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Tip revision: 851214e7fac6fb15020e33fcf2b88b2e88382956 authored by Syd Bauman on 05 July 2013, 23:14:00 UTC
Re-tagging the 2.4.0 release of P5.
Re-tagging the 2.4.0 release of P5.
Tip revision: 851214e
testmathml.xml
<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0">
<teiHeader>
<fileDesc>
<titleStmt>
<title>A sample article</title>
</titleStmt>
<publicationStmt>
<p> </p>
</publicationStmt>
<sourceDesc>
<p/>
</sourceDesc>
</fileDesc>
<revisionDesc>
<list type="simple">
<item>
<date>23 Oct 1999</date> SR converted from LaTeX</item>
</list>
</revisionDesc>
</teiHeader>
<text>
<front>
<docTitle>
<titlePart type="main">Simulation of Energy Loss Straggling</titlePart>
</docTitle>
<docAuthor>Maria Physicist</docAuthor>
<docDate>January 17, 1999</docDate>
</front>
<body>
<div xml:id="intro" org="uniform" part="N" sample="complete">
<head>Introduction</head>
<p>Due to the statistical nature of ionisation energy loss, large fluctuations can occur in
the amount of energy deposited by a particle traversing an absorber element.
Continuous processes such as multiple scattering and energy loss play a relevant role
in the longitudinal and lateral development of electromagnetic and hadronic
showers, and in the case of sampling calorimeters the measured resolution
can be significantly affected by such fluctuations in their active layers. The
description of ionisation fluctuations is characterised by the significance parameter
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
</math>
</formula>,
which is proportional to the ratio of mean energy loss to the maximum
allowed energy transfer in a single collision with an atomic electron
<formula notation="MathML" rend="display">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mrow>
<mi>κ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>ξ</mi>
</mrow>
<mrow>
<msub>
<mi>E</mi>
<mrow>
<mtext>max</mtext>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</math>
</formula>
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msub>
<mi>E</mi>
<mrow>
<mtext>max</mtext>
</mrow>
</msub>
</math>
</formula> is the
maximum transferable energy in a single collision with an atomic electron.
<formula notation="MathML" rend="display">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mrow>
<msub>
<mi>E</mi>
<mrow>
<mtext>max</mtext>
</mrow>
</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>
</math>
</formula> where
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>γ</mi>
<mo>=</mo>
<mi>E</mi>
<mo>/</mo>
<msub>
<mi>m</mi>
<mrow>
<mi>x</mi>
</mrow>
</msub>
</math>
</formula>,
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>E</mi>
</math>
</formula> is energy and
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msub>
<mi>m</mi>
<mrow>
<mi>x</mi>
</mrow>
</msub>
</math>
</formula> the mass of the
incident particle, <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msup>
<mi>β</mi>
<mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo>-</mo>
<mn>1</mn>
<mo>/</mo>
<msup>
<mi>γ</mi>
<mrow>
<mn>2</mn>
</mrow>
</msup>
</math>
</formula>
and <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msub>
<mi>m</mi>
<mrow>
<mi>e</mi>
</mrow>
</msub>
</math>
</formula> is the
electron mass. <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ξ</mi>
</math>
</formula>
comes from the Rutherford scattering crosss section and is defined as:
<table rend="inline">
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ξ</mi>
<mo>=</mo>
<mfrac>
<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>=</mo>
<mn>1</mn>
<mn>5</mn>
<mn>3</mn>
<mo>.</mo>
<mn>4</mn>
<mfrac>
<mrow>
<msup>
<mi>z</mi>
<mrow>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>β</mi>
<mrow>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac>
<mrow>
<mi>Z</mi>
</mrow>
<mrow>
<mi>A</mi>
</mrow>
</mfrac>
<mi>ρ</mi>
<mi>δ</mi>
<mi>x</mi>
<mspace width="12pt"/>
<mi>keV </mi>
<mo>,</mo>
<mtext/>
</math>
</formula>
</cell>
</row>
</table>
where
</p>
<p>
<table rend="inline">
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>z</mi>
</math>
</formula>
</cell>
<cell role="data" cols="1" rows="1">charge of the incident particle </cell>
</row>
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msub>
<mi>N</mi>
<mrow>
<mi>A</mi>
<mi>v</mi>
</mrow>
</msub>
</math>
</formula>
</cell>
<cell role="data" cols="1" rows="1">Avogadro's number </cell>
</row>
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>Z</mi>
</math>
</formula>
</cell>
<cell role="data" cols="1" rows="1">atomic number of the material</cell>
</row>
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>A</mi>
</math>
</formula>
</cell>
<cell role="data" cols="1" rows="1">atomic weight of the material </cell>
</row>
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ρ</mi>
</math>
</formula>
</cell>
<cell role="data" cols="1" rows="1">density </cell>
</row>
<row role="data">
<cell role="data" cols="1" rows="1">
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>δ</mi>
<mi>x</mi>
</math>
</formula>
</cell>
<cell role="data" cols="1" rows="1">thickness of the material </cell>
</row>
<row role="data">
<cell role="data" cols="1" rows="1"> </cell>
</row>
</table>
</p>
<p>
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
</math>
</formula>
measures the contribution of the collisions with energy transfer close to
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msub>
<mi>E</mi>
<mrow>
<mtext>max</mtext>
</mrow>
</msub>
</math>
</formula>. For a given absorber,
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
</math>
</formula> tends towards large
values if <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>δ</mi>
<mi>x</mi>
</math>
</formula> is large
and/or if <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>β</mi>
</math>
</formula> is small.
Likewise, <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
</math>
</formula> tends
towards zero if <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>δ</mi>
<mi>x</mi>
</math>
</formula> is
small and/or if <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>β</mi>
</math>
</formula>
approaches 1.
</p>
<p>The value of <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
</math>
</formula>
distinguishes two regimes which occur in the description of ionisation fluctuations
:
</p>
<list type="enumerate">
<item>
<p>A
large
number
of
collisions
involving
the
loss
of
all
or
most
of
the
incident
particle
energy
during
the
traversal
of
an
absorber.
</p>
<p>As
the
total
energy
transfer
is
composed
of
a
multitude
of
small
energy
losses,
we
can
apply
the
central
limit
theorem
and
describe
the
fluctuations
by
a
Gaussian
distribution.
This
case
is
applicable
to
non-relativistic
particles
and
is
described
by
the
inequality
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
<mo>></mo>
<mn>1</mn>
<mn>0</mn>
</math>
</formula>
(i.e.
when
the
mean
energy
loss
in
the
absorber
is
greater
than
the
maximum
energy
transfer
in
a
single
collision).
</p>
</item>
<item>
<p>Particles
traversing
thin
counters
and
incident
electrons
under
any
conditions.
</p>
<p>The
relevant
inequalities
and
distributions
are
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mn>0</mn>
<mo>.</mo>
<mn>0</mn>
<mn>1</mn>
<mo><</mo>
<mi>κ</mi>
<mo><</mo>
<mn>1</mn>
<mn>0</mn>
</math>
</formula>, Vavilov distribution,
and
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>κ</mi>
<mo><</mo>
<mn>0</mn>
<mo>.</mo>
<mn>0</mn>
<mn>1</mn>
</math>
</formula>,
Landau
distribution.</p>
</item>
</list>
<p>An additional regime is defined by the contribution of the collisions
with low energy transfer which can be estimated with the relation
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ξ</mi>
<mo>/</mo>
<msub>
<mi>I</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
</math>
</formula>,
where <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<msub>
<mi>I</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
</math>
</formula>
is the mean ionisation potential of the atom. Landau theory assumes that
the number of these collisions is high, and consequently, it has a restriction
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ξ</mi>
<mo>/</mo>
<msub>
<mi>I</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
<mo>≫</mo>
<mn>1</mn>
</math>
</formula>. In <code>GEANT</code> (see
URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has
been set at <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ξ</mi>
<mo>/</mo>
<msub>
<mi>I</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>5</mn>
<mn>0</mn>
</math>
</formula>.
Below this limit special models taking into account the atomic structure of the material are
used. This is important in thin layers and gaseous materials. Figure <ptr target="#fg_phys332-1" /> shows the behaviour
of <formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>ξ</mi>
<mo>/</mo>
<msub>
<mi>I</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
</math>
</formula> as
a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic
energy in Argon, Silicon and Uranium.
</p>
</div>
</body>
</text>
</TEI>
