Comparison of proton shower developments in the BGO calorimeter of the Dark Matter Particle Explorer between GEANT4 and FLUKA simulations
Wei Jiang, Chuan Yue, Ming-Yang Cui, Xiang Li, Qiang Yuan, Francesca Alemanno, Paolo Bernardini, Giovanni Catanzani, Zhan-Fang Chen, Ivan De Mitri, Tie-Kuang Dong, Giacinto Donvito, David Francois Droz, Piergiorgio Fusco, Fabio Gargano, Dong-Ya Guo, Dimitrios Kyratzis, Shi-Jun Lei, Yang Liu, Francesco Loparco, Peng-Xiong Ma, Giovanni Marsella, Mario Nicola Mazziotta, Xu Pan, Wen-Xi Peng, Antonio Surdo, Andrii Tykhonov, Yi-Yeng Wei, Yu-Hong Yu, Jing-Jing Zang, Ya-Peng Zhang, Yong-Jie Zhang, Yun-Long Zhang
CComparison of proton shower developments in the BGO calorimeter of the Dark Matter ParticleExplorer between GEANT4 and FLUKA simulations
Wei Jiang( 蒋 维 ) , , Chuan Yue( 岳 川 ) ∗ , Ming-Yang Cui( 崔 明 阳 ) † , Xiang Li( 李 翔 ) , Qiang Yuan( 袁 强 ) , ,Francesca Alemanno , , Paolo Bernardini , , Giovanni Catanzani , , Zhan-Fang Chen( 陈 占 方 ) , , Ivan DeMitri , , Tie-Kuang Dong( 董 铁 矿 ) , Giacinto Donvito , David Francois Droz , Piergiorgio Fusco , , FabioGargano , Dong-Ya Guo( 郭 东亚 ) , Dimitrios Kyratzis , , Shi-Jun Lei( 雷 仕 俊 ) , Yang Liu( 刘 杨 ) , FrancescoLoparco , , Peng-Xiong Ma( 马 鹏 雄 ) , , Giovanni Marsella , , Mario Nicola Mazziotta , Xu Pan( 潘 旭 ) , ,Wen-Xi Peng( 彭 文 溪 ) , Antonio Surdo , Andrii Tykhonov , Yi-Yeng Wei( 魏 逸 丰 ) , Yu-Hong Yu( 余 玉 洪 ) ,Jing-Jing Zang( 藏 京京 ) , Ya-Peng Zhang( 张 亚 鹏 ) , Yong-Jie Zhang( 张 永 杰 ) , and Yun-Long Zhang( 张 云 龙 ) Key Laboratory of Dark Matter and Space Astronomy,Purple Mountain Observatory,Chinese Academy of Sciences, Nanjing 210023, China School of Astronomy and Space Science,University of Science and Technology of China, Hefei 230026, China Gran Sasso Science Institute (GSSI),Viale F. Crispi 7, I-67100, L’Aquila, Italy Istituto Nazionale di Fisica Nucleare,Laboratori Nazionali del Gran Sasso,Via G.Acitelli 22, I-67100, Assergi, L’Aquila, Italy Dipartimento di Matematica e Fisica E. De Giorgi,Universit`a del Salento, I-73100 Lecce, Italy Istituto Nazionale di Fisica Nucleare (INFN)–Sezione di Lecce, I-73100 Lecce, Italy INFN Section of Perugia, I-06100 Perugia, Italy University of Perugia, I-06100 Perugia, Italy Istituto Nazionale di Fisica Nucleare (INFN)–Sezione di Bari, I-70125, Bari, Italy Department of Nuclear and Particle Physics,University of Geneva, CH-1211, Switzerland Dipartimento Interateneo “M. Merlin” dell’Universit`a degliStudi di Bari e del Politecnico di Bari, I-70125, Bari, Italy Dipartimento di Fisica e Chimica “E. Segr`e”,via delle Scienze Edificio 17,Universit`a Degli Studi di Palermo, I-90128, Palermo, Italy Institute of High Energy Physics,Chinese Academy of Sciences,YuquanLu 19B, Beijing 100049, China State Key Laboratory of Particle Detection and Electronics,University of Science and Technology of China, Hefei 230026, China Institute of Modern Physics, Chinese Academy of Sciences,Nanchang Road 59, Lanzhou 730000, China (Dated: September 29, 2020)The DArk Matter Particle Explorer (DAMPE) is a satellite-borne detector for high-energy cosmic rays and γ -rays. To fully understand the detector performance and obtain reliable physical results, extensive simulationsof the detector are necessary. The simulations are particularly important for the data analysis of cosmic raynuclei, which relies closely on the hadronic and nuclear interactions of particles in the detector material. Widelyadopted simulation softwares include the G EANT
PACS numbers: 96.50.S-, 13.85.Tp, 13.85.-t, 95.55.-n ∗ Corresponding author: [email protected] † Corresponding author: [email protected]
I. INTRODUCTION
The magnetic spectrometer experiments such as PAMELAand AMS-02 have pushed the precise measurements of en-ergy spectra of cosmic rays (CRs) to rigidities of ∼ TV (e.g.,[1–3]). At higher energies, the measurements still have largeuncertainties, which hinder a better understanding of the ori- a r X i v : . [ phy s i c s . i n s - d e t ] S e p gin and propagation of CRs [4]. In recent years, a numberof space calorimeter experiments have been launched, suchas the CALET [5], NUCLEON [6], DAMPE [7, 8], and ISS-CREAM [9], which have already or are expected to improvethe direct measurements of CR spectra remarkably.The Dark Matter Particle Explorer, is the first Chinese satel-lite for astroparticle physics studies. It was launched on De-cember 17, 2015, and has operated in a sun-synchronous orbitfor more than 4 years ever since. The DAMPE is dedicatedto indirectly detect the annihilation or decay products of darkmatter via high-energy-resolution measurements of CR elec-trons plus positrons and γ -rays. As a CR particle detector, theDAMPE can also explore the origin of CRs, as well as thetransient high-energy γ -ray sky [8, 10].The DAMPE detector is made up of four sub-detectors, in-cluding a Plastic Scintillator Detector (PSD; [11]), a SiliconTungsten tracKer-converter (STK; [12]), a Bismuth Germa-nium Oxide imaging calorimeter (BGO; [13]), and a NeUtronDetector (NUD; [14]). These four sub-detectors cooperate togive high-precision measurements of the charge, direction, en-ergy, and identity of each incident particle (see Ref. [8] formore details). The on-orbit calibration shows that the detec-tor is quite stable with time after the launch [15]. Up to now,high-precision measurements of the CR electron plus positronspectrum and the proton spectrum in wide energy ranges havebeen reported by the DAMPE collaboration [16, 17].Dedicated Monte Carlo (MC) simulations of the particleresponse in the DAMPE detector, including the impacts ofthe modules of the satellite platform, are important for under-standing the detector performance, such as the evaluations ofe ffi ciencies, the energy and direction responses, and the back-ground contaminations. For the hadronic CR analysis simula-tions are even more crucial since the calorimeter only recordsa fraction of the particle’s energy and the full energy responsecan only be obtained by simulations. Two of the leading soft-wares widely used for particle simulations are G EANT
EANT ++ toolkit to simu-late the passage of particles through matter. It has a large setof physics processes handling the complicated interactions ofparticles in the matter up to 100 TeV energies. FLUKA is aFORTRAN based, fully integrated particle physics simulationpackage for calculations of particle transport and interactionswith matter in the energy range from MeV up to PeV.The hadronic showers are essentially hybrid cascades ofhadronic processes and electromagnetic processes. Theinelastic hadronic interactions produce secondary particles(mainly pions), and charged pions may induce additionalhadronic interactions, while the neutral pions would mostlikely decay into photons which experience electromagneticcascades further. Other physical processes governing thehadronic showers include nuclear fragmentation, ionization,elastic scattering, nuclear de-excitation, and so on [4]. TheHARP-CDP experiments reported the comparison of the pro-duction yields of the interactions of protons and charged pi-ons with beryllium, copper, and tantalum nuclei between thesetwo software tools with the momentum up to 15 GeV / c [21].A poor agreement between the G EANT
EANT
EANT ff erencecan be considered as an estimate of the systematic uncertain-ties of the proton spectrum measurements [17]. II. DETECTOR SIMULATIONSA. Geometry configuration
Both G
EANT ff erent ele-ments in the detector for precise hadronic and electromagneticshower cascade simulation. For most parts of the satellite, themanufacturers provide detailed element components. For theremaining filling materials and electronic components whosecompositions were unknown, we send their samples to ana-lytical laboratories for detailed measurements to get the exactmass fractions of atoms. Thus, the geometric model of theentire satellite is established, which accurately describes thedetection units, the supporting structure and the filling cush-ioning materials of the sub-detectors, as well as the frame andelectronic components of the satellite, as shown in Fig. 1.The BGO is the core sub-detector of the satellite payload.The characteristics of its interaction with protons are empha-sized in this paper. The detailed structure of the BGO is shownin Fig. 2. We precisely configure the geometry of the BGOwhich consists of 14 layers, each with 22 crystal bars. Eachbar is an independent detection unit with independent read-out circuits at the two ends, assembled in a braced frame withcushioning material filling all the internal gaps. After a greatdeal of measurements and verifications, we configure a pre-cise BGO model in the geometry.Following the configuration and validation of the designedgeometry, we call the Geometry Description Markup Lan-guage (GDML) [27] interface for the specific program imple-mentation of G EANT
PSDSTK BGO NUD Braced Frame Star-trackers
High-voltage power supply , Circuit boards , Electronic components
Satellite ba ffl es FIG. 1: The geometric model of the entire satellite, including thedetector payload and the satellite platform.
X Layer (22 BGO bars)Y Layer 14 Layers
FIG. 2: The geometric model of the BGO calorimeter. rived from the designed documents and measured results andare checked repeatedly to confirm that this geometry reflectsthe real situation of the satellite accurately. The GDML geom-etry is integrated into the DAMPE o ffl ine software framework[28], which performs a series of standardized procedures (cal-ibration, reconstruction and analysis) from the real “raw data”,the original signal collected by each prob cell of DAMPE, toscientific results. Therefore, the GDML geometry is appliedas a unified interface for the data analysis of DAMPE. Onthe other hand, the FLUKA simulation is fully integrated andclosed source code. It only allows the Combinatorial Geom-etry (CG) [29] interface to develop the geometry. The geom-etry is rewritten from the GDML for the FLUKA simulation,called the “CG geometry”. After careful and repeated checksof these two geometry models in every detail, we are confi-dent that they are identical between each other and consistentwith the real satellite geometry, although there are slight dif- ferences in some micro components of the satellite which arenegligible during the simulation. B. Data process
The data flow for the complete simulation process is shownin Fig. 3, including primary generation, MC simulations, dig-itization, reconstruction and analysis. The first package, pri-mary generation, creates the incident particles feeding the MCsimulation including various distributions of incident posi-tions, directions and energies. In this work, we generate aflux of primary protons distributed isotropically with a sin-gle power-law energy spectrum with the index -1 from 10GeV to 100 TeV. The G
EANT ffl ine software framework so thatthe G EANT
EANT
EANT • the PEANUT package is activated in the whole energyrange for any reaction; • the minimum kinetic energy for DPMJET-III is set to5 GeV / n (applying only to reactions between two nu-clei heavier than a proton); • the minimum kinetic energy for RQMD is set to0 .
125 GeV / n (applying only to reactions between twonuclei heavier than a proton); • the same output format as the G EANT
PrimaryGeneration
Monte CarloSimulationDigitization Reconstruction Analysis
GDMLGeometry G EANT
CGGeometry
FIG. 3: General scheme for the full simulation data process.
Following numerous tests and validations to the developedsimulation package including a set of algorithms which areresponsible for generating the interactions of particles withthe detector based on both the G
EANT and FLUKA2011.2x , we allocate massive computing resources to runthese programs, producing the simulation data of billions ofprotons.Then, we run the digitization package to convert the phys-ical information into the digital signal of each detection unitassigning a digital ID. In such a way the digital information ofthe simulation is in the same format as the real “raw data”. Ac-cordingly, we can run the reconstruction package which con-tains large amounts of code for the for building up the phys-ical signals including deposited energy, reconstructed tracksand charge of each event from the “raw data”. This package isorganized as a series of algorithms that act successively to pro-cess the on-orbit data on a daily basis [15]. The massive codeto obtain the scientific results and the instrument performanceof the detector based on the reconstructed data is collectivelyreferred to as the analysis package, which is the result of col-lective e ff orts of many researchers. All the code in the pack-age undergoes continues enhancement and version update asthe detector comprehension improves with time. Major pub-lished results were also obtained using the package to analyzethe on-orbit data and simulation data. In these analysis pack-ages, the event selection packages including a list of selectionconditions for target particles are fundamental for the analy-ses. In this work, we focus on figuring out some features toanalyze the response of protons in the BGO calorimeter. Allthe below results are obtained based on the selected protonsamples following the event selections in Ref. [17]. III. RESULTSA. Tigger e ffi ciency Firstly, we investigate the tigger e ffi ciencies for simulationsusing G EANT ff erent trig-gers implemented on orbit: the Unbiased trigger, the Mini-mum Ionizing Particle (MIP) trigger, the Low-Energy (LE)trigger, and the High-Energy (HE) trigger [31]. The Unbiasedand MIP triggers are designed for the detector calibration [15],while the LE and HE triggers correspond to low threshold andhigh threshold triggering signals respectively. In the protonanalysis, the events are required to meet the HE trigger condi-tion in order to guarantee that the shower development startsabove or at the top of the calorimeter. The HE trigger e ffi -ciency is one of the most important factors related to the e ff ec-tive acceptance estimation. For di ff erent hadronic integrationmodels, the shower start-point and the secondaries from thefirst inelastic interaction would be di ff erent. As a result, we http: // geant4.web.cern.ch https: // would consider the di ff erence of the HE trigger e ffi cienciesbetween G EANT ffi ciency is estimated by means of the Un-biased trigger samples. The Unbiased trigger events are pre-scaled by a factor of 512 at latitudes ≤ ◦ and 2048 at lat-itudes > ◦ . The HE trigger e ffi ciency for protons is com-puted as ε trigger = N HE&Unb N Unb , (1)where N Unb is the number of proton events passing the Un-biased trigger condition and N HE&Unb is the number of oneswhich both pass the HE and Unbiased trigger conditions. Fig.4 shows the comparison of HE trigger e ffi ciencies among theflight data, G EANT
EANT ∼ −
5% comparedwith the G
EANT H E T r i gge r E ff i c i en cy Flight DataSimu-GEANT4Simu-FLUKA
BGO Energy [GeV] S i m u / D a t a FIG. 4:
The HE tigger e ffi ciencies for protons from FLUKA,G EANT
The top panel shows the HE tigger ef-ficiencies defined by Eq. 1. The bottom panel shows the e ffi ciencyratio of FLUKA and G EANT
B. Total energy deposit
The energy of an incident proton is measured by the sumof energy deposits of all BGO crystals in the calorimeter, i.e.the total energy deposit. Due to the limited vertical thick-ness of the BGO calorimeter ( ∼ . Energy Deposit Ratio N u m be r o f E v en t s Beam DataSimu-GEANT4Simu-FLUKA
Incident Energy [GeV] E ne r g y D epo s i t R a t i o Simu-GEANT4Simu-FLUKA
FIG. 5:
The energy response for protons from G
EANT
The top panel shows the distribution of the ratio of totalenergy deposit with respect to the incident energy for on-axis in-cident proton beams with 400 GeV / c momentum. Black, red andblue histograms correspond to Beam Data, G EANT
EANT E − . power-law spectrum. Di ff erent hadronic interaction models would present di ff er-ent energy response matrices [17], thereby leading to di ff er-ent deconvolutions for the initial spectrum of cosmic-ray pro-ton. Before launch, the Engineering Qualification Model ofDAMPE was extensively tested using test beams at the Eu-ropean Organization for Nuclear Research (CERN) in 2014-2015. To compare with the test beam data, we generate MCsamples follwing closely the settings of the test beams, suchas the incident energies, hit points, and directions. We alsoapply the same event selections to both the beam test data andMC data as those in the flight data analysis [17], includingthe HE trigger, the track selection, the geometric cut, and thecharge selection. The energy response of DAMPE for the on-axis incident proton beam with the momenta of 400 GeV / c iscompared with the results from G EANT
EANT / c. To further compare the en-ergy responses from G EANT E − . spectrum from 10 GeV to 100 TeV is generated for the sim-ulations. In the analysis, the spectra are re-weighted to E − . to be consistence with the CR flux. The most probable valuesof the deposited energies obtained by fitting the energy ra-tio probabilities with an asymmetric gaussian function, alongwith the incident energy, are shown in the bottom panel of Fig.5. The energy responses of G EANT ff erence from 10 GeV to 100 TeV, in con-sequence, the deconvoluted proton spectra based on G EANT ff erent spectral indices. C. Longitudinal development
The longitudinal development of a hadronic shower ishighly determined by the first inelastic interaction point, i.e.the inelastic scattering cross-section between the incident pro-ton and the detector material. We calculate the ratios of the en-ergy deposits in di ff erent BGO layers with the total energy de-posit to describe the longitudinal shower development. Fig. 6shows the comparisons of layer energy ratios among flightdata, G EANT
EANT
D. Transverse development
The transverse shower development, however, is intimatelyassociated with the distribution of the types of subsidiaryparticles created through the interactions. We calculate theshower spread to characterize the transverse development,expressed by the energy-weighted root-mean-square (RMS)value of hit positions in the calorimeter. The RMS value ofthe fired i th layer is calculated as: RMS i = (cid:115) Σ j ( x j , i − x c , i ) E j , i Σ j E j , i (2)Where x j , i and E j , i are the coordinates and energy deposit ofthe j th bar in the i th layer, and x c , i is the energy-weighted cen-tre coordinate of the i th layer. Fig. 7 show the comparisonsof RMS values in di ff erent layers among flight data, G EANT ff erences among the FLUKA, G EANT
EANT
EANT
Energy Ratio -4 -3 -2 -1
10 1 N o r m a li z ed N u m be r o f E v en t s Layer-1
Energy Ratio -4 -3 -2 -1
10 1 N o r m a li z ed N u m be r o f E v en t s Layer-5
Energy Ratio -4 -3 -2 -1
10 1 N o r m a li z ed N u m be r o f E v en t s Layer-10
Energy Ratio -4 -3 -2 -1
10 1 N o r m a li z ed N u m be r o f E v en t s Layer-14
Bgo Layer Number E ne r g y R a t i o /GeV < 63 dep
40 < E
Bgo Layer Number E ne r g y R a t i o /GeV < 398 dep
251 < E
Bgo Layer Number E ne r g y R a t i o /GeV < 2512 dep Bgo Layer Number E ne r g y R a t i o /GeV < 15849 dep FIG. 6:
The longitudinal shower development for protons from G
EANT
The 4 plots on the left side show the energy ratiodistributions in 4 typical BGO layers for total energy deposit between 1000 GeV and 1580 GeV. Black, red and blue histograms correspond toflight data, G
EANT
Shower Width (RMS)
20 40 60 80 100 120 140 N o r m a li z ed N u m be r o f E v en t s Layer-1
Shower Width (RMS)
20 40 60 80 100 120 140 N o r m a li z ed N u m be r o f E v en t s Layer-5
Shower Width (RMS)
20 40 60 80 100 120 140 N o r m a li z ed N u m be r o f E v en t s Layer-10
Shower Width (RMS)
20 40 60 80 100 120 140 N o r m a li z ed N u m be r o f E v en t s Layer-14
Bgo Layer Number S ho w e r W i d t h /GeV < 63 dep
40 < E
Bgo Layer Number S ho w e r W i d t h /GeV < 398 dep
251 < E
Bgo Layer Number S ho w e r W i d t h /GeV < 2512 dep Bgo Layer Number S ho w e r W i d t h /GeV < 15849 dep FIG. 7:
The transverse shower development for protons from G
EANT
The 4 plots on the left side show the RMS distribu-tions in 4 typical BGO layers for total energy deposit between 1000 GeV and 1580 GeV. Black, red and blue histograms correspond to Flightdata, G
EANT
E. E ff ect on the proton spectrum The absolute proton flux F in an incident energy bin[ E i , E i + ∆ E i ] can be calculated as F ( E i , E i + ∆ E i ) = N inc , i A e ff , i ∆ E i T exp ; N inc , i = n (cid:88) j = M i j N dep , j , (3)where N inc , i is the number of events in the i th incident energybin, N dep , j is the number of events in the j th deposited energybin, M i j is the response matrix, A e ff , i is the e ff ective accep-tance, ∆ E i is the width of the energy bin, and T exp is the ex-posure time. N inc , i in each incident energy bin can be obtainedvia the unfolding procedure based on the Bayes theorem [32].The proton spectrum depends closely on the e ff ective ac-ceptance and the energy response matrix, both are obtainedfrom MC simulations. The acceptance is obtained throughcalculating the fraction of events in each incident energy binsurvived from the whole selection procedure, and the responsematrix is obtained by counting the fraction of events in the deposited energy bin j for given incident energy bin i . Weapplied the same selections as the flight data analysis [17] toobtain the corresponding e ff ective acceptances and energy re-sponse matrices for protons. The e ff ective acceptance fromthe FLUKA sample is lower than that from the G EANT ∼ ffi ciency dif-ference (see Fig. 4). On the other hand, the energy responsedi ff erence between two MC softwares (see Fig. 5) results ina complex e ff ect on the fluxes after the spectrum deconvo-lution. The overall proton flux di ff erence between G EANT ff erence can be large as 10%, the global spectral structuresare consistent with each other. Based on the comparisonsof shower development, we chose the G EANT ff erence between G EANT ff erence varies from − .
6% to 9 . ff erent hadronic interac-tion models [17]. Kinetic Energy [GeV] S ys t e m a t i c U n c e r t a i n t y -0.4-0.200.20.4 Uncertainty
Geant4 F )/ Geant4 F - FLUKA F ( FIG. 8:
Energy dependence of the proton flux di ff erence betweenG EANT
The blue points show the di ff erence of mea-sured proton spectrum assuming FLUKA simulation with respect tothe spectrum based on G EANT
IV. CONCLUSION
As a calorimeter-based experiment, DAMPE depends heav-ily on the precise simulation of the interactions between theincident particle and the detector. Due to the limited verti-cal thickness of the DAMPE calorimeter and the large un-certainty for the hadronic interactions, the proton measure-ment is highly associated with the simulation of the shower development. The comparison of the proton simulations ofDAMPE between G
EANT ff erences. For the overall energy deposition, the FLUKAresults are higher by (3 ∼ ∼ ff ect the trigger e ffi ciency evaluation of pro-tons, which is leading to a deviation about 5% between theresults of these two simulation softwares. The overall uncer-tainties due to the hadronic models are estimated to be about10%. Acknowledgments
This work is partly supported by theNational Key Research and Development Program of China(Grant No. 2016YFA0400200), and the National Natural Sci-ence Foundation of China (Grant Nos. 11722328, 11773085,U1738127, U1738138, U1738205, U1738207, 11851305),the 100 Talents Program of Chinese Academy of Sciences, theYouth Innovation Promotion Association CAS, and the Pro-gram for Innovative Talents and Entrepreneur in Jiangsu. InEurope the activities are supported by the Swiss National Sci-ence Foundation (SNSF), Switzerland, and the National Insti-tute for Nuclear Physics (INFN), Italy. [1] O. Adriani et al. (PAMELA), Science , 69 (2011),1103.4055.[2] M. Aguilar et al. (AMS), Phys. Rev. Lett. , 171103 (2015).[3] M. Aguilar et al. (AMS), Phys. Rev. Lett. , 251101 (2017).[4] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D ,030001 (2018).[5] S. Torii and P. S. Marrocchesi (CALET), Adv. Space Res. ,2531 (2019).[6] E. Atkin et al., EPJ Web Conf. , 01002 (2015).[7] J. Chang, Chinese Journal of Space Science , 550 (2014).[8] J. Chang et al. (DAMPE), Astroparticle Physics , 6 (2017).[9] S. Kang et al., Adv. Space Res. , 2564 (2019).[10] Q. Yuan and L. Feng, Sci. China Phys. Mech. Astron. ,101002 (2018), 1807.11638.[11] Y. Yu et al., Astroparticle Physics , 1 (2017), 1703.00098.[12] P. Azzarello et al., Nuclear Instruments and Methods in PhysicsResearch A , 378 (2016).[13] Z. Zhang et al., Nuclear Instruments and Methods in PhysicsResearch A , 98 (2016).[14] Y.-Y. Huang, T. Ma, C. Yue, Y. Zhang, J. Chang, T.-K. Dong,and Y.-Q. Zhang, Research in Astronomy and Astrophysics (inpress) (2020), 2005.07828.[15] G. Ambrosi et al. (DAMPE), Astroparticle Physics , 18(2019), 1907.02173.[16] G. Ambrosi et al. (DAMPE), Nature , 63 (2017).[17] Q. An et al. (DAMPE), Science Advances , eaax3793 (2019),1909.12860.[18] S. Agostinelli, J. Allison, K. Amako, et al., Nuclear Instrumentsand Methods in Physics Research A , 250 (2003).[19] A. Ferrari, P. Sala, A. Fass¨o, and J. Ranft,FLUKA : A multi-particle transport code, CERN Yellow Re-ports: Monographs (CERN, Geneva, 2005), ISBN 9290832606, URL https://cds.cern.ch/record/898301 .[20] T. T. B¨ohlen, F. Cerutti, M. P. W. Chin, et al. (FLUKA Collab-oration), Nuclear Data Sheets , 211 (2014).[21] A. Bolshakova, I. Boyko, G. Chelkov, et al., European PhysicalJournal C , 543 (2010), 1006.3429.[22] J. Allison, K. Amako, J. Apostolakis, P. Arce, et al., Nuclear In-struments and Methods in Physics Research A , 186 (2016).[23] B. Andersson, G. Gustafson, and B. Nilsson-Almqvist, NuclearPhysics B , 289 (1987), ISSN 0550-3213.[24] B. Nilsson-Almqvist and E. Stenlund, Computer Physics Com-munications , 387 (1987), ISSN 0010-4655.[25] GEANT4 Collaboration, Geant4 physics list guide, http://geant4-userdoc.web.cern.ch/geant4-userdoc/UsersGuides/PhysicsListGuide/html/index.html .[26] S. Roesler, R. Engel, and J. Ranft, Advanced Monte Carlo forRadiation Physics, Particle Transport Simulation and Applica-tions pp. 1033—-1038 (2001).[27] R. Chytracek, J. Mccormick, W. Pokorski, and G. Santin, IEEETransactions on Nuclear Science , 2892 (2006).[28] C. Wang et al., Chinese Physics C , 106201 (2017).[29] M. B. Emmett, Tech. Rep. ORNL-4972, Oak Ridge NationalLaboratory, United States (1975).[30] GEANT4 Collaboration, Geant4 physics reference manual, https://geant4-userdoc.web.cern.ch/geant4-userdoc/UsersGuides/PhysicsReferenceManual/html/index.html .[31] Y.-Q. Zhang, J.-H. Guo, Y. Liu, et al., Research in Astronomyand Astrophysics (RAA) , 123 (2019), ISSN 2397-6209.[32] G. D’Agostini, Nuclear Instruments and Methods in PhysicsResearch A362