Simulation studies of the Moliere radius for EM calorimeter materials
The Monte-Carlo calculations of the Moliere radius (RM) for some homogeneous and heterogeneous media used in electromagnetic calorimetry in the energy range from 50 MeV to 10 GeV are presented in detail. The obtained results, the uncertainties in determining RM, estimations of the absorbed energy, m...
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Gavrishchuk, O.P. Kovtun, V.E. Malykhina, T.V. 2023-12-07T10:36:18Z 2023-12-07T10:36:18Z 2021 Simulation studies of the Moliere radius for EM calorimeter materials / O.P. Gavrishchuk, V.E. Kovtun, T.V. Malykhina // Problems of Atomic Science and Technology. — 2021. — № 6. — С. 171-174. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 02.70.Uu, 7.05.Tp, 29.40.Vj DOI: https://doi.org/10.46813/2021-136-171 https://nasplib.isofts.kiev.ua/handle/123456789/195804 The Monte-Carlo calculations of the Moliere radius (RM) for some homogeneous and heterogeneous media used in electromagnetic calorimetry in the energy range from 50 MeV to 10 GeV are presented in detail. The obtained results, the uncertainties in determining RM, estimations of the absorbed energy, methods for approximating the absorbed energy, and the accuracy of the results are discussed as well. Some RM are shown for calorimeter prototypes of the Spin Physics Detector experiment (SPD). A one-parameter function of the Moliere radius dependence on the absorber-scintillator thickness ratio is obtained. Представлені детальні обчислення методом Монте-Карло радіуса Мольєра (RM) для деяких гомогенних і гетерогенних середовищ, що використовуються в електромагнітній калориметрії в діапазоні енергій від 50 МеВ до 10 ГеВ. Обговорюються отримані результати, невизначеності опису RM, оцінки поглиненої енергії, методи апроксимації поглиненої енергії, точність результатів. Наведено RM для прототипів калориметра експерименту SPD. Отримано однопараметричну функцію залежності радіуса Мольєра від співвідношення товщин поглинача та сцинтилятора. Представлены детальные вычисления методом Монте-Карло радиуса Мольера (RM) для некоторых гомогенных и гетерогенных сред, используемых в электромагнитной калориметрии в диапазоне энергий от 50 МэВ до 10 ГэВ. Обсуждаются полученные результаты, неопределенности описания RM, оценки поглощенной энергии, методы аппроксимации поглощенной энергии,точность результатов. Приведены RM для прототипов калориметра эксперимента SPD. Получена однопараметрическая функция зависимости радиуса Мольера от соотношения толщин поглотителя и сцинтиллятора. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Experimental methods and processing of data Simulation studies of the Moliere radius for EM calorimeter materials Дослідження радіуса Мольєра для матеріалів електромагнітного калориметра методом комп'ютерного моделювання Исследование радиуса Мольера для материалов электромагнитного калориметра методом компьютерного моделирования Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Simulation studies of the Moliere radius for EM calorimeter materials |
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Simulation studies of the Moliere radius for EM calorimeter materials Gavrishchuk, O.P. Kovtun, V.E. Malykhina, T.V. Experimental methods and processing of data |
| title_short |
Simulation studies of the Moliere radius for EM calorimeter materials |
| title_full |
Simulation studies of the Moliere radius for EM calorimeter materials |
| title_fullStr |
Simulation studies of the Moliere radius for EM calorimeter materials |
| title_full_unstemmed |
Simulation studies of the Moliere radius for EM calorimeter materials |
| title_sort |
simulation studies of the moliere radius for em calorimeter materials |
| author |
Gavrishchuk, O.P. Kovtun, V.E. Malykhina, T.V. |
| author_facet |
Gavrishchuk, O.P. Kovtun, V.E. Malykhina, T.V. |
| topic |
Experimental methods and processing of data |
| topic_facet |
Experimental methods and processing of data |
| publishDate |
2021 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Дослідження радіуса Мольєра для матеріалів електромагнітного калориметра методом комп'ютерного моделювання Исследование радиуса Мольера для материалов электромагнитного калориметра методом компьютерного моделирования |
| description |
The Monte-Carlo calculations of the Moliere radius (RM) for some homogeneous and heterogeneous media used in electromagnetic calorimetry in the energy range from 50 MeV to 10 GeV are presented in detail. The obtained results, the uncertainties in determining RM, estimations of the absorbed energy, methods for approximating the absorbed energy, and the accuracy of the results are discussed as well. Some RM are shown for calorimeter prototypes of the Spin Physics Detector experiment (SPD). A one-parameter function of the Moliere radius dependence on the absorber-scintillator thickness ratio is obtained.
Представлені детальні обчислення методом Монте-Карло радіуса Мольєра (RM) для деяких гомогенних і гетерогенних середовищ, що використовуються в електромагнітній калориметрії в діапазоні енергій від 50 МеВ до 10 ГеВ. Обговорюються отримані результати, невизначеності опису RM, оцінки поглиненої енергії, методи апроксимації поглиненої енергії, точність результатів. Наведено RM для прототипів калориметра експерименту SPD. Отримано однопараметричну функцію залежності радіуса Мольєра від співвідношення товщин поглинача та сцинтилятора.
Представлены детальные вычисления методом Монте-Карло радиуса Мольера (RM) для некоторых гомогенных и гетерогенных сред, используемых в электромагнитной калориметрии в диапазоне энергий от 50 МэВ до 10 ГэВ. Обсуждаются полученные результаты, неопределенности описания RM, оценки поглощенной энергии, методы аппроксимации поглощенной энергии,точность результатов. Приведены RM для прототипов калориметра эксперимента SPD. Получена однопараметрическая функция зависимости радиуса Мольера от соотношения толщин поглотителя и сцинтиллятора.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/195804 |
| citation_txt |
Simulation studies of the Moliere radius for EM calorimeter materials / O.P. Gavrishchuk, V.E. Kovtun, T.V. Malykhina // Problems of Atomic Science and Technology. — 2021. — № 6. — С. 171-174. — Бібліогр.: 9 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2021. № 6(136) 171
https://doi.org/10.46813/2021-136-171
SIMULATION STUDIES OF THE MOLIERE RADIUS
FOR EM CALORIMETER MATERIALS
O.P. Gavrishchuk
1
, V.E. Kovtun
2
, T.V. Malykhina
2
1
Joint Institute for Nuclear Research (JINR), Dubna, Moscow Region, Russia;
2
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: vladimir.e.kovtun@univer.kharkov.ua
The Monte-Carlo calculations of the Moliere radius (RM) for some homogeneous and heterogeneous media used
in electromagnetic calorimetry in the energy range from 50 MeV to 10 GeV are presented in detail. The obtained
results, the uncertainties in determining RM, estimations of the absorbed energy, methods for approximating the ab-
sorbed energy, and the accuracy of the results are discussed as well. Some RM are shown for calorimeter prototypes
of the Spin Physics Detector experiment (SPD). A one-parameter function of the Moliere radius dependence on the
absorber-scintillator thickness ratio is obtained.
PACS: 02.70.Uu, 7.05.Tp, 29.40.Vj
INTRODUCTION
Requirements for the electromagnetic ECal calorim-
eter emerge from the physical problems of the SPD
NICA experiment [1]. A good energy resolution of 5%
must be ensured as well as effective π
0
-γ separation in
the energy range from 50 MeV to 10 GeV. The calorim-
eter module is a sampling lead-scintillator structure,
which has been investigated and significantly improved
for experiments KOPIO [2] and COMPASS II [3]. The
polystyrene scintillator thickness is the same for all pro-
totypes and is DPS=1.5 mm. The main prototype of the
ECaL SPD module has 4 cells and the silicon photomul-
tiplier (SiPM) with fiber readout. The final design ver-
sion will take into account the price factor and a detailed
calculation by the Monte-Carlo method for the im-
provement of the SPD calorimeter parameters [4, 5].
In this work we present the results of Monte-Carlo
simulation of the transverse evolution of an electromag-
netic shower in the ECal SPD module. The final goal is
to obtain reasonable values of the Moliere radius for the
heterogeneous structure of the prototypes of the SPD
calorimeter module.
1. RM CALCULATIONS METHODS
By definition [6], the Moliere radius RM is found
from the transverse dimension of the electromagnetic
(EM) shower absorbed by the medium according to the
formula:
( )
0.9
( )
ME R R
=
E R
. (1)
The results of the numerical solution of equation (1)
based on the first principles of the EM shower propaga-
tion in a medium by the Monte-Carlo method [7]
showed a significant difference with the estimates ac-
cording to the known formulas for RM both for homoge-
neous media and for heterogeneous media.
We consider various methods for solving equation
(1) by calorimeter modeling and subsequent processing
of the obtained data.
The first standard step consists of the generation of a
large number (in our case, N=10
4
) events using the
Geant4 toolkit for simulation of electromagnetic show-
ers from electrons with primary energy of 1 GeV. The
simulation of an infinite calorimeter was performed in
order to avoid shower energy leakage. At the next step,
90% of the deposited energy is summed up in accord-
ance with the chosen method.
1.1. RM CALCULATION FROM THE
TRANSVERSE SHAPE OF THE EM SHOWER
There are many parameterizations of the transverse
shape of the EM shower. The LumiCal collaboration
function was chosen as an example [8]. This function
was used to process experimental data from experiments
specifically devoted to finding the Moliere radius.
Comparisons (Fig. 1) were also made with Monte-Carlo
calculations.
Fig. 1. Parametrization of the shower transverse profile.
The function is taken from [8]
It was assumed that the shape of the EM shower has
an axial symmetry, and the distribution function consists
of a narrow Gaussian kernel and a wide hyperbola tail.
The function has four parameters that are found when
fitting. The result of this method depends on the region of
convergence and has, as a rule, a large parameter error.
1.2. RM CALCULATION BY FRACTION
OF DEPOSITED ENERGY
As a rule, the method for calculating the deposited
energy is based on the fact that the calorimeter is divid-
ed into pads with coordinates xiyj. In our case the pad
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 172
size is selected 0.150.15 mm. The deposited energy is
equal E90%=ΣEij provided R<RM. Fig. 2 shows a frag-
ment of the geometrical calculation of the Moliere radi-
us for some homogeneous materials that are used in
calorimetry.
Fig. 2. Geometric method of RM calculation
for some homogeneous media
The Moliere radius is determined by the intersection
of the E=E(R) function and the straight line E= 0.9·E0,
using this method. Table 1 presents the numerical val-
ues of the Moliere radius for some homogeneous mate-
rials. The presented results were obtained by the method
described above.
Table 1
Moliere radii for some homogeneous materials
Material Pb W U Cu Fe PS
RM , mm 19.4 11.9 11.14 25.1 28.1 137.5
The systematic error of the method is obviously de-
termined by the pad size and in our case is equal
ΔRm(syst.)~±0.2 mm.
1.3. RM CALCULATION FROM ENERGY
SPECTRUM
We propose a method for calculating RM directly
from the 90% peak of the total absorption of an EM
shower in the cylindrical volume of the calorimeter. An
example of an energy spectrum is shown in Fig. 3.
In this case, the shape of the peak will differ from
the Gaussian due to the energy leakage of the shower
into the outer cylinder.
The total absorption peak has the form of a δ-
function under the condition R<∞ since Emean=E0. Thus,
the problem is reduced to the correct determination of
the peak maximum when 90% of the shower energy is
absorbed. It should be noted that this effect of volume
energy leakage also exists for the previous case of cal-
culations. This effect is usually ignored when calculat-
ing the Moliere radius. The question is what value of the
deposited energy should be chosen. Whether the most
probable value of the deposited energy or the average
deposited energy value should be chosen.
In our case (E0=1 GeV), the difference between the
most probable energy distribution of events and the av-
erage value is ~4 MeV (0.5% of E0). The difference is
not very large for practical use, so we use the most
probable value.
Fig. 3. Energy spectrum of 90% of the deposited energy
in a lead cylinder. Vertical lines are the average
and the most probable value of deposited energies
The peak was approximated by the Das function [9],
from which the distribution parameters were obtained.
The most probable peak value ES and the standard devi-
ation σ are shown in Fig. 3.
Thus, we consider the most probable value of the re-
sulting distribution of the deposited energy in the cylin-
der, and not the average value. Such energy spectra are
close to the Gaussian distribution. Therefore, we have
χ
2
/ndf~1 for the used approximation as a rule [9].
The difference in the results of the two previous
methods is negligible, so the subsequent results were
obtained by the pads method.
2. RM FOR IDEAL SPD ECAL
The obtained values of Moliere radii for ideal proto-
types of calorimeters with Pb and W absorbers are pre-
sented in Tables 2 and 3.
Table 2
Moliere radii for ideal prototypes of calorimeters
with Pb absorbers
DPb[mm]+DPS(1.5 mm) 0.3 0.4 0.5
RM [mm] 74.0 65.0 58.5
Table 3
Moliere radii for ideal prototypes of calorimeters
with W absorbers
DW[mm]+DPS(1.5 mm) 0.3 0.4 0.5
RM [mm] 53.2 45.3 40.0
DPS, DPb, DW – thickness of the active and passive parts
of the calorimeter.
We present the result of RM calculations for the het-
erogeneous structure of the SPD calorimeter prototype
(Fig. 4) for comparison with the homogeneous structure
(see Fig. 2).
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 173
Fig. 4. Geometric method of RM calculation
for ECal prototypes with lead absorber
3. RM DEPENDENCE ON THE ABSORBER
THICKNESS
The Monte-Carlo method was used to investigate the
Moliere radius in more detail, depending on any ratio of
the thicknesses of the passive and active substances of
the calorimeter. A universal approximation function is
found:
1
1 1
PS abs
M M M
x x+a x
R (x)= R +R
+a x +a x
, (2)
where the variable x varies within [0,1]:
abs
abs PS
D
x =
D + D
, (3)
0 PS
M M
R ( )= R , 1 abs
M M
R ( )= R , (4)
DPS, Dаbs – the thicknesses of the active and passive
parts of the calorimeter.
PS
M
R ,
abs
MR – Moliere radii of
the plastic scintillator and absorber, MR (x) – the Mo-
liere radius of the sampling calorimeter as a function of
the ratio of the thicknesses x. The ‘a’ parameter is the
only dimensionless fitting parameter in the formula (2).
Fig. 5 shows functions (2)-(4) for a sampling calo-
rimeter with lead and tungsten absorbers and the ‘a’
parameter values obtained by fitting.
4. THE RESULTS TESTING
In order to obtain the test results and identify sys-
tematic errors, the parameters of computer models were
varied. Fig. 6 shows that in a wide range of RangeCut
values (from 1 m to 1 mm) the RM calculation results
do not differ significantly. The calculations were also
performed with a statistical accuracy of 1% for incident
electrons or photons in energy range from 100 MeV to
10 GeV.
During a series of calculations, the most suitable
model of physical processes emstandard_opt4 was cho-
sen instead of QGSP_BERT model in the Physics List
class.
Fig. 5. Universal function from (2)-(4) and RM depend-
ence on variable thickness x and fitting parameter ‘a’
Fig. 6. RM dependence on the cut-off value
Fig. 7. RM dependence on the energy
of primary particles
This allows to focus on purely electromagnetic pro-
cesses without taking into account nuclear reactions,
which appear in a small quantity when using
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 174
QGSP_BERT model. It is also shown (Fig. 7) that the
Moliere radius does not depend on the energy of the
incident particle.
The increase of RM in function RM=RM(E0) at the
beginning of the curve can be explained by the actual
absence of multiple processes of an electromagnetic
shower in the region of very low energies. It is also dif-
ficult to find the correct definition of RM in this area.
Errors in RM calculations are determined by the statisti-
cal error associated with the number of events and sys-
tematic errors due to data processing methods.
To determine the latter, series of calculations were
carried out. We estimate the total errors in calculating
RM as ΔRM±0.5 mm for the data from Tables 1-3.
CONCLUSIONS
A simulation study of the Moliere radius for an ideal
sampling calorimeter is presented. Various practical
methods are shown for solving the equation for the Mo-
liere radius, which follows from the Moliere radius def-
inition. The data were obtained by the Monte-Carlo
method. A convenient approximation of the curve of the
Moliere radius dependence on any possible thicknesses
of the active and passive parts of the sampling calorime-
ter is found. Thus, the formula is suitable for use in both
homogeneous and heterogeneous environments. An
estimate of the calculation accuracy for the obtained
results is made, which can be used in the development
of a sampling calorimeter of the SPD NICA setup. The
obtained results are practically independent of some
Geant4 parameters, such as cut, the Physics Lists model,
and the energy range of electrons or gamma quanta. The
methodology described in this paper for Moliere radius
calculation can be easily adapted to any sampling calo-
rimeter.
REFERENCES
1. Conceptual design of the Spin Physics Detector,
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arXiv::2102.00442v1, 31 Jan 2021, p. 191.
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3. I. Chirikov-Zorin, Z. Krumshtein, and A. Olchevski.
The design of a photodetector unit of a new
Shashlyk EM calorimeter for COMPASS II // Nucl.
Instrum. Meth. A. 2016, v. 824, p. 674.
4. O.P. Gavrishchuk, V.E. Kovtun, T.V. Malykhina.
Simulation Study of Energy Resolution of the Elec-
tromagnetic Shashlyk Calorimeter for Different of
Layers and Absorber Combinations // East Eur. J.
Phys. 2020, v. 3, p. 73-80, https://doi.org/10.26565/
2312-4334-2020-3-09.
5. O.P. Gavrishchuk, V.E. Kovtun, T.V. Malykhina.
Effect of energy leakage on the energy resolution of
E.M. sampling calorimeters // Problems of Atomic
Science and Technology. 2021, № 3(133), p. 76-80,
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8. H. Abramowicz et al. Measurement of shower de-
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Article received 04.10.2021
ИССЛЕДОВАНИЕ РАДИУСА МОЛЬЕРА ДЛЯ МАТЕРИАЛОВ ЭЛЕКТРОМАГНИТНОГО
КАЛОРИМЕТРА МЕТОДОМ КОМПЬЮТЕРНОГО МОДЕЛИРОВАНИЯ
О.П. Гаврищук, В.Е. Ковтун, Т.В. Малыхина
Представлены детальные вычисления методом Монте-Карло радиуса Мольера (RM) для некоторых гомо-
генных и гетерогенных сред, используемых в электромагнитной калориметрии в диапазоне энергий от
50 МэВ до 10 ГэВ. Обсуждаются полученные результаты, неопределенности описания RM, оценки погло-
щенной энергии, методы аппроксимации поглощенной энергии, точность результатов. Приведены RM для
прототипов калориметра эксперимента SPD. Получена однопараметрическая функция зависимости радиуса
Мольера от соотношения толщин поглотителя и сцинтиллятора.
ДОСЛІДЖЕННЯ РАДІУСА МОЛЬЄРА ДЛЯ МАТЕРІАЛІВ ЕЛЕКТРОМАГНІТНОГО
КАЛОРИМЕТРА МЕТОДОМ КОМП’ЮТЕРНОГО МОДЕЛЮВАННЯ
О.П. Гаврищук, В.Є. Ковтун, Т.В. Малихіна
Представлені детальні обчислення методом Монте-Карло радіуса Мольєра (RM) для деяких гомогенних і
гетерогенних середовищ, що використовуються в електромагнітній калориметрії в діапазоні енергій від
50 МеВ до 10 ГеВ. Обговорюються отримані результати, невизначеності опису RM, оцінки поглиненої енер-
гії, методи апроксимації поглиненої енергії, точність результатів. Наведено RM для прототипів калориметра
експерименту SPD. Отримано однопараметричну функцію залежності радіуса Мольєра від співвідношення
товщин поглинача та сцинтилятора.
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https://doi.org/10.1093/ptep/ptaa104
https://doi.org/10.1140/epjc/s10052-018-5611-9
https://arxiv.org/abs/1603.08591
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