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author | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
commit | a683c3d7e9fac38ec713f23fb6b9d2c7143aea82 (patch) | |
tree | 424ab223921f85fd3f167a4ccd0e2d37d05c2927 /Master/texmf-dist/doc/latex/latex-web-companion/apa | |
parent | 5beb5368a684995153c8566797ba054f21c666af (diff) |
move english latex doc out of texmf-doc
git-svn-id: svn://tug.org/texlive/trunk@13412 c570f23f-e606-0410-a88d-b1316a301751
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diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/README.apa b/Master/texmf-dist/doc/latex/latex-web-companion/apa/README.apa new file mode 100644 index 00000000000..0b972b5b1aa --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/README.apa @@ -0,0 +1,18 @@ +Files in Appendix A of the LaTeX Web Companion + +Section A.1 ++++++++++++ + +latexexa.tex Complete LaTeX source of example file +phys332-1.eps EPS file used in latexexa.tex +phys332-2.eps " " +latexexa.dtd DTD for representing LaTeX and MathML +latexexa-raw.xml raw XML file ontained from LaTeX-->XML translation +latexexa.xml nicely formated excerpt shown in Section A.1.2 + +Section A.2 ++++++++++++ + +teched.html techexplorer scripting examples (Section A.2.1) +teched.java " " " (Section A.2.2) + 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> + diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.dtd b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.dtd new file mode 100644 index 00000000000..53857510041 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.dtd @@ -0,0 +1,190 @@ +<!-- latex.dtd: XML version of LaTeX + MathML --> + +<!ENTITY % fontchange "emph|textit|textbf|textsf|textsl|texttt" > +<!ENTITY % misc "url|quad|hspace|vspace|includegraphics|footnote|tag|ent"> +<!ENTITY % xref "ref|cite|pageref"> +<!ENTITY % chunk "lalist|par|tabular|figure|table|align|bibliography"> +<!ENTITY % mathobj "displaymath|inlinemath|equation|eqnarray" > +<!ENTITY % inline "#PCDATA|%fontchange;|%chunk;|%misc;|%xref;|%mathobj;"> + +<!ELEMENT document (frontmatter?,bodymatter)> +<!ATTLIST document class CDATA "article"> + +<!ELEMENT frontmatter (title,author,date?,abstract?,keywords?)> +<!ELEMENT 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+%isoamsoe.dtd; +%isoamsre.dtd; +%isoteche.dtd; +%isonume.dtd; +%mmaliase.dtd; +%isogrk3e.dtd; +%isomscre.dtd; +%isomopfe.dtd; +%mmlent.dtd; +<!ENTITY aacute "á"> +<!ENTITY OverBar "[OverBar]"> +<!ENTITY negationslash "/"> + + +<!-- end of latex.dtd --> diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx new file mode 100644 index 00000000000..2e334aa4cc5 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx @@ -0,0 +1,109 @@ +\documentclass{article} +\usepackage{graphicx} +\usepackage{url} +\title{Simulation of Energy Loss Straggling} +\author{Maria Physicist} +\newcommand{\Emax}{\ensuremath{E_{\mathrm{max}}}} +\newcommand{\GEANT}{\texttt{GEANT}} +\begin{document} +\maketitle + +\section{Introduction} + +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 $\kappa$, which is +proportional to the ratio of mean energy loss to the maximum allowed +energy transfer in a single collision with an atomic electron +\[ +\kappa =\frac{\xi}{\Emax} +\] +\Emax{} +is the maximum transferable energy in a single collision with +an atomic electron. + +....... + +\section{Vavilov theory} +\label{vavref} + +Vavilov\cite{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 $ \Emax = \infty $. +Now we can write\cite{bib-SCH1}: +\begin{eqnarray*} +f \left ( \epsilon, \delta s \right ) & = & \frac{1}{\xi} \phi_{v} +\left ( \lambda_{v}, \kappa, \beta^{2} \right ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi_{v} \left ( \lambda_{v}, \kappa, \beta^{2} \right ) & = & +\frac{1}{2 \pi i} \int^{c+i\infty}_{c-i\infty}\phi \left( s \right ) +e^{\lambda s} ds \hspace{2cm} c \geq 0 \\ +\phi \left ( s \right ) & = & +\exp \left [ \kappa ( 1 + \beta^{2}\gamma ) \right ] +~ \exp \left [ \psi \left ( s \right ) \right ], \\ +\psi \left ( s \right ) & = & s \ln \kappa + ( s + \beta^{2} \kappa ) +\left [ \ln (s/\kappa) + E_{1} (s/\kappa) \right ] - \kappa e^{-s/\kappa}, +\end{eqnarray*} +and +\begin{eqnarray*} +E_{1}(z) & = & \int^{\infty}_{z} t^{-1} e^{-t} dt +\mbox{\hspace{1cm} (the exponential integral)} \\ +\lambda_v & = & \kappa \left [ \frac{\epsilon - \bar{\epsilon}}{\xi} +- \gamma' - \beta^2 \right] +\end{eqnarray*} + +The Vavilov parameters are simply related to the Landau parameter by +$\lambda_L = \lambda_v/\kappa - \ln\kappa $. It can be shown that as +$\kappa \rightarrow 0$, the distribution of the variable $\lambda_L$ +approaches that of Landau. For $\kappa \leq 0.01$ the two +distributions are already practically identical. Contrary to what many +textbooks report, the Vavilov distribution \emph{does not} approximate +the Landau distribution for small $\kappa$, but rather the +distribution of $\lambda_L$ defined above tends to the distribution of +the true $\lambda$ from the Landau density function. Thus the routine +\texttt{GVAVIV} samples the variable $\lambda_L$ rather than +$\lambda_v$. For $\kappa \geq 10$ the Vavilov distribution tends to a +Gaussian distribution (see next section). +.... + +\begin{thebibliography}{10} +\bibitem{bib-LAND} +L.Landau. +\newblock On the Energy Loss of Fast Particles by Ionisation. +\newblock Originally published in \emph{J. Phys.}, 8:201, 1944. +\newblock Reprinted in D.ter Haar, Editor, \emph{L.D.Landau, Collected + papers}, page 417. Pergamon Press, Oxford, 1965. + +\bibitem{bib-SCH1} +B.Schorr. +\newblock Programs for the Landau and the Vavilov distributions and the + corresponding random numbers. +\newblock \emph{Comp. Phys. Comm.}, 7:216, 1974. + +\bibitem{bib-SELT} +S.M.Seltzer and M.J.Berger. +\newblock Energy loss straggling of protons and mesons. +\newblock In \emph{Studies in Penetration of Charged Particles in + Matter}, Nuclear Science Series~39, Nat. Academy of Sciences, + Washington DC, 1964. + +\bibitem{bib-TALM} +R.Talman. +\newblock On the statistics of particle identification using ionization. +\newblock \emph{Nucl. Inst. Meth.}, 159:189, 1979. + +\bibitem{bib-VAVI} +P.V.Vavilov. +\newblock Ionisation losses of high energy heavy particles. +\newblock \emph{Soviet Physics JETP}, 5:749, 1957. +\end{thebibliography} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.tex b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.tex new file mode 100644 index 00000000000..50a0b576332 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.tex @@ -0,0 +1,546 @@ +%% version September 23 Sep 1998 +\documentclass{article} +\usepackage{graphicx} +\usepackage{url} +\title{Simulation of Energy Loss Straggling} +\author{Maria Physicist} +\newcommand{\Emax}{\ensuremath{E_{\mathrm{max}}}} +\newcommand{\GEANT}{\texttt{GEANT}} +\begin{document} +\maketitle + +\section{Introduction} + +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 $\kappa$, which is +proportional to the ratio of mean energy loss to the maximum allowed +energy transfer in a single collision with an atomic electron +\[ +\kappa =\frac{\xi}{\Emax} +\] +\Emax{} +is the maximum transferable energy in a single collision with +an atomic electron. +\[ +\Emax =\frac{2 m_e \beta^2\gamma^2 } +{1 + 2\gamma m_e/m_x + \left ( m_e/m_x\right)^2}, +\] +where $\gamma = E/m_x$, $E$ is energy and +$m_x$ the mass of the incident particle, +$\beta^2 = 1 - 1/\gamma^2$ and $m_e$ is the electron mass. +$\xi$ comes from the Rutherford scattering cross section +and is defined as: +\begin{eqnarray*} \xi = \frac{2\pi z^2 e^4 N_{Av} Z \rho \delta x} + {m_e \beta^2 c^2 A} = 153.4 \frac{z^2} {\beta^2} \frac{Z}{A} + \rho \delta x \quad\mathrm{keV}, +\end{eqnarray*} +where + +\begin{tabular}{ll} +$z$ & charge of the incident particle \\ +$N_{Av}$ & Avogadro's number \\ +$Z$ & atomic number of the material \\ +$A$ & atomic weight of the material \\ +$\rho$ & density \\ +$ \delta x$ & thickness of the material \\ +\end{tabular} + +$\kappa$ measures the contribution of the collisions with energy +transfer close to \Emax. For a given absorber, $\kappa$ tends +towards large values if $\delta x$ is large and/or if $\beta$ is +small. Likewise, $\kappa$ tends towards zero if $\delta x $ is small +and/or if $\beta$ approaches 1. + +The value of $\kappa$ distinguishes two regimes which occur in the +description of ionisation fluctuations: + +\begin{enumerate} +\item A large number of collisions involving the loss of all or most + of the incident particle energy during the traversal of an absorber. + + 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 $\kappa > 10 $ (i.e. when the mean energy loss in the + absorber is greater than the maximum energy transfer in a single + collision). + +\item Particles traversing thin counters and incident electrons under + any conditions. + + The relevant inequalities and distributions are $ 0.01 < \kappa < 10 + $, Vavilov distribution, and $\kappa < 0.01 $, Landau distribution. +\end{enumerate} + +An additional regime is defined by the contribution of the collisions +with low energy transfer which can be estimated with the relation +$\xi/I_0$, where $I_0$ 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 $\xi/I_0 \gg 1$. In \GEANT{} +(see URL \url{http://wwwinfo.cern.ch/asdoc/geant/geantall.html}), the +limit of Landau theory has been set at $\xi/I_0 = 50$. Below this +limit special models taking into account the atomic structure of the +material are used. This is important in thin layers and gaseous +materials. Figure \ref{fg:phys332-1} shows the behaviour of $\xi/I_0$ +as a function of the layer thickness for an electron of 100 keV and 1 +GeV of kinetic energy in Argon, Silicon and Uranium. + +\begin{figure} + \centering + \includegraphics[width=.6\linewidth]{phys332-1} + \caption{The variable $\xi/I_0$ can be used to measure the + validity range of the Landau theory. It depends + on the type and energy of the particle, $Z$, $A$ + and the ionisation potential of the material and + the layer thickness. + } + \label{fg:phys332-1} +\end{figure} + +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. + +\section{Landau theory} +\label{sec:phys332-1} + +For a particle of mass $m_x$ traversing a thickness of material +$\delta x $, the Landau probability distribution may be written in +terms of the universal Landau function $\phi(\lambda)$ +as\cite{bib-LAND}: +\begin{eqnarray*} +f( \epsilon , \delta x ) & = &\frac{1}{\xi} \phi ( \lambda ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi(\lambda )& = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty} +\exp \left ( u \ln u + \lambda u \right ) du \hspace{2cm} c \geq 0 \\ +\lambda & = & \frac{\epsilon -\bar{\epsilon} }{\xi} + - \gamma' - \beta^2 - \ln \frac{\xi}{\Emax} \\ +\gamma' & = & 0.422784\dots = 1 - \gamma \\ +\gamma & = & 0.577215\dots \mbox{(Euler's constant)} \\ +\bar{\epsilon} & = & \mbox{average energy loss} \\ +\epsilon & = & \mbox{actual energy loss} +\end{eqnarray*} + +\subsection{Restrictions} + +The Landau formalism makes two restrictive assumptions: +\begin{enumerate} +\item 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{vavref}). +\item 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{bib-TALM} is necessary to accurately simulate data + distributions. In \GEANT, a parameterised model by L. Urb\'{a}n is + used (see section \ref{urban}). +\end{enumerate} + +In addition, the average value of the Landau distribution is infinite. +Summing the Landau fluctuation obtained to the average energy from the +$dE/dx$ 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. + +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} +routine is: +\[ +\delta dE/dx = \epsilon - \bar{\epsilon} = \xi ( \lambda - \gamma' ++\beta^2 +\ln \frac{\xi}{\Emax}) +\] +In order for this to have average 0, we must impose that: +\[ +\bar{\lambda} = -\gamma' - \beta^2 -\ln \frac{\xi}{\Emax} +\] + +This is realised introducing a $\lambda_{\mathrm{max}}(\bar{\lambda})$ +such that if only values of $\lambda \leq \lambda_{\mathrm{max}}$ are +accepted, the average value of the distribution is $\bar{\lambda}$. + +A parametric fit to the universal Landau distribution has been +performed, with following result: +\[ +\lambda_{\mathrm{max}} = 0.60715 + + 1.1934\bar{\lambda}+(0.67794+0.052382\bar{\lambda}) + \exp(0.94753+0.74442\bar{\lambda}) +\] +only values smaller than $\lambda_{\mathrm{max}}$ are accepted, otherwise the +distribution is resampled. + + + +\section{Vavilov theory} +\label{vavref} + +Vavilov\cite{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 $ \Emax = \infty $. +Now we can write\cite{bib-SCH1}: +\begin{eqnarray*} +f \left ( \epsilon, \delta s \right ) & = & \frac{1}{\xi} \phi_{v} +\left ( \lambda_{v}, \kappa, \beta^{2} \right ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi_{v} \left ( \lambda_{v}, \kappa, \beta^{2} \right ) & = & +\frac{1}{2 \pi i} \int^{c+i\infty}_{c-i\infty}\phi \left( s \right ) +e^{\lambda s} ds \hspace{2cm} c \geq 0 \\ +\phi \left ( s \right ) & = & +\exp \left [ \kappa ( 1 + \beta^{2}\gamma ) \right ] +~ \exp \left [ \psi \left ( s \right ) \right ], \\ +\psi \left ( s \right ) & = & s \ln \kappa + ( s + \beta^{2} \kappa ) +\left [ \ln (s/\kappa) + E_{1} (s/\kappa) \right ] - \kappa e^{-s/\kappa}, +\end{eqnarray*} +and +\begin{eqnarray*} +E_{1}(z) & = & \int^{\infty}_{z} t^{-1} e^{-t} dt +\mbox{\hspace{1cm} (the exponential integral)} \\ +\lambda_v & = & \kappa \left [ \frac{\epsilon - \bar{\epsilon}}{\xi} +- \gamma' - \beta^2 \right] +\end{eqnarray*} + +The Vavilov parameters are simply related to the Landau parameter by +$\lambda_L = \lambda_v/\kappa - \ln\kappa $. It can be shown that as +$\kappa \rightarrow 0$, the distribution of the variable $\lambda_L$ +approaches that of Landau. For $\kappa \leq 0.01$ the two +distributions are already practically identical. Contrary to what many +textbooks report, the Vavilov distribution \emph{does not} approximate +the Landau distribution for small $\kappa$, but rather the +distribution of $\lambda_L$ defined above tends to the distribution of +the true $\lambda$ from the Landau density function. Thus the routine +\texttt{GVAVIV} samples the variable $\lambda_L$ rather than +$\lambda_v$. For $\kappa \geq 10$ the Vavilov distribution tends to a +Gaussian distribution (see next section). + +\section{Gaussian Theory} + +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{bib-SELT} that +for $\kappa \geq 10 $ the Vavilov distribution can be replaced by a +Gaussian of the form: +\begin{eqnarray*} +f( \epsilon , \delta s) \approx \frac{1} +{\xi \sqrt{\frac{2 \pi}{\kappa} \left( 1 - \beta^2/2 \right)}} + \exp \left [ \frac{( \epsilon - \bar{\epsilon} )^2}{2} \frac{\kappa} + {\xi^2 (1- \beta^2/2)}\right ] +\end{eqnarray*} +thus implying +\begin{eqnarray*} +\mathrm{mean} & = & \bar{\epsilon} \\ +\sigma^2 & = & \frac{\xi^2}{\kappa} (1-\beta^2/2) = \xi + \Emax (1-\beta^2/2) +\end{eqnarray*} + +\section{Urb\'an model} +\label{urban} + +The method for computing restricted energy losses with $\delta$-ray +production above given threshold energy in \GEANT{} 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{fg:phys332-2}. +\begin{figure} + \centering + \includegraphics[width=.6\linewidth]{phys332-2} + \caption{Energy loss distribution for a 3 GeV electron in + Argon as given by standard \GEANT. The width of the layers is + given in centimeters.} + \label{fg:phys332-2} +\end{figure} + +It is assumed that the atoms have only two energy levels with binding +energy $E_1$ and $E_2$. The particle--atom interaction will then be +an excitation with energy loss $E_1$ or $E_2$, or an ionisation with +an energy loss distributed according to a function $g(E) \sim 1/E^2$: +\begin{equation} +g(E) = \frac{(\Emax + I)I}{\Emax} \frac{1}{E^2} +\end{equation} + +The macroscopic cross-section for excitations ($i=1,2$) is +\begin{equation} +\label{eq:sigex} +\Sigma_i = C \frac{f_i}{E_i} \frac{\ln (2 m \beta^2 \gamma^2/E_i) - \beta^2} + {\ln (2 m \beta^2 \gamma^2/ I) - \beta^2}(1-r) +\end{equation} +and the macroscopic cross-section for ionisation is +\begin{equation} +\label{eq:sigion} +\Sigma_3 = C \frac{\Emax}{I(\Emax+I) \ln(\frac{\Emax+I}{I})} + ~ r +\end{equation} +\Emax{} is the \GEANT{} cut for $\delta$-production, or the maximum +energy transfer minus mean ionisation energy, if it is smaller than +this cut-off value. The following notation is used: + +\begin{tabular}{ll} +$r, C$ & parameters of the model \\ +$E_i$ & atomic energy levels \\ +$I$ & mean ionisation energy \\ +${f_i}$ & oscillator strengths +\end{tabular} + +The model has the parameters $f_i$, $E_i$, $C$ and $r ~(0\leq r\leq +1)$. The oscillator strengths $f_i$ and the atomic level energies +$E_i$ should satisfy the constraints +\begin{eqnarray} +f_1 + f_2 & = & 1 \label{eq:fisum}\\ +f_1 \ln E_1 + f_2 \ln E_2 & = & \ln I \label{eq:flnsum} +\end{eqnarray} +The parameter $C$ can be defined with the help of the mean energy loss +$dE/dx$ in the following way: The numbers of collisions ($n_i$, i = +1,2 for the excitation and 3 for the ionisation) follow the Poisson +distribution with a mean number $ \langle n_i \rangle $. In a step +$\Delta x$ the mean number of collisions is +\begin{equation} +\langle n_i \rangle = \Sigma_i \Delta x +\end{equation} +The mean energy loss $dE/dx$ in a step is the sum of the excitation +and ionisation contributions +\begin{equation} +\frac{dE}{dx} \Delta x = \left[ \Sigma_1 E_1 + \Sigma_2 E_2 + + \Sigma_3 \int_{I}^{\Emax+I} E~g(E)~dE \right] + \Delta x +\end{equation} +From this, using the equations (\ref{eq:sigex}), (\ref{eq:sigion}), +(\ref{eq:fisum}) and (\ref{eq:flnsum}), one can define the parameter +$C$ +\begin{equation} +C = \frac{dE}{dx} +\end{equation} + +The following values have been chosen in \GEANT{} for the other +parameters: +\[ +\begin{array}{lcl} +f_2 = \left\{ \begin{array}{ll} + 0 & \mathrm{if}\, Z \leq 2 \\ + 2/Z & \mathrm{if}\, Z > 2 \\ + \end{array} \right. & \Rightarrow & f_1 = 1 - f_2 \\ +E_2 = 10 Z^2 \mathrm{eV} & \Rightarrow & E_1 = \left(\frac{I}{E_{2}^{f_2}} + \right)^{\frac{1}{f_1}} \\ +r = 0.4 & & \\ +\end{array} +\] +With these values the atomic level $E_2$ corresponds approximately +the K-shell energy of the atoms and $Zf_2$ the number of K-shell +electrons. $r$ is the only variable which can be tuned freely. It +determines the relative contribution of ionisation and +excitation to the energy loss. + +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 +\begin{equation} +\Delta E_e = n_1 E_1 + n_2 E_2 +\end{equation} +where $n_1$ and $n_2$ are sampled from Poisson distribution as +discussed above. The loss due to the ionisation can be generated from +the distribution $g(E)$ by the inverse transformation method: +\begin{eqnarray} +u = F(E) & = & \int_{I}^E g(x) dx \nonumber \\ +E = F^{-1}(u) & = & \frac{I}{1 - u \frac {\Emax}{\Emax+I}} \\ +\end{eqnarray} +where $u$ is a uniform random number between $F(I)=0$ and +$F(\Emax+I)=1$. The contribution from the ionisations will be +\begin{equation} +\Delta E_i = \sum_{j=1}^{n_3} \frac{I} + {1 - u_j \frac {\Emax}{\Emax + I}} +\end{equation} +where $n_3$ is the number of ionisation (sampled from Poisson +distribution). The energy loss in a step will then be $\Delta E = +\Delta E_e + \Delta E_i$. + +\subsection{Fast simulation for $n_3 \geq 16$} + +If the number of ionisation $n_3$ 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 $[I, \alpha I]$ the interval A, and the latter +$[\alpha I,\Emax]$ the interval B. $\alpha$ lies between 1 and +$\Emax/I$. A collision with a loss in the interval A happens with +the probability +\begin{equation} +\label{eq:phys332-5} +P(\alpha) = \int_I^{\alpha I} g(\!E\!) \, dE = + \frac {( \Emax + I) (\alpha - 1)}{\Emax \alpha} +\end{equation} +The mean energy loss and the standard deviation for this type +of collision are +\begin{equation} +\langle \Delta E(\alpha) \rangle = \frac{1}{P(\alpha)} + \int_I^{\alpha I} E \, g(\!E\!) \, dE = + \frac{I \alpha \ln \alpha}{\alpha - 1} +\end{equation} +and +\begin{equation} +\sigma^2(\alpha) = \frac{1}{P(\alpha)} + \int_I^{\alpha I} E^2 \, g(\!E\!) \, dE = + I^2 \alpha \left( 1 - \frac{\alpha \ln \! ^2 \alpha}{(\alpha - 1)^2} \right) +\end{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: +\begin{eqnarray} +\label{eq:phys332-1} +\langle n_A \rangle & = & n_3 P(\alpha) \\ +\label{eq:phys332-2} +\sigma_A^2 & = & n_3 P(\alpha) ( 1 - P(\alpha)) +\end{eqnarray} +It is further assumed that the energy loss in these collisions +has a Gaussian distribution with +\begin{eqnarray} +\label{eq:phys332-3} +\langle \Delta E_A \rangle & = & n_A \langle \Delta E(\alpha) \rangle \\ +\label{eq:phys332-4} +\sigma_{E,A}^2 & = & n_A \sigma^2(\alpha) +\end{eqnarray} +The energy loss of these collision can then be sampled from the +Gaussian distribution. + +The collisions where the energy loss is in the interval B are sampled +directly from +\begin{equation} +\Delta E_B = \sum_{i=1}^{n_3 - n_A} \frac{\alpha I} + {1 - u_i \frac{\Emax + I - \alpha I}{\Emax + I}} +\end{equation} +The total energy loss is the sum of these two types of collisions: +\begin{equation} +\Delta E = \Delta E_A + \Delta E_B +\end{equation} + +The approximation of equations (\ref{eq:phys332-1}), +(\ref{eq:phys332-2}), (\ref{eq:phys332-3}) and (\ref{eq:phys332-4}) +can be used under the following conditions: +\begin{eqnarray} +\label{eq:phys332-6} +\langle n_A \rangle - c \, \sigma_A & \geq & 0 \\ +\label{eq:phys332-7} +\langle n_A \rangle + c \, \sigma_A & \leq & n_3 \\ +\label{eq:phys332-8} +\langle \Delta E_A \rangle - c \, \sigma_{E,A} & \geq & 0 +\end{eqnarray} +where $c \geq 4$. From the equations (\ref{eq:phys332-5}), +(\ref{eq:phys332-1}) and (\ref{eq:phys332-3}) and from the conditions +(\ref{eq:phys332-6}) and (\ref{eq:phys332-7}) the following limits can +be derived: +\begin{equation} +\alpha_{\mathrm{min}} = \frac{(n_3 + c^2)(\Emax+I)} + {n_3 (\Emax + I) + c^2 I} \; \leq \alpha \; \leq +\alpha_{\mathrm{max}} = \frac{(n_3 + c^2)(\Emax+I)} + {c^2 (\Emax + I) + n_3 I} +\end{equation} +This conditions gives a lower limit to number of the ionisations $n_3$ +for which the fast sampling can be done: +\begin{equation} +n_3 \; \geq \; c^2 +\end{equation} +As in the conditions (\ref{eq:phys332-6}), (\ref{eq:phys332-7}) and +(\ref{eq:phys332-8}) the value of $c$ is as minimum 4, one gets $n_3 +\; \geq 16$. In order to speed the simulation, the maximum value is +used for $\alpha$. + +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 $n_3$. The maximum number +of these collisions can be estimated as +\begin{equation} +n_{B,max} = n_3 - n_{A,min} \approx n_3 (\langle n_A \rangle + - \sigma_A) +\end{equation} +From the previous expressions for $\langle n_A \rangle$ and $\sigma_A$ +one can derive the condition +\begin{equation} +n_B \; \leq \; n_{B,max} = \frac{2 n_3 c^2}{n_3+c^2} +\end{equation} +The following values are obtained with $c=4$: + +\begin{tabular}{llcrr} +$n_3$ & $n_{B,max}$ & & $n_3$ & $n_{B,max}$\\ \hline +16 & 16 & & 200 & 29.63\\ +20 & 17.78 & & 500 & 31.01 \\ +50 & 24.24 & & 1000 & 31.50 \\ +100 & 27.59 & & $\infty$ & 32.00 +\end{tabular} + +\subsection{Special sampling for lower part of the spectrum} + +If the step length is very small ($\leq 5$ mm in gases, $\leq$ 2-3 +$\mu$m in solids) the model gives 0 energy loss for some events. To +avoid this, the probability of 0 energy loss is computed +\begin{equation} +P( \Delta E=0) = e^{-( \langle n_1 \rangle + \langle n_2 \rangle + + \langle n_3 \rangle )} +\end{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 +$E_0 = 10$ eV is chosen to correspond to the outer electrons. The mean +number of collisions can be calculated from +\begin{equation} +\langle n \rangle = \frac{1}{E_0} \frac{dE}{dx} \Delta x +\end{equation} +The number of collisions $n$ 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 +\begin{equation} +\Delta E = \sum_{i=1}^n \frac{E_0} + {1 - \frac {\Emax}{\Emax + E_0} u_i} +\end{equation} + +\begin{thebibliography}{10} +\bibitem{bib-LAND} +L.Landau. +\newblock On the Energy Loss of Fast Particles by Ionisation. +\newblock Originally published in \emph{J. Phys.}, 8:201, 1944. +\newblock Reprinted in D.ter Haar, Editor, \emph{L.D.Landau, Collected + papers}, page 417. Pergamon Press, Oxford, 1965. + +\bibitem{bib-SCH1} +B.Schorr. +\newblock Programs for the Landau and the Vavilov distributions and the + corresponding random numbers. +\newblock \emph{Comp. Phys. Comm.}, 7:216, 1974. + +\bibitem{bib-SELT} +S.M.Seltzer and M.J.Berger. +\newblock Energy loss straggling of protons and mesons. +\newblock In \emph{Studies in Penetration of Charged Particles in + Matter}, Nuclear Science Series~39, Nat. Academy of Sciences, + Washington DC, 1964. + +\bibitem{bib-TALM} +R.Talman. +\newblock On the statistics of particle identification using ionization. +\newblock \emph{Nucl. Inst. Meth.}, 159:189, 1979. + +\bibitem{bib-VAVI} +P.V.Vavilov. +\newblock Ionisation losses of high energy heavy particles. +\newblock \emph{Soviet Physics JETP}, 5:749, 1957. +\end{thebibliography} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.xml b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.xml new file mode 100644 index 00000000000..9d23dd47752 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.xml @@ -0,0 +1,171 @@ +<?xml version="1.0"?> +<!DOCTYPE document SYSTEM "latexexa.dtd" []> +<document> +<frontmatter> + <title>Simulation of Energy Loss Straggling</title> + <author>Maria Physicist</author> + <date>January 14, 1999</date> +</frontmatter> +<bodymatter> +<section id="intro"> <stitle>Introduction</stitle> +<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. + +.... + +</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> +<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> diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/phys332-1.eps b/Master/texmf-dist/doc/latex/latex-web-companion/apa/phys332-1.eps new file mode 100644 index 00000000000..8299292087f --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/phys332-1.eps @@ -0,0 +1,419 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 0 0 567 567 +%%Title: regime.eps +%%Creator: HIGZ Version 1.20/02 +%%CreationDate: 93/11/12 09.41 +%%EndComments +80 dict begin +/s {stroke} def /l {lineto} def /m {moveto} def /t { translate} def +/sw {stringwidth} def /r {rotate} def /rl {roll} def +/d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def +/c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def +/cl {closepath} def /sf {scalefont setfont} def +/box {m dup 0 exch d exch 0 d 0 exch neg d cl} 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definefont pop end } def +/accvec [ 176 /agrave 181 /Agrave 190 /acircumflex 192 /Acircumflex +201 /adieresis 204 /Adieresis 209 /ccedilla 210 /Ccedilla 211 /eacute +212 /Eacute 213 /egrave 214 /Egrave 215 /ecircumflex 216 /Ecircumflex +217 /edieresis 218 /Edieresis 219 /icircumflex 220 /Icircumflex +221 /idieresis 222 /Idieresis 223 /ntilde 224 /Ntilde 226 /ocircumflex +228 /Ocircumflex 229 /odieresis 230 /Odieresis 231 /ucircumflex 236 /Ucircumflex +237 /udieresis 238 /Udieresis 239 /aring 242 /Aring 243 /ydieresis +244 /Ydieresis 246 /aacute 247 /Aacute 252 /ugrave 253 /Ugrave] def +/Times-Roman /Times-Roman accvec ReEncode +/Times-Italic /Times-Italic accvec ReEncode +/Times-Bold /Times-Bold accvec ReEncode +/Times-BoldItalic /Times-BoldItalic accvec ReEncode +/Helvetica /Helvetica accvec ReEncode +/Helvetica-Oblique /Helvetica-Oblique accvec ReEncode +/Helvetica-Bold /Helvetica-Bold accvec ReEncode +/Helvetica-BoldOblique /Helvetica-BoldOblique accvec ReEncode +/Courier /Courier 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Sutor. All rights reserved. --> +<HEAD> + <META HTTP-EQUIV="Content-Type" + CONTENT="text/html; charset=iso-8859-1"> + <META NAME="GENERATOR" + CONTENT="Mozilla/4.01 [en] (Win95; I) [Netscape]"> + <TITLE>teched Sample LaTeX Editor</TITLE> +</HEAD> + +<!-- This is a very simple LaTeX editor built using the --> +<!-- IBM techexplorer Hypermedia Browser and a Java applet. --> + +<BODY> +<CENTER> + +<!-- The upper window is controlled by techexplorer. We --> +<!-- give the name 'teInput' to this window. We are --> +<!-- using a table to put a frame around the window. --> + +<TABLE BORDER=1> + <TR> + <TD> + <EMBED TYPE="application/x-techexplorer" + TEXDATA="\(\)" + NAME="teInput" WIDTH=600 HEIGHT=150> + </TD> + </TR> +</TABLE> + +<!-- The lower window is handled by the 'teched' Java --> +<!-- applet. Like the techexplorer window, it is 600 --> +<!-- pixels wide. --> + +<TABLE BORDER=1> + <TR> + <TD> + <APPLET CODE="teched.class" + NAME="teched" ALIGN=CENTER + WIDTH=600 HEIGHT=130 MAYSCRIPT></APPLET> + </TD> + </TR> +</TABLE> +</CENTER> +</BODY> +</HTML>
\ No newline at end of file diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/teched.java b/Master/texmf-dist/doc/latex/latex-web-companion/apa/teched.java new file mode 100644 index 00000000000..4f494f1ad74 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/teched.java @@ -0,0 +1,187 @@ +// teched.java +// (C) Copyright 1998 by Robert S. Sutor. All rights reserved. + +// We first import the classes we need from the standard Java +// distribution. This will work with Java 1.0 or higher. + +import java.awt.*; +import java.awt.event.*; +import java.lang.*; +import java.applet.Applet; + +// The following brings in the Netscape LiveConnect classes +// that we will use. + +import netscape.javascript.JSObject; + +// These are the classes that we use that are exposed by +// techexplorer. The first is the basic interface to the +// plug-in. The others are the event and listener classes. + +import ibm.techexplorer.plugin.techexplorerPlugin; +import ibm.techexplorer.awt.AWTEvent; +import ibm.techexplorer.awt.event.KeyListener; +import ibm.techexplorer.awt.event.KeyEvent; + +public class teched + extends java.applet.Applet + implements KeyListener +{ + // The JavaScript window object + JSObject Window = null; + // The JavaScript document object + JSObject Document = null; + + // The techexplorer plug-in instance + techexplorerPlugin tePlugin = null; + // The editable text area for the markup source + TextArea markupInputArea = null; + // The 'Clear input' button + Button clearInputButton = null; + + // A utility buffer for holding the markup. + StringBuffer markupString = new StringBuffer(""); + + public boolean action(Event evt, Object arg) { + // We only handle the 'Clear input' action. + + boolean result = false; + + if ( evt.target == clearInputButton ) { + // Empty the markup edit area. + markupInputArea.setText( "" ); + + // Reinitialize the techexplorer document. This + // needs to be a non-empty string to actually + // updated the document, so we give it some + // non-visible input. + + tePlugin.reloadFromTeXString( "\\(\\)" ); + + result = true; + } + + return result; + } + + public void init() { + // Initialize the components we are displaying + // with this Java applet. + + clearInputButton = new Button("Clear input"); + markupInputArea = new TextArea( 5, 80 ); + + this.setLayout( new FlowLayout() ); + this.add( markupInputArea ); + this.add( clearInputButton ); + } + + public void keyPressed( ibm.techexplorer.awt.event.KeyEvent e ) { + // We don't do anything with this event given us by + // techexplorer. But see 'keyTyped'. + } + + public void keyTyped( ibm.techexplorer.awt.event.KeyEvent e ) { + // This is a naive (but effective!) way of dealing with + // keys coming to us from techexplorer. We grab the key + // that was pressed and put it on the end of our markup. + // Then we update the techexplorer window. + + if ( e.getSource() == tePlugin ) { + markupInputArea.appendText( + ( new Character( e.getKeyChar() )).toString() ); + + // This replaces the document within the techexplorer + // window with that gotten by parsing the string + // passed to it. + + tePlugin.reloadFromTeXString( markupInputArea.getText() ); + } + } + + public void keyReleased( ibm.techexplorer.awt.event.KeyEvent e ) { + // This is where we deal with key release events coming to + // us from the techexplorer window. + + switch ( e.getKeyCode() ) { + case KeyEvent.VK_DELETE: + // When we see a 'delete' key, we remove the last character + // in the markup. + + if ( e.getSource() == tePlugin ) { + markupString = new StringBuffer( markupInputArea.getText() ); + int length = markupString.length(); + if ( length > 0 ) + --length; + markupString.setLength( length ); + markupInputArea.setText( markupString.toString() ); + tePlugin.reloadFromTeXString( markupInputArea.getText() ); + } + break; + + case KeyEvent.VK_ENTER: + // When we see that the 'enter' key has been pressed, we + // insert a newline in the markup. This improves readability. + + if ( e.getSource() == tePlugin ) { + markupInputArea.appendText( "\n" ); + tePlugin.reloadFromTeXString( markupInputArea.getText() ); + } + break; + + default: + break; + } + } + + public boolean keyUp( Event evt, int key ) + { + // This key is one from the markup input area. + // When a key is released, update the techexplorer + // document with the current markup. + + boolean result = false; + + if ( evt.target == markupInputArea ) { + if ( evt.id == Event.KEY_RELEASE ) { + int length = markupInputArea.getText().length(); + if ( length > 0 ) + tePlugin.reloadFromTeXString( + markupInputArea.getText() ); + else + tePlugin.reloadFromTeXString( "\\(\\)" ); + } + + result = true; + } + + return result; + } + + public void start() { + // Initialize the Netscape JavaScript objects. + + Window = (JSObject) JSObject.getWindow(this); + Document = (JSObject) Window.getMember("document"); + + // Try to get the techexplorer plug-in object. + tePlugin = (techexplorerPlugin) Document.getMember("teInput"); + + if ( tePlugin == null ) + // If we didn't get it, print a debug message. + System.out.println("teched: start(): null teched"); + else + // Otherwise add the listener for techexplorer keys. + tePlugin.addKeyListener( (KeyListener) this ); + } + + public void stop() { + if ( tePlugin == null ) + // If we don't have the techexplorer plug-in object, + // print a debug message. + System.out.println("teched: stop(): null teched"); + else + // Otherwise remove the listener for techexplorer keys. + tePlugin.removeKeyListener( (KeyListener) this ); + } +} |