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-<?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>&kappa;</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>&kappa;</mi><mo>=</mo> <mfrac> <mrow>
-<mi>&xi;</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>&infin;</mi></math></inlinemath>. Now we can write<cite
-refid="bib-SCH1"/>: <eqnarray><subeqn><math><mi>f</mi> <mfenced
-open='('
-close=')'><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>s</mi></mfenced>
-<mo>=</mo>
-<mfrac><mrow><mn>1</mn></mrow><mrow><mi>&xi;</mi></mrow></mfrac>
-<msub><mi>&phi;</mi><mrow><mi>v</mi></mrow>
-</msub> <mfenced open='('
-close=')'><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow> </msub>
-<mo>,</mo><mi>&kappa;</mi><mo>,</mo><msup><mi>&beta;</mi><mrow><mn>2</mn>
-</mrow> </msup> </mfenced> <mtext></mtext> </math></subeqn></eqnarray>
-where
-<eqnarray><subeqn><math><msub><mi>&phi;</mi><mrow><mi>v</mi></mrow>
-</msub> <mfenced open='('
-close=')'><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow> </msub>
-<mo>,</mo><mi>&kappa;</mi><mo>,</mo><msup><mi>&beta;</mi><mrow><mn>2</mn>
-</mrow> </msup> </mfenced> <mo>=</mo>
-<mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>&pi;</mi><mi>i</mi></mrow>
-</mfrac><msubsup><mo>&int;</mo>
-<mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>&infin;</mi></mrow>
-<mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>&infin;</mi>
-</mrow></msubsup><mi>&phi;</mi><mfenced
-open='('
-close=')'><mi>s</mi></mfenced><msup><mi>e</mi><mrow><mi>&lambda;</mi><mi>s</mi>
-</mrow> </msup> <mi>d</mi><mi>s</mi><mspace
-width='2cm'/><mi>c</mi><mo>&geq;</mo><mn>0</mn> <mtext></mtext>
-</math></subeqn><subeqn><math> </math></subeqn><subeqn
-><math><mi>&phi;</mi><mfenced open='(' close=')'><mi>s</mi></mfenced>
-<mo>=</mo> <mo>exp</mo><mfenced open='['
-close=']'><mi>&kappa;</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo>
-<msup><mi>&beta;</mi><mrow><mn>2</mn>
-</mrow> </msup>
-<mi>&gamma;</mi><mo>)</mo></mrow></mfenced><mo>exp</mo><mfenced
-open='[' close=']'><mi>&psi;</mi> <mfenced open='('
-close=')'><mi>s</mi></mfenced></mfenced><mo>,</mo> <mtext></mtext>
-</math></subeqn><subeqn><math> </math></subeqn><subeqn
-><math><mi>&psi;</mi> <mfenced open='(' close=')'><mi>s</mi></mfenced>
-<mo>=</mo> <mi>s</mi><mo>ln</mo>
-<mi>&kappa;</mi><mo>+</mo><mrow><mo>(</mo><mi>s</mi><mo>+</mo><msup>
-<mi>&beta;</mi><mrow><mn>2</mn>
-</mrow> </msup> <mi>&kappa;</mi><mo>)</mo></mrow><mfenced open='['
-close=']'><mo>ln</mo>
-<mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>&kappa;</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>&kappa;</mi><mo>)</mo>
-</mrow></mfenced><mo>-</mo><mi>&kappa;</mi><msup><mi>e</mi><mrow>
-<mo>-</mo><mi>s</mi><mo>/</mo><mi>&kappa;</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>&int;</mo>
-<mrow><mi>&infin;</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>&lambda;</mi><mrow><mi>v</mi></mrow> </msub>
-<mo>=</mo> <mi>&kappa;</mi><mfenced open='['
-close=']'><mfrac><mrow><mi>&epsi;</mi><mo>-</mo><munderover
-accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover>
-</mrow> <mrow><mi>&xi;</mi></mrow></mfrac>
-<mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi>
-<mo>-</mo><msup><mi>&beta;</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>&lambda;</mi><mrow><mi>L</mi> </mrow>
-</msub> <mo>=</mo><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow>
-</msub> <mo>/</mo><mi>&kappa;</mi><mo>-</mo><mo>ln</mo>
-<mi>&kappa;</mi></math></inlinemath>. It can be shown that as
-<inlinemath> <math>
-<mi>&kappa;</mi><mo>&rarr;</mo><mn>0</mn></math></inlinemath>, the
-distribution of the variable <inlinemath> <math>
-<msub><mi>&lambda;</mi><mrow><mi>L</mi> </mrow> </msub>
-</math></inlinemath> approaches that of Landau. For <inlinemath>
-<math>
-<mi>&kappa;</mi><mo>&leq;</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>&kappa;</mi></math></inlinemath>, but rather the
-distribution of <inlinemath> <math>
-<msub><mi>&lambda;</mi><mrow><mi>L</mi> </mrow> </msub>
-</math></inlinemath> defined above tends to the distribution of the
-true <inlinemath><math><mi>&lambda;</mi></math></inlinemath> from the
-Landau density function. Thus the routine <texttt> GVAVIV</texttt>
-samples the variable <inlinemath>
-<math><msub><mi>&lambda;</mi><mrow><mi>L</mi> </mrow> </msub>
-</math></inlinemath> rather than <inlinemath> <math>
-<msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow> </msub>
-</math></inlinemath>. For <inlinemath> <math>
-<mi>&kappa;</mi><mo>&geq;</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>