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diff --git a/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml b/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml deleted file mode 100644 index af9fbb686c4..00000000000 --- a/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml +++ /dev/null @@ -1,972 +0,0 @@ -<?xml version="1.0"?> -<!DOCTYPE TEI.2 SYSTEM "http://www.oucs.ox.ac.uk/dtds/tei-oucs.dtd" [ -<!ENTITY aacute "á"> -<!ENTITY ap "≈"> -<!ENTITY alpha "α"> -<!ENTITY barwed "⊼"> -<!ENTITY beta "β"> -<!ENTITY commaspace " "> -<!ENTITY gamma "γ"> -<!ENTITY delta "δ"> -<!ENTITY Delta "Δ"> -<!ENTITY dots "…"> -<!ENTITY epsi "ε"> -<!ENTITY gg "⋙"> -<!ENTITY Gt "≫"> -<!ENTITY geq "≥"> -<!ENTITY infin "∞"> -<!ENTITY infty "∞"> -<!ENTITY int "∫"> -<!ENTITY kappa "κ"> -<!ENTITY lambda "λ"> -<!ENTITY langle "〈"> -<!ENTITY leq "≤"> -<!ENTITY mu "μ"> -<!ENTITY nbsp " "> -<!ENTITY phi "φ"> -<!ENTITY pi "π"> -<!ENTITY psi "ψ"> -<!ENTITY rangle "〉"> -<!ENTITY rho "ρ"> -<!ENTITY sigma "σ"> -<!ENTITY Sigma "Σ"> -<!ENTITY sim "∼"> -<!ENTITY thinspace " "> -<!ENTITY thickspace " "> -<!ENTITY xi "ξ"> -<!ENTITY prime "′"> -<!ENTITY isinv "∈"> -<!ENTITY macr "̄"> -<!ENTITY Emax "<msub><mi>E</mi><mi>max</mi></msub>"> -<!ENTITY exp "E<mtext>exp</mtext>"> -<!ENTITY ln "E<mtext>ln</mtext>"> -<!ENTITY Rarr "⇒"> -<!ENTITY rarr "→"> -<!ENTITY GEANT "GEANT"> -<!ENTITY sum "∑"> -]> -<TEI.2> - <teiHeader> - <fileDesc> - <titleStmt> - <title>A sample article</title> - </titleStmt> - <publicationStmt> - <availability><p>Converted from LaTeX by Sebastian Rahtz</p> </availability> - </publicationStmt> - <sourceDesc> - <p></p> - </sourceDesc> - </fileDesc> - <revisionDesc> - <list> - <item> - <date>23 Oct 1999</date> SR converted from LaTeX</item> - </list> - </revisionDesc> - </teiHeader> - <text> - <front> - <docTitle> - <titlePart type="main">Simulation of Energy Loss Straggling</titlePart> - </docTitle> - <docAuthor>Maria Physicist</docAuthor> - <docDate>January 17, 1999</docDate> - </front> - <body> <div id="intro"> <head>Introduction</head> <p>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 <formula><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math></formula>, which is -proportional to the ratio of mean energy loss to the maximum allowed -energy transfer in a single collision with an atomic electron - <formula -type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow> <mi>κ</mi><mo>=</mo> -<mfrac> - <mrow><mi>ξ</mi></mrow> - <mrow> - &Emax; - </mrow> -</mfrac> -</mrow> -</math></formula> -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" ->&Emax;</math></formula> is the -maximum transferable energy in a single collision with an atomic electron. -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow> - &Emax;<mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup></mrow><!--____________ ---><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>γ</mi><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub><mo>+</mo><msup><mfenced -open='(' close=')'><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub> </mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo> -</mrow></math></formula> where -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>γ</mi><mo>=</mo><mi>E</mi><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula>, -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>E</mi></math></formula> is energy and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> the mass of the -incident particle, <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>/</mo><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup></math></formula> -and <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub></math></formula> is the -electron mass. <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ξ</mi></math></formula> -comes from the Rutherford scattering crosss section and is defined as: - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mn>4</mn></mrow></msup><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub><mi>Z</mi><mi>ρ</mi><mi>δ</mi><mi>x</mi></mrow><!-- - --><mrow><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>A</mi></mrow></mfrac> <mo>=</mo><mn>1</mn><mn>5</mn><mn>3</mn><mo>.</mo><mn>4</mn> <mfrac><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- ---><mrow><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mfrac><mrow><mi>Z</mi></mrow><!-- ---><mrow><mi>A</mi></mrow></mfrac><mi>ρ</mi><mi>δ</mi><mi>x</mi><mspace width='12pt'/><mi>keV </mi><mo>,</mo> <mtext></mtext> -</math></formula></cell></row></table> -where -</p><p><table rend="inline"><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>z</mi></math></formula></cell><cell ->charge of the incident particle </cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></math></formula></cell><cell ->Avogadro's number </cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>Z</mi></math></formula></cell><cell ->atomic number of the material</cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>A</mi></math></formula></cell><cell ->atomic weight of the material </cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ρ</mi></math></formula></cell><cell ->density </cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>δ</mi><mi>x</mi></math></formula></cell><cell ->thickness of the material </cell> -</row><row><cell -> </cell> -</row></table> -</p><p><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi></math></formula> -measures the contribution of the collisions with energy transfer close to -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" ->&Emax;</math></formula>. For a given absorber, -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi></math></formula> tends towards large -values if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>δ</mi><mi>x</mi></math></formula> is large -and/or if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>β</mi></math></formula> is small. -Likewise, <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi></math></formula> tends -towards zero if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>δ</mi><mi>x</mi></math></formula> is -small and/or if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>β</mi></math></formula> -approaches 1. -</p><p>The value of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi></math></formula> -distinguishes two regimes which occur in the description of ionisation fluctuations -: -</p><list type="enumerate"> -<item> -<p>A -large -number -of -collisions -involving -the -loss -of -all -or -most -of -the -incident -particle -energy -during -the -traversal -of -an -absorber. -</p><p>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 -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi><mo>></mo><mn>1</mn><mn>0</mn></math></formula> -(i.e. -when -the -mean -energy -loss -in -the -absorber -is -greater -than -the -maximum -energy -transfer -in -a -single -collision). -</p></item> -<item> -<p>Particles -traversing -thin -counters -and -incident -electrons -under -any -conditions. -</p><p>The -relevant -inequalities -and -distributions -are -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn><mo><</mo><mi>κ</mi><mo><</mo><mn>1</mn><mn>0</mn></math></formula>, -Vavilov -distribution, -and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula>, -Landau -distribution.</p></item></list> -<p>An additional regime is defined by the contribution of the collisions -with low energy transfer which can be estimated with the relation -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>, -where <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> -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 -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>≫</mo><mn>1</mn></math></formula>. In <code>GEANT</code> (see -URL <xptr url="http://wwwinfo.cern.ch/asdoc/geant/geantall.html"/>), the limit of Landau theory has -been set at <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mn>0</mn></math></formula>. -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 <ptr target="fg:phys332-1"/> shows the behaviour -of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> as -a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic -energy in Argon, Silicon and Uranium. -</p> -<p><figure file="phys332-1" id="fg:phys332-1"> -<head>The variable <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> -can be used to measure the validity range of the Landau -theory. It depends on the type and energy of the particle, -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>Z</mi></math></formula>, -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>A</mi></math></formula> -and the ionisation potential of the material and the layer thickness. </head> -</figure></p> -<p>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. -</p> -</div> -<div id="sec:phys332-1"> -<head>Landau theory</head> -<p>For a particle of mass <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> traversing -a thickness of material <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>δ</mi><mi>x</mi></math></formula>, -the Landau probability distribution may be written in terms of the universal Landau -function <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></math></formula> -as<ptr target="bib-LAND"/>: - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>x</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!-- ---><mrow><mi>ξ</mi></mrow></mfrac><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow> <mtext></mtext> -</math></formula></cell></row></table> -where - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_ ---><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup><mo>∫</mo> - <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup><mo>exp</mo><mfenced -open='(' close=')'><mi>u</mi><mo>ln</mo><mi>u</mi><mo>+</mo><mi>λ</mi><mi>u</mi></mfenced><mi>d</mi><mi>u</mi><mspace width='2cm'/><mi>c</mi><mo>≥</mo><mn>0</mn> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi> <mo>=</mo> <mfrac><mrow><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover></mrow><!-- - --><mrow><mi>ξ</mi></mrow></mfrac> <mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>ξ</mi></mrow><!-- ___ ---><mrow>&Emax;</mrow></mfrac> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mi>′</mi> <mo>=</mo> <mn>0</mn><mo>.</mo><mn>4</mn><mn>2</mn><mn>2</mn><mn>7</mn><mn>8</mn><mn>4</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><mn>1</mn><mo>-</mo><mi>γ</mi> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi> <mo>=</mo> <mn>0</mn><mo>.</mo><mn>5</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>5</mn><mo>.</mo><mo>.</mo><mo>.</mo><mtext>(Eulers constant)</mtext> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover> <mo>=</mo> <mtext>average energy loss</mtext> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi> <mo>=</mo> <mtext>actual energy loss</mtext> <mtext></mtext> -</math></formula></cell></row></table> -</p> -<div > -<head>Restrictions</head> -<p>The Landau formalism makes two restrictive assumptions : -</p><list type="enumerate"> -<item> -<p>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 -<ptr target="vavref"/>). -</p></item> -<item> -<p>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<ptr target="bib-TALM"/> -is -necessary -to -accurately -simulate -data -distributions. -In -<code>GEANT</code>, -a -parameterised -model -by -L. -Urbán -is -used -(see -section -<ptr target="urban"/>).</p></item></list> -<p>In addition, the average value of the Landau distribution is infinite. -Summing the Landau fluctuation obtained to the average energy from the -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> -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. -</p><p>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 <code>GLANDO</code> routine is: -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow> - <mi>δ</mi><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover><mo>=</mo><mi>ξ</mi><mrow><mo>(</mo><mi>λ</mi><mo>-</mo><mi>γ</mi><mi>′</mi><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>ln</mo> <mfrac><mrow><mi>ξ</mi></mrow><!-- ___ ---><mrow>&Emax;</mrow></mfrac> <mo>)</mo></mrow> -</mrow></math></formula> -In order for this to have average 0, we must impose that: -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow> - <munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover><mo>=</mo><mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>ξ</mi></mrow><!-- ___ ---><mrow>&Emax;</mrow></mfrac> -</mrow></math></formula> -</p><p>This is realised introducing a <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover><mo>)</mo></mrow></math></formula> -such that if only values of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>λ</mi><mo>≤</mo><msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub></math></formula> -are accepted, the average value of the distribution is -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover></math></formula>. -</p><p>A parametric fit to the universal Landau distribution has been performed, with following result: -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow> - <msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>0</mn><mn>7</mn><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mo>.</mo><mn>1</mn><mn>9</mn><mn>3</mn><mn>4</mn><munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover><mo>+</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>7</mn><mn>7</mn><mn>9</mn><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>5</mn><mn>2</mn><mn>3</mn><mn>8</mn><mn>2</mn><munderover accent='true'><mi>λ</mi><mrow></mrow><mo>¯</mo></munderover><mo>)</mo></mrow><mo>exp</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>4</mn><mn>7</mn><mn>5</mn><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn><mn>4</mn><mn>4</mn><mn>4</mn><mn>2</mn><munderover accent='true'><mi>λ</mi><mrow></mrow><mo>¯</mo></munderover><mo>)</mo></mrow> -</mrow></math></formula> only values -smaller than <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub></math></formula> -are accepted, otherwise the distribution is resampled. -</p> -</div> -</div> -<div id="vavref"> -<head>Vavilov theory</head> -<p>Vavilov<ptr target="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 -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" ->&Emax;<mo>=</mo><mi>∞</mi></math></formula>. Now -we can write<ptr target="bib-SCH1"/>: - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi> <mfenced -open='(' close=')'><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!-- ---><mrow><mi>ξ</mi></mrow></mfrac><msub><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced -open='(' close=')'><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mtext></mtext> -</math></formula></cell></row></table> -where - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced -open='(' close=')'><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_ ---><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup><mo>∫</mo> - <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup><mi>φ</mi><mfenced -open='(' close=')'><mi>s</mi></mfenced><msup><mi>e</mi><mrow><mi>λ</mi><mi>s</mi></mrow></msup><mi>d</mi><mi>s</mi><mspace width='2cm'/><mi>c</mi><mo>≥</mo><mn>0</mn> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mfenced -open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo> <mo>exp</mo><mfenced -open='[' close=']'><mi>κ</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mi>γ</mi><mo>)</mo></mrow></mfenced><mo>exp</mo><mfenced -open='[' close=']'><mi>ψ</mi> <mfenced -open='(' close=')'><mi>s</mi></mfenced></mfenced><mo>,</mo> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ψ</mi> <mfenced -open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo> <mi>s</mi><mo>ln</mo><mi>κ</mi><mo>+</mo><mrow><mo>(</mo><mi>s</mi><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mi>κ</mi><mo>)</mo></mrow><mfenced -open='[' close=']'><mo>ln</mo><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>κ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>E</mi><mrow> -<mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>κ</mi><mo>)</mo></mrow></mfenced><mo>-</mo><mi>κ</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>s</mi><mo>/</mo><mi>κ</mi></mrow></msup><mo>,</mo> <mtext></mtext> -</math></formula></cell></row></table> -and - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow> <mo>=</mo><msubsup> <mo>∫</mo> - <mrow><mi>∞</mi></mrow><mrow><mi>z</mi></mrow></msubsup><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mspace width='1cm'/><mtext>(the exponential integral)</mtext> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub> <mo>=</mo> <mi>κ</mi><mfenced -open='[' close=']'><mfrac><mrow><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover></mrow><!-- - --><mrow><mi>ξ</mi></mrow></mfrac> <mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mtext></mtext> -</math></formula></cell></row></table> -</p><p>The Vavilov parameters are simply related to the Landau parameter by -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>/</mo><mi>κ</mi><mo>-</mo><mo>ln</mo><mi>κ</mi></math></formula>. It can be shown that -as <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi><mo>→</mo><mn>0</mn></math></formula>, the distribution of -the variable <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></formula> approaches -that of Landau. For <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula> -the two distributions are already practically identical. Contrary to what many textbooks -report, the Vavilov distribution <emph>does not</emph> approximate the Landau distribution for small -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi></math></formula>, but rather the -distribution of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></formula> -defined above tends to the distribution of the true -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>λ</mi></math></formula> from -the Landau density function. Thus the routine <code>GVAVIV</code> samples the variable -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></formula> rather -than <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub></math></formula>. -For <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></formula> -the Vavilov distribution tends to a Gaussian distribution (see next section). -</p> -</div> -<div > -<head>Gaussian Theory</head> -<p>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<ptr target="bib-SELT"/> -that for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>κ</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></formula> -the Vavilov distribution can be replaced by a Gaussian of the form: - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi><mo>)</mo></mrow><mo>≈</mo> <mfrac><mrow><mn>1</mn></mrow><!--________ ---><mrow><mi>ξ</mi><msqrt><!--<mi>&radical;</mi> - ______________--><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><!-- - --><mrow><mi>κ</mi></mrow></mfrac> <mfenced -open='(' close=')'><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn></mfenced></msqrt></mrow></mfrac><mo>exp</mo><mfenced -open='[' close=']'><mfrac><mrow><msup><mrow><mo>(</mo><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><!-- - --><mrow><mn>2</mn></mrow></mfrac> <mfrac><mrow><mi>κ</mi></mrow><!-- _______ ---><mrow><msup><mi>ξ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mfenced> <mtext></mtext> -</math></formula></cell></row></table> -thus implying - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>mean</mi> <mo>=</mo> <munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mrow><mn>2</mn></mrow></msup> <mo>=</mo> <mfrac><mrow><msup><mi>ξ</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- - --><mrow><mi>κ</mi></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><mi>ξ</mi><msub><mi>E</mi><mrow><mi> -max</mi></mrow></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow> <mtext></mtext> -</math></formula></cell></row></table> -</p> -</div> -<div id="urban"> -<head>Urbán model</head> -<p>The method for computing restricted energy losses with -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>δ</mi></math></formula>-ray -production above given threshold energy in <code>GEANT</code> 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 <ptr target="fg:phys332-2"/>. -</p> -<p><figure file="phys332-2" id="fg:phys332-2"> -<head>Energy loss distribution for a 3 GeV electron in Argon as given by -standard GEANT. The width of the layers is given in centimeters.</head> -</figure></p> -<p>It is assumed that the atoms have only two energy levels with binding energy -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>. -The particle--atom interaction will then be an excitation with energy loss -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> or -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>, or -an ionisation with an energy loss distributed according to a function -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>∼</mo><mn>1</mn><mo>/</mo><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></math></formula>: -<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><!-- - --><mrow>&Emax;</mrow></mfrac> -<mfrac><mrow><mn>1</mn></mrow><!-- _ ---><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> -</math></formula></p><p>The -macroscopic cross-section for excitations (<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></formula>) -is <formula id="eq:sigex" type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <msub><mi>Σ</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></mrow><!-- ---><mrow><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac> <mfrac><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- - --><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>I</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo></mrow> </math></formula>and -the macroscopic cross-section for ionisation is -<formula id="eq:sigion" type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <msub><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow>&Emax;</mrow><!-- ________________ ---><mrow><mi>I</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow><!-- - --><mrow><mi>I</mi></mrow></mfrac> <mo>)</mo></mrow></mrow></mfrac><mi>r</mi> </math></formula> -<formula><math xmlns="http://www.w3.org/1998/Math/MathML">&Emax;</math></formula> -is the <code>GEANT</code> cut for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>δ</mi></math></formula>-production, -or the maximum energy transfer minus mean ionisation energy, if it is smaller than -this cut-off value. The following notation is used: -</p><p><table rend="inline"><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>r</mi><mo>,</mo><mi>C</mi></math></formula></cell><cell ->parameters of the model</cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell ->atomic energy levels </cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>I</mi></math></formula></cell><cell ->mean ionisation energy </cell> -</row><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell ->oscillator strengths </cell> -</row></table> -</p><p>The model has the parameters <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula>, -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>, -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>C</mi></math></formula> and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>r</mi><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></math></formula>. The oscillator -strengths <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula> and the -atomic level energies <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula> -should satisfy the constraints - <table rend="inline"><row><cell><formula type="subeqn" id="eq:fisum"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub> <mo>=</mo> <mn>1</mn> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:flnsum"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub> <mo>=</mo> <mo>ln</mo><mi>I</mi> -</math></formula></cell></row></table> -The parameter <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>C</mi></math></formula> -can be defined with the help of the mean energy loss -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> in the following way: The -numbers of collisions (<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></math></formula>, -i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean -number <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></formula>. In a step -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>Δ</mi><mi>x</mi></math></formula> the mean number -of collisions is <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow><mo><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow><mo>=</mo><msub><mi>Σ</mi><mrow><mi>i</mi></mrow></msub><mi>Δ</mi><mi>x</mi> -</math></formula>The -mean energy loss <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> -in a step is the sum of the excitation and ionisation contributions -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!-- ---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>Δ</mi><mi>x</mi><mo>=</mo><mfenced -open='[' close=']'><msub><mi>Σ</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>Σ</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><msubsup><mo>∫</mo> - <mrow><mi>I</mi></mrow><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi></mfenced><mi>Δ</mi><mi>x</mi> -</math></formula>From -this, using the equations (<ptr target="eq:sigex"/>), (<ptr target="eq:sigion"/>), (<ptr target="eq:fisum"/>) and (<ptr target="eq:flnsum"/>), one can define the parameter -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>C</mi></math></formula> -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>C</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!-- ---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> -</math></formula> -</p><p>The following values have been chosen in <code>GEANT</code> for the other parameters: -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow> - <mtable equalrows='false' equalcolumns='false'><mtr><mtd><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfenced -open='{' ><mtable equalrows='false' equalcolumns='false'><mtr><mtd><mn>0</mn> </mtd><mtd><mi>if</mi><mi>Z</mi><mo>≤</mo><mn>2</mn></mtd> -</mtr><mtr><mtd><mn>2</mn><mo>/</mo><mi>Z</mi></mtd><mtd><mi>if</mi><mi>Z</mi><mo>></mo><mn>2</mn></mtd> -</mtr><mtr><mtd> </mtd></mtr></mtable> </mfenced></mtd><mtd><mo>↠</mo></mtd><mtd><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>-</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub> </mtd> - </mtr><mtr><mtd><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn><msup><mi>Z</mi><mrow><mn>2</mn></mrow></msup><mi>eV </mi> </mtd><mtd><mo>↠</mo></mtd><mtd><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><msup><mfenced -open='(' close=')'> <mfrac><mrow><mi>I</mi></mrow><!--___ ---><mrow><msubsup><mi>E</mi><mrow><mn>2</mn></mrow><mrow><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></mrow></msubsup></mrow></mfrac> </mfenced><mrow> <mfrac><mrow><mn>1</mn></mrow><!-- _ ---><mrow><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac> </mrow></msup></mtd> - </mtr><mtr><mtd><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn> </mtd><mtd> </mtd><mtd> </mtd> - </mtr><mtr><mtd> </mtd></mtr></mtable> -</mrow></math></formula> With these values -the atomic level <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula> -corresponds approximately the K-shell energy of the atoms and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>Z</mi><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></math></formula> the number of -K-shell electrons. <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>r</mi></math></formula> -is the only variable which can be tuned freely. It determines the relative contribution -of ionisation and excitation to the energy loss. -</p><p>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 -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>Δ</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub> -</math></formula>where -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></math></formula> and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></math></formula> -are sampled from Poisson distribution as discussed above. The -loss due to the ionisation can be generated from the distribution -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math></formula> by -the inverse transformation method: - <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow> <mo>=</mo><msubsup> <mo>∫</mo> - <mrow><mi>I</mi></mrow><mrow><mi>E</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi> <mtext></mtext> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msup><mi>F</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mi>I</mi></mrow><!--____ ---><mrow><mn>1</mn><mo>-</mo><mi>u</mi> <mfrac><mrow>&Emax;</mrow><!-- ___ ---><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> -</math></formula></cell></row></table> -where <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>u</mi></math></formula> is a uniform random -number between <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>F</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></formula> and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>F</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></formula>. The contribution from the -ionisations will be <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>Δ</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mo>∑</mo> - <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></mrow></msubsup> <mfrac><mrow><mi>I</mi></mrow><!--________ ---><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>j</mi></mrow></msub> <mfrac><mrow>&Emax;</mrow><!-- ___ ---><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> -</math></formula>where -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> is the -number of ionisation (sampled from Poisson distribution). The energy loss in a step will -then be <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>Δ</mi><mi>E</mi><mo>=</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>. -</p> -<div > -<head>Fast simulation for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mn>6</mn></math></formula></head> -<p>If the number of ionisation <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> -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 -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>α</mi><mi>I</mi><mo>]</mo></mrow></math></formula> the interval A, -and the latter <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow><mo>[</mo><mi>α</mi><mi>I</mi><mo>,</mo>&Emax;<mo>]</mo></mrow></math></formula> the -interval B. <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>α</mi></math></formula> lies -between 1 and <formula><math xmlns="http://www.w3.org/1998/Math/MathML" ->&Emax;<mo>/</mo><mi>I</mi></math></formula>. -A collision with a loss in the interval A happens with the probability -<formula type="display" id="eq:phys332-5"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∫</mo> - <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!-- - --><mrow>&Emax;<mi>α</mi></mrow></mfrac> -</math></formula>The -mean energy loss and the standard deviation for this type of collision are -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow><mo><</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>>;</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___ ---><mrow><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>∫</mo> - <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mi>I</mi><mi>α</mi><mo>ln</mo><mi>α</mi></mrow><!-- - --><mrow><mi>α</mi><mo>-</mo><mn>1</mn></mrow></mfrac> -</math></formula>and <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___ ---><mrow><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>∫</mo> - <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><msup><mi>I</mi><mrow><mn>2</mn></mrow></msup><mi>α</mi><mfenced -open='(' close=')'><mn>1</mn><mo>-</mo> <mfrac><mrow><mi>α</mi><msup><mo>ln</mo><mrow><mn>2</mn></mrow></msup><mi>α</mi></mrow><!--_ ---><mrow><msup><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mfenced> -</math></formula>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: - <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-1"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow> <mo>=</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-2"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup> <mo>=</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>)</mo></mrow> -</math></formula></cell></row></table> -It is further assumed that the energy loss in these collisions has a Gaussian -distribution with - <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-3"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow> <mo>=</mo> <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mrow><mo><</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>>;</mo></mrow> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-4"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup> <mo>=</mo> <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><msup><mi>σ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow> -</math></formula></cell></row></table> -The energy loss of these collision can then be sampled from the Gaussian -distribution. -</p><p>The collisions where the energy loss is in the interval B are sampled directly from -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>Δ</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub><mo>=</mo><msubsup><mo>∑</mo> - <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub></mrow></msubsup> <mfrac><mrow><mi>α</mi><mi>I</mi></mrow><!--_________ ---><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>i</mi></mrow></msub> <mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi><mo>-</mo><mi>α</mi><mi>I</mi></mrow><!-- - --><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> -</math></formula>The -total energy loss is the sum of these two types of collisions: -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> - <mi>Δ</mi><mi>E</mi><mo>=</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub> -</math></formula></p> -<p>The approximation of equations (<ptr target="eq:phys332-1"/>), (<ptr target="eq:phys332-2"/>), (<ptr target="eq:phys332-3"/>) and (<ptr target="eq:phys332-4"/>) can be used under the following -conditions: - <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-6"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub> <mo>≥</mo> <mn>0</mn> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-7"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mi>c</mi><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub> <mo>≤</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub> - </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> - </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-8"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>σ</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow></msub> <mo>≥</mo> <mn>0</mn> -</math></formula></cell></row></table> -where <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>c</mi><mo>≥</mo><mn>4</mn></math></formula>. From -the equations (<ptr target="eq:phys332-5"/>), (<ptr target="eq:phys332-1"/>) and (<ptr target="eq:phys332-3"/>) and from the conditions (<ptr target="eq:phys332-6"/>) and (<ptr target="eq:phys332-7"/>) the following limits can be -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mrow><mi>min</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!-- ---><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>I</mi></mrow></mfrac> <mo>≤</mo><mi>α</mi><mo>≤</mo><msub><mi>α</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!-- ---><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>I</mi></mrow></mfrac> -</math></formula>This -conditions gives a lower limit to number of the ionisations -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> for which the fast -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup> </math></formula>As -in the conditions (<ptr target="eq:phys332-6"/>), (<ptr target="eq:phys332-7"/>) and (<ptr target="eq:phys332-8"/>) the value of -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>c</mi></math></formula> is as minimum -4, one gets <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mn>6</mn></math></formula>. -In order to speed the simulation, the maximum value is used for -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>α</mi></math></formula>. -</p><p>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 -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula>. -The maximum number of these collisions can be estimated as -<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi><mo>,</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>≈</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub><mo>)</mo></mrow> -</math></formula>From the previous -expressions for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></formula> and -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub></math></formula> one can derive the -<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi>B</mi></mrow></msub><mo>≤</mo><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow><!--_ ---><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> -</math></formula>The following -values are obtained with <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>c</mi><mo>=</mo><mn>4</mn></math></formula>: -</p><p><table rend="inline"><row><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell><cell -></cell><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell> -</row><row><cell ->16 </cell><cell ->16 </cell><cell -></cell><cell -> 200</cell><cell -> 29.63</cell> -</row><row><cell ->20 </cell><cell ->17.78 </cell><cell -></cell><cell -> 500</cell><cell -> 31.01</cell> -</row><row><cell ->50 </cell><cell ->24.24 </cell><cell -></cell><cell -> 1000</cell><cell -> 31.50</cell> -</row><row><cell ->100 </cell><cell ->27.59 </cell><cell -></cell><cell -><formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>∞</mi></math></formula></cell><cell -> 32.00</cell> -</row></table> -</p> -</div> -<div > -<head>Special sampling for lower part of the spectrum</head> -<p>If the step length is very small (<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mo>≤</mo><mn>5</mn></math></formula> -mm in gases, <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mo>≤</mo></math></formula> -2-3 <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>μ</mi></math></formula>m in solids) -the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mrow><mo>(</mo><mi>Δ</mi><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mrow><mo>(</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>></mo></mrow><mo>)</mo></mrow></mrow></msup> -</math></formula>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 -<formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn></math></formula> eV is chosen -to correspond to the outer electrons. The mean number of collisions can be calculated from -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><mi>n</mi><mo>></mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac> -</math></formula>The number -of collisions <formula><math xmlns="http://www.w3.org/1998/Math/MathML" -><mi>n</mi></math></formula> -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 -<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Δ</mi><mi>E</mi><mo>=</mo><msubsup><mo>∑</mo> - <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup> -<mfrac><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mn>1</mn><mo>-</mo> -<mfrac><mrow>&Emax;</mrow> -<mrow>&Emax;<mo>+</mo><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac> -<msub><mi>u</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac> -</math></formula> -</p> </div> </div> <div -type="star"> <head>References</head> <list type="bibliography"> <item -id="bib-LAND"> <p>L.Landau. On the Energy Loss of Fast Particles by -Ionisation. Originally published in <emph>J. Phys.</emph>, 8:201, -1944. Reprinted in D.ter Haar, Editor, <emph>L.D.Landau, Collected -papers</emph>, page 417. Pergamon Press, Oxford, 1965. -</p></item> <item id="bib-SCH1"> <p>B.Schorr. Programs for -the Landau and the Vavilov distributions and the corresponding random -numbers. <emph>Comp. Phys. Comm.</emph>, 7:216, 1974. -</p></item> <item id="bib-SELT"> <p>S.M.Seltzer and -M.J.Berger. Energy loss straggling of protons and mesons. In -<emph>Studies in Penetration of Charged Particles in Matter</emph>, -Nuclear Science Series 39, Nat. Academy of Sciences, Washington DC, -1964. </p></item> <item id="bib-TALM"> <p>R.Talman. On the -statistics of particle identification using ionization. <emph>Nucl. -Inst. Meth.</emph>, 159:189, 1979. </p></item> <item -id="bib-VAVI"> <p>P.V.Vavilov. Ionisation losses of high energy -heavy particles. <emph>Soviet Physics JETP</emph>, 5:749, -1957.</p></item></list> </div> - </body> </text> </TEI.2> - - |