Improving the energy uniformity for large liquid scintillator detectors
Guihong Huang, Yifang Wang, Wuming Luo, Liangjian Wen, Zeyuan Yu, Weidong Li, Guofu Cao, Ziyan Deng, Tao Lin, Jiaheng Zou, Miao Yu
IImproving the energy uniformity for large liquid scintillator detectors
Guihong Huang a , Yifang Wang a , Wuming Luo a, ∗ , Liangjian Wen a, ∗ , Zeyuan Yu a , Weidong Li a , Guofu Cao a , Ziyan Deng a , TaoLin a , Jiaheng Zou a , Miao Yu b a Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China b Wuhan University, Wuhan 430072, China
Abstract
It is challenging to achieve high precision energy resolution for large liquid scintillator detectors. Energy non-uniformity is one ofthe main obstacles. To surmount it, a calibration-data driven method was developed previously to reconstruct event energy in theJUNO experiment. In this paper, we investigated the choice of calibration sources thoroughly, optimized the calibration positionsand corrected the residual detector azimuthal asymmetry. All these e ff orts lead to a reduction of the energy non-uniformity near thedetector boundary, from about 0.64% to 0.38%. And within the fiducial volume of the detector it is improved from 0.3% to 0.17%.As a result the energy resolution could be further improved. Keywords:
JUNO, Liquid scintillator detector, Neutrino experiment, Energy reconstruction, Energy uniformity
1. Introduction
Liquid scintillator (LS) detectors with ultra-low background havebeen widely used in neutrino experiments. Just to name a few:KamLAND [1], Borexino [2], Double Chooz [3], Daya Bay [4]and RENO [5]. Instead of dwelling on the outstanding scientificachievements made by these experiments in recent decades [6–10], and how LS detectors will continue to play a crucial rolein neutrino physics in the future, let us do a quick comparisonof the LS detectors from above. The detector size varies from10 tons to 10 tons in target mass, and the detector energy res-olution ranges from 5% to 8%. On the other hand JUNO [11]will be the largest LS detector in the world with a target massof 20 kton upon completion, and its designed energy resolutionis ∼ / √ E , which is much more precise with respect to earlierexperiments. Even though it is rather challenging to achievesuch high precision energy resolution for a large LS detector, acomprehensive calibration program [12] demonstrated that therequired energy resolution of JUNO could be achieved, by ac-curately modeling the energy non-linearity and correcting forthe energy non-uniformity. The residual non-uniformity in [12]is less than 0.3% within the fiducial volume of the detector.Since the energy resolution has such significant impact, and oneof the main contributing factors is the energy non-uniformity,we wanted to further reduce it, especially for regions near thedetector boundary which amounts to more than 20% of thewhole detector volume.Due to the complicated optical processes, an optical modelindependent method [13] was developed previously to recon-struct the energy of positrons from inverse β -decay (IBD) events ∗ Corresponding author
Email addresses: [email protected] (Wuming Luo), [email protected] (Liangjian Wen) in the Central Detector (CD) of JUNO. The basic idea is to con-struct the maps of expected photoelectrons (PEs) for Photomul-tiplier tubes (PMTs) from calibration data, and then use thesemaps to build a maximum likelihood function to reconstructthe event energy. In this paper we will further improve the en-ergy uniformity in the CD of JUNO, by taking into account theasymmetry of a realistic detector and optimizing the calibrationstrategy. Due to lack of real data, Monte Carlo (MC) simulationdata generated by JUNO o ffl ine software [14] are used instead.The ideas and methods discussed here are also applicable toother experiments using large LS detectors.The structure of this paper is as follows: in Sec. 2, we willbriefly describe the CD of JUNO and its calibration systems.All the MC samples used will be listed in Sec. 3. From Sec. 4 toSec. 6, we will present an update on the maps of expected PEs,a thorough study on the calibration sources, and an optimizationon calibration points respectively. In Sec. 7 we will discuss theresidual energy non-uniformity. And finally we will give thesummary in Sec. 8.
2. JUNO CD and Calibration Systems
The CD of JUNO is made up of an acrylic sphere containing20,000 ton LS. The acrylic sphere is supported by a stainless-steel shell submerged in pure water. About 17,600 20-inchPMTs and 25,600 3-inch PMTs are installed on the stainless-steel shell to collect photons. JUNO also has a complex calibra-tion system [12] which consists of four sub-systems, namely theAutomated Calibration Unit (ACU) , the Cable Loop System(CLS), the Guide Tube (GT) and the remotely operated vehicle(ROV). Only the former three are used for energy reconstruc-tion in this paper. It should be emphasized that each sub-systemcould cover a di ff erent detector region: ACU can move alongthe Z-axis of the CD, CLS is able to reach those points permit- Preprint submitted to Nuclear Instruments and Methods in Physics Research A February 9, 2021 a r X i v : . [ phy s i c s . i n s - d e t ] F e b ed by the mechanics of the loop system within X-Z plane. GTis mounted on the outer surface of the acrylic sphere, designedto calibrate the detector in the edge region complementary toACU and CLS.
3. Monte Carlo Samples
Various calibration data samples with di ff erent sources takenfrom Ref. [12] are produced. For the prompt signal of IBDevents, a set of positron samples are also prepared. The infor-mation of the calibration samples and the positron samples usedin this paper are summarized in Tab. 1 and Tab. 2, respectively.For all these samples, the detector simulation is done based onGeant4 [15]. LS properties [16] and optical processes of pho-tons propagating in LS are implemented [17, 18]. Realistic de-tector geometry such as the arrangement of the PMTs and thesupporting structures is also deployed. For simplicity the elec-tronics simulation which includes various PMT characteristicsis disabled.The calibration samples are used to construct the maps ofexpected PEs per unit energy for PMTs, referred to as ˆ µ here-after and described in detail in Sec. 4. Calibration sources withdi ff erent types and energies are compared in order to select themost suitable one. Nine sets of positron samples with kineticenergy E k = (0, 1, 2, ..., 8) MeV are used to evaluate the per-formance of energy reconstruction. The events in each positronsample are uniformly distributed in the CD. Table 1: Information of the calibration samples. For the Laser source, “op”stands for optical photon and 1 MeV corresponds to 11522 optical photons.
Source Type Energy [MeV] Statistics / position Ge γ × Co γ + γ Table 2: List of the positron samples.
Source Kinetic energy Statistics / MeV Position e + (0,1,2, ..., 8) MeV 450k uniform in CD
4. Energy Reconstruction and ˆ µ As described in Ref. [13], an optical model independent methodwas developed to reconstruct the energy of positrons in theJUNO CD. The observables for each positron are { k i } , where k i represents the number of detected PEs for the i th PMT and isexpected to follow a Poisson distribution. The mean value ofthe Poisson distribution µ i is the product of the positron visibleenergy E vis and ˆ µ i from Sec. 3. So the probability of observing { k i } for all PMTs can be constructed as Eqn. 1 when an event deposits energy at position ( r , θ, φ ). L ( { k i }| r , θ, φ, E vis ) = (cid:89) i L ( k i | r , θ, φ, E vis ) = (cid:89) i e − µ i · µ k i i k i µ i = E vis · ˆ µ i (1)where the index i runs over all PMTs. After obtaining ˆ µ i , theevent energy can be fitted by maximizing this likelihood func-tion. In order to decouple the influence of the vertex uncertaintyon the energy reconstruction, the event vertex is assumed to beknown in this study.The key component of the energy reconstruction methoddiscussed above is ˆ µ . In Ref. [13], it is derived from the ACUcalibration data, under the assumption that the JUNO CD hasgood spherical symmetry. If the calibration source position isdefined as (cid:126) r = ( r , θ, φ =
0) and the i th PMT position as (cid:126) R i , asshown in Fig. 1, then ˆ µ can be calculated as:ˆ µ ( r , θ PMT ) = µ ( r , θ PMT ) E vis = ( 1 M M (cid:88) i = ¯ n i DE i ) · E vis E vis = PE total / Y (2)where E vis is the visible energy of the calibration source, PE total is the total number of PEs, Y is the constant light yield definedin Ref. [12], the index i runs over the PMTs with the same θ PMT ,¯ n i is the average number of detected PEs and DE i is the relativedetection e ffi ciency. Given there are only finite ACU calibrationpositions, ˆ µ ( r = z , θ PMT ) from these positions are extrapolatedthrough linear interpolation to the entire ( r , θ PMT ) phase space. sourceθ PMT i θ PMT r xz R i r Figure 1: Definition of three parameters of ˆ µ . (cid:126) r is the calibration source posi-tion, (cid:126) R i is the i th PMT position and θ PMT is the angle between (cid:126) r and (cid:126) R i . Fig. 2 compares the ˆ µ ( r , θ PMT ) maps for calibration posi-tions with the same radius but di ff erent zenith angle. The ap-parent di ff erences, which are mainly caused by the shadowinge ff ect of the supporting structures when θ varies, indicate thatthe detector is not symmetric along the θ direction, and this θ dependence for ˆ µ ( r , θ PMT ) must be taken into account. Since2he CLS system can move in the X-Z plane, we could combinethe CLS and ACU calibration data to construct ˆ µ ( r , θ, θ PMT ) inthe same way as before. e h e ] (cid:176) [ PMT q [ p . e . ] m (cid:176) = 90 q (cid:176) = 60 q (cid:176) = 30 q r = 10.0 m Figure 2: Comparison of ˆ µ for calibration positions with di ff erent zenith angle θ and fixed radius r =
10 m. The spikes are caused by the shadowing e ff ectof the supporting structures. The unit of ˆ µ is p.e. which stands for 1 photonelectron. A few examples of the ˆ µ ( r , θ, θ PMT ) maps at fixed θ PMT val-ues are shown in Fig. 3. The Delaunay triangles based cubicspline interpolation has been applied to ˆ µ ( r , θ, θ PMT ), so that itcould be extrapolated to the whole ( r , θ ) phase space from fi-nite calibration positions. At this point, it is quite natural to askwhether there is any φ dependence for ˆ µ , which could be causedby any detector asymmetry along the φ direction. We will leavethis discussion to Sec. 7.
5. Comparison of Calibration Sources
Our energy reconstruction method heavily relies on the usageof calibration data. Given all the available calibration sources,which one gives the best energy reconstruction performance?Bearing this question in mind, we thoroughly investigated thesesources: other than the energy, what else could be di ff erent forthese sources? How do the ˆ µ maps compare? And eventuallyhow does the energy reconstruction performance compare?In Fig. 4 we compared the distribution of the distance ∆ R be-tween energy-deposit center (cid:126) r edep and initial calibration position (cid:126) r init for di ff erent sources. Laser source is set to be point-like inthe MC simulation, and in reality it is approximately point-likedue to the di ff use ball which absorbs the optical photons fromLaser and re-emits them isotropically [19]. For illustration pur-pose, a virtual mono-energetic electron source is depicted in-stead of Laser. Electron source is very close to point-like and (cid:126) r edep is almost identical to (cid:126) r init . While for the gamma sourcesthere is a clear deviation of (cid:126) r edep from (cid:126) r init , since a gamma de-posits its energy in LS mainly through multiple Compton scat-tering. For Ge source, the two gammas from positron-electronannihilation tend to be back to back directionally. On the otherhand, Co radiates two gammas (1.173 MeV and 1.333 MeV)without any direction correlation, leading to a wider spread of ] [ m r [r a d ] q [ p . e . ] m (cid:176) = 10 PMT q ] [ m r [r a d ] q [ p . e . ] m (cid:176) = 30 PMT q ] [ m r [r a d ] q [ p . e . ] m (cid:176) = 150 PMT q Figure 3: Examples of the ˆ µ ( r , θ, θ PMT ) maps at three θ PMT angles. ∆ R . AmC is a neutron source, the neutron will travel some dis-tance before being captured by hydrogen and then emitting a2.22 MeV photon, thus its ∆ R has the widest spread among allthe sources. | [mm] init r- edep r| - - - - -
10 1 A . U . Ge AmCCo e- Figure 4: Distribution of the distance ∆ R between energy-deposit center (cid:126) r edep and initial calibration position (cid:126) r init for di ff erent sources. The CD is divided into regions I, II and III: namely the cen-tral region ( r < < r < < r ),and six representative calibration positions are picked: r = (10,15.6, 16.1, 16.2, 17.2, 17.4) m, θ = ◦ , φ = ◦ . The com-parison of the ˆ µ maps among di ff erent sources at these pointsis shown in Fig. 5. The ˆ µ maps had been smoothened and itwas checked that this smoothening has negligible impact on theenergy reconstruction. A few observations immediately standout:1. In region I, the ˆ µ maps are nearly monotonic and alsosimilar for all sources.2. In regions II and III, the ˆ µ maps have a kink. And thereare noticeable discrepancies among the sources aroundthe kink.In region I where the sources are relatively far away from thePMTs, all sources could be approximately regarded as point-like, thus leading to similar ˆ µ maps. Moreover, since total re-flection won’t occur in region I, as θ PMT increases, smaller solidangle leads to decreased ˆ µ . While in regions II and III, thereis always a total reflection zone for any given source position.For those PMTs in the total reflection zone (as indicated by theshadowed region in the plots), one would expect a large de-crease of detected PEs due to photon loss. In addition ∆ R alsobecomes relevant for these PMTs, because any small deviationof the source position would partially mitigate the impact oftotal reflection. And the more spread ∆ R is, the stronger themitigation e ff ect is, which is illustrated by the enlarged figuresin Fig. 5 and Fig. 4 from above.After obtaining the ˆ µ maps using di ff erent sources and thecalibration points from Case 2 in Sec. 6 , we applied them indi-vidually to the energy reconstruction of positron samples listedin Tab. 2. The uniformity of the reconstructed energy E rec withrespect to r for two di ff erent energies was plotted in Fig. 6. Thetwo vertical lines indicate the boundaries of the three regions.Each curve is normalized by its average value within region I.The results of E rec are quite consistent among the sources inregion I for all positron energies, which is not surprising giventhat the ˆ µ maps from di ff erent sources are almost the same. Thenon-uniformity in region II could be traced back to the featuresof the ˆ µ maps, caused by total reflection as mentioned before.Take the bump peak in the E k = r = µ , resulting in larger E rec . The size of the non-uniformity for each source is posi-tively correlated to its ∆ R spread. Another important thing weshould note is that using the ˆ µ maps from Ge source yieldsthe best uniformity at E k = µ maps from Laser source perform the best.By comparing the sources thoroughly, we aimed to pick outone that gives good energy reconstruction performance acrossthe entire positron energy range. Based on the studies above,none of the sources is satisfactory. If one single source won’tdo, is it possible to use a combined source? Let us dive backto the energy deposition of positron in LS again. The wholeprocess can be naturally broken down into two parts: almost all positrons will fully deposit their kinetic energy first, this partcan be treated as a point-like source. There is a small prob-ability that positrons will annihilate in flight, but this can besafely ignored. The second part is the positron-electron anni-hilation producing two gammas, which is almost the same asthe Ge source. This explains why the Ge source performsthe best for E k = E vis , wepropose the following combined ˆ µ comb ( r , θ, θ PMT ):ˆ µ comb = E vis · ( E Gevis · ˆ µ Ge ( r , θ, θ PMT ) + E k · ˆ µ L ( r , θ, θ PMT )) E vis = E Gevis + E k (3)where ˆ µ Ge ( r , θ, θ PMT ) and ˆ µ L ( r , θ, θ PMT ) correspond to the an-nihilation part and kinetic energy part of positron respectively, E k is the kinetic energy of positron and E Gevis (1.022 MeV) is thevisible energy of Ge.To validate the combined maps ˆ µ comb , they were comparedto those produced with positron samples listed in Tab. 2. Acrossthe whole energy range, ˆ µ comb are able to match the positron ˆ µ maps. A few examples are shown in Fig. 7. Note that it isassumed the kinetic energy part of the combined maps is lin-early proportional to the kinetic energy. Energy non-linearity isnot considered and has tiny impact on ˆ µ comb . Replacing Laserwith other point-like sources such as electron works as well anddoes not make any big di ff erence for ˆ µ comb . Laser is chosen butnot electron simply due to the lack of mono-energetic electronsources in reality.After ˆ µ comb were produced and validated, their energy re-construction performance was evaluated as before. The energyuniformity at various energies is shown in Fig. 8, which clearlyhas very small dependence on the energy. More importantly, itis largely improved in the total reflection region comparing toFig. 6. For each positron sample with di ff erent kinetic energy,the distribution of the reconstructed visible energy is fitted witha Gaussian function, and the corresponding energy resolutionis defined as the ratio of the Gaussian sigma to the Gaussianmean. The energy resolution with respect to the mean valueof the reconstructed visible energy for all the positron samplesin the three regions is shown in Fig. 9. Di ff erent colors corre-spond to di ff erent sources that are used to produce the ˆ µ maps.The dots represent the energy resolution at di ff erent energies.Please note that energy non-linearity is not corrected here. Inregions I and III, the energy resolution is almost identical whenusing ˆ µ maps from di ff erent sources. In region II, the combinedmaps ˆ µ comb yield the best energy resolution especially at highenergies, which is a direct consequence of improved energy uni-formity. The energy resolution is fitted with an empirical modelbelow: σ E = (cid:114) a E + b (4)where a is related to the photon poisson statistics, b is the con-stant term and the unit of energy is MeV. The fitted results are4
20 40 60 80 100 120 140 160 -
10 110 [ p . e . ] m Ge AmCCo Laser r = 10.0 m
20 40 60 80 100 120 140 160 -
10 110 r = 15.6 m
20 40 60 80 100 120 140 160 180 ] (cid:176) [ PMT q -
10 110 r = 16.1 m -
10 110 r = 16.2 m
20 40 60 80 100 120 140 160 -
10 110 r = 17.2 m
20 40 60 80 100 120 140 160 180 ] (cid:176) [ PMT q -
10 110 r = 17.4 m
30 35 40 45 50 55 60 65 70 75 800.020.040.060.080.10.120.140.160.18 30 35 40 45 50 55 60 65 70 75 800.020.040.060.080.10.120.140.160.1830 35 40 45 50 55 60 65 70 75 800.020.040.060.080.10.120.140.160.18 30 35 40 45 50 55 60 65 70 75 800.020.040.060.080.10.120.140.160.18 30 35 40 45 50 55 60 65 70 75 800.020.040.060.080.10.120.140.160.18
Figure 5: Comparison of ˆ µ for di ff erent sources at a few representative calibration positions. From left to right and top to bottom, r = (10, 15.6, 16.1, 16.2, 17.2,17.4) m, θ = ◦ , φ = ◦ . The shaded region indicates the total reflection zone. [m r0.9850.990.99511.0051.011.015 r ec N o r m a li ze d E Ge AmCCo Laser = 0 MeV k E0 1000 2000 3000 4000 5000 ] [m r0.980.9850.990.99511.0051.01 r ec N o r m a li ze d E = 5 MeV k E Figure 6: Uniformity of reconstructed energy E rec with respect to r using ˆ µ maps from di ff erent sources. For each r bin, the mean value of E rec is plotted.Top and bottom plots correspond to E k =
0, 5 MeV e + samples respectively. summarized in Tab. 3. In regions I and III, there is no big dif-ference for a and b . But in region II, it is clear that energy ] (cid:176) [ PMT q -
10 1 [ p . e . ] m Ge + 2 MeV Laser r = 16.2 m ] (cid:176) [ PMT q - -
10 1 Ge + 5 MeV Laser r = 17.2 m Figure 7: Validation of the combined maps ˆ µ comb . Top and bottom plots are for E k =
2, 5 MeV and r = [m r0.980.9850.990.99511.0051.011.015 r ec N o r m a li ze d E = 0 MeV k E = 1 MeV k E = 2 MeV k E = 3 MeV k E = 4 MeV k E = 5 MeV k E = 6 MeV k E = 7 MeV k E = 8 MeV k E Figure 8: Uniformity of E rec with respect to r using combined maps ˆ µ comb atvarious energies. For each r bin, the mean value of E rec is plotted.Table 3: Comparison of fitted parameters for the empirical energy resolutionmodel among various sources. The unit for a (b) is % × MeV (%). The fituncertainties for a and b are less than 0.005 (0.02) in regions I and II (III). Region I II III a b a b a bAmC 2.76 0.626 2.74 0.926 3.02 1.09 Co 2.76 0.624 2.76 0.853 3.01 1.11 Ge 2.76 0.623 2.77 0.784 3.01 1.10Laser 2.76 0.623 2.80 0.711 3.00 1.11 Ge + Laser 2.76 0.622 2.79 0.715 3.01 1.11non-uniformity contributes to the b term. Compared to theAmC source which has the worst non-uniformity, the combined Ge + Laser source improves the b term by 22.8%.
6. Optimization of Calibration Positions
In Sec. 5 we have looked into various calibration sources andfound that Ge + Laser combined source is the best choice forpositrons in the kinetic energy range of [0-10] MeV. In ad-dition to the source, the number and positions of calibrationpoints should also be carefully considered. As a simple mea-sure of the detector energy response, the contours of total num-ber of detected PEs on the θ − r plane for positron samplesare drawn in Fig. 10. They clearly show that the detector en-ergy response heavily depends on the position. Through finitecalibration points, we try to capture the features of the detectorenergy response and then extrapolate to all areas as accuratelyas possible. The better we could do this, the more we will beable to improve the energy uniformity of the detector. Howeverthe calibration points from Ref. [12], as represented by the opencircles in Fig. 10, do not have enough coverage near the detec-tor boundary and the arrangement is somewhat random. Weproposed to further optimize the calibration points utilizing thecontours. The arrangement could be more e ffi cient by assigningmore (less) points in areas where the detector energy responsechanges dramatically (slowly).For completeness, we considered a few di ff erent scenarios: • Case 1: 2000 points semi-randomly chosen, more densein regions II and III • Case 2: 275 points selected based on the contours of totalnumber of PEs • Case 3: replacing those unreachable points in Case 2 withadjacent points on the CLS boundaries • Case 4: Case 3 plus additional 19 points from GTThe arrangement of the calibration points on the θ − r planefor the above cases 2-4 are also shown in Fig. 10. The threepurple curves represent the CLS boundaries and the areas theysemi-enclose are not reachable. The points are the selected cal-ibration positions. The black dots are common for Cases 2, 3and 4. While the cross marks are for Case 2, the blue squaresare for Case 3 and the red triangles are the positions from GT.Most of the selected points are at the intersections of the con-tours and fixed θ or r lines. In areas where the contours varyrapidly, more points are assigned.We chose the combined Ge + Laser source, constructed theˆ µ maps for the 4 di ff erent cases above, and then compared theirenergy reconstruction performance. Fig. 11 shows the E rec uni-formity comparison at E k = ff erence between Case 1and Case 2 is marginal, indicating that we could largely re-duce the total number of calibration points without jeopardizingthe reconstruction performance. After replacing those unreach-able points in Case 2, the energy uniformity becomes worse forCase 3 near the detector border. The GT system is originallydesigned to calibrate the detector energy response near the de-tector border, complementary to CLS. After adding the pointsfrom the GT, the energy uniformity in Case 4 slightly improvedwith respect to Case 3.The energy resolution was also compared for the four casesas shown in Fig. 12. Overall, the performance is close. Dif-ferences of energy resolution in regions II and III are consis-tent with the energy uniformity comparison in Fig. 11, wheresmaller energy non-uniformity leads to better energy resolution.The fitted energy resolution using Eqn. 4 are listed in Tab. 4.The b term is larger for Case 3 compared to Case 2, and withthe help of the GT, the b term was slightly reduced from Case 3to Case 4.
7. Residual Azimuthal Asymmetry
After optimizing the calibration source and positions, there isstill some residual energy non-uniformity near the CD edge.While the ˆ µ maps have already taken into account the depen-dence on r and θ , the φ dependence has not been considereddue to limitation of the calibration system. To check the φ de-pendence, we compared the distribution of total number of PEsas a function of θ for two di ff erent φ planes at three di ff erentradius. From the left plot to the right plot in Fig. 13, one can6 rec E11.522.533.5 E n e r gy r e s o l u ti on [ % ] < 3800 m r Ge AmCCo LaserGe + Laser rec E E n e r gy r e s o l u ti on [ % ] < FV < r rec E11.522.533.5 E n e r gy r e s o l u ti on [ % ] < 5237 m FV < r
Figure 9: Comparison of the resolution of the reconstructed visible energy for positrons in the three regions of the CD. Di ff erent colors correspond to the varioussources which are used to construct the ˆ µ maps. The dots correspond to positrons with di ff erent kinetic energies. E rec is the mean value of the reconstructed visibleenergy. Please note that energy non-linearity is not corrected here. ] [m r ] (cid:176) [ q Figure 10: Contours of total number of PEs on θ - r plane and the arrangementof calibration points for di ff erent cases. [m r0.9850.990.99511.0051.01 r ec N o r m a li ze d E Case 1Case 2Case 3Case 4Case 5
Figure 11: Uniformity of E rec with respect to r from di ff erent cases of calibra-tion points at E k = σ variation of the E rec along the θ direction for Cases 3 and 5 respectively.Case 5 will be discussed separately in Sec. 7. Table 4: Comparison of fitted parameters for the empirical energy resolutionfor the five cases. The unit for a (b) is % × MeV (%). The fit uncertainties fora and b are less than 0.005 (0.02) in regions I and II (III). Region I II III a b a b a bCase 1 2.76 0.622 2.80 0.711 3.02 1.07Case 2 2.76 0.622 2.79 0.715 3.01 1.11Case 3 2.76 0.622 2.81 0.730 2.98 1.15Case 4 2.76 0.622 2.81 0.730 3.00 1.10Case 5 2.76 0.622 2.79 0.698 2.99 1.07easily see that as the position gets closer to the CD edge, the φ dependence becomes more and more prominent.The CD is only approximately symmetric along the φ direc-tion, since neither the PMTs nor the supporting structures haveperfect azimuthal symmetry. Near the CD edge, the shadowinge ff ect of the supporting structures can not be ignored any more.The dominant contribution comes from the numerous acrylicnodes. Although imperfect, the distribution of these acrylicnodes is roughly periodic every 6 degrees along the φ direction.So we can apply a φ dependent correction to the ˆ µ maps within6 degrees and extrapolate to the full φ range. Due to the me-chanical limitation of CLS, we have to rely on MC simulationfor this φ correction. Three φ planes with φ = ◦ , 4 ◦ , 6 ◦ wereselected, together with the CLS plane where φ = ◦ , a correc-tion function f ( φ ) was produced and applied to the ˆ µ maps ofCase 3 from previous section for the edge region ( r > . E rec at E k = rec E11.522.533.5 E n e r gy r e s o l u ti on [ % ] < 3800 m r Case 1Case 2Case 3Case 4Case 5 rec E E n e r gy r e s o l u ti on [ % ] < FV < r rec E11.522.533.5 E n e r gy r e s o l u ti on [ % ] < 5237 m FV < r
Figure 12: Comparison of the resolution of the reconstructed visible energy for positrons in the three regions of the CD. Di ff erent colors correspond to the variouscases described in the text. The dots correspond to positrons with di ff erent kinetic energies. E rec is the mean value of the reconstructed visible energy. Please alsonote that energy non-linearity is not corrected here. (cid:176) [ q T o t a l nu m b e r o f P E s (cid:176) = 0 f (cid:176) = 87.4 f Laser at r = 16.0 m 0 20 40 60 80 100 120 140 160 180] (cid:176) [ q T o t a l nu m b e r o f P E s (cid:176) = 0 f (cid:176) = 87.4 f Laser at r = 17.2 m 0 20 40 60 80 100 120 140 160 180] (cid:176) [ q T o t a l nu m b e r o f P E s (cid:176) = 0 f (cid:176) = 87.4 f Laser at r = 17.5 m
Figure 13: Comparison of the distribution of total number of PEs as a function of θ between two di ff erent φ planes. From left to right, the radius is fixed at r =
16 m,17.2 m and 17.5 m respectively. decrease for the b term in regions II and III respectively.The f ( φ ) correction derived above is able to reduce the resid-ual energy non-uniformity in the CD edge region. One caveatis that this correction is derived from MC simulation, whichhas to be validated against real data. There are several possibleways to do this. The calibration data from GT could be usedto check this correction. Another approach is to use spallationneutron events, which are abundant and uniformly distributedin the detector. In the future, if we were able to reconstruct theevent vertex with good precision, we could use spallation neu-tron events to obtain the f ( φ ) correction, instead of relying onMC simulation.
8. Conclusion
It is rather challenging to achieve high precision energy resolu-tion for large LS detectors such as JUNO. Lots of studies havebeen done previously to address the energy uniformity utiliz-ing calibration data in JUNO. In those studies, the residual en-ergy non-uniformity is about 0.3% within the fiducial volume,and gets much worse near the detector boundary due to compli-cated optical processes like total reflections, shadowing e ff ectsof opaque materials. In this paper we expanded the ˆ µ mapsof expected PEs for PMTs to include the θ dependence. Thechoice of the calibration source was thoroughly investigated, and we found that Ge + Laser combined source outperformsany single source across the entire energy range of interest.We also optimized the number and positions of the calibrationpoints based on a novel strategy, which utilizes the contoursof the total number of detected PEs. The small residual non-uniformity caused by the azimuthal asymmetry of the detectorwas handled by a φ dependent correction. As a result, we wereable to reduce the energy non-uniformity from about 0.64% to0.38% for regions near the detector boundary. And the energynon-uniformity within the fiducial volume (across the whole de-tector) could be well constrained under 0.17% (0.23%). As adirect consequence of the improved energy uniformity, betterenergy resolution was achieved.Another interesting finding is that the energy non-uniformityin the central region of the detector is not particularly a ff ectedby the choice of the calibration source, nor by the asymmetryof the detector. This allows for more flexibility on the cali-bration strategy in this region. Moreover the detector energyresponse changes relatively slowly in this region so that only asmall amount of calibration points are needed, which could alsoserve as a guideline for the calibration strategy.In addition to the calibration sources, there will be vari-ous physics events occurring inside the LS detector as well,we should also be able to use them to obtain a better under-standing of the detector response. Assuming we could select8ut some specific events, which are distributed across the entiredetector, and have reasonably well known energy and vertex,it would be straight forward to use them to construct ˆ µ mapsto include the φ dependence as well. And to go one step fur-ther, if we could have huge amount of these events, we couldtry novel techniques such as Machine Learning to study the de-tector energy response. As we are entering the precision era ofneutrino experiments, which demand much better energy res-olution than before, every bit of improvement counts. All theideas and methods in this paper improved the energy uniformityand consequently the energy resolution in JUNO. They could beapplied to other experiments with large LS detectors. Acknowledgements
This work was partially supported by the National RecruitmentProgram for Young Professionals, by the National Key R&DProgram of China under Grant No. 2018YFA0404100, by theStrategic Priority Research Program of the Chinese Academyof Sciences under Grant No. XDA10010100, and by the CASCenter for Excellence in Particle Physics (CCEPP). We wouldlike to thank Feiyang Zhang for the useful information and dis-cussion about the calibration systems of JUNO.
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