Calculated final state probability distributions for T2 β-decay measurements A Thesis submitted for the Degree of Doctor of Philosophy of the University of London by Natasha Doss Department of Physics & Astronomy University College London August 2007 The question of a possible finite neutrino mass is one of the most investigated and controversial topics in particle physics and cosmology. The most promising direct ex- periments for determining the neutrino mass are based on the study of the β-decay of molecular tritium. The mass of the neutrino is deduced by analysing the shape of the continuous energy spectrum of the electrons emitted in the β-decay. In this thesis, the molecular physics issues facing these experiments are investigated. Theoretical final state probability distributions of the daughter molecule are calculated to satisfy the higher resolution requirements and increased sensitivity of the future ex- periments. Transition probabilities to the six lowest electronically bound states of 3HeT+ are calculated. Rotational excitation of the daughter molecule is considered and probabilities obtained for the β-decay of T2 in the first four rotational states. Isotope contamination from DT and HT molecules is also investigated, and the probability distributions for 3HeD+ and 3HeH+ are calculated. The sensitivity of the initial temperature, ortho:para ratio and isotopic composition of the source is considered. Estimates of the error in the value of the neutrino mass deduced from fitting theoretical spectra, due to uncertainties in temperature, ortho:para ratio of T2 and percentage of DT molecules, are obtained. The R-matrix method is used to treat the electronic continuum of 3HeT+. Reso- nances converging to the first eight excited target states are obtained, and the transition probabilities to these resonances and background continuum are calculated. Endpoint effects due to the decay of other possible species in the source – T−, T, + + + + T , T2 , T3 and T5 – are also investigated. It is hoped that this data will be used as part of the forthcoming KATRIN experi- ment. 1 Contents 1 Introduction 11 1.1 Neutrinos - A brief history . 11 1.2 The neutrino mass problem - Implications of a non-zero neutrino rest mass 13 1.3 Neutrino mixing . 13 1.4 Neutrino oscillation experiments - Evidence for massive neutrinos . 14 1.4.1 Atmospheric neutrinos . 15 1.4.2 Solar neutrinos . 15 1.4.3 Reactor neutrinos . 16 1.4.4 Accelerator neutrinos . 17 1.5 Mixing parameters and neutrino mass hierarchy . 17 1.5.1 Sterile neutrinos . 18 1.6 Absolute mass searches . 19 1.6.1 Neutrinoless double beta decay . 19 1.6.2 Cosmological constraints . 21 1.6.3 Kinematical/direct measurements . 22 1.7 Tritium β-decay experiments . 22 1.7.1 Summary of early experiments . 24 1.7.2 Unphysical results . 25 1.7.3 The Mainz and Troitsk experiments . 27 1.7.4 KATRIN . 31 1.8 Objectives of this work . 34 2 Theoretical background I - Molecular excitations in the β-decay pro- cess 36 2 CONTENTS 2.1 The physics of β-decay . 36 2.2 The β spectrum and the effect of a non-zero neutrino mass on the endpoint 37 2.3 The final state probability distribution . 40 2.3.1 Bound rovibrational states . 40 2.3.2 The rovibrational (nuclear motion) continuum and predissociative resonance states . 45 2.3.3 Sum rules . 45 3 Theoretical background II - Approximations and programs 47 3.1 Sudden approximation . 47 3.2 Born-Oppenheimer approximation . 49 3.3 The Hartree-Fock approximation and configuration interaction . 51 3.4 Solving the radial Schr¨odinger equation of diatomic molecules - The LEVEL and BCONT programs . 53 3.5 Molecular R-matrix theory and implementation . 56 3.5.1 The inner region . 58 3.5.2 The outer region . 59 3.5.3 Resonances . 60 3.5.4 The UK Molecular R-matrix package (diatomic code) . 62 4 Previous studies 68 4.1 The final state distribution of Fackler et al. 68 4.2 The final state distribution of Saenz et al. 72 5 The final state probability distribution for the electronically bound states 76 5.1 The electronic ground state of 3HeT+ . 77 5.1.1 Probability distribution of bound and quasibound states . 77 5.1.2 Continuous probability density distribution of the nuclear motion continuum . 84 5.1.3 Total ground state distribution . 89 5.2 Rotational excitation of T2 . 91 5.3 Excited electronic states of 3HeT+ . 93 5.4 Isotopomers - 3HeH+ and 3HeD+ . 95 5.5 Sum rules . 96 3 CONTENTS 6 The error budget 99 6.1 Determination of the neutrino rest mass and the integral β-decay spectrum 99 6.2 Rotational temperature effect - thermal source . 101 6.3 Ortho:para ratio - non thermal source . 104 6.4 Isotope effects . 106 6.5 Summary . 108 7 The final state probability distribution for the electronic continuum of 3HeT+ 110 7.1 R-matrix calculation of the resonance states of 3HeT+ . 111 7.1.1 Target calculation . 111 7.1.2 Scattering calculation . 113 7.1.3 Resonances . 116 7.2 Final state distribution results . 126 7.2.1 Probability distribution of the resonances . 126 7.2.2 Probability distribution of the background continuum . 133 8 Endpoint considerations of other tritium species 136 9 Conclusions 146 A Publications 153 Bibliography 154 4 List of Figures 1.1 The fundamental matter particles of the Standard Model. 12 1.2 The normal and inverted hierarchy mass schemes for the neutrino masses. 18 1.3 Feynmann diagram for the neutrinoless double beta decay process. 19 1.4 The electron energy spectrum of tritium β-decay. 24 1.5 Principle of the MAC-E-Filter. 28 1.6 Mainz experiment fit results. 30 1.7 Design of the KATRIN experiment. 32 3.1 Graphic representation of the effective potential energy curve and predis- sociative resonances . 56 3.2 Splitting of configuration space in the R-matrix model. 57 3.3 Flow diagram of the inner region target state calculation. 64 3.4 Flow diagram of the inner region (N+1) scattering calculation. 65 3.5 Flow diagram of the outer region scattering calculation. 67 5.1 Electronic energy curves for the first six electronic states of 3HeT+ as a function of the internuclear separation R. Eh is the Hartree unit of energy. 79 5.2 Example of a narrow, isolated resonance. 89 5.3 Example of a wide non-isolated resonance. 90 5.4 The final state probability distribution of the electronic ground state of 3 + HeT for different initial rotational states, Ji = 0, 1, 2 and 3, of T2. 91 5.5 The Gaussian distributions for the electronic ground state of 3HeT+ for initial Ji = 0, 1, 2 and 3. 93 5 LIST OF FIGURES 5.6 The final state probability density distribution for the first five electron- 3 + ically excited states of HeT for different initial rotational states, Ji = 0, 1, 2 and 3, of T2. 94 5.7 The final state probability distributions of the electronic ground states of 3HeD+ and 3HeH+ for the β-decay of DT and HT molecules in initial rovibrational states Ji = 0 and 1. 97 5.8 The Gaussian distributions for the electronic ground states of 3HeT+, 3 + 3 + HeD and HeH for the β-decay of T2, DT and HT molecules in initial rovibrational states Ji = 0 and 1. 98 5.9 The final state probability density distribution for the first five excited 3 + 3 + 3 + electronic states of HeT , HeD and HeH for initial Ji=0 . 98 6.1 Response function of the KATRIN spectrometer for isotropically emitted electrons with fixed energy E as a function of the retarding energy qU. 100 6.2 Distribution of T2 molecules in each J state as a function of temperature 102 6.3 Overall final state distributions for a pure T2 source at different temper- atures between 5 - 50 K . 103 6.4 Error in the neutrino mass squared deduced from fitting, due to uncer- tainties in the source temperature. 104 6.5 Distribution of T2 molecules in each J state as a function of the ortho:para ratio, λ . 105 6.6 Error in the neutrino mass squared deduced from fitting, due to uncer- tainties in the ortho:para ratio. 106 6.7 Overall final state distribution for a source at temperature T = 30 K for different DT:T2 ratios . 107 6.8 Error in the neutrino mass squared deduced from fitting, due to uncer- tainties in the amount of DT molecules in the source. 108 7.1 Error in the target state energies from [173] when compared to the ex- act energies computed using the code by Power [176], as a function of internuclear separation. Eh represents the Hartree unit of energy . 112 7.2 Error in the target state energies obtained in this work when compared to the exact energies computed using the code by Power [176], as a function of internuclear separation. 114 6 LIST OF FIGURES 7.3 3HeT+ resonance curves and 3HeT2+ target potentials as a function of internuclear separation R. 118 7.4 Final state probability density distribution for nine resonance states of 3 + HeT resulting from the β-decay of a T2 molecule in the ground rovi- brational state of the n = 1 electronic state. 131 7.5 Discrete final state probability distribution of the background continuum and remaining resonances. 134 7.6 Final state probability density distribution for the electronic continuum 3 + of HeT resulting from the β-decay of a T2 molecule in the ground rovibrational state of the n = 1 electronic state. 135 3 ++ 8.1 ZPE effect on the repulsive curve of HeT2 .