(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 535421, 18549]*) (*NotebookOutlinePosition[ 536380, 18582]*) (* CellTagsIndexPosition[ 536336, 18578]*) (*WindowFrame->Normal*) Notebook[{ Cell["Corrected for symmetry", "Subsubtitle", FontColor->RGBColor[1, 0, 0]], Cell["\<\ Approximate angular scattering functions for Ar-Ar collisions\ \>", "Subtitle", Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center], Cell["\<\ We wish to generate a set of approximate angular distributions for use in \ Monte Carlo calculations. For that purpose we have used our modified \ screened Coulomb (or Rutherford) formula as described below. This notebook \ is based on our previous work on electrons in N2\ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell["Setup notebook enviroment ", "SmallText", PageWidth->Infinity], 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["The following does not work", "SmallText"], Cell["Off[Solve::ifun]", "Input", PageWidth->Infinity], Cell["<< Graphics`Graphics`", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(now\ = \ StringForm["\<``/``/`` ``:``:``\>", \(Date[]\)[\([2]\)], \(Date[]\)[\([3]\)], \(Date[]\)[\([1]\)], \(Date[]\)[\([4]\)], \(Date[]\)[\([5]\)], \(Date[]\)[\([6]\)]]\)], "Input"], Cell[BoxData[ InterpretationBox[ "\<\"\\!\\(11\\)/\\!\\(8\\)/\\!\\(1998\\) \ \\!\\(21\\):\\!\\(22\\):\\!\\(53\\)\"\>", StringForm[ "``/``/`` ``:``:``", 11, 8, 1998, 21, 22, 53], Editable->False]], "Output"] }, Open ]], Cell["Modified screened Coulomb scattering relations.", "Subsection", Evaluatable->False, FontFamily->"Helv", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ The discussion presented here begins with a detailed review of the . Note \ that the assumed angular distribution is different than the modified screened \ Coulomb formula summarized on the anonymous ftp site JILA.COLORADO.EDU in the \ directory COLLISION_DATA and the file ELECTRON.TXT. The change to has been \ made because I hope to get simpler results for the cumulative scattering \ probability and because the symmetry requirement for identical particles \ means that one automatically has a representation of large angle \ scattering.\ \>", "SmallText", Evaluatable->False], Cell["\<\ The modified differential scattering cross section i(theta,beta,enrel) in \ center-of-mass coordinates and energy is assumed to be\ \>", "SmallText", Evaluatable->False], Cell["\<\ i := (a/2)*(1 + 2*beta - Cos[theta])^-2 +b + (a/2)*(1 + 2*beta - Cos[Pi - \ theta])^-2\ \>", "Input", PageWidth->Infinity], Cell["\<\ where theta is the scattering angle and beta is the \"modified screening \ parameter\". Here both beta and a are functions of energy. For beta = 0 we \ have Coulomb or rutherford scattering. For beta = 0.5 the scattering is \ isotropic. Note that this function is symmetric about Pi/2 as required for \ identical particles.\ \>", "Text", Evaluatable->False, FontFamily->"Helv", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ My favorite reference on the conventional form is Strickland et al, J. \ Geophys. Res. 81, 2755 (1976).\ \>", "SmallText", Evaluatable->False], Cell["Total cross section qt", "Subsubsection", Evaluatable->False], Cell["\<\ A problem with the total cross section is that at high energies the \ scattering is expected to become more nearly that for a Coulomb potential. In \ this limit the total scattering cross section is well known to become \ undefined because of the large amount of small angle scattering. If this \ problem became serious, it might be necessary to adopt the Monte Carlo \ techniques devised by Kushner, et. al. for electron-electron collisions. \ With our modified screened Coulomb form and b = 0. Our total cross section \ qt is \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(qt\ = \ 2*Pi*\ Integrate[Sin[theta]*\ i\ \ (* /. \ b\ -> \ 0*) , \ \ {theta, \ 0, \ Pi}]\ // \ Together\)], "Input", Evaluatable->True], Cell[BoxData[ \(\(\((a + 4\ b\ beta + 4\ b\ beta\^2)\)\ \[Pi]\)\/\(beta\ \((1 + beta)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Apart[qt, a]\ \)\)], "Input"], Cell[BoxData[ \(4\ b\ \[Pi] + \(a\ \[Pi]\)\/\(beta\ \((1 + beta)\)\)\)], "Output"] }, Open ]], Cell["For b = 0 ", "SmallText"], Cell[CellGroupData[{ Cell["qt0 = Limit[qt,b -> 0]", "Input"], Cell[BoxData[ \(\(a\ \[Pi]\)\/\(beta\ \((1 + beta)\)\)\)], "Output"] }, Open ]], Cell["so that for b = 0 the coefficient a is ", "SmallText", Evaluatable->False], Cell["a := qt0/Pi*(1- beta)*beta;", "Input", PageWidth->Infinity, Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ Correction of experimental total cross section for detector resolution\ \>", "Subsubsection", Evaluatable->False, FontFamily->"Helv", FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["Not changed for symmetry", "Subsubtitle"], Cell["\<\ This cross section is defined as the cross section for scattering beyond an \ angle thetamin. It is calculated from the differential scattering cross \ section for b = 0 by\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ qtexp = 2*Pi* Integrate[Sin[theta]* i /. b -> 0, {theta, thetamin, Pi}] // \ Together\ \>", "Input", PageWidth->Infinity, Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ This cross section is that given by experiments like those of Amdur, et. al. \ In some of his 1954 experiments the detector acceptance angle was roughly \ thetamin =2* 0.00028 rad. The ratio of this experimental total cross section \ to the true total cross section obtained in the limit of thetamin -> 0 is:\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["qtexpoqt = qtexp/Limit[qtexp, thetamin -> 0]", "Input", PageWidth->Infinity, Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ For the next plot we compare this ratio as a function of beta for a large \ thetamin with that at for small thetamin so as to illustrate the possible \ error\ \>", "SmallText", Evaluatable->False], Cell["\<\ LogLogPlot[{qtexpoqt /. {thetamin -> 0.1, b ->0}, qtexpoqt /. {thetamin -> \ 2*0.00028,b -> 0}}, {beta,0.0001,0.99999}, PlotStyle -> {Dashing[{ }] (*thetamin = 0.01*), Dashing[{0.02,0.02}]} (*thetamin =2* \ 0.000028*), PlotRange -> {{0.000999,1},{0.1,1}}, AxesLabel -> {\"beta\",\"Qtexp/Qt\"}];\ \>", "Input", PageWidth->Infinity, Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ The upper curve is for Amdur's thetamin = 2*0.00028, while the lower curve is \ for thetamin = 0.1.\ \>", "SmallText", Evaluatable->False], Cell["\<\ This graph shows that if thetamin = 0.1 rad our minimum beta value of 0.005 \ would lead to a 40% error when we use the measured \"total\" cross section \ instead of an extrapoled total cross section. Of course this estimate \ assumes the validity of the screened Coulmb approximation to the differential \ cross section. A procedure that avoids this error is to use formulas for \ Qv/Qtexp instead of Qv/Qt when determining beta. Because of Amdur's small \ thetamin we do not make a significant error when using the thetamin = 0 \ formula. Thus, this section of the notebook is unnecessary.\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }, Closed]], Cell["The diffusion transfer cross section qd for isotopic Ar", "Subsubsection", Evaluatable->False], Cell["\<\ Here we make use of the conventional formula for diffusion. Since Greene \ pointed out that because of symmetry the formula for true self-diffusion \ gives the total cross section qt, we will note this this result applies only \ for diffusion of one isotope of Ar though another isotope of Ar. We assume \ that the isotope scattering is equal to twice the first terms of the i(theta) \ given above plus the constant term. When we took the fist term (a-term) \ equal to the first term of the identical particle term we got very low values \ for qd at high energies. We need to check with Greene and Burke on this \ assumption.\ \>", "SmallText"], Cell[CellGroupData[{ Cell["iso = a*(1 + 2*beta - Cos[theta])^-2 +b", "Input", PageWidth->Infinity], Cell[BoxData[ \(b + a\/\((1 + 2\ beta - Cos[theta])\)\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(qd1\ = \ \(2*Pi*\ Integrate[ Sin[theta]*iso*\((1 - Cos[theta])\), {theta, \ 0, \ Pi}]\ // \ PowerExpand\)\ // Simplify\)], "Input"], Cell[BoxData[ \(\(-\(\(2\ \[Pi]\ \((a - 2\ b - 2\ b\ beta + a\ \((1 + beta)\)\ Log[beta] - a\ \((1 + beta)\)\ Log[1 + beta])\)\)\/\(1 + beta\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(qd2\ = \ Apart[%\ , a]\)], "Input"], Cell[BoxData[ \(4\ b\ \[Pi] - \(2\ a\ \[Pi]\ \((1 + Log[beta] + beta\ Log[beta] - Log[1 + beta] - beta\ Log[1 + beta])\)\)\/\(1 + beta\)\)], "Output"] }, Open ]], Cell["Therefore", "SmallText"], Cell[BoxData[ \(qd\ := \ 4*Pi*\ b\ - \ a*2*Pi/\((1 + beta)\) + a*2*Pi*Log[\((1 + beta)\)/beta]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(qd - qd2\ // \ PowerExpand\)\ // \ Simplify\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ In the case of electrons in N2, we used the ratio qd/qt = qdoqt to determine \ beta. See the ftp file. Here the ratio is unity.\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["Viscosity cross section", "Subsubsection", Evaluatable->False, FontFamily->"Helv", FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["The viscosity cross section is ", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["qv1 = 2*Pi* Integrate[Sin[theta]^3*i,{theta,0,Pi}] ", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(2\ \[Pi]\ \((\(-4\)\ a + \(4\ b\)\/3 + \((a + 2\ a\ beta)\)\ Log[\(-1\) - beta] - \((a + 2\ a\ beta)\)\ Log[\(-beta\)] - \((a + 2\ a\ beta)\)\ Log[beta] + \((a + 2\ a\ beta)\)\ Log[1 + beta])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((Coefficient[qv1, a]/\((2*Pi)\)\ + 4\ )\)/\((1 + 2*beta)\)\ // \ Cancel\ \)\)], "Input"], Cell[BoxData[ \(Log[\(-1\) - beta] - Log[\(-beta\)] - Log[beta] + Log[1 + beta]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((Cancel[\((Apart[qv1, a]\ - 8*Pi*b/3)\)/\((2*\ a*\ Pi)\)\ ]\ + 4) \)/\((1 + 2 beta)\)\ \ // \ PowerExpand\)\ // \ Simplify\)], "Input"], Cell[BoxData[ \(\(-I\)\ \[Pi] + Log[\(-1\) - beta] - 2\ Log[beta] + Log[1 + beta]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\ \(( \(\((%\ /. \ beta\ -> \ \(-1\)\ + \ eta)\)\ // \ PowerExpand \)\ // \ Simplify)\)\ /. \ eta\ -> \ 1 + beta\ \)\)], "Input"], Cell[BoxData[ \(\(-2\)\ Log[beta] + 2\ Log[1 + beta]\)], "Output"] }, Open ]], Cell[BoxData[ \(qv\ := \ 8/3*\ Pi\ *b\ \ - \ 8\ *Pi*\ a\ \ + \ 4*Pi*a\ *\((1 + 2*beta)\)*\ Log[\((1 + beta)\)/beta]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((\(qv - qv1\ \ // \ PowerExpand\)\ // \ Simplify)\)\ /. \ beta\ -> \ \(-1\) - eta\ \ \ // \ PowerExpand\)\ \ // \ Simplify \)], "Input"], Cell[BoxData[ \(\(-4\)\ I\ a\ \((1 + 2\ eta)\)\ \[Pi]\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Log[\(-1\)]\)], "Input"], Cell[BoxData[ \(I\ \[Pi]\)], "Output"] }, Open ]], Cell["\<\ It would appear that this result differs from 0 by a I*2*Pi and should be \ considered as zero.\ \>", "SmallText"], Cell["Cumulative probability needed for Monte Carlo calculatioms.", "Subsection", Evaluatable->False], Cell["\<\ The probability of scattering through an angle less than theta0 is then\ \>", "SmallText", Evaluatable->False], Cell["\<\ cumprob1 = Integrate[2*Pi*Sin[theta]*i/qt, {theta, 0, theta0}] //Together;\ \>", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell[BoxData[ \(cumprob\ = \ cumprob1\ \ /. \ Cos[2*theta0]\ -> \ 2*Cos[theta0]^2\ - 1\)], "Input"], Cell[BoxData[ \(\((a + 4\ a\ beta + 4\ b\ beta + 4\ a\ beta\^2 + 20\ b\ beta\^2 + 32\ b\ beta\^3 + 16\ b\ beta\^4 - 4\ a\ beta\ Cos[theta0] - 2\ b\ beta\ Cos[theta0] - 4\ a\ beta\^2\ Cos[theta0] - 18\ b\ beta\^2\ Cos[theta0] - 32\ b\ beta\^3\ Cos[theta0] - 16\ b\ beta\^4\ Cos[theta0] - a\ Cos[theta0]\^2 - 4\ b\ beta\ Cos[theta0]\^2 - 4\ b\ beta\^2\ Cos[theta0]\^2 + 2\ b\ beta\ Cos[theta0]\ \((\(-1\) + 2\ Cos[theta0]\^2)\) + 2\ b\ beta\^2\ Cos[theta0]\ \((\(-1\) + 2\ Cos[theta0]\^2)\))\)/ \((2\ \((a + 4\ b\ beta + 4\ b\ beta\^2)\)\ \((1 + 2\ beta - Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\)) \)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cumprob3\ = \ Apart[cumprob, a]\)], "Input"], Cell[BoxData[ \(\(-\(\(\((\(-1\) + Cos[theta0])\)\ \((1 + 4\ beta + 4\ beta\^2 + Cos[theta0])\)\)\/\(2\ \((1 + 2\ beta - Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\)\)\)\) + \(2\ \((\(-b\)\ beta\ Cos[theta0] - b\ beta\^2\ Cos[theta0] + b\ beta\ Cos[theta0]\^3 + b\ beta\^2\ Cos[theta0]\^3)\)\)\/\(\(( a + 4\ b\ beta + 4\ b\ beta\^2)\)\ \((1 + 2\ beta - Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cumprob4\ = \ Apart[cumprob, b]\)], "Input"], Cell[BoxData[ \(1\/2\ \((1 - Cos[theta0])\) + \(a\ Cos[theta0] - a\ Cos[theta0]\^3\)\/\(2\ \((a + 4\ b\ beta + 4\ b\ beta\^2)\)\ \((1 + 2\ beta - Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[cumprob]\)], "Input"], Cell[BoxData[ \(\(-\(\(( \((\(-1\) + Cos[theta0])\)\ \((a + 4\ a\ beta + 4\ b\ beta + 4\ a\ beta\^2 + 20\ b\ beta\^2 + 32\ b\ beta\^3 + 16\ b\ beta\^4 + a\ Cos[theta0] - 4\ b\ beta\ Cos[theta0]\^2 - 4\ b\ beta\^2\ Cos[theta0]\^2)\)) \)/\((2\ \((a + 4\ b\ beta + 4\ b\ beta\^2)\)\ \((1 + 2\ beta - Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\)) \)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cumprob3 - cumprob4\ // Simplify\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["For b = 0 this gives", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", 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Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["cumprob1 = cumprob /. {b -> 1, a -> 2}", "Input", PageWidth->Infinity], Cell[BoxData[ \(\((2 + 12\ beta + 28\ beta\^2 + 32\ beta\^3 + 16\ beta\^4 - 10\ beta\ Cos[theta0] - 26\ beta\^2\ Cos[theta0] - 32\ beta\^3\ Cos[theta0] - 16\ beta\^4\ Cos[theta0] - 2\ Cos[theta0]\^2 - 4\ beta\ Cos[theta0]\^2 - 4\ beta\^2\ Cos[theta0]\^2 + 2\ beta\ Cos[theta0]\ \((\(-1\) + 2\ Cos[theta0]\^2)\) + 2\ beta\^2\ Cos[theta0]\ \((\(-1\) + 2\ Cos[theta0]\^2)\))\)/ \((2\ \((2 + 4\ beta + 4\ beta\^2)\)\ \((1 + 2\ beta - Cos[theta0])\)\ \((1 + 2\ beta + Cos[theta0])\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cumprob\ /. \ {beta\ -> \ 0.1, \ b\ -> \ 0, \ theta0\ -> \ 1}\ // \ Simplify\)], "Input"], Cell[BoxData[ \(0.396464367697200437`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cumprob\ /. \ {beta\ -> \ 0.1, \ b\ -> \ 1, \ a\ -> \ 1, \ theta0\ -> \ 1} \ // \ 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Evaluatable->False], Cell[BoxData[ \(cos1a\ = \ cos1\ \ \ //. \ {\ a\ -> \ 1, \ b\ -> 1, \ beta\ -> \ 0.1, randomnum\ -> \ 0.7} \)], "Input", Evaluatable->False], Cell[BoxData[ \(cos2\ = \ Cos[theta0]\ /. \ Part[rule1, 2]\)], "Input", Evaluatable->False], Cell[BoxData[ \(cos2a\ = \ cos2\ \ \ //. \ {a\ -> \ 1, \ b\ -> 1, \ beta\ -> \ 0.1, randomnum\ -> \ 0.7} \)], "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(cos3\ = \ Cos[theta0]\ /. \ Part[rule1, 3]\ \ // Simplify\)], "Input"], Cell[BoxData[ \(\(-\(1\/\(48\ b\ beta\ \((1 + beta)\)\)\(( 4\ \((a + 4\ b\ beta\ \((1 + beta)\))\)\ \((\(-1\) + 2\ randomnum)\) + 2\^\(2/3\)\ \((1 + I\ \@3)\)\ \((2\ \[Sqrt]\(( \((\(-48\)\ b\ beta\^2\ \((1 + beta)\)\^2\ \((a + b\ \((1 + 2\ beta)\)\^2)\) - \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((1 - 2\ randomnum)\)\^2)\)^3 + \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((\(-216\)\ b\^2\ beta\^2\ \((1 + beta)\)\^2\ \((1 + 2\ beta)\)\^2 + 72\ b\ beta\^2\ \((1 + beta)\)\^2\ \((a + b\ \((1 + 2\ beta)\)\^2)\) + \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((1 - 2\ randomnum)\)\^2)\)^2\ \((1 - 2\ randomnum)\)\^2)\) + 432\ b\^2\ beta\^2\ \((1 + beta)\)\^2\ \((1 + 2\ beta)\)\^2\ \((a + 4\ b\ beta\ \((1 + beta)\))\)\ \((\(-1\) + 2\ randomnum)\) - 144\ b\ beta\^2\ \((1 + beta)\)\^2\ \((a + 4\ b\ beta\ \((1 + beta)\))\)\ \((a + b\ \((1 + 2\ beta)\)\^2)\)\ \((\(-1\) + 2\ randomnum)\) - 2\ \((a + 4\ b\ beta\ \((1 + beta)\))\)\^3\ \((\(-1\) + 2\ randomnum)\)\^3)\)^\((1/3)\) - \((2\ \((1 - I\ \@3)\)\ \((\(-48\)\ b\ beta\^2\ \((1 + beta)\)\^2\ \((a + b\ \((1 + 2\ beta)\)\^2)\) - \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((1 - 2\ randomnum)\)\^2)\))\)/ \((\[Sqrt]\(( \((\(-48\)\ b\ beta\^2\ \((1 + beta)\)\^2\ \((a + b\ \((1 + 2\ beta)\)\^2)\) - \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((1 - 2\ randomnum)\)\^2)\)^3 + \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((\(-216\)\ b\^2\ beta\^2\ \((1 + beta)\)\^2\ \((1 + 2\ beta)\)\^2 + 72\ b\ beta\^2\ \((1 + beta)\)\^2\ \((a + b\ \((1 + 2\ beta)\)\^2)\) + \((a + 4\ b\ beta\ \((1 + beta)\))\)\^2\ \((1 - 2\ randomnum)\)\^2)\)^2\ \((1 - 2\ randomnum)\)\^2)\) - 12\ a\^2\ b\ beta\ \((1 + beta)\)\ \((6\ beta + 6\ beta\^2 + \((1 - 2\ randomnum)\)\^2)\)\ \((\(-1\) + 2\ randomnum)\) - a\^3\ \((\(-1\) + 2\ randomnum)\)\^3 + 96\ a\ b\^2\ beta\^2\ \((1 + beta)\)\^2\ \((\(-1\) + 2\ randomnum)\)\ \((1 + 3\ beta + 3\ beta\^2 + 2\ randomnum - 2\ randomnum\^2)\) + 256\ b\^3\ beta\^3\ \((1 + beta)\)\^3\ \((\(-1\) + 2\ randomnum)\)\ \((2 + 9\ beta + 9\ beta\^2 + randomnum - randomnum\^2)\)) \)^\((1/3)\))\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cos3a\ = \ cos3\ \ \ //. \ {a\ -> \ 1, \ b\ -> 1, \ beta\ -> \ 0.1, randomnum\ -> \ 0.7} \)], "Input"], Cell[BoxData[ \(\(-0.657187262243460601`\) + 3.3643121958338078`*^-16\ I\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ArcCos[Re[cos3a]]\)], "Input"], Cell[BoxData[ \(2.28787721699469681`\)], "Output"] }, Open ]], Cell["For b = 0 this gives", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["rule2 = Solve[cumprob0 == randomnum, Cos[theta0]]", "Input", PageWidth->Infinity], Cell[BoxData[ \({{Cos[theta0] \[Rule] \(1\/\(2\ \((\(-1\) + 2\ randomnum)\)\)\(( \(-\((\(-4\) - 4\ beta)\)\)\ beta - \[Sqrt]\(( \((\(-4\) - 4\ beta)\)\^2\ beta\^2 - 4\ \((\(-1\) + 2\ randomnum)\)\ \((1 + 4\ beta + 4\ beta\^2 - 2\ randomnum - 8\ beta\ randomnum - 8\ beta\^2\ randomnum)\))\)) \)\)}, {Cos[theta0] \[Rule] \(1\/\(2\ \((\(-1\) + 2\ randomnum)\)\)\(( \(-\((\(-4\) - 4\ beta)\)\)\ beta + \[Sqrt]\(( \((\(-4\) - 4\ beta)\)\^2\ beta\^2 - 4\ \((\(-1\) + 2\ randomnum)\)\ \((1 + 4\ beta + 4\ beta\^2 - 2\ randomnum - 8\ beta\ randomnum - 8\ beta\^2\ randomnum)\))\)) \)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["costheta0e = Cos[theta0] /. 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Part[rule2,2]", "Input", PageWidth->Infinity], Cell[BoxData[ \(\(1\/\(2\ \((\(-1\) + 2\ randomnum)\)\)\(( \(-\((\(-4\) - 4\ beta)\)\)\ beta + \[Sqrt]\(( \((\(-4\) - 4\ beta)\)\^2\ beta\^2 - 4\ \((\(-1\) + 2\ randomnum)\)\ \((1 + 4\ beta + 4\ beta\^2 - 2\ randomnum - 8\ beta\ randomnum - 8\ beta\^2\ randomnum)\))\))\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(costheta0e\ /. \ {beta\ -> \ 0.1, \ randomnum\ -> \ 0.3}\)], "Input"], Cell[BoxData[ \(0.770037878244408524`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(costheta0f\ /. \ {beta\ -> \ 0.1, \ randomnum\ -> \ 0.3}\)], "Input"], Cell[BoxData[ \(\(-1.8700378782444087`\)\)], "Output"] }, Open ]], Cell["\<\ This test suggests that only the first solution is of use to us.\ \>", "SmallText"], Cell["\<\ Determination of differential scattering cross section parameters from cross \ sections\ \>", "Subsection", Evaluatable->False], Cell["\<\ Now that we have increased the number of fitting parameters from two (a and \ beta) to three (a,b,and beta) we need to use all three cross sections to get \ a unique set of parameters. One problem is to avoid input cross sections \ that are inconsistent with the general form of our assumed differential \ scattering formula.\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["Elastic scattering of Ar by Ar", "Subsection", Evaluatable->False], Cell["\<\ From our tabulation of Ar-Ar cross sections and the equations used to fit the \ cross sections using QuatroPro we have\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["!! c:\\mathbook\\argon\\ararcollisions\\arqmqvqt.txt", "Input", PageWidth->Infinity], Cell["\<\ %Fits to Robinson's (1979) calculated Qm and Qv. As discussed in C:\\MATHBOOK\\ARGON\\MODSCRCL.MA we should force Qv/Qm to approach 2 at high energies. Robinson best Robinson best Amdur &diff.exp fit &visc.exp fit MM theor fit rel E Qm Qm Qv Qv Qt Qt eV m2 m2 m2 m2 m2 m2 % 0.01 5.41E-19 5.48E-19 4.48E-19 4.61E-19 6.91E-18 7.21E-18 0.02 4.61E-19 4.67E-19 3.82E-19 3.93E-19 6.02E-18 6.15E-18 0.05 3.74E-19 3.78E-19 3.1E-19 3.19E-19 5.01E-18 4.97E-18 0.1 3.19E-19 3.21E-19 2.64E-19 2.71E-19 4.36E-18 4.23E-18 0.2 2.72E-19 2.69E-19 2.25E-19 2.3E-19 3.8E-18 3.55E-18 0.5 2.2E-19 1.96E-19 1.82E-19 1.78E-19 3.16E-18 2.62E-18 1 7.1E-20 1.3E-19 1.12E-19 1.37E-19 1E-21 1.83E-18 2 6E-20 7.79E-20 8.5E-20 1.02E-19 1E-21 1.18E-18 5 4.5E-20 4.66E-20 7.2E-20 7.35E-20 1E-21 7.71E-19 10 3.7E-20 3.58E-20 6E-20 5.96E-20 1E-21 6.12E-19 20 2.9E-20 2.87E-20 4.4E-20 4.89E-20 1E-21 4.99E-19 50 2.1E-20 2.17E-20 3.7E-20 3.78E-20 1E-21 3.85E-19 100 1.7E-20 1.76E-20 3.1E-20 3.11E-20 1E-21 3.17E-19 200 1.4E-20 1.43E-20 2.5E-20 2.56E-20 2.5E-19 2.61E-19 500 1.1E-20 1.08E-20 2E-20 1.98E-20 2.1E-19 2.02E-19 1000 9E-21 8.81E-21 1.7E-20 1.63E-20 1.75E-19 1.66E-19 2000 1E-21 7.16E-21 1E-21 1.35E-20 1E-21 1.37E-19 5000 1E-21 5.44E-21 1E-21 1.04E-20 1E-21 1.06E-19 10000 1E-21 4.42E-21 1E-21 8.57E-21 1E-21 8.72E-20\ \>", "Print"] }, Open ]], Cell["\<\ Note that entries of 1E-21 are used instead of a blank entry or zero so that \ our plotting program ProPlot will work for log plots.\ \>", "SmallText", Evaluatable->False], Cell["\<\ Read in the data from this file and prepare plots for use near the end of \ this notebook\ \>", "Text"], Cell["\<\ inputfile := \ \"c:\\\\mathbook\\\\argon\\\\ararcollisions\\\\arqmqvqt.txt\"; stream = OpenRead[inputfile]; Find[stream,\"%\"]; Find[stream,\"%\"]; data1 = ReadList[stream, Number, RecordLists ->True]; Close[stream]; enrel2 = Part[Transpose[data1],1]; qd2 = Part[Transpose[data1],2]*10^20; qdbfit2 = Part[Transpose[data1],3]*10^20; qv2 = Part[Transpose[data1],4]*10^20; qvbfit2 = Part[Transpose[data1],5]*10^20; qt2 = Part[Transpose[data1],6]*10^20; qtbfit2 = Part[Transpose[data1],7]*10^20; dataqd = Transpose[{enrel2,qd2}]; dataqv = Transpose[{enrel2,qv2}]; dataqt = Transpose[{enrel2,qt2}]; bfitqd = Transpose[{enrel2,qdbfit2}]; bfitqv = Transpose[{enrel2,qvbfit2}]; bfitqt = Transpose[{enrel2,qtbfit2}]; dataqdplot = LogLogListPlot[Part[dataqd,Range[7,16]], \ (*Robinson qd - isotope*) PlotStyle ->{PointSize[0.025],Hue[0.55]}, PlotRange -> {{0.01,10000},{0.1,1000}}, (*PlotRange -> {0.1,1000},*) DisplayFunction ->Identity]; dataqvplot = LogLogListPlot[Part[dataqv,Range[7,16]], \ (*Robinson qv*) PlotStyle ->{PointSize[0.025],Hue[1]}, PlotRange -> {{0.01,10000},{0.1,1000}}, DisplayFunction -> Identity]; dataqtplot = LogLogListPlot[Part[dataqt,Range[14,16]], (*Amdur \ qt*) PlotStyle ->{PointSize[0.025],Hue[0.4]}, PlotRange -> {{0.01,10000},{0.1,1000}}, DisplayFunction -> Identity]; bfitqdplot = LogLogListPlot[bfitqd, (*qd \ best fit QPro*) PlotJoined -> True, PlotStyle -> {Hue[0.55],Dashing[{0.04,0.02}]}, PlotRange -> {{0.01,1000},{0.1,1000}}, DisplayFunction ->Identity]; bfitqvplot = LogLogListPlot[bfitqv, (*qv \ best fit QPro*) PlotJoined ->True, PlotStyle -> {Hue[1],Dashing[{0.04,0.02}]}, PlotRange -> {{0.01,10000},{0.1,1000}}, DisplayFunction -> Identity]; bfitqtplot = LogLogListPlot[bfitqt, (*qt \ best fit QPro*) PlotJoined -> True, PlotStyle -> {Hue[.4],Dashing[{0.04,0.02}]}, PlotRange -> {{0.01,1000},{0.1,1000}}, DisplayFunction ->Identity]; \ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell[BoxData[ \(dataqd\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ StyleBox["0.01`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "54.1000000000000014`"}], "}"}], ",", RowBox[{"{", RowBox[{ StyleBox["0.02`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "46.1000000000000031`"}], "}"}], ",", RowBox[{"{", RowBox[{ StyleBox["0.05`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "37.4000000000000065`"}], "}"}], ",", RowBox[{"{", RowBox[{ 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Open ]], Cell["\<\ Experimental diffusion (momentum transfer) cross section qd for isotopic \ Ar.\ \>", "Subsubsection", Evaluatable->False], Cell["\<\ Because of the symmetry in Ar-Ar collisions of a single isotope, Cris Greene \ points out that the total and diffusion cross sections are the same. We \ therfore limit qd to the diffusion of one isotope of Ar through a different \ isotope of Ar. The cross section then should pass through the self-diffusion \ data at low energies and to pass through the diffusion cross section \ calculated by Robinson at high energies.\ \>", "SmallText"], Cell["\<\ The following analytic empirical expression for qd is obtained by passing a \ smooth curve through the qd values derived in notebook ARTHERM.MA from \ thermal transport data at low energies and the momentum transfer cross \ sections calculated by Robinson (1978) in the higher energy region. Note \ these cross sections have been drawn through the average of the oscillations \ predicted by theory and found in experiment.\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ Note that in the JPCRD reference I assumed that the data of Robinson's Figs. \ 3 and 4 are plotted versus laboratory energy, while in the J. Appl. Phys. \ reference I assumed that they are plotted against relative energy. I have \ not been able to get a reply from Robinson, and will assume that they are \ plotted against relative energy.\ \>", "SmallText"], Cell["\<\ qdiso := 1.9*10^(-19)*enrel^(-0.23)/(1+enrel^2) + 7.*10^(-20)*enrel^(-0.3)/(1+enrel^(-2)); \ (*m^2*)\ \>", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["\<\ LogLogPlot[1.*^20*qdiso,{enrel,1,10000},PlotRange -> \ {{0.01,10000},{0.1,100}}];\ \>", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % 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They probably are caused \ by too large a ratio of qv to qt.\ \>", "SmallText"], Cell["\<\ outputbetaplot = LogLinearListPlot[Transpose[{energies,outputbeta}], PlotRange -> {{0.01,10000},{0,0.012}},PlotStyle -> \ PointSize[.03], AxesLabel -> {\"rel. energy (eV)\",\"beta\"}, DisplayFunction ->Identity];\ \>", "Input", PageWidth->Infinity], Cell["\<\ outputaplot = LogLinearListPlot[Transpose[{energies,outputa}], PlotRange -> {{0.01,10000},{0,0.5}},PlotStyle -> \ PointSize[.03], AxesLabel -> {\"rel. energy (eV)\",\"a\"}, DisplayFunction ->Identity];\ \>", "Input", PageWidth->Infinity], Cell["\<\ outputbplot = LogLinearListPlot[Transpose[{energies,outputb}], PlotRange -> {{0.01,10000},{0,5}},PlotStyle -> \ PointSize[.03], AxesLabel -> {\"rel. energy (eV)\",\"b\"}, DisplayFunction ->Identity];\ \>", "Input", PageWidth->Infinity], Cell["\<\ Empirical expressions for the parameters beta[enrel], a, and b are\ \>", "SmallText", Evaluatable->False, FontFamily->"Helv", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->RGBColor[0.501961, 1, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["We make the following choice of beta[enrel] . 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Cell[CellGroupData[{ Cell[BoxData[ \(qv\)], "Input"], Cell[BoxData[ \(\(-8\)\ a\ \[Pi] + \(8\ b\ \[Pi]\)\/3 + 4\ a\ \((1 + 2\ beta)\)\ \[Pi]\ Log[\(1 + beta\)\/beta]\)], "Output"] }, Open ]], Cell["From our fits to a, b, and beta", "Text"], Cell["\<\ qvfitlist = Table[(qv /. {a -> Part[empiricalalist,j],b -> \ Part[empiricalblist,j], beta -> Part[empiricalbetalist,j]}), {j,1,Length[energies]}];\ \>", "Input", PageWidth->Infinity], Cell["\<\ qvfitlistplot = LogLogListPlot[Transpose[{energies,qvfitlist}], \ PlotJoined -> True, PlotStyle -> Hue[0.55], PlotRange -> {0.1,1000}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ From our fit to Robinson's calculated momentum transfer (diffusion cross) \ sections\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ qdisolist = Table[10^20*(qdiso /. enrel -> Part[energies,j]), {j,1,Length[energies]}]\ \>", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \({54.7939059190074928`, 42.0265016613646125`, 32.0854433255653104`, 23.4081318834814666`, 13.0000000000000004`, 5.83055210952401914`, 3.58434745535491838`, 2.48978919837977219`, 1.75880342173039299`, 1.24483369246645758`, 0.88125078630722573`, 0.623875891988736075`, 0.441670159562457964`}\)], "Output"] }, Open ]], Cell["\<\ qdisolistplot = LogLogListPlot[Transpose[{energies,qdisolist}], \ PlotJoined -> True, PlotRange -> {0.1,1000}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["From our fits to a, b, and beta", "Text"], Cell[CellGroupData[{ Cell["\<\ qdfitlist = Table[(qd /. {a -> Part[empiricalalist,j],b -> \ Part[empiricalblist,j], beta -> Part[empiricalbetalist,j]}), {j,1,Length[energies]}]\ \>", "Input", PageWidth->Infinity], Cell[BoxData[ \({55.0745286651801002`, 42.2527025338504369`, 32.3462306856819603`, 24.2205934095300978`, 15.2694675985416283`, 6.90384780652185447`, 4.00483138665432925`, 2.80707824007170181`, 2.02600968601750697`, 1.46712243399638193`, 1.06278994139308302`, 0.769919634026741661`, 0.557757187025994394`}\)], "Output"] }, Open ]], Cell["\<\ qdfitlistplot = LogLogListPlot[Transpose[{energies,qdfitlist}], \ PlotJoined -> True, PlotStyle -> Hue[0.35], PlotRange -> {0.1,1000}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ From our fit to theory and Amdur and Mason's total cross sections\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ qtdatalist = Table[10^20*(qtdatafit /. enrel -> Part[energies,j]), {j,1,Length[energies]}]\ \>", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \({502.368803397874863`, 398.955987825019064`, 316.009820116713369`, 243.326819456151532`, 157.500000000000017`, 90.1788182755541001`, 61.0047253640437858`, 43.7782397838062031`, 31.6784243893610462`, 22.9459196184779612`, 16.6225909787158148`, 12.0419810889118661`, 8.72364234257782733`}\)], "Output"] }, Open ]], Cell["\<\ qtdatalistplot = LogLogListPlot[Transpose[{energies,qtdatalist}], \ PlotJoined -> True, PlotRange -> {0.1,1000}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["From our fits to a, b, and beta", "Text"], Cell["\<\ qtfitlist = Table[(qt /. {a -> Part[empiricalalist,j],b -> \ Part[empiricalblist,j], beta -> Part[empiricalbetalist,j]}), {j,1,Length[energies]}];\ \>", "Input", PageWidth->Infinity], Cell["\<\ qtfitlistplot = LogLogListPlot[Transpose[{energies,qtfitlist}], \ PlotJoined -> True, PlotStyle -> {Hue[1], Dashing[{0.03,0.03}]}, PlotRange -> {0.1,1000}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ thermqdplot = LogLogListPlot[qdrlist, PlotStyle -> {Hue[.65], PointSize[0.025]}, PlotRange -> {{0.01,10000},{0.1,1000}}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ thermqvplot = LogLogListPlot[qvslist, PlotStyle -> {Hue[0.1], PointSize[0.025]}, PlotRange -> {{0.01,10000},{0.1,1000}}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ The following is from the recent resuts of Serikov and Nanbu, J. Vac. Sci. \ Technol. A 14, 3108 (1996). We multiply their isotropic scattering (VHS) \ cross section (in 10^-20 m^2) for Ar-Ar collisions from K. Nanbu (private \ communication) by 2/3 and show their resultant viscosity cross section by the \ dashed black line in the graph below is given by\ \>", "Text"], Cell[BoxData[ \(qvhs\ := \ 2/3*60.3*\((0.0235/enrel)\)^0.311\)], "Input"], Cell["\<\ We would expect this viscosity cross section gives a very good fit to our \ viscosity cross section over the whole energy range.\ \>", "Text"], Cell["\<\ qvhsvalues = Table[(qvhs /. enrel -> Part[energies,j]), {j,1,Length[energies]}];\ \>", "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ qvhsplot = LogLogListPlot[Transpose[{energies,qvhsvalues}], \ PlotJoined -> True, PlotStyle -> Dashing[{0.03,0.03}], (* PlotStyle -> Hue[0.55],*) PlotRange -> {0.1,1000}, DisplayFunction -> Identity];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ The following is a calculation of the total cross section based on the theory \ of Massey and others as described by Bernstein. It calculates the forward \ scattering based on quantum mechanics for an inverse power law potential. We \ apply it to Ar-Ar collisions using the long range portion of the potential \ given by Barker, Fisher, and Watts, Molecular Physics 21, 657 (1971). See \ the notebook VANDERWALLS.NB. The result is \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(qtth\ = \ 1.924*^-18\/urev\^\(1/5\)\)], "Input"], Cell[BoxData[ \(1.92400000000000019`*^-18\/urev\^\(1/5\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(qtthtable\ = \ qtth\ /. \ urev\ -> \ thermenergies\)], "Input"], Cell[BoxData[ \({4.2072572605337335`*^-18, 3.50276552662217799`*^-18, 3.04933450229518232`*^-18, 2.6545998686513772`*^-18, 2.21009563501429528`*^-18}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[{thermenergies, 10^20*qtthtable}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ StyleBox["0.02`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "420.725726053373438`"}], "}"}], ",", RowBox[{"{", RowBox[{ StyleBox["0.05`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "350.276552662217888`"}], "}"}], ",", RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "304.933450229518321`"}], "}"}], ",", 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"Input"], Cell[BoxData[ \(\(jmax\ = \ 60; \)\)], "Input"], Cell[BoxData[ \(\(\ \(itheta001 = \ Table[\((\(f[j*Pi/jmax]\ /. \ rules\)\ /. \ enrel\ \ -> \ 0.01)\), { \ j, 0, jmax}]; \)\)\)], "Input"], Cell[BoxData[ \(\(\ \(itheta01 = \ Table[\((\(f[j*Pi/jmax]\ /. \ rules\)\ /. \ enrel\ \ -> \ 0.1)\), {\ j, 0, jmax}]; \)\)\)], "Input"], Cell[BoxData[ \(\(\ \(itheta1 = \ Table[\((\(f[j*Pi/jmax]\ /. \ rules\)\ /. \ enrel\ \ -> \ 1)\), {\ j, 0, jmax}]; \)\)\)], "Input"], Cell[BoxData[ \(\(\ \(itheta10 = \ Table[\((\(f[j*Pi/jmax]\ /. \ rules\)\ /. \ enrel\ \ -> \ 10)\), {\ j, 0, jmax}]; \)\)\)], "Input"], Cell[BoxData[ \(\(\ \(itheta100 = \ Table[\((\(f[j*Pi/jmax]\ /. \ rules\)\ /. \ enrel\ \ -> \ 100)\), {\ j, 0, jmax}]; \)\)\)], "Input"], Cell[BoxData[ \(\(\ \(itheta1000 = \ Table[\((\(f[j*Pi/jmax]\ /. \ rules\)\ /. \ enrel\ \ -> \ 1000)\), { \ j, 0, jmax}]; \)\)\)], "Input"], Cell[BoxData[ \(\(angles\ = \ Table[180/jmax*j, {j, 0, jmax}]; \)\)], "Input"], Cell[BoxData[ \(\(ithetatable\ = \ Transpose[{angles, itheta001, itheta01, itheta1, itheta10, itheta100, itheta1000}]; \)\)], "Input"], Cell["\<\ ithetatableout = TableForm[ithetatable,TableHeadings -> {None, {\"%theta\",\"0.01 eV\",\"0.1 eV\",\"1 eV\",\"10 \ eV\",\"100 eV\",\"1000 eV\"}}, TableSpacing -> {0,1}];\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ Block[ {stream1,theoryfile,today}, SetOptions[$Output,PageWidth->120]; outputfile := \"c:\\\\proplot\\\\arctprop\\\\itheta.dat\"; stream1 = OpenWrite[outputfile,PageWidth -> Infinity]; WriteString[stream1, \"%\", StringForm[\"Diffferential scattering I(theta)\"], \"\\n\", StringForm[\"Our fit to experiment and theory\"], \"\\n\", ToString[ithetatableout]]; Close[stream1]; SetOptions[$Output,PageWidth->53]; ]\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["!!c:\\proplot\\arctprop\\itheta.dat", "Input", PageWidth->Infinity], Cell["\<\ %Diffferential scattering I(theta) Our fit to experiment and theory %theta 0.01 eV 0.1 eV 1 eV 10 eV 100 eV 1000 eV 0 9376.63 5396.02 1148.45 258.841 133.44 70.017 3 5214.4 3027.42 860.187 223.605 115.519 60.6151 6 1681.11 987.091 437.03 152.358 79.0959 41.5052 9 570.85 337.257 200.418 91.5089 47.7525 25.0592 12 229.618 136.074 95.5369 52.9963 27.7719 14.5745 15 107.868 64.0197 49.2538 31.0726 16.3327 8.57158 18 57.5575 34.1805 27.537 18.8246 9.91508 5.20365 21 34.122 20.2629 16.5781 11.8548 6.25194 3.2812 24 22.1172 13.1277 10.6586 7.75814 4.09403 2.14867 27 15.4832 9.18242 7.26506 5.26199 2.77702 1.45747 30 11.5851 6.86322 5.21771 3.68661 1.9448 1.02068 33 9.1759 5.42945 3.92743 2.65934 1.4016 0.735588 36 7.62293 4.50502 3.08327 1.96932 1.03646 0.543948 39 6.58566 3.88746 2.51291 1.49335 0.784447 0.411679 42 5.87154 3.46224 2.11665 1.15717 0.606357 0.318208 45 5.36695 3.16175 1.8346 0.914652 0.47784 0.250755 48 5.00231 2.94458 1.62956 0.736417 0.383357 0.201165 51 4.73365 2.78457 1.47776 0.603259 0.312751 0.164107 54 4.5324 2.66469 1.36358 0.502345 0.25923 0.136016 57 4.3795 2.57361 1.27653 0.424929 0.218165 0.114463 60 4.26198 2.50361 1.20944 0.36494 0.186338 0.0977581 63 4.17084 2.44931 1.15728 0.318093 0.161481 0.0847115 66 4.09972 2.40694 1.1165 0.281326 0.141969 0.0744707 69 4.04407 2.37379 1.08454 0.252419 0.126628 0.0664187 72 4.00062 2.3479 1.05955 0.229755 0.114599 0.060105 75 3.96697 2.32785 1.04017 0.212146 0.105252 0.0551993 78 3.94139 2.31261 1.02543 0.198723 0.0981268 0.0514595 81 3.92263 2.30143 1.0146 0.188855 0.0928885 0.0487101 84 3.90979 2.29379 1.0072 0.182097 0.0893012 0.0468272 87 3.90232 2.28933 1.00288 0.178155 0.0872082 0.0457287 90 3.89986 2.28787 1.00146 0.176859 0.0865203 0.0453676 93 3.90232 2.28933 1.00288 0.178155 0.0872082 0.0457287 96 3.90979 2.29379 1.0072 0.182097 0.0893012 0.0468272 99 3.92263 2.30143 1.0146 0.188855 0.0928885 0.0487101 102 3.94139 2.31261 1.02543 0.198723 0.0981268 0.0514595 105 3.96697 2.32785 1.04017 0.212146 0.105252 0.0551993 108 4.00062 2.3479 1.05955 0.229755 0.114599 0.060105 111 4.04407 2.37379 1.08454 0.252419 0.126628 0.0664187 114 4.09972 2.40694 1.1165 0.281326 0.141969 0.0744707 117 4.17084 2.44931 1.15728 0.318093 0.161481 0.0847115 120 4.26198 2.50361 1.20944 0.36494 0.186338 0.0977581 123 4.3795 2.57361 1.27653 0.424929 0.218165 0.114463 126 4.5324 2.66469 1.36358 0.502345 0.25923 0.136016 129 4.73365 2.78457 1.47776 0.603259 0.312751 0.164107 132 5.00231 2.94458 1.62956 0.736417 0.383357 0.201165 135 5.36695 3.16175 1.8346 0.914652 0.47784 0.250755 138 5.87154 3.46224 2.11665 1.15717 0.606357 0.318208 141 6.58566 3.88746 2.51291 1.49335 0.784447 0.411679 144 7.62293 4.50502 3.08327 1.96932 1.03646 0.543948 147 9.1759 5.42945 3.92743 2.65934 1.4016 0.735588 150 11.5851 6.86322 5.21771 3.68661 1.9448 1.02068 153 15.4832 9.18242 7.26506 5.26199 2.77702 1.45747 156 22.1172 13.1277 10.6586 7.75814 4.09403 2.14867 159 34.122 20.2629 16.5781 11.8548 6.25194 3.2812 162 57.5575 34.1805 27.537 18.8246 9.91508 5.20365 165 107.868 64.0197 49.2538 31.0726 16.3327 8.57158 168 229.618 136.074 95.5369 52.9963 27.7719 14.5745 171 570.85 337.257 200.418 91.5089 47.7525 25.0592 174 1681.11 987.091 437.03 152.358 79.0959 41.5052 177 5214.4 3027.42 860.187 223.605 115.519 60.6151 180 9376.63 5396.02 1148.45 258.841 133.44 70.017\ \>", "Print"] }, Open ]], Cell["\<\ Differential scattering cross section for Ar-Ar from Parson, Siska, and Lee, \ J. Chem. Phys. 56, 1511 (1972) for comparison with our fit.\ \>", "Subsection"], Cell["\<\ This notebook imports a graph read by the HP scanner. It then allows me to \ read off points and prepare an output file of the points read.\ \>", "Subsubsection", Evaluatable->False], Cell["\<\ One way to scan and import the image to be analyzed is to: 1) enlarge image as much as possible with copier if it can be done without \ distortion 2) use the HP Deskscan II to Preview the image 3) select the portion of the image to be analyzed 4) use Edit, then Copy to put the the image in the Clipboard 5) start the desired Mathematica notebook 6) in Mathematica use Edit, then Paste and Discard to place the image in the \ notebook. This procedure does not allow rotation of the image.\ \>", "SmallText", Evaluatable->False], Cell["\<\ Alternatively one can 1) use Image Assistant to scan and rotate the image. 2) the image can be resized if needed 3) remove grey using Bilevel under Mode. 4) save as a *.tif file without thumbnail plot. This is a bitmap file. 5) use Import under File in Mathematica. This creates a new notebook. Use \ Edit, Copy, Paste and Discard to move image to desired notebook. 6) resize the image in Mathematica.\ \>", "SmallText", Evaluatable->False, FontFamily->"Arial", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ To read the points from the graph 1) enlarge the graph as much as possible while keeping all of image on \ screen. 2) move mouse pointer to first point and control-click on the desired data \ point. If one then holds control and clicks at the desired points in succession, the final list will be in the correct list \ format. Release control. 3) select Copy from the Edit menu 4) select the text insertion point and click on Paste and Discard from the \ Edit menu 5) note that it seems to work better if one uses only two points for the \ xscale and two for the yscale 6) after reading points make the graph cells inactive\ \>", "SmallText", Evaluatable->False, FontFamily->"Arial", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["", "Graphics", PageWidth->Infinity, Evaluatable->False, ImageSize->{652, 466}, ImageMargins->{{28, Inherited}, {Inherited, 0}}, FontFamily->"Courier New", FontSize->8, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["Differential scattering cross section", "Section", Evaluatable->False], Cell["Data 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oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00 oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00 oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00oolQo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-10.5063, -1.43421, 0.352895, 0.0342598}}] }, Open ]], Cell["\<\ This conparison suggests that our choice of cross section at low energies is \ reasonable.\ \>", "Text"], Cell["Export data to file for use with ProPlot", "Subsection", Evaluatable->False], Cell["Parameters and and resultant cross sections. ", "Subsubsection"], Cell[BoxData[ \(\(fittedqdqvqt\ = \ Transpose[{energies, empiricalbetalist, empiricalalist, empiricalblist, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ qvfitlist, qdfitlist, qtfitlist}]; \)\)], "Input"], Cell[BoxData[ \(\(fittedresult\ = \ Map[ToString[SetPrecision[#, 4.2], FormatType -> FortranForm]&, fittedqdqvqt, {2}]; \)\)], "Input"], Cell[CellGroupData[{ Cell["\<\ fitteddataout1 = TableForm[fittedresult,TableHeadings -> {None, {\"%rel. en.\",\"beta\",\"a\",\"b\",\"Qv\", \ \"Qd\",\"Qt\"}}, TableSpacing -> {0,1}]\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ TagBox[GridBox[{ {"\<\"%rel. en.\"\>", "\<\"beta\"\>", "\<\"a\"\>", "\<\"b\"\>", "\<\"Qv\"\>", "\<\"Qd\"\>", "\<\"Qt\"\>"}, {"\<\"0.01000\"\>", "\<\"0.002000\"\>", "\<\"0.3000\"\>", "\<\"3.600\"\>", "\<\"46.15\"\>", "\<\"55.1\"\>", "\<\"516.\"\>"}, {"\<\"0.03162\"\>", "\<\"0.002004\"\>", "\<\"0.2305\"\>", "\<\"2.761\"\>", "\<\"35.41\"\>", "\<\"42.25\"\>", "\<\"395.4\"\>"}, {"\<\"0.1000\"\>", "\<\"0.002036\"\>", "\<\"0.1788\"\>", "\<\"2.109\"\>", "\<\"27.16\"\>", "\<\"32.35\"\>", "\<\"301.9\"\>"}, {"\<\"0.3162\"\>", "\<\"0.002340\"\>", "\<\"0.1500\"\>", "\<\"1.548\"\>", "\<\"20.67\"\>", "\<\"24.22\"\>", "\<\"220.4\"\>"}, {"\<\"1.000\"\>", "\<\"0.004365\"\>", "\<\"0.1749\"\>", "\<\"0.827\"\>", "\<\"14.59\"\>", "\<\"15.27\"\>", "\<\"135.7\"\>"}, {"\<\"3.162\"\>", "\<\"0.00785\"\>", "\<\"0.2032\"\>", "\<\"0.1570\"\>", "\<\"8.80\"\>", "\<\"6.90\"\>", "\<\"82.6\"\>"}, {"\<\"10.00\"\>", "\<\"0.00887\"\>", "\<\"0.1627\"\>", "\<\"0.01413\"\>", "\<\"5.88\"\>", "\<\"4.005\"\>", "\<\"57.3\"\>"}, {"\<\"31.62\"\>", "\<\"0.00899\"\>", "\<\"0.1192\"\>", "\<\"0.001103\"\>", "\<\"4.212\"\>", "\<\"2.807\"\>", "\<\"41.31\"\>"}, {"\<\"100.0\"\>", "\<\"0.00900\"\>", "\<\"0.0864\"\>", "\<\"0.0000848\"\>", "\<\"3.047\"\>", "\<\"2.026\"\>", "\<\"29.91\"\>"}, {"\<\"316.2\"\>", "\<\"0.00900\"\>", "\<\"0.0626\"\>", "\<\"6.51e-6\"\>", "\<\"2.207\"\>", "\<\"1.467\"\>", "\<\"21.67\"\>"}, {"\<\"1000.\"\>", "\<\"0.00900\"\>", "\<\"0.04537\"\>", "\<\"4.995e-7\"\>", "\<\"1.599\"\>", "\<\"1.063\"\>", "\<\"15.69\"\>"}, {"\<\"3162.\"\>", "\<\"0.00900\"\>", "\<\"0.03287\"\>", "\<\"3.833e-8\"\>", "\<\"1.158\"\>", "\<\"0.770\"\>", "\<\"11.37\"\>"}, {"\<\"1.000e4\"\>", "\<\"0.00900\"\>", "\<\"0.02381\"\>", "\<\"2.941e-9\"\>", "\<\"0.839\"\>", "\<\"0.558\"\>", "\<\"8.24\"\>"} }, RowSpacings->0, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableHeadings -> {None, {"%rel. en.", "beta", "a", "b", "Qv", "Qd", "Qt"}}, TableSpacing -> {0, 1}]&)]], "Output"] }, Open ]], Cell["\<\ Block[ {stream1,theoryfile,today}, SetOptions[$Output,PageWidth->120]; outputfile := \"c:\\\\proplot\\\\arctprop\\\\ararxsec.dat\"; stream1 = OpenWrite[outputfile,PageWidth -> Infinity]; WriteString[stream1, \"%\", StringForm[\"Diffusion, viscosity, and forward peak \ cross sections\"], \"\\n\", ToString[fitteddataout1]] ; Close[stream1]; SetOptions[$Output,PageWidth->53]; ]\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["!!c:\\proplot\\arctprop\\ararxsec.dat", "Input", PageWidth->Infinity], Cell["\<\ %Diffusion, viscosity, and forward peak cross sections %rel. en. beta a b Qv Qd Qt 0.01000 0.002000 0.3000 3.600 46.15 55.1 516. 0.03162 0.002004 0.2305 2.761 35.41 42.25 395.4 0.1000 0.002036 0.1788 2.109 27.16 32.35 301.9 0.3162 0.002340 0.1500 1.548 20.67 24.22 220.4 1.000 0.004365 0.1749 0.827 14.59 15.27 135.7 3.162 0.00785 0.2032 0.1570 8.80 6.90 82.6 10.00 0.00887 0.1627 0.01413 5.88 4.005 57.3 31.62 0.00899 0.1192 0.001103 4.212 2.807 41.31 100.0 0.00900 0.0864 0.0000848 3.047 2.026 29.91 316.2 0.00900 0.0626 6.51e-6 2.207 1.467 21.67 1000. 0.00900 0.04537 4.995e-7 1.599 1.063 15.69 3162. 0.00900 0.03287 3.833e-8 1.158 0.770 11.37 1.000e4 0.00900 0.02381 2.941e-9 0.839 0.558 8.24\ \>", "Print"] }, Open ]], Cell["Diffferential cross sections ", "Subsubsection"], Cell[BoxData[ \(\(ithetadatafunction\ = \ Interpolation[ithetadataout]; \)\)], "Input"], Cell[BoxData[ \(\(ithetadatatable\ = \ Table[ithetadatafunction[2*j], {j, 2, 44/2}]; \)\)], "Input"], Cell[BoxData[ \(\(thetatable\ = \ Table[2*j, {j, 2, 44/2}]; \)\)], "Input"], Cell[BoxData[ \(\(ithetafittable\ = \ \(i\ \ //. \ \ {\ beta\ -> \ empiricalbeta, \ a\ -> \ empiricala, \ b\ -> \ empiricalb\ }\)\ //. \ \ \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { enrel\ \ -> \ 0.03, \ theta\ -> \ thetatable*Pi/90}; \)\)], "Input"], Cell[BoxData[ \(\(fittedithetaresult\ = \ Transpose[{thetatable, ithetadatatable, ithetafittable}]; \)\)], "Input"], Cell[CellGroupData[{ Cell["\<\ fittedithetadataout = TableForm[fittedithetaresult,TableHeadings -> {None, {\"%theta\",\"i expt.\",\"i fit\"}}, TableSpacing -> {0,1}]\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ TagBox[GridBox[{ {"\<\"%theta\"\>", "\<\"i expt.\"\>", "\<\"i fit\"\>"}, {"4", "80.5458866670324979`", "620.84182183851686`"}, {"6", "55.3777180867882279`", "177.267306392614294`"}, {"8", "55.6257397895800753`", "66.6669177948990654`"}, {"10", "31.281047680877454`", "31.0266073423358434`"}, {"12", "16.0988273872905001`", "17.0809845486993854`"}, {"14", "8.6750554490010483`", "10.7870011189048731`"}, {"16", "6.01587804237450729`", "7.6246139658988028`"}, {"18", "4.66875873942566776`", "5.89832991932403238`"}, {"20", "3.8937458298110128`", "4.89206716646155026`"}, {"22", "3.69931003488864718`", "4.27362161161104214`"}, {"24", "3.44860746968234099`", "3.87672825913985885`"}, {"26", "3.41712779625315787`", "3.61281344398487735`"}, {"28", "3.21255388658091511`", "3.43217047087984816`"}, {"30", "3.02135902178967441`", "3.30565642926068825`"}, {"32", "2.80838323591085483`", "3.21554487045245407`"}, {"34", "2.97766283697427613`", "3.15072131969817981`"}, {"36", "2.94717224074570527`", "3.10405058779656517`"}, {"38", "3.05757287567015634`", "3.07088203946276205`"}, {"40", "2.86223032287814014`", "3.0481770582719827`"}, {"42", "2.82927280750063303`", "3.033991187476337`"}, {"44", "2.8023311146461328`", "3.02716807732830517`"} }, RowSpacings->0, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableHeadings -> {None, {"%theta", "i expt.", "i fit"}}, TableSpacing -> {0, 1}]&)]], "Output"] }, Open ]], Cell["\<\ Block[ {stream1,theoryfile,today}, SetOptions[$Output,PageWidth->120]; outputfile := \"c:\\\\proplot\\\\arctprop\\\\parson.dat\"; (*DeleteFile[outputfile];*) stream1 = OpenWrite[outputfile,PageWidth -> Infinity]; WriteString[stream1, \"%\", StringForm[\"Diffferential scattering I(theta)\"], \"\\n\", StringForm[\"Parson,Sioska, and Lee, J. Chem. Phys. 56, \ 1511 (1972)\"], \"\\n\", ToString[fittedithetadataout]]; Close[stream1]; SetOptions[$Output,PageWidth->53]; ]\ \>", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->11, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["!!c:\\proplot\\arctprop\\parson.dat", "Input", PageWidth->Infinity], Cell["\<\ %Diffferential scattering I(theta) Parson,Sioska, and Lee, J. Chem. Phys. 56, 1511 (1972) %theta i expt. i fit 4 80.5459 620.842 6 55.3777 177.267 8 55.6257 66.6669 10 31.281 31.0266 12 16.0988 17.081 14 8.67506 10.787 16 6.01588 7.62461 18 4.66876 5.89833 20 3.89375 4.89207 22 3.69931 4.27362 24 3.44861 3.87673 26 3.41713 3.61281 28 3.21255 3.43217 30 3.02136 3.30566 32 2.80838 3.21554 34 2.97766 3.15072 36 2.94717 3.10405 38 3.05757 3.07088 40 2.86223 3.04818 42 2.82927 3.03399 44 2.80233 3.02717\ \>", "Print"] }, Open ]], Cell["\<\ Data from Cavalini, Gallinaro,Meneghetti, and Valbusa, Chem. Phys. Lett. 7, \ 303 (1970) for comparison with our fit.\ \>", "Subsection"], Cell["\<\ These data are relative differential cross sections only at 420 K and 655 K. \ Other temperature data are given in the lettter.\ \>", "SmallText"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoaH05OlC00?o EP1Zo`007_l<0003o`3o01;o1P02o`800_oo0?l09P1Zo`007ol6033o2`03 o`D01Ol?00[o4P0Coa001_l200?o0`000ol0000I00;o1@06ojd00_mW00[o 1`05o`D05_l401Oo0`0?o`P0Jol0023o1@0ao`8000Co00006ol201[o0P0J o`<06_l201[o0P0Jo`806_l301[o0`0Io`806_l301[o0P0Jo`806Ol301_o 0P0Ko`030?oo01So0`0Ho`806Ol00`3oo`0?o`<0Kol0023o100Ao`030?oo 027o0`0Ko`030?oo01Wo0P0Ko`806_l201[o0P0Jo`806_l303Ko0P0Ko`80 6_l203Ko00<0ool06_l203Oo00<0ool06?l00`3oo`0Go`030?oo087o000Q o`<0=Ol301_o0P0Jo`806ol00`3oo`0Io`806_l201[o0P0Jo`<0=_l201_o 00<0ool06Ol205?o0P0go`030?oo01So00<0ool05ol00`3oo`21o`008Ol3 03Go0`0Ko`806_l201_o0P0Jo`806_l201[o0P0Jo`<0=_l201_o00<0ool0 6Ol205?o0P0go`030?oo01Oo0P0Io`030?oo087o000Qo`<0=Ol301_o0P0J 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0.882041}, \ \n{0.375542, \ 0.882041}, \ {0.3948, \ 0.876264}, \ \n{0.414059, \ 0.866635}, \ {0.433317, \ 0.857005}, \ \n{0.475686, \ 0.831969}, \ {0.516553, \ 0.79923}, \ \n{0.554646, \ 0.760713}, \ {0.591237, \ 0.727973}, \ \n{0.668272, \ 0.650939}, \ {0.745306, \ 0.606644}, \ \n{0.812711, \ 0.55272}}; \)\)], "Input"], Cell["Determine relation of screen scale to graph scale for x axis", "Subsubsection", Evaluatable->False], Cell[BoxData[ \(\(x2data\ = \ {{0.121329, \ 0.182956}, \ {0.818488, \ 0.182956}}; \)\)], "Input"], Cell[BoxData[ \(\(x2graph\ = \ {2, 20}; \)\)], "Input"], Cell["x2data1 = Flatten[Take[Transpose[x2data],1]];", "Input", PageWidth->Infinity], Cell["x2data2 = Transpose[{x2data1,x2graph}];", "Input", PageWidth->Infinity], Cell["x2fit = Fit[x2data2,{1,x},x];", "Input", PageWidth->Infinity, FontFamily->"Arial", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["Determine relation of screen scale to graph scale for y axis", 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We prefer fitting our results (smooth \ curves) to the unnormalized Cavalini et al experimental values at large \ angles because our results were chosen to fit experimental viscosity and \ diffusivity data that emphasize large angle scattering. These Cavalini et al \ results are roughly consistent with those of Parson et al. discussed above.\ \>", "Text"], Cell["\<\ We have not attempted to compare the differential cross section data of \ Brenot et al. Phys. Rev. A 11, 1245 (1975) at 0.5 to 4 keV with our model \ because of the extremely small scattering angles of the experiment, i.e., \ less than 0.015 degree, and the importance of inelastic scattering at the \ larger angles.\ \>", "Text"], Cell["Discussion", "Subsubsection", Evaluatable->False], Cell["\<\ 1. The objective of this notebook is to use measured and/or calculated cross \ sections to infer the effectve width of the forward and backward peaked \ differential cross section for the elastic scattering of Ar by Ar. 2. At high energies the width is small and we have shown that the isotope \ diffusion and viscosity cross sections weigh the differential cross section \ the same way, i.e., theta^2. Note that that the ratio of the cross sections \ at high energies approaches 2:1. Therefore we learn very little from their \ ratio and rely largey on the ratio of the viscosity cross section to the \ total cross section. 3. We are trying to fit three observables, i.e., Qv(enrel), Qd(enrel), and \ Qt(enrel) with a formula having three parameters a(enrel), b(enrel), and \ beta(enrel). 4. At low energies one one cannot asssume that the differential cross section \ approaches isotropic. In fact our analysis give a higher degree of \ anisotropy, i.e., smaller beta, at small energies than at high energies. 5. We have emphasized the fit to the experimental viscosity coefficient data \ at low energies and to the theoretical viscosity cross section at high \ energies because we plan to use the results for the modeling of the transport \ of Ar atoms through Ar where the viscosity angular weighting is the important \ consideration, i.e., it determines the energy transport. 6. At high energies our decision to set the isotropic component of the \ differential cross section to zero is a compromize between the unrealistic \ negative values obtained by a three parameter fit to our input cross sections \ and the inability of our final set of parameters to fit the both the \ viscosity and isotope diffusion cross sections calculated by Robinson.\ \>", "SmallText", Evaluatable->False], Cell["Summary", "Subsubsection", Evaluatable->False, FontFamily->"Helv", FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ We have given a procedure for determining the three parameters of a screened \ Coulonb related differential scattering cross section for Ar-Ar scattering \ based on the available data for the viscosity, diffusion, and the total cross \ sections.\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell["runtime = SessionTime[] - startclock", "Input", PageWidth->Infinity], Cell[BoxData[ \(190.000000000000017`\)], "Output"] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{694, 433}, WindowMargins->{{2, Automatic}, {Automatic, 5}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> "AvpStyle.nb" ] (*********************************************************************** Cached data follows. 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