Improving the energy uniformity for large liquid scintillator detectors

Due to the complicated optical processes, an optical model independent method [13] was developed previously to reconstruct the energy of positrons from inverse $\beta$-decay (IBD) events in the Central Detector (CD) of JUNO. The basic idea is to construct the maps of expected photoelectrons (PEs) for Photomultiplier tubes (PMTs) from calibration data, and then use these maps to build a maximum likelihood function to reconstruct the event energy. In this paper we will further improve the energy uniformity in the CD of JUNO, by taking into account the asymmetry of a realistic detector and optimizing the calibration strategy. Due to lack of real data, Monte Carlo (MC) simulation data generated by JUNO offline software [14] are used instead. The ideas and methods discussed here are also applicable to other experiments using large LS detectors.

The structure of this paper is as follows: in Sec. 2, we will briefly describe the CD of JUNO and its calibration systems. All the MC samples used will be listed in Sec. 3. From Sec. 4 to Sec. 6, we will present an update on the maps of expected PEs, a thorough study on the calibration sources, and an optimization on calibration points respectively. In Sec. 7 we will discuss the residual energy non-uniformity. And finally we will give the summary in Sec. 8.

The calibration samples are used to construct the maps of expected PEs per unit energy for PMTs, referred to as $\hat{\mu}$ hereafter and described in detail in Sec. 4. Calibration sources with different types and energies are compared in order to select the most suitable one. Nine sets of positron samples with kinetic energy $E_{k}$ = (0, 1, 2, …, 8) MeV are used to evaluate the performance of energy reconstruction. The events in each positron sample are uniformly distributed in the CD.

The key component of the energy reconstruction method discussed above is $\hat{\mu}$. In Ref. [13], it is derived from the ACU calibration data, under the assumption that the JUNO CD has good spherical symmetry. If the calibration source position is defined as $\vec{r}$ = ($r,\theta,\phi$ = 0) and the $i^{th}$ PMT position as $\vec{R_{i}}$, as shown in Fig. 1, then $\hat{\mu}$ can be calculated as:

where $E_{vis}$ is the visible energy of the calibration source, $PE_{total}$ is the total number of PEs, $Y_{0}$ is the constant light yield defined in Ref. [12], the index $i$ runs over the PMTs with the same $\theta_{PMT}$, $\bar{n}_{i}$ is the average number of detected PEs and $DE_{i}$ is the relative detection efficiency. Given there are only finite ACU calibration positions, $\hat{\mu}(r=z,\theta_{PMT})$ from these positions are extrapolated through linear interpolation to the entire $(r,\theta_{PMT})$ phase space.

Fig. 2 compares the $\hat{\mu}(r,\theta_{PMT})$ maps for calibration positions with the same radius but different $\theta$ angle, as could be collected by the CLS calibration sub-system. The apparent differences, which are mainly caused by the shadowing effect of the acrylic nodes and connecting bars when $\theta$ varies, indicate that the detector is not symmetric along the $\theta$ direction, and this $\theta$ dependence for $\hat{\mu}(r,\theta_{PMT})$ must be taken into account. Since the CLS system can move in the X-Z plane, we could combine the CLS and ACU calibration data to construct $\hat{\mu}(r,\theta,\theta_{PMT})$ in the same way as before.

A few examples of the $\hat{\mu}(r,\theta,\theta_{PMT})$ maps at fixed $\theta_{PMT}$ values are shown in Fig. 3. And they are also affected by the same shadowing effect. The Delaunay triangles based cubic spline interpolation has been applied to $\hat{\mu}(r,\theta,\theta_{PMT})$, so that it could be extrapolated to the whole ($r,\theta$) phase space from finite calibration positions. At this point, it is quite natural to ask whether there is any $\phi$ dependence for $\hat{\mu}$, which could be caused by any detector asymmetry along the $\phi$ direction. We will leave this discussion to Sec. 7.

In Fig. 4 we compared the distribution of the distance $\Delta R$ between energy-deposit center $\vec{r}_{edep}$ and initial calibration position $\vec{r}_{init}$ for different sources. Laser source is set to be point-like in the MC simulation, and in reality it is approximately point-like due to the diffuse ball which absorbs the optical photons from Laser and re-emits them isotropically [19]. For illustration purpose, a virtual mono-energetic electron source is depicted instead of Laser. A hypothetical position source with $E_{k}$ = 5 MeV is also drawn for comparison. Electron source is very close to point-like and $\vec{r}_{edep}$ is almost identical to $\vec{r}_{init}$. While for the gamma sources there is a clear deviation of $\vec{r}_{edep}$ from $\vec{r}_{init}$, since a gamma deposits its energy in LS mainly through multiple Compton scattering. For ⁶⁸Ge source, the two gammas from positron-electron annihilation tend to be back to back directionally. On the other hand, ⁶⁰Co radiates two gammas (1.173 MeV and 1.333 MeV) without any direction correlation, leading to a wider spread of $\Delta R$. AmC is a neutron source, the neutron will travel some distance before being captured by hydrogen and then emitting a 2.22 MeV photon, thus its $\Delta R$  has the widest spread among all the sources.

The CD is divided into regions I, II and III: namely the central region ($r<$ 15.6 m), the total reflection region (15.6 m $<r<$ 17.2 m) and the outer-FV (fiducial volume) region (17.2 m $<r$), and six representative calibration positions are picked: $r$ = (10, 15.6, 16.1, 16.2, 17.2, 17.4) m, $\theta$ = 90^∘, $\phi$ = 0^∘. The comparison of the $\hat{\mu}$ maps among different sources at these points is shown in Fig. 5. The $\hat{\mu}$ maps had been smoothened and it was checked that this smoothening has negligible impact on the energy reconstruction. A few observations immediately stand out:

• 1.

  In region I, the $\hat{\mu}$ maps are nearly monotonic and also similar for all sources.

• 2.

  In regions II and III, the $\hat{\mu}$ maps have a kink. And there are noticeable discrepancies among the sources around the kink.

After obtaining the $\hat{\mu}$ maps using different sources and the calibration points from Case 2 in Sec. 6 , we applied them individually to the energy reconstruction of positron samples listed in Tab. 2. The uniformity of the reconstructed energy $E_{rec}$ with respect to $r^{3}$ for two different energies was plotted in Fig. 6. The two 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 in region I for all positron energies, which is not surprising given that the $\hat{\mu}$ maps from different sources are almost the same. The non-uniformity in region II could be traced back to the features of the $\hat{\mu}$ maps, caused by total reflection as mentioned before. Take the bump peak in the $E_{k}$ = 5 MeV case as an example. The corresponding radius is $r$ = 16.1 m which is the same as the top right plot in Fig. 5. Comparing to the Laser source, the other sources will give smaller expected $\hat{\mu}$, resulting in larger $E_{rec}$. The size of the non-uniformity for each source is positively correlated to its $\Delta R$ spread. Another important thing we should note is that using the $\hat{\mu}$ maps from ⁶⁸Ge source yields the best uniformity at $E_{k}$ = 0 MeV, while at high energies the $\hat{\mu}$ maps from Laser source perform the best.

By comparing the sources thoroughly, we aimed to pick out one that gives good energy reconstruction performance across the entire positron energy range. Based on the studies above, none of the sources is satisfactory. If one single source won’t do, is it possible to use a combined source? Let us dive back to the energy deposition of positron in LS again. The whole process can be naturally broken down into two parts: almost all positrons will fully deposit their kinetic energy first, this part can be treated as a point-like source. There is a small probability that positrons will annihilate in flight, but this can be safely ignored. The second part is the positron-electron annihilation producing two gammas, which is almost the same as the ⁶⁸Ge source. This explains why the ⁶⁸Ge source performs the best for $E_{k}$ = 0 MeV positron events. With increasing kinetic energy, positron becomes more and more point-like. Consequently point-like source such as Laser is more suitable at higher energies. Thus for positrons with visible energy $E_{vis}$, we propose the following combined $\hat{\mu}^{comb}(r,\theta,\theta_{PMT})$:

where $\hat{\mu}^{Ge}(r,\theta,\theta_{PMT})$ and $\hat{\mu}^{L}(r,\theta,\theta_{PMT})$ correspond to the annihilation part and kinetic energy part of positron respectively, $E_{k}$ is the kinetic energy of positron and $E^{Ge}_{vis}$ (1.022 MeV) is the visible energy of ⁶⁸Ge.

To validate the combined maps $\hat{\mu}^{comb}$, they were compared to those produced with positron samples listed in Tab. 2. Across the whole energy range, $\hat{\mu}^{comb}$ are able to match the positron $\hat{\mu}$ maps. A few examples are shown in Fig. 7. Note that it is assumed the kinetic energy part of the combined maps is linearly proportional to the kinetic energy. Energy non-linearity is not considered and has tiny impact on $\hat{\mu}^{comb}$. Replacing Laser with other point-like sources such as electron works as well and does not make any big difference for $\hat{\mu}^{comb}$. Laser is chosen but not electron simply due to the lack of mono-energetic electron sources in reality. For the Laser source, the emitted photons are assumed to be isotropic. In reality the non-uniformity of photon emission for the Laser source in JUNO is expected to be about a few percent. Alternative Laser samples were produced where an arbitrary 5% non-uniformity is added by hand. The resulting $\hat{\mu}^{L}$ maps do not change much compared with the default Laser samples, indicating that our calibration procedure is not particularly sensitive to the non-uniformity of the Laser source.

After $\hat{\mu}^{comb}$ were produced and validated, their energy reconstruction performance was evaluated as before. The energy uniformity at various energies is shown in Fig. 9,

which clearly has very small dependence on the energy. More importantly, it is largely improved in the total reflection region comparing to Fig. 6. For each positron sample with different kinetic energy, the distribution of the reconstructed visible energy is fitted with a Gaussian function, and the corresponding energy resolution is defined as the ratio of the Gaussian sigma to the Gaussian mean. The energy resolution with respect to the mean value of the reconstructed visible energy for all the positron samples in the three regions is shown in Fig. 8. Different colors correspond to different sources that are used to produce the $\hat{\mu}$ maps. The dots represent the energy resolution at different energies. Please note that energy non-linearity is not corrected here. In regions I and III, the energy resolution is almost identical when using $\hat{\mu}$ maps from different sources. In region II, the combined maps $\hat{\mu}^{comb}$ yield the best energy resolution especially at high energies, which is a direct consequence of improved energy uniformity. The energy resolution is fitted with an empirical model below:

In regions I and III, there is no big difference for $a$ and $b$. But in region II, it is clear that energy non-uniformity contributes to the $b$ term. Compared to the AmC source which has the worst non-uniformity, the combined ⁶⁸Ge+Laser source improves the $b$ term by 22.8%.

For completeness, we considered a few different scenarios:

• 1.

  Case 1: mimics the ideal case with infinite calibration points. 2000 points are randomly chosen in the ($r^{5},\theta$) plane to allow more points in regions II and III

• 2.

  Case 2: represents the optimal case with 275 points selected based on the contours of total number of PEs

• 3.

  Case 3: corresponds to a more realistic case where those unreachable points in Case 2 are replaced with adjacent points on the CLS boundaries

• 4.

  Case 4: includes additional 19 points from GT on top of Case 3

We chose the combined ⁶⁸Ge+Laser source, constructed the $\hat{\mu}$ maps for the 4 different cases above, and then compared their energy reconstruction performance. Fig. 11 shows the $E_{rec}$ uniformity comparison at $E_{k}$ = 5 MeV, similar results were observed at other energies given that energy uniformity has negligible energy dependence after using the combined source. Case 1 has the smallest non-uniformity. The difference between Case 1 and Case 2 is marginal, indicating that we could largely reduce the total number of calibration points without jeopardizing the reconstruction performance. After replacing those unreachable points in Case 2, the energy uniformity becomes worse for Case 3 near the detector border. The GT system is originally designed to calibrate the detector energy response near the detector border, complementary to CLS. After adding the points from the GT, the energy uniformity in Case 4 slightly improved with respect to Case 3.

The energy resolution was also compared for the four cases as shown in Fig. 12. Overall, the performance is close. Differences of energy resolution  in regions II and III are consistent with the energy uniformity comparison in Fig. 11, where smaller 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 with the help of the GT, the $b$ term was slightly reduced from Case 3 to Case 4.

The CD is only approximately symmetric along the $\phi$ direction, since neither the PMTs nor the supporting structures have perfect azimuthal symmetry. Near the CD edge, the shadowing effect of the supporting structures can not be ignored any more. The dominant contribution comes from the numerous acrylic nodes. Although imperfect, the distribution of these acrylic nodes is roughly periodic every 6 degrees along the $\phi$ direction. So we can apply a $\phi$ dependent correction to the $\hat{\mu}$ maps within 6 degrees and extrapolate to the full $\phi$ range. Due to the mechanical limitation of CLS, we have to rely on MC simulation for this $\phi$ correction. Three $\phi$ planes with $\phi$ = $2^{\circ}$, $4^{\circ}$, $6^{\circ}$ were selected, together with the CLS plane where $\phi$ = $0^{\circ}$, a correction function $f(\phi$) was produced and applied to the $\hat{\mu}$ maps of Case 3 from previous section for the edge region ($r>15.6$ m) only. With this correction, the uniformity of $E_{rec}$ at $E_{k}$ = 5 MeV is plotted as Case 5 in Fig. 11. The improvement of the energy non-uniformity is outstanding in the edge region. After the correction, the residual non-uniformity is about 0.17% within the fiducial volume and 0.23% across the entire detector. The energy resolution after the correction is plotted in Fig. 12, and the fitted results are listed in Tab. 4, both referred to as Case 5. Comparing to Case 3, the improved energy uniformity propagates to the energy resolution and leads to about 4.4% and 7% decrease for the $b$ term in regions II and III respectively.

The $f(\phi)$ correction derived above is able to reduce the residual energy non-uniformity in the CD edge region. One caveat is that this correction is derived from MC simulation, which has to be validated against real data. There are several possible ways to do this. The calibration data from GT could be used to check this correction. Another approach is to use spallation neutron events, which are abundant and uniformly distributed in the detector [20]. In the future, if we were able to reconstruct the event vertex with good precision, we could use spallation neutron events to obtain the $f(\phi)$ correction, instead of relying on MC simulation.

Another interesting finding is that the energy non-uniformity in the central region of the detector is not particularly affected by the choice of the calibration source, nor by the asymmetry of the detector. This allows for more flexibility on the calibration strategy in this region. Moreover the detector energy response changes relatively slowly in this region so that only a small amount of calibration points are needed, which could also serve as a guideline for the calibration strategy.

In addition to the calibration sources, there will be various physics events occurring inside the LS detector as well, we should also be able to use them to obtain a better understanding of the detector response. Assuming we could select out some specific events, which are distributed across the entire detector, and have reasonably well known energy and vertex, it would be straight forward to use them to construct $\hat{\mu}$ maps to include the $\phi$ dependence as well. And to go one step further, if we could have huge amount of these events, we could try novel techniques such as Machine Learning to study the detector energy response. As we are entering the precision era of neutrino experiments, which demand much better energy resolution than before, every bit of improvement counts. All the ideas and methods in this paper improved the energy uniformity and consequently the energy resolution in JUNO. They could be applied to other experiments with large LS detectors.