(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 31023, 1761]*) (*NotebookOutlinePosition[ 57570, 2724]*) (* CellTagsIndexPosition[ 57526, 2720]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData["Alpha Decay"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "by\nGerard P. Gilfoyle\nDepartment of Physics\nUniversity of Richmond\n\ Richmond, VA 23173\ngilfoyle@urvax.urich.edu\n"], "Subtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["Published in:\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], StyleBox["Mathematica in Education and Research\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], StyleBox["Vol.4 No.1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData["Copyright 1995 TELOS/Springer-Verlag"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["1. Introduction"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "One of the earliest and most convincing verifications of the validity of \ quantum mechanics was the resolution of the paradox of alpha decay observed \ in heavy nuclei. Gamow, Condon, and Gurney treated the problem as a barrier \ penetration problem and were able to provide a reasonable explanation of the \ decay rates for a broad range of alpha-emitting nuclei. In this exercise you \ will compile some of the existing data on alpha decay and make a comparison \ with a theoretical calculation based on the transfer matrix method."], "Text",\ Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["2. Constants, Commands, and Functions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "In this Section the ingredients necessary for the calculation of the \ transmission coefficient for the alpha potential barrier are generated. The \ commands in each Section should be executed since they will be used in the \ calculation of the transmission coefficient and they demonstrate the plotting \ of data with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["2.1 Constants:"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The mass and charge of the emitted particle are defined by \"Aout\" and \ \"Zout\" in units of \natomic mass units electron charges respectively. The \ mass and charge of the parent nucleus\nis defined by \"Anuc\" and \"Znuc\". \ The other quantities below are necessary constants. To define the constants \ place the pointer in the cell, click, and then hit \"Enter\". The emitted \ particle is an alpha particle."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Note: Mathematica will warn you about \"possible\" spelling errors in some \ of the definitions below because of the similarity of some of the names used. \ You can usually, safely ignore this warning.\n\nList of constants.\n \n \ Aout - emitted particle mass \n Zout - emitted particle charge\n \ Anuc - atomic number of parent nucleus \n Mout - mass of parent\n \ Znuc - charge of parent\n R0 - radius of parent\n Rmax \ - classical turning point of emitted\n particle\n Ndiv \ - Number of segments\n step - width of an individual barrier\n \ Ve - array of (radius, potential energy)\n "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Aout = 4.0; \nZout = 2.0; \nAnuc = 212;\nZnuc = 84;\nNdiv = \ 10;\nhbarc = 197.0; (* h_bar times c *)\ne2 = hbarc/137; (* electron \ charge squared *)\n"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["2.2 Necessary commands:"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Some of the commands needed to calculate the transmission coefficient for a \ given emitted particle energy are demonstrated here. The technique for \ generating an array of numbers is shown first. You must determine what values \ R0, Rmax, and step should have in terms of the constants defined above. The \ values given below are for demonstration purposes only."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "R0 = 8.0;\nRmax = 30.0;\nstep = 1.0;\nVe = Table[{r, Zout*Znuc*e2/r},\n \ {r, R0, Rmax-step,step}]; "], "Input", AspectRatioFixed->True], Cell[TextData[ "Put zeros at each end of the array since we are treating the potential as if \ it is zero in those regions."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "PrependTo[Ve, {R0-step, 0}];\nAppendTo[Ve, {Rmax, 0}];"], "Input", AspectRatioFixed->True], Cell[TextData["Initialize the transfer matrix to the unit matrix."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["test = { {1,0}, \n {0,1} };"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Use the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Do", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " command to loop over the segments and calculate the transfer matrix by \ multiplying successive matrices together. The commands for extracting \ elements from the array Ve and multiplying matrices are also demonstrated \ here within the loop.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Do[\n Vtest = Ve[[i,2]];\n mat1 = {{ Vtest, 0},\n { 0, \ Vtest} };\n mat2 = {{-Vtest, 0},\n { 0, Vtest} };\n \ test = test.mat1.mat2;,\n {i,1,Ndiv+1}\n ];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["2.3 Plotting commands"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["2.3.1 Plotting data."], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " can be used plots of data with the following commands that define an \ array of x \nand y ordered pairs (\"Stuff\") that is then plotted with the \ command \"ListPlot\" as demonstrated below.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Stuff= { {1,2},\n {2,4},\n {3,7},\n {4,6} }\n\ ListPlot[Stuff,\n PlotStyle->PointSize[0.025],\n \ AxesLabel->{FontForm[\"x\",{\"Times-Italic\",12}],\n \ FontForm[\"y\",{\"Times-Italic\",12}]}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{1, 2}, {2, 4}, {3, 7}, {4, 6}}\ \>", "\<\ {{1, 2}, {2, 4}, {3, 7}, {4, 6}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! 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Do NOT use the numbers displayed below. \ They are an example only.\n\n \n \ Energy Theoretical Lifetime", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["theory = { { ", AspectRatioFixed->True], StyleBox["8.78", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox[" , Log[10,", AspectRatioFixed->True], StyleBox["5.9 10^-5", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox[ "] },\n { 9.00 , Log[10,1.2 10^-12]},\n { 12.0 , \ Log[10,2.0 10^-7 ] } }\n", AspectRatioFixed->True] }], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{8.78, -4.229147988357856}, {9., -11.92081875395237}, {12., -6.698970004336018}}\ \>", "\<\ {{8.78, -4.22915}, {9., -11.9208}, {12., -6.69897}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The measured lifetimes taken from the table of isotopes are stored here in a \ manner similar to the table above. 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It defines two graphical objects, \"dataplot\" and \"theoryplot\" \ and then combines them. Place the pointer anywhere in the cell, click, and \ hit \"Enter\"."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "dataplot = ListPlot[\n data,\n Frame->True,\n \ FrameLabel->\n \ {FontForm[\"E(MeV)\",{\"Helvetica-Bold\",10}],\n \ FontForm[\"Log(t)\",{\"Helvetica-Bold\",10}],\n \ FontForm[\"Points->Data, Line->Theory\",{\"Helvetica-Bold\",10}],\n \ \" \"},\n Prolog->AbsolutePointSize[5],\n \ RotateLabel->False, \n DisplayFunction->Identity\n \ ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ The Unformatted text for this cell was not generated. 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }], MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@00000000000000060001T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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