(*********************************************************************** 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[ 14332, 500]*) (*NotebookOutlinePosition[ 15100, 527]*) (* CellTagsIndexPosition[ 15056, 523]*) (*WindowFrame->Normal*) Notebook[{ Cell["Cross sections for Van der Waals scattering", "Subtitle"], Cell[CellGroupData[{ Cell["Setup notebook environment", "Text"], Cell["ClearAll[\"Global`*\"]; ", "Input"], Cell[CellGroupData[{ Cell["Remove[\"Global`*\"]; ", "Input"], Cell[BoxData[ \(Remove::"rmnsm" \( : \ \) "There are no symbols matching \"\!\(\"Global`*\"\)\"."\)], "Message"] }, Open ]], Cell["startclock = SessionTime[];", "Input"], Cell["Off[General::spell]", "Input"], Cell["Off[General::spell1]", "Input"], Cell["Off[NumberForm::sigz]", "Input"], Cell["<< Graphics`Graphics`", "Input"], Cell[BoxData[ \(<< Miscellaneous`ChemicalElements`\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(now\ = \ StringForm["\<``/``/`` ``:``:``\>", \(Date[]\)[\([2]\)], \(Date[]\)[\([3]\)], \(Date[]\)[\([1]\)], \(Date[]\)[\([4]\)], \(Date[]\)[\([5]\)], \(Date[]\)[\([6]\)]]\)], "Input"], Cell[BoxData[ InterpretationBox[ "\<\"\\!\\(11\\)/\\!\\(6\\)/\\!\\(1998\\) \ \\!\\(19\\):\\!\\(37\\):\\!\\(39\\)\"\>", StringForm[ "``/``/`` ``:``:``", 11, 6, 1998, 19, 37, 39], Editable->False]], "Output"] }, Open ]], Cell["\<\ The objective of this notebook is to establish the relations among the \ various transport cross sections for atoms interacting via the van der Waals \ intereaction, i.e., a force varying as r^-6. \ \>", "SmallText"], Cell["Differential scattering cross section", "Subsection"], Cell["\<\ From McDaniel, Mitchell, and Rudd, Atomic Collisions: Heavy Particle \ Collisions (Wiley, New York, 1993), p. 91, the classical differential \ scattering cross section for small angles is given by:\ \>", "SmallText"], Cell[BoxData[ \(\(ismall\ = \ \((15*Pi/2)\)^\((1/3)\)/12* \((c/\((u*theta)\))\)^\((1/3)\)/\((theta*Sin[theta])\); \)\)], "Input"], Cell["\<\ This has a singularity at small angles and must be replaced by the quantum \ formulation of Massey et al.\ \>", "SmallText"], Cell["Total cross section", "Subsection"], Cell[BoxData[ \(\(qs\ := \ gammaLL*\((c/\((hbar*vo)\))\)^\((2/\((n - 1)\))\)\ \ \ \)\)], "Input"], Cell["\<\ where c is the van der Waals constant in V[r] = -c/r^n, vo is the relative \ velocity of approach, and the Landau and Lifshitz expression for gammaLL is\ \>", "SmallText"], Cell[BoxData[ \(\(gammaMM\ := \ N[Pi*\((2*n - 3)\)/\((n - 2)\)*\((2 fn)\)^\((2/\((n - 1)\))\)]\ \)\)], "Input"], Cell[BoxData[ \(\(gammaLL\ := \ N[Pi^2*\((2 fn)\)^\((2/\((n - 1)\))\)* Csc[Pi/\((n - 1)\)]/Gamma[2/\((n - 1)\)]]\ \)\)], "Input"], Cell["\<\ Note that this formula is different than that given in Massey, Burhop, and \ Gilbody (1971) or in McDaniel, Mitchell, and Rudd (1993), but gives numbers \ in exact agreement with Massey et al Table 16.2.\ \>", "SmallText"], Cell[BoxData[ \(\(vo\ := \ Sqrt[2*ur/mr]\ \ \ m/s\ (*\ ur\ in\ joules, \ mr\ in\ kgm\ *) \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(N[Sin[Pi/5]]\)], "Input"], Cell[BoxData[ \(0.587785252292473181`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Gamma[2/5]]\)], "Input"], Cell[BoxData[ \(2.2181595437576882`\)], "Output"] }, Open ]], Cell["For a van der Waals potential ", "SmallText"], Cell[BoxData[ \(\(n\ = \ 6; \)\)], "Input"], Cell[BoxData[ \(\(fn\ = \ \((6 - 3)\)/\((6 - 2)\)*\((6 - 5)\)/\((6 - 4)\)*Pi/2; \)\)], "Input"], Cell[BoxData[ \(\(qe\ = \ 1.602*^-19\ \ ; \)\)], "Input"], Cell[BoxData[ \(\(mr\ = \ 20*1.661*^-27\ \ ; \)\)], "Input"], Cell["and", "SmallText"], Cell[BoxData[ \(\(hbar\ = \ \(6.63*^-34/2\)/Pi\ \ \ \ \ joule\ \ s; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(gammaMM\)], "Input"], Cell[BoxData[ \(7.54752998327271029`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(gammaLL\)], "Input"], Cell[BoxData[ \(8.08277959060096762`\)], "Output"] }, Open ]], Cell["Massey, Mitchell, and Gilbody give gammaLL = 8.083 for s = 6", "SmallText"], Cell[CellGroupData[{ Cell["qs // PowerExpand ", "Input"], Cell[BoxData[ \(\(2.19938469538109204`*^9\ c\^\(2/5\)\)\/\(joule\^\(2/5\)\ m\^\(2/5\)\ ur\^\(1/5\)\)\)], "Output"] }, Open ]], Cell["\<\ The value of c is seldom given, but Barker, Fisher, and Watts, Molecular \ Physics 21, 657 (1971) show u/k = 42+-2 K at r = 5*^-8 cm so that\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(ubfw = \ 42*1.381*^-23/\((r/\((5*^-10)\))\)^6\ \ \ joule\)], "Input"], Cell[BoxData[ \(\(9.06281249999999616`*^-78\ joule\)\/r\^6\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cbfw = \ ubfw*\ \((r\ m)\)^6\)], "Input"], Cell[BoxData[ \(9.06281249999999616`*^-78\ joule\ m\^6\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(qsbfw\ = \ qs\ //. \ {c\ -> \ cbfw, ur\ -> \ qe*urev}\ \ \ // PowerExpand\ (* \ m^2\ *) \)\)], "Input"], Cell[BoxData[ \(\(1.92430493794192578`*^-18\ m\^2\)\/urev\^\(1/5\)\)], "Output"] }, Open ]], Cell["Here m^2 says the result is in sq meters.", "SmallText"], Cell["\<\ The value of 2.9*^-14 cm^2 = 290 A^2 at 1 eV seems very large. Try again \ using my notes in the margin of p. 72 of McDaniel, Mitchell, and Rudd (1993), \ i.e., qs = gammaLL*(V(ro)*ro/hbar/vo)^2/5*ro^2.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{hbar, qe, mr}, \nqe\ = \ 1.602*^-19\ ; \n mr\ = \ 20*1.661*^-27; \nhbar\ = \ \(6.63*^-34/2\)/Pi; \n\t qs1\ = N[\ gammaLL*\((\ 42*1.381*^-23/\((hbar*Sqrt[2*qe*urev/mr]/5*^-10)\)) \)^{2/5}*\((5*^-10)\)^2\ ]\ \n\t\t\t\t // \ PowerExpand\n\t] \)], "Input"], Cell[BoxData[ \({1.92430493794193004`*^-18\/urev\^\(1/5\)}\)], "Output"] }, Open ]], Cell["\<\ This magnitude is to be compared to the total cross section when the hard \ sphere contribution is dominant , e.g., Pi*(3.4*^-10)^2.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(qhs\ = \ Pi*\((3.4*^-10)\)^2\ \ (*\ m^2\ *) \)\)], "Input"], Cell[BoxData[ \(3.63168110754980055`*^-19\)], "Output"] }, Open ]], Cell["\<\ This says that the hard sphere contribution is small in accordance with the \ low isotropic portion of the differential cross section found by Parson, \ Siska, and Lee, J. Chem. Phys. 58, 1511 (1972).\ \>", "SmallText"], Cell["\<\ Massey and Burhop (1952), p. 387 give cmb = 100*1*^60 erg cm^6 = \ 1*^-58*1*^-7*1*^-12 = 1*^-77 joules m^6, which agrees well with the Barker et \ al. value.\ \>", "SmallText"], Cell["\<\ Comparison of our calculated total cross sections with those of Bernstein\ \>", "Subsubsection"], Cell["\<\ We now compare our calculated total cross sections with those of Rothe and \ Bernstein, J. Chem Phys.31, 1619 (1959). Here we try to make use of their \ results for K and Cs collisions with varous gases, including Ar, to infer the \ utility of the Massey and Mohr formalism that we wish to aply to Ar - Ar \ collisions. For K - Ar collisions they use c = 223*^-60 erg cm^6 = \ 223*^-60/10*7/10^12 = 223*^-79 joule m^6 and get a cross section of 461 A^2 \ at a relative velocity of vr = 0.506*^5 cm/s = 5.06*^2 m/s. They measure \ about 50% larger.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(AtomicWeight[Potassium]\)], "Input"], Cell[BoxData[ StyleBox["39.0983000000000036`", StyleBoxAutoDelete->True, PrintPrecision->6]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AtomicWeight[Argon]\)], "Input"], Cell[BoxData[ StyleBox["39.9480000000000003`", StyleBoxAutoDelete->True, PrintPrecision->5]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{vrel, c, \ mr, vo}, \n\t mr\ = \ \(( AtomicWeight[Argon]^\(-1\) + \ AtomicWeight[Potassium]^\(-1\)) \)^\(-1\)*1.661*^-27; \nvrel\ = \ 7.27*^2\ \ \ \ m/s; \n c\ = \ 223*^-79\ \ \ joule\ m^6; \n urelbernstein\ = \ mr*vrel^2/2\ \ ; \n\t qsbernstein\ = \ qs\ /. \ vo\ -> \ vrel\ \ // \ PowerExpand\n\t] \)], "Input"], Cell[BoxData[ \(4.93100511506005112`*^-18\ m\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(urelbernstein/qe\)], "Input"], Cell[BoxData[ \(\(0.054139883423759505`\ m\^2\)\/s\^2\)], "Output"] }, Open ]], Cell["\<\ By \"correcting\" the Landau and Lifschitz formula for gammaLL given by \ Massey et al we get rough agreement with Berstein's value of 4.61*^-18 \ m^2.\ \>", "SmallText"], Cell["\<\ Landau and Lifschitz, Quantum Mechanics (Pergamon,Oxford, 1965), p. 488 give \ the qLL written below\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{vrel, c}, \nvrel\ = \ 7.27*^2\ \ \ \ m/s; \n c\ = \ 223*^-79\ \ \ joule\ m^6; \n qLL\ = \ 2*Pi^\((n/\((n - 1)\))\)*Sin[Pi/2*\((n - 3)\)/\((n - 1)\)] Gamma[\((n - 3)\)/\((n - 1)\)]*\n\t\ \ \ \ \ \ \ \ \ \ \ \((Gamma[\((n/2 - 1/2)\)]/Gamma[n/2])\)^\((2/\((n - 1)\))\) \((\(c/hbar\)/vrel)\)^\((2/\((n - 1)\))\) /. \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ vo\ -> \ vrel\ \ // \ PowerExpand\n\t]\)], "Input"], Cell[BoxData[ \(4.93100511506004934`*^-18\ m\^2\)], "Output"] }, Open ]], Cell["Diffusion and viscosity cross sections", "Subsection"], Cell["\<\ Here we try to make use of the numerical calculations of Mason, J. Chem. \ Phys. 22, 169 (1954) for a 6-12 potential at zero temperature. The \ calculations are also tabulated in Table VII-B of Hirshfelder, Curtis, and \ Bird (1954). In priciple, one can also extrapolate to zero temperature the \ results of Kotani, Proc. Phys. - Math. Soc. Japan (3) 24, 76 (1942) for a \ Sutherland potential with an inverse 6th power attraction. In practice, the \ extrapolation looks risky.\ \>", "SmallText"], Cell["Consider the potential", "SmallText"], Cell[BoxData[ \(\(vr\ := \ epsilon/\((1 - 6/alpha)\)* \((6/alpha*Exp[alpha*\((1 - r/rm)\)] - \((rm/r)\)^6)\); \)\)], "Input"], Cell["\<\ The van der Waals constant here is c = epsilon/(1-6/alpha)*(r m)^6\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[vr\ , {r, 0, 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ \(-\(\(epsilon\ rm\^6\)\/\(\((1 - 6\/alpha)\)\ r\^6\)\)\), "+", \(\(6\ E\^alpha\ epsilon\)\/\(\((1 - 6\/alpha)\)\ alpha\)\), "-", \(\(6\ \((E\^alpha\ epsilon)\)\ r\)\/\(\((1 - 6\/alpha)\)\ rm\)\), "+", InterpretationBox[\(O[r]\^2\), SeriesData[ r, 0, {}, -6, 2, 1]]}], SeriesData[ r, 0, { Times[ -1, Power[ Plus[ 1, Times[ -6, Power[ alpha, -1]]], -1], epsilon, Power[ rm, 6]], 0, 0, 0, 0, 0, Times[ 6, Power[ Plus[ 1, Times[ -6, Power[ alpha, -1]]], -1], Power[ alpha, -1], Power[ E, alpha], epsilon], Times[ -6, Power[ Plus[ 1, Times[ -6, Power[ alpha, -1]]], -1], Power[ E, alpha], epsilon, Power[ rm, -1]]}, -6, 2, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[vr, alpha -> Infinity, Analytic\ -> True]\)], "Input"], Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ \(\(epsilon\ \((\(6\ E\^\(alpha\ \((1 - r\/rm)\)\)\)\/alpha - rm\^6\/r\^6) \)\)\/\(1 - 6\/alpha\)\), ",", RowBox[{"alpha", "\[Rule]", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], ",", \(Analytic \[Rule] True\)}], "]"}]], "Output"] }, Open ]], Cell["From Hirshfelder et al Table VII-B at zero temperature", "SmallText"], Cell[BoxData[ \(\(z11\ = \ 1.187; \)\)], "Input"], Cell[BoxData[ \(\(z22\ = \ 1.195; \)\)], "Input"], Cell["where these results are independent of alpha and ", "SmallText"], Cell[BoxData[ \(omega11star\ := \ \(z11/\((tempstar*\((1 - 6/alpha)\))\)^{1/3}\)/\((sigma/rm)\)^2\)], "Input"], Cell[BoxData[ \(omega22star\ := \ \(z22/\((tempstar*\((1 - 6/alpha)\))\)^{1/3}\)/\((sigma/rm)\)^2\)], "Input"], Cell[BoxData[ \(tempstar\ := \ k*temp/epsilon\)], "Input"], Cell[BoxData[ \(omega11\ := \ Pi*sigma^2\ *Sqrt[k*temp/\((2*Pi*mr)\)]*omega11star\)], "Input"], Cell[BoxData[ \(omega22\ := \ 2*Pi*sigma^2\ *Sqrt[k*temp/\((2*Pi*mr)\)]*omega22star\)], "Input"], Cell["This never got finished.", "SmallText"] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, CellGrouping->Manual, WindowSize->{756, 406}, WindowMargins->{{2, Automatic}, {Automatic, 5}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, StyleDefinitions -> "AvpStyle.nb" ] (*********************************************************************** Cached data follows. 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