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a65a7e70 | 1 | \RequirePackage{ifpdf} |
2 | ||
3 | \documentclass[a4paper,12pt]{article} | |
4 | \ifpdf | |
5 | \usepackage[pdftex]{graphicx} | |
6 | \else | |
7 | \usepackage[dvips]{graphicx} | |
8 | \fi | |
9 | \usepackage{epsfig} | |
10 | \usepackage{rotating} | |
11 | \usepackage{listings} | |
12 | \usepackage{booktabs} | |
13 | \usepackage{fancyhdr} | |
14 | \usepackage{float} | |
15 | \floatplacement{figure}{H} | |
16 | \floatplacement{table}{H} | |
17 | ||
18 | ||
19 | ||
20 | \begin{document} | |
21 | ||
22 | ||
23 | \section{Accuracy of local coordinate measurement} | |
24 | ||
25 | ||
26 | \begin{figure}[t] | |
27 | %\centering | |
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} | |
32 | \end{figure} | |
33 | ||
34 | The accuracy of the coordinate measurement is limited by a track | |
35 | angle which spreads ionization and by diffusion which amplifies | |
36 | this spread. | |
37 | ||
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 | |
44 | (fig.~\ref{figTPC}). | |
45 | ||
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 | \[L_{\rm{a}}=L_{\rm{pad}}\tan\alpha\] | |
54 | ||
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. | |
62 | ||
63 | ||
64 | ||
65 | \subsection{Gas gain fluctuation effect} | |
66 | ||
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. | |
72 | ||
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 | |
79 | $X_{\rm{COG}}$ | |
80 | \begin{eqnarray} | |
81 | X_{\rm{COG}}=\frac{\sum_{i=1}^{N}{g_ix_i}}{\sum_{i=1}^N{g_i}}, | |
82 | \label{eqCOGdefGG} | |
83 | \end{eqnarray} | |
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 | |
86 | electron. | |
87 | ||
88 | The mean value of $X_{\rm{COG}}$ is equal to the mean value | |
89 | $\overline{x}$ of the original distribution of electrons | |
90 | \begin{eqnarray} | |
91 | \overline{X_{COG}}= | |
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}. | |
95 | \label{eqCOGMeanGG} | |
96 | \end{eqnarray} | |
97 | ||
98 | However, the same is not true for the dispersion of the position, | |
99 | %$\sigma^2_{X_{COG}}\sigma_x^2$: | |
100 | %\begin {center} | |
101 | \begin{eqnarray} | |
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= | |
106 | \nonumber\\ | |
107 | &&{}=\overline{\frac{{\sum\sum{x_ix_jg_ig_j}}}{{\sum\sum{g_ig_j}}}}- | |
108 | \overline{x}^2= | |
109 | \nonumber\\&&{}= | |
110 | \overline{x^2}\overline{\frac{\sum_i{g_i^2}}{\sum\sum{g_ig_j}}}- | |
111 | \overline{x}^2 | |
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}}}= | |
117 | \nonumber\\ | |
118 | &&{}=\frac{\sigma_x^2}{N}{\times}G_{\rm{gfactor}}^2 | |
119 | \label{eqCOGSigmaGG} | |
120 | \end{eqnarray} | |
121 | ||
122 | where | |
123 | \begin{eqnarray} | |
124 | G_{\rm{gfactor}}^2 = N\overline{\frac{\sum{g_i^2}}{\sum\sum{g_ig_j}}} | |
125 | \label{eqCOGGGfactor0} | |
126 | \end{eqnarray} | |
127 | ||
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 | |
132 | the following | |
133 | ||
134 | \begin{eqnarray} | |
135 | \lefteqn{G_{\rm{gfactor}}^2 \approx | |
136 | N\frac{\overline{\sum{g_i^2}}} | |
137 | {\overline{\sum\sum{g_ig_j}}}}\nonumber\\ | |
138 | &&{} = | |
139 | N\frac{N\overline{g^2}}{N(N-1)\overline{g}^2+N\overline{g^2}}= | |
140 | \nonumber\\ | |
141 | &&{} =N\frac{ \left(\sigma_g^2/\overline{g}^2+1 \right)} | |
142 | {N+\sigma_g^2/\overline{g}^2} | |
143 | \label{eqCOGGGfactorE} | |
144 | \end{eqnarray} | |
145 | ||
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. | |
149 | \begin{eqnarray} | |
150 | \sigma_{\rm{g}} =\bar{g} | |
151 | \label{eqSigmaexp} | |
152 | \end{eqnarray} | |
153 | ||
154 | Substituting $\sigma_{\rm{g}}$ into equation | |
155 | (\ref{eqCOGGGfactorE}) | |
156 | \begin{eqnarray} | |
157 | G_{\rm{gfactor}}^2 =\frac{2N}{N+1}. | |
158 | \label{eqCOGGGfactorR} | |
159 | \end{eqnarray} | |
160 | ||
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 | |
164 | the simulation. | |
165 | ||
166 | ||
167 | \subsection{Secondary ionization effect} | |
168 | ||
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 $g_j$. | |
173 | ||
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: | |
177 | ||
178 | \begin{eqnarray} | |
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} | |
182 | \end{eqnarray} | |
183 | A new variable $G_n$ is introduced as the total electron gain: | |
184 | \begin{eqnarray} | |
185 | G_n=\sum_{j=1}^{n}{g_j}. | |
186 | \label{eqGNdef} | |
187 | \end{eqnarray} | |
188 | ||
189 | ||
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: | |
193 | ||
194 | \begin{eqnarray} | |
195 | \lefteqn{ | |
196 | \overline{G_n}=\overline{n}\overline{g}} \\ | |
197 | &&{} | |
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}} | |
202 | \label{eqGNsigma} | |
203 | \end{eqnarray} | |
204 | ||
205 | Inserting $G_n$ into equation (\ref{eqCOGdefGGPIO}) results in an | |
206 | equation similar to the equation (\ref{eqCOGdefGG}). | |
207 | ||
208 | Multiplicative factor $G_{\rm{Lfactor}}$ is defined as an analog | |
209 | of $G_{\rm{gfactor}}$, from the equation (\ref{eqCOGGGfactor0}) | |
210 | \begin{eqnarray} | |
211 | G_{\rm{Lfactor}}^2 = N\frac{\overline{\sum{G_i^2}}} | |
212 | {\overline{\sum\sum{G_iG_j}}}. | |
213 | \label{eqCOGLfactor0} | |
214 | \end{eqnarray} | |
215 | ||
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 | |
226 | effect. | |
227 | ||
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 $G_{\rm{Lfactor}}$. | |
234 | ||
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}}. | |
246 | ||
247 | ||
248 | ||
249 | \section{Center-of-gravity error parametrization} | |
250 | ||
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 | |
260 | fluctuation. | |
261 | ||
262 | Furthermore we will use following assumptions: | |
263 | \begin{itemize} | |
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 | |
268 | is neglected. | |
269 | \end{itemize} | |
270 | ||
271 | Each of electrons is characterized by a random vector | |
272 | $\vec{z}^i_j$ | |
273 | \begin{eqnarray} | |
274 | \vec{z}^i_j =\vec{x}^i+\vec{y}^i_j, | |
275 | \label{eqZtot} | |
276 | \end{eqnarray} | |
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. | |
283 | ||
284 | The center of gravity of the electron cloud is given: | |
285 | \begin{eqnarray} | |
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}=} | |
289 | \nonumber\\ | |
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\\ | |
297 | &&{} | |
298 | \vec{x}_{\rm{COG}}+\vec{y}_{\rm{COG}}. | |
299 | \label{eqCOGSec} | |
300 | \end{eqnarray} | |
301 | ||
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 | $\overline{\vec{y}_{\rm{COG}}}$. | |
305 | ||
306 | The sigma of COG in one of the dimension of vector | |
307 | $\vec{z}_{1COG}$ is given by following equation | |
308 | \begin{eqnarray} | |
309 | \lefteqn{\sigma_{z_{\rm{1COG}}}^2=\sigma_{x_{\rm{1COG}}}^2+ | |
310 | \sigma_{y_{\rm{1COG}}}^2+}\nonumber\\ | |
311 | &&{} | |
312 | 2\left(\overline{x_{\rm{1COG}}y_{\rm{1COG}}}-\bar{x}_{\rm{1COG}} | |
313 | \bar{y}_{1COG}\right). | |
314 | \label{eqCOGSigSec} | |
315 | \end{eqnarray} | |
316 | ||
317 | If the vectors $\vec{x}$ and $\vec{y}$ are independent random | |
318 | variables the last term in the equation (\ref{eqCOGSigSec}) is | |
319 | equal to zero. | |
320 | \begin{eqnarray} | |
321 | \sigma_{z_{1COG}}^2=\sigma_{x_{\rm{1COG}}}^2+ | |
322 | \sigma_{y_{\rm{1COG}}}^2, | |
323 | \label{eqCOGSigSecIn} | |
324 | \end{eqnarray} | |
325 | r.m.s. of COG distribution is given by the sum of r.m.s of | |
326 | {\it{x}} and {\it{y}} components. | |
327 | ||
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. | |
331 | ||
332 | ||
333 | \begin{eqnarray} | |
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). | |
337 | \label{eqZtotplus} | |
338 | \end{eqnarray} | |
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 | |
344 | previous one. | |
345 | ||
346 | Combining previous equation and neglecting $\mathbf{E{\times}B}$ | |
347 | and unisochronity | |
348 | effects, the COG distortion parametrization appears as:\\ | |
349 | {$\sigma_{z}$} of cluster center in {\it{z}} (time) direction | |
350 | \begin{eqnarray}\ | |
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}}, | |
355 | \label{eqResZ1} | |
356 | \end{eqnarray} | |
357 | ||
358 | and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction | |
359 | \begin{eqnarray} | |
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}}, | |
363 | \label{eqResY1} | |
364 | \end{eqnarray} | |
365 | where | |
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. | |
372 | ||
373 | \section{Precision of cluster COG determination using measured | |
374 | amplitude} | |
375 | ||
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 | |
383 | instead. | |
384 | ||
385 | The ${N_{\rm{ch}}}$ and ${N_{\rm{chprim}}}$ are random variables | |
386 | described by a Landau distribution, and Poisson distribution | |
387 | respectively . | |
388 | ||
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:\\ | |
395 | ||
396 | {$\sigma_{z}$} of cluster center in {\it{z}} (time) direction: | |
397 | \begin{eqnarray} | |
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\\ | |
400 | &&{} | |
401 | \frac{\tan^2\alpha~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}} | |
402 | \label{eqZtotAmp} | |
403 | \end{eqnarray} | |
404 | ||
405 | and {$\sigma_{y}$} of cluster center in {\it{y}}(pad) direction: | |
406 | \begin{eqnarray} | |
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\\ | |
409 | &&{} | |
410 | \frac{\tan^2\beta~L_{\rm{pad}}^2}{12A}{\times}\frac{G_{Lfactor}(A)}{k_{\rm{prim}}}+\sigma^2_{\rm{noise}} | |
411 | \label{eqYtotAmp} | |
412 | \end{eqnarray} | |
413 | ||
414 | \section{Estimation of the precision of cluster position | |
415 | determination using measured cluster shape} | |
416 | ||
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 | |
424 | parametrization | |
425 | ||
426 | \begin{equation} | |
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), | |
429 | \label{eq:GaussTP} | |
430 | \end{equation} | |
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 | |
435 | distribution. | |
436 | ||
437 | The mean width of the cluster distribution is given by: | |
438 | \begin{equation} | |
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}}, | |
441 | \end{equation} | |
442 | ||
443 | ||
444 | \begin{equation} | |
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}}, | |
447 | \end{equation} | |
448 | where ${\sigma_{\rm{preamp}}}$ and ${\sigma_{\rm{PRF}}}$ are the | |
449 | r.m.s. of the time response function and pad response function, | |
450 | respectively. | |
451 | ||
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. | |
457 | ||
458 | The measured r.m.s of the cluster is influenced by a threshold | |
459 | effect. | |
460 | \begin{equation} | |
461 | \sigma_{\rm{t}}^2 = \sum_{A(t,p)>\rm{threshold}}{(t-t_{\rm{0}})^2{\times}A(t,p)} | |
462 | \end{equation} | |
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. | |
467 | ||
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. | |
475 | ||
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: | |
480 | \begin{eqnarray} | |
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} | |
485 | \end{eqnarray} | |
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. | |
490 | ||
491 | ||
492 | ||
493 | ||
494 | ||
495 | ||
496 | \section{TPC cluster finder} | |
497 | ||
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 | |
507 | about 0.75 bins. | |
508 | ||
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. | |
516 | ||
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. | |
526 | ||
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 | |
535 | during clustering. | |
536 | ||
537 | ||
538 | \subsection{Cluster unfolding} | |
539 | ||
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. | |
547 | ||
548 | ||
549 | \begin{figure}[t] | |
550 | \centering | |
551 | \includegraphics[width=60mm,angle=-90]{picCluster/unfolding1.eps} | |
552 | \caption{ | |
553 | Schematic view of unfolding principle.} \label{figUnfolding1} | |
554 | \end{figure} | |
555 | \begin{figure}[t] | |
556 | \centering | |
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} | |
560 | \end{figure} | |
561 | ||
562 | The unfolding algorithm has the following steps: | |
563 | \begin{itemize} | |
564 | ||
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. | |
569 | ||
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 | |
576 | $A_{\rm{L3}}=C_3)$. | |
577 | ||
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 | |
582 | \begin{eqnarray} | |
583 | A_{\rm{L4}} \rightarrow | |
584 | C_{\rm{4}}\frac{A_{\rm{L4}}}{A_{\rm{L4}}+A_{\rm{R4}}} | |
585 | \end{eqnarray} | |
586 | and | |
587 | \begin{eqnarray} | |
588 | A_{\rm{R4}} \rightarrow | |
589 | C_{\rm{4}}\frac{A_{\rm{R4}}}{A_{\rm{L4}}+A_{\rm{R4}}}. | |
590 | \end{eqnarray} | |
591 | \end{itemize} | |
592 | ||
593 | ||
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. | |
599 | ||
600 | ||
601 | \subsection{Cluster characteristics} | |
602 | ||
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. | |
614 | ||
615 | ||
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. | |
623 | ||
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 $^2\mu_i$). | |
629 | ||
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 | \[d = {^1\mu_1} -{^2\mu_1}.\] | |
633 | ||
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: | |
637 | \begin{eqnarray} | |
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})} | |
641 | \label{eqMeas} | |
642 | \end{eqnarray} | |
643 | ||
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. | |
648 | ||
649 | ||
650 | ||
651 | \subsection{Space point resolution parameterization} | |
652 | ||
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. | |
655 | ||
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 $p_A$ | |
658 | \begin{eqnarray}\ | |
659 | \sigma^2_{{\rm{COG}}} \approx p^2_0+p^2_{L}L_{\rm{Drift}}+p^2_{A}\tan^2\alpha | |
660 | \label{eqResCOG0} | |
661 | \nonumber\\ | |
662 | p^2_L \approx \frac{\sigma^2_DG_{\rm{g}}}{N_{\rm{ch}}} | |
663 | \nonumber\\ | |
664 | p^2_A \approx \frac{L_{\rm{pad}}^2G_{\rm{Lfactor}}}{N_{\rm{chprim}}} | |
665 | \end{eqnarray} | |
666 | ||
667 | ||
668 | ||
669 | \begin{table} | |
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 | |
679 | ||
680 | \end{tabular} | |
681 | \label{table:PointResolFitParam} | |
682 | \end{table} | |
683 | ||
684 | ||
685 | \begin{eqnarray}\ | |
686 | N_{\rm{ch}} \approx {L_{\rm{pad}}} \nonumber \\ | |
687 | N_{\rm{chprim}} \approx {L_{\rm{pad}}} \nonumber \\ | |
688 | \nonumber\\ | |
689 | p_L \approx \frac{1}{\sqrt{L_{\rm{pad}}}} | |
690 | \nonumber\\ | |
691 | p_A \approx \sqrt{L_{\rm{pad}}} | |
692 | \label{eq:ResolScaling} | |
693 | \end{eqnarray} | |
694 | ||
695 | ||
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. | |
700 | ||
701 | ||
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: | |
705 | \begin{eqnarray}\ | |
706 | \nonumber\\ | |
707 | p_L \approx p_{L0}p_{LC}=p_{L0}(1+p_{L1}L_{\rm{Drift}}+p_{L2}\tan^2\alpha) | |
708 | \nonumber\\ | |
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} | |
711 | \end{eqnarray} | |
712 | ||
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 | |
714 | ||
715 | \begin{figure} | |
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. | |
721 | } | |
722 | ||
723 | \end{figure} | |
724 | ||
725 | ||
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\%. | |
727 | ||
728 | ||
729 | ||
730 | ||
731 | \begin{figure} | |
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} | |
737 | \end{figure} | |
738 | ||
739 | \begin{figure} | |
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} | |
745 | \end{figure} | |
746 | ||
747 | ||
748 | ||
749 | ||
750 | \end{document} |