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+<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"> &gt; </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">&lt; <img
+src="emmi10-14.gif"alt="k"class="10--14"> &lt; </span>10, Vavilov
+ distribution, and <span
+class="emmi-10"><img
+src="emmi10-14.gif"alt="k"class="10--14"> &lt; </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">&gt;</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. Urb&aacute;n
+ 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">&lt;</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">&gt;</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="--&gt;"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">&lt;</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">&gt;</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">&gt;</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>Urb&aacute;n 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">&lt;</span> </span><span
+class="emmi-10">r </span><span
+class="cmsy-10"><span
+class="underline">&lt;</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">&lt;</span><span
+class="emmi-10">n</span><sub ><span
+class="emmi-7">i</span></sub><span
+class="cmsy-10">&gt;</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="&lt;n&gt; = 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 &lt; 2
+f2 = 2/Z ifZ &gt; 2 ==&gt; f1 = 1- f2
+ ( )f11
+E2 = 10Z2eV ==&gt; 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">&gt;</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-
+&lt; E(a)&gt; = 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">&lt;</span><span
+class="emmi-10">n</span><sub ><span
+class="emmi-7">A</span></sub><span
+class="cmsy-10">&gt;</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">&lt;</span><span
+class="emmi-10">E</span><sub ><span
+class="emmi-7">A</span></sub><span
+class="cmsy-10">&gt;</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">&lt;</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">&gt;</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">&lt;</span><span
+class="emmi-10">n</span><sub ><span
+class="emmi-7">A</span></sub><span
+class="cmsy-10">&gt; - </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">&gt;</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">&lt;</span><span
+class="emmi-10">n</span><sub ><span
+class="emmi-7">A</span></sub><span
+class="cmsy-10">&gt; </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">&lt;</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">&lt;</span><span
+class="emmi-10">E</span><sub ><span
+class="emmi-7">A</span></sub><span
+class="cmsy-10">&gt; - </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">&gt;</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">&gt;</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- &lt; a &lt; 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 &gt; 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">&gt;</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(&lt;nA&gt; - 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">&lt;</span><span
+class="emmi-10">n</span><sub ><span
+class="emmi-7">A</span></sub><span
+class="cmsy-10">&gt; </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 &lt; 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">&lt;</span> </span>5 mm in gases, <span
+class="cmsy-10"><span
+class="underline">&lt;</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=" -(&lt;n &gt;+&lt;n &gt;+&lt;n &gt;)
+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
+&lt;n&gt; = ----- 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>
+
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