3 \documentclass[a4paper,12pt]{article}
5 \usepackage[pdftex]{graphicx}
7 \usepackage[dvips]{graphicx}
15 \floatplacement{figure}{H}
16 \floatplacement{table}{H}
23 \section{Accuracy of local coordinate measurement}
28 \includegraphics[width=60mm,angle=-90]{picCluster/pic2.eps}
29 \includegraphics[width=60mm,angle=-90]{picCluster/pic1.eps}
30 \caption{Schematic view of the detection process in TPC (upper
31 part - perspective view, lower part - side view).} \label{figTPC}
34 The accuracy of the coordinate measurement is limited by a track
35 angle which spreads ionization and by diffusion which amplifies
38 The track direction with respect to pad plane is given by two
39 angles $\alpha$ and $\beta$ (see fig.~\ref{figTPC}). For the
40 measurement along the pad-row, the angle $\alpha$ between the
41 track projected onto the pad plane and pad-row is relevant. For
42 the measurement of the the drift coordinate ({\it{z}}--direction)
43 it is the angle $\beta$ between the track and {\it{z}} axis
46 The ionization electrons are randomly distributed along the
47 particle trajectory. Fixing the reference {\it{x}} position of an
48 electron at the middle of pad-row, the {\it{y}} (resp. {\it{z}})
49 position of the electron is a random variable characterized by
50 uniform distribution with the width $L_{\rm{a}}$, where
51 $L_{\rm{a}}$ is given by the pad length $L_{\rm{pad}}$ and the
52 angle $\alpha$ (resp. $\beta$):
53 \[L_{\rm{a}}=L_{\rm{pad}}\tan\alpha\]
55 The diffusion smears out the position of the electron with
56 gaussian probability distribution with $\sigma_{\rm{D}}$.
57 Contribution of the $\mathbf{E{\times}B}$ and unisochronity
58 effects for the Alice TPC are negligible. The typical resolution
59 in the case of ALICE TPC is on the level of
60 $\sigma_{y}\sim$~0.8~mm and $\sigma_{z}\sim$~1.0~mm integrating
61 over all clusters in the TPC.
65 \subsection{Gas gain fluctuation effect}
67 Being collected on sense wire, electron is "multiplied" in strong
68 electric field. This multiplication is subject of a large
69 fluctuations, contributing to the cluster position resolution.
70 Because of these fluctuations the center of gravity of the
71 electron cloud can be shifted.
73 Each electron is amplified independently. However, in the
74 reconstruction electrons are not treated separately. The Centre Of
75 Gravity (COG) of the cluster is usually used as an estimation for
76 the local track position. The influence of the gas gain
77 fluctuation to the reconstructed point characteristic can be
78 described by a simple model, introducing a weighted COG
81 X_{\rm{COG}}=\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}},
84 where {\it{N}} is the total number of electrons in the cluster and
85 $g_i$ is a random variable equal to a gas amplification for given
88 The mean value of $X_{\rm{COG}}$ is equal to the mean value
89 $\overline{x}$ of the original distribution of electrons
92 \overline{\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}}}
93 =\overline{x}\overline{\frac{\sum_{i=1}^{N}{g_i}}
94 {\sum_{i=1}^N{g_i}}} =\overline{x}.
98 However, the same is not true for the dispersion of the position,
99 %$\sigma^2_{X_{COG}}\sigma_x^2$:
102 \lefteqn{ \sigma^2_{X_{\rm{COG}}}
103 =\overline{X_{\rm{COG}}^2}-\overline{X_{\rm{COG}}}^2=}\nonumber\\&&{}
104 =\overline{\left(\frac{1}{\sum_{i=1}^N{g_i}}\sum_{i=1}^{N}{g_ix_i}
105 \right)^2}-\overline{x}^2=
107 &&{}=\overline{\frac{{\sum\sum{x_ix_jg_ig_j}}}{{\sum\sum{g_ig_j}}}}-
110 \overline{x^2}\overline{\frac{\sum_i{g_i^2}}{\sum\sum{g_ig_j}}}-
112 \overline{\frac{\sum\sum{g_ig_j}-\sum\sum_{i\ne{j}}{g_ig_j}}
113 {\sum\sum{g_ig_j}}}= \nonumber\\&&
114 =\left(\overline{x^2}-\overline{x}^2\right)
115 \overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=
116 \sigma_x^2\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=
118 &&{}=\frac{\sigma_x^2}{N}{\times}G_{\rm{gfactor}}^2
124 G_{\rm{gfactor}}^2 = N\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}
125 \label{eqCOGGGfactor0}
128 The diffusion term is effectively multiplied by gas gain factor
129 $G_{\rm{gfactor}}$. For sufficiently large number of electrons,
130 when $g_i^2$ and $\sum\sum{g_ig_j}$ are quasi independent
131 variables, equation (\ref{eqCOGGGfactor0}) can be transformed to
135 \lefteqn{G_{\rm{gfactor}}^2 \approx
136 N\frac{\overline{\sum{g_i^2}}}
137 {\overline{\sum\sum{g_ig_j}}}}\nonumber\\
139 N\frac{N\overline{g^2}}{N(N-1)\overline{g}^2+N\overline{g^2}}=
141 &&{} =N\frac{ \left(\sigma_g^2/\overline{g}^2+1 \right)}
142 {N+\sigma_g^2/\overline{g}^2}
143 \label{eqCOGGGfactorE}
146 Gas gain fluctuation of the gas detector working in proportional
147 regime is described with the exponential distribution with the
148 mean value $\bar{g}$ and r.m.s.
150 \sigma_{\rm{g}} =\bar{g}
154 Substituting $\sigma_{\rm{g}}$ into equation
155 (\ref{eqCOGGGfactorE})
157 G_{\rm{gfactor}}^2 =\frac{2N}{N+1}.
158 \label{eqCOGGGfactorR}
161 Gas multiplication fluctuation in chamber deteriorates
162 $\sigma_{X_{\rm{COG}}}$ by a factor of about ${\sqrt{2}}$. The
163 prediction of this model is in good agreement with results from
167 \subsection{Secondary ionization effect}
169 Charged particle penetrating the gas of the detector produces
170 {\it{N}} primary electrons. Primary electron {\it{i}} produces
171 $n_{\rm{s}}^i-1$ secondary electrons. Each of these electrons is
172 amplified in the electric field by a factor of $g_j$.
174 Each primary cluster is characterized by a position $x_i$ with
175 mean value $\overline{x}$ and $\sigma_x$. The COG given by
176 equation (\ref{eqCOGdefGG}) is modified to the following form:
179 X_{\rm{COG}}=\frac{1}{\sum_{i=1}^N\sum_{j=1}^{n_i}{g_j^{i}}}
180 \sum_{i=1}^{N}{x_i}\sum_{j=1}^{n_i}{g_j^{i}}.
181 \label{eqCOGdefGGPIO}
183 A new variable $G_n$ is introduced as the total electron gain:
185 G_n=\sum_{j=1}^{n}{g_j}.
190 Knowing the distribution of {\it{n}} and {\it{g}} and assuming
191 that {\it{n}} and {\it{g}} are independent variables the mean
192 value and variance of the $G_n$ can be expressed as:
196 \overline{G_n}=\overline{n}\overline{g}} \\
198 \frac{\sigma^2_{G_n}}{\overline{G_n^2}}=
199 \frac{\sigma^2_n}{\overline{n}^2}+
200 \frac{\sigma^2_g}{\overline{g}^2}
201 \frac{1}{\overline{n}}
205 Inserting $G_n$ into equation (\ref{eqCOGdefGGPIO}) results in an
206 equation similar to the equation (\ref{eqCOGdefGG}).
208 Multiplicative factor $G_{\rm{Lfactor}}$ is defined as an analog
209 of $G_{\rm{gfactor}}$, from the equation (\ref{eqCOGGGfactor0})
211 G_{\rm{Lfactor}}^2 = N\frac{\overline{\sum{G_i^2}}}
212 {\overline{\sum\sum{G_iG_j}}}.
213 \label{eqCOGLfactor0}
216 Using the new variable $G_n$ and simply replacing gas gain
217 {\it{g}} by $G_n$ in the similar way as in equation
218 (\ref{eqCOGGGfactorE}) does not work. For $1/E^{2}$
219 parametrization of secondary ionization process
220 $\sigma^2_{G_n}/\overline{G_n}$ goes to infinity and thus
221 $\sigma^2_{X_{COG}}=\sigma_x^2$. Moreover $G_i^2$ and
222 $\sum\sum{G_iG_j}$ are not quasi independent as the sum
223 $\sum\sum{G_iG_j}$ could be given by one "exotic" electron
224 cluster. Approximations used for deriving the equation
225 (\ref{eqCOGGGfactorE}) are not valid for secondary ionization
228 In order to estimate the impact of this effect on COG equation
229 (\ref{eqCOGLfactor0}) has to be solved numerically. Simulation
230 showed that $G_{\rm{Lfactor}}$ does not depend strongly on the cut
231 used for maximum number of electrons created in the process of
232 secondary ionization. A change of the cut, from 1000 electrons up
233 produces a change of about 3\% in $G_{\rm{Lfactor}}$.
235 Equation (\ref{eqCOGGGfactorE}) is not applicable in this
236 situation because of the infinity of the $\sigma_G$. According to
237 the simulation, the threshold on the number of electrons in the
238 cluster has a little influence to the resulting
239 $G_{\rm{Lfactor}}$. Therefore we fit simulated $G_{\rm{Lfactor}}$
240 with formula (\ref{eqCOGGGfactorE}) where
241 $\sigma_G^2/\overline{G}^2$ was a free parameter. However, this
242 parametrization does not describe the data for wide enough range
243 of {\it{N}}. In further study the linear parametrization of the
244 COG factor was used. This parametrization was validated on
245 reasonable interval of {\it{N}}.
249 \section{Center-of-gravity error parametrization}
251 Detected position of charged particle is a random variable given
252 by several stochastic processes: diffusion, angular effect, gas
253 gain fluctuation, Landau fluctuation of the secondary ionization,
254 $\mathbf{E{\times}B}$ effect, electronic noise and systematic
255 effects (like space charge, etc.). The relative influence of these
256 processes to the resulting distortion of position determination
257 depends on the detector parameters. In the big drift detectors
258 like the ALICE TPC the main contribution is given by diffusion,
259 gas gain fluctuation, angular effect and secondary ionization
262 Furthermore we will use following assumptions:
264 \item $N_{\rm{prim}}$ primary electrons are produced at a random
265 positions $x_i$ along the particle trajectory. \item $n_i-1$
266 electrons are produced in the process of secondary ionization.
267 \item Displacement of produced electrons due to the thermalization
271 Each of electrons is characterized by a random vector
274 \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j,
277 where {\it{i}} is the index of primary electron cluster and
278 {\it{j}} is the index of the secondary electron inside of the
279 primary electron cluster. Random variable $\vec{x}^i$ is a
280 position where the primary electron was created. The position
281 $\vec{y}^i_j$ is a random variable specific for each electron. It
282 is given mainly by a diffusion.
284 The center of gravity of the electron cloud is given:
286 \lefteqn{\vec{z}_{\rm{COG}}=\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
287 \sum_{j=1}^{n_i}{g_j^i}}
288 \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{z}_j^i}=}
290 &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
291 \sum_{j=1}^{n_i}{g_j^i}}
292 \sum_{i=1}^{N_{\rm{prim}}}\vec{x}^i\sum_{j=1}^{n_i}{g_j^i}+\nonumber\\
293 &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
294 \sum_{j=1}^{n_i}{g_j^i}}
295 \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{y}_j^i}=
296 \nonumber\\ \nonumber\\
298 \vec{x}_{\rm{COG}}+\vec{y}_{\rm{COG}}.
302 The mean value $\overline{\vec{z}_{\rm{COG}}}$ is equal to the sum
303 of mean values $\overline{\vec{x}_{\rm{COG}}}$ and
304 $\overline{\vec{y}_{\rm{COG}}}$.
306 The sigma of COG in one of the dimension of vector
307 $\vec{z}_{1COG}$ is given by following equation
309 \lefteqn{\sigma_{z_{\rm{1COG}}}^2=\sigma_{x_{\rm{1COG}}}^2+
310 \sigma_{y_{\rm{1COG}}}^2+}\nonumber\\
312 2\left(\overline{x_{\rm{1COG}}y_{\rm{1COG}}}-\bar{x}_{\rm{1COG}}
313 \bar{y}_{1COG}\right).
317 If the vectors $\vec{x}$ and $\vec{y}$ are independent random
318 variables the last term in the equation (\ref{eqCOGSigSec}) is
321 \sigma_{z_{1COG}}^2=\sigma_{x_{\rm{1COG}}}^2+
322 \sigma_{y_{\rm{1COG}}}^2,
323 \label{eqCOGSigSecIn}
325 r.m.s. of COG distribution is given by the sum of r.m.s of
326 {\it{x}} and {\it{y}} components.
328 In order to estimate the influence of the $\mathbf{E{\times}B}$
329 and unisochronity effect to the space resolution two additional
330 random vectors are added to the initial electron position.
334 \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j+
335 \vec{X}_{\mathbf{E{\times}B}}(\vec{x}^i+\vec{y}^i_j)+
336 \vec{X}_{\rm{Unisochron}}(\vec{x}^i+\vec{y}^i_j).
339 The probability distributions of $\vec{X}_{\mathbf{E{\times}B}}$
340 and $\vec{X}_{\rm{Unisochron}}$ are functions of random vectors
341 $\vec{x^i}$ and $\vec{y^i_j}$, and they are strongly correlated.
342 However, simulation indicates that in large drift detectors
343 distortions, due to these effects, are negligible compared with a
346 Combining previous equation and neglecting $\mathbf{E{\times}B}$
348 effects, the COG distortion parametrization appears as:\\
349 {$\sigma_{z}$} of cluster center in {\it{z}} (time) direction
351 \lefteqn{\sigma^2_{{z_{\rm{COG}}}} = \frac{D^2_{\rm{L}}
352 L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}
353 \frac{{\tan^2\alpha}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+
354 \sigma^2_{\rm{noise}},
358 and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction
360 \lefteqn{\sigma^2_{y_{\rm{COG}}} = \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}
361 \frac{{\tan^2\beta}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+
362 \sigma^2_{\rm{noise}},
366 ${N_{\rm{ch}}}$ is the total number of electrons in the cluster,
367 ${N_{\rm{chprim}}}$ is the number of primary electrons in the
368 cluster, ${G_{\rm{g}}}$ is the gas gain fluctuation factor,
369 ${G_{\rm{Lfactor}}}$ is the secondary ionization fluctuation
370 factor and $\sigma_{\rm{noise}}$ describe the contribution of the
371 electronic noise to the resulting sigma of the COG.
373 \section{Precision of cluster COG determination using measured
376 We have derived parametrization using as parameters the total
377 number of electrons ${N_{\rm{ch}}}$ and the number of primary
378 electrons ${N_{\rm{chprim}}}$. This parametrization is in good
379 agreement with simulated data, where the ${N_{\rm{ch}}}$ and
380 ${N_{\rm{chprim}}}$ are known. It can be used as an estimate for
381 the limits of accuracy, if the mean values
382 $\overline{N}_{\rm{ch}}$ and $\overline{N}_{\rm{chprim}}$ are used
385 The ${N_{\rm{ch}}}$ and ${N_{\rm{chprim}}}$ are random variables
386 described by a Landau distribution, and Poisson distribution
389 In order to use previously derived formulas (\ref{eqResZ1},
390 \ref{eqResY1}), the number of electrons can be estimated assuming
391 their proportionality to the total measured charge $A$ in the
392 cluster. However, it turns out that an empirical parametrization
393 of the factors $G(N)/N=G(A)/(kA)$ gives better results.
394 Formulas (\ref{eqResZ1}) and (\ref{eqResY1}) are transformed to following form:\\
396 {$\sigma_{z}$} of cluster center in {\it{z}} (time) direction:
398 \lefteqn{\sigma^2_{z_{\rm{COG}}} =
399 \frac{D^2_{\rm{L}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\
401 \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}
405 and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction:
407 \lefteqn{\sigma_{y_{\rm{COG}}} =
408 \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\
410 \frac{\tan^2\beta~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}
414 \section{Estimation of the precision of cluster position
415 determination using measured cluster shape}
417 The shape of the cluster is given by the convolution of the
418 responses to the electron avalanches. The time response function
419 and the pad response function are almost gaussian, as well as the
420 spread of electrons due to the diffusion. The spread due to the
421 angular effect is uniform. Assuming that the contribution of the
422 angular spread does not dominate the cluster width, the cluster
423 shape is not far from gaussian. Therefore, we can use the
427 f(t,p) = K_{\rm{Max}}.\exp\left(-\frac{(t-t_{\rm{0}})^2}{2\sigma_{\rm{t}}^2}-
428 \frac{(p-p_{\rm{0}})^2}{2\sigma_{\rm{p}}^2}\right),
431 where ${K_{\rm{Max}}}$ is the normalization factor, $t$ and $p$
432 are time and pad bins, $t_0$ and $p_0$ are centers of the cluster
433 in time and pad direction and $\sigma_{\rm{t}}$ and
434 $\sigma_{\rm{p}}$ are the r.m.s. of the time and pad cluster
437 The mean width of the cluster distribution is given by:
439 \sigma_{\rm{t}} = \sqrt{D{\rm{^2_L}}L_{\rm{drift}}+\sigma^2_{\rm{preamp}}+
440 \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12}},
445 \sigma_{\rm{p}} = \sqrt{D{\rm{^2_T}}L_{\rm{drift}}+\sigma^2_{\rm{PRF}}+
446 \frac{\tan^2\beta~L_{\rm{pad}}^2}{12}},
448 where ${\sigma_{\rm{preamp}}}$ and ${\sigma_{\rm{PRF}}}$ are the
449 r.m.s. of the time response function and pad response function,
452 The fluctuation of the shape depends on the contribution of the
453 random diffusion and angular spread, and on the contribution given
454 by a gas gain fluctuation and secondary ionization. The
455 fluctuation of the time and pad response functions is small
456 compared with the previous one.
458 The measured r.m.s of the cluster is influenced by a threshold
461 \sigma_{\rm{t}}^2 = \sum_{A(t,p)>\rm{threshold}}{(t-t_{\rm{0}})^2{\times}A(t,p)}
463 The threshold effect can be eliminated using two dimensional
464 gaussian fit instead of the simple COG method. However, this
465 approach is slow and, moreover, the result is very sensitive to
466 the gain fluctuation.
468 To eliminate the threshold effect in r.m.s. method, the bins
469 bellow threshold are replaced with a virtual charge using
470 gaussian interpolation of the cluster shape. The introduction of
471 the virtual charge improves the precision of the COG measurement.
472 Large systematic shifts in the estimate of the cluster position
473 (depending on the local track position relative to pad--time) due
474 to the threshold are no longer observed.
476 Measuring the r.m.s. of the cluster, the local diffusion and
477 angular spread of the electron cloud can be estimated. This
478 provides additional information for the estimation of
479 distortions. A simple additional correction function is used:
481 \sigma_{\rm{COG}} \rightarrow
482 \sigma_{\rm{COG}}(A){\times}(1+{\rm{const} {\times}\frac{\delta
483 \rm{RMS}}{\rm{teorRMS}}}),
484 \label{eqResUsingRMS}
486 where $\sigma_{\rm{COG}}(A)$ is calculated according formulas
487 \ref{eqResY1} and \ref{eqResZ1}, and the
488 $\delta\rm{RMS}/\rm{teorRMS}$ is the relative distortion of the
489 signal shape from the expected one.
496 \section{TPC cluster finder}
498 The classical approach for the beginning of the tracking was
499 chosen. Before the tracking itself, two-dimensional clusters in
500 pad-row--time planes are found. Then the positions of the
501 corresponding space points are reconstructed, which are
502 interpreted as the crossing points of the tracks and the centers
503 of the pad rows. We investigate the region 5$\times$5 bins in
504 pad-row--time plane around the central bin with maximum amplitude.
505 The size of region, 5$\times$5 bins, is bigger than typical size
506 of cluster as the $\sigma_{\rm{t}}$ and $\sigma_{\rm{pad}}$ are
509 The COG and r.m.s are used to characterize cluster. The COG and
510 r.m.s are affected by systematic distortions induced by the
511 threshold effect. Depending on the number of time bins and pads in
512 clusters the COG and r.m.s. are affected in different ways.
513 Unfortunately, the number of bins in cluster is the function of
514 local track position. To get rid of this effect, two-dimensional
515 gaussian fitting can be used.
517 Similar results can be achieved by so called r.m.s. fitting using
518 virtual charge. The signal below threshold is replaced by the
519 virtual charge, its expected value according a interpolation. If
520 the virtual charge is above the threshold value, then it is
521 replaced with amplitude equal to the threshold value. The signal
522 r.m.s is used for later error estimation and as a criteria for
523 cluster unfolding. This method gives comparable results as
524 gaussian fit of the cluster but is much faster. Moreover, the COG
525 position is less sensitive to the gain fluctuations.
527 The cluster shape depends on the track parameters. The response
528 function contribution and diffusion contribution to the cluster
529 r.m.s. are known during clustering. This is not true for a angular
530 contribution to the cluster width. The cluster finder should be
531 optimised for high momentum particle coming from the primary
532 vertex. Therefore, a conservative approach was chosen, assuming
533 angle $\alpha$ to be zero. The tangent of the angle $\beta$ is
534 given by {\it{z}}-position and pad-row radius, which is known
538 \subsection{Cluster unfolding}
540 The estimated width of the cluster is used as criteria for cluster
541 unfolding. If the r.m.s. in one of the directions is greater then
542 critical r.m.s, cluster is considered for unfolding. The fast
543 spline method is used here. We require the charge to be conserved
544 in this method. Overlapped clusters are supposed to have the same
545 r.m.s., which is equivalent to the same track angles. If this
546 assumption is not fulfilled, tracks diverge very rapidly.
551 \includegraphics[width=60mm,angle=-90]{picCluster/unfolding1.eps}
553 Schematic view of unfolding principle.} \label{figUnfolding1}
557 \includegraphics[width=60mm,angle=-90]{picCluster/unfoldingres.eps}
558 \caption{ Dependence of the position residual as function of the
559 distance to the second cluster.} \label{figUnfoldingRes}
562 The unfolding algorithm has the following steps:
565 \item Six amplitudes $C_i$ are investigated (see fig.
566 \ref{figUnfolding1}). First (left) local maxima, corresponding to
567 the first cluster is placed at position 3, second (right) local
568 maxima corresponding to the second cluster is at position 5.
570 \item In the first iteration, amplitude in bin 4 corresponding to
571 the cluster on left side $A_{\rm{L4}}$ is calculated using
572 polynomial interpolation, assuming virtual amplitude at
573 $A_{\rm{L5}}$ and derivation at $A_{\rm{L5}}^{'}$ to be 0.
574 Amplitudes $A_{\rm{L2}}$ and $A_{\rm{L3}}$ are considered to be
575 not influenced by overlap ($A_{\rm{L2}}=C_2$ and
578 \item The amplitude $A_{\rm{R4}}$ is calculated in similar way. In
579 the next iteration the amplitude $A_{\rm{L4}}$ is calculated
580 requiring charge conservation
581 $C_{\rm{4}}=A_{\rm{R4}}+A_{\rm{L4}}$. Consequently
583 A_{\rm{L4}} \rightarrow
584 C_{\rm{4}}\frac{A_{\rm{L4}}}{A_{\rm{L4}}+A_{\rm{R4}}}
588 A_{\rm{R4}} \rightarrow
589 C_{\rm{4}}\frac{A_{\rm{R4}}}{A_{\rm{L4}}+A_{\rm{R4}}}.
594 Two cluster resolution depends on the distance between the two
595 tracks. Until the shape of cluster triggers unfolding, there is a
596 systematic shifts towards to the COG of two tracks (see fig.
597 \ref{figUnfoldingRes}), only one cluster is reconstructed.
598 Afterwards, no systematic shift is observed.
601 \subsection{Cluster characteristics}
603 The cluster is characterized by the COG in {\it{y}} and {\it{z}}
604 directions (fY and fZ) and by the cluster width (fSigmaY,
605 fSigmaZ). The deposited charge is described by the signal at
606 maximum (fMax), and total charge in cluster (fQ). The cluster type
607 is characterized by the data member fCType which is defined as a
608 ratio of the charge supposed to be deposited by the track and
609 total charge in cluster in investigated region 5$\times$5. The
610 error of the cluster position is assigned to the cluster only
611 during tracking according formulas
612 (\ref{eqZtotAmp}) and (\ref{eqYtotAmp}), when track
613 angles $\alpha$ and $\beta$ are known with sufficient precision.
616 Obviously, measuring the position of each electron separately the
617 effect of the gas gain fluctuation can be removed, however this is
618 not easy to implement in the large TPC detectors. Additional
619 information about cluster asymmetry can be used, but the resulting
620 improvement of around 5\% in precision on simulated data is
621 negligible, and it is questionable, how successful will be such
622 correction for the cluster asymmetry on real data.
624 However, a cluster asymmetry can be used as additional criteria
625 for cluster unfolding. Let's denote $\mu_i$ the {\it{i}}-th
626 central momentum of the cluster, which was created by overlapping
627 from two sub-clusters with unknown positions and deposited energy
628 (with momenta $^1\mu_i$ and $^2\mu_i$).
630 Let $r_1$ is the ratio of two clusters amplitudes:
631 \[r_1={^1\mu_0}/({^1\mu_0}+{^2\mu_0})\] and the track distance {\it{d}} is equal to
632 \[d = {^1\mu_1} -{^2\mu_1}.\]
634 Assuming that the second moments for both sub-clusters are the
635 same (${^0\mu_2}={^1\mu_2}={^2\mu_2}$), two sub-clusters distance
636 {\it{d}} and amplitude ratio $r_1$ can be estimated:
638 R = \frac{(\mu_3^6)}{(\mu_2^2-{^0\mu_2^2})^3}\\
639 r_{\rm{1}} =0.5\pm0.5{\times}\sqrt{\frac{1}{1-4/R}} \\
640 d = \sqrt{(4+R){\times}(\mu_2^2-{^0\mu_2^2})}
644 In order to trigger unfolding using the shape information
645 additional information about track and mean cluster shape over
646 several pad-rows are needed. This information is available only
647 during tracking procedure.
651 \subsection{Space point resolution parameterization}
653 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.
654 The space point resolution was extracted from the data in bins of these variables.
656 In the first approximation the angular part and diffusion part are independent. The
657 paramaterization is obtained fitting parameters $p_{0}$,$p_L$ and $p_A$
659 \sigma^2_{{\rm{COG}}} \approx p^2_0+p^2_{L}L_{\rm{Drift}}+p^2_{A}\tan^2\alpha
662 p^2_L \approx \frac{\sigma^2_DG_{\rm{g}}}{N_{\rm{ch}}}
664 p^2_A \approx \frac{L_{\rm{pad}}^2G_{\rm{Lfactor}}}{N_{\rm{chprim}}}
670 \caption{Resolution parameterization}
671 \begin{tabular}{|l|l|l|l|} \hline
672 Pad size & 0.75x0.4 $cm^2$ & 1.0x0.6$cm^2$ & 1.5x0.6$cm^2$ \\ \hline
673 $p_{0y}$ & 0.026 cm & 0.031 cm & 0.023 cm \\ \hline
674 $p_{0z}$ & 0.032 cm & 0.032 cm & 0.028 cm \\ \hline
675 $p_{Ly}\sqrt{L_{pad}}$ & 0.0051 & 0.0060 & 0.0059 \\ \hline
676 $p_{Lz}\sqrt{L_{pad}}$ & 0.0056 & 0.0056 & 0.0059 \\ \hline
677 $p_{Ay}/\sqrt{L_{pad}}$ & 0.13 $cm^{1/2}$ & 0.15 $cm^{1/2}$ & 0.15 $cm^{1/2}$ \\ \hline
678 $p_{Az}/\sqrt{L_{pad}}$ & 0.15 $cm^{1/2}$ & 0.16 $cm^{1/2}$ & 0.17 $cm^{1/2}$ \\ \hline
681 \label{table:PointResolFitParam}
686 N_{\rm{ch}} \approx {L_{\rm{pad}}} \nonumber \\
687 N_{\rm{chprim}} \approx {L_{\rm{pad}}} \nonumber \\
689 p_L \approx \frac{1}{\sqrt{L_{\rm{pad}}}}
691 p_A \approx \sqrt{L_{\rm{pad}}}
692 \label{eq:ResolScaling}
696 The TPC space resolution is scaling with the number of contributed electrons
697 $N_{\rm{chprim}}$ and ${N_{\rm{ch}}}$, therefore is scaling with pad length.
698 In ALICE TPC three different pad gemetries are used.
699 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.
702 The agreement between previously mentioned fit and the data is on thel level of the
703 $\approx10-20\%$. 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
704 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:
707 p_L \approx p_{L0}p_{LC}=p_{L0}(1+p_{L1}L_{\rm{Drift}}+p_{L2}\tan^2\alpha)
709 p_A \approx p_{A0}p_{AC}=p_{A0}(1+p_{A1}L_{\rm{Drift}}+p_{A2}\tan^2\alpha)
710 \label{eq:PointResolFitCorrection}
713 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
716 \centering\epsfig{figure=picClusterResol/QresolY_mag.eps,width=0.7\linewidth}
717 \centering\epsfig{figure=picClusterResol/QresolY.eps,width=0.7\linewidth}
718 \label{figPointResolYQ}
719 \caption{Space point resolution in Y direcition as function of deposited charge $Q_{max}$.
720 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.
726 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\%.
732 \centering\epsfig{figure=picClusterResol/YResol_Pad0.eps,width=0.7\linewidth}
733 \centering\epsfig{figure=picClusterResol/YResol_Pad1.eps,width=0.7\linewidth}
734 \centering\epsfig{figure=picClusterResol/YResol_Pad2.eps,width=0.7\linewidth}
735 \caption{Space point resolution in Y direcition as function of the drift length and the inlination angle.}
736 \label{figPointResolYDRTAN}
740 \centering\epsfig{figure=picClusterResol/ZResol_Pad0.eps,width=0.7\linewidth}
741 \centering\epsfig{figure=picClusterResol/ZResol_Pad1.eps,width=0.7\linewidth}
742 \centering\epsfig{figure=picClusterResol/ZResol_Pad2.eps,width=0.7\linewidth}
743 \caption{Space point resolution in Z direcition as function of the drift length and the inlination angle.}
744 \label{figPointResolZDRTAN}