detailed description of the algorithm used for the simulation of the cluster counting
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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)