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Efficient Cluster Counting Algorithm through Monte Carlo Simulations

This study presents the detailed algorithm used for cluster counting simulation, leveraging programs like MAGBOLTZ, GARFIELD, and HEED along with custom C++/Root Monte Carlo simulations. Complemented with literature data, the study explores ionization detectors and impact parameter resolutions to enhance gas tracking devices.

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Efficient Cluster Counting Algorithm through Monte Carlo Simulations

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  1. Detailed description of the algorithm used for the simulation of the cluster counting For the studies of CluCou we have used standard programs like MAGBOLTZ, GARFIELD, HEED plus our own C++/Root Montecarlo. Whenever necessary, we have complemented the simulations with data taken from the literature. (for example: the distribution of the number of electrons per cluster is not well simulated in the standard programs; many data on Helium have better recent measurements). Details in G.F. Tassielli - A gas tracking device based on Cluster Counting for future colliders. PhD Thesis, Lecce, 2007. (Available as detached appendix to the 4th LOI).

  2. [3] http://www.le.infn.it¥ ∼chiodini¥allow listing¥chipclucou¥tesivarlamava. V. Varlamava. Tesi di Laurea in microelettronica: “Circuito di interfaccia per camera a drift in tecnologia integrata CMOS 0.13 µm”. Universit` a del Salento (2006-2007). [4] http://www.le.infn.it¥ ∼chiodini¥tesi¥Tesi Mino Pierri.pdf. C. Pierri. Tesi di Laurea in microelettronica: “Caratterizzazione di un dis- positivo VLSI Custom per l’acquisizione di segnali veloci da un rivelatore di particelle”. Universit` a del Salento (2007-2008). [1] A. Baschirotto, S. D’Amico, M. De Matteis, F. Grancagnolo, M. Panareo, R. Perrino, G. Chiodini and G.Tassielli. “A CMOS high-speed front-end for cluster counting techniques in ionization detectors”. Proc. of IWASI 2007. A 0.13µm CMOS Front-End for Cluster Counting Technique in Ionization Detectors S. D’Amico1,3, A. Baschirotto2, M. De Matteis1, F. Grancagnolo3, M. Panareo1,3, R. Perrino3, G. Chiodini3, A.Corvaglia3 A CMOS high-speed front-end for cluster counting techniques in ionization detectors A. Baschirotto1, S. D’Amico1, M. De Matteis1, F. Grancagnolo2, M. Panareo1,2, R. Perrino2, G. Chiodini2, G. Tassielli2,3

  3. s tj+1-tj Impact parameter Cluster number

  4. ionizing track drift tube electron . drift distance sense wire impact parameter b  ionization clusters ionization act Impact Parameter Resolution mV threshold drift time t1 [0.5 ns units] The impact parameter b is generally defined as: where t1 - t0 is the arrival time of the first (few) e–. b is, with this approach, therefore, systematically overestimated by the quantity: with: ranging from to 1st cluster 2nd cluster

  5. N =12.5/cm r =0.5cm N =12.5/cm r =1cm N =12.5/cm r =2cm N =50/cm r =1cm How large is bmax? Systematic overestimate of b: Usually, though improperly, referred as ionization statistics contribution to the impact parameter resolution

  6. A short note on  and  Poisson statistics tells us that the number N of ionization acts fluctuates with a variance 2(N) = N. The corresponding variance of  = 1/Nis 2() = 1/N42(N) = 1/N3= 3. For a gas with a density of 12.5 clusters/cm and an ionization length of 1 cm, N = 12.5 and  = 0.080, with (N) = 3.54 and () = 0.023, or (N)/N = ()/ = 28% Same gas but 2 cm cell gives a factor  smaller for both (20%); 0.5 cm cell gives (N)/N = ()/ = 40%. Obviously, in this last case, the error is more asymmetric. COROLLARY 1 For a round (or hexagonal) cell, when the impact parameter grows and approaches the edge of the cell, the length of the chord shortens and the relative fluctuations of N and  increase accordingly. COROLLARY 2 Tracks at an angle  with respect to the sense wire reduce the error by a factor (sin )-1/2 (e.g. 20% for =45). COROLLARY 3 Sense wires at alternating stereo angles , even at  = 0, reduce the error by a factor (cos 2)-1/2 (a few %). In our case, N ionizations are distributed over half chord: 1/(2N) = (/2), and, therefore, (/2) =(/2)3/2= 1/(22)3/2= 1/(22)(). Eventhough < 1> = /4, we’ll assume, conservatively, (1) =(/2)

  7. extreme solutions as defined by the first cluster only “real” track 5 5 5 4 4 4 3 3 3 2 2 2 1 1 sense wire 1 “equi-drift” Can we do any better in He gas mixtures and small cells? First of all, let’s get rid of the systematic overestimate of b by calculating b and 1 from d1 and d2 and assume, for simplicity, that thedi’s are not affected by error(no diffusion, no electronics): from which one gets: and: By generalizing this result with the contribution of the i-th (i2) cluster: the impact parameter can then be calculated by a weighted average with its proper variance: as opposed to:

  8. N = 12.5/cm r = 0.5 cm with <i> b/r with max i 61 m 40 m 28 m Relative gain of (b) as a function of the number of clusters used max i <i> “Real” statistics contribution to (b) From: and its generalization: since

  9. He/iC4H10 = 90/10 (N = 12.5 / cm) r = 1.0 cm Magboltz our exp. points What about diffusion? So far, so good! We have reduced the contribution to theimpact parameter resolutiondue to the ionization statistics at small impact parameter b (where this contribution is dominant since the uncertainty on the drift distance due to electron diffusion is negligible: we have, in fact, assumed so far no error on di’s). What happens as b increases?

  10. (b) with first 2 clusters (b) with first 4 clusters (b) with all clusters 69 m 56 m 49 m Can we do any better? Our previous generalization has brought to the result: b = 0.1 cm b = 0.5 cm b = 0.9 cm

  11. (b) with first 2 clusters (b) with first 4 clusters (b) with all clusters 48 m 41 m 38 m 0 0.5 0.3 0.4 0.2 0.1 Impact parameter resolution with CLUSTER COUNTING first cluster only all clusters in cylindrical drift tubes r = 1.0 cm r = 0.5 cm (N = 12.5 clusters/cm) 145 m 49 m 116 m 38 m

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