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diff --git a/Master/texmf-dist/doc/latex/classicthesis/Chapters/Chapter03.tex b/Master/texmf-dist/doc/latex/classicthesis/Chapters/Chapter03.tex new file mode 100644 index 00000000000..e960dafbc64 --- /dev/null +++ b/Master/texmf-dist/doc/latex/classicthesis/Chapters/Chapter03.tex @@ -0,0 +1,110 @@ +%************************************************ +\myChapter{Math Test Chapter}\label{ch:mathtest} +%************************************************ +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.} +\[ +\kappa =\frac{\xi}{E_{\mathrm{max}}} +\] +$E_{\mathrm{max}}$ 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. Indeed, we have +\[ + t[u_1,\dots,u_n] = \sum_{k=1}^n{{n-1} \choose {k-1}} (1-t)^{n-k} + t^{k-1}u_k, +\] +a Bernstein polynomial of order $n-1$. + +%***************************************** +%***************************************** +%***************************************** +%***************************************** +%***************************************** |