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diff --git a/Master/texmf-dist/doc/xmltex/passivetex/latextei-fo.xml b/Master/texmf-dist/doc/xmltex/passivetex/latextei-fo.xml deleted file mode 100644 index bfc33c1f6bb..00000000000 --- a/Master/texmf-dist/doc/xmltex/passivetex/latextei-fo.xml +++ /dev/null @@ -1,647 +0,0 @@ -<?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">></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"><</mo><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>, -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"><</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><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</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><</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</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>></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><</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><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><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><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><mi>Δ</mi><mi>E</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>>;</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><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><mi>Δ</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</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><</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo><</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>></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><</mo><mi>n</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><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>
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