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+<?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>&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.
+<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>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--____________
+--><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>&gamma;</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>&gamma;</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>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>/</mo><msup><mi>&gamma;</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>&xi;</mi></math></inlinemath>
+comes from the Rutherford scattering cross section and is defined as:
+
+ <eqnarray ><subeqn ><math><mi>&xi;</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>&pi;</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>&rho;</mi><mi>&delta;</mi><mi>x</mi></mrow><!--
+ --><mrow><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>&beta;</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>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mfrac><mrow><mi>Z</mi></mrow><!--
+--><mrow><mi>A</mi></mrow></mfrac><mi>&rho;</mi><mi>&delta;</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>&rho;</mi></math></inlinemath></cell><cell
+>density </cell>
+</row><row><cell
+><inlinemath><math
+><mi>&delta;</mi><mi>x</mi></math></inlinemath></cell><cell
+>thickness of the material </cell>
+</row><row><cell
+> </cell>
+</row></tabular>
+</par><par><inlinemath><math
+><mi>&kappa;</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>&kappa;</mi></math></inlinemath> tends towards large
+values if <inlinemath><math
+><mi>&delta;</mi><mi>x</mi></math></inlinemath> is large
+and/or if <inlinemath><math
+><mi>&beta;</mi></math></inlinemath> is small.
+Likewise, <inlinemath><math
+><mi>&kappa;</mi></math></inlinemath> tends
+towards zero if <inlinemath><math
+><mi>&delta;</mi><mi>x</mi></math></inlinemath> is
+small and/or if <inlinemath><math
+><mi>&beta;</mi></math></inlinemath>
+approaches 1.
+</par><par>The value of <inlinemath><math
+><mi>&kappa;</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>&kappa;</mi><mo>&gt;</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>&lt;</mo><mi>&kappa;</mi><mo>&lt;</mo><mn>1</mn><mn>0</mn></math></inlinemath>,
+Vavilov
+distribution,
+and
+<inlinemath><math
+><mi>&kappa;</mi><mo>&lt;</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>&xi;</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>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>&Gt;</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>&xi;</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>&xi;</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>&xi;</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>&delta;</mi><mi>x</mi></math></inlinemath>,
+the Landau probability distribution may be written in terms of the universal Landau
+function <inlinemath><math
+><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow></math></inlinemath>
+as<cite refid="bib-LAND"/>:
+
+ <eqnarray ><subeqn ><math><mi>f</mi><mrow><mo>(</mo><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>x</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--
+--><mrow><mi>&xi;</mi></mrow></mfrac><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow> <mtext></mtext>
+</math></subeqn></eqnarray>
+where
+
+ <eqnarray ><subeqn ><math><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow> <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><mo>exp</mo><mfenced
+open='(' close=')'><mi>u</mi><mo>ln</mo><mi>u</mi><mo>+</mo><mi>&lambda;</mi><mi>u</mi></mfenced><mi>d</mi><mi>u</mi><mspace width='2cm'/><mi>c</mi><mo>&geq;</mo><mn>0</mn> <mtext></mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn ><math><mi>&lambda;</mi> <mo>=</mo> <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><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>&xi;</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>&gamma;</mi><mi>&prime;</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>&gamma;</mi> <mtext></mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn ><math><mi>&gamma;</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>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover> <mo>=</mo> <mtext>average energy loss</mtext> <mtext></mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn ><math><mi>&epsi;</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.
+Urb&aacute;n
+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>&delta;</mi><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>=</mo><mi>&xi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>ln</mo> <mfrac><mrow><mi>&xi;</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>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>=</mo><mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>&xi;</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>&lambda;</mi><mrow><mi>max</mi></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow></math></inlinemath>
+such that if only values of <inlinemath><math
+><mi>&lambda;</mi><mo>&leq;</mo><msub><mi>&lambda;</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath>
+are accepted, the average value of the distribution is
+<inlinemath><math
+><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</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>&lambda;</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>&lambda;</mi><mrow></mrow><mo>&barwed;</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>&lambda;</mi><mrow></mrow><mo>&barwed;</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>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow>
+</mrow></math></displaymath> only values
+smaller than <inlinemath><math
+><msub><mi>&lambda;</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>&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 >
+<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>&kappa;</mi><mo>&geq;</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>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>s</mi><mo>)</mo></mrow><mo>&ap;</mo> <mfrac><mrow><mn>1</mn></mrow><!--________
+--><mrow><mi>&xi;</mi><msqrt><!--<mi>&radical;</mi>
+ ______________--><mfrac><mrow><mn>2</mn><mi>&pi;</mi></mrow><!--
+ --><mrow><mi>&kappa;</mi></mrow></mfrac> <mfenced
+open='(' close=')'><mn>1</mn><mo>-</mo><msup><mi>&beta;</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>&epsi;</mi><mo>-</mo><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><!--
+ --><mrow><mn>2</mn></mrow></mfrac> <mfrac><mrow><mi>&kappa;</mi></mrow><!-- _______
+--><mrow><msup><mi>&xi;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</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>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover> <mtext></mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn ><math><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup> <mo>=</mo> <mfrac><mrow><msup><mi>&xi;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
+ --><mrow><mi>&kappa;</mi></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><mi>&xi;</mi><msub><mi>E</mi><mrow><mi>
+max</mi></mrow></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</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>
+Urb&aacute;n model</stitle>
+<par>The method for computing restricted energy losses with
+<inlinemath><math
+><mi>&delta;</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>&sim;</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>&Sigma;</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>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</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>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
+ --><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>I</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>&beta;</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>&Sigma;</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>&delta;</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>&leq;</mo><mi>r</mi><mo>&leq;</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>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></inlinemath>. In a step
+<inlinemath><math
+><mi>&Delta;</mi><mi>x</mi></math></inlinemath> the mean number
+of collisions is <equation ><math>
+ <mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow><mo>=</mo><msub><mi>&Sigma;</mi><mrow><mi>i</mi></mrow></msub><mi>&Delta;</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>&Delta;</mi><mi>x</mi><mo>=</mo><mfenced
+open='[' close=']'><msub><mi>&Sigma;</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>&Sigma;</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>&Sigma;</mi><mrow><mn>3</mn></mrow></msub><msubsup><mo>&int;</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>&Delta;</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>&leq;</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>&gt;</mo><mn>2</mn></mtd>
+</mtr><mtr><mtd> </mtd></mtr></mtable> </mfenced></mtd><mtd><mo>&Rarr;</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>&Rarr;</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>&Delta;</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>&int;</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>&Delta;</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mo>&sum;</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>&Delta;</mi><mi>E</mi><mo>=</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>+</mo><mi>&Delta;</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>&geq;</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>&alpha;</mi><mi>I</mi><mo>]</mo></mrow></math></inlinemath> the interval A,
+and the latter <inlinemath><math
+><mrow><mo>[</mo><mi>&alpha;</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>&alpha;</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>&alpha;</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>&int;</mo>
+ <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</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>&alpha;</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!--
+ --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mi>&alpha;</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>&lt;</mo><mi>&Delta;</mi><mi>E</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>>;</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___
+--><mrow><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>&int;</mo>
+ <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</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>&alpha;</mi><mo>ln</mo><mi>&alpha;</mi></mrow><!--
+ --><mrow><mi>&alpha;</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>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___
+--><mrow><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>&int;</mo>
+ <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</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>&alpha;</mi><mfenced
+open='(' close=')'><mn>1</mn><mo>-</mo> <mfrac><mrow><mi>&alpha;</mi><msup><mo>ln</mo><mrow><mn>2</mn></mrow></msup><mi>&alpha;</mi></mrow><!--_
+--><mrow><msup><mrow><mo>(</mo><mi>&alpha;</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>&lt;</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>&alpha;</mi><mo>)</mo></mrow> <mtext>(16)</mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn id="eq:phys332-2"><math><msubsup><mi>&sigma;</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>&alpha;</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</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>&lt;</mo><mi>&Delta;</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>&lt;</mo><mi>&Delta;</mi><mi>E</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>>;</mo></mrow> <mtext>(18)</mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn id="eq:phys332-4"><math><msubsup><mi>&sigma;</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>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>&alpha;</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>&Delta;</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub><mo>=</mo><msubsup><mo>&sum;</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>&alpha;</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>&alpha;</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>&Delta;</mi><mi>E</mi><mo>=</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>+</mo><mi>&Delta;</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>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub> <mo>&geq;</mo> <mn>0</mn> <mtext>(22)</mtext>
+ </math></subeqn><subeqn ><math>
+ </math></subeqn><subeqn id="eq:phys332-7"><math><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub> <mo>&leq;</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>&lt;</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow></msub> <mo>&geq;</mo> <mn>0</mn> <mtext>(24)</mtext>
+</math></subeqn></eqnarray>
+where <inlinemath><math
+><mi>c</mi><mo>&geq;</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>&alpha;</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>&leq;</mo><mi>&alpha;</mi><mo>&leq;</mo><msub><mi>&alpha;</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>&geq;</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>&geq;</mo><mn>1</mn><mn>6</mn></math></inlinemath>.
+In order to speed the simulation, the maximum value is used for
+<inlinemath><math
+><mi>&alpha;</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>&ap;</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><msub><mi>&sigma;</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>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></inlinemath> and
+<inlinemath><math
+><msub><mi>&sigma;</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>&leq;</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>&infin;</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>&leq;</mo><mn>5</mn></math></inlinemath>
+mm in gases, <inlinemath><math
+><mo>&leq;</mo></math></inlinemath>
+2-3 <inlinemath><math
+><mi>&mu;</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>&Delta;</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>&lt;</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mrow><mo>&lt;</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>&lt;</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>&Delta;</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>&Delta;</mi><mi>E</mi><mo>=</mo><msubsup><mo>&sum;</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>
+