Precision Measurement of Neutrino Oscillation Parameters and Investigation of Nuclear Georeactor Hypothesis with KamLAND Citation Zhang, Chao (2011) Precision Measurement of Neutrino Oscillation Parameters and Investigation of Nuclear Georeactor Hypothesis with KamLAND. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ME5M-F406. https://resolver.caltech.edu/CaltechTHESIS:06102010-171034096 Abstract A combined analysis of examining the neutrino oscillation parameters and investigation of nuclear georeactor hypothesis with the KamLAND experiement is presented. With a total exposure of 2.75 kton-years, 930 anti-electron-neutrino candidate events above 3.4 MeV neutrino energy threshold were detected, with estimated 109±13 events from backgrounds. Assuming CPT invariance by combining with solar neutrino results, the best-fit value of georeactor fission power is 4.9 +3.8 -4.8 TW. The 90% upper limit on the georeactor power is determined to be 11.2 TW. This result has put a significant constraint on the contribution of a possible georeactor to the total heat from the Earth. The best-fit values of the neutrino oscillation parameters, including the georeactor power as a free parameter, is consistent with KamLAND's previously published results with null-georeactor assumption. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: KamLAND; Georeactor; Neutrino Oscillation Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): McKeown, Robert D. Thesis Committee: McKeown, Robert D. (chair) Filippone, Bradley W. Golwala, Sunil Vogel, Petr Defense Date: 10 June 2010 Record Number: CaltechTHESIS:06102010-171034096 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06102010-171034096 DOI: 10.7907/ME5M-F406 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5946 Collection: CaltechTHESIS Deposited By: Chao Zhang Deposited On: 14 Sep 2010 21:14 Last Modified: 09 Oct 2019 17:05 Thesis Files Preview PDF (Complete Thesis) - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

Precision Measurement of Neutrino Oscillation Parameters and Investigation of Nuclear Georeactor Hypothesis with KamLAND Thesis by Chao Zhang In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2011 (Defended June 10, 2010) ii c 2011 ° Chao Zhang All Rights Reserved iii Acknowledgments I have a lot of people to thank without whom I would not have made this far. I am deeply grateful to my advisor Bob McKeown who has supported me over the past years and brought me into the KamLAND project. His patience, encouragement and wisdom have been guiding me through my Ph.D. study. I owe my special thanks to Dan Dwyer for his constant advice and guidance to make this thesis analysis even possible. His knowledge and diligence always set an example for me. I am greatly thankful to Christopher Mauger who has been working together with me since the first day I joined KamLAND, taught me all the techniques in experimental nuclear physics, and helped me go though all the difficulties living in a foreign country, both in the United States and in Japan. I would like to thank my fellow graduate students for all the days we spent together to get the work done. Tommy O’Donnell deserves my many thanks for the countless discussions we had about the analysis. His encouragement is essential to keep me moving on. Greg Keefer and Chris Grant are my kind roommates during my long stays in Japan. Their presence is the only thing that kept me sane. I would like to thank Kyohei Nakajima, Yuri Shimizu and Yukie Minekawa for always being my accompany in the underground mine, for making my life in Mozumi bearable. I would like to thank all my KamLAND collaborators for their hard work. I thank Atsuto Suziki, Kunio Inoue and Stuart Freedman for leading this experiment to such a success today. Many thanks to the people who have lent me valuable knowledges about KamLAND: Brian Fujikawa, Patrick Decowski, Jason Detwiler, Lindley Winslow, Bruce Berger, Glenn Horton-Smith, Itaru Shimizu, Koichi Ichimura, Yasuhiro Kishimoto, Sei Yoshida, Yoshihito Gando, Tadao Mitsui and many others. I iv would like to specially thank Masayuki Koga and Kengo Nakamura for their guidance during my work in Japan and for introducing me to the wonderful Japanese food and culture. I would like to thank Bob Carr, Jianglai Liu, Bei Cai, Fenfang Wu and the rest of Kellogg research group for their daily help and for bearing with me for so long. Special thanks goes to Jianglai Liu who read my thesis and provided valuable feedback and discussion. The ‘Ernie’s Lunch Time’ with the group is the most enjoyable time during my everyday life at CalTech. Finally I would like to thank my parents Mengqiang Zhang and Renzhu Shen and my brother Gang Shen. No words can describe their love and endless support to me throughout my life for which I owe my deepest thanks. v Abstract A combined analysis of examining the neutrino oscillation parameters and investigation of nuclear georeactor hypothesis with the KamLAND experiement is presented. With a total exposure of 2.75 kton-years, 930 ν̄e candidate events above 3.4 MeV ν̄e energy threshold were detected, with estimated 109±13 events from backgrounds. Assuming CP T invariance by combining with solar neutrino results, the best-fit value of georeactor fission power is 4.9+3.8 −4.8 TW. The 90% upper limit on the georeactor power is determined to be 11.2 TW using Feldman and Cousins’ approach. This result has put a significant constraint on the contribution of a possible georeactor to the total heat from the Earth. The best-fit values of the neutrino oscillation parameters, including the georeactor power as a free parameter, are tan2 θ12 = 0.44+0.05 −0.05 and −5 ∆m221 = 7.59+0.27 eV2 . All oscillation parameter spaces are excluded at 99.73% −0.27 ×10 C.L. except the LMA-I region. The result is consistent with KamLAND’s previously published results with null-georeactor assumption. vi Contents Acknowledgments iii Abstract v 1 Introduction 1 1.1 1.2 1.3 1.4 Introduction to Neutrino Oscillation Theory . . . . . . . . . . . . . . 1 1.1.1 General Properties of the Neutrinos . . . . . . . . . . . . . . . 2 1.1.2 Neutrino Oscillation in Vacuum . . . . . . . . . . . . . . . . . 10 1.1.3 Neutrino Oscillation in Matter . . . . . . . . . . . . . . . . . . 11 History of Neutrino Oscillation Experiments . . . . . . . . . . . . . . 13 1.2.1 Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . . 13 1.2.2 Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.3 Reactor Neutrinos 28 . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Nuclear Georeactor Hypothesis . . . . . . . . . . . . 34 1.3.1 Nuclear Earth Model and Substructure of the Inner Core . . . 34 1.3.2 Georeactor Feasibility . . . . . . . . . . . . . . . . . . . . . . 36 1.3.3 Georeactor Implication . . . . . . . . . . . . . . . . . . . . . . 40 Motivation for a Combined Analysis . . . . . . . . . . . . . . . . . . . 42 2 The KamLAND Experiment 44 2.1 Anti-neutrino Detection Method . . . . . . . . . . . . . . . . . . . . . 44 2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.1 Detector Design Overview . . . . . . . . . . . . . . . . . . . . 49 2.2.2 Detector Properties . . . . . . . . . . . . . . . . . . . . . . . . 49 vii 2.3 2.2.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.4 Calibration System . . . . . . . . . . . . . . . . . . . . . . . . 58 KamLAND History and Status . . . . . . . . . . . . . . . . . . . . . 60 2.3.1 Achievements in Reactor Phase . . . . . . . . . . . . . . . . . 60 2.3.2 Solar Phase and Purification of Liquid Scintillator . . . . . . . 65 3 Event Reconstruction 70 3.1 Event Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Energy Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4.3 Detector Energy Response . . . . . . . . . . . . . . . . . . . . 83 Muon Track Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 90 3.4 3.5 4 Anti-neutrino Candidate Selection 92 4.1 Run Selection and Live Time . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Reconstruction Quality . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Muon Spallation Cut and Live Time Correction . . . . . . . . . . . . 96 4.4 Fiducial Volume Cut and Number of Target Protons . . . . . . . . . 98 4.5 Prompt and Delayed Energy Cut . . . . . . . . . . . . . . . . . . . . 107 4.6 Time Correlation Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.7 Spacial Correlation Cut . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.8 Probability Density Estimator (PDE) Cut . . . . . . . . . . . . . . . 114 4.8.1 Accidental Coincidence Background . . . . . . . . . . . . . . . 114 viii 4.8.2 4.9 PDE Cut Definition . . . . . . . . . . . . . . . . . . . . . . . 117 Summary of ν̄e Candidate Selection Efficiency . . . . . . . . . . . . . 122 4.10 Final Sample of ν̄e Candidate . . . . . . . . . . . . . . . . . . . . . . 123 5 Signals and Backgrounds 127 5.1 Anti-neutrino from Nuclear Reactor . . . . . . . . . . . . . . . . . . . 127 5.2 Anti-neutrino from Nuclear Georeactor . . . . . . . . . . . . . . . . . 132 5.3 Accidental Coincidence Background . . . . . . . . . . . . . . . . . . . 137 5.4 13 5.5 9 5.6 Fast Neutron Background . . . . . . . . . . . . . . . . . . . . . . . . 150 5.7 Atmospheric Neutrino Background . . . . . . . . . . . . . . . . . . . 153 5.8 Summary of Signals and Backgrounds . . . . . . . . . . . . . . . . . . 154 C(α,n)16 O Background . . . . . . . . . . . . . . . . . . . . . . . . . 138 Li Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6 Analysis and Results 156 6.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.2 Analysis Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.3 Best Fit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.4 Goodness-of-fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.5 Significance of Energy Spectrum Distortion . . . . . . . . . . . . . . . 168 6.6 Correlation with Reactor ν̄e Flux . . . . . . . . . . . . . . . . . . . . 169 6.7 Results on Georeactor Fission Power . . . . . . . . . . . . . . . . . . 173 6.8 Results on Neutrino Oscillation Parameters . . . . . . . . . . . . . . . 179 7 Conclusions 183 A KamLAND Kr/Ar Monitor 185 A.1 Motivation and Methodology . . . . . . . . . . . . . . . . . . . . . . . 185 A.2 Detector Design and Operation . . . . . . . . . . . . . . . . . . . . . 187 ix A.3 On-site Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B Decay Chains of Selected Isotopes 197 C Event Category 201 C.1 Low Level Event Classification . . . . . . . . . . . . . . . . . . . . . . 201 C.2 Radioactive Calibration Source Events . . . . . . . . . . . . . . . . . 204 C.3 Alpha Particle Events . . . . . . . . . . . . . . . . . . . . . . . . . . 205 C.3.1 214 Po Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . . 206 C.3.2 212 Po Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . . 208 C.3.3 210 Po Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . . 210 C.4 Muon Spallation Events . . . . . . . . . . . . . . . . . . . . . . . . . 210 C.4.1 Spallation Neutron . . . . . . . . . . . . . . . . . . . . . . . . 211 C.4.2 Spallation 12 B . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 C.4.3 Spallation 9 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Bibliography 217 x List of Tables 1.1 Summary of the pp Chain in the Sun . . . . . . . . . . . . . . . . . . 19 1.2 Summary of Simulation of Georeactor . . . . . . . . . . . . . . . . . . 39 2.1 Radioactivities of Selected Materials in KamLAND . . . . . . . . . . 51 2.2 Summary of Radioactive Calibration Sources at KamLAND . . . . . 59 2.3 Purification Goal for KamLAND Solar Phase . . . . . . . . . . . . . . 67 3.1 Summary of Vertex Reconstruction Status . . . . . . . . . . . . . . . 77 3.2 Data Points for Estimating Energy Scale Parameters . . . . . . . . . 87 3.3 Best-fit Energy Scale Parameters . . . . . . . . . . . . . . . . . . . . 87 4.1 Summary of Live Time . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Summary of Fiducial Volume Measurements from 4π Calibrations . . 101 4.3 Summary of Uncertainties on Target Protons . . . . . . . . . . . . . . 107 4.4 Target Nuclei for (n,γ) Reaction in KamLAND . . . . . . . . . . . . 110 4.5 Summary of ν̄e Selection Efficiency . . . . . . . . . . . . . . . . . . . 122 5.1 Summary of Japanese and Korean Reactors . . . . . . . . . . . . . . 128 5.2 Summary of Reactor Related Systematic Uncertainties . . . . . . . . 133 5.3 Target Nuclei for (α,n) Reaction in KamLAND . . . . . . . . . . . . 138 5.4 Summary of Signal and Background Events . . . . . . . . . . . . . . 155 5.5 Summary of Systematic Uncertainties . . . . . . . . . . . . . . . . . . 155 6.1 Summary of Model Parameters . . . . . . . . . . . . . . . . . . . . . 157 6.2 Summary of Best-fit Values of Free Model Parameters . . . . . . . . . 163 A.1 Characteristic Properties of 85 Kr and 39 Ar . . . . . . . . . . . . . . . 186 xi A.2 Systematic Uncertainties of KamKAM . . . . . . . . . . . . . . . . . 195 B.1 238 U Decay Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 B.2 232 Th Decay Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 B.3 40 K Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 B.4 85 Kr Decay Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 B.5 39 Ar Decay Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 xii List of Figures 1.1 Flavor Composition and Mass Splitting of Neutrinos . . . . . . . . . . 5 1.2 Neutrino Flux at the Surface of the Earth . . . . . . . . . . . . . . . 7 1.3 Regions of Neutrino Oscillation Parameters . . . . . . . . . . . . . . . 14 1.4 Atmospheric Neutrinos Observed at Super-Kamiokande . . . . . . . . 16 1.5 Regions of Atmospheric Neutrino Oscillation Parameters . . . . . . . 18 1.6 Solar Neutrino Energy Spectrum . . . . . . . . . . . . . . . . . . . . 20 1.7 Solar Neutrino measured at Super-Kamiokande . . . . . . . . . . . . 23 1.8 SNO Measured 8 B Solar Neutrino Flux . . . . . . . . . . . . . . . . . 24 1.9 Parameter Space Determined by CHOOZ, Palo Verde and SK . . . . 30 1.10 Location of Nuclear Reactors in the World . . . . . . . . . . . . . . . 31 1.11 ν̄e Survival Probability Measured by Reactor Experiments . . . . . . 32 1.12 Spectrum Distortion Observed by KamLAND . . . . . . . . . . . . . 33 1.13 Mass Ratio of Various Chondrites . . . . . . . . . . . . . . . . . . . . 36 1.14 Sub-core Structure of the Earth in the Nuclear Earth Model . . . . . 37 1.15 Illustration of the Nuclear Earth Model . . . . . . . . . . . . . . . . . 38 2.1 Illustration of the ν̄e Signals in KamLAND . . . . . . . . . . . . . . . 45 2.2 The KamLAND Site . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 KamLAND Detector Schematics . . . . . . . . . . . . . . . . . . . . . 48 2.4 KamLAND DAQ Schematics . . . . . . . . . . . . . . . . . . . . . . . 54 2.5 Evidence of Neutrino Oscillation in KamLAND . . . . . . . . . . . . 61 2.6 Allowed Region of Neutrino Oscillation Parameters in KamLAND . . 62 2.7 Geo-neutrinos in KamLAND . . . . . . . . . . . . . . . . . . . . . . . 64 2.8 Energy Spectrum of Singles Events in KamLAND Reactor Phase . . . 66 xiii 2.9 85 Kr and 210 Bi Distribution in KamLAND after the First Purification 68 3.1 Schematic Diagram of Event Reconstruction Process . . . . . . . . . 71 3.2 WaveForm Analysis Example . . . . . . . . . . . . . . . . . . . . . . 73 3.3 Waveform from LS Muon . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4 Vertex Reconstruction Performance . . . . . . . . . . . . . . . . . . . 78 3.5 Energy Reconstruction Performance . . . . . . . . . . . . . . . . . . . 82 3.6 Light Yield Before and After Purification . . . . . . . . . . . . . . . . 83 3.7 Energy Reconstruction Bias for Spallation Neutron Events . . . . . . 84 3.8 Time Variation of Energy Scale from Calibration Sources . . . . . . . 85 3.9 Energy Scale Calibration in KamLAND . . . . . . . . . . . . . . . . . 88 3.10 Energy Resolution Before and After Purification . . . . . . . . . . . . 89 3.11 Light Yield of Muon . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1 Reconstruction Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Trigger Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Live Time Fraction After Spallation Cut . . . . . . . . . . . . . . . . 99 4.4 Vertex Distribution of Singles Events in KamLAND . . . . . . . . . . 100 4.5 Illustration of the 4π Calibration System . . . . . . . . . . . . . . . . 101 4.6 Radial Distribution of 4π Calibration Sources . . . . . . . . . . . . . 102 4.7 Spacial Distribution of the Spallation 12 B Events 4.8 Fiducial Volume Ratio vs. Energy from Spallation 12 B Events . . . . 105 4.9 ν̄e Energy Spectrum from Natural Radioactivity In the Earth . . . . . 108 . . . . . . . . . . . 104 4.10 Energy Spectrum of Neutron Capture on 1 H . . . . . . . . . . . . . . 109 4.11 Distribution of Neutron Capture Time from Spallation Neutron Events 111 4.12 Spacial Correlation of Prompt and Delayed Events from Simulation . 113 4.13 Singles Event Rate vs. Run . . . . . . . . . . . . . . . . . . . . . . . 115 4.14 Properties of Accidental Backgrounds . . . . . . . . . . . . . . . . . 116 xiv 4.15 Pdf’s for ν̄e Signal and Accidental Background Events . . . . . . . . . 118 4.16 PDE Distribution for Signal and Background Events . . . . . . . . . 119 4.17 Figure of Merit for PDE cut . . . . . . . . . . . . . . . . . . . . . . . 121 4.18 Radial Dependence of PDE Cut Efficiency . . . . . . . . . . . . . . . 122 4.19 Vertex Distributions of ν̄e Candidate Events . . . . . . . . . . . . . . 124 4.20 Energy Distribution of ν̄e Candidate Events . . . . . . . . . . . . . . 125 4.21 Distribution of Neutron Capture Time from ν̄e Candidates . . . . . . 126 5.1 ν̄e Flux at KamLAND v.s. Baseline . . . . . . . . . . . . . . . . . . . 129 5.2 ν̄e Energy Spectra From Fission Isotopes . . . . . . . . . . . . . . . . 131 5.3 Time Variation of Reactor ν̄e Flux at KamLAND . . . . . . . . . . . 132 5.4 Expected Prompt Visible Energy Spectrum . . . . . . . . . . . . . . . 133 5.5 Prompt Energy Spectrum of Georeactor ν̄e . . . . . . . . . . . . . . . 134 5.6 Oscillation of Georeactor ν̄e . . . . . . . . . . . . . . . . . . . . . . . 135 5.7 Prompt Energy Spectrum of Georeactor and Man-made reactors . . . 136 5.8 Energy Spectra of Accidental Backgrounds . . . . . . . . . . . . . . . 137 5.9 Nsummax Distribution of 210 Po Events . . . . . . . . . . . . . . . . . 140 5.10 Time Dependence of 210 Po Decay Rate . . . . . . . . . . . . . . . . . 141 5.11 Radial Distribution of 210 Po Activities . . . . . . . . . . . . . . . . . 141 5.12 Cross Section of 13 C(α,n)16 O Reaction . . . . . . . . . . . . . . . . . 143 5.13 dE/dX of α particles in KamLAND . . . . . . . . . . . . . . . . . . . 144 5.14 Differential Cross Section of 13 C(α,n)16 O Reaction . . . . . . . . . . . 144 5.15 Neutron Energy Spectrum from 13 C(α,n)16 O Reaction . . . . . . . . . 145 5.16 Visible Prompt Energy Spectrum of 13 C(α,n)16 O Background . . . . . 146 5.17 Time Between 9 Li Events and Associated Muons . . . . . . . . . . . . 147 5.18 Distance Between 12 B Events and Associated Muon Track . . . . . . 149 5.19 Time between ν̄e Candidate Events and Previous Muons . . . . . . . 149 5.20 OD Inefficiency in Tagging ID Muons . . . . . . . . . . . . . . . . . . 151 xv 5.21 Energy Spectrum and Capture Time of Fast Neutron . . . . . . . . . 151 5.22 Fast Neutron Vertex Distribution . . . . . . . . . . . . . . . . . . . . 152 5.23 Energy Spectrum of Atmospheric Neutrino Background . . . . . . . . 154 6.1 SNO-only Contour Plot . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2 Best-fit Prompt Visible Energy Spectrum . . . . . . . . . . . . . . . . 164 6.3 Best-fit Time Variation of Event Rate . . . . . . . . . . . . . . . . . . 166 6.4 Goodness-of-fit of Energy Spectrum . . . . . . . . . . . . . . . . . . . 167 6.5 Goodness-of-fit of Time Variation Spectrum . . . . . . . . . . . . . . 167 6.6 Distortion of Prompt Energy Spectrum . . . . . . . . . . . . . . . . . 170 6.7 Goodness-of-fit of No-oscillation Analysis . . . . . . . . . . . . . . . . 171 6.8 Correlation with Reactor ν̄e Flux . . . . . . . . . . . . . . . . . . . . 172 6.9 Confidence Region of (Pgeo , tan2 θ12 ) . . . . . . . . . . . . . . . . . . 174 6.10 1-D χ2 Distribution of Pgeo . . . . . . . . . . . . . . . . . . . . . . . . 176 6.11 Period-I-only 1-D χ2 Distribution of Pgeo . . . . . . . . . . . . . . . . 178 6.12 Heat Balance of the Earth . . . . . . . . . . . . . . . . . . . . . . . . 178 6.13 Allowed Region of (∆m221 , tan2 θ12 ) in Null Georeactor Analysis . . . 180 6.14 Allowed Region of (∆m221 , tan2 θ12 ) in Combined Analysis . . . . . . . 182 A.1 KamKAM Detector Design . . . . . . . . . . . . . . . . . . . . . . . . 188 A.2 Picture of KamKAM Setup . . . . . . . . . . . . . . . . . . . . . . . 189 A.3 A Sample RGA Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 192 A.4 KamKAM Measurements and Calibration . . . . . . . . . . . . . . . 193 C.1 Low Level Event Classification . . . . . . . . . . . . . . . . . . . . . . 202 C.2 Reconstructed Energies of Calibration Sources . . . . . . . . . . . . . 205 C.3 Reconstructed vertices of Calibration Sources . . . . . . . . . . . . . 206 C.4 Energy Spectrum and Decay Time of 214 Po . . . . . . . . . . . . . . . 207 C.5 Reconstructed Energy Bias of 214 Po α Events . . . . . . . . . . . . . 207 xvi C.6 Energy Spectrum and Decay Time of 212 Po . . . . . . . . . . . . . . . 209 C.7 Reconstructed Energy Bias of 212 Po α Events . . . . . . . . . . . . . 209 C.8 Event Selection and Vertex Distribution of Spallation Neutron . . . . 212 C.9 Energy Spectrum and Capture Time of Spallation Neutron . . . . . . 213 C.10 Reconstructed Energy Bias of Spallation Neutron Capture on 1 H . . . 213 C.11 Reconstructed Energy Bias of Spallation Neutron Capture on 12 C . . 214 C.12 Energy Spectrum and Decay Time of Spallation 12 B . . . . . . . . . . 215 C.13 Energy Spectrum and Vertex Distribution of Spallation 9 Li . . . . . . 216 1 Chapter 1 Introduction KamLAND (Kamioka Liquid Scintillator Anti-Neutrino Detector) is the largest liquid scintillator detector built to study the neutrino oscillation phenomenon, in particular to explore the regions found by the solar neutrino experiments, but with anti-neutrinos produced by the nuclear reactors on the Earth. Meanwhile, a hypothetical nuclear georeactor [1] at the center of the Earth is also producing anti-neutrinos that will be detected by KamLAND. A combined analysis of georeactor and neutrino oscillation, performed in this dissertation, is not only going to investigate the georeactor hypothesis, but will also provide a stronger test on the current best-known neutrino oscillation theories. In this chapter we will first review the neutrino oscillation theory and past neutrino oscillation experiments, followed by an introduction of the nuclear georeactor theory and the motivation of a combined analysis. Details of the KamLAND experiment will be discussed in Chapter 2. Analysis procedures and results will be presented in Chapter 3, 4, 5 and 6. Finally we will state the conclusion in Chapter 7. 1.1 Introduction to Neutrino Oscillation Theory The existence of neutrinos was first proposed by Pauli [2] in 1930 to rescue the energy and angular momentum conservation in the beta decay process. It is not until 1956 that Reines and Cowan first discovered the (anti-)neutrinos νe in the experiment [3]. Subsequent experiments discovered two more types of neutrinos νµ [4] and ντ [5]. All three neutrinos are elementary particles in the standard model of electroweak interactions. The idea of the possible existence of neutrino oscillation was first introduced 2 by Pontecorvo [6], now supported by many strong experimental evidences since the late 1990s. 1.1.1 General Properties of the Neutrinos Neutrino Flavors Neutrinos are “elusive” particles that are neutral, do not feel the strong interactions and only interact weakly through coupling with W ± (charge current) and Z 0 (neutral current) bosons. The flavor of a neutrino is defined by the corresponding charged lepton that is connected to the same charge current vertex. Three flavors of neutrino have been discovered in the lab, together with their lepton partners they form the doublets of the group SU(2) in the standard model: µ ¶ νe , e− µ ¶ νµ , µ− µ ντ τ− ¶ (1.1) The number of light (mν < MZ /2) active neutrino species is studies by the e+ e− annihilation at the Z-resonance peak. The Z → ν ν̄ decay channel has partial width Γν ν̄ = 166.9 MeV, but with invisible final states. Subtracting the visible channels, interpretation of the measured Z-resonance width from the LEP experiments gives the total number of neutrino active flavors Nν = 2.984 ± 0.008 [7]. However sterile neutrinos which do not participate in the weak interactions might still exist. Neutrino Mass and Mixing The neutrinos with a definite flavor are not necessarily states of a definite mass. Similar to the Cabibbo-Kobayashi-Maskawa (CKM) mixing for the quark flavors in the weak charged current, the neutrino flavor states are in general superpositions of 3 their mass eigenstates. For 3 neutrino flavor case, it follows: |να i = 3 X i=1 ∗ Uαi |νi i. α = e, µ, τ (1.2) where the unitary mixing matrix Uαi is also called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. The PMNS matrix is conventionally parameterized by three mixing angles θ12 , θ13 , θ23 , one CP violating phase δ and two Majorana phases α1 , α2 , as follows:     −iδ 0 0   c13 0 s13 e  νe   1      ν  =  0 c  s23  0 1 0 23  µ        ντ 0 −s23 c23 −s13 eiδ 0 c13    iα1 /2  c12 s12 0   ν1 e      −s  iα2 /2  12 c12 0   ν2 e      0 0 1 ν3       (1.3) where the notations cij = cos θij and sij = sin θij are used. The subscripts of the three mixing angles are conventionally fixed by the associated experiments which measure them, and our current best knowledge is [7]: • sin2 (2θ12 ) = 0.87 ± 0.03 (solar) • sin2 (2θ23 ) > 0.92 (90% C.L.) (atmospheric) • sin2 (2θ13 ) < 0.19 (90% C.L.) (short-baseline reactor) Except θ13 , the other two mixing angles are measured to be large, which is interestingly very different from the small mixings of quark flavors as one finds in the CKM matrix [7]. The absolute mass of the neutrino mass eigenstate ν1 , ν2 , ν3 is still unknown. The best upper limit (mν < 2 eV [7]) comes from Tritium beta decay experiments, which 4 measure the effective anti-electron-neutrino mass as = m2(eff) νe X i |Uei |2 m2νi (1.4) It is worth mentioning that from the current cosmological data and some cosmological assumptions, it has been concluded [8, 9] that X i mi < (0.17 − 2.0) eV (1.5) where the sum is over all light neutrino mass eigenstates that may exist and were in thermal equilibrium in the early universe. Neutrino oscillation experiments, on the other hand, are able to measure the mass-squared differences between the mass eigenstates (Section 1.1.2). In the case of three neutrino mass eigenstates, there are only two independent ∆m2 . The current best-measured values are [7]: • ∆m221 = (7.59 ± 0.20) × 10−5 eV2 (long-baseline reactor, solar) • ∆m232 = (2.43 ± 0.13) × 10−3 eV2 (atmospheric, long-baseline accelerator) As one can see, the scales of the two ∆m2 differ much. Conventionally we assign the labels such that m2 > m1 . The low mass scale ∆m221 is associated with the solar neutrino oscillation (often called ∆m2sol ), while the high mass scale ∆m232 (or ∆m231 ) is associated with the atmospheric neutrino oscillation (often called ∆m2atm ). The sign of ∆m232 , however, is still unknown. One could either have m3 > m2 > m1 , in which case is similar to the quark sector, called “normal hierarchy,” or m2 > m1 > m3 , in which case is called “inverse hierarchy.” Future long baseline accelerator experiments are aiming to determine which one is the correct mass hierarchy, and at the same time to measure yet another unknown parameter, the CP violating phase δ. 5 Figure 1.1: The three-neutrino mass-squared spectrum that accounts for the current experimental results (in the normal hierarchy case). The three levels correspond to the mass eigenstate ν1 , ν2 , ν3 from bottom to top respectively (in the inverse hierarchy case, relative positions of ν3 and (ν1 , ν2 ) would exchange). The νe fraction of each mass eigenstate is crosshatched green, the νµ fraction is indicated by red right-leaning hatching, and the ντ fraction by blue left-leaning hatching. The absolute mass scale of the lowest mass eigenstate is yet unknown. (Figure taken from [7].) As a summary, our current knowledges of the flavor budget of each neutrino mass eigenstate and the mass splitting between them is graphically illustrated in Fig. 1.1 (for normal hierarchy case). There are about equal fractions of νµ and ντ in each mass eigenstate, determined by the atmospheric neutrino experiments. The νe fraction of ν2 is about 1/3, measured from the solar neutrino experiments. The νe fraction of ν3 is known to be small from the null-oscillation observations of the short-baseline reactor experiments, which then leaves about 2/3 for the νe fraction in ν1 . Dirac or Majorana Neutrino It has been observed in many experiments and demonstrated in the standard model that the neutrino has (almost) purely negative helicity, while the anti-neutrino has (almost) purely positive helicity. The amplitude of the “wrong” helicity component is suppressed by a factor of mν /E. This brings up an interesting question that whether or not the differences we have observed between neutrino and anti-neutrino simply come from the strong dynamical suppression (of order (mν /E)2 ) from the helicity, and that the neutrino and anti-neutrino are in fact the same particle. If this is the case the neutrinos are “Majorana” particles, otherwise they are “Dirac” particles, 6 same as the quarks and charged leptons. The two Majorana phases α1 , α2 introduced in Eqn. 1.3 are only non-zero if neutrinos are Majorana particles, otherwise they can be absorbed into the mass eigenstates by re-phasing. The most promising way to distinguish between Dirac and Majorana neutrinos is the neutrino-less double beta decay (0νββ) experiment through the following process: (Z, A) → (Z + 2, A) + 2e− (1.6) which obviously violates lepton number conservation and can only proceed if the neutrino is its own anti-particle. The amplitude for 0νββ decay is proportional to the effective Majorana mass: A(0νββ) ∝ | X i Uei2 mi | ≡ hmββ i (1.7) 0νββ decay has not been observed so far1 . The current best upper limit of hmββ i is from Heidelerg-Moscow experiment on 76 Ge, which reports hmββ i < 0.35 eV at 90% C.L. [11]. Neutrino Sources Neutrinos are in fact very common particles in the universe. Fig. 1.2 shows the energy spectrum of neutrinos that reach the surface of the Earth. The spectra extend a large range in both the energy and the intensity. Excluding the nuclear reactors and accelerators, whose neutrino flux depends on how close one can put the detector at, from approximately high to low flux the neutrino sources are: Cosmological neutrinos or big-bang relic neutrinos since they are the leftovers from the early universe similar to the CMB photons. They have a large number 1 There was one controversial claim of evidence of 0νββ decay [10], which later received criticisms from various literatures. 7 Figure 1.2: Flux of neutrinos at the surface of the Earth. The three arrows show the energy thresholds for charge current interactions on a free proton target. The geoneutrinos include the 238 U and 232 Th decay chains. The atmospheric neutrino fluxes are calculated at the Kamioka location. (Figure taken from [12].) 8 density of ∼56 cm−3 for each neutrino species (3 flavors and their anti-particles) with a black-body spectrum of very low temperature Tν = 1.9 K. The flux of cosmological neutrinos depends on the mass of the neutrinos. With the upper bound of mν < 2 eV [7], the flux can be calculated to be at least ∼4 ×1010 cm−2 s−1 for each neutrino species. Solar neutrinos from the fusion reactions inside the sun. They are born as νe ’s and their flux at the Earth surface is ∼6 × 1010 cm−2 s−1 , spanning an energy range up to ∼15 MeV. Details of solar neutrinos are discussed in Section 1.2.2. Some theories predict a ν̄e component in the solar flux through processes such as spin-flavor precession combined with neutrino oscillation. KamLAND has put an upper limit of such ν̄e flux to be less than 3.7 × 102 cm−2 s−1 at 90% C.L. in the energy window 8.3 – 14.8 MeV [13]. There are also theoretical predictions of a thermal flux of low-energy (< 5 keV) solar ν and ν̄’s from a variety of neutrino pair production processes. The predicted flux per flavor is ∼106 cm−2 s−1 [14]. Supernova neutrinos from type II supernovae explosions in the Milky Way. These are rare events happening about once per 40 years. When such SN happens, enormous amount of energy (∼1053 erg) is released in the form of all neutrino flavors in a timescale of ∼10 s, spanning an energy range up to ∼100 MeV. Up until now the only detected SN neutrinos are from SN1987A (55 kpc away from the Earth) on 23 February 1987. The predicted instantaneous neutrino flux from SN1987A is ∼109 cm−2 s−1 . Kamiokande [15] and IMB [16] together detected 20 events in a time interval of 13 s. Cosmological theory also predicts a Diffuse Supernova Neutrino Background (DSNB), from an integration of estimated core-collapse supernovae rate history, to be on the order of ∼1 cm−2 s−1 . Super-Kamiokande has put an upper limit on DSNB to be less than 1.2 ν̄e cm−2 s−1 (Eν > 19.3 MeV) at 90% C.L. [17]. 9 Geoneutrinos from the decay chains of natural radioactivities such as 238 U, 232 Th and 40 K in the Earth’s crust and mantle. They are born mostly as ν̄e ’s from the β − decays with energies up to 3.3 MeV. Geological models predict the radiogenic heat composes ∼19 TW or more of the total Earth heat flow [18, 19, 20]. Geoneutrinos have been observed at KamLAND [21, 22] and the ν̄e flux from 238 U and 232 Th decay chains are measured to be (4.4 ± 1.6) × 106 cm−2 s−1 . There is also theory predicting a natural nuclear reactor at the center of the Earth [1], called “georeactor,” which will yield ν̄e ’s as a byproduct of the fission process, with energies up to ∼10 MeV. A hypothetical 10 TW georeactor will output a ν̄e flux of ∼4×105 cm−2 s−1 at the surface of the Earth. Atmospheric neutrinos from the cosmic-ray interactions and decays in the Earth’s atmosphere. They are born as νµ ’s and νe ’s (and their antiparticles), mostly from the decays of cosmic ray pions and muons, with energies up to ∼10 TeV. The flux falls down rapidly at high energy [23] following an approximate power law. At 1 GeV the νµ flux is approximately 10−2 cm−2 s−1 sr−1 . More details of atmospheric neutrinos are discussed in Section 1.2.1. Astrophysical neutrinos with ultra high energies (> TeV) from galactic or extragalactic sources. Possible sources of such neutrinos are extreme acceleration environments such as AGN and GRB, GZK neutrinos from proton interaction with CMB photons, and possible annihilation or decay of heavy particles. Flux of UHE neutrinos is extremely low (∼1 km−2 y−1 for GZK neutrinos), so kilometer-scale neutrino telescopes such as IceCube [24], Antares [25], Anita [26] are needed to detect them. 10 1.1.2 Neutrino Oscillation in Vacuum The phenomenon of neutrino oscillation is the consequence of quantum mechanics from Eqn. 1.2. As the neutrino propagates from the source to the detector in the vacuum, it follows: |να (t)i = where Ei = p 3 X i=1 ∗ −iEi t Uαi e |νi i. α = e, µ, τ (1.8) p2 + m2i ≃ E + m2i /2E is the total energy of the mass eigenstate νi . Dropping the common phases to all three mass eigenstates, the probability for neutrino flavor state να to oscillate into flavor state νβ after traveling a distance L is ¯2 ¯ ¯ ¯X 2 ¯ ¯ ∗ P (να → νβ ) = ¯ Uβi Uαi E −imi L/(2E) ¯ ¯ ¯ i X ∗ ∗ = δαβ − 4 R(Uαi Uβi Uαj Uβj ) sin2 [1.27∆m2ij (L/E)] +2 X (1.9) i>j ∗ ∗ I(Uαi Uβi Uαj Uβj ) sin[2.54∆m2ij (L/E)] i>j where ∆m2ij ≡ m2i − m2j , L is in km and E is in GeV. A few observations can be made from Eqn. 1.9 • Oscillation probability is independent of the Majorana phase α1 , α2 . • For P (ν̄α → ν̄β ) one replaces the mixing matrix U → U ∗ . Oscillation can violate CP if U is complex, that is if the CP violating phase δ is different from 0 or π2. • For P (νβ → να ) one exchanges the subscript α ↔ β. Oscillation can violate T if δ is different from 0 or π. 2 To violate CP it also requires that all three mixing angles and the mass splittings are different from zero. In particular θ13 must not be zero, since the other mixing angles have already been measured to be non-zero. 11 • For P (ν̄β → ν̄α ) one exchanges both the subscript α ↔ β and U → U ∗ . One can see that oscillation probability remains the same and CP T is conserved independent of δ, as it should be. One application is that P (νe → νe ) = P (ν̄e → ν̄e ) under the assumption of CP T invariance. The complicated equation 1.9 can be considerably simplified given the fact that one mass splitting ∆m2atm is much bigger than the other mass splitting ∆m2sol (“one mass scale dominance”). As a consequence, in many neutrino oscillation experiments only two mass eigenstates are relevant (“quasi-two-neutrino oscillation”). The oscillation probability simply reduces to 2 P (να → νβ ) = sin 2θ sin 2 µ L [km] 1.27∆m [eV ] E [GeV] 2 µ 2 2 ¶ (1.10) with β 6= α (appearance channel), and 2 P (να → να ) = 1 − sin 2θ sin L [km] 1.27∆m [eV ] E [GeV] 2 2 ¶ (1.11) in the disappearance channel. Here ∆m2 is the mass splitting relevant to the experiment such that ∆m2 L/E = O(1), and θ is the mixing angle relevant to the experiment, for example θ12 for solar and long-baseline reactor neutrino experiments, θ13 for short-baseline reactor neutrino experiments, and θ23 for atmospheric neutrino experiments. Eqn. 1.11 will be extensively used in this dissertation. 1.1.3 Neutrino Oscillation in Matter The presence of matter provides a coherent effect of forward scattering from particles in the matter on the propagation of a neutrino. Such an effect is also known as the “MSW effect” as it is first suggested by Mikheyev, Smirnov and Wolfenstein [27, 28]. The effective potential introduced by the MSW effect receives contribution from all 12 target particles, i.e. electrons, protons and neutrons for ordinary matter. However, since only potential differences between the neutrino flavors play a role in the neutrino oscillation, the effective MSW potential can be written as √ V ≡ Vνe − Vνµ = Vνe − Vντ = + 2GF Ne (1.12) which simply comes from νe scatters off electrons through charge current interaction, as the neutral current contribution is the same for νe , νµ and ντ . Here GF is the Fermi constant and Ne is the number density of electrons. For anti-neutrinos the MSW potential reverses its sign. The MSW effect on the neutrino oscillation is best illustrated in a two neutrino oscillation scenario (e.g. νe → νx as is the case for solar neutrinos). The effective Hamiltonian H, in the flavor space, is (dropping multiples of the identity matrix): H = Hvacuum + Hmatter       2 0   cos θ − sin θ   V 0   cos θ sin θ   m1 /2E =  +    2 sin θ cos θ 0 m2 /2E − sin θ cos θ 0 0   sin 2θ  ∆m2  − cos 2θ + ξ =   4E sin 2θ cos 2θ − ξ   (∆m2 )m  − cos 2θm sin 2θm  (1.13) =   4E sin 2θm cos 2θm where we define 2V E ≃ ξ= ∆m2 µ ±8 × 10−5 ∆m2 [eV2 ] ¶µ ρ g/cm3 ¶µ E GeV ¶µ Ye 0.5 ¶ (1.14) which is positive for νe and negative for ν̄e . Here ρ is the density of the matter and Ye is the electron number per nucleon (∼0.5 on average). From Eqn. 1.13 one can see 13 that the presence of MSW effect can be accommodated into the vacuum oscillation by redefining θ → θm and ∆m2 → (∆m2 )m such that sin2 2θ sin 2θm = sin2 2θ + (cos 2θ − ξ)2 (1.15) q (∆m )m = ∆m sin2 2θ + (cos 2θ − ξ)2 (1.16) 2 and 2 2 As can be easily seen, when MSW effect is small, ξ → 0 and vacuum oscillation is recovered. On the other hand, from Eqn. 1.14, MSW effect plays an important role for solar neutrinos since density is high (∼100 g/cm2 ) at the center of the sun where the neutrinos are created, and for long-baseline accelerator neutrinos traversing the Earth, with a beam energy higher than a few tens of GeV. 1.2 History of Neutrino Oscillation Experiments There is a long history and a large number of neutrino oscillation experiments starting as early as the 1960s, using various types of neutrino sources. Early experiments were not specifically designed to look for neutrino oscillations, so when the neutrino oscillation phenomenon emerged from the data it was quite surprising, and turned out to be an even more interesting field of study. Fig. 1.3 shows a summary of regions of mass-squared splitting and mixing angle favored or excluded by various experiments, taken from PDG 2008 [7]. Some of those experiments will be discussed in this section. 1.2.1 Atmospheric Neutrinos Despite some of the hints in the earlier experiments, the first strong evidence of neutrino oscillation comes from the study of atmospheric neutrinos by Kamiokande experiment in 1988 [29]. Atmospheric neutrinos are produced by the cosmic rays 14 Figure 1.3: Regions of neutrino oscillation parameters favored or excluded by various experiments. Figures taken from PDG 2008 [7]. Data used in this figure can be found at http://hitoshi.berkeley.edu/neutrino/ . 15 interacting with the Earth atmosphere. The primary cosmic rays, mostly protons, have a spectrum approximately following a power law Φ(E) ∝ E −2.7 , extending up to extremely high energies ∼1020 eV. The primary cosmic ray interacts with Earth atmosphere creating abundant secondary particles, called “air showers.” Among them the dominant source of neutrinos are the π ± through their decay chains, such as: π + → µ + + νµ , µ+ → e+ + νe + ν̄µ (1.17) The calculations of the absolute atmospheric neutrino flux usually have large uncertainties ∼20% [23] due to the many factors involved, such as primary cosmic ray flux, solar modulation, geomagnetic cutoff, cosmic ray interaction models, etc. However, some properties have more robust estimations: • The flux ratio of (νµ + ν̄µ )/(νe + ν̄e ) is approximately two at low energy (E < 3 GeV), which is the simple consequence of the pion decay chain in Eqn. 1.17. The statement is also true for the differential flux at each neutrino energy, since the three neutrinos produced in the pion decay chain have approximately the same average energy. The flux ratio increases at higher energy because some of the muons would reach the ground before decaying away. The flux ratio is calculated to be accurate to ∼5%. • The neutrino flux has an up-down symmetry: φνα (E, θ) = φνα (E, π − θ) (1.18) because the Earth is a well-defined sphere and the primary cosmic rays are isotropic. The statement is more true for high energy neutrinos (E > a few GeV) where the influences from geomagnetic field is minimal. 16 Figure 1.4: Zenith-angle distributions observed in Super-Kamiokande based on 61 kton-yr data: (a) sub-GeV e-like events; (b) sub-GeV µ-like events; (c) multi-GeV elike events; (d) multi-GeV µ-like events including partially contained events. cos θ = 1 means upward going and cos θ = −1 means downward going. Solid lines show the Monte Carlo prediction without oscillation (normalization varied in the fit). Dashed lines show the prediction with oscillation (∆m2 = 2.8 × 10−3 eV2 , sin2 2θ = 1.0). Figure taken from Ref. [30]. 17 The largest sample of atmospheric neutrino data is from the Super-Kamiokande experiment. The detector contains 50 ktons of pure water to look for the Cherenkov light produced by the charged particles passing through the detector. The atmospheric neutrinos produce e’s and µ’s inside the detector through CC interaction, and the directions of the leptons well represent the directions of the neutrinos at energies higher than ∼1 GeV. The detector is able to distinguish an electron event from a muon event by the pattern of hit PMTs: a “sharp” ring for µ-like or a “fuzzy” ring for e-like event. The events are further grouped into 4 categories: sub-GeV e-like, sub-GeV µ-like, multi-GeV e-like, multi-GeV µ-like including the partially contained events, and their zenith angle distributions are shown in Fig. 1.4 together with the Monte Carlo predictions without neutrino oscillation. It is surprising that all e-like events agree well with the Monte Carlo simulation, while µ-like events show significant deficit for up-going events only. On the other hand, the data can be well explained with neutrino oscillation because the up-going neutrinos travel much longer distances though the Earth to the detector (∼104 km) compared with the down-going neutrinos which only travel through the atmosphere (∼20 km). Since null ν̄e → ν̄µ (thus νµ → νe assuming CP T invariance) oscillations have been observed by the short-baseline reactor neutrino experiments, the data agree well with the νµ → ντ oscillation explanation3 . From Eqn. 1.11 one can immediately estimate the mixing angle θ23 : P (νµ → νµ ) = 1 − 0.5 sin2 2θ23 ≃ ⇒ sin2 2θ23 ∼ 1 (θ23 ∼ 45◦ ) 3 Nup ∼ 0.5 Ndown (1.19) The atmospheric neutrino data by themselves do not rule out νµ → νe oscillation. The observed null flux change of e-like events can be a simple cancellation between νe oscillating into νµ and ντ with equal probability, and νu oscillating into νe with the same probability but twice the flux. 18 2 |∆matm | (eV2/c 4) MINOS Preliminary 0.006 MINOS Best Fit MINOS 68% C.L. MINOS 90% C.L. SK 90% C.L. SK (L/E) 90% C.L. K2K 90% C.L. 0.005 0.004 0.003 0.002 0.001 0 0.4 0.5 0.6 0.7 0.8 0.9 1 sin2(2θatm) Figure 1.5: The regions of the atmospheric neutrino oscillation parameters allowed by Super-Kamiokande (SK), K2K and MINOS experiment. Figure taken from [7]. and the ∆m223 : ∆m223 ∼ 3 GeV E [GeV] ∼ ∼ 2.4 × 10−3 eV2 2.54 · L [km] 2.54 · 500 km (1.20) Here the average energy hEν i of ∼3 GeV and the horizontal path-length of ∼500 km are used. The same oscillation parameter space has been explored by the long-baseline accelerator neutrino experiments such as K2K [31] and MINOS [32]. The flux of νµ beams is measured at a near detector before any oscillation is expected, and at a far detector, 250 km away for K2K and 735 km away for MINOS from the neutrino source. Both K2K and MINOS observed significant disappearance of the expected νu flux at the far detector, and distortion of the expected νµ energy spectrum consistent with the νµ → ντ oscillation. Fig. 1.5 shows the allowed regions by combining the Super-Kamiokande (SK), K2K and MINOS data. The constrain on ∆m2 mostly comes from the analysis of the νu energy spectrum distortion, while the constrain on sin2 2θ mostly comes from the large data set of SK atmospheric neutrino samples. 19 1.2.2 Solar Neutrinos The Sun is a huge neutrino source for the Earth. Bethe [33] first pointed out in 1939 that the sun shines by the internal fusion process: 4p →4 He + 2e+ + 2νe + 26.73 MeV (1.21) The general fusion reaction 1.21 can proceed with different reaction cycles that produce the same final particles, but result in different energy distribution of neutrinos. In the Sun ∼98.5% of the energy is produced by the “pp” chain, while only ∼1.5% by the “CNO” chain. The pp chain reactions and the νe flux predictions from the Standard Solar Model (BP04 [34]) are summarized in Table 1.1. The energy spectrum of different solar neutrinos are shown in Fig. 1.6. Here we neglect “CNO” neutrinos for simplicity, as they typically have relatively low flux (∼5 × 108 cm−2 s−1 ) and have energies below 1 MeV, which made them difficult to be detected. The theoretical prediction for “CNO” neutrinos also have large uncertainties of ∼40% [34]. Table 1.1: The νe production from pp chain in the Sun. The termination percentage is the fraction of terminations of the pp chain. The predicted νe flux is from BP04 [34]. Source Reaction Termination νe Energy νe Flux [%] [MeV] [cm−2 s−1 ] 2 + pp p + p → H + e + νe 99.6 < 0.43 5.94 × 1010 (±1%) pep p + e− + p → 2 H + νe 0.4 1.44 1.40 × 108 (±2%) 2 3 H + p → He + γ 100 3 He+ 3 He → 4 He + 2p 85 3 4 7 He+ He → Be + γ 15 7 7 Be Be + e− → 7 Li + νe 15 0.86 (90%) 4.86 × 109 (±12%) 0.38 (10%) 7 4 Li + p → 2 He 15 7 Be + p → 8 B + γ 0.02 8 8 B B → 8 Be∗ + e+ + νe 0.02 < 15 5.79 × 106 (±23%) ∗ 8 Be → 2 4 He 0.02 3 hep He + p → 4 He + e+ + νe 2 × 10−5 < 18.8 7.88 × 103 (±16%) Up to present there are five main types of solar neutrino experiments using dif- 20 Figure 1.6: The energy spectra of solar neutrinos from the pp Chain, predicted from BP04 [34] solar model. For continuum sources, the neutrino fluxes are given in cm−2 s−1 MeV−1 at the Earth’s surface. For line sources, the units are cm−2 s−1 . The sensitive regions of the Gallium, Chlorine, Water and Heavy Water experiment are also shown. 21 ferent detector targets: Chlorine, Gallium, water, heavy water and liquid scintillator. The first two are radiochemical measurements typically involving collecting and counting the final-state radioactive isotopes periodically, while the last three are real-time measurements of the neutrino energy spectrum. Chlorine Experiments In the late 1960s began the first solar neutrino experiment led by Ray Davis [35] in the Homestake gold mine (depth of 4300 m.w.e) in South Dakota. The experiment uses ∼600 tons of liquid C2 Cl4 to detect solar neutrinos through the radiochemical reaction νe + 37 Cl → 37 Ar + e− (1.22) The reaction has a threshold energy of 0.814 MeV and is mostly sensitive to 8 B neutrinos, but there is a substantial contribution from 7 Be neutrinos as well. The experiment was taking data for over 30 years, during when the 37 Ar atoms were flushed out with helium gas every ∼100 days and condensed into proportional counters for counting. As a result, the average neutrino flux observed by Homestake Chlorine experiment is 2.56 ± 0.23 solar neutrino units (SNU)4 , while the standard solar model (BP04) predicts 8.5 ± 1.8 SNU. The incompatibility between theory and experiment is well known as “The Solar Neutrino Problem.” Gallium Experiments A lower energy threshold can be obtained with a Gallium target through the reaction νe + 71 Ga → 71 Ge + e− 4 1 SNU = 10−36 interactions per target atom per second. (1.23) 22 which has a threshold energy of 0.214 MeV and is sensitive to the most abundant pp neutrinos. Such experiments have been carried out by SAGE [36] in the Baksan laboratory (depth of 4700 m.w.e) in Russia with 60 tons of liquid Ga metal, and by GALLEX/GNO [37] in the Gran Sasso laboratory (depth of 3300 m.w.e) in Italy with 30 tons of Ga in a concentrated GaCl3 -HCl solution. In both experiments the 71 Ge is extracted monthly and converted into the form of GeH4 and counted. The measured +7.2+3.5 average flux is 67.2−7.0−3.0 SNU for SAGE and 69.3 ± 5.5 SNU for GALLEX+GNO. Both experiments only observed about a half of the predicted 131 SNU from the standard solar model. Water Experiments The first real-time solar neutrino measurement is from the Kamiokande/Super-Kamiokande experiment located in the Kamioka mine (depth of 2700 w.m.e) in Japan. As described in Section 1.2.1, the detector contains 50 ktons of pure water to look for the Cherenkov light produced by the charged particles passing through the detector, in particular for solar neutrinos through the elastic scattering on electrons: νx + e− → νx + e− (1.24) The elastic scattering is sensitive to all neutrino flavors, however, the cross section for νe (∼1.0 × 10−44 E [MeV] cm2 ) is approximately 7 times larger than for νµ and ντ (∼0.15×10−44 E [MeV] cm2 ). The direction of the electron from the elastic scattering in strongly correlated with the neutrino, pointing back to the Sun (Fig. 1.7(a)). The energy threshold is about 7 MeV for Kamiokande and 5 MeV for Super-Kamiokande, thus the sensitivity of the measurement is almost entirely to the 8 B neutrinos. As a result, the measured 8 B neutrino flux is (2.8±0.4)×106 cm−2 s−1 by Kamiokande [38], and (2.35 ± 0.08) × 106 cm−2 s−1 by Super-Kamiokande [39, 40]. Both are significantly 0.8 5-20 MeV Super-Kamiokande 2 Data/SSMBP2004 Event/day/bin 23 θSun 0.6 1 0 -1.0 0.4 0.2 -0.5 0.0 0.5 1.0 5 10 cos θSun (a) 15 20 Energy(MeV) (b) Figure 1.7: The solar neutrinos measured at Super-Kamiokande: (a) Angular distribution relative to the Sun. The shaded area indicates the elastic scattering peak, the dotted area is the contribution from the background. (b) Ratio of observed and expected energy spectrum. The purple lines represent 1σ systematic error region. Figures taken from Ref. [39, 40] less than the solar model predicted 8 B neutrino flux of 5.79(1 ± 0.23) × 106 cm−2 s−1 . Furthermore, no significant distortion of the 8 B neutrino energy spectrum has been observed (Fig. 1.7(b)). Heavy Water Experiments The SNO experiment located in the Sudbury mine (depth of 6200 m.w.e) in Canada uses 1 kton of heavy water (D2 O) as a Cherenkov detector. With heavy water, three different reactions can be registered by the detector: CC : νe + d → p + p + e− (1.25) N C : νx + d → p + n + νx (1.26) ES : νx + e− → νx + e− (1.27) The CC reaction is sensitive only to νe ’s, while the N C reaction is sensitive to all neutrino flavors. The ES reaction is the same as in the water experiments such as Super-Kamiokande, mostly sensitive to νe ’s with reduced sensitivity to νµ ’s and ντ ’s. φµτ (× 10 6 cm -2 s-1) 24 BS05 φSSM 68% C.L. 6 NC φµ τ 68%, 95%, 99% C.L. 5 4 3 SNO 2 φCC 68% C.L. 1 φES 68% C.L. SNO φNC 68% C.L. SNO SK φES 68% C.L. 0 0 0.5 1 1.5 2 2.5 3 3.5 φe (× 106 cm-2 s-1) Figure 1.8: SNO measured flux of µ + τ neutrinos versus flux of electron neutrinos. CC, N C and ES flux measurements are indicated by the filled bands. The total 8 B solar neutrino flux predicted by the standard solar model [34] is shown as dashed lines. Super-Kamiokande result is shown as the grey band. Figure taken from. [41]. The energy threshold in SNO for CC and ES reactions is about 5.5 MeV and for N C reaction is the reaction threshold of 2.2 MeV, thus the solar neutrino contribution is also almost entirely from the 8 B solar neutrinos. The different angular and energy distribution allows disentangling the three reactions on a statistical basis. The SNO experiment has been running in three phases with increasing ability of detecting the neutrons from the N C reactions. In the first phase with pure heavy water, neutrons are captured on deuterium giving 6.25 MeV γ-rays. In the second phase 2.7 tons of salt were added to the heavy water, which increased the neutron detection efficiency through capture on Cl and γ-rays with total energy ∼8.6 MeV are released. In the third phase the salt was removed and an array of 3 He proportional counters was installed in the detector, enabling neutron detection independent of the PMT array. The measured flux in each channel, in units of 106 cm−2 s−1 , is [34]: 25 +0.06+0.08 φCC = φνe = 1.68−0.06−0.09 (1.28) +0.22+0.15 φES = φνe + 0.155(φνµ + φντ ) = 2.35−0.22−0.15 (1.29) +0.21+0.38 φN C = φνe + φνµ + φντ = 4.94−0.21−0.34 (1.30) where the statistical error is indicated before the systematic error. The flux in CC and ES channel shows a similar large deficit as other solar neutrino experiments. On the other hand, the N C flux measurement is in good agreement with the standard solar model prediction of 5.79(1 ± 0.23) × 106 cm−2 s−1 . The SNO result is considered as the “smoking-gun” evidence that the neutrino oscillation is the key to solve the “Solar Neutrino Problem” that lasted for more than 30 years. Liquid Scintillator Experiments Liquid scintillator detector, due to its large scintillation light output, has a much lower energy threshold than the Cherenkov detectors. Carbon-based organic liquid scintillators are used in the KamLAND experiment (∼1 kton), located in the Kamioka mine (depth of 2700 w.m.e) in Japan, and in the Borexino experiment (∼300 ton) in the Gran Sasso laboratory (depth of 3300 m.w.e) in Italy. They are sensitive to the solar neutrinos through the elastic scattering process similar to Super-Kamiokande and SNO, but with energy threshold as low as a few hundred keV, thus are sensitive to the more abundant 7 Be solar neutrinos. One disadvantage of most liquid scintillator detectors is the lack of directional information about the neutrino event, since the scintillation light is isotropic. The greatest obstacles for low energy measurements are the radioactive backgrounds from naturally occurring or cosmogenical radioisotopes in the detector (14 C, 238 U, 232 Th, 40 K, 210 Pb, 85 Kr, 39 Ar, 11 C, etc.), thus advanced purification techniques and careful operation of the detector are needed in order to remove 26 the initial radioactive contaminants and to prevent introducing new contaminants. The 7 Be solar neutrino flux measured by Borexino [42] is (2.8 ± 0.2stat ± 0.3syst ) × 109 cm−2 s−1 , which is again significantly lower than the standard solar model prediction of (5.08 ± 0.25) × 109 cm−2 s−1 . Interpretation of Experimental Data All of the measurements from various solar neutrino experiments can be nicely explained by the neutrino oscillation theory, in particular the Large Mixing Angle MSW (LMA-MSW) solution. Given the small size of Ue3 , Solar neutrino propagation is approximately a two-flavor oscillation νe → νx , where νx is a 50-50 mixture of νµ and ντ if θatm = 45◦ . The survival probability of νe in the case of 3-neutrino oscillation can be simply related to the 2-neutrino case by Pν3νe →νe = sin4 θ13 + cos4 θ13 · Pν2νe →νe (∆m221 , θ12 ; Ne → cos2 θ13 Ne ) (1.31) where Ne is the electron density at the center of the sun where the neutrinos are created. From Eqn. 1.15 and 1.16, MSW effect is important when ξ ≫ cos 2θ12 , and negligible when ξ is small. ξ can be calculated from the standard solar model (assuming a reasonable θ12 ) for each type of solar neutrinos, and the result is that the MSW potential dominates for the 8 B neutrino, while the vacuum potential dominates for the 7 Be and pp neutrinos. The νe survival probability Pee measured by the Gallium experiments and Borexino 7 Be experiment is ∼0.55, from which the mixing angle θ12 can be immediately estimated with the vacuum oscillation formula 27 Pee (7 Be, pp) ≃ 1 − 0.5 sin2 2θ12 ∼ 0.55 ⇒ sin2 2θ12 ∼ 0.9 (θ12 ∼ 36◦ ) (1.32) For 8 B neutrinos at the center of the sun, since the MSW potential dominates, the vacuum potential can be neglected as a first approximation in Eqn. 1.13. The Hamiltonian can be then written as:   H= √  2GF Ne 0   0 0 (1.33) The diagonal Hamiltonian means that the 8 B νe is born as the higher-energy mass eigenstate. The propagation of a 8 B νe to the outer edge of the Sun is adiabatic since the electron density Ne changes slowly over the route, which means that the neutrino keeps its status as the higher-energy mass eigenstate5 of the slowly-varying H until it reaches the outer edge of the Sun, where H becomes Hvacuum . The neutrino then emerges out of the sun as the higher-energy mass eigenstate of Hvacuum , which by definition is ν2 , until it reaches the Earth (during the journey no oscillation happens since it is a mass eigenstate) being detected as a νe again. The νe survival probability 2 Pee is then simply Ue2 ∼ sin2 θ12 . The pure Pee for 8 B neutrinos is measured by the SNO CC reaction to be ∼0.3, from which θ12 can be again estimated Pee (8 B) ≃ sin2 θ12 ∼ 0.3 (θ12 ∼ 33◦ ) (1.34) which agrees well with the vacuum oscillation case for 7 Be and pp neutrinos. The Pee for 8 B neutrinos measured by the water experiments and SNO ES reaction is a little higher, because they have reduced sensitivity to νµ and ντ flux as well. Finally, 5 It can be shown that the eigenvalues of H never cross as Ne changes with r. 28 the SNO N C reaction measured all three neutrinos and is consistent with the standard solar model prediction. The mystery of “The Solar Neutrino Problem” is solved — Solar neutrinos are not disappearing, they are simply redistributing themselves among the three flavors. More rigorous global analysis of the solar neutrino data has been done, for example, in Ref.[43, 44]. Historically there were four allowed regions in the (∆m221 , θ12 ) space, namely LMA (large mixing angle, ∆m2 ∼ 10−5 eV2 ), SMA (small mixing angle, ∆m2 ∼ 10−5 eV2 ), LOW (∆m2 ∼ 10−7 eV2 ) and VACUUM (θ12 ∼ 45◦ , ∆m221 ∼ 10−10 eV2 ) regions. It becomes clear later by combing with the KamLAND [45, 46, 22] data that the LMA-MSW solution is the solely correct solution for solar neutrinos. 1.2.3 Reactor Neutrinos The fission processes inside the nuclear reactors typically release ν̄e ’s with energies up to ∼10 MeV. ν̄e ’s are usually detected through the inverse beta decay reaction ν̄e + p → n + e+ (1.35) The relatively large cross section (∼10−42 cm2 ) combined with the clean signal (prompt positron events in coincidence with delayed neutron capture event) makes the detection of ν̄e relatively easy and historically that was how the neutrinos were first detected [3]. Reactor neutrino oscillation experiments are typically disappearance experiments that measure the ν̄e flux from the reactors and compare with the theoretical predictions. The ν̄e survival probability is directly related to the νe survival probability assuming CP T invariance: P (ν̄e → ν̄e ) = P (νe → νe ). 29 Short Baseline Reactor Experiments Omitting a few very short baseline (< 100 m) reactor neutrino experiments in the early times (Savannah River [3], ILL [47], Gösgen [48], Rovno [49], Krasnoyarsk [50], Bugey [51], etc.), the first short baseline reactor neutrino experiments specifically look for oscillation parameters found by the atmospheric neutrino measurements are CHOOZ [52, 53] and Palo Verde [54]. Both experiments have a baseline ∼1 km and are suitable to look for ν̄e disappearance corresponding to 2 P (ν̄e → ν̄e ) = 1 − sin 2θ13 sin 2 µ 1.27∆m2atm [eV2 ] L [km] E [GeV] ¶ (1.36) Both detectors use liquid scintillator loaded with 0.1% natural gadolinium, which has a high thermal neutron capture cross section and releases an 8 MeV γ-ray cascade that is above all natural radioactivity. The CHOOZ detector was built at distances of 1115 m and 998 m from the two reactors of the CHOOZ power plant in France. The plant had a total thermal power of 8.5 GW, and was only commissioned after the start of the data-taking of the experiment. This gives CHOOZ a special opportunity to observe the background during the zero-power period and the slow power-ramp-up period. The CHOOZ detector consisted of a central volume of scintillator with mass of ∼5 tons, and was placed in an underground cavity with a relatively deep overburden (300 m.w.e). The measured ratio between ν̄e detected and expected is RCHOOZ = 1.01 ± 0.028(stat) ± 0.027(syst) (1.37) The Palo Verde experiment was built at distances of 890m, 890m and 750m from the three reactors of the Palo Verde Nuclear Generating Station in Arizona, US. The total thermal power was 11.6 GW. The detector is located in a relatively shallow 30 Figure 1.9: Exclusion contours on (∆m231 , sin2 2θ13 ) parameter space determined by CHOOZ, Palo Verde along with the allowed region obtained by Super-Kamiokande. Figure taken from. [53]. bunker (32 m.w.e). The rather large muon flux (∼2 kHz) produced a substantial number of spallation neutrons, so a segmented detector design is used to take full advantage of the triple coincidence given by the positron and the subsequent annihilation γ-rays. The fiducial mass of the detector is 12 tons, but the more elaborate topological signature reduced the detection efficiency and increased the systematic errors. The measured ratio between ν̄e detected and expected is RPalo Verde = 1.01 ± 0.024(stat) ± 0.053(syst) (1.38) Since no ν̄e disappearance has been observed by either experiment, only upper limit of θ13 can be set. Fig. 1.9 shows the exclusion region on (∆m231 , sin2 2θ13 ) parameter space determined by CHOOZ, Palo Verde along with the allowed region obtained by Super-Kamiokande 3-neutrino analysis of the atmospheric neutrino data. The combined analysis sets the upper limit of sin2 (2θ13 ) < 0.19 at 90% C.L. 31 (a) (b) Figure 1.10: The locations of nuclear power plants (a) in the world, (b) in Japan, Korea and Far East Russia. Substantial concentrations of reactors are found in Europe, Eastern US and Japan. KamLAND detector is located at (36.42 ◦ N, 137.31 ◦ E) in the middle of Japan. Maps can be found at http://www.insc.anl.gov/ from International Nuclear Safety Center at Argonne National Laboratory. Long Baseline Reactor Experiments To examine the oscillation parameters found by the solar neutrino experiments, a baseline of at least ∼100 km is needed, as can be easily seen from equation 2 P (ν̄e → ν̄e ) = 1 − sin 2θ12 sin 2 µ 1.27∆m2sol [eV2 ] L [km] E [GeV] ¶ (1.39) since ∆m2sol is approximately 10−5 eV2 for LMA solution and even lower for other solutions (Section 1.2.2). On the other hand, the reactor ν̄e flux decreases as 1/r2 , which means powerful nuclear reactors are needed in order to get good statistics. A quick look at the locations of nuclear power plants in the world (Fig. 1.10(a)) reveals that such a detector can be only placed in Europe, Eastern US or Japan. Historically, there was already an existing underground cavity (depth of 2700 m.w.e) built for the old Kamiokande detector, which is located in the middle of Japan surrounded by ∼55 reactor cores with total thermal power of ∼130 GW and flux averaged baseline of 32 1.4 1.2 Nobs/Nexp 1.0 0.8 ILL Savannah River Bugey Rovno Goesgen Krasnoyarsk Palo Verde Chooz 0.6 0.4 0.2 KamLAND 0.0 101 102 103 104 Distance to Reactor (m) 105 Figure 1.11: The ν̄e survival probability measured by various reactor neutrino experiments as a function of baseline distance. The shaded area indicates the 95% C.L. LMA flux predictions found in a global analysis of the solar neutrino data [43]. The dotted curve corresponds to sin2 2θ = 0.833 and ∆m2 = 5.5 × 10−5 eV2 , while the dashed curve shows the case of small mixing angle (or no oscillation). Figure taken from [45]. ∼180 km (Fig. 1.10(b)). Naturally, this site was selected to house the KamLAND experiment to specifically explore the regions of neutrino oscillation parameters found by the solar neutrino experiments, but on the Earth. The KamLAND detector consists of ∼1 kton of liquid scintillator and detects ν̄e through the usual inverse beta decay reaction. The predicted no-oscillation ν̄e event rate is ∼1 event/day/kton, modulated by the refueling and maintenance of the nuclear power plants in Japan. Details of the KamLAND experiment are discussed in Chapter 2. KamLAND for the first time observed significant deficit of the expected ν̄e flux. The measured ratio between ν̄e detected and expected is [45] RKamLAND = 0.611 ± 0.085(stat) ± 0.041(syst) (1.40) Fig. 1.11 summarizes the ν̄e survival probability measured by various reactor neutrino experiments as a function of the baseline distance. As discussed in the previous 33 80 no-oscillation accidentals 16 10-5 20 0 0 2 Events / 0.425 MeV 40 10-4 ∆m2 (eV ) C(α,n) O spallation best-fit oscillation + BG KamLAND data 13 60 Solar KamLAND 95% C.L. 95% C.L. 99% C.L. 99% C.L. 99.73% C.L. 99.73% C.L. solar best fit 1 2 3 4 5 Eprompt (MeV) (a) 6 7 8 KamLAND best fit 10-1 1 10 tan2 θ (b) Figure 1.12: (a) Prompt event energy spectrum of ν̄e candidate events at KamLAND with associated background spectra, and the predicted spectrum in case of no oscillation. (b) Neutrino oscillation parameter allowed region from KamLAND (shaded area) and solar neutrino experiments (lines). The three isolated shaded regions from bottom to top are historically called LMA-0, LMA-I and LMA-II region respectively. Figure taken from [46]. sections, the null-oscillation observation of short-baseline reactor neutrino experiments is the consequence of the smallness of θ13 , while the observed ν̄e flux deficit by KamLAND agrees well with the LMA-MSW oscillation parameters found by the solar neutrino experiments [43]. KamLAND has also observed significant distortion of ν̄e energy spectrum [46] as shown in Fig. 1.12(a). The observed spectrum distortion enabled KamLAND to make a very precise measurement of ∆m221 . Fig. 1.12(b) shows the allowed regions of oscillation parameters from KamLAND and solar neutrino experiments [46]. The three isolated KamLAND allowed regions from bottom to top are historically called LMA-0, LMA-I and LMA-II region respectively. The recent released KamLAND result [22] shows that only the LMA-I region is allowed at 99.73% C.L. Combining with the results of solar experiments assuming CP T invariance, the neutrino oscil−5 lation parameters are measured to be ∆m221 = 7.59+0.21 eV2 and tan2 θ12 = −0.21 × 10 0.47+0.06 −0.05 [22]. 34 1.3 Introduction to Nuclear Georeactor Hypothesis It has been proposed by J.M.Herndon [1] that there could be enough 235 U and 238 U at the center of the Earth that started a natural fast breeder type nuclear reactor from 4.5 billion years ago, and sustained until present. This “georeactor” provides a source of energy that can last for a geological time scale, and possibly powers the geomagnetic fields. The georeactor model also provides alternative ways to explain the irregular geomagnetic field reversal and the 3 He/4 He anomaly problem [55]. On the other hand, this model is in contradictory with the traditional Bulk Silicate Earth (BSE) model, which disfavors any significant Uranium content in the Earth core. Although not popular in the Earth science literature, the georeactor hypothesis itself is self-consistent and is possible to be tested with a large ν̄e detector such as KamLAND. 1.3.1 Nuclear Earth Model and Substructure of the Inner Core The Earth’s chemical composition has been studied with various methods such as direct sampling from bore-holes and xenolith from lava flows, but these samples are all from above the upper mantle of the Earth. It is assumed that the the composition of non-volatile elements in the Earth should be similar to the composition of the Sun’s outer layer (solar abundance), since they are formed in the same process as the entire solar system. Studies of the meteorites showed that the type I carbonaceous chondrites have very similar composition as to the solar abundance, thus they should also represent the composition of the bulk Earth in its early formation stage. This formed the basis of the traditional BSE model. In this model, the inner core of the 35 Earth consists of partially crystallized iron and nickel, and it is surrounded by a fluid outer core of iron, nickel and some light elements such as S, O or Si. The bulk of the Earth has the similar composition as the the type I carbonaceous chondrites, which is formed in oxygen-rich environment, and most elements formed oxide. The U and Th, being lithophile elements, like to bond with oxygen in silicates and oxides but not with metallic iron, so they can only exist in the mantle and crust. An alternative model called “Nuclear Earth Model” was proposed by Herndon [1, 56], and the central idea is that the main composition of the bulk Earth is more similar to the enstatite chondrites (also known as E-type chondrites), which is formed under oxygen-deficient conditions and has much less oxidation. Only five elements (Fe, Mg, Si, O, S) constitute approximately 95% of the mass of each of the hundreds of known anhydrous chondrites. Herndon plotted [56] the mass ratio of metal+sulfides to silicates as a function of whole-rock molar ratio of oxygen to the three major elements (Fe, Mg, Si) with which oxygen forms compounds in anhydrous chondrites, as shown in Fig. 1.13. The core-to-mantle ratios for the earth, calculated from seismic-based data [57], show that the Earth as a whole has a state of oxidation similar to certain highly reduced enstatite chondrites (particularly the Abee enstatite chondrite) and unlike most other types of chondrites. One prediction of the “Nuclear Earth Model” that Earth being of enstatite chondrite composition, contradictory of of the BSE model, is that the inner core of the Earth is not composed of iron-nickel metal but of nickel silicide (mainly Ni2 Si, also known as perryite), as has been found in enstatite meteorites. The normal lithophile elements such as Si, Mg, Ca and U under oxygen-deficient conditions now occur in part as non-oxides, for example sulfides, and together with S, Fe and Ni constitute the alloy portion correspond to the Earth’s core. These alloy elements precipitate under gravity and one would expect the low-density precipitates such as CaS and MgS being at the core-mantle boundary, and high-density, high-temperature precipitates, most 36 Figure 1.13: Mass ratio of (metal+sulfides) to silicates as a function of whole-rock molar ratio of oxygen to the three major elements with which oxygen forms compounds in anhydrous chondrites. the core to lower mantle ratio and the core to (upper+lower) mantle ratio are indicated by the broken lines. Figure taken from [56]. importantly uranium or its compound (mainly monosulfides), being collected at the center of the Earth. This creates an inner core substructure as shown in Fig. 1.14. The difference of the chemical composition of the Earth between the traditional BSE model and the “Nuclear Earth Model” is graphically shown in Fig. 1.15, taken from the August 2002 issue of Discover magazine. 1.3.2 Georeactor Feasibility The concentration of Uranium at the center of the earth provides the feasibility of the existence of a natural self-sustaining nuclear fission reactor, or “georeactor,” as suggested by Herndon [1]. The 235 U/238 U ratio, being ∼0.7% now, was much higher at the beginning life of the Earth 4.5 billion years ago because 235 U has a shorter halflife (0.7 billion years) than 238 U (4.5 billion years). With enough uranium collected at the center, a self-sustaining chain of nuclear fission reactions is possible to happen. 37 Figure 1.14: (A) Schematic representation of inner-core substructure, calculated with uranium assumed to be the mono-sulfide, present in the same relative proportion as in the alloy portion of the Abee enstatite chondrite. (B) Shell structure of the interior of the Earth. [57], shown for reference. Figure taken from [56]. Lacking of low atomic mass elements severed as neutron moderators at the center, the georeactor operates more similar to a fast breeder reactor. The fission production is mostly from 235 U, and partially from 238 U fast neutron fissions. The 235 U fuel is bred by 238 U by the following fuel production reactions, which regenerate 235 U that would otherwise be lost due to α decay: 238 U(n, γ)239 U(β − )239 Np(β − )239 Pu(α)235 U (1.41) Numerical simulations has been performed [60, 59] using the SAS2 sequence contained in the SCALE Code Package form Oak Ridge National Laboratory [61]. The code performs a 1-D transport analysis at selected time intervals, calculating the energy flux spectrum, updating the time-dependent weighted cross sections from the depletion analysis, and calculating the neutron multiplication factor Keff of the system. The nuclear georeactor is assumed to start 4.5 billion years ago, and would cease operation when Keff < 1. All fission products were assumed to be removed upon for- 38 Figure 1.15: The difference of shell structure of the Earth between the traditional Earth model and the “Nuclear Earth Model.” Figure taken from [58]. 39 Table 1.2: Summary of Simulation of Georeactor with SAS2 Code. Table reformatted from [59]. Nuclide 4 TW Simulation 30 TW Simulation Initial Mass (4.5 billion years ago) [kg] 235 U 4.8672 × 1015 2.1039 × 1016 238 16 U 1.6060 × 10 6.9427 × 1016 Present Day Fission Fraction 235 U 7.46 × 10−1 7.60 × 10−1 238 −1 U 2.49 × 10 2.28 × 10−1 233 U 2.60 × 10−3 6.90 × 10−3 236 −3 U 1.38 × 10 3.97 × 10−3 232 Th 5.63 × 10−4 5.77 × 10−4 239 Pu 1.70 × 10−4 3.53 × 10−4 241 −17 Pu 1.11 × 10 1.97 × 10−16 mation. As a result from the simulation, Hollenbach and Herndon showed that [60] the Keff > 1 condition can be sustained for the whole Earth’s age, and that Keff and 235 U/238 U ratio are stabilized after reaching some minimum values at about 3 billion years ago. The maximum steady-state fission power is determined to be 30 TW, using the maximum uranium content at 4.5 billion years ago. A 4 TW simulation is also performed for reference, using a smaller amount of initial uranium. The input parameters to the simulation and the resulting present day fission fractions of selected isotopes are summarized in Table 1.2. One notable observation is that the 235 U and 238 U together contribute to ∼99% of the present-day georeactor fission6 . This is very different from the common thermal neutron fission reactors. For example, the relative fission rate of typical reactor cores relevant to KamLAND experiment (mostly BWR and PWR) is 235 U : 238 U : 239 Pu : 241 Pu = 0.56 : 0.08 : 0.30 : 0.06, as will be discussed in Section 5.1. As a supporting evidence, the uranium ore deposits found at Oklo (in Republic of Gabon, Africa) in 1972 are believed to be functioning at 2 billion years ago as a natural nuclear fission reactor (Bodu et al.,1972). The 235 U concentration in these 6 There are other simulations that produce different fission fraction predictions, for example in Ref. [62] 40 samples is only 0.7171%, while the normal concentration should be 0.7202 ± 0.0006%. The depletion of 235 U resembles the used nuclear fuel from nuclear reactors. It has been shown [63] that besides functioning as a thermal neutron reactor moderated by underground water, the Oklo reactor also functioned as a fast-neutron breeder reactor. The existence of Oklo reactor demonstrates the possibility of naturally occurring nuclear fission reactors. 1.3.3 Georeactor Implication Herndon suggested a few implications of the existence of a georeactor at the Earth’s core. One prediction is that the georeactor serves as the the energy source to sustain the geomagnetic field, and interruption of georeactor operation is the cause of the geomagnetic field reversals. The Earth’s magnetic field is known to have existed for at least 2.7 billion years from the studies of the oldest rock samples. It extends a few tens of thousands of kilometers into space, known as magnetosphere, and shields the the surface of the Earth from charged particles of the solar wind. The geomagnetic field is approximately a magnetic dipole inclined by ∼11.3◦ relative to Earth’s axis of rotation. The cause of the geomagnetic field is traditionally explained by the dynamo theory [64], stating that the rotating, convecting and electrically conducting fluid iron outer core acts to maintain the geomagnetic fields. The energy needed to sustain such geodynamo is at least 0.02 TW [65]. Herndon proposed [1] that this energy is primarily produced by the georeactor at the Earth center, with the inner core serving as heat sink, through which heat is transferred to the base of the fluid shell, creating thermal convection in the electrically conducting fluid core that generates the geomagnetic field. The direction of the geomagnetic field, based on the study of lava flows of basalt, is known to reverse very irregularly, with average intervals of ∼250,000 years. The last reversal happened about 780,000 years ago. Simulation of the dynamo theory [66] 41 showed that the convection of the fluid core is constantly trying to reverse the field, while the inner core tries to prevent it. The frequency of the the simulated reversal, however, is ∼100,000 years and seems to be slightly too high. Herndon proposed an alternative explanation [1, 60] from the georeactor hypothesis, stating that the georeactor could be temporarily shut down if some fission fragments such as 149 Sm, which have a high neutron capture cross section and would poison the reactor, are not removed out of the sub-core region readily. After a period of time for the reactor poisons to diffuse to regions of lower density, the reactor output would resume, and the Earth’s magnetic field would re-establish itself, either in the same direction or in the reverse direction. Herndon also proposed that the georeactor is the origin of the 3 He/4 He anomaly problem [60, 55]. The atmospheric 3 He/4 He ratio is measured to be RA = 1.4 × 10−6 . It is discovered that Helium is venting from the Earth’s interior [67], but the 3 He/4 He ratio of the helium released to the oceans at mid-oceanic ridges is about 8 RA . Iceland plume 3 He/4 He ratio is found to be as high as 37 RA . The extra 3 He from the Earth’s interior is traditionally believed to be of primordial origin, that the 3 He was trapped within the mantle at the time that the Earth formed and not well outgassed. The primordial 3 He/4 He ratio however is about 100 RA , inferred from gas-rich meteorites [68], which is too high compared with the helium released from the Earth’s interior. The dilution is generally assumed to be from the 4 He produced by the natural radioactive decay of U and Th in the mantle and the crust. Hollenbach and Herndon instead suggested [60] that the extra 3 He is the product and evidence of the georeactor, since 3 H being one of the fission products decays to 3 He, which could possibly travel out of the core region and being outgassed. A simulation with the SAS2 code [61] yields the similar present-day 3 He/4 He as those measured from the basalts along the global spreading ridge system, at about 4 – 8 RA [55]. 42 1.4 Motivation for a Combined Analysis One inevitable consequence of the georeactor hypothesis is the ν̄e flux coming out from the Earth core. Being the largest ν̄e detector on the Earth, KamLAND is well suited to measure this flux, or setting a limit of the georeactor power if the flux is not observed. A 10 TW hypothetical georeactor produces ∼30 events/year (oscillated) at KamLAND with the detector’s efficiency (∼0.88) and ∼700 tons of fiducial mass (details discussed in Section 5.2). On the other hand, KamLAND is optimized to measure the ν̄e flux from the man-made reactors, which unfortunately becomes background source for georeactor ν̄e flux measurements. However, the neutrino oscillation causes a difference in ν̄e spectrum shape between man-made reactors and georeactor, allowing disentanglement of the two. Furthermore, the total man-made reactor power varies with time from the maintenance and refueling of different reactors. In Fig. 5.3 we can see that the maximum variation can be as large as a factor of three. A notable period of low reactor power since the summer of 2007 is due to the shutdown of the 25 GWth Kashiwazaki nuclear power plants after an earthquake of magnitude 6.8 striking in Niigata prefecture. The time variation of ν̄e flux from man-made reactors can be distinguished from the georeactor, which is assumed to provide a constant power output over the time scale of the KamLAND experiment. The measured total heat flow from the Earth puts an upper limit on the georeactor thermal power to be 44.2 ± 1.0 TW [69]. A more recent evaluation of the same data (assuming much lower hydrothermal heat flow near mid-ocean ridges) however led to a lower value of 31 ± 1 TW [70]. From the study of chondrites the calculated radiogenic power (from the decay chains of natural radioactivities of 238 U, 232 Th and 40 K in the Earth) is thought to be 19 TW [18], a value that can be measured by KamLAND but currently limited by statistics [21]. Some other models of mantle convection suggest that radiogenic power is a larger fraction of the total power [19, 20]. On the other 43 hand, the georeactor theory itself does not have a well-predicted output power. The maximum steady-state fission power is determined to be 30 TW, using the maximum uranium content at 4.5 billion years ago [59]. The current poor constrain on the georeactor power can be much improved with KamLAND’s large sample of ν̄e events from about 5 years of data-taking. The ν̄e ’s coming out of the center of Earth follows the same neutrino oscillation physics laws as all other neutrinos do. A combined analysis of georeactor and neutrino oscillation is thus inevitable. On the other hand, this analysis is in turn going to put a stronger test of the neutrino oscillation theory, and demonstrating whether or not a significant change in the current best-fit neutrino oscillation results is needed by including the georeactor hypothesis. Similar work has been done before with either fewer statistics or lack of full treatment of systematic uncertainties [71, 72], issues that this dissertation is trying to resolve and improve. 44 Chapter 2 The KamLAND Experiment KamLAND (Kamioka Liquid Scintillator Anti-Neutrino Detector) experiment is located in the Kamioka Underground Laboratory under the peak of Mt. Ikenoyama (36.42 ◦ N, 137.31 ◦ E) in Gifu Prefecture, Japan, about 50 km south of the city of Toyama. The experiment site is in the middle of Japan and surrounded by ∼55 nuclear reactor cores with total thermal power ∼130 GW and a flux weighted average distance ∼180 km. This puts KamLAND in an optimal position to directly observe the neutrino oscillation in vacuum associated with LMA-MSW solution (Section 1.2.2). KamLAND has been taking data since January 2002, and has made major discoveries in various scopes of science ever since. 2.1 Anti-neutrino Detection Method KamLAND is a liquid scintillator detector for detecting energetic particles. The energy loss mechanism in matter varies with particle types [7], but eventually (part of) the energy of the particle is transfered to the free valence electrons of the scintillator molecules, and excite them from the lowest states (S0 ) to the excited states (S∗ , S∗∗ ). The excitations decay immediately to the S∗ state without the emission of radiation (“internal degradation”). The S∗ state then decays to one of the vibrational states of S0 , with emission of “fluorescence” photons. The fluorescence photon travels through the detector with the probability of incident on a photomultiplier tube (PMT) in the end. The PMT converts the light into a weak current of photoelectrons which is then further amplified by and electron-multiplier system. The resulting current signal is then analyzed by the dedicated electronics system. 45 Prompt Event Delayed Event γ (0.511 MeV) _ νe ee+ γ (0.511 MeV) p d n n p γ (2.223 MeV) ~ 200 μs Figure 2.1: Illustration of how KamLAND detects ν̄e from inverse beta decay reaction. The prompt event from positron annihilation and the delayed events from neutron capture are correlated in ∼ 200µs. Due to the extremely weak interactions between (anti-)neutrinos and matter, KamLAND only detects ν̄e through a specific channel called inverse beta decay: ν̄e + p → n + e+ (2.1) The reaction generates a neutron and a positron. The positron quickly deposits its energy (together with two annihilation γ’s) in the detector and gets detected as a “prompt” event. The neutron loses its energy primarily by elastically scattering on 1 H (“thermalizes”) until reaching thermal energy, then gets captured by 1 H and emits a 2.2 MeV γ-ray. The capture time follows an exponential distribution with mean capture time ∼206 µs (Section 4.6), so this 2.2 MeV γ-ray is detected as a “delayed” event. Fig. 2.1 shows a visual description of this process. KamLAND detects ν̄e ’s from reactors at a very low rate less than 1 per day, however, the “prompt-delay” coincidence signature is unique to ν̄e events and makes ν̄e detection possible even with background event rate millions times higher. Details of the ν̄e signals and backgrounds will be discussed in Chapters 4 and 5 46 The energy of the ν̄e must be above a threshold for the inverse beta decay reaction to happen, and is given by: (Mn + Me )2 − Mp2 Ethreshold = = 1.806 [MeV] 2Mp (2.2) The cross section of inverse beta decay is given in Ref. [73] and is shown as a function of ν̄e energy in Fig. 5.2. The cross section can be expressed, neglecting terms of R order of Eν /M , in terms of the neutron lifetime and the phase-space factor fp.s. = 1.7152 [74] as 2π 2 /m5e (0) (0) (0) σtot = R E p fp.s. τn e e (2.3) In this way, the cross section is tied directly to the neutron lifetime, known to 0.1% [7]. The small (order Eν /M ) energy-dependent outer radiative corrections to σtot [75, 76] are not negligible and are included in the calculation shown in Fig. 5.2. The total systematic uncertainty from the inverse beta decay cross section is estimated to be 0.2%. The energy of the ν̄e can be calculated from the total energy of the prompt event, which includes the kinetic energy of the positron and two 0.511 MeV gamma-rays, as follows: Eν̄e = Mn + Tn + me + Te+ − Mp = Te+ + 2me + Tn + (Mn − Mp − me ) = Eprompt + Tn + 0.782 [MeV] (2.4) The neutron recoil energy Tn depends on the scattering angle, and in current analysis the average hcos θi from the positron angular distribution of inverse beta decay [73] is used. Tn is in average only ∼10 KeV though so it can be safely neglected. Detection of ν̄e events at KamLAND gives a real-time measurement of the ν̄e flux 47 and its energy spectrum, which can be expressed by: d2 Nν̄e (Ep , t) = dEp dt Z ∞ · ′ ′ ′ dEν̄e R(Ep , Eν̄e )ε(Ep )ntarget σ(Eν̄e ) · 0 sources X ′ ′ fi (Eν̄e , t)Pν̄e →ν̄e (Eν̄e ; ∆m221 , θ12 ) (2.5) i ′ where Ep is the prompt energy deposited in the detector by the ν̄e event. R(Ep , Eν̄e ) is the detector energy response function, including both the conversion function from ν̄e energy to prompt energy (Eqn. 2.4) and the detector energy resolution function. ε(Ep ) is the possible energy dependent detection efficiency. ntarget is the total number ′ ′ of target protons. σ(Eν̄e ) is the inverse beta decay cross section. fi (Eν̄e , t) is the instantaneous flux of ν̄e source i at KamLAND, which includes ν̄e ’s from nuclear reactors, the hypothetical georeactor and various background events that mimic the ′ ν̄e event (Chapter 5). Pν̄e →ν̄e (Eν̄e ; ∆m221 , θ12 ) is the energy dependent ν̄e survival probability given by Eqn. 1.11. Equation 2.5 will be used to extract the neutrino oscillation parameters ∆m221 and tan2 θ12 from the KamLAND data. 2.2 Detector KamLAND detector is located in the Kamioka Underground Laboratory under the peak of Mt. Ikenoyama in the existing cavity built for the old Kamiokande detector. The average vertical rock overburden is 2700 meter water equivalent (m.w.e), which reduces the cosmic-ray muon rate by a factor of ∼ 105 compared to at the surface [77] to about 0.34 Hz in the detector. Besides the detector itself, other facilities such as liquid scintillator purification system, water system, material counting facilities, etc, are also located onsite as shown in Fig. 2.2. 48 Mt. Ikenoyama 1000 m Electronics Hut KamLAND HV Room R Control oom ystem ation S Purific r e t a W Liquid Sc intillator Pu rification Detector System Figure 2.2: KamLAND Site Calibration Device Containmenet Vessel (diam. 18m) Balloon (diam. 13m) Liquid Scintillator (1 kton) Outer Detector (Water Cherenkov) ID PMT Buffer Oil OD PMT Figure 2.3: Schematic Diagram of the KamLAND detector. 49 2.2.1 Detector Design Overview Fig. 2.3 shows the the schematic diagram of the KamLAND detector. The primary active region for neutrino detection is ∼1 kton of ultra-pure liquid scintillator (LS) at the center of the detector contained in a 13m diameter spherical balloon made of 135 µm thick transparent nylon/EVOH (ethylene vinyl alcohol copolymer) composite film and supported by a network of Kevlar ropes. The region between the balloon and the surrounding 18m diameter spherical stainless steel outer vessel is filled with ∼ 1800 m3 buffer oil (BO) comprising of 57% isoparaffin and 43% dodecane to shield the LS from external radiations. An array of 1325 17-inch photomultiplier tubes (PMTs) and 554 20-inch PMTs are mounted on the inner surface of the outer containment vessel, providing 34% photocathode coverage (22% with 17-inch PMTs only). A 3mm thick spherical acrylic barrier at 16.6m diameter helps preventing Rn emanating from the PMT glass from entering the BO. The LS and BO regions together make up the inner detector (ID). The stainless steel vessel is surrounded by a 3.2 kton water Cherenkov detector instrumented with 225 20-inch PMTs, which is called outer detector (OD). The OD absorbs external γ-rays and neutrons from surrounding rock and enables the tagging and veto of cosmic-ray muons. On the top of the detector is the calibration device. Calibration sources can be lowered through the concentric chimney region into the LS. 2.2.2 Detector Properties Liquid Scintillator KamLAND liquid scintillator comprises of 80.2% normal-paraffin (Dodecane, C12 H26 , 0.7526 g/cm3 at 15 ◦ C) and 19.8% pseudocumene (1,2,4-Trimetthylbenzene, C9 H12 , 0.8796 g/cm3 at 15 ◦ C) by volume, and 1.36 ± 0.03 g/liter of the fluor PPO (2,5Diphyenyloxazole, C15 H11 NO). The density of the LS is measured to be 0.7775 g/cm3 50 at 15 ◦ C (further discussion in Section 4.4). Pseudocumene is an aromatic solvent with flashing point at 54 ◦ C. The adding of normal-paraffin increases the flashing point of the LS to 64 ◦ C, above the (60 ◦ C) requirement mandated by the mine operations. High purity nitrogen flow is kept on the top of the LS to prevent the oxygenation. The LS composition is adjusted to maximize the light yield for central events. The actual light output is 49% anthracene (8300 photons/MeV). The attenuation length of the LS is measured to be 10 meter at 400 nm wavelength. The light yield is ∼300 p.e./MeV for 17-inch PMTs only and ∼500 p.e./MeV for 17+20-inch PMTs. For more details of the LS properties refer to Ref. [78]. The LS is designed to have good pulse-shape discrimination (PSD) between scintillation produced by α particles, γ-rays and neutron recoils at the Test Bench Facility [78], so that further suppression of background is possible. Unfortunately, probably due to the large size of the KamLAND detector and the absorption and re-emission of the light, the PSD is not obvious from the actual KamLAND data [79] and is not implemented in the analysis described in this dissertation. Buffer Oil The buffer oil comprises of 57% isoparaffin (Cn H2n , n∼14) and 43% normal-paraffin and is contained in a 18m diameter stainless steel vessel, separated from the LS region by the balloon. The BO shields the LS from external radiations such as from PMT glass and the stainless steel, and attenuates spallation products outside of the LS. The density difference between the LS and the BO is less than 0.04 % to maintain the spherical balloon shape. The index of refraction is 1.46 in the LS and 1% lower in the BO at 400 nm. To reduce the internal radioactivities mainly from decay chains of 238 U, 232 Th and 40 K (Appendix B), both LS and BO are purified before filling into the detector. Tech- niques such as ultra-filtration, water extraction and nitrogen purge are utilized [80]. 51 Table 2.1 summarizes the measured radioactivities of the LS and BO after the purification. Activities of 238 U, 232 Th and 40 K in the LS are measured by KamLAND detector itself with event tagging. Radioactivities of the BO are measured by the neutron activation analysis (NAA) [81]. Both meet the low background requirement for ν̄e detection. Table 2.1: Radioactivities of 238 U, 232 Th and 40 K for selected materials in KamLAND. 238 232 40 Material U Th K Concentration LS [g/g] 2.7 × 10−18 6.1 × 10−17 1.7 × 10−16 −15 −15 BO [g/g] < 8 × 10 < 2.8 × 10 < 1.3 × 10−15 Balloon [ppb] 0.018 0.014 0.27 Rope [ppb] 0.08 0.8 1.2 PMT Glass [ppb] 150 240 10 SUS Tank [ppb] 0.5 0.7 0.1 Rock [ppb] ∼1000 ∼2500 ∼2000 Activity 3 LS [Bq/m ] 2.6 × 10−8 1.2 × 10−7 3.5 × 10−5 3 −5 −6 BO [Bq/m ] < 7.7 × 10 < 5.5 × 10 < 2.7 × 10−4 Balloon [Bq] 0.02 0.006 7.2 Rope [Bq] 0.1 0.33 31 Balloon and Ropes The KamLAND balloon, which separates the LS region and BO region, is made of a five-layer composite film as EVOH/nylon/nylon/nylon/EVOH, with total thickness of 135 µm. The nylon provides strength and compatibility with the oil and scintillator, and the EVOH has very small gas permeability so it is used to protect the LS against the diffusion of ambient radon from the surrounding components. The light transparency of the balloon is more than 90%. There are 44 longitudinal and 30 lateral Kevlar (a light strong para-aramid synthetic fiber) ropes to support the balloon and keep the balloon’s spherical shape. The tension of the ropes is monitored by a set of load cells at the top of the detector. The liquid levels of the LS and the BO are also 52 carefully monitored to keep enough pressure inside the balloon to maintain its shape. The radioactivities of the balloon and the Kevlar ropes are measured by inductively coupled plasma mass spectrometry (ICP-MS), and are summarized in Table 2.1. Thermometers Three thermometers were initially attached at the top, center and bottom of a vertical 3.5 mm diameter teflon cable slightly off the central z-axis of the detector, reaching into the 5.5m radius inside the LS. The cable and the attached thermometers were later removed on 19 April 2004 because of their slightly high radioactivities seen by the KamLAND detector. Photomultiplier Tubes The ID has 1325 specially developed fast PMTs (Hamamatsu R7250) masked to 17inch diameter and 554 older 20-inch diameter PMTs (Hamamatsu R3600) reused from the Kamiokande experiment [82]. The 20-inch PMTs were not operated until after 27 February 2003. The 20-inch PMT uses venetian-blind type dynode shape and its time-transit-spread (TTS) is ∼5.5 ns. The 17-inch PMT uses line-focus type dynode shape and improves the TTS to ∼3 ns. The peak-to-valley ratio is also improved from ∼1.5 to ∼3.0. The quantum efficiency is about 22% in the wavelength from 350 nm to 400 nm. Earth’s magnetic field (∼0.5 Gauss) affects the electrons’ trajectories and degrades the PMT gain. A set of compensating coils is placed in the cavern surrounding the detector to cancel the magnetic field to a level well below the 50-mGauss limit necessary for proper operation of the PMTs. 53 Water Cherenkov Outer Detector The OD water comes from a fountain spot and goes through the “natural” filter of the 1000m rock of Mt. Ikenoyama to reach KamLAND. It is constantly circulated at ∼5 tons/hour by the water purification system with ultra-filtration, reverse osmosis (RO) membrane, and UV sterilizer system before sending to KamLAND. The 238 U and 232 Th concentration in the water has been measured by ICP-MS to be both less than 0.5 ppt [81]. The Rn activity however is high in the water at ∼10,000 Bq/ton. The temperature of the water is quite stable (∼ 10 ◦ C), and circulation of OD water removes excess heat produced by the PMTs in the ID and OD. The OD is divided by the stainless steel containment vessel and Tyvek plastic sheets into four optically separate sections: top, upper, lower and bottom, mounted with 50, 60, 60 and 55 20-inch PMTs (same type 20-inch PMTs as in the ID) respectively. The OD detects cosmic-ray muons by the Cherenkov light they produced when passing through the water. The Tyvek sheets are highly reflective to optimize the light collection in each section. The tagging of muon by the OD provide extra veto against muon spallation background (Section 4.3). Muon Tracker A muon tracking system is installed in September 2008 on top of the KamLAND detector to track a subset of the muons incident onto the KamLAND detector. This helps to measure the OD muon tagging efficiency, and calibrate the muon reconstruction algorithms (Section 3.5). The muon tracker consists of 48 multiwire proportional chambers (MWPCs) with an active area of 5.4 m2 . It is triggered by coincidence of two plastic scintillator planes mounted on top of the higher and lower planes of MWPCs, whose relative positions are set up to be movable to allow a greater range of impact parameters relative to the center of KamLAND detector. 54 2.2.3 Data Acquisition Fig. 2.4 shows the schematic diagram of the KamLAND data acquisition flow. The PMTs (or “channels”) send their analog signals to the Front End Electronics (FEE) boards, and the decision of whether or not to digitize the signal is made by the trigger system based on the number of “hit” channels (Nchannels ) sent to the trigger system by each FEE board. The trigger system also sends a 40 MHz clock signal to each FEE board for synchronization. The DAQ system then reads out the digitized waveforms from the FEE and the trigger records from the trigger system asynchronously, compresses the data and stores them onto the disks. The DAQ also sends the run conditions to the FEE and Trigger system and can change their behaviors based on different situations. N channels PMTs Signals FEE Trigger Digitize? Run Conditions Nsum Waveforms Disks Data DAQ Figure 2.4: Schematic diagram of the KamLAND data acquisition process. Front End Electronics The KamLAND FEE system consists of 200 FEE boards hosted in 10 VME crates, and each board supports 12 channels. All FEE boards are running at 40 MHz with the internal clock distributed by the trigger system. The analog PMT signal is split and one copy is sent to a discriminator whose threshold is set at 0.15 p.e. (∼0.3 mV, single p.e. efficiency >95%). The other copy is sent to a delayed line which leads to an Analog Transient Waveform Digitizer (ATWD) [83] chip on the board. If the 55 signal crosses the discriminator threshold, the ATWD starts to capture the PMT signal in 128 samples at ∼0.65 GHz frequency, resulting in about 200 ns of data in each waveform. The ATWD holds the waveform for 175 ns while awaiting for the trigger decision, based on which the ATWD either erases the waveform within ∼1 µs or digitizes the waveform with a 10-bit ADC in ∼27 µs. To reduce the dead time, each PMT is connected to two ATWDs, so that one can acquire a waveform when the other is busy1 . Each ATWD has three gains (×20, ×4, ×0.5) so that a wide dynamic range of signals can be recorded. While only the high gain waveform is recorded for a typical low energy event, a high energy particle such as a muon could produce a signal as large as 1 V and saturate the high and medium gain, in which case all three waveforms are recorded2 . Each waveform is associated with a time-stamp as the number of clock ticks (25 ns) between the start of the run and the the issuing of trigger digitization command. Each waveform is also associated with a clock tick offset of the launching of the ATWD to the trigger command (Nlaunch , which is always ≥ 0 meaning before the trigger). This time information allows for synchronizing all different waveforms and trigger records at a later offline event reconstruction stage. The digitized waveforms from all 12 channels on a FEE board are buffered onto a 32 MB on-board memory and read out through the VME bus by the DAQ system. Trigger System The ID PMT’s dark rate is on the order of tens of kHz at the set threshold. Since each waveform requires 256 Byte of memory, it is impossible to record all the data from all channels with the current computer technology and resources. Instead, a trigger system was designed at Stanford Univ. with Field Programmable Gate Arrays 1 A secondary electronics system named MoGURA is designed to have zero dead time through the use of free-running high-speed flash ADCs, and is installed to KamLAND in August 2009. 2 After Oct 2007, a decision has been made that only the lowest-gain waveform is kept, in order to reduce the data transfer overhead after muon events. 56 (FPGAs) to allow varying complexity of triggering algorithms, and is responsible for making decisions of whether or not to record the data. Every clock tick (25 ns) each FEE board sums up the total number of channels (Nchannels ) on the board that exceed the discriminator threshold in the past 5 clock ticks (125 ns), and sends to the trigger system. The trigger module calculates the sum of all Nchannels in each section of the detector: ID (nsumID 3 ) and OD (nsumODtop , nsumODupper , nsumODlower , nsumODbottom ), compares with different trigger threshold (“trigger type”), and issues digitization command to the ATWD’s if the criteria of the trigger type is met. At the same time a trigger record is generated with the timestamp, trigger type and nsum information, which is read out by the DAQ system separately. The primary trigger type in KamLAND normal runs is the “prompt trigger,” which issues a global acquisition command to FEE boards if nsumID exceeds 200 or 180 (before and after 13 April 2004, respectively)4 . This trigger threshold corresponds to ∼0.8 MeV deposited energy. The trigger module also issues “OD trigger” when nsumOD exceeds the threshold set for each OD section (6, 5, 6, 7 for top, upper, lower and bottom section respectively). A special trigger type is the “supernova trigger” for optimizing the detection of the rare supernova events. If the trigger module sees 8 events with nsumID > 772 or 1100 (before and after February 2004, respectively) in ∼0.84 s , it then sets optimal trigger parameters to collect as many supernova candidate events as possible for 1 min before returning to the initial trigger settings, during which the possible manual stopping of the run is disabled. Other common trigger types are listed below. For a complete description of trigger types and trigger electronics refer to Ref. [84, 79]. 3 nsumID only includes the 17-inch PMTs in the sum. The trigger thresholds listed are taken from the nominal values used in this analysis periods only. In later runs particularly during the KamLAND solar phase, the thresholds are typically lowered, allowing for more efficiency in acquiring low energy events. 4 57 • Delayed Trigger: A global acquisition command to FEE boards if nsumID > 120 and a prompt trigger has occurred within 1 ms. This is useful for more efficiency in capturing coincidence events. • Prescale Trigger: A global acquisition command to FEE boards if nsumID is above threshold and the absolute time is within the prescale time from the last 1PPS signal. This is useful when the event rate is too high for the DAQ to handle, such as in background runs when very low energy events are recorded. • 1PPS Trigger: A global acquisition command to FEE boards (can be disabled by DAQ) when the trigger module receives a One Pulse Per Second (1PPS) input from the GPS module. Some trigger types only output trigger records without issuing acquisition commands to the FEE boards. One example is the “history” trigger record which is generated for up to 8 clock ticks (200 ns) when nsumID exceeds 120. The maximum value of nsumID in the train of the history records accompanying an event is called nsummax, and is highly correlated with the energy of the event. This nsummax value is very useful in a low-level data analysis independent of the complexities of the event reconstruction algorithms, as will be discussed in later chapters. KiNOKO DAQ System The DAQ system is controlled by the KiNOKO software, which stands for “Kinoko Is Network distributed Object-oriented KamLAND Online-system”. KiNOKO initializes all the hardware components at the beginning of the run, and sends run conditions to the FEE and trigger systems. KiNOKO asynchronously reads out the data from the FEE and trigger VME crates, compresses the data by a factor of 3 with lossless Huffman encoding [85], and stores the data onto the disks. KiNOKO also does simple online analysis and provides user interfaces for controlling and monitoring the 58 run behaviors in real time. For detailed information about KiNOKO system refer to Ref. [86]. Other Electronics Systems In addition to the three main components of data acquisition system , there are some other necessary sub-systems for operating the KamLAND experiment including: • Trigger Backup System: trigger records are duplicated and independently stored by the trigger backup system. This is useful in case the main DAQ system is overwhelmed by the possible high data rate, for example caused by the neutrino burst from a supernova explosion. • Absolute Time Acquisition System: the absolute time is obtained from a Global Positioning System (GPS) receiver located outside of the experimental tunnel, accurate to ∼150 ns. The encoded time is sent to the GPS VME interface through optical fibers and a One Pulse Per Second (1PPS) signal is sent to the trigger system. The absolute time of an event can be calculated with the difference in the time-stamps between an event and its nearest 1PPS trigger. This is useful for (dis-)associating an event with other known astronomical transient events. For details of the trigger backup system and absolute time acquisition system refer to Ref. [84]. 2.2.4 Calibration System Radioactive sources and light flashers (laser and LED) are periodically deployed into the KamLAND detector by the calibration system in order to understand the properties of the detector such as light output and propagation and to check the detector’s stabilities. The calibrations also provide references for the event energy and position 59 reconstruction algorithms, and are essential to understand the event reconstruction’s performances. Table 2.2 lists the radioactive sources used for KamLAND calibrations. Among them 68 Ge, 60 Co, 203 Hg, 137 Cs and 65 Zn are γ sources at various energies. Unmoderated 241 Am9 Be and 210 Po13 C sources serve to calibrate the detector response to gammas and neutrons. 210 Po13 C source [87] is also specially used to understand the 13 C(α,n)16 O reaction, one of the largest background for ν̄e detection, in KamLAND (Section 5.4). Table 2.2: Summary of Radioactive Calibration Sources at KamLAND Source (half life) Scheme Particle Type & Energy [MeV] 68 68 Ge (271 d) Ge + e− −→ 68 Ga + νe 68 Ga → 68 Zn + e+ + νe 2γ (0.511 ×2) 60 60 60 − Co (5.3 y) Co → Ni + e + ν̄e 2γ (1.173 + 1.332) 203 203 Hg (46.6 d) Hg → 203 Ti + e− + ν̄e γ (0.279) 137 137 137 − Cs (30.1 y) Cs → Ba + e + ν̄e γ (0.661) 65 65 Zn (244.3 d) Zn + e− −→ 65 Cu + νe γ (1.116) 241 241 Am9 Be (432.2 y) Am → 237 Np + α α + 9 Be → X + n neutrons (< 10) γ (2.223, 4.945) from n capture γ (4.439, 7.654, 9.641) from 12 C∗ 210 13 210 206 Po C (138.4 d) Po → Pb + α 13 α + 13 C → 16 O + n C(α,n)16 O reaction (Section 5.4) All parts in the calibration systems that go into the KamLAND liquid scintillator need to be certified as free of radioactivities. This is done at the on-site counting facilities with rigorous certification procedures. Usually parts are soaked in acid solution and samples of the acid are counted with the on-site germanium detector. For details of the on-site material certification refer to Ref. [88]. 60 Z-Axis System There are two calibration systems used in the KamLAND experiment5 . Before December 2005, all calibration sources were deployed with the Z-Axis system, which positions the sources along the central z-axis from -6m to 6m relative to the center of KamLAND. The system used a motor to unspool a stainless steel cable and a pulley encoder to measure the amount of cable payed out. The accuracy and the reproducibility of the source position is about 2 – 3 mm. 4π Full Volume System The 4π full volume system was installed in December 2005, with the purpose of positioning radioactive sources throughout the active volume of the detector and testing the reconstruction performances in the full volume. This is achieved by attaching the sources to a segmented pole which is suspended by two cables near each end of the pole. By changing the relative length of the two cables a sweep in zenith angle can be performed. By rotating the entire glovebox (mounted on a rotary stage) on top of the detector, where the calibration system is stored, a sweep in azimuthal angle can be performed. The absolute position of the source is accurate to less than 5 cm, and the relative position between the sources in the pole segments is accurate to a few mm. For details of the 4π system refer to Ref. [88]. 2.3 KamLAND History and Status 2.3.1 Achievements in Reactor Phase KamLAND finished installing major components and started taking data in January 2002. The period between January 2002 and April 2007 is called reactor phase, 5 A third calibration system named MiniCal was installed in February 2009 to meet more stringent low background requirements in the KamLAND solar phase. 61 (a) (b) Figure 2.5: Evidence of neutrino oscillation in KamLAND [22] (a) Observed prompt energy spectrum in KamLAND compared with no-oscillation prediction. (b) The ratio of background-subtracted ν̄e events to no-oscillation expectation as a function of L0 /Eν̄e , where L0 = 180 km is the flux-weighted effective baseline. during when the primary goal is to detect ν̄e ’s from the reactors and study the neutrino oscillation theory. KamLAND has made major discoveries in the reactor phase contributing to various scopes of science including neutrino oscillation, solar antineutrino, earth radiogenic heat model, nucleon decay, muon spallation products, etc. Precision Measurement of Neutrino Oscillation Parameters KamLAND has established strong evidences of neutrino oscillation by observing a more than 5σ disappearance of reactor ν̄e flux and distortion of the reactor ν̄e spectrum [45, 46, 22]. Fig. 2.5 ([22]) shows the energy spectrum observed in KamLAND compared with no-oscillation prediction, and the ratio of background-subtracted ν̄e events to no-oscillation expectation as a function of L/Eν̄e . The evidence of spectrum distortion and the oscillatory behavior of the survival probability is clear. Fig. 2.6 shows the allowed contours in (∆m221 , tan2 θ12 ) space, assuming that the neutrino oscillation is the explanation of the observed results. KamLAND alone disfavors the other solar neutrino solutions at 4σ, allowing only the LMA-I region (Section 1.2.2). Combined with the results of solar experiments assuming CP T in- 62 Figure 2.6: Allowed region for neutrino oscillation parameters in KamLAND and solar experiments [22]. The side-panels show the ∆χ2 -profiles for KamLAND (dashed line), solar experiments (dotted line) and the combination of the two (solid line). variance, a precision measurement of the neutrino oscillation parameters is achieved −5 eV2 and tan2 θ12 = 0.47+0.06 as ∆m221 = 7.59+0.21 −0.21 × 10 −0.05 . High Sensitivity Search for Solar ν̄e KamLAND has done a high sensitivity search for ν̄e from the sun [13]. There are several conceivable mechanisms which would lead to a ν̄e component in the solar flux incident on Earth, such as spin-flavor precession combined with neutrino oscillations and neutrino decay. However the observed ν̄e events in the high energy window 8.3 MeV < Eν̄e < 14.8 MeV is consistent with the background expectations, which leads to an upper limit of solar ν̄e flux of 3.7 × 102 cm−2 s−1 at 90% C.L. Combined with the 63 expected solar 8 B νe flux from Standard Solar Model, this ν̄e flux limit corresponds to an upper limit on the neutrino conversion probability of 2.8 × 10−4 at 90% C.L. and represents a factor of 30 improvement over the best previous measurement [89]. Investigation of Terrestrial Anti-neutrino KamLAND has shown the first indications of the geologically produced ν̄e ’s, or geoneutrinos, from the decay chains of natural radioactivities (mainly 238 U and 232 Th) in the Earth [21]. Fig. 2.7 ([21]) shows the observed ν̄e energy spectrum in KamLAND especially in the low energy window 1.8 MeV < ν̄e < 3.4 MeV where the geo-neutrinos contribute. The data fit well with the expectations from a reference model of 16 TW radiogenic power form 238 U and 232 Th, which is approximately half of the total measured heat dissipation rate from the Earth. From the data an upper limit of 60 TW for the radiogenic power of U and Th in the Earth is put, a quantity that was previously poorly constrained. Search for Invisible Neutron Decay Some of the possible nucleon decay modes are called invisible modes, such as n → ν’s and nn → ν’s, by observing which would indicate new physics beyond the Standard Model. KamLAND has good sensitivity for these invisible modes by searching for single neutron or two neutron intra-nuclear disappearance that would produce holes in the s-shell level of 12 C nuclei. The de-excitation of the corresponding daughter nucleus results in a sequence of time and space correlated events. The observed number of such events is consistent with the expectation from accidental coincidence background, which leads to the current best lifetime limits of the invisible nucleon decay mode τ (n → inv) > 5.8 × 1029 years and τ (nn → inv) > 1.4 × 1030 years [90]. 30 Events / 0.17MeV 64 a Events / 0.17MeV 25 20 b 15 10 20 5 0 2 4 6 8 Anti-neutrino energy, Eν (MeV) 15 10 5 0 1.8 2 2.2 2.4 2.6 2.8 Anti-neutrino energy, Eν (MeV) 3 3.2 3.4 (a) 100 CL 68.3% CL 95.4% CL 99.7% 5 60 Δχ2 NU+NTh a 80 4 CL 95.4% 3 CL 90.0% 40 2 20 1 0 -1 -0.5 0 0.5 (NU-NTh)/(NU+NTh) 0 0 1 b CL 68.3% 20 40 60 NU+NTh 80 (b) Figure 2.7: Geo-neutrinos in KamLAND [21]: (a) Observed ν̄e energy spectrum in KamLAND (black dots) with the total expectation (thin dotted black line) and total expectation excluding the geoneutrino signal (thick solid black line). The colored lines are expected 238 U geoneutrinos (dot-dashed red), expected 232 Th geoneutrinos (dotted green), reactor ν̄e background (dashed light blue), 13 C(α,n)16 O background (dotted brown) and random coincidence background (dashed purple). The inset panel extends the spectra to higher energy. (b) Confidence levels for detected 238 U and 232 Th geoneutrinos (left) and ∆χ2 -profiles for the total number of 238 U and 232 Th geoneutrinos (right). The grey band gives the prediction by geophysical model. 65 Study of Cosmic Muon Spallation Products KamLAND has provided valuable data for the production yields of the isotopes from cosmic muon initiated spallation [91]. The neutron production yield is measured to be (2.8 ± 0.3) × 10−4 n/(µ · (g/cm2 )), and other nuclear isotopes’ production yields are listed in Table V of Ref [91]. The understanding of cosmic muon spallation products is essential for future rare event detection such as neutrino experiments, double beta decay experiments and dark matter searches. 2.3.2 Solar Phase and Purification of Liquid Scintillator After completion of the primary goal in the reactor phase, KamLAND went through major purification of the liquid scintillator from May 2007, with the goal of reducing the intrinsic radioactive background in the LS mainly from 210 Pb and 85 Kr, and make a direct measurement of 7 Be solar neutrinos. The period from May 2007 to present is referred to as solar phase, or low background phase. It is worth mentioning that all the previous programs in the reactor phase will also benefit from the lower radioactive background, and will co-exist with the solar neutrino program with increasing sensitivities and precisions. The 7 Be νe from the sun is mono-energetic at 862 KeV. It elastically scatters on an electron through both CC and N C interactions6 : νe + e− −→ νe + e− (2.6) and the recoiled electron gives a “Compton” spectrum with maximum energy of 665 KeV. The νe elastic scattering channel, unlike the ν̄e inverse beta decay, only gives a single e− event, and is indistinguishable from the β and γ decays of intrinsic 6 Same as in Super-Kamiokande experiment, the νµ and ντ from the oscillation of solar νe can only elastically scatter on electrons through N C interaction, with a cross section approximately 7 times smaller the νe . 66 Figure 2.8: The observed singles event rate as a function of deposited energy (Evis ) in KamLAND in the reactor phase, overlaid with the expected un-oscillated solar 7 Be ν̄e spectrum from Standard Solar Model. In the expected 7 Be νe signal window 0.3MeV < Evis < 0.8MeV, the beta decays of 85 Kr (green), 210 Bi (magenta) and alpha decays of 210 Po (cyan) dominate the event rate. radioactive impurities. Because of the lack of a good pulse shape discrimination ability, the e− event is also indistinguishable from a radioactive α decay event. Fig. 2.8 shows the observed singles event rate as a function of deposited energy (Evis ) in KamLAND in the reactor phase, overlaid with the expected un-oscillated solar 7 Be ν̄e spectrum from the Standard Solar Model. The expected un-oscillated 7 Be νe event rate in the signal window 0.3MeV < Evis < 0.8MeV is ∼400/day/kton, but the background event rate from beta decays of 85 Kr, 210 Bi and alpha decays of 210 Po overwhelms. 210 Bi and 210 Po are daughter nuclei of 210 Pb (Appendix B.1), who were introduced by means of 222 Rn contamination during the initial filling. The 85 Kr was also introduced at the initial filling, whose activity in the LS is ∼0.6 Bq/m3 , comparable to the 85 Kr activity in the atmosphere (Appendix A). Both 210 Pb (τ1/2 = 22 y) and 85 Kr (τ1/2 = 10.8 y) have long half-lives, so waiting for them to decay away is not practical. Table 2.3 lists the radioimpurity levels in KamLAND LS in the reactor phase, and the requirement needed in the solar phase in order to make a 7 Be solar neutrino measurement. While 238 U and 232 Th already meet the solar phase low 67 Isotope 85 Kr 210 Pb 40 K 39 Ar 232 Th 238 U Concentration [g/g] 5.5 × 10−20 2.3 × 10−20 1.7 × 10−16 < 4.3 × 10−21 6.1 × 10−17 2.7 × 10−18 Purification Goal [g/g] 10−26 10−25 10−18 10−24 10−16 10−17 Purification Factor ∼ 5 × 106 ∼ 2 × 105 ∼ 200 ∼ 1000 O.K. O.K. Table 2.3: Concentration of radioimpurities in KamLAND liquid scintillator in the reactor phase and purification goal for solar phase in order to make a 7 Be solar neutrino measurement. background requirement, 85 Kr and 210 Pb need to be reduced by a factor of ∼106 and ∼105 respectively, together with a factor of 1000 and 200 reduction in 39 Ar and 40 K. Following intensive R&D of purification method, the collaboration discovered that distillation effectively removes 210 Pb and 40 K, while ultra-pure nitrogen purging effectively removes gas contaminants such as 85 Kr, 39 Ar and 222 Rn. A custom LS distillation system and a nitrogen purge system (with an ultra-pure nitrogen production facility) were constructed at the KamLAND site to implement these purification techniques. Various auxiliary monitoring systems, such as Kr/Ar monitor (Appendix A), Rn monitor, light yield and attenuation length monitor, were developed to insure the purity and quality of the final purified LS products before filling them back into the KamLAND detector. The first full-scale purification campaign started from 20 May 2007, and continued until 1 August 2007. LS was drawn from the bottom of the detector, passed through the purification system and filled back into the top of the detector. Temperature and density of the new LS were controlled (warmer and less dense) to insure the boundary between the new and old LS to reduce the mixing effects. A total of 1700 m3 of LS was purified, among which 500 m3 in the KamLAND LS volume was effectively purified twice. A clear boundary between the once- and twice-purified LS volume can be seen from the data as shown in Fig. 2.9. The reduction factor for 210 Bi, 85 Kr and 210 Po in the twice-purified volume is measured to be ∼ 200, 40 and 5 respectively. Z [m] 68 103 8 6 4 2 102 0 -2 -4 -6 -8 0 10 0.2 0.4 0.6 0.8 1 1.2 1.4 (ρ/6.5m) 2 Figure 2.9: The vertex distribution of 85 Kr and 210 Bi events (0.4MeV < Evis < 1.2MeV) after the first purification campaign, from a one-day physics run 7830. The black line indicates the balloon position at 6.5m radius. The red line indicates the edge of the 6m fiducial volume used in this analysis. A clear boundary between the once- and twice-purified LS volume can be seen. The reduction factor does not agree well with the results from the monitoring system which samples LS emerging right after the distillation system, and shows that the 85 Kr and 222 Rn concentrations in those LS already meet the required levels for a 7 Be solar neutrino measurement (Appendix A). Further investigations revealed possible causes of the discrepancy including leaks around the detector chimney and convection of the LS during and after the filling. Following important improvements of the distillation system, including an ultrapure nitrogen over-pressured tent around the detector chimney, a heat exchanger to precisely control the LS temperature, and a new bottom-filling scheme to minimize the LS convection, the second purification campaign started on 16 June 2008 and continued until 29 January 2009. A full solar neutrino analysis is on the way and the result is promising [92]. 69 Data Set Used in this Dissertation The analysis described in this dissertation incorporates all the KamLAND data taken between April 2002 and June 2008, except the period between May 2007 and September 2007 during when the first purification campaign was on-going and the detector properties were unstable7 . The first purification campaign naturally divides the whole data set into two periods: Period I (before purification) and Period II (after purification). Due to the difference in liquid scintillator properties in the two periods, such as radiopurity, light yield, uniformity, etc. the analysis treats the two periods separately and combines the results together in the end. The total live time is 1432 days in Period I and 200 days in Period II. The full definition of the data set is summarized in Table 4.1. 7 The data taken after June 2008 is not used in this analysis, which includes the period during and after the second purification campaign. 70 Chapter 3 Event Reconstruction The raw data acquired by the DAQ system (Section 2.2.3) need to be reconstructed into meaningful events before doing any physics analysis. Fig. 3.1 shows the schematic flow diagram of the KamLAND event reconstruction process. The asynchronously collected raw waveforms and trigger records are first sorted in time and grouped into events (“event building”). The analysis then extracts the charge and time information of the event from its waveforms, which are further used in the vertex and energy reconstruction algorithms (“event reconstruction”). The reconstruction algorithms use the data from calibration runs such as 60 Co center deployments as references. Based on either the low level (algorithm-independent) information such as charge, time and trigger record, or high level information such as reconstructed energy and position, events are further categorized (“event tagging”) for future physics analysis. The properties of various events are used to check the reconstruction algorithms’ performances. 3.1 Event Building As described in Section 2.2.3, the DAQ system asynchronously reads out waveforms and trigger records and stores them onto disks. The event builder program then sorts the data and groups them into events with the same time-stamp. Given the size of each waveform (256 Byte) and the data transfer rate at a few MB/second, the memory usage of the event builder program is non-trivial. Waveforms that are read out a few minutes out-of-order (typically during noisy periods) cannot be sorted and have to be thrown away. The efficiency of the event building can be calculated from 71 simply counting the waveforms and trigger records for each time-stamp, and is found to be greater than 99.999%. 3.2 Waveform Analysis The ATWD samples a PMT signal at a frequency of ∼0.65 GHz, and stores them onto an array of 128 internal capacitors. The amount of charge on each capacitor is then digitized with a 10-bit ADC. This gives an array of 128 ADC values, spanning ∼200 ns (1.5 ns ×128), which is referred to as the waveform of the PMT signal. Pedestal Subtraction The pedestal of an ATWD refers to the base charge variation of the ATWD’s capacitors in absence of any photon signal. The pedestal adds structured-offset to the waveform and needs to be subtracted before any further waveform analysis. At the beginning of each run, 50 force-acquisition waveforms are collected for each ATWD. The pedestal waveform is obtained by simply averaging these waveforms, since most Raw Data Time Sorting Muon Events Events Spallation Events Waveform Analysis Events with Charge & Timing Radioactive Decay Events Reconstruction Calibration Data Reference Events with Vertex & Energy Event Categorization Anti-Neutrino Events Physics Analysis Figure 3.1: Schematic diagram of event reconstruction process 72 of them have zero-pulses given the PMT dark rate at ∼10 kHz1 . A common overall offset is subtracted so that only the variation between the samples is saved. An example of a raw waveform and the pedestal waveform of corresponding ATWD is shown in Fig. 3.2. Baseline Subtraction After subtracting the pedestal waveform, each waveform is found to be floating above a non-zero baseline. This baseline value is related to the activities of the board at the acquiring time, and has to be calculated for each waveform and subtracted from it. For small pulses the baseline value is obtained by iteratively calculating the mean value of the waveform and throwing away large deviation samples, until the sample deviation is smaller than 2.5%. For large pulses such as from muons, the sample values might not return to the baseline before the end of the waveform window, so the baseline value is simply calculated from the mean value of the first 10 samples. Pulse Finding After the pedestal and baseline subtraction, the waveform is smoothed with SavitzkyGolay filter [93] to remove high-frequency noises. The pulse finding algorithm then defines a candidate pulse as any contiguous area above zero in a waveform. Noise pulses are iteratively removed if the pulse’s total charge is less than 15% of the summed charges of all candidate pulses. An example of a two-pulse waveform is shown in Fig. 3.2(a). This algorithm however will count multiple pulses that overlaps above zero as one single pulse. One example is shown in Fig. 3.2(b). This algorithm is nevertheless used because of its simplicity and robustness, and further reconstruction algorithms are written in a way to minimize its drawbacks. 1 Samples that are 4σ away from the mean of the waveform are thrown away, which removes occasional real pulses. Raw Waveform 250 Pedestal Waveform 200 ADC Value ADC Value 73 Raw Waveform 250 Pedestal Waveform 200 Processed Waveform Processed Waveform 150 150 100 100 50 50 0 0 20 40 60 80 100 120 Sample 0 0 20 40 (a) 60 80 100 120 Sample (b) Figure 3.2: Waveform analysis example: Raw waveform (black), Pedestal waveform (Blue). The processed waveform (Red) is obtained by subtracting the pedestal and baseline from the raw waveform and smoothing the waveform with Savitzky-Golay filter. (a) Two pulses are found in this waveform. (b) A waveform with two overlapping pulses. The pulse finding algorithm will only count one pulse in this case since the first pulse does not return to zero before the second pulse arrives. Pulse Time Extraction The time between each sample ∆tsample in a waveform is set by the DAQ system at the beginning of the run and is typically ∼1.5 ns. The sampling time however varies slightly from ATWD to ATWD and needs to be precisely obtained in order to extract the arriving time of each pulse. This is done by collecting 50 clock waveforms, which are the signals from the internal 40 MHz clock sent out by the trigger module, at the beginning of the run for each ATWD. After subtracting the pedestal waveform and baseline, a Fast Fourier Transform (FFT) is performed to the clock waveforms and the mean sampling frequency is calculated for each ATWD. For a small pulse the time of the pulse, in units of number of samples, is defined as the peak position of the pulse (Ppulse ), which comes from a quadratic fit to the three points closest to the peak. For a large pulse such as from a muon, the pulse time is estimated from its rising edge, defined as where the lowest gain waveform (Fig. 3.3) crosses 50 ADC units, since the peak comes at a much later time. A small 74 ADC Value Waveform from LS Muon 1000 800 600 gain: x20 gain: x4 gain: x0.5 400 200 0 0 20 40 60 80 100 120 Sample Figure 3.3: One example of raw waveforms from an LS muon event. The large muon signal saturated the two high gain channels, and the time and charge information are extracted from the lowest gain waveform. correction δt(q) is applied to compensate for the possible charge correlated bias in the time extraction algorithm, which is estimated from the laser calibration data where the timing and intensity of the signals are precisely controlled. Each ATWD has a constant time offset T0 relative to the absolute photon arriving time on the PMT, due to the different cable length or FEE channel behavior. T0 is calculated and updated for each ATWD from the 60 Co center deployment every few weeks, and has a spread of ∼20 ns over all ATWDs. Combining with the launch offset Nlaunch (Section 2.2.3) of the ATWD recorded with each waveform, the relative timing of each pulse with respect to the the time-stamp of the event is given by: tpulse = Ppulse × ∆tsample − Nlaunch × 25 [ns] − T0 − δt(q) (3.1) Pulse Charge Extraction The charge of a pulse is defined as the area of the pulse, in units of ADC×Nsamples . For large signals the charge is calculated from the lowest gain waveform (Fig. 3.3). The 75 charges are then normalized to number of photo-electrons (Np.e. ) for each ATWD by dividing the corresponding single p.e. charge (Q0 ). Same as T0 , the Q0 is calculated and updated for each ATWD from the 60 Co center deployment every few weeks. The distribution of the time and charge information of the pulses of an event across all PMTs in the detector is used as an input to the later event reconstruction algorithms. 3.3 Vertex Reconstruction The vertex reconstruction algorithm uses the distribution of pulse arriving time (Eqn. 3.1) of an event to reconstruct its interaction position. Only pulses from 17-inch PMTs are used in the reconstruction because of their better timing resolution. It is assumed that the scintillation light is isotropic and that the light travels in a straight line until it reaches a PMT. Second order effects such as absorption and re-emission of the photons, or reflection of the photons off the balloon are ignored. The vertex reconstruction algorithm however only selects the pulses around the peak arriving time to minimize the drawbacks of the simplified optical model. The algorithm is thus called peak time fitter. 3.3.1 Algorithm The peak time fitter finds the event vertex iteratively: 1. The starting vertex of the iteration is estimated from a “charge weighted average position” P i Np.e.~ri ~r0 = 1.62 Pi i i Np.e. (3.2) i where i runs over all the pulses of the event. Np.e. is the normalized charge (number of photoelectrons) of pulse i and ~ri is the position of the PMT who registered pulse i. The overall scaling factor 1.62 is empirically tuned from 60 Co 76 z-axis calibration data. ′ 2. The vertex-corrected pulse arriving time ti is calculated by subtracting the expected photon travel time ttravel,i , given the event’s current reconstructed vertex, from the raw pulse arriving time ti (Eqn. 3.1) as follows: ′ ti = ti − ttravel,i = ti − ( rLS,i rBO,i + ) cLS cBO (3.3) where rLS , rBO is the expected photon traveling distance in LS and BO region, and cLS = 196.1 mm/ns, cBO = 220.0 mm/ns is the empirical effective speed of light in the LS and BO tuned from 60 Co z-axis calibration data, which does not correspond to the true speed of light in the medium due to the simplified optical model used in the algorithm. 3. The distribution of vertex-corrected pulse arriving time and the peak position ′ tpeak in the distribution is calculated. A “push-pull” vector δ~r is then calculated ′ using only pulses inside the (-10ns, +5ns) window around tpeak : δ~r = X t′i − t′peak i ttravel,i · (~r − ~ri ) (3.4) where ~r is the current reconstructed vertex of the event and ~ri is the position of the PMT that registered pulse i. δ~r is then added to the current reconstructed vertex of the event, which essentially pushes the vertex closer the the PMTs who were hit earlier by the scintillation light. 4. Step 2 and 3 are iterated until δ~r is less than 1 mm or 100 iterations have been performed. A vertex reconstruction status is then assigned to each event. The statuses are summarized in Table 3.1. 77 3.3.2 Performance The performance of the vertex reconstruction is checked by the z-axis calibration data from different radioactive sources at various energies, as shown in Fig. 3.4. The peak time fitter works very well for events with energies higher than 0.6 MeV, with the typical bias less than 3 cm between -6 m and 6 m along the z-axis. The peak time fitter’s performance decreases at lower energies since the number of pulses in the peak time window becomes smaller and the contribution of pulses from dark hits, which are uncorrelated with the event, becomes larger. This can be clearly seen from 203 Hg (0.279 MeV γ-ray) data, where the inward vertex bias is as large as 10 cm at z = 6m. The vertex reconstruction bias is slightly worse in Period II (after purification). The effective speed of light cLS and cBO are tuned in early Period I and fixed afterwards. The change of LS properties and the non-uniformity of the LS after purification (Fig. 2.9) could have caused the change of cLS and cBO and consequently reduced the peak time fitter’s performance in Period II. ′ ′ Table 3.1: Summary of vertex reconstruction status. tRM S , tpeakRM S and fpulse refer to the RMS of the vertex-corrected pulse time distribution, the RMS of the vertexcorrected pulse time distribution in the peak time window, and the fraction of pulses in the peak time window, respectively. An event can have multiple statuses. Status Description Valid Fit is successful Unknown Event has less than 4 pulses Not Valid Reconstructed vertex is unphysical (r > 10 m) Bad Fit Fit does not converge ′ ′ Bad RMS tRM S < 35 ns or tRM S > 90 ns ′ ′ Bad Peak RMS tpeakRM S < 1.7 ns or tpeakRM S > 4 ns Bad Pulse Ratio fpulse < 0.22 or fpulse > 0.55 Zfit - Zsource [mm] 78 203 Hg, 0.28 MeV Cs, 0.66 MeV Ge, 1.02 MeV 241 Am 9Be, 2.2 MeV 60 Co, 2.50 MeV 210 Po13C, 4.44 MeV 9 241 Am Be, 4.95 MeV 137 100 68 50 0 -50 -100 -6000 -4000 -2000 0 2000 4000 6000 Zsource [mm] Zfit - Zsource [mm] (a) in Period I (before purification) 203 Hg, 0.28 MeV Cs, 0.66 MeV Ge, 1.02 MeV 241 Am 9Be, 2.2 MeV 60 Co, 2.50 MeV 241 Am 9Be, 4.95 MeV 137 100 68 50 0 -50 -100 -6000 -4000 -2000 0 2000 4000 6000 Zsource [mm] (b) in Period II (after purification) Figure 3.4: The vertex reconstruction performance in (a) Period I and (b) Period II, checked with z-axis calibration data from different radioactive sources at various energies. 79 3.4 Energy Reconstruction The KamLAND detector can be viewed as a calorimeter that measures the total energies deposited by a particle, which is partially transferred to the scintillation light-output, regardless of the particle’s type. It is then convenient to work with the visible energy of an event, instead of its real energy. The scale of the visible energy is defined such that the sum of the two gamma-ray energies from 60 Co at the detector center has the same value (2.506 MeV) in real and reconstructed energy. The energy reconstruction algorithm used in this analysis is called multi-photoelectron (MPE) fitter, which uses the distribution of number of photoelectrons on each PMT to estimate the event’s energy. The MPE fitter relies on the reconstructed position of the event as an input, so its performance is directly correlated with the vertex reconstruction. 3.4.1 Algorithm The MPE fitter first constructs an energy likelihood function from all active inner detector 17-inch PMTs: L(E) = PMTs Y Li (E; qi ) (3.5) i where Li (E; qi ) is the probability of PMTi producing charge qi from an event having visible energy E at reconstructed position ~r. This probability follows Poisson statistics f (n; µ) where n is the observed number of photoelectrons and µ is the expected number of photoelectrons on the PMT. The probability of charge qi being n photoelectrons is assumed to follow Gaussian distribution: P (n|qi ) = p 1 2πnσi2 2 2 e−(qi −n·q0i ) /(2nσi ) (3.6) where q0i and σi are the mean and variance of the single photoelectron charge dis- 80 tribution, calculated and updated for each PMT from 60 Co center deployment about once a month. Li (E; qi ) can then be written as:    f (0; µi ) + P∞ f (n; µi ) · P (qi < qth |n) if not hit. n=1 Li (E; qi ) = P   ∞ if hit. n=1 f (n; µi ) · P (n|qi ) (3.7) where the second term in the “not-hit” case accounts for the probability of photoelectrons not producing enough charge to cross the discriminator’s threshold qth . The expected number of photoelectrons µi is directly related to the energy of the event and the vertex ~r of the event relative to the PMT position. The factors that can affect µi are: • ηi : number of photoelectrons per MeV for events at the detector center. • Λ: effective attenuation length of the LS and BO, which is fixed from 60 Co data to be 30 m. • δi : number of dark photoelectrons uncorrelated with the event. • Ωi (~r): solid angle extended from the event position to the PMT cathode area. • Si (~r): the shadowing correction due to the balloon and ropes, calculated from KamLAND detector simulation package (KLG4sim). µi can be then written as: µ i = ηi · Ωi (~r) · Si (~r) · e−li (~r)/Λ · E + δi Ωi (0) · Si (0) · e−li (0)/Λ (3.8) where all the geometry factors are normalized to the detector center. The two constants ηi and δi are calculated and updated from 60 Co center deployments for each PMT, by free floating ηi and δi until Li (ηi , δi ; E = 2.506 MeV) is maximized. 81 With the full likelihood function L(E) in Eqn. 3.5 defined, the effective χ2 = −2 ln L is minimized using the Minuit Package [94] with respect to the unknown parameter E. The reconstructed visible energy of the event is taken to be the bestfit value of E if the minimization succeeds. If the minimization fails the energy reconstruction status is set to “not valid.” 3.4.2 Performance The performance of the energy reconstruction is again checked by the z-axis calibration data from different radioactive sources at various energies, as shown in Fig. 3.5. The reconstructed energy bias relative to the detector center is generally less than 3% between -6 m and 6 m along the z-axis. The shape of the bias however indicates some systematic biases in the fitter’s algorithm, and is similar at all energies. This shape is different in Period II and is believed to be caused by the change of LS light yield (Fig. 3.6) and the non-uniformity of the LS (Fig. 2.9) after purification. Spallation neutron events are uniformly distributed in the detector volume and can be used to check the energy reconstruction bias in the full volume, as shown in Fig. 3.7. Spallation neutron capture events are selected along the x-, y-, z-axis, and similar shapes of reconstruction bias are observed. The angular dependence of the bias is also checked and the variation is only ∼0.3%. Combing these observations the MPE fitter’s bias is considered to be mainly coming from the the radial dependence of the events’ positions, and the volume weighted variation is within ∼1% at all calibration energies. Details of selections of spallation neutron events are discussed in Appendix C.4.1. The 60 Co source is deployed to the center of the detector about twice a month. Half of these runs are used for updating the energy reconstruction’s constants, and the other half are used to monitor the stability of reconstruction’s performance in time. The time variation of the reconstructed energy is checked with calibration sources Efit /E center 82 1.06 203 Hg, 0.28 MeV Cs, 0.66 MeV Ge, 1.02 MeV 9 241 Am Be, 2.2 MeV 60 Co, 2.50 MeV 9 241 Am Be, 4.95 MeV 137 68 1.04 1.02 1 0.98 0.96 -6000 -4000 -2000 0 2000 4000 6000 Zsource [mm] Efit /E center (a) in Period I (before purification) 1.06 203 Hg, 0.28 MeV Cs, 0.66 MeV Ge, 1.02 MeV 9 241 Am Be, 2.2 MeV 60 Co, 2.50 MeV 9 241 Am Be, 4.95 MeV 137 68 1.04 1.02 1 0.98 0.96 -6000 -4000 -2000 0 2000 4000 6000 Zsource [mm] (b) in Period II (after purification) Figure 3.5: The energy reconstruction performance in (a) Period I and (b) Period II, checked with z-axis calibration data from different radioactive sources at various energies. 83 60 Events / Bin Light Yield from Co at Detector Center Period I 600 Period II 500 400 300 200 100 0 400 500 600 700 800 900 Np.e. on 17-inch PMTs Figure 3.6: The distribution of total number of photoelectrons on all 17-inch PMTs from 60 Co events (2.506 MeV) at the detector center in Period I (black) and Period II (red). There is about 10% light yield decrease in the LS after the purification. and is found to be within 0.5%, as shown in Fig. 3.8. Spallation neutrons and alpha decay events are also used to check the time variation of reconstructed energy. Details can be found in Appendix C. 3.4.3 Detector Energy Response The reconstructed visible energy of an event is related to the scintillation light output from the energies deposited by the particle in the detector and in general is not equalling or proportional to the real energy of the particle. The conversion from real energy to visible energy is called “energy scale,” and is parametrized by the following equation: Evis = a0 [1 − δB (E, kB ) + kc δc (E) + k0 δ0 (E)] E (3.9) where each term corresponds to a process which would lead to the non-linearity of the energy scale: • δB (E, kB ): The light-output suppression due to the quenching interactions be- 84 Evis [MeV] Energy Reconstruction Bias of Spallation Neutron Events 2.48 X Dependence Y Dependence Z Dependence R Dependence 2.46 2.44 2.42 2.4 2.38 2.36 2.34 2.32 -6 -4 -2 0 2 4 6 Distance to Detector Center [m] 2.48 Evis [MeV] Evis [MeV] (a) Energy Reconstruction Bias in X, Y, Z and R 2.46 2.48 2.46 2.44 2.44 2.42 2.42 2.4 2.4 2.38 2.38 2.36 2.36 2.34 2.34 2.32 -1 -0.5 0 0.5 1 cosθ (b) Energy Reconstruction Bias in cos θ 2.32 -3 -2 -1 0 1 2 3 Azimuth [rad] (c) Energy Reconstruction Bias in φ Figure 3.7: Energy reconstruction bias in the full volume checked by spallation neutron capture events (in Period I only): (a) Bias in X, Y, Z and R for events in a 2.5 m radius cylinder along the corresponding axis. (b) Bias in cos θ for events inside 6m radius. (c) Bias in φ for events in a 2.5 m radius cylinder along z-axis (−5m < z < 5m). Combining all three figures, the major systematic bias of the energy reconstruction comes from the radial dependence of the events’ positions. vis Energy Scale (E /E real) 85 241 9 Am Be Co Ge 137 Cs 203 Hg 60 1.1 68 1.05 1 0.95 0.9 0.85 Period I 0.8 0 1000 2000 3000 4000 Period II 5000 6000 7000 8000 Run Number Figure 3.8: The time variation of energy scale from calibration sources. The reconstructed visible energy is divided by the expected real energy of the γ-rays from the source. All the center calibration runs throughout the analysis periods are included. The average time variation is ∼0.5%. tween the excited molecules along the path of the incident particle, which drain energy that would otherwise go into fluorescence photons [95]. This quenching effect is more obvious for high ionizing power particles such as α particles, since they would produce a higher density of excited molecules. The Birk’s formula introduces a positive Birk’s constant kB to semi-empirically correct for the quenching effect as follows: dE/dx dL ∝ dx 1 + kB · (dE/dx) (3.10) where dL/dx and dE/dx are the light output per unit length and the particle stopping power respectively. For small dE/dx Eqn. 3.10 reduces to a linear relationship, but for large dE/dx such as for α particles the light output is saturated and is only related to the particle range. The Birk’s formula Eqn. 3.10 is used to calculate δB (E, kB ) for α particles and protons, together with the SRIM package [96] to calculate dE/dx of the particle given KamLAND LS’s composition. For γ and e± particles, the EGSnrc 86 package [97] is used, which does a Monte Carlo simulation and tracks the energy deposited per tracking length. The Birk’s constant kB is limited to 5 – 15 mg/(cm2 MeV) in calculating and extrapolating the δB (E, kB ) function. kB outside of the range would be unphysical for KamLAND LS. • kc δc (E): The light output from Cherenkov radiation, which is emitted when a particle travels in a medium faster than the speed of light in the same medium. The Cherenkov radiation is an electromagnetic shock wave, whose coherent wavefront is conical in shape and is emitted at a well-defined angle θc cos θc = c nv (3.11) with respect to the track of the particle, where c is the speed of light in vacuum and n is the index refraction of the medium. δc (E) is again calculated from the EGSnrc Monte Carlo package and is proportional to sin2 θc . The constant kc represents the detector’s response in collecting the Cherenkov photons. • k0 δ0 (E): A byproduct from the EGSnrc Monte Carlo simulation, which keeps track of the energy that is lost when it is below the simulation tracking threshold. The constant k0 allows for the recovery of some of this energy. • a0 : An overall normalization factor such that the sum of the two γ-ray energy from 60 Co at the detector center is 2.506 MeV in both visible energy and real energy. The four energy scale parameters a0 , kB , k0 and kc and their error matrices have to be extracted from the KamLAND data. The data points that are used are taken from various known energy events and are summarized in Table 3.2. Besides the events from radioactive calibration sources, the γ-rays from spallation neutron capture on 1 H and 12 C and the α decays from 214 Po and 212 Po are also used. Details of the 87 Table 3.2: Data Points for Determine Energy Scale Parameters Events Ereal [MeV] Evis [MeV] Period I Period II α γ 203 0.279 0.244 ± 0.003 0.238 ± 0.003 Hg 68 Ge 1.022/2 0.941 ± 0.012 0.913 ± 0.010 137 Cs 0.662 0.637 ± 0.008 0.616 ± 0.010 65 Zn 1.116 1.120 ± 0.013 60 Co 1.173 + 1.332 2.539 ± 0.030 2.462 ± 0.030 1 2.225 2.414 ± 0.030 2.436 ± 0.048 H(n,γ)2 H 12 C(n,γ)13 C 4.946 5.485 ± 0.056 5.585 ± 0.056 214 Po 7.687 0.630 ± 0.016 0.640 ± 0.016 212 Po 8.785 0.827 ± 0.023 0.806 ± 0.023 Table 3.3: Best-fit Energy Scale Parameters Parameter Period I Period II a0 1.08 ± 0.02 1.00 ± 0.03 kB [mg/(cm2 MeV)] 9.8 ± 0.3 9.0 ± 0.3 k0 0.85 ± 0.10 1.01 ± 0.13 kc 0.43 ± 0.10 0.81 ± 0.15 selection and properties of these events can be found in Appendix C. The Evis value in Table 3.2 is the volume weighted mean value for the correspond events inside a 6 m radius sphere from the detector center, which is the fiducial volume used in this analysis (Section 4.4). Evis ’s error comes from a combination of its spacial variation inside the fiducial volume and time variation over each period. The errors on the alpha events are manually enlarged by a factor of 1.2 to partially compensate the higher order quenching effects [98] which are not included in the current energy response model. These data points are then fitted to the energy scale function (Eqn. 3.9) as shown in Fig. 3.9, where the shaded area indicates the 68.3% C.L. region of the fit. Because of the change of LS properties after purification, the energy scale parameters are estimated in Period I and Period II separately. The χ2 /ndf from the fit is 4.8/5 for Period I and 3.9/4 for Period II. The best-fit values are summarized in Table 3.3. 88 (Visible Energy)/(Real Energy) [arbitrary] γ Energy Quenching Factor 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0 1 2 3 4 5 6 Real Energy [MeV] (Visible Energy)/(Real Energy) [arbitrary] α Energy Quenching Factor 0.1 0.095 0.09 0.085 0.08 0.075 0.07 0.065 5 6 7 8 9 10 Real Energy [MeV] (a) in Period I (before purification) (Visible Energy)/(Real Energy) [arbitrary] γ Energy Quenching Factor 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0 1 2 3 4 5 6 Real Energy [MeV] (Visible Energy)/(Real Energy) [arbitrary] α Energy Quenching Factor 0.1 0.095 0.09 0.085 0.08 0.075 0.07 0.065 5 6 7 8 9 10 Real Energy [MeV] (b) in Period II (after purification) Figure 3.9: The fit to the energy scale function (Eqn. 3.9) in (a) Period I and (b) Period II. The top panel is for γ particles and the bottom panel is for α particles. The shaded area is the 68.3% C.L. region from the fit. The χ2 /ndf from the fit is 4.8/5 for Period I and 3.9/4 for Period II. 89 σ [MeV] Energy Resolution 0.14 0.12 0.1 0.08 0.06 Period I 0.04 Period II 0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 Evis [MeV] Figure 3.10: The detector energy resolution as a function of reconstructed energy in Period I (black) and Period II (red). The data points are from radioactive γ-ray sources’ reconstructed energy widths. The worse energy resolution in Period II is caused by the decrease of light yield in the LS after the purification. The energy resolution of the detector is dominated by the Poisson statistics of the total number of photoelectrons collected. It can be approximated by a Gaussian function with width σEvis σEvis = r σ02 + σ12 · Evis 1MeV (3.12) where σ0 accounts for the background photoelectrons uncorrelated with the event and σ1 accounts for the photoelectrons from the event. Eqn. 3.12 is then fitted from calibration γ-rays’ reconstructed energy widths (e.g. Fig. C.2) for Period I and Period II separately. Fig. 3.10 shows the fitting results, which are σ0 = 0.0169 ± 0.0016 MeV, σ1 = 0.0681 ± 0.0004 MeV in Period I, and σ0 = 0.0189 ± 0.0016 MeV, σ1 = 0.0763 ± 0.0004 MeV in Period II. The light yield decrease after purification (Fig. 3.6) caused the worse energy resolution in Period II. OD NsumMax 90 120 105 100 104 80 103 60 102 40 (b) (a) (c) 10 20 0 -1 0 1 2 3 4 5 6 7 log (Ntotal ) p.e. 1 10 Figure 3.11: The light yields of muon events plotted as ODnsummax v.s. total number of photoelectrons collected by the ID 17-inch PMTs. Three distinct regions can be identified: (a) muons that only pass through the OD; (b) muons that pass through the BO without entering LS; (c) muons that pass through the LS, which produce the most light outputs. The dotted line indicates the OD muon tagging threshold at ODnsummax > 10. 3.5 Muon Track Reconstruction The muons that incident on KamLAND detector have typical energies above 10 GeV (average energy ∼268 GeV [99]). Most of them traverse the detector without stopping and produce scintillation light and Cherenkov light along their tracks. At such energies the muons are approximately minimum ionizing with stopping power of ∼2 MeV/(g cm−2 ) [7]. This causes high light yields for typical muon events, as can be seen in Fig. 3.11. The muons also produce secondary spallation products along their tracks which are potential backgrounds for ν̄e detection. It is thus necessary to reconstruct the tracks of muons for efficient vetoing of the spallation events after the muons (Section 4.3). The high light yield of the muon event results in photoelectrons on almost every PMT. The muon track reconstruction algorithm use the fastest-light model, which 91 takes advantage of the fact that the first-arriving photons on a PMT are always coming from the direct light on the muon track at the Cherenkov angle θc , even true for scintillation light. The algorithm then construct a likelihood function L(~x) = PMTs Y i P (ti − tfastest,i (~x)) (3.13) where ~x is the vector of muon track, i runs over all ID 17-inch PMTs, ti is the earliest hit time of PMTi , and tfastest,i is the predicted earliest hit time of PMTi given the muon track geometry. The hit time distribution P (∆t) is modeled by a Gaussian with an exponential tail from the KamLAND data. The effective χ2 = −2 ln L is then minimized using the Minuit Package [94] to extract the best-fit value of the muon track vector ~x. If the minimization converges with a good χ2 , a “good track” status is assigned. The “good track” reconstruction efficiency is 99.7% for both LS and Oil muons. The result of the muon track reconstruction is compared with the MUSIC and MUSUN Monte Carlo simulations [100]. The simulations use the detailed shape of the KamLAND overburden to calculate the muon flux at KamLAND detector. The distribution of muon tracks in zenith angle and azimuth angle from the simulation shows good agreement with KamLAND data [99]. 92 Chapter 4 Anti-neutrino Candidate Selection In order to efficiently select the real ν̄e events and reject irrelevant background events, selection rules for ν̄e candidate events are defined as follows : • The event must be from a good quality run period • The event must have a good reconstruction status • The event must pass the muon spallation cut ~p + R ~ d )/2| < 6 m • |(R • 2.7 MeV < Ep < 15 MeV • 2.04 MeV < Ed < 2.82 MeV • 0.5 µs < ∆Tp−d < 660 µs • ∆Rp−d < 1.6 m • The event must pass the probability density estimator (PDE) cut The definitions and efficiencies of these selection rules will be discussed in detail in the subsequent sections in this chapter. The remaining background events after the selection will be discussed in Chapter 5. 4.1 Run Selection and Live Time As discussed in Section 2.3, two periods are defined in this analysis as before the purification and after the purification. Due to the unstable condition of the detector, the data during the purification period are not used in this analysis. The run range 93 Period I Period II Total Table 4.1: Summary of Run Range and Live Time. Run Range Time Range Live Time Live Time [Days] [Days] (After Spallation Cut) 394 – 6756 2002/04 – 2007/04 1432.09 1261.35 7116 – 7830 2007/10 – 2008/06 200.41 178.52 1632.50 1439.87 and time range of each period are summarized in Table 4.1. Some runs are specially taken only to understand the properties and status of the detector, such as calibration runs and background runs. Data from these runs are not included in the ν̄e analysis. Much effort has been put by KamLAND collaborators to maximize the the total live time of the detector. Except for a few minutes between the daily run change, and the occasional trouble-shooting of the detector when problem occurs, the data-taking is almost continuous. In addition to increasing the size of the data for analysis, maximizing the live time is also idea for catching rare events such as a type II supernova in the Milky Way, which happens only about once per 40 years. After the raw data of each run has been reconstructed into “physics events” (Chapter 3), rigorous checks of the quality of the data are performed by qualified collaborators. Based on the results of the quality checks, the entire run or sub-periods of the run could be vetoed as “bad” and are not used in the analysis. Examples of the causes of a “bad run” are high voltage failures, abnormal DAQ errors, unstable event rate, unusual noisy periods, etc. The live time of each run is then calculated by adding up the data taking time from all the clean periods, minus the short trigger disable gaps (if no trigger records are issued for more than 0.1s) when the data flow rate is too high. Given the normal trigger rate of over ∼100 Hz, this subtraction wrongly removes less than 0.05% of the live time and is treated as a systematic uncertainty. Summing over all the selected good runs, the total live time is 1432.09 days in Period I and 200.41 days in Period II, which is ∼80% of the possible maximum time. Muon spallation cut at the analysis stage effectively reduces the total live time and will be 94 discussed in section 4.3. 4.2 Reconstruction Quality The event reconstruction algorithms described in Chapter 3 occasionally fail to return a good status on either the energy or the position of the event. The reason of the failure varies from the noisy unphysical events to issues that are not very-well addressed by the algorithms such as the dark hits on the PMTs. Those bad-reconstructed events are not used in this analysis. The percentage of those bad-reconstructed events vary with energy, and is more prominent at energies below 1 MeV where the “dark hits” contribute a non-negligible amount to the total PMT hits. For the energy range of ν̄e events between 2 MeV and 15 MeV though, the reconstruction is very efficient. Fig. 4.1 shows the nsummax distribution from a typical physics run for all reconstructed events and bad-reconstructed events. The typical ν̄e events have nsummax distribution from 250 to 700. An integration in this range results a conservative estimation of reconstruction efficiency of physics events being 0.9939±0.0001. The reconstruction efficiency is also calculated with 60 Co events at 2.5 MeV and 68 Ge events at 1.0 MeV from calibration runs, which gives 0.9994 ± 0.0001 and 0.9989 ± 0.0001 respectively. The cumulative reconstruction efficiency for physics events is taken to be the average of the two as 0.9992 ± 0.0001. The event is also required to have a minimum nsummax 200 to avoid occasional low nsummmax events that are acquired with special trigger types and unknown trigger efficiency. The trigger efficiency at nsummax ≥ 200 is energy dependent and can be calculated from special low-trigger-threshold runs for each energy bin as: ε200 (E) = number of events with nsummmax ≥ 200 number of total events (4.1) Fig. 4.2 shows the trigger efficiency at nsummax ≥ 200 as a function of visible energy, events / nsummax 95 105 All Reconstructed Events Bad-reconstrcted Events 104 103 102 10 1 0 200 400 600 800 1000 1200 nsummax Figure 4.1: An example of the nsummax distribution from run 4274 for all reconstructed events (black) and bad-reconstructed events (red). The two vertical lines at nsummax 120 and 200 correspond to the delayed and prompt trigger thresholds. The peak at very low nsummax is due to calibration triggers and is not related to physics events. the peaks above nsummax 1000 are from muon events and post-muon noises. The efficiency of good reconstructed events for nsummax from 250 to 700 is 0.9939 ± 0.0001. Trigger Efficiency (nsummax > 200) 96 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 Evis [MeV] Figure 4.2: Trigger efficiency at nsummax ≥ 200 as a function of visible energy, calculated from a special low-threshold run 3888 with prompt trigger set at nsummax = 35. calculated from a special low-threshold run 3888 with prompt trigger set at nsumID = 35. Above 2.0 MeV, which is the lowest energy of the ν̄e events (from delayed neutron capture events), the trigger efficiency is essentially 100%, with negligible errors. 4.3 Muon Spallation Cut and Live Time Correction Cosmic muons entering the KamLAND detector produce large amount of spallation products [91]. Among them the spallation neutrons and the spallation 9 Li’s could give prompt-delayed signals and mimic the ν̄e events, so they are potential background for ν̄e detection. The high energy spallation neutrons predominantly lose energies by elastically scattering on 1 H nuclei. The recoiled protons inside the liquid scintillator could give prompt signals with visible energies above the analysis threshold at 2.7 MeV. The thermalized neutrons then capture on protons and give delayed signals. It is also possible that a pair of neutron captures will coincide in time given that 97 the neutrons are essentially created at the same time. To remove those spallation neutrons the following muon spallation cut rule is applied: • For all inner and outer detector muons1 the whole detector is vetoed for 2 milliseconds after the muon. Since the neutron capture time is only ∼206 µs (Section 4.6), this cut essentially removes all the correlated spallation neutron background after identified muons. The remaining fast neutron background after unidentified muons will be discussed in Section 5.6. The loss of live time due to the 2 ms veto is only 0.1%. This 2 ms veto also removes the short post-muon noisy period due to the electronic effects, typically less than 10µs. The spallation 9 Li beta decays (τ1/2 = 178.3 ms, Q = 13.6 MeV) to 9 Be and the beta particle gives a prompt signal. The excited states of 9 Be however are neutron unstable and could decay to 8 Be with emitting a neutron. This neutron then thermalizes, captures on 1 H and gives a delayed signal. The beta particles are not as penetrating as fast neutrons thus the 9 Li events are predominantly produced by muons traversing the liquid scintillator. Due to the long decay time of 9 Li, a 2 second veto is needed to remove 99.96% of the 9 Li events. Given the LS muon rate of ∼0.2 Hz, vetoing the whole detector after every LS muon will greatly reduce the live time of the experiment. However, as will be discussed in Section 5.5, the majority of the 9 Li’s are produced by shower muons (a subset of LS Muons, which have the highest light yields. Appendix. C.1) and along the tracks of the muons. Thus more efficient muon spallation cut rules are applied as follows: • For all shower muons and muons with mis-reconstructed tracks, the whole detector is vetoed for 2 seconds after the muon. 1 Refer to Section 3.5 for definition of the muon types 98 • For all LS muons with well-reconstructed tracks, a 2-second veto after the muon is applied only to the 3-m cylinder volume along the muon track The remaining number of 9 Li background events after these cuts will be discussed in Section 5.5. The loss of live time due to the muon spallation cut, because of the complicated geometry dependent cut efficiency, is calculated from a simple Monte Carlo simulation on a run-by-run basis. Simulated ν̄e events are generated uniformly inside the detector volume, and events after and before the muon spallation cut are calculated. The ratio between the two is defined as “muon spallation cut efficiency” and treated as a correction to the live time of run. Fig. 4.3 shows the time variation of the muon spallation cut efficiency and it is 0.881 ± 0.004 in Period I and 0.891 ± 0.003 in Period II. The efficiency is fairly constant over time except for the jump from Period I to Period II, which is due to the decrease of rate of “shower muons” (defined as total Np.e > 7 × 105 ) because of the ∼ 10% loss of the light yield after the purification of the liquid scintillator. The total live time after the muon spallation cut, summing up all the runs used in this analysis, is 1261.35 days in Period I and 178.52 days in Period II. 4.4 Fiducial Volume Cut and Number of Target Protons Fig. 4.4 shows the typical event vertex distribution from a normal physics run. The high event rate near the balloon edge is clearly visible. These events mostly come from the radioactive impurities on the balloon material, or from outside of the liquid scintillator such as in the buffer oil or on the PMTs. To avoid the accidental coincidence (Section 4.8.1) from these high rate background events, we take advantage of the large detector itself as a shield and only use the very inside spherical liquid 99 Live Time Fraction After Spallation Cut 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0 1000 2000 3000 4000 5000 6000 7000 8000 Run Number Figure 4.3: The live time fraction after spallation cut for each run. The increase in Period II is due to the decrease of number of “shower muons” because of the ∼ 10% loss of the light yield after the purification of the liquid scintillator, which effectively reduced the spallation cut vetoing time. scintillator volume with a 6 meter radius, defined as the fiducial volume for the analysis. Since each ν̄e detection refers to a prompt-delayed event pair, we require that the average reconstructed position of the event pair be within the fiducial volume: ~p + R ~ d )/2| < 6m • |(R The total number of target protons depend on the the physical size of the fiducial volume, which could be different from the geometric size of a 6-m radius sphere due to the energy dependent reconstruction biases and uncertainties on event positions. Two methods are combined to estimate the size and the uncertainty of the fiducial volume, one with the 4π calibration data, another with the spallation 12 B beta decay events. The two methods are described in the following sections. 4π Calibration Measurements The 4π calibration system, introduced in Section 2.2.4, is specifically designed to test the vertex reconstruction performance across the full detector volume. Fig. 4.5 shows 100 Z [m] 8 104 6 4 103 2 0 102 -2 -4 10 -6 -8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (ρ/6.5m)2 1 Figure 4.4: The vertex distribution of all the events with reconstructed energy higher than 1 MeV, from a one-day physics run 6756. The black line indicates the balloon position at 6.5m radius. The red line indicates the edge of the 6m fiducial volume used in this analysis. an illustration of the 4π calibration system and an example of 60 Co data taken with a 4π deployment. Excluding the initial commissioning period, sets of of 4π calibration runs were performed around Oct 2006 to systematically scan the full detector volume. For each pole configuration, data were taken at a range of zenith positions while pivoting the pole about the detector center. Fig. 4.6 shows the radial distribution of all off-axis source deployments with the composite 60 Co68 Ge source in a single azimuthal plane. The maximum radial reach at the equator was 4.6 m. Additional points were probed along the equatorial plane and near the polar regions by translating the pole up and down, allowing a maximum radial reach of 5.5 m. To study the energy dependence of reconstruction deviations, deployments were also made to a subset of the points using other sources. The 4π calibration data were analyzed in detail in Ref. [85]. Table 4.2 summarizes the radial position reconstruction bias and the corresponding volume bias at 5.5 101 (a) (b) Figure 4.5: (a) Illustration of the 4π calibration system in the KamLAND detector. A radioactive source was attached to one end of the pole, and was positioned throughout the fiducial volume by adjusting the orientation and the length of the pole. Additional 60 Co pin sources were located along the pole for monitoring the pole position. Two cables and a spool system manipulated the pole position. (b) An example of 60 Co data taken with a 4π deployment (run 6167). Source Radial Bias [mm] 60 Co, 2.5 MeV -5 ± 17 68 Ge, 1.0 MeV 15 ± 20 241 Am9 Be, 2.2 MeV -5 ± 18 241 Am9 Be, 4.5–7 MeV 25 ± 26 210 Po13 C, 4.4 MeV 25 ± 28 210 13 Po C, 6.1 MeV 105 ± 34 Volume Deviation [%] 0.3 ± 0.9 -0.8 ± 1.1 0.3 ± 1.0 -1.4 ± 1.5 -1.4 ± 1.6 -5.7 ± 1.9 Table 4.2: The fiducial volume measurements from 4π calibrations with different sources. The radial bias and volume deviation is evaluated at 5.5 m radius. 102 Figure 4.6: The radial distribution of all off-axis source deployments with the composite 60 Co68 Ge source excluding the initial commissioning runs, in a single azimuthal plane. 103 m radius, measured with different sources at different energies. The mean fiducial volume hf i and its uncertainty hσf i for ν̄e events is obtained by integrating f (E) across the expected unoscillated ν̄e spectrum S(E). Using the best fit oscillated ν̄e energy spectrum resulted in negligible changes in the results. hf i = R E1 hσf i = R E1 and E0 E0 f (E) · S(E)dE R E1 S(E)dE E0 σf (E) · S(E)dE R E1 S(E)dE E0 (4.2) (4.3) f (E) is obtained from a linear fit to the data summarized in Table 4.2. It is found that the result is insensitive to the choice of model of f (E). Comparing with geometric size of a 5.5 m radius sphere, the fiducial volume at radius 5.5 m is reduced by 1.0% to be 689.9 ± 13.8 m3 . The relative uncertainty is 2.0%. Spallation 12 B Measurements The maximum radial reach of the 4π calibration is 5.5 m. This is determined mainly to ensure the safety of the balloon, which is located at radius of 6.5 m. The fiducial volume used in this analysis is however at radius 6 m. It is non-trivial to extrapolate the 4π calibration data directly to 6 m. Instead, we use a separate method to evaluate the fiducial volume between 5.5 m and 6 m. The spallation 12 B events are generated by the cosmic muons uniformly over the detector volume. We define the fiducial volume ratio Rfv as the number of the spallation 12 B beta decay events inside the fiducial volume divided by the number of the spallation 12 B events in the whole detector. The fiducial volume Vf v can then be simply expressed as: Vf v = VLS · Rfv (4.4) 104 Events / 10.8 m3 Spallation 12B Events 900 5.5m 6m 800 700 600 500 400 300 200 100 0 0 50 100 150 200 250 300 350 400 R3 [m3] Figure 4.7: The background subtracted spacial distribution of spallation 12 B Events. The dotted line indicates the fiducial volume boundary at radius 5.5 m and 6 m. where VLS is the total volume of the liquid scintillator measured with flow-meter during the filling of the LS to be 1171 ± 25 m3 . One crucial assumption of the 12 B method is that KamLAND detects the 12 B events with 100% efficiency inside liquid scintillator volume and zero efficiency outside. Simulation shows that the portion of the beta particles from 12 B decay that are generated outside of the liquid scintillator but deposit part of the energies inside the LS near the boundary introduces a 0.5% systematic uncertainty. The selection of spallation 12 B events is summarized in Appendix C.4.2. The spacial distribution of all 12 B events in Period I with visible energies between 4 MeV and 14 MeV are shown in Fig. 4.7. The distortion from uniform distribution near the balloon edge is caused by the reconstruction resolution, imperfect shape of the balloon, and decrease of efficiency at the very edge of the balloon. The fiducial volume 5.5m ratio Rfv for radius less than 5.5 m is calculated to be 0.602 · (1 ± 1.2%) where the uncertainty is statistical only. This results a fiducial volume of 704.9 m3 for R < 5.5 m. Comparing with the 4π calibration results (689.9 m3 ), the two measurements 105 Fiducial Volume Ratio Spallation 12B Events 0.21 0.2 0.19 0.18 0.17 0.16 0.15 4 6 8 10 12 14 Evis [MeV] Figure 4.8: The ratio of fiducial volume between radius 5.5 m and 6 m to all liquid scintillator volume as a function of reconstructed energy, calculated from spallation 12 B events. agreed to within 2.2%. The fiducial volume between radius 5.5 m and 6 m is measured with the 12 B 5.5m−6m method only. The fiducial volume ratio Rfv is calculated to be 0.173 · (1 ± 0.02stat. ). To further investigate the energy dependent systematic uncertainties on 5.5m−6m fiducial volume, the spallation 12 B events are binned into 1 MeV bins. Rfv is calculated for each bin and is shown in Fig. 4.8. The systematic uncertainties from energy dependency is calculated to be 4.5% using Eqn. 4.3 with a linear fit to the data. The uncertainty is found to be insensitive to the choice of the fit functions. Combining together, the fiducial volume between radius 5.5 m and 6 m from the 12 B measurement is 202.6 ± 10.9 m3 . The relative uncertainty is 5.4%. The total fiducial volume inside 6 m radius is obtained by combining the 4π calibration measurements and the spallation 12 B measurement to be 892.5 ± 17.6 m3 . The relative uncertainty is 2.0%. Comparing with geometric size of a 6 m radius sphere, the fiducial volume is reduced by 1.4%. In Period II, the change of liquid scintillator properties after the purification sys- 106 tematically affects the quality of the event reconstructions. For this reason the results from the 4π calibrations, which were only deployed in Period I, are not applied to Period II. The fiducial volume inside 6 m radius is measured by the 12 B data in Period II only. The fiducial volume in Period II is measured to be 907.0 ± 59.7 m3 . The relative uncertainty is 6.6%. Number of Target Protons The total number of target protons can be calculated as Np = I1 H · nH · ρLS · Vfv (4.5) where I1 H = 0.99985 is the isotopic abundance of 1 H, nH is the number density of hydrogen atoms per unit mass, ρLS is the density of the liquid scintillator, and Vfv is the fiducial volume defined by the analysis and measured with the 4π calibration and spallation 12 B events. The number density of hydrogen atoms per unit mass is calculated as follows: nH = NA mH + mC /(H/C) + mN /(H/N ) + mO /(H/O) = 8.4708 × 1022 /g (4.6) where the masses are referring to the atomic mass of H, C, N and O. The ratio of H/C , H/N and H/O are calculated from the composition of the liquid scintillator to be 1.96908, 17842.0 and 17842.0 where the small amount of nitrogen and oxygen come from the fluor PPO. The H to C ratio was verified by elemental analysis to within ±2%, which translates into ±0.1% in the uncertainty of nH . The density of KamLAND liquid scintillator is measured to be 0.7775 ± 0.0001 g/cm3 at 15 ◦ C in both periods. The actual temperature distribution in KamLAND is 107 not uniform, and is measured to be 11.5 ± 1.5 ◦ C with the instrument units during the 4π calibration system deployments [88]. The temperature of the liquid scintillator in Period II is a little higher during the filling to keep the new LS in separate with the old LS, and an conservative estimation of ±4 ◦ C is assigned to the systematic uncertainty. Given the temperature coefficient of density of the liquid scintillator measured to be −7.41 × 10−4 g/cm3 /K, the actual density of KamLAND liquid scintillator is 0.7796 g/cm3 with 0.1% uncertainty in Period I and 0.4% uncertainty in Period II. Putting all the values and uncertainties into Eqn. 4.5, the total number of target protons is calculated to be 5.89×1031 ·(1±2.0%) in Period I and 5.99×1031 ·(1±6.6%) in Period II. The contributions of uncertainties are summarized in Table 4.3. Table 4.3: Summary of Uncertainties on Target Protons Description Uncertainty [%] Period I Period II Fiducial Volume 2.0 6.6 Density variation from composition 0.1 0.1 Density variation from temperature 0.1 0.4 Total 2.0 6.6 Combining with the live time measurement discussed in Section 4.1 and 4.3, the total exposure for this analysis is 8.50 × 1034 proton-days, or ∼2.75 kiloton-years. 4.5 Prompt and Delayed Energy Cut The following selection rules on the visible energy of prompt and delayed events are applied in this analysis: • 2.7 MeV < Ep < 15 MeV • 2.04 MeV < Ed < 2.82 MeV The prompt events are required to have visible energies higher than 2.7 MeV to specifically avoid the ν̄e background from the decay chains of natural radioactivities, Number of anti-neutrinos per MeV per parent 108 101 238 U series Th series K 232 40 100 10-1 10-2 0.5 1 1.5 2 2.5 Anti-neutrino energy, Eν (MeV) 3 3.5 Figure 4.9: The energy spectrum of ν̄e from 238 U, 232 Th and 40 K decay chains, which are the natural radioactivities in the Earth. The dotted line indicates the inverse beta decay threshold at 1.8 MeV. KamLAND is insensitive of detecting ν̄e ’s below this threshold. Figure taken from [21]. mainly 238 U, 232 Th and 40 K, in the Earth. These ν̄e ’s are often referred to as “geoneutrinos.” Fig. 4.9 shows the energy spectra of ν̄e from 238 U, 232 Th and 40 K decay chains. The maximum energy of these ν̄e ’s is 3.3 MeV, corresponding to 2.5 MeV for the prompt positron event. Given the energy response model and detector resolution (Section 3.4), a cut at Ep > 2.7MeV removes the contribution from geo-neutrinos with 100% efficiency and negligible uncertainties. The geo-neutrinos are however very interesting and important topics in geology. Geological models predict the radiogenic heat composes ∼19 TW or more of the total Earth’s heat flow [18, 19, 20]. A dedicated analysis on KamLAND low energy data [21] is now putting good constrains on the radiogenic power of uranium and thorium in the Earth. The ν̄e flux from the fission reactions from nuclear reactors and the georeactor is negligible above 8.5 MeV (Fig. 5.2). We, however, include higher energy events up to Ep = 15 MeV in this analysis to constrain the number of fast neutron and atmospheric neutrino background events from the shape of the spectrum at high energies. Details of those backgrounds will be discussed in Section 5.6 and 5.7. 109 Events / 0.05 MeV Spallation Neutron Events 35000 30000 χ2 / ndf constant mean sigma bkgd slope 15.37 / 10 3.591e+04 ± 107 2.415 ± 0.000 0.1205 ± 0.0004 8469 ± 289.3 -2985 ± 111.8 2.6 2.8 25000 20000 15000 10000 5000 0 2 2.2 2.4 3 Evis [MeV] Figure 4.10: The reconstructed energy distribution of the spallation neutron capture events. The efficiency of the delayed energy cut 2.04MeV < Ed < 2.82MeV is calculated from the fitted Gaussian peak to be 0.9987 ± 0.0003. The dotted lines indicate the delayed energy cut thresholds. The delayed event from neutron capture on 1 H is a monochromatic gamma-ray at 2.2 MeV. The visible energy cut on the delayed events is set to be 2.04MeV < Ed < 2.82MeV for historical reasons to agree with the older analysis [85]. The efficiency of this cut is calculated from the reconstructed energy distribution of the spallation neutron capture events (Fig. 4.10) to be 0.9987 ± 0.0003. The neutrons can, however, capture on other nuclei besides 1 H. Table 4.4 lists the possible target nuclei in KamLAND liquid scintillator for (n,γ) reaction and the corresponding cross sections and γ-ray energies, in decreasing order of number densities of the nuclei. Besides 1 H, only 12 C contributes non-negligibly to the neutron capture process. The efficiency of only selecting neutron capture on 1 H is calculated to be 0.9946 ± 0.0010. 110 Table 4.4: Target Nuclei for (n,γ) Reaction in KamLAND Target Nuclei Cross Section [mb] Eγ [MeV] Density [nuclei/cm3 ] 1 H 332.6 (7) 2.22 6.603 × 1022 12 C 3.53 (5) 4.95 3.316 × 1022 13 C 1.37 (4) 8.17 3.723 × 1020 2 H 0.519 (7) 6.25 9.905 × 1018 16 O 0.19 (2) 4.14 3.692 × 1018 14 N 80 (2) 10.83 3.687 × 1018 4.6 Time Correlation Cut The temporal correlation between the prompt and delayed event from the inverse beta decay is a very powerful tool to reject background from the ν̄e events. The following selection rule on the time correlation is applied in this analysis: • 0.5 µs < ∆Tp−d < 660 µs The probability of a thermalized neutron get captured in a time interval dt is constant over time (“lack of aging”), thus the neutron capture time follows an exponential distribution. The mean neutron capture time τ can be estimated on the first order as following: X v 1 = =v· ni σi (v) τ λ i (4.7) where v is the velocity of the neutron, λ is the mean free path of the neutron, ni and σi are the number density and cross section of target nuclei i as listed in Table 4.4. The cross sections in Table 4.4 are measured at neutron velocity v0 = 2200 m/s [101], however, if we assume that the neutron capture cross section is inversely proportional to its velocity (which is true at neutron energies below ∼10 keV), then v · σ(v) = v0 · σ(v0 ) (4.8) and the capture time becomes independent of the neutron energy. The time it takes to slow down an energetic neutron to below 10 keV is less than 0.1 µs and can Events / 10µs 111 105 104 103 0.5 1 1.5 2 2.5 3 3.5 4 Time since previous muon [ms] Figure 4.11: The distribution of time to the previous muon for all spallation neutrons inside 6m fiducial volume. A fit from 0.6 ms to 3 ms to exponential distribution plus a flat background results the mean neutron capture time τ = 206.4 ± 1.3µs. The χ2 /ndf from the fit is 227/237. be neglected here. Using the values from Table 4.4, this simple estimation gives τ ∼ 206 µs. A better estimation comes from the study of spallation neutron events in the detector. Fig. 4.11 shows the distribution of time to the previous muon for all spallation neutrons in the analysis periods inside 6m fiducial volume. As shown in Fig. C.8(a), muons with high multiplicity (number of events within 0.15 – 10 ms following a muon) result in severe loss of waveforms and spallation events immediately following the muon. For this reason only muons with multiplicity less than 75 are included. We fit the distribution to an exponential distribution plus a flat background from 0.6 ms to 3 ms and avoid the earlier time since the loss of neutron events immediately after muon will distort the distribution. This yields the mean neutron capture time τ = 206.4 ± 1.3µs. The efficiency of time correlation cut can be then calculated from: ε∆T = Z 660µs 0.5µs 1 −t/τ e dt τ (4.9) 112 to be 0.9567 ± 0.0008. The neutron capture time from spallation neutron study is cross-checked with the data from 241 Am9 Be z-axis calibration runs. The measured neutron capture time from combining all the 241 Am9 Be data is 205.2±0.5 µs [91]. The discrepancy between this value and the one from the spallation neutron measurement is not completely understood, but is suspected to be caused by neutrons from the 241 Am9 Be source that capture on the stainless-steel source capsule, which has mostly iron whose neutron capture cross section is ∼10 times higher than 1 H. The efficiency of the time correlation cut is insensitive to this discrepancy though. 4.7 Spacial Correlation Cut The prompt positron event and delayed neutron capture gamma event are also correlated in space. The following spacial correlation cut is applied to further reject background events: • ∆Rp−d < 1.6 m The distribution of the distance of the prompt/delayed event position to the ν̄e inverse beta decay interaction position is approximated from the 68 Ge and AmBe z-axis calibration data respectively. The 68 Ge event radial spread may be taken as an upper limit for prompt event since the absorption length is much shorter for positrons than for the (annihilation) γ’s. Fig. 4.12 shows the distribution of the distance of the 68 Ge event (two annihilation γ’s) position to the 68 Ge source position (black line), and the AmBe neutron capture gamma event position to the AmBe source position (blue line), from summing up all the corresponding z-axis calibration runs across the liquid scintillator volume in the analysis periods. These distributions are then taken as an input into a simple simulation where the positions of the prompt and delayed events are generated assuming no angular correlation. The distribution of the distance Probability Density / m3 113 10 68 Prompt Event ( Ge) Delayed Event (AmBe) 1 ( ∆Rp-d) (MC) 3 10 -1 10 -2 10 -30 1 2 3 4 5 6 7 8 (∆R)3 (m3) Figure 4.12: The distribution of the distance of the 68 Ge event position to the 68 Ge source position (black line) and the AmBe neutron capture gamma event position to the AmBe source position (blue line), calculated from z-axis calibration runs. Also shown is the distribution of the distances between prompt and delayed events (red line) from monte carlo simulations. The dotted line indicates the spacial correlation cut at (∆Rp−d )3 = 4.096 m3 . between the two is plotted in Fig. 4.12 (red line). Since the 68 Ge event distribution gives a maximal absorption length for the prompt event, the fraction of the simulated events within 1.6 m of each other can be taken as an upper limit of the efficiency. The lower limit is obtained by assuming no smearing of prompt event, in other words by integrating the AmBe distribution. The efficiency of the ∆Rp−d < 1.6 m cut is calculated in this way to be 0.9881 ± 0.0057. The efficiency on spacial correlation cut is cross-checked with the result using GEANT4 [102] based KLG4sim (KamLAND Monte Carlo Simulation) full detector simulation package, where inverse beta decay events were simulated uniformly inside the detector and distribution of ∆Rp−d is calculated. KLG4sim simulation yields the efficiency being 0.9887 ± 0.0051 [84]. The results of the two methods agree well with each other. 114 4.8 Probability Density Estimator (PDE) Cut The so far discussed “box-cuts” combined all together are very efficient in selecting ν̄e signals and rejecting background events, and were actually all one needed for a 5.5mradius or less fiducial volume analysis, for example in Ref. [85, 84]. However, when we enlarge the fiducial volume to 6m radius, the accidental coincidence background starts to become significant and further rejection method is needed. 4.8.1 Accidental Coincidence Background The normal trigger rate of KamLAND is just over ∼100 Hz. Most of these events are uncorrelated in time and do not cause background for ν̄e detection and are referred to as “singles events.” Fig. 4.13 shows the singles event rate of each run for all goodreconstructed events that have nsummax higher than 200 (or 180 for runs before 3753) and have passed the muon spallation cuts (Section 4.3). The jumps in singles rate are mostly correlated with electronics upgrades, while the sudden drop between Period I and II comes from the decrease of the radioactive impurities are purification. Although singles events themselves do not cause background for ν̄e detection, because of the high event rate, two events can accidentally coincide in time and be wrongly tagged as a prompt-delayed pair. The probability of seeing two random singles events inside the time correlation cut window ∆Tp−d follows Poisson distribution and can be calculated on a run-by-run basis as follows: Pcoincidence = f (n; ν) = f (1; Rsingles · ∆Tp−d ) (4.10) where ν is the expected number of singles events inside the ∆Tp−d window, and Rsingles is the singles rate of the run. For a 20 Hz singles rate, Pcoincidence is ∼1.3%. The accidentally temporal correlated events need to pass the other “box-cuts” selection rules in order to be considered as “ν̄e candidate events.” The probability Singles Event Rate [Hz] 115 22 21 20 19 18 17 16 15 0 1000 2000 3000 4000 5000 6000 7000 8000 Run Number Figure 4.13: The singles event rate per run, where “singles” refers to all goodreconstructed events in the detector with nsummax > 200 and have passed the muon spallation cuts. that this could happen is calculated on a run-by-run basis. Singles events in each run are randomly paired with each other and the probability Pcuts is defined as the number the event pairs that satisfy the prompt and delayed energy cut, spacial correlation cut and fiducial volume cut, divided by the total number of event pairs. To avoid the possible correlations during event pairing, each event is used once and only once, so the total number of event pairs is just half of the total number of singles events. The total number of accidental background events can be then calculated as: Naccidentals = X run Trun · Rsingles · Pcoincidence · Pcuts (4.11) where Trun is the live time of each run and we sum over all the runs in the analysis period. For the 6m fiducial volume, the number of accidental background events is calculated to be 438.7 ± 3.8 in Period I and 84.7 ± 1.5 in Period II. The accidental background is about half of the number of expected reactor ν̄e ’s (Section 5.1) and 116 Events / 0.05MeV Accidentals Prompt Energy Spectrum 104 103 102 10 1 2 4 6 8 10 12 14 Evis (MeV) (a) 900 Events / 0.2m3 Accidentals Spacial Correlation Events / 2m3 Accidentals Radial Distribution 2500 2000 800 700 600 1500 500 400 1000 300 200 500 100 0 0 50 100 150 (b) 200 250 300 |(Rp+Rd)/2|3 [m3] 0 0 2 4 6 8 10 3 3 (∆Rp-d) [m ] (c) Figure 4.14: Statistically measured properties of accidental coincidence background: (a) Prompt visible energy spectrum. The dotted line indicates the analysis threshold at 2.7 MeV. (b) Radial distribution of average position of prompt and delayed events. The dotted line indicates the fiducial volume boundary at 6m. (c) Distribution of distance between prompt and delayed events. The dotted line indicates the spacial correlation cut at (∆Rp−d )3 = 4.096 m3 . requires further reduction in order to get a good signal to background ratio 2 . Fig. 4.14 shows some of the properties of the accidental coincidence background from the statistical measurement. The prompt visible energy spectrum is shown in Fig. 4.14(a). The peak near the analysis threshold 2.7 MeV is caused by the 2 For a 5.5m fiducial volume, the number of the accidental coincidence events is greatly reduced to 22.5 ± 0.9 in Period I and 4.3 ± 0.3 in Period II. This is the reason why smaller fiducial volumes are chosen in the earlier KamLAND analysis, for example in Ref. [45, 46]. 117 external gamma-ray from 208 Tl concentrated on the balloon surface and outside of liquid scintillator, which beta decays (Q = 5.0 MeV) to 208 Pb and emits a 2.6 MeV gamma-ray. The accidentals event rate drops sharply as energy increases and becomes negligible above 4 MeV. Fig. 4.14(b) shows the radial distribution of the average position of the prompt and delayed events from the accidental coincidence background. The distribution follows a Gaussian shape with the mean position at the balloon surface, which is very different from the uniformly distributed ν̄e events. The accidentals event rate becomes negligible at radius less than 5m, but is still significant near the 6m fiducial volume boundary. Fig. 4.14(c) shows the distribution of distance between prompt and delayed events. Compared with the ν̄e events (Fig. 4.12), the temporal correlated accidental coincidence events are only weakly correlated in space. The differences in the spacial distributions between accidental coincidence events and ν̄e events make further background rejection possible with a multivariate analysis method, as will be discussed in Section 4.8.2. 4.8.2 PDE Cut Definition Compared with traditional “box cut” methods, multivariate data analysis tries to extract a maximum of the available information from the data by combining different properties (“variables”) of the data. The systematic uncertainties from the multivariate analysis, however, are generally complicated and extensive Monte Carlo simulations are needed to extract the full error matrix, if a lot of variables are used. To avoid the complication, we construct the probability density estimator (PDE) using only two spacial variables: • Rmean : The mean radial position of the prompt and delayed events. • ∆R : The distance between the prompt and delayed events. 0.05 Probability Density / m3 Probability Density / m3 118 ν e Signal 0.04 Accidentals B.G. 0.03 7 ν e Signal 6 Accidentals B.G. 5 4 3 0.02 2 0.01 1 0 0 20 40 60 80 100 120 140 160 180 200 R3mean [m3] 0 0 0.5 1 1.5 2 2.5 3 3.5 4 (∆R)3 [m3] (a) (b) Figure 4.15: Probability density function for ν̄e signal and accidental background events. (a) Pdf’s of mean position of the prompt and delayed events, normalized inside the 6m fiducial volume. (b) Pdf’s of distance between the prompt and delayed events, normalized insidethe ∆Rp−d < 1.6m cut. and define the PDE, following the likelihood ratio method, as: PDE = Lsignal Lsignal + Lbackground (4.12) where Lsignal or Lbackground is the likelihood (probability) that an event is a ν̄e signal event or an accidental coincidence background event. The likelihood can be calculated by: Lα = Y (pdf)αi α = Signal or B.G. (4.13) i where the subscript i runs over all the variables used in the multivariate analysis and (pdf)αi stands for the probability density function of variable i, for signal events or background events respectively. With this definition, a signal-like event will have a PDE close to 1 and a background-like event will have a PDE close to 0. 3 The probability density functions of Rmean and (∆R)3 for signal and background events are shown in Fig. 4.15. The ν̄e signal events are taken to be uniformly dis3 3 tributed in the detector so their Rmean distribution is flat, while the Rmean distribu- tion for accidental backgrounds is calculated from a fit to a Gaussian function from Probability Density / 0.01 PDE Unit 119 1 ν e Signal Accidentals B.G. 10-1 10-2 -3 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability Density Estimator Figure 4.16: The distribution of probability density estimator (PDE) for simulated ν̄e signal and accidentals background events. Signal and background events are strongly peaked at 1 and 0 respectively. The dotted line indicates the PDE cut at PDE > 0.4. Fig. 4.14(b) (χ2 /ndf = 34.1/44). These distributions are then normalized to unity inside the 6m fiducial volume. (∆R)3 distribution for ν̄e signal events has been shown in Fig. 4.12, and for accidental backgrounds is calculated from a linear fit to the distribution shown in Fig. 4.14(c) (χ2 /ndf = 13.6/19). The (∆R)3 distributions are then normalized to unity inside the ∆Rp−d < 1.6 m cut. With the pdf of each variable defined, the PDE can be calculated from Eqn. 4.12 for each simulated ν̄e event and accidental coincidence event. The normalized distribution of PDE is shown in Fig. 4.16. As expected, the ν̄e signal events’ PDEs are strongly peaked at 1 and the accidental background events’ PDEs are strongly peaked at 0. To determine where the PDE cut threshold should be placed, three “figure of merit” rules are considered: 1. High signal acceptance efficiency. 2. High background rejection efficiency. 120 √ 3. High S/ S + B for good statistical performance. Rule 1 and 2 are in general contradictory with each other and some prior decisions need to be made. The decision in this analysis is set such that the PDE cut will reduce the accidentals background to equal or below the background from other sources (∼40 events, Table 5.4), while keeping the signal efficiency above 90%. Fig. 4.17(a) plots the signal efficiency and number of accidental background events as a function of PDE cut threshold. A PDE cut placed at 0.4 will have a 93.30% signal efficiency, while reducing the accidentals background from 438.7 to 42.0 events (in Period I), a 90.43% background rejection efficiency. This satisfies our prior requirements. √ It’s always a good idea to maximize S/ S + B (rule 3) to reduce the statistical uncertainties. The total number of signal events, however, is unclear in prior because of the suppression by neutrino oscillation and unknown amount of georeactor ν̄e events. If we take the oscillation parameters from Ref. [22] and assume null geore√ actor contribution, the S/ S + B can be estimated and used as a cross-check on the √ PDE cut decision. Fig. 4.17(b) shows S/ S + B as a function of PDE cut threshold, √ and a plateau exists for optimum performance of S/ S + B. The PDE cut placed at 0.4 will put us in the middle of the plateau and satisfies all three “figure of merit” rules, thus is used in this analysis: • Events must have probability density estimator (PDE) higher than 0.4 The total efficiency of PDE > 0.4 cut is calculated to be 0.9330 ± 0.0047 where the uncertainty comes from allowing the ∆R distribution of ν̄e events to vary within its uncertainty from the Monte Carlo simulation (Section 4.7). Fig. 4.18. shows the 3 PDE cut efficiency as a function of Rmean . The efficiency drops quickly near the fiducial volume boundary as the accidental backgrounds start to dominate. The total number of accidental coincidence events, after the PDE cut, is 40.2 ± 1.2 in Period I and 8.2 ± 0.5 in Period II. It is worthing emphasizing that the efficiency of the PDE 1 500 0.9 450 0.8 400 0.7 350 0.6 300 0.5 250 0.4 200 0.3 150 0.2 100 0.1 50 0 0 0.2 Number of Accidental Coincidence Events νe Signal Efficiency 121 0 0.4 0.6 0.8 1 Probability Density Estimator Cut Threshold 1 30 0.9 29 0.8 28 0.7 27 0.6 26 0.5 25 0.4 24 0.3 23 0.2 22 0.1 21 0 0 0.2 Estimated S / S+B νe Signal Efficiency (a) 20 0.4 0.6 0.8 1 Probability Density Estimator Cut Threshold (b) Figure 4.17: Figure of Merit in determining the PDE cut: (a) signal efficiency and number√of accidental background events vs. PDE cut threshold. (b) signal efficiency and S/ S + B vs. PDE cut threshold. The optimum PDE cut threshold is set to 0.4, indicated by the dotted red line. Signal Efficiency from PDE cut 122 1 0.9 0.8 0.7 0.6 0.5 0 20 40 60 80 100 120 140 160 180 200 R3mean [m3] 3 Figure 4.18: The PDE cut efficiency as a function of Rmean . The efficiency drops 3 quickly near the fiducial volume boundary (Rmean = 216m) as the accidental backgrounds start to dominate. 3 cut is energy independent, as long as the probability density functions of Rmean and (∆R)3 for ν̄e events do not depend on the ν̄e energy, which is assumed to be the case. 4.9 Summary of ν̄e Candidate Selection Efficiency Table 4.5 summarizes the ν̄e candidate selection efficiency from each selection rule. The total efficiency is 0.875(1 ± 0.9%) and the uncertainty is mostly coming from the PDE cut and the ∆Rp−d cut. Table 4.5: Summary of ν̄e Selection Efficiency Description Efficiency Uncertainty Reconstruction Quality 0.9992 0.0001 2.04MeV < Ed < 2.82MeV 0.9987 0.0003 1 Neutron Capture on H 0.9946 0.0010 0.5µs < ∆Tp−d < 660µs 0.9567 0.0008 ∆Rp−d < 1.6 m 0.9881 0.0057 PDE > 0.4 0.9330 0.0047 Total 0.8754 0.0075 123 4.10 Final Sample of ν̄e Candidate After applying all the selection rules discussed in this section, 930 ν̄e candidate events were extracted out of the total ∼12 billion raw events from the 1439.87 live-day’s data. This apparent 107 : 1 background rejection ratio shows the power of inverse beta decay coincidence technique. Fig. 4.19 shows the vertex distribution of the ν̄e candidate events, by applying all the cuts except the fiducial volume cut. We emphasize that the PDE cut is only 3 applied to events inside the 6m fiducial volume since the pdf of Rmean distribution is normalized inside the fiducial volume only (Fig. 4.15(a)). As can be seen, the ν̄e candidates appear to be uniformly distributed inside the fiducial volume, and the accidental coincidence background dominates outside of the fiducial volume. Fig. 4.20 shows the energy distributions of ν̄e candidate events, by applying all the cuts except the prompt and/or delayed energy cuts. The events with prompt energy below the analysis threshold 2.7 MeV are mostly accidental coincidence events, with some contributions from 13 C(α,n)16 O events (Section 5.4) and possible geoneutrino events from natural radio-activities in the Earth (Section 4.5). The events with delayed energy below 2 MeV are also from accidentals. The 5 events with delayed visible energy around 5.5 MeV agree in number with neutron capture on 12 C, comparing with the number of neutron capture on 1 H events (Table 4.4). Fig. 4.21 shows the ∆Tp−d distribution of ν̄e candidate events, by applying all the cuts except the time correlation cut. A fit to an exponential function plus a flat background yields the neutron capture time being 199.7 ± 12.4µs. This agrees with the neutron capture time measurement from spallation neutrons (206.4 ± 1.3µs, Section 4.6). The flat background comes from the accidental coincidence events since they have no natural time correlation. Within the 0.5µs < ∆Tp−d < 660µs window, the fitted total number of accidental events is 40.7 ± 19, which agrees with the 124 z [m] νe Candidate Vertex Distribution 8 Balloon Surface 6 6m F.V. Boundary 4 2 0 -2 -4 -6 -8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (ρ/6.5m)2 150 200 250 R3mean [m3] (a) Events / 10.8m3 νe Candidate Radial Distribution 600 500 400 300 200 100 0 0 50 100 (b) Figure 4.19: (a) Scatter plot of ν̄e candidate’s vertex. (b) Radial distribution of ν̄e candidate. All vertexes refer to the average positions of the prompt and delayed event pairs. Outside of the 6m fiducial volume (dotted line), the PDE cut is not applied, so the accidental coincidence background dominates. Inside the fiducial volume the ν̄e candidates appear to be uniformly distributed. 125 Edelayed [MeV] νe Candidate Energy Distribution 6 5 4 3 2 2 4 6 8 10 12 14 16 18 20 Eprompt [MeV] (a) ν e Candidate Delayed Visible Energy Spectrum Events / 0.1MeV Events / 0.425MeV ν e Candidate Prompt Visible Energy Spectrum 102 102 10 10 1 1 2 4 6 8 (b) 10 12 14 Evis [MeV] 1 2 3 4 5 6 7 Evis [MeV] (c) Figure 4.20: The energy distributions of ν̄e candidate events: (a) Delayed energy vs. prompt energy. The red box is the candidate window. (b) Prompt visible energy spectrum. Events below 2.7 MeV are dominated by accidentals and have contributions from geoneutrinos. (c) Delayed visible energy spectrum. The 5 events around 5.5 MeV agree with neutron capture on 12 C. The dotted lines indicate the prompt and delayed energy threshold used in this analysis. 126 Events / 25µs νe Candidate Neutron Capture Time 100 χ2 / ndf 45.8 / 37 constant 111.3 ± 5.8 decay constant 199.7 ± 12.4 bkgd 1.542 ± 0.720 80 60 40 20 0 0 100 200 300 400 500 600 700 800 900 1000 ∆Tp-d [µs] Figure 4.21: The ∆Tp−d distribution of ν̄e candidate events. A fit to an exponential function plus a flat background yields the neutron capture time being 199.7 ± 12.4µs. The χ2 /ndf of the fit is 45.8/37. The dotted line indicates the time correlation cut at 660 µs. The fitted total flat background within the cut is 40.7 ± 19. estimation (48.4 ± 1.3) from the statistical calculation in Section 4.8. 127 Chapter 5 5.1 Signals and Backgrounds Anti-neutrino from Nuclear Reactor Most of the ν̄e ’s detected in KamLAND come from the man-made nuclear reactors. Besides providing ν̄e ’s for scientists, these nuclear power plants produce a large amount of electric power. In Japan, nuclear power contributes to 34.5% of the total electricity. Since the ν̄e flux goes down as inverse square of the distance, only reactors nearby KamLAND, namely reactors in Japan and Korea, contribute significantly. Table 5.1 summarizes the information of the 57 reactor cores in Japan and the 20 reactor cores in Korea, including their types, distances to the KamLAND detector and thermal powers. Most of the reactors are boiling water reactors (BWR) and pressurized water reactors (PWR). There are two research-type test reactors including an advanced thermal reactor (ATR) and a fast breeder reactor (FBR). The contribution of Korean reactors to the total ν̄e flux is estimated to be 3.2 ± 0.3%, where the uncertainty comes from the conservative uncertainty (10%) of converting reported electric power to thermal power. Reactors from other countries are included assuming one effective reactor at one distance for each country, and is estimated to contribute 1.0 ± 0.5% of the total ν̄e flux. The positions of Japanese reactors were given by Tokyo Electric Power Company (TEPCO), and are accurate to within 70 m. The distances range from 80 km to 900 km, with a flux-weighted average baseline of ∼180 km (Fig. 5.1). The ν̄e flux uncertainty from baseline is less than 0.1% . The reactor thermal power is monitored at each reactor from the heat balance of the reactor cores and the uncertainty is conservatively assigned as 2.0%, which is dominated 128 by the accuracy of the feed-water flowmeters1 . Table 5.1: Summary of Japanese and Korean Reactors Site Distance [km] Cores Type Power [GWth] Japan Hamaoka 214 5 BWR 14.5 Shimane 401 2 BWR 3.8 Tokai2 295 1 BWR 3.3 Tsuruga-1 138 1 BWR 1.1 Tsuruga-2 138 1 PWR 3.4 Tomari 783 2 PWR 3.3 Shika 88 2 BWR 5.5 Fugen 139 1 ATR 0.6 Monju 142 1 FBR 0.7 Mihama 146 3 PWR 4.9 Ohi 179 4 PWR 13.7 Takahama 191 4 PWR 10.2 Genkai 754 4 PWR 10.1 Sendai 830 2 PWR 5.3 Ikata 561 3 PWR 6.0 Onagawa 431 3 BWR 6.4 Higashidori 636 1 BWR 3.3 Fukushima1 349 6 BWR 14.2 Fukushima2 345 4 BWR 13.2 KashiwazakiKariwa 160 7 BWR 24.3 Korea Kori 736 4 PWR 9.2 Ulchin 712 6 PWR 17.3 Wolsong 709 4 PWR 8.2 Yonggwang 986 6 PWR 17.7 Four main fissile nuclei 235 U, 238 U, 239 Pu and 241 Pu contribute to 99.9% of the ν̄e ’s from the nuclear reactors. 235 U is the initial fuel enriched to ∼3–4% of the uranium. It fissions with thermal neutrons as follows: 235 U + n −→ X1 + X2 + 6.1e− + 6.1ν̄e + xn + 202MeV (5.1) where X1 an X2 are the daughter fission fragment nuclei such as 94 Zn and 140 Ce. 238 U, 1 The reactor thermal power uncertainty might be reduced to ∼0.8%, from the random cancellation as described in Ref. [103]. 129 no-osc νe (events) 150 100 50 0 0 250 500 750 1000 distance (km) Figure 5.1: Baseline distribution of the expected on-oscillation anti-neutrino event rate (arbitrary unit) at KamLAND. The flux-weighted average baseline is ∼180 km. Figure taken from [46]. on the other hand, can only fission with fast neutrons: 238 238 U + n(> 1MeV) −→ X3 + X4 + 5 ∼ 7e− + 5 ∼ 7ν̄e + yn + 205MeV (5.2) U can also capture a neutron and through subsequent β decays breeds another fissionable isotope 239 Pu: 238 239 U(n, γ)239 U(β − )239 Np(β − )239 Pu (5.3) Pu fissions with thermal neutrons as follows: 239 Pu + n −→ X5 + X6 + 5.6e− + 5.6ν̄e + zn + 210MeV (5.4) Or it could sequentially capture two neutrons and produce another fissionable isotope 241 Pu, which then fissions with thermal neutron as follows: 241 Pu + n −→ X7 + X8 + 6.4e− + 6.4ν̄e + wn + 212MeV (5.5) 130 On average, 6 ν̄e ’s and 200 MeV energy are released in each fission. A typical 3 GWth reactor produces ∼5 × 1020 ν̄e ’s per second. The ν̄e energy spectra from fissions of 235 U, 239 Pu, 241 Pu are deduced from the measured beta spectra at ILL high flux reactor in Grenoble [104, 105]. The ν̄e spectrum from fission of 238 U is calculated theoretically [106], since 238 U only fissions with fast neutrons and lacks experimental data. These ν̄e spectra are shown in Fig. 5.2. In typical reactor cores relevant to KamLAND, the relative fission rate of the isotopes is 235 U : 238 U : 239 Pu : 241 Pu = 0.56 : 0.08 : 0.30 : 0.06. The uncertainty of the total ν̄e spectrum shape is estimated from the rate-weighted sum of the uncertainty of each spectrum of the isotope [104, 105, 106] to be 2.5%. The shapes of ν̄e spectra are checked by the Bugey [51, 107] experiment and agree well with the measurement within 1.4%. The existence of 1.8 MeV threshold in the inverse beta decay detection process ensures that only ν̄e ’s from large-Q-valued (short half life) β decays can be detected. Two long-lived isotopes 106 Ru (T1/2 = 373.6 days) and 144 Ce (T1/2 = 284.9 days) are from the fission products and their daughters 106 Rh and 144 Pr can beta decay and produce ν̄e ’s above inverse beta decay threshold. They are important for the periods immediately following reactor shutdowns, and for the contribution from spent fuels which are stored near the reactors for a long period of time. The contribution from these long-lived isotopes is estimated to be ∼0.04% above the analysis energy threshold 2.7 MeV [108] and is negligible. To accurately calculate the total ν̄e rates at KamLAND site, it is necessary to trace the time variation of fission rate of all the reactors. Detailed simulations exist that can calculate the fuel component in accordance with the burn-up, but it is very computationally intensive. Instead, a simple model of reactor cores [109] is constructed by TEPCO and Tohoku University to calculate the instantaneous fission rate of different isotopes. The input to this model are the initial fuel composition and the instantaneous and integrated thermal power of the reactor. These data for ×10-42 10 2 1.8 1.6 235 U 238 U 239 Pu 8 241 Pu 1.4 cross section 1.2 6 1 0.8 4 0.6 0.4 2 inverse beta decay cross section [cm-2] number of νe’s [MeV -1 fission-1 131 0.2 0 1 2 3 4 5 6 7 8 9 10 Eν [MeV] 0 e Figure 5.2: The ν̄e energy spectrum from different fission isotope 235 U, 238 U, 239 Pu and 241 Pu. The inverse beta decay cross section as a function of ν̄e is show in blue line. all 57 reactor cores are provided weekly (hourly on reactor starting or stopping) by TEMPCO on a special agreement exchanged between Tohoku university and the power agencies as a member of KamLAND. The accuracy of this simple model is estimated to be 1% compared to the detailed simulations. The instantaneous ν̄e energy spectrum at KamLAND site can be calculated as follows: X P (E, Lreactor ) X dNν̄isotope (E) isotope dNν̄e (E, t) e = freactor (t) · 2 dE 4πLreactor dE reactors isotopes (5.6) where E is the energy of the ν̄e , Lreactor is the distance from the reactor to KamLAND, isotope P (E, Lreactor ) is the survival probability of the ν̄e , and freactor (t) is the instantaneous fission rate of each isotope in each reactor. KamLAND then detects these ν̄e through the inverse beta decay process described in Section 2.1. The expected un-oscillated ν̄e event rate at KamLAND as a function of time is shown in Fig. 5.3. The noticeable drop of event rate during Period II is due to the shutdown of Kashiwazaki Kariwa expected no-osci. ν e rate [proton-1 Sec -1] 132 ×10 -36 0.3 0.25 0.2 0.15 0.1 0.05 Period I 2001/12 2004/01 Period II 2005/12 2008/01 Year Figure 5.3: The expected un-oscillated ν̄e event rate at KamLAND as a function of time. reactors after an earthquake of magnitude 6.8 striking in Niigata prefecture on 16 July 2007. The expected prompt visible energy spectrum at KamLAND from reactors can then be calculated with Eqn. 2.5, and is shown in Fig. 5.4 in case of no-oscillation, and oscillation with LMA-0, LMA-I and LMA-II solutions (Section 1.2.3). The total expected un-oscillated reactor ν̄e ’s within the analysis energy window and fiducial volume, with the detection efficiency of 87.5% (Section 4.9), is 1267.6 events in Period I and 104.8 events in Period II. Table 5.2 summarizes the reactor related systematic uncertainties discussed in this section. 5.2 Anti-neutrino from Nuclear Georeactor As described in Section 1.3, a hypothetical georeactor at the core of the earth could produce sizable amount of ν̄e ’s and be detected at KamLAND. The georeactor model predicts that 235 U comprises 76% of the present-day georeactor fission and 238 U comprises 24% [59]. Based on this fuel composition, the normalized prompt visible energy Expected Events [proton-1MeV-1] 133 ×10-30 No Oscillation 10 LMA 0 LMA I LMA II 8 6 4 2 0 0 2 4 6 8 10 Evis [MeV] Figure 5.4: The expected prompt visible energy spectrum in Period I in case of nooscillation, and oscillation with LMA-0 (∆m221 = 1.0 × 10−5 eV2 ), LMA-I (∆m221 = 7.59 × 10−5 eV2 ) and LMA-II (∆m221 = 2.0 × 10−4 eV2 ) solutions. tan2 θ12 is fixed at 0.47 in the calculation. The dotted line indicates the analysis threshold at 2.7 MeV. Table 5.2: Summary of Reactor Related Systematic Uncertainties Description Uncertainty Thermal Power Japanese Reactors 2.0% Korean Reactors 0.3% Other Reactors 0.5% ν̄e Spectra 2.5% Distance 0.1% Fuel Composition 1.0% Total 3.4% Probability density / MeV 134 235 235 0.25 235 238 U(76%) U(100%) U(0) 238 U(24%) 238 U(0) U(100%) 0.2 0.15 0.1 0.05 0 0 2 4 6 8 10 Evis [MeV] Figure 5.5: The prompt visible energy spectrum at KamLAND from georeactor ν̄e ’s. The georeactor model predicts the present-day fuel composition of 76% 235 U and 24% 238 U. Two other extreme case of 100% 235 U or 100% 238 U are also calculated. All three spectra are normalized to unity to compare their shapes. The shape difference due to the fuel composition uncertainty is small. spectrum from georeactor ν̄e is calculated similarly as in Section 5.1 and shown in Fig. 5.5. Two other extreme cases of 100% 235 U or 100% 238 U are also calculated to compare their shapes. Even in the extreme cases, the fuel composition uncertainty only causes a small difference in the spectrum shape and is ignored in the subsequent analysis. The ν̄e ’s from georeactor undergo the same oscillation as described in Section 1.1 as they travel through the Earth. The matter effect, as can be calculated from Eqn. 1.14, is small because of the low energy of the neutrinos and the relative low density of the Earth and is neglected here. The survival probability of the ν̄e is: ¸ · 2 R⊕ (km) 2 P (ν̄e → ν̄e ) = 1 − sin θ12 sin 1.27∆m21 (eV ) Eν̄e (GeV) 2 2 (5.7) where R⊕ = 6370 km is the radius of the Earth. The size of the model-predicted georeactor is ∼5 km in radius [59], thus can be treated as a point source compared with the Probability Density / MeV 135 Ereal Evis 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 Energy [MeV] Figure 5.6: The calculated oscillation effect (∆m221 = 7.59 × 10−5 eV2 [22]) on the prompt energy spectrum at KamLAND from georeactor ν̄e . The fine structure of the oscillation (Ereal spectrum, black line) is smeared out after applying detector resolution (Evis spectrum, red line). oscillation length (∼100 km) if we take the LMA solutions for ∆m221 . Because of the long distance of R⊕ , the energy related oscillatory part undergoes many oscillations within the georeactor ν̄e energy range from 1 to 10 MeV. The fine structure of the oscillated energy spectrum, however, is smeared out by the finite size of the energy resp olution (∼ 7%/ E/1MeV, Section 3.4) of KamLAND detector. This smearing effect can be visually seen from Fig. 5.6, where ∆m221 = 7.59 × 10−5 eV2 is used for the simulation. The survival probability is then simply given by P (ν̄e → ν̄e ) = 1 − 0.5 sin2 θ12 and is used in the subsequent analysis. The oscillation only results in a total rate suppression and does not distort the ν̄e spectrum shape of the georeactor. To calculate the georeactor ν̄e event rate at KamLAND, we assume that each georeactor fission reaction releases 200 MeV energy and that the fission rate is constant over the time scale of the KamLAND experiment. The event rate of georeactor ν̄e can be then calculated from the fission rate similarly as described in Section 5.1. Fig. 5.7 shows the expected prompt visible energy spectrum in Period I from a hypothetical 10 TW georeactor, in comparison with the expected ν̄e events from the Expected Events [proton-1MeV-1] 136 6 ×10-30 man made reactor geo reactor (10TW) 5 4 3 2 1 0 0 2 4 6 8 10 Evis [MeV] Figure 5.7: The expected prompt visible energy spectrum at KamLAND in Period I from a hypothetical 10 TW georeactor (red line), in comparison with the expected ν̄e events from the man-made reactor (black line). The oscillation parameters are taken from Ref. [22]. The dotted line indicates the analysis threshold at 2.7 MeV. man-made reactors (Section 5.1), calculated with the oscillation parameters taken from Ref. [22]. With the same detection efficiency and analysis threshold as reactor ν̄e , for a 10 TW georeactor we expect to see 102 events in Period I and 14.5 events in Period II within the 6m fiducial volume. This is ∼13.4% of the expected oscillated man-made reactor ν̄e signal. The normalized oscillated georeactor ν̄e event rate is ∼0.012 event/(day·kton·TW). The georeactor model [59] only loosely constrains its fission power from 3 TW to 30 TW. Thus, the number of georeactor ν̄e events is treated as a free parameter in the analysis. Its spectrum shape and time variation signatures are however very different from the reactor ν̄e , which helps to distinguish them from the large reactor ν̄e background and a well constraint on the georeactor power can be put. 137 Events / 0.05MeV / day Accidentals Prompt Energy Spectrum 10-1 Period I Period II 10-2 10-3 10-4 1 1.5 2 2.5 3 3.5 4 4.5 5 Evis [MeV] Figure 5.8: The prompt visible energy spectrum of accidental background in Period I (blue) and Period II (red). The dotted line indicates the analysis threshold at 2.7 MeV. 5.3 Accidental Coincidence Background Accidental coincidence background has already been discussed in detail in Section 4.8. In summary, the total number of estimated accidental background events is 42.0 ± 1.2 in Period I, and 8.2 ± 0.5 in Period II. Fig. 5.8 shows the expected prompt visible energy spectrum of accidental coincidence events from simulation (Section 4.8). Comparing Period I and II, after the purification the accidental event rate below 1 MeV was reduced by a factor of ∼5, which is dominated by the 210 Bi decay in the liquid scintillator. Above the analysis threshold at 2.7 MeV though, the event rate did not decrease after purification, probably because it’s dominated by the 208 Tl decay on the balloon surface and outside of the liquid scintillator. In fact, due to the change in energy scale, the accidental event rate slightly increased in Period II. 138 5.4 13 C(α,n)16O Background Radioactive impurities in KamLAND liquid scintillator, during their decay chains, emit α particles (Appendix B). The visible energies of these α particles are heavily quenched down to below 1 MeV, however, the second order (α,n) reactions could produce fast neutrons. The thermalization and capture of those fast neutrons then give prompt-delayed signals and mimic the ν̄e events. More than 99% of the α activity in KamLAND comes from the decay of 210 Po [108], who is the daughter nucleus of 210 Pb (Table B.1) and was introduced by means of 222 Rn contamination during the initial filling. 210 Po has a half life of 138 days and emits an α particle with kinetic energy of 5.304 MeV. Table 5.3 lists the possible target nuclei for (α,n) reaction, in decreasing order of number densities of the nuclei. The maximum α energy in KamLAND is 8.785 MeV from 212 Po. Below this threshold, the only dominant target nucleus is 13 C. Therefore only 13 C(α,n)16 O reactions from 210 Po decay is considered in the following analysis. Table 5.3: Target Nuclei for (α,n) Reaction in KamLAND Target Nuclei Q value [MeV] Threshold [MeV] Density [nuclei/cm3 ] 1 H -23.68 115.4 6.603 × 1022 12 C -8.50 11.34 3.316 × 1022 13 C 2.22 0 3.723 × 1020 2 H -4.19 12.5 9.905 × 1018 16 O -12.13 15.17 3.692 × 1018 14 N -4.74 6.09 3.687 × 1018 210 Po Decay Rate in KamLAND The α particle from 210 Po decay has a kinetic energy 5.304 MeV, but is quenched down to ∼0.3 MeV, which is below the trigger threshold during normal run (∼0.7 MeV), so the 210 Po activity is estimated from special low threshold background runs taken at about once a week during the running of the KamLAND experiment. The 139 energy reconstruction algorithm (Section 3.4) has known issues at such low energy and often returns invalid reconstruction status. To avoid this inefficiency, the lowlevel nsummax distribution is used for the estimation of 210 Po decay rate instead of the reconstructed energy distribution. The nsummax of an event , however, does not tell the vertex of the event, which is needed to estimate 210 Po’s activity inside the 6m fiducial volume. To reduce the reconstruction inefficiency, a more loose “acceptably reconstructed event” selection is used, which includes events with somewhat degraded position reconstruction quality unless “unknown,” “not valid” or “bad” position reconstruction status is set (Table 3.1). Fig. 5.9 shows the nsummax distribution of the good reconstructed, acceptably reconstructed and all events near 210 Po α energy in the whole KamLAND detector. As can be seen, the “good reconstructed events” have energy dependent inefficiency and distort the nsummax spectrum, so they are not used in the analysis. The efficiency of “acceptably reconstructed event” is 98% ± 2% by comparing with all the events in the detector. To estimate the 210 Po activity, the nsummax spectrum for events inside 6m fiducial volume is modeled by a Gaussian function for the 210 Po peak around nsummax 85, plus a second order polynomial for the background under the 210 Po peak (mainly from 85 Kr β decay), plus an exponential decay function for the rise at low nsummax (mainly from 14 C β decay). The systematic error from the fitting is estimated to be 16%, which comes from trying different fitting ranges and assuming different background functions. The low threshold background runs are usually as short as 5 minutes2 . To study the time dependence of 210 Po decay rate with enough statistics, those background runs are grouped into 8 periods with running time ∼8 hours each. The nsummax spectra inside 6m fiducial volume are fitted in each period and the result is shown in Fig. 5.10. The first 7 points are from Period I. The 210 Po activity inside the fiducial volume in Period I is consistent with a fitted constant 40.7 ± 7.2 Bq over time. The 2 There are three special 8-hour background runs (run 3888, 5757 and 5767). Events/NsumMax 140 30000 All events acceptably reconstructed events good reconstructed events 25000 20000 15000 10000 50 60 70 80 90 100 110 120 130 140 150 NsumMax Figure 5.9: Nsummax distribution of good reconstructed, acceptably reconstructed and all events in KamLAND detector near 210 Po alpha energy (5.304 MeV, quenched down to ∼0.3 MeV in visible energy), taken from a special low threshold background run 3888. last point shows the 210 Po activity in Period II after the purification. The 210 Po activity is reduced by about a factor of four to 11.3 ± 2.6 Bq inside the 6m fiducial volume. To study the position distribution of 210 Po decay rate, the nsummax spectrum is fitted in different equal-volume regions of the detector and the result is shown in Fig. 5.11. As can be seen, the 210 Po activity rises near the balloon boundary, which indicates the contamination from the balloon surface. But inside the 6m fiducial volume the 210 Po activity is consistent with constant over the volume. Since the 210 Po α events have the same radial distribution as the ν̄e events, the detection efficiency of 13 C(α,n)16 O events is taken to be the same as ν̄e events, which is 0.875 (1±0.9%) (Section 4.9). 210 Po α Decay rate [Bq] 141 50 40 30 20 10 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Time since 2001/9/1 [days] 210 Po α decay rate [Bq] / 10.8 m3 Figure 5.10: Time dependence of 210 Po decay rate inside 6m fiducial volume, estimated from the low threshold background runs throughout the KamLAND running time. The 210 Po activity in Period I is consistent with a fitted constant 40.7 ± 7.2 Bq over time. In Period II, due to the purification, the 210 Po activity is reduced to 11.3 ± 2.6 Bq inside the 6m fiducial volume. 6 5 4 3 2 1 0 0 50 100 150 200 250 300 350 R3 [m3] Figure 5.11: Radial distribution of 210 Po Activities. The dashed line indicates the 6m fiducial volume boundary. Inside the fiducial volume, the 210 Po activity is consistent with constant over the radius. The activity rises near the balloon boundary, which indicates the contamination from the balloon surface. 142 13 C(α,n)16 O Reaction Rate and Prompt Energy Spectrum Fig. 5.12 shows the cross section of 13 C(α,n)16 O reaction as a function of α energy [110]. At 210 Po α particle energy of 5.304 MeV, the final state of 16 O can be its ground state, first excited state or second excited state. The prompt signal from 13 C(α,n)16 O background thus consists of the following events: • The neutrons from the 13 C(α,n)16 Og.s. reaction typically have energies from 3 MeV to 7 MeV. Most of them lose energy by elastically scattering on 1 H nuclei during thermalization, and the recoiled protons could give visible energies above the analysis threshold of 2.7 MeV. The neutrons from reactions involving 16 O 1st or 2nd excited state typically have energies below 1 MeV and do not contribute to the prompt signal. • High energy neutrons sometimes inelastically scatter on 12 C and excite the nucleus to its first excited state. The 12 C∗ then emits a 4.438 MeV γ particle during de-excitation. • If the final state of 16 O is its 1st exited state, the 16 O∗ returns to its ground state by emitting an e+ e− pair with total energy of 6.049 MeV. • If the final state of 16 O is its second exited state, the 16 O∗ returns to its ground state by emitting a 6.129 MeV γ ray. The energy loss function of α particles in KamLAND detector is calculated from the SRIM software package [96], with the composition of KamLAND liquid scintillator given in Section 2.2. The result is shown in Fig. 5.13. The 13 C(α,n)16 O reaction rate is then calculated through: Rni = Z 0 µ dX dEα ntarget Rα σ (Eα ) − dEα E0 i ¶ , i = 0, 1, 2 (5.8) Cross Section [cm2] 143 10-24 13 C(α,n) O (g.s.) 13 16 C(α,n) O (6.049) 13 C(α,n) O (6.130) 16 16 10-25 10-26 10-27 10-28 0 2 4 6 8 10 Alpha Energy [MeV] Figure 5.12: Cross section of 13 C(α,n)16 O reaction [110] in case of 16 O ground state (red), first excited state (green) and second excited state (blue). The dashed line indicates the 210 Po α particle energy at 5.304 MeV. where Eα is the energy of the α particle, E0 is the initial 210 Po α energy 5.304 MeV, ntarget is 13 C density in the KamLAND liquid scintillator (3.723 × 1020 atoms/cm3 ), Rα is the 210 Po α decay rate in the fiducial volume, σ i (Eα ) is the cross section of 13 C(α,n)16 O reaction with 16 O at ith final state, and dEα /dX is the energy loss function of α particles in the KamLAND detector. The energy of the neutron coming out from the 13 C(α,n)16 O reaction in the lab frame is calculated from the kinematics of the two body scattering 13 C + 4 He −→ n + 16 O Enlab = γEnCM + γβPnCM cos θCM β = Pαlab /(Eαlab + MC ) , EnCM = (5.9) p γ = 1/ 1 − β 2 Mα2 + MC2 + 2MC Eαlab + Mn2 − MO2 p 2 Mα2 + MC2 + 2MC Eαlab As seen from Eqn. 5.9, the energy of the neutron depends on the scattering angle. In this analysis, the differential cross section of 13 C(α,n)16 O reaction is parameterized Energy Loss [MeV/cm] 144 2200 2000 1800 1600 1400 1200 1000 800 600 400 0 2 4 6 8 10 Alpha Energy [MeV] Figure 5.13: dE/dX function of α particles in KamLAND, calculated from the SRIM software package [96]. dσ/dΩ [arbitrary scale] Lengedre Coefficient a1/a 0 Eα = 5.03 MeV 1 0.5 0 4.5 4 3.5 3 2.5 2 -0.5 1.5 1 -1 0.5 1 2 3 4 (a) 5 6 Alpha Energy [MeV] -1 -0.5 0 0.5 1 cosθ (b) Figure 5.14: (a) Legendre Polynomial coefficient a1 /a0 as a function of α energy. (b) 13 C(α,n)16 O differential cross section at α energy 5.03 MeV. p.d.f. / MeV 145 0.45 13 16 13 16 13 16 C(α,n) O (g.s.) 0.4 C(α,n) O (6.049) C(α,n) O (6.130) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 Neutron Energy [MeV] Figure 5.15: Normalized Neutron energy spectrum from 13 C(α,n)16 O reaction in the KamLAND detector. by Legendre Polynomials to the 8th power as: 8 X dσ = al (Eα )Pl (cos θ) dΩ l=0 (5.10) where the coefficient al is taken from Ref. [111, 112]. Fig. 5.14 shows an example of a1 as a function of α energy and the differential cross section at α energy 5.03 MeV. The neutron energy spectrum is then calculated from: Rni (En ) = Z 0 E0 dEα Z dσ i dΩ δ(En ; Ω, Eα )ntarget Rα dΩ µ dX − dEα ¶ , i = 0, 1, 2 (5.11) where the parameters are the same as in Eqn. 5.8, with the δ function δ(En ; Ω, Eα ) calculated from Eqn. 5.9. The resulting normalized neutron energy spectrum is shown in Fig. 5.15. Finally, to calculate the prompt visible energy spectrum, a monte carlo simulation of 13 C(α,n)16 O interactions are performed according to the cross section of 13 C(α,n)16 O reaction and the p.d.f of the neutron energy spectrum. The neutrons are p.d.f. / MeV 146 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Evis [MeV] Figure 5.16: Normalized visible prompt energy Spectrum of 13 C(α,n)16 O background in KamLAND. The dashed line indicates the analysis energy threshold 2.7 MeV. then thermalized and the visible energies of the various prompt events discussed at the beginning of this subsection are summed together. After applying the detector resolution, the normalized prompt energy spectrum is shown in Fig. 5.16. The first peak in the spectrum comes from the recoiled protons during neutron thermalization, the second peak comes from the inelastic scattering of neutrons on 12 C and subsequent de-excitation of 12 C∗ nuclei, and the last peak comes from the de-excitation of 16 O first and second excited states. Because of the different systematic errors of the cross sections, the number of events under each peak is estimated separately in the analysis. The cross section of 13 C(α,n)16 O∗ reaction is not very well measured and a 50% uncertainty is assigned. Given the 210 Po α decay rate, the detection efficiency and the exposure of Period I and II, the total number of 13 C(α,n)16 O background events under each peak above the analysis threshold 2.7 MeV is estimated to be 2.7 ± 0.5 , 4.8 ± 2.4, 22.4 ± 11.2 for Period I, and 0.1 ± 0.0, 0.2 ± 0.1, 0.9 ± 0.4 for Period II respectively. 147 9 9 Spallation Li Events 180 160 χ2 / ndf 39.94 / 37 constant 221.4 ± 7.7 bkgd 2.718 ± 0.398 140 Events/0.1sec Events/0.05sec Spallation Li Events 30 25 χ2 / ndf 18.37 / 17 constant 26.55 ± 6.07 bkgd 5.824 ± 0.696 20 120 100 15 80 60 10 40 5 20 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time since shower and misreconstructed muons [sec] 0 0.2 (a) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time since well-tracked LS muons [sec] (b) Figure 5.17: (a) Time between spallation 9 Li events and associated shower or misreconstructed muons. (b) Time between spallation 9 Li events and associated welltracked non-show LS muons within 3-m of the muon track. Both distributions are fitted to an exponential decay function with fixed half life of 178.3 ms. The fitted functions are integrated from 0 sec to infinity, resulting a total number of 1139.3 ± 39.8 9 Li events after shower and misreconstructed muons, and 68.3 ± 15.6 9 Li events after well-tracked non-shower LS muons. 5.5 9 Li Background As discussed in Section 4.3, 9 Li produced by muon spallation could give a promptdelayed signal and mimic the ν̄e event. The muon spallation cuts are designed to reject these events, however, a fraction of 9 Li events still remains due to inefficiency of the cuts. A 2-second veto on the full detector is applied after a shower muon or a muon with misreconstructed track. Given the half life of 9 Li (178.3 msec), this veto rejects 99.96% of the 9 Li background. To estimate the remaining, 9 Li events after showering or misreconstructed muon are selected by applying the same ν̄e selection cuts except inverting the muon spallation cuts. The distribution of time between 9 Li event and associated muon is shown in Fig. 5.17(a). This time distribution is fitted to an exponential decay function with a fixed half life of 178.3 ms, resulting a total number of 1139.3 ± 39.8 events integrated from 0 second to infinity. After the 2-second veto, the remaining 9 Li background events are 0.48 ± 0.02. 148 For well-tracked non-shower LS muons, a 2-second veto is applied only to the 3-m cylinder volume along the muon track. The distribution of time between spallation 9 Li events inside the 3-m cylinder and associated well-tracked non-shower LS muons is shown in Fig. 5.17(b). This time distribution is again fitted to an exponential decay function with a fixed half life of 178.3 ms, resulting a total number of 68.3 ± 15.6 events from the fit. After the 2-second veto, the remaining 9 Li background events are 0.03 ± 0.01. For the volume outside of the 3-m cylinder along the muon track, only a 2ms veto is applied (Section 4.3). Given the half life of 9 Li (178.3 msec), this veto only rejects 0.77% of the 9 Li background. The inefficiency of the 3-m cylinder cut is investigated through the 12 B spallation events. The selection of 12 B events is described in Appendix C.4.2. The distribution of distance between 12 B events and the associated LS muon track is shown in Fig. 5.18. 11.7% ± 2.8% of the 12 B events have reconstructed positions outside of the 3-m cylinder along the muon track. The result agrees with a FLUKA [113] based simulation in Ref. [114]. Assuming a similar vertex distribution for 9 Li events, given 68.3 ± 15.6 9 Li events inside the 3-m cylinder along the muon track, 9.0 ± 3.0 9 Li events lie outside of the 3-m cylinder and are not rejected by the muon spallation cuts. The above analysis is performed for Period I and II separately because of the observed changes in muon spallation cut efficiency between the two periods (Fig. 4.3), total which is caused by the decrease of “shower muons” (defined as Np.e > 7 × 105 ) rate because of the ∼ 10% loss of the light yield after the purification of the liquid scintillator. Combining all contributions together, the total number of spallation 9 Li background events is estimated to be 9.5 ± 3.1 in Period I and 1.3 ± 0.6 in Period II. A separate estimation of 9 Li background is performed by studying the distribution of time between the ν̄e candidate and the previous muon event, shown in Fig. 5.19. 149 Events/0.1m Spallation 12B Events 250 200 150 100 50 0 0 1 2 3 4 5 6 7 8 Distance to muon track [m] Figure 5.18: The distance between spallation 12 B events and associated muon track. The percentage of the 12 B events inside the 3-m cylinder along the muon track is 88.3% ± 2.8%. Events/0.1sec νe Candidate Events χ2 / ndf 13.48 / 18 18 constant 2.228 ± 3.353 16 bkgd 8.941 ± 0.785 14 12 10 8 6 4 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time since LS muons [sec] Figure 5.19: Time between ν̄e candidate events and previous LS muons. The distribution is fitted to an exponential decay function with fixed 9 Li half life of 178.3 ms. The fitted function is integrated from 0 sec to infinity, resulting a total number of 5.7 ± 8.6 9 Li events in the ν̄e candidates. 150 Any time correlation with the muon should be coming from the spallation 9 Li background. This time distribution is fitted to an exponential decay function with a fixed 9 Li half life of 178.3 ms, resulting in a total number of 5.7 ± 8.9 events from the fit. This number is consistent with the previous more precise estimation and is only used as a consistency cross-check. 5.6 Fast Neutron Background Fast neutrons produced by muons could enter the KamLAND detector and give coincidence events that mimic the ν̄e events (Section 4.3). They are easily removed by the 2 ms spallation cut after each muon, if the associated muon is detected. However, due to the OD inefficiency, muons going through OD only might not be efficiently tagged. Furthermore, muons passing outside the detector, for example in the surrounding rocks, produce fast neutrons that could travel inside the detector without being detected by the OD. The property of the fast neutron background is studied by applying all the cuts same as the ν̄e candidate selection, except reversing the 2 ms OD muon cut by requiring that the prompt event must happen within 2 ms of an OD-only muon. This predominantly selects the fast neutrons that are produced by muons passing through outer detector only. As a result, a total of 39 fast-neutron-like events were tagged by the OD. The OD inefficiency in tagging ID muons can be studied by counting ID muons that do not register enough OD hits (OD nsummmax > 10), as shown in Fig. 5.20. The inefficiency is in average ∼1%, but is increasing with time due to the dying of OD PMTs. However, It can not be assumed that this inefficiency is the same as for tagging non-ID muons, since they might produce less light in the OD, especially for muons only passing through the rock. For this reason the number of fast neutron events in the ν̄e candidates can not be estimated directly, and is treated as a free 151 Inefficiency OD Inefficiency in Tagging ID muon 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 1000 2000 3000 4000 5000 6000 7000 8000 Run Number Figure 5.20: The OD inefficiency in tagging ID muons. The increasing of inefficiency over time is due to the wear-out of OD PMTs. Fast Neutron Events Events / 20 µ s Events / MeV Fast Neutron Events 9 8 7 30 25 6 20 5 15 4 3 10 2 5 1 0 5 10 15 20 25 30 35 40 45 50 Evis [MeV] (a) Fast Neutron Energy Spectrum 0 0 100 200 300 400 500 600 700 800 900 1000 ∆Tp-d [µ s] (b) Neutron Capture Time of Fast Neutron Figure 5.21: The fast neutron events are selected the same as ν̄e events except requiring that the prompt event happens within 2 ms of an OD-only muon. (a) Prompt visible energy spectrum (Evis > 2.7 MeV) of fast neutron events. There is no apparent structure in the spectrum and the spectrum is assumed to be flat in the analysis. (b) The distribution of time between prompt and delayed events. The distribution is fitted to an exponential decay function plus a flat background. The neutron capture time from the fit is 193.5 ± 21.9 µs. The χ2 /ndf from the fit is 29.7/40. parameter in this analysis, confined only by the shape of its energy spectrum. Fig. 5.21(a) shows the energy spectrum of the OD-tagged fast neutrons after applying all ν̄e selection rules but reversing the 2 ms OD muon cut. The spectrum is rather flat in the energy region up to ∼30 MeV, and a flat spectrum is assumed for 3 Fast Neutron Events Vertex Distribution 8 Balloon Surface 6 6m F.V. Boundary 4 Events / 10.8m Z [m] 152 2 Fast Neutron Events Radial Distribution 50 45 40 2.7 MeV < E < 15 MeV 35 30 0 25 20 -2 15 -4 10 -6 5 0.2 0.4 0.6 0.8 1 3 Fast Neutron Events Radial Distribution 30 25 00 1.2 1.4 (ρ/6.5m) 2 Events / 10.8m Events / 10.8m 3 -80 15 MeV < E < 30 MeV 20 50 100 150 200 250 3 3 R mean [m ] 200 250 3 3 R mean [m ] Fast Neutron Events Radial Distribution 20 18 16 30 MeV < E < 45 MeV 14 12 10 15 8 10 6 4 5 2 0 0 50 100 150 200 250 3 3 R mean [m ] 0 0 50 100 150 Figure 5.22: The vertex distribution of OD-tagged fast neutron events (PDE cut is not applied). In the top left figure the prompt energy is only required to be higher than 2.7 MeV. The radial distribution of the fast neutrons are shown in three different energy ranges: 2.7 – 15 MeV, 15 – 30 MeV and 30 - 45 MeV. The vertical dotted line indicate the 6m fiducial volume boundary. the OD-missed fast neutrons as well in the following analysis. Fig. 5.21(b) shows the distribution of time between prompt and delayed events. To increase the statistics the prompt energy is only required to be higher than 2.7 MeV and no upper limit is required. The distribution is fitted to an exponential decay function plus a flat background and the neutron capture time from the fit is 193.5 ± 21.9 µs. A study of the vertex distribution of fast neutron events is shown in Fig. 5.22. Here the PDE cut is not applied since it preferentially removes events near the fiducial volume boundary. The prompt energy is only required to be higher than 2.7 MeV, and the radial distribution of the fast neutrons are shown in three different energy ranges: 2.7 – 15 MeV, 15 – 30 MeV and 30 – 45 MeV. The attenuation of fast neutrons 153 by the liquid scintillator is clear, and higher energy neutrons are more penetrating than the lower energy ones. In the analysis energy range 2.7 – 15 MeV, the radial distribution is very similar to the accidental background (Fig. 4.14(b)), thus the PDE cut is expected to be removing a lot of fast neutron events as well (Fig. 4.18, 66 events → 39 events after applying the PDE cut). Nevertheless, we treat the number of fast neutron background events as a free parameter and do not estimate the efficiency directly. 5.7 Atmospheric Neutrino Background The quasi-elastic neutral current interactions between atmospheric neutrinos and neutrons in the liquid scintillator produce energetic neutrons and could become a potential background source. This background has been conservatively estimated in Ref. [115], by using the calculated maximum atmospheric neutrino flux (solar minimum case) dφ/dEν at KamLAND site from Ref. [23] and neutrino-neutron elastic scattering cross section dσ/dQ2 (modified from Ref. [116] by changing the isovector form factor) and neglecting the nuclear effects of carbon (including which lowers the cross-section by ∼15%). The recoiled neutron energy spectrum can be then calculated from X dN = Nn · Tlive · dT i Z dφi dσ i dQ2 dEν dEν dQ2 dT (5.12) where Nn = 1.8 × 1032 is the number of the neutrons (from 12 C) in the 6m fiducial volume, and Tlive = 1440 days is the total live time. The sum is over the 4 types of atmospheric neutrinos νµ , ν̄µ , νe and ν̄e , and the neutron recoil energy T is related to the squared momentum transfer Q2 simply by T = Q2 /2Mn . This neutron energy spectrum is then used as an input to a GEANT4 based simulation of KamLAND detector and the expected prompt visible energy spectrum is generated as shown in Fig. 5.23. Within the analysis threshold from 2.7 MeV to 15 MeV, a maximum 154 Events [arbitrary unit] Prompt Energy Spectrum from Atmospheric Neutrinos (MC) 1.2 1 0.8 0.6 0.4 0.2 5 10 15 20 25 30 Evis [MeV] Figure 5.23: The prompt visible energy spectrum from atmospheric neutrino interacting in KamLAND detector, simulated by KLG4sim in Ref [115]. number of 9 events are expected. In the current analysis, the atmospheric neutrino background is treated the same as the fast neutron background by allowing the total number free floating but assuming a flat energy spectrum in the analysis energy window 2.7 – 15 MeV. Assuming the correctness of the Monte Carlo spectrum in Fig. 5.23, this treatment systematically underestimates the number of atmospheric neutrinos at the lower energy. The maximum deviation is less than 2 events though, assuming that the total number of atmospheric neutrino events is less than 9 as estimated above. 5.8 Summary of Signals and Backgrounds Table 5.4 summarizes the number of expected signal and background events discussed in this chapter. Accidentals are the dominant background in both periods, while 13 C(α,n)16 O background, being significant in Period I, is significantly reduced in Period II due to the purification of the liquid scintillator. Table 5.5 summarizes the estimated sources of systematic uncertainties on deter- 155 mining the number of expected reactor ν̄e events. The systematic uncertainty in the energy scale (Section 3.4.3) at the 2.7 MeV prompt visible energy threshold is 1.2%, which corresponds to a 1.2% uncertainty in integrating the number of events in an un-oscillated reactor ν̄e spectrum. The total systematic uncertainty is 4.2% and is dominated by the uncertainties of ν̄e spectra, reactor power and fiducial volume. Table 5.4: Summary of Signal and Background Events Period I Period II live time (days) 1261.35 178.52 expected reactor ν̄e (no osci.) 1267.6 104.8 georeactor ν̄e free parameter observed ν̄e candidate events 844 86 Background Events accidentals 40.2 ± 1.2 8.2 ± 0.5 13 16 C(α,n) O (recoiled protons) 2.7 ± 0.5 0.1 ± 0.0 13 C(α,n)16 O (12 C∗ de-excitation) 4.8 ± 2.4 0.2 ± 0.1 13 C(α,n)16 O∗ (16 O∗ de-excitation) 22.4 ± 11.2 0.9 ± 0.4 9 Li 9.5 ± 3.1 1.3 ± 0.6 fast neutron and atmospheric ν̄e free parameter Total Backgrounds 79.6 ± 11.9 10.7 ± 0.9 Table 5.5: Summary of estimated systematic uncertainties (%). Fiducial volume 2.0 Reactor power 2.1 Energy threshold 1.2 Fuel composition 1.0 Efficiency of cuts 0.9 ν̄e spectra 2.5 Cross section 0.2 Total Systematic Uncertainty 4.2 156 Chapter 6 Analysis and Results An unbinned maximum likelihood (ML) method is used in this dissertation to estimate the set of model parameters θ = (θ1 , . . . , θm ) whose values are unknown. The model parameters are divided into two sets: free parameters whose values are allowed to be free floating in the analysis, and nuisance parameters whose values are constrained by independent measurements. Table 6.1 summarizes all the free model parameters and nuisance model parameters used in this analysis. The neutrino oscillation parameters (tan2 θ12 , ∆m221 ) and the georeactor fission power Pgeo are treated as free parameters, and a combined fit of them is performed. ǫreactor denotes the efficiency of detecting reactor ν̄e events, summarized in Table 4.5. The uncertainties on ǫreactor is summarized in Table 5.2. ǫcommon denotes the common efficiency of detecting all signal and background events. Its uncertainty includes those on the number of target protons, and is summarized in Table 4.3. αij denotes the four detector energy response paramj,k j j eters in period j, discussed in Section 3.4. Nacc , N(α,n) and NLi9 indicate the number of corresponding background events in period j, summarized in Section 5.8. Nfn,atm , the total number of fast neutron and atmospheric ν background events is allowed to be free floating but assuming a flat energy spectrum (Section 5.6, 5.7). 6.1 Likelihood Function At the beginning of the analysis, a likelihood function L(θ) = f (~x; θ) is constructed, which defines the joint probability density function (p.d.f.) of the data ~x = (x1 , . . . , xn ), evaluated with the measurements obtained from the experiments, predicted by the set of model parameters θ. The likelihood function is regarded as a function of the 157 Table 6.1: Model parameters used in this analysis. The number in the parenthesis indicates the total number of parameters in the corresponding category Free Model Parameters 2 tan θ12 Neutrino oscillation mixing angle ∆m221 Neutrino oscillation mass square difference Pgeo Georeactor fission power Nfn,atm Number of fast neutron and atmospheric ν background events Nuisance Model Parameters ǫcommon Common Efficiency for ν̄e and background events detection ǫreactor Efficiency for reactor and georeactor ν̄e detection only αij Detector energy response parameters in period j (8) j Nacc Number of accidental coincidence background events in period j (2) j,k N(α,n) Number of type k 13 C(α,n)16 O background events in period j (6) j NLi9 Number of 9 Li background events in period j (2) b to be those values of model parameters, and the ML method takes the estimators θ θ that maximize L(θ). Unbinned ML method is often used when the experiment is limited by low statistics, especially when a multi-dimensional binning is needed for b known as maximum likelihood estimator (MLE) the binned analysis. Meanwhile, θ, has the nice properties of being asymptotically consistent, unbiased and efficient. The likelihood function L(θ) can be written as a product of independent pdfs, and evaluated separately: L(θ) = Y Ll (θ; ~x) (6.1) l = Lpenalty · Lrate · Lshape · Ltime · Lsolar (6.2) In this analysis, the likelihood function consists of several pdf terms including penalty terms for nuisance parameters, absolute event rate, shape of the prompt visible energy spectrum, time variation of event rate and solar experiment term. Details will be discussed in the following subsections. It is equivalent and often easier to work with ln L instead of L. It can be shown that in the large sample limit, L has a Gaussian shape and ln L is parabolic. In this 158 case, one often defines χ2 (θ) = −2[ln L − ln Lmax ] (6.3) = χ2penalty + χ2rate + χ2shape + χ2time + χ2solar − χ2min (6.4) b The variance where Lmax is the value of maximized likelihood function, or, L(θ). of the MLE θbi can then be numerically determined by the fact that the contour in parameter space defined by χ2 (θ) = s2 has tangent planes located at s-standarddeviation away from the MLE θbi . Penalty Term Each nuisance parameter θi in Table 6.1 is treated as a Gaussian distributed random variable, whose mean µi and standard deviation σi is determined from independent measurements. They then introduce a χ2penalty term in Eqn. 6.4 χ2penalty = X µ θi − µ i ¶2 i σi (6.5) where the subscript i runs over all the independent nuisance parameters. Some nuisance parameters, namely the four detector energy response parameters αi , are correlated. In this case, the covariance error matrix is applied, and the χ2penalty becomes χ2penalty (α) = X (αi − µi )(V −1 )ij (αj − µj ) (6.6) i,j where the covariance matrix Vij = cov[αi , αj ] is calculated in Section 3.4 from the fit to the detector energy response model. 159 Rate Term Lrate defines the probability of detecting n events, when expecting ν events. This simply follows the Poisson distribution: Lrate (θ) = f (n; ν) = ν n e−ν n! (6.7) where n = 930 is the total number of observed ν̄e candidate events in the two analysis periods. The total number of expected events ν is the sum of the number of expected reactor ν̄e events, the number of expected georeactor ν̄e events and the number of all other background events in both analysis periods, predicted by model parameters θ. Shape Term The visible energy spectrum of prompt events is also model parameter θ dependent. Lshape defines the probability of detecting the event set with prompt visible energies ~ = (E1 , . . . , En ). We consider the normalized prompt visible energy spectrum as a E pdf f (E; θ), and define Lshape (θ) = = events Y f (Ei ; θ) i à events Y i II 1 X X dNkj (E; θ) ν j=I k dE (6.8) !¯ ¯ ¯ ¯ ¯ (6.9) E=Ei where i runs over all the observed 930 event pairs, j runs over Period I and II and k runs over all the signal and background components of the energy spectrum (because of the change of the detector energy response between Period I and II, the visible energy spectrum of each component has to be evaluated separately in each period). ν is the same total number of expected events as in the Lrate term. In particular, the 160 prompt visible energy spectrum is normalized in such a way that Z Emax Emin and Z Emax dE Emin dNkj (E; θ) = νkj dE dE II X X dN j (E; θ) k j=I k dE (6.10) =ν (6.11) where (Emin , Emax ) = (2.7MeV, 15MeV) is the analysis range of prompt energy, and νkj is the expected number of signal or background events νk in period j. Shape + Time Term The time information of the observed events provides another powerful tool for evaluating the model parameters θ. In particular, the reactor ν̄e events are correlated with reactor power and varies with time, while the georeactor and other background type events are typically constant over time. On the other hand, the two terms Ltime and Lshape are not independent of each other. This is because the relative of change of signal and background event rate with time will also affect the normalization of the energy spectrum, thus makes the shape time-dependent. In this case, a joint likelihood function Lshape+time is defined as the probability of detecting the event set with ~ = (E1 , . . . , En ) at time ~t = (t1 , . . . , tn ). We can then write prompt visible energies E down the normalized instantaneous prompt energy spectrum as a joint pdf f (E, t; θ), and define Lshape+time as Lshape+time (θ) = = 930 Y f (Ei , ti ; θ) i=1 à 930 Y i II 1 X X dNkj (E, t; θ) ν j=I k dEdt (6.12) !¯ ¯ ¯ ¯ ¯ (6.13) E=Ei , t=ti 161 ×10-3 ∆ m 2 (eV 2) -3 0.2 0.2 68% CL 0.18 95% CL 0.16 0.15 0.14 99.73% CL 0.12 0.1 0.1 0.08 0.06 0.05 0.04 0.02 0 0.2 0.4 0.6 0.8 1 tan 2θ Figure 6.1: The SNO-only contour plot on neutrino oscillation parameters, which includes a combined fit from SNO D2O & salt phase day and night spectra and NCD phase flux information [117] where the subscripts are the same as in Eqn. 6.9 and the instantaneous reactor ν̄e rate is evaluated from Eqn. 5.6. Georeactor ν̄e rate and all other background ground event rates, on the other hand, are considered to be time-invariant over period j, such that: 1 dNkj (E; θ) dNkj (E, t; θ) = dEdt Tj dE (6.14) where Tj is the total live time of period j, and k runs over all the signal and background components except the reactor ν̄e component. The normalization then, after integrating over the live time in each period, becomes the same as in Eqn. 6.10 and 6.11 Solar Term Solar neutrino experiments, discussed in Section 1.2.2, probe the same neutrino oscillation parameters region as KamLAND does, assuming CPT invariance. In particular, as will be shown, they provide better constraint on tan2 θ12 than KamLAND-only analysis. 162 The SNO-only χ2 table [118] is used in this analysis, which includes a combined fit from SNO D2O & salt phase day and night spectra and NCD phase flux information [117]. Fig. 6.1 shows the SNO-only contour plot on (∆m221 , tan2 θ12 ) parameter space. Since including other solar neutrino experiment results into the global fit does not improve significantly on the contour, while introducing non-trivia correlated systematic errors, it is avoided in this analysis. The solar term then introduces a χ2solar term which only depends on model parameter tan2 θ12 and ∆m221 : χ2solar (θ) = χ2SNO (tan2 θ12 , ∆m221 ) 6.2 (6.15) Analysis Modes Four analysis modes are performed in this dissertation by including all or part of the likelihood function terms defined in Eqn. 6.2. The four modes are defined as follows: KL-RS KamLAND-only (rate + shape) LKL−RS = Lpenalty · Lrate · Lshape (6.16) KL-RST KamLAND-only (rate + shape + time) LKL−RST = Lpenalty · Lrate · Lshape+time (6.17) KL-RS-Solar KamLAND (rate + shape) + Solar LKL−RS−Solar = Lpenalty · Lrate · Lshape · Lsolar (6.18) 163 KL-RST-Solar KamLAND (rate + shape + time) + Solar LKL−RST−Solar = Lpenalty · Lrate · Lshape+time · Lsolar 6.3 (6.19) Best Fit Parameters The best-fit model parameters θ were extracted from the two KamLAND-only analysis modes “KL-RS” and “KL-RST”. In each analysis mode, the effective χ2 = −2 ln L is minimized using the Minuit Package [94], which is equivalent to maximize the likelihood function. The best-fit values of free model parameters are summarized in Table 6.2. Notice that in the “KL-RS” analysis mode, the best-fit value of georeactor power comes out to be zero (with large uncertainties). Fig. 6.2 shows the prompt visible energy spectra from signal and background components, predicted by the best-fit model parameters from each analysis mode. Also shown on the plot is the binned observed events’ energy spectrum for visual comparison. Fig. 6.3 shows the time variation of event rate from signal and background components, predicted by the best-fit model parameters. KamLAND data are binned into 8 equal live-time bins with 180 live-days each. The filled area shows the expected total events in each bin (reactor + georeactor + all backgrounds). The best-fit average event rate is ∼0.6/day from oscillated reactor ν̄e , and ∼0.05/day from oscillated Table 6.2: Summary of best-fit values of free model parameters, in the “rate + shape” (KL-RS) and “rate + shape + time” (KL-RST) analysis modes. Shown in the parenthesis is the uncertainty of each parameter returned by the Minuit package. Parameter KL-RS KL-RST Description tan2 θ12 0.40(07) 0.51(20) ν oscillation mixing angle ∆m221 [×10−5 eV2 ] 7.58(24) 7.56(27) ν oscillation mass square difference Pgeo [TW] 0(5.4) 6.6(7.4) Georeactor fission power Nfn,atm 19(6) 19(6) fast neutron + atmospheric ν events events / 0.425 MeV 164 KamLAND Data best-fit all reactor ν e accidental 13 C(α,n)16O 9 Li fast neutron 102 10 1 0 2 4 6 8 10 12 14 Evis [MeV] events / 0.425 MeV (a) Rate + Shape KamLAND Data best-fit all geo reactor ν e reactor ν e accidental 13 16 C(α,n) O 9 Li fast neutron 102 10 1 0 2 4 6 8 10 12 14 Evis [MeV] (b) Rate + Shape + Time Figure 6.2: The prompt visible energy spectra predicted from the best-fit model parameters, with binned ν̄e candidate data overlaid, in (a) “rate + shape” analysis mode; (b) “rate + shape + time” analysis mode. The energy spectrum components shown on the graph are: best-fit total energy spectrum (black solid), oscillated georeactor ν̄e (solid red), oscillated reactor ν̄e (solid blue), accidentals (dashed green), 13 C(α,n)16 O (dashed magenta), 9 Li (dashed light blue) and fast neutron (dashed cyan). The bestfit georeactor power is zero in “rate + shape” analysis mode. The vertical long-dashed line indicates the analysis threshold at 2.7 MeV. 165 georeactor ν̄e events in the KL-RST analysis mode. 6.4 Goodness-of-fit Test The maximum likelihood method, however, does not provide a direct way to evaluate the goodness of fit. In particular, the effective χ2 = −2 ln L does not indicate the quality of the fit. Instead, a binned goodness-of-fit test is performed following the statistical techniques described in Ref. [119]. Goodness-of-fit of Energy Spectrum A goodness-of-fit test is performed to the best-fit prompt visible energy spectrum (from the KL-RS mode) as follows. First, N equal probability bins are constructed from the expected total energy spectrum (Fig. 6.2(a)) in the analysis range 2.7 – 15 MeV. The Pearson χ2 is then defined by : χ2Person (shape) = N bins X i=1 (ni − µi )2 /µi (6.20) where ni is the number of observed events in the ith bin, and µi is the number of expected events in the ith bin. Then, the χ2penalty term from nuisance model parameters (Eqn. 6.5) is added to the total Pearson χ2 : χ2test = χ2Person (shape) + χ2penalty (6.21) The remaining four free model parameters (tan2 θ12 , ∆m221 , Pgeo , Nfn,atm ) decrease the total number of degree of freedom (ndf) by 4. The p-value is then looked up from the χ2 distribution with N − 4 degree of freedom, and plotted against the number of bins, shown in Fig. 6.4. events / 180 live-days 166 200 KamLAND Data expected total. 180 reactor ν e 160 accidental C(α,n)16O 13 140 120 100 80 60 40 20 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 run number events / 180 live-days (a) Rate + Shape 200 KamLAND Data expected total. reactor ν e geo reactor ν e accidental 13 C(α,n)16O 180 160 140 120 100 80 60 40 20 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 run number (b) Rate + Shape + Time Figure 6.3: Time variation of event rate predicted by the best-fit model parameters in (a) “rate + shape” analysis mode; (b) “rate + shape + time” analysis mode. KamLAND data (solid red) are binned into 8 equal live-time bins with 180 livedays each. The filled light blue area shows the expected total events in each bin (reactor + georeactor + all BG). Also shown on the figures are : oscillated reactor ν̄e (dotted black), oscillated georeactor ν̄e (dotted red), accidentals (dashed blue) and 13 C(α,n)16 O (dashed magenta). 9 Li and fast neutron background are too small to be shown. The best-fit georeactor power is zero in the “rate + shape” analysis mode. 167 p-value Goodness of Fit (shape) 0.25 0.2 0.15 0.1 0.05 0 10 15 20 25 30 35 40 45 50 number of bins Figure 6.4: Goodness-of-fit of the prompt visible energy spectrum. The p-value is plotted against the number of bins used to construct the goodness-of-fit test. p-value Goodness of Fit (time) 0.3 0.25 0.2 0.15 0.1 0.05 0 10 15 20 25 30 35 40 45 50 number of bins Figure 6.5: Goodness-of-fit of the time variation spectrum. The p-value is plotted against the number of bins used to construct the goodness-of-fit test. 168 Goodness-of-fit of Time Variation Spectrum A goodness-of-fit test is also performed to the best-fit time variation spectrum (from the KL-RST mode). N equal probability bins are constructed from the expected total event rate time-variation spectrum shown in Fig. 6.3(b). The χ2 test is then defined by : χ2test = χ2Person (time) + χ2penalty (6.22) where χ2penalty is defined the same as in goodness-of-fit test of energy spectrum. The four free model parameters (tan2 θ12 , ∆m221 , Pgeo , Nfn,atm ) decrease ndf by 4. The p-value is then looked up from the χ2 distribution with N − 4 degree of freedom, and plotted against the number of bins, shown in Fig. 6.5. The number of equal probability bins used to construct the χ2 test is subjective and need to be chosen based on the data set. From Eqn 25.62 in Ref. [119], the recommended number of bins is approximately proportional to n0.4 , where n is the total number of events. For this analysis, a recommended range of 25 – 50 bins is calculated following Ref. [119]. Increasing the number of bins would make the fit more sensitive to the high frequency components of the spectrum. If we choose 40 equal probability bins, the p-value is 0.18 for the energy spectrum χ2 test and 0.11 for the time variation spectrum χ2 test. The goodness-of-fit test finds no issues with the fit and the result for the unbinned likelihood analysis is reasonable. 6.5 Significance of Energy Spectrum Distortion In Fig. 6.6(a) we plot the observed prompt visible energy spectrum together with the expected spectrum without neutrino oscillation. As can be seen the two spectra are inconsistent with each other, in particular due to the large deficit of the observed event rate. One alternative explanation of this inconsistence is that some unknown factors 169 cause a larger decrease of ν̄e detection efficiency than expected. A no-oscillation shape-only analysis is performed to test this hypothesis, by fixing θ12 at zero but allowing the efficiency ǫreactor to float freely. The ν̄e ’s from man-made reactors and from the georeactor are treated together since their spectra essentially become the same in the no-oscillation case. The resulting best-fit spectra are shown in Fig. 6.6(b). As can be visually seen, the best-fit prompt energy spectrum in the on-oscillation case does not agree well with the data. A goodness-of-fit χ2 test is constructed the same as in Section 6.4. The two free model parameters (ǫreactor , Nfn,atm ) decrease the ndf by 2. The resulting p-value is plotted against the number of bins used to construct the χ2 test, shown in Fig. 6.7. One can compare this with the goodness-offit test from oscillation analysis (Fig. 6.4). Although the p-values are sensitive to the number of bins chosen, the majority of them are below 0.1%, corresponding to the no-oscillation hypothesis being disfavored at more than 3σ from the distortion of the shape of prompt energy spectrum. 6.6 Correlation with Reactor ν̄e Flux In Fig. 6.8(b) we show the correlation of the data with the reactor ν̄e flux by plotting the observed event rate against the expected reactor ν̄e rate without oscillation. A positive correlation is clearly visible. If the survival probability and background rate is constant over time, then one would expect a perfect linear correlation between the two. A linear fit and the 1σ C.L. region from the fit is shown as the gray shaded area in Fig. 6.8(b). The fitting result seems to suggest an extra event rate above the expected background level at zero reactor power, a hint of the existence of nonnegligible georeactor ν̄e flux. However, the above conclusion is incorrect because the survival probability of ν̄e varies with time too. From Eqn. 1.11, the survival probability depends on the events / 0.425 MeV 170 300 KamLAND Data reactor ν e (no osci.) best-fit all. reactor ν e accidental 13 C(α ,n)16O 250 200 150 100 50 0 2 4 6 8 10 12 14 Evis [MeV] (a) Oscillation Analysis events / 0.425 MeV No Oscillation Analysis 300 KamLAND Data reactor ν e (100% eff.) best-fit all. reactor + georeactor ν e accidental 13 C(α,n)16O 250 200 150 100 50 0 2 4 6 8 10 12 14 Evis [MeV] (b) No Oscillation Analysis Figure 6.6: The best-fit prompt visible energy spectra from (a) KamLAND-only oscillation (rate + shape) analysis, (b) KamLAND-only no-oscillation (shape-only) analysis. In both cases, 9 Li and fast neutron backgrounds are too small to be shown. The vertical long-dashed line indicates the analysis threshold at 2.7 MeV. The best-fit total prompt energy spectrum from no-oscillation analysis does not agree well with the data. 171 p-value Goodness of Fit (no-oscillation) 10-2 10-3 10-4 10 15 20 25 30 35 40 45 50 number of bins Figure 6.7: Goodness-of-fit of no-oscillation analysis. The p-value is plotted against the number of bins used to construct the χ2Person test. distances of the reactors to the detector. With different reactors turned on and off at different times, the relative contribution of ν̄e flux from each reactor changes, which causes the survival probability to be time-dependent. With the oscillation parameters from Ref. [22], the survival probability is calculated as a function of time as shown in Fig. 6.8(a). On average, a ∼10% variation of the survival probability is seen over time. In particular, the Shika reactors are at a much closer distance (88 km) than the average (180 km). When they were in full operation, a noticeable decrease of the survival probability was observed. We can account for the time variation of survival probability by plotting the observed event rate against the expected oscillated reactor ν̄e rate, as shown in Fig. 6.8(c). A linear fit yields χ2 /ndf = 9.8/15. The allowed 68.3% C.L. region from the fit is shown as the grey shaded area. The correlation coefficient (slope) from the fit is 0.784 ± 0.211. The rather large uncertainty comes from the limited statistics, and reflects the fact that KamLAND by itself does not constrain the θ12 very well. ×10-36 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 observed rate [events/day] 0.7 survivial probability expected no-osci. event rate [proton-1 Sec-1] 172 1 KamLAND Data 0.9 linear fit. expected BG rate w/o georeactor 0.8 0.7 0.6 0.5 0.4 0.3 0.2 2001/12 2004/01 2005/12 0.1 0 2008/01 0 0 0.2 Year observed rate [events/day] (a) Time Variation of Survival Probability 0.4 0.6 0.8 1 1.2 1.4 expected reactor νe rate (no osci.) [events/day] (b) Correlation with Un-oscillated Reactor ν̄e Flux KamLAND Data 1 expected BG w/o georeactor expected BG w/ best fit georeactor 0.8 0.6 0.4 best linear fit linear fit 68% C.L. 0.2 best linear fit (slope fixed to 1) linear fit (slope fixed to 1) 68% C.L. 0 0 0.2 0.4 0.6 0.8 1 expected oscillated reactor νe rate [events/day] (c) Correlation with Oscillated Reactor ν̄e Flux Figure 6.8: (a) Time variation of survival probability calculated with the oscillation parameters from Ref. [22]. (b) Correlation of observed event rate with expected un-oscillated reactor ν̄e flux. (c) Correlation of observed event rate with expected oscillated reactor ν̄e flux. In case (c), a linear fit yields χ2 /ndf = 9.8/15, with the allowed 68.3% C.L. region shown as the grey shaded area. The correlation coefficient (slope) from the fit is 0.784 ± 0.211. A linear fit with the slope fixed to 1 yields χ2 /ndf = 10.8/15 with the allowed 68.3% C.L. region shown as the red shaded area. The event rate at zero reactor ν̄e flux is 0.09 ± 0.02 from the fit. The expected background event rate without georeactor is 0.07 ± 0.01 events/day (blue marker). The expected background event rate including best fit georeactor ν̄e is 0.12 ± 0.04 events/day (black marker) from the “rate + shape + time” unbinned likelihood analysis. 173 If we take as a priori the precision of the oscillation parameters from Ref. [22], a linear fit can be performed with the slope fixed to expected value 1. The result from the fit is χ2 /ndf = 10.8/16 with the allowed 68.3% C.L. region shown as the red shaded area in Fig. 6.8(c). The expected background event rate without georeactor is 0.07 ± 0.01 events/day (blue marker), which agrees with the fitted event rate at zero reactor ν̄e flux (0.09 ± 0.02 events/day) within 1σ, and no clear evidence of the existence of the georeactor is observed. As a comparison, the expected background event rate including best fit georeactor ν̄e is 0.12 ± 0.04 events/day (black marker) from the KL-RST unbinned likelihood analysis. 6.7 Results on Georeactor Fission Power All four analysis modes described in Section 6.2 have been performed on the data to estimate the georeactor fission power. In each analysis mode, a global scan on (Pgeo , tan2 θ12 ) parameter space is performed, where for each pair of values of (Pgeo , tan2 θ12 ), a best fit is obtained by varying other model parameters until the effective χ2 is minimized. The 68.3%, 90% and 99% confidence regions are given as all points that have a χ2 within 2.30, 4.61 and 9.21 of the global minimum value. The results are shown in Fig. 6.9. A positive correlation between Pgeo and tan2 θ12 is clear in all four analyses. The positive correlation mainly comes from the following two facts: 1. The higher the tan2 θ12 is, the more georeactor ν̄e events disappeared by oscillation before getting to the detector, thus for the same number of observed georeactor ν̄e events, the higher the georeactor power is. 2. The higher the tan2 θ12 is, the more reactor ν̄e events disappear by oscillation. For the same number of total observed events, this implies more among them could be georeactor ν̄e events, which leads to a higher georeactor power. Geo-reactor Power [TW] Geo-reactor Power [TW] 174 68.3% C.L. 90% C.L. 25 99% C.L. best fit 20 15 15 5 5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 tan2θ12 0.1 68.3% C.L. 90% C.L. 25 99% C.L. best fit 20 15 0.5 0.6 0.7 0.8 0.9 tan2θ12 (c) KamLAND (rate + shape) + Solar 0.6 0.7 0.8 0.9 tan2θ12 90% C.L. 99% C.L. 15 5 0.4 0.5 best fit 5 0.3 0.4 20 10 0.2 0.3 68.3% C.L. 25 10 0 0.1 0.2 (b) KamLAND-only (rate + shape + time) Geo-reactor Power [TW] (a) KamLAND-only (rate + shape) 99% C.L. best fit 10 0.2 90% C.L. 20 10 0 0.1 Geo-reactor Power [TW] 68.3% C.L. 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 tan2θ12 (d) KamLAND (rate + shape + time) + Solar Figure 6.9: The 68.3%, 90% and 99% confidence regions of (Pgeo , tan2 θ12 ) from (a) KamLAND-only (rate + shape + time) analysis; (b) KamLAND-only (rate + shape + time); (c) KamLAND (rate + shape + time) + Solar analysis; (d) KamLAND (rate + shape + time) + Solar analysis. 175 The positive correlation between Pgeo and tan2 θ12 implies that a better constraint on tan2 θ12 will also give a better constraint on Pgeo . KamLAND-only analysis, while giving the best constraint on ∆m221 (from distortion of energy spectrum), does not constrain tan2 θ12 very well (Section 6.8), especially on the higher end of tan2 θ12 . The situation becomes worse when including the georeactor in the analysis since too large a deficit caused by a large tan2 θ12 can now be partially compensated for by the georeactor ν̄e flux. On the other hand, the solar neutrino experiments give a much better constraint on tan2 θ12 [117], especially on the higher end of tan2 θ12 . By including the solar χ2 term in the analysis, the constraint on georeactor power is largely improved. The χ2 distribution of georeactor fission power is obtained by performing a onedimensional scan on Pgeo . For each value of Pgeo , a best fit is obtained by varying other model parameters. The global minimum χ2min is subtracted from each best-fit χ2 , and the ∆χ2 profile is shown in Fig. 6.10. As expected, the best constraint comes from including the solar term in the analysis. The best fit value of georeactor fission +3.8 power is at 1.0+4.8 −1.0 TW from the KL-RS-Solar analysis, and 4.9−4.8 TW from the KL- RST-Solar analysis. By including the time variation information of expected event rate, the best-fit value of georeactor power moved from 1.0 TW to 4.9 TW which is approximately 1σ away from zero. The reason is best illustrated from the best-fit time variation spectrum in Fig. 6.3. The few extra observed events in bin 1, 2 and 8 slightly prefer a sizable georeactor ν̄e flux which is constant over time. Nevertheless zero georeactor power is still allowed at 1σ even in the KL-RST-Solar analysis, indicating a small signal. For this reason we follow Feldman and Cousins’ unified approach [120] to construct the confidence intervals. The 68.3%, 90%, and 99% confidence intervals of georeactor fission power in units of TW are (0, 5.8), (0, 8.8) and (0, 13.3) from the KL-RS-Solar analysis and (1.1, 8.7), (0, 11.2), and (0, 14.7) from KL-RST-Solar analysis, respectively. ∆ χ2 176 10 KamLAND(E) 3σ KamLAND(E,t) KamLAND(E) + Solar 8 KamLAND(E,t) + Solar 6 2σ 4 2 1σ 0 0 5 10 15 20 25 30 Georeactor Power [TW] Figure 6.10: 1-D χ2 distribution of georeactor fission power. The results comes from 4 different analysis modes: “KamLAND-only (rate + shape)” (dashed blue), “KamLAND-only (rate + shape + time)” (dashed red), “KamLAND (rate + shape) + solar” (solid blue) and “KamLAND (rate + shape + time) + solar” (solid red). The best fit value of georeactor fission power is at 1.0+4.8 −1.0 TW with “KamLAND (rate + shape) + solar” analysis, and 4.9+3.8 TW with “KamLAND (rate + shape + time) −4.8 + solar” analysis. 177 Period-I-Only Analysis As mentioned in the previous chapters, the ∼180-live-day data from Period II is important to the analysis because of the reduced reactor ν̄e flux (Fig. 5.3) due to the shut-down of Kashiwazaki reactors. However, due to the change of liquid scintillator properties after the purification, the data from Period II have different systematics from Period I. Even though they are properly treated as described in the previous chapters, as a consistency check, a period-I-only analysis is performed by only including the data from Period I. The results are shown in Fig. 6.11. The best fit value of georeactor fission power is at 2.8+4.6 −2.8 TW in the KL-RST-Solar analysis. The 68.3%, 90%, and 99% confidence intervals of georeactor fission power in units of TW are (0.3, 7.4), (0, 10.3) and (0, 14.6) following Feldman and Cousins’ unified approach [120]. Comparing with the results from full-data-set analysis, the best-fit georeactor signal is less than 1σ from zero in the period-I-only analysis. The result is quite interesting and suggests that for future improvement on the georeactor detection, a longer low-reactor-power period is necessary. If possible, a new experiment set up far away from any man-made nuclear reactors is preferred. Heat Balance of the Earth Fig. 6.12 shows the heat balance of the Earth plotted as georeactor heat output v.s. radiogenic (and other residual) heat output. The 90% C.L region of total heat flow of the Earth comes from a combination of two models [69, 70], which predicts 44.2 ± 1.0 TW and 31 ± 1 TW respectively. The expected radiogenic heat from Ref. [18, 19, 20] predicts 19 TW to 31 TW. The geoneutrino measurement by KamLAND[21, 22] infers the radiogenic heat from uranium and thorium decay chains to be within (7, 27) TW at 90% C.L. if fixing the Th/U mass ratio to 3.9 as derived from chondritic meteorites [121]. The allowed region of heat flow generated by georeactor, derived from KL-RST-Solar analysis in this dissertation, is shown as the red shaded area. 178 ∆χ2 Period I only 10 KamLAND(E) 3σ KamLAND(E,t) KamLAND(E) + Solar 8 KamLAND(E,t) + Solar 6 2σ 4 2 1σ 0 0 5 10 15 20 25 30 Georeactor Power [TW] Figure 6.11: 1-D χ2 distribution of georeactor fission power, including only the data from Period I. Pgeoreactor [TW] Heat Balance of the Earth 60 Total Heat Flow 90% C.L. Georeactor 90% C.L. 50 Expected Radiogenic Heat 40 30 20 10 0 0 10 20 30 40 50 60 Prad + residual [TW] Figure 6.12: Heat balance of the Earth plotted as georeactor heat output v.s. radiogenic (and other residual) heat output. The light blue filled area is the total heat flow of the Earth at 90% C.L. [69, 70], the green hatched area is the expected radiogenic heat output [18, 19, 20], and the red hatched area is the allowed 90% region from the analysis described in this dissertation. 179 6.8 Results on Neutrino Oscillation Parameters Null Georeactor Analysis KamLAND’s previous published results [45, 46, 22] on neutrino oscillation all assumed non-existence of georeactor in the analysis. To cross-check with these results, a nullgeoreactor analysis is performed with the same analysis method as described in this thesis except fixing the georeactor power at zero. In the absence of ν̄e disappearance, we expect to observe 1372.4 ± 57.6(syst.) reactor ν̄e events above 2.7 MeV, where the systematic uncertainty is summarized in Table 5.5. We observed 930 events, among which the total number of background events is 109.4 ± 13.5. The average survival probability of ν̄e is calculated to be R= Nobs − NBG = 0.598 ± 0.022(stat.) ± 0.027(syst.) Nexpected (6.23) The probability of no-oscillation is rejected at (1 − 5 × 10−17 ) C.L. (8.3σ). The allowed region in (∆m221 , tan2 θ12 ) space from null-georeactor analysis is shown in Fig. 6.13. In KL-RST analysis, there are still small allowed regions in the LMA-II region at 99% C.L. The recent released KamLAND results [22] lowered the analysis energy threshold to 0.9 MeV, which helped to exclude LMA-0 and LMA-II at 99.73% C.L. with “KamLAND only” analysis. In either case, when combining with solar results, only LMA-I solution is allowed at 99.73% C.L. From the null-georeactor analysis, the best-fit values of neutrino oscillation pa+0.28 2 −5 rameters are tan2 θ12 = 0.38+0.08 eV2 in KL-RST analysis, −0.06 , ∆m21 = 7.59−0.27 × 10 +0.26 2 −5 and tan2 θ12 = 0.42+0.04 eV2 in KL-RST-Solar analysis. −0.04 , ∆m21 = 7.50−0.21 × 10 This is consistent with KamLAND’s previous published results. 2 ∆m21 [eV 2] 180 10-4 KamLAND 95% C.L. SNO 95% C.L. 99% C.L. 99.73% C.L. best fit 99.73% C.L. 99.73% C.L. 10-5 -1 10 1 tan2θ12 2 ∆m21 [10 -5 eV 2] (a) KamLAND-only, Null Georeactor 95% C.L. 11 99% C.L. 99.73% C.L. 10 best fit 9 8 7 6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 tan2θ12 (b) KamLAND + Solar, Null Georeactor Figure 6.13: The 95%, 99% and 99.73% confidence regions of (∆m221 , tan2 θ12 ) from (a) KamLAND-only (rate + shape + time) analysis, (b) KamLAND (rate + shape + time) + Solar analysis. In both analyses, a null georeactor hypothesis is assumed. 181 Combined Analysis of Neutrino Oscillation and Georeactor The Best-fit georeactor fission power from the KL-RST-Solar analysis is 4.9+3.8 −4.8 TW, +45.4 which corresponds to 58.5−56.1 oscillated georeactor ν̄e events at KamLAND. The +47.3 total number of background events, including the georeactor ν̄e ’s, is 167.9−57.7 . The ν̄e survival probability is calculated to be R= Nobs − NBG = 0.555 ± 0.022(stat.) ± 0.048(syst.) Nexpected (6.24) The probability of no-oscillation is rejected at (1 − 10−12 ) C.L. (7.0σ). Fig. 6.14 shows the allowed region in (∆m221 , tan2 θ12 ) space from the combined analysis of neutrino oscillation and georeactor with the method described in this thesis. Comparing with null-georeactor analysis, in KL-RST case, because of the greater freedom introduced by the existence of georeactor ν̄e , more areas are allowed especially in the LMA-0 (lower) and LMA-II (upper) regions. In particular, the uncertainty in the large tan2 θ12 region becomes larger for the reason discussed in Section 6.7. However by including solar neutrino results in the analysis, still only LMA-I solution is allowed at 99.73% confidence level. From the combined analysis, the best-fit values of neutrino oscillation parameters +0.29 −5 2 eV2 in KL-RST analysis, and are tan2 θ12 = 0.50+0.14 −0.14 , ∆m21 = 7.59−0.26 × 10 +0.27 2 −5 tan2 θ12 = 0.44+0.05 eV2 in KL-RST-Solar analysis. This −0.05 , ∆m21 = 7.59−0.27 × 10 is consistent with KamLAND’s previously published results with null-georeactor hypothesis, and shows that including the georeactor does not significantly change the current best measured values of neutrino oscillation parameters. 2 ∆m21 [eV 2] 182 10-4 KamLAND 95% C.L. SNO 95% C.L. 99% C.L. 99.73% C.L. best fit 99.73% C.L. 99.73% C.L. 10-5 -1 10 1 tan2θ12 2 ∆m21 [10 -5 eV 2] (a) KamLAND-only 95% C.L. 11 99% C.L. 99.73% C.L. 10 best fit 9 8 7 6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 tan2θ12 (b) KamLAND + Solar Figure 6.14: The 95%, 99% and 99.73% confidence regions of (∆m221 , tan2 θ12 ) from (a) KamLAND-only (rate + shape + time) analysis, (b) KamLAND (rate + shape + time) + Solar analysis. The result comes from the combined analysis of neutrino oscillation and georeactor with the method described in this thesis. 183 Chapter 7 Conclusions The KamLAND experiment, being on the surface of the Earth, provides a unique tool to investigate the georeactor hypothesis which predicts a natural fast breeder type nuclear reactor at the center of the Earth. The anti-electron-neutrinos from a georeactor have a constant flux over time that is rather distinguishable from manmade reactors whose powers may vary over time due to the operations. They also have a different energy spectrum from the man-made reactors when reaching the KamLAND detector due to the neutrino oscillation effects. Combining with the solar neutrino experiments, KamLAND for the first time can provide a strong test on the georeactor hypothesis, measuring or setting a limit on its fission power. The analysis presented in this dissertation includes all the data taken from Apr 2002 to Jun 2008, except the six months from Apr 2007 to Oct 2007 during which the detector upgrading work was on-going. Since July 2007, the 25 GWth Kashiwazaki nuclear plant has been shut down due to damage from an earthquake, which resulted in about 25% of the reduction of reactor anti-neutrino signal in KamLAND when it is fully operational. The reduced event rate enables a more precise determination of the extrapolated zero power rate, which helps to set the limit on the georeactor power. The total live time for the analysis is 1439.87 days after muon veto. The fiducial volume for the analysis has been extended to 6 meter radius compared with the previous US analysis (5.5 m), which enlarges the fiducial mass to 0.7 ktons and gives a total exposure of 2.75 kton-years (8.50 × 1034 proton-days). From the data set 930 ν̄e candidate events above 3.4 MeV ν̄e energy threshold were extracted, with estimated 109 ± 13 events from backgrounds. The overall ν̄e detection efficiency is estimated to be 87.5 ± 0.8%. An unbinned maximum likelihood analysis 184 was performed using the event rate, spectrum shape and time variation information. Combined with the solar neutrino experiments assuming CP T invariance, the best-fit value of georeactor power is 4.9+3.8 −4.8 TW. Using Feldman and Cousins’ approach, the 90% upper limit on georeactor power is determined to be 11.2 TW. Since the total heat from the Earth is known to be between 30 and 45 TW, this result has put a significant constraint on the contribution of a possible georeactor to the total power. The 1σ georeactor signal requires additional future statistics for verification. Periods with low man-made reactor power will be especially helpful. A combined fit to the neutrino oscillation parameters is also performed, including the georeactor power as a free parameter. Combining with the solar neutrino experiments, the best-fit values of the oscillation parameters are tan2 θ12 = 0.44+0.05 −0.05 −5 and ∆m221 = 7.59+0.27 eV2 . All parameter spaces are excluded at 99.73% −0.27 × 10 C.L. except the LMA-I region. The result is consistent with KamLAND’s previously published results with null-georeactor analysis, and shows that including the georeactor does not significantly change the current best measured values of neutrino oscillation parameters. 185 Appendix A A.1 KamLAND Kr/Ar Monitor Motivation and Methodology Since 2007, the KamLAND collaboration started to upgrade the detector by purifying the liquid scintillator to reduce low energy backgrounds due to the radioactive contaminants, with the primary goal of observing solar 7 Be neutrinos. As discussed in Section 2.3.2, the signal from a νe elastic scattering is a single recoiled e− event with maximum energy of 665 keV which is indistinguishable from a radioactive β − decay event at similar energies. Table 2.3 lists the most concerned radioimpurity levels in KamLAND LS before the purification, and the requirement needed in solar phase in order to make a 7 Be solar neutrino measurement. Among them the 85 Kr and 39 Ar are especially harmful through their β − decays 85 85 − 36 Kr −→ 37 Rb + e + ν̄e Q = 687 keV, t1/2 = 10.76 y (A.1) 39 39 − 18 Ar −→ 18 K + e + ν̄e Q = 565 keV, t1/2 = 269 y (A.2) and because of their long half-lives and the end-point Q values that are similar to endpoint of the recoiled e− spectrum from solar 7 Be νe elastic scattering. It is therefore necessary to have a Kr/Ar monitoring system that operates in-line with the purification system to ensure that the required purification factors are always achieved (∼106 for 85 Kr and ∼103 for 39 Ar). Such a system must meet the following requirements: • A high detection sensitivity. The system should be able to detect 1 – 10 µBq/m3 level of activity for dissolved 85 Kr and 39 Ar in the liquid scintillator, in order to 186 Isotope 85 Kr 39 Ar T1/2 [yr] 10.76 269 Aair [Bq/m3 ] 1.4 16.6 × 10−3 nat Nair [mol/mol] iso N/nat N 1.099 × 10−6 2.8 × 10−11 −2 0.933 × 10 8.1 × 10−16 Table A.1: Summary of characteristic properties of 85 Kr and 39 Ar. The symbols stand for: T1/2 : half life; Aair : activity in the atmosphere; nat Nair : natural gas abundance in the atmosphere; iso N/nat N: isotopic abundance. The values are taken from [122, 123, 124, 125]. ensure the required purification factors. • A fast measurement cycle. The system should provide measurement result in a relatively short time scale, in order for timely operation to be performed on the purification system. • A low need for liquid scintillator sample volume. The system should be able to achieve the required sensitivity with a relatively small LS sample volume, in order to reduce the time and cost of LS production. The above requirements make any radioassay techniques impractical for evaluating the reduction factors because of the low activities of 85 Kr and 39 Ar. However, by making a reasonable assumption that the purification techniques (distillation plus nitrogen purging) removes all isotopes of Kr or Ar with equal efficiency, it is more practical to measure the concentration of nat Kr and nat Ar in the LS, and infer the 85 Kr and 39 Ar activity provided that the 85 Kr/nat Kr and 39 Ar/nat Ar ratios are known. The natural argon concentration in the air is measured to be 0.933 ± 0.003% [122], and for natural krypton it is 1.099 ± 0.009 ppm [123]. In nature, 39 Ar is mainly produced by cosmic-ray induced 40 Ar(n,2n)39 Ar reactions in the atmosphere. The activity of 39 Ar in the atmosphere is measured to be 16.6 ± 0.6 mBq/m3 [124], which corresponds to the 39 Ar/nat Ar ratio of (8.1 ± 0.3) × 10−16 . 85 Kr, on the other hand, is mostly anthropogenic. It is amply produced by the fissions of uranium and plutonium in the nuclear reactors and is released to the atmosphere when spent fuels are repro- 187 cessed. The anthropogenic production of 85 Kr has resulted in an order of magnitude increase of the isotope in the atmosphere since the early 1950s and a non-uniform geological distribution. The 85 Kr activity in the atmosphere at several locations in Japan has been monitored monthly from 1996–2001 [125], and an annual increasing of ∼0.03 Bq/m3 has been observed. The averaged activity 1.4 Bq/m3 with an uncertainty of 50% is adopted in our work, which corresponds to the 85 Kr/nat Kr ratio of 2.8 × 10−11 . These characteristic properties are summarized in Table A.1. Extensive work has been done at CalTech to develop such a detecting system that meets all the requirement. The central idea is to turn the measurement of the low concentration of nat Kr and nat Ar into a partial pressure measurement, and use a residual gas analyzer (RGA) as a detector to take advantage of its high sensitivity and fine mass resolution. The schematic design of KamKAM (KamLAND Kr/Ar Monitor) is shown in Fig. A.1. A picture of the on-site setup is shown in Fig. A.2. Details will be discussed in Section A.2. The relatively low cost and fast operation cycle makes this detecting system competitive to other alternative systems such as ATTA (Atom Trap Trace Analysis) [126]. It is also worth noting that there are growing interests and active R&D’s in using liquid scintillator detectors, in particular the noble gas detectors such as Xenon and Argon, for dark matter search, solar neutrino and double beta decay experiments. All these experiments demand low contaminations from 85 Kr and 39 Ar, for which having a Kr/Ar monitoring system such as KamKAM will be very beneficial. A.2 Detector Design and Operation The first part of the KamKAM system is the Kr and Ar extraction and trapping system. The method is inspired by Ray Davis’ famous solar neutrino experiment [35] where the 37 Ar atoms produced by solar neutrinos were flushed out with helium 188 Figure A.1: Schematic design of the KamLAND Kr/Ar monitor. 189 Helium Input LS Input RGA Bubbler Cold Trap Sublimation Pump Figure A.2: A picture of the on-site KamKAM setup. Various components are labeled. 190 gas every ∼100 days and condensed into proportional counters for counting. In the KamKAM system, the Kr/Ar extraction part (The Bubbler) is a stainless steel vacuum chamber with a volume of 8.3 liters and is pumped down to below 0.1 torr before any measurement starts. At the beginning of each measurement, 5 liters of liquid scintillator sample is input either from a direct line right after the purification system, or from a sampling port at the top of the bubbler. A helium carrier gas (99.9999% pure) is then inserted from the gas line on top of the liquid until pressure reaches 800 torr. Next we use a peristaltic pump (Masterflex I/P system) to pump the liquid from the bottom to the top and sparge into the helium gas at 0.2 liter/min for 2 hours in a closed re-circulation loop. This procedure establishes equilibrium concentrations of the dissolved gas in the helium carrier gas and the liquid scintillator. The percentage of dissolved Kr and Ar gas extracted from the LS at equilibrium is calibrated to be 79 ± 6%. After the bubbling stage is finished the helium gas is released into the bottom of a molecular sieve trap held in a liquid nitrogen bath at 77K to freeze the krypton (BP = 120 K) and argon (BP = 87 K) gas. The molecular sieve trap contains type 13X (pore size of 8 Angström) synthetic zeolite beads which effectively enlarges the surface area for trapping large molecules. Many other gases and organic vapors with a high boiling point are also trapped and become potential background in later measurements. On the other hand, most of the helium carrier gas (BP = 4 K) simply passes the trap and is pumped out. After all the helium carrier gas has been passed through the cold trap, the bubbler sub-system is closed and the cold trap is heated up to release the trapped gases into the the second part of the system, which is a vacuum line that leads to a Residual Gas Analyzer (RGA) for measuring the partial pressure of Kr and Ar in the vacuum system. This part of the system is initially under ultra-high vacuum below 10−7 torr and is continuously pumped down by a turbo-molecular pump (Varian Turbo-V 70). 191 The pressure goes up when the trapped gases are released into the system. The partial pressure of each gas species integrated over the release time is directly proportional to the amount of the gas that has been released from the trap, and hence is proportional to the concentration of the gas which is originally dissolved in the liquid scintillator. The partial pressure is continuously measured by the RGA (SRS RGA100), which is essentially a mass-spectrometer that ionizes the gases, separates the ion paths of different gas species by their mass-to-charge ratio through a quadrupole filter, and collects the ions at the end of the filter rods. The RGA can be operated in two modes depending on which type of ion collector being used: the faraday cup (FC) mode which has a dynamic range of (10−9 torr – 10−4 torr), or the electron multiplier (EM) mode which has a dynamic range of (10−12 torr – 10−6 torr). By combining the two modes a wide dynamic range and a high sensitivity can be achieved. The RGA signal is in general directly proportional to the partial pressure. However, at total pressure higher than 10−5 torr, this linear relationship breaks. The nonlinearity arises from the space charge effect in the ion source which causes the difficulty of transmitting ions from the source to the detector and decreases the signal strength at high pressures [127]. The gas release flow rate from the cold trap increases as the the temperature goes up. Because of the multitude of trapped gases including all background gases such as N2 (from the N2 purge tower), O2 (from the residual air), and oil vapors, the total pressure of the system at a certain point would exceed the RGA linearity region. The problem is solved by inserting a titanium sublimation pump (NEC sublimators) between the cold trap and the RGA. The pump contains a titanium filament which under high current sublimates and deposits highly reactive thin films of titanium on the inner surface of the chamber surrounding the filaments. Most of the N2 , O2 and oil vapors are chemically absorbed by the titanium films before they reach the RGA, while the noble gases such as Kr and Ar are not affected. In this way the resulting gas being sampled by the RGA is primarily the trace amount 192 Figure A.3: Partial-pressure vs. mass-to-charge-ratio measured by the RGA (analog scan mode) from a relatively air-rich LS sample. The gases passed through a sublimator before being sampled by the RGA. The various residual gas peaks are identified and labeled. of noble gasses of interest. Fig. A.3 shows a plot of partial-pressure versus mass-to-charge-ratio measured by the RGA from a relatively air-rich LS sample. The various residual gas peaks are identified and labeled in the plot. Since the gases passed through the sublimator before being sampled by the RGA, the partial pressure of N2 , O2 and other Ti-active background gases are greatly reduced, and the total pressure of the system is well within the RGA linearity region (below 10−5 torr). The peaks from krypton and argon are easily identifiable because of the rich isotopic feature of nat Kr (84 Kr 57%, 86 Kr 17.3%, 83 Kr 11.5%, 82 Kr 11.6%) and nat Ar (40 Ar 99.6%, 36 Ar 0.34%, 38 Ar 0.06%), which provide unambiguous signals over background gases at similar mass-to-chargeratios. After the measurement procedure is finished, the partial pressures of krypton and argon together with other background gases recorded by the RGA are plotted against Partial Pressure [torr] 193 10-6 84 Kr 40 10-7 Ar N2 H2O O2 CO 2 Oil 10-8 10-9 10-10 10-11 10-12 0 10000 20000 30000 40000 50000 60000 70000 Time [sec] (a) Partial Pressure Trend Integrated Measured Signal [torr sec] Calibration of KamKAM 10-1 10-2 x -- 85Kr y -- 84Kr -3 x -- 39Ar y -- 40Ar 10 10-4 10-5 10-6 10-7 10-8 -1 10 1 10 102 103 104 Inferred Input Activity [µBq/m3] (b) Calibration Figure A.4: (a) The partial pressure trend of different gases since the cold trap is heated up. The integrated peak areas of 40 Ar and 84 Kr are directly proportional to the amount of krypton and argon gas which were released from the cold trap. (b) The calibration of KamKAM . Known mole-amount of calibration gas is inserted into the system and the measured signals are plotted against the inferred input 39 Ar and 85 Kr activities. A linear relation is assumed and the fitted linear constant is used for conversion. 194 the time since the the cold trap is heated up. An example plot is shown in Fig. A.4(a). The building-up and pumping-down of 40 Ar and 84 Kr partial pressure is clear, while the rather constant pressure of other gases shows the effectiveness of the sublimation pumping. The integrated areas of 40 Ar and 84 Kr (in units of torr·sec) are directly proportional to the amount of krypton and argon gas which were originally dissolved in the liquid scintillator and the proportionality constant is periodically calibrated. Pure nat Kr and nat Ar gas are used for large signal calibrations. For small signal calibrations, however, precise low-mole pure nat Kr and nat Argon samples are difficult to prepare. Instead we use air for calibration and assume that the nat Kr and nat Ar concentration in the atmosphere are 1.099 ppm and 0.933%, as shown in Table A.1. The mole-amount of nat Kr and nat Ar are converted to the activity of 85 Kr and 39 Ar in units of µBq/m3 (the LS sample used for each measurement is always 5 liter), a quantity that most concerns KamLAND, from the values given in Table A.1. The calibration plot is shown in Fig. A.4(b), from which the activity of 85 Kr and 39 Ar in the measured LS sample can be directly interpreted. The uncertainty from the fitted linear constant is 26% for krypton and 4% for argon. Another systematic uncertainty comes from the reproducibility of each measurement, which incorporates uncertainties from gas trapping efficiency, gas releasing efficiency, hysteresis effects etc. Same mole-amount of calibration gases are measured several times and the variation on the results shows the uncertainty being 53% for krypton and 1% for argon. Combining all together, the total systematic uncertainty of the KamKAM measurements is 78% for 85 Kr and 10% for 39 Ar. The contributions to the systematic uncertainty are summarized in Table A.2. The sensitivity of the KamKAM system scales up linearly with the amount of liquid scintillator sample taken and is essentially limited by the air leak rate of the sampler and gas sparging system. With five liters of LS sample for each measurement, the on-site KamKAM system achieved the sensitivity of 10−13 g/g for nat Kr in LS and 195 Table A.2: Estimated systematic uncertainties (%) of KamKAM measurements. 85 Description Kr 39 Ar Gas extraction efficiency 8 8 Linear calibration 26 4 reproducibility 53 1 Isotope abundance 50 4 Total systematic uncertainty 78 10 4 × 10−8 g/g for nat Ar in LS. This corresponds to the sensitivity of 30 µBq/m3 for 85 Kr in LS and 0.3 µBq/m3 for 39 Ar in LS. A.3 On-site Measurements KamKAM was installed as a part of the KamLAND purification facilities in the spring of 2007 and started daily operation. During the first purification campaign, all the KamKAM measurement results from the LS samples right after the purification system showed no sign of krypton and argon at the sensitivity level (30µBq/m3 for 85 Kr in LS and 0.3µBq/m3 for 39 Ar in LS). This is the first strong indication that a great reduction factor was achieved by the purification (> 2 × 104 for 85 Kr). Fitting the data from KamLAND, on the other hand, only showed a reduction factor of ∼40 for 85 Kr. The inconsistency led us to believe that there are possible air leaks around the KamLAND detector chimney region, and that the convection of the LS during and after the filling is strong which caused the mixing of the new and old liquid scintillators. A second purification campaign was then started in Jun 2008 following important improvements to minimizing these effects, including an ultra-pure nitrogen overpressured tent around the detector chimney, a heat exchanger to precisely control the LS temperature, and a new bottom-filling scheme to minimize the LS convection. All the KamKAM measurement results during the second campaign again showed no sign of krypton and argon at the sensitivity level from the LS samples right after 196 the purification system. The results from the spectral fitting of the KamLAND data showed the decreasing activity of 85 Kr after each full-detector-volume LS transfer, indicating the effects of LS convection. Three full-detector-volume LS transfers were done at the end of the second purification. The 85 Kr activity in the cleanest region (∼100 m3 in the center) of the KamLAND detector from the spectrum fitting is on the order of 10 µBq/m3 . Detailed data analysis is on the way [92]. During the second purification, a new KamKAM LS sampling system was constructed which can sample the LS from the return line before it goes into the purification system. This LS comes directly out of the KamLAND detector and a measurement from this would provide an independent check on the 85 Kr and 39 Ar activity from the spectral fitting methods. One measurement has been performed on such a sample on Aug 30, 2008. The result (11.4 mBq/m3 for 85 Kr in LS and 38.0µBq/m3 for 39 Ar in LS) agrees well with the spectral fitting result around that time period. Unfortunately because of the maintenance work of the KamKAM system, no following measurements were performed in later periods. Nevertheless, this shows that an independent physical measurement of 85 Kr and 39 Ar background level in the detector is possible, which is very important to low energy solar neutrino and other low background experiments. 197 Appendix B 238 Decay Chains of Selected Isotopes U, 232 Th and 40 K have long half-lives and are natural radioactive background for KamLAND experiment. Tables B.1 and B.2 summarize the decay chains of 238 U and 232 Th. The α decays are indicated with symbol “ւ” and β − decays with “→.” Only decays with visible energy higher than 100 KeV or branching ratio more than 0.5% are listed. Effects of internal conversions are not included as they have negligible effects for KamLAND. The most intensive transition γ-rays accompanying the decays are also listed. Some isotopes have multiple β decay branches and in those cases only the highest Q-value decay branch and the (next) most probable decay branch are shown, with a “†” symbol attached to the end reminding one to look up the nuclear database (e.g. Ref. [128]) for the full decay branches. The branching ratio is normalized to per 238 U or 232 Th decay, assuming the state of secular equilibrium being kept during the decaying chains. Care should be taken when doing calculation since possible thermal or chemical processing of the intermediate daughter could result in a break in the secular equilibrium. An example is the concentration of radon in ore-bearing rocks. The decays of 40 K are summarized in Table B.3. The electron capture decay branch dominates the β + decay branch (0.001%) and is indicated by the symbol “← (EC).” 85 Kr and 39 Ar also have long half-lives and are potential backgrounds for solar 7 Be neutrino detection. Their decay schemes are summarized in Table B.4 and Table B.4. 198 Table B.1: 238 U Decay Chain Isotope (half life) & Eα Qβ Eγ Branching Ratio Decay Mode [KeV] [KeV] [KeV] [/ 238 U Decay] 238 U (4.5 × 109 y) ւ 4198 0.790 4151 50 0.210 ւ 234 Th (24.1 d) → 273 0.703 → 181 92 0.192† 234 Pa (1.2 m) → 2290 0.977 → 1224 1001 0.010† 234 U (2.5 × 105 y) ւ 4775 0.714 ւ 4722 53 0.286 230 4 Th (7.5 × 10 y) ւ 4688 0.763 ւ 4621 68 0.237 226 Ra (1600 y) ւ 4784 0.944 4602 186 0.056 ւ 222 Rn (3.8 d) ւ 5490 1.000 218 Po (3.1 m) ւ 6002 1.000 214 Pb (26.8 m) → 1019 0.110 → 667 352 0.460† 214 Bi (19.9 m) → 3272 0.191 1540 1120 0.176† → 214 Po (164 µs) ւ 6902 800 0.0001 ւ 7687 0.9999 210 Pb (22 y) → 64 0.160 → 17 47 0.840 210 Bi (5.0 d) → 1163 1.000 210 Po (138 d) ւ 5304 1.000 206 Pb (stable) 199 Table B.2: 232 Th Decay Chain Isotope (half life) & Eα Qβ Eγ [KeV] [KeV] [KeV] Decay Mode 232 Th (1.4 × 1010 y) ւ 4013 ւ 3954 64 228 Ra (5.8 y) → 46 228 Ac (6.1 h) → 2069 58 → 1158 969 228 Th (1.9 y) ւ 5423 ւ 5340 84 224 Ra (3.7 d) ւ 5685 ւ 5449 241 220 Rn (56 s) ւ 6288 216 Po (0.145 s) ւ 6778 212 Pb (11 h) → 574 → 335 239 212 Bi (61 m) 64.1% 35.9% → 2252 1525 727 → ւ 6090 ւ 6050 40 212 Po (299 ns) ւ 8784 208 Tl (3.1 m) → 1801 2615 → 1525 2615 208 Pb (stable) Branching Ratio [/ 232 Th Decay] 0.779 0.221 1.000 0.008 0.299† 0.715 0.285 0.949 0.051 1.000 1.000 0.123 0.825† 0.554 0.045† 0.098 0.262 0.640 0.175 0.026† 200 Table B.3: 40 K Decay Isotope (half life) & Eα Qβ Eγ Decay Mode [KeV] [KeV] [KeV] 40 9 K (1.3 × 10 y) 89% 11% → 1311 ← (EC) 1461 40 Ar (stable) 40 Ca (stable) Table B.4: 85 Kr Decay Isotope (half life) & Eα Qβ Eγ [KeV] [KeV] [KeV] Decay Mode 85 Kr (10.8 y) → 687 → 173 514 85 Rb (stable) Table B.5: 39 Ar Decay Isotope (half life) & Eα Qβ Eγ Decay Mode [KeV] [KeV] [KeV] 39 Ar (269 y) → 565 39 K (stable) Branching Ratio [/ 40 K Decay] 0.893 0.106 Branching Ratio [/ 85 Kr Decay] 0.9956 0.0043 Branching Ratio [/ 39 Ar Decay] 1.000 201 Appendix C C.1 Event Category Low Level Event Classification Events can be first classified by some low level information independent of their energy and position reconstructions. Two especially useful variables are the total number of total photoelectrons on all 17-inch PMTs (Np.e ) and the standard deviation of the number of photoelectrons between PMTs (σNp.e. ), defined as: 2 σN = p.e. 1 total Np.e PMTs X i i (Np.e − hNp.e. i)2 (C.1) total A plot of σNp.e. v.s. Np.e for a typical one-day’s data in KamLAND is shown in Fig. C.1. There are some distinct regions on this plot, based on which a few event categories can be defined: total 2 > 2. These are events with very large signals PMT Flasher Np.e > 103 and σN p.e. collected in a single PMT or a small group of PMTs, and are not correlated with physical events. total 2 Oil Muon 103 < Np.e < 104.8 and σN > 0.015. These are muons that pass p.e. through the buffer oil region without entering the LS region. The lights collected from oil muons are mostly from Cherenkov radiation. total LS Muon 104.8 < Np.e < 7 × 105 . These are muons that deposit a significant amount of energy in the LS region, and are associated with high light yields. total Shower Muon Np.e > 7 × 105 . These are a subset of LS Muons, which have the highest light yields. Shower muons are expected to deposit more energy than 2 104 10 p.e. log (σN ) 202 1 (e) 103 0 (f) -1 (d) (a) (g) (h) 102 -2 (c) 10 -3 (b) -4 -1 0 1 2 3 4 5 6 7 ) log (Ntotal p.e. 1 Figure C.1: Distribution of events in a typical one-day’s data (run 6756) (events occurring within 50 µs after a LS muon are not included). The two variables σNp.e. total and Np.e discriminates a few event classes: (a) background light in the ID, from low threshold ID trigger events; (b) low energy physics events; (c) higher energy physics events, mostly from 12 B; (d) residual post-muon noise; (e) PMT flasher events; (f) muons pass through BO only; (g) muons pass through LS; (h) shower muons which are a subset of LS muons that have highest light output. 203 minimum ionizing, and are found to create more spallation products. ID Muon Events tagged as either “Oil Muon,” “LS Muon” or “Shower Muon.” All of these muons deposit a significant amount of energy in the inner detector. Low Threshold Background Events Events in region (a) of Fig. C.1. These events are acquired from lower-threshold ID trigger for monitoring the background in KamLAND. These events include the muons that pass though outer detector only, acquired from the OD-to-ID trigger [79]. Candidate Physics Events Events in region (b) and (c) of Fig. C.1. These events are mostly from low energy (< 15 MeV) particle interactions, and are candidate for ν̄e events. Region (b) are mostly from radioactive decays of impurities in the LS such as 210 Pb and 85 Kr. Region (c) are mostly from higher energy events as spallation 12 B decays. A few event categories can be classified with other low-level variables as well: OD Muon OD nsummax > 10. These are muons events that are detected by the outer detector PMTs. OD muon tagging is independent of inner detector. Muon Events that are tagged as either ID muon or OD muon. Post Muon Noise Events that occurring within 50 µs after LS muons. These are mostly from after-pulses from electronics after a large muon signal. Candidate Spallation Events Events that occur within 2 seconds after ID muons. These are mostly from decays of spallation radioactive isotopes created by the muons, and can be further categorized into short-lived (∆t < 2 ms, e.g. neutrons), medium-lived (∆t < 200 ms, e.g. 12 B) and long-lived (∆t < 2 s, e.g. 9 Li). 204 Coincidence Multiplet Events are added into a multiplet if it occurs within 1.5 ms of the previous event. A multiplet group thus contains events with temporal coincidences, including the ν̄e inverse beta decay events. C.2 Radioactive Calibration Source Events The radioactive calibration source events are trivially selected spatially by requiring their reconstructed positions to be within 1 m from the deployed source position, which is known accurate to 3 mm. Other selection rules for each source are defined below: • 203 Hg: 0.1MeV < Evis < 0.4MeV • 137 Cs: 0.4MeV < Evis < 0.9MeV • 68 Ge: 0.7MeV < Evis < 1.1MeV • 65 Zn: 0.9MeV < Evis < 1.3MeV • 60 Co: 2MeV < Evis < 3MeV • 241 Am9 Be: require a prompt-delay event pair, ∆Tp−d < 1.2 ms, ∆Rp−d < 3 m, ∆Rp−source < 2 m, 1.9MeV < Ed < 2.7MeV (neutron capture on 1 H), 4.5MeV < Ed < 6.0MeV (neutron capture on 12 C) • 210 Po13 C: require a prompt-delay event pair, ∆Tp−d < 1.2 ms, ∆Rp−d < 3 m, ∆Rp−source < 2 m, 1.9MeV < Ed < 2.7MeV (neutron capture on 1 H), 4.3MeV < Ep < 5.5MeV (12 C excited state), 6.1MeV < Ep < 7.2MeV (16 O excited states) Fig. C.2 shows the reconstructed energy of various calibration sources from se√ lected typical center calibration runs. The energy resolution is 6.8%/ Evis in Period 205 Events [arbitrary unit] Calibration Sources 1000 203 Hg Cs Ge 9 241 Am Be 60 Co 210 13 Po C 900 137 68 800 700 600 500 400 300 200 100 0 0 1 2 3 4 5 6 7 Evis [MeV] Figure C.2: The reconstructed energy of calibration sources from selected center calibration runs. The energy resolution is 6.8% in Period I and 7.6% in Period II. √ I and 7.6%/ Evis in Period II. The energy reconstruction variation along the z-axis has been shown in Fig. 3.5 and its volume average is ∼1%. The time variation of the source reconstructed energy has been shown in Fig. 3.8 and the average time variation is ∼0.5%. The source reconstructed energies are used to determine the energy scale parameters of the detector, except for the 241 Am9 Be and 210 Po13 C due to the biases caused by the capsule shadowing. The γ’s from spallation neutrons are used instead at similar energies. Details have been discussed in Section 3.4. Fig. C.3 shows an example of reconstructed positions from a series of z-axis 60 Co calibration runs performed on 09/21/2006. The vertex reconstruction resolution is ∼20 cm along the z-axis. The reconstructed vertex bias has been shown in Fig. 3.4, and is typically less than 3 cm along the z-axis. Details have been discussed in Section 3.3. C.3 Alpha Particle Events Alpha particles in KamLAND detector mostly come from the decay chains of 232 Th and 238 U (Appendix B), or their out-of-equilibrium daughter nuclei. α particles gen- 206 60 Events [arbitrary unit] Co Z-Axis Scan 700 600 500 400 300 200 100 0 -6000 -4000 -2000 0 2000 4000 6000 Z [mm] Figure C.3: The reconstructed positions from a series of z-axis 60 Co calibration runs (performed on 09/21/2006). The vertex resolution is ∼20 cm. erally have energies higher than 4 MeV. However, due to their high stopping power, thery are highly quenched down to below 1 MeV in visible energy, as can be calculated from Eqn. 3.10. Alpha events are hardly identifiable in singles spectrum except for 210 Po α decays because of 210 Po’s higher concentrations (not in equilibrium with 238 U). Coincidence techniques however can be used to cleanly select 214 Po and 212 Po α decay events. C.3.1 214 214 Po Alpha Decay Po alpha decays with a half-life τ1/2 = 164µs. The energy of the α particle is 7.687 MeV, but is quenched down to ∼0.63 MeV in visible energy. It is preceded by the beta decay from 214 Bi with an end point at 3.272 MeV. This creates a prompt-delayed coincidence event pair which can be selected by: • 5µs < ∆Tp−d < 505µs (signal window) • 505µs < ∆Tp−d < 905µs (background window for subtraction) • ∆Rp−d < 1 m 207 214 Po Alpha Decay Events Events / 20 µ s Events / 0.01 MeV 214 3500 3000 2500 Po Alpha Decay Events 7000 6000 5000 2000 4000 1500 3000 1000 2000 500 1000 0 0.4 0.5 0.6 0.7 0.8 0.9 1 Evis [MeV] 0 100 (a) 214 Po α Energy Spectrum 200 300 400 500 600 700 800 900 1000 ∆Tp-d [µ s] (b) 214 Po Decay Time Figure C.4: The energy and time distribution of 214 Po α decay events (from all runs in Period I inside 6m fiducial volume). (a) The background subtracted (from background time window) energy spectrum. A fit to Gaussian plus a linear background yields mean energy at 0.6281 ± 0.0004 MeV, with χ2 /ndf = 39.5/45. (b) The distribution of time to previous 214 Bi β decay event. A fit to an exponential decay function plus a flat background yields the half-life 162.2 ± 2.7 µs, with χ2 /ndf = 34.0/43. This agrees with the 214 Po half-life (τ1/2 = 164µs). 214 Po Alpha Decay Events 0.65 Evis [MeV] Evis [MeV] 214 0.64 Po Alpha Decay Events 0.65 0.64 0.63 0.63 0.62 0.62 0.61 0.61 0.6 0 50 100 150 200 (a) Volume Variation of Evis 250 R3 [m3] 0.6 1000 2000 3000 4000 5000 6000 run number (b) Time Variation of Evis Figure C.5: The position and time variation of reconstructed energy of 214 Po α decay events (from all runs in Period I). (a) Each bin has a same volume of 10.8 m3 . The radial variation of Evis variation inside the 6m fiducial volume (dotted line) is 0.61%. (b) Each bin has the same integrated live time. The time variation of Evis in Period I is 0.67%. • 1.8MeV < Ep < 4.0MeV • 0.4MeV < Ed < 0.9MeV 208 The energy and time distribution of 214 Po α decay events are shown in Fig. C.4. Events in the background time window are statistically subtracted from the events in the signal window in order to get the correct energy spectrum of 214 Po α decay, as shown in Fig. C.4(a). The fit to the distribution of time to previous 214 Bi β decay event, shown in Fig. C.4(b), yields the the half-life 162.2 ± 2.7 µs, which agrees with 214 Po half-life (164 µs). The 214 Po α reconstructed energy is used to calibrate the energy scale parameters discussed in Section 3.4.3. Its mean value in the fiducial volume, volume variation and time variation (e.g. Fig. C.5) are estimated for Period I and II separately and are summarized in Table 3.2 C.3.2 212 212 Po Alpha Decay Po alpha decays with a half-life τ1/2 = 0.299µs. The energy of the α particle is 8.784 MeV, but is quenched down to ∼0.83 MeV in visible energy. It is preceded by the beta decay (branching ratio 64.1%) from 212 Bi with an end point at 2.252 MeV. This creates a prompt-delayed coincidence event pair which can be selected by: • 0.5µs < ∆Tp−d < 2.5µs (signal window) • 2.5µs < ∆Tp−d < 4.5µs (background window for subtraction) • ∆Rp−d < 1 m • 1.0MeV < Ep < 3.0MeV • 0.6MeV < Ed < 1.0MeV The energy and time distribution of 212 Po α decay events are shown in Fig. C.6. Events in the background time window are statistically subtracted from the events in the signal window in order to get the correct energy spectrum of 212 Po α decay, as shown in Fig. C.6(a). The fit to the distribution of time to previous 212 Bi β decay 209 212 212 Po Alpha Decay Events Events / 20 µ s Events / 0.02 MeV Po Alpha Decay Events 300 250 200 450 400 350 300 150 250 200 100 150 50 100 0 0.5 50 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0 0.95 1 Evis [MeV] 0 0.5 (a) 212 Po α Energy Spectrum 1 1.5 2 2.5 3 3.5 4 4.5 5 ∆Tp-d [µ s] (b) 212 Po Decay Time Figure C.6: The energy and time distribution of 212 Po α decay events (from all runs in Period I inside 6m fiducial volume). (a) The background subtracted (from background time window) energy spectrum. A fit to Gaussian plus a linear background yields mean energy at 0.826 ± 0.002 MeV, with χ2 /ndf = 16.2/15. (b) The distribution of time to previous 212 Bi β decay event. A fit to an exponential decay function plus a flat background yields the half-life 0.305 ± 0.008 µs, with χ2 /ndf = 82.1/41. The relatively high χ2 probably reflects the electronics biases at such short time scale. The fitted half-life agrees with the 212 Po half-life (τ1/2 = 0.299µs). 212 212 0.86 0.86 Po Alpha Decay Events Efit [MeV] Evis [MeV] Po Alpha Decay Events 0.84 0.84 0.82 0.82 0.8 0.8 0.78 0.78 0 50 100 150 200 250 R3 [m3] (a) Volume Variation of Evis 1000 2000 3000 4000 5000 6000 run number (b) Time Variation of Evis Figure C.7: The position and time variation of reconstructed energy of 212 Po α decay events (from all runs in Period I). (a) Each bin has a same volume of 21.6 m3 . The radial variation of Evis variation inside the 6m fiducial volume (dotted line) is 0.90%. (b) Each bin has the same integrated live time. The time variation of Evis in Period I is 1.21%. 210 event, shown in Fig. C.6(b), yields the the half-life 0.305 ± 0.008 µs, which agrees with 212 Po half-life (0.299 µs). The 212 Po α reconstructed energy is also used to calibrate the energy scale parameters discussed in Section 3.4.3. Its mean value in the fiducial volume, volume variation and time variation (e.g. Fig. C.7) are estimated for Period I and II separately and are summarized in Table 3.2. C.3.3 210 210 Po Alpha Decay Po alpha decays with a half-life of 138 days. The energy of the α particle is 5.304 MeV, but is quenched down to ∼0.3 MeV in visible energy. The 210 Po contribute to more than 99% of the α activity in KamLAND detector [108], and is visible from daily singles spectrum (e.g. Fig. 2.8). Because of known poor reconstruction efficiency and bias at visible energies below 0.4 MeV, the 210 Po alpha decay events are not used to calibrate the energy scale parameters discussed in Section 3.4.3. The absolute 210 Po decay rate however need to be measured because its high activity makes 13 C(α,n)16 O background significant in KamLAND. The rate of 210 Po decay inside the 6m fiducial volume is measured to be 0.19 ± 0.03 Bq/m3 in Period I and 0.05 ± 0.01 Bq/m3 in Period II. Its high activity even after the purification of LS, which removed orders of magnitude more 210 Bi, suggests that it is out of equilibrium from not only the 238 U decay chain but also the 210 Pb decay chain1 . Details of 210 Po event selection and estimation of 13 C(α,n)16 O background has been discussed in Section 5.4. C.4 Muon Spallation Events As described in Appendix C.1, events that occurring within 2 seconds after ID muons are considered candidate spallation events. These events are mostly from decays of 1 Similar effect has been observed in the Borexino experiment [42] 211 spallation radioactive isotopes created by the muons, and can be further categorized into short-lived (∆t < 2 ms, e.g. neutrons), medium-lived (∆t < 200 ms, e.g. 12 B) and long-lived (∆t < 2 s, e.g. 9 Li) spallation products. C.4.1 Spallation Neutron KamLAND receives spallation neutrons in the LS volume at a rate of ∼3000/day. The large daily sample of spallation neutron events and their spacial uniformity make them ideal for monitoring the detector stability over time and scintillator properties over the entire active volume of the detector. The neutron capture events happen relatively close to the preceding muons (∼206 µs). However, a large muon signal is often correlated with high event multiplicity following the muon, which causes the overload on individual electronics resulting waveform loss and dead time of the detector, extending as long as a few hundred micro-seconds after the large muon signal. This could cause biases in the energy reconstruction or neutron capture time study. Following selection rules are applied to select a relatively clean sample of spallation neutrons: • multiplicity (number of evets within 0.15 – 10 ms following a muon) < 75 • 0.4ms < ∆Tn−µ < 1.4ms (signal window) • 1.4ms < ∆Tn−µ < 5.4ms (background window for subtraction) This selection window is shown in Fig. C.8(a). High multiplicity muons or events too close in time after a muon are rejected to avoid the electronics biases. The distribution of spallation neutron events is uniform inside the LS volume, as can be seen from Fig. C.8(b). This uniformity can be used to calculate the 6m fiducial volume size, as discussed in Section 4.4. The fiducial volume ratio Rfv from spallation neutrons is calculated to be 0.784(±0.27%), and is used as a cross check 212 104 450 400 103 350 300 Events / 10.8 m3 Multiplicity Spallation Neutron Events 500 16000 14000 12000 10000 250 102 8000 200 6000 150 10 100 4000 2000 50 0 0 0.5 1 1.5 2 2.5 3 Time to Last Muon [ms] (a) Spallation Neutron Event Selection 1 0 0 50 100 150 200 250 300 R3 [m3] (b) Spallation Neutron Vertex Distribution Figure C.8: (a) Spallation neutron event selection window (red box). High multiplicity (> 75) muons or events too close in time after a muon (< 0.4 ms) are rejected to avoid the electronics bias. Events inside 1.4 – 5.4 ms window are used for background subtraction. (b) The vertex distribution of spallation neutron (background subtracted). The distribution is flat inside the 6m fiducial volume (dashed line). with the result from spallation 12 B, which is free from electronic bias and covers a larger energy range. The uniformity and abundance of spallation neutron events are utilized to check the energy reconstruction bias over the whole scintillator volume. Details have been discussed in Section 3.4. Spallation neutron events are also used for determining the neutron capture time discussed in Section 4.6. The mean neutron capture time from the spallation neutron is 206.4 ± 1.3 µs (Fig. C.9(b)). Both spallation neutron capture γ-rays (on 1 H and 12 C) are evident from the energy spectrum shown in Fig. C.9(a). Their reconstructed energies are used to calibrate the energy scale parameters discussed in Section 3.4.3. The mean value in the fiducial volume, volume variation and time variation (e.g. Fig. C.10, C.11) are estimated for Period I and II separately and are summarized in Table 3.2. 213 Events / 10µs Events / 0.05 MeV Spallation Neutron Events 104 103 105 104 102 10 103 1 2 4 6 8 10 12 0.5 14 Evis [MeV] (a) Spallation Neutron Energy Spectrum 1 1.5 2 2.5 3 3.5 4 Time since previous muon [ms] (b) Spallation Neutron Capture Time Figure C.9: Energy spectrum and capture Time of spallation neutron events (from all runs in Period I inside 6m fiducial volume): (a) Energy spectrum in the signal window (solid line) and background window (dashed line, statistically scaled). The two peaks correspond to γ-rays from neutron capture on 1 H and on 12 C. The fit to a Gaussian plus linear background yield the mean energy of the peak at 2.424 ± 0.001 MeV and 5.499 ± 0.008 MeV. (b) The distribution of time to the previous muon. A fit from 0.6 ms to 3 ms to exponential distribution plus a flat background results the mean neutron capture time τ = 206.4 ± 1.3µs. The χ2 /ndf from the fit is 227/237. Spallation Neutron Capture on 1H Evis [MeV] Evis [MeV] Spallation Neutron Capture on 1H 2.46 2.46 2.44 2.44 2.42 2.42 2.4 2.4 2.38 2.38 2.36 2.36 2.34 2.34 0 50 100 150 200 250 (a) Volume Variation of Evis 300 R3 [m3] 1000 2000 3000 4000 5000 6000 run number (b) Time Variation of Evis Figure C.10: The position and time variation of reconstructed energy of spallation neutron capture on 1 H events (from all runs in Period I). (a) Each bin has a same volume of 10.8 m3 . The radial variation of Evis variation inside the 6m fiducial volume (dotted line) is 0.60%. (b) Each bin has the same integrated live time. The time variation of Evis in Period I is 0.52%. 214 Spallation Neutron Capture on 12C Evis [MeV] Efit [MeV] Spallation Neutron Capture on 12C 5.6 5.6 5.5 5.5 5.4 5.4 5.3 5.3 0 50 100 150 200 250 300 R3 [m3] (a) Volume Variation of Evis 1000 2000 3000 4000 5000 6000 run number (b) Time Variation of Evis Figure C.11: The position and time variation of reconstructed energy of spallation neutron capture on 12 C events (from all runs in Period I). (a) Each bin has a same volume of 21.6 m3 . The radial variation of Evis variation inside the 6m fiducial volume (dotted line) is 0.50%. (b) Each bin has the same integrated live time. The time variation of Evis in Period I is 0.88%. C.4.2 Spallation 12 B The spallation product 12 B beta decays with half-life τ1/2 = 20.2 ms. The end point of the β spectrum is 13.4 MeV. They are the most abundant spallation isotopes at high energies above 4 MeV. They can be selected from the time correlation with preceding muon events as follows: • The preceding muon is an isolated muon (previous and next muon are at least 1 second away). • 2ms < ∆TB−µ < 52ms (signal window) • 300ms < ∆TB−µ < 500ms (background window for subtraction) • OD nsummax < 10 (Event itself is not an OD muon) • 4MeV < EB < 14MeV (only high energy part of the β spectrum) The distribution of time of 12 B to the previous muon is shown in Fig. C.12(b). The distribution is fitted to the sum of two exponential decay functions, including a Spallation 12B Events Spallation 12B Events Events / 0.4 MeV Events / 2ms 215 800 103 600 102 400 200 10 0 0 2 4 6 8 10 12 14 16 18 Evis [Mev] (a) Spallation 12 B Energy Spectrum 0 100 200 300 400 500 Time since previous muon [ms] (b) Spallation 12 B Decay Time Figure C.12: Energy spectrum and decay Time of spallation 12 B events (from all runs in Period I inside 6m fiducial volume): (a) Energy spectrum fitted to the expected 12 B beta decay visible energy spectrum from 4–14 MeV. The χ2 /ndf from the fit is 37.4/25. (b) The distribution of time to the previous muon. The distribution is fitted to the sum of two exponential decay functions, including a long-lived 9 Li component with fixed half-life τ1/2 = 178.3 ms, and a flat background. The 12 B half life from the fit is 20.0 ± 0.2 ms. The χ2 /ndf from the fit is 383/394. long-lived 9 Li component with fixed half-life τ1/2 = 178.3 ms, and a flat background. The 12 B half time from the fit is 20.0 ± 0.2 ms, which agrees with the expected value (20.2 ms). The visible energy spectrum of spallation 12 B is shown in Fig. C.12(a). This spectrum is fitted to the expected 12 B beta decay visible energy spectrum from 4–14 MeV. The χ2 /ndf from the fit is 37.4/25. The 12 B visible energy spectrum is used to cross-check the validity of energy scale at high energies, since the other highest calibration point is only 4.9 MeV (from spallation neutron capture on 12 C). The spallation 12 B events are used to determine the fiducial volume of the analysis and its systematic errors, details discussed in Section 4.4. Spallation 12 B events are also used to determine the inefficiency of the 3-m cylinder cut along the LS muon track, which has been discussed in Section 5.5. 216 Events / 21.6 m3 9 Spallation Li Events Events / 1 MeV 9 Spallation Li Events 120 100 120 100 80 80 60 60 40 40 20 20 0 0 2 4 6 8 10 12 14 16 Evis [Mev] (a) Spallation 9 Li Energy Spectrum 0 0 50 100 150 200 250 300 R3 [m3] (b) Spallation 9 Li Radial Distribution Figure C.13: Spallation 9 Li events are selected from all runs in Period I inside 6m fiducial volume: (a) Background subtracted energy spectrum fitted to the expected 9 Li beta decay visible energy spectrum from 2.6–14.6 MeV. The χ2 /ndf from the fit is 19.1/11. (b) The vertex distribution of spallation 9 Li (background subtracted). The dotted line indicate the 6m fiducial volume boundary. C.4.3 Spallation 9 Li The spallation product 9 Li beta decays with half-life τ1/2 = 178.3 ms. The end point of the β spectrum is 13.6 MeV. 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