Chapter03.lyx

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\begin_body

\begin_layout Chapter
Rad50
\end_layout

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\begin_inset CommandInset label
LatexCommand label
name "ch:Rad50"

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Ei choro aeterno antiopam mea, labitur bonorum pri no.
 His no decore nemore graecis.
 In eos meis nominavi, liber soluta vim cu.
 Sea commune suavitate interpretaris eu, vix eu libris efficiantur.
\end_layout

\begin_layout Section
Some Formulas
\end_layout

\begin_layout Standard
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
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Examples taken from Walter Schmidt's great gallery: 
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http://home.vrweb.de/~was/mathfonts.html
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.
 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 
\begin_inset Formula $\kappa$
\end_inset

, which is proportional to the ratio of mean energy loss to the maximum
 allowed energy transfer in a single collision with an atomic electron:
 
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You might get unexpected results using math in chapter or section heads.
 Consider the 
\family typewriter
pdfspacing
\family default
 option.
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\end_inset

 
\begin_inset Formula 
\[
\kappa=\frac{\xi}{E_{\textrm{max}}}
\]

\end_inset

 
\begin_inset Formula $E_{\textrm{max}}$
\end_inset

 is the maximum transferable energy in a single collision with an atomic
 electron.
 
\begin_inset Formula 
\[
E_{\textrm{max}}=\frac{2m_{\textrm{e}}\beta^{2}\gamma^{2}}{1+2\gamma m_{\textrm{e}}/m_{\textrm{x}}+\left(m_{\textrm{e}}/m_{\textrm{x}}\right)^{2}}\ ,
\]

\end_inset

 where 
\begin_inset Formula $\gamma=E/m_{\textrm{x}}$
\end_inset

, 
\begin_inset Formula $E$
\end_inset

 is energy and 
\begin_inset Formula $m_{\textrm{x}}$
\end_inset

 the mass of the incident particle, 
\begin_inset Formula $\beta^{2}=1-1/\gamma^{2}$
\end_inset

 and 
\begin_inset Formula $m_{\textrm{e}}$
\end_inset

 is the electron mass.
 
\begin_inset Formula $\xi$
\end_inset

 comes from the Rutherford scattering cross section and is defined as: 
\begin_inset Formula 
\begin{eqnarray*}
\xi=\frac{2\pi z^{2}e^{4}N_{\textrm{Av}}Z\rho\delta x}{m_{\textrm{e}}\beta^{2}c^{2}A}=153.4\frac{z^{2}}{\beta^{2}}\frac{Z}{A}\rho\delta x\quad\textrm{keV},
\end{eqnarray*}

\end_inset

 where
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\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="2">
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<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $z$
\end_inset

 
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</cell>
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charge of the incident particle 
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</row>
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\begin_inset Formula $N_{\textrm{Av}}$
\end_inset

 
\end_layout

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</cell>
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\begin_layout Plain Layout
Avogadro's number 
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</row>
<row>
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\begin_inset Formula $Z$
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atomic number of the material 
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\begin_inset Formula $A$
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\end_layout

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</cell>
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atomic weight of the material 
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\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $\rho$
\end_inset

 
\end_layout

\end_inset
</cell>
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density 
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</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $\delta x$
\end_inset

 
\end_layout

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</cell>
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thickness of the material 
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\begin_layout Standard
\begin_inset Formula $\kappa$
\end_inset

 measures the contribution of the collisions with energy transfer close
 to 
\begin_inset Formula $E_{\textrm{max}}$
\end_inset

.
 For a given absorber, 
\begin_inset Formula $\kappa$
\end_inset

 tends towards large values if 
\begin_inset Formula $\delta x$
\end_inset

 is large and/or if 
\begin_inset Formula $\beta$
\end_inset

 is small.
 Likewise, 
\begin_inset Formula $\kappa$
\end_inset

 tends towards zero if 
\begin_inset Formula $\delta x$
\end_inset

 is small and/or if 
\begin_inset Formula $\beta$
\end_inset

 approaches 
\begin_inset Formula $1$
\end_inset

.
\end_layout

\begin_layout Standard
The value of 
\begin_inset Formula $\kappa$
\end_inset

 distinguishes two regimes which occur in the description of ionisation
 fluctuations:
\end_layout

\begin_layout Enumerate
A large number of collisions involving the loss of all or most of the incident
 particle energy during the traversal of an absorber.
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
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 
\begin_inset Formula $\kappa>10$
\end_inset

 (i.
\begin_inset space \thinspace{}
\end_inset

e., when the mean energy loss in the absorber is greater than the maximum
 energy transfer in a single collision).
\end_layout

\end_deeper
\begin_layout Enumerate
Particles traversing thin counters and incident electrons under any conditions.
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\end_inset


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\begin_deeper
\begin_layout Standard
The relevant inequalities and distributions are 
\begin_inset Formula $0.01<\kappa<10$
\end_inset

, Vavilov distribution, and 
\begin_inset Formula $\kappa<0.01$
\end_inset

, Landau distribution.
 
\end_layout

\end_deeper
\begin_layout Section
Various Mathematical Examples
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $n>2$
\end_inset

, the identity 
\begin_inset Formula 
\[
t[u_{1},\dots,u_{n}]=t\bigl[t[u_{1},\dots,u_{n_{1}}],t[u_{2},\dots,u_{n}]\bigr]
\]

\end_inset

 defines 
\begin_inset Formula $t[u_{1},\dots,u_{n}]$
\end_inset

 recursively, and it can be shown that the alternative definition 
\begin_inset Formula 
\[
t[u_{1},\dots,u_{n}]=t\bigl[t[u_{1},u_{2}],\dots,t[u_{n-1},u_{n}]\bigr]
\]

\end_inset

 gives the same result.
\end_layout

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