Comparison of proton shower developments in the BGO calorimeter of the Dark Matter Particle Explorer between GEANT4 and FLUKA simulations

Both GEANT4 and FLUKA simulations are based on an accurate geometric model including both the payload and the satellite platform, which is designed to study various characteristics and performance improvement of the detector. The sizes, shapes, positions of all components of the satellite in the designed geometry are measured and validated in detail during the assembly of the satellite. Meanwhile, the materials information of all the parts is appended into the geometry. We take into account accurate atomic composition of different elements in the detector for precise hadronic and electromagnetic shower cascade simulation. For most parts of the satellite, the manufacturers provide detailed element components. For the remaining filling materials and electronic components whose compositions were unknown, we send their samples to analytical laboratories for detailed measurements to get the exact mass fractions of atoms. Thus, the geometric model of the entire satellite is established, which accurately describes the detection units, the supporting structure and the filling cushioning materials of the sub-detectors, as well as the frame and electronic 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 emphasized in this paper. The detailed structure of the BGO is shown in Fig. 2. We precisely configure the geometry of the BGO which consists of 14 layers, each with 22 crystal bars. Each bar is an independent detection unit with independent readout circuits at the two ends, assembled in a braced frame with cushioning material filling all the internal gaps. After a great deal of measurements and verifications, we configure a precise BGO model in the geometry.

Following the configuration and validation of the designed geometry, we call the Geometry Description Markup Language (GDML) Chytracek et al. (2006) interface for the specific program implementation of GEANT4 simulation, called the “GDML geometry’. The parameters of the GDML geometry are directly derived from the designed documents and measured results and are checked repeatedly to confirm that this geometry reflects the real situation of the satellite accurately. The GDML geometry is integrated into the DAMPE offline software framework Wang et al. (2017), which performs a series of standardized procedures (calibration, reconstruction and analysis) from the real “raw data”, the original signal collected by each prob cell of DAMPE, to scientific results. Therefore, the GDML geometry is applied as a unified interface for the data analysis of DAMPE. On the other hand, the FLUKA simulation is fully integrated and closed source code. It only allows the Combinatorial Geometry (CG) Emmett (1975) interface to develop the geometry. The geometry is rewritten from the GDML for the FLUKA simulation, called the “CG geometry”. After careful and repeated checks of these two geometry models in every detail, we are confident that they are identical between each other and consistent with the real satellite geometry, although there are slight differences in some micro components of the satellite which are negligible during the simulation.

The data flow for the complete simulation process is shown in Fig. 3, including primary generation, MC simulations, digitization, reconstruction and analysis. The first package, primary generation, creates the incident particles feeding the MC simulation including various distributions of incident positions, directions and energies. In this work, we generate a flux of primary protons distributed isotropically with a single power-law energy spectrum with the index -1 from 10 GeV to 100 TeV. The GEANT4 simulation package is integrated into the DAMPE offline software framework so that the GEANT4 simulation can be managed and coordinated as standardized configuration. We choose the FTFP_BERT Allison et al. (2016) from the GEANT4 Physics Lists GEANT4 Collaboration (a) with the default configuration recommended in the GEANT4 documents GEANT4 Collaboration (b). The FLUKA simulation runs in a separate operating environment, and is performed with the following settings:

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  the PEANUT package is activated in the whole energy range for any reaction;

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  the minimum kinetic energy for DPMJET-III is set to $5~{}\mathrm{GeV}/n$ (applying only to reactions between two nuclei heavier than a proton);

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  the minimum kinetic energy for RQMD is set to $0.125~{}\mathrm{GeV}/n$ (applying only to reactions between two nuclei heavier than a proton);

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  the same output format as the GEANT4 simulation.

Following numerous tests and validations to the developed simulation package including a set of algorithms which are responsible for generating the interactions of particles with the detector based on both the GEANT4.10.03³³3http://geant4.web.cern.ch and FLUKA 2011.2x⁴⁴4https://www.fluka.org, we allocate massive computing resources to run these programs, producing the simulation data of billions of protons.

Then, we run the digitization package to convert the physical information into the digital signal of each detection unit assigning a digital ID. In such a way the digital information of the simulation is in the same format as the real “raw data”. Accordingly, we can run the reconstruction package which contains large amounts of code for the for building up the physical signals including deposited energy, reconstructed tracks and charge of each event from the “raw data”. This package is organized as a series of algorithms that act successively to process the on-orbit data on a daily basis Ambrosi et al. (2019). The massive code to obtain the scientific results and the instrument performance of the detector based on the reconstructed data is collectively referred to as the analysis package, which is the result of collective efforts of many researchers. All the code in the package undergoes continues enhancement and version update as the detector comprehension improves with time. Major published results were also obtained using the package to analyze the on-orbit data and simulation data. In these analysis packages, the event selection packages including a list of selection conditions for target particles are fundamental for the analyses. In this work, we focus on figuring out some features to analyze the response of protons in the BGO calorimeter. All the below results are obtained based on the selected proton samples following the event selections in Ref. An et al. (2019).

Firstly, we investigate the tigger efficiencies for simulations using GEANT4 and FLUKA. DAMPE has four different triggers implemented on orbit: the Unbiased trigger, the Minimum Ionizing Particle (MIP) trigger, the Low-Energy (LE) trigger, and the High-Energy (HE) trigger Zhang et al. (2019). The Unbiased and MIP triggers are designed for the detector calibration Ambrosi et al. (2019), while the LE and HE triggers correspond to low threshold and high threshold triggering signals respectively. In the proton analysis, the events are required to meet the HE trigger condition in order to guarantee that the shower development starts above or at the top of the calorimeter. The HE trigger efficiency is one of the most important factors related to the effective acceptance estimation. For different hadronic integration models, the shower start-point and the secondaries from the first inelastic interaction would be different. As a result, we would consider the difference of the HE trigger efficiencies between GEANT4 and FLUKA simulations.

The HE trigger efficiency is estimated by means of the Unbiased trigger samples. The Unbiased trigger events are pre-scaled by a factor of 512 at latitudes $\leq 20^{\circ}$ and 2048 at latitudes $>20^{\circ}$. The HE trigger efficiency for protons is computed as

where $N_{\rm Unb}$ is the number of proton events passing the Unbiased trigger condition and $N_{\rm HE\&Unb}$ is the number of ones which both pass the HE and Unbiased trigger conditions. Fig. 4 shows the comparison of HE trigger efficiencies among the flight data, GEANT4 and FLUKA. Despite of the limited statistics of flight data, it suggests that the GEANT4 achieves a good agreement with flight data in the whole energy range, while FLUKA presents a systematic deviation of $\sim-5\%$ compared with the GEANT4 and flight data .

The energy of an incident proton is measured by the sum of energy deposits of all BGO crystals in the calorimeter, i.e. the total energy deposit. Due to the limited vertical thickness of the BGO calorimeter ($\sim 1.6$ nuclear interaction length) and the missing energy due to muon and neutrino components in hadronic showers, the total energy deposit measured by DAMPE would underestimate the intrinsic kinetic energy of an incident proton. In order to deconvolute the measured deposit spectrum into the initial spectrum, we need a good knowledge of the energy response, which is determined by the MC simulations.

Different hadronic interaction models would present different energy response matrices An et al. (2019), thereby leading to different deconvolutions for the initial spectrum of cosmic-ray proton. Before launch, the Engineering Qualification Model of DAMPE was extensively tested using test beams at the European Organization for Nuclear Research (CERN) in 2014-2015. To compare with the test beam data, we generate MC samples follwing closely the settings of the test beams, such as the incident energies, hit points, and directions. We also apply the same event selections to both the beam test data and MC data as those in the flight data analysis An et al. (2019), including the HE trigger, the track selection, the geometric cut, and the charge selection. The energy response of DAMPE for the on-axis incident proton beam with the momenta of 400 GeV/c is compared with the results from GEANT4 and FLUKA simulations (see the top panel of Fig. 5). Both GEANT4 and FLUKA achieve good agreements with the beam test data, specifically at the momenta of 400 GeV/c. To further compare the energy responses from GEANT4 and FLUKA in the entire energy range of interest, an isotropic proton source with $E^{-1.0}$ spectrum from 10 GeV to 100 TeV is generated for the simulations. In the analysis, the spectra are re-weighted to $E^{-2.7}$ to be consistence with the CR flux. The most probable values of the deposited energies obtained by fitting the energy ratio probabilities with an asymmetric gaussian function, along with the incident energy, are shown in the bottom panel of Fig. 5. The energy responses of GEANT4 and FLUKA show an energy dependent difference from 10 GeV to 100 TeV, in consequence, the deconvoluted proton spectra based on GEANT4 and FLUKA should have different spectral indices.

The longitudinal development of a hadronic shower is highly determined by the first inelastic interaction point, i.e. the inelastic scattering cross-section between the incident proton and the detector material. We calculate the ratios of the energy deposits in different BGO layers with the total energy deposit to describe the longitudinal shower development. Fig. 6 shows the comparisons of layer energy ratios among flight data, GEANT4 and FLUKA. While the DAMPE calorimeter is the thickest one in space, still the hadronic showers at these energies cannot be fully contained and a proportion of energy leaks in the bottom, as shown in the 4 plots on the right side of Fig. 6. Both simulations (GEANT4 and FLUKA) show good agreements of the longitudinal shower development with the flight data.

The transverse shower development, however, is intimately associated with the distribution of the types of subsidiary particles created through the interactions. We calculate the shower 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 of the fired $i$th layer is calculated as:

Where $x_{j,i}$ and $E_{j,i}$ are the coordinates and energy deposit of the $j$th bar in the $i$th layer, and $x_{c,i}$ is the energy-weighted centre coordinate of the $i$th layer. Fig. 7 show the comparisons of RMS values in different layers among flight data, GEANT4 and FLUKA. The differences among the FLUKA, GEANT4 and flight data suggest some systematical uncertainties. Overall, the results of GEANT4 show a better agreement with the flight data than the FULKA. Based on the comparisons, we conclude that GEANT4 carries out a more reliable simulation for the transverse development of the proton shower.

The absolute proton flux $F$ in an incident energy bin $[E_{i},\,E_{i}+\Delta E_{i}]$ can be calculated as

where $N_{{\rm inc},i}$ is the number of events in the $i$th incident energy bin, $N_{{\rm dep},j}$ is the number of events in the $j$th deposited energy bin, $\mathbf{M}_{ij}$ is the response matrix, $A_{{\rm eff},i}$ is the effective acceptance, $\Delta E_{i}$ is the width of the energy bin, and $T_{\rm exp}$ is the exposure time. $N_{{\rm inc},i}$ in each incident energy bin can be obtained via the unfolding procedure based on the Bayes theorem D’Agostini (1995).

The proton spectrum depends closely on the effective acceptance and the energy response matrix, both are obtained from MC simulations. The acceptance is obtained through calculating the fraction of events in each incident energy bin survived from the whole selection procedure, and the response matrix is obtained by counting the fraction of events in the deposited energy bin $j$ for given incident energy bin $i$. We applied the same selections as the flight data analysis An et al. (2019) to obtain the corresponding effective acceptances and energy response matrices for protons. The effective acceptance from the FLUKA sample is lower than that from the GEANT4 sample by $\sim 5\%$, which is dominated by the trigger efficiency difference (see Fig. 4). On the other hand, the energy response difference between two MC softwares (see Fig. 5) results in a complex effect on the fluxes after the spectrum deconvolution. The overall proton flux difference between GEANT4 and FLUKA is shown in Fig. 8. Even though the maximum difference can be large as 10%, the global spectral structures are consistent with each other. Based on the comparisons of shower development, we chose the GEANT4 spectrum as the benchmark, and take the difference between GEANT4 and FLUKA as the uncertainty. As shown in Fig. 8, the proton flux difference varies from $-6.6\%$ to $9.8\%$, which is taken as the systematic uncertainty due to different hadronic interaction models An et al. (2019).