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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

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slide1

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).

slide2

[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

slide3

s

tj+1-tj

Impact parameter

Cluster number

impact parameter resolution

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

how large is b max

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

a short note on and
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)

can we do any better in he gas mixtures and small cells

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:

real statistics contribution to b

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

what about diffusion

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?

can we do any better

(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

impact parameter resolution with cluster counting

(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