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author | Karl Berry <karl@freefriends.org> | 2012-08-26 23:07:28 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2012-08-26 23:07:28 +0000 |
commit | d3535e838cefea309221187bb9e62698c0df8c51 (patch) | |
tree | dbee8197e1bfecefe547d53e843f0acff729a3fe /Master/texmf-dist/doc/latex/latex-web-companion/ch4 | |
parent | 33353453d22fc5877a8ced58c673c7008ce8c00a (diff) |
latex-web-companion non-update
git-svn-id: svn://tug.org/texlive/trunk@27531 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/latex-web-companion/ch4')
16 files changed, 4234 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.aux b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.aux new file mode 100644 index 00000000000..96bb466013e --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.aux @@ -0,0 +1,29 @@ +\relax +\ifx\rEfLiNK\UnDef\gdef \xRef#1#2{#2}\fi +\newlabel{fg:phys332-1}{{\rEfLiNK{1-10061}{1}}{\rEfLiNK{1-10061}{6}}} +\citation{bib-LAND} +\citation{bib-TALM} +\newlabel{sec:phys332-1}{{\rEfLiNK{1-20002}{2}}{\rEfLiNK{1-20002}{7}}} +\citation{bib-VAVI} +\citation{bib-SCH1} +\newlabel{vavref}{{\rEfLiNK{1-40003}{3}}{\rEfLiNK{1-40003}{9}}} +\citation{bib-SELT} +\newlabel{urban}{{\rEfLiNK{1-60005}{5}}{\rEfLiNK{1-60005}{10}}} +\newlabel{fg:phys332-2}{{\rEfLiNK{1-60012}{2}}{\rEfLiNK{1-60012}{12}}} +\newlabel{eq:sigex}{{\rEfLiNK{1-6003r2}{2}}{\rEfLiNK{1-6003r2}{13}}} +\newlabel{eq:sigion}{{\rEfLiNK{1-6004r3}{3}}{\rEfLiNK{1-6004r3}{13}}} +\newlabel{eq:fisum}{{\rEfLiNK{1-6005r4}{4}}{\rEfLiNK{1-6005r4}{14}}} +\newlabel{eq:flnsum}{{\rEfLiNK{1-6005r5}{5}}{\rEfLiNK{1-6005r5}{14}}} +\newlabel{eq:phys332-5}{{\rEfLiNK{1-7001r13}{13}}{\rEfLiNK{1-7001r13}{17}}} +\newlabel{eq:phys332-1}{{\rEfLiNK{1-7004r16}{16}}{\rEfLiNK{1-7004r16}{17}}} +\newlabel{eq:phys332-2}{{\rEfLiNK{1-7004r17}{17}}{\rEfLiNK{1-7004r17}{17}}} +\newlabel{eq:phys332-3}{{\rEfLiNK{1-7005r18}{18}}{\rEfLiNK{1-7005r18}{18}}} +\newlabel{eq:phys332-4}{{\rEfLiNK{1-7005r19}{19}}{\rEfLiNK{1-7005r19}{18}}} +\newlabel{eq:phys332-6}{{\rEfLiNK{1-7008r22}{22}}{\rEfLiNK{1-7008r22}{18}}} +\newlabel{eq:phys332-7}{{\rEfLiNK{1-7008r23}{23}}{\rEfLiNK{1-7008r23}{18}}} +\newlabel{eq:phys332-8}{{\rEfLiNK{1-7008r24}{24}}{\rEfLiNK{1-7008r24}{19}}} +\bibcite{bib-LAND}{1} +\bibcite{bib-SCH1}{2} +\bibcite{bib-SELT}{3} +\bibcite{bib-TALM}{4} +\bibcite{bib-VAVI}{5} diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.css b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.css new file mode 100644 index 00000000000..333d950a156 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.css @@ -0,0 +1,45 @@ + +/* start css.sty */ +.cmsy-7{font-size:70%;} +.emr-17{font-size:170%;} +.emr-12{font-size:120%;} +.emtt-10{font-family: monospace;} +.emr-7{font-size:70%;} +.emr-5{font-size:50%;} +.emmi-7{font-size:70%;} +.emmi-5{font-size:50%;} +.emti-10{font-style: italic;} +.small-caps{font-variant: small-caps; } +p.noindent { text-indent: 0em } +p.indent{ text-indent: 1.5em } +.sub, .sup {font-size:110%} +.Canvas { position:relative; } +img.mathdisplay{ margin-top: 1em; margin-bottom: 1em; } +li p.indent, li p.indent{ text-indent: 0em } +p.bibitem { text-indent: -2em; margin-left: 2em; } +.quote {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; } +DIV.td00{ margin-left:0; margin-right:0; } +DIV.td01{ margin-left:0; margin-right:5; } +DIV.td10{ margin-left:5; margin-right:0; } +DIV.td11{ margin-left:5; margin-right:5; } +.hline hr, .cline hr{ height : 1px; } +.verbatim {margin-bottom:0.5em; margin-top:0.5em; } +span.footnotetext{ font-size:75%; font-style:italic; } +span.TEX {letter-spacing: -0.125em; } +span.TEX span.E{ position:relative;top:0.5ex;left:-0.0417em;} +a span.TEX span.E {text-decoration: none; } +span.LATEX span.A{ position:relative; top:-0.5ex; left:-0.4em; font-size:85%;} +span.LATEX span.TEX{ position:relative; left: -0.4em; } +.marginpar {width:20%; float:right; text-align:left; margin-left:1em; margin-top:0.5em; font-size:85%; text-decoration:underline;} +.equation TD{text-align:center; } +.obeylines-h,.obeylines-v {white-space: nowrap; } +span.pmatrix img{vertical-align:middle;} +.underline{ text-decoration:underline; } +.overline{ text-decoration:overline; } +td.caption{white-space: nowrap; } +h2.titleHead{text-align:center;} +div.maketitle{ margin-bottom: 2em; } +h1.partHead{text-align: center} +.paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;} +.subparagraphHead, .likesubparagraphHead { font-weight: bold;} +/* end css.sty */ diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.dvi b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.dvi Binary files differnew file mode 100644 index 00000000000..49e057e59c8 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.dvi diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.html b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.html new file mode 100644 index 00000000000..7461e32259f --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.html @@ -0,0 +1,1510 @@ + + +<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> +<html +><head><!--latexexa.html from latexexa.tex (TeX4ht, 1999-03-31 09:15:00)--> +<title>Simulation of Energy Loss Straggling</title><link +rel="stylesheet" type="text/css" href="latexexa.css"></head><body +> <div align="center" class="maketitle"> +<h2 class="titleHead">Simulation of Energy Loss Straggling</h2> +<div class="author" align="center"><span +class="emr-12">Maria Physicist</span></div> +<br> +<div class="date" align="center"><span +class="emr-12">March 31, 1999</span></div> + <span class="thanks"></span></div> + <h2 class="sectionHead">1 <a + name="1-10001"></a><a + name="QQ1-1-1"></a>Introduction</h2> +<!--16--><p class="noindent">Due to the statistical nature of ionisation energy loss, large fluctuations can occur in +the amount of energy deposited by a particle traversing an absorber element. +Continuous processes such as multiple scattering and energy loss play a +relevant role in the longitudinal and lateral development of electromagnetic and +hadronic showers, and in the case of sampling calorimeters the measured +resolution can be significantly affected by such fluctuations in their active +layers. The description of ionisation fluctuations is characterised by the +significance parameter <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"></span>, which is proportional to the ratio of mean energy loss to +the maximum allowed energy transfer in a single collision with an atomic +electron + <center> +<img +src="latexexa0x.gif"alt=" q +k = E---- + max"class="mathdisplay"></center> <span +class="emmi-10">E</span><span +class="emr-7">max</span> +is the maximum transferable energy in a single collision with an atomic +electron. + <center> +<img +src="latexexa1x.gif"alt=" 2m b2g2 +Emax = ----------e----------2-, + 1 +2gme/mx + (me/mx)"class="mathdisplay"></center> where +<span +class="emmi-10"><img +src="emmi10-d.gif"alt="g"class="10--d"> </span>= <span +class="emmi-10">E/m</span><sub ><span +class="emmi-7">x</span></sub> , <span +class="emmi-10">E </span>is energy and <span +class="emmi-10">m</span><sub ><span +class="emmi-7">x</span></sub> the mass of the incident particle, <span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup> = 1 <span +class="cmsy-10">- </span>1<span +class="emmi-10">/<img +src="emmi10-d.gif"alt="g"class="10--d"></span><sup ><span +class="emr-7">2</span></sup> and +<span +class="emmi-10">m</span><sub ><span +class="emmi-7">e</span></sub> is the electron mass. <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"> </span>comes from the Rutherford scattering cross section and is +defined as: <div align="center" class="eqnarray"><a + name="1-1001r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"> </span>= 2<span +class="emmi-10"><img +src="emmi10-19.gif"alt="p"class="10--19">z</span><sup ><span +class="emr-7">2</span></sup><span +class="emmi-10">e</span><sup ><span +class="emr-7">4</span></sup><span +class="emmi-10">N</span><sub ><span +class="emmi-7">Av</span></sub><span +class="emmi-10">Z<img +src="emmi10-1a.gif"alt="r"class="emmi-10--1a"align="middle"><img +src="emmi10-e.gif"alt="d"class="10--e">x</span> + <span +class="emmi-10">m</span><sub ><span +class="emmi-7">e</span></sub><span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup><span +class="emmi-10">c</span><sup ><span +class="emr-7">2</span></sup><span +class="emmi-10">A</span> = 153<span +class="emmi-10">.</span>4 <span +class="emmi-10">z</span><sup ><span +class="emr-7">2</span></sup> +<span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup> <span +class="emmi-10">Z</span> +<span +class="emmi-10">A</span><span +class="emmi-10"><img +src="emmi10-1a.gif"alt="r"class="emmi-10--1a"align="middle"><img +src="emmi10-e.gif"alt="d"class="10--e">x</span> keV<span +class="emmi-10">,</span></td><td +align="center"nowrap +class="eqnarray2"></td><td +align="left"nowrap +class="eqnarray3"></td></tr></table> +</div>where + <div align="center"><table class="tabular" +cellspacing="0pt" cellpadding="0" +frame="void" ><colgroup><col +id="TBL-2-1"><col +id="TBL-2-2"></colgroup><tr +valign="baseline" id="TBL-2-1-"><td align="left"nowrap id="TBL-2-1-1" +><div class="td11"><span +class="emmi-10">z </span></div></td><td align="left"nowrap id="TBL-2-1-2" +><div class="td11">charge of the incident particle </div></td> +</tr><tr +valign="baseline" id="TBL-2-2-"><td align="left"nowrap id="TBL-2-2-1" +><div class="td11"><span +class="emmi-10">N</span><sub ><span +class="emmi-7">Av</span></sub></div></td><td align="left"nowrap id="TBL-2-2-2" +><div class="td11">Avogadro's number </div></td> +</tr><tr +valign="baseline" id="TBL-2-3-"><td align="left"nowrap id="TBL-2-3-1" +><div class="td11"><span +class="emmi-10">Z </span></div></td><td align="left"nowrap id="TBL-2-3-2" +><div class="td11">atomic number of the material</div></td> +</tr><tr +valign="baseline" id="TBL-2-4-"><td align="left"nowrap id="TBL-2-4-1" +><div class="td11"><span +class="emmi-10">A </span></div></td><td align="left"nowrap id="TBL-2-4-2" +><div class="td11">atomic weight of the material </div></td> +</tr><tr +valign="baseline" id="TBL-2-5-"><td align="left"nowrap id="TBL-2-5-1" +><div class="td11"><span +class="emmi-10"><img +src="emmi10-1a.gif"alt="r"class="emmi-10--1a"align="middle"> </span></div></td><td align="left"nowrap id="TBL-2-5-2" +><div class="td11">density </div></td> +</tr><tr +valign="baseline" id="TBL-2-6-"><td align="left"nowrap id="TBL-2-6-1" +><div class="td11"><span +class="emmi-10"><img +src="emmi10-e.gif"alt="d"class="10--e">x </span></div></td><td align="left"nowrap id="TBL-2-6-2" +><div class="td11">thickness of the material </div></td> +</tr><tr +valign="baseline" id="TBL-2-7-"><td align="left"nowrap id="TBL-2-7-1" +><div class="td11"> </div></td> </tr></table> +</div> +<!--57--><p class="indent"> <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span>measures the contribution of the collisions with energy transfer close to <span +class="emmi-10">E</span><span +class="emr-7">max</span>. +For a given absorber, <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span>tends towards large values if <span +class="emmi-10"><img +src="emmi10-e.gif"alt="d"class="10--e">x </span>is large and/or if <span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"> </span>is +small. Likewise, <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span>tends towards zero if <span +class="emmi-10"><img +src="emmi10-e.gif"alt="d"class="10--e">x </span>is small and/or if <span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"> </span>approaches +1. +<!--63--><p class="indent"> The value of <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span>distinguishes two regimes which occur in the description of +ionisation fluctuations : + <ol type="1"class="enumerate1" +> + <li class="enumerate"><a + name="1-1003x1"></a>A large number of collisions involving the loss of all or most of the incident + particle energy during the traversal of an absorber. + <!--70--><p class="noindent">As the total energy transfer is composed of a multitude of small energy + losses, we can apply the central limit theorem and describe the fluctuations + by a Gaussian distribution. This case is applicable to non-relativistic + particles and is described by the inequality <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> > </span>10 (i.e. when the mean + energy loss in the absorber is greater than the maximum energy transfer + in a single collision). + </li> + <li class="enumerate"><a + name="1-1005x2"></a>Particles traversing thin counters and incident electrons under any + conditions. + <!--81--><p class="noindent">The relevant inequalities and distributions are 0<span +class="emmi-10">.</span>01 <span +class="emmi-10">< <img +src="emmi10-14.gif"alt="k"class="10--14"> < </span>10, Vavilov + distribution, and <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> < </span>0<span +class="emmi-10">.</span>01, Landau distribution.</li></ol> +<!--83--><p class="noindent"> +<!--85--><p class="indent"> An additional regime is defined by the contribution of the collisions with low +energy transfer which can be estimated with the relation <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle">/I</span><sub ><span +class="emr-7">0</span></sub>, where <span +class="emmi-10">I</span><sub ><span +class="emr-7">0</span></sub> is the mean +ionisation potential of the atom. Landau theory assumes that the number of these +collisions is high, and consequently, it has a restriction <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle">/I</span><sub ><span +class="emr-7">0</span></sub> <span +class="cmsy-10">» </span>1. In <span +class="emtt-10">GEANT </span>(see URL +<span +class="emtt-10">http://wwwinfo.cern.ch/asdoc/geant/geantall.html</span>), the limit of Landau +theory has been set at <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle">/I</span><sub ><span +class="emr-7">0</span></sub> = 50. Below this limit special models taking into account +the atomic structure of the material are used. This is important in thin layers and +gaseous materials. Figure <a + href="#1-10061">1</a> shows the behaviour of <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle">/I</span><sub ><span +class="emr-7">0</span></sub> as a function of the layer +thickness for an electron of 100 keV and 1 GeV of kinetic energy in Argon, Silicon +and Uranium. +<a + name="1-10061"></a> + <hr class="float"><div align="center" class="float" +><table class="float"><tr class="float"><td class="float" +> +<img +src="latexexa2x.gif"alt="PIC"> +<br><div align="center"class="caption"><table class="caption" +><tr valign="baseline" class="caption"><td class="id">Figure 1</td><td +class="content">The variable <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle">/I</span><sub ><span +class="emr-7">0</span></sub> can be used to measure the validity range of the +Landau theory. It depends on the type and energy of the particle, <span +class="emmi-10">Z </span>, <span +class="emmi-10">A </span>and the +ionisation potential of the material and the layer thickness. </td></tr></table></div> + </td></tr></table></div><hr class="endfloat"> +<!--110--><p class="indent"> In the following sections, the different theories and models for the energy loss +fluctuation are described. First, the Landau theory and its limitations are discussed, +and then, the Vavilov and Gaussian straggling functions and the methods in the thin +layers and gaseous materials are presented. + <h2 class="sectionHead">2 <a + name="1-20002"></a><a + name="QQ1-1-3"></a>Landau theory</h2> +<!--119--><p class="noindent">For a particle of mass <span +class="emmi-10">m</span><sub ><span +class="emmi-7">x</span></sub> traversing a thickness of material <span +class="emmi-10"><img +src="emmi10-e.gif"alt="d"class="10--e">x</span>, the Landau +probability distribution may be written in terms of the universal Landau function +<span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span>(<span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span>) as[<a + href="#Xbib-LAND">1</a>]: <div align="center" class="eqnarray"><a + name="1-2001r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">f</span>(<span +class="emmi-10"><img +src="emmi10-f.gif"alt="e"class="10--f">, <img +src="emmi10-e.gif"alt="d"class="10--e">x</span>)</td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">1 + <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"></span> <span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span>(<span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span>)</td></tr></table> +</div>where <div align="center" class="eqnarray"><a + name="1-2002r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span>(<span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span>)</td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"> 1_ +2<span +class="emmi-10"><img +src="emmi10-19.gif"alt="p"class="10--19">i</span> <span +class="cmex-10"><img +src="cmex10-5a.gif"alt=" integral "class="cmex-10--5a"align="middle"></span> + <sub> <span +class="emmi-10">c </span><span +class="cmsy-10">- </span><span +class="emmi-10">i</span><span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span></sub><sup><span +class="emmi-10">c </span>+ <span +class="emmi-10">i</span><span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span></sup> exp <img +src="latexexa3x.gif"alt="(u ln u+ cu)"class="left" align="middle"> <span +class="emmi-10">du</span><span +class="emmi-10"> </span><span +class="emmi-10"> </span><span +class="emmi-10"> </span><span +class="emmi-10"> </span><span +class="emmi-10"> </span> <span +class="emmi-10">c </span><span +class="cmsy-10"><span +class="underline">></span> </span>0</td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10"><img +src="emmi10-f.gif"alt="e"class="10--f"> </span><span +class="cmsy-10">-</span> <img +src="latexexa4x.gif"alt="e"class="bar" > + <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"></span> <span +class="cmsy-10">- </span><span +class="emmi-10"><img +src="emmi10-d.gif"alt="g"class="10--d"></span><span +class="cmsy-7">'</span> <span +class="cmsy-10">- </span><span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup> <span +class="cmsy-10">-</span> ln <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"></span> ___ +<span +class="emmi-10">E</span><span +class="emr-7">max</span> </td> </tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-d.gif"alt="g"class="10--d"></span><span +class="cmsy-7">'</span></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">0<span +class="emmi-10">.</span>422784 <span +class="emmi-10">. . .</span> = 1 <span +class="cmsy-10">- </span><span +class="emmi-10"><img +src="emmi10-d.gif"alt="g"class="10--d"></span></td> </tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-d.gif"alt="g"class="10--d"></span></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">0<span +class="emmi-10">.</span>577215 <span +class="emmi-10">. . .</span> (Euler's constant)</td> </tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><img +src="latexexa5x.gif"alt="e"class="bar" ></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">average energy loss</td> </tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-f.gif"alt="e"class="10--f"></span></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">actual energy loss</td> </tr></table> +</div> + <h3 class="subsectionHead">2.1 <a + name="1-30002.1"></a><a + name="QQ1-1-4"></a>Restrictions</h3> +<!--140--><p class="noindent">The Landau formalism makes two restrictive assumptions : + <ol type="1"class="enumerate1" +> + <li class="enumerate"><a + name="1-3002x1"></a>The typical energy loss is small compared to the maximum energy loss in + a single collision. This restriction is removed in the Vavilov theory (see + section <a + href="#1-40003">3</a>). + </li> + <li class="enumerate"><a + name="1-3004x2"></a>The typical energy loss in the absorber should be large compared to the + binding energy of the most tightly bound electron. For gaseous detectors, + typical energy losses are a few keV which is comparable to the binding + energies of the inner electrons. In such cases a more sophisticated approach + which accounts for atomic energy levels[<a + href="#Xbib-TALM">4</a>] is necessary to accurately + simulate data distributions. In <span +class="emtt-10">GEANT</span>, a parameterised model by L. Urbán + is used (see section <a + href="#1-60005">5</a>).</li></ol> +<!--153--><p class="noindent"> +<!--155--><p class="indent"> In addition, the average value of the Landau distribution is infinite. Summing the +Landau fluctuation obtained to the average energy from the <span +class="emmi-10">dE/dx </span>tables, we +obtain a value which is larger than the one coming from the table. The +probability to sample a large value is small, so it takes a large number of steps +(extractions) for the average fluctuation to be significantly larger than zero. This +introduces a dependence of the energy loss on the step size which can affect +calculations. +<!--164--><p class="indent"> A solution to this has been to introduce a limit on the value of the variable +sampled by the Landau distribution in order to keep the average fluctuation to 0. +The value obtained from the <span +class="emtt-10">GLANDO </span>routine is: + <center> +<img +src="latexexa6x.gif"alt="ddE/dx = e- e = q(c -g'+ b2 + ln--q-) + Emax"class="mathdisplay"></center> In +order for this to have average 0, we must impose that: + <center> +<img +src="latexexa7x.gif"alt="c = -g'- b2 -ln -q--- + Emax"class="mathdisplay"></center> +<!--177--><p class="indent"> This is realised introducing a <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emr-7">max</span></sub>(<img +src="latexexa8x.gif"alt="c"class="bar" >) such that if only values of <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"> </span><span +class="cmsy-10"><span +class="underline"><</span> </span><span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emr-7">max</span></sub> are +accepted, the average value of the distribution is <img +src="latexexa9x.gif"alt="c"class="bar" >. +<!--181--><p class="indent"> A parametric fit to the universal Landau distribution has been performed, with +following result: + <center> +<img +src="latexexa10x.gif"alt="cmax = 0.60715+ 1.1934c + (0.67794+ 0.052382c)exp(0.94753 +0.74442c)"class="mathdisplay"></center> +only values smaller than <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emr-7">max</span></sub> are accepted, otherwise the distribution is +resampled. + <h2 class="sectionHead">3 <a + name="1-40003"></a><a + name="QQ1-1-5"></a>Vavilov theory</h2> +<!--197--><p class="noindent">Vavilov[<a + href="#Xbib-VAVI">5</a>] derived a more accurate straggling distribution by introducing the +kinematic limit on the maximum transferable energy in a single collision, rather than +using <span +class="emmi-10">E</span><span +class="emr-7">max</span> = <span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span>. Now we can write[<a + href="#Xbib-SCH1">2</a>]: <div align="center" class="eqnarray"><a + name="1-4001r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">f</span><img +src="latexexa11x.gif"alt="(e,ds)"class="left" align="middle"></td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">1 + <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"></span> <span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span><sub ><span +class="emmi-7">v</span></sub><img +src="latexexa12x.gif"alt="( ) + cv,k,b2"class="left" align="middle"></td></tr></table> +</div>where <div align="center" class="eqnarray"><a + name="1-4002r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span><sub ><span +class="emmi-7">v</span></sub><img +src="latexexa13x.gif"alt="(cv,k,b2)"class="left" align="middle"></td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"> 1_ +2<span +class="emmi-10"><img +src="emmi10-19.gif"alt="p"class="10--19">i</span> <span +class="cmex-10"><img +src="cmex10-5a.gif"alt=" integral "class="cmex-10--5a"align="middle"></span> + <sub> <span +class="emmi-10">c </span><span +class="cmsy-10">- </span><span +class="emmi-10">i</span><span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span></sub><sup><span +class="emmi-10">c </span>+ <span +class="emmi-10">i</span><span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span></sup><span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span><img +src="latexexa14x.gif"alt="(s)"class="left" align="middle"> <span +class="emmi-10">e</span><sup ><span +class="emmi-7"><img +src="emmi7-15.gif"alt="c"class="7--15">s</span></sup><span +class="emmi-10">ds</span><span +class="emmi-10"> </span><span +class="emmi-10"> </span><span +class="emmi-10"> </span><span +class="emmi-10"> </span><span +class="emmi-10"> </span> <span +class="emmi-10">c </span><span +class="cmsy-10"><span +class="underline">></span> </span>0</td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-1e.gif"alt="f"class="10--1e"></span><img +src="latexexa15x.gif"alt="(s)"class="left" align="middle"></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">exp <img +src="latexexa16x.gif"alt="[ 2 ] + k(1+ b g)"class="left" align="middle"> <span +class="emmi-10"> </span> exp <img +src="latexexa17x.gif"alt="[y(s)]"class="left" align="middle"> <span +class="emmi-10">,</span> </td> </tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-20.gif"alt="y"class="10--20"></span><img +src="latexexa18x.gif"alt="(s)"class="left" align="middle"></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10">s</span> ln <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span>+ (<span +class="emmi-10">s </span>+ <span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup><span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"></span>)<img +src="latexexa19x.gif"alt="[ln(s/k)+ E1(s/k)]"class="left" align="middle"> <span +class="cmsy-10">- </span><span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14">e</span><sup ><span +class="cmsy-7">-</span><span +class="emmi-7">s/<img +src="emmi7-14.gif"alt="k"class="7--14"></span></sup><span +class="emmi-10">,</span></td> </tr></table> +</div>and <div align="center" class="eqnarray"><a + name="1-4003r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">E</span><sub ><span +class="emr-7">1</span></sub>(<span +class="emmi-10">z</span>)</td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="cmex-10"><img +src="cmex10-5a.gif"alt=" integral "class="cmex-10--5a"align="middle"></span> + <sub> <span +class="emmi-10">z</span></sub><sup><span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span></sup><span +class="emmi-10">t</span><sup ><span +class="cmsy-7">-</span><span +class="emr-7">1</span></sup><span +class="emmi-10">e</span><sup ><span +class="cmsy-7">-</span><span +class="emmi-7">t</span></sup><span +class="emmi-10">dt</span> (the exponential integral)</td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">v</span></sub></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"></span><img +src="latexexa20x.gif"alt="[ ] + e--e - g'- b2 + q"class="left" align="middle"></td> </tr></table> +</div> +<!--224--><p class="indent"> The Vavilov parameters are simply related to the Landau parameter by +<span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">L</span></sub> = <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">v</span></sub><span +class="emmi-10">/<img +src="emmi10-14.gif"alt="k"class="10--14"> </span><span +class="cmsy-10">-</span> ln <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"></span>. It can be shown that as <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span><span +class="cmsy-10"><img +src="cmsy10-21.gif"alt="-->"class="10--21"> </span>0, the distribution of the variable <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">L</span></sub> +approaches that of Landau. For <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span><span +class="cmsy-10"><span +class="underline"><</span> </span>0<span +class="emmi-10">.</span>01 the two distributions are already practically +identical. Contrary to what many textbooks report, the Vavilov distribution <span +class="emti-10">does not</span> +approximate the Landau distribution for small <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"></span>, but rather the distribution of <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">L</span></sub> +defined above tends to the distribution of the true <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"> </span>from the Landau density +function. Thus the routine <span +class="emtt-10">GVAVIV </span>samples the variable <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">L</span></sub> rather than <span +class="emmi-10"><img +src="emmi10-15.gif"alt="c"class="10--15"></span><sub ><span +class="emmi-7">v</span></sub>. For +<span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span><span +class="cmsy-10"><span +class="underline">></span> </span>10 the Vavilov distribution tends to a Gaussian distribution (see next +section). + <h2 class="sectionHead">4 <a + name="1-50004"></a><a + name="QQ1-1-6"></a>Gaussian Theory</h2> +<!--239--><p class="noindent">Various conflicting forms have been proposed for Gaussian straggling functions, but +most of these appear to have little theoretical or experimental basis. However, it has +been shown[<a + href="#Xbib-SELT">3</a>] that for <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"> </span><span +class="cmsy-10"><span +class="underline">></span> </span>10 the Vavilov distribution can be replaced by a Gaussian +of the form : <div align="center" class="eqnarray"><a + name="1-5001r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">f</span>(<span +class="emmi-10"><img +src="emmi10-f.gif"alt="e"class="10--f">, <img +src="emmi10-e.gif"alt="d"class="10--e">s</span>) <span +class="cmsy-10"><img +src="cmsy10-19.gif"alt=" ~~ "class="10--19"></span> 1 __________ +<span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"></span><img +src="latexexa21x.gif"alt=" V~ ------------ + 2pk (1- b2/2)"class="sqrtsign" > exp <img +src="latexexa22x.gif"alt="[(e- e)2 k ] + --------2-----2--- + 2 q (1- b /2)"class="left" align="middle"></td><td +align="center"nowrap +class="eqnarray2"></td><td +align="left"nowrap +class="eqnarray3"></td></tr></table> +</div>thus implying <div align="center" class="eqnarray"><a + name="1-5002r1"></a> +<table +class="eqnarray-star"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1">mean</td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><img +src="latexexa23x.gif"alt="e"class="bar" ></td> </tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sup ><span +class="emr-7">2</span></sup></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle"></span><sup ><span +class="emr-7">2</span></sup> + <span +class="emmi-10"><img +src="emmi10-14.gif"alt="k"class="10--14"></span> (1 <span +class="cmsy-10">- </span><span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup><span +class="emmi-10">/</span>2) = <span +class="emmi-10"><img +src="emmi10-18.gif"alt="q"class="emmi-10--18"align="middle">E</span> +<span +class="emr-7">max</span>(1 <span +class="cmsy-10">- </span><span +class="emmi-10"><img +src="emmi10-c.gif"alt="b"class="emmi-10--c"align="middle"></span><sup ><span +class="emr-7">2</span></sup><span +class="emmi-10">/</span>2)</td></tr></table> +</div> + <h2 class="sectionHead">5 <a + name="1-60005"></a><a + name="QQ1-1-7"></a>Urbán model</h2> +<!--260--><p class="noindent">The method for computing restricted energy losses with <span +class="emmi-10"><img +src="emmi10-e.gif"alt="d"class="10--e"></span>-ray production +above given threshold energy in <span +class="emtt-10">GEANT </span>is a Monte Carlo method that can +be used for thin layers. It is fast and it can be used for any thickness of +a medium. Approaching the limit of the validity of Landau's theory, the +loss distribution approaches smoothly the Landau form as shown in Figure +<a + href="#1-60012">2</a>. +<a + name="1-60012"></a> + <hr class="float"><div align="center" class="float" +><table class="float"><tr class="float"><td class="float" +> +<img +src="latexexa24x.gif"alt="PIC"> +<br><div align="center"class="caption"><table class="caption" +><tr valign="baseline" class="caption"><td class="id">Figure 2</td><td +class="content">Energy loss distribution for a 3 GeV electron in Argon as given by +standard <span +class="emtt-10">GEANT</span>. The width of the layers is given in centimeters.</td></tr></table></div> + </td></tr></table></div><hr class="endfloat"> +<!--275--><p class="indent"> It is assumed that the atoms have only two energy levels with binding energy <span +class="emmi-10">E</span><sub ><span +class="emr-7">1</span></sub> +and <span +class="emmi-10">E</span><sub ><span +class="emr-7">2</span></sub>. The particle--atom interaction will then be an excitation with energy loss <span +class="emmi-10">E</span><sub ><span +class="emr-7">1</span></sub> +or <span +class="emmi-10">E</span><sub ><span +class="emr-7">2</span></sub>, or an ionisation with an energy loss distributed according to a function +<span +class="emmi-10">g</span>(<span +class="emmi-10">E</span>) <span +class="cmsy-10">~ </span>1<span +class="emmi-10">/E</span><sup ><span +class="emr-7">2</span></sup>: <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa25x.gif"alt=" (Emax +-I)I-1 +g(E) = Emax E2"class="mathdisplay"><a + name="1-6002r1"></a></center></td><td width="5%">(1)</td></tr></table> +<!--283--><p class="indent"> The macroscopic cross-section for excitations (<span +class="emmi-10">i </span>= 1<span +class="emmi-10">, </span>2) is <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa26x.gif"alt=" f ln(2mb2g2/E )- b2 +i = C-i-------2-2-i----2 (1- r) + Ei ln(2mb g/I) - b"class="mathdisplay"><a + name="1-6003r2"></a></center></td><td width="5%">(2)</td></tr></table> +and the macroscopic cross-section for ionisation is <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa27x.gif"alt="3 = C-------Emax---------r + I(Emax + I)ln(EmaIx+I)"class="mathdisplay"><a + name="1-6004r3"></a></center></td><td width="5%">(3)</td></tr></table> +<span +class="emmi-10">E</span><span +class="emr-7">max</span> is the <span +class="emtt-10">GEANT </span>cut for <span +class="emmi-10"><img +src="emmi10-e.gif"alt="d"class="10--e"></span>-production, or the maximum energy transfer minus mean +ionisation energy, if it is smaller than this cut-off value. The following notation is +used: + <div align="center"><table class="tabular" +cellspacing="0pt" cellpadding="0" +frame="void" ><colgroup><col +id="TBL-3-1"><col +id="TBL-3-2"></colgroup><tr +valign="baseline" id="TBL-3-1-"><td align="left"nowrap id="TBL-3-1-1" +><div class="td11"><span +class="emmi-10">r, C</span></div></td><td align="left"nowrap id="TBL-3-1-2" +><div class="td11">parameters of the model</div></td> +</tr><tr +valign="baseline" id="TBL-3-2-"><td align="left"nowrap id="TBL-3-2-1" +><div class="td11"><span +class="emmi-10">E</span><sub ><span +class="emmi-7">i</span></sub> </div></td><td align="left"nowrap id="TBL-3-2-2" +><div class="td11">atomic energy levels </div></td> +</tr><tr +valign="baseline" id="TBL-3-3-"><td align="left"nowrap id="TBL-3-3-1" +><div class="td11"><span +class="emmi-10">I </span></div></td><td align="left"nowrap id="TBL-3-3-2" +><div class="td11">mean ionisation energy </div></td> +</tr><tr +valign="baseline" id="TBL-3-4-"><td align="left"nowrap id="TBL-3-4-1" +><div class="td11"><span +class="emmi-10">f</span><sub ><span +class="emmi-7">i</span></sub> </div></td><td align="left"nowrap id="TBL-3-4-2" +><div class="td11">oscillator strengths </div></td> </tr></table> +</div> +<!--306--><p class="indent"> The model has the parameters <span +class="emmi-10">f</span><sub ><span +class="emmi-7">i</span></sub> , <span +class="emmi-10">E</span><sub ><span +class="emmi-7">i</span></sub> , <span +class="emmi-10">C </span>and <span +class="emmi-10">r</span><span +class="emmi-10"> </span>(0 <span +class="cmsy-10"><span +class="underline"><</span> </span><span +class="emmi-10">r </span><span +class="cmsy-10"><span +class="underline"><</span> </span>1). The oscillator +strengths <span +class="emmi-10">f</span><sub ><span +class="emmi-7">i</span></sub> and the atomic level energies <span +class="emmi-10">E</span><sub ><span +class="emmi-7">i</span></sub> should satisfy the constraints +<div align="center" class="eqnarray"><a + name="1-6005r4"></a> +<table +class="eqnarray"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">f</span><sub ><span +class="emr-7">1</span></sub> + <span +class="emmi-10">f</span><sub ><span +class="emr-7">2</span></sub></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">1</td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(4)<a + name="1-6005r5"></a></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">f</span><sub ><span +class="emr-7">1</span></sub> ln <span +class="emmi-10">E</span><sub ><span +class="emr-7">1</span></sub> + <span +class="emmi-10">f</span><sub ><span +class="emr-7">2</span></sub> ln <span +class="emmi-10">E</span><sub ><span +class="emr-7">2</span></sub></td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3">ln <span +class="emmi-10">I</span></td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(5)<a + name="1-6005r6"></a></td></tr></table> +</div>The parameter <span +class="emmi-10">C </span>can be defined with the help of the mean energy loss <span +class="emmi-10">dE/dx </span>in the +following way: The numbers of collisions (<span +class="emmi-10">n</span><sub ><span +class="emmi-7">i</span></sub> , i = 1,2 for the excitation and 3 for the +ionisation) follow the Poisson distribution with a mean number <span +class="cmsy-10"><</span><span +class="emmi-10">n</span><sub ><span +class="emmi-7">i</span></sub><span +class="cmsy-10">></span>. In a step <span +class="emmi-10">x</span> +the mean number of collisions is <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa28x.gif"alt="<n> = x + i i"class="mathdisplay"><a + name="1-6006r6"></a></center></td><td width="5%">(6)</td></tr></table> +The mean energy loss <span +class="emmi-10">dE/dx </span>in a step is the sum of the excitation and ionisation +contributions <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa29x.gif"alt=" [ ] +dE integral Emax+I +dx- x = 1E1 + 2E2 + 3 E g(E) dE x + I"class="mathdisplay"><a + name="1-6007r7"></a></center></td><td width="5%">(7)</td></tr></table> +From this, using the equations (<a + href="#1-6003r2">2</a>), (<a + href="#1-6004r3">3</a>), (<a + href="#1-6005r4">4</a>) and (<a + href="#1-6005r5">5</a>), one can define the parameter <span +class="emmi-10">C</span> +<table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa30x.gif"alt="C = dE- + dx"class="mathdisplay"><a + name="1-6008r8"></a></center></td><td width="5%">(8)</td></tr></table> +<!--335--><p class="indent"> The following values have been chosen in <span +class="emtt-10">GEANT </span>for the other parameters: + <center> +<img +src="latexexa31x.gif"alt=" { + 0 ifZ < 2 +f2 = 2/Z ifZ > 2 ==> f1 = 1- f2 + ( )f11 +E2 = 10Z2eV ==> E1 = EIf2 +r = 0.4 2"class="mathdisplay"></center> With +these values the atomic level <span +class="emmi-10">E</span><sub ><span +class="emr-7">2</span></sub> corresponds approximately the K-shell energy of the +atoms and <span +class="emmi-10">Zf</span><sub ><span +class="emr-7">2</span></sub> the number of K-shell electrons. <span +class="emmi-10">r </span>is the only variable which can be +tuned freely. It determines the relative contribution of ionisation and excitation to +the energy loss. +<!--354--><p class="indent"> The energy loss is computed with the assumption that the step length (or the +relative energy loss) is small, and---in consequence---the cross-section can be +considered constant along the path length. The energy loss due to the excitation is +<table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa32x.gif"alt="Ee = n1E1 +n2E2"class="mathdisplay"><a + name="1-6009r9"></a></center></td><td width="5%">(9)</td></tr></table> +where <span +class="emmi-10">n</span><sub ><span +class="emr-7">1</span></sub> and <span +class="emmi-10">n</span><sub ><span +class="emr-7">2</span></sub> are sampled from Poisson distribution as discussed above. The loss +due to the ionisation can be generated from the distribution <span +class="emmi-10">g</span>(<span +class="emmi-10">E</span>) by the inverse +transformation method: <div align="center" class="eqnarray"><a + name="1-6010r10"></a> +<table +class="eqnarray"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">u </span>= <span +class="emmi-10">F</span>(<span +class="emmi-10">E</span>)</td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="cmex-10"><img +src="cmex10-5a.gif"alt=" integral "class="cmex-10--5a"align="middle"></span> + <sub> <span +class="emmi-10">I</span></sub><sup><span +class="emmi-10">E</span></sup><span +class="emmi-10">g</span>(<span +class="emmi-10">x</span>)<span +class="emmi-10">dx</span></td> <td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4"></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10">E </span>= <span +class="emmi-10">F</span><sup ><span +class="cmsy-7">-</span><span +class="emr-7">1</span></sup>(<span +class="emmi-10">u</span>)</td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"> <span +class="emmi-10">I</span>_____ +1 <span +class="cmsy-10">- </span><span +class="emmi-10">u</span> <span +class="emmi-7">E</span><span +class="emr-5">max</span>__ +<span +class="emmi-7">E</span><span +class="emr-5">max</span><span +class="emr-7">+</span><span +class="emmi-7">I</span> </td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(10)<a + name="1-6010r11"></a></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"></td> <td +align="center"nowrap +class="eqnarray2"></td> <td +align="left"nowrap +class="eqnarray3"></td> <td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(11)<a + name="1-6010r12"></a></td></tr></table> +</div>where <span +class="emmi-10">u </span>is a uniform random number between <span +class="emmi-10">F</span>(<span +class="emmi-10">I</span>) = 0 and <span +class="emmi-10">F</span>(<span +class="emmi-10">E</span><span +class="emr-7">max</span> + <span +class="emmi-10">I</span>) = 1. The +contribution from the ionisations will be <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa33x.gif"alt=" n sum 3 -----I------ +Ei = 1 -uj -Emax-- + j=1 Emax+I"class="mathdisplay"><a + name="1-6011r12"></a></center></td><td width="5%">(12)</td></tr></table> +where <span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub> is the number of ionisation (sampled from Poisson distribution). The +energy loss in a step will then be <span +class="emmi-10">E </span>= <span +class="emmi-10">E</span><sub ><span +class="emmi-7">e</span></sub> + <span +class="emmi-10">E</span><sub ><span +class="emmi-7">i</span></sub>. + <h3 class="subsectionHead">5.1 <a + name="1-70005.1"></a><a + name="QQ1-1-9"></a>Fast simulation for <span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub> <span +class="cmsy-10"><span +class="underline">></span> </span>16</h3> +<!--380--><p class="noindent">If the number of ionisation <span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub> is bigger than 16, a faster sampling method can be +used. The possible energy loss interval is divided in two parts: one in which the +number of collisions is large and the sampling can be done from a Gaussian +distribution and the other in which the energy loss is sampled for each collision. Let +us call the former interval [<span +class="emmi-10">I, <img +src="emmi10-b.gif"alt="a"class="10--b">I</span>] the interval A, and the latter [<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b">I, E</span><span +class="emr-7">max</span>] the interval +B. <span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"> </span>lies between 1 and <span +class="emmi-10">E</span><span +class="emr-7">max</span><span +class="emmi-10">/I</span>. A collision with a loss in the interval A happens +with the probability <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa34x.gif"alt=" integral aI +P(a) = g(E)dE = (Emax-+-I)(a---1) + I Emaxa"class="mathdisplay"><a + name="1-7001r13"></a></center></td><td width="5%">(13)</td></tr></table> +The mean energy loss and the standard deviation for this type of collision are +<table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa35x.gif"alt=" --1-- integral aI Ia-lna- +< E(a)> = P (a) I E g(E) dE = a- 1"class="mathdisplay"><a + name="1-7002r14"></a></center></td><td width="5%">(14)</td></tr></table> +and <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa36x.gif"alt=" integral aI ( 2 ) +s2(a) =--1-- E2 g(E) dE = I2a 1 - -aln-a2 + P (a) I (a- 1)"class="mathdisplay"><a + name="1-7003r15"></a></center></td><td width="5%">(15)</td></tr></table> +If the collision number is high , we assume that the number of the type A collisions +can be calculated from a Gaussian distribution with the following mean value and +standard deviation: <div align="center" class="eqnarray"><a + name="1-7004r16"></a> +<table +class="eqnarray"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="cmsy-10"><</span><span +class="emmi-10">n</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10">></span></td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub><span +class="emmi-10">P</span>(<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"></span>)</td> <td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(16)<a + name="1-7004r17"></a></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sub><span +class="emmi-10">A</span></sub><sup>2</sup></td> <td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub><span +class="emmi-10">P</span>(<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"></span>)(1 <span +class="cmsy-10">- </span><span +class="emmi-10">P</span>(<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"></span>))</td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(17)<a + name="1-7004r18"></a></td></tr></table> +</div>It is further assumed that the energy loss in these collisions has a Gaussian +distribution with <div align="center" class="eqnarray"><a + name="1-7005r18"></a> +<table +class="eqnarray"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="cmsy-10"><</span><span +class="emmi-10">E</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10">></span></td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10">n</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10"><</span><span +class="emmi-10">E</span>(<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"></span>)<span +class="cmsy-10">></span></td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(18)<a + name="1-7005r19"></a></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="emmi-10"><img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sub><span +class="emmi-10">E, A</span></sub><sup>2</sup></td><td +align="center"nowrap +class="eqnarray2">=</td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10">n</span><sub ><span +class="emmi-7">A</span></sub><span +class="emmi-10"><img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sup ><span +class="emr-7">2</span></sup>(<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"></span>)</td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(19)<a + name="1-7005r20"></a></td></tr></table> +</div>The energy loss of these collision can then be sampled from the Gaussian +distribution. +<!--427--><p class="indent"> The collisions where the energy loss is in the interval B are sampled directly from +<table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa37x.gif"alt=" n3- sum nA aI +EB = 1--u-Emax+I-aI + i=1 i Emax+I"class="mathdisplay"><a + name="1-7006r20"></a></center></td><td width="5%">(20)</td></tr></table> +The total energy loss is the sum of these two types of collisions: <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa38x.gif"alt="E = EA + EB"class="mathdisplay"><a + name="1-7007r21"></a></center></td><td width="5%">(21)</td></tr></table> +<!--438--><p class="indent"> The approximation of equations ((<a + href="#1-7004r16">16</a>), (<a + href="#1-7004r17">17</a>), (<a + href="#1-7005r18">18</a>) and (<a + href="#1-7005r19">19</a>) can be used under the +following conditions: <div align="center" class="eqnarray"><a + name="1-7008r22"></a> +<table +class="eqnarray"><tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="cmsy-10"><</span><span +class="emmi-10">n</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10">> - </span><span +class="emmi-10">c <img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sub ><span +class="emmi-7">A</span></sub></td> <td +align="center"nowrap +class="eqnarray2"><span +class="cmsy-10"><span +class="underline">></span></span></td><td +align="left"nowrap +class="eqnarray3">0</td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(22)<a + name="1-7008r23"></a></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="cmsy-10"><</span><span +class="emmi-10">n</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10">> </span>+ <span +class="emmi-10">c <img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sub ><span +class="emmi-7">A</span></sub></td> <td +align="center"nowrap +class="eqnarray2"><span +class="cmsy-10"><span +class="underline"><</span></span></td><td +align="left"nowrap +class="eqnarray3"><span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub></td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(23)<a + name="1-7008r24"></a></td></tr> +<tr valign="middle" class="eqnarray"><td +align="right"nowrap +class="eqnarray1"><span +class="cmsy-10"><</span><span +class="emmi-10">E</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10">> - </span><span +class="emmi-10">c <img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sub ><span +class="emmi-7">E,A</span></sub></td><td +align="center"nowrap +class="eqnarray2"><span +class="cmsy-10"><span +class="underline">></span></span></td><td +align="left"nowrap +class="eqnarray3">0</td><td +width="10" class="eqnarray4"></td><td +align="right"nowrap +class="eqnarray4">(24)<a + name="1-7008r25"></a></td></tr></table> +</div>where <span +class="emmi-10">c </span><span +class="cmsy-10"><span +class="underline">></span> </span>4. From the equations (<a + href="#1-7001r13">13</a>), (<a + href="#1-7004r16">16</a>) and (<a + href="#1-7005r18">18</a>) and from the conditions (<a + href="#1-7008r22">22</a>) +and (<a + href="#1-7008r23">23</a>) the following limits can be derived: <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa39x.gif"alt=" (n3 + c2)(Emax + I) (n3 + c2)(Emax +I) +amin =---------------2- < a < amax = -2--------------- + n3(Emax + I)+ c I c(Emax + I)+ n3I"class="mathdisplay"><a + name="1-7009r25"></a></center></td><td width="5%">(25)</td></tr></table> +This conditions gives a lower limit to number of the ionisations <span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub> for which the fast +sampling can be done: <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa40x.gif"alt="n3 > c2"class="mathdisplay"><a + name="1-7010r26"></a></center></td><td width="5%">(26)</td></tr></table> +As in the conditions (<a + href="#1-7008r22">22</a>), (<a + href="#1-7008r23">23</a>) and (<a + href="#1-7008r24">24</a>) the value of <span +class="emmi-10">c </span>is as minimum 4, one gets +<span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub> <span +class="cmsy-10"><span +class="underline">></span> </span>16. In order to speed the simulation, the maximum value is used for +<span +class="emmi-10"><img +src="emmi10-b.gif"alt="a"class="10--b"></span>. +<!--469--><p class="indent"> The number of collisions with energy loss in the interval B (the number of +interactions which has to be simulated directly) increases slowly with the total +number of collisions <span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub>. The maximum number of these collisions can be estimated +as <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa41x.gif"alt="nB,max = n3 -nA,min ~~ n3(<nA> - sA)"class="mathdisplay"><a + name="1-7011r27"></a></center></td><td width="5%">(27)</td></tr></table> +From the previous expressions for <span +class="cmsy-10"><</span><span +class="emmi-10">n</span><sub ><span +class="emmi-7">A</span></sub><span +class="cmsy-10">> </span>and <span +class="emmi-10"><img +src="emmi10-1b.gif"alt="s"class="10--1b"></span><sub ><span +class="emmi-7">A</span></sub> one can derive the condition +<table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa42x.gif"alt=" -2n3c2- +nB < nB,max = n3 + c2"class="mathdisplay"><a + name="1-7012r28"></a></center></td><td width="5%">(28)</td></tr></table> +The following values are obtained with <span +class="emmi-10">c </span>= 4: + <div align="center"><table class="tabular" +cellspacing="0pt" cellpadding="0" +frame="void" ><colgroup><col +id="TBL-6-1"><col +id="TBL-6-2"><col +id="TBL-6-3"><col +id="TBL-6-4"><col +id="TBL-6-5"></colgroup><tr +valign="baseline" id="TBL-6-1-"><td align="left"nowrap id="TBL-6-1-1" +><div class="td11"><span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub> </div></td><td align="left"nowrap id="TBL-6-1-2" +><div class="td11"><span +class="emmi-10">n</span><sub ><span +class="emmi-7">B,max</span></sub></div></td><td align="center"nowrap id="TBL-6-1-3" +><div class="td11"></div></td><td align="right"nowrap id="TBL-6-1-4" +><div class="td11"> <span +class="emmi-10">n</span><sub ><span +class="emr-7">3</span></sub></div></td><td align="right"nowrap id="TBL-6-1-5" +><div class="td11"><span +class="emmi-10">n</span><sub ><span +class="emmi-7">B,max</span></sub></div></td> +</tr><tr +class="hline"><td><hr></td><td><hr></td><td><hr></td><td><hr></td><td><hr></td></tr><tr +valign="baseline" id="TBL-6-2-"><td align="left"nowrap id="TBL-6-2-1" +><div class="td11">16 </div></td><td align="left"nowrap id="TBL-6-2-2" +><div class="td11">16 </div></td><td align="center"nowrap id="TBL-6-2-3" +><div class="td11"></div></td><td align="right"nowrap id="TBL-6-2-4" +><div class="td11"> 200</div></td><td align="right"nowrap id="TBL-6-2-5" +><div class="td11"> 29.63</div></td> +</tr><tr +valign="baseline" id="TBL-6-3-"><td align="left"nowrap id="TBL-6-3-1" +><div class="td11">20 </div></td><td align="left"nowrap id="TBL-6-3-2" +><div class="td11">17.78 </div></td><td align="center"nowrap id="TBL-6-3-3" +><div class="td11"></div></td><td align="right"nowrap id="TBL-6-3-4" +><div class="td11"> 500</div></td><td align="right"nowrap id="TBL-6-3-5" +><div class="td11"> 31.01</div></td> +</tr><tr +valign="baseline" id="TBL-6-4-"><td align="left"nowrap id="TBL-6-4-1" +><div class="td11">50 </div></td><td align="left"nowrap id="TBL-6-4-2" +><div class="td11">24.24 </div></td><td align="center"nowrap id="TBL-6-4-3" +><div class="td11"></div></td><td align="right"nowrap id="TBL-6-4-4" +><div class="td11">1000</div></td><td align="right"nowrap id="TBL-6-4-5" +><div class="td11"> 31.50</div></td> +</tr><tr +valign="baseline" id="TBL-6-5-"><td align="left"nowrap id="TBL-6-5-1" +><div class="td11">100</div></td><td align="left"nowrap id="TBL-6-5-2" +><div class="td11">27.59 </div></td><td align="center"nowrap id="TBL-6-5-3" +><div class="td11"></div></td><td align="right"nowrap id="TBL-6-5-4" +><div class="td11"> <span +class="cmsy-10"><img +src="cmsy10-31.gif"alt=" oo "class="10--31"></span></div></td><td align="right"nowrap id="TBL-6-5-5" +><div class="td11"> 32.00</div></td> </tr></table> +</div> + <h3 class="subsectionHead">5.2 <a + name="1-80005.2"></a><a + name="QQ1-1-10"></a>Special sampling for lower part of the spectrum</h3> +<!--494--><p class="noindent">If the step length is very small (<span +class="cmsy-10"><span +class="underline"><</span> </span>5 mm in gases, <span +class="cmsy-10"><span +class="underline"><</span> </span>2-3 <span +class="emmi-10"><img +src="emmi10-16.gif"alt="m"class="emmi-10--16"align="middle"></span>m in solids) the model gives +0 energy loss for some events. To avoid this, the probability of 0 energy loss is +computed <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa43x.gif"alt=" -(<n >+<n >+<n >) +P( E = 0) = e 1 2 3"class="mathdisplay"><a + name="1-8001r29"></a></center></td><td width="5%">(29)</td></tr></table> +If the probability is bigger than 0.01 a special sampling is done, taking into +account the fact that in these cases the projectile interacts only with the outer +electrons of the atom. An energy level <span +class="emmi-10">E</span><sub ><span +class="emr-7">0</span></sub> = 10 eV is chosen to correspond to +the outer electrons. The mean number of collisions can be calculated from +<table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa44x.gif"alt=" 1 dE +<n> = ----- x + E0 dx"class="mathdisplay"><a + name="1-8002r30"></a></center></td><td width="5%">(30)</td></tr></table> +The number of collisions <span +class="emmi-10">n </span>is sampled from Poisson distribution. In the case of the +thin layers, all the collisions are considered as ionisations and the energy loss is +computed as <table width="100%" +class="equation"><tr><td> + <center> +<img +src="latexexa45x.gif"alt=" n + sum -----E0------ +E = 1 - EEmax+E-ui + i=1 max 0"class="mathdisplay"><a + name="1-8003r31"></a></center></td><td width="5%">(31)</td></tr></table> + <h2 class="likesectionHead"><a + name="1-9000"></a><a + name="QQ1-1-11"></a>References</h2> + <div class="thebibliography"><p class="bibitem"> + [1] <a + name="Xbib-LAND"></a>L.Landau. On the Energy Loss of Fast Particles by Ionisation. Originally + published in <span +class="emti-10">J. Phys.</span>, 8:201, 1944. Reprinted in D.ter Haar, Editor, + <span +class="emti-10">L.D.Landau, Collected papers </span>, page 417. Pergamon Press, Oxford, 1965. + </p><p class="bibitem"> + [2] <a + name="Xbib-SCH1"></a>B.Schorr. Programs for the Landau and the Vavilov distributions and + the corresponding random numbers. <span +class="emti-10">Comp. Phys. Comm.</span>, 7:216, 1974. + </p><p class="bibitem"> + [3] <a + name="Xbib-SELT"></a>S.M.Seltzer and M.J.Berger. Energy loss straggling of protons and + mesons. In <span +class="emti-10">Studies in Penetration of Charged Particles in Matter </span>, Nuclear + Science Series 39, Nat. Academy of Sciences, Washington DC, 1964. + </p><p class="bibitem"> + [4] <a + name="Xbib-TALM"></a>R.Talman. On the statistics of particle identification using ionization. + <span +class="emti-10">Nucl. Inst. Meth.</span>, 159:189, 1979. + </p><p class="bibitem"> + [5] <a + name="Xbib-VAVI"></a>P.V.Vavilov. Ionisation losses of high energy heavy particles. <span +class="emti-10">Soviet</span> + <span +class="emti-10">Physics JETP </span>, 5:749, 1957.</p></div> + +</body> +</html> + diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.idv b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.idv Binary files differnew file mode 100644 index 00000000000..b2eeb02a051 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.idv diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.lg b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.lg new file mode 100644 index 00000000000..4740e4922d9 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.lg @@ -0,0 +1,121 @@ +File: latexexa.html +File: latexexa.css +File: tex4ht.tmp +Font_Class(1,"1"): <img src=""alt=""class="%s-%d--%x"> +Font_Class(3,"3"): <img src=""alt=""class="%s-%s-%d--%x"align="middle"> +File: tex4ht.tmp 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characters out of 72703 + 101183 words of memory out of 263001 + 6117 multiletter control sequences out of 10000+0 + 20729 words of font info for 41 fonts, out of 200000 for 1000 + 14 hyphenation exceptions out of 1000 + 28i,14n,24p,338b,560s stack positions out of 300i,100n,500p,30000b,4000s + +Output written on latexexa.dvi (22 pages, 73912 bytes). diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.otc b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.otc new file mode 100644 index 00000000000..43614e1f94f --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.otc @@ -0,0 +1,13 @@ +\expandafter\ifx\csname doTocEntry\endcsname\relax \expandafter\endinput\fi +\doTocEntry\tocsection{1}{\csname a:TocEntry\endcsname{1}{QQ1-1-1}{QQ2-1-1}{Introduction}}{2}\relax +\doTocEntry\toclof{1}{\Link{1-10061}{}\ignorespaces The variable $\xi /I_0$ can be used to measure the validity range of the Landau theory. It depends on the type and energy of the particle, $Z$, $A$ and the ionisation potential of the material and the layer thickness. \EndLink}{figure} +\doTocEntry\tocsection{2}{\csname a:TocEntry\endcsname{1}{QQ1-1-3}{QQ2-1-3}{Landau theory}}{7}\relax +\doTocEntry\tocsubsection{2.1}{\csname a:TocEntry\endcsname{1}{QQ1-1-4}{QQ2-1-4}{Restrictions}}{7}\relax +\doTocEntry\tocsection{3}{\csname a:TocEntry\endcsname{1}{QQ1-1-5}{QQ2-1-5}{Vavilov theory}}{9}\relax +\doTocEntry\tocsection{4}{\csname a:TocEntry\endcsname{1}{QQ1-1-6}{QQ2-1-6}{Gaussian Theory}}{10}\relax +\doTocEntry\tocsection{5}{\csname a:TocEntry\endcsname{1}{QQ1-1-7}{QQ2-1-7}{Urb\protect\' an model}}{10}\relax +\doTocEntry\toclof{2}{\Link{1-60012}{}\ignorespaces Energy loss distribution for a 3 GeV electron in Argon as given by standard \protect\texttt {GEANT}. The width of the layers is given in centimeters.\EndLink}{figure} +\doTocEntry\tocsubsection{5.1}{\csname a:TocEntry\endcsname{1}{QQ1-1-9}{QQ2-1-9}{Fast simulation for $n_3 \geq 16$}}{16}\relax +\doTocEntry\tocsubsection{5.2}{\csname a:TocEntry\endcsname{1}{QQ1-1-10}{QQ2-1-10}{Special sampling for lower part of the spectrum}}{20}\relax +\doTocEntry\toclikesection{}{\csname a:TocEntry\endcsname{1}{QQ1-1-11}{QQ2-1-11}{References}}{21}\relax +\par diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.tex b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.tex new file mode 100644 index 00000000000..faa7158a0ca --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.tex @@ -0,0 +1,549 @@ +\documentclass{article} +\usepackage{graphicx} +\usepackage{url} +\title{Simulation of Energy Loss Straggling} +\author{Maria Physicist} +\newcommand{\Emax}{\ensuremath{E_{\mathrm{max}}}} +\newcommand{\GEANT}{\texttt{GEANT}} +\usepackage[T1]{fontenc} +\usepackage{em} +\usepackage{tex4ht} +\begin{document} +\maketitle + +\section{Introduction} + +Due to the statistical nature of ionisation energy loss, large +fluctuations can occur in the amount of energy deposited by a particle +traversing an absorber element. Continuous processes such as multiple +scattering and energy loss play a relevant role in the longitudinal +and lateral development of electromagnetic and hadronic +showers, and in the case of sampling calorimeters the +measured resolution can be significantly affected by such fluctuations +in their active layers. The description of ionisation fluctuations is +characterised by the significance parameter $\kappa$, which is +proportional to the ratio of mean energy loss to the maximum allowed +energy transfer in a single collision with an atomic electron +\[ +\kappa =\frac{\xi}{\Emax} +\] +\Emax{} +is the maximum transferable energy in a single collision with +an atomic electron. +\[ +\Emax =\frac{2 m_e \beta^2\gamma^2 } +{1 + 2\gamma m_e/m_x + \left ( m_e/m_x\right)^2}, +\] +where $\gamma = E/m_x$, $E$ is energy and +$m_x$ the mass of the incident particle, +$\beta^2 = 1 - 1/\gamma^2$ and $m_e$ is the electron mass. +$\xi$ comes from the Rutherford scattering cross section +and is defined as: +\begin{eqnarray*} \xi = \frac{2\pi z^2 e^4 N_{Av} Z \rho \delta x} + {m_e \beta^2 c^2 A} = 153.4 \frac{z^2} {\beta^2} \frac{Z}{A} + \rho \delta x \quad\mathrm{keV}, +\end{eqnarray*} +where + +\begin{tabular}{ll} +$z$ & charge of the incident particle \\ +$N_{Av}$ & Avogadro's number \\ +$Z$ & atomic number of the material \\ +$A$ & atomic weight of the material \\ +$\rho$ & density \\ +$ \delta x$ & thickness of the material \\ +\end{tabular} + +$\kappa$ measures the contribution of the collisions with energy +transfer close to \Emax. For a given absorber, $\kappa$ tends +towards large values if $\delta x$ is large and/or if $\beta$ is +small. Likewise, $\kappa$ tends towards zero if $\delta x $ is small +and/or if $\beta$ approaches 1. + +The value of $\kappa$ distinguishes two regimes which occur in the +description of ionisation fluctuations : + +\begin{enumerate} +\item A large number of collisions involving the loss of all or most + of the incident particle energy during the traversal of an absorber. + + As the total energy transfer is composed of a multitude of small + energy losses, we can apply the central limit theorem and describe + the fluctuations by a Gaussian distribution. This case is + applicable to non-relativistic particles and is described by the + inequality $\kappa > 10 $ (i.e. when the mean energy loss in the + absorber is greater than the maximum energy transfer in a single + collision). + +\item Particles traversing thin counters and incident electrons under + any conditions. + + The relevant inequalities and distributions are $ 0.01 < \kappa < 10 + $, Vavilov distribution, and $\kappa < 0.01 $, Landau distribution. +\end{enumerate} + +An additional regime is defined by the contribution of the collisions +with low energy transfer which can be estimated with the relation +$\xi/I_0$, where $I_0$ is the mean ionisation potential of the atom. +Landau theory assumes that the number of these collisions is high, and +consequently, it has a restriction $\xi/I_0 \gg 1$. In \GEANT{} +(see URL \url{http://wwwinfo.cern.ch/asdoc/geant/geantall.html}), the +limit of Landau theory has been set at $\xi/I_0 = 50$. Below this +limit special models taking into account the atomic structure of the +material are used. This is important in thin layers and gaseous +materials. Figure \ref{fg:phys332-1} shows the behaviour of $\xi/I_0$ +as a function of the layer thickness for an electron of 100 keV and 1 +GeV of kinetic energy in Argon, Silicon and Uranium. + +\begin{figure} + \centering + \includegraphics[width=.6\linewidth]{phys332-1} + \caption{The variable $\xi/I_0$ can be used to measure the + validity range of the Landau theory. It depends + on the type and energy of the particle, $Z$, $A$ + and the ionisation potential of the material and + the layer thickness. + } + \label{fg:phys332-1} +\end{figure} + +In the following sections, the different theories and models for the +energy loss fluctuation are described. First, the Landau theory and +its limitations are discussed, and then, the Vavilov and Gaussian +straggling functions and the methods in the thin layers and gaseous +materials are presented. + +\section{Landau theory} +\label{sec:phys332-1} + +For a particle of mass $m_x$ traversing a thickness of material +$\delta x $, the Landau probability distribution may be written in +terms of the universal Landau function $\phi(\lambda)$ +as\cite{bib-LAND}: +\begin{eqnarray*} +f( \epsilon , \delta x ) & = &\frac{1}{\xi} \phi ( \lambda ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi(\lambda )& = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty} +\exp \left ( u \ln u + \lambda u \right ) du \hspace{2cm} c \geq 0 \\ +\lambda & = & \frac{\epsilon -\bar{\epsilon} }{\xi} + - \gamma' - \beta^2 - \ln \frac{\xi}{\Emax} \\ +\gamma' & = & 0.422784\dots = 1 - \gamma \\ +\gamma & = & 0.577215\dots \mbox{(Euler's constant)} \\ +\bar{\epsilon} & = & \mbox{average energy loss} \\ +\epsilon & = & \mbox{actual energy loss} +\end{eqnarray*} + +\subsection{Restrictions} + +The Landau formalism makes two restrictive assumptions : +\begin{enumerate} +\item The typical energy loss is small compared to the maximum energy + loss in a single collision. This restriction is removed in the + Vavilov theory (see section \ref{vavref}). +\item The typical energy loss in the absorber should be large compared + to the binding energy of the most tightly bound electron. For + gaseous detectors, typical energy losses are a few keV which is + comparable to the binding energies of the inner electrons. In such + cases a more sophisticated approach which accounts for atomic energy + levels\cite{bib-TALM} is necessary to accurately simulate data + distributions. In \GEANT, a parameterised model by L. Urb\'{a}n is + used (see section \ref{urban}). +\end{enumerate} + +In addition, the average value of the Landau distribution is infinite. +Summing the Landau fluctuation obtained to the average energy from the +$dE/dx$ tables, we obtain a value which is larger than the one coming +from the table. The probability to sample a large value is small, so +it takes a large number of steps (extractions) for the average +fluctuation to be significantly larger than zero. This introduces a +dependence of the energy loss on the step size which can affect +calculations. + +A solution to this has been to introduce a limit on the value of the +variable sampled by the Landau distribution in order to keep the +average fluctuation to 0. The value obtained from the \texttt{GLANDO} +routine is: +\[ +\delta dE/dx = \epsilon - \bar{\epsilon} = \xi ( \lambda - \gamma' ++\beta^2 +\ln \frac{\xi}{\Emax}) +\] +In order for this to have average 0, we must impose that: +\[ +\bar{\lambda} = -\gamma' - \beta^2 -\ln \frac{\xi}{\Emax} +\] + +This is realised introducing a $\lambda_{\mathrm{max}}(\bar{\lambda})$ +such that if only values of $\lambda \leq \lambda_{\mathrm{max}}$ are +accepted, the average value of the distribution is $\bar{\lambda}$. + +A parametric fit to the universal Landau distribution has been +performed, with following result: +\[ +\lambda_{\mathrm{max}} = 0.60715 + + 1.1934\bar{\lambda}+(0.67794+0.052382\bar{\lambda}) + \exp(0.94753+0.74442\bar{\lambda}) +\] +only values smaller than $\lambda_{\mathrm{max}}$ are accepted, otherwise the +distribution is resampled. + + + +\newpage +\section{Vavilov theory} +\label{vavref} + +Vavilov\cite{bib-VAVI} derived a more accurate straggling distribution +by introducing the kinematic limit on the maximum transferable energy +in a single collision, rather than using $ \Emax = \infty $. +Now we can write\cite{bib-SCH1}: +\begin{eqnarray*} +f \left ( \epsilon, \delta s \right ) & = & \frac{1}{\xi} \phi_{v} +\left ( \lambda_{v}, \kappa, \beta^{2} \right ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi_{v} \left ( \lambda_{v}, \kappa, \beta^{2} \right ) & = & +\frac{1}{2 \pi i} \int^{c+i\infty}_{c-i\infty}\phi \left( s \right ) +e^{\lambda s} ds \hspace{2cm} c \geq 0 \\ +\phi \left ( s \right ) & = & +\exp \left [ \kappa ( 1 + \beta^{2}\gamma ) \right ] +~ \exp \left [ \psi \left ( s \right ) \right ], \\ +\psi \left ( s \right ) & = & s \ln \kappa + ( s + \beta^{2} \kappa ) +\left [ \ln (s/\kappa) + E_{1} (s/\kappa) \right ] - \kappa e^{-s/\kappa}, +\end{eqnarray*} +and +\begin{eqnarray*} +E_{1}(z) & = & \int^{\infty}_{z} t^{-1} e^{-t} dt +\mbox{\hspace{1cm} (the exponential integral)} \\ +\lambda_v & = & \kappa \left [ \frac{\epsilon - \bar{\epsilon}}{\xi} +- \gamma' - \beta^2 \right] +\end{eqnarray*} + +The Vavilov parameters are simply related to the Landau parameter by +$\lambda_L = \lambda_v/\kappa - \ln\kappa $. It can be shown that as +$\kappa \rightarrow 0$, the distribution of the variable $\lambda_L$ +approaches that of Landau. For $\kappa \leq 0.01$ the two +distributions are already practically identical. Contrary to what many +textbooks report, the Vavilov distribution \emph{does not} approximate +the Landau distribution for small $\kappa$, but rather the +distribution of $\lambda_L$ defined above tends to the distribution of +the true $\lambda$ from the Landau density function. Thus the routine +\texttt{GVAVIV} samples the variable $\lambda_L$ rather than +$\lambda_v$. For $\kappa \geq 10$ the Vavilov distribution tends to a +Gaussian distribution (see next section). + +\section{Gaussian Theory} + +Various conflicting forms have been proposed for Gaussian straggling +functions, but most of these appear to have little theoretical or +experimental basis. However, it has been shown\cite{bib-SELT} that +for $\kappa \geq 10 $ the Vavilov distribution can be replaced by a +Gaussian of the form : +\begin{eqnarray*} +f( \epsilon , \delta s) \approx \frac{1} +{\xi \sqrt{\frac{2 \pi}{\kappa} \left( 1 - \beta^2/2 \right)}} + \exp \left [ \frac{( \epsilon - \bar{\epsilon} )^2}{2} \frac{\kappa} + {\xi^2 (1- \beta^2/2)}\right ] +\end{eqnarray*} +thus implying +\begin{eqnarray*} +\mathrm{mean} & = & \bar{\epsilon} \\ +\sigma^2 & = & \frac{\xi^2}{\kappa} (1-\beta^2/2) = \xi + \Emax (1-\beta^2/2) +\end{eqnarray*} + +\section{Urb\'an model} +\label{urban} + +The method for computing restricted energy losses with $\delta$-ray +production above given threshold energy in \GEANT{} is a Monte +Carlo method that can be used for thin layers. It is fast and it can +be used for any thickness of a medium. Approaching the limit of the +validity of Landau's theory, the loss distribution approaches smoothly +the Landau form as shown in Figure \ref{fg:phys332-2}. +\begin{figure} + \centering + \includegraphics[width=.6\linewidth]{phys332-2} + \caption{Energy loss distribution for a 3 GeV electron in + Argon as given by standard \GEANT. The width of the layers is + given in centimeters.} + \label{fg:phys332-2} +\end{figure} + +It is assumed that the atoms have only two energy levels with binding +energy $E_1$ and $E_2$. The particle--atom interaction will then be +an excitation with energy loss $E_1$ or $E_2$, or an ionisation with +an energy loss distributed according to a function $g(E) \sim 1/E^2$: +\begin{equation} +g(E) = \frac{(\Emax + I)I}{\Emax} \frac{1}{E^2} +\end{equation} + +The macroscopic cross-section for excitations ($i=1,2$) is +\begin{equation} +\label{eq:sigex} +\Sigma_i = C \frac{f_i}{E_i} \frac{\ln (2 m \beta^2 \gamma^2/E_i) - \beta^2} + {\ln (2 m \beta^2 \gamma^2/ I) - \beta^2}(1-r) +\end{equation} +and the macroscopic cross-section for ionisation is +\begin{equation} +\label{eq:sigion} +\Sigma_3 = C \frac{\Emax}{I(\Emax+I) \ln(\frac{\Emax+I}{I})} + ~ r +\end{equation} +\Emax{} is the \GEANT{} cut for $\delta$-production, or the maximum +energy transfer minus mean ionisation energy, if it is smaller than +this cut-off value. The following notation is used: + +\begin{tabular}{ll} +$r, C$ & parameters of the model \\ +$E_i$ & atomic energy levels \\ +$I$ & mean ionisation energy \\ +${f_i}$ & oscillator strengths +\end{tabular} + +The model has the parameters $f_i$, $E_i$, $C$ and $r ~(0\leq r\leq +1)$. The oscillator strengths $f_i$ and the atomic level energies +$E_i$ should satisfy the constraints +\begin{eqnarray} +f_1 + f_2 & = & 1 \label{eq:fisum}\\ +f_1 \ln E_1 + f_2 \ln E_2 & = & \ln I \label{eq:flnsum} +\end{eqnarray} +The parameter $C$ can be defined with the help of the mean energy loss +$dE/dx$ in the following way: The numbers of collisions ($n_i$, i = +1,2 for the excitation and 3 for the ionisation) follow the Poisson +distribution with a mean number $ \langle n_i \rangle $. In a step +$\Delta x$ the mean number of collisions is +\begin{equation} +\langle n_i \rangle = \Sigma_i \Delta x +\end{equation} +The mean energy loss $dE/dx$ in a step is the sum of the excitation +and ionisation contributions +\begin{equation} +\frac{dE}{dx} \Delta x = \left[ \Sigma_1 E_1 + \Sigma_2 E_2 + + \Sigma_3 \int_{I}^{\Emax+I} E~g(E)~dE \right] + \Delta x +\end{equation} +From this, using the equations (\ref{eq:sigex}), (\ref{eq:sigion}), +(\ref{eq:fisum}) and (\ref{eq:flnsum}), one can define the parameter +$C$ +\begin{equation} +C = \frac{dE}{dx} +\end{equation} + +The following values have been chosen in \GEANT{} for the other +parameters: +\[ +\begin{array}{lcl} +f_2 = \left\{ \begin{array}{ll} + 0 & \mathrm{if}\, Z \leq 2 \\ + 2/Z & \mathrm{if}\, Z > 2 \\ + \end{array} \right. & \Rightarrow & f_1 = 1 - f_2 \\ +E_2 = 10 Z^2 \mathrm{eV} & \Rightarrow & E_1 = \left(\frac{I}{E_{2}^{f_2}} + \right)^{\frac{1}{f_1}} \\ +r = 0.4 & & \\ +\end{array} +\] +With these values the atomic level $E_2$ corresponds approximately +the K-shell energy of the atoms and $Zf_2$ the number of K-shell +electrons. $r$ is the only variable which can be tuned freely. It +determines the relative contribution of ionisation and +excitation to the energy loss. + +The energy loss is computed with the assumption that the step length +(or the relative energy loss) is small, and---in consequence---the +cross-section can be considered constant along the path length. The +energy loss due to the excitation is +\begin{equation} +\Delta E_e = n_1 E_1 + n_2 E_2 +\end{equation} +where $n_1$ and $n_2$ are sampled from Poisson distribution as +discussed above. The loss due to the ionisation can be generated from +the distribution $g(E)$ by the inverse transformation method: +\begin{eqnarray} +u = F(E) & = & \int_{I}^E g(x) dx \nonumber \\ +E = F^{-1}(u) & = & \frac{I}{1 - u \frac {\Emax}{\Emax+I}} \\ +\end{eqnarray} +where $u$ is a uniform random number between $F(I)=0$ and +$F(\Emax+I)=1$. The contribution from the ionisations will be +\begin{equation} +\Delta E_i = \sum_{j=1}^{n_3} \frac{I} + {1 - u_j \frac {\Emax}{\Emax + I}} +\end{equation} +where $n_3$ is the number of ionisation (sampled from Poisson +distribution). The energy loss in a step will then be $\Delta E = +\Delta E_e + \Delta E_i$. + +\subsection{Fast simulation for $n_3 \geq 16$} + +If the number of ionisation $n_3$ is bigger than 16, a faster sampling +method can be used. The possible energy loss interval is divided in +two parts: one in which the number of collisions is large and the +sampling can be done from a Gaussian distribution and the other in +which the energy loss is sampled for each collision. Let us call the +former interval $[I, \alpha I]$ the interval A, and the latter +$[\alpha I,\Emax]$ the interval B. $\alpha$ lies between 1 and +$\Emax/I$. A collision with a loss in the interval A happens with +the probability +\begin{equation} +\label{eq:phys332-5} +P(\alpha) = \int_I^{\alpha I} g(\!E\!) \, dE = + \frac {( \Emax + I) (\alpha - 1)}{\Emax \alpha} +\end{equation} +The mean energy loss and the standard deviation for this type +of collision are +\begin{equation} +\langle \Delta E(\alpha) \rangle = \frac{1}{P(\alpha)} + \int_I^{\alpha I} E \, g(\!E\!) \, dE = + \frac{I \alpha \ln \alpha}{\alpha - 1} +\end{equation} +and +\begin{equation} +\sigma^2(\alpha) = \frac{1}{P(\alpha)} + \int_I^{\alpha I} E^2 \, g(\!E\!) \, dE = + I^2 \alpha \left( 1 - \frac{\alpha \ln \! ^2 \alpha}{(\alpha - 1)^2} \right) +\end{equation} +If the collision number is high , we assume that the number of the +type A collisions can be calculated from a Gaussian distribution +with the following mean value and standard deviation: +\begin{eqnarray} +\label{eq:phys332-1} +\langle n_A \rangle & = & n_3 P(\alpha) \\ +\label{eq:phys332-2} +\sigma_A^2 & = & n_3 P(\alpha) ( 1 - P(\alpha)) +\end{eqnarray} +It is further assumed that the energy loss in these collisions +has a Gaussian distribution with +\begin{eqnarray} +\label{eq:phys332-3} +\langle \Delta E_A \rangle & = & n_A \langle \Delta E(\alpha) \rangle \\ +\label{eq:phys332-4} +\sigma_{E,A}^2 & = & n_A \sigma^2(\alpha) +\end{eqnarray} +The energy loss of these collision can then be sampled from the +Gaussian distribution. + +The collisions where the energy loss is in the interval B are sampled +directly from +\begin{equation} +\Delta E_B = \sum_{i=1}^{n_3 - n_A} \frac{\alpha I} + {1 - u_i \frac{\Emax + I - \alpha I}{\Emax + I}} +\end{equation} +The total energy loss is the sum of these two types of collisions: +\begin{equation} +\Delta E = \Delta E_A + \Delta E_B +\end{equation} + +The approximation of equations ((\ref{eq:phys332-1}), +(\ref{eq:phys332-2}), (\ref{eq:phys332-3}) and (\ref{eq:phys332-4}) +can be used under the following conditions: +\begin{eqnarray} +\label{eq:phys332-6} +\langle n_A \rangle - c \, \sigma_A & \geq & 0 \\ +\label{eq:phys332-7} +\langle n_A \rangle + c \, \sigma_A & \leq & n_3 \\ +\label{eq:phys332-8} +\langle \Delta E_A \rangle - c \, \sigma_{E,A} & \geq & 0 +\end{eqnarray} +where $c \geq 4$. From the equations (\ref{eq:phys332-5}), +(\ref{eq:phys332-1}) and (\ref{eq:phys332-3}) and from the conditions +(\ref{eq:phys332-6}) and (\ref{eq:phys332-7}) the following limits can +be derived: +\begin{equation} +\alpha_{\mathrm{min}} = \frac{(n_3 + c^2)(\Emax+I)} + {n_3 (\Emax + I) + c^2 I} \; \leq \alpha \; \leq +\alpha_{\mathrm{max}} = \frac{(n_3 + c^2)(\Emax+I)} + {c^2 (\Emax + I) + n_3 I} +\end{equation} +This conditions gives a lower limit to number of the ionisations $n_3$ +for which the fast sampling can be done: +\begin{equation} +n_3 \; \geq \; c^2 +\end{equation} +As in the conditions (\ref{eq:phys332-6}), (\ref{eq:phys332-7}) and +(\ref{eq:phys332-8}) the value of $c$ is as minimum 4, one gets $n_3 +\; \geq 16$. In order to speed the simulation, the maximum value is +used for $\alpha$. + +The number of collisions with energy loss in the interval B (the +number of interactions which has to be simulated directly) increases +slowly with the total number of collisions $n_3$. The maximum number +of these collisions can be estimated as +\begin{equation} +n_{B,max} = n_3 - n_{A,min} \approx n_3 (\langle n_A \rangle + - \sigma_A) +\end{equation} +From the previous expressions for $\langle n_A \rangle$ and $\sigma_A$ +one can derive the condition +\begin{equation} +n_B \; \leq \; n_{B,max} = \frac{2 n_3 c^2}{n_3+c^2} +\end{equation} +The following values are obtained with $c=4$: + +\begin{tabular}{llcrr} +$n_3$ & $n_{B,max}$ & & $n_3$ & $n_{B,max}$\\ \hline +16 & 16 & & 200 & 29.63\\ +20 & 17.78 & & 500 & 31.01 \\ +50 & 24.24 & & 1000 & 31.50 \\ +100 & 27.59 & & $\infty$ & 32.00 +\end{tabular} + +\subsection{Special sampling for lower part of the spectrum} + +If the step length is very small ($\leq 5$ mm in gases, $\leq$ 2-3 +$\mu$m in solids) the model gives 0 energy loss for some events. To +avoid this, the probability of 0 energy loss is computed +\begin{equation} +P( \Delta E=0) = e^{-( \langle n_1 \rangle + \langle n_2 \rangle + + \langle n_3 \rangle )} +\end{equation} +If the probability is bigger than 0.01 a special sampling is done, +taking into account the fact that in these cases the projectile +interacts only with the outer electrons of the atom. An energy level +$E_0 = 10$ eV is chosen to correspond to the outer electrons. The mean +number of collisions can be calculated from +\begin{equation} +\langle n \rangle = \frac{1}{E_0} \frac{dE}{dx} \Delta x +\end{equation} +The number of collisions $n$ is sampled from Poisson distribution. +In the case of the thin layers, all the collisions are considered as +ionisations and the energy loss is computed as +\begin{equation} +\Delta E = \sum_{i=1}^n \frac{E_0} + {1 - \frac {\Emax}{\Emax + E_0} u_i} +\end{equation} + +\begin{thebibliography}{10} +\bibitem{bib-LAND} +L.Landau. +\newblock On the Energy Loss of Fast Particles by Ionisation. +\newblock Originally published in \emph{J. Phys.}, 8:201, 1944. +\newblock Reprinted in D.ter Haar, Editor, \emph{L.D.Landau, Collected + papers}, page 417. Pergamon Press, Oxford, 1965. + +\bibitem{bib-SCH1} +B.Schorr. +\newblock Programs for the Landau and the Vavilov distributions and the + corresponding random numbers. +\newblock \emph{Comp. Phys. Comm.}, 7:216, 1974. + +\bibitem{bib-SELT} +S.M.Seltzer and M.J.Berger. +\newblock Energy loss straggling of protons and mesons. +\newblock In \emph{Studies in Penetration of Charged Particles in + Matter}, Nuclear Science Series~39, Nat. Academy of Sciences, + Washington DC, 1964. + +\bibitem{bib-TALM} +R.Talman. +\newblock On the statistics of particle identification using ionization. +\newblock \emph{Nucl. Inst. Meth.}, 159:189, 1979. + +\bibitem{bib-VAVI} +P.V.Vavilov. +\newblock Ionisation losses of high energy heavy particles. +\newblock \emph{Soviet Physics JETP}, 5:749, 1957. +\end{thebibliography} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.toc b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.toc new file mode 100644 index 00000000000..15064d9096e --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.toc @@ -0,0 +1,12 @@ +\expandafter\ifx\csname doTocEntry\endcsname\relax \expandafter\endinput\fi +\doTocEntry\tocsection{1}{\csname a:TocEntry\endcsname{1}{QQ1-1-1}{QQ2-1-1}{Introduction}}{2}\relax +\doTocEntry\toclof{1}{\Link{1-10061}{}\ignorespaces The variable $\xi /I_0$ can be used to measure the validity range of the Landau theory. It depends on the type and energy of the particle, $Z$, $A$ and the ionisation potential of the material and the layer thickness. \EndLink}{figure} +\doTocEntry\tocsection{2}{\csname a:TocEntry\endcsname{1}{QQ1-1-3}{QQ2-1-3}{Landau theory}}{7}\relax +\doTocEntry\tocsubsection{2.1}{\csname a:TocEntry\endcsname{1}{QQ1-1-4}{QQ2-1-4}{Restrictions}}{7}\relax +\doTocEntry\tocsection{3}{\csname a:TocEntry\endcsname{1}{QQ1-1-5}{QQ2-1-5}{Vavilov theory}}{9}\relax +\doTocEntry\tocsection{4}{\csname a:TocEntry\endcsname{1}{QQ1-1-6}{QQ2-1-6}{Gaussian Theory}}{10}\relax +\doTocEntry\tocsection{5}{\csname a:TocEntry\endcsname{1}{QQ1-1-7}{QQ2-1-7}{Urb\protect\' an model}}{10}\relax +\doTocEntry\toclof{2}{\Link{1-60012}{}\ignorespaces Energy loss distribution for a 3 GeV electron in Argon as given by standard \protect\texttt {GEANT}. The width of the layers is given in centimeters.\EndLink}{figure} +\doTocEntry\tocsubsection{5.1}{\csname a:TocEntry\endcsname{1}{QQ1-1-9}{QQ2-1-9}{Fast simulation for $n_3 \geq 16$}}{16}\relax +\doTocEntry\tocsubsection{5.2}{\csname a:TocEntry\endcsname{1}{QQ1-1-10}{QQ2-1-10}{Special sampling for lower part of the spectrum}}{20}\relax +\doTocEntry\toclikesection{}{\csname a:TocEntry\endcsname{1}{QQ1-1-11}{QQ2-1-11}{References}}{21}\relax diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.xref b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.xref new file mode 100644 index 00000000000..4c7a40b1849 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/latexexa.xref @@ -0,0 +1,81 @@ +\:CrossWord{)F1F-}{latexexa.html}{1}% +\:CrossWord{)title)}{Simulation of Energy Loss Straggling}{1}% +\:CrossWord{)M1x0}{;}{2}% +\:CrossWord{)Q1-10001}{1}{2}% +\:CrossWord{)QQQ1-1-1}{1}{2}% +\:CrossWord{)Q1-1001r1}{1}{3}% +\:CrossWord{)M2x1}{;}{3}% 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b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/phys332-2.eps new file mode 100644 index 00000000000..e255ddadf55 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/phys332-2.eps @@ -0,0 +1,556 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 0 0 567 567 +%%Title: curves.eps +%%Creator: HIGZ Version 1.19/01 +%%CreationDate: 93/07/16 12.02 +%%EndComments +80 dict begin +/s {stroke} def /l {lineto} def /m {moveto} def /t { translate} def +/sw {stringwidth} def /r {rotate} def /rl {roll} def +/d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def +/c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def +/cl {closepath} def /sf {scalefont setfont} def +/box {m dup 0 exch d exch 0 d 0 exch neg d cl} def +/bl {box s} def /bf {box f} def +/mp {newpath /y exch def /x exch def} def +/side {[w .77 mul w .23 mul] .385 w mul sd w 0 l currentpoint t -144 r} def +/mr {mp x y w2 0 360 arc} def /m24 {mr s} def /m20 {mr f} def +/mb {mp x y w2 add m w2 neg 0 d 0 w neg d w 0 d 0 w d cl} def +/mt {mp x y w2 add m w2 neg w neg d w 0 d cl} def +/m21 {mb f} def /m25 {mb s} def /m22 {mt f} def /m26 {mt s} def +/m23 {mp x y w2 sub m w2 w d w neg 0 d cl f} def + /m27 {mp x y w2 add m w3 neg w2 neg d w3 w2 neg d w3 w2 d cl s} def + /m28 {mp x w2 sub y w2 sub w3 add m w3 0 d 0 w3 neg d w3 0 d 0 w3 d w3 0 d + 0 w3 d w3 neg 0 d 0 w3 d w3 neg 0 d 0 w3 neg d w3 neg 0 d cl s } def + /m29 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t + 4 {side} repeat cl fill gr} def + /m30 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t + 5 {side} repeat s gr} def /m31 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d + x w2 sub y w2 add m w w neg d x w2 sub y w2 + sub m w w d s} def +/m2 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d s} def +/m5 {mp x w2 sub y w2 sub m w w d x w2 sub y w2 add m w w neg d s} def +/reencdict 24 dict def /ReEncode {reencdict begin /nco&na exch def +/nfnam exch def /basefontname exch def /basefontdict basefontname findfont def +/newfont basefontdict maxlength dict def basefontdict {exch dup /FID ne +{dup /Encoding eq {exch dup length array copy newfont 3 1 roll put} {exch +newfont 3 1 roll put} ifelse} {pop pop} ifelse } forall newfont +/FontName nfnam put nco&na aload pop nco&na length 2 idiv {newfont +/Encoding get 3 1 roll put} repeat nfnam newfont definefont pop end } def +/accvec [ 176 /agrave 181 /Agrave 190 /acircumflex 192 /Acircumflex +201 /adieresis 204 /Adieresis 209 /ccedilla 210 /Ccedilla 211 /eacute +212 /Eacute 213 /egrave 214 /Egrave 215 /ecircumflex 216 /Ecircumflex +217 /edieresis 218 /Edieresis 219 /icircumflex 220 /Icircumflex +221 /idieresis 222 /Idieresis 223 /ntilde 224 /Ntilde 226 /ocircumflex +228 /Ocircumflex 229 /odieresis 230 /Odieresis 231 /ucircumflex 236 /Ucircumflex +237 /udieresis 238 /Udieresis 239 /aring 242 /Aring 243 /ydieresis +244 /Ydieresis 246 /aacute 247 /Aacute 252 /ugrave 253 /Ugrave] def +/Times-Roman /Times-Roman accvec ReEncode +/Times-Italic /Times-Italic accvec ReEncode +/Times-Bold /Times-Bold accvec ReEncode +/Times-BoldItalic /Times-BoldItalic accvec ReEncode +/Helvetica /Helvetica accvec ReEncode +/Helvetica-Oblique /Helvetica-Oblique accvec ReEncode +/Helvetica-Bold /Helvetica-Bold accvec ReEncode +/Helvetica-BoldOblique /Helvetica-BoldOblique accvec ReEncode +/Courier /Courier accvec ReEncode +/Courier-Oblique /Courier-Oblique accvec ReEncode +/Courier-Bold /Courier-Bold accvec ReEncode +/Courier-BoldOblique /Courier-BoldOblique accvec ReEncode +/oshow {gsave [] 0 sd true charpath stroke gr} def +/stwn { /fs exch def /fn exch def /text exch def fn findfont fs sf + text sw pop xs add /xs exch def} def +/stwb { /fs exch def /fn exch def /nbas exch def /textf exch def +textf length /tlen exch def nbas tlen gt {/nbas tlen def} if +fn findfont fs sf textf dup length nbas sub nbas getinterval sw +pop neg xs add /xs exch def} def +/accspe [ 65 /plusminus 66 /bar 67 /existential 68 /universal +69 /exclam 70 /numbersign 71 /greater 72 /question 73 /integral +74 /colon 75 /semicolon 76 /less 77 /bracketleft 78 /bracketright +79 /greaterequal 80 /braceleft 81 /braceright 82 /radical +83 /spade 84 /heart 85 /diamond 86 /club 87 /lessequal +88 /multiply 89 /percent 90 /infinity 48 /circlemultiply 49 /circleplus +50 /emptyset 51 /lozenge 52 /bullet 53 /arrowright 54 /arrowup +55 /arrowleft 56 /arrowdown 57 /arrowboth 48 /degree 44 /comma 43 /plus + 45 /angle 42 /angleleft 47 /divide 61 /notequal 40 /equivalence 41 /second + 97 /approxequal 98 /congruent 99 /perpendicular 100 /partialdiff 101 /florin + 102 /intersection 103 /union 104 /propersuperset 105 /reflexsuperset + 106 /notsubset 107 /propersubset 108 /reflexsubset 109 /element 110 /notelement + 111 /gradient 112 /logicaland 113 /logicalor 114 /arrowdblboth + 115 /arrowdblleft 116 /arrowdblup 117 /arrowdblright 118 /arrowdbldown + 119 /ampersand 120 /omega1 121 /similar 122 /aleph ] def +/Symbol /Special accspe ReEncode +gsave .25 .25 scale +%%EndProlog + gsave 0 0 t 0 setgray [] 0 sd 1 lw 4 lw 2268 2268 0 0 bl + gsave 771 1813 + t 0 r 0 0 m + /Helvetica-Bold findfont 47 sf 0 0 m +(Landau) + show + gr + /xs 0 def +(40) + /Helvetica-Bold 54 stwn + gsave 862 1533 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 54 sf 0 0 m +(40) + show + gr + /xs 0 def +(20) + /Helvetica-Bold 54 stwn + gsave 789 1347 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 54 sf 0 0 m +(20) + show + gr + /xs 0 def +(10) + /Helvetica-Bold 54 stwn + gsave 771 1011 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 54 sf 0 0 m +(10) + show + gr + /xs 0 def +(5) + /Helvetica-Bold 54 stwn + gsave 699 824 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 54 sf 0 0 m +(5) + show + gr + /xs 0 def +(1) + /Helvetica-Bold 54 stwn + gsave 517 740 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 54 sf 0 0 m +(1) + show + gr + /xs 0 def +(0.5) + /Helvetica-Bold 54 stwn + gsave 336 525 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 54 sf 0 0 m +(0.5) + show + gr + /xs 0 def 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726 1860 l 748 1860 l 748 + 1697 l 771 1697 l 771 1393 l 794 1393 l 794 992 l 816 992 l 816 831 l 839 831 l + 839 636 l 862 636 l 862 516 l 885 516 l 885 481 l 907 481 l 907 443 l 930 443 l + 930 380 l 953 380 l 953 343 l 975 343 l 975 324 l 998 324 l 998 283 l 1021 283 + l 1021 301 l 1043 301 l 1043 288 l 1066 288 l 1066 294 l 1089 294 l 1089 272 l + 1111 272 l 1111 262 l 1134 262 l 1134 272 l 1157 272 l 1157 251 l 1179 251 l + 1179 253 l 1202 253 l 1202 255 l 1225 255 l 1225 257 l 1247 257 l 1247 245 l + 1270 245 l 1270 244 l 1293 244 l 1293 247 l 1315 247 l 1315 249 l 1338 249 l + 1338 240 l 1361 240 l 1361 231 l 1383 231 l 1383 244 l 1406 244 l 1406 234 l + 1429 234 l 1429 231 l 1452 231 l 1452 238 l 1474 238 l 1497 238 l 1497 234 l + 1520 234 l 1520 236 l 1542 236 l 1542 231 l 1565 231 l 1565 234 l 1588 234 l + 1588 232 l 1610 232 l 1610 231 l 1633 231 l 1633 229 l 1656 229 l 1656 234 l + 1678 234 l 1701 234 l 1701 232 l 1724 232 l 1724 229 l 1746 229 l 1769 229 l + 1769 234 l 1792 234 l 1792 227 l 1814 227 l 1814 232 l 1837 232 l 1837 234 l + 1860 234 l 1860 229 l 1882 229 l 1882 231 l 1905 231 l 1905 227 l 1928 227 l + 1928 229 l 1950 229 l 1950 227 l 1973 227 l 1973 229 l 1996 229 l 1996 231 l + 2019 231 l 2041 231 l 2041 227 l s 1 lw 227 227 m 227 2041 l s 4 lw 261 227 m + 227 227 l s 244 264 m 227 264 l s 244 301 m 227 301 l s 244 339 m 227 339 l s + 244 376 m 227 376 l s 261 413 m 227 413 l s 244 451 m 227 451 l s 244 488 m 227 + 488 l s 244 525 m 227 525 l s 244 563 m 227 563 l s 261 600 m 227 600 l s 244 + 637 m 227 637 l s 244 675 m 227 675 l s 244 712 m 227 712 l s 244 749 m 227 749 + l s 261 787 m 227 787 l s 244 824 m 227 824 l s 244 861 m 227 861 l s 244 899 m + 227 899 l s 244 936 m 227 936 l s 261 973 m 227 973 l s 244 1011 m 227 1011 l s + 244 1048 m 227 1048 l s 244 1085 m 227 1085 l s 244 1123 m 227 1123 l s 261 + 1160 m 227 1160 l s 244 1197 m 227 1197 l s 244 1235 m 227 1235 l s 244 1272 m + 227 1272 l s 244 1309 m 227 1309 l s 261 1347 m 227 1347 l s 244 1384 m 227 + 1384 l s 244 1421 m 227 1421 l s 244 1459 m 227 1459 l s 244 1496 m 227 1496 l + s 261 1533 m 227 1533 l s 244 1570 m 227 1570 l s 244 1608 m 227 1608 l s 244 + 1645 m 227 1645 l s 244 1682 m 227 1682 l s 261 1720 m 227 1720 l s 244 1757 m + 227 1757 l s 244 1794 m 227 1794 l s 244 1832 m 227 1832 l s 244 1869 m 227 + 1869 l s 261 1906 m 227 1906 l s 261 1906 m 227 1906 l s 244 1944 m 227 1944 l + s 244 1981 m 227 1981 l s 244 2018 m 227 2018 l s + /xs 0 def +(0) + /Helvetica-Bold 43 stwn + gsave 181 211 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0) + show + gr + /xs 0 def +(100) + /Helvetica-Bold 43 stwn + gsave 181 398 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(100) + show + gr + /xs 0 def +(200) + /Helvetica-Bold 43 stwn + gsave 181 584 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(200) + show + gr + /xs 0 def +(300) + /Helvetica-Bold 43 stwn + gsave 181 771 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(300) + show + gr + /xs 0 def +(400) + /Helvetica-Bold 43 stwn + gsave 181 957 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(400) + show + gr + /xs 0 def +(500) + /Helvetica-Bold 43 stwn + gsave 181 1144 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(500) + show + gr + /xs 0 def +(600) + /Helvetica-Bold 43 stwn + gsave 181 1331 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(600) + show + gr + /xs 0 def +(700) + /Helvetica-Bold 43 stwn + gsave 181 1517 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(700) + show + gr + /xs 0 def +(800) + /Helvetica-Bold 43 stwn + gsave 181 1704 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(800) + show + gr + /xs 0 def +(900) + /Helvetica-Bold 43 stwn + gsave 181 1891 + t 0 r xs neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(900) + show + gr + 1 lw 227 227 m 2041 227 l s 4 lw 227 261 m 227 227 l s 263 244 m 263 227 l s + 299 244 m 299 227 l s 336 244 m 336 227 l s 372 244 m 372 227 l s 408 261 m 408 + 227 l s 445 244 m 445 227 l s 481 244 m 481 227 l s 517 244 m 517 227 l s 553 + 244 m 553 227 l s 590 261 m 590 227 l s 626 244 m 626 227 l s 662 244 m 662 227 + l s 699 244 m 699 227 l s 735 244 m 735 227 l s 771 261 m 771 227 l s 807 244 m + 807 227 l s 844 244 m 844 227 l s 880 244 m 880 227 l s 916 244 m 916 227 l s + 953 261 m 953 227 l s 989 244 m 989 227 l s 1025 244 m 1025 227 l s 1061 244 m + 1061 227 l s 1098 244 m 1098 227 l s 1134 261 m 1134 227 l s 1170 244 m 1170 + 227 l s 1207 244 m 1207 227 l s 1243 244 m 1243 227 l s 1279 244 m 1279 227 l s + 1315 261 m 1315 227 l s 1352 244 m 1352 227 l s 1388 244 m 1388 227 l s 1424 + 244 m 1424 227 l s 1461 244 m 1461 227 l s 1497 261 m 1497 227 l s 1533 244 m + 1533 227 l s 1569 244 m 1569 227 l s 1606 244 m 1606 227 l s 1642 244 m 1642 + 227 l s 1678 261 m 1678 227 l s 1715 244 m 1715 227 l s 1751 244 m 1751 227 l s + 1787 244 m 1787 227 l s 1823 244 m 1823 227 l s 1860 261 m 1860 227 l s 1896 + 244 m 1896 227 l s 1932 244 m 1932 227 l s 1969 244 m 1969 227 l s 2005 244 m + 2005 227 l s 2041 261 m 2041 227 l s + /xs 0 def +(0) + /Helvetica-Bold 43 stwn + gsave 227 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0) + show + gr + /xs 0 def +(0.01) + /Helvetica-Bold 43 stwn + gsave 408 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.01) + show + gr + /xs 0 def +(0.02) + /Helvetica-Bold 43 stwn + gsave 590 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.02) + show + gr + /xs 0 def +(0.03) + /Helvetica-Bold 43 stwn + gsave 771 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.03) + show + gr + /xs 0 def +(0.04) + /Helvetica-Bold 43 stwn + gsave 953 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.04) + show + gr + /xs 0 def +(0.05) + /Helvetica-Bold 43 stwn + gsave 1134 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.05) + show + gr + /xs 0 def +(0.06) + /Helvetica-Bold 43 stwn + gsave 1315 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.06) + show + gr + /xs 0 def +(0.07) + /Helvetica-Bold 43 stwn + gsave 1497 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.07) + show + gr + /xs 0 def +(0.08) + /Helvetica-Bold 43 stwn + gsave 1678 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.08) + show + gr + /xs 0 def +(0.09) + /Helvetica-Bold 43 stwn + gsave 1860 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.09) + show + gr + /xs 0 def +(0.1) + /Helvetica-Bold 43 stwn + gsave 2041 172 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(0.1) + show + gr + /xs 0 def +(x 10) + /Helvetica-Bold 43 stwn + gsave 2041 109 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 43 sf 0 0 m +(x 10) + show + gr + /xs 0 def +(-4) + /Helvetica-Bold 35 stwn + gsave 2105 141 + t 0 r xs 2 div neg 0 t 0 0 m + /Helvetica-Bold findfont 35 sf 0 0 m +(-4) + show + gr + [12 12] 0 sd 227 227 m 227 227 l 249 227 l 272 227 l 295 227 l 318 227 l 340 + 227 l 363 227 l 386 227 l 408 227 l 431 227 l 454 227 l 476 227 l 499 227 l 522 + 227 l 544 227 l 544 229 l 567 229 l 567 232 l 590 232 l 590 270 l 612 270 l 612 + 352 l 635 352 l 635 546 l 658 546 l 658 901 l 680 901 l 680 1199 l 703 1199 l + 703 1490 l 726 1490 l 726 1628 l 748 1628 l 748 1533 l 771 1533 l 771 1404 l + 794 1404 l 794 1048 l 816 1048 l 816 738 l 839 738 l 839 572 l 862 572 l 862 + 417 l 885 417 l 885 331 l 907 331 l 907 272 l 930 272 l 930 244 l 953 244 l 953 + 231 l 975 231 l 998 231 l 998 227 l 1021 227 l 1043 227 l 1066 227 l 1089 227 l + 1111 227 l 1134 227 l 1157 227 l 1179 227 l 1202 227 l 1225 227 l 1247 227 l + 1270 227 l 1293 227 l 1315 227 l 1338 227 l 1361 227 l 1361 227 l 1383 227 l + 1406 227 l 1429 227 l 1452 227 l 1474 227 l 1497 227 l 1520 227 l 1542 227 l + 1565 227 l 1588 227 l 1610 227 l 1633 227 l 1656 227 l 1678 227 l 1701 227 l + 1724 227 l 1746 227 l 1769 227 l 1792 227 l 1814 227 l 1837 227 l 1860 227 l + 1882 227 l 1905 227 l 1928 227 l 1928 227 l 1950 227 l 1973 227 l 1996 227 l + 2019 227 l 2041 227 l s [4 8] 0 sd 227 227 m 227 227 l 249 227 l 272 227 l 295 + 227 l 318 227 l 340 227 l 363 227 l 386 227 l 408 227 l 431 227 l 454 227 l 476 + 227 l 499 227 l 499 231 l 522 231 l 522 244 l 544 244 l 544 257 l 567 257 l 567 + 315 l 590 315 l 590 447 l 612 447 l 612 613 l 635 613 l 635 833 l 658 833 l 658 + 1029 l 680 1029 l 680 1190 l 703 1190 l 703 1212 l 726 1212 l 726 1285 l 748 + 1285 l 748 1141 l 771 1141 l 771 1085 l 794 1085 l 794 884 l 816 884 l 816 775 + l 839 775 l 839 619 l 862 619 l 862 492 l 885 492 l 885 413 l 907 413 l 907 369 + l 930 369 l 930 294 l 953 294 l 953 290 l 975 290 l 975 260 l 998 260 l 998 244 + l 1021 244 l 1021 238 l 1043 238 l 1043 234 l 1066 234 l 1066 229 l 1089 229 l + 1089 231 l 1111 231 l 1111 227 l 1134 227 l 1134 229 l 1157 229 l 1157 227 l + 1179 227 l 1202 227 l 1225 227 l 1247 227 l 1270 227 l 1293 227 l 1315 227 l + 1338 227 l 1361 227 l 1361 227 l 1383 227 l 1406 227 l 1429 227 l 1452 227 l + 1474 227 l 1497 227 l 1520 227 l 1542 227 l 1565 227 l 1588 227 l 1610 227 l + 1633 227 l 1656 227 l 1678 227 l 1701 227 l 1724 227 l 1746 227 l 1769 227 l + 1792 227 l 1814 227 l 1837 227 l 1860 227 l 1882 227 l 1905 227 l 1928 227 l + 1928 227 l 1950 227 l 1973 227 l 1996 227 l 2019 227 l 2041 227 l s + [12 15 4 15] 0 sd 227 227 m 227 227 l 249 227 l 272 227 l 295 227 l 318 227 l + 340 227 l 363 227 l 386 227 l 408 227 l 431 227 l 431 229 l 454 229 l 454 234 l + 476 234 l 476 242 l 499 242 l 499 277 l 522 277 l 522 335 l 544 335 l 544 412 l + 567 412 l 567 580 l 590 580 l 590 749 l 612 749 l 612 889 l 635 889 l 635 953 l + 658 953 l 658 984 l 680 984 l 680 1009 l 703 1009 l 703 942 l 726 942 l 726 962 + l 748 962 l 748 869 l 771 869 l 771 820 l 794 820 l 794 725 l 816 725 l 816 677 + l 839 677 l 839 580 l 862 580 l 862 492 l 885 492 l 885 458 l 907 458 l 907 404 + l 930 404 l 930 341 l 953 341 l 953 328 l 975 328 l 975 315 l 998 315 l 998 277 + l 1021 277 l 1021 270 l 1043 270 l 1043 255 l 1066 255 l 1066 262 l 1089 262 l + 1089 236 l 1111 236 l 1134 236 l 1134 238 l 1157 238 l 1157 231 l 1179 231 l + 1179 229 l 1202 229 l 1202 227 l 1225 227 l 1225 231 l 1247 231 l 1247 227 l + 1270 227 l 1293 227 l 1315 227 l 1338 227 l 1338 229 l 1361 229 l 1361 227 l + 1383 227 l 1406 227 l 1429 227 l 1452 227 l 1474 227 l 1497 227 l 1520 227 l + 1542 227 l 1565 227 l 1588 227 l 1610 227 l 1633 227 l 1656 227 l 1678 227 l + 1701 227 l 1724 227 l 1746 227 l 1769 227 l 1792 227 l 1814 227 l 1837 227 l + 1860 227 l 1882 227 l 1905 227 l 1928 227 l 1928 227 l 1950 227 l 1973 227 l + 1996 227 l 2019 227 l 2041 227 l s [] 0 sd 227 227 m 227 227 l 249 227 l 272 + 227 l 295 227 l 318 227 l 340 227 l 363 227 l 386 227 l 408 227 l 408 232 l 431 + 232 l 431 247 l 454 247 l 454 277 l 476 277 l 476 324 l 499 324 l 499 419 l 522 + 419 l 522 544 l 544 544 l 544 600 l 567 600 l 567 738 l 590 738 l 590 742 l 612 + 742 l 612 830 l 635 830 l 635 798 l 658 798 l 658 789 l 680 789 l 680 764 l 703 + 764 l 703 772 l 726 772 l 726 766 l 748 766 l 748 691 l 771 691 l 771 596 l 794 + 596 l 794 619 l 816 619 l 816 565 l 839 565 l 839 553 l 862 553 l 862 479 l 885 + 479 l 885 430 l 907 430 l 907 423 l 930 423 l 930 456 l 953 456 l 953 393 l 975 + 393 l 975 361 l 998 361 l 998 359 l 1021 359 l 1021 346 l 1043 346 l 1043 329 l + 1066 329 l 1066 307 l 1089 307 l 1089 285 l 1111 285 l 1111 294 l 1134 294 l + 1134 281 l 1157 281 l 1157 270 l 1179 270 l 1179 260 l 1202 260 l 1202 247 l + 1225 247 l 1247 247 l 1247 245 l 1270 245 l 1270 234 l 1293 234 l 1315 234 l + 1338 234 l 1361 234 l 1361 244 l 1383 244 l 1383 229 l 1406 229 l 1406 232 l + 1429 232 l 1429 229 l 1452 229 l 1452 231 l 1474 231 l 1474 227 l 1497 227 l + 1497 229 l 1520 229 l 1520 227 l 1542 227 l 1542 231 l 1565 231 l 1565 229 l + 1588 229 l 1610 229 l 1610 227 l 1633 227 l 1656 227 l 1678 227 l 1701 227 l + 1724 227 l 1746 227 l 1769 227 l 1792 227 l 1814 227 l 1837 227 l 1860 227 l + 1882 227 l 1905 227 l 1928 227 l 1928 227 l 1950 227 l 1973 227 l 1996 227 l + 2019 227 l 2041 227 l s [] 0 sd 227 227 m 227 231 l 249 231 l 249 234 l 272 234 + l 272 249 l 295 249 l 295 266 l 318 266 l 318 270 l 340 270 l 340 300 l 363 300 + l 363 384 l 386 384 l 386 460 l 408 460 l 408 499 l 431 499 l 431 570 l 454 570 + l 454 667 l 476 667 l 476 691 l 499 691 l 499 706 l 522 706 l 522 699 l 544 699 + l 544 706 l 567 706 l 567 660 l 590 660 l 590 671 l 612 671 l 612 585 l 635 585 + l 635 563 l 658 563 l 658 559 l 680 559 l 680 555 l 703 555 l 703 490 l 726 490 + l 726 434 l 748 434 l 748 454 l 771 454 l 771 423 l 794 423 l 794 395 l 816 395 + l 816 393 l 839 393 l 839 389 l 862 389 l 862 331 l 885 331 l 885 348 l 907 348 + l 907 328 l 930 328 l 930 331 l 953 331 l 953 315 l 975 315 l 975 326 l 998 326 + l 998 303 l 1021 303 l 1021 309 l 1043 309 l 1043 287 l 1066 287 l 1066 303 l + 1089 303 l 1089 313 l 1111 313 l 1111 292 l 1134 292 l 1134 279 l 1157 279 l + 1157 272 l 1179 272 l 1179 292 l 1202 292 l 1202 305 l 1225 305 l 1225 257 l + 1247 257 l 1247 268 l 1270 268 l 1270 273 l 1293 273 l 1293 268 l 1315 268 l + 1315 270 l 1338 270 l 1338 255 l 1361 255 l 1361 259 l 1383 259 l 1383 255 l + 1406 255 l 1406 266 l 1429 266 l 1429 249 l 1452 249 l 1452 251 l 1474 251 l + 1474 268 l 1497 268 l 1497 244 l 1520 244 l 1520 238 l 1542 238 l 1542 244 l + 1565 244 l 1588 244 l 1588 247 l 1610 247 l 1633 247 l 1633 242 l 1656 242 l + 1656 238 l 1678 238 l 1678 240 l 1701 240 l 1701 245 l 1724 245 l 1724 244 l + 1746 244 l 1746 234 l 1769 234 l 1792 234 l 1814 234 l 1837 234 l 1837 238 l + 1860 238 l 1860 236 l 1882 236 l 1882 238 l 1905 238 l 1905 232 l 1928 232 l + 1928 232 l 1950 232 l 1950 236 l 1973 236 l 1973 234 l 1996 234 l 2019 234 l + 2019 236 l 2041 236 l 2041 227 l s [12 12] 0 sd 227 227 m 227 326 l 249 326 l + 249 268 l 272 268 l 272 354 l 295 354 l 295 404 l 318 404 l 318 406 l 340 406 l + 340 473 l 363 473 l 363 516 l 386 516 l 386 542 l 408 542 l 408 652 l 431 652 l + 431 654 l 454 654 l 454 593 l 476 593 l 476 667 l 499 667 l 499 568 l 522 568 l + 522 637 l 544 637 l 544 682 l 567 682 l 567 587 l 590 587 l 590 598 l 612 598 l + 612 496 l 635 496 l 635 522 l 658 522 l 658 527 l 680 527 l 680 466 l 703 466 l + 703 464 l 726 464 l 726 440 l 748 440 l 748 436 l 771 436 l 771 371 l 794 371 l + 794 400 l 816 400 l 816 384 l 839 384 l 839 341 l 862 341 l 862 335 l 885 335 l + 885 301 l 907 301 l 907 324 l 930 324 l 930 313 l 953 313 l 953 316 l 975 316 l + 975 296 l 998 296 l 998 285 l 1021 285 l 1021 288 l 1043 288 l 1043 279 l 1066 + 279 l 1066 268 l 1089 268 l 1089 260 l 1111 260 l 1111 283 l 1134 283 l 1134 + 255 l 1157 255 l 1157 266 l 1179 266 l 1179 255 l 1202 255 l 1202 259 l 1225 + 259 l 1225 245 l 1247 245 l 1247 251 l 1270 251 l 1270 245 l 1293 245 l 1293 + 257 l 1315 257 l 1315 251 l 1338 251 l 1338 245 l 1361 245 l 1361 242 l 1383 + 242 l 1383 255 l 1406 255 l 1406 236 l 1429 236 l 1429 242 l 1452 242 l 1452 + 249 l 1474 249 l 1474 257 l 1497 257 l 1497 238 l 1520 238 l 1542 238 l 1542 + 240 l 1565 240 l 1588 240 l 1588 242 l 1610 242 l 1610 244 l 1633 244 l 1633 + 253 l 1656 253 l 1656 247 l 1678 247 l 1678 245 l 1701 245 l 1701 253 l 1724 + 253 l 1724 240 l 1746 240 l 1746 238 l 1769 238 l 1769 242 l 1792 242 l 1792 + 240 l 1814 240 l 1814 249 l 1837 249 l 1837 236 l 1860 236 l 1860 240 l 1882 + 240 l 1905 240 l 1905 242 l 1928 242 l 1928 240 l 1950 240 l 1950 231 l 1973 + 231 l 1973 240 l 1996 240 l 1996 247 l 2019 247 l 2019 244 l 2041 244 l 2041 + 227 l s +gr gr showpage +end +%%EOF diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/tex4ht.env b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/tex4ht.env new file mode 100644 index 00000000000..4787cd8341c --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/tex4ht.env @@ -0,0 +1,70 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% tex4ht.env / .tex4ht % +% % +% Notes: % +% 1. empty lines are harmful % +% 2. place this file in your work % +% directory and/or root directory % +% and/or in directory `xxx' of your % +% choice. In the latest case, compile % +% tex4ht.c with `#define HTFDIR xxx' % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Replace the directories `candy/...' % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +tcandy/tex/texmf/fonts/tfm/! +tcandy/tex/texmf/fonts/vf/! +icandy/tex4ht.dir/ +icandy/tex4ht.dir/ht-fonts/iso88591/! +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% aliases for .htf fonts % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +acmbsy cmsy +acmbx cmr +acmitt cmtt +acmsl cmr +acmss cmti +acmtcsc cmtt +acmtex cmtt +acmu cmti +aptmb cmr +aptmr7 cmr +aptmri cmti +% To activate the following aliases, +% remove the leading space character + adcbx ectt + adccc ecsx + adcit ectt + adcr ectt + adcsi ecsx + adcsl ecsx + adcss ecsx + adctc ecsx + adcti ectt + aecbx ec + aecsl ec + aecsltt ec + aectt ec + apcrb7t pcrr7t + apcrro7t pcrr7t + aphvr7t ptmr7t + aptmb7t ptmr7t + aptmr8t ectt + aptmri8t ectt + aptmro7t ptmr7t + awncyi wncyr +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Default scripts, shifted rightward 1 pos +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +s--- needs --- %%1.idv[%%2] ==> /tmp/%%3 --- + b--- characters --- + g.gif +Gdvips -Pem -f %%1 -pp %%2 > tmp.ps +Gconvert -crop 0x0 -density 110x110 -transparency '#FFFFFF' tmp.ps %%3 +S* +% t4ht -d%%2 +Mmv %%1 %%2%%3 +Ccp %%1 %%2%%3 +% t4ht -d%%2 -m%%1 +Achmod %%1 %%2%%3 +Hgold/tex4ht.dir/ + diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch4/tex4ht.tmp b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/tex4ht.tmp new file mode 100644 index 00000000000..41b58e2cf47 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch4/tex4ht.tmp @@ -0,0 +1,2 @@ + +/* css.sty */
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+% version = "1.1",
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