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author | Karl Berry <karl@freefriends.org> | 2006-01-09 00:45:48 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2006-01-09 00:45:48 +0000 |
commit | 5dc602d16c5be2fd035b254ca23484a90aebd6dc (patch) | |
tree | 72efb15fba318cc2096a8cc6999ed3fa0bff317d /Master/texmf-dist/doc/xmltex/passivetex/latextei.xml | |
parent | b4fc5f639874db951177ec539299d20908adb654 (diff) |
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diff --git a/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml b/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml new file mode 100644 index 00000000000..af9fbb686c4 --- /dev/null +++ b/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml @@ -0,0 +1,972 @@ +<?xml version="1.0"?> +<!DOCTYPE TEI.2 SYSTEM "http://www.oucs.ox.ac.uk/dtds/tei-oucs.dtd" [ +<!ENTITY aacute "á"> +<!ENTITY ap "≈"> +<!ENTITY alpha "α"> +<!ENTITY barwed "⊼"> +<!ENTITY beta "β"> +<!ENTITY commaspace " "> +<!ENTITY gamma "γ"> +<!ENTITY delta "δ"> +<!ENTITY Delta "Δ"> +<!ENTITY dots "…"> +<!ENTITY epsi "ε"> +<!ENTITY gg "⋙"> +<!ENTITY Gt "≫"> +<!ENTITY geq "≥"> +<!ENTITY infin "∞"> +<!ENTITY infty "∞"> +<!ENTITY int "∫"> +<!ENTITY kappa "κ"> +<!ENTITY lambda "λ"> +<!ENTITY langle "〈"> +<!ENTITY leq "≤"> +<!ENTITY mu "μ"> +<!ENTITY nbsp " "> +<!ENTITY phi "φ"> +<!ENTITY pi "π"> +<!ENTITY psi "ψ"> +<!ENTITY rangle "〉"> +<!ENTITY rho "ρ"> +<!ENTITY sigma "σ"> +<!ENTITY Sigma "Σ"> +<!ENTITY sim "∼"> +<!ENTITY thinspace " "> +<!ENTITY thickspace " "> +<!ENTITY xi "ξ"> +<!ENTITY prime "′"> +<!ENTITY isinv "∈"> +<!ENTITY macr "̄"> +<!ENTITY Emax "<msub><mi>E</mi><mi>max</mi></msub>"> +<!ENTITY exp "E<mtext>exp</mtext>"> +<!ENTITY ln "E<mtext>ln</mtext>"> +<!ENTITY Rarr "⇒"> +<!ENTITY rarr "→"> +<!ENTITY GEANT "GEANT"> +<!ENTITY sum "∑"> +]> +<TEI.2> + <teiHeader> + <fileDesc> + <titleStmt> + <title>A sample article</title> + </titleStmt> + <publicationStmt> + <availability><p>Converted from LaTeX by Sebastian Rahtz</p> </availability> + </publicationStmt> + <sourceDesc> + <p></p> + </sourceDesc> + </fileDesc> + <revisionDesc> + <list> + <item> + <date>23 Oct 1999</date> SR converted from LaTeX</item> + </list> + </revisionDesc> + </teiHeader> + <text> + <front> + <docTitle> + <titlePart type="main">Simulation of Energy Loss Straggling</titlePart> + </docTitle> + <docAuthor>Maria Physicist</docAuthor> + <docDate>January 17, 1999</docDate> + </front> + <body> <div id="intro"> <head>Introduction</head> <p>Due to +the statistical nature of ionisation energy loss, large fluctuations +can occur in the amount of energy deposited by a particle traversing +an absorber element. Continuous processes such as multiple scattering +and energy loss play a relevant role in the longitudinal and lateral +development of electromagnetic and hadronic showers, and in the case +of sampling calorimeters the measured resolution can be significantly +affected by such fluctuations in their active layers. The description +of ionisation fluctuations is characterised by the significance +parameter <formula><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math></formula>, which is +proportional to the ratio of mean energy loss to the maximum allowed +energy transfer in a single collision with an atomic electron + <formula +type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow> <mi>κ</mi><mo>=</mo> +<mfrac> + <mrow><mi>ξ</mi></mrow> + <mrow> + &Emax; + </mrow> +</mfrac> +</mrow> +</math></formula> +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +>&Emax;</math></formula> is the +maximum transferable energy in a single collision with an atomic electron. +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow> + &Emax;<mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup></mrow><!--____________ +--><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>γ</mi><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub><mo>+</mo><msup><mfenced +open='(' close=')'><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub> </mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo> +</mrow></math></formula> where +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>γ</mi><mo>=</mo><mi>E</mi><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula>, +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>E</mi></math></formula> is energy and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> the mass of the +incident particle, <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>/</mo><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup></math></formula> +and <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub></math></formula> is the +electron mass. <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ξ</mi></math></formula> +comes from the Rutherford scattering crosss section and is defined as: + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mn>4</mn></mrow></msup><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub><mi>Z</mi><mi>ρ</mi><mi>δ</mi><mi>x</mi></mrow><!-- + --><mrow><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>A</mi></mrow></mfrac> <mo>=</mo><mn>1</mn><mn>5</mn><mn>3</mn><mo>.</mo><mn>4</mn> <mfrac><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- +--><mrow><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mfrac><mrow><mi>Z</mi></mrow><!-- +--><mrow><mi>A</mi></mrow></mfrac><mi>ρ</mi><mi>δ</mi><mi>x</mi><mspace width='12pt'/><mi>keV </mi><mo>,</mo> <mtext></mtext> +</math></formula></cell></row></table> +where +</p><p><table rend="inline"><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>z</mi></math></formula></cell><cell +>charge of the incident particle </cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></math></formula></cell><cell +>Avogadro's number </cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>Z</mi></math></formula></cell><cell +>atomic number of the material</cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>A</mi></math></formula></cell><cell +>atomic weight of the material </cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ρ</mi></math></formula></cell><cell +>density </cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>δ</mi><mi>x</mi></math></formula></cell><cell +>thickness of the material </cell> +</row><row><cell +> </cell> +</row></table> +</p><p><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi></math></formula> +measures the contribution of the collisions with energy transfer close to +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +>&Emax;</math></formula>. For a given absorber, +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi></math></formula> tends towards large +values if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>δ</mi><mi>x</mi></math></formula> is large +and/or if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>β</mi></math></formula> is small. +Likewise, <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi></math></formula> tends +towards zero if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>δ</mi><mi>x</mi></math></formula> is +small and/or if <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>β</mi></math></formula> +approaches 1. +</p><p>The value of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi></math></formula> +distinguishes two regimes which occur in the description of ionisation fluctuations +: +</p><list type="enumerate"> +<item> +<p>A +large +number +of +collisions +involving +the +loss +of +all +or +most +of +the +incident +particle +energy +during +the +traversal +of +an +absorber. +</p><p>As +the +total +energy +transfer +is +composed +of +a +multitude +of +small +energy +losses, +we +can +apply +the +central +limit +theorem +and +describe +the +fluctuations +by +a +Gaussian +distribution. +This +case +is +applicable +to +non-relativistic +particles +and +is +described +by +the +inequality +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi><mo>></mo><mn>1</mn><mn>0</mn></math></formula> +(i.e. +when +the +mean +energy +loss +in +the +absorber +is +greater +than +the +maximum +energy +transfer +in +a +single +collision). +</p></item> +<item> +<p>Particles +traversing +thin +counters +and +incident +electrons +under +any +conditions. +</p><p>The +relevant +inequalities +and +distributions +are +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn><mo><</mo><mi>κ</mi><mo><</mo><mn>1</mn><mn>0</mn></math></formula>, +Vavilov +distribution, +and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula>, +Landau +distribution.</p></item></list> +<p>An additional regime is defined by the contribution of the collisions +with low energy transfer which can be estimated with the relation +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>, +where <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> +is the mean ionisation potential of the atom. Landau theory assumes that +the number of these collisions is high, and consequently, it has a restriction +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>≫</mo><mn>1</mn></math></formula>. In <code>GEANT</code> (see +URL <xptr url="http://wwwinfo.cern.ch/asdoc/geant/geantall.html"/>), the limit of Landau theory has +been set at <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mn>0</mn></math></formula>. +Below this limit special models taking into account the atomic structure of the material are +used. This is important in thin layers and gaseous materials. Figure <ptr target="fg:phys332-1"/> shows the behaviour +of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> as +a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic +energy in Argon, Silicon and Uranium. +</p> +<p><figure file="phys332-1" id="fg:phys332-1"> +<head>The variable <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>ξ</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> +can be used to measure the validity range of the Landau +theory. It depends on the type and energy of the particle, +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>Z</mi></math></formula>, +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>A</mi></math></formula> +and the ionisation potential of the material and the layer thickness. </head> +</figure></p> +<p>In the following sections, the different theories and models for the energy loss +fluctuation are described. First, the Landau theory and its limitations are discussed, +and then, the Vavilov and Gaussian straggling functions and the methods in the thin +layers and gaseous materials are presented. +</p> +</div> +<div id="sec:phys332-1"> +<head>Landau theory</head> +<p>For a particle of mass <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> traversing +a thickness of material <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>δ</mi><mi>x</mi></math></formula>, +the Landau probability distribution may be written in terms of the universal Landau +function <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></math></formula> +as<ptr target="bib-LAND"/>: + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>x</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!-- +--><mrow><mi>ξ</mi></mrow></mfrac><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow> <mtext></mtext> +</math></formula></cell></row></table> +where + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_ +--><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup><mo>∫</mo> + <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup><mo>exp</mo><mfenced +open='(' close=')'><mi>u</mi><mo>ln</mo><mi>u</mi><mo>+</mo><mi>λ</mi><mi>u</mi></mfenced><mi>d</mi><mi>u</mi><mspace width='2cm'/><mi>c</mi><mo>≥</mo><mn>0</mn> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi> <mo>=</mo> <mfrac><mrow><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover></mrow><!-- + --><mrow><mi>ξ</mi></mrow></mfrac> <mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>ξ</mi></mrow><!-- ___ +--><mrow>&Emax;</mrow></mfrac> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mi>′</mi> <mo>=</mo> <mn>0</mn><mo>.</mo><mn>4</mn><mn>2</mn><mn>2</mn><mn>7</mn><mn>8</mn><mn>4</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><mn>1</mn><mo>-</mo><mi>γ</mi> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi> <mo>=</mo> <mn>0</mn><mo>.</mo><mn>5</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>5</mn><mo>.</mo><mo>.</mo><mo>.</mo><mtext>(Eulers constant)</mtext> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover> <mo>=</mo> <mtext>average energy loss</mtext> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi> <mo>=</mo> <mtext>actual energy loss</mtext> <mtext></mtext> +</math></formula></cell></row></table> +</p> +<div > +<head>Restrictions</head> +<p>The Landau formalism makes two restrictive assumptions : +</p><list type="enumerate"> +<item> +<p>The +typical +energy +loss +is +small +compared +to +the +maximum +energy +loss +in +a +single +collision. +This +restriction +is +removed +in +the +Vavilov +theory +(see +section +<ptr target="vavref"/>). +</p></item> +<item> +<p>The +typical +energy +loss +in +the +absorber +should +be +large +compared +to +the +binding +energy +of +the +most +tightly +bound +electron. +For +gaseous +detectors, +typical +energy +losses +are +a +few +keV +which +is +comparable +to +the +binding +energies +of +the +inner +electrons. +In +such +cases +a +more +sophisticated +approach +which +accounts +for +atomic +energy +levels<ptr target="bib-TALM"/> +is +necessary +to +accurately +simulate +data +distributions. +In +<code>GEANT</code>, +a +parameterised +model +by +L. +Urbán +is +used +(see +section +<ptr target="urban"/>).</p></item></list> +<p>In addition, the average value of the Landau distribution is infinite. +Summing the Landau fluctuation obtained to the average energy from the +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> +tables, we obtain a value which is larger than the one coming from the table. The +probability to sample a large value is small, so it takes a large number of steps +(extractions) for the average fluctuation to be significantly larger than zero. This +introduces a dependence of the energy loss on the step size which can affect +calculations. +</p><p>A solution to this has been to introduce a limit on the value of the +variable sampled by the Landau distribution in order to keep the average +fluctuation to 0. The value obtained from the <code>GLANDO</code> routine is: +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow> + <mi>δ</mi><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover><mo>=</mo><mi>ξ</mi><mrow><mo>(</mo><mi>λ</mi><mo>-</mo><mi>γ</mi><mi>′</mi><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>ln</mo> <mfrac><mrow><mi>ξ</mi></mrow><!-- ___ +--><mrow>&Emax;</mrow></mfrac> <mo>)</mo></mrow> +</mrow></math></formula> +In order for this to have average 0, we must impose that: +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow> + <munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover><mo>=</mo><mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>ξ</mi></mrow><!-- ___ +--><mrow>&Emax;</mrow></mfrac> +</mrow></math></formula> +</p><p>This is realised introducing a <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover><mo>)</mo></mrow></math></formula> +such that if only values of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>λ</mi><mo>≤</mo><msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub></math></formula> +are accepted, the average value of the distribution is +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover></math></formula>. +</p><p>A parametric fit to the universal Landau distribution has been performed, with following result: +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow> + <msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>0</mn><mn>7</mn><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mo>.</mo><mn>1</mn><mn>9</mn><mn>3</mn><mn>4</mn><munderover accent='true'><mo>¯</mo><mi>λ</mi><mrow></mrow></munderover><mo>+</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>7</mn><mn>7</mn><mn>9</mn><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>5</mn><mn>2</mn><mn>3</mn><mn>8</mn><mn>2</mn><munderover accent='true'><mi>λ</mi><mrow></mrow><mo>¯</mo></munderover><mo>)</mo></mrow><mo>exp</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>4</mn><mn>7</mn><mn>5</mn><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn><mn>4</mn><mn>4</mn><mn>4</mn><mn>2</mn><munderover accent='true'><mi>λ</mi><mrow></mrow><mo>¯</mo></munderover><mo>)</mo></mrow> +</mrow></math></formula> only values +smaller than <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub></math></formula> +are accepted, otherwise the distribution is resampled. +</p> +</div> +</div> +<div id="vavref"> +<head>Vavilov theory</head> +<p>Vavilov<ptr target="bib-VAVI"/> derived a more accurate straggling distribution by introducing the kinematic +limit on the maximum transferable energy in a single collision, rather than using +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +>&Emax;<mo>=</mo><mi>∞</mi></math></formula>. Now +we can write<ptr target="bib-SCH1"/>: + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi> <mfenced +open='(' close=')'><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!-- +--><mrow><mi>ξ</mi></mrow></mfrac><msub><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced +open='(' close=')'><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mtext></mtext> +</math></formula></cell></row></table> +where + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced +open='(' close=')'><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_ +--><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup><mo>∫</mo> + <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup><mi>φ</mi><mfenced +open='(' close=')'><mi>s</mi></mfenced><msup><mi>e</mi><mrow><mi>λ</mi><mi>s</mi></mrow></msup><mi>d</mi><mi>s</mi><mspace width='2cm'/><mi>c</mi><mo>≥</mo><mn>0</mn> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mfenced +open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo> <mo>exp</mo><mfenced +open='[' close=']'><mi>κ</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mi>γ</mi><mo>)</mo></mrow></mfenced><mo>exp</mo><mfenced +open='[' close=']'><mi>ψ</mi> <mfenced +open='(' close=')'><mi>s</mi></mfenced></mfenced><mo>,</mo> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ψ</mi> <mfenced +open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo> <mi>s</mi><mo>ln</mo><mi>κ</mi><mo>+</mo><mrow><mo>(</mo><mi>s</mi><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mi>κ</mi><mo>)</mo></mrow><mfenced +open='[' close=']'><mo>ln</mo><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>κ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>E</mi><mrow> +<mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>κ</mi><mo>)</mo></mrow></mfenced><mo>-</mo><mi>κ</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>s</mi><mo>/</mo><mi>κ</mi></mrow></msup><mo>,</mo> <mtext></mtext> +</math></formula></cell></row></table> +and + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow> <mo>=</mo><msubsup> <mo>∫</mo> + <mrow><mi>∞</mi></mrow><mrow><mi>z</mi></mrow></msubsup><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mspace width='1cm'/><mtext>(the exponential integral)</mtext> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub> <mo>=</mo> <mi>κ</mi><mfenced +open='[' close=']'><mfrac><mrow><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover></mrow><!-- + --><mrow><mi>ξ</mi></mrow></mfrac> <mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mtext></mtext> +</math></formula></cell></row></table> +</p><p>The Vavilov parameters are simply related to the Landau parameter by +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>/</mo><mi>κ</mi><mo>-</mo><mo>ln</mo><mi>κ</mi></math></formula>. It can be shown that +as <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi><mo>→</mo><mn>0</mn></math></formula>, the distribution of +the variable <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></formula> approaches +that of Landau. For <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula> +the two distributions are already practically identical. Contrary to what many textbooks +report, the Vavilov distribution <emph>does not</emph> approximate the Landau distribution for small +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi></math></formula>, but rather the +distribution of <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></formula> +defined above tends to the distribution of the true +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>λ</mi></math></formula> from +the Landau density function. Thus the routine <code>GVAVIV</code> samples the variable +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mi>L</mi></mrow></msub></math></formula> rather +than <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub></math></formula>. +For <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></formula> +the Vavilov distribution tends to a Gaussian distribution (see next section). +</p> +</div> +<div > +<head>Gaussian Theory</head> +<p>Various conflicting forms have been proposed for Gaussian straggling functions, but most +of these appear to have little theoretical or experimental basis. However, it has been shown<ptr target="bib-SELT"/> +that for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>κ</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></formula> +the Vavilov distribution can be replaced by a Gaussian of the form: + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi><mo>)</mo></mrow><mo>≈</mo> <mfrac><mrow><mn>1</mn></mrow><!--________ +--><mrow><mi>ξ</mi><msqrt><!--<mi>&radical;</mi> + ______________--><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><!-- + --><mrow><mi>κ</mi></mrow></mfrac> <mfenced +open='(' close=')'><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn></mfenced></msqrt></mrow></mfrac><mo>exp</mo><mfenced +open='[' close=']'><mfrac><mrow><msup><mrow><mo>(</mo><mi>ε</mi><mo>-</mo><munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><!-- + --><mrow><mn>2</mn></mrow></mfrac> <mfrac><mrow><mi>κ</mi></mrow><!-- _______ +--><mrow><msup><mi>ξ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mfenced> <mtext></mtext> +</math></formula></cell></row></table> +thus implying + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>mean</mi> <mo>=</mo> <munderover accent='true'><mo>¯</mo><mi>ε</mi><mrow></mrow></munderover> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mrow><mn>2</mn></mrow></msup> <mo>=</mo> <mfrac><mrow><msup><mi>ξ</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- + --><mrow><mi>κ</mi></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><mi>ξ</mi><msub><mi>E</mi><mrow><mi> +max</mi></mrow></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow> <mtext></mtext> +</math></formula></cell></row></table> +</p> +</div> +<div id="urban"> +<head>Urbán model</head> +<p>The method for computing restricted energy losses with +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>δ</mi></math></formula>-ray +production above given threshold energy in <code>GEANT</code> is a Monte Carlo method that +can be used for thin layers. It is fast and it can be used for any thickness of a +medium. Approaching the limit of the validity of Landau's theory, the loss +distribution approaches smoothly the Landau form as shown in Figure <ptr target="fg:phys332-2"/>. +</p> +<p><figure file="phys332-2" id="fg:phys332-2"> +<head>Energy loss distribution for a 3 GeV electron in Argon as given by +standard GEANT. The width of the layers is given in centimeters.</head> +</figure></p> +<p>It is assumed that the atoms have only two energy levels with binding energy +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>. +The particle--atom interaction will then be an excitation with energy loss +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> or +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>, or +an ionisation with an energy loss distributed according to a function +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>∼</mo><mn>1</mn><mo>/</mo><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></math></formula>: +<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><!-- + --><mrow>&Emax;</mrow></mfrac> +<mfrac><mrow><mn>1</mn></mrow><!-- _ +--><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> +</math></formula></p><p>The +macroscopic cross-section for excitations (<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></formula>) +is <formula id="eq:sigex" type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub><mi>Σ</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></mrow><!-- +--><mrow><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac> <mfrac><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- + --><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>I</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo></mrow> </math></formula>and +the macroscopic cross-section for ionisation is +<formula id="eq:sigion" type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow>&Emax;</mrow><!-- ________________ +--><mrow><mi>I</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow><!-- + --><mrow><mi>I</mi></mrow></mfrac> <mo>)</mo></mrow></mrow></mfrac><mi>r</mi> </math></formula> +<formula><math xmlns="http://www.w3.org/1998/Math/MathML">&Emax;</math></formula> +is the <code>GEANT</code> cut for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>δ</mi></math></formula>-production, +or the maximum energy transfer minus mean ionisation energy, if it is smaller than +this cut-off value. The following notation is used: +</p><p><table rend="inline"><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>r</mi><mo>,</mo><mi>C</mi></math></formula></cell><cell +>parameters of the model</cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell +>atomic energy levels </cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>I</mi></math></formula></cell><cell +>mean ionisation energy </cell> +</row><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell +>oscillator strengths </cell> +</row></table> +</p><p>The model has the parameters <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula>, +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>, +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>C</mi></math></formula> and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>r</mi><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></math></formula>. The oscillator +strengths <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula> and the +atomic level energies <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula> +should satisfy the constraints + <table rend="inline"><row><cell><formula type="subeqn" id="eq:fisum"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub> <mo>=</mo> <mn>1</mn> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:flnsum"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub> <mo>=</mo> <mo>ln</mo><mi>I</mi> +</math></formula></cell></row></table> +The parameter <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>C</mi></math></formula> +can be defined with the help of the mean energy loss +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> in the following way: The +numbers of collisions (<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></math></formula>, +i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean +number <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></formula>. In a step +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>Δ</mi><mi>x</mi></math></formula> the mean number +of collisions is <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow><mo><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow><mo>=</mo><msub><mi>Σ</mi><mrow><mi>i</mi></mrow></msub><mi>Δ</mi><mi>x</mi> +</math></formula>The +mean energy loss <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> +in a step is the sum of the excitation and ionisation contributions +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!-- +--><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>Δ</mi><mi>x</mi><mo>=</mo><mfenced +open='[' close=']'><msub><mi>Σ</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>Σ</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><msubsup><mo>∫</mo> + <mrow><mi>I</mi></mrow><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi></mfenced><mi>Δ</mi><mi>x</mi> +</math></formula>From +this, using the equations (<ptr target="eq:sigex"/>), (<ptr target="eq:sigion"/>), (<ptr target="eq:fisum"/>) and (<ptr target="eq:flnsum"/>), one can define the parameter +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>C</mi></math></formula> +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>C</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!-- +--><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> +</math></formula> +</p><p>The following values have been chosen in <code>GEANT</code> for the other parameters: +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow> + <mtable equalrows='false' equalcolumns='false'><mtr><mtd><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfenced +open='{' ><mtable equalrows='false' equalcolumns='false'><mtr><mtd><mn>0</mn> </mtd><mtd><mi>if</mi><mi>Z</mi><mo>≤</mo><mn>2</mn></mtd> +</mtr><mtr><mtd><mn>2</mn><mo>/</mo><mi>Z</mi></mtd><mtd><mi>if</mi><mi>Z</mi><mo>></mo><mn>2</mn></mtd> +</mtr><mtr><mtd> </mtd></mtr></mtable> </mfenced></mtd><mtd><mo>↠</mo></mtd><mtd><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>-</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub> </mtd> + </mtr><mtr><mtd><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn><msup><mi>Z</mi><mrow><mn>2</mn></mrow></msup><mi>eV </mi> </mtd><mtd><mo>↠</mo></mtd><mtd><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><msup><mfenced +open='(' close=')'> <mfrac><mrow><mi>I</mi></mrow><!--___ +--><mrow><msubsup><mi>E</mi><mrow><mn>2</mn></mrow><mrow><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></mrow></msubsup></mrow></mfrac> </mfenced><mrow> <mfrac><mrow><mn>1</mn></mrow><!-- _ +--><mrow><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac> </mrow></msup></mtd> + </mtr><mtr><mtd><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn> </mtd><mtd> </mtd><mtd> </mtd> + </mtr><mtr><mtd> </mtd></mtr></mtable> +</mrow></math></formula> With these values +the atomic level <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula> +corresponds approximately the K-shell energy of the atoms and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>Z</mi><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></math></formula> the number of +K-shell electrons. <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>r</mi></math></formula> +is the only variable which can be tuned freely. It determines the relative contribution +of ionisation and excitation to the energy loss. +</p><p>The energy loss is computed with the assumption that the step length (or the relative +energy loss) is small, and---in consequence---the cross-section can be considered +constant along the path length. The energy loss due to the excitation is +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>Δ</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub> +</math></formula>where +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></math></formula> and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></math></formula> +are sampled from Poisson distribution as discussed above. The +loss due to the ionisation can be generated from the distribution +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math></formula> by +the inverse transformation method: + <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow> <mo>=</mo><msubsup> <mo>∫</mo> + <mrow><mi>I</mi></mrow><mrow><mi>E</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi> <mtext></mtext> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msup><mi>F</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mi>I</mi></mrow><!--____ +--><mrow><mn>1</mn><mo>-</mo><mi>u</mi> <mfrac><mrow>&Emax;</mrow><!-- ___ +--><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> +</math></formula></cell></row></table> +where <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>u</mi></math></formula> is a uniform random +number between <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>F</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></formula> and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>F</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></formula>. The contribution from the +ionisations will be <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>Δ</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mo>∑</mo> + <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></mrow></msubsup> <mfrac><mrow><mi>I</mi></mrow><!--________ +--><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>j</mi></mrow></msub> <mfrac><mrow>&Emax;</mrow><!-- ___ +--><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> +</math></formula>where +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> is the +number of ionisation (sampled from Poisson distribution). The energy loss in a step will +then be <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>Δ</mi><mi>E</mi><mo>=</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>. +</p> +<div > +<head>Fast simulation for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mn>6</mn></math></formula></head> +<p>If the number of ionisation <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> +is bigger than 16, a faster sampling method can be used. The possible energy loss +interval is divided in two parts: one in which the number of collisions is large and the +sampling can be done from a Gaussian distribution and the other in which +the energy loss is sampled for each collision. Let us call the former interval +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>α</mi><mi>I</mi><mo>]</mo></mrow></math></formula> the interval A, +and the latter <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow><mo>[</mo><mi>α</mi><mi>I</mi><mo>,</mo>&Emax;<mo>]</mo></mrow></math></formula> the +interval B. <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>α</mi></math></formula> lies +between 1 and <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +>&Emax;<mo>/</mo><mi>I</mi></math></formula>. +A collision with a loss in the interval A happens with the probability +<formula type="display" id="eq:phys332-5"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∫</mo> + <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!-- + --><mrow>&Emax;<mi>α</mi></mrow></mfrac> +</math></formula>The +mean energy loss and the standard deviation for this type of collision are +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow><mo><</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>>;</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___ +--><mrow><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>∫</mo> + <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mi>I</mi><mi>α</mi><mo>ln</mo><mi>α</mi></mrow><!-- + --><mrow><mi>α</mi><mo>-</mo><mn>1</mn></mrow></mfrac> +</math></formula>and <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___ +--><mrow><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>∫</mo> + <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><msup><mi>I</mi><mrow><mn>2</mn></mrow></msup><mi>α</mi><mfenced +open='(' close=')'><mn>1</mn><mo>-</mo> <mfrac><mrow><mi>α</mi><msup><mo>ln</mo><mrow><mn>2</mn></mrow></msup><mi>α</mi></mrow><!--_ +--><mrow><msup><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mfenced> +</math></formula>If the +collision number is high, we assume that the number of the type A collisions can be +calculated from a Gaussian distribution with the following mean value and standard +deviation: + <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-1"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow> <mo>=</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-2"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup> <mo>=</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>)</mo></mrow> +</math></formula></cell></row></table> +It is further assumed that the energy loss in these collisions has a Gaussian +distribution with + <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-3"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow> <mo>=</mo> <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mrow><mo><</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>>;</mo></mrow> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-4"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup> <mo>=</mo> <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><msup><mi>σ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow> +</math></formula></cell></row></table> +The energy loss of these collision can then be sampled from the Gaussian +distribution. +</p><p>The collisions where the energy loss is in the interval B are sampled directly from +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>Δ</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub><mo>=</mo><msubsup><mo>∑</mo> + <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub></mrow></msubsup> <mfrac><mrow><mi>α</mi><mi>I</mi></mrow><!--_________ +--><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>i</mi></mrow></msub> <mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi><mo>-</mo><mi>α</mi><mi>I</mi></mrow><!-- + --><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> +</math></formula>The +total energy loss is the sum of these two types of collisions: +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>Δ</mi><mi>E</mi><mo>=</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub> +</math></formula></p> +<p>The approximation of equations (<ptr target="eq:phys332-1"/>), (<ptr target="eq:phys332-2"/>), (<ptr target="eq:phys332-3"/>) and (<ptr target="eq:phys332-4"/>) can be used under the following +conditions: + <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-6"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub> <mo>≥</mo> <mn>0</mn> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-7"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mi>c</mi><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub> <mo>≤</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub> + </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"> + </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-8"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>σ</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow></msub> <mo>≥</mo> <mn>0</mn> +</math></formula></cell></row></table> +where <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>c</mi><mo>≥</mo><mn>4</mn></math></formula>. From +the equations (<ptr target="eq:phys332-5"/>), (<ptr target="eq:phys332-1"/>) and (<ptr target="eq:phys332-3"/>) and from the conditions (<ptr target="eq:phys332-6"/>) and (<ptr target="eq:phys332-7"/>) the following limits can be +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mrow><mi>min</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!-- +--><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>I</mi></mrow></mfrac> <mo>≤</mo><mi>α</mi><mo>≤</mo><msub><mi>α</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!-- +--><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>I</mi></mrow></mfrac> +</math></formula>This +conditions gives a lower limit to number of the ionisations +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> for which the fast +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup> </math></formula>As +in the conditions (<ptr target="eq:phys332-6"/>), (<ptr target="eq:phys332-7"/>) and (<ptr target="eq:phys332-8"/>) the value of +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>c</mi></math></formula> is as minimum +4, one gets <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mn>6</mn></math></formula>. +In order to speed the simulation, the maximum value is used for +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>α</mi></math></formula>. +</p><p>The number of collisions with energy loss in the interval B (the number of interactions +which has to be simulated directly) increases slowly with the total number of collisions +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula>. +The maximum number of these collisions can be estimated as +<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi><mo>,</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>≈</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub><mo>)</mo></mrow> +</math></formula>From the previous +expressions for <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mrow><mo><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></formula> and +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub></math></formula> one can derive the +<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi>B</mi></mrow></msub><mo>≤</mo><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow><!--_ +--><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> +</math></formula>The following +values are obtained with <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>c</mi><mo>=</mo><mn>4</mn></math></formula>: +</p><p><table rend="inline"><row><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell><cell +></cell><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell> +</row><row><cell +>16 </cell><cell +>16 </cell><cell +></cell><cell +> 200</cell><cell +> 29.63</cell> +</row><row><cell +>20 </cell><cell +>17.78 </cell><cell +></cell><cell +> 500</cell><cell +> 31.01</cell> +</row><row><cell +>50 </cell><cell +>24.24 </cell><cell +></cell><cell +> 1000</cell><cell +> 31.50</cell> +</row><row><cell +>100 </cell><cell +>27.59 </cell><cell +></cell><cell +><formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>∞</mi></math></formula></cell><cell +> 32.00</cell> +</row></table> +</p> +</div> +<div > +<head>Special sampling for lower part of the spectrum</head> +<p>If the step length is very small (<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mo>≤</mo><mn>5</mn></math></formula> +mm in gases, <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mo>≤</mo></math></formula> +2-3 <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>μ</mi></math></formula>m in solids) +the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mrow><mo>(</mo><mi>Δ</mi><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mrow><mo>(</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>></mo></mrow><mo>)</mo></mrow></mrow></msup> +</math></formula>If the +probability is bigger than 0.01 a special sampling is done, taking into account the fact that in +these cases the projectile interacts only with the outer electrons of the atom. An energy level +<formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn></math></formula> eV is chosen +to correspond to the outer electrons. The mean number of collisions can be calculated from +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo><</mo><mi>n</mi><mo>></mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac> +</math></formula>The number +of collisions <formula><math xmlns="http://www.w3.org/1998/Math/MathML" +><mi>n</mi></math></formula> +is sampled from Poisson distribution. In the case of the thin layers, all the +collisions are considered as ionisations and the energy loss is computed as +<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Δ</mi><mi>E</mi><mo>=</mo><msubsup><mo>∑</mo> + <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup> +<mfrac><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mn>1</mn><mo>-</mo> +<mfrac><mrow>&Emax;</mrow> +<mrow>&Emax;<mo>+</mo><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac> +<msub><mi>u</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac> +</math></formula> +</p> </div> </div> <div +type="star"> <head>References</head> <list type="bibliography"> <item +id="bib-LAND"> <p>L.Landau. On the Energy Loss of Fast Particles by +Ionisation. Originally published in <emph>J. Phys.</emph>, 8:201, +1944. Reprinted in D.ter Haar, Editor, <emph>L.D.Landau, Collected +papers</emph>, page 417. Pergamon Press, Oxford, 1965. +</p></item> <item id="bib-SCH1"> <p>B.Schorr. Programs for +the Landau and the Vavilov distributions and the corresponding random +numbers. <emph>Comp. Phys. Comm.</emph>, 7:216, 1974. +</p></item> <item id="bib-SELT"> <p>S.M.Seltzer and +M.J.Berger. Energy loss straggling of protons and mesons. In +<emph>Studies in Penetration of Charged Particles in Matter</emph>, +Nuclear Science Series 39, Nat. Academy of Sciences, Washington DC, +1964. </p></item> <item id="bib-TALM"> <p>R.Talman. On the +statistics of particle identification using ionization. <emph>Nucl. +Inst. Meth.</emph>, 159:189, 1979. </p></item> <item +id="bib-VAVI"> <p>P.V.Vavilov. Ionisation losses of high energy +heavy particles. <emph>Soviet Physics JETP</emph>, 5:749, +1957.</p></item></list> </div> + </body> </text> </TEI.2> + + |