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<?xml version="1.0" encoding="UTF-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0">
<info xmlns="http://docbook.org/docbook-ng">
<title>Simulation of Energy Loss Straggling</title>
<author><personname>
<surname>Physicist</surname>
<firstname>Maria</firstname></personname></author>
<pubdate>2004-03-11</pubdate>
<copyright>
<year>1999</year>
<holder>CERN</holder>
</copyright>
</info>
<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" type="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" type="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" type="subeqn">
<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>
<p>
<figure>
<graphic url="phys332-1.jpg" xml:id="fg_phys332-1"/>
<head>The variable <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>
can be used to measure the validity range of the Landau
theory. It depends on the type and energy of the
particle,
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>Z</mi>
</math>
</formula>,
<formula notation="MathML">
<math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mi>A</mi>
</math>
</formula>
and the ionisation potential of the material and the layer thickness. </head>
</figure>
</p>
<p>In the following sections, the different theories and models for the energy loss
fluctuation are described. First, the Landau theory and its limitations are discussed,
and then, the Vavilov and Gaussian straggling functions and the methods in the thin
layers and gaseous materials are presented.
</p>
</div>
</body>
</text>
</TEI>
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