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, HEEDplus our own C++/Root Montecarlo.

Whenever necessary, we have complemented the simulations withdata taken from the literature. (for example: the distribution of the number of electrons per clusteris 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).

[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

Cluster number

tj+1-tjs

Impact parameter

Impact Parameter Resolution

threshold

drifttime

t1

mV

[0.5 ns units]

1st cluster

2nd cluster

2 1

1

b

1

d1

d2

b

b

2

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

b vdrift x(t) dtt0

t1

bd1 b b2 12 b

1 0, 2

bmin 0

bmax d1 d12 2 2

ionizingtrack

drift tube

.sensewire

drift distance

impact parameterb

ionizationact

electron

ionizationclusters

How large is bmax?

bmaxr

br

N =50/cmr =1cm

N =12.5/cmr =2cm

N =12.5/cmr =1cm

N =12.5/cmr =0.5cm

1 N

bmax b2 2 2 b

bmax 61m

bmax 3m

bmax 20m

bmax 3517 10m

Systematic overestimate of b:

Usually, though improperly, referred as ionization statistics contribution to the impact parameter resolution

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/N42(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 1For 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 2Tracks at an angle with respect to the sense wire reduce the error by a factor (sin )-1/2 (e.g. 20% for =45).

COROLLARY 3Sense 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/(22) 3/2 = 1/(22) ().

Eventhough < 1> = /4, we’ll assume, conservatively, (1) = (/2)

1

1

1 3 2

1 1 3 2 2 3 2

12 2

12 2 3 2

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 the di’s are not affected by error (no diffusion, no electronics):

12 d1

2 b2

22 1 2 d2

2 b2

from which one gets:

1(2) 2

d2

2 d12

2

2 1(2) 1(2)

2

and:

b22 f2

2 ,d1,d2

2 b 1(2)

b2

1(2) 1(2) 1(2)

b2

1(2)2

b2

By generalizing this result with the contribution of the i-th (i2) cluster:

bi2 fi

2 ,d1,di

i bi 1(i)

bi i 1(i)

1(i)ibi

the impact parameter can then be calculated by a weighted average with its proper variance:

bj

2

i

b j j

2 b j j

2

i

1

j2 b j

i int i 2 1 i1

sensewire

“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:

b1 bmax

1(i) 1 i

2int i 2 di

2 d12

int i 2

i 1(i) i

“Real” statistics contribution to (b)

1(2) 2

d2

2 d12

2

2 1(2) 1(2)

2

1(i) 1 i1int i 2

2di

2 d12

2 i 1

i 1(i) i

From: and its generalization:

since

b 1

b 1

i b 1ib

1ib

1 1/4 1/2

2 3/4 3/16 1/2 1/4

3 5/4 5/16 3/2 3/4

4 7/4 7/16 3/2 3/4

5 9/4 9/16 5/2 5/4

6 11/4 11/16 5/2 5/4

i

maxi

i b 5 2 b

i b 5 2 b

i

br

b r

N = 12.5/cmr = 0.5 cm

61 m

40 m

28 m

b/rwith <i>

with max i

Relative gain of (b)

as a function of thenumber of clusters used

<i>

max i

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 di’s).What happens as b increases?

1 kV /cm

our exp.points

Magboltz

1e @1cm m

E Voltcmtorr

vs

diff x cm

x, drift distance cm

He/iC4H10 = 90/10(N = 12.5 / cm)

r = 1.0 cm

diff diff x

rw

rt

dx

rt rw127m

Can we do any better?

bi2 fi

2 ,d1,di di

2 d12

2 int i 2

2 2

di

2 d12

2 int i 2

2

, bj

2

i

b j j

2 b j j

2

i

1

j2 b j

, j b j 1( j ) jb j

and i2 b

1

j

2

i

1

j2 b j

Our previous generalization has brought to the result:

Now, i2 bi i

2 ,d1,di d1

2 ,d1,di di

2 ,d1,di , where:

i2 ,d1,di

1

16b2int i 2

1

3

di2 d1

2

int i 2

2

2

2

d1

2 ,d1,di d1

2

4b21

1

int i 2 2

di2 d1

2

int i 2

2

diff2 d1

di

2 ,d1,di di2

4b21 1

int i 2 2

di2 d1

2

int i 2

2

diff2 di

b = 0.1 cm b = 0.5 cm b = 0.9 cm

69 m

56 m49 m

(b) with first 2 clusters

(b) with first 4 clusters

(b) with all clusters

Impact parameter resolution with CLUSTER COUNTING

145 m

49 m

116 m

38 m

b cm

b cm

(b) with first 2 clusters(b) with first 4 clusters

(b) with all clusters

48 m41 m38 m

0.1 0.2 0.3 0.4 0.50

b cm

b cm

b vs b using

first cluster only all clusters

in cylindrical drift tubes

r = 1.0 cmr = 0.5 cm

(N = 12.5 clusters/cm)