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Calculated Final State Probability Distributions for T2 /3-Decay

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Calculated Final State Probability Distributions for T2 /3-Decay Calculated final state probability distributions for T 2 /3-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 UMI Number: U592743 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U592743 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 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 /3-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 /3-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 /?-decay of T 2 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 T 2 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+, T+ T+ and T+ - 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 n eu trin o s ........................................................................................ 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 /5-decay ex p erim en ts ...............................................................................22 1.7.1 Summary of early experiments ................................................................24 1.7.2 Unphysical r e s u lts ......................................................................................25 1.7.3 The Mainz and Troitsk experiments ...................................................... 27 1.7.4 K A T R IN ..................................................................................................... 31 1.8 Objectives of this work ........................................................................................... 34 2 Theoretical background I - Molecular excitations in the /5-decay pro­ cess 36 2 CONTENTS 2.1 The physics of (3-d ecay ............................................................................................ 36 2.2 The (3 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 r u le s ................................................................................... 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 Schrodinger equation of diatomic molecules - The LEVEL and BCONT programs ............................................................................................ 53 3.5 Molecular R-matrix theory and implementation ................................................ 56 3.5.1 The inner re g io n .........................................................................................58 3.5.2 The outer re g io n .........................................................................................59 3.5.3 R esonances ................................................................................................... 60 3.5.4 The UK Molecular R-matrix package (diatomic code) ........................62 4 P rev io u s 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 s ta te s .....................77 5.1.2 Continuous probability density distribution of the nuclear motion continuum ......................................................................................................84 5.1.3 Total ground state d istrib u tio n .................................................................89 5.2 Rotational excitation of T 2 .................................................................................. 91 5.3 Excited electronic states of 3HeT+ ................................................................. 93 5.4 Isotopomers - 3HeH+ and 3HeD+ .........................................................................95 5.5 Sum r u le s ............................................................................... ..............................96 3 CONTENTS 6 The error budget 99 6.1 Determination of the neutrino rest mass and the integral /?-decay spectrum 99 6.2 Rotational temperature effect - thermal s o u r c e ................................................101 6.3 Ortho:para ratio - non thermal source ............................................................... 104 6.4 Isotope effects .......................................................................................................... 106 6.5 S u m m a ry .................................................................................................................108 7 The final state probability distribution for the electronic continuum of 3H eT+ 110 7.1 R-matrix calculation of the resonance states of 3HeT+ ..................................I ll 7.1.1 Target c alc u la tio n ..................................................................................... I l l 7.1.2 Scattering c alcu latio n ............................................................................... 113 7.1.3 R esonances ..................................................................................................116 7.2 Final state distribution results ...........................................................................126 7.2.1 Probability distribution of the resonances ........................................... 126 7.2.2 Probability distribution of the
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