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      <fileDesc> 
         <titleStmt> 
            <title>A sample article</title> 
	    <author>Maria Physicist</author>
         </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> 
        <p>This text should be read with a pinch of salt </p>
         <div 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>
                  <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 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>
                  <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 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>
                  <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>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>E</mi>
                  </math>
               </formula> is energy and
<formula>
                  <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>
                  <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>
                  <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>
                  <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 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>
                           <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>
                           <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>
                           <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>
                           <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>
                           <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>
                           <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>
                  <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>
                  <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>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>κ</mi>
                  </math>
               </formula> tends towards large
values if <formula>
                  <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>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>β</mi>
                  </math>
               </formula> is small.
Likewise, <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>κ</mi>
                  </math>
               </formula> tends
towards zero if <formula>
                  <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>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>β</mi>
                  </math>
               </formula>
approaches 1.
</p>
            <p>The value of <formula>
                  <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>
                        <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                           <mi>κ</mi>
                           <mo>&gt;</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>
                        <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                           <mn>0</mn>
                           <mo>.</mo>
                           <mn>0</mn>
                           <mn>1</mn>
                           <mo>&lt;</mo>
                           <mi>κ</mi>
                           <mo>&lt;</mo>
                           <mn>1</mn>
                           <mn>0</mn>
                        </math>
                     </formula>, Vavilov distribution,
and
<formula>
                        <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                           <mi>κ</mi>
                           <mo>&lt;</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>
                  <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>
                  <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>
                  <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>
                  <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. <ptr target="fg:phys332-1" targOrder="U"/> shows the behaviour
of <formula>
                  <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 file="phys332-1" id="fg:phys332-1">
                  <head>The variable <formula>
                        <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>
                        <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                           <mi>Z</mi>
                        </math>
                     </formula>,
<formula>
                        <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>
         <div id="landau" org="uniform" part="N" sample="complete">
            <head>
Landau theory</head>
            <p>For a particle of mass <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>m</mi>
                        <mrow>
                           <mi>x</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula> traversing
a thickness of material <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>δ</mi>
                     <mi>x</mi>
                  </math>
               </formula>,
the Landau probability distribution may be written in terms of the universal Landau
function <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>φ</mi>
                     <mrow>
                        <mo>(</mo>
                        <mi>λ</mi>
                        <mo>)</mo>
                     </mrow>
                  </math>
               </formula>
as<ptr target="bib-LAND" targOrder="U"/>:
                         <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>f</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>ε</mi>
                                 <mo>,</mo>
                                 <mi>δ</mi>
                                 <mi>x</mi>
                                 <mo>)</mo>
                              </mrow>  
                              <mo>=</mo>  
                              <mfrac>
                                 <mrow>
                                    <mn>1</mn>
                                 </mrow>
                                 <mrow>
                                    <mi>ξ</mi>
                                 </mrow>
                              </mfrac>
                              <mi>φ</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>λ</mi>
                                 <mo>)</mo>
                              </mrow>                         
                              <mtext/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
where
             <table rend="eqnarray">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>φ</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>λ</mi>
                                 <mo>)</mo>
                              </mrow>  
                              <mo>=</mo>   
                              <mfrac>
                                 <mrow>
                                    <mn>1</mn>
                                 </mrow>
                                 <mrow>
                                    <mn>2</mn>
                                    <mi>π</mi>
                                    <mi>i</mi>
                                 </mrow>
                              </mfrac>
                              <msubsup>
                                 <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>exp</mo>
                              <mfenced open="(" close=")">
                                 <mi>u</mi>
                                 <mo>ln</mo>
                                 <mi>u</mi>
                                 <mo>+</mo>
                                 <mi>λ</mi>
                                 <mi>u</mi>
                              </mfenced>
                              <mi>d</mi>
                              <mi>u</mi>
                              <mspace width="2cm"/>
                              <mi>c</mi>
                              <mo>≥</mo>
                              <mn>0</mn>             
                              <mtext/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
               
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>λ</mi>  
                              <mo>=</mo>  
                              <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>
                              <mo>-</mo>
                              <mo>ln</mo>   
                              <mfrac>
                                 <mrow>
                                    <mi>ξ</mi>
                                 </mrow>
                                 <mrow>
                                    <msub>
                                       <mi>E</mi>
                                       <mrow>
                                          <mtext>max</mtext>
                                       </mrow>
                                    </msub>
                                 </mrow>
                              </mfrac>                          
                              <mtext/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
               
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>γ</mi>
                              <mi>′</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>γ</mi>                              
                              <mtext/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
               
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>γ</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/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <munderover accent="true">
                                 <mo>¯</mo>
                                 <mi>ε</mi>
                                 <mrow/>
                              </munderover>  
                              <mo>=</mo>  
                              <mtext>average energy loss</mtext>                            
                              <mtext/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>ε</mi>  
                              <mo>=</mo>  
                              <mtext>actual energy loss</mtext>                             
                              <mtext/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
            </p>
            <div org="uniform" part="N" sample="complete">
               <head>
Restrictions</head>
               <p>The Landau formalism makes two restrictive assumptions :
</p>
               <list type="enumerate">
                  <item>
                     <p>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
<ptr target="vavref" targOrder="U"/>).
</p>
                  </item>
                  <item>
                     <p>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<ptr target="bib-TALM" targOrder="U"/>
is
necessary
to
accurately
simulate
data
distributions.
In
<code>GEANT</code>,
a
parameterised
model
by
L.
Urbán
is
used
(see
section
<ptr target="urban" targOrder="U"/>).</p>
                  </item>
               </list>
               <p>In addition, the average value of the Landau distribution is infinite.
Summing the Landau fluctuation obtained to the average energy from the
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>d</mi>
                        <mi>E</mi>
                        <mo>/</mo>
                        <mi>d</mi>
                        <mi>x</mi>
                     </math>
                  </formula>
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.
</p>
               <p>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 <code>GLANDO</code> routine is:
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
                  
                           <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>
                                       <mrow>
                                          <mtext>max</mtext>
                                       </mrow>
                                    </msub>
                                 </mrow>
                              </mfrac> 
                              <mo>)</mo>
                           </mrow>
                        </mrow>
                     </math>
                  </formula>
In order for this to have average 0, we must impose that:
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
                        
                           <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>
                                    <mrow>
                                       <mtext>max</mtext>
                                    </mrow>
                                 </msub>
                              </mrow>
                           </mfrac>
                        </mrow>
                     </math>
                  </formula>
               </p>
               <p>This is realised introducing a <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>λ</mi>
                           <mrow>
                              <mtext>max</mtext>
                           </mrow>
                        </msub>
                        <mrow>
                           <mo>(</mo>
                           <munderover accent="true">
                              <mo>¯</mo>
                              <mi>λ</mi>
                              <mrow/>
                           </munderover>
                           <mo>)</mo>
                        </mrow>
                     </math>
                  </formula>
such that if only values of <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>λ</mi>
                        <mo>≤</mo>
                        <msub>
                           <mi>λ</mi>
                           <mrow>
                              <mtext>max</mtext>
                           </mrow>
                        </msub>
                     </math>
                  </formula>
are accepted, the average value of the distribution is
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <munderover accent="true">
                           <mo>¯</mo>
                           <mi>λ</mi>
                           <mrow/>
                        </munderover>
                     </math>
                  </formula>.
</p>
               <p>A parametric fit to the universal Landau distribution has been performed, with following result:
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
    
                           <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>
                     </math>
                  </formula> only values
smaller than <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>λ</mi>
                           <mrow>
                              <mtext>max</mtext>
                           </mrow>
                        </msub>
                     </math>
                  </formula>
are accepted, otherwise the distribution is resampled.
</p>
            </div>
         </div>
         <div id="vavref" org="uniform" part="N" sample="complete">
            <head>
Vavilov theory</head>
            <p>Vavilov<ptr target="bib-VAVI" targOrder="U"/> derived a more accurate straggling distribution by introducing the kinematic
limit on the maximum transferable energy in a single collision, rather than using
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mtext>max</mtext>
                        </mrow>
                     </msub>
                     <mo>=</mo>
                     <mi>∞</mi>
                  </math>
               </formula>. Now
we can write<ptr target="bib-SCH1" targOrder="U"/>:
                       <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>f</mi> 
                              <mfenced open="(" close=")">
                                 <mi>ε</mi>
                                 <mo>,</mo>
                                 <mi>δ</mi>
                                 <mi>s</mi>
                              </mfenced>  
                              <mo>=</mo>  
                              <mfrac>
                                 <mrow>
                                    <mn>1</mn>
                                 </mrow>
                                 <mrow>
                                    <mi>ξ</mi>
                                 </mrow>
                              </mfrac>
                              <msub>
                                 <mi>φ</mi>
                                 <mrow>
                                    <mi>v</mi>
                                 </mrow>
                              </msub> 
                              <mfenced open="(" close=")">
                                 <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/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
where
         <table rend="eqnarray">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <msub>
                                 <mi>φ</mi>
                                 <mrow>
                                    <mi>v</mi>
                                 </mrow>
                              </msub> 
                              <mfenced open="(" close=")">
                                 <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>=</mo>   
                              <mfrac>
                                 <mrow>
                                    <mn>1</mn>
                                 </mrow>
                                 <mrow>
                                    <mn>2</mn>
                                    <mi>π</mi>
                                    <mi>i</mi>
                                 </mrow>
                              </mfrac>
                              <msubsup>
                                 <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>φ</mi>
                              <mfenced open="(" close=")">
                                 <mi>s</mi>
                              </mfenced>
                              <msup>
                                 <mi>e</mi>
                                 <mrow>
                                    <mi>λ</mi>
                                    <mi>s</mi>
                                 </mrow>
                              </msup>
                              <mi>d</mi>
                              <mi>s</mi>
                              <mspace width="2cm"/>
                              <mi>c</mi>
                              <mo>≥</mo>
                              <mn>0</mn>                  
                              <mtext/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
               
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>φ</mi>
                              <mfenced open="(" close=")">
                                 <mi>s</mi>
                              </mfenced>  
                              <mo>=</mo>  
                              <mo>exp</mo>
                              <mfenced open="[" close="]">
                                 <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>exp</mo>
                              <mfenced open="[" close="]">
                                 <mi>ψ</mi> 
                                 <mfenced open="(" close=")">
                                    <mi>s</mi>
                                 </mfenced>
                              </mfenced>
                              <mo>,</mo>                         
                              <mtext/>
                 
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
               
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>ψ</mi> 
                              <mfenced open="(" close=")">
                                 <mi>s</mi>
                              </mfenced>  
                              <mo>=</mo>  
                              <mi>s</mi>
                              <mo>ln</mo>
                              <mi>κ</mi>
                              <mo>+</mo>
                              <mrow>
                                 <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="]">
                                 <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>-</mo>
                              <mi>κ</mi>
                              <msup>
                                 <mi>e</mi>
                                 <mrow>
                                    <mo>-</mo>
                                    <mi>s</mi>
                                    <mo>/</mo>
                                    <mi>κ</mi>
                                 </mrow>
                              </msup>
                              <mo>,</mo>         
                              <mtext/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
and
           <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <msub>
                                 <mi>E</mi>
                                 <mrow>
                                    <mn>1</mn>
                                 </mrow>
                              </msub>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>z</mi>
                                 <mo>)</mo>
                              </mrow>  
                              <mo>=</mo>
                              <msubsup>  
                                 <mo>∫</mo>
                       
                                 <mrow>
                                    <mi>∞</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/>
                
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
              
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <msub>
                                 <mi>λ</mi>
                                 <mrow>
                                    <mi>v</mi>
                                 </mrow>
                              </msub>  
                              <mo>=</mo>  
                              <mi>κ</mi>
                              <mfenced open="[" close="]">
                                 <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/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
            </p>
            <p>The Vavilov parameters are simply related to the Landau parameter by
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>λ</mi>
                        <mrow>
                           <mi>L</mi>
                        </mrow>
                     </msub>
                     <mo>=</mo>
                     <msub>
                        <mi>λ</mi>
                        <mrow>
                           <mi>v</mi>
                        </mrow>
                     </msub>
                     <mo>/</mo>
                     <mi>κ</mi>
                     <mo>-</mo>
                     <mo>ln</mo>
                     <mi>κ</mi>
                  </math></formula>. It can be shown that
as <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>κ</mi>
                     <mo>→</mo>
                     <mn>0</mn>
                  </math>
               </formula>, the distribution of
the variable <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>λ</mi>
                        <mrow>
                           <mi>L</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula> approaches
that of Landau. For <formula>
                  <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>
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
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>κ</mi>
                  </math>
               </formula>, but rather the
distribution of <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>λ</mi>
                        <mrow>
                           <mi>L</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>
defined above tends to the distribution of the true
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>λ</mi>
                  </math>
               </formula> from
the Landau density function. Thus the routine <code>GVAVIV</code> samples the variable
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>λ</mi>
                        <mrow>
                           <mi>L</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula> rather
than <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>λ</mi>
                        <mrow>
                           <mi>v</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>.
For <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>κ</mi>
                     <mo>≥</mo>
                     <mn>1</mn>
                     <mn>0</mn>
                  </math>
               </formula>
the Vavilov distribution tends to a Gaussian distribution (see next section).
</p>
         </div>
         <div org="uniform" part="N" sample="complete">
            <head>
Gaussian Theory</head>
            <p>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<ptr target="bib-SELT" targOrder="U"/>
that for <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>κ</mi>
                     <mo>≥</mo>
                     <mn>1</mn>
                     <mn>0</mn>
                  </math>
               </formula>
the Vavilov distribution can be replaced by a Gaussian of the form:
            <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>f</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>ε</mi>
                                 <mo>,</mo>
                                 <mi>δ</mi>
                                 <mi>s</mi>
                                 <mo>)</mo>
                              </mrow>
                              <mo>≈</mo>         
                              <mfrac>
                                 <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>exp</mo>
                              <mfenced open="[" close="]">
                                 <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/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
thus implying
                <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>mean</mi>  
                              <mo>=</mo>  
                              <munderover accent="true">
                                 <mo>¯</mo>
                                 <mi>ε</mi>
                                 <mrow/>
                              </munderover>                                       
                              <mtext/>
                    
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                  
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <msup>
                                 <mi>σ</mi>
                                 <mrow>
                                    <mn>2</mn>
                                 </mrow>
                              </msup>  
                              <mo>=</mo>  
                              <mfrac>
                                 <mrow>
                                    <msup>
                                       <mi>ξ</mi>
                                       <mrow>
                                          <mn>2</mn>
                                       </mrow>
                                    </msup>
                                 </mrow>
                                 <mrow>
                                    <mi>κ</mi>
                                 </mrow>
                              </mfrac> 
                              <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>
                              <mo>=</mo>
                              <mi>ξ</mi>
                              <msub>
                                 <mi>E</mi>
                                 <mrow>
                                    <mi>
max</mi>
                                 </mrow>
                              </msub>
                              <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>                
                              <mtext/>
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
            </p>
         </div>
         <div id="urban" org="uniform" part="N" sample="complete">
            <head>
Urbán model</head>
            <p>The method for computing restricted energy losses with
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>δ</mi>
                  </math>
               </formula>-ray
production above given threshold energy in <code>GEANT</code> 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 <ptr target="fg:phys332-2" targOrder="U"/>.
</p>
            <p>
               <figure file="phys332-2" id="fg:phys332-2">
                  <head>Energy loss distribution for a 3 GeV electron in Argon as given by
standard GEANT. The width of the layers is given in centimeters.</head>
               </figure>
            </p>
            <p>It is assumed that the atoms have only two energy levels with binding energy
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mn>1</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula> and
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mn>2</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula>.
The particle--atom interaction will then be an excitation with energy loss
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mn>1</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula> or
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mn>2</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula>, or
an ionisation with an energy loss distributed according to a function
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>g</mi>
                     <mrow>
                        <mo>(</mo>
                        <mi>E</mi>
                        <mo>)</mo>
                     </mrow>
                     <mo>∼</mo>
                     <mn>1</mn>
                     <mo>/</mo>
                     <msup>
                        <mi>E</mi>
                        <mrow>
                           <mn>2</mn>
                        </mrow>
                     </msup>
                  </math>
               </formula>:
<formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        
                     <mi>g</mi>
                     <mrow>
                        <mo>(</mo>
                        <mi>E</mi>
                        <mo>)</mo>
                     </mrow>
                     <mo>=</mo>
                     <mfrac>
                        <mrow>
                           <mrow>
                              <mo>(</mo>
                              <msub>
                                 <mi>E</mi>
                                 <mrow>
                                    <mtext>max</mtext>
                                 </mrow>
                              </msub>
                              <mo>+</mo>
                              <mi>I</mi>
                              <mo>)</mo>
                           </mrow>
                           <mi>I</mi>
                        </mrow>
                        <mrow>
                           <msub>
                              <mi>E</mi>
                              <mrow>
                                 <mtext>max</mtext>
                              </mrow>
                           </msub>
                        </mrow>
                     </mfrac>
                     <mfrac>
                        <mrow>
                           <mn>1</mn>
                        </mrow>
                        <mrow>
                           <msup>
                              <mi>E</mi>
                              <mrow>
                                 <mn>2</mn>
                              </mrow>
                           </msup>
                        </mrow>
                     </mfrac>
                  </math>
               </formula>
            </p>
            <p>The
macroscopic cross-section for excitations (<formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>i</mi>
                     <mo>=</mo>
                     <mn>1</mn>
                     <mo>,</mo>
                     <mn>2</mn>
                  </math>
               </formula>)
is <formula id="eq:sigex" type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                   
                     <msub>
                        <mi>Σ</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>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>
                        <mo>(</mo>
                        <mn>1</mn>
                        <mo>-</mo>
                        <mi>r</mi>
                        <mo>)</mo>
                     </mrow>               
                  </math>
               </formula>and
the macroscopic cross-section for ionisation is
<formula id="eq:sigion" type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                    
                     <msub>
                        <mi>Σ</mi>
                        <mrow>
                           <mn>3</mn>
                        </mrow>
                     </msub>
                     <mo>=</mo>
                     <mi>C</mi>            
                     <mfrac>
                        <mrow>
                           <msub>
                              <mi>E</mi>
                              <mrow>
                                 <mtext>max</mtext>
                              </mrow>
                           </msub>
                        </mrow>
                        <mrow>
                           <mi>I</mi>
                           <mrow>
                              <mo>(</mo>
                              <msub>
                                 <mi>E</mi>
                                 <mrow>
                                    <mtext>max</mtext>
                                 </mrow>
                              </msub>
                              <mo>+</mo>
                              <mi>I</mi>
                              <mo>)</mo>
                           </mrow>
                           <mo>ln</mo>
                           <mrow>
                              <mo>(</mo>
                              <mfrac>
                                 <mrow>
                                    <msub>
                                       <mi>E</mi>
                                       <mrow>
                                          <mtext>max</mtext>
                                       </mrow>
                                    </msub>
                                    <mo>+</mo>
                                    <mi>I</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>I</mi>
                                 </mrow>
                              </mfrac>      
                              <mo>)</mo>
                           </mrow>
                        </mrow>
                     </mfrac>
                     <mi>r</mi>                 
                  </math>
               </formula>
               <formula>
                  <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 <code>GEANT</code> cut for <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>δ</mi>
                  </math>
               </formula>-production,
or the maximum energy transfer minus mean ionisation energy, if it is smaller than
this cut-off value. The following notation is used:
</p>
            <p>
               <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula>
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>r</mi>
                              <mo>,</mo>
                              <mi>C</mi>
                           </math>
                        </formula>
                     </cell>
                     <cell role="data" cols="1" rows="1">parameters of the model</cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula>
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <msub>
                                 <mi>E</mi>
                                 <mrow>
                                    <mi>i</mi>
                                 </mrow>
                              </msub>
                           </math>
                        </formula>
                     </cell>
                     <cell role="data" cols="1" rows="1">atomic energy levels      </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula>
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>I</mi>
                           </math>
                        </formula>
                     </cell>
                     <cell role="data" cols="1" rows="1">mean ionisation energy  </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula>
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <msub>
                                 <mi>f</mi>
                                 <mrow>
                                    <mi>i</mi>
                                 </mrow>
                              </msub>
                           </math>
                        </formula>
                     </cell>
                     <cell role="data" cols="1" rows="1">oscillator strengths       </cell>
                  </row>
               </table>
            </p>
            <p>The model has the parameters <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>f</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>,
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>,
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>C</mi>
                  </math>
               </formula> and
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>r</mi>
                     <mrow>
                        <mo>(</mo>
                        <mn>0</mn>
                        <mo>≤</mo>
                        <mi>r</mi>
                        <mo>≤</mo>
                        <mn>1</mn>
                        <mo>)</mo>
                     </mrow>
                  </math>
               </formula>. The oscillator
strengths <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>f</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula> and the
atomic level energies <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>
should satisfy the constraints
                              <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn" id="eq:fisum">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <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>                     
                                   
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                      
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn" id="eq:flnsum">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <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>                   
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
The parameter <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>C</mi>
                  </math>
               </formula>
can be defined with the help of the mean energy loss
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>d</mi>
                     <mi>E</mi>
                     <mo>/</mo>
                     <mi>d</mi>
                     <mi>x</mi>
                  </math>
               </formula> in the following way: The
numbers of collisions (<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>n</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>,
i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean
number <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mrow>
                        <mo>&lt;</mo>
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mi>i</mi>
                           </mrow>
                        </msub>
                        <mo>&gt;;</mo>
                     </mrow>
                  </math>
               </formula>. In a step
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>Δ</mi>
                     <mi>x</mi>
                  </math>
               </formula> the mean number
of collisions is <formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                            
                     <mrow>
                        <mo>&lt;</mo>
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mi>i</mi>
                           </mrow>
                        </msub>
                        <mo>&gt;;</mo>
                     </mrow>
                     <mo>=</mo>
                     <msub>
                        <mi>Σ</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                     <mi>Δ</mi>
                     <mi>x</mi>                                        
                  </math>
               </formula>The
mean energy loss <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>d</mi>
                     <mi>E</mi>
                     <mo>/</mo>
                     <mi>d</mi>
                     <mi>x</mi>
                  </math>
               </formula>
in a step is the sum of the excitation and ionisation contributions
<formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
            
                     <mfrac>
                        <mrow>
                           <mi>d</mi>
                           <mi>E</mi>
                        </mrow>
                        <mrow>
                           <mi>d</mi>
                           <mi>x</mi>
                        </mrow>
                     </mfrac> 
                     <mi>Δ</mi>
                     <mi>x</mi>
                     <mo>=</mo>
                     <mfenced open="[" close="]">
                        <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>
                                 <mrow>
                                    <mtext>max</mtext>
                                 </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>Δ</mi>
                     <mi>x</mi>        
                  </math>
               </formula>From
this, using the equations (<ptr target="eq:sigex" targOrder="U"/>), (<ptr target="eq:sigion" targOrder="U"/>), (<ptr target="eq:fisum" targOrder="U"/>) and (<ptr target="eq:flnsum" targOrder="U"/>), one can define the parameter
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>C</mi>
                  </math>
               </formula>
               <formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              
                     <mi>C</mi>
                     <mo>=</mo>
                     <mfrac>
                        <mrow>
                           <mi>d</mi>
                           <mi>E</mi>
                        </mrow>
                        <mrow>
                           <mi>d</mi>
                           <mi>x</mi>
                        </mrow>
                     </mfrac>                           
                  </math>
               </formula>
            </p>
            <p>The following values have been chosen in <code>GEANT</code> for the other parameters:
<formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <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>≤</mo>
                                             <mn>2</mn>
                                          </mtd>
                                       </mtr>
                                       <mtr>
                                          <mtd>
                                             <mn>2</mn>
                                             <mo>/</mo>
                                             <mi>Z</mi>
                                          </mtd>
                                          <mtd>
                                             <mi>if</mi>
                                             <mi>Z</mi>
                                             <mo>&gt;</mo>
                                             <mn>2</mn>
                                          </mtd>
                                       </mtr>
                                       <mtr>
                                          <mtd>     </mtd>
                                       </mtr>
                                    </mtable>         
                                 </mfenced>
                              </mtd>
                              <mtd>
                                 <mo>↠</mo>
                              </mtd>
                              <mtd>
                                 <msub>
                                    <mi>f</mi>
                                    <mrow>
                                       <mn>1</mn>
                                    </mrow>
                                 </msub>
                                 <mo>=</mo>
                                 <mn>1</mn>
                                 <mo>-</mo>
                                 <msub>
                                    <mi>f</mi>
                                    <mrow>
                                       <mn>2</mn>
                                    </mrow>
                                 </msub>     
                              </mtd>
                     
                           </mtr>
                           <mtr>
                              <mtd>
                                 <msub>
                                    <mi>E</mi>
                                    <mrow>
                                       <mn>2</mn>
                                    </mrow>
                                 </msub>
                                 <mo>=</mo>
                                 <mn>1</mn>
                                 <mn>0</mn>
                                 <msup>
                                    <mi>Z</mi>
                                    <mrow>
                                       <mn>2</mn>
                                    </mrow>
                                 </msup>
                                 <mi>eV </mi>    
                              </mtd>
                              <mtd>
                                 <mo>↠</mo>
                              </mtd>
                              <mtd>
                                 <msub>
                                    <mi>E</mi>
                                    <mrow>
                                       <mn>1</mn>
                                    </mrow>
                                 </msub>
                                 <mo>=</mo>
                                 <msup>
                                    <mfenced open="(" close=")">  
                                       <mfrac>
                                          <mrow>
                                             <mi>I</mi>
                                          </mrow>
                                          <mrow>
                                             <msubsup>
                                                <mi>E</mi>
                                                <mrow>
                                                   <mn>2</mn>
                                                </mrow>
                                                <mrow>
                                                   <msub>
                                                      <mi>f</mi>
                                                      <mrow>
                                                         <mn>2</mn>
                                                      </mrow>
                                                   </msub>
                                                </mrow>
                                             </msubsup>
                                          </mrow>
                                       </mfrac>  
                                    </mfenced>
                                    <mrow> 
                                       <mfrac>
                                          <mrow>
                                             <mn>1</mn>
                                          </mrow>
                                          <mrow>
                                             <msub>
                                                <mi>f</mi>
                                                <mrow>
                                                   <mn>1</mn>
                                                </mrow>
                                             </msub>
                                          </mrow>
                                       </mfrac> 
                                    </mrow>
                                 </msup>
                              </mtd>
                     
                           </mtr>
                           <mtr>
                              <mtd>
                                 <mi>r</mi>
                                 <mo>=</mo>
                                 <mn>0</mn>
                                 <mo>.</mo>
                                 <mn>4</mn>            
                              </mtd>
                              <mtd>  </mtd>
                              <mtd>          </mtd>
                     
                           </mtr>
                           <mtr>
                              <mtd>             </mtd>
                           </mtr>
                        </mtable>
                     </mrow>
                  </math>
               </formula> With these values
the atomic level <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mn>2</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula>
corresponds approximately the K-shell energy of the atoms and
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>Z</mi>
                     <msub>
                        <mi>f</mi>
                        <mrow>
                           <mn>2</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula> the number of
K-shell electrons. <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>r</mi>
                  </math>
               </formula>
is the only variable which can be tuned freely. It determines the relative contribution
of ionisation and excitation to the energy loss.
</p>
            <p>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
<formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                         
                     <mi>Δ</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>                      
                  </math>
               </formula>where
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>n</mi>
                        <mrow>
                           <mn>1</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula> and
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>n</mi>
                        <mrow>
                           <mn>2</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula>
are sampled from Poisson distribution as discussed above. The
loss due to the ionisation can be generated from the distribution
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>g</mi>
                     <mrow>
                        <mo>(</mo>
                        <mi>E</mi>
                        <mo>)</mo>
                     </mrow>
                  </math>
               </formula> by
the inverse transformation method:
                       <table rend="inline">
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <mi>u</mi>
                              <mo>=</mo>
                              <mi>F</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>E</mi>
                                 <mo>)</mo>
                              </mrow>  
                              <mo>=</mo>
                              <msubsup>  
                                 <mo>∫</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/>
                              
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              <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>
                                                <mtext>max</mtext>
                                             </mrow>
                                          </msub>
                                       </mrow>
                                       <mrow>
                                          <msub>
                                             <mi>E</mi>
                                             <mrow>
                                                <mtext>max</mtext>
                                             </mrow>
                                          </msub>
                                          <mo>+</mo>
                                          <mi>I</mi>
                                       </mrow>
                                    </mfrac> 
                                 </mrow>
                              </mfrac>                
                              
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                              
                           </math>
                        </formula>
                     </cell>
                  </row>
                  <row role="data">
                     <cell role="data" cols="1" rows="1">
                        <formula type="subeqn">
                           <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">                                
                           </math>
                        </formula>
                     </cell>
                  </row>
               </table>
where <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>u</mi>
                  </math>
               </formula> is a uniform random
number between <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>F</mi>
                     <mrow>
                        <mo>(</mo>
                        <mi>I</mi>
                        <mo>)</mo>
                     </mrow>
                     <mo>=</mo>
                     <mn>0</mn>
                  </math>
               </formula> and
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>F</mi>
                     <mrow>
                        <mo>(</mo>
                        <msub>
                           <mi>E</mi>
                           <mrow>
                              <mtext>max</mtext>
                           </mrow>
                        </msub>
                        <mo>+</mo>
                        <mi>I</mi>
                        <mo>)</mo>
                     </mrow>
                     <mo>=</mo>
                     <mn>1</mn>
                  </math>
               </formula>. The contribution from the
ionisations will be <formula type="display">
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 
                     <mi>Δ</mi>
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                     <mo>=</mo>
                     <msubsup>
                        <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>
                        <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>
                                       <mtext>max</mtext>
                                    </mrow>
                                 </msub>
                              </mrow>
                              <mrow>
                                 <msub>
                                    <mi>E</mi>
                                    <mrow>
                                       <mtext>max</mtext>
                                    </mrow>
                                 </msub>
                                 <mo>+</mo>
                                 <mi>I</mi>
                              </mrow>
                           </mfrac> 
                        </mrow>
                     </mfrac>                                      
                  </math>
               </formula>where
<formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <msub>
                        <mi>n</mi>
                        <mrow>
                           <mn>3</mn>
                        </mrow>
                     </msub>
                  </math>
               </formula> is the
number of ionisation (sampled from Poisson distribution). The energy loss in a step will
then be <formula>
                  <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     <mi>Δ</mi>
                     <mi>E</mi>
                     <mo>=</mo>
                     <mi>Δ</mi>
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mi>e</mi>
                        </mrow>
                     </msub>
                     <mo>+</mo>
                     <mi>Δ</mi>
                     <msub>
                        <mi>E</mi>
                        <mrow>
                           <mi>i</mi>
                        </mrow>
                     </msub>
                  </math>
               </formula>.
</p>
            <div org="uniform" part="N" sample="complete">
               <head>
Fast simulation for <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                        <mo>≥</mo>
                        <mn>1</mn>
                        <mn>6</mn>
                     </math>
                  </formula>
               </head>
               <p>If the number of ionisation <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                     </math>
                  </formula>
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
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
                           <mo>[</mo>
                           <mi>I</mi>
                           <mo>,</mo>
                           <mi>α</mi>
                           <mi>I</mi>
                           <mo>]</mo>
                        </mrow>
                     </math>
                  </formula> the interval A,
and the latter <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
                           <mo>[</mo>
                           <mi>α</mi>
                           <mi>I</mi>
                           <mo>,</mo>
                           <msub>
                              <mi>E</mi>
                              <mrow>
                                 <mtext>max</mtext>
                              </mrow>
                           </msub>
                           <mo>]</mo>
                        </mrow>
                     </math>
                  </formula> the
interval B. <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>α</mi>
                     </math>
                  </formula> lies
between 1 and <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>E</mi>
                           <mrow>
                              <mtext>max</mtext>
                           </mrow>
                        </msub>
                        <mo>/</mo>
                        <mi>I</mi>
                     </math>
                  </formula>.
A collision with a loss in the interval A happens with the probability
<formula type="display" id="eq:phys332-5">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                
                        <mi>P</mi>
                        <mrow>
                           <mo>(</mo>
                           <mi>α</mi>
                           <mo>)</mo>
                        </mrow>
                        <mo>=</mo>
                        <msubsup>
                           <mo>∫</mo>
   
                           <mrow>
                              <mi>I</mi>
                           </mrow>
                           <mrow>
                              <mi>α</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>
                                       <mtext>max</mtext>
                                    </mrow>
                                 </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>
                                 <mrow>
                                    <mtext>max</mtext>
                                 </mrow>
                              </msub>
                              <mi>α</mi>
                           </mrow>
                        </mfrac>                
                     </math>
                  </formula>The
mean energy loss and the standard deviation for this type of collision are
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                
                        <mrow>
                           <mo>&lt;</mo>
                           <mi>Δ</mi>
                           <mi>E</mi>
                           <mrow>
                              <mo>(</mo>
                              <mi>α</mi>
                              <mo>)</mo>
                           </mrow>
                           <mo>&gt;;</mo>
                        </mrow>
                        <mo>=</mo>   
                        <mfrac>
                           <mrow>
                              <mn>1</mn>
                           </mrow>
                           <mrow>
                              <mi>P</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>α</mi>
                                 <mo>)</mo>
                              </mrow>
                           </mrow>
                        </mfrac>
                        <msubsup>
                           <mo>∫</mo>
          
                           <mrow>
                              <mi>I</mi>
                           </mrow>
                           <mrow>
                              <mi>α</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>α</mi>
                              <mo>ln</mo>
                              <mi>α</mi>
                           </mrow>
                           <mrow>
                              <mi>α</mi>
                              <mo>-</mo>
                              <mn>1</mn>
                           </mrow>
                        </mfrac>            
                     </math>
                  </formula>and <formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msup>
                           <mi>σ</mi>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msup>
                        <mrow>
                           <mo>(</mo>
                           <mi>α</mi>
                           <mo>)</mo>
                        </mrow>
                        <mo>=</mo>   
                        <mfrac>
                           <mrow>
                              <mn>1</mn>
                           </mrow>
                           <mrow>
                              <mi>P</mi>
                              <mrow>
                                 <mo>(</mo>
                                 <mi>α</mi>
                                 <mo>)</mo>
                              </mrow>
                           </mrow>
                        </mfrac>
                        <msubsup>
                           <mo>∫</mo>
          
                           <mrow>
                              <mi>I</mi>
                           </mrow>
                           <mrow>
                              <mi>α</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>α</mi>
                        <mfenced open="(" close=")">
                           <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>         
                     </math>
                  </formula>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:
                     <table rend="inline">
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-1">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <msub>
                                       <mi>n</mi>
                                       <mrow>
                                          <mi>A</mi>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>  
                                 <mo>=</mo>  
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mn>3</mn>
                                    </mrow>
                                 </msub>
                                 <mi>P</mi>
                                 <mrow>
                                    <mo>(</mo>
                                    <mi>α</mi>
                                    <mo>)</mo>
                                 </mrow>                         
                         
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                     
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-2">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <msubsup>
                                    <mi>σ</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>α</mi>
                                    <mo>)</mo>
                                 </mrow>
                                 <mrow>
                                    <mo>(</mo>
                                    <mn>1</mn>
                                    <mo>-</mo>
                                    <mi>P</mi>
                                    <mrow>
                                       <mo>(</mo>
                                       <mi>α</mi>
                                       <mo>)</mo>
                                    </mrow>
                                    <mo>)</mo>
                                 </mrow>                 
                              </math>
                           </formula>
                        </cell>
                     </row>
                  </table>
It is further assumed that the energy loss in these collisions has a Gaussian
distribution with
                       <table rend="inline">
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-3">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <mi>Δ</mi>
                                    <msub>
                                       <mi>E</mi>
                                       <mrow>
                                          <mi>A</mi>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>  
                                 <mo>=</mo>  
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mi>A</mi>
                                    </mrow>
                                 </msub>
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <mi>Δ</mi>
                                    <mi>E</mi>
                                    <mrow>
                                       <mo>(</mo>
                                       <mi>α</mi>
                                       <mo>)</mo>
                                    </mrow>
                                    <mo>&gt;;</mo>
                                 </mrow>                  
                            
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                       
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-4">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <msubsup>
                                    <mi>σ</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>σ</mi>
                                    <mrow>
                                       <mn>2</mn>
                                    </mrow>
                                 </msup>
                                 <mrow>
                                    <mo>(</mo>
                                    <mi>α</mi>
                                    <mo>)</mo>
                                 </mrow>                     
                              </math>
                           </formula>
                        </cell>
                     </row>
                  </table>
The energy loss of these collision can then be sampled from the Gaussian
distribution.
</p>
               <p>The collisions where the energy loss is in the interval B are sampled directly from
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                 
                        <mi>Δ</mi>
                        <msub>
                           <mi>E</mi>
                           <mrow>
                              <mi>B</mi>
                           </mrow>
                        </msub>
                        <mo>=</mo>
                        <msubsup>
                           <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>
                           <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>
                                       <mrow>
                                          <mtext>max</mtext>
                                       </mrow>
                                    </msub>
                                    <mo>+</mo>
                                    <mi>I</mi>
                                    <mo>-</mo>
                                    <mi>α</mi>
                                    <mi>I</mi>
                                 </mrow>
                                 <mrow>
                                    <msub>
                                       <mi>E</mi>
                                       <mrow>
                                          <mtext>max</mtext>
                                       </mrow>
                                    </msub>
                                    <mo>+</mo>
                                    <mi>I</mi>
                                 </mrow>
                              </mfrac>    
                           </mrow>
                        </mfrac>             
                     </math>
                  </formula>The
total energy loss is the sum of these two types of collisions:
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                          
                        <mi>Δ</mi>
                        <mi>E</mi>
                        <mo>=</mo>
                        <mi>Δ</mi>
                        <msub>
                           <mi>E</mi>
                           <mrow>
                              <mi>A</mi>
                           </mrow>
                        </msub>
                        <mo>+</mo>
                        <mi>Δ</mi>
                        <msub>
                           <mi>E</mi>
                           <mrow>
                              <mi>B</mi>
                           </mrow>
                        </msub>                     
                     </math>
                  </formula>
               </p>
               <p>The approximation of equations (<ptr target="eq:phys332-1" targOrder="U"/>), (<ptr target="eq:phys332-2" targOrder="U"/>), (<ptr target="eq:phys332-3" targOrder="U"/>) and (<ptr target="eq:phys332-4" targOrder="U"/>) can be used under the following
conditions:
                           <table rend="inline">
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-6">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <msub>
                                       <mi>n</mi>
                                       <mrow>
                                          <mi>A</mi>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>
                                 <mo>-</mo>
                                 <mi>c</mi>
                                 <msub>
                                    <mi>σ</mi>
                                    <mrow>
                                       <mi>A</mi>
                                    </mrow>
                                 </msub>  
                                 <mo>≥</mo>  
                                 <mn>0</mn>                    
                                   
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                           
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-7">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <msub>
                                       <mi>n</mi>
                                       <mrow>
                                          <mi>A</mi>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>
                                 <mo>+</mo>
                                 <mi>c</mi>
                                 <msub>
                                    <mi>σ</mi>
                                    <mrow>
                                       <mi>A</mi>
                                    </mrow>
                                 </msub>  
                                 <mo>≤</mo>  
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mn>3</mn>
                                    </mrow>
                                 </msub>                   
                                   
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                       
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula type="subeqn" id="eq:phys332-8">
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <mi>Δ</mi>
                                    <msub>
                                       <mi>E</mi>
                                       <mrow>
                                          <mi>A</mi>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>
                                 <mo>-</mo>
                                 <mi>c</mi>
                                 <msub>
                                    <mi>σ</mi>
                                    <mrow>
                                       <mi>E</mi>
                                       <mo>,</mo>
                                       <mi>A</mi>
                                    </mrow>
                                 </msub>  
                                 <mo>≥</mo>  
                                 <mn>0</mn>                    
                              </math>
                           </formula>
                        </cell>
                     </row>
                  </table>
where <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>c</mi>
                        <mo>≥</mo>
                        <mn>4</mn>
                     </math>
                  </formula>. From
the equations (<ptr target="eq:phys332-5" targOrder="U"/>), (<ptr target="eq:phys332-1" targOrder="U"/>) and (<ptr target="eq:phys332-3" targOrder="U"/>) and from the conditions (<ptr target="eq:phys332-6" targOrder="U"/>) and (<ptr target="eq:phys332-7" targOrder="U"/>) the following limits can be
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>α</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>
                                       <mtext>max</mtext>
                                    </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>
                                       <mtext>max</mtext>
                                    </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>≤</mo>
                        <mi>α</mi>
                        <mo>≤</mo>
                        <msub>
                           <mi>α</mi>
                           <mrow>
                              <mtext>max</mtext>
                           </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>
                                       <mtext>max</mtext>
                                    </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>
                                       <mtext>max</mtext>
                                    </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>         
                     </math>
                  </formula>This
conditions gives a lower limit to number of the ionisations
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                     </math>
                  </formula> for which the fast
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                        <mo>≥</mo>
                        <msup>
                           <mi>c</mi>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msup>                                                  
                     </math>
                  </formula>As
in the conditions (<ptr target="eq:phys332-6" targOrder="U"/>), (<ptr target="eq:phys332-7" targOrder="U"/>) and (<ptr target="eq:phys332-8" targOrder="U"/>) the value of
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>c</mi>
                     </math>
                  </formula> is as minimum
4, one gets <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                        <mo>≥</mo>
                        <mn>1</mn>
                        <mn>6</mn>
                     </math>
                  </formula>.
In order to speed the simulation, the maximum value is used for
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>α</mi>
                     </math>
                  </formula>.
</p>
               <p>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
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                     </math>
                  </formula>.
The maximum number of these collisions can be estimated as
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <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>≈</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>&gt;;</mo>
                           </mrow>
                           <mo>-</mo>
                           <msub>
                              <mi>σ</mi>
                              <mrow>
                                 <mi>A</mi>
                              </mrow>
                           </msub>
                           <mo>)</mo>
                        </mrow>              
                     </math>
                  </formula>From the previous
expressions for <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
                           <mo>&lt;</mo>
                           <msub>
                              <mi>n</mi>
                              <mrow>
                                 <mi>A</mi>
                              </mrow>
                           </msub>
                           <mo>&gt;;</mo>
                        </mrow>
                     </math>
                  </formula> and
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>σ</mi>
                           <mrow>
                              <mi>A</mi>
                           </mrow>
                        </msub>
                     </math>
                  </formula> one can derive the
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>n</mi>
                           <mrow>
                              <mi>B</mi>
                           </mrow>
                        </msub>
                        <mo>≤</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>                                       
                     </math>
                  </formula>The following
values are obtained with <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>c</mi>
                        <mo>=</mo>
                        <mn>4</mn>
                     </math>
                  </formula>:
</p>
               <p>
                  <table rend="inline">
                     <row role="data">
                        <cell role="data" cols="1" rows="1">
                           <formula>
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mn>3</mn>
                                    </mrow>
                                 </msub>
                              </math>
                           </formula>
                        </cell>
                        <cell role="data" cols="1" rows="1">
                           <formula>
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mi>B</mi>
                                       <mo>,</mo>
                                       <mi>m</mi>
                                       <mi>a</mi>
                                       <mi>x</mi>
                                    </mrow>
                                 </msub>
                              </math>
                           </formula>
                        </cell>
                        <cell role="data" cols="1" rows="1"/>
                        <cell role="data" cols="1" rows="1">
                           <formula>
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mn>3</mn>
                                    </mrow>
                                 </msub>
                              </math>
                           </formula>
                        </cell>
                        <cell role="data" cols="1" rows="1">
                           <formula>
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <msub>
                                    <mi>n</mi>
                                    <mrow>
                                       <mi>B</mi>
                                       <mo>,</mo>
                                       <mi>m</mi>
                                       <mi>a</mi>
                                       <mi>x</mi>
                                    </mrow>
                                 </msub>
                              </math>
                           </formula>
                        </cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">16                                                                                                     </cell>
                        <cell role="data" cols="1" rows="1">16                                                                                                     </cell>
                        <cell role="data" cols="1" rows="1"/>
                        <cell role="data" cols="1" rows="1">                                                               200</cell>
                        <cell role="data" cols="1" rows="1">                                                             29.63</cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">20                                                                                                     </cell>
                        <cell role="data" cols="1" rows="1">17.78                                                                                                 </cell>
                        <cell role="data" cols="1" rows="1"/>
                        <cell role="data" cols="1" rows="1">                                                               500</cell>
                        <cell role="data" cols="1" rows="1">                                                             31.01</cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">50                                                                                                     </cell>
                        <cell role="data" cols="1" rows="1">24.24                                                                                                 </cell>
                        <cell role="data" cols="1" rows="1"/>
                        <cell role="data" cols="1" rows="1">                                                              1000</cell>
                        <cell role="data" cols="1" rows="1">                                                             31.50</cell>
                     </row>
                     <row role="data">
                        <cell role="data" cols="1" rows="1">100                                                                                                   </cell>
                        <cell role="data" cols="1" rows="1">27.59                                                                                                 </cell>
                        <cell role="data" cols="1" rows="1"/>
                        <cell role="data" cols="1" rows="1">
                           <formula>
                              <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                                 <mi>∞</mi>
                              </math>
                           </formula>
                        </cell>
                        <cell role="data" cols="1" rows="1">                                                             32.00</cell>
                     </row>
                  </table>
               </p>
            </div>
            <div org="uniform" part="N" sample="complete">
               <head>
Special sampling for lower part of the spectrum</head>
               <p>If the step length is very small (<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mo>≤</mo>
                        <mn>5</mn>
                     </math>
                  </formula>
mm in gases, <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mo>≤</mo>
                     </math>
                  </formula>
2-3 <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>μ</mi>
                     </math>
                  </formula>m in solids)
the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>P</mi>
                        <mrow>
                           <mo>(</mo>
                           <mi>Δ</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>&gt;;</mo>
                                 </mrow>
                                 <mo>+</mo>
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <msub>
                                       <mi>n</mi>
                                       <mrow>
                                          <mn>2</mn>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>
                                 <mo>+</mo>
                                 <mrow>
                                    <mo>&lt;</mo>
                                    <msub>
                                       <mi>n</mi>
                                       <mrow>
                                          <mn>3</mn>
                                       </mrow>
                                    </msub>
                                    <mo>&gt;;</mo>
                                 </mrow>
                                 <mo>)</mo>
                              </mrow>
                           </mrow>
                        </msup>                                
                     </math>
                  </formula>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
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <msub>
                           <mi>E</mi>
                           <mrow>
                              <mn>0</mn>
                           </mrow>
                        </msub>
                        <mo>=</mo>
                        <mn>1</mn>
                        <mn>0</mn>
                     </math>
                  </formula> eV is chosen
to correspond to the outer electrons. The mean number of collisions can be calculated from
<formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mrow>
                           <mo>&lt;</mo>
                           <mi>n</mi>
                           <mo>&gt;;</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>
   	                   </mrow>
                           <mrow>
                              <mi>E</mi>
                           </mrow>
                        </mfrac>
                     </math>
                  </formula>The number
of collisions <formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>n</mi>
                     </math>
                  </formula>
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
<formula type="display">
                     <math xmlns="http://www.w3.org/1998/Math/MathML" overflow="scroll">
                        <mi>Δ</mi>
                        <mi>E</mi>
                        <mo>=</mo>
                        <msubsup>
                           <mo>∑</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>
                                          <mtext>max</mtext>
                                       </mrow>
                                    </msub>
                                 </mrow>
                                 <mrow>
                                    <msub>
                                       <mi>E</mi>
                                       <mrow>
                                          <mtext>max</mtext>
                                       </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>
                     </math>
                  </formula> 
               </p> 
            </div> 
         </div> 
         <div type="star" org="uniform" part="N" sample="complete"> 
            <head> References</head> 
            <list type="bibliography"> 
               <item id="bib-LAND"> 
                  <p>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 Haar, Editor, <emph>L.D.Landau, Collected
papers</emph>, page 417.  Pergamon Press, Oxford, 1965.
</p>
               </item> 
               <item id="bib-SCH1"> 
                  <p>B.Schorr.  Programs for
the Landau and the Vavilov distributions and the corresponding random
numbers.  <emph>Comp.  Phys.  Comm.</emph>, 7:216, 1974.
</p>
               </item> 
               <item id="bib-SELT"> 
                  <p>S.M.Seltzer and
M.J.Berger.  Energy loss straggling of protons and mesons.  In
<emph>Studies in Penetration of Charged Particles in Matter</emph>,
Nuclear Science Series 39, Nat.  Academy of Sciences, Washington DC,
1964.  </p>
               </item> 
               <item id="bib-TALM"> 
                  <p>R.Talman.  On the
statistics of particle identification using ionization.  <emph>Nucl.
Inst.  Meth.</emph>, 159:189, 1979.  </p>
               </item> 
               <item id="bib-VAVI"> 
                  <p>P.V.Vavilov.  Ionisation losses of high energy
heavy particles.  <emph>Soviet Physics JETP</emph>, 5:749,
1957.</p>
               </item>
            </list> 
         </div>
 
      </body> 
   </text> 
</TEI.2>
