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-<?xml version="1.0"?>
-
-<!DOCTYPE document SYSTEM "latex.xmldtd"
-[
-<!ENTITY % MathML "INCLUDE">
-<!ENTITY % LaTeXEntShort "IGNORE">
-<!ENTITY % LaTeXMath "IGNORE">
-<!ENTITY % LaTeXEnt "IGNORE">
-]>
- <document>
-<frontmatter>
-<title>Simulation of Energy Loss Straggling</title>
-<author>Maria Physicist</author>
-<date>
-January 17, 1999</date>
-</frontmatter>
-<bodymatter>
-<section id="intro">
-<stitle>
-Introduction</stitle>
-<displaymath>
-<math
->
-<mrow>
-<par>
- <msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup>
-</mrow></math></displaymath>
-</par><par>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
-<inlinemath><math
-><mi>&kappa;</mi></math></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
-<displaymath><math
-><mrow>
- <mi>&kappa;</mi><mo>=</mo> <mfrac><mrow><mi>&xi;</mi></mrow><!--___
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac>
-</mrow></math></displaymath>
-<inlinemath><math
-><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath> is the
-maximum transferable energy in a single collision with an atomic electron.
-<displaymath><math
-><mrow>
- <msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--____________
---><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>&gamma;</mi><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub><mo>+</mo><msup><mfenced
-open='(' close=')'><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub> </mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo>
-</mrow></math></displaymath> where
-<inlinemath><math
-><mi>&gamma;</mi><mo>=</mo><mi>E</mi><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></inlinemath>,
-<inlinemath><math
-><mi>E</mi></math></inlinemath> is energy and
-<inlinemath><math
-><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></inlinemath> the mass of the
-incident particle, <inlinemath><math
-><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>/</mo><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup></math></inlinemath>
-and <inlinemath><math
-><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub></math></inlinemath> is the
-electron mass. <inlinemath><math
-><mi>&xi;</mi></math></inlinemath>
-comes from the Rutherford scattering cross section and is defined as:
-
- <eqnarray ><subeqn ><math><mi>&xi;</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>&pi;</mi><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mn>4</mn></mrow></msup><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub><mi>Z</mi><mi>&rho;</mi><mi>&delta;</mi><mi>x</mi></mrow><!--
- --><mrow><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>A</mi></mrow></mfrac> <mo>=</mo><mn>1</mn><mn>5</mn><mn>3</mn><mo>.</mo><mn>4</mn> <mfrac><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
---><mrow><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mfrac><mrow><mi>Z</mi></mrow><!--
---><mrow><mi>A</mi></mrow></mfrac><mi>&rho;</mi><mi>&delta;</mi><mi>x</mi><mspace width='12pt'/><mi>keV </mi><mo>,</mo> <mtext></mtext>
-</math></subeqn></eqnarray>
-where
-</par><par><tabular preamble="ll"><row><cell
-><inlinemath><math
-><mi>z</mi></math></inlinemath></cell><cell
->charge of the incident particle </cell>
-</row><row><cell
-><inlinemath><math
-><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></math></inlinemath></cell><cell
->Avogadro's number </cell>
-</row><row><cell
-><inlinemath><math
-><mi>Z</mi></math></inlinemath></cell><cell
->atomic number of the material</cell>
-</row><row><cell
-><inlinemath><math
-><mi>A</mi></math></inlinemath></cell><cell
->atomic weight of the material </cell>
-</row><row><cell
-><inlinemath><math
-><mi>&rho;</mi></math></inlinemath></cell><cell
->density </cell>
-</row><row><cell
-><inlinemath><math
-><mi>&delta;</mi><mi>x</mi></math></inlinemath></cell><cell
->thickness of the material </cell>
-</row><row><cell
-> </cell>
-</row></tabular>
-</par><par><inlinemath><math
-><mi>&kappa;</mi></math></inlinemath>
-measures the contribution of the collisions with energy transfer close to
-<inlinemath><math
-><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath>. For a given absorber,
-<inlinemath><math
-><mi>&kappa;</mi></math></inlinemath> tends towards large
-values if <inlinemath><math
-><mi>&delta;</mi><mi>x</mi></math></inlinemath> is large
-and/or if <inlinemath><math
-><mi>&beta;</mi></math></inlinemath> is small.
-Likewise, <inlinemath><math
-><mi>&kappa;</mi></math></inlinemath> tends
-towards zero if <inlinemath><math
-><mi>&delta;</mi><mi>x</mi></math></inlinemath> is
-small and/or if <inlinemath><math
-><mi>&beta;</mi></math></inlinemath>
-approaches 1.
-</par><par>The value of <inlinemath><math
-><mi>&kappa;</mi></math></inlinemath>
-distinguishes two regimes which occur in the description of ionisation fluctuations
-:
-</par><lalist class="enumerate">
-<item>
-<par>A
-large
-number
-of
-collisions
-involving
-the
-loss
-of
-all
-or
-most
-of
-the
-incident
-particle
-energy
-during
-the
-traversal
-of
-an
-absorber.
-</par><par>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
-<inlinemath><math
-><mi>&kappa;</mi><mo>&gt;</mo><mn>1</mn><mn>0</mn></math></inlinemath>
-(i.e.
-when
-the
-mean
-energy
-loss
-in
-the
-absorber
-is
-greater
-than
-the
-maximum
-energy
-transfer
-in
-a
-single
-collision).
-</par></item>
-<item>
-<par>Particles
-traversing
-thin
-counters
-and
-incident
-electrons
-under
-any
-conditions.
-</par><par>The
-relevant
-inequalities
-and
-distributions
-are
-<inlinemath><math
-><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn><mo>&lt;</mo><mi>&kappa;</mi><mo>&lt;</mo><mn>1</mn><mn>0</mn></math></inlinemath>,
-Vavilov
-distribution,
-and
-<inlinemath><math
-><mi>&kappa;</mi><mo>&lt;</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></inlinemath>,
-Landau
-distribution.</par></item></lalist>
-<par>An additional regime is defined by the contribution of the collisions
-with low energy transfer which can be estimated with the relation
-<inlinemath><math
-><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></inlinemath>,
-where <inlinemath><math
-><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></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
-<inlinemath><math
-><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>&Gt;</mo><mn>1</mn></math></inlinemath>. In <texttt>GEANT</texttt> (see
-URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has
-been set at <inlinemath><math
-><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mn>0</mn></math></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 <ref refid="fg:phys332-1"/> shows the behaviour
-of <inlinemath><math
-><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></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.
-</par>
-<figure>
-<includegraphics file="phys332-1"/>
-<!--Figure 1--><caption id="fg:phys332-1">The variable <inlinemath><math
-><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></inlinemath>
-can be used to measure the validity range of the Landau
-theory. It depends on the type and energy of the particle,
-<inlinemath><math
-><mi>Z</mi></math></inlinemath>,
-<inlinemath><math
-><mi>A</mi></math></inlinemath>
-and the ionisation potential of the material and the layer thickness. </caption>
-</figure>
-<par>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.
-</par>
-</section>
-<section id="sec:phys332-1">
-<stitle>
-Landau theory</stitle>
-<par>For a particle of mass <inlinemath><math
-><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></inlinemath> traversing
-a thickness of material <inlinemath><math
-><mi>&delta;</mi><mi>x</mi></math></inlinemath>,
-the Landau probability distribution may be written in terms of the universal Landau
-function <inlinemath><math
-><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow></math></inlinemath>
-as<cite refid="bib-LAND"/>:
-
- <eqnarray ><subeqn ><math><mi>f</mi><mrow><mo>(</mo><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>x</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--
---><mrow><mi>&xi;</mi></mrow></mfrac><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow> <mtext></mtext>
-</math></subeqn></eqnarray>
-where
-
- <eqnarray ><subeqn ><math><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_
---><mrow><mn>2</mn><mi>&pi;</mi><mi>i</mi></mrow></mfrac><msubsup><mo>&int;</mo>
- <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>&infin;</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>&infin;</mi></mrow></msubsup><mo>exp</mo><mfenced
-open='(' close=')'><mi>u</mi><mo>ln</mo><mi>u</mi><mo>+</mo><mi>&lambda;</mi><mi>u</mi></mfenced><mi>d</mi><mi>u</mi><mspace width='2cm'/><mi>c</mi><mo>&geq;</mo><mn>0</mn> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>&lambda;</mi> <mo>=</mo> <mfrac><mrow><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover></mrow><!--
- --><mrow><mi>&xi;</mi></mrow></mfrac> <mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>&xi;</mi></mrow><!-- ___
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>&gamma;</mi><mi>&prime;</mi> <mo>=</mo> <mn>0</mn><mo>.</mo><mn>4</mn><mn>2</mn><mn>2</mn><mn>7</mn><mn>8</mn><mn>4</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><mn>1</mn><mo>-</mo><mi>&gamma;</mi> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>&gamma;</mi> <mo>=</mo> <mn>0</mn><mo>.</mo><mn>5</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>5</mn><mo>.</mo><mo>.</mo><mo>.</mo><mtext>(Euler's constant)</mtext> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover> <mo>=</mo> <mtext>average energy loss</mtext> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>&epsi;</mi> <mo>=</mo> <mtext>actual energy loss</mtext> <mtext></mtext>
-</math></subeqn></eqnarray>
-</par>
-<subsection >
-<stitle>
-Restrictions</stitle>
-<par>The Landau formalism makes two restrictive assumptions :
-</par><lalist class="enumerate">
-<item>
-<par>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
-<ref refid="vavref"/>).
-</par></item>
-<item>
-<par>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<cite refid="bib-TALM"/>
-is
-necessary
-to
-accurately
-simulate
-data
-distributions.
-In
-<texttt>GEANT</texttt>,
-a
-parameterised
-model
-by
-L.
-Urb&aacute;n
-is
-used
-(see
-section
-<ref refid="urban"/>).</par></item></lalist>
-<par>In addition, the average value of the Landau distribution is infinite.
-Summing the Landau fluctuation obtained to the average energy from the
-<inlinemath><math
-><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></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.
-</par><par>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 <texttt>GLANDO</texttt> routine is:
-<displaymath><math
-><mrow>
- <mi>&delta;</mi><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>=</mo><mi>&xi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>ln</mo> <mfrac><mrow><mi>&xi;</mi></mrow><!-- ___
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> <mo>)</mo></mrow>
-</mrow></math></displaymath>
-In order for this to have average 0, we must impose that:
-<displaymath><math
-><mrow>
- <munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>=</mo><mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo> <mfrac><mrow><mi>&xi;</mi></mrow><!-- ___
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac>
-</mrow></math></displaymath>
-</par><par>This is realised introducing a <inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>max</mi></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow></math></inlinemath>
-such that if only values of <inlinemath><math
-><mi>&lambda;</mi><mo>&leq;</mo><msub><mi>&lambda;</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath>
-are accepted, the average value of the distribution is
-<inlinemath><math
-><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover></math></inlinemath>.
-</par><par>A parametric fit to the universal Landau distribution has been performed, with following result:
-<displaymath><math
-><mrow>
- <msub><mi>&lambda;</mi><mrow><mi>max</mi></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'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>+</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>7</mn><mn>7</mn><mn>9</mn><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>5</mn><mn>2</mn><mn>3</mn><mn>8</mn><mn>2</mn><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow><mo>exp</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>4</mn><mn>7</mn><mn>5</mn><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn><mn>4</mn><mn>4</mn><mn>4</mn><mn>2</mn><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow>
-</mrow></math></displaymath> only values
-smaller than <inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath>
-are accepted, otherwise the distribution is resampled.
-</par>
-</subsection>
-</section>
-<section id="vavref">
-<stitle>
-Vavilov theory</stitle>
-<par>Vavilov<cite refid="bib-VAVI"/> derived a more accurate straggling distribution by introducing the kinematic
-limit on the maximum transferable energy in a single collision, rather than using
-<inlinemath><math
-><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo><mi>&infin;</mi></math></inlinemath>. Now
-we can write<cite refid="bib-SCH1"/>:
-
- <eqnarray ><subeqn ><math><mi>f</mi> <mfenced
-open='(' close=')'><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>s</mi></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--
---><mrow><mi>&xi;</mi></mrow></mfrac><msub><mi>&phi;</mi><mrow><mi>v</mi></mrow></msub> <mfenced
-open='(' close=')'><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>&kappa;</mi><mo>,</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mtext></mtext>
-</math></subeqn></eqnarray>
-where
-
- <eqnarray ><subeqn ><math><msub><mi>&phi;</mi><mrow><mi>v</mi></mrow></msub> <mfenced
-open='(' close=')'><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>&kappa;</mi><mo>,</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_
---><mrow><mn>2</mn><mi>&pi;</mi><mi>i</mi></mrow></mfrac><msubsup><mo>&int;</mo>
- <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>&infin;</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>&infin;</mi></mrow></msubsup><mi>&phi;</mi><mfenced
-open='(' close=')'><mi>s</mi></mfenced><msup><mi>e</mi><mrow><mi>&lambda;</mi><mi>s</mi></mrow></msup><mi>d</mi><mi>s</mi><mspace width='2cm'/><mi>c</mi><mo>&geq;</mo><mn>0</mn> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>&phi;</mi><mfenced
-open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo> <mo>exp</mo><mfenced
-open='[' close=']'><mi>&kappa;</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mi>&gamma;</mi><mo>)</mo></mrow></mfenced><mo>exp</mo><mfenced
-open='[' close=']'><mi>&psi;</mi> <mfenced
-open='(' close=')'><mi>s</mi></mfenced></mfenced><mo>,</mo> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>&psi;</mi> <mfenced
-open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo> <mi>s</mi><mo>ln</mo><mi>&kappa;</mi><mo>+</mo><mrow><mo>(</mo><mi>s</mi><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mi>&kappa;</mi><mo>)</mo></mrow><mfenced
-open='[' close=']'><mo>ln</mo><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>&kappa;</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>E</mi><mrow>
-<mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>&kappa;</mi><mo>)</mo></mrow></mfenced><mo>-</mo><mi>&kappa;</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>s</mi><mo>/</mo><mi>&kappa;</mi></mrow></msup><mo>,</mo> <mtext></mtext>
-</math></subeqn></eqnarray>
-and
-
- <eqnarray ><subeqn ><math><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow> <mo>=</mo><msubsup> <mo>&int;</mo>
- <mrow><mi>&infin;</mi></mrow><mrow><mi>z</mi></mrow></msubsup><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mspace width='1cm'/><mtext>(the exponential integral)</mtext> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub> <mo>=</mo> <mi>&kappa;</mi><mfenced
-open='[' close=']'><mfrac><mrow><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover></mrow><!--
- --><mrow><mi>&xi;</mi></mrow></mfrac> <mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mtext></mtext>
-</math></subeqn></eqnarray>
-</par><par>The Vavilov parameters are simply related to the Landau parameter by
-<inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub><mo>=</mo><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub><mo>/</mo><mi>&kappa;</mi><mo>-</mo><mo>ln</mo><mi>&kappa;</mi></math></inlinemath>. It can be shown that
-as <inlinemath><math
-><mi>&kappa;</mi><mo>&rarr;</mo><mn>0</mn></math></inlinemath>, the distribution of
-the variable <inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></inlinemath> approaches
-that of Landau. For <inlinemath><math
-><mi>&kappa;</mi><mo>&leq;</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></inlinemath>
-the two distributions are already practically identical. Contrary to what many textbooks
-report, the Vavilov distribution <emph>does not</emph> approximate the Landau distribution for small
-<inlinemath><math
-><mi>&kappa;</mi></math></inlinemath>, but rather the
-distribution of <inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></inlinemath>
-defined above tends to the distribution of the true
-<inlinemath><math
-><mi>&lambda;</mi></math></inlinemath> from
-the Landau density function. Thus the routine <texttt>GVAVIV</texttt> samples the variable
-<inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></inlinemath> rather
-than <inlinemath><math
-><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub></math></inlinemath>.
-For <inlinemath><math
-><mi>&kappa;</mi><mo>&geq;</mo><mn>1</mn><mn>0</mn></math></inlinemath>
-the Vavilov distribution tends to a Gaussian distribution (see next section).
-</par>
-</section>
-<section >
-<stitle>
-Gaussian Theory</stitle>
-<par>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<cite refid="bib-SELT"/>
-that for <inlinemath><math
-><mi>&kappa;</mi><mo>&geq;</mo><mn>1</mn><mn>0</mn></math></inlinemath>
-the Vavilov distribution can be replaced by a Gaussian of the form:
-
- <eqnarray ><subeqn ><math><mi>f</mi><mrow><mo>(</mo><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>s</mi><mo>)</mo></mrow><mo>&ap;</mo> <mfrac><mrow><mn>1</mn></mrow><!--________
---><mrow><mi>&xi;</mi><msqrt><!--<mi>&radical;</mi>
- ______________--><mfrac><mrow><mn>2</mn><mi>&pi;</mi></mrow><!--
- --><mrow><mi>&kappa;</mi></mrow></mfrac> <mfenced
-open='(' close=')'><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn></mfenced></msqrt></mrow></mfrac><mo>exp</mo><mfenced
-open='[' close=']'><mfrac><mrow><msup><mrow><mo>(</mo><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><!--
- --><mrow><mn>2</mn></mrow></mfrac> <mfrac><mrow><mi>&kappa;</mi></mrow><!-- _______
---><mrow><msup><mi>&xi;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mfenced> <mtext></mtext>
-</math></subeqn></eqnarray>
-thus implying
-
- <eqnarray ><subeqn ><math><mi>mean</mi> <mo>=</mo> <munderover accent='true'><mi>&epsi;</mi><mrow></mrow><mo>&barwed;</mo></munderover> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup> <mo>=</mo> <mfrac><mrow><msup><mi>&xi;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
- --><mrow><mi>&kappa;</mi></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><mi>&xi;</mi><msub><mi>E</mi><mrow><mi>
-max</mi></mrow></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow> <mtext></mtext>
-</math></subeqn></eqnarray>
-</par>
-</section>
-<section id="urban">
-<stitle>
-Urb&aacute;n model</stitle>
-<par>The method for computing restricted energy losses with
-<inlinemath><math
-><mi>&delta;</mi></math></inlinemath>-ray
-production above given threshold energy in <texttt>GEANT</texttt> 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 <ref refid="fg:phys332-2"/>.
-</par><figure>
-<includegraphics file="phys332-2"/>
-<!--Figure 2--><caption id="fg:phys332-2">Energy loss distribution for a 3 GeV electron in Argon as given by
-standard <texttt>GEANT</texttt>. The width of the layers is given in centimeters.</caption>
-</figure>
-<par>It is assumed that the atoms have only two energy levels with binding energy
-<inlinemath><math
-><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></inlinemath> and
-<inlinemath><math
-><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath>.
-The particle--atom interaction will then be an excitation with energy loss
-<inlinemath><math
-><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></inlinemath> or
-<inlinemath><math
-><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath>, or
-an ionisation with an energy loss distributed according to a function
-<inlinemath><math
-><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&sim;</mo><mn>1</mn><mo>/</mo><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></math></inlinemath>:
-<equation ><math>
- <mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><!--
- --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac> <mfrac><mrow><mn>1</mn></mrow><!-- _
---><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mi>(</mi><mi>1</mi><mi>)</mi>
-</math></equation>
-</par><par>The macroscopic cross-section for excitations
-(<inlinemath><math
-><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></inlinemath>) is
-<equation id="eq:sigex"><math>
- <msub><mi>&Sigma;</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></mrow><!--
---><mrow><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac> <mfrac><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
- --><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>I</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo></mrow> <mi>(</mi><mi>2</mi><mi>)</mi>
-</math></equation>and
-the macroscopic cross-section for ionisation is
-<equation id="eq:sigion"><math>
- <msub><mi>&Sigma;</mi><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!-- ________________
---><mrow><mi>I</mi><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow><!--
- --><mrow><mi>I</mi></mrow></mfrac> <mo>)</mo></mrow></mrow></mfrac><mi>r</mi> <mi>(</mi><mi>3</mi><mi>)</mi>
-</math></equation><inlinemath><math
-><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></math></inlinemath>
-is the <texttt>GEANT</texttt> cut for <inlinemath><math
-><mi>&delta;</mi></math></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:
-</par><par><tabular preamble="ll"><row><cell
-><inlinemath><math
-><mi>r</mi><mo>,</mo><mi>C</mi></math></inlinemath></cell><cell
->parameters of the model</cell>
-</row><row><cell
-><inlinemath><math
-><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath></cell><cell
->atomic energy levels </cell>
-</row><row><cell
-><inlinemath><math
-><mi>I</mi></math></inlinemath></cell><cell
->mean ionisation energy </cell>
-</row><row><cell
-><inlinemath><math
-><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath></cell><cell
->oscillator strengths </cell>
-</row></tabular>
-</par><par>The model has the parameters <inlinemath><math
-><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>,
-<inlinemath><math
-><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>,
-<inlinemath><math
-><mi>C</mi></math></inlinemath> and
-<inlinemath><math
-><mi>r</mi><mrow><mo>(</mo><mn>0</mn><mo>&leq;</mo><mi>r</mi><mo>&leq;</mo><mn>1</mn><mo>)</mo></mrow></math></inlinemath>. The oscillator
-strengths <inlinemath><math
-><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath> and the
-atomic level energies <inlinemath><math
-><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>
-should satisfy the constraints
-
- <eqnarray ><subeqn id="eq:fisum"><math><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub> <mo>=</mo> <mn>1</mn> <mtext>(4)</mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn id="eq:flnsum"><math><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub> <mo>=</mo> <mo>ln</mo><mi>I</mi> <mtext>(5)</mtext>
-</math></subeqn></eqnarray>
-The parameter <inlinemath><math
-><mi>C</mi></math></inlinemath>
-can be defined with the help of the mean energy loss
-<inlinemath><math
-><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></inlinemath> in the following way: The
-numbers of collisions (<inlinemath><math
-><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>,
-i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean
-number <inlinemath><math
-><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></inlinemath>. In a step
-<inlinemath><math
-><mi>&Delta;</mi><mi>x</mi></math></inlinemath> the mean number
-of collisions is <equation ><math>
- <mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow><mo>=</mo><msub><mi>&Sigma;</mi><mrow><mi>i</mi></mrow></msub><mi>&Delta;</mi><mi>x</mi> <mi>(</mi><mi>6</mi><mi>)</mi>
-</math></equation>The
-mean energy loss <inlinemath><math
-><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></inlinemath>
-in a step is the sum of the excitation and ionisation contributions
-<equation ><math>
- <mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!--
---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>&Delta;</mi><mi>x</mi><mo>=</mo><mfenced
-open='[' close=']'><msub><mi>&Sigma;</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>&Sigma;</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>&Sigma;</mi><mrow><mn>3</mn></mrow></msub><msubsup><mo>&int;</mo>
- <mrow><mi>I</mi></mrow><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></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>&Delta;</mi><mi>x</mi> <mi>(</mi><mi>7</mi><mi>)</mi>
-</math></equation>From
-this, using the equations (<ref refid="eq:sigex"/>), (<ref refid="eq:sigion"/>), (<ref refid="eq:fisum"/>) and (<ref refid="eq:flnsum"/>), one can define the parameter
-<inlinemath><math
-><mi>C</mi></math></inlinemath>
-<equation ><math>
- <mi>C</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!--
---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>(</mi><mi>8</mi><mi>)</mi>
-</math></equation>
-</par><par>The following values have been chosen in <texttt>GEANT</texttt> for the other parameters:
-<displaymath><math
-><mrow>
- <mtable equalrows='false' equalcolumns='false'><mtr><mtd><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfenced
-open='{' ><mtable equalrows='false' equalcolumns='false'><mtr><mtd><mn>0</mn> </mtd><mtd><mi>if</mi><mi>Z</mi><mo>&leq;</mo><mn>2</mn></mtd>
-</mtr><mtr><mtd><mn>2</mn><mo>/</mo><mi>Z</mi></mtd><mtd><mi>if</mi><mi>Z</mi><mo>&gt;</mo><mn>2</mn></mtd>
-</mtr><mtr><mtd> </mtd></mtr></mtable> </mfenced></mtd><mtd><mo>&Rarr;</mo></mtd><mtd><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>-</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub> </mtd>
- </mtr><mtr><mtd><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn><msup><mi>Z</mi><mrow><mn>2</mn></mrow></msup><mi>eV </mi> </mtd><mtd><mo>&Rarr;</mo></mtd><mtd><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><msup><mfenced
-open='(' close=')'> <mfrac><mrow><mi>I</mi></mrow><!--___
---><mrow><msubsup><mi>E</mi><mrow><mn>2</mn></mrow><mrow><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></mrow></msubsup></mrow></mfrac> </mfenced><mrow> <mfrac><mrow><mn>1</mn></mrow><!-- _
---><mrow><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac> </mrow></msup></mtd>
- </mtr><mtr><mtd><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn> </mtd><mtd> </mtd><mtd> </mtd>
- </mtr><mtr><mtd> </mtd></mtr></mtable>
-</mrow></math></displaymath> With these values
-the atomic level <inlinemath><math
-><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath>
-corresponds approximately the K-shell energy of the atoms and
-<inlinemath><math
-><mi>Z</mi><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath> the number of
-K-shell electrons. <inlinemath><math
-><mi>r</mi></math></inlinemath>
-is the only variable which can be tuned freely. It determines the relative contribution
-of ionisation and excitation to the energy loss.
-</par><par>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
-<equation ><math>
- <mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub> <mi>(</mi><mi>9</mi><mi>)</mi>
-</math></equation>where
-<inlinemath><math
-><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></math></inlinemath> and
-<inlinemath><math
-><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></math></inlinemath>
-are sampled from Poisson distribution as discussed above. The
-loss due to the ionisation can be generated from the distribution
-<inlinemath><math
-><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math></inlinemath> by
-the inverse transformation method:
-
- <eqnarray ><subeqn ><math><mi>u</mi><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow> <mo>=</mo><msubsup> <mo>&int;</mo>
- <mrow><mi>I</mi></mrow><mrow><mi>E</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi> <mtext></mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math><mi>E</mi><mo>=</mo><msup><mi>F</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow> <mo>=</mo> <mfrac><mrow><mi>I</mi></mrow><!--____
---><mrow><mn>1</mn><mo>-</mo><mi>u</mi> <mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!-- ___
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> <mtext>(10)</mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn ><math> <mtext>(11)</mtext>
-</math></subeqn></eqnarray>
-where <inlinemath><math
-><mi>u</mi></math></inlinemath> is a uniform random
-number between <inlinemath><math
-><mi>F</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></inlinemath> and
-<inlinemath><math
-><mi>F</mi><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></inlinemath>. The contribution from the
-ionisations will be <equation ><math>
- <mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mo>&sum;</mo>
- <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></mrow></msubsup> <mfrac><mrow><mi>I</mi></mrow><!--________
---><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>j</mi></mrow></msub> <mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!-- ___
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> <mi>(</mi><mi>1</mi><mi>2</mi><mi>)</mi>
-</math></equation>where
-<inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath> is the
-number of ionisation (sampled from Poisson distribution). The energy loss in a step will
-then be <inlinemath><math
-><mi>&Delta;</mi><mi>E</mi><mo>=</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>+</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></inlinemath>.
-</par>
-<subsection >
-<stitle>
-Fast simulation for <inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><mn>1</mn><mn>6</mn></math></inlinemath></stitle>
-<par>If the number of ionisation <inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></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
-<inlinemath><math
-><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>&alpha;</mi><mi>I</mi><mo>]</mo></mrow></math></inlinemath> the interval A,
-and the latter <inlinemath><math
-><mrow><mo>[</mo><mi>&alpha;</mi><mi>I</mi><mo>,</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>]</mo></mrow></math></inlinemath> the
-interval B. <inlinemath><math
-><mi>&alpha;</mi></math></inlinemath> lies
-between 1 and <inlinemath><math
-><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><mi>I</mi></math></inlinemath>.
-A collision with a loss in the interval A happens with the probability
-<equation id="eq:phys332-5"><math>
- <mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>&int;</mo>
- <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</mi><mi>I</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>&alpha;</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!--
- --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mi>&alpha;</mi></mrow></mfrac> <mi>(</mi><mi>1</mi><mi>3</mi><mi>)</mi>
-</math></equation>The
-mean energy loss and the standard deviation for this type of collision are
-<equation ><math>
- <mrow><mo>&lt;</mo><mi>&Delta;</mi><mi>E</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>>;</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___
---><mrow><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>&int;</mo>
- <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</mi><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mi>I</mi><mi>&alpha;</mi><mo>ln</mo><mi>&alpha;</mi></mrow><!--
- --><mrow><mi>&alpha;</mi><mo>-</mo><mn>1</mn></mrow></mfrac> <mi>(</mi><mi>1</mi><mi>4</mi><mi>)</mi>
-</math></equation>and
-<equation ><math>
- <msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--___
---><mrow><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>&int;</mo>
- <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</mi><mi>I</mi></mrow></msubsup><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><msup><mi>I</mi><mrow><mn>2</mn></mrow></msup><mi>&alpha;</mi><mfenced
-open='(' close=')'><mn>1</mn><mo>-</mo> <mfrac><mrow><mi>&alpha;</mi><msup><mo>ln</mo><mrow><mn>2</mn></mrow></msup><mi>&alpha;</mi></mrow><!--_
---><mrow><msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mfenced> <mi>(</mi><mi>1</mi><mi>5</mi><mi>)</mi>
-</math></equation>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:
-
- <eqnarray ><subeqn id="eq:phys332-1"><math><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow> <mo>=</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow> <mtext>(16)</mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn id="eq:phys332-2"><math><msubsup><mi>&sigma;</mi><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup> <mo>=</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>)</mo></mrow> <mtext>(17)</mtext>
-</math></subeqn></eqnarray>
-It is further assumed that the energy loss in these collisions has a Gaussian
-distribution with
-
- <eqnarray ><subeqn id="eq:phys332-3"><math><mrow><mo>&lt;</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow> <mo>=</mo> <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mrow><mo>&lt;</mo><mi>&Delta;</mi><mi>E</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>>;</mo></mrow> <mtext>(18)</mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn id="eq:phys332-4"><math><msubsup><mi>&sigma;</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup> <mo>=</mo> <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow> <mtext>(19)</mtext>
-</math></subeqn></eqnarray>
-The energy loss of these collision can then be sampled from the Gaussian
-distribution.
-</par><par>The collisions where the energy loss is in the interval B are sampled directly from
-<equation ><math>
- <mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub><mo>=</mo><msubsup><mo>&sum;</mo>
- <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub></mrow></msubsup> <mfrac><mrow><mi>&alpha;</mi><mi>I</mi></mrow><!--_________
---><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>i</mi></mrow></msub> <mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi><mo>-</mo><mi>&alpha;</mi><mi>I</mi></mrow><!--
- --><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac> <mi>(</mi><mi>2</mi><mi>0</mi><mi>)</mi>
-</math></equation>The
-total energy loss is the sum of these two types of collisions:
-<equation ><math>
- <mi>&Delta;</mi><mi>E</mi><mo>=</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>+</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub> <mi>(</mi><mi>2</mi><mi>1</mi><mi>)</mi>
-</math></equation>
-</par><par>The approximation of equations (<ref refid="eq:phys332-1"/>), (<ref refid="eq:phys332-2"/>), (<ref refid="eq:phys332-3"/>) and (<ref refid="eq:phys332-4"/>) can be used under the following
-conditions:
-
- <eqnarray ><subeqn id="eq:phys332-6"><math><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub> <mo>&geq;</mo> <mn>0</mn> <mtext>(22)</mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn id="eq:phys332-7"><math><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub> <mo>&leq;</mo> <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub> <mtext>(23)</mtext>
- </math></subeqn><subeqn ><math>
- </math></subeqn><subeqn id="eq:phys332-8"><math><mrow><mo>&lt;</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow></msub> <mo>&geq;</mo> <mn>0</mn> <mtext>(24)</mtext>
-</math></subeqn></eqnarray>
-where <inlinemath><math
-><mi>c</mi><mo>&geq;</mo><mn>4</mn></math></inlinemath>. From
-the equations (<ref refid="eq:phys332-5"/>), (<ref refid="eq:phys332-1"/>) and (<ref refid="eq:phys332-3"/>) and from the conditions (<ref refid="eq:phys332-6"/>) and (<ref refid="eq:phys332-7"/>) the following limits can be
-derived: <equation ><math>
- <msub><mi>&alpha;</mi><mrow><mi>min</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></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><mrow><mi>max</mi></mrow></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>&leq;</mo><mi>&alpha;</mi><mo>&leq;</mo><msub><mi>&alpha;</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></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><mrow><mi>max</mi></mrow></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> <mi>(</mi><mi>2</mi><mi>5</mi><mi>)</mi>
-</math></equation>This
-conditions gives a lower limit to number of the ionisations
-<inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath> for which the fast
-sampling can be done: <equation ><math>
- <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup> <mi>(</mi><mi>2</mi><mi>6</mi><mi>)</mi>
-</math></equation>As
-in the conditions (<ref refid="eq:phys332-6"/>), (<ref refid="eq:phys332-7"/>) and (<ref refid="eq:phys332-8"/>) the value of
-<inlinemath><math
-><mi>c</mi></math></inlinemath> is as minimum
-4, one gets <inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><mn>1</mn><mn>6</mn></math></inlinemath>.
-In order to speed the simulation, the maximum value is used for
-<inlinemath><math
-><mi>&alpha;</mi></math></inlinemath>.
-</par><par>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
-<inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath>.
-The maximum number of these collisions can be estimated as
-<equation ><math>
- <msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi><mo>,</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>&ap;</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub><mo>)</mo></mrow> <mi>(</mi><mi>2</mi><mi>7</mi><mi>)</mi>
-</math></equation>From the previous
-expressions for <inlinemath><math
-><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></inlinemath> and
-<inlinemath><math
-><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub></math></inlinemath> one can derive the
-condition <equation ><math>
- <msub><mi>n</mi><mrow><mi>B</mi></mrow></msub><mo>&leq;</mo><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow><!--_
---><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mi>(</mi><mi>2</mi><mi>8</mi><mi>)</mi>
-</math></equation>The following
-values are obtained with <inlinemath><math
-><mi>c</mi><mo>=</mo><mn>4</mn></math></inlinemath>:
-</par><par><tabular preamble="llcrr"><row><cell
-><inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath></cell><cell
-><inlinemath><math
-><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></inlinemath></cell><cell
-></cell><cell
-><inlinemath><math
-><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></inlinemath></cell><cell
-><inlinemath><math
-><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></inlinemath></cell>
-</row><row><cell
->16 </cell><cell
->16 </cell><cell
-></cell><cell
-> 200</cell><cell
-> 29.63</cell>
-</row><row><cell
->20 </cell><cell
->17.78 </cell><cell
-></cell><cell
-> 500</cell><cell
-> 31.01</cell>
-</row><row><cell
->50 </cell><cell
->24.24 </cell><cell
-></cell><cell
-> 1000</cell><cell
-> 31.50</cell>
-</row><row><cell
->100 </cell><cell
->27.59 </cell><cell
-></cell><cell
-><inlinemath><math
-><mi>&infin;</mi></math></inlinemath></cell><cell
-> 32.00</cell>
-</row></tabular>
-</par>
-</subsection>
-<subsection >
-<stitle>
-Special sampling for lower part of the spectrum</stitle>
-<par>If the step length is very small (<inlinemath><math
-><mo>&leq;</mo><mn>5</mn></math></inlinemath>
-mm in gases, <inlinemath><math
-><mo>&leq;</mo></math></inlinemath>
-2-3 <inlinemath><math
-><mi>&mu;</mi></math></inlinemath>m in solids)
-the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is
-computed <equation ><math>
- <mi>P</mi><mrow><mo>(</mo><mi>&Delta;</mi><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mrow><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>>;</mo></mrow><mo>)</mo></mrow></mrow></msup> <mi>(</mi><mi>2</mi><mi>9</mi><mi>)</mi>
-</math></equation>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
-<inlinemath><math
-><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn></math></inlinemath> eV is chosen
-to correspond to the outer electrons. The mean number of collisions can be calculated from
-<equation ><math>
- <mrow><mo>&lt;</mo><mi>n</mi><mo>>;</mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><!--_
---><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac> <mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!--
---><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>&Delta;</mi><mi>x</mi> <mi>(</mi><mi>3</mi><mi>0</mi><mi>)</mi>
-</math></equation>The number
-of collisions <inlinemath><math
-><mi>n</mi></math></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
-<equation ><math>
- <mi>&Delta;</mi><mi>E</mi><mo>=</mo><msubsup><mo>&sum;</mo>
- <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup> <mfrac><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow><!--_________
---><mrow><mn>1</mn><mo>-</mo> <mfrac><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><!--_____
---><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></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> <mi>(</mi><mi>3</mi><mi>1</mi><mi>)</mi>
-</math></equation>
-</par>
-</subsection>
-</section>
-<section class="star">
-<stitle>
-References</stitle>
-<bibliography >
-<bibitem id="bib-LAND">
-<par>L.Landau.
-On
-the
-Energy
-Loss
-of
-Fast
-Particles
-by
-Ionisation.
-Originally
-published
-in
-<emph>J.
-Phys.</emph>,
-8:201,
-1944.
-Reprinted
-in
-D.ter
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