summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml
diff options
context:
space:
mode:
Diffstat (limited to 'Master/texmf-dist/doc/xmltex/passivetex/latextei.xml')
-rw-r--r--Master/texmf-dist/doc/xmltex/passivetex/latextei.xml972
1 files changed, 972 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml b/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml
new file mode 100644
index 00000000000..af9fbb686c4
--- /dev/null
+++ b/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml
@@ -0,0 +1,972 @@
+<?xml version="1.0"?>
+<!DOCTYPE TEI.2 SYSTEM "http://www.oucs.ox.ac.uk/dtds/tei-oucs.dtd" [
+<!ENTITY aacute "&#x00E1;">
+<!ENTITY ap "&#x2248;">
+<!ENTITY alpha "&#x03B1;">
+<!ENTITY barwed "&#x22BC;">
+<!ENTITY beta "&#x03B2;">
+<!ENTITY commaspace " ">
+<!ENTITY gamma "&#x03B3;">
+<!ENTITY delta "&#x03B4;">
+<!ENTITY Delta "&#x0394;">
+<!ENTITY dots "&#x2026;">
+<!ENTITY epsi "&#x03B5;">
+<!ENTITY gg "&#x22D9;">
+<!ENTITY Gt "&#x226B;">
+<!ENTITY geq "&#x2265;">
+<!ENTITY infin "&#x221E;">
+<!ENTITY infty "&#x221E;">
+<!ENTITY int "&#x222B;">
+<!ENTITY kappa "&#x03BA;">
+<!ENTITY lambda "&#x03BB;">
+<!ENTITY langle "&#x2329;">
+<!ENTITY leq "&#x2264;">
+<!ENTITY mu "&#x03BC;">
+<!ENTITY nbsp "&#x00A0;">
+<!ENTITY phi "&#x03C6;">
+<!ENTITY pi "&#x3C0;">
+<!ENTITY psi "&#x3C8;">
+<!ENTITY rangle "&#x232A;">
+<!ENTITY rho "&#x3C1;">
+<!ENTITY sigma "&#x3C3;">
+<!ENTITY Sigma "&#x03A3;">
+<!ENTITY sim "&#x223C;">
+<!ENTITY thinspace "&#x2009;">
+<!ENTITY thickspace "&#x2005;">
+<!ENTITY xi "&#x03BE;">
+<!ENTITY prime "&#x2032;">
+<!ENTITY isinv "&#x2208;">
+<!ENTITY macr "&#x0304;">
+<!ENTITY Emax "<msub><mi>E</mi><mi>max</mi></msub>">
+<!ENTITY exp "E<mtext>exp</mtext>">
+<!ENTITY ln "E<mtext>ln</mtext>">
+<!ENTITY Rarr "&#x21D2;">
+<!ENTITY rarr "&#x2192;">
+<!ENTITY GEANT "GEANT">
+<!ENTITY sum "&#x2211;">
+]>
+<TEI.2>
+ <teiHeader>
+ <fileDesc>
+ <titleStmt>
+ <title>A sample article</title>
+ </titleStmt>
+ <publicationStmt>
+ <availability><p>Converted from LaTeX by Sebastian Rahtz</p> </availability>
+ </publicationStmt>
+ <sourceDesc>
+ <p></p>
+ </sourceDesc>
+ </fileDesc>
+ <revisionDesc>
+ <list>
+ <item>
+ <date>23 Oct 1999</date> SR converted from LaTeX</item>
+ </list>
+ </revisionDesc>
+ </teiHeader>
+ <text>
+ <front>
+ <docTitle>
+ <titlePart type="main">Simulation of Energy Loss Straggling</titlePart>
+ </docTitle>
+ <docAuthor>Maria Physicist</docAuthor>
+ <docDate>January 17, 1999</docDate>
+ </front>
+ <body> <div id="intro"> <head>Introduction</head> <p>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 <formula><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&kappa;</mi></math></formula>, which is
+proportional to the ratio of mean energy loss to the maximum allowed
+energy transfer in a single collision with an atomic electron
+ <formula
+type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow> <mi>&kappa;</mi><mo>=</mo>
+<mfrac>
+ <mrow><mi>&xi;</mi></mrow>
+ <mrow>
+ &Emax;
+ </mrow>
+</mfrac>
+</mrow>
+</math></formula>
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+>&Emax;</math></formula> is the
+maximum transferable energy in a single collision with an atomic electron.
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow>
+ &Emax;<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></formula> where
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&gamma;</mi><mo>=</mo><mi>E</mi><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula>,
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>E</mi></math></formula> is energy and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> the mass of the
+incident particle, <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula>
+and <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub></math></formula> is the
+electron mass. <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&xi;</mi></math></formula>
+comes from the Rutherford scattering crosss section and is defined as:
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row></table>
+where
+</p><p><table rend="inline"><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>z</mi></math></formula></cell><cell
+>charge of the incident particle </cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></math></formula></cell><cell
+>Avogadro's number </cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>Z</mi></math></formula></cell><cell
+>atomic number of the material</cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>A</mi></math></formula></cell><cell
+>atomic weight of the material </cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&rho;</mi></math></formula></cell><cell
+>density </cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&delta;</mi><mi>x</mi></math></formula></cell><cell
+>thickness of the material </cell>
+</row><row><cell
+> </cell>
+</row></table>
+</p><p><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi></math></formula>
+measures the contribution of the collisions with energy transfer close to
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+>&Emax;</math></formula>. For a given absorber,
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi></math></formula> tends towards large
+values if <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&delta;</mi><mi>x</mi></math></formula> is large
+and/or if <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&beta;</mi></math></formula> is small.
+Likewise, <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi></math></formula> tends
+towards zero if <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&delta;</mi><mi>x</mi></math></formula> is
+small and/or if <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&beta;</mi></math></formula>
+approaches 1.
+</p><p>The value of <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi></math></formula>
+distinguishes two regimes which occur in the description of ionisation fluctuations
+:
+</p><list type="enumerate">
+<item>
+<p>A
+large
+number
+of
+collisions
+involving
+the
+loss
+of
+all
+or
+most
+of
+the
+incident
+particle
+energy
+during
+the
+traversal
+of
+an
+absorber.
+</p><p>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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi><mo>&gt;</mo><mn>1</mn><mn>0</mn></math></formula>
+(i.e.
+when
+the
+mean
+energy
+loss
+in
+the
+absorber
+is
+greater
+than
+the
+maximum
+energy
+transfer
+in
+a
+single
+collision).
+</p></item>
+<item>
+<p>Particles
+traversing
+thin
+counters
+and
+incident
+electrons
+under
+any
+conditions.
+</p><p>The
+relevant
+inequalities
+and
+distributions
+are
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula>,
+Vavilov
+distribution,
+and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi><mo>&lt;</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula>,
+Landau
+distribution.</p></item></list>
+<p>An additional regime is defined by the contribution of the collisions
+with low energy transfer which can be estimated with the relation
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>,
+where <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>
+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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>&Gt;</mo><mn>1</mn></math></formula>. In <code>GEANT</code> (see
+URL <xptr url="http://wwwinfo.cern.ch/asdoc/geant/geantall.html"/>), the limit of Landau theory has
+been set at <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mn>0</mn></math></formula>.
+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 <ptr target="fg:phys332-1"/> shows the behaviour
+of <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> as
+a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic
+energy in Argon, Silicon and Uranium.
+</p>
+<p><figure file="phys332-1" id="fg:phys332-1">
+<head>The variable <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>
+can be used to measure the validity range of the Landau
+theory. It depends on the type and energy of the particle,
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>Z</mi></math></formula>,
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>A</mi></math></formula>
+and the ionisation potential of the material and the layer thickness. </head>
+</figure></p>
+<p>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.
+</p>
+</div>
+<div id="sec:phys332-1">
+<head>Landau theory</head>
+<p>For a particle of mass <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> traversing
+a thickness of material <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&delta;</mi><mi>x</mi></math></formula>,
+the Landau probability distribution may be written in terms of the universal Landau
+function <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow></math></formula>
+as<ptr target="bib-LAND"/>:
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row></table>
+where
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&lambda;</mi> <mo>=</mo> <mfrac><mrow><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></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>&Emax;</mrow></mfrac> <mtext></mtext>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>(Eulers constant)</mtext> <mtext></mtext>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover> <mo>=</mo> <mtext>average energy loss</mtext> <mtext></mtext>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&epsi;</mi> <mo>=</mo> <mtext>actual energy loss</mtext> <mtext></mtext>
+</math></formula></cell></row></table>
+</p>
+<div >
+<head>Restrictions</head>
+<p>The Landau formalism makes two restrictive assumptions :
+</p><list type="enumerate">
+<item>
+<p>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
+<ptr target="vavref"/>).
+</p></item>
+<item>
+<p>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<ptr target="bib-TALM"/>
+is
+necessary
+to
+accurately
+simulate
+data
+distributions.
+In
+<code>GEANT</code>,
+a
+parameterised
+model
+by
+L.
+Urb&aacute;n
+is
+used
+(see
+section
+<ptr target="urban"/>).</p></item></list>
+<p>In addition, the average value of the Landau distribution is infinite.
+Summing the Landau fluctuation obtained to the average energy from the
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula>
+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.
+</p><p>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 <code>GLANDO</code> routine is:
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></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>&Emax;</mrow></mfrac> <mo>)</mo></mrow>
+</mrow></math></formula>
+In order for this to have average 0, we must impose that:
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow>
+ <munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></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>&Emax;</mrow></mfrac>
+</mrow></math></formula>
+</p><p>This is realised introducing a <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></munderover><mo>)</mo></mrow></math></formula>
+such that if only values of <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&lambda;</mi><mo>&leq;</mo><msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub></math></formula>
+are accepted, the average value of the distribution is
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></munderover></math></formula>.
+</p><p>A parametric fit to the universal Landau distribution has been performed, with following result:
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow>
+ <msub><mi>&lambda;</mi><mrow><mtext>max</mtext></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'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></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>&macr;</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>&macr;</mo></munderover><mo>)</mo></mrow>
+</mrow></math></formula> only values
+smaller than <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub></math></formula>
+are accepted, otherwise the distribution is resampled.
+</p>
+</div>
+</div>
+<div id="vavref">
+<head>Vavilov theory</head>
+<p>Vavilov<ptr target="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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+>&Emax;<mo>=</mo><mi>&infin;</mi></math></formula>. Now
+we can write<ptr target="bib-SCH1"/>:
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row></table>
+where
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row></table>
+and
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></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></formula></cell></row></table>
+</p><p>The Vavilov parameters are simply related to the Landau parameter by
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula>. It can be shown that
+as <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi><mo>&rarr;</mo><mn>0</mn></math></formula>, the distribution of
+the variable <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></formula> approaches
+that of Landau. For <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi><mo>&leq;</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula>
+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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi></math></formula>, but rather the
+distribution of <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></formula>
+defined above tends to the distribution of the true
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&lambda;</mi></math></formula> from
+the Landau density function. Thus the routine <code>GVAVIV</code> samples the variable
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></formula> rather
+than <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub></math></formula>.
+For <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi><mo>&geq;</mo><mn>1</mn><mn>0</mn></math></formula>
+the Vavilov distribution tends to a Gaussian distribution (see next section).
+</p>
+</div>
+<div >
+<head>Gaussian Theory</head>
+<p>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<ptr target="bib-SELT"/>
+that for <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&kappa;</mi><mo>&geq;</mo><mn>1</mn><mn>0</mn></math></formula>
+the Vavilov distribution can be replaced by a Gaussian of the form:
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></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></formula></cell></row></table>
+thus implying
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>mean</mi> <mo>=</mo> <munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover> <mtext></mtext>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row></table>
+</p>
+</div>
+<div id="urban">
+<head>Urb&aacute;n model</head>
+<p>The method for computing restricted energy losses with
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&delta;</mi></math></formula>-ray
+production above given threshold energy in <code>GEANT</code> 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 <ptr target="fg:phys332-2"/>.
+</p>
+<p><figure file="phys332-2" id="fg:phys332-2">
+<head>Energy loss distribution for a 3 GeV electron in Argon as given by
+standard GEANT. The width of the layers is given in centimeters.</head>
+</figure></p>
+<p>It is assumed that the atoms have only two energy levels with binding energy
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>.
+The particle--atom interaction will then be an excitation with energy loss
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> or
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>, or
+an ionisation with an energy loss distributed according to a function
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula>:
+<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><!--
+ --><mrow>&Emax;</mrow></mfrac>
+<mfrac><mrow><mn>1</mn></mrow><!-- _
+--><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac>
+</math></formula></p><p>The
+macroscopic cross-section for excitations (<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></formula>)
+is <formula id="eq:sigex" type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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> </math></formula>and
+the macroscopic cross-section for ionisation is
+<formula id="eq:sigion" type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub><mi>&Sigma;</mi><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow>&Emax;</mrow><!-- ________________
+--><mrow><mi>I</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow><!--
+ --><mrow><mi>I</mi></mrow></mfrac> <mo>)</mo></mrow></mrow></mfrac><mi>r</mi> </math></formula>
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML">&Emax;</math></formula>
+is the <code>GEANT</code> cut for <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&delta;</mi></math></formula>-production,
+or the maximum energy transfer minus mean ionisation energy, if it is smaller than
+this cut-off value. The following notation is used:
+</p><p><table rend="inline"><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>r</mi><mo>,</mo><mi>C</mi></math></formula></cell><cell
+>parameters of the model</cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell
+>atomic energy levels </cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>I</mi></math></formula></cell><cell
+>mean ionisation energy </cell>
+</row><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell
+>oscillator strengths </cell>
+</row></table>
+</p><p>The model has the parameters <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula>,
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>,
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>C</mi></math></formula> and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula>. The oscillator
+strengths <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula> and the
+atomic level energies <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>
+should satisfy the constraints
+ <table rend="inline"><row><cell><formula type="subeqn" id="eq:fisum"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:flnsum"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula></cell></row></table>
+The parameter <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>C</mi></math></formula>
+can be defined with the help of the mean energy loss
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> in the following way: The
+numbers of collisions (<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></math></formula>,
+i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean
+number <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></formula>. In a step
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&Delta;</mi><mi>x</mi></math></formula> the mean number
+of collisions is <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>
+</math></formula>The
+mean energy loss <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula>
+in a step is the sum of the excitation and ionisation contributions
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>&Emax;<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>
+</math></formula>From
+this, using the equations (<ptr target="eq:sigex"/>), (<ptr target="eq:sigion"/>), (<ptr target="eq:fisum"/>) and (<ptr target="eq:flnsum"/>), one can define the parameter
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>C</mi></math></formula>
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>C</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!--
+--><mrow><mi>d</mi><mi>x</mi></mrow></mfrac>
+</math></formula>
+</p><p>The following values have been chosen in <code>GEANT</code> for the other parameters:
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula> With these values
+the atomic level <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>
+corresponds approximately the K-shell energy of the atoms and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>Z</mi><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></math></formula> the number of
+K-shell electrons. <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>r</mi></math></formula>
+is the only variable which can be tuned freely. It determines the relative contribution
+of ionisation and excitation to the energy loss.
+</p><p>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
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>
+</math></formula>where
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></math></formula> and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></math></formula>
+are sampled from Poisson distribution as discussed above. The
+loss due to the ionisation can be generated from the distribution
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math></formula> by
+the inverse transformation method:
+ <table rend="inline"><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>&Emax;</mrow><!-- ___
+--><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+</math></formula></cell></row></table>
+where <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>u</mi></math></formula> is a uniform random
+number between <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>F</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></formula> and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>F</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></formula>. The contribution from the
+ionisations will be <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>&Emax;</mrow><!-- ___
+--><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>
+</math></formula>where
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> is the
+number of ionisation (sampled from Poisson distribution). The energy loss in a step will
+then be <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><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></formula>.
+</p>
+<div >
+<head>Fast simulation for <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><mn>1</mn><mn>6</mn></math></formula></head>
+<p>If the number of ionisation <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula>
+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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>&alpha;</mi><mi>I</mi><mo>]</mo></mrow></math></formula> the interval A,
+and the latter <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow><mo>[</mo><mi>&alpha;</mi><mi>I</mi><mo>,</mo>&Emax;<mo>]</mo></mrow></math></formula> the
+interval B. <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&alpha;</mi></math></formula> lies
+between 1 and <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+>&Emax;<mo>/</mo><mi>I</mi></math></formula>.
+A collision with a loss in the interval A happens with the probability
+<formula type="display" id="eq:phys332-5"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>&alpha;</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!--
+ --><mrow>&Emax;<mi>&alpha;</mi></mrow></mfrac>
+</math></formula>The
+mean energy loss and the standard deviation for this type of collision are
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>
+</math></formula>and <formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula>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:
+ <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-1"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-2"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula></cell></row></table>
+It is further assumed that the energy loss in these collisions has a Gaussian
+distribution with
+ <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-3"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-4"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula></cell></row></table>
+The energy loss of these collision can then be sampled from the Gaussian
+distribution.
+</p><p>The collisions where the energy loss is in the interval B are sampled directly from
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>&Emax;<mo>+</mo><mi>I</mi><mo>-</mo><mi>&alpha;</mi><mi>I</mi></mrow><!--
+ --><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>
+</math></formula>The
+total energy loss is the sum of these two types of collisions:
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ <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>
+</math></formula></p>
+<p>The approximation of equations (<ptr target="eq:phys332-1"/>), (<ptr target="eq:phys332-2"/>), (<ptr target="eq:phys332-3"/>) and (<ptr target="eq:phys332-4"/>) can be used under the following
+conditions:
+ <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-6"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-7"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+ </math></formula></cell></row><row><cell><formula type="subeqn"><math xmlns="http://www.w3.org/1998/Math/MathML">
+ </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-8"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula></cell></row></table>
+where <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>c</mi><mo>&geq;</mo><mn>4</mn></math></formula>. From
+the equations (<ptr target="eq:phys332-5"/>), (<ptr target="eq:phys332-1"/>) and (<ptr target="eq:phys332-3"/>) and from the conditions (<ptr target="eq:phys332-6"/>) and (<ptr target="eq:phys332-7"/>) the following limits can be
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!--
+--><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo>&Emax;<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><mtext>max</mtext></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>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!--
+--><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo>&Emax;<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>
+</math></formula>This
+conditions gives a lower limit to number of the ionisations
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> for which the fast
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup> </math></formula>As
+in the conditions (<ptr target="eq:phys332-6"/>), (<ptr target="eq:phys332-7"/>) and (<ptr target="eq:phys332-8"/>) the value of
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>c</mi></math></formula> is as minimum
+4, one gets <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><mn>1</mn><mn>6</mn></math></formula>.
+In order to speed the simulation, the maximum value is used for
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&alpha;</mi></math></formula>.
+</p><p>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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula>.
+The maximum number of these collisions can be estimated as
+<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula>From the previous
+expressions for <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></formula> and
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub></math></formula> one can derive the
+<formula type="equation"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula>The following
+values are obtained with <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>c</mi><mo>=</mo><mn>4</mn></math></formula>:
+</p><p><table rend="inline"><row><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell><cell
+></cell><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></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
+><formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&infin;</mi></math></formula></cell><cell
+> 32.00</cell>
+</row></table>
+</p>
+</div>
+<div >
+<head>Special sampling for lower part of the spectrum</head>
+<p>If the step length is very small (<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mo>&leq;</mo><mn>5</mn></math></formula>
+mm in gases, <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mo>&leq;</mo></math></formula>
+2-3 <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>&mu;</mi></math></formula>m in solids)
+the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula>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
+<formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn></math></formula> eV is chosen
+to correspond to the outer electrons. The mean number of collisions can be calculated from
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>
+</math></formula>The number
+of collisions <formula><math xmlns="http://www.w3.org/1998/Math/MathML"
+><mi>n</mi></math></formula>
+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
+<formula type="display"><math xmlns="http://www.w3.org/1998/Math/MathML"><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>&Emax;</mrow>
+<mrow>&Emax;<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>
+</math></formula>
+</p> </div> </div> <div
+type="star"> <head>References</head> <list type="bibliography"> <item
+id="bib-LAND"> <p>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.
+</p></item> <item id="bib-SCH1"> <p>B.Schorr. Programs for
+the Landau and the Vavilov distributions and the corresponding random
+numbers. <emph>Comp. Phys. Comm.</emph>, 7:216, 1974.
+</p></item> <item id="bib-SELT"> <p>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. </p></item> <item id="bib-TALM"> <p>R.Talman. On the
+statistics of particle identification using ionization. <emph>Nucl.
+Inst. Meth.</emph>, 159:189, 1979. </p></item> <item
+id="bib-VAVI"> <p>P.V.Vavilov. Ionisation losses of high energy
+heavy particles. <emph>Soviet Physics JETP</emph>, 5:749,
+1957.</p></item></list> </div>
+ </body> </text> </TEI.2>
+
+