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23 \section{Accuracy of local coordinate measurement}
\r
28 \includegraphics[width=60mm,angle=-90]{picCluster/pic2.eps}
\r
29 \includegraphics[width=60mm,angle=-90]{picCluster/pic1.eps}
\r
30 \caption{Schematic view of the detection process in TPC (upper
\r
31 part - perspective view, lower part - side view).} \label{figTPC}
\r
34 The accuracy of the coordinate measurement is limited by a track
\r
35 angle which spreads ionization and by diffusion which amplifies
\r
38 The track direction with respect to pad plane is given by two
\r
39 angles $\alpha$ and $\beta$ (see fig.~\ref{figTPC}). For the
\r
40 measurement along the pad-row, the angle $\alpha$ between the
\r
41 track projected onto the pad plane and pad-row is relevant. For
\r
42 the measurement of the the drift coordinate ({\it{z}}--direction)
\r
43 it is the angle $\beta$ between the track and {\it{z}} axis
\r
44 (fig.~\ref{figTPC}).
\r
46 The ionization electrons are randomly distributed along the
\r
47 particle trajectory. Fixing the reference {\it{x}} position of an
\r
48 electron at the middle of pad-row, the {\it{y}} (resp. {\it{z}})
\r
49 position of the electron is a random variable characterized by
\r
50 uniform distribution with the width $L_{\rm{a}}$, where
\r
51 $L_{\rm{a}}$ is given by the pad length $L_{\rm{pad}}$ and the
\r
52 angle $\alpha$ (resp. $\beta$):
\r
53 \[L_{\rm{a}}=L_{\rm{pad}}\tan\alpha\]
\r
55 The diffusion smears out the position of the electron with
\r
56 gaussian probability distribution with $\sigma_{\rm{D}}$.
\r
57 Contribution of the $\mathbf{E{\times}B}$ and unisochronity
\r
58 effects for the Alice TPC are negligible. The typical resolution
\r
59 in the case of ALICE TPC is on the level of
\r
60 $\sigma_{y}\sim$~0.8~mm and $\sigma_{z}\sim$~1.0~mm integrating
\r
61 over all clusters in the TPC.
\r
65 \subsection{Gas gain fluctuation effect}
\r
67 Being collected on sense wire, electron is "multiplied" in strong
\r
68 electric field. This multiplication is subject of a large
\r
69 fluctuations, contributing to the cluster position resolution.
\r
70 Because of these fluctuations the center of gravity of the
\r
71 electron cloud can be shifted.
\r
73 Each electron is amplified independently. However, in the
\r
74 reconstruction electrons are not treated separately. The Centre Of
\r
75 Gravity (COG) of the cluster is usually used as an estimation for
\r
76 the local track position. The influence of the gas gain
\r
77 fluctuation to the reconstructed point characteristic can be
\r
78 described by a simple model, introducing a weighted COG
\r
81 X_{\rm{COG}}=\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}},
\r
84 where {\it{N}} is the total number of electrons in the cluster and
\r
85 $g_i$ is a random variable equal to a gas amplification for given
\r
88 The mean value of $X_{\rm{COG}}$ is equal to the mean value
\r
89 $\overline{x}$ of the original distribution of electrons
\r
92 \overline{\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}}}
\r
93 =\overline{x}\overline{\frac{\sum_{i=1}^{N}{g_i}}
\r
94 {\sum_{i=1}^N{g_i}}} =\overline{x}.
\r
98 However, the same is not true for the dispersion of the position,
\r
99 %$\sigma^2_{X_{COG}}\sigma_x^2$:
\r
102 \lefteqn{ \sigma^2_{X_{\rm{COG}}}
\r
103 =\overline{X_{\rm{COG}}^2}-\overline{X_{\rm{COG}}}^2=}\nonumber\\&&{}
\r
104 =\overline{\left(\frac{1}{\sum_{i=1}^N{g_i}}\sum_{i=1}^{N}{g_ix_i}
\r
105 \right)^2}-\overline{x}^2=
\r
107 &&{}=\overline{\frac{{\sum\sum{x_ix_jg_ig_j}}}{{\sum\sum{g_ig_j}}}}-
\r
110 \overline{x^2}\overline{\frac{\sum_i{g_i^2}}{\sum\sum{g_ig_j}}}-
\r
112 \overline{\frac{\sum\sum{g_ig_j}-\sum\sum_{i\ne{j}}{g_ig_j}}
\r
113 {\sum\sum{g_ig_j}}}= \nonumber\\&&
\r
114 =\left(\overline{x^2}-\overline{x}^2\right)
\r
115 \overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=
\r
116 \sigma_x^2\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}=
\r
118 &&{}=\frac{\sigma_x^2}{N}{\times}G_{\rm{gfactor}}^2
\r
119 \label{eqCOGSigmaGG}
\r
124 G_{\rm{gfactor}}^2 = N\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}}
\r
125 \label{eqCOGGGfactor0}
\r
128 The diffusion term is effectively multiplied by gas gain factor
\r
129 $G_{\rm{gfactor}}$. For sufficiently large number of electrons,
\r
130 when $g_i^2$ and $\sum\sum{g_ig_j}$ are quasi independent
\r
131 variables, equation (\ref{eqCOGGGfactor0}) can be transformed to
\r
135 \lefteqn{G_{\rm{gfactor}}^2 \approx
\r
136 N\frac{\overline{\sum{g_i^2}}}
\r
137 {\overline{\sum\sum{g_ig_j}}}}\nonumber\\
\r
139 N\frac{N\overline{g^2}}{N(N-1)\overline{g}^2+N\overline{g^2}}=
\r
141 &&{} =N\frac{ \left(\sigma_g^2/\overline{g}^2+1 \right)}
\r
142 {N+\sigma_g^2/\overline{g}^2}
\r
143 \label{eqCOGGGfactorE}
\r
146 Gas gain fluctuation of the gas detector working in proportional
\r
147 regime is described with the exponential distribution with the
\r
148 mean value $\bar{g}$ and r.m.s.
\r
150 \sigma_{\rm{g}} =\bar{g}
\r
154 Substituting $\sigma_{\rm{g}}$ into equation
\r
155 (\ref{eqCOGGGfactorE})
\r
157 G_{\rm{gfactor}}^2 =\frac{2N}{N+1}.
\r
158 \label{eqCOGGGfactorR}
\r
161 Gas multiplication fluctuation in chamber deteriorates
\r
162 $\sigma_{X_{\rm{COG}}}$ by a factor of about ${\sqrt{2}}$. The
\r
163 prediction of this model is in good agreement with results from
\r
167 \subsection{Secondary ionization effect}
\r
169 Charged particle penetrating the gas of the detector produces
\r
170 {\it{N}} primary electrons. Primary electron {\it{i}} produces
\r
171 $n_{\rm{s}}^i-1$ secondary electrons. Each of these electrons is
\r
172 amplified in the electric field by a factor of $g_j$.
\r
174 Each primary cluster is characterized by a position $x_i$ with
\r
175 mean value $\overline{x}$ and $\sigma_x$. The COG given by
\r
176 equation (\ref{eqCOGdefGG}) is modified to the following form:
\r
179 X_{\rm{COG}}=\frac{1}{\sum_{i=1}^N\sum_{j=1}^{n_i}{g_j^{i}}}
\r
180 \sum_{i=1}^{N}{x_i}\sum_{j=1}^{n_i}{g_j^{i}}.
\r
181 \label{eqCOGdefGGPIO}
\r
183 A new variable $G_n$ is introduced as the total electron gain:
\r
185 G_n=\sum_{j=1}^{n}{g_j}.
\r
190 Knowing the distribution of {\it{n}} and {\it{g}} and assuming
\r
191 that {\it{n}} and {\it{g}} are independent variables the mean
\r
192 value and variance of the $G_n$ can be expressed as:
\r
196 \overline{G_n}=\overline{n}\overline{g}} \\
\r
198 \frac{\sigma^2_{G_n}}{\overline{G_n^2}}=
\r
199 \frac{\sigma^2_n}{\overline{n}^2}+
\r
200 \frac{\sigma^2_g}{\overline{g}^2}
\r
201 \frac{1}{\overline{n}}
\r
205 Inserting $G_n$ into equation (\ref{eqCOGdefGGPIO}) results in an
\r
206 equation similar to the equation (\ref{eqCOGdefGG}).
\r
208 Multiplicative factor $G_{\rm{Lfactor}}$ is defined as an analog
\r
209 of $G_{\rm{gfactor}}$, from the equation (\ref{eqCOGGGfactor0})
\r
211 G_{\rm{Lfactor}}^2 = N\frac{\overline{\sum{G_i^2}}}
\r
212 {\overline{\sum\sum{G_iG_j}}}.
\r
213 \label{eqCOGLfactor0}
\r
216 Using the new variable $G_n$ and simply replacing gas gain
\r
217 {\it{g}} by $G_n$ in the similar way as in equation
\r
218 (\ref{eqCOGGGfactorE}) does not work. For $1/E^{2}$
\r
219 parametrization of secondary ionization process
\r
220 $\sigma^2_{G_n}/\overline{G_n}$ goes to infinity and thus
\r
221 $\sigma^2_{X_{COG}}=\sigma_x^2$. Moreover $G_i^2$ and
\r
222 $\sum\sum{G_iG_j}$ are not quasi independent as the sum
\r
223 $\sum\sum{G_iG_j}$ could be given by one "exotic" electron
\r
224 cluster. Approximations used for deriving the equation
\r
225 (\ref{eqCOGGGfactorE}) are not valid for secondary ionization
\r
228 In order to estimate the impact of this effect on COG equation
\r
229 (\ref{eqCOGLfactor0}) has to be solved numerically. Simulation
\r
230 showed that $G_{\rm{Lfactor}}$ does not depend strongly on the cut
\r
231 used for maximum number of electrons created in the process of
\r
232 secondary ionization. A change of the cut, from 1000 electrons up
\r
233 produces a change of about 3\% in $G_{\rm{Lfactor}}$.
\r
235 Equation (\ref{eqCOGGGfactorE}) is not applicable in this
\r
236 situation because of the infinity of the $\sigma_G$. According to
\r
237 the simulation, the threshold on the number of electrons in the
\r
238 cluster has a little influence to the resulting
\r
239 $G_{\rm{Lfactor}}$. Therefore we fit simulated $G_{\rm{Lfactor}}$
\r
240 with formula (\ref{eqCOGGGfactorE}) where
\r
241 $\sigma_G^2/\overline{G}^2$ was a free parameter. However, this
\r
242 parametrization does not describe the data for wide enough range
\r
243 of {\it{N}}. In further study the linear parametrization of the
\r
244 COG factor was used. This parametrization was validated on
\r
245 reasonable interval of {\it{N}}.
\r
249 \section{Center-of-gravity error parametrization}
\r
251 Detected position of charged particle is a random variable given
\r
252 by several stochastic processes: diffusion, angular effect, gas
\r
253 gain fluctuation, Landau fluctuation of the secondary ionization,
\r
254 $\mathbf{E{\times}B}$ effect, electronic noise and systematic
\r
255 effects (like space charge, etc.). The relative influence of these
\r
256 processes to the resulting distortion of position determination
\r
257 depends on the detector parameters. In the big drift detectors
\r
258 like the ALICE TPC the main contribution is given by diffusion,
\r
259 gas gain fluctuation, angular effect and secondary ionization
\r
262 Furthermore we will use following assumptions:
\r
264 \item $N_{\rm{prim}}$ primary electrons are produced at a random
\r
265 positions $x_i$ along the particle trajectory. \item $n_i-1$
\r
266 electrons are produced in the process of secondary ionization.
\r
267 \item Displacement of produced electrons due to the thermalization
\r
271 Each of electrons is characterized by a random vector
\r
274 \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j,
\r
277 where {\it{i}} is the index of primary electron cluster and
\r
278 {\it{j}} is the index of the secondary electron inside of the
\r
279 primary electron cluster. Random variable $\vec{x}^i$ is a
\r
280 position where the primary electron was created. The position
\r
281 $\vec{y}^i_j$ is a random variable specific for each electron. It
\r
282 is given mainly by a diffusion.
\r
284 The center of gravity of the electron cloud is given:
\r
286 \lefteqn{\vec{z}_{\rm{COG}}=\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
\r
287 \sum_{j=1}^{n_i}{g_j^i}}
\r
288 \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{z}_j^i}=}
\r
290 &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
\r
291 \sum_{j=1}^{n_i}{g_j^i}}
\r
292 \sum_{i=1}^{N_{\rm{prim}}}\vec{x}^i\sum_{j=1}^{n_i}{g_j^i}+\nonumber\\
\r
293 &&{}\frac{1}{\sum_{i=1}^{N_{\rm{prim}}}
\r
294 \sum_{j=1}^{n_i}{g_j^i}}
\r
295 \sum_{i=1}^{N_{\rm{prim}}}\sum_{j=1}^{n_i}{g_j^i\vec{y}_j^i}=
\r
296 \nonumber\\ \nonumber\\
\r
298 \vec{x}_{\rm{COG}}+\vec{y}_{\rm{COG}}.
\r
302 The mean value $\overline{\vec{z}_{\rm{COG}}}$ is equal to the sum
\r
303 of mean values $\overline{\vec{x}_{\rm{COG}}}$ and
\r
304 $\overline{\vec{y}_{\rm{COG}}}$.
\r
306 The sigma of COG in one of the dimension of vector
\r
307 $\vec{z}_{1COG}$ is given by following equation
\r
309 \lefteqn{\sigma_{z_{\rm{1COG}}}^2=\sigma_{x_{\rm{1COG}}}^2+
\r
310 \sigma_{y_{\rm{1COG}}}^2+}\nonumber\\
\r
312 2\left(\overline{x_{\rm{1COG}}y_{\rm{1COG}}}-\bar{x}_{\rm{1COG}}
\r
313 \bar{y}_{1COG}\right).
\r
314 \label{eqCOGSigSec}
\r
317 If the vectors $\vec{x}$ and $\vec{y}$ are independent random
\r
318 variables the last term in the equation (\ref{eqCOGSigSec}) is
\r
321 \sigma_{z_{1COG}}^2=\sigma_{x_{\rm{1COG}}}^2+
\r
322 \sigma_{y_{\rm{1COG}}}^2,
\r
323 \label{eqCOGSigSecIn}
\r
325 r.m.s. of COG distribution is given by the sum of r.m.s of
\r
326 {\it{x}} and {\it{y}} components.
\r
328 In order to estimate the influence of the $\mathbf{E{\times}B}$
\r
329 and unisochronity effect to the space resolution two additional
\r
330 random vectors are added to the initial electron position.
\r
334 \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j+
\r
335 \vec{X}_{\mathbf{E{\times}B}}(\vec{x}^i+\vec{y}^i_j)+
\r
336 \vec{X}_{\rm{Unisochron}}(\vec{x}^i+\vec{y}^i_j).
\r
339 The probability distributions of $\vec{X}_{\mathbf{E{\times}B}}$
\r
340 and $\vec{X}_{\rm{Unisochron}}$ are functions of random vectors
\r
341 $\vec{x^i}$ and $\vec{y^i_j}$, and they are strongly correlated.
\r
342 However, simulation indicates that in large drift detectors
\r
343 distortions, due to these effects, are negligible compared with a
\r
346 Combining previous equation and neglecting $\mathbf{E{\times}B}$
\r
348 effects, the COG distortion parametrization appears as:\\
\r
349 {$\sigma_{z}$} of cluster center in {\it{z}} (time) direction
\r
351 \lefteqn{\sigma^2_{{z_{\rm{COG}}}} = \frac{D^2_{\rm{L}}
\r
352 L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}
\r
353 \frac{{\tan^2\alpha}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+
\r
354 \sigma^2_{\rm{noise}},
\r
358 and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction
\r
360 \lefteqn{\sigma^2_{y_{\rm{COG}}} = \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{N_{\rm{ch}}}G_{\rm{g}}+}\nonumber\\&&{}
\r
361 \frac{{\tan^2\beta}~L_{\rm{pad}}^2G_{\rm{Lfactor}}(N_{\rm{chprim}})}{12N_{\rm{chprim}}}+
\r
362 \sigma^2_{\rm{noise}},
\r
366 ${N_{\rm{ch}}}$ is the total number of electrons in the cluster,
\r
367 ${N_{\rm{chprim}}}$ is the number of primary electrons in the
\r
368 cluster, ${G_{\rm{g}}}$ is the gas gain fluctuation factor,
\r
369 ${G_{\rm{Lfactor}}}$ is the secondary ionization fluctuation
\r
370 factor and $\sigma_{\rm{noise}}$ describe the contribution of the
\r
371 electronic noise to the resulting sigma of the COG.
\r
373 \section{Precision of cluster COG determination using measured
\r
376 We have derived parametrization using as parameters the total
\r
377 number of electrons ${N_{\rm{ch}}}$ and the number of primary
\r
378 electrons ${N_{\rm{chprim}}}$. This parametrization is in good
\r
379 agreement with simulated data, where the ${N_{\rm{ch}}}$ and
\r
380 ${N_{\rm{chprim}}}$ are known. It can be used as an estimate for
\r
381 the limits of accuracy, if the mean values
\r
382 $\overline{N}_{\rm{ch}}$ and $\overline{N}_{\rm{chprim}}$ are used
\r
385 The ${N_{\rm{ch}}}$ and ${N_{\rm{chprim}}}$ are random variables
\r
386 described by a Landau distribution, and Poisson distribution
\r
389 In order to use previously derived formulas (\ref{eqResZ1},
\r
390 \ref{eqResY1}), the number of electrons can be estimated assuming
\r
391 their proportionality to the total measured charge $A$ in the
\r
392 cluster. However, it turns out that an empirical parametrization
\r
393 of the factors $G(N)/N=G(A)/(kA)$ gives better results.
\r
394 Formulas (\ref{eqResZ1}) and (\ref{eqResY1}) are transformed to following form:\\
\r
396 {$\sigma_{z}$} of cluster center in {\it{z}} (time) direction:
\r
398 \lefteqn{\sigma^2_{z_{\rm{COG}}} =
\r
399 \frac{D^2_{\rm{L}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\
\r
401 \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}
\r
405 and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction:
\r
407 \lefteqn{\sigma_{y_{\rm{COG}}} =
\r
408 \frac{D^2_{\rm{T}}L_{\rm{Drift}}}{A}{\times}\frac{G_g(A)}{k_{\rm{ch}}}+}\nonumber\\
\r
410 \frac{\tan^2\beta~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}}
\r
414 \section{Estimation of the precision of cluster position
\r
415 determination using measured cluster shape}
\r
417 The shape of the cluster is given by the convolution of the
\r
418 responses to the electron avalanches. The time response function
\r
419 and the pad response function are almost gaussian, as well as the
\r
420 spread of electrons due to the diffusion. The spread due to the
\r
421 angular effect is uniform. Assuming that the contribution of the
\r
422 angular spread does not dominate the cluster width, the cluster
\r
423 shape is not far from gaussian. Therefore, we can use the
\r
427 f(t,p) = K_{\rm{Max}}.\exp\left(-\frac{(t-t_{\rm{0}})^2}{2\sigma_{\rm{t}}^2}-
\r
428 \frac{(p-p_{\rm{0}})^2}{2\sigma_{\rm{p}}^2}\right),
\r
431 where ${K_{\rm{Max}}}$ is the normalization factor, $t$ and $p$
\r
432 are time and pad bins, $t_0$ and $p_0$ are centers of the cluster
\r
433 in time and pad direction and $\sigma_{\rm{t}}$ and
\r
434 $\sigma_{\rm{p}}$ are the r.m.s. of the time and pad cluster
\r
437 The mean width of the cluster distribution is given by:
\r
439 \sigma_{\rm{t}} = \sqrt{D{\rm{^2_L}}L_{\rm{drift}}+\sigma^2_{\rm{preamp}}+
\r
440 \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12}},
\r
445 \sigma_{\rm{p}} = \sqrt{D{\rm{^2_T}}L_{\rm{drift}}+\sigma^2_{\rm{PRF}}+
\r
446 \frac{\tan^2\beta~L_{\rm{pad}}^2}{12}},
\r
448 where ${\sigma_{\rm{preamp}}}$ and ${\sigma_{\rm{PRF}}}$ are the
\r
449 r.m.s. of the time response function and pad response function,
\r
452 The fluctuation of the shape depends on the contribution of the
\r
453 random diffusion and angular spread, and on the contribution given
\r
454 by a gas gain fluctuation and secondary ionization. The
\r
455 fluctuation of the time and pad response functions is small
\r
456 compared with the previous one.
\r
458 The measured r.m.s of the cluster is influenced by a threshold
\r
461 \sigma_{\rm{t}}^2 = \sum_{A(t,p)>\rm{threshold}}{(t-t_{\rm{0}})^2{\times}A(t,p)}
\r
463 The threshold effect can be eliminated using two dimensional
\r
464 gaussian fit instead of the simple COG method. However, this
\r
465 approach is slow and, moreover, the result is very sensitive to
\r
466 the gain fluctuation.
\r
468 To eliminate the threshold effect in r.m.s. method, the bins
\r
469 bellow threshold are replaced with a virtual charge using
\r
470 gaussian interpolation of the cluster shape. The introduction of
\r
471 the virtual charge improves the precision of the COG measurement.
\r
472 Large systematic shifts in the estimate of the cluster position
\r
473 (depending on the local track position relative to pad--time) due
\r
474 to the threshold are no longer observed.
\r
476 Measuring the r.m.s. of the cluster, the local diffusion and
\r
477 angular spread of the electron cloud can be estimated. This
\r
478 provides additional information for the estimation of
\r
479 distortions. A simple additional correction function is used:
\r
481 \sigma_{\rm{COG}} \rightarrow
\r
482 \sigma_{\rm{COG}}(A){\times}(1+{\rm{const} {\times}\frac{\delta
\r
483 \rm{RMS}}{\rm{teorRMS}}}),
\r
484 \label{eqResUsingRMS}
\r
486 where $\sigma_{\rm{COG}}(A)$ is calculated according formulas
\r
487 \ref{eqResY1} and \ref{eqResZ1}, and the
\r
488 $\delta\rm{RMS}/\rm{teorRMS}$ is the relative distortion of the
\r
489 signal shape from the expected one.
\r
496 \section{TPC cluster finder}
\r
498 The classical approach for the beginning of the tracking was
\r
499 chosen. Before the tracking itself, two-dimensional clusters in
\r
500 pad-row--time planes are found. Then the positions of the
\r
501 corresponding space points are reconstructed, which are
\r
502 interpreted as the crossing points of the tracks and the centers
\r
503 of the pad rows. We investigate the region 5$\times$5 bins in
\r
504 pad-row--time plane around the central bin with maximum amplitude.
\r
505 The size of region, 5$\times$5 bins, is bigger than typical size
\r
506 of cluster as the $\sigma_{\rm{t}}$ and $\sigma_{\rm{pad}}$ are
\r
509 The COG and r.m.s are used to characterize cluster. The COG and
\r
510 r.m.s are affected by systematic distortions induced by the
\r
511 threshold effect. Depending on the number of time bins and pads in
\r
512 clusters the COG and r.m.s. are affected in different ways.
\r
513 Unfortunately, the number of bins in cluster is the function of
\r
514 local track position. To get rid of this effect, two-dimensional
\r
515 gaussian fitting can be used.
\r
517 Similar results can be achieved by so called r.m.s. fitting using
\r
518 virtual charge. The signal below threshold is replaced by the
\r
519 virtual charge, its expected value according a interpolation. If
\r
520 the virtual charge is above the threshold value, then it is
\r
521 replaced with amplitude equal to the threshold value. The signal
\r
522 r.m.s is used for later error estimation and as a criteria for
\r
523 cluster unfolding. This method gives comparable results as
\r
524 gaussian fit of the cluster but is much faster. Moreover, the COG
\r
525 position is less sensitive to the gain fluctuations.
\r
527 The cluster shape depends on the track parameters. The response
\r
528 function contribution and diffusion contribution to the cluster
\r
529 r.m.s. are known during clustering. This is not true for a angular
\r
530 contribution to the cluster width. The cluster finder should be
\r
531 optimised for high momentum particle coming from the primary
\r
532 vertex. Therefore, a conservative approach was chosen, assuming
\r
533 angle $\alpha$ to be zero. The tangent of the angle $\beta$ is
\r
534 given by {\it{z}}-position and pad-row radius, which is known
\r
538 \subsection{Cluster unfolding}
\r
540 The estimated width of the cluster is used as criteria for cluster
\r
541 unfolding. If the r.m.s. in one of the directions is greater then
\r
542 critical r.m.s, cluster is considered for unfolding. The fast
\r
543 spline method is used here. We require the charge to be conserved
\r
544 in this method. Overlapped clusters are supposed to have the same
\r
545 r.m.s., which is equivalent to the same track angles. If this
\r
546 assumption is not fulfilled, tracks diverge very rapidly.
\r
551 \includegraphics[width=60mm,angle=-90]{picCluster/unfolding1.eps}
\r
553 Schematic view of unfolding principle.} \label{figUnfolding1}
\r
557 \includegraphics[width=60mm,angle=-90]{picCluster/unfoldingres.eps}
\r
558 \caption{ Dependence of the position residual as function of the
\r
559 distance to the second cluster.} \label{figUnfoldingRes}
\r
562 The unfolding algorithm has the following steps:
\r
565 \item Six amplitudes $C_i$ are investigated (see fig.
\r
566 \ref{figUnfolding1}). First (left) local maxima, corresponding to
\r
567 the first cluster is placed at position 3, second (right) local
\r
568 maxima corresponding to the second cluster is at position 5.
\r
570 \item In the first iteration, amplitude in bin 4 corresponding to
\r
571 the cluster on left side $A_{\rm{L4}}$ is calculated using
\r
572 polynomial interpolation, assuming virtual amplitude at
\r
573 $A_{\rm{L5}}$ and derivation at $A_{\rm{L5}}^{'}$ to be 0.
\r
574 Amplitudes $A_{\rm{L2}}$ and $A_{\rm{L3}}$ are considered to be
\r
575 not influenced by overlap ($A_{\rm{L2}}=C_2$ and
\r
576 $A_{\rm{L3}}=C_3)$.
\r
578 \item The amplitude $A_{\rm{R4}}$ is calculated in similar way. In
\r
579 the next iteration the amplitude $A_{\rm{L4}}$ is calculated
\r
580 requiring charge conservation
\r
581 $C_{\rm{4}}=A_{\rm{R4}}+A_{\rm{L4}}$. Consequently
\r
583 A_{\rm{L4}} \rightarrow
\r
584 C_{\rm{4}}\frac{A_{\rm{L4}}}{A_{\rm{L4}}+A_{\rm{R4}}}
\r
588 A_{\rm{R4}} \rightarrow
\r
589 C_{\rm{4}}\frac{A_{\rm{R4}}}{A_{\rm{L4}}+A_{\rm{R4}}}.
\r
594 Two cluster resolution depends on the distance between the two
\r
595 tracks. Until the shape of cluster triggers unfolding, there is a
\r
596 systematic shifts towards to the COG of two tracks (see fig.
\r
597 \ref{figUnfoldingRes}), only one cluster is reconstructed.
\r
598 Afterwards, no systematic shift is observed.
\r
601 \subsection{Cluster characteristics}
\r
603 The cluster is characterized by the COG in {\it{y}} and {\it{z}}
\r
604 directions (fY and fZ) and by the cluster width (fSigmaY,
\r
605 fSigmaZ). The deposited charge is described by the signal at
\r
606 maximum (fMax), and total charge in cluster (fQ). The cluster type
\r
607 is characterized by the data member fCType which is defined as a
\r
608 ratio of the charge supposed to be deposited by the track and
\r
609 total charge in cluster in investigated region 5$\times$5. The
\r
610 error of the cluster position is assigned to the cluster only
\r
611 during tracking according formulas
\r
612 (\ref{eqZtotAmp}) and (\ref{eqYtotAmp}), when track
\r
613 angles $\alpha$ and $\beta$ are known with sufficient precision.
\r
616 Obviously, measuring the position of each electron separately the
\r
617 effect of the gas gain fluctuation can be removed, however this is
\r
618 not easy to implement in the large TPC detectors. Additional
\r
619 information about cluster asymmetry can be used, but the resulting
\r
620 improvement of around 5\% in precision on simulated data is
\r
621 negligible, and it is questionable, how successful will be such
\r
622 correction for the cluster asymmetry on real data.
\r
624 However, a cluster asymmetry can be used as additional criteria
\r
625 for cluster unfolding. Let's denote $\mu_i$ the {\it{i}}-th
\r
626 central momentum of the cluster, which was created by overlapping
\r
627 from two sub-clusters with unknown positions and deposited energy
\r
628 (with momenta $^1\mu_i$ and $^2\mu_i$).
\r
630 Let $r_1$ is the ratio of two clusters amplitudes:
\r
631 \[r_1={^1\mu_0}/({^1\mu_0}+{^2\mu_0})\] and the track distance {\it{d}} is equal to
\r
632 \[d = {^1\mu_1} -{^2\mu_1}.\]
\r
634 Assuming that the second moments for both sub-clusters are the
\r
635 same (${^0\mu_2}={^1\mu_2}={^2\mu_2}$), two sub-clusters distance
\r
636 {\it{d}} and amplitude ratio $r_1$ can be estimated:
\r
638 R = \frac{(\mu_3^6)}{(\mu_2^2-{^0\mu_2^2})^3}\\
\r
639 r_{\rm{1}} =0.5\pm0.5{\times}\sqrt{\frac{1}{1-4/R}} \\
\r
640 d = \sqrt{(4+R){\times}(\mu_2^2-{^0\mu_2^2})}
\r
644 In order to trigger unfolding using the shape information
\r
645 additional information about track and mean cluster shape over
\r
646 several pad-rows are needed. This information is available only
\r
647 during tracking procedure.
\r
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