Neutrino Oscillation Measurement
7.1 Signal prediction
For a simplified case with only one reactor and one detector, the total number of IBD events observed can be expressed as:
NIBD =
N Bins
X
i
σi· φi· 1
4πl2 · t · NP · fH · ε, (7.1) where σ is the IBD reaction cross section; φ is the neutrino flux from the reactor which is energy dependent and suppressed by the baseline, l, summarized in Table 7.3; t is the data-taking time, shown in Table 7.1; NP is the number of protons; fH is the fraction of captures on hydrogen and ε is the detection efficiency.
The neutron capture on hydrogen process can occur in GdLS, LS and Acrylic regions.
AD1AD2AD3AD4AD5AD6AD7AD8 Fulltime(d)565.4361565.4361568.0185568.0185562.4139562.4139562.4139562.4139 Livetime(d)449.4485447.8241473.3918483.4036551.9756551.9621551.8261553.1068 Rµ(Hz)200.3232200.3233150.0765150.086615.747815.747915.747815.7478 Veto(d)115.99117.6194.6384.6110.4410.4510.599.31 εµ0.79490.79200.83340.85100.98140.98140.98120.9835 εm0.98440.98450.98460.99080.98440.98410.98390.9912 εµεm0.78250.77970.82060.84320.96610.96580.96540.9748 Candidates217613±466219721±469208606±457136718±37056880±23856106±23759230±24338037±195 Acc.Bkg.26227.61±49.3725736.30±48.4725318.10±42.8716364.88±28.5129894.74±21.3630039.52±21.5532206.76±22.7620430.99±16.25 RACC(1/d)59.28±0.1158.37±0.1154.32±0.0934.17±0.0655.02±0.0455.30±0.0459.32±0.0437.27±0.03 Can.-Acc.191385±469193985±471183288±459120353±37126985±25526066±25327023±26017606±204 RFastN(1/d)2.18±0.762.18±0.762.02±1.031.32±0.670.14±0.060.14±0.060.14±0.060.09±0.04 RLi9(1/d)1.70±0.731.70±0.731.46±0.720.95±0.470.15±0.050.15±0.050.15±0.050.10±0.03 RAmC(1/d)0.08±0.040.08±0.040.08±0.040.05±0.030.03±0.020.03±0.020.03±0.020.02±0.01 IBD189632±662192238±662181631±744119242±54026809±25925892±25626849±26417492±206 RIBD(1/d)428.61±1.50436.03±1.50389.69±1.60248.96±1.1349.34±0.4547.67±0.4549.45±0.4631.91±0.36
Table 7.1: Summary of the hydrogen capture data sample.
Date
12-31 03-02 05-01 07-01 08-31 10-31 12-31 03-02 05-02 07-01 08-31 10-31 12-31
IBD Rate [day]
Figure 7.1: IBD rate for each detector from Dec 24, 2011 to Nov 30, 2013.
GdLS LS Acrylic
Capture fraction 0.1502 0.9580 0.0288
εep 0.9518 ± 0.0128(cor.) ± 0.0048 (unc.) 0.9297 ± 0.0145(cor.) ± 0.0046 (unc.) εed 0.9695 ± 0.0018(cor.) ± 0.0048 (unc.) 0.6799 ± 0.0023(cor.) ± 0.0034 (unc.) εT 0.9889 ± 0.0014 0.8403 ± 0.0014
εd 0.7527 ± 0.004 0.0046
Table 7.2: Summary of the efficiencies.
The fraction of neutron captured by hydrogen and the detection efficiency in the above three zones are different. Therefore, the above formula is changed into
NIBD =
For the real case, there are 6 reactors which contribute to IBD events detected by each detector. Hence the predicted number of IBD events can be written as
Npredict =
Reactors
X
r
NIBD,r. (7.3)
Combining Eq (6.1) and Eq (7.1), (7.2) and (7.4), the number of IBD events is
AD 1 362.38 371.763 903.466 817.158 1353.62 1265.32 AD 2 357.94 368.414 903.347 816.896 1354.23 1265.89 AD 3 1332.48 1358.15 467.574 489.577 557.579 499.207 AD 4 1337.43 1362.88 472.971 495.346 558.707 501.071 AD 5 1919.63 1894.34 1533.18 1533.63 1551.38 1524.94 AD 6 1917.52 1891.98 1534.92 1535.03 1554.77 1528.05 AD 7 1925.26 1899.86 1538.93 1539.47 1556.34 1530.08 AD 8 1923.15 1897.51 1540.67 1540.87 1559.72 1533.18
Table 7.3: The baseline [m] between ADs and reactor cores. Reactors are grouped into three reactor complex: Daya Bay(D1,D2), Ling Ao I (L1,L2) and Ling Ao II(L3, L4)
fH, hydrogen capture fraction
Both of fH,GdLS and fH,LS are obtained by MC simulations. The former, fH,GdLS, is (15.63 ± 0.12)% and the later, fH,LS, is (95.81 ± 0.006)%, as shown in Table 6.1.
Cross section
The differential cross section of inverse beta decay process is from Ref. [34], dσ
f = 1, g = 1.26, ∆ = Mn− Mp, Ee(0) = Eν− ∆,
The cross section of inverse beta decay process is shown in Figure 7.2, which is given by integrating Eq. 7.5.
[MeV]
_e ν
E
0 1 2 3 4 5 6 7 8 9 10
]-2 IBD cross section [cm
0
Figure 7.2: The cross section of inverse beta decay.
Flux
Generally, a fresh fuel rod contains four main fissile nuclei, 235U, 238U, 239Pu and 241Pu.
On average, each fission releases about 200 MeV of energy and six neutrinos. For a typical reactor ( 3 GWth), it emits (5 ∼ 6) × 1020 antineutrinos per second.
In the real case, the neutrino flux from the reactor is time dependent. The instanta-neous anti-neutrino flux φ(Eν) for a reactor core is
φ(Eν) = Wth
where Wth is the reactor thermal power, for Daya Bay experiment is about 2.85 GWth, fi/F is the fractional contribution for each isotope, ei is the average energy released by each isotope in a fission, as shown in Table 7.4, and si(Eν) is the average energy spectrum from each isotope in a fission. There are many experimental groups to measure the anti-neutrino spectrum for each isotope, such as ILL [35] [36] [37], Bugey3 [38] and Huber [39].
For the theoretical calculation, Vogel and Mueller et al. [40] have a good prediction on
238U.
To precisely determine the neutrino flux, one needs the instantaneous fission fraction and the thermal power information. Such information is provided by the power plant company. During the analysis process, this information is blinded.
235U 238U 239Pu 241Pu
Fraction of Isotope 64 % 8 % 25 % 3 %
Energy per Fission [MeV] 201.92 ±0.46 205.52±0.96 209.99±0.60 213.60 ±0.65 Table 7.4: The fraction and energy per fission of each isotope for the fuel rod [41].
Number of target protons
The number of target protons is calculated by NT P = mtarget · NP, where mtarget is the target mass. Table 7.5 shows the target mass of GdLS and LS for each AD. These are recorded in Daya Bay Target Proton Service. During the liquid filling, there were measured by a Coriolis meter. NP is the proton density and defined by
NP = fH
mH · NA, (7.9)
where NAis Avogadro’s Number, fH is the hydrogen mass fraction and mH = 1.00794 g/mol is the average proton mass.
mtarget,GdLS NT P,GdLS mtarget,LS NT P,LS
AD1 19941±0.9 1.42957029e30 21573.5±42 1.53517025e30 AD2 19966±0.9 1.43136254e30 21519.6±42 1.53133474e30 AD3 19891± 3 1.42584241e30 21587.2±42 1.53614515e30 AD4 19944± 1.42978536e30 21499.9± 1.52637488e30 AD5 19913±1 1.42756297e30 21566.2±42 1.53465079e30 AD6 19991±1.9 1.43301141e30 21408.8±42 1.52345020e30 AD7 19892±0.9 1.42598579e30 21652.6±42 1.54079902e30 AD8 19931 1.42885339e30 21524.5 1.52812541e30
Table 7.5: The target mass [kg] for each AD.
Volumn Proton density (1025protons/kg)
GdLS 7.169±0.034
LS 7.116±0.043
Acrylic 4.78
Table 7.6: Proton density.
7.2 Extraction of θ
13The χ2 function with the correlated systematic errors is quoted from Ref. [23] and the form is where Md are the measured IBD candidates of the dth AD with accidental background subtracted, Bd is the sum of other background, σMd is the corresponding error for Md, σdB error for Bd, Td is the prediction from neutrino flux, proton number, cross section, effi-ciency, and neutrino oscillation parameter etc. σr is the uncorrelated reactor uncertainty, σdis the uncorrelated uncertainty of the detection efficiency which is the quadratic sum of all uncorrelated uncertainties of the efficiency. The parameter σBd is the quadratic sum of the background uncertainties. The corresponding pull parameters are (αr,εdand ηd). The detector- and reactor-related correlated uncertainties were not included in the analysis.
The absolute normalization ε was determined from the fit to the data. The parameter ωdr is the fraction of IBD event contribution from the rth reactor to the dth AD determined by baselines and reactor fluxes. In all these parameters, only σdM and σd are larger than the nGd analysis. σdM is limited by the accidental background and statistics. σd may be
Figure 7.3: Top: The detected prompt energy spectrum by the far hall ADs (blue) and the expected prompt energy spectrum from extrapolating the near hall measurements without the effect of neutrino oscillation (open circle). Bottom: The near-to-far ratio with best fit θ13. Insert: The ratio of the measured to the predicted rates in each AD.
improved by further studies.
Results
Using the analysis package, M IN U IT , to fit this model with the six-detector period (Dec 24, 2011 to July 28, 2012), the best-fit value for θ13 is
sin22θ13= 0.083 ± 0.018, χ2/ndf = 4.6/4. (7.11) where |∆m231| is taken from MINOS [18]. The near-to-far ratio with the best fit θ13 is shown in Figure 7.3.
In nH fit, if only statistical uncertainties are considered, the uncertainty of sin22θ13 in nH analysis is 0.015 which is the same as the neutron captured by Gd analysis; if one considers all uncoupled errors with respect to nGd analysis, the uncertainty of sin22θ13 is 0.017.
The mixing angle measurement by combining nH and nGd analyses is
sin22θ13 = 0.089 ± 0.008. (7.12)
By using the data from Dec 24, 2011 to Nov 30, 2013 and the updated background estimation, the preliminary result for the best-fit value of θ13 is
sin22θ13= 0.0707 ± 0.0113, χ2/ndf = 5.2/4. (7.13) where |∆m231| is taken from MINOS [18].
Chapter 8 Conclusion
With the optimized baseline and functionally identical detectors and the extensive analysis efforts, the mixing angle θ13 has been measured precisely at Daya Bay experiment.
This study used different and independent methods to measure the mixing angle θ13 with the neutron captured on hydrogen signals. The clear electron anti-neutrino disap-pearance is observed! The best-fit value θ13 mixing angle is
sin22θ13= 0.083 ± 0.018, χ2/ndf = 4.6/4. (8.1) where |∆m231| is taken from MINOS [18].
This results gives a strong confidence for the earlier results from Daya Bay.
The mixing angle measurement is improved by combining nH and nGd measurements, which gives
sin22θ13 = 0.089 ± 0.008. (8.2)
Results with more statistics and the prediction on the future sensitivity With more statistics ( data set from Dec 24, 2011 to Nov 30, 2013) and the updated background estimation, the preliminary result for the best-fit value of θ13 is
sin22θ13= 0.0707 ± 0.0113, χ2/ndf = 5.2/4. (8.3) where |∆m231| is taken from MINOS [18].
The current error for the Daya Bay results is dominated by the statistical uncertainty.
Assuming the reactor thermal power is 2.85 GWth and all the efficiencies are the same as
Experiment time(day)
0 200 400 600 800 1000 1200 1400
Sensitivity (90% C.L.)
0.0089 0.0115 0.0141 0.0167 0.0193 0.022 0.0246 0.0272 0.0298 0.0324 0.035 0.0377 0.0403
6AD to 8AD 6AD only 8AD only
Figure 8.1: Projected sensitivity to sin22θ13.
before, Figure 8.1 shows the projected uncertainty.
Future plan
As it is discussed in Chapter 3, the energy distribution depends on the vertex. The energy leakage is very serious in LS region. It is worth understanding the non-uniformity in the LS region. On the other hand, the energy non-linearity also affects the energy cuts and impacts on the number of IBD events. The analysis of the missing angle θ13 based on the non-linearity and non-uniformity is knows as “spectral measurement” or “shape anslysis”.
The Daya Bay collaboration has published results on the spectral measurement with nGd signals [42]. The spectral measurement with nH signals is on the progress.
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