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<?xml version="1.0" encoding="utf-8"?>
<fo:root xmlns:fo="http://www.w3.org/XSL/Format/1.0" xmlns:fop="http://www.jtauber.com/fop" xmlns:fotex="http://www.tug.org/fotex"><fo:layout-master-set><fo:simple-page-master page-master-name="left" margin-top="75pt" margin-bottom="100pt" margin-left="80pt" margin-right="150pt"><fo:region-body margin-bottom="24pt" margin-top="24pt"/><fo:region-after extent="25pt"/><fo:region-before extent="25pt"/></fo:simple-page-master><fo:simple-page-master page-master-name="right" margin-top="75pt" margin-bottom="100pt" margin-left="80pt" margin-right="150pt"><fo:region-body margin-bottom="24pt" margin-top="24pt"/><fo:region-after extent="25pt"/><fo:region-before extent="25pt"/></fo:simple-page-master><fo:simple-page-master page-master-name="first" margin-top="75pt" margin-bottom="100pt" margin-left="80pt" margin-right="150pt"><fo:region-body margin-bottom="24pt" margin-top="24pt"/><fo:region-after extent="25pt"/><fo:region-before extent="25pt"/></fo:simple-page-master></fo:layout-master-set><fo:page-sequence><fo:static-content flow-name="xsl-after"/><fo:static-content flow-name="xsl-before"/><fo:sequence-specification><fo:sequence-specifier-single page-master="right"/></fo:sequence-specification><fo:flow> 
		<fo:block font-size="18pt" space-after="8pt" text-align-last="centered"> 
		  Simulation of Energy Loss Straggling 
		</fo:block> 
		<fo:block space-after="6pt" font-size="14pt" text-align-last="centered"><fo:inline-sequence font-style="italic">Maria Physicist</fo:inline-sequence></fo:block> 
		<fo:block space-after="6pt" font-size="16pt" text-align-last="centered">January 17, 1999</fo:block> 
	 </fo:flow></fo:page-sequence><fo:page-sequence initial-page-number="1"><fo:sequence-specification><fo:sequence-specifier-alternating page-master-first="first" page-master-odd="right" page-master-even="left"/></fo:sequence-specification><fo:static-content flow-name="xsl-after" fop:master="right"><fo:block font-size="10pt"><fo:inline-rule rule-thickness="0pt"/><fo:page-number/></fo:block></fo:static-content><fo:static-content flow-name="xsl-after" fop:master="left"><fo:block font-size="10pt"><fo:page-number/><fo:inline-rule rule-thickness="0pt"/></fo:block></fo:static-content><fo:static-content flow-name="xsl-before" fop:master="right"><fo:block text-align-last="centered" font-size="10pt"><!--Running Head: title-->Simulation of Energy Loss Straggling</fo:block></fo:static-content><fo:static-content flow-name="xsl-before" fop:master="left"><fo:block text-align-last="centered" font-size="10pt"><!--Running Head: author-->Maria Physicist</fo:block></fo:static-content><fo:static-content flow-name="xsl-before" fop:master="first"/><fo:static-content flow-name="xsl-after" fop:master="first"><fo:block font-size="10pt"><fo:inline-rule rule-thickness="0pt"/><fo:page-number/><fo:inline-rule rule-thickness="0pt"/></fo:block></fo:static-content><fo:flow font-family="Times Roman" font-size="10pt"> <fo:block keep-with-next="true" id="intro" text-align="start" font-size="18pt" text-indent="-3em" font-weight="bold" space-after="6pt" space-before.optimum="12pt">1. Introduction<fotex:bookmark fotex-bookmark-level="0" fotex-bookmark-label="intro">1. Introduction</fotex:bookmark></fo:block>  <fo:block font-size="10pt" text-align="justified">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 <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex: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
 <fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"> <mi>κ</mi><mo>=</mo>
<mfrac>
 <mrow><mi>ξ</mi></mrow>
 <mrow>
   <msub><mi>E</mi><mi>max</mi></msub>
 </mrow>
</mfrac>
</mrow>
</fotex:displaymath>
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>max</mi></msub></fotex:inlinemath> is the
maximum transferable energy in a single collision with an atomic electron.
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML">
                    <msub><mi>E</mi><mi>max</mi></msub><mo>=</mo>          <mfrac><mrow><mn>2</mn><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>γ</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></fotex:displaymath> where
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">γ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mrow><mi>x</mi></mrow></msub></fotex:inlinemath>,
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi></fotex:inlinemath> is energy and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mrow><mi>x</mi></mrow></msub></fotex:inlinemath> the mass of the
incident particle, <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mrow><mn>2</mn></mrow></msup></fotex:inlinemath>
and <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mrow><mi>e</mi></mrow></msub></fotex:inlinemath> is the
electron mass. <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi></fotex:inlinemath>
comes from the Rutherford scattering crosss section and is defined as:
              <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>π</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>ρ</mi><mi>δ</mi><mi>x</mi></mrow><mrow><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>A</mi></mrow></mfrac>     <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">5</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">3</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">4</mn> <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Z</mi></mrow><mrow><mi>A</mi></mrow></mfrac><mi xmlns="http://www.w3.org/1998/Math/MathML">ρ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi><mspace width="12pt" xmlns="http://www.w3.org/1998/Math/MathML"/><mi xmlns="http://www.w3.org/1998/Math/MathML">keV </mi><mo xmlns="http://www.w3.org/1998/Math/MathML">,</mo>                  <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
where
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt"><fo:inline-included-container><fo:table id="N610"><fo:table-body><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">z</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>charge of the incident particle </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>Avogadro's number               </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">Z</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>atomic number of the material</fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">A</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>atomic weight of the material </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ρ</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>density                               </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>thickness of the material        </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence>                                                                  </fo:inline-sequence></fo:table-cell></fo:table-row></fo:table-body></fo:table></fo:inline-included-container>
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt"><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex:inlinemath>
measures the contribution of the collisions with energy transfer close to
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>max</mi></msub></fotex:inlinemath>. For a given absorber,
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex:inlinemath> tends towards large
values if <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath> is large
and/or if <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">β</mi></fotex:inlinemath> is small.
Likewise, <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex:inlinemath> tends
towards zero if <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath> is
small and/or if <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">β</mi></fotex:inlinemath>
approaches 1.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The value of <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex:inlinemath>
distinguishes two regimes which occur in the description of ionisation fluctuations
:
</fo:block><fo:list-block font-size="10pt" margin-right="10pt" space-before.optimum="10pt" space-after.optimum="10pt" margin-left="15pt"><fo:list-item><fo:list-item-label><fo:block margin-right="2.5pt" text-align="centered">•</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal">
<fo:block font-size="10pt" text-align="justified">A
large
number
of
collisions
involving
the
loss
of
all
or
most
of
the
incident
particle
energy
during
the
traversal
of
an
absorber.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">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
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">&gt;</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath>
(i.e.
when
the
mean
energy
loss
in
the
absorber
is
greater
than
the
maximum
energy
transfer
in
a
single
collision).
</fo:block></fo:block></fo:list-item-body></fo:list-item><fo:list-item><fo:list-item-label><fo:block margin-right="2.5pt" text-align="centered">•</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal">
<fo:block font-size="10pt" text-align="justified">Particles
traversing
thin
counters
and
incident
electrons
under
any
conditions.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The
relevant
inequalities
and
distributions
are
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">&lt;</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">&lt;</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath>,
Vavilov
distribution,
and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">&lt;</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn></fotex:inlinemath>,
Landau
distribution.</fo:block></fo:block></fo:list-item-body></fo:list-item></fo:list-block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">An additional regime is defined by the contribution of the collisions
with low energy transfer which can be estimated with the relation
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>0</mn></mrow></msub></fotex:inlinemath>,
where <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>0</mn></mrow></msub></fotex: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
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">≫</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn></fotex:inlinemath>. In <fo:inline-sequence font-family="Computer-Modern-Typewriter">GEANT</fo:inline-sequence> (see
URL <fo:inline-sequence font-family="Computer-Modern-Typewriter" color="green"><fo:simple-link external-destination="http://wwwinfo.cern.ch/asdoc/geant/geantall.html">http://wwwinfo.cern.ch/asdoc/geant/geantall.html</fo:simple-link></fo:inline-sequence>), the limit of Landau theory has
been set at <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">5</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex: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 <fo:inline-sequence color="green"><fo:simple-link internal-destination="fg:phys332-1">1</fo:simple-link></fo:inline-sequence> shows the behaviour
of <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>0</mn></mrow></msub></fotex: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.
</fo:block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt"><fo:float id="fg:phys332-1"><fo:block text-align-last="centered"><fo:display-graphic href="phys332-1"/></fo:block><fo:block text-align-last="centered">Figure 1. The variable <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>0</mn></mrow></msub></fotex:inlinemath>
can    be    used    to    measure    the    validity    range    of    the    Landau
theory.    It    depends    on    the    type    and    energy    of    the    particle,
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">Z</mi></fotex:inlinemath>,
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">A</mi></fotex:inlinemath>
and the ionisation potential of the material and the layer thickness. </fo:block></fo:float></fo:block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">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.
</fo:block>

<fo:block keep-with-next="true" id="sec:phys332-1" text-align="start" font-size="18pt" text-indent="-3em" font-weight="bold" space-after="6pt" space-before.optimum="12pt">2. Landau theory<fotex:bookmark fotex-bookmark-level="0" fotex-bookmark-label="sec:phys332-1">2. Landau theory</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">For a particle of mass <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mrow><mi>x</mi></mrow></msub></fotex:inlinemath> traversing
a thickness of material <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath>,
the Landau probability distribution may be written in terms of the universal Landau
function <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">φ</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></fotex:inlinemath>
as<fo:inline-sequence color="green"><fo:simple-link internal-destination="bib-LAND"> [1]</fo:simple-link></fo:inline-sequence>:
                         <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">f</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>x</mi><mo>)</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mi>ξ</mi></mrow></mfrac><mi xmlns="http://www.w3.org/1998/Math/MathML">φ</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>λ</mi><mo>)</mo></mrow>                         <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
where
             <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">φ</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>λ</mi><mo>)</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>   <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo>
                           <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup><mo xmlns="http://www.w3.org/1998/Math/MathML">exp</mo><mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>ln</mo><mi>u</mi><mo>+</mo><mi>λ</mi><mi>u</mi></mfenced><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">u</mi><mspace width="2cm" xmlns="http://www.w3.org/1998/Math/MathML"/><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn>             <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
               </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">λ</mi>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ε</mi><mo>-</mo><munderover accent="true"><mo>̄</mo><mi>ε</mi><mrow/></munderover></mrow><mrow><mi>ξ</mi></mrow></mfrac> <mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">γ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">′</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">ln</mo>   <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ξ</mi></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow></mfrac>                          <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
               </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">γ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">′</mi>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">4</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">2</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">2</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">7</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">8</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">4</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">γ</mi>                              <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
               </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">γ</mi>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">5</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">7</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">7</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">2</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">5</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mtext xmlns="http://www.w3.org/1998/Math/MathML">(Eulers constant)</mtext>                    <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><munderover accent="true" xmlns="http://www.w3.org/1998/Math/MathML"><mo>̄</mo><mi>ε</mi><mrow/></munderover>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mtext xmlns="http://www.w3.org/1998/Math/MathML">average energy loss</mtext>                            <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ε</mi>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mtext xmlns="http://www.w3.org/1998/Math/MathML">actual energy loss</mtext>                             <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
</fo:block>
<fo:block keep-with-next="true" id="N1783" text-align="start" font-size="14pt" text-indent="-3em" font-weight="bold" space-after="3pt" space-before.optimum="9pt">2.1. Restrictions<fotex:bookmark fotex-bookmark-level="1" fotex-bookmark-label="N1783">2.1. Restrictions</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">The Landau formalism makes two restrictive assumptions :
</fo:block><fo:list-block font-size="10pt" margin-right="10pt" space-before.optimum="10pt" space-after.optimum="10pt" margin-left="15pt"><fo:list-item><fo:list-item-label><fo:block margin-right="2.5pt" text-align="centered">•</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal">
<fo:block font-size="10pt" text-align="justified">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
<fo:inline-sequence color="green"><fo:simple-link internal-destination="vavref">3 (Vavilov theory)</fo:simple-link></fo:inline-sequence>).
</fo:block></fo:block></fo:list-item-body></fo:list-item><fo:list-item><fo:list-item-label><fo:block margin-right="2.5pt" text-align="centered">•</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal">
<fo:block font-size="10pt" text-align="justified">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<fo:inline-sequence color="green"><fo:simple-link internal-destination="bib-TALM"> [4]</fo:simple-link></fo:inline-sequence>
is
necessary
to
accurately
simulate
data
distributions.
In
<fo:inline-sequence font-family="Computer-Modern-Typewriter">GEANT</fo:inline-sequence>,
a
parameterised
model
by
L.
Urbán
is
used
(see
section
<fo:inline-sequence color="green"><fo:simple-link internal-destination="urban">5 (Urbán model)</fo:simple-link></fo:inline-sequence>).</fo:block></fo:block></fo:list-item-body></fo:list-item></fo:list-block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">In addition, the average value of the Landau distribution is infinite.
Summing the Landau fluctuation obtained to the average energy from the
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex: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.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">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 <fo:inline-sequence font-family="Computer-Modern-Typewriter">GLANDO</fo:inline-sequence> routine is:
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML">
                  <mi>δ</mi><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>ε</mi><mo>-</mo><munderover accent="true"><mo>̄</mo><mi>ε</mi><mrow/></munderover><mo>=</mo><mi>ξ</mi><mrow><mo>(</mo><mi>λ</mi><mo>-</mo><mi>γ</mi><mi>′</mi><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>ln</mo>   <mfrac><mrow><mi>ξ</mi></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow></mfrac> <mo>)</mo></mrow>
</mrow></fotex:displaymath>
In order for this to have average 0, we must impose that:
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML">
                        <munderover accent="true"><mo>̄</mo><mi>λ</mi><mrow/></munderover><mo>=</mo><mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo>   <mfrac><mrow><mi>ξ</mi></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow></mfrac>
</mrow></fotex:displaymath>
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">This is realised introducing a <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><munderover accent="true"><mo>̄</mo><mi>λ</mi><mrow/></munderover><mo>)</mo></mrow></fotex:inlinemath>
such that if only values of <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">λ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub></fotex:inlinemath>
are accepted, the average value of the distribution is
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><munderover accent="true" xmlns="http://www.w3.org/1998/Math/MathML"><mo>̄</mo><mi>λ</mi><mrow/></munderover></fotex:inlinemath>.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">A parametric fit to the universal Landau distribution has been performed, with following result:
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML">
    <msub><mi>λ</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>̄</mo><mi>λ</mi><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>λ</mi><mrow/><mo>̄</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>λ</mi><mrow/><mo>̄</mo></munderover><mo>)</mo></mrow>
</mrow></fotex:displaymath> only values
smaller than <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mtext>max</mtext></mrow></msub></fotex:inlinemath>
are accepted, otherwise the distribution is resampled.
</fo:block>


<fo:block keep-with-next="true" id="vavref" text-align="start" font-size="18pt" text-indent="-3em" font-weight="bold" space-after="6pt" space-before.optimum="12pt">3. Vavilov theory<fotex:bookmark fotex-bookmark-level="0" fotex-bookmark-label="vavref">3. Vavilov theory</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">Vavilov<fo:inline-sequence color="green"><fo:simple-link internal-destination="bib-VAVI"> [5]</fo:simple-link></fo:inline-sequence> derived a more accurate straggling distribution by introducing the kinematic
limit on the maximum transferable energy in a single collision, rather than using
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>max</mi></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">∞</mi></fotex:inlinemath>. Now
we can write<fo:inline-sequence color="green"><fo:simple-link internal-destination="bib-SCH1"> [2]</fo:simple-link></fo:inline-sequence>:
                       <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">f</mi> <mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi></mfenced>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mi>ξ</mi></mrow></mfrac><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced>                       <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
where
         <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>   <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo>
                            <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup><mi xmlns="http://www.w3.org/1998/Math/MathML">φ</mi><mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></mfenced><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mrow><mi>λ</mi><mi>s</mi></mrow></msup><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">s</mi><mspace width="2cm" xmlns="http://www.w3.org/1998/Math/MathML"/><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn>                  <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
               </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">φ</mi><mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></mfenced>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mo xmlns="http://www.w3.org/1998/Math/MathML">exp</mo><mfenced open="[" close="]" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mi>γ</mi><mo>)</mo></mrow></mfenced><mo xmlns="http://www.w3.org/1998/Math/MathML">exp</mo><mfenced open="[" close="]" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ψ</mi> <mfenced open="(" close=")"><mi>s</mi></mfenced></mfenced><mo xmlns="http://www.w3.org/1998/Math/MathML">,</mo>                         <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                 </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
               </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">ψ</mi> <mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></mfenced>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mi xmlns="http://www.w3.org/1998/Math/MathML">s</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">ln</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>s</mi><mo>+</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mi>κ</mi><mo>)</mo></mrow><mfenced open="[" close="]" xmlns="http://www.w3.org/1998/Math/MathML"><mo>ln</mo><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>κ</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>κ</mi><mo>)</mo></mrow></mfenced><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mrow><mo>-</mo><mi>s</mi><mo>/</mo><mi>κ</mi></mrow></msup><mo xmlns="http://www.w3.org/1998/Math/MathML">,</mo>         <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
and
           <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>1</mn></mrow></msub><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>z</mi><mo>)</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msubsup xmlns="http://www.w3.org/1998/Math/MathML">  <mo>∫</mo>
                       <mrow><mi>∞</mi></mrow><mrow><mi>z</mi></mrow></msubsup><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">t</mi><mspace width="1cm" xmlns="http://www.w3.org/1998/Math/MathML"/><mtext xmlns="http://www.w3.org/1998/Math/MathML">(the exponential integral)</mtext>           <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
              </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>v</mi></mrow></msub>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mfenced open="[" close="]" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>ε</mi><mo>-</mo><munderover accent="true"><mo>̄</mo><mi>ε</mi><mrow/></munderover></mrow><mrow><mi>ξ</mi></mrow></mfrac>  <mo>-</mo><mi>γ</mi><mi>′</mi><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced>                                 <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The Vavilov parameters are simply related to the Landau parameter by
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>L</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mo xmlns="http://www.w3.org/1998/Math/MathML">ln</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex:inlinemath>. It can be shown that
as <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">→</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath>, the distribution of
the variable <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>L</mi></mrow></msub></fotex:inlinemath> approaches
that of Landau. For <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">.</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn></fotex:inlinemath>
the two distributions are already practically identical. Contrary to what many textbooks
report, the Vavilov distribution <fo:inline-sequence font-style="italic">does not</fo:inline-sequence> approximate the Landau distribution for small
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi></fotex:inlinemath>, but rather the
distribution of <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>L</mi></mrow></msub></fotex:inlinemath>
defined above tends to the distribution of the true
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">λ</mi></fotex:inlinemath> from
the Landau density function. Thus the routine <fo:inline-sequence font-family="Computer-Modern-Typewriter">GVAVIV</fo:inline-sequence> samples the variable
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>L</mi></mrow></msub></fotex:inlinemath> rather
than <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mrow><mi>v</mi></mrow></msub></fotex:inlinemath>.
For <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath>
the Vavilov distribution tends to a Gaussian distribution (see next section).
</fo:block>

<fo:block keep-with-next="true" id="N3325" text-align="start" font-size="18pt" text-indent="-3em" font-weight="bold" space-after="6pt" space-before.optimum="12pt">4. Gaussian Theory<fotex:bookmark fotex-bookmark-level="0" fotex-bookmark-label="N3325">4. Gaussian Theory</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">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<fo:inline-sequence color="green"><fo:simple-link internal-destination="bib-SELT"> [3]</fo:simple-link></fo:inline-sequence>
that for <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">κ</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath>
the Vavilov distribution can be replaced by a Gaussian of the form:
            <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">f</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">≈</mo>         <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mi>ξ</mi><msqrt><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>κ</mi></mrow></mfrac>  <mfenced open="(" close=")"><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn></mfenced></msqrt></mrow></mfrac><mo xmlns="http://www.w3.org/1998/Math/MathML">exp</mo><mfenced open="[" close="]" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mrow><mo>(</mo><mi>ε</mi><mo>-</mo><munderover accent="true"><mo>̄</mo><mi>ε</mi><mrow/></munderover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac>           <mfrac><mrow><mi>κ</mi></mrow><mrow><msup><mi>ξ</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mfenced>                <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
thus implying
                <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">mean</mi>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <munderover accent="true" xmlns="http://www.w3.org/1998/Math/MathML"><mo>̄</mo><mi>ε</mi><mrow/></munderover>                                       <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                    </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                  </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mn>2</mn></mrow></msup>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>ξ</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>κ</mi></mrow></mfrac> <mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">ξ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>
max</mi></mrow></msub><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow>                <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
</fotex:subeqn></fotex:eqnarray>
</fo:block>

<fo:block keep-with-next="true" id="urban" text-align="start" font-size="18pt" text-indent="-3em" font-weight="bold" space-after="6pt" space-before.optimum="12pt">5. Urbán model<fotex:bookmark fotex-bookmark-level="0" fotex-bookmark-label="urban">5. Urbán model</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">The method for computing restricted energy losses with
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi></fotex:inlinemath>-ray
production above given threshold energy in <fo:inline-sequence font-family="Computer-Modern-Typewriter">GEANT</fo:inline-sequence> 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 <fo:inline-sequence color="green"><fo:simple-link internal-destination="fg:phys332-2">2</fo:simple-link></fo:inline-sequence>.
</fo:block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt"><fo:float id="fg:phys332-2"><fo:block text-align-last="centered"><fo:display-graphic href="phys332-2"/></fo:block><fo:block text-align-last="centered">Figure 2. Energy loss distribution for a 3 GeV electron in Argon as given by
standard GEANT. The width of the layers is given in centimeters.</fo:block></fo:float></fo:block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">It is assumed that the atoms have only two energy levels with binding energy
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>1</mn></mrow></msub></fotex:inlinemath> and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msub></fotex:inlinemath>.
The particle--atom interaction will then be an excitation with energy loss
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>1</mn></mrow></msub></fotex:inlinemath> or
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msub></fotex:inlinemath>, or
an ionisation with an energy loss distributed according to a function
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">∼</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msup></fotex:inlinemath>:
<fotex:equation xmlns:m="http://www.w3.org/1998/Math/MathML">
                        <mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>(</mo><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow></mfrac>
<mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac>
</fotex:equation></fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The
macroscopic cross-section for excitations (<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">i</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mo xmlns="http://www.w3.org/1998/Math/MathML">,</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">2</mn></fotex:inlinemath>)
is <fotex:equation id="eq:sigex" xmlns:m="http://www.w3.org/1998/Math/MathML">
                   <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>Σ</mi><mrow><mi>i</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi> <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><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 xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</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>β</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup><msup><mi>γ</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>I</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo></mrow>               </fotex:equation>and
the macroscopic cross-section for ionisation is
<fotex:equation id="eq:sigion" xmlns:m="http://www.w3.org/1998/Math/MathML">
                    <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi>            <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow><mrow><mi>I</mi><mrow><mo>(</mo><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi></mrow><mrow><mi>I</mi></mrow></mfrac>      <mo>)</mo></mrow></mrow></mfrac><mi xmlns="http://www.w3.org/1998/Math/MathML">r</mi>                 </fotex:equation>
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>max</mi></msub></fotex:inlinemath>
is the <fo:inline-sequence font-family="Computer-Modern-Typewriter">GEANT</fo:inline-sequence> cut for <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">δ</mi></fotex: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:
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt"><fo:inline-included-container><fo:table id="N4332"><fo:table-body><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">r</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">,</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>parameters of the model</fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>atomic energy levels      </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">I</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>mean ionisation energy  </fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>oscillator strengths       </fo:inline-sequence></fo:table-cell></fo:table-row></fo:table-body></fo:table></fo:inline-included-container>
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The model has the parameters <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath>,
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath>,
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi></fotex:inlinemath> and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">r</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></fotex:inlinemath>. The oscillator
strengths <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath> and the
atomic level energies <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath>
should satisfy the constraints
                              <fotex:eqnarray><fotex:subeqn id="eq:fisum" xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mn>2</mn></mrow></msub>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn>                     
                                   </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                      </fotex:subeqn><fotex:subeqn id="eq:flnsum" xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">ln</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">ln</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msub>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <mo xmlns="http://www.w3.org/1998/Math/MathML">ln</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">I</mi>                   
</fotex:subeqn></fotex:eqnarray>
The parameter <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi></fotex:inlinemath>
can be defined with the help of the mean energy loss
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath> in the following way: The
numbers of collisions (<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath>,
i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean
number <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>&gt;;</mo></mrow></fotex:inlinemath>. In a step
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath> the mean number
of collisions is <fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                                            <mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>&gt;;</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>Σ</mi><mrow><mi>i</mi></mrow></msub><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi>                                        
</fotex:displaymath>The
mean energy loss <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi></fotex:inlinemath>
in a step is the sum of the excitation and ionisation contributions
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
            <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>d</mi><mi>E</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfenced open="[" close="]" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Σ</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>Σ</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>Σ</mi><mrow><mn>3</mn></mrow></msub><msubsup><mo>∫</mo>
    <mrow><mi>I</mi></mrow><mrow><msub><mi>E</mi><mi>max</mi></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 xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi>        
</fotex:displaymath>From
this, using the equations (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:sigex">2</fo:simple-link></fo:inline-sequence>), (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:sigion">3</fo:simple-link></fo:inline-sequence>), (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:fisum">1</fo:simple-link></fo:inline-sequence>) and (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:flnsum">1</fo:simple-link></fo:inline-sequence>), one can define the parameter
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi></fotex:inlinemath>
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                              <mi xmlns="http://www.w3.org/1998/Math/MathML">C</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>d</mi><mi>E</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac>                           
</fotex:displaymath>
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The following values have been chosen in <fo:inline-sequence font-family="Computer-Modern-Typewriter">GEANT</fo:inline-sequence> for the other parameters:
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML">
                     <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>≤</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>⇒</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>⇒</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></fotex:displaymath> With these values
the atomic level <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msub></fotex:inlinemath>
corresponds approximately the K-shell energy of the atoms and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">Z</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mn>2</mn></mrow></msub></fotex:inlinemath> the number of
K-shell electrons. <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">r</mi></fotex:inlinemath>
is the only variable which can be tuned freely. It determines the relative contribution
of ionisation and excitation to the energy loss.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">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
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                         <mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>1</mn></mrow></msub><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>2</mn></mrow></msub><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msub>                      
</fotex:displaymath>where
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>1</mn></mrow></msub></fotex:inlinemath> and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>2</mn></mrow></msub></fotex:inlinemath>
are sampled from Poisson distribution as discussed above. The
loss due to the ionisation can be generated from the distribution
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow></fotex:inlinemath> by
the inverse transformation method:
                       <fotex:eqnarray><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">u</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">F</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msubsup xmlns="http://www.w3.org/1998/Math/MathML">  <mo>∫</mo>
                                     <mrow><mi>I</mi></mrow><mrow><mi>E</mi></mrow></msubsup><mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">x</mi>                     <mtext xmlns="http://www.w3.org/1998/Math/MathML"/>
                              </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                     </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>u</mi><mo>)</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>       <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>I</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>u</mi>  <mfrac><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>                
                              </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                              </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">                                
</fotex:subeqn></fotex:eqnarray>
where <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">u</mi></fotex:inlinemath> is a uniform random
number between <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">F</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath> and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">F</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn></fotex:inlinemath>. The contribution from the
ionisations will be <fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                                 <mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</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 xmlns="http://www.w3.org/1998/Math/MathML"><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><mi>max</mi></msub></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>                                      
</fotex:displaymath>where
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub></fotex:inlinemath> is the
number of ionisation (sampled from Poisson distribution). The energy loss in a step will
then be <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>i</mi></mrow></msub></fotex:inlinemath>.
</fo:block>
<fo:block keep-with-next="true" id="N5956" text-align="start" font-size="14pt" text-indent="-3em" font-weight="bold" space-after="3pt" space-before.optimum="9pt">5.1. Fast simulation for <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">6</mn></fotex:inlinemath><fotex:bookmark fotex-bookmark-level="1" fotex-bookmark-label="N5956">5.1. Fast simulation for n3≥16</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">If the number of ionisation <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub></fotex: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
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>I</mi><mo>,</mo><mi>α</mi><mi>I</mi><mo>]</mo></mrow></fotex:inlinemath> the interval A,
and the latter <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>α</mi><mi>I</mi><mo>,</mo><msub><mi>E</mi><mi>max</mi></msub><mo>]</mo></mrow></fotex:inlinemath> the
interval B. <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">α</mi></fotex:inlinemath> lies
between 1 and <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>max</mi></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">/</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">I</mi></fotex:inlinemath>.
A collision with a loss in the interval A happens with the probability
<fotex:displaymath id="eq:phys332-5" xmlns:m="http://www.w3.org/1998/Math/MathML">
                <mi xmlns="http://www.w3.org/1998/Math/MathML">P</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo>
   <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>(</mo><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub><mi>α</mi></mrow></mfrac>                
</fotex:displaymath>The
mean energy loss and the standard deviation for this type of collision are
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                <mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>&gt;;</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>   <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo>
          <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>I</mi><mi>α</mi><mo>ln</mo><mi>α</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn>1</mn></mrow></mfrac>            
</fotex:displaymath>and <fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mn>2</mn></mrow></msup><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>   <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo>
          <mrow><mi>I</mi></mrow><mrow><mi>α</mi><mi>I</mi></mrow></msubsup><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>2</mn></mrow></msup><mi xmlns="http://www.w3.org/1998/Math/MathML">g</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi xmlns="http://www.w3.org/1998/Math/MathML">d</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mn>2</mn></mrow></msup><mi xmlns="http://www.w3.org/1998/Math/MathML">α</mi><mfenced open="(" close=")" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo> <mfrac><mrow><mi>α</mi><msup><mo>ln</mo><mrow><mn>2</mn></mrow></msup><mi>α</mi></mrow><mrow><msup><mrow><mo>(</mo><mi>α</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mfenced>         
</fotex:displaymath>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:
                     <fotex:eqnarray><fotex:subeqn id="eq:phys332-1" xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi xmlns="http://www.w3.org/1998/Math/MathML">P</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo></mrow>                         
                         </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                     </fotex:subeqn><fotex:subeqn id="eq:phys332-2" xmlns:m="http://www.w3.org/1998/Math/MathML"><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi xmlns="http://www.w3.org/1998/Math/MathML">P</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>-</mo><mi>P</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>)</mo></mrow>                 
</fotex:subeqn></fotex:eqnarray>
It is further assumed that the energy loss in these collisions has a Gaussian
distribution with
                       <fotex:eqnarray><fotex:subeqn id="eq:phys332-3" xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>A</mi></mrow></msub><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>&gt;;</mo></mrow>                  
                            </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                       </fotex:subeqn><fotex:subeqn id="eq:phys332-4" xmlns:m="http://www.w3.org/1998/Math/MathML"><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup>  <mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo>  <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>A</mi></mrow></msub><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mn>2</mn></mrow></msup><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>)</mo></mrow>                     
</fotex:subeqn></fotex:eqnarray>
The energy loss of these collision can then be sampled from the Gaussian
distribution.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The collisions where the energy loss is in the interval B are sampled directly from
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                 <mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>B</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</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 xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</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><mi>max</mi></msub><mo>+</mo><mi>I</mi><mo>-</mo><mi>α</mi><mi>I</mi></mrow><mrow><msub><mi>E</mi><mi>max</mi></msub><mo>+</mo><mi>I</mi></mrow></mfrac>    </mrow></mfrac>             
</fotex:displaymath>The
total energy loss is the sum of these two types of collisions:
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML">
                          <mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mi>B</mi></mrow></msub>                     
</fotex:displaymath></fo:block>
<fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">The approximation of equations (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-1">1</fo:simple-link></fo:inline-sequence>), (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-2">1</fo:simple-link></fo:inline-sequence>), (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-3">1</fo:simple-link></fo:inline-sequence>) and (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-4">1</fo:simple-link></fo:inline-sequence>) can be used under the following
conditions:
                           <fotex:eqnarray><fotex:subeqn id="eq:phys332-6" xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mi>A</mi></mrow></msub>  <mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo>  <mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn>                    
                                   </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                           </fotex:subeqn><fotex:subeqn id="eq:phys332-7" xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">+</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mi>A</mi></mrow></msub>  <mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo>  <msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub>                   
                                   </fotex:subeqn><fotex:subeqn xmlns:m="http://www.w3.org/1998/Math/MathML">
                       </fotex:subeqn><fotex:subeqn id="eq:phys332-8" xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow></msub>  <mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo>  <mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn>                    
</fotex:subeqn></fotex:eqnarray>
where <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">4</mn></fotex:inlinemath>. From
the equations (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-5">6</fo:simple-link></fo:inline-sequence>), (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-1">1</fo:simple-link></fo:inline-sequence>) and (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-3">1</fo:simple-link></fo:inline-sequence>) and from the conditions (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-6">1</fo:simple-link></fo:inline-sequence>) and (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-7">1</fo:simple-link></fo:inline-sequence>) the following limits can be
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mrow><mi>min</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><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><mi>max</mi></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><mi>max</mi></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 xmlns="http://www.w3.org/1998/Math/MathML">≤</mo><mi xmlns="http://www.w3.org/1998/Math/MathML">α</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mrow><mtext>max</mtext></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mfrac xmlns="http://www.w3.org/1998/Math/MathML"><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><mi>max</mi></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><mi>max</mi></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>         
</fotex:displaymath>This
conditions gives a lower limit to number of the ionisations
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub></fotex:inlinemath> for which the fast
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mrow><mn>2</mn></mrow></msup>                                                  </fotex:displaymath>As
in the conditions (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-6">1</fo:simple-link></fo:inline-sequence>), (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-7">1</fo:simple-link></fo:inline-sequence>) and (<fo:inline-sequence color="green"><fo:simple-link internal-destination="eq:phys332-8">1</fo:simple-link></fo:inline-sequence>) the value of
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi></fotex:inlinemath> is as minimum
4, one gets <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">≥</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">6</mn></fotex:inlinemath>.
In order to speed the simulation, the maximum value is used for
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">α</mi></fotex:inlinemath>.
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt">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
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub></fotex:inlinemath>.
The maximum number of these collisions can be estimated as
<fotex:equation xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">-</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>A</mi><mo>,</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">≈</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow><mo>-</mo><msub><mi>σ</mi><mrow><mi>A</mi></mrow></msub><mo>)</mo></mrow>              
</fotex:equation>From the previous
expressions for <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>&gt;;</mo></mrow></fotex:inlinemath> and
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mi>A</mi></mrow></msub></fotex:inlinemath> one can derive the
<fotex:equation xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>B</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo> <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><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>                                       
</fotex:equation>The following
values are obtained with <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">c</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">4</mn></fotex:inlinemath>:
</fo:block><fo:block font-size="10pt" text-align="justified" text-indent="1em" space-before="0pt"><fo:inline-included-container><fo:table id="N7981"><fo:table-body><fo:table-row><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence/></fo:table-cell><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mn>3</mn></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></fotex:inlinemath></fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence>16                                                                                                     </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>16                                                                                                     </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence/></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                               200</fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                             29.63</fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence>20                                                                                                     </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>17.78                                                                                                 </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence/></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                               500</fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                             31.01</fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence>50                                                                                                     </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>24.24                                                                                                 </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence/></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                              1000</fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                             31.50</fo:inline-sequence></fo:table-cell></fo:table-row><fo:table-row><fo:table-cell><fo:inline-sequence>100                                                                                                   </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>27.59                                                                                                 </fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence/></fo:table-cell><fo:table-cell><fo:inline-sequence><fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">∞</mi></fotex:inlinemath></fo:inline-sequence></fo:table-cell><fo:table-cell><fo:inline-sequence>                                                             32.00</fo:inline-sequence></fo:table-cell></fo:table-row></fo:table-body></fo:table></fo:inline-included-container>
</fo:block>

<fo:block keep-with-next="true" id="N8234" text-align="start" font-size="14pt" text-indent="-3em" font-weight="bold" space-after="3pt" space-before.optimum="9pt">5.2. Special sampling for lower part of the spectrum<fotex:bookmark fotex-bookmark-level="1" fotex-bookmark-label="N8234">5.2. Special sampling for lower part of the spectrum</fotex:bookmark></fo:block>

<fo:block font-size="10pt" text-align="justified">If the step length is very small (<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">5</mn></fotex:inlinemath>
mm in gases, <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mo xmlns="http://www.w3.org/1998/Math/MathML">≤</mo></fotex:inlinemath>
2-3 <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">μ</mi></fotex:inlinemath>m in solids)
the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">P</mi><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>Δ</mi><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msup xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mrow><mo>-</mo><mrow><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>&gt;</mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&gt;</mo></mrow><mo>)</mo></mrow></mrow></msup>                                
</fotex:displaymath>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
<fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><msub xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><mn xmlns="http://www.w3.org/1998/Math/MathML">1</mn><mn xmlns="http://www.w3.org/1998/Math/MathML">0</mn></fotex:inlinemath> eV is chosen
to correspond to the outer electrons. The mean number of collisions can be calculated from
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mrow xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><mi>n</mi><mo>&gt;</mo></mrow><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo> <mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac>
</fotex:displaymath>The number
of collisions <fotex:inlinemath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">n</mi></fotex: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
<fotex:displaymath xmlns:m="http://www.w3.org/1998/Math/MathML"><mi xmlns="http://www.w3.org/1998/Math/MathML">Δ</mi><mi xmlns="http://www.w3.org/1998/Math/MathML">E</mi><mo xmlns="http://www.w3.org/1998/Math/MathML">=</mo><msubsup xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo>
    <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup>
<mfrac xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mn>1</mn><mo>-</mo>
<mfrac><mrow><msub><mi>E</mi><mi>max</mi></msub></mrow>
<mrow><msub><mi>E</mi><mi>max</mi></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>
</fotex:displaymath> 
</fo:block>   <fo:block keep-with-next="true" id="N8570" text-align="start" font-size="18pt" text-indent="-3em" font-weight="bold" space-after="6pt" space-before.optimum="12pt">6. References<fotex:bookmark fotex-bookmark-level="0" fotex-bookmark-label="N8570">6. References</fotex:bookmark></fo:block>  <fo:list-block font-size="10pt" margin-right="10pt" space-before.optimum="10pt" space-after.optimum="10pt" margin-left="15pt"><fo:list-item><fo:list-item-label id="bib-LAND"><fo:block margin-right="2.5pt" text-align="end"> [1]</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal"> <fo:block font-size="10pt" text-align="justified">L.Landau.  On the Energy Loss of Fast Particles by
Ionisation.  Originally published in <fo:inline-sequence font-style="italic">J.  Phys.</fo:inline-sequence>, 8:201,
1944.  Reprinted in D.ter Haar, Editor, <fo:inline-sequence font-style="italic">L.D.Landau, Collected
papers</fo:inline-sequence>, page 417.  Pergamon Press, Oxford, 1965.
</fo:block></fo:block></fo:list-item-body></fo:list-item><fo:list-item><fo:list-item-label id="bib-SCH1"><fo:block margin-right="2.5pt" text-align="end"> [2]</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal"> <fo:block font-size="10pt" text-align="justified">B.Schorr.  Programs for
the Landau and the Vavilov distributions and the corresponding random
numbers.  <fo:inline-sequence font-style="italic">Comp.  Phys.  Comm.</fo:inline-sequence>, 7:216, 1974.
</fo:block></fo:block></fo:list-item-body></fo:list-item><fo:list-item><fo:list-item-label id="bib-SELT"><fo:block margin-right="2.5pt" text-align="end"> [3]</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal"> <fo:block font-size="10pt" text-align="justified">S.M.Seltzer and
M.J.Berger.  Energy loss straggling of protons and mesons.  In
<fo:inline-sequence font-style="italic">Studies in Penetration of Charged Particles in Matter</fo:inline-sequence>,
Nuclear Science Series 39, Nat.  Academy of Sciences, Washington DC,
1964.  </fo:block></fo:block></fo:list-item-body></fo:list-item><fo:list-item><fo:list-item-label id="bib-TALM"><fo:block margin-right="2.5pt" text-align="end"> [4]</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal"> <fo:block font-size="10pt" text-align="justified">R.Talman.  On the
statistics of particle identification using ionization.  <fo:inline-sequence font-style="italic">Nucl.
Inst.  Meth.</fo:inline-sequence>, 159:189, 1979.  </fo:block></fo:block></fo:list-item-body></fo:list-item><fo:list-item><fo:list-item-label id="bib-VAVI"><fo:block margin-right="2.5pt" text-align="end"> [5]</fo:block></fo:list-item-label><fo:list-item-body><fo:block font-weight="normal"> <fo:block font-size="10pt" text-align="justified">P.V.Vavilov.  Ionisation losses of high energy
heavy particles.  <fo:inline-sequence font-style="italic">Soviet Physics JETP</fo:inline-sequence>, 5:749,
1957.</fo:block></fo:block></fo:list-item-body></fo:list-item></fo:list-block>  
 </fo:flow></fo:page-sequence></fo:root>