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author | Karl Berry <karl@freefriends.org> | 2012-08-26 23:07:28 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2012-08-26 23:07:28 +0000 |
commit | d3535e838cefea309221187bb9e62698c0df8c51 (patch) | |
tree | dbee8197e1bfecefe547d53e843f0acff729a3fe /Master/texmf-dist/doc/latex/latex-web-companion/ch2 | |
parent | 33353453d22fc5877a8ced58c673c7008ce8c00a (diff) |
latex-web-companion non-update
git-svn-id: svn://tug.org/texlive/trunk@27531 c570f23f-e606-0410-a88d-b1316a301751
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diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.aux b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.aux new file mode 100644 index 00000000000..7df7d7030da --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.aux @@ -0,0 +1,55 @@ +\relax +\ifx\hyper@anchor\@undefined +\global \let \oldcontentsline\contentsline +\gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global \let \oldnewlabel\newlabel +\gdef \newlabel#1#2{\newlabelxx{#1}#2} +\gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\let \contentsline\oldcontentsline +\let \newlabel\oldnewlabel} +\else +\global \let \hyper@last\relax +\fi + +\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}} +\citation{bib-LAND} +\@writefile{toc}{\contentsline {section}{\numberline {2}Landau theory}{2}{section.2}} +\newlabel{sec:phys332-1}{{2}{2}{Landau theory\relax }{section.2}{}} +\@writefile{brf}{\backcite{bib-LAND}{{2}{2}{section.2}}} +\citation{bib-TALM} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The variable $\xi /I_0$ can be used to measure the validity range of the Landau theory. It depends on the type and energy of the particle, $Z$, $A$ and the ionisation potential of the material and the layer thickness. }}{3}{figure.1}} +\newlabel{fg:phys332-1}{{1}{3}{Introduction\relax }{figure.1}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Restrictions}{3}{subsection.2.1}} +\@writefile{brf}{\backcite{bib-TALM}{{3}{2}{Item.4}}} +\citation{bib-VAVI} +\citation{bib-SCH1} +\citation{bib-SELT} +\@writefile{toc}{\contentsline {section}{\numberline {3}Vavilov theory}{5}{section.3}} +\newlabel{vavref}{{3}{5}{Vavilov theory\relax }{section.3}{}} +\@writefile{brf}{\backcite{bib-VAVI}{{5}{3}{section.3}}} +\@writefile{brf}{\backcite{bib-SCH1}{{5}{3}{section.3}}} +\@writefile{toc}{\contentsline {section}{\numberline {4}Gaussian Theory}{5}{section.4}} +\@writefile{brf}{\backcite{bib-SELT}{{5}{4}{section.4}}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Energy loss distribution for a 3 GeV electron in Argon as given by standard \texttt {GEANT}. The width of the layers is given in centimeters.}}{6}{figure.2}} +\newlabel{fg:phys332-2}{{2}{6}{Urb\'an model\relax }{figure.2}{}} +\@writefile{toc}{\contentsline {section}{\numberline {5}Urb\'an model}{6}{section.5}} +\newlabel{urban}{{5}{6}{Urb\'an model\relax }{section.5}{}} +\newlabel{eq:sigex}{{2}{6}{Urb\'an model\relax }{equation.2}{}} +\newlabel{eq:sigion}{{3}{6}{Urb\'an model\relax }{equation.3}{}} +\newlabel{eq:fisum}{{4}{7}{Urb\'an model\relax }{equation.4}{}} +\newlabel{eq:flnsum}{{5}{7}{Urb\'an model\relax }{equation.4}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Fast simulation for $n_3 \geq 16$}{8}{subsection.5.1}} +\newlabel{eq:phys332-5}{{13}{8}{Fast simulation for $n_3 \geq 16$\relax }{equation.13}{}} +\newlabel{eq:phys332-1}{{16}{8}{Fast simulation for $n_3 \geq 16$\relax }{equation.16}{}} +\newlabel{eq:phys332-2}{{17}{8}{Fast simulation for $n_3 \geq 16$\relax }{equation.16}{}} +\newlabel{eq:phys332-3}{{18}{8}{Fast simulation for $n_3 \geq 16$\relax }{equation.18}{}} +\newlabel{eq:phys332-4}{{19}{8}{Fast simulation for $n_3 \geq 16$\relax }{equation.18}{}} +\newlabel{eq:phys332-6}{{22}{9}{Fast simulation for $n_3 \geq 16$\relax }{equation.22}{}} +\newlabel{eq:phys332-7}{{23}{9}{Fast simulation for $n_3 \geq 16$\relax }{equation.22}{}} +\newlabel{eq:phys332-8}{{24}{9}{Fast simulation for $n_3 \geq 16$\relax }{equation.22}{}} +\bibcite{bib-LAND}{1} +\bibcite{bib-SCH1}{2} +\bibcite{bib-SELT}{3} +\bibcite{bib-TALM}{4} +\bibcite{bib-VAVI}{5} +\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Special sampling for lower part of the spectrum}{10}{subsection.5.2}} diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.brf b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.brf new file mode 100644 index 00000000000..dc4d5c318ee --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.brf @@ -0,0 +1,5 @@ +\backcite {bib-LAND}{{2}{2}{section.2}} +\backcite 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a/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.pdf b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.pdf Binary files differnew file mode 100644 index 00000000000..19c6a108f19 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.pdf diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.tex b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.tex new file mode 100644 index 00000000000..21cf0c8b4cd --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/latexexa.tex @@ -0,0 +1,569 @@ +\documentclass{article} +\usepackage{graphicx} +\usepackage{url} +\title{Simulation of Energy Loss Straggling} +\author{Maria Physicist} +\usepackage{ae} +\usepackage[T1]{fontenc} +\usepackage[latin1]{inputenc} +\usepackage{fancyhdr,pifont} +\pagestyle{fancy} +\cfoot{\NavigationBar} +\def\NavigationBar{{\Large + \Acrobatmenu{FontsInfo}{\reflectbox{\ding{227}}} + \Acrobatmenu{NextPage}{\ding{227}} + \Acrobatmenu{FirstPage}{\reflectbox{\ding{224}}} + \Acrobatmenu{LastPage}{\ding{224}} + \Acrobatmenu{GoBack}{\reflectbox{\ding{249}}} + \Acrobatmenu{Quit}{\ding{54}}% +}} +\usepackage[colorlinks, backref, + pdfpagemode=FullScreen, + pdfauthor={Maria Physicist}, + pdftitle={Simulation of Energy Loss Straggling}, + pdfcreator={pdfTeX}, + pdfsubject={Energy Loss}, + pdfkeywords={physics,energy}, + pdfpagetransition={Blinds /Dm /V} +]{hyperref} +\newcommand{\Emax}{\ensuremath{E_{\mathrm{max}}}} +\newcommand{\GEANT}{\texttt{GEANT}} +\begin{document} +\maketitle + +\section{Introduction} + +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 $\kappa$, which is +proportional to the ratio of mean energy loss to the maximum allowed +energy transfer in a single collision with an atomic electron +\[ +\kappa =\frac{\xi}{\Emax} +\] +\Emax{} +is the maximum transferable energy in a single collision with +an atomic electron. +\[ +\Emax =\frac{2 m_e \beta^2\gamma^2 } +{1 + 2\gamma m_e/m_x + \left ( m_e/m_x\right)^2}, +\] +where $\gamma = E/m_x$, $E$ is energy and +$m_x$ the mass of the incident particle, +$\beta^2 = 1 - 1/\gamma^2$ and $m_e$ 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_{Av} Z \rho \delta x} + {m_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 \\ +$N_{Av}$ & 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 \Emax. 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 $ (i.e. 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 $, Landau distribution. +\end{enumerate} + +An additional regime is defined by the contribution of the collisions +with low energy transfer which can be estimated with the relation +$\xi/I_0$, where $I_0$ 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 $\xi/I_0 \gg 1$. In \GEANT{} +(see URL \url{http://wwwinfo.cern.ch/asdoc/geant/geantall.html}), the +limit of Landau theory has been set at $\xi/I_0 = 50$. Below this +limit special models taking into account the atomic structure of the +material are used. This is important in thin layers and gaseous +materials. Figure \ref{fg:phys332-1} shows the behaviour of $\xi/I_0$ +as a function of the layer thickness for an electron of 100 keV and 1 +GeV of kinetic energy in Argon, Silicon and Uranium. + +\begin{figure} + \centering + \includegraphics[width=.6\linewidth]{phys332-1} + \caption{The variable $\xi/I_0$ can be used to measure the + validity range of the Landau theory. It depends + on the type and energy of the particle, $Z$, $A$ + and the ionisation potential of the material and + the layer thickness. + } + \label{fg:phys332-1} +\end{figure} + +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. + +\section{Landau theory} +\label{sec:phys332-1} + +For a particle of mass $m_x$ traversing a thickness of material +$\delta x $, the Landau probability distribution may be written in +terms of the universal Landau function $\phi(\lambda)$ +as\cite{bib-LAND}: +\begin{eqnarray*} +f( \epsilon , \delta x ) & = &\frac{1}{\xi} \phi ( \lambda ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi(\lambda )& = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty} +\exp \left ( u \ln u + \lambda u \right ) du \hspace{2cm} c \geq 0 \\ +\lambda & = & \frac{\epsilon -\bar{\epsilon} }{\xi} + - \gamma' - \beta^2 - \ln \frac{\xi}{\Emax} \\ +\gamma' & = & 0.422784\dots = 1 - \gamma \\ +\gamma & = & 0.577215\dots \mbox{(Euler's constant)} \\ +\bar{\epsilon} & = & \mbox{average energy loss} \\ +\epsilon & = & \mbox{actual energy loss} +\end{eqnarray*} + +\subsection{Restrictions} + +The Landau formalism makes two restrictive assumptions : +\begin{enumerate} +\item The typical energy loss is small compared to the maximum energy + loss in a single collision. This restriction is removed in the + Vavilov theory (see section \ref{vavref}). +\item The typical energy loss in the absorber should be large compared + to the binding energy of the most tightly bound electron. For + gaseous detectors, typical energy losses are a few keV which is + comparable to the binding energies of the inner electrons. In such + cases a more sophisticated approach which accounts for atomic energy + levels\cite{bib-TALM} is necessary to accurately simulate data + distributions. In \GEANT, a parameterised model by L. Urb\'{a}n is + used (see section \ref{urban}). +\end{enumerate} + +In addition, the average value of the Landau distribution is infinite. +Summing the Landau fluctuation obtained to the average energy from the +$dE/dx$ 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. + +A solution to this has been to introduce a limit on the value of the +variable sampled by the Landau distribution in order to keep the +average fluctuation to 0. The value obtained from the \texttt{GLANDO} +routine is: +\[ +\delta dE/dx = \epsilon - \bar{\epsilon} = \xi ( \lambda - \gamma' ++\beta^2 +\ln \frac{\xi}{\Emax}) +\] +In order for this to have average 0, we must impose that: +\[ +\bar{\lambda} = -\gamma' - \beta^2 -\ln \frac{\xi}{\Emax} +\] + +This is realised introducing a $\lambda_{\mathrm{max}}(\bar{\lambda})$ +such that if only values of $\lambda \leq \lambda_{\mathrm{max}}$ are +accepted, the average value of the distribution is $\bar{\lambda}$. + +A parametric fit to the universal Landau distribution has been +performed, with following result: +\[ +\lambda_{\mathrm{max}} = 0.60715 + + 1.1934\bar{\lambda}+(0.67794+0.052382\bar{\lambda}) + \exp(0.94753+0.74442\bar{\lambda}) +\] +only values smaller than $\lambda_{\mathrm{max}}$ are accepted, otherwise the +distribution is resampled. + + + +\newpage +\section{Vavilov theory} +\label{vavref} + +Vavilov\cite{bib-VAVI} derived a more accurate straggling distribution +by introducing the kinematic limit on the maximum transferable energy +in a single collision, rather than using $ \Emax = \infty $. +Now we can write\cite{bib-SCH1}: +\begin{eqnarray*} +f \left ( \epsilon, \delta s \right ) & = & \frac{1}{\xi} \phi_{v} +\left ( \lambda_{v}, \kappa, \beta^{2} \right ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi_{v} \left ( \lambda_{v}, \kappa, \beta^{2} \right ) & = & +\frac{1}{2 \pi i} \int^{c+i\infty}_{c-i\infty}\phi \left( s \right ) +e^{\lambda s} ds \hspace{2cm} c \geq 0 \\ +\phi \left ( s \right ) & = & +\exp \left [ \kappa ( 1 + \beta^{2}\gamma ) \right ] +~ \exp \left [ \psi \left ( s \right ) \right ], \\ +\psi \left ( s \right ) & = & s \ln \kappa + ( s + \beta^{2} \kappa ) +\left [ \ln (s/\kappa) + E_{1} (s/\kappa) \right ] - \kappa e^{-s/\kappa}, +\end{eqnarray*} +and +\begin{eqnarray*} +E_{1}(z) & = & \int^{\infty}_{z} t^{-1} e^{-t} dt +\mbox{\hspace{1cm} (the exponential integral)} \\ +\lambda_v & = & \kappa \left [ \frac{\epsilon - \bar{\epsilon}}{\xi} +- \gamma' - \beta^2 \right] +\end{eqnarray*} + +The Vavilov parameters are simply related to the Landau parameter by +$\lambda_L = \lambda_v/\kappa - \ln\kappa $. It can be shown that as +$\kappa \rightarrow 0$, the distribution of the variable $\lambda_L$ +approaches that of Landau. For $\kappa \leq 0.01$ the two +distributions are already practically identical. Contrary to what many +textbooks report, the Vavilov distribution \emph{does not} approximate +the Landau distribution for small $\kappa$, but rather the +distribution of $\lambda_L$ defined above tends to the distribution of +the true $\lambda$ from the Landau density function. Thus the routine +\texttt{GVAVIV} samples the variable $\lambda_L$ rather than +$\lambda_v$. For $\kappa \geq 10$ the Vavilov distribution tends to a +Gaussian distribution (see next section). + +\section{Gaussian Theory} + +Various conflicting forms have been proposed for Gaussian straggling +functions, but most of these appear to have little theoretical or +experimental basis. However, it has been shown\cite{bib-SELT} that +for $\kappa \geq 10 $ the Vavilov distribution can be replaced by a +Gaussian of the form : +\begin{eqnarray*} +f( \epsilon , \delta s) \approx \frac{1} +{\xi \sqrt{\frac{2 \pi}{\kappa} \left( 1 - \beta^2/2 \right)}} + \exp \left [ \frac{( \epsilon - \bar{\epsilon} )^2}{2} \frac{\kappa} + {\xi^2 (1- \beta^2/2)}\right ] +\end{eqnarray*} +thus implying +\begin{eqnarray*} +\mathrm{mean} & = & \bar{\epsilon} \\ +\sigma^2 & = & \frac{\xi^2}{\kappa} (1-\beta^2/2) = \xi + \Emax (1-\beta^2/2) +\end{eqnarray*} + +\section{Urb\'an model} +\label{urban} + +The method for computing restricted energy losses with $\delta$-ray +production above given threshold energy in \GEANT{} is a Monte +Carlo method that can be used for thin layers. It is fast and it can +be used for any thickness of a medium. Approaching the limit of the +validity of Landau's theory, the loss distribution approaches smoothly +the Landau form as shown in Figure \ref{fg:phys332-2}. +\begin{figure} + \centering + \includegraphics[width=.6\linewidth]{phys332-2} + \caption{Energy loss distribution for a 3 GeV electron in + Argon as given by standard \GEANT. The width of the layers is + given in centimeters.} + \label{fg:phys332-2} +\end{figure} + +It is assumed that the atoms have only two energy levels with binding +energy $E_1$ and $E_2$. The particle--atom interaction will then be +an excitation with energy loss $E_1$ or $E_2$, or an ionisation with +an energy loss distributed according to a function $g(E) \sim 1/E^2$: +\begin{equation} +g(E) = \frac{(\Emax + I)I}{\Emax} \frac{1}{E^2} +\end{equation} + +The macroscopic cross-section for excitations ($i=1,2$) is +\begin{equation} +\label{eq:sigex} +\Sigma_i = C \frac{f_i}{E_i} \frac{\ln (2 m \beta^2 \gamma^2/E_i) - \beta^2} + {\ln (2 m \beta^2 \gamma^2/ I) - \beta^2}(1-r) +\end{equation} +and the macroscopic cross-section for ionisation is +\begin{equation} +\label{eq:sigion} +\Sigma_3 = C \frac{\Emax}{I(\Emax+I) \ln(\frac{\Emax+I}{I})} + ~ r +\end{equation} +\Emax{} is the \GEANT{} cut for $\delta$-production, or the maximum +energy transfer minus mean ionisation energy, if it is smaller than +this cut-off value. The following notation is used: + +\begin{tabular}{ll} +$r, C$ & parameters of the model \\ +$E_i$ & atomic energy levels \\ +$I$ & mean ionisation energy \\ +${f_i}$ & oscillator strengths +\end{tabular} + +The model has the parameters $f_i$, $E_i$, $C$ and $r ~(0\leq r\leq +1)$. The oscillator strengths $f_i$ and the atomic level energies +$E_i$ should satisfy the constraints +\begin{eqnarray} +f_1 + f_2 & = & 1 \label{eq:fisum}\\ +f_1 \ln E_1 + f_2 \ln E_2 & = & \ln I \label{eq:flnsum} +\end{eqnarray} +The parameter $C$ can be defined with the help of the mean energy loss +$dE/dx$ in the following way: The numbers of collisions ($n_i$, i = +1,2 for the excitation and 3 for the ionisation) follow the Poisson +distribution with a mean number $ \langle n_i \rangle $. In a step +$\Delta x$ the mean number of collisions is +\begin{equation} +\langle n_i \rangle = \Sigma_i \Delta x +\end{equation} +The mean energy loss $dE/dx$ in a step is the sum of the excitation +and ionisation contributions +\begin{equation} +\frac{dE}{dx} \Delta x = \left[ \Sigma_1 E_1 + \Sigma_2 E_2 + + \Sigma_3 \int_{I}^{\Emax+I} E~g(E)~dE \right] + \Delta x +\end{equation} +From this, using the equations (\ref{eq:sigex}), (\ref{eq:sigion}), +(\ref{eq:fisum}) and (\ref{eq:flnsum}), one can define the parameter +$C$ +\begin{equation} +C = \frac{dE}{dx} +\end{equation} + +The following values have been chosen in \GEANT{} for the other +parameters: +\[ +\begin{array}{lcl} +f_2 = \left\{ \begin{array}{ll} + 0 & \mathrm{if}\, Z \leq 2 \\ + 2/Z & \mathrm{if}\, Z > 2 \\ + \end{array} \right. & \Rightarrow & f_1 = 1 - f_2 \\ +E_2 = 10 Z^2 \mathrm{eV} & \Rightarrow & E_1 = \left(\frac{I}{E_{2}^{f_2}} + \right)^{\frac{1}{f_1}} \\ +r = 0.4 & & \\ +\end{array} +\] +With these values the atomic level $E_2$ corresponds approximately +the K-shell energy of the atoms and $Zf_2$ the number of K-shell +electrons. $r$ is the only variable which can be tuned freely. It +determines the relative contribution of ionisation and +excitation to the energy loss. + +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 +\begin{equation} +\Delta E_e = n_1 E_1 + n_2 E_2 +\end{equation} +where $n_1$ and $n_2$ are sampled from Poisson distribution as +discussed above. The loss due to the ionisation can be generated from +the distribution $g(E)$ by the inverse transformation method: +\begin{eqnarray} +u = F(E) & = & \int_{I}^E g(x) dx \nonumber \\ +E = F^{-1}(u) & = & \frac{I}{1 - u \frac {\Emax}{\Emax+I}} \\ +\end{eqnarray} +where $u$ is a uniform random number between $F(I)=0$ and +$F(\Emax+I)=1$. The contribution from the ionisations will be +\begin{equation} +\Delta E_i = \sum_{j=1}^{n_3} \frac{I} + {1 - u_j \frac {\Emax}{\Emax + I}} +\end{equation} +where $n_3$ is the number of ionisation (sampled from Poisson +distribution). The energy loss in a step will then be $\Delta E = +\Delta E_e + \Delta E_i$. + +\subsection{Fast simulation for $n_3 \geq 16$} + +If the number of ionisation $n_3$ 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 $[I, \alpha I]$ the interval A, and the latter +$[\alpha I,\Emax]$ the interval B. $\alpha$ lies between 1 and +$\Emax/I$. A collision with a loss in the interval A happens with +the probability +\begin{equation} +\label{eq:phys332-5} +P(\alpha) = \int_I^{\alpha I} g(\!E\!) \, dE = + \frac {( \Emax + I) (\alpha - 1)}{\Emax \alpha} +\end{equation} +The mean energy loss and the standard deviation for this type +of collision are +\begin{equation} +\langle \Delta E(\alpha) \rangle = \frac{1}{P(\alpha)} + \int_I^{\alpha I} E \, g(\!E\!) \, dE = + \frac{I \alpha \ln \alpha}{\alpha - 1} +\end{equation} +and +\begin{equation} +\sigma^2(\alpha) = \frac{1}{P(\alpha)} + \int_I^{\alpha I} E^2 \, g(\!E\!) \, dE = + I^2 \alpha \left( 1 - \frac{\alpha \ln \! ^2 \alpha}{(\alpha - 1)^2} \right) +\end{equation} +If the collision number is high , we assume that the number of the +type A collisions can be calculated from a Gaussian distribution +with the following mean value and standard deviation: +\begin{eqnarray} +\label{eq:phys332-1} +\langle n_A \rangle & = & n_3 P(\alpha) \\ +\label{eq:phys332-2} +\sigma_A^2 & = & n_3 P(\alpha) ( 1 - P(\alpha)) +\end{eqnarray} +It is further assumed that the energy loss in these collisions +has a Gaussian distribution with +\begin{eqnarray} +\label{eq:phys332-3} +\langle \Delta E_A \rangle & = & n_A \langle \Delta E(\alpha) \rangle \\ +\label{eq:phys332-4} +\sigma_{E,A}^2 & = & n_A \sigma^2(\alpha) +\end{eqnarray} +The energy loss of these collision can then be sampled from the +Gaussian distribution. + +The collisions where the energy loss is in the interval B are sampled +directly from +\begin{equation} +\Delta E_B = \sum_{i=1}^{n_3 - n_A} \frac{\alpha I} + {1 - u_i \frac{\Emax + I - \alpha I}{\Emax + I}} +\end{equation} +The total energy loss is the sum of these two types of collisions: +\begin{equation} +\Delta E = \Delta E_A + \Delta E_B +\end{equation} + +The approximation of equations ((\ref{eq:phys332-1}), +(\ref{eq:phys332-2}), (\ref{eq:phys332-3}) and (\ref{eq:phys332-4}) +can be used under the following conditions: +\begin{eqnarray} +\label{eq:phys332-6} +\langle n_A \rangle - c \, \sigma_A & \geq & 0 \\ +\label{eq:phys332-7} +\langle n_A \rangle + c \, \sigma_A & \leq & n_3 \\ +\label{eq:phys332-8} +\langle \Delta E_A \rangle - c \, \sigma_{E,A} & \geq & 0 +\end{eqnarray} +where $c \geq 4$. From the equations (\ref{eq:phys332-5}), +(\ref{eq:phys332-1}) and (\ref{eq:phys332-3}) and from the conditions +(\ref{eq:phys332-6}) and (\ref{eq:phys332-7}) the following limits can +be derived: +\begin{equation} +\alpha_{\mathrm{min}} = \frac{(n_3 + c^2)(\Emax+I)} + {n_3 (\Emax + I) + c^2 I} \; \leq \alpha \; \leq +\alpha_{\mathrm{max}} = \frac{(n_3 + c^2)(\Emax+I)} + {c^2 (\Emax + I) + n_3 I} +\end{equation} +This conditions gives a lower limit to number of the ionisations $n_3$ +for which the fast sampling can be done: +\begin{equation} +n_3 \; \geq \; c^2 +\end{equation} +As in the conditions (\ref{eq:phys332-6}), (\ref{eq:phys332-7}) and +(\ref{eq:phys332-8}) the value of $c$ is as minimum 4, one gets $n_3 +\; \geq 16$. In order to speed the simulation, the maximum value is +used for $\alpha$. + +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 $n_3$. The maximum number +of these collisions can be estimated as +\begin{equation} +n_{B,max} = n_3 - n_{A,min} \approx n_3 (\langle n_A \rangle + - \sigma_A) +\end{equation} +From the previous expressions for $\langle n_A \rangle$ and $\sigma_A$ +one can derive the condition +\begin{equation} +n_B \; \leq \; n_{B,max} = \frac{2 n_3 c^2}{n_3+c^2} +\end{equation} +The following values are obtained with $c=4$: + +\begin{tabular}{llcrr} +$n_3$ & $n_{B,max}$ & & $n_3$ & $n_{B,max}$\\ \hline +16 & 16 & & 200 & 29.63\\ +20 & 17.78 & & 500 & 31.01 \\ +50 & 24.24 & & 1000 & 31.50 \\ +100 & 27.59 & & $\infty$ & 32.00 +\end{tabular} + +\subsection{Special sampling for lower part of the spectrum} + +If the step length is very small ($\leq 5$ mm in gases, $\leq$ 2-3 +$\mu$m in solids) the model gives 0 energy loss for some events. To +avoid this, the probability of 0 energy loss is computed +\begin{equation} +P( \Delta E=0) = e^{-( \langle n_1 \rangle + \langle n_2 \rangle + + \langle n_3 \rangle )} +\end{equation} +If the probability is bigger than 0.01 a special sampling is done, +taking into account the fact that in these cases the projectile +interacts only with the outer electrons of the atom. An energy level +$E_0 = 10$ eV is chosen to correspond to the outer electrons. The mean +number of collisions can be calculated from +\begin{equation} +\langle n \rangle = \frac{1}{E_0} \frac{dE}{dx} \Delta x +\end{equation} +The number of collisions $n$ 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 +\begin{equation} +\Delta E = \sum_{i=1}^n \frac{E_0} + {1 - \frac {\Emax}{\Emax + E_0} u_i} +\end{equation} + +\begin{thebibliography}{10} +\bibitem{bib-LAND} +L.Landau. +\newblock On the Energy Loss of Fast Particles by Ionisation. +\newblock Originally published in \emph{J. Phys.}, 8:201, 1944. +\newblock Reprinted in D.ter Haar, Editor, \emph{L.D.Landau, Collected + papers}, page 417. Pergamon Press, Oxford, 1965. + +\bibitem{bib-SCH1} +B.Schorr. +\newblock Programs for the Landau and the Vavilov distributions and the + corresponding random numbers. +\newblock \emph{Comp. Phys. Comm.}, 7:216, 1974. + +\bibitem{bib-SELT} +S.M.Seltzer and M.J.Berger. +\newblock Energy loss straggling of protons and mesons. +\newblock In \emph{Studies in Penetration of Charged Particles in + Matter}, Nuclear Science Series~39, Nat. Academy of Sciences, + Washington DC, 1964. + +\bibitem{bib-TALM} +R.Talman. +\newblock On the statistics of particle identification using ionization. +\newblock \emph{Nucl. Inst. Meth.}, 159:189, 1979. + +\bibitem{bib-VAVI} +P.V.Vavilov. +\newblock Ionisation losses of high energy heavy particles. +\newblock \emph{Soviet Physics JETP}, 5:749, 1957. +\end{thebibliography} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch2/phys332-1.pdf b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/phys332-1.pdf Binary files differnew file mode 100644 index 00000000000..bbc8ac5dfa5 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/phys332-1.pdf diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/ch2/phys332-2.pdf b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/phys332-2.pdf Binary files differnew file mode 100644 index 00000000000..a058beb73f2 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/ch2/phys332-2.pdf |