Optimization of electromagnetic sampling calorimeters

II

Nuclear Instruments and Methods in Physics Resdarch A.314 (1992) 21-25 North-Holland

-

TR IIENTS A METNO« IN PMYSS RE$EARCM SectionA

Optimization of electromagnetic sampling calorimeters A. Prabhaharan and L. Bugge Department of Physics, P.O. Box 1048 Univcrsih~

of Oslv, N-0310 Oslo 3, Norway

Received 14 October 1991

Allowing variable sampling steps, we show that an electromagnetic calorimeter can be optimized to give small overall energy fluctuations from sampling and leakage. Compared to the case of constant sampling frequency the benefits are: significantly better energy resolution for a given number of samplers, a significant reduction of the number of samplers needed to obtain a given energy resolution, and a significant compactification of the calorimeter.

l. Introduction

where Ei is the deposited energy in sampler i, and

Sampling calorimeters are normally constructed with constant sampling steps throughout the device . Recently the question has been raised whether improvements can be obtained by mixing different absorber materials or by allowing variable sampling steps inside a calorimeter [1]. In the present report we argue that the answer to the latter question is yes: By optimizing the positions of the sampling layers (samplers), one can achieve a better energy resolution for a given number of samplers, or reduce the number of samplers (and hence the number of readout channels) needed to obtain a given energy resolution . The optimization procedure finds the optimum position of the sampling planes and an optimum effective length of the calorimeter (given by the position of the last sampling plane). We show that the optimized calorimeter, at least for the energy range studied here, is shorter than the constant sampling one, i.e. the optimization also leads to a compactification . In the present work we have studied the optimization problem for the case of incoming energies up to 45 GeV, resembling the situation for a luminosity monitor at LEP. We show that the optimized calorimeter has a linear energy response in this energy range.

Var

N

N

E Ei = F Var(Ei ) + 2 1: Cov( Ei,

(i=l

)

i=t

<<

Ej .

The relative fluctuation a-E/Ei of energy deposited by ionization in an active layer depends strongly on the depth of the sampler inside the calorimeter. At the same time the covariances Cov (Ei, Ej ) depend on the relative positions of samplers i and j. In general the covariance of two samplers which are placed close to each other is positive. It may become negative, however, when the distance between the samplers is increased and their absolute positions are chosen in a

2. The energy resolution The fluctuation of the total deposited energy for a calorimeter with N sampling layers is : =

Var

N i=1

Ei

)

,

, I6 20 ; ti î Depth of active rried , um inside the Coic-mete,

Fig. 1 . Longitudinal variation of relative fluctuation (GEANT simulation).

0168-9002/92/$05 .00 C 199,2 - Elsevier Science Publishers B.V . All rights reserved

A. Prabhaharan, L . Bagge / Electromagnetic sampling calorimeters

22 ~,

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on leakage of the position of the last sampling plane . Therefore, we present below a straightforward numerical algorithm for such an optimization . Consider a sampling calorimeter of total depth

cl

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negative .

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An

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pairs of samplers at different ,r (04675), o the value, pairs of samples (A, B) and

in

the

first

part of the The

part

shower,

fluctuations

quickly becomes large, CPU power will be a limiting factor .

of

the

shower,

Naturally the covariance of two samplers placed in the comes

a

sampling step T , . One expects the results to improve

can be defined as before (after) the shower maximum . latter

in

pling step will be an integer multiple of the minimum

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part of the shower . The first (latter) part of the shower

and

simulating

then

obvious constraint from the method is that each sam-

deposit less (more) energy than average in the latter

first

of

represents

denotes the

average

and

optimum positions of N out of M samplers defined as

suitable manner . Events which deposit more (less) enthan

consists

samplers

(C, D), respectively.

ergy

L

radiation lengths . With M samplers and constant sam. The method pling, the sampling step is T, =L/M

11 11 11 I:1 1, 1, L1 11 I,

u

relative fluctuations near shower

small

in the first and last part of the shower, and the effect

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maximum, the negative covariances between samplers

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optimally

of

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bede-

posited in a sampling layer are dominated by fluctuations in the number of charged particles crossing the

In the optimized calorimeter (M - N) layers are not used . It should be noted that the simulated optimized calorimeter thus

from a

is slightly different

practical

realization of it in the sense that the latter would not have the extra

partitioning of the

absorber

and

the

corresponding extra silicon layers. The excess of silicon layers in the simulated case affects the shower development, but we expect this effect to be total

thickness of the

additional

small since the

silicon layers repre-

sents only a few percent of a radiation length .

layer . Therefore the relative fluctuation of the energy deposited in

a sampling layer

has a minimum in the

vicinity of the shower maximum .

4. Results

In order to better understand how the relationship between covariances and variances of individual samplers affects the total energy resolution, we consider a simple case of two samplers . The fluctuation of Etl_,t = E, + E, is 1 ,'Var(E i ) + Var(E,) + 2Cov(E, , E,) . Consider now two combinations of two samplers, (A, B) and (C, D) . From figs. 1 and 2 we can conclude that for the pair (A, B) the relative fluctuation terms are small and the covariance term

is large, whereas for (C, D)

the opposite is true . A third posited

Our results

is the

in

the

electrons . The

3. The optimization procedure It is not easy (if possible at all) analytically to find the positions of N samplers in a calorimeter combining

table

1 . The table

type

codes are : C :

Constant

GeV

incoming

sampling,

and

O(NI M ): Optimized N out of M sampling planes, where the M

sampling

planes

are

placed

equidistantly

in

the

test

calorimeter . "Effective depth" denotes the calorimeter length after the optimization Test

No . of

calorimeter

samplers

Type

Effective vE /E depth

depth (X0 ~)

longitudinal

energy leakage .

in

Results for various test calorimeters for 45

Case

fluctuation

summarised

Table 1

important source of fluctuation in the de-

energy,

are

reports the results of 7 cases . Each case is character-

[%]

(X0 )

1

24

12

C

24

3 .77+0.08

2

24

12

0(12/24)

17

3 .36±0.08

3

17

12

C

17

3.94±0.09

4

24

10

000/24)

17

3 .75 ±0 .08

5

17

10

C

17

4 .48+0 .10

6

17

10

000/24)

17

3 .53 ±0 .08

7

24

8

0(8/24)

16

4 .26±0 .10

A. Prahhaharan, Its Bugge / Electrornagneticsampling calorimeters

ized by the length of the test calorimeter, the number of samplers, the type (constant sampling ("C") or optimized N out of M samplers ("O(N/M)")), and the effective calorimeter depth (position of the last sampler). For each case the resolution at 45 GeV is reported . As will be discussed below, the optimized calorimeter is linear in its response, and its resolution scales as 11F, so the results in the table can be scaled to any energy below 45 GeV . Using electron and photon energy cutoffs of 10-kcV a simulation [3] of a Si/W electromagnetic sampling calorimeter with constant ;`ampling frequency 2 (case 1) gave an energy resolution of crhonstant /E = 3.77 + 0.08% for a sample of 1000 incoming positrons of 45 GeV . This was obtained with a calorimeter of a total depth of 24X, i.e. with N = 12. The thickness of the silicon sampling layers was 500 gm. The calorimeter had the shape of a cylindrical shell with inner and outer radii of 6 and 15 cm, respectively. The incoming particles were generated at a fixed angle of 2.8° with the calorimeter axis. We then (case 2) applied the optimization procedure with M = 24 and consequently T, = I X,), and N = 12, i.e. the same number of samplers as in the constant sampling case. The constant step size result is significantly improved from 3.77 ± 0.08% to a value of C.Lptimized E = 3.36 ± 0.08%. The positions of the 12 / active samplers in this optimized case turned out to be after 2,l,2,1,2,2,1,l,2,1,1,1 X , respectively . It is interesting to note that the last sampler is at 17X. This means that the optimization has also resulted in a compactification of the calorimeter. The increased fluctuations from energy leakage is compensated for by the reduced contributions from the samplers with large sampling fluctuations in the last part of the shower . In fig . 3 we show the distribution of total energy deposit in the active layers . It is evident from the figure

Fig. 3. Distribution of total energy deposit in 12 sampling layers of a constant sampling calorimeter and an optimized calorimeter .

23

that an important result of the optimization is a significant increase in the total deposited energy . To check if the compact ification is a genuine effect of the optimization, we also simulated a test calorimeter of 17X depth and 12 sampling layers with constant sampling frequency. The result (case 3) is a relative energy resolution of 3.94%, showing that the compactification is indeed a consequence of the optimization . The next step was to investigate whether the optimization procedure could reduce the number of samplers needed to obtain a given energy resolution. To this end an optimization of 10 out of 24 samplers for a 24X,, test calorimeter was performed, i.e. a 17% reduction in the number of Si planes. The result, case 4 in table 1, was an energy resolution of (3.75 ± 0.08)%, i.e. comparable to the 24X 12 samplers constant sampling (case 1). Again the optimization lead to a compactification to an effective depth of 17X. For completeness a simulation of a 17X deep calorimeter with 10 samplers constant sampling was also run (case 5). It gave an energy resolution of (4.48 ± 0.10)%. Case 6 (T, = 17/24 = 0.71) compared to case 4 (T, = 24/24 = 1) shows that reducing the sampling step improves the optimization. Finally, case 7 shows that reducing the number of samplers to 8, i.e. a reduction of 33% in the number of silicon layers, worsens the resolution with only 13% compared to the 12 samplers constant sampling case, and at the same time gives a compactification to 16X.

5. Performance of the optimized calorimeter In this section we discuss the performance of the optimized 12 samplers calorimeter (case 2 in table 1) for energies below 45 GeV . A priori one might expect deviations from a linear energy response and from a 1/ FE behaviour of o-FJE due to the variable sampling steps . However, from fig . 4 which shows the response as a function of incoming electron energy for the 12 samplers constant and the 12 and 10 samplers optimized calorimeters, one sees no indication of deteriorated linearity properties in the optimized calorimeters as compared to the constant sampling one . The reason for the optimized calorimeter obeying linearity can be understood from fig . 5 which shows the fractional energy deposited in each sampling plane for 1 and 45 GeV showers, for the 12 samplers constant and optimized calorimeters . For a linear calorimeter the sum of fractional energy deposits should be the same for any energy . From the figure one sees that linearity is maintained in the optimized calorimeter because when the incoming energy increases, the increased fractional leakage is compensated for by increased relative energy response due to the nonuni-

A. Prahliaharan, L. Bugge / Electromagnetic sampling calorimeters

24

Table 2 Fit results for fits of energy response to Ed,,,. = a + h x E and 04

w w

w

energy resolution to or,.., / E _ 2, and 4 (see table 1)

pi /F `+ p2 for cases 1,

Case

Type

a [MeV]

h 10 --,

p,

p2

1

C

2

002/24) 000/24)

7.841 ±0.005 10 .158 f0.006 8.822 ±0 .006

26.4 ±0 .9 24 .5 f0.4 26 .0 ±0.6

0 f 1 .5 0

4

-0 .37 ±0 .06 -0.29 . f0.07 -1 .11 f 0.06

w

ti)

15

2(1

.~

'0

3h

a;1

.n

Fig. 4. Calorimeter energy (GeV) response as a function of incoming energy. 0 12 samplers constant sampling calorime ter. o 10 samplers optimized calorimeter. 0 12 samplers optimized calorimeter .

[%]

f 1 .1 0 ±2.6

coming energy [2], changing only by = 0.8X from 45 to 100 GeV. Fig . 6 shows the energy dependence of the energy resolution of the calorimeters . All three energy resolutions give good fits to: QF

form longitudinal distribution of the samplers . (The sampling frequency is higher in the second half of the calorimeter (8-17X,)) than in the first half.) The fit parameters which are given in table 2, are valid below 45 GeV . However, we do not expect big departures from linearity even for energies up to 100 GeV because the longitudinal shower profile changes little in the 45 to 100 GeV range. The position of the shower maximum depends logarithmically on the in-

[%n]

./E =

P ~(

l

)2

+P ; .

Therefore the energy resolutions in table 1 can be scaled to any energy below 45 GeV applying the parameters of table 2. The linearity and energy resolution properties of this particular example of an optimized calorimeter are not general results in the sense that all optimized calorimeters optimized at any energy would share these properties. In general there will exist many nearly equivalent solutions to the optimization problem with

.. .. . . . ... ....

Fig. 5 . Fraction of incoming energy deposited in each sampling layer at 1 and 45 GeV. The x-axis gives the sampler positions in units of X. (a) 12 samplers constant sampling calorimeter . (b) 12 samplers optimized calorimeter.

Fig. 6. Energy resolution as a function of incoming energy (GeV). The solid line and the given fit paramete rs are results of fits toQ,,./E=V(Pt/F ) +P; .

A. Prahltaharan, L. Buggc / Elcctrontagttctic satttpling calorimeters respect to energy resolution, and one should use the one with best linearity properties. 6. Conclusions We have described a method of optimizing the positions of sampling layers in an electromagnetic calorimeter to improve the energy resolution. The proccdare optimizes the contributions to the energy resolutiou from three sources : 1) individual fluctuation of energy deposit in each sampling layer. 2) Correlations between all pairs of sampling layers. 3) Leakage. We have shown that the method leads to a significant imp .-ovcment of the energy resolution for a given number of sampling layers, that the calorimeter can be compactified, and that the number of samplers needed to obtain a given energy resolution can be significantly reduced. The compact ification is a result of factor 1) above by avoiding the contributions from samplers in the last part of the shower where individual fluctuations are large.

25

We consider the possible reduction in the number of sampling planes (and the corresponding reduction in the number of readout channels) and the compactification (allowing savings in absorber material) as particularly important : An optimized calorimeter is shorter, lighter, and cheaper than a conventional, constant sampling one . We have also shown that the particular optimized calorimeter discussed in this article has a linear energy response and that its energy resolution obeys the 1 / law characteristic for constant sampling calorimeters. The authors arc currently looking into ways of improving the optimization algorithm .

References [ 1 ] C.W . Fabian and T. Ludlam. Ann . Rev . Nucl . Part . Sci . (1982) 32. [2) U. Amaldi, Phys . Scripta 23 (1981) 4119 . [3] The GÉANT Monte Carlo code. Version 3.14. CERN Program Library.