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%************************************************
\chapter{Math Test Chapter}\label{ch:mathtest} % $\mathbb{ZNR}$
%************************************************
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.
\section{Some Formulas}
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\footnote{Examples taken from Walter
Schmidt's great gallery: \\
\url{http://home.vrweb.de/~was/mathfonts.html}}. 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 $\kappa$, which is
proportional to the ratio of mean energy loss to the maximum allowed
energy transfer in a single collision with an atomic electron:
\graffito{You might get unexpected results using math in chapter or
section heads. Consider the \texttt{pdfspacing} option.}
\[
\kappa =\frac{\xi}{E_{\mathrm{max}}} %\mathbb{ZNR}
\]
$E_{\mathrm{max}}$ is the maximum transferable energy in a single
collision with
an atomic electron.
\[
E_{\mathrm{max}} =\frac{2 m_{\mathrm{e}} \beta^2\gamma^2 }{1 +
2\gamma m_{\mathrm{e}}/m_{\mathrm{x}} + \left ( m_{\mathrm{e}}
/m_{\mathrm{x}}\right)^2}\ ,
\]
where $\gamma = E/m_{\mathrm{x}}$, $E$ is energy and
$m_{\mathrm{x}}$ the mass of the incident particle,
$\beta^2 = 1 - 1/\gamma^2$ and $m_{\mathrm{e}}$ is the electron mass.
$\xi$ comes from the Rutherford scattering cross section
and is defined as:
\begin{eqnarray*} \xi = \frac{2\pi z^2 e^4 N_{\mathrm{Av}} Z \rho
\delta x}{m_{\mathrm{e}} \beta^2 c^2 A} = 153.4 \frac{z^2}{\beta^2}
\frac{Z}{A}
\rho \delta x \quad\mathrm{keV},
\end{eqnarray*}
where
\begin{tabular}{ll}
$z$ & charge of the incident particle \\
$N_{\mathrm{Av}}$ & Avogadro's number \\
$Z$ & atomic number of the material \\
$A$ & atomic weight of the material \\
$\rho$ & density \\
$ \delta x$ & thickness of the material \\
\end{tabular}
$\kappa$ measures the contribution of the collisions with energy
transfer close to $E_{\mathrm{max}}$. For a given absorber, $\kappa$
tends
towards large values if $\delta x$ is large and/or if $\beta$ is
small. Likewise, $\kappa$ tends towards zero if $\delta x $ is small
and/or if $\beta$ approaches $1$.
The value of $\kappa$ distinguishes two regimes which occur in the
description of ionisation fluctuations:
\begin{enumerate}
\item A large number of collisions involving the loss of all or most
of the incident particle energy during the traversal of an absorber.
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 $\kappa > 10 $ (\ie, when the mean energy loss in the
absorber is greater than the maximum energy transfer in a single
collision).
\item Particles traversing thin counters and incident electrons under
any conditions.
The relevant inequalities and distributions are $ 0.01 < \kappa < 10
$,
Vavilov distribution, and $\kappa < 0.01 $, Landau distribution.
\end{enumerate}
\section{Various Mathematical Examples}
If $n > 2$, the identity
\[
t[u_1,\dots,u_n] = t\bigl[t[u_1,\dots,u_{n_1}], t[u_2,\dots,u_n]
\bigr]
\]
defines $t[u_1,\dots,u_n]$ recursively, and it can be shown that the
alternative definition
\[
t[u_1,\dots,u_n] = t\bigl[t[u_1,u_2],\dots,t[u_{n-1},u_n]\bigr]
\]
gives the same result.
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