A semi-analytical energy response model for low-energy events in JUNO
PPrepared for submission to JINST
A semi-analytical energy response model for low-energyevents in JUNO
P. Kampmann, a , b , Y. Cheng, and a , c , L. Ludhova a , b a Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany b RWTH Aachen University, 52062 Aachen, Germany c Institute of High Energy Physics, 100094 Beijing, China
E-mail: [email protected]
Abstract: The Jiangmen Underground Neutrino Observatory (JUNO) is a next-generation neutrinoexperiment under construction in China expected to be completed in 2022. As the main goal it aimsto determine the neutrino mass ordering with 3-4 σ significance using a 20 kton liquid scintillatordetector. It will measure the oscillated energy spectrum of electron anti-neutrinos from two nuclearpower plants at about 53 km baseline with an unprecedented energy resolution of 3% at 1 MeV. Arequirement of the JUNO experiment is the knowledge of the energy non-linearity of the detectorwith a sub-percent precision. As the light yield of the liquid scintillator is not fully linear to theenergy of the detected particle and dependent on the particle type, a model for this light yield ispresented in this paper. Based on an energy non-linearity model of electrons, this article providesthe conversion to the more complex energy response of positrons and gammas. This conversionuses a fast and simple algorithm to calculate the spectrum of secondary electrons generated by agamma, which is introduced here and made open access to potential users. It is also discussed howthe positron non-linearity can be obtained from the detector calibration with gamma sources usingthe results presented in this article.Keywords: Neutrino detectors; Detector modelling and simulations I; Detector modelling andsimulations II; Ionization and excitation processes; Liquid detectors; Photoemission; Scintillators,scintillation and light emission processes; Simulation methods and programs;ArXiv ePrint: xxxx Corresponding author. Beijing Institute of Spacecraft Environment Engineering, Beijing 100094, China a r X i v : . [ phy s i c s . i n s - d e t ] J un ontents The Jiangmen Underground Neutrino Observatory (JUNO) is a neutrino experiment under con-struction in China expected to be completed in 2022 [1]. As a next generation neutrino experimentit aims to address current challenges in neutrino and astroparticle physics. The detector, a 20 ktliquid scintillator tank instrumented with 18,000 photomultiplier tubes (PMTs), will be placed ata distance of 53 km from the Taishan and the Yangjiang nuclear power plants. It will be placedunderground with an overburden of about 700 m rock.The 53 km distance optimizes the sensitivity to the neutrino Mass Ordering (MO) determi-nation, which is the main goal. JUNO aims to address it with 3-4 σ significance. In addition tothat, JUNO aims to measure the oscillation parameters θ , ∆ m , and | ∆ m | with a sub-percentprecision. These measurements rely on measuring the coincident signal from Inverse Beta Decay(IBD) events which consist of a prompt positron and a delayed gamma signal due to neutron capture.– 1 –part from measuring the neutrinos from the nuclear reactors, JUNO plans to measure theneutrinos from various other natural sources. These include the measurement of geo-neutrinos,solar neutrinos, atmospheric neutrinos, supernova neutrinos, and the diffuse supernova neutrinobackground. JUNO will also search for the proton decay. The physics programme is described indetail in [1].To reach its goals, the JUNO experiment needs to fulfill several requirements. Among themost challenging ones are those on the energy scale and resolution. It is required to reach a relativeuncertainty on the energy scale of less than 1% and an energy resolution better than 3% at 1 MeV.The light emission in a liquid scintillator is not linear with the deposited energy of particles anddepends on the detected initial particle type. As inappropriate modeling of the non-linearity biasesthe determination of the MO [1], it is of imminent importance to address this topic carefully. As theJUNO experiment will use gamma sources for the energy non-linearity calibration, it is importantto develop methods how to derive non-linearity model for electrons and positrons based on thecalibration data.This article focuses on the description of the energy non-linearity in the scintillation medium.It is a topic of general interest in organic scintillator physics with numerous publications on it [2–5].The non-linearity will be determined from the ratio of the visible energy over deposited energy.The term visible energy is used here to describe the expected amount of detectable light producedin the scintillation medium in the detector.The article is organized as follows. Section 2 briefly reviews the physics motivation of thereactor anti-neutrino program (Sec. 2.1) and the detector concept (Sec. 2.2) of the JUNO experiment.In Section 3 the modelling of the energy non-linearity effects is described. The part 3.1 deals with thelight non-linearity in the measurement of electrons and of the kinetic energy loss for positrons (thusexcluding the annihilation part) due to the ionisation quenching (Sec. 3.1.1) and Cherenkov lightproduction (Sec. 3.1.2). Since electrons directly produce scintillation photons through ionization,their model is simpler than for gammas or positrons, which also produce other secondary particles.The non-linearity model is based on the Birks’ empirical formula [6]. Section 3.2 describes theconversion of the electron non-linearity model to the more complex non-linearity model for gammasand positrons including the annihilation. Gammas loose their energy in the scintillation mediumunder the production of several secondary electrons responsible for the emission of scintillationlight. An algorithm is presented to generate the energy distributions of the secondary electrons,as discussed in Sec. 3.2.1. These are then used to calculate the full non-linearity of the gammasby adding up the contributions from each secondary electron. This algorithm is validated andcompared to the JUNO Geant4 simulation [7] in Sec. 3.2.2. It is shown, that it reproduces theresults from the Geant4 simulation and has computational benefits due to its easy use and fastcalculation (Sec. 3.2.3). Resulting non-linearity model for gammas is then presented in Sec. 3.2.4.As positrons produce scintillation light directly in the deposition of their kinetic energy as well asthrough the annihilation producing two gammas as secondary particles, their non-linearity modelcan be constructed through the combination of these. This is shown in Sec. 3.3. In part 3.3.1, thepositron non-linearity is evaluated for the case of positron annihilation at rest into two photons.To evaluate the accuracy of this simplification, the resulting non-linearity is compared to the fullJUNO Geant4 simulation with its more comprehensive physics description (Sec. 3.3.2). The finalsummary and outlook is then discussed in Sec. 4.– 2 – ~sin θ ~sin θ ~Δm Figure 1 . Expected energy spectra of the prompt IBD candidates in the JUNO experiment in no oscillation(dashed line) and three-neutrino oscillation models assuming normal (solid blue line) and inverted (solid redline) MO. The calculation was performed with the GNA software [9] assuming 3% / (cid:112) E prompt [MeV] energyresolution and normal ordering oscillation parameters from [10]. To display the inverted ordering, onlythe sign of ∆ m was inverted. The insert shows in detail the energy region between 2 - 4 MeV, which isthe most dependent on the MO. The features of the spectra sensitive to θ and θ mixing angles are alsodemonstrated. Nuclear reactors are a powerful source of electron anti-neutrinos ( ¯ ν e ) with MeV energies. In severalhistorical experiments they played a key role, that they preserve also today in the quest for answeringopen questions of neutrino physics. JUNO is the only experiment that aims to determine the yetunknown sign of ∆ m mass splitting based on the oscillation of reactor anti-neutrinos in the vacuumdominated regime. We refer to this sign as the Mass Ordering (MO). The MO is of importancein the determination of the CP-violation phase δ , of the Majorana phases in case the neutrino is aMajorana particle, and of the θ -octant [8].To detect reactor anti-neutrinos, JUNO will use the Inverse Beta Decay (IBD) reaction onprotons [11]: ¯ ν e + p → e + + n (2.1)sensitive only to electron flavour and having a kinematic threshold of 1.806 MeV. The cross sectionof the IBD interaction can be calculated precisely with an uncertainty of 0.4% [12]. In this process,– 3 – positron and a neutron are emitted as reaction products. The positron promptly comes to restand annihilates emitting two 511 keV γ -rays, yielding a prompt signal. The neutron, after beingthermalized, is captured ( τ ∼ µ s) mostly on a proton, releasing a gamma with the constantenergy of the deuteron binding energy of 2.2 MeV, providing a delayed signal. A space and timecoincidence between the prompt and delayed signals significantly suppresses the background.The JUNO sensitivity to MO is based on the dependence of the oscillation pattern of reactoranti-neutrino energy spectrum on it. In the IBD interaction, the incident anti-neutrino energy E ¯ ν e is directly correlated with a visible energy of the prompt signal E prompt : E prompt ∼ E ¯ ν e − .
784 MeV . (2.2)This dependence is then exploited in the measurement of the E prompt spectrum, as depicted in Fig. 1.Consequently, the precision and accuracy of the reconstructed neutrino energy depends directly onthe reconstruction of the positron energy. The distinction of the E prompt spectra corresponding to thenormal and inverted MO allows its determination. Strict requirements on the energy reconstructionmust be imposed to increase the significance of MO determination. These include not only aresolution of less than 3% at 1 MeV, but also a sub-percent uncertainty on the energy scale. The scheme of the JUNO detector setup is shown in Fig. 2. The core central detector (CD) iscomposed of the
Acrylic Sphere with 17.7 m radius filled with 20 kt of liquid organic scintillator(LS). The CD is equipped with about 18,000 large, 20” and about 25,000 small, 3” PMTs.The detection medium in the scintillator is linear alkylbenzene (LAB), a straight alkyl chainof 10-13 carbon atoms attached to a benzene ring [1]. As additives, 2.5 g/l of PPO (2,5-diphenyloxazole) as fluor and 1-3 mg/l bis-MSB as wavelength shifter will be used. The density ofthe liquid scintillator mixture is expected to be 0.859 g/ml [1].The Acrylic Sphere is surrounded by a spherical tank with 20 m radius containing ultrapurewater and the PMT array mounted on the
Stainless Steel Lattice Shell (SSLS). Around 5,000 ofthe large 20”-PMTs are dynode PMTs while the larger part of around 13,000 are PMTs with amicrochannel plate instead of a dynode structure [13]. To reach the goal of a highly precise energyresolution of 3% at 1 MeV, the PMT array has a geometrical coverage of about 78%. Moreover,each large PMT is required to have a high and homogeneous photon detection efficiency at a levelof 28%. This leads to a total light collection of about 1200 photoelectrons (p.e.) at 1 MeV kineticelectron energy. Scintillation photons from events with energies of a few MeV are detected usuallyin the single p.e. regime of each PMT. This minimizes non-linearity effects in the PMT readoutelectronics.The steel sphere of the CD is contained in a
Water Pool , which is equipped with around 2,40020”-PMTs, and serves as a muon veto. In addition to that, a
Top Tracker (TT) made up of plasticscintillator strips, covers the water cylinder partially on top to measure the incoming direction ofmuons. – 4 –
Central Detector
SSLSAcrylic sphere (20 kt LS)~18,000 20” PMT~25,000 3” PMT
Water Pool ~2000 20” PMT
Acrylic sphere
Diameter = 35.4 mThickness = 120 mm
SSLS
I-Diameter = 40.1 mO-Diameter = 41.1 m
Water Pool
Diameter = 43.5 mHeight = 44 mWater Depth = 43.5 m
SSLS = Stainless Steel Lattice Shell
Acrylic sphere diameter = 35.4 mSSLS diameter = 40.1 m
Figure 2 . The scheme of the JUNO detector.
The non-linearity of the scintillation light yield with energy deposit is known as ionization quench-ing. Empirically, the light yield per unit depth d X is expressed by the Birks’ formula [6]: (cid:28) d L d X (cid:29) = L (cid:10) d E d X (cid:11) + k B (cid:10) d E d X (cid:11) , (3.1)where L is the light yield, L is the scintillation light yield normalization, (cid:104) d E / d X (cid:105) is the averageenergy loss of the particle per unit depth, and k B is the Birks’ material constant [6]. The parameter L describes therefore the scintillation light yield under the absence of the quenching effect. Theoverall amount of the scintillation light (cid:104) L ( E kin )(cid:105) from the deposition of the kinetic energy E kin ofparticle in the LS can then be expressed as (cid:104) L ( E kin )(cid:105) = ∫ path (cid:28) d L d X (cid:29) d X . (3.2)The latter equation can be then reformulated using d X = (cid:104) d E / d X (cid:105) − d E as (cid:104) L ( E kin )(cid:105) = L ∫ E kin + k B (cid:10) d E d X (cid:11) ( E (cid:48) ) d E (cid:48) . (3.3)In the following, the contribution to the visible energy E vis from scintillation light will be determinedfrom this equation. The non-linearity of the energy scale will be then defined as the dependence ofthe ratio E vis / E dep on the particle’s kinetic energy E kin . In the case of electrons and gammas, theparticle’s deposited energy E dep is equal to its kinetic energy.– 5 –o describe the suppression of the light emission in the LS, one needs to know the energydependent energy loss per unit depth in the scintillation medium. For electrons, it is described bythe Møller model, while the Bhabha model describes the energy loss of positrons [10]. The fullformulas for these models can be found in Appendix A.To apply the energy loss equations for the JUNO scintillator, one needs to find appropriatevalues for the density correction δ ( E ) , the mean excitation energy I , and for the ratio of the atomicnumber to the atomic mass number Z / A . The LS volume is dominated by LAB, so the followingcalculations assume LAB as the only component of the scintillator, having a chemical compositionC H . The parameter Z / A is therefore approximated by (cid:104) Z / A (cid:105) = (cid:104) Z (cid:105) /(cid:104) A (cid:105) by taking the atom-abundancy weighted mean of the LAB-molecule, as it is suggested in [10]. The value is calculatedto be (cid:104) Z / A (cid:105) = . molg . The mean excitation energy I = . (cid:104) d E / d X (cid:105) calculatedusing the Møller and the Bhabha models, and evaluated via the EStar-tool. The same densitycorrection term was applied in all three cases based on the evaluation in [15]. Then, using theEq. 3.3, one can obtain the respective E vis / E kin = f ( E kin ) non-linearity curves. The integration wasevaluated numerically under the usage of the Gauss-Legendre method implemented in the ROOTframework [16]. A typical value of k B = .
01 cm /MeV/g was used here and will be used in the laterevaluations in this paper. The exact value for the JUNO scintillator was not conclusively measuredso far. For the better visibility of the quenching impact, the scintillation light yield normalizationis set to L =
1. With this normalization, the visible energy from scintillation light only is equal tothe deposited energy under the assumption of no quenching.The resulting three non-linearity curves, shown in the right plot of Fig. 3, are similar for allthree different (cid:104) d E / d X (cid:105) energy loss models. We note, that the validity of the chosen quenchingmodel needs to be evaluated based on the future calibration data. Additionally to scintillation light, also Cherenkov light is produced by charged particles in thescintillation medium. As it is usually created at small photon wavelengths [17], it is mostlyabsorbed and re-emitted by the scintillation medium [18]. Due to this conversion, it can not beseparated easily from the scintillation light and must be included in the non-linearity model. Theamount of Cherenkov light depends on the kinetic energy and the mass of the particle. Therefore,the same amount of Cherenkov light is assumed for electrons and positrons.To model the amount of detected Cherenkov photons in the detector, a simple empiricalexpression is applied, which was used [19] in the solar-neutrino analysis of the Borexino experimentusing Pseudocumene-based LS: N Cherenkov ( E kin ) E kin = (cid:32) (cid:213) n = A n x n (cid:33) · (cid:18) E kin + A (cid:19) (3.4)with x = ln (cid:18) + E kin E (cid:19) . – 6 – kin E1.81.922.12.22.32.42.5 / g ) < d E / d X > ( M e V c m dEdX curvesMoller model (e )Bhabha model (e+)EStar tool (e ) −
10 1 (MeV) kin
E0.90.920.940.960.981 k i n / E v i s E Energy loss curvesMoller model (e )Bhabha model (e+)EStar tool (e )
Figure 3 . Comparison between the three different energy loss models (Møller, Bhabha, and EStar-tool) asa function of the kinetic energy. Left: The energy loss per unit depth (cid:104) d E / d X (cid:105) . Right: The respective non-linearity curves E vis / E kin , obtained via integration of the Birks’ formula (Eq. 3.3) with k B = .
01 cm /MeV/gand a light yield normalization of L = To evaluate the model of detected Cherenkov light and to estimate the parameters A i , Eq. 3.4 isfitted to the detected number of Cherenkov photons based on the JUNO Geant4 simulation [7] forelectrons. As the parameters of Eq. 3.4 were found to be highly correlated in the fit, the parameters A and A were fixed to 0. The best fit values for the parameters A i with i ∈ { , , } , which areshown with the fit on Fig. 4, are used in the following analysis. The Cherenkov threshold was alsoestimated from the simulation output to be E = 0.2 MeV.While the amount of Cherenkov light can be fully determined from Eq. 3.4 after the determina-tion of its parameters, the normalization of scintillation light and therefore the relative contributionof Cherenkov light still needs to be determined. This is also taken from the JUNO Geant4 simulation.A simulation of electrons without quenching at 1 MeV yields a ratio of L Cherenkov L Scintillation ( ) = . ± . . ± . = ( . ± . ) % , with a pure statistical uncertainty originating in the amount of simulated data. This number isused here to fix the relative contribution of Cherenkov light to the visible energy at 1 MeV kineticenergy. Due to this choice of normalization, the non-linearity ratio E vis / E kin can be larger than 1,if Cherenkov light is included in the following.The JUNO Geant4 simulation is not assumed here to yield the amount of Cherenkov light witha high precision. However, it is used here to obtain parameter values of Eq. 3.4 and a normalization.These values can be seen illustrative here, as they need to be evaluated in future calibration studies.– 7 – kin E0102030405060708090 ( p . e ./ M e V ) k i n / E C he r en k o v N Cherenkov fit to Geant4Geant4 data pointsCherenkov model fit 0.08 ± = 7.46 A 0.04 ± = 12.66 A 0.002 1/MeV ± = 0.037 A = 0.2 MeV E −
10 1 (MeV) kin
E0.90.9511.051.1 k i n / E v i s E Non linearity of cherenkov lightMoller modelMoller model+cherenkov light
Figure 4 . Left: Number of detected Cherenkov photons per E kin as a function of kinetic energy based onthe JUNO Geant4 simulation of electrons (black points). The solid red line shows the fit of Eq. 3.4 to theMonte Carlo data points. Right: Effect of the Cherenkov contribution to the LS non-linearity E vis / E kin , as afunction of kinetic energy, using the Møller model for the ionisation loss calculation. The solid line showsthe case with only the scintillation light, while the dashed line the case including also the Cherenkov light. To develop a non-linearity model for positrons based on the electron non-linearity model, oneneeds to combine the non-linearity model of the positron itself and the non-linearity model of thetwo annihilation gammas. The annihilation gammas interact with electrons of the scintillator withseveral different processes creating secondary electrons. To describe the energy loss of gammas, analgorithm was developed to obtain the energies of the secondary electrons, which are responsible forthe production of scintillation light. In the regime of gamma energies below E γ = · m e c , these arethe photoelectric effect, Rayleigh-scattering, and Compton-scattering [10]. Above E γ = · m e c ,the gamma can also produce an electron-positron pair in an interaction with a nucleus or an electron.To simulate the behaviour of the initial gamma, the scattering process is determined first. Thiswas done by using the relative contribution of the process to the total interaction cross sectionas the probability to undergo that process. The relative cross sections are shown in Fig. 5. Thisprovides the probabilities of the gamma to interact with the respective process in the medium. Forhigher energies above E = · m e c , pure Compton-scattering was assumed as the cross section forelectron-positron pair production is small compared to the cross section for Compton scattering fortypical energies of a few MeV in JUNO [20].In the case of Rayleigh-scattering, the gamma does not loose energy and is just re-emitted ina different direction. As the simulation does not consider the spatial behaviour of the gammas butonly the energy, the process of Rayleigh scattering is not considered.In the case of the photoelectric effect, the full energy of the gamma is transferred to theelectron of the scintillation medium. The gamma gets absorbed and the algorithm is stopped, if the– 8 – − − −
10 1 (MeV) γ E00.20.40.60.81 t o t σ / σ Relative interaction cross sectionsCompton scatteringPhotoelectric effectRayleigh scatteringPair production
Figure 5 . Relative cross sections of gamma scattering processes in LAB from [20]. photoelectric effect occurs.Compton-scattering is the dominant process for initial gammas with an energy above ≈
20 keVin LAB. For the process of Compton-scattering on an electron at rest, the distribution of scatteringangles is calculated first. The distribution of scattering angles is determined via the Klein-Nishinadifferential cross section [21]:d σ d Ω = d σ d φ d cos θ = α m e (cid:18) E (cid:48) γ E γ (cid:19) (cid:20) E (cid:48) γ E γ + E γ E (cid:48) γ − sin θ (cid:21) . (3.5)In this formula, α ≈ /
137 is the electromagnetic fine structure constant, E γ is the gamma energybefore scattering, E (cid:48) γ is the gamma energy after scattering, and θ is the scattering angle. The energyloss ratio follows E (cid:48) γ E γ = + E γ m e ( − cos θ ) . (3.6)Since the total cross section in an scattering angle interval is proportional to the scattering prob-ability, the normalized Klein-Nishina differential cross section is the probability density function(PDF) for the angular distribution shown in Fig. 6.The energy of the gamma after scattering is obtained from Eq. 3.6 by inserting a random anglefollowing the angular distribution shown in Fig. 6. The kinetic energy of the scattered electronfollows energy conservation: E (cid:48) e,kin = E γ − E (cid:48) γ . (3.7)After each Compton scattering the calculation is repeated until the gamma is absorbed via thephotoelectric effect or has an energy of less than E min =
250 eV. This minimal energy was chosento be the same as the default one used by the Geant4-software for electromagnetic processes [22].
The algorithm for calculating the distributions and energies of secondary electrons, as it waspresented in the previous Section, is validated against the JUNO Geant4 simulation. We have– 9 – π (rad)/ θ − × ( no r m a li z ed ) Ω d σ d ) θ s i n ( = 0.0 MeV γ E = 2.0 MeV γ E = 4.0 MeV γ E = 6.0 MeV γ E = 8.0 MeV γ E = 10.0 MeV γ E Figure 6 . The evaluation of the Klein-Nishina cross section Eq. 3.5. As it is used as a probability distribution,it is normalized to 1 and weighted with sin ( θ ) as the integration is evaluated over d cos θ . chosen gammas with E = m e c =
511 keV energy simulated in the detector center, in order to avoidborder effects at the acrylic vessel. Figure 7 compares the distribution of the number of secondaryelectrons and their full energy spectrum for the case of our algorithm (solid blue line) and theJUNO Geant4 simulation (solid red line). The complete physics list of the JUNO Geant simulationcan be found in Appendix B. For a better comparison, the histograms obtained with our algorithmwere scaled down to match the amount of data in the Geant4 simulation. In general, the numberof secondary electrons is not a reliable number for comparison, as it can depend on the so called production cuts , i.e. the lower energy cut, at which the tracking of the mother particle is stopped [23].As the lower energy limit here is given by the dominance of the photoelectric absorption at lowenergies, these production cuts are not reached and the number of secondary electrons is a reliablequantity for comparison. As we can see in the left part of Fig. 7, the distribution of the numberof secondary electrons shows a reasonable agreement with only slight differences of the meanand the standard deviation. The right part of Fig. 7 demonstrates the excellent agreement of theoverall energy spectrum of the secondary electrons. This comparison approves the reliability of thepresented algorithm. In the process of calculating the NL from gammas or positrons (see Sec. 3.3) using the resultsof this article, the production of a representative sample of secondary electrons, dominates thecomputing time. To estimate the gain in computation time of this algorithm with respect to theevaluation by Geant4, both algorithms were run for different initial gamma energies and number ofgenerated events. All of these runs were executed on the same machine by an AMD Opteron TM Processor 6238. In the JUNO Geant4-framework, the computation time is usually dominated by thepropagation of optical photons, when the default settings are used. The production of these opticalphotons has been disabled. For the computation time one would expect, that each algorithm needsa certain amount of start-up time T for its setup before each event needs about the same time T event – 10 – e N00.010.020.030.040.050.060.070.08 no r m a li z ed en t r i e s JUNO Geant4 0.022 ± Mean = 17.195 0.015 ± StdDev = 5.066 This algorithm 0.004 ± Mean = 17.107 0.002 ± StdDev = 5.044
Number of secondary electrons e kin E − − − − − no r m a li z ed en t r i e s JUNO Geant4This algorithm
Spectrum of secondary electrons
Figure 7 . Comparison of the presented algorithm for secondary electrons (Sec. 3.2.1, solid blue line) with theJUNO Geant4 simulation [7] (solid red line) for E γ = 511 keV. The left side shows the number of secondaryelectrons and the right side shows their overall energy spectrum. c o m pu t a t i on t i m e ( s ) E = 0.1 MeVE = 3.3 MeVE = 6.7 MeVE = 10.0 MeVThis algorithmGeant4
Figure 8 . Comparison of the presented algorithm for secondary electrons (Sec. 3.2.1, solid lines) with theJUNO Geant4 simulation [7] (dashed lines) for E γ = 0.1, 3.3, 6.7, and 10.0 MeV in terms of the computationaltime. The lines represent estimations for the parameters T and T event from Eq. 3.8 for each E γ energy, obtainedby a fit with equal weights for each data point. One can clearly see a large gain on the CPU time using ouralgorithm, due to differences both in the start-up time T as well as in the T event time needed per event. to be processed. The total computation time is expected to follow T comp = T + T event · N , (3.8)with N being the number of processed events. For both algorithms, the computation times forthe energies of 0.1 MeV, 3.3 MeV, 6.7 MeV, and 10.0 MeV, as well as for the event numbers of– 11 –, 500, 1000, 1500, 2000, and 2500 events were evaluated. The parameters T and T event wereestimated then for each energy separately. The results are then summarized graphically in Fig. 8and numerically in Table 1. The large gain in computation time is directly visible. As describedin [7], the JUNO Geant4 framework contains a comprehensive description of the detector geometryand the physics processes, which are relevant in the JUNO experiment. It serves the purpose ofbeing a general tool for simulating events in a broad energy range to study analysis methods andparticle interactions with a precise model of the detection effects in the JUNO detector. The highercomplexity, caused by the higher universality, results especially in a higher start-up time, but as wellin a higher time needed per event. Extrapolating the values from Table 1 one would need around4.5 h to simulate 100 000 gammas at 0.1 MeV using the JUNO Geant4 simulation and only about1 h using the presented algorithm.Apart from the gain in computational time, the presented algorithm has benefits in easiermaintenance. While the JUNO Geant4 framework is a complex framework built upon a softwarestack of external programs, the presented algorithm consists of single C++ class, which uses methodsfrom the ROOT framework [16]. Table 1 . Comparison of the presented algorithm for secondary electrons (Sec. 3.2.1) with the JUNO Geant4simulation [7] for E γ = 0.1, 3.3, 6.7, and 10.0 MeV in terms of the computational time. The estimated valuesfor the start-up time T and the computational time needed per event T event as estimated from Fig. 8. This algorithm JUNO Geant4Energy (MeV) T (s) T event (s) T (s) T event (s)0.1 1.41 0.0386 181.23 0.16243.3 1.38 0.0604 178.02 0.16506.7 -1.10 0.0640 189.77 0.171210.0 2.40 0.0629 190.93 0.1631 To evaluate the scintillator non-linearity model for gammas from the simulation of the secondaryelectron spectrum, the sum of all light emissions from secondary electrons was taken. The non-linearity of the secondary electrons was evaluated using the Birks’ law (Eq. 3.3) and the energyloss via the Møller model (Eq. A.1). It was assumed that there are no correlated effects in the lightproduction between different secondary electrons and each of them can be treated individually.In Figure 9 one can see in yellow the evaluation of E vis / E kin for about 1 . · gammassimulated without the Cherenkov light in the energy range E kin = E γ from 1 keV to 9 MeV. Theright plot shows the zoom of the left plot in the energy range below E kin < . kin E0.50.60.70.80.911.1 k i n / E v i s E Non linearity of gamma photonsMoller model (e )Moller model (e ) + cherenkovGamma scintillationGamma scintillation + cherenkov kin
E0.50.60.70.80.911.1 k i n / E v i s E Figure 9 . The results for the non-linearity E vis / E kin model for gammas calculated with the presentedalgorithm. The distribution of the non-linearity of all gammas without the Cherenkov light is shown inyellow, while the corresponding average non-linearity is shown in solid red. The red dashed curve showsthe average non-linearity of gammas with the Cherenov light included. For comparison, the electron non-linearity curve resulting from the Møller model is shown in blue, without (solid) and with (dashed) lines.The right plot shows the zoom of the left plot in the energy range below E kin < . the gammas only create visible energy due to secondary electrons, having lower energies and thus,higher quenching (Fig. 3 right), with respect to the electron of the same kinetic energy as the originalgamma. At very low energies below 20 keV, the photoelectric effect becomes dominant, as it can beseen in Fig. 5. This causes the non-linearity curves for electrons and for gammas to be the same, asthe full kinetic energy of the gamma is transferred to the electron. In the transition region at around70 keV, a local minimum of the nonlinearity can be seen in the right plot in Fig. 9. The energy deposition of positrons at energies of a few MeV happens usually in two steps. First,positrons deposit energy in the scintillator due to ionization and create scintillation light similarto the energy loss of electrons. Additionally, a positron annihilates afterwards with an electron ofthe detector material to produce gammas. In this section is is assumed, that positrons annihilate atrest after depositing their total kinetic energy due to ionization. The resulting two gammas havetherefore a total energy of E γ = m e c =
511 keV each. To calculate the non-linearity curve forthese positrons, one needs to combine the non-linearity curve resulting from the Bhabha model inFig. 3 and the non-linearity of the two gammas at E γ = m e c : E e+ vis ( E kin ) = E ionBhabha ( E kin ) + · E Gammavis ( m e c ) . (3.9)The result of this evaluation together with the models for the electron non-linearity and thegamma non-linearity are shown in Fig. 10. Again, the dashed lines show the full deposited energy– 13 –f scintillation light and Cherenkov light combined, while the solid lines represent the scintillationlight only. Here, the non-linearity is expressed as E vis / E dep , where the variable E dep is used forthe total deposited energy. For electrons and gammas it is E dep = E kin , while for positrons it is E dep = E kin + m e c . kin E0.60.650.70.750.80.850.90.9511.051.1 dep / E v i s E Non linearity modelse scintillatione scintillation + cherenkov scintillation γ scintillation + cherenkov γ e+ scintillatione+ scintillation + cherenkov Figure 10 . The non-linearity E vis / E dep ( E dep is the total deposited energy) models for positrons (purple),electrons (blue), and gammas (red). The solid curves show the non-linearity curves for scintillation lightonly, while the dashed curves include also the Cherenkov light. The calculation of the previous section assumes that positrons always annihilate at rest after loosingtheir kinetic energy completely in the scintillation medium via ionization. The annihilation in flight,the forming of positronium, and the creation of gammas via Bremsstrahlung are not considered.If positronium is formed, there is a chance to form para-positronium (p-Ps) or ortho-positronium(o-Ps). If o-Ps is formed in vacuum, it can not decay into two gammas, as o-Ps has a total spin of1 and therefore needs to decay into an odd number of photons. However in matter, several effectscause o-Ps to decay into two gammas instead of three [24, 25]. These are e.g. magnetic effectswhich cause a spin-flip or positron pick-off by surrounding electrons. Moreover, the creation ofelectron-positron pairs by gammas was not considered in our model. To study the impact of theseeffects, the JUNO Geant4 simulation was used to generate a comprehensive set of particles and theirenergy depositions created by an initial positron. Also here, the detectable light is entirely producedby positrons and electrons. To evaluate the amount of visible energy, the amount of scintillation lightwas evaluated from the integration of Birks’ formula (Eq. 3.3) and the amount of Cherenkov light– 14 –as evaluated from Eq. 3.4 for each positron and electron. These contributions were summed up tothe visible energy of the full event. To cross-check the validity of the simple positron model fromSec. 3.3.1, a sub-sample of these events was selected, which follows the assumptions of Sec. 3.3.1.These are the annihilation at rest into two gammas after the total kinetic energy is deposited due toionization. The production of Bremsstrahlung as well as the o-Ps decay into three gammas was notconsidered in Sec. 3.3.1. In total 970 000 events were simulated in the JUNO Geant4 simulation,from which 78 884 events were contained in the sub-sample following the assumptions of Sec. 3.3.1.The comparison of presented model from Sec. 3.3.1 with the full JUNO Geant4 simulation as wellas the selected sub-sample can be seen in Fig. 11. (MeV) kin E dep / E v i s E e+ NL validatione+ NL modelFull Geant4 simulationSelected Geant4 eventse+ NL model with CherenkovFull Geant4 simulation with CherenkovSelected Geant4 events with Cherenkov (MeV) kin E S i m ua t i on / M ode l r a t i o e+ NL ratio to modelFull Geant4 simulationSelected eventsFull Geant4 simulation with CherenkovSelected events with Cherenkov Figure 11 . Comparison of the positron NL curve from Sec. 3.3.1 to the NL obtained by using the particle setcreated by the JUNO Geant4 simulation [7] under usage of the Møller- and the Bhabha model. The graphson the left show the comparison of the non-linearity model from Sec. 3.3.1 (purple) to all simulation events(red) and selected simulation events, which followed the assumptions of Sec. 3.3.1 (blue). In the right graphsthe ratios of the NL evaluated using simulated particles by Geant4 to the NL curve from Sec. 3.3.1 are shown.
One can clearly see the difference between the positron non-linearity curve and the full simu-lation, while the selected sub-sample shows no clear difference to the positron non-linearity curve.This approves that the selected sub-sample is well described by the results of Sec. 3.3.1, like it isexpected from the validation results of Sec. 3.2.2. It can be further seen, that the deviation of thefull simulation to the model barely exceeds 1%, which is the requirement for the accuracy of thenon-linearity model in JUNO. Nevertheless, it is expected to have additional sources of uncertaintiesas the limited range of calibration sources, as well as limited calibration data. Therefore, the effectsof the annihilation in flight and Bremsstrahlung should to be studied further and be included in themodel.The averaged deviation of the Geant4 evaluation compared to the simple model in dependenceof the Birks’ constant is shown in Fig. 12. The average was taken over the expected JUNO promptspectrum under the assumption of the normal neutrino mass ordering shown in Fig. 1. One can see,that the average deviation is less than 0.5% for the large evaluated range of k B .– 15 – kB (MeV cm00.00050.0010.00150.0020.00250.0030.00350.0040.0045 > m ode l N L f u ll N L < with Cherenkovwithout Cherenkov Figure 12 . The deviation shown in Fig. 11 of the simple model to the JUNO Geant4 evaluation versus theBirks’ constant averaged over the JUNO prompt spectrum under the assumption of normal neutrino massordering.
As the accurate knowledge of the non-linearity is crucial for JUNO, it is of imminent importance todevelop a good model for the positron non-linearity. It was shown in this paper, that it is possiblewith easy methods to obtain a non-linearity model for electrons and gammas. For this an algorithmwas shown to evaluate a precise spectrum of Compton electrons.As only gamma sources are planned for the calibration of the JUNO experiment [1], thisevaluation can be used to determine the gamma non-linearity from the calibration data. For this, thescintillation light normalization, the Birks’s constant k B , as well as the Cherenkov curve parametersof Eq. 3.4 need to be determined. This can be used with the results of Sec. 3.3 to evaluate thepositron non-linearity.If the positron annihilates at rest without emitting Bremsstrahlung, the presented model showsgood agreement with the presented Geant4 simulation. However, if the positron annihilates in flightor produces Bremsstrahlung, there are clear differences of the model to the Geant4 simulation.As the JUNO experiment has very stringent requirements of an accurate energy non-linearitydescription, these effects should be treated in further studies to be used in the later data-analysis.Nevertheless, for sensitivity studies, the deviations are in a acceptable range to obtain a reasonableeffect on the energy spectrum.
Acknowledgements
This work was funded through the Recruitment Initiative by the Helmholtz Association of GermanResearch Centers and through Jülich-OCPC Programme for the Involvement of Postdocs in BilateralCollaboration Projects with China. We thank Maxim Gonchar, Dmitry Naumov, and KonstantinTreskov from Dzhelepov Laboratory of Nuclear Problems of JINR, Dubna for the fruitful discussions– 16 –nd the help in the reviewing process. We acknowledge the support by the JUNO collaboration forproviding us their software framework.
A Energy loss models
For a quick reference, here are explicitly given the well-known energy loss equations from [10].The Møller model (electron-electron scattering): (cid:28) − dEdX (cid:29) = K ZA β (cid:34) ln m e c β γ m e c ( γ − ) I + ( − β ) − γ − γ ln 2 + (cid:18) γ − γ (cid:19) − δ ( E ) (cid:35) (A.1)The Bhabha model (positron-electron scattering): (cid:28) − dEdX (cid:29) = K ZA β (cid:20) ln m e c β γ m e c ( γ − ) I + − β (cid:18) + γ + + ( γ + ) + ( γ + ) (cid:19) − δ ( E ) (cid:21) (A.2) B JUNO Geant4 Simulation: Physics List
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