On the Calibration of the Antineutrino Detectors

Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (September 30, 2006)

Abstract

The requirement that the energy scale of scintillation in the antineutrino detectors be known to 1% at 6 MeV leads to a requirement that the “gain” of each of the 224 PMTs in a detector be known to 10%. This surprisingly modest requirement could be meet by a calibration based on as few as 40 thermal neutron captures from reactor antineutrino interactions,permitting continual recalibration of the detectors every 10-12 hours. Occasional source-based calibrations will provide an important check on the antineutrino calibrations.

1 How Well Do We Need to Know the PMT Gains?

In the Daya Bay Reactor Neutrino experiment, \overline{ν}_{e}’s of energy ≈ 1-10 MeV will be detected via the inverse β-decay reaction,

\overline{ν}_{e} + p → n + e^{+},  (1)

on free protons in Gd-loaded liquid scintillator in 8 antineutrino detectors. The energy E\overline{ν} of the antineutrino is inferred from the observed scintillation energy E_{e^{+}} > 1.022 MeV of the positron and its decay products after annihilation. The signature of inverse β-decay is confirmed by the observation of the delayed capture of the (thermalized) neutron on a Gd nucleus, which releases ≈ 8 MeV energy in the form of 3-4 γ’s. A simulated spectrum of the scintillation energy observed following a neutron capture is shown below. The tail to low energies is due to events in which one or more of the γ’s from the neutron capture deposits energy outside the active scintillation volume of the detector.

The strongest requirement on the calibration of the scintillation process is that the energy scale at 6 MeV be known to an accuracy of 1% so that the effect of a cut at this energy on the efficiency of neutron detection is less than 0.2%.

Each antineutrino detector will have N = 224 8′′ photomultiplier tubes (PMTs), arrayed in 7 rings of 32 PMTs on the inside of the vertical surface of the detector tank, as sketched below. The effective photocathode coverage is about 12% of the surface area of the detectors, resulting in about 75 photoelectrons per MeV of deposited energy. The number n of photoelectrons per PMT is,

n ≈ \frac{75}{224}E ≈ \frac{E(MeV)}{3} .  (2)

We label the PMTs by the index i which runs from 1 to N = 224. The number of photoelectrons observed (during some appropriately narrow time window) in PMT i is denoted n_{i}, and the “gain” of PMT i in units of MeV per photoelectron is G_{i}.^{1} The signal S in MeV from a scintillation process such as neutron capture is,

S = ∑_{i=1}^{N} G_{i}n_{i} ≈ G ∑_{i=1}^{N} n_{i} ≈ \frac{GNE(MeV)}{3} ≈ 75GE(MeV),  (3)

where G is the typical value of the PMT gains G_{i}, and we use eq. (2) to approximate n_{i}. The root-mean-square uncertainty σ_{S} on the signal (3) due to the uncertainties σ_{G_{i}} in the PMT gains and due to the fluctuations in the numbers n_{i} of observed photoelectrons from a given energy deposition E is related by,

σ_{S} = \sqrt{∑_{i=1}^{N}[σ_{G_{i}}^{2}n_{i}^{2} + G_{i}^{2}σ_{n_{i}}^{2}]} ≈ \sqrt{σ_{G}^{2} ∑_{i=1}^{N} n_{i}^{2} + G^{2} ∑_{i=1}^{N} n_{i}}  (4)

≈ \sqrt{σ_{G}^{2}N (\frac{E^{2}(MeV)}{9} + \frac{E(MeV)}{3}) + \frac{G^{2}NE(MeV)}{3}},

1In more detail, a PMT “multiplies” each photoelectron by a factor of g_{i} ≈ 10^{6} so that the observed charge is g_{i}n_{i}e, where e is the charge of an electron. Each PMT must be associated with a calibration constant k_{i} in units of MeV per electron charge, such that G_{i} = k_{i}g_{i}. In practice, there is no need to know g_{i} and k_{i} separately, so we discuss only the single “gain” factor G_{i}.

where σ_{G} is the typical uncertainty in the PMT gains G_{i}, and we use the estimate σ_{n_{i}} = \sqrt{n_{i}} for the fluctuations in the number of observed photoelectrons.^{2} The relative uncertainty in the signal S due only to the uncertainty σ_{G} in the PMT gains is,

\frac{σ_{S}}{S} |_{Gain} = \frac{\sqrt{∑_{i=1}^{N} σ_{G_{i}}^{2}n_{i}^{2}}}{∑_{i=1}^{N} G_{i}n_{i}} ≈ \frac{1}{\sqrt{N}} \frac{σ_{G}}{G} \sqrt{1 + \frac{3}{E(MeV)}} ≈ 0.08\frac{σ_{G}}{G} ,  (5)

while the relative uncertainty in S due to fluctuations in the numbers of photoelectrons is,

\frac{σ_{S}}{S} |_{statistics} = \frac{\sqrt{∑_{i=1}^{N} G_{i}^{2}σ_{n_{i}}^{2}}}{∑_{i=1}^{N} G_{i}n_{i}} ≈ \sqrt{\frac{3}{NE(MeV)}} ≈ \frac{0.12}{\sqrt{E(MeV)}} .  (6)

The energy resolution of the detector is 12%/\sqrt{E(MeV)} according to eq. (6), assuming that the gains of the PMTs are well known. For example, at the 8-MeV peak in the scintillation spectrum from neutron capture on Gd, the energy resolution can be σ_{S}/S = 0.042, provided that the uncertainty (5) due to uncertainty in the PMT gains is small compared to this. However, this consideration leads only to the weak requirement that σ_{G}/G < ∼ 0.5.

A stronger requirement is that the uncertainty in the scintillation energy scale be less than 1% at the 6-MeV cut on the neutron-capture spectrum shown on p. 1. From eq. (5), we deduce the requirement that,

\frac{σ_{G}}{G} < ∼ 0.1  (7)

for each of the 8 × 224 = 1792 PMTs in the antineutrino detectors.^{3}

2 PMT Calibration via Neutron Capture on Gd Nuclei

One way to calibrate the PMT gains G_{i} is via observation of thermal neutron capture on Gd nuclei in the liquid scintillator, either from localized neutrons sources, of from the uniform spatial distribution of inverse β-decay due to reactor antineutrinos.

Either way, we must have additional knowledge of the expected spatial variation of the signal S_{i}(x) in PMT i due to a neutron emitted (or created) at location x, such that the total signal,

S = ∑_{i=1}^{N} S_{i}(x),  (8)

is independent of x. The design of the antineutrino detector includes an oil buffer around the PMTs so that the expected signal S_{i}(x) in PMT i is a weak function of x, and we can rely on a Monte Carlo simulation for a calculation of S_{i}(x).

2Equation (5) includes the approximation that ∑_{i=1}^{N} n_{i}^{2} = N ⟨n_{i}^{2}⟩ = N(⟨n_{i}⟩^{2} + σ_{n_{i}}^{2}) ≈ N(⟨n_{i}⟩^{2} + ⟨n_{i}⟩).

Thanks to L. Littenberg for pointing this out.

3The requirement (7) is based on the assumption that the signal is roughly equal in all 224 PMTs of a detector. For a neutron capture close to a particular PMT, the signal in that PMT will be larger than that of eq. (2) by a factor k. In the approximation that this PMT is the only one whose signal departs from eq. (2), the resulting modification to eq. (5) is (σ_{S}/S)_{Gain} ≈ (1/\sqrt{N})(σ_{G}/G)(1 + k^{2}/2N). For example, k = 5 results in a requirement that is 5% stronger than eq. (7).

If reactor antineutrinos are used as the source of thermal neutron captures, then the location x of the capture must first be reconstructed from the PMT data before the S_{i}(x) can be calculated. Simulations indicate that the spatial resolution in the reconstruction of the location of the neutron capture will be σ_{x} ≈ 15 cm, which is small compared to the length scale of variation of S_{i}(x).

Suppose that a total of M neutron captures are observed at known or reconstructed positions x_{j}, j = 1, ...M, and that S_{ij} = S_{i}(x_{j}) are the expected energy signals in PMT i. The actual signal associated with PMT i is G_{i}n_{ij}, where n_{ij} is the number of photoelectrons observed in PMT i during event j. To extract the gains G_{i} from this data we can form a χ^{2} for each PMT,

χ_{i}^{2} = ∑_{j=1}^{M} \frac{(S_{ij} − G_{i}n_{ij})^{2}}{σ_{ij}^{2}} = ∑_{j=1}^{M} \frac{(S_{ij} − G_{i}n_{ij})^{2}}{G_{i}^{2}n_{ij}} = \frac{1}{G_{i}^{2}} ∑_{j=1}^{M} \frac{S_{ij}^{2}}{n_{ij}} − \frac{2}{G_{i}} ∑_{j=1}^{M} S_{ij} + ∑_{j=1}^{M} n_{ij}. (9)

The needed derivatives are,

\frac{dχ_{i}^{2}/2}{dG_{i}} = −\frac{1}{G_{i}^{3}} ∑_{j=1}^{M} \frac{S_{ij}^{2}}{n_{ij}} + \frac{1}{G_{i}^{2}} ∑_{j=1}^{M} S_{ij},  (10)

and,

\frac{d^{2}χ_{i}^{2}/2}{dG_{i}^{2}} = \frac{3}{G_{i}^{4}} ∑_{j=1}^{M} \frac{S_{ij}^{2}}{n_{ij}} − \frac{2}{G_{i}^{3}} ∑_{j=1}^{M} S_{ij}.  (11)

Setting the first derivative, eq. (10), to zero, we obtain the best estimate of the gain G_{i},

G_{i} = \frac{∑{M \atop j=1} \frac{S_{ij}^{2}}{n_{ij}}}{∑_{j=1}^{M} S_{ij}} .  (12)

Using the fit value for G_{i} in eq. (11), we obtain the estimate of the uncertainty σ_{G_{i}},

\frac{1}{σ_{G_{i}}^{2}} = \frac{d^{2}χ_{i}^{2}/2}{dG_{i}^{2}} = \frac{(∑_{j=1}^{M} S_{ij})^{4}}{(∑{M \atop j=1} \frac{S_{ij}^{2}}{n_{ij}} )3} ,  (13)

so that,

\frac{σ_{G_{i}}}{G_{i}} = \frac{\sqrt{∑{M \atop j=1} \frac{S_{ij}^{2}}{n_{ij}}}}{∑_{j=1}^{M} S_{ij}} ≈ \frac{1}{\sqrt{Mn_{i}}} ≈ \sqrt{\frac{3}{ME(MeV)}} = \frac{0.61}{\sqrt{M}} ,  (14)

where the last form holds for thermal neutron capture on Gd with E = 8 MeV. A calibration of the gains to accuracy σ_{G_{i}}/G_{i} = 0.1 can be obtained with only M = 37 events.

The expected rate of neutron captures from antineutrino interactions in a single detector module at the Far site is 90 per day. Hence, a calibration of the gains of all 224 PMTs in each module can be obtained every 10 hours to the desired accuracy of σ_{G_{i}}/G_{i} = 0.1. This calibration depends on the reconstruction of the position x_{j} of each neutron capture, and reliable modeling of the spatial dependence S_{ij} = S_{i}(x_{j}) of the expected signal in PMT i. Also, the data used in this calibration will include a few percent of background events with nearly flat energy distribution in the needed window around the 8-MeV neutron-capture peak.

We can also perform occasional calibrations in which a neutron source is lowered into the detector and placed at a fixed location. These calibrations will have a much smaller background component than those based on reactor antineutrinos, as well as a different dependence on the model for the PMT signals S_{i}(x_{j}), and will provide a cross check on the continual energy calibration based on the antineutrino data itself.