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diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa-raw.xml b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa-raw.xml new file mode 100644 index 00000000000..095b37c922d --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa-raw.xml @@ -0,0 +1,1054 @@ +<?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> + |