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<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>
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