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)