-\RequirePackage{ifpdf}\r
-\r
-\documentclass[a4paper,12pt]{article}\r
-\ifpdf\r
- \usepackage[pdftex]{graphicx}\r
-\else\r
- \usepackage[dvips]{graphicx}\r
-\fi\r
-\usepackage{epsfig}\r
-\usepackage{rotating}\r
-\usepackage{listings}\r
-\usepackage{booktabs}\r
-\usepackage{fancyhdr}\r
-\usepackage{float}\r
-\floatplacement{figure}{H}\r
-\floatplacement{table}{H}\r
-\r
-\r
-\r
-\begin{document}\r
-\r
-\r
-\section{Accuracy of local coordinate measurement}\r
-\r
-\r
-\begin{figure}[t]\r
-%\centering\r
-\includegraphics[width=60mm,angle=-90]{picCluster/pic2.eps}\r
-\includegraphics[width=60mm,angle=-90]{picCluster/pic1.eps}\r
-\caption{Schematic view of the detection process in TPC (upper\r
-part - perspective view, lower part - side view).} \label{figTPC}\r
-\end{figure}\r
-\r
-The accuracy of the coordinate measurement is limited by a track\r
-angle which spreads ionization and by diffusion which amplifies\r
-this spread.\r
-\r
-The track direction with respect to pad plane is given by two\r
-angles $\alpha$ and $\beta$ (see fig.~\ref{figTPC}). For the\r
-measurement along the pad-row, the angle $\alpha$ between the\r
-track projected onto the pad plane and pad-row is relevant. For\r
-the measurement of the the drift coordinate ({\it{z}}--direction)\r
-it is the angle $\beta$ between the track and {\it{z}} axis\r
-(fig.~\ref{figTPC}).\r
-\r
-The ionization electrons are randomly distributed along the\r
-particle trajectory. Fixing the reference {\it{x}} position of an\r
-electron at the middle of pad-row, the {\it{y}} (resp. {\it{z}})\r
-position of the electron is a random variable characterized by\r
-uniform distribution with the width $L_{\rm{a}}$, where\r
-$L_{\rm{a}}$ is given by the pad length $L_{\rm{pad}}$ and the\r
-angle $\alpha$ (resp. $\beta$):\r
-\[L_{\rm{a}}=L_{\rm{pad}}\tan\alpha\]\r
-\r
-The diffusion smears out the position of the electron with\r
-gaussian probability distribution with $\sigma_{\rm{D}}$.\r
-Contribution of the $\mathbf{E{\times}B}$ and unisochronity\r
-effects for the Alice TPC are negligible. The typical resolution\r
-in the case of ALICE TPC is on the level of\r
-$\sigma_{y}\sim$~0.8~mm and $\sigma_{z}\sim$~1.0~mm integrating\r
-over all clusters in the TPC.\r
-\r
-\r
-\r
-\subsection{Gas gain fluctuation effect}\r
-\r
-Being collected on sense wire, electron is "multiplied" in strong\r
-electric field. This multiplication is subject of a large\r
-fluctuations, contributing to the cluster position resolution.\r
-Because of these fluctuations the center of gravity of the\r
-electron cloud can be shifted.\r
-\r
-Each electron is amplified independently. However, in the\r
-reconstruction electrons are not treated separately. The Centre Of\r
-Gravity (COG) of the cluster is usually used as an estimation for\r
-the local track position. The influence of the gas gain\r
-fluctuation to the reconstructed point characteristic can be\r
-described by a simple model, introducing a weighted COG\r
-$X_{\rm{COG}}$\r
-\begin{eqnarray}\r
- X_{\rm{COG}}=\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}},\r
-\label{eqCOGdefGG}\r
-\end{eqnarray}\r
-where {\it{N}} is the total number of electrons in the cluster and\r
-$g_i$ is a random variable equal to a gas amplification for given\r
-electron.\r
-\r
-The mean value of $X_{\rm{COG}}$ is equal to the mean value\r
-$\overline{x}$ of the original distribution of electrons\r
-\begin{eqnarray}\r
- \overline{X_{COG}}=\r
- \overline{\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}}}\r
- =\overline{x}\overline{\frac{\sum_{i=1}^{N}{g_i}}\r
- {\sum_{i=1}^N{g_i}}} =\overline{x}.\r
-\label{eqCOGMeanGG}\r
-\end{eqnarray}\r
-\r
-However, the same is not true for the dispersion of the position,\r
-%$\sigma^2_{X_{COG}}\sigma_x^2$:\r
-%\begin {center}\r
-\begin{eqnarray}\r
- \lefteqn{ \sigma^2_{X_{\rm{COG}}}\r
- =\overline{X_{\rm{COG}}^2}-\overline{X_{\rm{COG}}}^2=}\nonumber\\&&{}\r
- =\overline{\left(\frac{1}{\sum_{i=1}^N{g_i}}\sum_{i=1}^{N}{g_ix_i}\r
- \right)^2}-\overline{x}^2=\r
- \nonumber\\\r
- &&{}=\overline{\frac{{\sum\sum{x_ix_jg_ig_j}}}{{\sum\sum{g_ig_j}}}}-\r
- \overline{x}^2=\r
- \nonumber\\&&{}=\r
- \overline{x^2}\overline{\frac{\sum_i{g_i^2}}{\sum\sum{g_ig_j}}}-\r
- \overline{x}^2\r
- \overline{\frac{\sum\sum{g_ig_j}-\sum\sum_{i\ne{j}}{g_ig_j}}\r
- {\sum\sum{g_ig_j}}}= \nonumber\\&&\r
- =\left(\overline{x^2}-\overline{x}^2\right)\r
- \overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=\r
- \sigma_x^2\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=\r
- \nonumber\\\r
- &&{}=\frac{\sigma_x^2}{N}{\times}G_{\rm{gfactor}}^2\r
-\label{eqCOGSigmaGG}\r
-\end{eqnarray}\r
-\r
-where\r
-\begin{eqnarray}\r
- G_{\rm{gfactor}}^2 = N\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}\r
-\label{eqCOGGGfactor0}\r
-\end{eqnarray}\r
-\r
-The diffusion term is effectively multiplied by gas gain factor\r
-$G_{\rm{gfactor}}$. For sufficiently large number of electrons,\r
-when $g_i^2$ and $\sum\sum{g_ig_j}$ are quasi independent\r
-variables, equation (\ref{eqCOGGGfactor0}) can be transformed to\r
-the following\r
-\r
-\begin{eqnarray}\r
- \lefteqn{G_{\rm{gfactor}}^2 \approx\r
- N\frac{\overline{\sum{g_i^2}}}\r
- {\overline{\sum\sum{g_ig_j}}}}\nonumber\\\r
- &&{} =\r
- N\frac{N\overline{g^2}}{N(N-1)\overline{g}^2+N\overline{g^2}}=\r
- \nonumber\\\r
- &&{} =N\frac{ \left(\sigma_g^2/\overline{g}^2+1 \right)}\r
- {N+\sigma_g^2/\overline{g}^2}\r
-\label{eqCOGGGfactorE}\r
-\end{eqnarray}\r
-\r
-Gas gain fluctuation of the gas detector working in proportional\r
-regime is described with the exponential distribution with the\r
-mean value $\bar{g}$ and r.m.s.\r
-\begin{eqnarray}\r
- \sigma_{\rm{g}} =\bar{g}\r
-\label{eqSigmaexp}\r
-\end{eqnarray}\r
-\r
-Substituting $\sigma_{\rm{g}}$ into equation\r
-(\ref{eqCOGGGfactorE})\r
-\begin{eqnarray}\r
- G_{\rm{gfactor}}^2 =\frac{2N}{N+1}.\r
-\label{eqCOGGGfactorR}\r
-\end{eqnarray}\r
-\r
-Gas multiplication fluctuation in chamber deteriorates\r
-$\sigma_{X_{\rm{COG}}}$ by a factor of about ${\sqrt{2}}$. The\r
-prediction of this model is in good agreement with results from\r
-the simulation.\r
-\r
-\r
-\subsection{Secondary ionization effect}\r
-\r
-Charged particle penetrating the gas of the detector produces\r
-{\it{N}} primary electrons. Primary electron {\it{i}} produces\r
-$n_{\rm{s}}^i-1$ secondary electrons. Each of these electrons is\r
-amplified in the electric field by a factor of $g_j$.\r
-\r
-Each primary cluster is characterized by a position $x_i$ with\r
-mean value $\overline{x}$ and $\sigma_x$. The COG given by\r
-equation (\ref{eqCOGdefGG}) is modified to the following form:\r
-\r
-\begin{eqnarray}\r
- X_{\rm{COG}}=\frac{1}{\sum_{i=1}^N\sum_{j=1}^{n_i}{g_j^{i}}}\r
- \sum_{i=1}^{N}{x_i}\sum_{j=1}^{n_i}{g_j^{i}}.\r
-\label{eqCOGdefGGPIO}\r
-\end{eqnarray}\r
-A new variable $G_n$ is introduced as the total electron gain:\r
-\begin{eqnarray}\r
- G_n=\sum_{j=1}^{n}{g_j}.\r
-\label{eqGNdef}\r
-\end{eqnarray}\r
-\r
-\r
-Knowing the distribution of {\it{n}} and {\it{g}} and assuming\r
-that {\it{n}} and {\it{g}} are independent variables the mean\r
-value and variance of the $G_n$ can be expressed as:\r
-\r
-\begin{eqnarray}\r
- \lefteqn{\r
- \overline{G_n}=\overline{n}\overline{g}} \\\r
- &&{}\r
- \frac{\sigma^2_{G_n}}{\overline{G_n^2}}=\r
- \frac{\sigma^2_n}{\overline{n}^2}+\r
- \frac{\sigma^2_g}{\overline{g}^2}\r
- \frac{1}{\overline{n}}\r
-\label{eqGNsigma}\r
-\end{eqnarray}\r
-\r
-Inserting $G_n$ into equation (\ref{eqCOGdefGGPIO}) results in an\r
-equation similar to the equation (\ref{eqCOGdefGG}).\r
-\r
-Multiplicative factor $G_{\rm{Lfactor}}$ is defined as an analog\r
-of $G_{\rm{gfactor}}$, from the equation (\ref{eqCOGGGfactor0})\r
-\begin{eqnarray}\r
- G_{\rm{Lfactor}}^2 = N\frac{\overline{\sum{G_i^2}}}\r
- {\overline{\sum\sum{G_iG_j}}}.\r
-\label{eqCOGLfactor0}\r
-\end{eqnarray}\r
-\r
-Using the new variable $G_n$ and simply replacing gas gain\r
-{\it{g}} by $G_n$ in the similar way as in equation\r
-(\ref{eqCOGGGfactorE}) does not work. For $1/E^{2}$\r
-parametrization of secondary ionization process\r
-$\sigma^2_{G_n}/\overline{G_n}$ goes to infinity and thus\r
-$\sigma^2_{X_{COG}}=\sigma_x^2$. Moreover $G_i^2$ and\r
-$\sum\sum{G_iG_j}$ are not quasi independent as the sum\r
-$\sum\sum{G_iG_j}$ could be given by one "exotic" electron\r
-cluster. Approximations used for deriving the equation\r
-(\ref{eqCOGGGfactorE}) are not valid for secondary ionization\r
-effect.\r
-\r
-In order to estimate the impact of this effect on COG equation\r
-(\ref{eqCOGLfactor0}) has to be solved numerically. Simulation\r
-showed that $G_{\rm{Lfactor}}$ does not depend strongly on the cut\r
-used for maximum number of electrons created in the process of\r
-secondary ionization. A change of the cut, from 1000 electrons up\r
-produces a change of about 3\% in $G_{\rm{Lfactor}}$.\r
-\r
-Equation (\ref{eqCOGGGfactorE}) is not applicable in this\r
-situation because of the infinity of the $\sigma_G$. According to\r
-the simulation, the threshold on the number of electrons in the\r
-cluster has a little influence to the resulting\r
-$G_{\rm{Lfactor}}$. Therefore we fit simulated $G_{\rm{Lfactor}}$\r
-with formula (\ref{eqCOGGGfactorE}) where\r
-$\sigma_G^2/\overline{G}^2$ was a free parameter. However, this\r
-parametrization does not describe the data for wide enough range\r
-of {\it{N}}. In further study the linear parametrization of the\r
-COG factor was used. This parametrization was validated on\r
-reasonable interval of {\it{N}}.\r
-\r
-\r
-\r
-\section{Center-of-gravity error parametrization}\r
-\r
-Detected position of charged particle is a random variable given\r
-by several stochastic processes: diffusion, angular effect, gas\r
-gain fluctuation, Landau fluctuation of the secondary ionization,\r
-$\mathbf{E{\times}B}$ effect, electronic noise and systematic\r
-effects (like space charge, etc.). The relative influence of these\r
-processes to the resulting distortion of position determination\r
-depends on the detector parameters. In the big drift detectors\r
-like the ALICE TPC the main contribution is given by diffusion,\r
-gas gain fluctuation, angular effect and secondary ionization\r
-fluctuation.\r
-\r
-Furthermore we will use following assumptions:\r
-\begin{itemize}\r
-\item $N_{\rm{prim}}$ primary electrons are produced at a random\r
-positions $x_i$ along the particle trajectory. \item $n_i-1$\r
-electrons are produced in the process of secondary ionization.\r
-\item Displacement of produced electrons due to the thermalization\r
-is neglected.\r
-\end{itemize}\r
-\r
-Each of electrons is characterized by a random vector\r
-$\vec{z}^i_j$\r
-\begin{eqnarray}\r
- \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j,\r
-\label{eqZtot}\r
-\end{eqnarray}\r
-where {\it{i}} is the index of primary electron cluster and\r
-{\it{j}} is the index of the secondary electron inside of the\r
-primary electron cluster. Random variable $\vec{x}^i$ is a\r
-position where the primary electron was created. The position\r
-$\vec{y}^i_j$ is a random variable specific for each electron. It\r
-is given mainly by a diffusion.\r
-\r
-The center of gravity of the electron cloud is given:\r
-\begin{eqnarray}\r
- \lefteqn{\vec{z}_{\rm{COG}}=\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}\r
- \sum_{j=1}^{n_i}{g_j^i}}\r
- \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{z}_j^i}=}\r
- \nonumber\\\r
- &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}\r
- \sum_{j=1}^{n_i}{g_j^i}}\r
- \sum_{i=1}^{N_{\rm{prim}}}\vec{x}^i\sum_{j=1}^{n_i}{g_j^i}+\nonumber\\\r
- &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}\r
- \sum_{j=1}^{n_i}{g_j^i}}\r
- \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{y}_j^i}=\r
- \nonumber\\ \nonumber\\\r
- &&{}\r
- \vec{x}_{\rm{COG}}+\vec{y}_{\rm{COG}}.\r
-\label{eqCOGSec}\r
-\end{eqnarray}\r
-\r
-The mean value $\overline{\vec{z}_{\rm{COG}}}$ is equal to the sum\r
-of mean values $\overline{\vec{x}_{\rm{COG}}}$ and\r
-$\overline{\vec{y}_{\rm{COG}}}$.\r
-\r
-The sigma of COG in one of the dimension of vector\r
-$\vec{z}_{1COG}$ is given by following equation\r
-\begin{eqnarray}\r
- \lefteqn{\sigma_{z_{\rm{1COG}}}^2=\sigma_{x_{\rm{1COG}}}^2+\r
- \sigma_{y_{\rm{1COG}}}^2+}\nonumber\\\r
- &&{}\r
- 2\left(\overline{x_{\rm{1COG}}y_{\rm{1COG}}}-\bar{x}_{\rm{1COG}}\r
- \bar{y}_{1COG}\right).\r
-\label{eqCOGSigSec}\r
-\end{eqnarray}\r
-\r
-If the vectors $\vec{x}$ and $\vec{y}$ are independent random\r
-variables the last term in the equation (\ref{eqCOGSigSec}) is\r
-equal to zero.\r
-\begin{eqnarray}\r
- \sigma_{z_{1COG}}^2=\sigma_{x_{\rm{1COG}}}^2+\r
- \sigma_{y_{\rm{1COG}}}^2,\r
-\label{eqCOGSigSecIn}\r
-\end{eqnarray}\r
-r.m.s. of COG distribution is given by the sum of r.m.s of\r
-{\it{x}} and {\it{y}} components.\r
-\r
-In order to estimate the influence of the $\mathbf{E{\times}B}$\r
-and unisochronity effect to the space resolution two additional\r
-random vectors are added to the initial electron position.\r
-\r
-\r
-\begin{eqnarray}\r
-\vec{z}^i_j =\vec{x}^i+\vec{y}^i_j+\r
- \vec{X}_{\mathbf{E{\times}B}}(\vec{x}^i+\vec{y}^i_j)+\r
- \vec{X}_{\rm{Unisochron}}(\vec{x}^i+\vec{y}^i_j).\r
-\label{eqZtotplus}\r
-\end{eqnarray}\r
-The probability distributions of $\vec{X}_{\mathbf{E{\times}B}}$\r
-and $\vec{X}_{\rm{Unisochron}}$ are functions of random vectors\r
-$\vec{x^i}$ and $\vec{y^i_j}$, and they are strongly correlated.\r
-However, simulation indicates that in large drift detectors\r
-distortions, due to these effects, are negligible compared with a\r
-previous one.\r
-\r
-Combining previous equation and neglecting $\mathbf{E{\times}B}$\r
-and unisochronity\r
-effects, the COG distortion parametrization appears as:\\\r
-{$\sigma_{z}$} of cluster center in {\it{z}} (time) direction\r
-\begin{eqnarray}\\r
- \lefteqn{\sigma^2_{{z_{\rm{COG}}}} = \frac{D^2_{\rm{L}}\r
- L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}\r
- \frac{{\tan^2\alpha}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+\r
- \sigma^2_{\rm{noise}},\r
- \label{eqResZ1}\r
-\end{eqnarray}\r
-\r
-and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction\r
- \begin{eqnarray}\r
- \lefteqn{\sigma^2_{y_{\rm{COG}}} = \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}\r
- \frac{{\tan^2\beta}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+\r
- \sigma^2_{\rm{noise}},\r
- \label{eqResY1}\r
- \end{eqnarray}\r
- where\r
-${N_{\rm{ch}}}$ is the total number of electrons in the cluster,\r
-${N_{\rm{chprim}}}$ is the number of primary electrons in the\r
-cluster, ${G_{\rm{g}}}$ is the gas gain fluctuation factor,\r
-${G_{\rm{Lfactor}}}$ is the secondary ionization fluctuation\r
-factor and $\sigma_{\rm{noise}}$ describe the contribution of the\r
-electronic noise to the resulting sigma of the COG.\r
-\r
-\section{Precision of cluster COG determination using measured\r
-amplitude}\r
-\r
-We have derived parametrization using as parameters the total\r
-number of electrons ${N_{\rm{ch}}}$ and the number of primary\r
-electrons ${N_{\rm{chprim}}}$. This parametrization is in good\r
-agreement with simulated data, where the ${N_{\rm{ch}}}$ and\r
-${N_{\rm{chprim}}}$ are known. It can be used as an estimate for\r
-the limits of accuracy, if the mean values\r
-$\overline{N}_{\rm{ch}}$ and $\overline{N}_{\rm{chprim}}$ are used\r
-instead.\r
-\r
-The ${N_{\rm{ch}}}$ and ${N_{\rm{chprim}}}$ are random variables\r
-described by a Landau distribution, and Poisson distribution\r
-respectively .\r
-\r
-In order to use previously derived formulas (\ref{eqResZ1},\r
-\ref{eqResY1}), the number of electrons can be estimated assuming\r
-their proportionality to the total measured charge $A$ in the\r
-cluster. However, it turns out that an empirical parametrization\r
-of the factors $G(N)/N=G(A)/(kA)$ gives better results.\r
-Formulas (\ref{eqResZ1}) and (\ref{eqResY1}) are transformed to following form:\\\r
-\r
-{$\sigma_{z}$} of cluster center in {\it{z}} (time) direction:\r
- \begin{eqnarray}\r
- \lefteqn{\sigma^2_{z_{\rm{COG}}} =\r
- \frac{D^2_{\rm{L}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\\r
- &&{}\r
- \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}\r
- \label{eqZtotAmp}\r
- \end{eqnarray}\r
-\r
-and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction:\r
- \begin{eqnarray}\r
- \lefteqn{\sigma_{y_{\rm{COG}}} =\r
- \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\\r
- &&{}\r
- \frac{\tan^2\beta~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}\r
- \label{eqYtotAmp}\r
- \end{eqnarray}\r
-\r
-\section{Estimation of the precision of cluster position\r
-determination using measured cluster shape}\r
-\r
-The shape of the cluster is given by the convolution of the\r
-responses to the electron avalanches. The time response function\r
-and the pad response function are almost gaussian, as well as the\r
-spread of electrons due to the diffusion. The spread due to the\r
-angular effect is uniform. Assuming that the contribution of the\r
-angular spread does not dominate the cluster width, the cluster\r
-shape is not far from gaussian. Therefore, we can use the\r
-parametrization\r
-\r
-\begin{equation}\r
- f(t,p) = K_{\rm{Max}}.\exp\left(-\frac{(t-t_{\rm{0}})^2}{2\sigma_{\rm{t}}^2}-\r
- \frac{(p-p_{\rm{0}})^2}{2\sigma_{\rm{p}}^2}\right),\r
- \label{eq:GaussTP}\r
-\end{equation}\r
-where ${K_{\rm{Max}}}$ is the normalization factor, $t$ and $p$\r
-are time and pad bins, $t_0$ and $p_0$ are centers of the cluster\r
-in time and pad direction and $\sigma_{\rm{t}}$ and\r
-$\sigma_{\rm{p}}$ are the r.m.s. of the time and pad cluster\r
-distribution.\r
-\r
- The mean width of the cluster distribution is given by:\r
-\begin{equation}\r
- \sigma_{\rm{t}} = \sqrt{D{\rm{^2_L}}L_{\rm{drift}}+\sigma^2_{\rm{preamp}}+\r
- \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12}},\r
-\end{equation}\r
-\r
-\r
-\begin{equation}\r
- \sigma_{\rm{p}} = \sqrt{D{\rm{^2_T}}L_{\rm{drift}}+\sigma^2_{\rm{PRF}}+\r
- \frac{\tan^2\beta~L_{\rm{pad}}^2}{12}},\r
-\end{equation}\r
-where ${\sigma_{\rm{preamp}}}$ and ${\sigma_{\rm{PRF}}}$ are the\r
-r.m.s. of the time response function and pad response function,\r
-respectively.\r
-\r
-The fluctuation of the shape depends on the contribution of the\r
-random diffusion and angular spread, and on the contribution given\r
-by a gas gain fluctuation and secondary ionization. The\r
-fluctuation of the time and pad response functions is small\r
-compared with the previous one.\r
-\r
-The measured r.m.s of the cluster is influenced by a threshold\r
-effect.\r
-\begin{equation}\r
- \sigma_{\rm{t}}^2 = \sum_{A(t,p)>\rm{threshold}}{(t-t_{\rm{0}})^2{\times}A(t,p)}\r
-\end{equation}\r
-The threshold effect can be eliminated using two dimensional\r
-gaussian fit instead of the simple COG method. However, this\r
-approach is slow and, moreover, the result is very sensitive to\r
-the gain fluctuation.\r
-\r
-To eliminate the threshold effect in r.m.s. method, the bins\r
-bellow threshold are replaced with a virtual charge using\r
-gaussian interpolation of the cluster shape. The introduction of\r
-the virtual charge improves the precision of the COG measurement.\r
-Large systematic shifts in the estimate of the cluster position\r
-(depending on the local track position relative to pad--time) due\r
-to the threshold are no longer observed.\r
-\r
-Measuring the r.m.s. of the cluster, the local diffusion and\r
-angular spread of the electron cloud can be estimated. This\r
-provides additional information for the estimation of\r
-distortions. A simple additional correction function is used:\r
-\begin{eqnarray}\r
- \sigma_{\rm{COG}} \rightarrow\r
- \sigma_{\rm{COG}}(A){\times}(1+{\rm{const} {\times}\frac{\delta\r
- \rm{RMS}}{\rm{teorRMS}}}),\r
-\label{eqResUsingRMS}\r
-\end{eqnarray}\r
-where $\sigma_{\rm{COG}}(A)$ is calculated according formulas\r
-\ref{eqResY1} and \ref{eqResZ1}, and the\r
-$\delta\rm{RMS}/\rm{teorRMS}$ is the relative distortion of the\r
-signal shape from the expected one.\r
-\r
-\r
-\r
-\r
-\r
-\r
-\section{TPC cluster finder}\r
-\r
-The classical approach for the beginning of the tracking was\r
-chosen. Before the tracking itself, two-dimensional clusters in\r
-pad-row--time planes are found. Then the positions of the\r
-corresponding space points are reconstructed, which are\r
-interpreted as the crossing points of the tracks and the centers\r
-of the pad rows. We investigate the region 5$\times$5 bins in\r
-pad-row--time plane around the central bin with maximum amplitude.\r
-The size of region, 5$\times$5 bins, is bigger than typical size\r
-of cluster as the $\sigma_{\rm{t}}$ and $\sigma_{\rm{pad}}$ are\r
-about 0.75 bins.\r
-\r
-The COG and r.m.s are used to characterize cluster. The COG and\r
-r.m.s are affected by systematic distortions induced by the\r
-threshold effect. Depending on the number of time bins and pads in\r
-clusters the COG and r.m.s. are affected in different ways.\r
-Unfortunately, the number of bins in cluster is the function of\r
-local track position. To get rid of this effect, two-dimensional\r
-gaussian fitting can be used.\r
-\r
-Similar results can be achieved by so called r.m.s. fitting using\r
-virtual charge. The signal below threshold is replaced by the\r
-virtual charge, its expected value according a interpolation. If\r
-the virtual charge is above the threshold value, then it is\r
-replaced with amplitude equal to the threshold value. The signal\r
-r.m.s is used for later error estimation and as a criteria for\r
-cluster unfolding. This method gives comparable results as\r
-gaussian fit of the cluster but is much faster. Moreover, the COG\r
-position is less sensitive to the gain fluctuations.\r
-\r
-The cluster shape depends on the track parameters. The response\r
-function contribution and diffusion contribution to the cluster\r
-r.m.s. are known during clustering. This is not true for a angular\r
-contribution to the cluster width. The cluster finder should be\r
-optimised for high momentum particle coming from the primary\r
-vertex. Therefore, a conservative approach was chosen, assuming\r
-angle $\alpha$ to be zero. The tangent of the angle $\beta$ is\r
-given by {\it{z}}-position and pad-row radius, which is known\r
-during clustering.\r
-\r
-\r
-\subsection{Cluster unfolding}\r
-\r
-The estimated width of the cluster is used as criteria for cluster\r
-unfolding. If the r.m.s. in one of the directions is greater then\r
-critical r.m.s, cluster is considered for unfolding. The fast\r
-spline method is used here. We require the charge to be conserved\r
-in this method. Overlapped clusters are supposed to have the same\r
-r.m.s., which is equivalent to the same track angles. If this\r
-assumption is not fulfilled, tracks diverge very rapidly.\r
-\r
-\r
-\begin{figure}[t]\r
-\centering\r
-\includegraphics[width=60mm,angle=-90]{picCluster/unfolding1.eps}\r
-\caption{\r
-Schematic view of unfolding principle.} \label{figUnfolding1}\r
-\end{figure}\r
-\begin{figure}[t]\r
-\centering\r
-\includegraphics[width=60mm,angle=-90]{picCluster/unfoldingres.eps}\r
-\caption{ Dependence of the position residual as function of the\r
-distance to the second cluster.} \label{figUnfoldingRes}\r
-\end{figure}\r
-\r
-The unfolding algorithm has the following steps:\r
-\begin{itemize}\r
-\r
-\item Six amplitudes $C_i$ are investigated (see fig.\r
-\ref{figUnfolding1}). First (left) local maxima, corresponding to\r
-the first cluster is placed at position 3, second (right) local\r
-maxima corresponding to the second cluster is at position 5.\r
-\r
-\item In the first iteration, amplitude in bin 4 corresponding to\r
-the cluster on left side $A_{\rm{L4}}$ is calculated using\r
-polynomial interpolation, assuming virtual amplitude at\r
-$A_{\rm{L5}}$ and derivation at $A_{\rm{L5}}^{'}$ to be 0.\r
-Amplitudes $A_{\rm{L2}}$ and $A_{\rm{L3}}$ are considered to be\r
-not influenced by overlap ($A_{\rm{L2}}=C_2$ and\r
-$A_{\rm{L3}}=C_3)$.\r
-\r
-\item The amplitude $A_{\rm{R4}}$ is calculated in similar way. In\r
-the next iteration the amplitude $A_{\rm{L4}}$ is calculated\r
-requiring charge conservation\r
-$C_{\rm{4}}=A_{\rm{R4}}+A_{\rm{L4}}$. Consequently\r
-\begin{eqnarray}\r
- A_{\rm{L4}} \rightarrow\r
- C_{\rm{4}}\frac{A_{\rm{L4}}}{A_{\rm{L4}}+A_{\rm{R4}}}\r
-\end{eqnarray}\r
-and\r
-\begin{eqnarray}\r
- A_{\rm{R4}} \rightarrow\r
- C_{\rm{4}}\frac{A_{\rm{R4}}}{A_{\rm{L4}}+A_{\rm{R4}}}.\r
-\end{eqnarray}\r
-\end{itemize}\r
-\r
-\r
-Two cluster resolution depends on the distance between the two\r
-tracks. Until the shape of cluster triggers unfolding, there is a\r
-systematic shifts towards to the COG of two tracks (see fig.\r
-\ref{figUnfoldingRes}), only one cluster is reconstructed.\r
-Afterwards, no systematic shift is observed.\r
-\r
-\r
-\subsection{Cluster characteristics}\r
-\r
-The cluster is characterized by the COG in {\it{y}} and {\it{z}}\r
-directions (fY and fZ) and by the cluster width (fSigmaY,\r
-fSigmaZ). The deposited charge is described by the signal at\r
-maximum (fMax), and total charge in cluster (fQ). The cluster type\r
-is characterized by the data member fCType which is defined as a\r
-ratio of the charge supposed to be deposited by the track and\r
-total charge in cluster in investigated region 5$\times$5. The\r
-error of the cluster position is assigned to the cluster only\r
-during tracking according formulas\r
- (\ref{eqZtotAmp}) and (\ref{eqYtotAmp}), when track\r
- angles $\alpha$ and $\beta$ are known with sufficient precision.\r
-\r
-\r
-Obviously, measuring the position of each electron separately the\r
-effect of the gas gain fluctuation can be removed, however this is\r
-not easy to implement in the large TPC detectors. Additional\r
-information about cluster asymmetry can be used, but the resulting\r
-improvement of around 5\% in precision on simulated data is\r
-negligible, and it is questionable, how successful will be such\r
-correction for the cluster asymmetry on real data.\r
-\r
-However, a cluster asymmetry can be used as additional criteria\r
-for cluster unfolding. Let's denote $\mu_i$ the {\it{i}}-th\r
-central momentum of the cluster, which was created by overlapping\r
-from two sub-clusters with unknown positions and deposited energy\r
-(with momenta $^1\mu_i$ and $^2\mu_i$).\r
-\r
-Let $r_1$ is the ratio of two clusters amplitudes:\r
-\[r_1={^1\mu_0}/({^1\mu_0}+{^2\mu_0})\] and the track distance {\it{d}} is equal to\r
-\[d = {^1\mu_1} -{^2\mu_1}.\]\r
-\r
-Assuming that the second moments for both sub-clusters are the\r
-same (${^0\mu_2}={^1\mu_2}={^2\mu_2}$), two sub-clusters distance\r
-{\it{d}} and amplitude ratio $r_1$ can be estimated:\r
-\begin{eqnarray}\r
- R = \frac{(\mu_3^6)}{(\mu_2^2-{^0\mu_2^2})^3}\\\r
- r_{\rm{1}} =0.5\pm0.5{\times}\sqrt{\frac{1}{1-4/R}} \\\r
- d = \sqrt{(4+R){\times}(\mu_2^2-{^0\mu_2^2})}\r
-\label{eqMeas}\r
-\end{eqnarray}\r
-\r
-In order to trigger unfolding using the shape information\r
-additional information about track and mean cluster shape over\r
-several pad-rows are needed. This information is available only\r
-during tracking procedure.\r
-\r
-\r
-\r
-\subsection{Space point resolution parameterization}\r
-\r
-The space point resolution is the function of many parameters but for the ALICE TPC the dominant one are the diffusion, track inclination angle and deposited charge.\r
-The space point resolution was extracted from the data in bins of these variables.\r
-\r
-In the first approximation the angular part and diffusion part are independent. The\r
-paramaterization is obtained fitting parameters $p_{0}$,$p_L$ and $p_A$\r
-\begin{eqnarray}\\r
- \sigma^2_{{\rm{COG}}} \approx p^2_0+p^2_{L}L_{\rm{Drift}}+p^2_{A}\tan^2\alpha\r
- \label{eqResCOG0} \r
- \nonumber\\\r
- p^2_L \approx \frac{\sigma^2_DG_{\rm{g}}}{N_{\rm{ch}}}\r
- \nonumber\\\r
- p^2_A \approx \frac{L_{\rm{pad}}^2G_{\rm{Lfactor}}}{N_{\rm{chprim}}} \r
-\end{eqnarray}\r
-\r
-\r
-\r
-\begin{table}\r
-\caption{Resolution parameterization}\r
-\begin{tabular}{|l|l|l|l|} \hline\r
-Pad size & 0.75x0.4 $cm^2$ & 1.0x0.6$cm^2$ & 1.5x0.6$cm^2$ \\ \hline\r
-$p_{0y}$ & 0.026 cm & 0.031 cm & 0.023 cm \\ \hline\r
-$p_{0z}$ & 0.032 cm & 0.032 cm & 0.028 cm \\ \hline\r
-$p_{Ly}\sqrt{L_{pad}}$ & 0.0051 & 0.0060 & 0.0059 \\ \hline \r
-$p_{Lz}\sqrt{L_{pad}}$ & 0.0056 & 0.0056 & 0.0059 \\ \hline \r
-$p_{Ay}/\sqrt{L_{pad}}$ & 0.13 $cm^{1/2}$ & 0.15 $cm^{1/2}$ & 0.15 $cm^{1/2}$ \\ \hline \r
-$p_{Az}/\sqrt{L_{pad}}$ & 0.15 $cm^{1/2}$ & 0.16 $cm^{1/2}$ & 0.17 $cm^{1/2}$ \\ \hline \r
-\r
-\end{tabular}\r
-\label{table:PointResolFitParam}\r
-\end{table}\r
-\r
-\r
-\begin{eqnarray}\\r
- N_{\rm{ch}} \approx {L_{\rm{pad}}} \nonumber \\\r
- N_{\rm{chprim}} \approx {L_{\rm{pad}}} \nonumber \\ \r
- \nonumber\\\r
- p_L \approx \frac{1}{\sqrt{L_{\rm{pad}}}}\r
- \nonumber\\\r
- p_A \approx \sqrt{L_{\rm{pad}}}\r
-\label{eq:ResolScaling} \r
-\end{eqnarray}\r
-\r
-\r
-The TPC space resolution is scaling with the number of contributed electrons \r
-$N_{\rm{chprim}}$ and ${N_{\rm{ch}}}$, therefore is scaling with pad length.\r
-In ALICE TPC three different pad gemetries are used. \r
-The space point resolution was fitted for separatelly for each geometry. The fitted parameters $p_0$ $p_L$ and $p_A$ are shown in the table \ref{table:PointResolFitParam} rescaled with the pad length.\r
-\r
-\r
-\r
-The agreement between previously mentioned fit and the data is on thel level of the\r
-$\approx10-20\%$. In previous formula we assumed that all of the electrons created in ionization are contibuting to the measured signal. Because of the threshold effect the\r
-part of the signal is cut-off. The fraction of the signal bellow threshold is proportional to the response function witdth and is incresing with drift length and inclination angle. The following correction function are used: \r
-\begin{eqnarray}\\r
- \nonumber\\\r
- p_L \approx p_{L0}p_{LC}=p_{L0}(1+p_{L1}L_{\rm{Drift}}+p_{L2}\tan^2\alpha)\r
- \nonumber\\\r
- p_A \approx p_{A0}p_{AC}=p_{A0}(1+p_{A1}L_{\rm{Drift}}+p_{A2}\tan^2\alpha)\r
-\end{eqnarray}\r
-\r
-The measured reolution in Y and Z direction and corresponding fits are shown on picure \ref{figPointResolYDRTAN} and \ref{figPointResolZDRTAN}. The agrement with the data is on the level of about 2\%.\r
-\r
-\r
-\begin{figure}\r
- \centering\epsfig{figure=picClusterResol/YResol_Pad0.eps,width=0.7\linewidth}\r
- \centering\epsfig{figure=picClusterResol/YResol_Pad1.eps,width=0.7\linewidth}\r
- \centering\epsfig{figure=picClusterResol/YResol_Pad2.eps,width=0.7\linewidth}\r
- \caption{Space point resolution in Y direcition as function of the drift length and the inlination angle.}\r
- \label{figPointResolYDRTAN}\r
-\end{figure}\r
-\r
-\begin{figure}\r
- \centering\epsfig{figure=picClusterResol/ZResol_Pad0.eps,width=0.7\linewidth}\r
- \centering\epsfig{figure=picClusterResol/ZResol_Pad1.eps,width=0.7\linewidth}\r
- \centering\epsfig{figure=picClusterResol/ZResol_Pad2.eps,width=0.7\linewidth}\r
- \caption{Space point resolution in Z direcition as function of the drift length and the inlination angle.}\r
- \label{figPointResolZDRTAN}\r
-\end{figure}\r
-\r
-\r
-\r
-\r
-\end{document}\r
+\RequirePackage{ifpdf}
+
+\documentclass[a4paper,12pt]{article}
+\ifpdf
+ \usepackage[pdftex]{graphicx}
+\else
+ \usepackage[dvips]{graphicx}
+\fi
+\usepackage{epsfig}
+\usepackage{rotating}
+\usepackage{listings}
+\usepackage{booktabs}
+\usepackage{fancyhdr}
+\usepackage{float}
+\floatplacement{figure}{H}
+\floatplacement{table}{H}
+
+
+
+\begin{document}
+
+
+\section{Accuracy of local coordinate measurement}
+
+
+\begin{figure}[t]
+%\centering
+\includegraphics[width=60mm,angle=-90]{picCluster/pic2.eps}
+\includegraphics[width=60mm,angle=-90]{picCluster/pic1.eps}
+\caption{Schematic view of the detection process in TPC (upper
+part - perspective view, lower part - side view).} \label{figTPC}
+\end{figure}
+
+The accuracy of the coordinate measurement is limited by a track
+angle which spreads ionization and by diffusion which amplifies
+this spread.
+
+The track direction with respect to pad plane is given by two
+angles $\alpha$ and $\beta$ (see fig.~\ref{figTPC}). For the
+measurement along the pad-row, the angle $\alpha$ between the
+track projected onto the pad plane and pad-row is relevant. For
+the measurement of the the drift coordinate ({\it{z}}--direction)
+it is the angle $\beta$ between the track and {\it{z}} axis
+(fig.~\ref{figTPC}).
+
+The ionization electrons are randomly distributed along the
+particle trajectory. Fixing the reference {\it{x}} position of an
+electron at the middle of pad-row, the {\it{y}} (resp. {\it{z}})
+position of the electron is a random variable characterized by
+uniform distribution with the width $L_{\rm{a}}$, where
+$L_{\rm{a}}$ is given by the pad length $L_{\rm{pad}}$ and the
+angle $\alpha$ (resp. $\beta$):
+\[L_{\rm{a}}=L_{\rm{pad}}\tan\alpha\]
+
+The diffusion smears out the position of the electron with
+gaussian probability distribution with $\sigma_{\rm{D}}$.
+Contribution of the $\mathbf{E{\times}B}$ and unisochronity
+effects for the Alice TPC are negligible. The typical resolution
+in the case of ALICE TPC is on the level of
+$\sigma_{y}\sim$~0.8~mm and $\sigma_{z}\sim$~1.0~mm integrating
+over all clusters in the TPC.
+
+
+
+\subsection{Gas gain fluctuation effect}
+
+Being collected on sense wire, electron is "multiplied" in strong
+electric field. This multiplication is subject of a large
+fluctuations, contributing to the cluster position resolution.
+Because of these fluctuations the center of gravity of the
+electron cloud can be shifted.
+
+Each electron is amplified independently. However, in the
+reconstruction electrons are not treated separately. The Centre Of
+Gravity (COG) of the cluster is usually used as an estimation for
+the local track position. The influence of the gas gain
+fluctuation to the reconstructed point characteristic can be
+described by a simple model, introducing a weighted COG
+$X_{\rm{COG}}$
+\begin{eqnarray}
+ X_{\rm{COG}}=\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}},
+\label{eqCOGdefGG}
+\end{eqnarray}
+where {\it{N}} is the total number of electrons in the cluster and
+$g_i$ is a random variable equal to a gas amplification for given
+electron.
+
+The mean value of $X_{\rm{COG}}$ is equal to the mean value
+$\overline{x}$ of the original distribution of electrons
+\begin{eqnarray}
+ \overline{X_{COG}}=
+ \overline{\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}}}
+ =\overline{x}\overline{\frac{\sum_{i=1}^{N}{g_i}}
+ {\sum_{i=1}^N{g_i}}} =\overline{x}.
+\label{eqCOGMeanGG}
+\end{eqnarray}
+
+However, the same is not true for the dispersion of the position,
+%$\sigma^2_{X_{COG}}\sigma_x^2$:
+%\begin {center}
+\begin{eqnarray}
+ \lefteqn{ \sigma^2_{X_{\rm{COG}}}
+ =\overline{X_{\rm{COG}}^2}-\overline{X_{\rm{COG}}}^2=}\nonumber\\&&{}
+ =\overline{\left(\frac{1}{\sum_{i=1}^N{g_i}}\sum_{i=1}^{N}{g_ix_i}
+ \right)^2}-\overline{x}^2=
+ \nonumber\\
+ &&{}=\overline{\frac{{\sum\sum{x_ix_jg_ig_j}}}{{\sum\sum{g_ig_j}}}}-
+ \overline{x}^2=
+ \nonumber\\&&{}=
+ \overline{x^2}\overline{\frac{\sum_i{g_i^2}}{\sum\sum{g_ig_j}}}-
+ \overline{x}^2
+ \overline{\frac{\sum\sum{g_ig_j}-\sum\sum_{i\ne{j}}{g_ig_j}}
+ {\sum\sum{g_ig_j}}}= \nonumber\\&&
+ =\left(\overline{x^2}-\overline{x}^2\right)
+ \overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=
+ \sigma_x^2\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=
+ \nonumber\\
+ &&{}=\frac{\sigma_x^2}{N}{\times}G_{\rm{gfactor}}^2
+\label{eqCOGSigmaGG}
+\end{eqnarray}
+
+where
+\begin{eqnarray}
+ G_{\rm{gfactor}}^2 = N\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}
+\label{eqCOGGGfactor0}
+\end{eqnarray}
+
+The diffusion term is effectively multiplied by gas gain factor
+$G_{\rm{gfactor}}$. For sufficiently large number of electrons,
+when $g_i^2$ and $\sum\sum{g_ig_j}$ are quasi independent
+variables, equation (\ref{eqCOGGGfactor0}) can be transformed to
+the following
+
+\begin{eqnarray}
+ \lefteqn{G_{\rm{gfactor}}^2 \approx
+ N\frac{\overline{\sum{g_i^2}}}
+ {\overline{\sum\sum{g_ig_j}}}}\nonumber\\
+ &&{} =
+ N\frac{N\overline{g^2}}{N(N-1)\overline{g}^2+N\overline{g^2}}=
+ \nonumber\\
+ &&{} =N\frac{ \left(\sigma_g^2/\overline{g}^2+1 \right)}
+ {N+\sigma_g^2/\overline{g}^2}
+\label{eqCOGGGfactorE}
+\end{eqnarray}
+
+Gas gain fluctuation of the gas detector working in proportional
+regime is described with the exponential distribution with the
+mean value $\bar{g}$ and r.m.s.
+\begin{eqnarray}
+ \sigma_{\rm{g}} =\bar{g}
+\label{eqSigmaexp}
+\end{eqnarray}
+
+Substituting $\sigma_{\rm{g}}$ into equation
+(\ref{eqCOGGGfactorE})
+\begin{eqnarray}
+ G_{\rm{gfactor}}^2 =\frac{2N}{N+1}.
+\label{eqCOGGGfactorR}
+\end{eqnarray}
+
+Gas multiplication fluctuation in chamber deteriorates
+$\sigma_{X_{\rm{COG}}}$ by a factor of about ${\sqrt{2}}$. The
+prediction of this model is in good agreement with results from
+the simulation.
+
+
+\subsection{Secondary ionization effect}
+
+Charged particle penetrating the gas of the detector produces
+{\it{N}} primary electrons. Primary electron {\it{i}} produces
+$n_{\rm{s}}^i-1$ secondary electrons. Each of these electrons is
+amplified in the electric field by a factor of $g_j$.
+
+Each primary cluster is characterized by a position $x_i$ with
+mean value $\overline{x}$ and $\sigma_x$. The COG given by
+equation (\ref{eqCOGdefGG}) is modified to the following form:
+
+\begin{eqnarray}
+ X_{\rm{COG}}=\frac{1}{\sum_{i=1}^N\sum_{j=1}^{n_i}{g_j^{i}}}
+ \sum_{i=1}^{N}{x_i}\sum_{j=1}^{n_i}{g_j^{i}}.
+\label{eqCOGdefGGPIO}
+\end{eqnarray}
+A new variable $G_n$ is introduced as the total electron gain:
+\begin{eqnarray}
+ G_n=\sum_{j=1}^{n}{g_j}.
+\label{eqGNdef}
+\end{eqnarray}
+
+
+Knowing the distribution of {\it{n}} and {\it{g}} and assuming
+that {\it{n}} and {\it{g}} are independent variables the mean
+value and variance of the $G_n$ can be expressed as:
+
+\begin{eqnarray}
+ \lefteqn{
+ \overline{G_n}=\overline{n}\overline{g}} \\
+ &&{}
+ \frac{\sigma^2_{G_n}}{\overline{G_n^2}}=
+ \frac{\sigma^2_n}{\overline{n}^2}+
+ \frac{\sigma^2_g}{\overline{g}^2}
+ \frac{1}{\overline{n}}
+\label{eqGNsigma}
+\end{eqnarray}
+
+Inserting $G_n$ into equation (\ref{eqCOGdefGGPIO}) results in an
+equation similar to the equation (\ref{eqCOGdefGG}).
+
+Multiplicative factor $G_{\rm{Lfactor}}$ is defined as an analog
+of $G_{\rm{gfactor}}$, from the equation (\ref{eqCOGGGfactor0})
+\begin{eqnarray}
+ G_{\rm{Lfactor}}^2 = N\frac{\overline{\sum{G_i^2}}}
+ {\overline{\sum\sum{G_iG_j}}}.
+\label{eqCOGLfactor0}
+\end{eqnarray}
+
+Using the new variable $G_n$ and simply replacing gas gain
+{\it{g}} by $G_n$ in the similar way as in equation
+(\ref{eqCOGGGfactorE}) does not work. For $1/E^{2}$
+parametrization of secondary ionization process
+$\sigma^2_{G_n}/\overline{G_n}$ goes to infinity and thus
+$\sigma^2_{X_{COG}}=\sigma_x^2$. Moreover $G_i^2$ and
+$\sum\sum{G_iG_j}$ are not quasi independent as the sum
+$\sum\sum{G_iG_j}$ could be given by one "exotic" electron
+cluster. Approximations used for deriving the equation
+(\ref{eqCOGGGfactorE}) are not valid for secondary ionization
+effect.
+
+In order to estimate the impact of this effect on COG equation
+(\ref{eqCOGLfactor0}) has to be solved numerically. Simulation
+showed that $G_{\rm{Lfactor}}$ does not depend strongly on the cut
+used for maximum number of electrons created in the process of
+secondary ionization. A change of the cut, from 1000 electrons up
+produces a change of about 3\% in $G_{\rm{Lfactor}}$.
+
+Equation (\ref{eqCOGGGfactorE}) is not applicable in this
+situation because of the infinity of the $\sigma_G$. According to
+the simulation, the threshold on the number of electrons in the
+cluster has a little influence to the resulting
+$G_{\rm{Lfactor}}$. Therefore we fit simulated $G_{\rm{Lfactor}}$
+with formula (\ref{eqCOGGGfactorE}) where
+$\sigma_G^2/\overline{G}^2$ was a free parameter. However, this
+parametrization does not describe the data for wide enough range
+of {\it{N}}. In further study the linear parametrization of the
+COG factor was used. This parametrization was validated on
+reasonable interval of {\it{N}}.
+
+
+
+\section{Center-of-gravity error parametrization}
+
+Detected position of charged particle is a random variable given
+by several stochastic processes: diffusion, angular effect, gas
+gain fluctuation, Landau fluctuation of the secondary ionization,
+$\mathbf{E{\times}B}$ effect, electronic noise and systematic
+effects (like space charge, etc.). The relative influence of these
+processes to the resulting distortion of position determination
+depends on the detector parameters. In the big drift detectors
+like the ALICE TPC the main contribution is given by diffusion,
+gas gain fluctuation, angular effect and secondary ionization
+fluctuation.
+
+Furthermore we will use following assumptions:
+\begin{itemize}
+\item $N_{\rm{prim}}$ primary electrons are produced at a random
+positions $x_i$ along the particle trajectory. \item $n_i-1$
+electrons are produced in the process of secondary ionization.
+\item Displacement of produced electrons due to the thermalization
+is neglected.
+\end{itemize}
+
+Each of electrons is characterized by a random vector
+$\vec{z}^i_j$
+\begin{eqnarray}
+ \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j,
+\label{eqZtot}
+\end{eqnarray}
+where {\it{i}} is the index of primary electron cluster and
+{\it{j}} is the index of the secondary electron inside of the
+primary electron cluster. Random variable $\vec{x}^i$ is a
+position where the primary electron was created. The position
+$\vec{y}^i_j$ is a random variable specific for each electron. It
+is given mainly by a diffusion.
+
+The center of gravity of the electron cloud is given:
+\begin{eqnarray}
+ \lefteqn{\vec{z}_{\rm{COG}}=\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
+ \sum_{j=1}^{n_i}{g_j^i}}
+ \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{z}_j^i}=}
+ \nonumber\\
+ &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
+ \sum_{j=1}^{n_i}{g_j^i}}
+ \sum_{i=1}^{N_{\rm{prim}}}\vec{x}^i\sum_{j=1}^{n_i}{g_j^i}+\nonumber\\
+ &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
+ \sum_{j=1}^{n_i}{g_j^i}}
+ \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{y}_j^i}=
+ \nonumber\\ \nonumber\\
+ &&{}
+ \vec{x}_{\rm{COG}}+\vec{y}_{\rm{COG}}.
+\label{eqCOGSec}
+\end{eqnarray}
+
+The mean value $\overline{\vec{z}_{\rm{COG}}}$ is equal to the sum
+of mean values $\overline{\vec{x}_{\rm{COG}}}$ and
+$\overline{\vec{y}_{\rm{COG}}}$.
+
+The sigma of COG in one of the dimension of vector
+$\vec{z}_{1COG}$ is given by following equation
+\begin{eqnarray}
+ \lefteqn{\sigma_{z_{\rm{1COG}}}^2=\sigma_{x_{\rm{1COG}}}^2+
+ \sigma_{y_{\rm{1COG}}}^2+}\nonumber\\
+ &&{}
+ 2\left(\overline{x_{\rm{1COG}}y_{\rm{1COG}}}-\bar{x}_{\rm{1COG}}
+ \bar{y}_{1COG}\right).
+\label{eqCOGSigSec}
+\end{eqnarray}
+
+If the vectors $\vec{x}$ and $\vec{y}$ are independent random
+variables the last term in the equation (\ref{eqCOGSigSec}) is
+equal to zero.
+\begin{eqnarray}
+ \sigma_{z_{1COG}}^2=\sigma_{x_{\rm{1COG}}}^2+
+ \sigma_{y_{\rm{1COG}}}^2,
+\label{eqCOGSigSecIn}
+\end{eqnarray}
+r.m.s. of COG distribution is given by the sum of r.m.s of
+{\it{x}} and {\it{y}} components.
+
+In order to estimate the influence of the $\mathbf{E{\times}B}$
+and unisochronity effect to the space resolution two additional
+random vectors are added to the initial electron position.
+
+
+\begin{eqnarray}
+\vec{z}^i_j =\vec{x}^i+\vec{y}^i_j+
+ \vec{X}_{\mathbf{E{\times}B}}(\vec{x}^i+\vec{y}^i_j)+
+ \vec{X}_{\rm{Unisochron}}(\vec{x}^i+\vec{y}^i_j).
+\label{eqZtotplus}
+\end{eqnarray}
+The probability distributions of $\vec{X}_{\mathbf{E{\times}B}}$
+and $\vec{X}_{\rm{Unisochron}}$ are functions of random vectors
+$\vec{x^i}$ and $\vec{y^i_j}$, and they are strongly correlated.
+However, simulation indicates that in large drift detectors
+distortions, due to these effects, are negligible compared with a
+previous one.
+
+Combining previous equation and neglecting $\mathbf{E{\times}B}$
+and unisochronity
+effects, the COG distortion parametrization appears as:\\
+{$\sigma_{z}$} of cluster center in {\it{z}} (time) direction
+\begin{eqnarray}\
+ \lefteqn{\sigma^2_{{z_{\rm{COG}}}} = \frac{D^2_{\rm{L}}
+ L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}
+ \frac{{\tan^2\alpha}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+
+ \sigma^2_{\rm{noise}},
+ \label{eqResZ1}
+\end{eqnarray}
+
+and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction
+ \begin{eqnarray}
+ \lefteqn{\sigma^2_{y_{\rm{COG}}} = \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}
+ \frac{{\tan^2\beta}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+
+ \sigma^2_{\rm{noise}},
+ \label{eqResY1}
+ \end{eqnarray}
+ where
+${N_{\rm{ch}}}$ is the total number of electrons in the cluster,
+${N_{\rm{chprim}}}$ is the number of primary electrons in the
+cluster, ${G_{\rm{g}}}$ is the gas gain fluctuation factor,
+${G_{\rm{Lfactor}}}$ is the secondary ionization fluctuation
+factor and $\sigma_{\rm{noise}}$ describe the contribution of the
+electronic noise to the resulting sigma of the COG.
+
+\section{Precision of cluster COG determination using measured
+amplitude}
+
+We have derived parametrization using as parameters the total
+number of electrons ${N_{\rm{ch}}}$ and the number of primary
+electrons ${N_{\rm{chprim}}}$. This parametrization is in good
+agreement with simulated data, where the ${N_{\rm{ch}}}$ and
+${N_{\rm{chprim}}}$ are known. It can be used as an estimate for
+the limits of accuracy, if the mean values
+$\overline{N}_{\rm{ch}}$ and $\overline{N}_{\rm{chprim}}$ are used
+instead.
+
+The ${N_{\rm{ch}}}$ and ${N_{\rm{chprim}}}$ are random variables
+described by a Landau distribution, and Poisson distribution
+respectively .
+
+In order to use previously derived formulas (\ref{eqResZ1},
+\ref{eqResY1}), the number of electrons can be estimated assuming
+their proportionality to the total measured charge $A$ in the
+cluster. However, it turns out that an empirical parametrization
+of the factors $G(N)/N=G(A)/(kA)$ gives better results.
+Formulas (\ref{eqResZ1}) and (\ref{eqResY1}) are transformed to following form:\\
+
+{$\sigma_{z}$} of cluster center in {\it{z}} (time) direction:
+ \begin{eqnarray}
+ \lefteqn{\sigma^2_{z_{\rm{COG}}} =
+ \frac{D^2_{\rm{L}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\
+ &&{}
+ \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}
+ \label{eqZtotAmp}
+ \end{eqnarray}
+
+and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction:
+ \begin{eqnarray}
+ \lefteqn{\sigma_{y_{\rm{COG}}} =
+ \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\
+ &&{}
+ \frac{\tan^2\beta~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}
+ \label{eqYtotAmp}
+ \end{eqnarray}
+
+\section{Estimation of the precision of cluster position
+determination using measured cluster shape}
+
+The shape of the cluster is given by the convolution of the
+responses to the electron avalanches. The time response function
+and the pad response function are almost gaussian, as well as the
+spread of electrons due to the diffusion. The spread due to the
+angular effect is uniform. Assuming that the contribution of the
+angular spread does not dominate the cluster width, the cluster
+shape is not far from gaussian. Therefore, we can use the
+parametrization
+
+\begin{equation}
+ f(t,p) = K_{\rm{Max}}.\exp\left(-\frac{(t-t_{\rm{0}})^2}{2\sigma_{\rm{t}}^2}-
+ \frac{(p-p_{\rm{0}})^2}{2\sigma_{\rm{p}}^2}\right),
+ \label{eq:GaussTP}
+\end{equation}
+where ${K_{\rm{Max}}}$ is the normalization factor, $t$ and $p$
+are time and pad bins, $t_0$ and $p_0$ are centers of the cluster
+in time and pad direction and $\sigma_{\rm{t}}$ and
+$\sigma_{\rm{p}}$ are the r.m.s. of the time and pad cluster
+distribution.
+
+ The mean width of the cluster distribution is given by:
+\begin{equation}
+ \sigma_{\rm{t}} = \sqrt{D{\rm{^2_L}}L_{\rm{drift}}+\sigma^2_{\rm{preamp}}+
+ \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12}},
+\end{equation}
+
+
+\begin{equation}
+ \sigma_{\rm{p}} = \sqrt{D{\rm{^2_T}}L_{\rm{drift}}+\sigma^2_{\rm{PRF}}+
+ \frac{\tan^2\beta~L_{\rm{pad}}^2}{12}},
+\end{equation}
+where ${\sigma_{\rm{preamp}}}$ and ${\sigma_{\rm{PRF}}}$ are the
+r.m.s. of the time response function and pad response function,
+respectively.
+
+The fluctuation of the shape depends on the contribution of the
+random diffusion and angular spread, and on the contribution given
+by a gas gain fluctuation and secondary ionization. The
+fluctuation of the time and pad response functions is small
+compared with the previous one.
+
+The measured r.m.s of the cluster is influenced by a threshold
+effect.
+\begin{equation}
+ \sigma_{\rm{t}}^2 = \sum_{A(t,p)>\rm{threshold}}{(t-t_{\rm{0}})^2{\times}A(t,p)}
+\end{equation}
+The threshold effect can be eliminated using two dimensional
+gaussian fit instead of the simple COG method. However, this
+approach is slow and, moreover, the result is very sensitive to
+the gain fluctuation.
+
+To eliminate the threshold effect in r.m.s. method, the bins
+bellow threshold are replaced with a virtual charge using
+gaussian interpolation of the cluster shape. The introduction of
+the virtual charge improves the precision of the COG measurement.
+Large systematic shifts in the estimate of the cluster position
+(depending on the local track position relative to pad--time) due
+to the threshold are no longer observed.
+
+Measuring the r.m.s. of the cluster, the local diffusion and
+angular spread of the electron cloud can be estimated. This
+provides additional information for the estimation of
+distortions. A simple additional correction function is used:
+\begin{eqnarray}
+ \sigma_{\rm{COG}} \rightarrow
+ \sigma_{\rm{COG}}(A){\times}(1+{\rm{const} {\times}\frac{\delta
+ \rm{RMS}}{\rm{teorRMS}}}),
+\label{eqResUsingRMS}
+\end{eqnarray}
+where $\sigma_{\rm{COG}}(A)$ is calculated according formulas
+\ref{eqResY1} and \ref{eqResZ1}, and the
+$\delta\rm{RMS}/\rm{teorRMS}$ is the relative distortion of the
+signal shape from the expected one.
+
+
+
+
+
+
+\section{TPC cluster finder}
+
+The classical approach for the beginning of the tracking was
+chosen. Before the tracking itself, two-dimensional clusters in
+pad-row--time planes are found. Then the positions of the
+corresponding space points are reconstructed, which are
+interpreted as the crossing points of the tracks and the centers
+of the pad rows. We investigate the region 5$\times$5 bins in
+pad-row--time plane around the central bin with maximum amplitude.
+The size of region, 5$\times$5 bins, is bigger than typical size
+of cluster as the $\sigma_{\rm{t}}$ and $\sigma_{\rm{pad}}$ are
+about 0.75 bins.
+
+The COG and r.m.s are used to characterize cluster. The COG and
+r.m.s are affected by systematic distortions induced by the
+threshold effect. Depending on the number of time bins and pads in
+clusters the COG and r.m.s. are affected in different ways.
+Unfortunately, the number of bins in cluster is the function of
+local track position. To get rid of this effect, two-dimensional
+gaussian fitting can be used.
+
+Similar results can be achieved by so called r.m.s. fitting using
+virtual charge. The signal below threshold is replaced by the
+virtual charge, its expected value according a interpolation. If
+the virtual charge is above the threshold value, then it is
+replaced with amplitude equal to the threshold value. The signal
+r.m.s is used for later error estimation and as a criteria for
+cluster unfolding. This method gives comparable results as
+gaussian fit of the cluster but is much faster. Moreover, the COG
+position is less sensitive to the gain fluctuations.
+
+The cluster shape depends on the track parameters. The response
+function contribution and diffusion contribution to the cluster
+r.m.s. are known during clustering. This is not true for a angular
+contribution to the cluster width. The cluster finder should be
+optimised for high momentum particle coming from the primary
+vertex. Therefore, a conservative approach was chosen, assuming
+angle $\alpha$ to be zero. The tangent of the angle $\beta$ is
+given by {\it{z}}-position and pad-row radius, which is known
+during clustering.
+
+
+\subsection{Cluster unfolding}
+
+The estimated width of the cluster is used as criteria for cluster
+unfolding. If the r.m.s. in one of the directions is greater then
+critical r.m.s, cluster is considered for unfolding. The fast
+spline method is used here. We require the charge to be conserved
+in this method. Overlapped clusters are supposed to have the same
+r.m.s., which is equivalent to the same track angles. If this
+assumption is not fulfilled, tracks diverge very rapidly.
+
+
+\begin{figure}[t]
+\centering
+\includegraphics[width=60mm,angle=-90]{picCluster/unfolding1.eps}
+\caption{
+Schematic view of unfolding principle.} \label{figUnfolding1}
+\end{figure}
+\begin{figure}[t]
+\centering
+\includegraphics[width=60mm,angle=-90]{picCluster/unfoldingres.eps}
+\caption{ Dependence of the position residual as function of the
+distance to the second cluster.} \label{figUnfoldingRes}
+\end{figure}
+
+The unfolding algorithm has the following steps:
+\begin{itemize}
+
+\item Six amplitudes $C_i$ are investigated (see fig.
+\ref{figUnfolding1}). First (left) local maxima, corresponding to
+the first cluster is placed at position 3, second (right) local
+maxima corresponding to the second cluster is at position 5.
+
+\item In the first iteration, amplitude in bin 4 corresponding to
+the cluster on left side $A_{\rm{L4}}$ is calculated using
+polynomial interpolation, assuming virtual amplitude at
+$A_{\rm{L5}}$ and derivation at $A_{\rm{L5}}^{'}$ to be 0.
+Amplitudes $A_{\rm{L2}}$ and $A_{\rm{L3}}$ are considered to be
+not influenced by overlap ($A_{\rm{L2}}=C_2$ and
+$A_{\rm{L3}}=C_3)$.
+
+\item The amplitude $A_{\rm{R4}}$ is calculated in similar way. In
+the next iteration the amplitude $A_{\rm{L4}}$ is calculated
+requiring charge conservation
+$C_{\rm{4}}=A_{\rm{R4}}+A_{\rm{L4}}$. Consequently
+\begin{eqnarray}
+ A_{\rm{L4}} \rightarrow
+ C_{\rm{4}}\frac{A_{\rm{L4}}}{A_{\rm{L4}}+A_{\rm{R4}}}
+\end{eqnarray}
+and
+\begin{eqnarray}
+ A_{\rm{R4}} \rightarrow
+ C_{\rm{4}}\frac{A_{\rm{R4}}}{A_{\rm{L4}}+A_{\rm{R4}}}.
+\end{eqnarray}
+\end{itemize}
+
+
+Two cluster resolution depends on the distance between the two
+tracks. Until the shape of cluster triggers unfolding, there is a
+systematic shifts towards to the COG of two tracks (see fig.
+\ref{figUnfoldingRes}), only one cluster is reconstructed.
+Afterwards, no systematic shift is observed.
+
+
+\subsection{Cluster characteristics}
+
+The cluster is characterized by the COG in {\it{y}} and {\it{z}}
+directions (fY and fZ) and by the cluster width (fSigmaY,
+fSigmaZ). The deposited charge is described by the signal at
+maximum (fMax), and total charge in cluster (fQ). The cluster type
+is characterized by the data member fCType which is defined as a
+ratio of the charge supposed to be deposited by the track and
+total charge in cluster in investigated region 5$\times$5. The
+error of the cluster position is assigned to the cluster only
+during tracking according formulas
+ (\ref{eqZtotAmp}) and (\ref{eqYtotAmp}), when track
+ angles $\alpha$ and $\beta$ are known with sufficient precision.
+
+
+Obviously, measuring the position of each electron separately the
+effect of the gas gain fluctuation can be removed, however this is
+not easy to implement in the large TPC detectors. Additional
+information about cluster asymmetry can be used, but the resulting
+improvement of around 5\% in precision on simulated data is
+negligible, and it is questionable, how successful will be such
+correction for the cluster asymmetry on real data.
+
+However, a cluster asymmetry can be used as additional criteria
+for cluster unfolding. Let's denote $\mu_i$ the {\it{i}}-th
+central momentum of the cluster, which was created by overlapping
+from two sub-clusters with unknown positions and deposited energy
+(with momenta $^1\mu_i$ and $^2\mu_i$).
+
+Let $r_1$ is the ratio of two clusters amplitudes:
+\[r_1={^1\mu_0}/({^1\mu_0}+{^2\mu_0})\] and the track distance {\it{d}} is equal to
+\[d = {^1\mu_1} -{^2\mu_1}.\]
+
+Assuming that the second moments for both sub-clusters are the
+same (${^0\mu_2}={^1\mu_2}={^2\mu_2}$), two sub-clusters distance
+{\it{d}} and amplitude ratio $r_1$ can be estimated:
+\begin{eqnarray}
+ R = \frac{(\mu_3^6)}{(\mu_2^2-{^0\mu_2^2})^3}\\
+ r_{\rm{1}} =0.5\pm0.5{\times}\sqrt{\frac{1}{1-4/R}} \\
+ d = \sqrt{(4+R){\times}(\mu_2^2-{^0\mu_2^2})}
+\label{eqMeas}
+\end{eqnarray}
+
+In order to trigger unfolding using the shape information
+additional information about track and mean cluster shape over
+several pad-rows are needed. This information is available only
+during tracking procedure.
+
+
+
+\subsection{Space point resolution parameterization}
+
+The space point resolution is the function of many parameters but for the ALICE TPC the dominant one are the diffusion, track inclination angle and deposited charge.
+The space point resolution was extracted from the data in bins of these variables.
+
+In the first approximation the angular part and diffusion part are independent. The
+paramaterization is obtained fitting parameters $p_{0}$,$p_L$ and $p_A$
+\begin{eqnarray}\
+ \sigma^2_{{\rm{COG}}} \approx p^2_0+p^2_{L}L_{\rm{Drift}}+p^2_{A}\tan^2\alpha
+ \label{eqResCOG0}
+ \nonumber\\
+ p^2_L \approx \frac{\sigma^2_DG_{\rm{g}}}{N_{\rm{ch}}}
+ \nonumber\\
+ p^2_A \approx \frac{L_{\rm{pad}}^2G_{\rm{Lfactor}}}{N_{\rm{chprim}}}
+\end{eqnarray}
+
+
+
+\begin{table}
+\caption{Resolution parameterization}
+\begin{tabular}{|l|l|l|l|} \hline
+Pad size & 0.75x0.4 $cm^2$ & 1.0x0.6$cm^2$ & 1.5x0.6$cm^2$ \\ \hline
+$p_{0y}$ & 0.026 cm & 0.031 cm & 0.023 cm \\ \hline
+$p_{0z}$ & 0.032 cm & 0.032 cm & 0.028 cm \\ \hline
+$p_{Ly}\sqrt{L_{pad}}$ & 0.0051 & 0.0060 & 0.0059 \\ \hline
+$p_{Lz}\sqrt{L_{pad}}$ & 0.0056 & 0.0056 & 0.0059 \\ \hline
+$p_{Ay}/\sqrt{L_{pad}}$ & 0.13 $cm^{1/2}$ & 0.15 $cm^{1/2}$ & 0.15 $cm^{1/2}$ \\ \hline
+$p_{Az}/\sqrt{L_{pad}}$ & 0.15 $cm^{1/2}$ & 0.16 $cm^{1/2}$ & 0.17 $cm^{1/2}$ \\ \hline
+
+\end{tabular}
+\label{table:PointResolFitParam}
+\end{table}
+
+
+\begin{eqnarray}\
+ N_{\rm{ch}} \approx {L_{\rm{pad}}} \nonumber \\
+ N_{\rm{chprim}} \approx {L_{\rm{pad}}} \nonumber \\
+ \nonumber\\
+ p_L \approx \frac{1}{\sqrt{L_{\rm{pad}}}}
+ \nonumber\\
+ p_A \approx \sqrt{L_{\rm{pad}}}
+\label{eq:ResolScaling}
+\end{eqnarray}
+
+
+The TPC space resolution is scaling with the number of contributed electrons
+$N_{\rm{chprim}}$ and ${N_{\rm{ch}}}$, therefore is scaling with pad length.
+In ALICE TPC three different pad gemetries are used.
+The space point resolution was fitted for separatelly for each geometry. The fitted parameters $p_0$ $p_L$ and $p_A$ are shown in the table \ref{table:PointResolFitParam} rescaled with the pad length.
+
+
+The agreement between previously mentioned fit and the data is on thel level of the
+$\approx10-20\%$. In previous formula we assumed that all of the electrons created in ionization are contibuting to the measured signal. Because of the threshold effect the
+part of the signal is cut-off. The fraction of the signal bellow threshold is proportional to the response function witdth and is incresing with drift length and inclination angle. The following correction functions are used:
+\begin{eqnarray}\
+ \nonumber\\
+ p_L \approx p_{L0}p_{LC}=p_{L0}(1+p_{L1}L_{\rm{Drift}}+p_{L2}\tan^2\alpha)
+ \nonumber\\
+ p_A \approx p_{A0}p_{AC}=p_{A0}(1+p_{A1}L_{\rm{Drift}}+p_{A2}\tan^2\alpha)
+\label{eq:PointResolFitCorrection}
+\end{eqnarray}
+
+To estimate the number of electrons contibuted to creation of the signal, the cluster charge can be used. Additional correction was tested. Terms proportional to $1/Q$ can be added to the formula \ref{eq:PointResolFitCorrection}. However the space point resolution is improving only until some limit (see fig.\ref{figPointResolYQ}) determined by the range of the secondary delta electrons. Q dependent
+
+\begin{figure}
+ \centering\epsfig{figure=picClusterResol/QresolY_mag.eps,width=0.7\linewidth}
+ \centering\epsfig{figure=picClusterResol/QresolY.eps,width=0.7\linewidth}
+ \label{figPointResolYQ}
+\caption{Space point resolution in Y direcition as function of deposited charge $Q_{max}$.
+Upper part-with magnetic field, lower part without magnetic field. Space point resolution is improving increasing deposited charge $Q_{max}$. Starting from some critical charge the resolution is worsening. The effect can be explained to be due to the secondary electrons - delta rays. The range of the delta rays is much smaller in presence of the magnetic field.
+}
+
+\end{figure}
+
+
+The measured resolution in Y and Z direction and corresponding fits are shown on picure \ref{figPointResolYDRTAN} and \ref{figPointResolZDRTAN}. The agrement with the data is on the level of about 2\%.
+
+
+
+
+\begin{figure}
+ \centering\epsfig{figure=picClusterResol/YResol_Pad0.eps,width=0.7\linewidth}
+ \centering\epsfig{figure=picClusterResol/YResol_Pad1.eps,width=0.7\linewidth}
+ \centering\epsfig{figure=picClusterResol/YResol_Pad2.eps,width=0.7\linewidth}
+ \caption{Space point resolution in Y direcition as function of the drift length and the inlination angle.}
+ \label{figPointResolYDRTAN}
+\end{figure}
+
+\begin{figure}
+ \centering\epsfig{figure=picClusterResol/ZResol_Pad0.eps,width=0.7\linewidth}
+ \centering\epsfig{figure=picClusterResol/ZResol_Pad1.eps,width=0.7\linewidth}
+ \centering\epsfig{figure=picClusterResol/ZResol_Pad2.eps,width=0.7\linewidth}
+ \caption{Space point resolution in Z direcition as function of the drift length and the inlination angle.}
+ \label{figPointResolZDRTAN}
+\end{figure}
+
+
+
+
+\end{document}
git://git.uio.no / u / mrichter / AliRoot.git / blobdiff