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’Amico 1,3, A. Baschirotto 2, M. De Matteis 1, F. Grancagnolo 3, M. Panareo 1,3, R. Perrino 3, G. Chiodini 3, A.Corvaglia 3 A CMOS high-speed front-end for cluster counting techniques in ionization detectors A. Baschirotto 1, S. D’Amico 1, M. De Matteis 1, F. Grancagnolo 2, M. Panareo 1,2, R. Perrino 2, G. Chiodini 2, G. Tassielli 2,3

3. Cluster number t j+1 -t j ss Impact parameter

4. Impact Parameter Resolution threshold drift time t1t1 mV [0.5 ns units] 1 st cluster 2 nd cluster The impact parameter b is generally defined as: where t 1 - t 0 is the arrival time of the first (few) e –. b is, with this approach, therefore, systematically overestimated by the quantity: with: ranging from to ionizing track drift tube. sense wire drift distance impact parameter b ionization act electron ionization clusters

5. How large is  b max ? N =50/cm r =1cm N =12.5/cm r =2cm N =12.5/cm r =1cm N =12.5/cm r =0.5cm 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/N is  2 ( ) = 1/N 4  2 (N) = 1/N 3 = 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 = /4, we’ll assume, conservatively,  ( 1 ) =  ( /2)

7. 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 d 1 and d 2 and assume, for simplicity, that the d i ’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: sense wire “real” track extreme solutions as defined by the first cluster only 5 4 3 2 1 2 3 4 5 “equi-drift” 1 2 3 4 5 1 as opposed to:

8. “Real” statistics contribution to  (b) From:and its generalization: since 11/41/2 23/43/161/21/4 35/45/163/23/4 47/47/163/23/4 59/49/165/25/4 611/411/165/25/4 N = 12.5/cm r = 0.5 cm 61  m 40  m 28  m  b/r with with max i Relative gain of  (b) as a function of the number of clusters used max i

9. What about diffusion? So far, so good! We have reduced the contribution to the impact parameter resolution due 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 d i ’s). What happens as b increases? our exp. points Magboltz He/iC 4 H 10 = 90/10 (N = 12.5 / cm) r = 1.0 cm

10. 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 69  m 56  m 49  m  (b) with first 2 clusters  (b) with first 4 clusters  (b) with all clusters

11. Impact parameter resolution with CLUSTER COUNTING 145  m 49  m 116  m 38  m  (b) with first 2 clusters  (b) with first 4 clusters  (b) with all clusters 48  m 41  m 38  m 0.1 0.2 0.30.4 0.50 first cluster only all clusters in cylindrical drift tubes r = 1.0 cm r = 0.5 cm (N = 12.5 clusters/cm)