Empty Channel per Chip
Rudi Ganz August, 6th 1997
The is the problem of common noise detection in the octagon wafer. As a basis of the discussion I collected some numbers of how many channels on average we can use which are not occupied by a hit and therefore can by used for the common noise detection.
Here are some estimates for the number of not occupied channels. It assumes, that the number of hits on a single segment of the octagon wafer are Poisson distributed with a mean value equals the average occupancy as given in the charts. The chips have either 64 channel or 128 channel. The diagram shows the probability of having more the k channel not occupied versus k. For example in case of Nch=64 and occupancy of 3 the is about a 60% probability to have 2 or more (k=1 !) channel with no hit. (The probabilities for the number of empty channels are calculated by means of a Binomial-distribution)
HERE you can find a discussion of the proposal for a new octagon wafer design
DOWNLOAD THE EXCEL FILE HERE !
This is our current solution in case the occupancy is a factor 2 higher than what HIJET predicts:
Occup. |
Nch |
p |
Number of sensors |
||
total |
|||||
6 |
64 |
0.2% |
44 |
||
Criterium: |
|||||
k |
Exactly k |
k or less |
More than k |
Average number |
Percentage of |
empty channel |
empty channel |
empty channel |
of ch. with k hits |
acceptable events |
|
0 |
85.3135% |
85.3135% |
14.6865% |
0.2 |
0.0000% |
1 |
13.5678% |
98.8813% |
1.1187% |
1.0 |
0.0000% |
2 |
1.0620% |
99.9433% |
0.0567% |
2.9 |
0.0000% |
3 |
0.0545% |
99.9979% |
0.0021% |
5.7 |
0.0000% |
4 |
0.0021% |
99.9999% |
0.0001% |
8.6 |
0.0000% |
5 |
0.0001% |
100.0000% |
0.0000% |
10.3 |
0.0000% |
6 |
0.0000% |
100.0000% |
0.0000% |
10.3 |
0.0000% |
7 |
0.0000% |
100.0000% |
0.0000% |
8.8 |
0.0000% |
8 |
0.0000% |
100.0000% |
0.0000% |
6.6 |
0.0000% |
9 |
0.0000% |
100.0000% |
0.0000% |
4.4 |
0.0000% |
10 |
0.0000% |
100.0000% |
0.0000% |
2.6 |
0.0000% |
11 |
0.0000% |
100.0000% |
0.0000% |
1.4 |
0.0000% |
12 |
0.0000% |
100.0000% |
0.0000% |
0.7 |
0.0000% |
13 |
0.0000% |
100.0000% |
0.0000% |
0.3 |
0.0000% |
14 |
0.0000% |
100.0000% |
0.0000% |
0.1 |
0.0000% |
15 |
0.0000% |
100.0000% |
0.0000% |
0.1 |
0.0000% |
16 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
17 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
18 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
19 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
20 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
The following is our current solution with the occupancy 3 as predicted by HIJET for the central module of the octagon. So if one requests eg. At least 5 empty channels for a common noise determination one can do that in only 20 % of a central collision.
Occup. |
Nch |
p |
Number of sensors |
||
total |
|||||
3 |
64 |
5.0% |
44 |
||
Criterium: |
|||||
k |
Exactly k |
k or less |
More than k |
Average number |
Percentage of |
empty channel |
empty channel |
empty channel |
of ch. with k hits |
acceptable events |
|
0 |
3.8066% |
3.8066% |
96.1934% |
3.2 |
18.1296% |
1 |
12.7648% |
16.5715% |
83.4285% |
9.6 |
0.0345% |
2 |
21.0679% |
37.6394% |
62.3606% |
14.3 |
0.0000% |
3 |
22.8133% |
60.4527% |
39.5473% |
14.3 |
0.0000% |
4 |
18.2286% |
78.6813% |
21.3187% |
10.8 |
0.0000% |
5 |
11.4612% |
90.1425% |
9.8575% |
6.5 |
0.0000% |
6 |
5.9051% |
96.0476% |
3.9524% |
3.2 |
0.0000% |
7 |
2.5636% |
98.6112% |
1.3888% |
1.4 |
0.0000% |
8 |
0.9570% |
99.5682% |
0.4318% |
0.5 |
0.0000% |
9 |
0.3120% |
99.8802% |
0.1198% |
0.2 |
0.0000% |
10 |
0.0899% |
99.9701% |
0.0299% |
0.1 |
0.0000% |
11 |
0.0231% |
99.9933% |
0.0067% |
0.0 |
0.0000% |
12 |
0.0054% |
99.9986% |
0.0014% |
0.0 |
0.0000% |
13 |
0.0011% |
99.9997% |
0.0003% |
0.0 |
0.0000% |
14 |
0.0002% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
15 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
16 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
17 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
18 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
19 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
20 |
0.0000% |
100.0000% |
0.0000% |
0.0 |
0.0000% |
If now one would double the number of channels and on a single octagon wafer and reads it out with two 64 channel chip, one would get the following significantly improved picture for HIJET central collisions (the case above then would represent the double HIJET multiplicity case in this layout):
Occup. |
Nch |
p |
Number of sensors |
||
total |
|||||
1.5 |
64 |
22.3% |
44 |
||
Criterium: |
|||||
k |
Exactly k |
k or less |
More than k |
Average number |
Percentage of |
empty channel |
empty channel |
empty channel |
of ch. with k hits |
acceptable events |
|
0 |
0.0000% |
0.0000% |
100.0000% |
14.3 |
99.9996% |
1 |
0.0002% |
0.0002% |
99.9998% |
21.4 |
99.9918% |
2 |
0.0016% |
0.0018% |
99.9982% |
16.1 |
99.9216% |
3 |
0.0095% |
0.0113% |
99.9887% |
8.0 |
99.5058% |
4 |
0.0415% |
0.0528% |
99.9472% |
3.0 |
97.7042% |
5 |
0.1431% |
0.1958% |
99.8042% |
0.9 |
91.7361% |
6 |
0.4041% |
0.5999% |
99.4001% |
0.2 |
76.7392% |
7 |
0.9616% |
1.5615% |
98.4385% |
0.0 |
50.0326% |
8 |
1.9679% |
3.5294% |
96.4706% |
0.0 |
20.5768% |
9 |
3.5168% |
7.0462% |
92.9538% |
0.0 |
4.0156% |
10 |
5.5555% |
12.6018% |
87.3982% |
0.0 |
0.2668% |
11 |
7.8331% |
20.4349% |
79.5651% |
0.0 |
0.0043% |
12 |
9.9367% |
30.3716% |
69.6284% |
0.0 |
0.0000% |
13 |
11.4159% |
41.7875% |
58.2125% |
0.0 |
0.0000% |
14 |
11.9444% |
53.7318% |
46.2682% |
0.0 |
0.0000% |
15 |
11.4354% |
65.1672% |
34.8328% |
0.0 |
0.0000% |
16 |
10.0586% |
75.2258% |
24.7742% |
0.0 |
0.0000% |
17 |
8.1572% |
83.3830% |
16.6170% |
0.0 |
0.0000% |
18 |
6.1175% |
89.5005% |
10.4995% |
0.0 |
0.0000% |
19 |
4.2539% |
93.7544% |
6.2456% |
0.0 |
0.0000% |
20 |
2.7490% |
96.5035% |
3.4965% |
0.0 |
0.0000% |
In case we would go for a 128 channel chip and therefore double the number of channels on a single octagon wafer the situation for a multiplicity twice the HIJET prediction one get the following:
Occup. |
Nch |
p |
Number of sensors |
||
total |
|||||
3 |
128 |
5.0% |
44 |
||
Criterium: |
|||||
k |
Exactly k |
k or less |
More than k |
Average number |
Percentage of |
empty channel |
empty channel |
empty channel |
of ch. with k hits |
acceptable events |
|
0 |
0.1449% |
0.1449% |
99.8551% |
6.4 |
93.8189% |
1 |
0.9718% |
1.1167% |
98.8833% |
19.1 |
61.0106% |
2 |
3.2334% |
4.3501% |
95.6499% |
28.7 |
14.1293% |
3 |
7.1154% |
11.4655% |
88.5345% |
28.7 |
0.4709% |
4 |
11.6505% |
23.1160% |
76.8840% |
21.5 |
0.0009% |
5 |
15.1388% |
38.2549% |
61.7451% |
12.9 |
0.0000% |
6 |
16.2608% |
54.5157% |
45.4843% |
6.5 |
0.0000% |
7 |
14.8491% |
69.3647% |
30.6353% |
2.8 |
0.0000% |
8 |
11.7677% |
81.1324% |
18.8676% |
1.0 |
0.0000% |
9 |
8.2210% |
89.3534% |
10.6466% |
0.3 |
0.0000% |
10 |
5.1259% |
94.4793% |
5.5207% |
0.1 |
0.0000% |
11 |
2.8811% |
97.3604% |
2.6396% |
0.0 |
0.0000% |
12 |
1.4718% |
98.8322% |
1.1678% |
0.0 |
0.0000% |
13 |
0.6881% |
99.5203% |
0.4797% |
0.0 |
0.0000% |
14 |
0.2962% |
99.8164% |
0.1836% |
0.0 |
0.0000% |
15 |
0.1179% |
99.9344% |
0.0656% |
0.0 |
0.0000% |
16 |
0.0436% |
99.9780% |
0.0220% |
0.0 |
0.0000% |
17 |
0.0151% |
99.9931% |
0.0069% |
0.0 |
0.0000% |
18 |
0.0049% |
99.9980% |
0.0020% |
0.0 |
0.0000% |
19 |
0.0015% |
99.9994% |
0.0006% |
0.0 |
0.0000% |
20 |
0.0004% |
99.9998% |
0.0002% |
0.0 |
0.0000% |
And for a HIJET central event one gets:
Occup. |
Nch |
p |
Number of sensors |
||
total |
|||||
1.5 |
128 |
22.3% |
44 |
||
Criterium: |
|||||
k |
Exactly k |
k or less |
More than k |
Average number |
Percentage of |
empty channel |
empty channel |
empty channel |
of ch. with k hits |
acceptable events |
|
0 |
0.0000% |
0.0000% |
100.0000% |
28.6 |
100.0000% |
1 |
0.0000% |
0.0000% |
100.0000% |
42.8 |
100.0000% |
2 |
0.0000% |
0.0000% |
100.0000% |
32.1 |
100.0000% |
3 |
0.0000% |
0.0000% |
100.0000% |
16.1 |
100.0000% |
4 |
0.0000% |
0.0000% |
100.0000% |
6.0 |
100.0000% |
5 |
0.0000% |
0.0000% |
100.0000% |
1.8 |
100.0000% |
6 |
0.0000% |
0.0000% |
100.0000% |
0.5 |
99.9999% |
7 |
0.0000% |
0.0000% |
100.0000% |
0.1 |
99.9992% |
8 |
0.0001% |
0.0001% |
99.9999% |
0.0 |
99.9965% |
9 |
0.0002% |
0.0003% |
99.9997% |
0.0 |
99.9863% |
10 |
0.0008% |
0.0011% |
99.9989% |
0.0 |
99.9511% |
11 |
0.0025% |
0.0036% |
99.9964% |
0.0 |
99.8430% |
12 |
0.0069% |
0.0105% |
99.9895% |
0.0 |
99.5406% |
13 |
0.0177% |
0.0281% |
99.9719% |
0.0 |
98.7699% |
14 |
0.0417% |
0.0698% |
99.9302% |
0.0 |
96.9746% |
15 |
0.0910% |
0.1608% |
99.8392% |
0.0 |
93.1658% |
16 |
0.1845% |
0.3453% |
99.6547% |
0.0 |
85.8834% |
17 |
0.3491% |
0.6944% |
99.3056% |
0.0 |
73.5942% |
18 |
0.6184% |
1.3128% |
98.6872% |
0.0 |
55.9085% |
19 |
1.0283% |
2.3411% |
97.6589% |
0.0 |
35.2637% |
20 |
1.6096% |
3.9507% |
96.0493% |
0.0 |
16.9728% |