((ionProduction2\  /. \ 
                vd\  -> vdList3[\([j]\)])\), {p0dc, 5. }, 
          DampingFactor -> 0.01, MaxIterations -> 10000], {j, 1, 
          Length[vdList3]}];\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(Transpose[{Join[Reverse[vdList3], vdList3], 
          Join[Reverse[p0dc /. %%], p0dc\  /. %]}];\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(Export["\<pdList3.txt\>", %, "\<Table\>"];\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(!! pdList3.txt\)\)], "Input",
  Evaluatable->False],

Cell["Fit to \"average\" calculations for all pdc", "Section"],

Cell["\<\
The following is an empirical fit to the calculations of the number of ions \
produced per primary electron from Pitchford et al and from the low and high \
pd limit relations.  It is a function of p0dc and of the discharge voltage \
which here is called energy.  The fit was obtained by trial and error.\
\>", "SmallText"],

Cell[BoxData[
    \(\(productionFit\ \  := \ 
        lowpdproduction\ \((1/\((1 + \((p0dc/
                          1)\)^4)\) + \ \((p0dc/\((0.15*\((200/
                                energy)\)^0.5)\))\)^1.7/\((1 + \
\((p0dc/\((1.15*\((energy/200)\)^1.6)\))\)^\((3.7*\((energy/
                                200)\)^0.5)\))\))\);\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(mi := \ \((0.5)\)^\((\((vd/400)\)^0.5)\) + 0.15\)], "Input"],

Cell[BoxData[
    \(mi := \ \((0.5)\)^\((\((vd/400)\)^0.3)\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(productionFit\ \  := \((\((lowpdproduction3)\)^
                mi + \((ionProduction2)\)^mi)\)^\((1/mi)\)\ \  /. \ 
        energy -> vd\)], "Input"],

Cell["\<\
The following are plots of the number of ions produced per primary electron \
as given by various calculations\
\>", "SmallText"],

Cell[BoxData[
    \(\(vdtest = 200. ;\)\)], "Input"],

Cell["Local field results - red", "SmallText"],

Cell[BoxData[
    \(\(largePDPlota\  = \ 
        LogLogPlot[
          ionProduction2\  /. \ vd\  -> \ vdtest\ , {p0dc, 0.1, 100}, 
          PlotPoints -> 500, PlotRange\  -> \ {{0.001, 100}, {1, 10000}}, 
          PlotStyle\  -> \ Hue[1], \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["My fit to Cornbecher's beam results - dark blue", "SmallText"],

Cell[BoxData[
    \(\(lowPDPlot\  = \ 
        LogLogPlot[
          lowpdproduction\ *p0dc^0 /. \ energy\  -> \ vdtest, {p0dc, 0.001, 
            0.01}, PlotRange -> \ {{0.001, 100. }, {1, 1000. }}, 
          PlotStyle\  -> \ {{Hue[0.6], Thickness[0.02]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["Folding of low pd and local field - purple", "Text"],

Cell[BoxData[
    \(\(lowPDFoldedPlota\  = \ 
        LogLogPlot[\((\((lowpdproduction3\ \  /. \ 
                      energy -> vdtest)\)^0.5 + \((ionProduction2 /. \ 
                      vd -> vdtest)\)^0.5)\)^\((1/0.5)\), {p0dc, 0.001, 
            100.0}, PlotRange -> \ {{0.001, 100. }, {1, 1000. }}, 
          PlotPoints -> 500, 
          PlotStyle\  -> \ {{Hue[0.8], Thickness[0.01]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["Pitchford's MC results - black", "SmallText"],

Cell[BoxData[
    \(\(productionPlot2a = \ 
        LogLogPlot[\((prodn1\  /. \ {v\  -> \ vdtest, 
                ndc\  -> \ 3.54*10^16*p0dc})\), {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 1000. }}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["\<\
The following is from the fit for all voltages and p0dc values - green \
\>", "SmallText"],

Cell[BoxData[
    \(\(productionFitPlot1a\  = \ 
        LogLogPlot[productionFit\  /. \ vd -> vdtest, {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 10000. }}, 
          PlotStyle\  -> \ {Hue[0.3], Thickness[0.015]}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[lowPDPlot, largePDPlota, lowPDFoldedPlota\ , productionPlot2a, 
        productionFitPlot1a, \ GridLines\  -> \ Automatic, 
        PlotLabel -> "\<Ion pairs vs p0dc (Torr cm) at 200V\>", \ 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(vdtest = 300. ;\)\)], "Input"],

Cell["Local field results - red", "SmallText"],

Cell[BoxData[
    \(\(largePDPlotb\  = \ 
        LogLogPlot[
          ionProduction2\ \  /. \ vd\  -> \ vdtest\ , {p0dc, 0.1, 100}, 
          PlotPoints -> 500, PlotRange\  -> \ {{0.001, 100}, {1, 10000}}, 
          PlotStyle\  -> \ Hue[1], \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDPlot\  = \ 
        LogLogPlot[
          lowpdproduction\ \ *p0dc^0 /. \ energy\  -> \ vdtest, {p0dc, 0.001, 
            0.01}, PlotRange -> \ {{0.001, 100. }, {1, 1000. }}, 
          PlotStyle\  -> {{\ Hue[0.6], Thickness[0.02]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDFoldedPlotb\  = \ 
        LogLogPlot[\((\((lowpdproduction3\ \  /. \ 
                  energy -> vdtest)\) + \((ionProduction2 /. \ 
                  vd -> vdtest)\))\), {p0dc, 0.001, 100.0}, 
          PlotRange -> \ {{0.001, 100. }, {1, 1000. }}, 
          PlotStyle\  -> \ {{Hue[0.8], Thickness[0.01]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(productionPlot2b = \ 
        LogLogPlot[\((prodn1\  /. \ {v\  -> \ vdtest, 
                ndc\  -> \ 3.54*10^16*p0dc})\), {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 1000. }, {1, 1000. }}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["\<\
The following is from the fit for all voltages and p0dc values - green \
\>", "SmallText"],

Cell[BoxData[
    \(\(productionFitPlot1b\  = \ 
        LogLogPlot[
          productionFit\  /. \ vd\  -> \ vdtest, {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 10000. }}, 
          PlotStyle\  -> \ {Hue[0.3], Thickness[0.015]}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[lowPDPlot, largePDPlotb, lowPDFoldedPlotb, productionPlot2b, 
        productionFitPlot1b, \ GridLines\  -> \ Automatic, 
        PlotLabel -> "\<Ion pairs vs p0dc (Torr cm) at 300V\>", \ 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(vdtest = 400. ;\)\)], "Input"],

Cell["Local field results - red", "SmallText"],

Cell[BoxData[
    \(\(largePDPlotc\  = \ 
        LogLogPlot[
          ionProduction2\ \  /. \ vd\  -> \ vdtest\ , {p0dc, 0.1, 100}, 
          PlotPoints -> 500, PlotRange\  -> \ {{0.001, 100. }, {1, 10000}}, 
          PlotStyle\  -> \ Hue[1], \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDPlot\  = \ 
        LogLogPlot[
          lowpdproduction\ \ *p0dc^0 /. \ energy\  -> \ vdtest, {p0dc, 0.001, 
            0.01}, PlotRange -> \ {{0.001, 100. }, {1, 10000. }}, 
          PlotStyle\  -> \ {{Hue[0.6], Thickness[0.02]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDFoldedPlotc\  = \ 
        LogLogPlot[\((\((lowpdproduction3\ \  /. \ 
                  energy -> vdtest)\) + \((ionProduction2 /. \ 
                  vd -> vdtest)\))\), {p0dc, 0.001, 100.0}, 
          PlotRange -> \ {{0.001, 100. }, {1, 1000. }}, 
          PlotStyle\  -> \ {{Hue[0.8], Thickness[0.01]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(productionPlot2c = \ 
        LogLogPlot[\((prodn1\  /. \ {v\  -> \ vdtest, 
                ndc\  -> \ 3.54*10^16*p0dc})\), {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 10000. }}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["\<\
The following is from the fit for all voltages and p0dc values - green \
\>", "SmallText"],

Cell[BoxData[
    \(\(productionFitPlot1c\  = \ 
        LogLogPlot[productionFit\  /. vd -> \ vdtest, {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 10000. }}, 
          PlotStyle\  -> \ {Hue[0.3], Thickness[0.015]}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[lowPDPlot, largePDPlotc, lowPDFoldedPlotc, productionPlot2c, 
        productionFitPlot1c, \ GridLines\  -> \ Automatic, 
        PlotLabel -> "\<Ion pairs vs p0dc (Torr cm) at 400V\>", 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(vdtest = 700. ;\)\)], "Input"],

Cell["Local field results - red", "SmallText"],

Cell[BoxData[
    \(\(largePDPlotd\  = \ 
        LogLogPlot[
          ionProduction2\ \  /. \ vd\  -> \ vdtest\ , {p0dc, 0.1, 100}, 
          PlotPoints -> 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 1000000. }}, 
          PlotStyle\  -> \ Hue[1], \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDPlot\  = \ 
        LogLogPlot[
          lowpdproduction\ \ *p0dc^0 /. \ energy\  -> \ vdtest, {p0dc, 0.001, 
            0.01}, PlotRange -> \ {{0.001, 100. }, {1, 1000000. }}, 
          PlotStyle\  -> \ {{Hue[0.6], Thickness[0.02]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDFoldedPlotd\  = \ 
        LogLogPlot[\((\((lowpdproduction3\ \  /. \ 
                  energy -> vdtest)\) + \((ionProduction2 /. \ 
                  vd -> vdtest)\))\), {p0dc, 0.001, 100.0}, 
          PlotRange -> \ {{0.001, 100. }, {1, 1000. }}, 
          PlotStyle\  -> \ {{Hue[0.8], Thickness[0.01]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["\<\
We cannot use Pitchford's empirical formulas for high voltages, so the next \
plot is omitted.\
\>", "Text"],

Cell[BoxData[
    \(\(productionPlot2d = \ 
        LogLogPlot[\((prodn1\  /. \ {v\  -> \ vdtest, 
                ndc\  -> \ 3.54*10^16*p0dc})\), {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 1000000. }}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input",
  Evaluatable->False],

Cell["\<\
The following is from the fit for all voltages and p0dc values - green \
\>", "SmallText"],

Cell[BoxData[
    \(\(productionFitPlot1d\  = \ 
        LogLogPlot[
          productionFit\  /. \ vd\  -> \ vdtest, {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {1, 1000000. }}, 
          PlotStyle\  -> \ {Hue[0.3], Thickness[0.015]}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[lowPDPlot, largePDPlotd, 
        lowPDFoldedPlotd,  (*\(productionPlot2d\)\(,\)*) 
          productionFitPlot1d, \ GridLines\  -> \ Automatic, 
        PlotLabel -> "\<Ion pairs vs p0dc (Torr cm) at 700V\>", 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(vdtest = 1000. ;\)\)], "Input"],

Cell["Local field results - red", "SmallText"],

Cell[BoxData[
    \(\(largePDPlote\  = \ 
        LogLogPlot[
          ionProduction2\ \  /. \ vd\  -> \ vdtest\ , {p0dc, 0.1, 100}, 
          PlotPoints -> 500, PlotRange\  -> \ {{0.001, 100. }, {10, 10^9}}, 
          PlotStyle\  -> \ Hue[1], \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDPlot\  = \ 
        LogLogPlot[
          lowpdproduction\ \ *p0dc^0 /. \ energy\  -> \ vdtest, {p0dc, 0.001, 
            0.01}, PlotRange -> \ {{0.001, 100}, {10, 10^9}}, 
          PlotStyle\  -> \ {{Hue[0.6], Thickness[0.02]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(lowPDFoldedPlote\  = \ 
        LogLogPlot[\((\((lowpdproduction3\ *p0dc^0\  /. \ 
                      energy -> vdtest)\)^0.33 + \((ionProduction2 /. \ 
                      vd -> vdtest)\)^0.33)\)^3, {p0dc, 0.001, 100.0}, 
          PlotRange -> \ {{0.001, 100. }, {1, 10^9}}, PlotPoints -> 500, 
          PlotStyle\  -> \ {{Hue[0.8], Thickness[0.01]}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell["\<\
We cannot use Pitchford's empirical formulas for high voltages, so the next \
plot is omitted.\
\>", "Text"],

Cell[BoxData[
    \(\(productionPlot2e = \ 
        LogLogPlot[\((prodn1\  /. \ {v\  -> \ vdtest, 
                ndc\  -> \ 3.54*10^16*p0dc})\), {p0dc, 0.001, 100. }, 
          PlotPoints\  -> \ 500, 
          PlotRange\  -> \ {{0.001, 100. }, {10, 10^9}}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input",
  Evaluatable->False],

Cell["\<\
The following is from the fit for all voltages and p0dc values - green \
\>", "SmallText"],

Cell[BoxData[
    \(\(productionFitPlot1e\  = \ 
        LogLogPlot[
          productionFit\  /. \ vd\  -> \ vdtest, {p0dc, 0.001, 100. }, 
          PlotPoints\  -> 500, PlotRange\  -> \ {{0.001, 100. }, {10, 10^9}}, 
          PlotStyle\  -> \ {Hue[0.3], Thickness[0.015]}, \ 
          DisplayFunction\  -> \ Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[lowPDPlot, largePDPlote, 
        lowPDFoldedPlote,  (*\(productionPlot2e\)\(,\)*) 
          productionFitPlot1e, \ GridLines\  -> \ Automatic, 
        PlotLabel -> "\<Ion pairs vs p0dc (Torr cm) at 1000V\>", 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[largePDPlota, largePDPlotb, largePDPlotc, largePDPlotd, 
        largePDPlote, \ 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[productionPlot2a, productionPlot2b, 
        productionPlot2c,  (*productionPlot2d, productionPlot2e, *) \ 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[productionFitPlot1a, productionFitPlot1b, productionFitPlot1c, 
        productionFitPlot1d, productionFitPlot1e, \ 
        DisplayFunction\  -> \ $DisplayFunction];\)\)], "Input"],

Cell["Make 3D plot of production", "Text"],

Cell[BoxData[
    \(\(vdList;\)\)], "Input"],

Cell[BoxData[
    \(\(p0dcList\  = Table[0.1*10^\((k/4)\), {k, 1, 12}];\)\)], "Input"],

Cell[BoxData[
    \(productionLogTable\  = \ 
      Table[Log[10, productionFit]\  /. \ {vd\  -> \ Part[vdList, j], 
            p0dc\  -> \ 0.1*10^\((k/4)\)}, {j, 1, Length[vdList]}, {k, 1, 
          12}]\)], "Input"],

Cell["\<\
In the following plot the horizontal axes are integers indexing the ploints \
plotted and have no physical significance.\
\>", "Text"],

Cell[BoxData[
    \(\(ListPlot3D[productionLogTable, PlotRange -> All];\)\)], "Input"],

Cell[BoxData[
    \(\(ListContourPlot[productionLogTable, PlotRange -> All];\)\)], "Input"],

Cell[BoxData[
    \(\(productionLogTable4 = 
        Table[{vdList[\([j]\)], 
            p0dcList[\([k]\)], \((Log[10, 
                  productionFit]\  /. \ {vd\  -> \ Part[vdList, j], 
                  p0dc -> p0dcList[\([k]\)]})\)}, {j, 1, Length[vdList]}, {k, 
            1, Length[p0dcList]}];\)\)], "Input"],

Cell[BoxData[
    \(\(productionLogTable3\  = \ 
        Partition[
          Flatten[Table[{Log[10, vdList[\([j]\)]], 
                Log[10, 
                  p0dcList[\([k]\)]], \((Log[10, 
                      productionFit]\  /. \ {vd\  -> \ Part[vdList, j], 
                      p0dc -> p0dcList[\([k]\)]})\)}, {j, 1, 
                Length[vdList]}, {k, 1, Length[p0dcList]}]], 3];\)\)], "Input"],

Cell["<<Graphics`Graphics3D`", "Input",
  CellTags->"S5.34.1"],

Cell[BoxData[
    \(\(ScatterPlot3D[productionLogTable3, 
        BoxRatios -> {1, 1, 1}];\)\)], "Input"],

Cell[BoxData[
    \(\(ScatterPlot3D[productionLogTable3, BoxRatios -> {1, 1, 1}, 
        PlotJoined -> \ True];\)\)], "Input"],

Cell[BoxData[
    \(Options[ListSurfacePlot3D]\)], "Input"],

Cell[BoxData[
    \(\(ListSurfacePlot3D[productionLogTable4, BoxRatios -> {1, 1, 1}, \ 
        ViewPoint -> {\(-1\), \(-1\), 1}, PlotRange -> All];\)\)], "Input"],

Cell["Export data ", "Subsection"],

Cell["Export fit to production", "Subsubsection"],

Cell[BoxData[
    \(\(vdList\  = \ {100, 150, 200, 250, 300, 400, 500, 700, 1000, 1500, 
          2000, 3000};\)\)], "Input"],

Cell[BoxData[
    \(\(productionTable\  = \ 
        Table[productionFit /. \ {vd\  -> \ Part[vdList, j], 
              p0dc\  -> \ 0.01*10^\((k/4)\)}, {k, 1, 16}, {j, 1, 
            Length[vdList]}];\)\)], "Input"],

Cell[BoxData[
    \(\(productionTable2\  = \ Transpose[productionTable];\)\)], "Input"],

Cell[BoxData[
    \(\(p0dcList2 = Table[\ 0.01*10^\((k/4)\), {k, 1, 16}];\)\)], "Input"],

Cell[BoxData[
    \(\(productionTable3\  = \ 
        Transpose[Join[{p0dcList2}, productionTable2]];\)\)], "Input"],

Cell[BoxData[
    \(\(productionTable4\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          productionTable3, {2}];\)\)], "Input"],

Cell[BoxData[
    \(\(Export["\<multiplication.dat\>", 
        Join[{Join[{"\<%Fitted multiplication\>"}, {now}], 
            Join[{"\<%p0dc\Vc\>"}, vdList]}, 
          productionTable4], "\<Table\>"];\)\)], "Input"],

Cell[BoxData[
    \(\(!! multiplication.dat\)\)], "Input"],

Cell["\<\
Export Pitchford's fit to Monte Carlo result for ionization in CF to \
Proplot\
\>", "Subsubsection"],

Cell[BoxData[{
    \(\(p0dc = p0dc400\ ;\)\), "\[IndentingNewLine]", 
    \(\(v\  = 400;\)\)}], "Input"],

Cell[BoxData[
    \(\(snop0List = \ 
        Table[\((\(\((snon\ *3.54*10^16)\)\  //. \ {v\  -> \ 400. , 
                  ndc\  -> \ p0dc*3.54*10^16, 
                  nx\  -> p0x*3.54*10^16}\)\  //. \ p0x -> 0.02*i)\), {i, 0, 
            50}];\)\)], "Input"],

Cell[BoxData[
    \(\(snop0Table\  = \ 
        Transpose[{Table[0.02*j, {j, 0, 50}], snop0List\ }];\)\)], "Input"],

Cell[BoxData[
    \(\(dataout2\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          snop0Table, {2}];\)\)], "Input"],

Cell["\<\
dataout2 = TableForm[dataout2,TableHeadings -> {None,
                   {\"%Distance (m)\", \"Energy Distribution (m^-1)\"}},
                   TableSpacing -> {0,1}];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
SetOptions[$Output,PageWidth->120];
theoryfile := \"c:\\\\proplot\\\\argon\\\\arabnor\\\\ionproMC.txt\";
stream1 = OpenWrite[theoryfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Ionization rate in Ar cathode fall from \
Monte Carlo model\"],
                    \"\\n\",
                     StringForm[\"V = `` V,p0dc= ``,jt=`` A/m^2, gamma_i = \
0.025,d = 1 cm,p = 1 Torr\",v,p0dc,jt],
                    \"\\n\",
                    ToString[dataout2]];
Close[stream1];
SetOptions[$Output,PageWidth->53];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(\(!! c:\\proplot\\argon\\arabnor\\ionproMC.txt\)\)], "Input"],

Cell["Ionization in local field model", "Section"],

Cell["\<\
Here we wish to compare the ionization predicted by the local field model \
with that calculated by Pitchford et al using the Monte Carlo method applied \
above.  In order to do this we need normalize the local field results to the \
pressure and to the electron flux leaving the cathode.  \
\>", "Subsubtitle"],

Cell[BoxData[
    \(\(SetDirectory["\<c:\\siglo\\siglo-tr\>"];\)\)], "Input"],

Cell[BoxData[
    \(Clear[gammai, vd, p0dc, p0dc]\)], "Input"],

Cell["Read in electric field versus position at the steady-state", \
"Subsection"],

Cell["\<\
Fit the SIGLO-TR data for gammai = 0.01, v = 400 V, jt = 2.04 mA/cm^2\
\>", "SmallText"],

Cell[BoxData[
    \(\(gammai1\  = \ 0.01;\)\)], "Input"],

Cell[BoxData[
    \(\(vd\  = \ 400;\)\)], "Input"],

Cell[BoxData[
    \(\(p0 = 1.0*273/300;\)\)], "Input"],

Cell["\<\
The following file is from SIGLO-TR and is read to get the calculated \
electric field versus position in the steady state for the above \
conditions.\
\>", "SmallText"],

Cell[BoxData[
    \(\(!! result01.out\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"result01.out\";
            stream = OpenRead[inputfile];
            Find[stream,\"Number\"];
            current1 = Part[ReadList[stream, Number,2,RecordLists \
->True],1,1];          
            Find[stream,\"---\"];
            data1 = Flatten[ReadList[stream, Number,51,RecordLists ->True]];  \
        
            positionList = Drop[data1,-1]+  Part[data1,-1]/100;
            Do[Find[stream,\"---\"],{4}];
            fieldList = Flatten[ReadList[stream, Number,50,RecordLists \
->True]];
            Close[stream];
            fieldTable=Transpose[{positionList,fieldList}];
            fieldPlot = ListPlot[fieldTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.85]},
                         PlotRange -> {{0.,1.},{0.,1500.}}(*,
                         DisplayFunction ->Identity*)];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["Fitting near the cathode gives", "SmallText"],

Cell[BoxData[
    \(\(fittedLine\  = 
        Fit[Part[fieldTable, Range[1, 7]], {1, x}, x]\ \ ;\)\)], "Input"],

Cell[BoxData[
    \(\(Solve[fittedLine == 0, x];\)\)], "Input"],

Cell[BoxData[
    \(x1\  = \ x\  /. \ %\ [\([1]\)]\)], "Input"],

Cell[TextData[{
  "From the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook CURRENTDENSITY.NB the current density at the cathode is"
}], "SmallText"],

Cell[BoxData[
    \(\(epsilon0\  = \ 8.85*10^\(-12\);\)\)], "Input"],

Cell["\<\
jpMax = ((45197.7*epsilon0*p0^2*vd^2)/(p0dc^3*(1 + \
0.007864*(vd/p0dc)^1.5)^0.33)) //.
             p0dc -> x1*p0\
\>", "Input"],

Cell[BoxData[
    \(jt1\  = \ jpMax*\((1 + gammai1)\)\)], "Input"],

Cell["\<\
The current given by SIGLO-TR for gammai = 0.01 and V = 400 V is\
\>", "SmallText"],

Cell[BoxData[
    \(current1\)], "Input"],

Cell["\<\
Note that these current densities agree now that I have replaced the mobility \
table in SIGLO-TR with values consistent with Phelps and Petrovic (1999).\
\>", "Subsubtitle"],

Cell[BoxData[
    \(Clear[gammai, vd, p0dc, p0dc, x]\)], "Input",
  Evaluatable->False],

Cell["\<\
Fit the SIGLO-TR data for gammai = 0.13,  v = 400 V,  j = 1.76 mA/cm^2\
\>", "Subsubsection"],

Cell[BoxData[
    \(\(gammai3\  = \ 0.13;\)\)], "Input"],

Cell[BoxData[
    \(\(vd\  = \ 400;\)\)], "Input"],

Cell[BoxData[
    \(\(p0 = 1.0*273/300;\)\)], "Input"],

Cell[BoxData[
    \(\(!! result13.out\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"result13.out\";
            stream = OpenRead[inputfile];
            Find[stream,\"Number\"];
            current3 = Part[ReadList[stream, Number,2,RecordLists \
->True],1,1];          
            Find[stream,\"---\"];
            data1 = Flatten[ReadList[stream, Number,51,RecordLists ->True]];  \
        
            positionList = Drop[data1,-1]+  Part[data1,-1]/100;
            Do[Find[stream,\"---\"],{4}];
            fieldList = Flatten[ReadList[stream, Number,50,RecordLists \
->True]];
            Close[stream];
            fieldTable=Transpose[{positionList,fieldList}];
            fieldPlot = ListPlot[fieldTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.85]},
                         PlotRange -> {{0.,1.},{0.,5000.}}(*,
                         DisplayFunction ->Identity*)];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["Fitting near the cathode gives", "SmallText"],

Cell[BoxData[
    \(\(fittedLine\  = 
        Fit[Part[fieldTable, Range[1, 7]], {1, x}, x]\ \ ;\)\)], "Input"],

Cell[BoxData[
    \(\(Solve[fittedLine == 0, x];\)\)], "Input"],

Cell[BoxData[
    \(x3\  = \ Part[x\  /. \ %, 1]\)], "Input"],

Cell[TextData[{
  "From the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook CURRENTDENSITY.NB the current density at the cathode is"
}], "SmallText"],

Cell[BoxData[
    \(\(epsilon0\  = \ 8.85*10^\(-12\);\)\)], "Input"],

Cell["\<\
jpMax = ((45197.7*epsilon0*p0^2*vd^2)/(p0dc^3*(1 + \
0.007864*(vd/p0dc)^1.5)^0.33)) //.
             p0dc -> x3*p0\
\>", "Input"],

Cell[BoxData[
    \(jt3\  = \ jpMax*\((1 + gammai3)\)\)], "Input"],

Cell["\<\
The current given by SIGLO-TR for gammai = 0.13 and V = 400 V is\
\>", "SmallText"],

Cell[BoxData[
    \(current3\)], "Input"],

Cell["\<\
Finally, we read in results of local field calculations for pd = 1 Torr cm, V \
= 400 V, and gamma_i = 0.025, j = 0.272 mA/cm^2.\
\>", "Subsubsection"],

Cell[BoxData[
    \(Clear[gammai, vd, p0dc, p0dc, x]\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(gammai\  = \ 0.025;\)\)], "Input"],

Cell[BoxData[
    \(\(vd\  = \ 400;\)\)], "Input"],

Cell[BoxData[
    \(\(p0 = 1.0*273/300;\)\)], "Input"],

Cell[BoxData[
    \(\(!! result025.out\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"result025.out\";
            stream = OpenRead[inputfile];
            Find[stream,\"Number\"];
            current = Part[ReadList[stream, Number,2,RecordLists \
->True],1,1];          
            Find[stream,\"---\"];
            data1 = Flatten[ReadList[stream, Number,51,RecordLists ->True]];  \
        
            positionList = Drop[data1,-1]+  Part[data1,-1]/100;
            Do[Find[stream,\"---\"],{4}];
            fieldList = Flatten[ReadList[stream, Number,50,RecordLists \
->True]];
            Close[stream];
            fieldTable=Transpose[{positionList,fieldList}];
            fieldPlot = ListPlot[fieldTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.85]},
                         PlotRange -> {{0.,1.},{0.,2500.}}(*,
                         DisplayFunction ->Identity*)];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["Fitting near the cathode gives", "SmallText"],

Cell[BoxData[
    \(\(fittedLine\  = 
        Fit[Part[fieldTable, Range[1, 7]], {1, x}, x]\ \ ;\)\)], "Input"],

Cell[BoxData[
    \(\(Solve[fittedLine == 0, x];\)\)], "Input"],

Cell[BoxData[
    \(x\  = \ Part[x\  /. \ %, 1]\)], "Input"],

Cell[TextData[{
  "From the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook CURRENTDENSITY.NB the current density at the cathode is"
}], "SmallText"],

Cell[BoxData[
    \(\(epsilon0\  = \ 8.85*10^\(-12\);\)\)], "Input"],

Cell["\<\
jpMax = ((45197.7*epsilon0*p0^2*vd^2)/(p0dc^3*(1 + \
0.007864*(vd/p0dc)^1.5)^0.33)) //.
            p0dc -> x*p0\
\>", "Input"],

Cell[BoxData[
    \(jt\  = \ \(\(jpMax\)\(*\)\((1 + gammai)\)\(\ \)\)\)], "Input"],

Cell["\<\
The current given by SIGLO-TR for gammai = 0.025 and V = 400 V is\
\>", "SmallText"],

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

Cell["Read in ionization versus position at the steady-state", "Subsection"],

Cell["\<\
We read in results of ionization calculations for pd = 1 Torr cm, V = 400 V, \
and gamma_i = 0.025.\
\>", "SmallText"],

Cell[BoxData[
    \(\(!! result025.out\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"result025.out\";
            stream = OpenRead[inputfile];
            Find[stream,\"---\"];
            positionList = Flatten[ReadList[stream, Number,51,RecordLists \
->True]];  
            Do[Find[stream,\"---\"],{5}];
            ionizationList = Flatten[ReadList[stream, Number,51,RecordLists \
->True]];
            Close[stream];
            ionizationTable=Transpose[{positionList,ionizationList}];
            ionizationPlot = ListPlot[ionizationTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.85]},
                         PlotRange -> {{0.,1.},{0.,100.}},
                         DisplayFunction ->Identity];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["We compare this with Pitchford et al below.", "SmallText"],

Cell["Read in current and voltage transients", "Subsection"],

Cell["\<\
We read in results of current-voltage calculations for pd = 1 Torr cm, V = \
400 V, and gamma_i = 0.025.\
\>", "SmallText"],

Cell["\<\
Here we are primarily interested in the steady-state current.\
\>", "SmallText"],

Cell[BoxData[
    \(\(!! result025.out\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"result025.out\";
            stream = OpenRead[inputfile];
            Do[Find[stream,\"time (s)\"],{2}];
            data = Transpose[ReadList[stream,Number,RecordLists -> True]];
            Close[stream];
            timeList = Part[data,1];  
            currentList = Part[data,2];
            currentTable=Transpose[{timeList,currentList}];
            currentPlot = ListPlot[currentTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.85]},
                         PlotRange -> {{0.,25*10^-6},{0.,0.2}}(*,
                         DisplayFunction ->Identity*)];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(\(\(finalCurrentDen\)\(\ \)\(=\)\(\ \)\(Part[
        currentList, \(-1\)]\)\(\ \ \ \)\( (*mA/cm^2*) \)\)\)], "Input"],

Cell["\<\
To get the electron flux at the cathode, we prodply the total charged \
particle flux by gamma_i/(gamm_i+1).\
\>", "SmallText"],

Cell[BoxData[
    \(cathodeElectronFlux\  = \ 
      finalCurrentDen*10^\(-3\)/\((1.602*10^\(-19\))\)\ *\(\(gammai/\((gammai \
+ 1)\)\)\(\ \ \)\( (*\(electrons/cm^2\)/s*) \)\)\)], "Input"],

Cell["\<\
The ionization plot needs to be normalized to this electron flux.  Note that \
the calculations are already normalized to the pressure, because they were \
done for 1 Torr.  Similarly, the length scale is normalised to Torr cm \
because the calculations were done for 1 Tott and 1 cm.\
\>", "SmallText"],

Cell[BoxData[{
    \(\(ionizationTable2 = 
        Transpose[{positionList, 
            ionizationList/
              cathodeElectronFlux}];\)\), "\[IndentingNewLine]", 
    \(\(ionizationPlot2 = 
        ListPlot[ionizationTable2, 
          PlotStyle \[Rule] {PointSize[0.025], Hue[0.85]}, 
          PlotRange \[Rule] {{0. , 1. }, {0. , 300. }}, 
          DisplayFunction \[Rule] Identity];\)\)}], "Input"],

Cell[BoxData[
    \(\(Show[ionizationPlot2, snop0Plot, 
        DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input"],

Cell[BoxData[
    \(ionProduction\  = \ ListIntegrate[ionizationTable2, 2]\)], "Input"],

Cell["\<\
The areas look about the same and the integral is very close to the 39.5 for \
the fit to the Monte Carlo result using the same integration procedure.\
\>", "Subsubtitle"],

Cell["Export local field result for graphing", "Subsubsection"],

Cell[BoxData[
    \(\(dataout3\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          ionizationTable2, {2}];\)\)], "Input"],

Cell["\<\
dataout3 = TableForm[dataout3,TableHeadings -> {None,
                   {\"%Distance (cm)\", \"Ionization (cm^-1Torr^-1)\"}},
                   TableSpacing -> {0,1}];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
SetOptions[$Output,PageWidth->120];
theoryfile := \"c:\\\\proplot\\\\argon\\\\arabnor\\\\ionproLF.txt\";
stream1 = OpenWrite[theoryfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Ionization rate in Ar cathode fall from \
local-field model\"],
                    \"\\n\",
                     StringForm[\"V = `` V, gamma_i = ``, p0d  =  `` Torr cm, \
p0dc = `` Torr cm\",v,gammai,p0*1,p0*x],
                    \"\\n\",
                    ToString[dataout3]];
Close[stream1];
SetOptions[$Output,PageWidth->53];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(\(!! c:\\proplot\\argon\\arabnor\\ionproLF.txt\)\)], "Input"],

Cell["\<\
Export p0dc, jt, and gammai from preceding SIGLO-TR calculations\
\>", "Subsubsection"],

Cell[BoxData[
    \(\(p0dcdata\  = \ {{p0*x1, \((1 + 1/gammai1)\), 
            current1/p0^2}, {p0*x, \((1 + 1/gammai)\), 
            current/p0^2}, {p0*x3, \((1 + 1/gammai3)\), 
            current3/p0^2}};\)\)], "Input"],

Cell[BoxData[
    \(\(p0dcdata2\  = \ 
        Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm] &, 
          p0dcdata, {2}];\)\)], "Input"],

Cell[BoxData[
    \(\(Export["\<c:\\proplot\\argon\\arabnor\\SIGLO-TR.dat\>", 
        p0dcdata2, "\<Table\>"];\)\)], "Input"],

Cell[BoxData[
    \(\(!! c:\\proplot\\argon\\arabnor\\SIGLO-TR.dat\)\)], "Input"],

Cell["\<\
Read in Townsend ionization coefficient, alpha_i/p versus E/p\
\>", "Subsection"],

Cell["\<\
Here we are primarily interested in comparing SIGLO-TR with Kruithof (1940) \
and Phelps and Petrovic (1999).\
\>", "SmallText"],

Cell["\<\
The following was taken from the version of GASPAR.DAT, now with the filename \
GASPAR00.DAT, that was downloaded from the Kinema Web site on 5/31/00 and \
dated in 1998.  The version of GASPAR supplied with the CDROM by Pitchford in \
April 2000 appears to be the older, circa Fiala, version.\
\>", "SmallText"],

Cell[BoxData[
    \(\(!! sigloal2.txt\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"sigloal2.txt\";
            stream = OpenRead[inputfile];
            Do[Find[stream,\"%\"],{4}];
            data4 = ReadList[stream, Number,148];
            Close[stream];
            eonTdList = Part[data4,Range[1,74]]/.33;
            alphaonList=Part[data4,Range[75,148]]*100/(3.3*10^22);
            alphaonTable=Transpose[{eonTdList,alphaonList}];
            alphaonPlotNew = LogLogListPlot[alphaonTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.85]},
                         PlotRange -> {{10,10000},{10^-22,10^-19}},
                         DisplayFunction ->Identity];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
 inputfile := \"sigloal2.txt\";
            stream = OpenRead[inputfile];
            Do[Find[stream,\"%\"],{6}];
            data4b = ReadList[stream, Number,134];
            Close[stream];
            eonTdList = Part[data4b,Range[1,67]]/.33;
            alphaonList=Part[data4b,Range[68,134]]*100/(3.3*10^22);
            alphaonTable=Transpose[{eonTdList,alphaonList}];
            alphaonPlotOld = LogLogListPlot[Drop[alphaonTable,1],  
                         PlotStyle ->{PointSize[0.025],Hue[0.3]},
                         PlotRange -> {{10,10000},{10^-22,10^-19}},
                         DisplayFunction ->Identity];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["The Townsend ionization coefficient from Kruithof (1940) is", \
"Subsubsection"],

Cell[BoxData[
    \(\(!! kruithof.exp\)\)], "Input",
  Evaluatable->False],

Cell["\<\
 inputfile := \"Kruithof.exp\";
            stream = OpenRead[inputfile];
            Do[Find[stream,\"%\"],{2}];
            data5 = Transpose[ReadList[stream, Number,RecordLists ->True]];
            Close[stream];
            eonTdListK = Part[data5,3];
            alphaonListK=Part[data5,4];
            alphaonTableK=Transpose[{eonTdListK,alphaonListK}];
            alphaonPlotK = LogLogListPlot[Drop[alphaonTableK,1],  
                         PlotStyle ->{PointSize[0.025],Hue[0.55]},
                         PlotRange -> {{10,10000},{10^-22,10^-19}},
                         DisplayFunction ->Identity];            \
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
The Townsend ionization coefficient from Phelps and Petrovic (1999) is\
\>", "Subsubsection"],

Cell[BoxData[
    \(\(alphaion\  = \ \(+5.5\)*10^\(-21\)*Exp[\(-187\)/eonTd] + 
          3.2*10^\(-20\)*Exp[\(-700\)/eonTd] - 
          1.5*10^\(-20\)*Exp[\(-10000\)/eonTd] + 
          1.1*10^\(-22\)*Exp[\(-72\)/eonTd];\)\)], "Input"],

Cell[BoxData[
    \(\(alphaionPlot = 
        LogLogPlot[Evaluate[alphaion], {eonTd, 10, 10000}, 
          PlotRange \[Rule] {{10, 10000}, {10^\(-22\), 10^\(-19\)}}, 
          DisplayFunction \[Rule] Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(Show[alphaonPlotNew, alphaonPlotOld, alphaionPlot, alphaonPlotK, 
        DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input"],

Cell["\<\
This plot shows that the ionization coefficients for Ar in the file \
gaspar.dat of 9/12/99 are too high by a signifcant factor, i.e., they are \
about the same as in the Fiala, Pitchford, and Boeuf (1994) paper.  The \
values for Ar in gaspar.dat downloaded from Kinema on 5/31/00 and dated in \
1988 gives the \"new\" and \"old\" ionization corfficients.  The new values \
are in good agreement with Kruithof (1940) for E/n < 500 Td, but are rather \
high at 1500 Td.\
\>", "Subtitle"],

Cell["\<\
We did not find much change in the local-field result for the ionization when \
we changed the gaspar.dat file from the 1999 version to the 1998 version with \
the better alpha_i/n.  We did find a significant change when the ion mobility \
was extended using Phelps and Petrovic (1999).\
\>", "Subsubtitle"],

Cell["\<\
The normalized Ar+ mobility from Phelps and Petrovic (1999) is\
\>", "Subsubsection"],

Cell[BoxData[
    \(\(\(ionMobility = \(4\ /\((1 + \((0.007*eonTd)\)^{1.5})\)^{0.33}\)/
          10^\(-21\);\)\(\ \ \)\( (*m^\(-1\) V^\(-1\) 
        s^\(-1\)*) \)\)\)], "Input"],

Cell[BoxData[
    \(\(\(mobilityPlotFit = 
        LogLogPlot[Evaluate[ionMobility], {eonTd, 10, 10000}, 
          PlotRange \[Rule] {{10, 10000}, {10^20, 10^22}}, 
          DisplayFunction \[Rule] Identity];\)\(\ \ \ \ \)\)\)], "Input"],

Cell["\<\
The Ar+ mobility from the new Ar data in gaspar.dat (1988) is in units of \
Torr cm^2/V/s\
\>", "Subsubsection"],

Cell["\<\
 inputfile := \"sigloal2.txt\";
            stream = OpenRead[inputfile];
            Do[Find[stream,\"%\"],{2}];
            data4 = ReadList[stream, Number,58];
            Close[stream];
            eonTdList = Part[data4,Range[1,29]]/.322;
            ionMobilityList=Part[data4,Range[30,58]]/10^4*3.22*10^22;
            ionMobilityTable=Transpose[{eonTdList,ionMobilityList}];
            ionMobilityPlotNew = LogLogListPlot[ionMobilityTable,  
                         PlotStyle ->{PointSize[0.025],Hue[0.3]},
                         PlotRange -> {{10,10000},{10^20,10^22}},
                         DisplayFunction ->Identity];            \
\>", "Input",
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Cell["\<\
black line - Phelps and Petrovic (1999), green points - from Pitchford's 1998 \
revised GASPAR table \
\>", "Text"],

Cell["\<\
We need to extend the GASPAR.DAT tables of mobility and ionization \
coefficients\
\>", "SmallText"],

Cell[BoxData[
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The following data comes from the file C:\\SPREADSHEETS\\ARGON\\ARIONCOEF.WB3 \
(Quatro Pro8), where we compare various ionization coefficient sources.\
\>", "SmallText"],

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Cell["Ionization source term for uniform electric field.", "Section"],

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Cell[BoxData[
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Cell["\<\
Here I wish to plot the ionization source term assuming a spatially uniform \
electric field such that one gets the same production as in the above \
nonumifoem field cases.  Such a situation would correspond to breakdown.  We \
begin with the breakdown condition that ln(1+1/gammai) = alphai*d = \
(alphai/n)/(E/n)*(E*d) = eta*V or eta = 1/V*ln(1+1/gammai)\
\>", "SmallText"],

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Cell["Export uniform field result to Proplot", "Subsubsection"],

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dataout4 = TableForm[dataout4,TableHeadings -> {None,
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                   TableSpacing -> {0,1}];\
\>", "Input",
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SetOptions[$Output,PageWidth->120];
theoryfile := \"c:\\\\proplot\\\\argon\\\\arabnor\\\\ionproUN.txt\";
stream1 = OpenWrite[theoryfile,PageWidth -> Infinity];
WriteString[stream1, \"%\",
                     StringForm[\"Ionization rate in Ar cathode fall from \
uniform-field model\"],
                    \"\\n\",
                     StringForm[\"V = `` V, gamma_i = ``,d = 1 cm,p = 1 \
Torr\",v,gammai],
                    \"\\n\",
                    ToString[dataout4]];
Close[stream1];
SetOptions[$Output,PageWidth->53];\
\>", "Input",
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  FontFamily->"Arial",
  FontSize->11,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
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Cell["\<\
I should consider adding a plot of the ion production given by the \
single-beam model.\
\>", "Subtitle"],

Cell["\<\
Why don't we see the effects a decreasing ionization cross section with \
increasing energy at electron energies above about 100 eV?\
\>", "Subtitle"]
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*)



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