4.1 Refinements to the GEANT Simulation
In a reliable simulation all detector module materials, intrinsic properties of
the calorimeter, as well as details pertaining the trigger and readout have to
be accurately described. This includes for example electronic noise,
photoelectron statistics and the individual crystal non-uniformities. The
electronic noise was included after fitting a Gaussian distribution to the
pedestal of all channels under consideration. After fitting, the averaged
position and width was implemented into the simulation code. To this end a
randomly generated number - within the parameters of the obtained Gaussian -
becomes added to the energy deposited in a crystal volume.
As reported in sect. 3.2.4, the optical non-uniformity as well as the light
output of each crystal was measured. The photoelectron statistics is taken into
account in the simulation in the following way. The measured number of
photoelectrons per MeV for each crystal is used to generate a normalized
Poisson distribution in the simulation code for the detector module. The
calculated energy deposited in the crystal is then multiplied for each step
with the corresponding Poisson distribution.
The optical non-uniformity that maps the average energy deposition of minimum
ionizing particle per volume unit cells of each crystal resulted in a
characteristic response function reflecting the measured axial and transverse
non-uniformities. This function is convolved with the GEANT shower energy
distributions of positrons and photons in the PiBeta detector. This requires
some adaptation for the simulation, since the light output per MeV now depends
on the shower depths inside the crystal. Although most of the crystals show
positive non-uniformity, which indicates an increase of luminosity towards the
readout device, some have a negative coefficient. A CsI crystal with a positive
non-uniformity would `gain' energy, while a crystal with negative
non-uniformity would lose energy. The gains of the 240 CsI modules have to be
unified, therefore.
The gain matching procedure requires two sets of simulations. In one set
perfect scintillators are assumed, while in the other non-uniformity is
included in the simulations. 240 sum spectra are created for the comparison of
both cases. A `sum' histogram corresponding to a given crystal is incremented
only for those events in which this crystal receives at least 50% of the total
energy deposited in the calorimeter. The histogrammed variable is the summed
energy of all crystals that received more than 0.5 MeV of energy above the
noise threshold. The use of single spectra was not practical due to the great
lineshape differences for different crystals. The resulting individual sum
spectra show scaleable differences in their peak position. Software gain
factors are obtained through repetitive comparison of the ideal and `real'
spectra. Both Kolmogorov-[18] and
c 2-test were applied for comparison. After about four passes a
final set of software gains was found. Table 4-1 is showing a sample of 37 out
of 44 crystals that were used during the 1996 beamtime. Here 70 MeV positrons
were emitted uniformly onto an array of 44 crystals. This table indicates that
both methods result in similar values.
Channel
#
|
Zero
Non-uniformity
|
Unmatched
|
Kolmogorov-Test
|
c 2-Method
|
|
Peak Position
|
FWHM
|
Peak Position
|
FWHM
|
Peak Position
|
FWHM
|
Peak Position
|
FWHM
|
1
|
67.02
|
3.94
|
64.99
|
4.86
|
66.34
|
4.86
|
66.29
|
4.43
|
2
|
67.21
|
3.50
|
65.63
|
3.79
|
66.86
|
3.85
|
66.81
|
3.65
|
3
|
67.43
|
3.30
|
65.80
|
3.66
|
66.76
|
3.93
|
66.76
|
3.89
|
4
|
67.20
|
3.60
|
64.67
|
4.53
|
66.86
|
3.83
|
66.10
|
4.46
|
5
|
66.97
|
4.35
|
63.95
|
5.31
|
66.62
|
5.50
|
65.94
|
5.40
|
6
|
67.18
|
3.73
|
66.92
|
4.32
|
67.11
|
4.01
|
67.04
|
4.08
|
7
|
66.44
|
4.63
|
62.53
|
5.56
|
66.16
|
5.90
|
64.86
|
5.79
|
8, 9
|
|
|
|
|
|
|
|
|
10
|
66.30
|
4.46
|
67.01
|
4.33
|
65.88
|
4.45
|
66.40
|
4.56
|
11
|
67.82
|
3.54
|
65.27
|
3.92
|
67.68
|
3.93
|
67.28
|
3.76
|
12
|
68.02
|
3.10
|
67.45
|
3.32
|
67.48
|
3.52
|
67.95
|
3.32
|
13
|
67.97
|
3.16
|
67.69
|
3.03
|
67.85
|
3.13
|
67.87
|
3.27
|
14
|
67.90
|
3.22
|
67.93
|
3.30
|
67.46
|
3.25
|
68.10
|
3.18
|
15
|
67.84
|
3.35
|
69.75
|
4.04
|
67.19
|
4.00
|
67.82
|
3.87
|
16
|
67.46
|
3.65
|
66.56
|
4.01
|
67.47
|
3.65
|
67.18
|
3.68
|
17
|
67.61
|
3.24
|
67.41
|
3.42
|
67.19
|
3.26
|
67.30
|
3.09
|
18, 19
|
|
|
|
|
|
|
|
|
20
|
66.55
|
3.99
|
67.37
|
4.35
|
66.53
|
3.95
|
66.77
|
4.36
|
21
|
67.44
|
3.44
|
68.35
|
3.29
|
67.13
|
3.52
|
67.65
|
3.30
|
22
|
67.82
|
3.29
|
67.76
|
3.13
|
67.81
|
3.04
|
68.16
|
2.99
|
23
|
67.84
|
3.04
|
68.48
|
2.99
|
67.64
|
2.69
|
67.97
|
2.95
|
24
|
67.97
|
2.71
|
68.30
|
2.76
|
68.09
|
2.69
|
67.37
|
3.29
|
25
|
67.79
|
3.10
|
67.17
|
3.36
|
67.85
|
3.02
|
67.51
|
3.56
|
26
|
68.02
|
2.90
|
67.06
|
3.21
|
67.74
|
2.99
|
67.56
|
2.90
|
27
|
66.89
|
3.72
|
66.78
|
3.51
|
66.78
|
3.44
|
66.87
|
3.63
|
28, 29
|
|
|
|
|
|
|
|
|
30
|
66.97
|
3.57
|
68.06
|
3.65
|
67.09
|
3.29
|
67.24
|
3.27
|
31
|
67.67
|
3.15
|
67.90
|
2.91
|
67.40
|
2.94
|
67.75
|
2.98
|
32
|
67.83
|
3.06
|
67.52
|
2.92
|
67.78
|
2.79
|
67.81
|
2.80
|
33
|
67.37
|
3.17
|
67.43
|
3.21
|
67.25
|
3.24
|
67.37
|
2.99
|
34
|
66.91
|
3.82
|
67.21
|
3.75
|
66.88
|
3.63
|
67.04
|
3.70
|
35
|
|
|
|
|
|
|
|
|
36
|
64.03
|
6.89
|
62.43
|
9.28
|
64.56
|
6.47
|
64.30
|
7.46
|
37
|
66.65
|
4.46
|
64.06
|
5.78
|
65.80
|
4.93
|
65.13
|
5.25
|
38
|
67.50
|
2.95
|
66.36
|
3.35
|
67.54
|
3.12
|
67.11
|
3.29
|
39
|
66.81
|
3.99
|
67.82
|
5.90
|
66.52
|
4.71
|
67.29
|
4.78
|
40
|
64.77
|
5.64
|
57.94
|
9.69
|
64.30
|
5.55
|
63.00
|
6.80
|
41
|
68.88
|
2.94
|
|
|
68.98
|
3.11
|
69.03
|
3.23
|
42
|
60.57
|
4.01
|
|
|
59.39
|
4.01
|
58.70
|
4.02
|
43
|
60.82
|
3.96
|
|
|
60.96
|
4.08
|
60.67
|
|
44
|
67.49
|
3.50
|
|
|
67.46
|
3.32
|
67.50
|
3.56
|
Average
|
67.19±0.9
|
3.69±0.8
|
66.41±2.3
|
4.20±1.6
|
66.96±0.9
|
3.85±
1.0
|
66.90±1.2
|
3.96±1.1
|
Inner
Six
|
67.86±0.1
|
3.08±
0.2
|
68.19±0.9
|
3.23±0.5
|
67.67±0.3
|
3.07±
0.5
|
67.76±0.3
|
3.28±
0.4
|
Table 4-1
Table
demonstrating the influence of crystal non-uniformity to peak position and
resolution The first two rows refer two a simulation run without compensation
for the non-uniformity of the individual crystals. The others include a gain
correction factor obtained by comparing single spectra without non-uniformity
and with non-uniformity.
The gain matching has been repeated with several energies in order to check
consistency. Thus it was carried out for 129 MeV photons and for both 70 MeV
positrons and photons. Following the gain matching routine as described above
one settles with a set of software gain factors. Beside the consistency among
those three simulated energies the accuracy of the final set of software gains
can be cross checked by looking at the energy resolution.
[18] For a description see [Hbo95]