diff options
Diffstat (limited to 'Master/texmf-doc/doc/english/latex-web-companion/apa/latexexa-raw.xml')
-rw-r--r-- | Master/texmf-doc/doc/english/latex-web-companion/apa/latexexa-raw.xml | 1054 |
1 files changed, 0 insertions, 1054 deletions
diff --git a/Master/texmf-doc/doc/english/latex-web-companion/apa/latexexa-raw.xml b/Master/texmf-doc/doc/english/latex-web-companion/apa/latexexa-raw.xml deleted file mode 100644 index 095b37c922d..00000000000 --- a/Master/texmf-doc/doc/english/latex-web-companion/apa/latexexa-raw.xml +++ /dev/null @@ -1,1054 +0,0 @@ -<?xml version="1.0"?> - -<!DOCTYPE document SYSTEM "latex.xmldtd" -[ -<!ENTITY % MathML "INCLUDE"> -<!ENTITY % LaTeXEntShort "IGNORE"> -<!ENTITY % LaTeXMath "IGNORE"> -<!ENTITY % LaTeXEnt "IGNORE"> -]> - <document> -<frontmatter> -<title>Simulation of Energy Loss Straggling</title> -<author>Maria Physicist</author> -<date> -January 17, 1999</date> -</frontmatter> -<bodymatter> -<section id="intro"> -<stitle> -Introduction</stitle> -<displaymath> -<math -> -<mrow> -<par> - <msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup> -</mrow></math></displaymath> -</par><par>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 -<inlinemath><math -><mi>κ</mi></math></inlinemath>, -which is proportional to the ratio of mean energy loss to the maximum -allowed energy transfer in a single collision with an atomic electron -<displaymath><math -><mrow> - <mi>κ</mi><mo>=</mo> <mfrac><mrow><mi>ξ</mi></mrow><!--___ ---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> -</mrow></math></displaymath> -<inlinemath><math -><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath> is the -maximum transferable energy in a single collision with an atomic electron. -<displaymath><math -><mrow> - <msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><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></displaymath> where -<inlinemath><math -><mi>γ</mi><mo>=</mo><mi>E</mi><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></inlinemath>, -<inlinemath><math -><mi>E</mi></math></inlinemath> is energy and -<inlinemath><math -><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></inlinemath> the mass of the -incident particle, <inlinemath><math -><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></inlinemath> -and <inlinemath><math -><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub></math></inlinemath> is the -electron mass. <inlinemath><math -><mi>ξ</mi></math></inlinemath> -comes from the Rutherford scattering cross section and is defined as: - - <eqnarray ><subeqn ><math><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></subeqn></eqnarray> -where -</par><par><tabular preamble="ll"><row><cell -><inlinemath><math -><mi>z</mi></math></inlinemath></cell><cell ->charge of the incident particle </cell> -</row><row><cell -><inlinemath><math -><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></math></inlinemath></cell><cell ->Avogadro's number </cell> -</row><row><cell -><inlinemath><math -><mi>Z</mi></math></inlinemath></cell><cell ->atomic number of the material</cell> -</row><row><cell -><inlinemath><math -><mi>A</mi></math></inlinemath></cell><cell ->atomic weight of the material </cell> -</row><row><cell -><inlinemath><math -><mi>ρ</mi></math></inlinemath></cell><cell ->density </cell> -</row><row><cell -><inlinemath><math -><mi>δ</mi><mi>x</mi></math></inlinemath></cell><cell ->thickness of the material </cell> -</row><row><cell -> </cell> -</row></tabular> -</par><par><inlinemath><math -><mi>κ</mi></math></inlinemath> -measures the contribution of the collisions with energy transfer close to -<inlinemath><math -><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath>. For a given absorber, -<inlinemath><math -><mi>κ</mi></math></inlinemath> tends towards large -values if <inlinemath><math -><mi>δ</mi><mi>x</mi></math></inlinemath> is large -and/or if <inlinemath><math -><mi>β</mi></math></inlinemath> is small. -Likewise, <inlinemath><math -><mi>κ</mi></math></inlinemath> tends -towards zero if <inlinemath><math -><mi>δ</mi><mi>x</mi></math></inlinemath> is -small and/or if <inlinemath><math -><mi>β</mi></math></inlinemath> -approaches 1. -</par><par>The value of <inlinemath><math -><mi>κ</mi></math></inlinemath> -distinguishes two regimes which occur in the description of ionisation fluctuations -: -</par><lalist class="enumerate"> -<item> -<par>A -large -number -of -collisions -involving -the -loss -of -all -or -most -of -the -incident -particle -energy -during -the -traversal -of -an -absorber. -</par><par>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 -<inlinemath><math -><mi>κ</mi><mo>></mo><mn>1</mn><mn>0</mn></math></inlinemath> -(i.e. -when -the -mean -energy -loss -in -the -absorber -is -greater -than -the -maximum -energy -transfer -in -a -single -collision). -</par></item> -<item> -<par>Particles -traversing -thin -counters -and -incident -electrons -under -any -conditions. -</par><par>The -relevant -inequalities -and -distributions -are -<inlinemath><math -><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></inlinemath>, -Vavilov -distribution, -and -<inlinemath><math -><mi>κ</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></inlinemath>, -Landau -distribution.</par></item></lalist> -<par>An additional regime is defined by the contribution of the collisions -with low energy transfer which can be estimated with the relation -<inlinemath><math -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></inlinemath>, -where <inlinemath><math -><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></inlinemath> -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 -<inlinemath><math -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>≫</mo><mn>1</mn></math></inlinemath>. In <texttt>GEANT</texttt> (see -URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has -been set at <inlinemath><math -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mn>0</mn></math></inlinemath>. -Below this limit special models taking into account the atomic structure of the material are -used. This is important in thin layers and gaseous materials. Figure <ref refid="fg:phys332-1"/> shows the behaviour -of <inlinemath><math -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></inlinemath> as -a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic -energy in Argon, Silicon and Uranium. -</par> -<figure> -<includegraphics file="phys332-1"/> -<!--Figure 1--><caption id="fg:phys332-1">The variable <inlinemath><math -><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></inlinemath> -can be used to measure the validity range of the Landau -theory. It depends on the type and energy of the particle, -<inlinemath><math -><mi>Z</mi></math></inlinemath>, -<inlinemath><math -><mi>A</mi></math></inlinemath> -and the ionisation potential of the material and the layer thickness. </caption> -</figure> -<par>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. -</par> -</section> -<section id="sec:phys332-1"> -<stitle> -Landau theory</stitle> -<par>For a particle of mass <inlinemath><math -><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></inlinemath> traversing -a thickness of material <inlinemath><math -><mi>δ</mi><mi>x</mi></math></inlinemath>, -the Landau probability distribution may be written in terms of the universal Landau -function <inlinemath><math -><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></math></inlinemath> -as<cite refid="bib-LAND"/>: - - <eqnarray ><subeqn ><math><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></subeqn></eqnarray> -where - - <eqnarray ><subeqn ><math><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></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><mi>λ</mi> <mo>=</mo> <mfrac><mrow><mi>ε</mi><mo>-</mo><munderover accent='true'><mi>ε</mi><mrow></mrow><mo>⌅</mo></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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> <mtext></mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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>(Euler's constant)</mtext> <mtext></mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><munderover accent='true'><mi>ε</mi><mrow></mrow><mo>⌅</mo></munderover> <mo>=</mo> <mtext>average energy loss</mtext> <mtext></mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><mi>ε</mi> <mo>=</mo> <mtext>actual energy loss</mtext> <mtext></mtext> -</math></subeqn></eqnarray> -</par> -<subsection > -<stitle> -Restrictions</stitle> -<par>The Landau formalism makes two restrictive assumptions : -</par><lalist class="enumerate"> -<item> -<par>The -typical -energy -loss -is -small -compared -to -the -maximum -energy -loss -in -a -single -collision. -This -restriction -is -removed -in -the -Vavilov -theory -(see -section -<ref refid="vavref"/>). -</par></item> -<item> -<par>The -typical -energy -loss -in -the -absorber -should -be -large -compared -to -the -binding -energy -of -the -most -tightly -bound -electron. -For -gaseous -detectors, -typical -energy -losses -are -a -few -keV -which -is -comparable -to -the -binding -energies -of -the -inner -electrons. -In -such -cases -a -more -sophisticated -approach -which -accounts -for -atomic -energy -levels<cite refid="bib-TALM"/> -is -necessary -to -accurately -simulate -data -distributions. -In -<texttt>GEANT</texttt>, -a -parameterised -model -by -L. -Urbán -is -used -(see -section -<ref refid="urban"/>).</par></item></lalist> -<par>In addition, the average value of the Landau distribution is infinite. -Summing the Landau fluctuation obtained to the average energy from the -<inlinemath><math -><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></inlinemath> -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. -</par><par>A solution to this has been to introduce a limit on the value of the -variable sampled by the Landau distribution in order to keep the average -fluctuation to 0. The value obtained from the <texttt>GLANDO</texttt> routine is: -<displaymath><math -><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'><mi>ε</mi><mrow></mrow><mo>⌅</mo></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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> <mo>)</mo></mrow> -</mrow></math></displaymath> -In order for this to have average 0, we must impose that: -<displaymath><math -><mrow> - <munderover accent='true'><mi>λ</mi><mrow></mrow><mo>⌅</mo></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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> -</mrow></math></displaymath> -</par><par>This is realised introducing a <inlinemath><math -><msub><mi>λ</mi><mrow><mi>max</mi></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mi>λ</mi><mrow></mrow><mo>⌅</mo></munderover><mo>)</mo></mrow></math></inlinemath> -such that if only values of <inlinemath><math -><mi>λ</mi><mo>≤</mo><msub><mi>λ</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath> -are accepted, the average value of the distribution is -<inlinemath><math -><munderover accent='true'><mi>λ</mi><mrow></mrow><mo>⌅</mo></munderover></math></inlinemath>. -</par><par>A parametric fit to the universal Landau distribution has been performed, with following result: -<displaymath><math -><mrow> - <msub><mi>λ</mi><mrow><mi>max</mi></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'><mi>λ</mi><mrow></mrow><mo>⌅</mo></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></displaymath> only values -smaller than <inlinemath><math -><msub><mi>λ</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath> -are accepted, otherwise the distribution is resampled. -</par> -</subsection> -</section> -<section id="vavref"> -<stitle> -Vavilov theory</stitle> -<par>Vavilov<cite refid="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 -<inlinemath><math -><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo><mi>∞</mi></math></inlinemath>. Now -we can write<cite refid="bib-SCH1"/>: - - <eqnarray ><subeqn ><math><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></subeqn></eqnarray> -where - - <eqnarray ><subeqn ><math><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></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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></subeqn></eqnarray> -and - - <eqnarray ><subeqn ><math><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></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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'><mi>ε</mi><mrow></mrow><mo>⌅</mo></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></subeqn></eqnarray> -</par><par>The Vavilov parameters are simply related to the Landau parameter by -<inlinemath><math -><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></inlinemath>. It can be shown that -as <inlinemath><math -><mi>κ</mi><mo>→</mo><mn>0</mn></math></inlinemath>, the distribution of -the variable <inlinemath><math -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></inlinemath> approaches -that of Landau. For <inlinemath><math -><mi>κ</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></inlinemath> -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 -<inlinemath><math -><mi>κ</mi></math></inlinemath>, but rather the -distribution of <inlinemath><math -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></inlinemath> -defined above tends to the distribution of the true -<inlinemath><math -><mi>λ</mi></math></inlinemath> from -the Landau density function. Thus the routine <texttt>GVAVIV</texttt> samples the variable -<inlinemath><math -><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></inlinemath> rather -than <inlinemath><math -><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub></math></inlinemath>. -For <inlinemath><math -><mi>κ</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></inlinemath> -the Vavilov distribution tends to a Gaussian distribution (see next section). -</par> -</section> -<section > -<stitle> -Gaussian Theory</stitle> -<par>Various conflicting forms have been proposed for Gaussian straggling functions, but most -of these appear to have little theoretical or experimental basis. However, it has been shown<cite refid="bib-SELT"/> -that for <inlinemath><math -><mi>κ</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></inlinemath> -the Vavilov distribution can be replaced by a Gaussian of the form: - - <eqnarray ><subeqn ><math><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'><mi>ε</mi><mrow></mrow><mo>⌅</mo></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></subeqn></eqnarray> -thus implying - - <eqnarray ><subeqn ><math><mi>mean</mi> <mo>=</mo> <munderover accent='true'><mi>ε</mi><mrow></mrow><mo>⌅</mo></munderover> <mtext></mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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></subeqn></eqnarray> -</par> -</section> -<section id="urban"> -<stitle> -Urbán model</stitle> -<par>The method for computing restricted energy losses with -<inlinemath><math -><mi>δ</mi></math></inlinemath>-ray -production above given threshold energy in <texttt>GEANT</texttt> is a Monte Carlo method that -can be used for thin layers. It is fast and it can be used for any thickness of a -medium. Approaching the limit of the validity of Landau's theory, the loss -distribution approaches smoothly the Landau form as shown in Figure <ref refid="fg:phys332-2"/>. -</par><figure> -<includegraphics file="phys332-2"/> -<!--Figure 2--><caption id="fg:phys332-2">Energy loss distribution for a 3 GeV electron in Argon as given by -standard <texttt>GEANT</texttt>. The width of the layers is given in centimeters.</caption> -</figure> -<par>It is assumed that the atoms have only two energy levels with binding energy -<inlinemath><math -><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></inlinemath> and -<inlinemath><math -><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath>. -The particle--atom interaction will then be an excitation with energy loss -<inlinemath><math -><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></inlinemath> or -<inlinemath><math -><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath>, or -an ionisation with an energy loss distributed according to a function -<inlinemath><math -><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></inlinemath>: -<equation ><math> - <mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><!-- - --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> <mfrac><mrow><mn>1</mn></mrow><!-- _ ---><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mi>(</mi><mi>1</mi><mi>)</mi> -</math></equation> -</par><par>The macroscopic cross-section for excitations -(<inlinemath><math -><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></inlinemath>) is -<equation id="eq:sigex"><math> - <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> <mi>(</mi><mi>2</mi><mi>)</mi> -</math></equation>and -the macroscopic cross-section for ionisation is -<equation id="eq:sigion"><math> - <msub><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!-- ________________ ---><mrow><mi>I</mi><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow><!-- - --><mrow><mi>I</mi></mrow></mfrac> <mo>)</mo></mrow></mrow></mfrac><mi>r</mi> <mi>(</mi><mi>3</mi><mi>)</mi> -</math></equation><inlinemath><math -><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath> -is the <texttt>GEANT</texttt> cut for <inlinemath><math -><mi>δ</mi></math></inlinemath>-production, -or the maximum energy transfer minus mean ionisation energy, if it is smaller than -this cut-off value. The following notation is used: -</par><par><tabular preamble="ll"><row><cell -><inlinemath><math -><mi>r</mi><mo>,</mo><mi>C</mi></math></inlinemath></cell><cell ->parameters of the model</cell> -</row><row><cell -><inlinemath><math -><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath></cell><cell ->atomic energy levels </cell> -</row><row><cell -><inlinemath><math -><mi>I</mi></math></inlinemath></cell><cell ->mean ionisation energy </cell> -</row><row><cell -><inlinemath><math -><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath></cell><cell ->oscillator strengths </cell> -</row></tabular> -</par><par>The model has the parameters <inlinemath><math -><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>, -<inlinemath><math -><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>, -<inlinemath><math -><mi>C</mi></math></inlinemath> and -<inlinemath><math -><mi>r</mi><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></math></inlinemath>. The oscillator -strengths <inlinemath><math -><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath> and the -atomic level energies <inlinemath><math -><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath> -should satisfy the constraints - - <eqnarray ><subeqn id="eq:fisum"><math><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> <mtext>(4)</mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn id="eq:flnsum"><math><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> <mtext>(5)</mtext> -</math></subeqn></eqnarray> -The parameter <inlinemath><math -><mi>C</mi></math></inlinemath> -can be defined with the help of the mean energy loss -<inlinemath><math -><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></inlinemath> in the following way: The -numbers of collisions (<inlinemath><math -><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>, -i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean -number <inlinemath><math -><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></inlinemath>. In a step -<inlinemath><math -><mi>Δ</mi><mi>x</mi></math></inlinemath> the mean number -of collisions is <equation ><math> - <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> <mi>(</mi><mi>6</mi><mi>)</mi> -</math></equation>The -mean energy loss <inlinemath><math -><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></inlinemath> -in a step is the sum of the excitation and ionisation contributions -<equation ><math> - <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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><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> <mi>(</mi><mi>7</mi><mi>)</mi> -</math></equation>From -this, using the equations (<ref refid="eq:sigex"/>), (<ref refid="eq:sigion"/>), (<ref refid="eq:fisum"/>) and (<ref refid="eq:flnsum"/>), one can define the parameter -<inlinemath><math -><mi>C</mi></math></inlinemath> -<equation ><math> - <mi>C</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!-- ---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>(</mi><mi>8</mi><mi>)</mi> -</math></equation> -</par><par>The following values have been chosen in <texttt>GEANT</texttt> for the other parameters: -<displaymath><math -><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></displaymath> With these values -the atomic level <inlinemath><math -><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath> -corresponds approximately the K-shell energy of the atoms and -<inlinemath><math -><mi>Z</mi><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath> the number of -K-shell electrons. <inlinemath><math -><mi>r</mi></math></inlinemath> -is the only variable which can be tuned freely. It determines the relative contribution -of ionisation and excitation to the energy loss. -</par><par>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 -<equation ><math> - <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> <mi>(</mi><mi>9</mi><mi>)</mi> -</math></equation>where -<inlinemath><math -><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></math></inlinemath> and -<inlinemath><math -><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath> -are sampled from Poisson distribution as discussed above. The -loss due to the ionisation can be generated from the distribution -<inlinemath><math -><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math></inlinemath> by -the inverse transformation method: - - <eqnarray ><subeqn ><math><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></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math><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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!-- ___ ---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> <mtext>(10)</mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn ><math> <mtext>(11)</mtext> -</math></subeqn></eqnarray> -where <inlinemath><math -><mi>u</mi></math></inlinemath> is a uniform random -number between <inlinemath><math -><mi>F</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></inlinemath> and -<inlinemath><math -><mi>F</mi><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></inlinemath>. The contribution from the -ionisations will be <equation ><math> - <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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!-- ___ ---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> <mi>(</mi><mi>1</mi><mi>2</mi><mi>)</mi> -</math></equation>where -<inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath> is the -number of ionisation (sampled from Poisson distribution). The energy loss in a step will -then be <inlinemath><math -><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></inlinemath>. -</par> -<subsection > -<stitle> -Fast simulation for <inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mn>6</mn></math></inlinemath></stitle> -<par>If the number of ionisation <inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath> -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 -<inlinemath><math -><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>α</mi><mi>I</mi><mo>]</mo></mrow></math></inlinemath> the interval A, -and the latter <inlinemath><math -><mrow><mo>[</mo><mi>α</mi><mi>I</mi><mo>,</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>]</mo></mrow></math></inlinemath> the -interval B. <inlinemath><math -><mi>α</mi></math></inlinemath> lies -between 1 and <inlinemath><math -><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><mi>I</mi></math></inlinemath>. -A collision with a loss in the interval A happens with the probability -<equation id="eq:phys332-5"><math> - <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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!-- - --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mi>α</mi></mrow></mfrac> <mi>(</mi><mi>1</mi><mi>3</mi><mi>)</mi> -</math></equation>The -mean energy loss and the standard deviation for this type of collision are -<equation ><math> - <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> <mi>(</mi><mi>1</mi><mi>4</mi><mi>)</mi> -</math></equation>and -<equation ><math> - <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> <mi>(</mi><mi>1</mi><mi>5</mi><mi>)</mi> -</math></equation>If the -collision number is high, we assume that the number of the type A collisions can be -calculated from a Gaussian distribution with the following mean value and standard -deviation: - - <eqnarray ><subeqn id="eq:phys332-1"><math><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> <mtext>(16)</mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn id="eq:phys332-2"><math><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> <mtext>(17)</mtext> -</math></subeqn></eqnarray> -It is further assumed that the energy loss in these collisions has a Gaussian -distribution with - - <eqnarray ><subeqn id="eq:phys332-3"><math><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> <mtext>(18)</mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn id="eq:phys332-4"><math><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> <mtext>(19)</mtext> -</math></subeqn></eqnarray> -The energy loss of these collision can then be sampled from the Gaussian -distribution. -</par><par>The collisions where the energy loss is in the interval B are sampled directly from -<equation ><math> - <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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>-</mo><mi>α</mi><mi>I</mi></mrow><!-- - --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> <mi>(</mi><mi>2</mi><mi>0</mi><mi>)</mi> -</math></equation>The -total energy loss is the sum of these two types of collisions: -<equation ><math> - <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> <mi>(</mi><mi>2</mi><mi>1</mi><mi>)</mi> -</math></equation> -</par><par>The approximation of equations (<ref refid="eq:phys332-1"/>), (<ref refid="eq:phys332-2"/>), (<ref refid="eq:phys332-3"/>) and (<ref refid="eq:phys332-4"/>) can be used under the following -conditions: - - <eqnarray ><subeqn id="eq:phys332-6"><math><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> <mtext>(22)</mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn id="eq:phys332-7"><math><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> <mtext>(23)</mtext> - </math></subeqn><subeqn ><math> - </math></subeqn><subeqn id="eq:phys332-8"><math><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> <mtext>(24)</mtext> -</math></subeqn></eqnarray> -where <inlinemath><math -><mi>c</mi><mo>≥</mo><mn>4</mn></math></inlinemath>. From -the equations (<ref refid="eq:phys332-5"/>), (<ref refid="eq:phys332-1"/>) and (<ref refid="eq:phys332-3"/>) and from the conditions (<ref refid="eq:phys332-6"/>) and (<ref refid="eq:phys332-7"/>) the following limits can be -derived: <equation ><math> - <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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!-- ---><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><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><mi>max</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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!-- ---><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><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> <mi>(</mi><mi>2</mi><mi>5</mi><mi>)</mi> -</math></equation>This -conditions gives a lower limit to number of the ionisations -<inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath> for which the fast -sampling can be done: <equation ><math> - <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup> <mi>(</mi><mi>2</mi><mi>6</mi><mi>)</mi> -</math></equation>As -in the conditions (<ref refid="eq:phys332-6"/>), (<ref refid="eq:phys332-7"/>) and (<ref refid="eq:phys332-8"/>) the value of -<inlinemath><math -><mi>c</mi></math></inlinemath> is as minimum -4, one gets <inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mn>6</mn></math></inlinemath>. -In order to speed the simulation, the maximum value is used for -<inlinemath><math -><mi>α</mi></math></inlinemath>. -</par><par>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 -<inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath>. -The maximum number of these collisions can be estimated as -<equation ><math> - <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> <mi>(</mi><mi>2</mi><mi>7</mi><mi>)</mi> -</math></equation>From the previous -expressions for <inlinemath><math -><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></inlinemath> and -<inlinemath><math -><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub></math></inlinemath> one can derive the -condition <equation ><math> - <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> <mi>(</mi><mi>2</mi><mi>8</mi><mi>)</mi> -</math></equation>The following -values are obtained with <inlinemath><math -><mi>c</mi><mo>=</mo><mn>4</mn></math></inlinemath>: -</par><par><tabular preamble="llcrr"><row><cell -><inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath></cell><cell -><inlinemath><math -><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></inlinemath></cell><cell -></cell><cell -><inlinemath><math -><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath></cell><cell -><inlinemath><math -><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></inlinemath></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 -><inlinemath><math -><mi>∞</mi></math></inlinemath></cell><cell -> 32.00</cell> -</row></tabular> -</par> -</subsection> -<subsection > -<stitle> -Special sampling for lower part of the spectrum</stitle> -<par>If the step length is very small (<inlinemath><math -><mo>≤</mo><mn>5</mn></math></inlinemath> -mm in gases, <inlinemath><math -><mo>≤</mo></math></inlinemath> -2-3 <inlinemath><math -><mi>μ</mi></math></inlinemath>m in solids) -the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is -computed <equation ><math> - <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> <mi>(</mi><mi>2</mi><mi>9</mi><mi>)</mi> -</math></equation>If the -probability is bigger than 0.01 a special sampling is done, taking into account the fact that in -these cases the projectile interacts only with the outer electrons of the atom. An energy level -<inlinemath><math -><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn></math></inlinemath> eV is chosen -to correspond to the outer electrons. The mean number of collisions can be calculated from -<equation ><math> - <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> <mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!-- ---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>Δ</mi><mi>x</mi> <mi>(</mi><mi>3</mi><mi>0</mi><mi>)</mi> -</math></equation>The number -of collisions <inlinemath><math -><mi>n</mi></math></inlinemath> -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 -<equation ><math> - <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><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!--_____ ---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><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> <mi>(</mi><mi>3</mi><mi>1</mi><mi>)</mi> -</math></equation> -</par> -</subsection> -</section> -<section class="star"> -<stitle> -References</stitle> -<bibliography > -<bibitem id="bib-LAND"> -<par>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. -</par></bibitem> -<bibitem id="bib-SCH1"> -<par>B.Schorr. -Programs -for -the -Landau -and -the -Vavilov -distributions -and -the -corresponding -random -numbers. -<emph>Comp. -Phys. -Comm.</emph>, -7:216, -1974. -</par></bibitem> -<bibitem id="bib-SELT"> -<par>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. -</par></bibitem> -<bibitem id="bib-TALM"> -<par>R.Talman. -On -the -statistics -of -particle -identification -using -ionization. -<emph>Nucl. -Inst. -Meth.</emph>, -159:189, -1979. -</par></bibitem> -<bibitem id="bib-VAVI"> -<par>P.V.Vavilov. -Ionisation -losses -of -high -energy -heavy -particles. -<emph>Soviet -Physics -JETP</emph>, -5:749, -1957.</par></bibitem></bibliography> -</section> -</bodymatter></document> - |