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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>