HIGH-PRECISION HALF-LIFE MEASUREMENTS FOR. SUPERALLOWED FERMI β DECAYS. A Thesis. Presented to. The Faculty of Graduate Studies

Transcription

1 HIGH-PRECISION HALF-LIFE MEASUREMENTS FOR SUPERALLOWED FERMI β DECAYS A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph by GEOFFREY F. GRINYER In partial fulfilment of requirements for the degree of Doctor of Philosophy December, 2007 c Geoffrey F. Grinyer, 2008

2 ABSTRACT HIGH-PRECISION HALF-LIFE MEASUREMENTS FOR SUPERALLOWED FERMI β DECAYS Geoffrey F. Grinyer University of Guelph, 2007 Advisor: Professor C.E. Svensson High-precision measurements of the f t values for superallowed Fermi β decays between 0 + isobaric analogue states have, for decades, provided demanding tests of the Standard Model description of electroweak interactions. In order to significantly contribute to these tests experimentally, β decay half-lives and branching ratios must be determined to overall precisions of ± 0.05% or better, and β decay Q values must be deduced to at least ± 0.01%. For β decay half-lives in particular, this demanding requirement is generally accomplished using direct β counting techniques. This method was employed as part of this thesis in order to deduce the half-life of the superallowed β + emitter 62 Ga using mass-separated radioactive ion beams provided by the Isotope Seperator and Accelerator (ISAC) facility at TRIUMF. The result of this analysis, T 1/2 ( 62 Ga) β = ± ms, is now the single most precise superallowed half-life ever reported. In cases where there are large amounts of contaminant or daughter activities, one must instead rely on a measurement of the half-life using the γ-ray activity. Halflife measurements using the technique of γ-ray photopeak counting have, however, been previously limited by a systematic bias associated with detector pulse pile-up effects. While detector pulse pile-up has been qualitatively understood for decades, there has not been a quantitative description of its effects on half-life measurements

3 to the level of precision required (± 0.05%) for superallowed Fermi β decay studies. Using the 8π γ-ray spectrometer, a spherical array of 20 HPGe detectors at ISAC, a new method was developed that, for the first time, provides the necessary quantitative description of detector pulse pile-up to the required level of precision. This novel technique has been verified through both a detailed Monte-Carlo simulation and experimentally using radioactive beams of 26 Na. Following a correction of nearly 30 statistical standard deviations for pulse pile-up, the half-life of 26 Na deduced in this work, T 1/2 ( 26 Na) γ = ± s, is precise to the level of 0.05% and is in excellent agreement with the corresponding value, T 1/2 ( 26 Na) β = ± s, deduced from direct β counting. This study has demonstrated the feasibility of using the γ-ray counting technique to deduce β decay half-lives to the necessary level of ± 0.05% precision. As an extension to this work, the half-life of the superallowed β + emitter 18 Ne was determined to be, T 1/2 ( 18 Ne) γ = ± s, a result that is a factor of four times more precise than the previous world average.

4 Weaseling out of things is important to learn. It s what separates us from the animals...except the weasel. -H.S.

5 Acknowledgements There are a large number of people to whom I am deeply indebted and without whose help none of this would have been possible. I would first like to thank my supervisor, Carl Svensson, who has had the greatest influence on my fledgling scientific career. We have spent the better part of six years working together, and without his passion and tireless efforts, I would not be where I am today. I would also like to thank my pseudo-supervisors Paul Garrett at Guelph and Gordon Ball at TRIUMF. Paul s sense of humour provided the comic relief that is a necessity to making it this far. We will always remember the potatoes at the golf course restaurant! I have also had the unique opportunity to work with several people from around the world including my group at the University of Guelph: Corina Andreoiu, Dipa Bandyopadhyay, Greg Demand, Paul Finlay, Katie Green, Bronwyn Hyland, Jose Javier, Kyle Leach, Andrew Phillips, Mike Schumaker, and James Wong. As students working at a university that is 2000 km from the lab, we sometimes forget the tireless efforts of the on-site TRIUMF staff that make all of these experiments possible. I would therefore like to thank the TRIUMF crew: Gordon Ball, Greg Hackman, Colin Morton, Chris Pearson, and Scott Williams. I am grateful to other TRIUMF staff members including Pierre Bricault, Marik Dombsky, Matt Pearson, John Behr, Jens Lassen, in addition to all of the members of the beam development groups. I would i

6 also like to thank Martin Smith and Hamish Leslie, in particular, for performing parallel analysis to my own in the case of the 18 Ne and 26 Na & 62 Ga experiments, respectively. A special thanks to John Hardy and Ian Towner, the superallowed gurus, who provided me with up-to-the-minute data that were used in Chapter 1 of this thesis. Most of all I would like to thank my wife, Joanna, for her never ending love and support. My daughter, Daniela, who made the last year of my doctoral studies an unforgettable one to say the least! Finally, who could forget Cleo the husky? ii

7 Contents Acknowledgements i 1 Introduction Nuclear Beta Decay Beta-Decay Formalism Beta Decay Classification Isospin Fermi Theory of Beta Decay Statistical Rate Function f Superallowed Fermi Beta Decay Radiative Corrections Isospin Symmetry Breaking Corrected ft Values Free Neutron Beta Decay Pion Beta Decay CKM Quark Mixing Matrix Summary Experimental Facilities 41 iii

8 2.1 Isotope Separator and Accelerator (ISAC) The 8π γ-ray Spectrometer The SCEPTAR array Moving Tape Collector System General Purpose Station (GPS) Summary Detector Pulse Pile-up Quantitative Description of Pulse Pile-up Pile-up Probabilities for Radioactive Decay Rate-Independent Tests Detector Solid Angle Number of Detectors γ-ray Multiplicity Rate-Dependent Refinements Pile-up Time Resolution Trigger-Energy Threshold Pile-up Detection Energy Threshold Cosmic Rays Summary Half-life of 26 Na Compton Suppression Dead-Time Corrections Dead-Time Corrections for Multi-Detector Arrays Pile-up Probability Analysis iv

9 4.4 Half-life Analysis Tests of the Pile-up Method Systematic Uncertainties Summary Half-life of 18 Ne Pile-up Probability Analysis Half-life Analysis Systematic Uncertainties Diffusion and the Half-life of 23 Ne Comparison to Previous Results Present Status of the 18 Ne ft and Ft values Summary Half-life of 62 Ga Half-life Analysis Systematic Uncertainties Comparison to Previous Results Present Status of the 62 Ga ft and Ft values Summary Conclusion and Future Work Scalar Interactions in Superallowed Decay Half-life of 14 O Half-life of 10 C Half-life of 34 Ar Conclusion v

10 A Dead-time and Pile-up Corrected Decay-curve Fitting 216 Bibliography 219 vi

11 List of Tables 1.1 Radiative and isospin symmetry breaking corrections for the 13 most precisely determined superallowed decays Present status of all world-averaged data for the thirteen most precisely determined superallowed decays Present status of the world averaged f t and F t values for the thirteen most precisely determined superallowed decays Vector coupling constants G V, up-down CKM matrix elements V ud, and corresponding tests of top-row CKM unitarity deduced from nuclear, neutron, and pion β decays Run-by-run summary of experimental conditions employed in the 26 Na half-life experiment Half-life of 26 Na deduced on a run-by-run basis with statistical errors and resulting reduced χ 2 values for all 14 runs in this analysis Comparison of the half-life of 26 Na obtained as each correction to the pile-up method was applied to the raw γ-ray gated data Run-by-run summary of experimental conditions employed in the 18 Ne half-life experiment vii

12 5.2 Half-life of 18 Ne deduced on a run-by-run basis with statistical errors and resulting reduced χ 2 values for all 15 runs in this analysis Half-life of 23 Ne deduced on a cycle-by-cycle basis with statistical errors and resulting reduced χ 2 values for all 20 cycles in this analysis Summary of all high-precision 18 Ne half-life measurements. The new world average of T 1/2 = ± s with a reduced χ 2 value of 1.52 is obtained from a weighted average of these 5 measurements Single cycle timing sequence as determined by a 100 khz oscillator for a 3.0 s counting cycle Run-by-run summary of the electronic settings used in the 62 Ga experiment Run-by-run summary of the electronic settings used in the 62 Ga experiment Isobaric contamination in the A = 62 beam deduced from β-γ coincidences between the 20 HPGe detectors of the 8π spectrometer and the 20 plastic scintillators of the SCEPTAR array Differences T between the best-fit 62 Ga half-life in order to arrive at an estimate of the systematic uncertainty Summary of all high-precision 62 Ga half-life measurements. The new world average of T 1/2 = ± s with a reduced χ 2 value of is obtained from a weighted average of these 7 measurements viii

13 List of Figures 1.1 Feynman diagram for β + decay Experimental determinations of the f t values for thirteen superallowed Fermi β decays between A=10 and A= Values of the isospin symmetry breaking corrections in superallowed Fermi β decays Comparison of the ft values and Ft values for thirteen superallowed Fermi β decays between A=10 and A= The TRIUMF 520 MeV H cyclotron The ISAC experimental hall The 8π γ-ray spectrometer at TRIUMF-ISAC Schematic diagram of a single 8π germanium detector The SCEPTAR array at TRIUMF-ISAC The 8π moving tape collector system The GPS 4π gas counter and fast tape transport system Schematic of the 4π gas counter and fast tape transport system Schematic diagram of the 4π continuous gas flow proportional counter Observed count rate of a 90 Sr source Schematic representations of detector pulse pile-up ix

14 3.2 Analytic and simulated pile-up probabilities versus the dimensionless detector rate x Simulated decay data curves for a single exponential decay plus a constant background Simulated bin-by-bin probability of pile-up for a single exponential decay plus a constant background Simulated decay data set following corrections for pile-up and deadtime effects Simulation of 25 runs with and without the pile-up correction applied Pile-up probability versus the detector solid angle Pile-up probabilities versus the number of detectors employed Simulation of 25 runs comparing γ-ray multiplicities M γ = 1 and M γ = Pile-up probabilities versus the γ-ray decay multiplicity Simulated pile-up time τ p with respect to the master-trigger time for 2 different values of the pile-up time resolution τ r Simulated decay-curve data demonstrating the effect of a non-zero pileup time resolution Analytic and simulated pile-up probabilities versus the dimensionless detector rate x when the fraction of events that exceed the energy threshold α is varied Analytic functions describing the probability of obtaining a trigger event D and the probability of obtaining a not-piled-up trigger event PD as a function of the dimensionless detector rate x =Rτ p Simulated probability of pile-up data and best-fit curve when a nonzero CFD energy threshold is included x

15 3.16 Simulated decay-curve data data and best-fit curve when a pile-up efficiency that is less than unity is included Simulated probability of pile-up data when saturating cosmic-ray events are included Simulated decay-curve data data and best-fit curve when a saturating events due to cosmic-rays are included Simplifed 26 Na β decay scheme to the stable daughter 26 Mg Not piled-up γ-ray singles spectra from all 20 detectors in the 8π spectrometer following the β decay of 26 Na Sum HPGe time spectrum from all 20 detectors in the 8π spectrometer for a single run with the 56 Co source in the array Sum pile-up time spectra from all 20 detectors in the 8π spectrometer for a single run with the 56 Co source in the array Number of 1809-keV photopeaks recorded per-cycle for a single run Experimental probability of pile-up spectra for runs with and without the 56 Co source Run-by-run plot of the maximum probability of pile-up taken at the start of the decay for each run (t = beam off) in the 26 Na experiment Comparison of the 26 Na decay curves obtained from single runs via a direct β counting a γ-ray photopeak counting determination at the 8π spectrometer Pile-up and dead-time corrected decay curve obtained from a single run following a gate on the 1809-keV transition in 26 Mg Half-life of 26 Na with statistical errors versus run number for all 14 runs in this analysis xi

16 4.11 Half-life of 26 Na versus the number of leading channels removed Half-life of 26 Na versus the γ-ray photopeak gate width Half-life of 26 Na with statistical errors deduced from an average of all runs at each common adjustable setting Decay scheme for 18 Ne β + decay to 18 F Singles spectra of γ rays following the β decay of 18 Ne Experimental probability of pile-up spectra for the 18 Ne experiment Run-by-run plot of the maximum probability of pile-up taken at the start of the decay for each run (t = beam off) in the 18 Ne experiment Pile-up and dead-time corrected decay curve obtained from a single run following a gate on the 1042-keV transition in 18 F Half-life of 18 Ne versus individual run number Half-life measurements of 18 Ne sorted by adjustable electronic setting Calculated bremsstrahlung yield of 17 F and 18 Ne β particles in delrin Pile-up and dead-time corrected decay curve obtained from a single run following a gate directly above the 1042-keV transition in 18 F Deduced half-life of 18 Ne versus the number of leading channels removed Singles spectra of γ rays following the β decay of 23 Ne Experimental probability of pile-up spectra for the 23 Ne experiment Pile-up and dead-time corrected decay curve for 23 Ne obtained from a single run following a gate on the 440-keV transition in 23 Na Half-life of 23 Ne deduced on a cycle-by-cycle basis Deduced half-life of 23 Ne versus the number of leading channels removed Comparison of 18 Ne half-life measurements xii

17 6.1 Cycle-by-cycle data pre-selection criteria in the 62 Ga experiment Summed β-γ coincidence spectra collected in the 62 Ga experiment Typical dead-time corrected decay curve from a single 62 Ga run Half-life of 62 Ga versus the experimental run number Half-life measurements of 62 Ga sorted by adjustable electronic and experimental settings Deduced half-life of 62 Ga as a function of the number of leading channels removed from the analysis Half-life of 62 Ga versus the detector rate at t = Comparison of all high-precision 62 Ga half-life measurements. The new world average of T 1/2 = ± ms with a reduced χ 2 value of is obtained from a weighted average of these 7 measurements and is overlayed for comparison Current Ft values for the thirteen most precisely determined superallowed decays with a scalar interaction of ± 0.2% overlayed for comparison Decay scheme for 14 O β + decay to 14 N Previous 14 O half-life measurements separated by counting method Systematic effect between γ-ray and β counting half-life determinations on the 10 C and 14 O Ft values Decay scheme for 10 C β + decay to 10 B Total simulated β activities for 34 Ar and 34 Cl decay Decay scheme for 34 Ar β + decay to 34 Cl Initial attempt to deduce the 34 Ar via direct β counting with the 20 plastic scintillators of the SCEPTAR array xiii

18 7.9 Sum γ-ray singles spectrum obtained with the 8π spectrometer comprised of 10 hours of A = 34 beam Growth and decay curves highlighting the contamination due to 34 Cl bremsstrahlung radiation Typical growth and decay curve obtained from the sum of all 20 SCEP- TAR plastic scintillators for a single 34 Ar run xiv

19 Chapter 1 Introduction High-precision measurements of the ft values for superallowed Fermi β decays between isobaric analogue states have, for decades, provided demanding tests of the Standard Model description of electroweak interactions [1]. To first order, because neither spin nor orbital angular momentum can be transferred in these decays, the axial-vector current does not contribute and these transitions can be described solely in terms of the vector current. The ft values for these decays are therefore directly related to the vector coupling constant for semi-leptonic weak interactions G V under the conserved-vector-current (CVC) hypothesis which stipulates that G V is not re-normalized in the nuclear medium. Once corrected for small (of order 1%) radiative and isospin symmetry breaking effects, the corrected superallowed f t values (denoted Ft) have confirmed the CVC hypothesis at the level of [2], limited the existence of possible scalar interactions to the level of [3], and, together with data from kaon decays, have provided the most demanding test of the unitarity of the Cabbibo-Kobayashi-Maskawa (CKM) quark mixing matrix [4]. In order to significantly contribute to these tests experimentally, β decay halflives and branching ratios must be determined to precisions of 0.05% or better and 1

20 β decay Q values must be deduced to at least 0.01%. Similarly, because the theoretical corrections to the ft values are on the order of 1%, these calculations must be accurate to within 10% of their central value, a demanding requirement of any theoretical nuclear-structure dependent model. Even with these stringent demands, by 1990 eight Ft values ( 14 O, 26m Al, 34 Cl, 38m K, 42 Sc, 46 V, 50 Mn, and 54 Co) were determined to a precision of 0.1% or better [5] and their consistency to such a high degree of accuracy and over such a large mass range was a triumph for both theoretical and experimental nuclear physics. The eight well studied cases were, however, the most straightforward to measure because nearly 100% of these β decays proceed directly to ground state of the stable daughters and, due to their proximity to stability, all of these nuclides could be readily produced in statistically significant quantities. In the 17 years that have followed, new experimental techniques, combined with intense and exotic ion beams being developed at world-class radioactive beam facilities such as the Isotope Separator and Accelerator (ISAC) facility at TRIUMF (Tri University Meson Facility), have provided new opportunities in superallowed β decay studies. Five new cases 10 C (1995), 22 Mg (2003), 34 Ar (2002), 62 Ga (2006), and 74 Rb (2003) have since been added to the eight previous high-precision Ft values and together they establish the world-average value as F t = (12) s (see Sec ), a factor of 3 improvement in precision over the result from 1990 [5]. Experimentally these decays were previously considered some of the most difficult to measure. However, with the recent advances of on-line precision Penning traps (in the cases of 62 Ga [6] and 74 Rb [7]), painstaking absolute detector efficiency calibrations ( 22 Mg [8]), novel in situ efficiency calibration techniques ( 10 C [9]), ultra-high sensitivities to weak non-analogue branches ( 62 Ga [10], 74 Rb [11]), and high-efficiency mass separation techniques ( 34 Ar [12]) these previously inaccessible superallowed decays are now some of 2

21 the most precisely determined. All of these techniques, which are now widely applicable across all areas of nuclear physics research, were born out of the necessity to determine, with high precision, superallowed ft values. Before the accuracy of these methods could be trusted to such a high-degree of precision, years of calibration and simulation were performed to ensure that potential sources of systematic uncertainty were either corrected for, or minimized. A primary focus of this thesis is the presentation of a new technique that allows, for the first time, measurements of β decay half-lives using the method of γ-ray photopeak counting to the level of 0.05% precision. Many nuclear half-lives have been determined using γ-ray counting techniques, however, there are very few highprecision measurements (< 0.1 %) available because of the potentially large systematic effects resulting from detector pulse pile-up. While γ-ray detector pulse pile-up has been qualitatively understood for decades (and even electronic circuits available for the rejection of pile-up since the early 1960 s [13]) there has not been a quantitative description of its effects on half-life measurements to the level of precision (0.05%) necessary for the superallowed-fermi β decay program. In this chapter an introduction to β decay in general, and superallowed Fermi β decay in particular, is given to explain the need for high-precision measurements of the ft values for this special class of β decays. 3

22 1.1 Nuclear Beta Decay In the process of nuclear β decay an unstable nucleus, with atomic number Z and neutron number N, is transformed into a nucleus that is more stable, with Z ± 1 and N 1, and is accompanied by the emission of a β particle (e ) and a neutrino (ν). Because nuclear β decay describes the process of a neutron decaying into a proton (denoted β decay) or a proton into a neutron (β + decay), the atomic number A = Z + N is unchanged. These decay processes can be expressed as, A Z X N A Z X N A Z+1 Y N 1 + e + ν e, [ β decay ] (1.1) A Z 1 W N+1 + e + + ν e. [ β + decay ] (1.2) A third decay process, called electron capture or EC decay, competes with β + decay and involves the capture of an orbital atomic electron by a proton in the nucleus. This decay process can be expressed as, A Z X N + e A Z 1 W N+1 + ν e. [ EC decay ] (1.3) For nuclear β decay or orbital electron capture the Q value, or energy released in the decay process, can be expressed in terms of the atomic mass differences [14], Q β Q β + Q EC = [ m ( A Z X N ) m ( A Z+1 Y N 1)] c 2, (1.4) = [ m ( A Z X N ) m ( A Z 1W N+1 )] c 2 2m e c 2, (1.5) = [ m ( A Z X N ) m ( A Z 1 W N+1)] c 2. (1.6) Decay of an unstable parent nucleus via β + and EC decay yields the same daughter nucleus and the difference between the Q values for these decays is given by, Q EC Q β + 2m e c 2 = MeV. (1.7) 4

23 Nuclei in which β + decay is energetically possible may also undergo EC decay. The reverse is not true, however, as it is possible to have situations in which the Q EC value is positive while the Q β + value is negative. In these cases electron-capture decay is the only possible decay mode. The energy from the β decay of the unstable parent nucleus ( A Z X N) is released in the form of kinetic energy (T) to the decay products. Considering only the 3-body final-state β ± decays, conservation of energy yields, Q β = T D + T e + T νe, (1.8) where T D, T e, and T νe are the kinetic energies of the recoiling daughter nucleus, the β ± particle, and the neutrino, respectively. In the limit where the kinetic energy of the daughter is neglected (the recoiling daughter nucleus is at least 10 3 times more massive than the β particle), the Q value for β decay can be approximated by, Q β T e + T νe, (1.9) and it follows that the maximum kinetic energy of the β particle (denoted T max e ) occurs when the kinetic energy of the neutrino is zero. In this limit, Q β T max e. (1.10) To a good approximation, the β decay Q value is equal to the maximum kinetic energy of the emitted β particle (or neutrino). For high-precision measurements the β decay Q values are generally determined either through threshold energies for nuclear reactions connecting the parent and daughter nuclei of the β decay, or directly from Eqns through differences of high-precision atomic mass measurements. 5

24 1.1.1 Beta-Decay Formalism Nuclear β decay is a semi-leptonic process that is governed by the weak interaction and is mediated through the short-range exchange of massive intermediate charged bosons (W ± ). A Feynman diagram for a single proton in the nucleus undergoing β + decay is depicted in Fig In a nuclear system, the initial proton and final neutron may be one of many nucleons contributing to complex parent and daughter nuclear wavefunctions. The β decay matrix element between initial and final states M f,i, must then be calculated from the initial state nuclear wavefunctions ψ P corresponding to the parent nucleus, and the final state wavefunctions which contain contributions from the daughter nucleus ψ D, the emitted β particle φ e, and the neutrino φ ν. Denoting the weak-interaction operator as Ĥint, the transition matrix element is, M f,i = [ψd φ e φ ν ]Ĥintψ P d 3 r, (1.11) where the integral is over all nucleons as well as the β particle and neutrino coordinates. The weak-interaction operator Ĥint is described (in the minimal electroweak Standard Model) by an equal mixture of vector type (V) and axial-vector type (A) interactions and weak interaction theory is therefore universally known as V-A theory. Despite decades of searches for contributions from other interaction types (scalar, pseudoscalar, and tensor) with ever increasing precision there has thus far been no defect observed in V-A theory. The matrix element can be expressed as a reduced matrix element M f,i 2 by removing the strength parameter of the weak interaction, or weak interaction coupling constant g, M f,i 2 = g 2 M f,i 2. (1.12) In order to calculate the matrix element (or the reduced matrix element) for nuclear β decay, the neutrino wavefunctions can be expressed using free-particle plane-wave 6

25 n { d u d e e + gv ud W + g d u p u { Figure 1.1: Feynman diagram for β + decay of a single proton in the nucleus showing the exchange of an intermediate vector boson W +. The weak-interaction coupling constant is denoted g at the lepton vertex and gv ud at the hadronic vertex. solutions, ψ ν(r) = e iq r/, (1.13) while the β-particle wavefunction is a free-particle plane wave distorted by the Coulomb field of the daughter nucleus, ψ e (r) = [F(Z, p)]1 2 e ip r/. (1.14) In these expressions the Fermi function F(Z, p) describes the Coulomb distortion of the outgoing β particle wavefunction and p and q are the electron and neutrino momenta, respectively. In order to complete the calculation of the β decay matrix element and relate it to decay observables such as the β decay half-life (see Sec. 1.2), the parent ψ P and daughter ψ D nuclear wavefunctions must also be calculated. For this, one must rely on a model that predicts the complex nuclear wavefunctions. 7

26 1.1.2 Beta Decay Classification For a β particle of momentum p 1 MeV/c, for example, the quantity p/ in the electron wavefunction of Eqn is only fm 1. The wavefunctions for both the neutrino and the β particle can therefore be expanded yielding for the matrix element between initial and final states, M f,i = [ F(Z, p) ψ DĤintψ P d 3 r + i(p + q) ] ψdrĥintψ P d 3 r + (1.15) Note that the first term (zeroeth order expansion, L = 0) measures the overlap between the parent and daughter nuclear wavefunctions. Applying the parity operator, which is a reflection of the radial coordinates in 3-dimensions (r r), this term is positive with respect to parity and is therefore zero unless the parities of the parent and daughter wavefunctions were identical. The second term (first order, L = 1) is negative under the parity operator because of the additional factor of r in the integrand, and the parent and daughter wavefunctions must therefore have opposite parity for this term to contribute. In general the β decay selection rule is governed by π P = π D (-1) L where π P and π D are the parities of the parent and daughter states, respectively. Allowed decay is defined as L = 0 and must therefore occur between states of the same parity. Similarly if angular momentum is explicitly introduced into the above analysis by expanding the electron and neutrino plane waves into spherical harmonics, then the orbital angular momentum carried by the electron and neutrino pair L must satisfy the vector sum L P = L D +L. Furthermore, because the electron and neutrino are spin- 1 2 fermions, intrinsic spin angular momentum carried by the electron and neutrino pair S must also be included by introducing the Pauli spin matrices to Eqn The coupling of the electron and neutrino spins to S = 0 is called Fermi 8

27 (or vector) β decay, while the S = 1 coupling corresponds to Gamow-Teller (or axialvector) β decay. Defining the total angular momentum of the electron and neutrino pair J = L + S the corresponding angular momentum selection rule is therefore J P = J D + J. Allowed Fermi β decay therefore has L = 0, S = 0, and hence J = 0 and thus cannot change the total angular momentum of the nucleus thus J P = J D. A change in the total angular momentum of the nucleus by 1 following an allowed β decay (e.g ) must therefore be a pure Gamow-Teller (S = 1) β decay and is commonly observed. Far more rare are transitions of the pure Fermi type as these can only occur between states of J = 0 total angular momentum ( or 0 0 ), because Gamow-Teller decay is then strictly forbidden by the angular momentum selection rules ( ). For all other β decays between states of the same total spin and parity (e.g ), Gamow-Teller decay is not forbidden ( 1 = 1 + 1) and these transitions are mixtures of Fermi and Gamow-Teller decay. A further angular-correlation experiment is then required to distinguish the vector and axialvector contributions. This will be discussed in further detail in Sec. 1.4 where the mixture of Fermi and Gamow-Teller contributions is currently the main limitation in extracting a high-precision ft value from the β decay of the free neutron ( ) Isospin Unlike the electromagnetic interaction, the nuclear force (to a good approximation) does not distinguish between protons and neutrons. It is therefore convenient to describe these two particles as isobaric spin or isospin projections of the same particle, the nucleon. A nucleon can be described as being either isospin-up with isospin projection t z = + 1 (the neutron), or isospin-down with t 2 z = 1 (the proton), 2 9

28 both of which are measured with respect to an arbitrary z-axis in isospin space. A two nucleon system, for example, can have total isospin T = 0 or 1 corresponding to the anti-parallel or parallel coupling of the isospin vectors, respectively. Isospin algebra is thus equivalent to ordinary angular momentum algebra for a system of spin- 1 2 particles. The z component of the total isospin for a multi-nucleon system can be written in terms of the total number of neutrons N and protons Z in the nucleus, T z = 1 (N Z). (1.16) 2 The total isospin of the nucleus T is the vector sum of the isospins of the the individual nucleons and can therefore take on any value in integer steps between T z and N+Z 2, T = T z, T z + 1,, N + Z. (1.17) 2 As decided by the symmetry energy term in the semi-empirical formula for nuclear masses and binding energies, nuclear states with the lowest total isospin T = T z are generally more bound than states of higher isospin. Almost all nuclei thus have ground-state isospin T = T z = 1 (N Z), with the only known exceptions being 2 certain heavy (A 34) odd-odd N = Z nuclei for which the pairing energy overcomes the symmetry energy making the T = 1, I π = 0 + the ground state, rather than the lowest T = T z = 0 state [15]. Similar selection rules to those for angular momentum apply to isospin in nuclear β decay. Fermi decay, for example, is strictly forbidden unless the change in isospin between the parent and daughter states T is identically 0 while Gamow-Teller decay can occur when T = 1, 0, +1. The term superallowed Fermi β decay is used to describe allowed (L = 0) pure Fermi (S = 0) decay between states of the same isospin ( T = 0), which are isobaric analogue states. These states have identical nucleonic wavefunctions with the exception of one proton exchanged for a neutron between the 10

29 parent and the daughter and correspond to members of a multiplet of 2T + 1 states with the same total isospin T but different values of T z. Non-analogue pure Fermi branches are relatively weak (and would be forbidden if isospin was an exact nuclear symmetry) but are observed connecting states that are not isobaric analogues of each other, although they do generally have the same total isospin ( T = 0). The matrix elements for non-analogue Fermi β decay branches are typically reduced by 3 to 4 orders of magnitude compared to the superallowed Fermi β decay branch. Weak Fermi β decay branches to non-analogue 0 + states will be discussed in Sec Fermi Theory of Beta Decay The decay constant λ f,i for nuclear β decay can be derived from Fermi s Golden Rule, which is a calculation of the probability (per unit time) to observe a transition between some initial and final state [16], λ f,i = 2π M f,i 2 dn de, (1.18) where dn de is the density of final states, and M f,i 2 is the matrix element of the weak interaction Hamiltonian between the initial and final states given in Eqn Expanding the electron and neutrino free-particle exponential wavefunctions (under the assumption p r as discussed above in Sec ) and keeping only the zeroeth order (L = 0) term is known as the allowed approximation. The expectation values of the electron and neutrino wavefunctions under the allowed approximation and normalized to an arbitrary volume V can therefore be expressed as, ψ e (r) 2 1 V F(Z, p), ψ ν (r) 2 1 V. (1.19) Under the allowed approximation, the reduced matrix element M f,i 2 depends only on the electron momentum through the Fermi function F(Z, p) and is independent 11

30 of the neutrino momentum. It is customary to remove the Coulomb effects and the volume-normalization from the matrix element to define, M f,i 2 = g2 F(Z, p) V 2 M f,i 2. (1.20) The number of final states n for a particle of momentum p in 3-dimensions is given in standard texts [16, 14] and can be expressed as, n = 1 d 3 x d 3 p, (1.21) (2π ) 3 where the first integral in Eqn is over the arbitrary volume V that was chosen to normalize the wavefunctions, while the second can be expressed as the volume of a sphere in 3-dimensional momentum space ( 4π 3 p3 ). In differential form, the density of final states for the electron and neutrino are therefore, dn e de = V p2 2π 2 3 dp de, dn ν de = V q2 2π 2 3 dq de. (1.22) Nuclear β ± decay (as opposed to EC decay) results in a 3-body final state (electron, neutrino, and recoiling daughter nucleus), thus the neutrino momentum q (or energy E ν ) can be expressed in terms of the energies of both the electron and the daughter nucleus using energy conservation (E = E e +E ν +E D ) and the relativistic energy and momentum relation, E 2 ν = q2 c 2 + m 2 ν c4. Neglecting the mass of the neutrino and kinetic energy of the daughter recoil (as discussed in Sec. 1.1, Eqn. 1.9) one obtains, q 2 dq de = (E E e) 2 c 3. (1.23) Substituting this result, along with the wavefunctions for the electron and neutrino under the allowed approximation (Eqn. 1.19), and the density of final states (Eqn. 1.22) into the expression for Fermi s Golden Rule (Eqn. 1.18), and integrating over all β particle momenta yields, λ f,i = g2 M f,i 2 2π 3 7 c 3 p max e F(Z, p)p 2 (E E e ) 2 dp, (1.24) 0 12

31 where p max e is the maximum momentum of the β particle. The integral in Eqn is conventionally expressed as a dimensionless quantity through the substitution of the dimensionless variables W = Ee m ec and ρ = p 2 m ec, that satisfy the relativistic energymomentum relation W 2 = ρ Expressing the integrand solely in terms of the dimensionless energy of the β particle yields, where W o = Emax e m ec 2 λ f,i = g2 m 5 ec 4 M f,i 2 2π 3 7 can therefore be expressed as, W o 1 F(Z, W)W W 2 1(W o W) 2 dw, (1.25) Q β m ec The decay constant λ between initial and final states λ f,i = g2 m 5 ec 4 M f,i 2 2π 3 7 f, (1.26) where the dimensionless quantity f is known as the statistical rate function or phase space integral and replaces the integral in Eqn Evaluation of the statistical rate function will be discussed in Sec of this thesis. The decay constant λ f,i can also be written in terms of the partial half-life t for decay to the final state of interest which is defined as the total β decay half-life T 1/2 divided by the β branching ratio to the final state BR. For β + decays an additional factor is required to account for the fraction of decays that occur by electron-capture P EC. The partial half-life for β + decays is therefore, t = ln2 λ f,i = T 1/2 BR (1 + P EC). (1.27) This expression is also valid for β decays with P EC = 0 as electron-capture decay does not compete with β decays (see Sec. 1.1). Combining the expressions for f and t defines the f t value, which is a convenient way to describe the β decay transition. For allowed β decays the following result is obtained, ft = 1 g 2 M f,i 2 2π3 7 ln2 m 5 e c4. (1.28) 13

32 The right hand side of Eqn contains fundamental constants with the exception of the matrix element, M f,i 2. Differences between ft values for all allowed β decay transitions are due solely to differences in nuclear matrix elements and the relative contributions from Fermi and Gamow-Teller decays contained in them: g 2 M f,i 2 = G V 2 M f,i (F) 2 + G A 2 M f,i (GT) 2, (1.29) where G V and G A are the coupling constants for vector and axial-vector nuclear β decay, respectively. The f t values can be deduced experimentally, thus the coupling constants G V and G A, and the reduced matrix elements M f,i (F) 2 and M f,i (GT) 2 must be computed in order to complete Eqn Theoretical calculations of decay constants for allowed β decays, requires a model that predicts the nuclear wavefunctions from which the matrix element can be computed. Because any model to describe the nuclear many-body system is at best an approximation, the resulting calculations would only be as accurate as the approximations used to describe the wavefunctions. Superallowed Fermi β decays are, however, a special class of β decays that are of the pure Fermi type ( M f,i(gt) 2 = 0), and occur between nuclear isobaric analogue states for which the calculation of the Fermi matrix element M f,i (F) 2 is largely independent of the individual nuclear wavefunctions. If the reduced Fermi matrix element is known exactly, and the ft values are deduced from experiment, then it is possible to determine the fundamental coupling constant of the weak vector interaction G V Statistical Rate Function f Evaluation of the ft values for nuclear β decay requires the calculation of the statistical rate function f given by, f = W o 1 F(Z, W)W W 2 1(W o W) 2 dw, (1.30) 14

33 where the Fermi function F(Z, W) was introduced in Eqn to account for the Coulomb distortion as the liberated β particle interacts with the Coulomb field of the daughter nucleus. This function has been evaluated using a relativistic treatment of the electron wavefunction integrated through the nuclear charge distribution taken, in a first approximation, as a uniformly charged sphere of radius R and is given by [17], ( F(Z, W) 4L(Z, W) 2R ) 2(γ 1) W 2 1 e πη Γ(γ + iη) 2 Γ(2γ + 1) 2, (1.31) where γ = 1 (αz) 2, η = ±ZαW/ W 2 1 for β /β + decay and α is the fine structure constant. The function L(Z, W) replaces a point nucleus with the uniformly charged sphere of radius R, and accounts for the integration of the electron Dirac wavefunction through the resulting Coulomb field. For small values of Z an approximate expression for L(Z, W) is [17]: L(Z, W) = αzrw(41 26γ) αzrγ(17 2γ) 60 (αz)2 15(2γ 1) 30W(2γ 1), (1.32) although for high-precision determinations of the statistical rate function for superallowed Fermi β decay studies (see Sec. 1.3 below), still higher-order corrections to L(Z, W) are required. In addition, further corrections to the Fermi function F(Z, W) itself are required to account for screening by the atomic electron cloud, a more realistic nuclear charge distribution, higher-order interactions between the leptonic wavefunctions of the electron and neutrino with the nucleonic wavefunctions of the parent and daughter nuclei, and a daughter nucleus recoil correction. All of these have been calculated [1, 17] and are required for high-precision work. Inserting these results into Eqn provides a means to deduce the statistical rate function given measurements of β decay Q values. Note that in the extreme limit of high energy and 15

34 (or) low Z where F(Z, W) 1 the statistical rate function can be approximated by, f W o 1 W 2 (W o W) 2 dw 1 30 W 5 o 1 30 ( Qβ m e c 2 ) 5, (1.33) which scales with the Q value to the fifth power. In order to deduce f values to the same level of precision as partial half-lives, one therefore requires Q-value measurements to be performed with five times more precision than is required for half-life or branching-ratio measurements. 1.3 Superallowed Fermi Beta Decay Superallowed Fermi β decays occuring between 0 + isobaric analogue states have, for decades, provided stringent tests of the Standard Model s description of electroweak interactions. For this special class of β decays, the reduced matrix element M f,i (F) 2 can be calculated exactly in the limit that isospin is a good quantum number. As Fermi β decay involves changing a proton into a neutron (or a neutron into a proton) while leaving the spatial and spin wavefunctions of the nucleon unchanged, the Fermi transition operator is equivalent to the isospin ladder operator ˆT ± (for β ± decays). The matrix element is therefore given by [18], M f,i(f) 2 = (T T z )(T ± T z + 1), (1.34) where T z is the z-projection of the isospin of the parent nucleus (Eqn. 1.16). For superallowed Fermi β decays between members of a T = 1 multiplet (T z = 1, 0, +1), for example, the reduced matrix element is given by, M f,i(f) 2 = 2. (1.35) Note that this result is exact, to the extent that isospin is an exact symmetry of the nuclear Hamiltonian, and is independent of the complex parent and daughter 16

35 nuclear wavefunctions. Substituting this result into the expression for the f t value (Eqn. 1.28) and noting that, to first order, there is no axial-vector contribution to these pure Fermi decays one obtains, ft = 2π3 7 ln2 2G 2 = constant. (1.36) V m 5 ec4 The f t values for all superallowed Fermi β decays between T = 1 isobaric analogue states are therefore predicted to be constant and independent of nuclear structure. This is somewhat surprising considering the nucleon involved in the β decay is not a free particle but is embedded within the nucleus. In deriving this result it was implicitly assumed that the vector coupling G V is indeed a fundamental constant and is not re-normalized in the nuclear medium. This important assumption is known as the Conserved Vector Current (CVC) hypothesis [19], and measurements of the f t values for superallowed transitions set sensitive limits on its validity. Under the assumption that CVC is satisfied, these transitions can similarly be used to extract the fundamental weak interaction coupling constant G V to high precision without any model-dependent assumptions regarding the individual nucleon wavefunctions. Presently there are thirteen superallowed ft values that have been determined to experimental precisions of 0.3% or better ( 10 C, 14 O, 22 Mg, 26m Al, 34 Cl, 34 Ar, 38m K, 42 Sc, 46 V, 50 Mn, 54 Co, 62 Ga, and 74 Rb) and are plotted in Fig Although there does appear to be some fluctuations about a constant value, the ft values for these thirteen decays are all within approximately 1% of each other. The f t values for β decay in general span nearly 20 orders of magnitude and thus the small deviations in the f t values observed between these thirteen superallowed decays agree to first order with the CVC hypothesis. Corrections to the f t values resulting primarily from charge dependent sources such as radiative bremsstrahlung processes as the charged β particle is emitted from the positively charged nucleus, and the assumption that 17

36 Ga 74 Rb ft (s) Mg 34 Cl 34 Ar 38m K 46 V 54 Co C 14 O 26m Al 42 Sc 50 Mn Z of daughter Figure 1.2: Experimental determinations of the f t values for thirteen superallowed Fermi β decays between A=10 and A=74. isospin is an exact nuclear symmetry are required in order to further improve tests of the CVC hypothesis Radiative Corrections There are two main corrections that must be applied to the ft values resulting from the need to incorporate small ( 1%) charge-dependent radiative effects. The first correction (denoted δ R ) results from the fact that, once created, the charged β particle interacts with the Coulomb field of the daughter nucleus. This effect is a function only of the total charge of the daughter nucleus Z and the electron energy and thus, while it depends on the particular nucleus, it is independent of nuclear structure. Because no 18

37 model assumptions need to be made, the nucleus independent radiative corrections δ R have been calculated to order Zα2 and estimated to Z 2 α 3 [20, 21, 22]. These corrections are on the order of 1.5% [1] and are considered to be very reliable [23]. In deriving the superallowed ft values between 0 + isobaric analogue states it was shown in Sec that these decays are pure vector transitions as the axialvector interaction is forbidden to participate due to angular momentum selection rules. Although angular momentum is an exact nuclear symmetry (unlike isospin), it is possible for the axial-vector current to cause a nucleon to undergo a spin change. This can then be followed by virtual photon (or virtual Z 0 boson) exchange with the departing β particle, which can change it back. The entire process therefore has the appearance of a pure vector transition but was mediated through an axial-vector weak interaction in higher-order. This process can either act on the same nucleon, and is therefore transition and nucleus independent (discussed below), or can occur in different nucleons and will therefore depend on the structure of the particular nucleus. The nuclear-structure dependent part of the axial-vector corrections (denoted δ NS ) have been evaluated using shell-model calculations with a Woods-Saxon plus Coulomb potential [23] and are generally at least a factor of five smaller than δ R. The total radiative correction δ R is the sum of these two effects, δ R = δ R + δ NS. (1.37) A summary of the values δ R and δ NS from the most recent calculation [24] are listed in Table 1.1. A final correction is required to account for radiative effects that are not dependent on the specific nuclear decay. This correction is denoted V R and contains contributions from axial-vector spin-exchanges on the same nucleon as discussed above, as 19

38 well as higher-order loops involving virtual W ± and Z 0 boson exchange with the polarized fermion vacuum [25, 26, 27]. Recently, a new calculation of this correction factor (estimated to order Z 2 α 3 ) was published [28] that was able to better control hadronic uncertainties in the short distance loops. This result has led to an increase in the precision of V R by a factor of 2. The present value [24] is, V R = ± 0.038%, (1.38) and is independent of both the particular transition and the details of nuclear structure. Because this radiative correction describes the interaction of a single quark with the polarized fermion vacuum, this factor also applies to the decay of a free neutron, and pion decay which will be discussed in Secs. 1.4 and 1.5 of this thesis Isospin Symmetry Breaking Isospin is not an exact symmetry of the nuclear Hamiltonian and another small correction must therefore be applied to the matrix element of Eqn which describes superallowed Fermi β decay between isobaric analogue states of a T = 1 multiplet. Because the breaking of perfect isospin symmetry can only lead to a reduced overlap between the neutron and proton wavefunctions, the correction (denoted δ C ) must act to reduce the overall matrix element: M f,i (F) 2 = 2(1 δ C ). (1.39) Isospin symmetry breaking is also manifested as weak Fermi β decay branches to non-analogue 0 + states in the daughter and the total correction for isospin symmetry breaking δ C is typically expressed as a sum of two components [29], δ C = δ IM + δ RO. (1.40) 20

39 Table 1.1: Radiative and isospin symmetry breaking corrections for the 13 most precisely determined superallowed decays. Calculations using two distinct interactions were performed in Ref. [34] for A 62 and the ranges obtained are shown for 62 Ga and 74 Rb. Decay δ R (%) δ NS (%) δ R (%) (TH)δ C (%) (OB)δ C (%) 10 C 1.679(4) 0.357(35) 1.322(35) 0.180(18) 0.15(9) 14 O 1.543(8) 0.247(50) 1.296(51) 0.320(25) 0.15(9) 22 Mg 1.466(17) 0.237(20) 1.229(26) 0.265(14) 0.21(9) 26m Al 1.478(20) (20) 1.490(28) 0.270(14) 0.30(9) 34 Cl 1.443(32) 0.081(15) 1.362(35) 0.635(36) 0.57(9) 34 Ar 1.412(35) 0.181(15) 1.231(38) 0.640(41) 0.39(9) 38m K 1.440(39) 0.096(15) 1.344(42) 0.620(45) 0.59(9) 42 Sc 1.453(47) (20) 1.486(51) 0.490(42) 0.42(9) 46 V 1.445(54) 0.037(7) 1.408(54) 0.425(32) 0.38(9) 50 Mn 1.445(62) 0.038(7) 1.407(62) 0.505(36) 0.35(9) 54 Co 1.443(71) 0.025(7) 1.418(71) 0.610(43) 0.44(9) 62 Ga 1.459(87) 0.036(20) 1.423(89) 1.380(155) Rb 1.498(120) 0.061(20) 1.437(122) 1.430(404) The first term δ IM accounts for the fact that if isospin symmetry were exact then the only 0 + state populated in the daughter would be the isobaric analogue state while all other decays to 0 + states in the daughter would be strictly forbidden. The second term δ RO is the dominant contribution to the total isospin symmetry breaking correction and accounts for imperfect overlap of the single nucleon radial wavefunctions between the neutron and proton due to binding energy and Coulomb effects. As there can be many 0 + states in the daughter, the first term δ IM can be approximately expanded into the contributions from each of the non-analogue states, δ IM δ 1 IM + δ2 IM +, (1.41) where mixing of different isospins (T = 0,2,3, ) into the dominant T = 1, I π = 0 + states breaks the exact identity of the above expression. Calculations of δ IM are 21

40 performed using a shell model calculation which computes the nucleon wavefunctions and hence the matrix elements for the analogue and all non-analogue β branches. For 0 + states within the β decay Q value window, this correction can also be deduced experimentally by measuring the β branching ratios to the analogue (BR 0 ) and nonanalogue (BR i ) states. Considering a simple case where the daughter has only a single non-analogue 0 + state with a branching ratio BR 1 and squared matrix element 2δIM 1 (1 δ RO), in addition to the dominant decay branch to the analogue state BR 0 with squared matrix element 2(1 δim 1 )(1 δ RO), the β branching ratio to the nonanalogue state can be expressed using Eqn. 1.36, BR 1 = BR 0 t 0 t 1 BR 0 f 1 f 0 δ 1 IM 1 δ 1 IM BR 0 f 1 f 0 δ 1 IM, (1.42) where the assumption that δ 1 IM 1 was used to obtain the final simplified expression. This result can be generalized to the case of many non-analogue states yielding, δ i IM f 0 f i BR i BR 0. (1.43) In deriving the above expressions it was implicitly assumed that the total correction for isospin symmetry breaking δ C could be factorized using [30], 1 δ C = 1 δ IM δ RO (1 δ IM )(1 δ RO ), (1.44) which is valid for small (< 1%) values of δ IM and δ RO of interest here. Under these assumptions it is clear from Eqn that measurements of the non-analogue branching ratios BR i and the β decay Q values (which are required to calculate the statistical rate functions) to the analogue and non-analogue states, allow the δ i IM to be deduced approximately from experiment. Because theoretical calculations of isospin mixing δ IM and imperfect radial overlap δ RO corrections require predictions of the nucleon wavefunctions, both of these 22

41 2.0 Woods-Saxon (TH) Hartree-Fock (OB) 62 Ga 70 Br 1.5 Calculated δ C (%) Ne 30 S 38 38m K Ca 42 Ti 54 Co 66 As 74 Rb Mg 26 Si 10 C 14 O 26mAl 34 Cl 34 Ar 42 Sc 46 V 50 Mn Z of Daughter Figure 1.3: Values of the isospin symmetry breaking corrections in superallowed Fermi beta decays. There is qualitative agreement between the Woods-Saxon (TH) calculations and the Hartree-Fock model (OB), however, a small systematic effect between the two methods lead to an increased uncertainty in the absolute corrections. Woods-Saxon data were taken from [23], while the Hartree-Fock data are taken from [29, 32, 33, 34]. corrections, and thus the total theoretical corrections δ C, are highly dependent on the nuclear model employed. One set of calculations are performed by Towner and Hardy (TH) and use a shell-model diagonalization with a Woods-Saxon plus a Coulomb potential [23, 31]. A second and independent calculation was performed by Ormand and Brown (OB) and used a self-consistent Hartree-Fock method to deduce the single particle wavefunctions [29, 32, 33, 34]. A summary of the isospin symmetry breaking corrections δ C obtained from both models is listed in Table 1.1 for the thirteen most precisely determined superallowed β decay f t values. Although both models show 23

42 qualitative agreement over many of the superallowed decays, as shown in Fig. 1.3, there is a small systematic discrepancy in the absolute magnitude for these corrections with the OB δ C corrections systematically smaller than the TH values. Although the discrepancies between isospin symmetry breaking corrections arise primarily from the two different interactions used in the calculations, it is not clear which of these two models provides a better description of the effects of isospin symmetry breaking in superallowed Fermi β decay. Tests of the theoretical predictions can, however, be performed experimentally as was discussed above in the context of the isospin mixing δ IM corrections. These experimental measurements are criticial to discriminate between the two theoretical models for δ C and considerable effort has therefore recently focused on the nuclei in Fig. 1.3 that show the greatest model dependencies ( 14 O, 18 Ne, 30 S, 34 Ar, for example) or have the largest absolute corrections (all decays with A 62) Corrected f t Values With the small corrections for radiative and isospin symmetry breaking effects, the corrected ft values (denoted Ft) for superallowed Fermi β decays between T = 1 isobaric analogue states can be expressed as [1]: Ft = ft(1 + δ R )(1 δ C ) = 2π 3 7 ln2 2G 2 V (1 + V R )m = constant. (1.45) e 5 c4 As discussed in Sec. 1.2 of this thesis, experimental determination of the ft value requires measurements of the β decay half-life and the superallowed β branching ratios in order to deduce the partial half-life t, while the statistical rate function f depends on the decay energy or Q value. Because several new measurements have been published since the previous superallowed world data evaluation performed in 24

43 Table 1.2: Present status of all world-averaged data for the thirteen most precisely determined superallowed decays. Decay f T 1/2 BR P EC ft (ms) (%) (%) (s) 10 C (12) 19290(12) (19) (471) 14 O (23) 70620(14) (65) (265) 22 Mg (17) (24) 53.16(12) (722) 26m Al (94) (19) > (108) 34 Cl (41) (44) > (113) 34 Ar (14) (40) 94.45(25) (821) 38m K (33) (27) (44) (96) 42 Sc (12) (26) (14) (139) 46 V (33) (11) (42) (161) 50 Mn (17) (11) (30) (125) 54 Co (32) (62) (30) (148) 62 Ga (83) (21) (80) (114) 74 Rb 47285(108) (43) 99.50(10) (796) early 2005 [1], it was necessary to re-calculate the ft values for the thirteen most precise cases. The f t values were calculated from all of the experimental data given in the previous survey [1] and updated to include new measurements of the halflives of 14 O [35], 34 Cl [12], 34 Ar [12], 50 Mn [36], and 62 Ga [37, 38], β branching ratio measurements for 14 O [39], 38m K [40] and 62 Ga [10, 41], and Q-value measurements for 26m Al [42], 42 Sc [42], 46 V [42, 43], 62 Ga [6], and 74 Rb [44]. The ft value for 62 Ga decay can only now be added to the twelve previous high-precision ft values as new experimental measurements on all three necessary quantities have been published since the last survey. Inclusion of the new Q-value results for 46 V was critical in the present analysis because the value of Ref. [43], which was one of the first to use a precision Penning trap to deduce a superallowed Q value, obtained a result that disagreed with an older ( 3 He,t) transfer reaction experiment [45] at the level 25

44 Table 1.3: Present status of the world averaged ft and Ft values for the thirteen most precisely determined superallowed decays. Case-by-case uncertainties in the OB δ C corrections have been adopted to be equal to those of TH to avoid double counting an overall systematic uncertainty in these calculations already included in the values of Table 1.1. Decay ft δ R TH δ C OB δ C TH Ft OB Ft (s) (%) (%) (%) (s) (s) 10 C (47) 1.322(35) 0.180(18) 0.150(18) (49) (49) 14 O (27) 1.296(51) 0.320(25) 0.150(25) (32) (32) 22 Mg (72) 1.229(26) 0.265(14) 0.210(14) (73) (73) 26m Al (11) 1.490(28) 0.270(14) 0.300(14) (15) (15) 34 Cl (11) 1.362(35) 0.635(36) 0.570(36) (19) (19) 34 Ar (82) 1.231(38) 0.640(41) 0.380(41) (84) (85) 38m K (10) 1.344(42) 0.620(45) 0.590(45) (21) (21) 42 Sc (14) 1.486(51) 0.490(42) 0.420(42) (25) (24) 46 V (16) 1.408(54) 0.425(32) 0.380(32) (25) (25) 50 Mn (12) 1.407(62) 0.505(36) 0.350(36) (25) (25) 54 Co (15) 1.418(71) 0.610(43) 0.440(43) (29) (29) 62 Ga (11) 1.423(89) 1.380(155) 1.290(155) (55) (56) 74 Rb (80) 1.437(122) 1.430(404) 0.980(404) 3084(15) 3098(15) Average Ft (8) (8) χ 2 /ν of 3σ. Because the older result of Ref. [45] appeared in a publication that included Q value measurements for six other superallowed decays, an intense experimental effort has been launched [42, 46, 47] to re-measure all of the superallowed Q values using precision Penning traps. Recently, new measurements have been performed for 26m Al [42], 42 Sc [42], and 74 Rb [44], in addition to an independent confirmation of 46 V [42], all of which are in generally good agreement with previous results effectively ruling out a widespread systematic effect. The updated world averaged values for the statistical rate functions f, β decay half-lives T 1/2, superallowed branching ratios BR, electron-capture fractions P EC [1], and the ft values (calculated using the partial 26

45 half-life t defined in Eqn. 1.27) are presented in Table 1.2. With an updated data set for the thirteen most precisely determined superallowed f t values one can now apply the nucleus-dependent corrections for radiative bremsstrahlung processes and isospin symmetry breaking discussed in Sec and 1.3.2, respectively in order test the CVC hypothesis. The corrected F t values using both the isospin symmetry breaking corrections from the Woods-Saxon calculation (TH) and those of the self-consistent Hartree-Fock calculations (OB) are summarized in Table 1.3. The corrected F t values using the Woods-Saxon isospin symmetry breaking corrections are compared in Fig. 1.4 with the uncorrected experimental f t values. Note that because the uncertainties in the isospin symmetry breaking corrections of Ormand and Brown already included an estimate of the systematic uncertainty in the model, the OB corrected F t values were calculated from the OB central δ C values with the case-by-case TH δ C uncertainties. If one were to instead use the isospin symmetry breaking corrections of Ormand and Brown with their overall quoted uncertainties, the result would lead to an increased estimate of the systematic uncertainty between these calculations because of double counting the systematic effect. With the corrections applied to each of the thirteen superallowed decays the 1% fluctuations in the experimental f t values are removed providing a strong confirmation of the CVC hypothesis. In fact, a weighted average of the thirteen corrected Ft values using the Woods-Saxon corrections for isospin symmetry breaking (TH) yields, Ft(TH) = (75) s, χ 2 /ν = (1.46) This result is a powerful vindication of both the theoretical corrections and the CVC hypothesis as the ft values listed in Table 1.3 are comprised of more than 125 independent measurements [2] and encompass the large mass range 10 A 74. If 27

46 Ga 74 Rb ft (s) Mg 34 Cl 34 Ar 38m K 46 V 54 Co C 14 O 26m Al 42 Sc 50 Mn Z of daughter Ft = ± 0.8 s χ 2 /ν = Rb Ft (s) Mg 34 Ar 42 Sc 46 V 54 Co 62 Ga C 14 O 26m Al 34 Cl 38m K 50 Mn Z of daughter Figure 1.4: Comparison of the f t values and F t values for thirteen superallowed Fermi β decays between A=10 and A=74. 28

47 instead the isospin symmetry breaking corrections from the Hartree-Fock calculations (OB) are used the corrected Ft value is, Ft(OB) = (75) s, χ 2 /ν = 1.17, (1.47) which is an equally powerful result, albeit with a somewhat larger reduced χ 2 value. For twelve degrees of freedom, the probability of obtaining a reduced χ 2 value that exceeds 1.17 is approximately 25% [48] and thus one cannot discriminate between the theoretical isospin symmetry breaking corrections solely by statistical means. Unfortunately, while both of these results are in agreement with the CVC hypothesis and both are precise to the level of 0.03%, the systematic effect between the two calculations of the isospin symmetry breaking corrections has manifested itself in the corrected F t values as a difference of 1.75 s or nearly three times the statistical uncertainty. In order to address the systematic effect of the isospin symmetry breaking corrections in the overall average corrected F t value, a general procedure has been adopted [1]. This method involves taking the unweighted average of the corrected Ft values from both calculations as the overall mean Ft and combining this with a systematic uncertainty equal to half of the spread between them [1]. Adopting this method the overall average corrected Ft value is given by, Ft = (75) Ft (87) δc s, = (12) s (1.48) where the first uncertainty is due to the experimental ft values in addition to the caseby-case uncertainties from the radiative and isospin symmetry breaking corrections from the Woods-Saxon calculations of Towner and Hardy. The second uncertainty is the estimated systematic effect resulting from the discrepancy in the calculations for 29

48 isospin symmetry breaking effects. Combining the statistical and systematic uncertainties in quadrature yields an overall uncertainty of 1.2 s. It should be noted that in the previous superallowed data evaluation only the nine historically well-measured superallowed ft values were used to calculate the systematic uncertainty. The result yielded an estimated systematic of 0.9 s [1], which is statistically equivalent to the value derived here with the inclusion of 22 Mg, 34 Ar, 62 Ga, and 74 Rb. Removing these four decays from the calculation decreases the systematic uncertainty estimate only marginally from 0.87 s to 0.82 s. Using the fundamental constants from the Particle Data Group [4], one can express the constants contained in Eqn as, K ( c) 6 = 2π3 ln2 (m e c 2 ) 5 = ( ± 0.004) GeV 4 s. (1.49) With the transition-independent radiative correction V R = 2.361(38)% from Eqn. 1.38, the vector coupling constant G V deduced from superallowed Fermi β decay between isobaric analogue states is, G V ( c) 3 = (14) Ft (16) δc (21) V R 10 5 GeV 2, = (30) 10 5 GeV 2, (1.50) where the first uncertainty combines the experimental f t values with the case-bycase uncertainties in the radiative and isospin symmetry breaking corrections, the second is due to the systematic difference between calculations of isospin symmetry breaking, and the third is due to the 1.6% uncertainty associated with the transitionindependent radiative correction V R. The total uncertainty is obtained from the quadrature sum of these three contributions. Although the superallowed decays permit the extraction of the vector coupling constant at the level of 0.03%, this precision is not limited by experiment but rather by theory with the dominant contribution 30

49 coming from the uncertainty in the nucleus-independent radiative correction V R. This uncertainty is based on a conservative estimate [28] and can, in principle, be reduced further by refining estimates of the intermediate-energy QCD loop effects. The dominant uncertainty in the extraction of G V could therefore soon be due to the systematic discrepancy associated with the isospin symmetry breaking corrections. There is therefore considerable interest in studying pion and free neutron β decay for which there are no nuclear-structure dependent corrections. 1.4 Free Neutron Beta Decay The decay of a free neutron can also be used to deduce the weak vector coupling constant for pure Fermi decays G V. Because neutron β decay, n p + e + ν e (1.51) occurs between free spin- 1 2 fermions, the decay is not purely vector as in the superallowed Fermi decays discussed above. As a result, the total matrix element will be a mixture of vector and axial-vector components, ( g 2 M f,i 2 2 = G V M f,i(f) 2 + G 2 ) A 2 G M f,i(gt) 2 V, (1.52) where M f,i (F) 2 and M f,i (GT) 2 are the reduced matrix elements for the Fermi and Gamow-Teller contributions to the overall β decay and G A and G V are the axial-vector and vector coupling constants, respectively. The radiative corrections for neutron decay are similar to those used in superallowed nuclear β decay, however, because the neutron is a free particle there are no nuclear-structure dependent corrections to consider. The f t value for neutron decay can therefore be expressed analogously to 31

50 Eqn [2, 49]: fτ n (1 + δ R) = K/ln2 G V 2 (1 + V R )(1 + 3λ2 ), (1.53) where τ n is the neutron mean lifetime, δ R is the same transition-dependent radiative correction as discussed above evaluated for the case of a neutron [23] (note that there is no δ NS equivalent), K is the constant given by Eqn. 1.49, and λ contains the admixture of the vector and axial-vector contributions to neutron decay given by [49], λ = G A(1 + A R )1 2. (1.54) G V (1 + V R )1 2 The radiative correction V R (Eqn. 1.38) is the same as is used in the superallowed decays and corrects for the small contribution of the axial-vector current to pure vector transitions as well as QCD loop effects. Similarly the correction A R is required to correct for the influence of the vector current in pure axial-vector decays. The factor of 3 in front of the λ 2 term is obtained from summing over all possible spin orientiations for the Gamow-Teller matrix element. Although there are no corrections required for nuclear structure dependencies, the f t value for the case of neutron decay requires experimental determinations of the statistical rate function f, the neutron mean lifetime τ n, and the asymmetry parameter λ. Using the recommended values of λ = (28) [2], the neutron half-life T 1/2 = 885.7(8) s [4], and the phase space integral evaluated for neutron decay f(1 + δ R ) = (15) [50, 51] one obtains, G V ( c) 3(n) = (52) ft(21) V R (208) λ 10 5 GeV 2, = (22) 10 5 GeV 2, (1.55) for the vector coupling constant derived from free neutron decay, where the first uncertainty is obtained from the experimental ft value, the second is due to the 32

51 uncertainty in the radiative correction V R, and the third is from the experimental asymmetry parameter λ. This result agrees with the value of G V obtained from the superallowed decays (Eqn. 1.50) but is 7 times less precise. As the overall uncertainty in the G V deduced from neutron β decay is presently limited by experimental measurements of the asymmetry parameter λ, new and more precise measurements are warranted before a true comparison can be made to the value deduced from the superallowed decays. Furthermore, a recent measurement of the neutron mean lifetime τ n = 878.5(8) s [52] disagrees with the world average by nearly 9 statistical standard deviations. Although the Particle Data Group has simply rejected this measurement on the basis of its departure from the average, it should be noted that the world average is itself dominated by the single measurement of Ref. [53]. With the lack of precision in the asymmetry parameter λ combined with the recent ambiguities in the neutron half-life, extraction of G V from neutron decay, at least for the time being, cannot compete with superallowed nuclear β decay. 1.5 Pion Beta Decay A third method to extract the weak vector coupling constant G V is from pion β decay, π + π 0 + e + + ν e. (1.56) Pion decay is, like neutron decay, free of the nuclear structure dependent corrections that limit the uncertainty in the superallowed Ft values. Furthermore it is a 0 0 pure vector decay and there is therefore no axial-vector to vector ratio term λ which is the present limitation in extracting G V precisely from neutron decay. However, because the branching ratio for this decay mode is on the order of 10 8, extraction of G V from pion decay suffers from severe statistical limitations. The ft value for 33

52 pion decay can be expressed as [2, 49, 50]: ft(1 + δ R) = K 2G 2 (1.57) V (1 + V R ). Substitution of the radiative correction V R from Eqn. 1.38, the constant K from Eqn. 1.49, the value of the statistical rate function evaluated for the case of the pion f(1+δ R ) = (28) [50], the pion mean lifetime τ π = (5) 10 8 s [4], and the most precise measurement of the branching ratio BR = 1.040(6) 10 8 [54] yields, G V ( c) 3(π) = (91) ft(21) R (350) BR 10 5 GeV 2, = (36) 10 5 GeV 2, (1.58) for the vector coupling constant. This value agrees with, but is an order of magnitude less precise than, the value deduced from the superallowed Fermi β decays and is presently limited by the overall precision in the branching ratio. Although the vector coupling constant G V can be extracted by measurements of either nuclear, neutron, or pion β decays, superallowed Fermi β decays presently yield the most precise method. The weighted average of G V deduced by all three of these methods differs from the superallowed value by only 0.002%. It should be noted, however, that the precision in the superallowed result is presently limited by theoretical uncertainties whereas the neutron and pion decay results are limited by significant experimental difficulties. With a further reduction anticipated in the nucleus-independent correction V R, the systematic differences between the nuclearstructure dependent isospin symmetry breaking corrections must be resolved in order to further constrain the value of G V. Eventually, however, because V R is common to all three β decays this calculated correction may ultimately limit the overall precision with which G V can be determined by any method. 34

53 1.6 CKM Quark Mixing Matrix The ratio of the vector coupling constant G V to the Fermi coupling constant from purely leptonic muon decay G F defines the up-down element of the Cabibbo-Kobayashi- Maskawa (CKM) quark mixing matrix, a 3 3 matrix that relates the weak interaction quark eigenstates (d, s, b ) to the mass eigenstates (d, s, b) through the relation: V ud V us V ub d = V cd V cs V cb s. (1.59) V td V ts V tb b d s b In the Standard Model, the CKM matrix is a unitary transformation in three dimensional quark flavour space. One condition of unitarity is that the sum of the squares of the elements of any row or column must equal unity. The most precise test to date comes from the top row for which V ud, deduced from superallowed Fermi β decay, is the dominant term. With the Fermi coupling constant G F /( c) 3 = (1) 10 5 GeV 2 [4] from muon decay, and the value for the vector coupling constant G V deduced from superallowed decays (Eqn. 1.50) one obtains, V ud = G V G F = (12) Ft (14) δc (18) V R, = (26), (1.60) for the up-down element of the CKM matrix. From the vector coupling constants deduced from free neutron and pion decay given by Eqns and 1.58, respectively the corresponding values for V ud are, V ud (n) = G V (n) G F = (44) ft (18) V R (178) λ, = (18), (1.61) V ud (π) = G V (π) G F = (78) ft (18) V R (300) BR, = (31), (1.62) 35

54 which are less precise than, but in agreement with, the value of V ud deduced from superallowed Fermi β decay. The value of V ud must be combined with the values of V us = (21) [4], which is determined from semi-leptonic neutral and charged Kl 3 decays, K 0 π l + ν l and K + π 0 l + ν l, where the leptons l are electrons or muons, and V ub = (30) [4] whose value is small enough that when squared ( V ub ) is entirely negligible in the unitarity sum. Combining the value of V ud deduced from the superallowed decays (Eqn. 1.60) with the presently accepted world averages of V us and V ub, the unitarity test of the top row of the CKM quark mixing matrix yields, V ud 2 + V us 2 + V ub 2 = (5) Vud (9) Vus, = (11), (1.63) which satisfies unitarity at the level of 0.7 standard deviations. The unitarity test can also be performed using the values of V ud deduced from neutron and pion decays and the results, V ud (n) 2 + V us 2 + V ub 2 = (36) Vud (9) Vus, = (37), (1.64) V ud (π) 2 + V us 2 + V ub 2 = (60) Vud (9) Vus, = (61), (1.65) are both evidently in agreement with unity at the quoted level of precision. A summary of the values of V ud and the corresponding tests of CKM unitarity obtained in this analysis for all three β decay types is listed in Table 1.4. The present value of V us which dominates the uncertainty in the unitarity sum of the top row of CKM matrix elements (when G V deduced from the superallowed 36

55 Table 1.4: Vector coupling constants G V, up-down CKM matrix elements V ud, and corresponding tests of top-row CKM unitarity deduced from nuclear, neutron, and pion β decays. Decay G V 10 5 V ud V ui 2 GeV 2 i Nuclear (30) (26) (11) Neutron (22) (18) (37) Pion (36) (31) (61) data is used in the unitarity sum) has been under intense scrutiny over the past three years. In the 2004 evaluation of the Particle Data Group the adopted value of V us was V us = (26) [55] and, with the present analysis of V ud from the superallowed decays, would lead to a violation of the unitarity test i V ui 2 = ± by nearly 3 standard deviations. This apparent discrepancy prompted new measurements of Kl 3 decays [56, 57, 58], which were all consistent with each other, and have led to a 2.6% increase to the value V us = (21) that has now been adopted in the 2006 evaluation [4]. Although this new value apparently restores unitarity, a theoretical correction to V us to account for SU(3) quark-flavour symmetry breaking (similar to the SU(2) isospin symmetry breaking δ C correction in nuclear β decay) is required. The presently accepted calculated value for this symmetry breaking in the semi-leptonic Kl 3 decays is given by, f + (0) = ± [59], a calculation from 1984 that has been confirmed by a recent calculation performed in 2005 [60]. However, other calculations in the last 4 years obtained f + (0) = ± [61, 62]. If the latter calculations are accepted the value of V us would be decreased to the V us = ± and, when combined with the value of V ud from nuclear β decay, would lead to i V ui 2 = ± , again violating unitarity at 1.6 standard deviations. Although all three of the modern experimental measurements of V us are 37

56 in agreement, the calculation of f + (0) remains the dominant source of uncertainty in V us and which calculation is to be adopted has not yet been resolved. If the value of V ub = ± [4] is removed from the above calculations of the unitarity sums, the corresponding values listed in Table 1.4 would only be reduced by The up-bottom element is therefore insignificant at the present level of 0.1% precision. A deviation from CKM unitarity would have the important consequence of requiring new physics, such as an unknown fourth quark generation, right-handed weak interactions, or additional interactions types (scalar, psuedoscalar, or tensor interactions). Although the present data seem to suggest that unitarity is satisfied, further improvements in the precision of V ud, which is a by-far the most precisely determined CKM matrix element, clearly remain desirable. Independent of the extraction of V ud, the superallowed F t values provide the most stringent tests of the CVC hypothesis and set strict limits on the existence of scalar currents in the Standard Model. With the ambiguities in the neutron half-life and with no significant improvements to the dominant experimental uncertainties of the neutron asymmetry parameter λ and the pion branching ratio expected in the immediate future, the best chance for improving these demanding tests of the Standard Model rests with the superallowed decays. 1.7 Summary Improving the overall precision in V ud is a major pursuit in experimental nuclear physics. Experimentally this can be most readily accomplished through high-precision measurements of the f t values for superallowed Fermi β decays between isobaric analogue states. However, because the systematic discrepancy between two independent 38

57 calculations of the isospin symmetry breaking corrections is anticipated to soon dominate the overall uncertainty in G V deduced from the superallowed data, the first experimental objective is to attempt to discriminate between the two models employed. Recent work has therefore focused on the T z = 0 superallowed emitters such as 62 Ga [6, 10, 37, 103] and 74 Rb [7, 11, 64] in the A 62 region where large δ C corrections (> 1%) are predicted. Similar tests of the isospin symmetry breaking corrections can be achieved through measurements of the T z = 1 superallowed decays in the 14 A 42 region, where the discrepancy between the δ C calculations is either enhanced ( 14 O, 30 S, 34 Ar, see Fig. 1.3) or, in certain cases ( 18 Ne, 42 Ti), the Woods-Saxon calculations are anomalously large while the Hartree-Fock calculations are not available. Although the experimental f t values are the smallest contributor to the overall uncertainty in V ud, expanding the number of precisely determined superallowed Ft values will continue to have a significant impact. Of the thirteen most precisely determined Ft values only 4 are T z = 1 superallowed emitters ( 10 C, 14 O, 22 Mg, and 34 Ar) and, with the exception of 14 O, have all been added in the past 12 years. As the T z = 1 decays are further from stability than the well-known T z = 0 cases, in general have unstable daughters, and exhibit multiple Gamow-Teller decay branches, high precision measurements represent a considerable experimental challenge. Novel experimental techniques are, however, being developed to deal with these difficulties. In this thesis, a new method has been developed for the measurement of superallowed β decay half-lives to high-precision ( 0.05%) using the technique of γ-ray photopeak counting. Although γ-ray photopeak counting has been used to measure nuclear half-lives in the past, there are relatively few high-precision measurements available due to the systematic effects of detector pulse pile-up. Half-lives of many of 39

58 the T z = 0 superallowed emitters with A 62 have already been determined to the required level of precision through the technique of β counting which is favoured for these decays because there is generally no β contamination from the stable T z = +1 daughters. The half-life determinations for the T z = 1 superallowed emitters, however, have proven to be a considerable experimental challenge because the daughters are radioactive and yield unwanted grown-in and decay contamination in the resulting β-decay time spectrum. These problems are compounded further if the initial samples cannot be produced free of other radioactive species which can introduce further time-dependent β activities to the composite β-decay spectrum. A review of the experimental facilities employed during this thesis work is presented in Chapter 2. In Chapter 3 an introduction to detector pulse pile-up in γ-ray counting experiments is presented and leads to the first quantitative description of these effects to the level of precision (0.05%) required by the superallowed Fermi β decay program. This technique is demonstrated in Chapter 4 with a half-life determination of 26 Na using the 8π γ-ray spectrometer at TRIUMF s ISAC facility. In Chapter 5 the half-life determination of 18 Ne, the first high-precision superallowed half-life measurement via γ-ray photopeak counting is presented. In Chapter 6, an introduction to A 62 superallowed decays is discussed in the context of a high-precision half-life measurement for 62 Ga using the method of direct β counting. This work has led to the most precise superallowed half-life measurement ever reported, and makes important contributions to constraining the systematic discrepancy between two independent calculations of the isospin symmetry breaking corrections to the experimental f t values. Conclusions and future work are discussed in Chapter 7. 40

59 Chapter 2 Experimental Facilities 2.1 Isotope Separator and Accelerator (ISAC) The ISAC facility at the Tri-Universities Meson Facility (TRIUMF) in Vancouver B.C., is one of the premier experimental facilities in the world to conduct research with radioactive ion beams [65]. The primary driver is the TRIUMF 520 MeV cyclotron [66] which provides intense beams of up to 100µA of protons to ISAC, in addition to simultaneous secondary beams of π mesons and muons, for use in a variety of multidisciplinary experiments in chemistry, solid-state physics, biology, medicine, and nuclear physics. A photograph of the cyclotron, taken in the spring of 1972 during its construction, is shown in Fig A unique feature of this cyclotron is the acceleration of negative H ions. The main advantage for the use of H ions is the increased proton extraction efficiency which is simplified once the negative ions are passed through thin carbon foils which strip the two atomic electrons leaving positively charged protons that bend in the opposite direction in the applied magnetic field. This high extraction efficiency permits a continuous delivery of proton beams simultaneously delivered to multiple experimental stations at independent energies 41

60 Figure 2.1: The TRIUMF 520 MeV H cyclotron and staff during its construction in the spring of 1972 (Courtesy of TRIUMF). and intensities [66]. The ISAC radioactive ion-beam facility is located on a dedicated 100 µa beam line from the TRIUMF main cyclotron and has been delivering radioactive ion beams since 1998 [67]. The 500 MeV protons bombard thick layered-foil targets and produce radioisotopes through spallation of the target nuclei. The reaction products diffuse to the target surface and are ionized in a coupled ion source. The target and ion-source modules are housed 2 floors below the experimental hall and are encased within 2 m of steel and between 2 and 4 m of concrete shielding [67]. To date, various ion sources at ISAC have been employed to provide radioactive ion beams to the experimental facilities and include surface ion sources [68, 69], a 2.45 GHz electroncyclotron-resonance (ECR) source [70], the TRIUMF-ISAC Laser Ionization Source (TRILIS) [71, 72], and a Forced Electron Beam-Induced Arc Discharge (FEBIAD) ion 42

61 source [70]. The ionized reaction products are removed from the ion source with an electric field and are transported to a mass separator. The mass separator at ISAC is a 45 degree magnetic bend between two electrostatic quadrupoles [65, 68]. Ions are deflected in circular orbits based on their mass-to-charge ratio according to the classical expression, r = 1 B 2m V, (2.1) q where r is the radius of the circular orbit, B is the applied magnetic field in the separator, m is the mass of the particular ion (which can be alternatively expressed as the atomic mass number A), q is the ionic charge, and V is the voltage difference between the ion source and the mass separator. At ISAC the voltage difference is V = kv and thus singly charged 1 + ion beams have energies between 30 and 60 kev. Following the mass separator, the beam passes through a set of adjustable slits that are tuned to the radius of the particular singly-charged ion of interest. The mass separator at ISAC is generally operated in a low-resolution, high-transmission mode with resolving power m/m 1/1000. While able to distinguish between neighbouring isotopes (with different mass numbers A), this mode does generally leave the possibility for isobaric (and molecular) contamination within a given mass number. In the experiments that will be discussed in the following chapters of this thesis, careful attention was paid to isobaric contamination due to 26m Al in the 26 Na beam, 18 F and H 17 F in the 18 Ne beam, and 62 Cu, 62m Co, and 62g Co in the 62 Ga beam. Although, in principle, any atom (up to and including the target materials atomic mass) can be produced from the target and ionized in the coupled ion source, specific combinations of target material and ion source are more efficient at ionizing particular elements and this efficiency depends primarily on the elemental chemistry. For example, a surface ionization source at ISAC uses a rhenium filament heated to 2400 K [68] 43

62 Figure 2.2: Schematic representation of the ISAC experimental hall. The 8π γ- ray spectrometer and the SCEPTAR array of plastic scintillators, in addition to a 4π gas proportional counter and fast tape transport system (located under the TITAN platform), are the major experimental facilities used in this work (Courtesy of TRIUMF). which provides sufficient energy to ionize the alkali metal species such as Li, Na, K, Rb, and Cs. For noble gas beams, a 2.45 GHz ECR source was developed [70] which provided the necessary higher ionization energies for the noble gas species such as Ne, Ar, Kr, Xe, and Rn. The TRILIS laser ion source uses one laser tuned to a specific frequency of an atomic transition in the element of interest, followed by ionization to the continuum with a second laser and thus particular elements can be selectively ionized through resonant excitations [71, 72]. For the 62 Ga beams discussed in Chapter 6 of this thesis, the technique of resonant ionization improved the 62 Ga yield by a factor of 40 over surface ionization. Orders of magnitude changes in radioactive ionbeam yields can therefore be realized by choosing suitable combinations of production target and ion source. 44

63 Following ionization and mass separation, high-intensity radioactive beams are delivered to several experimental facilities in the ISAC hall. A schematic layout of the ISAC facility is depicted in Fig From the yield station, beams of kev can either be sent directly to several low-energy experiments such as the 8π spectrometer, or they can be accelerated (if the mass to charge ratio is A/q 30) to energies of up to 1.7 MeV per nucleon through a radio frequency quadrupole (RFQ) and drift-tube linear accelerators (DTL) for use in nuclear astrophysics experiments. Beams can also be sent to the ISAC-II facility, a recent upgrade to ISAC, that can accelerate beams to even higher energies of 6.5 MeV per nucleon. These energies are above the Coulomb barrier for many targets allowing for a variety of nuclear reaction experiments [73]. The three main experimental facilities employed in this thesis are a spherical array of 20 Compton-suppressed high-purity germanium (HPGe) detectors called the 8π γ-ray spectrometer. Located inside the 8π spectrometer and surrounding the target chamber is an array of 20 plastic scintillators called the Scintillating Electron Positron Tagging Array (SCEPTAR). For high-precision half-life experiments, a 4π gas proportional counter and fast tape transport system was utilized and is located under the TITAN platform in Fig. 2.2 at the general purpose station (GPS). The calibration and use of these instruments for performing high-precision half-life measurements for superallowed Fermi β decays form the basis of this thesis. 2.2 The 8π γ-ray Spectrometer The 8π γ-ray spectrometer [74, 75, 76] is a spherically symmetric array consisting of 20 Compton-suppressed HPGe detectors covering approximately 13% of the 4π solid angle. Each cylindrical HPGe crystal has a diameter of 5.3 cm and has a nominal 45

64 relative efficiency of 25% compared to that of a 7.5 cm 7.5 cm NaI crystal. The full array has an absolute photopeak efficiency of approximately 1.0% at 1.3 MeV. Photographs of the 8π spectrometer are shown in Fig The geometry of the 8π is that of a truncated icosahedron (soccer ball) with the 20 HPGe detectors located in the positions of the 20 hexagons. Each detector is equipped with tungsten heavy-metal (HM) collimators to prevent γ rays from directly striking the bismuth germanate (BGO) Compton-suppression shields. The collimators are covered with 2 cm thick plastic (delrin) to minimize bremsstrahlung production from high-energy β particles (up to 7.5 MeV for 26 Na β decay, see Chapter 4). A schematic of a single 8π detector is shown in Fig The data aquisition system of the 8π spectrometer has been upgraded [73] in order to provide event-by-event time stamping for every γ ray, an essential component in tracking and correcting the effects of detector pulse pile-up to high precision. Two identical signals from the HPGe detector preamplifiers are used. The energy signal is amplified by an Ortec 572 spectroscopy amplifier with shaping times adjusted between 1.0 and 6.0 µs. The outputs from the amplifiers are digitized with Ortec AD bit analogue-to-digital converters (ADC s). The ADC s are peak sensing, and are operated in a mode that yields zero-suppressed, non-overflow-suppressed data. The second signal from the preamplifier (the time signal) is amplified by a timing-filter amplifier and discriminated with Ortec 583b constant-fraction discriminators (CFD s). The fast outputs from the CFD s are subsequently used as input into LeCroy 4516 logic modules. One of the outputs from the logic modules is sent to a LeCroy channel multi-hit time-to-digital converter (TDC) that provides intra-event nanosecond timing of the γ-rays. A common stop to the TDC is derived from the delayed master trigger. Other elements of the time signal include separate 46

65 Figure 2.3: The 8π γ-ray spectrometer at TRIUMF-ISAC. (Top) An aerial view of the 20 HPGe detector array as seen from an overhead maintenance crane (courtesy of M.R. Pearson). (Bottom) One hemisphere (10 detectors) of the 8π γ-ray spectrometer. Two centimetre thick plastic (delrin) covers each detector and minimizes bremsstrahlung production from high energy β particles. 47

66 Figure 2.4: Schematic diagram of a single 8π germanium detector. LeCroy 3377 TDC s that are used for (i) a BGO time to provide Compton-suppression and (ii) a pile-up indicator that uses the inhibit outputs from the spectroscopy amplifiers. The presence of one or more hits in the pile-up TDC with a corresponding time in the TDC of the same HPGe detector is used to indicate that the event recorded by that detector was piled-up. The final crucial element in the aquisition system is a LeCroy 2367 universal logic module (ULM) which consists of three 32-bit latching scalars. The first scalar counts pulses from a Stanford Research Systems high-precision 10 MHz ± 0.1 Hz temperature-stabilized oscillator providing a global time stamp for every event. The second scalar counts pulses from the same 10 MHz oscillator that remain after a veto by the aquisition dead time which also veto s the 48

67 master trigger for the same period of time after each trigger event, and the third counts the number of master triggers. In this way the dead time can be calculated on an event-by-event basis using differences in the changes of the recorded number of clock ticks between the first two scalars in the ULM. When rates permit, the array can be operated in γ-singles mode where every event recorded by the HPGe, BGO suppressors, and pile-up TDC s are written to disk and all gating, suppression, coincidence timing, and pile-up rejection windows are set in software during the offline analysis to provide the maximum degree of flexibility. 2.3 The SCEPTAR array The Scintillating Electron Positron Tagging Array (SCEPTAR) is an array of 20 plastic scintillators covering approximately 80% of the 4π solid angle. The detectors are arranged into two rings of five trapezoidal pieces and two rings of five rectangular pieces and are positioned so that one plastic scintillator covers the same solid angle subtended by one of the HPGe detectors of the 8π [74]. Each plastic scintillator is 1.6 mm in thickness and thus β particles in excess of 500 kev are not stopped in the array and SCEPTAR therefore primarily acts as a E detector. Light is collected from the scintillator edges and transported using light guides of approximately 25 cm in length to the the photo-multiplier tubes that are located outside of the 8π array. The photomultiplier tubes for the upstream half of SCEPTAR are visible in Fig In spectroscopy experiments, coincidences between β particles recorded in SCEPTAR and γ rays registered in the HPGe detectors of the 8π provide a powerful tool to suppress room background allowing measurements of γ-ray transitions with beam rates as low as 2 ions per second [78]. Other experiments have been performed that have used 49

68 Figure 2.5: The SCEPTAR array at TRIUMF-ISAC. This picture shows the upstream hemisphere (10 detectors) of SCEPTAR which is located inside the 8π spectrometer and surrounds the beam implantation location. the γ-ray activity in coincidence with the β activity to measure β decay branches as low as 10 5 [10]. In this thesis, SCEPTAR was used solely to provide measurements of possible isobaric beam contaminants in the 18 Ne half-life determination presented in Chapter 5. The use of SCEPTAR as a β counter for high-precision half-life studies is also being investigated. A signal from each of the 20 SCEPTAR photomultiplier tubes is fed into a fast Phillips Scientific 776 preamplifier that provides a factor of 10 amplification of the pulse height. One of the outputs of the preamplifier is delayed by 720 ns and input into a LeCroy 4300 fast encoding read-out amplifier (FERA) for charge-to-digital conversion. The other output of the preamplifier is used for timing, and is sent to an Ortec 935 CFD. The CFD outputs are fed into individual channels of a 32-channel 50

69 input SIS3801 virtual machine environment (VME) multi-channel scaler (MCS), and into CAEN 894 discriminators which are then sent to 32-channel multi-hit LeCroy 3377 TDC s. As with the germanium data stream, a second LeCroy 2367 ULM is used to provide a global time stamp for every β event. In most applications, due primarily to the large solid angle subtended by the array, SCEPTAR is operated in a scaled-down β singles mode, where only a small fraction (typically 1 in 10) of the singles events are written to disk while the MCS s multiscale all singles events in addition to β-γ coincidence events individually. Both the SCEPTAR and 8π triggers are input into LeCroy 2365 octal logic units that can be programmed to handle the master trigger logic. Considering only the 8π and SCEPTAR arrays, users can choose to collect experimental data using γ singles (or scaled γ singles), β singles (or scaled β singles), β-γ coincidences, γ-γ coincidences, or any combination of these simultaneously. Trigger selections can be changed through input to the data aquisition computer interface. 2.4 Moving Tape Collector System Another recent addition to the 8π spectrometer was the installation of a moving continuous-loop (approximately 120 m in length) in-vacuum tape collector system [73, 74, 76, 77]. The radioactive ion beam is implanted in the centre of the 8π target chamber on a 40 µm thick mylar-backed iron-oxide collector tape as shown in Fig Beam pulsing and tape movement intervals are controlled through a Jorway controller in CAMAC (computer automated management and control databus). The Jorway is controlled from the main aquisition computer where tape cycle times, dwell times, tape movement length, and beam on and off times can all be modified at the aquisition 51

70 Figure 2.6: (Top) Photograph of the downstream half of SCEPTAR (10 detectors) with the moving collector tape passing through the centre of the target chamber. (Bottom) Longer-lived activities can be moved outside of the 8π array where the collector tape is stored in a loose pile in a shielded box. 52

71 computer terminal. In the 26 Na experiment discussed in Chapter 4 for example, tape cycling times of were used which corresponded to counting background activity for 1.0 s before turning the beam on for 1.0 s to build up a sample of 26 Na on the tape. The beam was then turned off and the decay of 26 Na was counted for 30.0 s ( 30 halflives). This was followed by a 1.0 s delay while the tape was moved bringing a fresh section of tape into the array while simultaneously removing any unwanted longlived contaminant activities that may have been present in the beam. The unwanted long-lived activities are stored outside the array, behind a lead wall, in a shielded tape-storage box shown in Fig To avoid unnecessary tension and unwanted tape breaks (which can only be fixed by breaking vacuum), the tape is not held on spools but rather is loosely piled as shown in Fig For the 18 Ne experiment discussed in Chapter 5 of this thesis, the tape system proved invaluable as the 18 Ne half-life is only T 1/2 = (19) s [79], while the daughter, 18 F, has a half-life of T 1/2 = (5) minutes [80]. Without the ability to cycle the tape after the decay of every 18 Ne sample, the 18 F decay activity inside the target chamber would have continually increased throughout the experiment. 2.5 General Purpose Station (GPS) High-precision half-life measurements using the technique of direct β counting are performed using a 4π gas proportional counter and fast tape transport system. A photograph and a schematic representation of the apparatus are shown in Figs. 2.7 and 2.8, respectively. Radioactive beams are implanted onto a 25 mm wide aluminized mylar tape (denoted by a solid circle in Fig 2.8) under vacuum. The beam 53

72 Figure 2.7: The GPS 4π gas counter and fast tape transport system. Figure 2.8: Schematic of the 4π gas counter and fast tape transport system. 54