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Notebook[{
Cell["A simplified model of ionization in abnormal glow discharges", \
"Subtitle"],

Cell["Setup notebook  enviroment ", "SmallText",
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  FontWeight->"Bold",
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Cell[CellGroupData[{

Cell[BoxData[
    \(now\  = \ 
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Cell[CellGroupData[{

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

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Cell["\<\
The objective of these notes is to outline a potential technique for \
calculating the ion production in a very low pressure abnormal cathode fall \
discharge, such as used for sputtering.  The principle assumptions are:
\
\>", "Subsubtitle"],

Cell["\<\
1. The current density is high enough so that the thickness of the cathode \
fall is short compared to an electron mean-free-path at an energy \
corresponding to the discharge voltage.  In this limit one can neglect the \
ionization produced in the high field region of the cathode fall and consider \
the electron injected into the essentiall zero electric field region of the \
negative glow and Faraday dark space.\
\>", "Subsubtitle"],

Cell["\<\

2. The ionization by electrons could  then be calculated using a version of \
the multiple scattering technique of Chantry, Phelps, and Schulz (1966).  \
Here the loss to the anode is treated as a boundary condition. \
\>", "Subsubtitle"],

Cell["\<\
3) A possible alternative to 2) is to integrate over the interelectrode space \
and treat the wall loss as a kind of inelastic collision.  This has the \
advantage of reducng the problem to one-dimension in energy, as in the usual \
electron Boltzmann equation.\
\>", "Subsubtitle"],

Cell["\<\
4) A potentially useful map for relating such modeling to other models is to \
plot pdc/electron mfp versus pd/electron range for constant gammai.  At low \
pdc/mfp (high j/p^2) one has the electron injection models of ionization, \
while at high pdc/mfp one approaches local-field models of ionization.  At \
low pd/range the ionization is inefficient and the voltage high (as desired \
for sputtering), while at high pd/range one has efficient ionization and \
lower voltages.  This latter could be shown on a 3-D plot.\
\>", "Subsubtitle"],

Cell["Change the following to p0 units", "Subtitle",
  FontColor->RGBColor[1, 0, 0]],

Cell["Electron range ", "Subsubsection"],

Cell["\<\
From Phelps, Jelenkovic and Pitchford (1987) the elecron range for energies \
above 200 eV is approximately:\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(normrange\  = \ 
      6.2*10^20*\(\(\((energy/400. )\)^1.56\)\(\ \)\( (*\ 
          normrange\ in\ Torr\ cm\ for\ energy\ in\ eV*) \)\)\)], "Input"],

Cell[BoxData[
    \(5.4097613035168496`*^16\ energy\^1.56`\)], "Output"]
}, Open  ]],

Cell["\<\
This range is slightly larger than calculated from the single beam model by \
Phelps et al (1987).  Expressed on p0 units\
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(normrangep0\  = \ normrange/\((3.54*10^22/100)\)\)], "Input"],

Cell[BoxData[
    \(0.00015281811591855507`\ energy\^1.56`\)], "Output"]
}, Open  ]],

Cell["\<\
We need to compare this range with the exponential fall-off distance found by \
Pitchford et al from fitting their Monte Carlo calculations.\
\>", "SmallText"],

Cell["Perez et al (1992) and Pitchford (private communication)", \
"Subsubsection"],

Cell["\<\
Following Peres 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,
  Evaluatable->False,
  ShowSpecialCharacters->False],

Cell["\<\
where pdc is in Torr cm, ndc in cm^-2, v in Volts, and kp in cm^-1Torr^-1 at \
300K.\
\>", "SmallText"],

Cell["kp =((a1*pdc+a2)*v+b1*pdc+b2)", "Input",
  PageWidth->Infinity,
  Evaluatable->False,
  ShowSpecialCharacters->False],

Cell["and", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(smax\  = \ \((c1*pdc + c2*v + c3)\)*v + d1*pdc + b2\)], "Input"],

Cell[BoxData[
    \(b2 + d1\ pdc + v\ \((c3 + c1\ pdc + c2\ v)\)\)], "Output"]
}, Open  ]],

Cell["\<\
From figure 3a of Perez et al (1992) for 250 V, the pdc and attenuation \
lengths read from the decreasing portion of the graphs are\
\>", "SmallText"],

Cell[BoxData[
    \(attenuationList\  = \ {{0.1, 0.1}, {0.175, 0.085}, {0.25, 
          0.07}}\)], "Input",
  Evaluatable->False],

Cell["\<\
This shows that my idea of an attenuation dependent only on the total voltage \
is of questionable value at low energies.\
\>", "SmallText"],

Cell["From Pitchford (1999) we get revised values of:", "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[BoxData[
    \(\(kp = \((\((a1*pdc + a2)\)*v + b1*pdc + b2)\);\)\)], "Input"],

Cell[CellGroupData[{

Cell["\<\
kop0 =((a1*pdc+a2)*v+b1*pdc+b2)*p/p0 /. {pdc -> p0dc*300/273,p->p0*300/273}\
\>", "Input",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(100\/91\ \((b2 + \(100\ b1\ p0dc\)\/91 + \((a2 + \(100\ a1\ p0dc\)\/91)\
\)\ v)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
smaxop0 =((c1*pdc+c2*v+c3)*v+d1*pdc+d2)*p/p0 /. {pdc -> \
p0dc*300/273,p->p0*300/273}\
\>", "Input",
  PageWidth->Infinity,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(100\/91\ \((d2 + \(100\ d1\ p0dc\)\/91 + 
          v\ \((c3 + \(100\ c1\ p0dc\)\/91 + c2\ v)\))\)\)], "Output"]
}, Open  ]],

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["\<\
To compare the normalized range and the reciprocal attenuation in Torr cm as \
a function of electron energy \
\>", "SmallText"],

Cell[CellGroupData[{

Cell[BoxData[
    \(attenuation1\  = \ 
      1/kop0\  /. \ {v\  -> \ energy, p0dc\  -> 10^\(-4\)}\)], "Input"],

Cell[BoxData[
    \(91\/\(100\ \((\(\(9.225339560439561`\)\(\[InvisibleSpace]\)\) - \
0.0145030989010989`\ energy)\)\)\)], "Output"]
}, Open  ]],

Cell["\<\
Note that this attenuation distance decreases with increasing pdc at fixed \
voltage.  Here we have used a very small pdc valure of 0.0001 Torr cm.\
\>", "SmallText"],

Cell["\<\
The following plots show nd values in units of Toee cm versus applied voltage \
in V or electron energy in eV.  The first plot is the extrapolated range \
based on Fig. 6 of Phelps, Jelenkovic and Pitchford (1987).\
\>", "Subsubtitle"],

Cell[BoxData[
    \(\(rangeplot\  = \ 
        LogLogPlot[normrange/\((3.54*10^22/100)\), \ {energy, 100, 10000}, 
          PlotStyle -> \ Thickness[0.007], 
          PlotRange -> {{100, 10000}, { .01, 100}}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
From Brewer and Westhaver, J. Appl. Phys. 8, 779 (1937), Fig. 1. Assume Tg = \
293K.\
\>", "Text"],

Cell[BoxData[
    \(\(negativeGlowPlot = 
        LogLogListPlot[{{250, 1. *273/293}, {400, 2.2*273/293}, {550, 
              3.3*273/293}}, PlotRange -> {{100, 10000}, { .01, 100}}, 
          PlotStyle -> {Hue[0.1], \ PointSize[0.02]}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
From Fig. 6 of Lehmann, Proc. Roy. Soc. (London) A115, 624 (1927)\
\>", "Text"],

Cell[BoxData[
    \(\(lehmannRangePlot = 
        LogLogListPlot[{{290, 1*273/293}, {430, 2*273/293}, {630, 
              3.7*273/293}}, PlotRange -> {{100, 10000}, { .01, 100}}, 
          PlotStyle -> {Hue[0.8], \ PointSize[0.02]}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
Range results are are also available from Vasenkov (1998) in units of \
\>", "SmallText"],

Cell[BoxData[
    \(\(vasenkovplot\  = \ 
        LogLogListPlot[{{100, 0.68*1.5*10^20/\((3.54*10^22/100)\)}, {500, 
              3.82*1.5*10^20/\((3.54*10^22/100)\)}, {1000, 
              9.60*1.5*10^20/\((3.54*10^22/100)\)}, {2000, 
              27.07*1.5*10^20/\((3.54*10^22/100)\)}, {5000, 
              131.7*1.5*10^20/\((3.54*10^22/100)\)}}, 
          PlotRange -> {{100, 10000}, { .01, 100}}, 
          PlotStyle -> \ {PointSize[0.02], Hue[1]}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
This plot is from the fit to MC calculations of Pitchford et al as discussed \
above for a representative ndc value \
\>", "SmallText"],

Cell[BoxData[
    \(\(perezplot\  = \ 
        LogLogPlot[attenuation1, \ {energy, 100, 400}, 
          PlotRange -> {{100, 10000}, { .01, 100}}, 
          PlotStyle -> {Dashing[{0.05, 0.02}], Thickness[0.007]}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
Here we try to estimate the extrapolated range of the electrons in figs. 3b \
and 4b of Belenguer and Pitchford (1999).  The value for 1000 eV is only a \
lower limit because of the limited available data.\
\>", "SmallText"],

Cell[BoxData[
    \(\(belenguerPlot\  = \ 
        LogLogListPlot[{{125, 0.05*273/300}, {500, 0.5*273/300}, {1000, 
              1. *273/300}}, PlotRange -> {{100, 10000}, { .01, 100}}, 
          PlotStyle -> \ PointSize[0.02], 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
The attenuation distances are ~0.05 cm at 125(?) eV, ~0.5 at 500 V and ~1 cm \
at 1000 V and range are 1.5 at 500 eV and > 1.9 at 1000 eV.\
\>", "Text"],

Cell["\<\
We need to convert the pressure units of the next four sets of data from \
Rozsa et al (1990)  from mbar at 300K to Torr cm at 273K.  The first set of a \
data from the lower curve of Fig. 8 gives a list of j/p^2 and pd values.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Take[Import["\<Rozsa95NegGlow.dat\>", "\<Table\>"], {5, 11}]\)], "Input"],

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        0.32086000000000003`}, {0.0022982`, 
        0.43752399999999997`}, {0.00476009`, 0.607683`}, {0.010069`, 
        0.9541069999999999`}}\)], "Output"]
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    \(LogLogListPlot[%]\)], "Input",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(rozsaList1a = 
      Interpolation[
        Table[\(Take[
                Import["\<Rozsa95NegGlow.dat\>", "\<Table\>"], {5, 
                  11}]\)[\([j]\)]*{\(1000/0.76\)/0.76, 0.76*273/300}, {j, 1, 
            7}], InterpolationOrder -> 1]\)], "Input"],

Cell[BoxData[
    TagBox[\(InterpolatingFunction[{{0.16398320637119115`, 
            17.43247922437673`}}, "<>"]\),
      False,
      Editable->False]], "Output"]
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Cell["\<\
The second set of data from Fig. 6 gives a list of j/p^2 and V values.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Take[Import["\<Rozsa95Volt.dat\>", "\<Table\>"], {3, 13}]\)], "Input"],

Cell[BoxData[
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        312.229`}, {0.000320429`, 324.104`}, {0.000484693`, 
        365.91`}, {0.000633528`, 390.612`}, {0.0009819140000000002`, 
        432.84200000000004`}, {0.00169121`, 475.182`}, {0.00198921`, 
        516.8180000000001`}, {0.0037765399999999997`, 
        605.674`}, {0.007839229999999999`, 677.448`}}\)], "Output"]
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    \(LogLogListPlot[%]\)], "Input",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(rozsaVoltInt = 
      Interpolation[
        Table[\(Take[
                Import["\<Rozsa95Volt.dat\>", "\<Table\>"], {3, 
                  13}]\)[\([j]\)]*{\(1000/ .76\)/ .76, 1}, {j, 1, 11}], 
        InterpolationOrder -> 1]\)], "Input"],

Cell[BoxData[
    TagBox[\(InterpolatingFunction[{{0.164237188365651`, 
            13.572074099722991`}}, "<>"]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(jopsqList1 = { .165,  .25,  .4,  .7, 1. , 2. , 3. , 5. , 7. , 10. , 
          13. };\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(rozsaTable1 = 
      Table[{rozsaVoltInt[jopsqList1[\([j]\)]], 
          rozsaList1a[jopsqList1[\([j]\)]]}, {j, 1, 
          Length[jopsqList1]}]\)], "Input"],

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        0.13373373086894413`}, {381.32968430812645`, 
        0.16060086219613584`}, {443.18595970088654`, 
        0.2275252584030931`}, {480.9928766442953`, 
        0.26545904491421596`}, {561.5007861894558`, 
        0.33078490622574364`}, {610.3849808624335`, 
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        0.5443206343417032`}}\)], "Output"]
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Cell[BoxData[
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        LogLogListPlot[
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            PlotStyle -> {Hue[0.5], \ PointSize[0.02]}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
 The first set of a data from the upper curve of Fig. 8 gives a list of j/p^2 \
and pd values.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Take[
      Import["\<Rozsa95NegGlow.dat\>", "\<Table\>"], {13, 16}]\)], "Input"],

Cell[BoxData[
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        0.385364`}, {0.0011463600000000001`, 0.549108`}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(rozsaList2 = 
      Interpolation[
        Table[\(Take[
                Import["\<Rozsa95NegGlow.dat\>", "\<Table\>"], {13, 
                  16}]\)[\([j]\)]*{\(1000/0.76\)/0.76, 0.76*273/300}, {j, 1, 
            4}], InterpolationOrder -> 1]\)], "Input"],

Cell[BoxData[
    TagBox[\(InterpolatingFunction[{{0.2846572022160665`, 
            1.984695290858726`}}, "<>"]\),
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      Editable->False]], "Output"]
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Cell[BoxData[
    \(\(jopsqList2 = { .29,  .4,  .7, 1. , 1.9};\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(rozsaTable2 = 
      Table[{rozsaVoltInt[jopsqList2[\([j]\)]], 
          rozsaList2[jopsqList2[\([j]\)]]}, {j, 1, 
          Length[jopsqList2]}]\)], "Input"],

Cell[BoxData[
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        0.2373471215579078`}, {439.73809251990707`, 
        0.3675096289798838`}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(rozsaPlot2\  = \ 
        LogLogListPlot[
          rozsaTable2,  (*\(PlotRange -> {{100, 10000}, { .01, 100}}\)\(,\)*) 
            PlotStyle -> {Hue[0.7], \ PointSize[0.02]}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell["\<\
Phelps' fit of attenuation data.  Note that I had to add a dependence on p0dc \
at low energies so as to fit both Perez et al and Belenguer and Pitchford.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(phelpsFit = \ \((energy/1000)\)^1.78*\((1 + 
              1/\((1 + \((p0dc/0.1)\)^2)\)^1. *\((300/
                      energy)\)^3.56)\)^0.5\)], "Input"],

Cell[BoxData[
    \(4.570881896148749`*^-6\ energy\^1.78`\ \((1 + \(6.584937644483684`*^8\ \
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}, Open  ]],

Cell[BoxData[
    \(\(phelpsFitPlot2\  = \ 
        LogLogPlot[phelpsFit\  /. \ energy -> 125, \ {p0dc, 0.01, 1. }, 
          PlotRange -> {{0.01, 1. }, { .01, 1. }}, 
          PlotStyle -> {{Hue[0.5], Thickness[0.007]}, {Hue[0.7], 
                Thickness[0.007]}}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell[BoxData[
    \(\(perezPlot2\  = \ 
        LogLogPlot[1/kop0\  /. v -> 125, \ {p0dc, 0.01, 1. }, 
          PlotRange -> {{0.01, 1. }, { .01, 1. }}, 
          PlotStyle -> {{Thickness[0.007]}, {Hue[0.7], Thickness[0.007]}}, 
          DisplayFunction -> Identity];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Show[perezPlot2, phelpsFitPlot2, GridLines -> Automatic, 
        PlotLabel -> "\<Attenuation length (Torr cm) versus p0dc (Torr \
cm)\>", DisplayFunction -> $DisplayFunction];\)\)], "Input"],

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Cell["\<\
red points - Vasenkov range theory; orange points - Brewer and Westhaver neg. \
glow; black points - Belenguer and Pitchford; black dashed curve - Perez et \
al fit to attenuation from MC calc.; black solid line - range from Phelps et \
al (1987); purple points - Lehmann range (1927); cyan and blue curves - \
Phelps' fit to attenuation length for p0dc of 0.4 and \
\>", "Text"],

Cell["\<\
This shows that the range data is much larger than the \"attenuation length\" \
or fall-off distance from the source term calculations.  Presumably part of \
reason the source term with the electric field falls off more rapidly than \
does the range is that the former includes significantly lower energy \
electrons formed in the high field region as well as promary electrons.  This \
is particularly so when the multiplication is large.  When the p0dc is large \
enough so that the local-field model is become valid the attenuation length \
becomes short compared to the cathode-fall length. Note that the range \
estimate from Fig. 3b of Belengeur and Pitchford (1999) is in rough \
agreenment with that of Vasenkov (1998).  The comparison with Vasenkov \
suggests that the range values from Phelps et al (1987) are too large at the \
higher energies shown. \
\>", "Subsubtitle"],

Cell["\<\
The data of Fig. 3a and 3b of Belenguer and Pitchford (1999) suggests to me \
that the ionization at large px is caused by those electrons that are \
accelerated through the applied voltage without significant energy loss.  If \
so, the \"maximum\" range and the ionization rate at large px will be \
independent of pdc.\
\>", "Subsubtitle"],

Cell["\<\
Pitchford (~5/10/00) points out that those ions procuced to the anode side of \
the potential minimum do not return to the cathode.  This reduces the ion \
current reaching the cathode and raise the voltage required to maintain the \
discharge.\
\>", "Subsubtitle"]
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End of Mathematica Notebook file.
*******************************************************************)