The Investigation of the β Decay of 46 K: Detailed Spectroscopy of the Low-Lying Structure of 46 Ca with the GRIFFIN Spectrometer

Transcription

1 The Investigation of the β Decay of 46 K: Detailed Spectroscopy of the Low-Lying Structure of 46 Ca with the GRIFFIN Spectrometer by Jennifer Louise Pore M.Sc., Simon Fraser University, 2013 B.Sc., Mills College, 2010 Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Chemistry Faculty of Science Jennifer Louise Pore 2016 SIMON FRASER UNIVERSITY Fall 2016 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, education, satire, parody, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately.

2 Approval Name: Degree: Title: Jennifer Louise Pore Doctor of Philosophy (Chemistry) The Investigation of the β Decay of 46 K: Detailed Spectroscopy of the Low-Lying Structure of 46 Ca with the GRIFFIN Spectrometer Examining Committee: Chair: Dr. Robert A. Britton Professor Dr. Corina Andreoiu Senior Supervisor Associate Professor Dr. Zuo-Guang Ye Supervisor Professor Dr. Adam Garnsworthy Supervisor Research Scientist TRIUMF Dr. Peter Kunz Supervisor Adjunct Professor Dr. Paul Schaffer Internal Examiner Adjunct Professor Dr. Patrick H. Regan External Examiner Professor Department of Physics University of Surrey Date Defended: November 25, 2016 ii

3 Abstract The calcium region is currently a new frontier for modern shell model calculations, and detailed experimental data from these nuclei is critical for a comprehensive understanding of the region. Due to its very low natural abundance of 0.004%, the structure of the magic nucleus 46 Ca has not been studied in great detail. Some excited states were previously identified by various reaction mechanisms, and few gamma rays were placed in the level scheme from results of beta-decay experiments equipped with limited detection capabilities. A high-statistics data set of the beta decay of the 46 K 2 ground state into the excited states of 46 Ca was measured with the GRIFFIN spectrometer located at TRIUMF-ISAC in December of A radioactive beam consisting almost entirely of 46 K was implanted at the center of the GRIFFIN array, and the emitted gamma rays were detected by 15 high-purity germanium clover detectors. From forty hours of data collection, 430 million gamma-gamma coincidences were observed and analysed to construct the 46 Ca level scheme. In total, 194 gamma rays were identified and placed into the level scheme; 150 of these transitions were observed for the first time. Angular correlations between pairs of gamma rays were analysed to investigate the spin assignments of the observed excited states. Correlations were investigated for 18 of the 42 observed excited states, and it was possible to confirm 7 previously reported spin assignments, and assign 3 new spins of 3, 2, and 3 for the 4435, 5052, and 5535 kev states, respectively. The measured half-life of the 96.41(10) s is in agreement with previous results. From the observed beta feeding intensities of this work, it is suggested that the 46 K 2 ground state may contain more πs 1/2 character than has been previously believed. This is due to the strong population of the 5052 kev 2 state and the absence of observed feeding to the 46 Ca ground state. Keywords: nuclear structure, radioactive decay, radiation detectors iii

4 Dedication for Mom and Dad, love you. iv

5 Acknowledgements I have put all of myself into this program over the past few years. There have been many low moments, but those have always been overshadowed by highs. So many people have kept me on track and focused when times were tough, and without them I never would have been able to complete my PhD. First, I would like to acknowledge my supervisor, Dr. Corina Andreoiu; I will always consider you a life-long friend. You have taught me so much about how to navigate myself through this field, and I greatly appreciate it. You are still fabulous! I would also like to thank my supervisory committee members, Dr. Adam Garnsworthy, Dr. Peter Kunz, and Dr. Zuo-Guang Ye. Unlike so many other graduate students I know, I was never nervous about an upcoming committee meeting. Your feedback was always constructive and positive. Thank you for taking a genuine interest in my work and encouraging me to stay in science. Thank you to all of people that are apart of the Gamma-Ray Spectroscopy (GRSI) group at TRIUMF. In particular I have to thank Dr. Jenna Smith. I learnt a lot from you and it was a pleasure to work so closely with you during the analysis of our data sets. Graduate school would not have been the same without my fellow nuclear science graduate students at SFU. From extended-celebratory lunch breaks to impromptu-ranting coffee breaks, I think that I have made some life-long friends; Fatima Garcia, Aaron Chester, Thomas Domingo, Jonathan Williams, and David Cross, thank you for taking this journey with me. I can t wait until we all meet up again at some conference somewhere in the future to reminisce about how crazy life was when we were in grad school, but how worth it each moment was. This degree has been a long time in the making. My family has always encouraged me and supported me through every decision I have ever made. Even when it meant picking up and moving to Canada on a whim. Mom, Dad, and Bob, I love you. We did this, it s amazing. To Doug (and the cats), we survived! I love you, and our future looks oh-so bright. v

6 Table of Contents Approval Abstract Acknowledgements Table of Contents List of Tables List of Figures ii iii v vi ix xiii 1 Introduction The Shell Model The Atomic Shell Model The Nuclear Shell Model The Evolution of Shell Structure in Nuclei Modern Nuclear Models The Calcium Isotopes Previous Investigations of 46 Ca Spectroscopy with The GRIFFIN Spectrometer Decay Spectroscopy Beta Decay Gamma Decay Rare-Isotope Beam Production at TRIUMF GRIFFIN and its Ancillary Detectors GRIFFIN HPGe Clover Detectors The Ancillary Detectors Beta Decay Scheme of 46 K Details of the Measurement vi

7 3.2 Gamma-Ray Energy Calibration Calibration for Energies Above MeV Gamma-Ray Efficiency Calibration High-Energy Efficiency Final Absolute Efficiency Fit Parameters Gamma-Ray Time Gates LED "Walk" Correction Gamma-Gamma Coincidence Timing The 46 Ca Level Scheme Determination of Gamma-Ray Intensities Intensities from the Gamma-Singles Data Intensities from the Gamma-Gamma Coincidence Data Absolute Intensity Normalization Beta Feeding Parity Assignments Comparison to Previous Measurements K Half-Life Measurement Previous Measurements Measurement from this Analysis Measurements of Other Gamma-Rays Angular Correlations of Successive Gamma Rays Theoretical Description GRIFFIN Angular Properties and Analysis Methods The kev Gamma-Ray Cascade Event Mixing Technique χ 2 /ν Minimization Analysis Spin Assignments for Excited States in 46 Ca The 2422 kev State The 2575 kev State The 3021 kev State The 3611 kev State The 3638 kev State The 3857 kev State The 4257 kev State The 4386 kev State The 4405 kev State The 4428 kev State The 4432 kev State vii

8 The 4487 kev State The 5052 kev State The 5375 kev State The 5414 kev State The 5535 kev State The 5712 kev State The 6111 kev State The Structure of 46 K Systematics of the Even-Even Ca Isotopes Evolution of the sd-shell in the Potassium Isotopes Shell Model Calculation of 46 Ca Suggestions for Future Work Conclusions 130 Bibliography 131 Appendix A Further Discussion of the Efficiency Calibration 134 A.1 Efficiency of the 60 Co Source A.2 Corrections to the 152 Eu Source A.3 Efficiency of the 152 Eu, 133 Ba and 56 Co Sources viii

9 List of Tables Table 1.1 Table 2.1 Table 2.2 Table 2.3 Table 2.4 Previous measurements of gamma-ray intensities observed following the decay of 46 K Selection rules and logf t values for beta decay transitions of varying forbiddenness, where J shows the allowed change in angular momentum between the decaying state of the parent and the populated state of the daughter, and π shows whether or not a change in parity is allowed [2] Selection rules and multipolarities for different types of gamma-ray transitions Weisskopf estimates for the transition rates (δl) for a given gamma ray with energy E γ of either electric (E) or magnetic character [2] The addback factors (F AB ) as calculated from Eqn for experimental source data taken of 152 Eu, 133 Ba, 60 Co, and 56 Co. The errors on these values are purely statistical Table 3.1 The different cycle conditions used during the measurement of the 46 K beta decay with GRIFFIN Table 3.2 Channel numbers that correspond to the energies taken from [34, 35, 36, 37] of the gamma-ray transitions from 56 Co, 60 Co, 133 Ba, and 152 Eu chosen for the purpose of the energy calibration Table 3.3 Fit parameters obtained from a linear fit of the data presented in Table 3.2 with their associated uncertainties Table 3.4 The difference between the measured (E M ) and accepted (E A ) [38, 39] gamma-ray energies. An additional systematic uncertainty of 0.25 kev is required to make all the measured values consistent with the accepted values of these transitions Table 3.5 Channel numbers for high-energy gamma-ray transitions that have energies that correspond to excited state energies. These excited state energies are from an initial χ 2 -minimization fit of the gamma-ray energies observed below MeV ix

10 Table 3.6 Fit parameters obtained from a second-order polynomial fit of the data presented in Table 3.5 with their associated uncertainties Table 3.7 Energy of the MeV transition determined from the low-energy (A) and high-energy (B) calibrations respectively. To ensure agreement between the two calibrations, a systematic uncertainty of 0.82 kev was adopted for values obtained from the high-energy calibration Table 3.8 Simulated high-energy absolute efficiency (ɛ sim ) data points calculated with the GRIFFIN efficiency calculator for the addback-clover data mode for 15 clover detectors Table 3.9 Parameters for the gamma-ray absolute efficiency fit for Equation 3.4 for the addback-clover data Table 3.10 Fit parameters obtained for the second-order polynomial fit used to describe the error on the efficiency for a given energy Table 3.11 Resultant fit parameters for Eqn These parameters were used to correct the timestamps of low-energy events to overcome the walk effect Table 3.12 There are 194 gamma rays and 42 excited states that were observed in the 46 Ca level scheme from the beta decay of 46 K 2 ground state. These are presented per level, where the intensity (I γ ) of each gamma ray has been determined relative to that of the 1346 kev transition. Absolute intensities can be obtained by multiplying the listed value by (6). The gamma-ray branching ratio (BR γ ) of a given gamma ray is the ratio of the intensity of that gamma ray relative to the intensity of the strongest gamma ray that depopulates the same level. Spinparity assignments (J π ) are from Ref. [24] unless otherwise stated. a spin (J) assignments are the result of the angular correlation analysis presented in Sec b parity (π) assignments suggested from determined logft values as discussed in Sec c spin and parity (J π ) assignments suggested from determined logft values Table 3.13 Determination of N from the intensity (I γ ) of the 1229 kev fit from the gamma-singles data, given the number of counts in the 1229 kev peak (N 1229 ) observed in a gate taken on the 1346 kev gamma ray. The efficiency of GRIFFIN at 1229 (ɛ 1229 ) and 1346 kev (ɛ 1346 ) are also included x

11 Table 3.14 For each of the following pairs of coincident gamma rays, the gammagamma intensity was determined with the gating from below method, where a gate was taken on the draining gamma ray so that the intensity of the feeding gamma ray could be determined. In each case the gamma-gamma intensity is equivalent to the measured intensity of the feeding gamma ray from the gamma-singles data Table 3.15 The total gamma ray intensity observed feeding the 46 Ca ground state from 15 gamma rays Table 3.16 The total gamma-ray intensity that populates and depopulates each excited state in 46 Ca is presented with any unobserved feeding. Any unobserved intensity populating an excited state is attributed to beta feeding from the 46 K 2 ground state Table 3.17 The beta feeding observed in this data set compared to the two previous measurements. In total, 42 excited states were observed and 39 of them are populated by the beta decay of the 46 K 2 ground state. The J π values shown here are those that have been previously reported from Ref. [24] Table 3.18 A comparison of select gamma rays observed in this data set to the three previous measurements. The intensities are reported relative to the 1346 kev transition Table 4.1 Previous measurements of the half-life (T 1/2 ) of the beta decay of 46 K 46 Ca. Currently the Nuclear Data Sheets for A = 46 reports T 1/2 = 105 s [24], as an average of the two earliest measurements from Parsa and Gordon [25] and from Yagi et al. [26] Table 4.2 Resultant fit parameters for Eqn. 4.1 with χ 2 /ν= Table 4.3 The final parameters fit with Eqn. 4.2, with χ 2 =0.97, for the decay of the 1346 kev gamma ray from 300 to 1200 s Table 4.4 Chop analysis performed for the fit of the decay of the 1346 kev gamma ray with varying start times and a constant stop time of 1200 s to look for possible systematic effects associated with the start time of the fit. This analysis was performed with time binned into 1, 2, and 4 s bins. It is clear that the resultant T 1/2 is dependent on the start time of the fit, as it systematically decreases with increasing start time. The data presented here is plotted in Fig. 4.5 for visual inspection of the systematic behaviour of the data xi

12 Table 4.5 Table 4.6 Chop analysis performed for the fit of the decay of the 1346 kev gamma ray with a constant start time of 300 s and a varying stop time to look for possible systematic effects associated with the end time of the fit. This analysis was performed with time binned into 1, 2, and 4 s bins. This shows that there is no systematic behaviour of the resultant T 1/2 associated with the end time chosen for the fit The resultant T 1/2 fit for the strongest gamma ray depopulating each excited state observed in 46 Ca (when statistically possible to fit). The listed numbers (#) correspond each fit to the plot shown in Fig Table 5.1 Table 5.2 The specific angular distribution W mi m f (θ) for emitted radiation between pairs of m-states in Fig Theoretical A 2 and A 4 coefficients used to describe the angular correlation of each listed cascade as per Eqn Table 5.3 The 52 angular indexes (labelled 0 to 51) that exist between the 64 Table 5.4 GRIFFIN HPGe crystals. The angle in degrees and the number of detector pairs (weights) at each angle are listed A summary of the angular correlation analysis from this work. The spin assignments of 18 excited states were investigated by looking at angular correlations from a cascade of two coincidence gamma rays γ 1, and γ 2. There are 12 reported spin assignments (J P ) from previous measurements [24]. The spin assignments (J) reported from this work are shown in comparison. It was assumed that γ 1 was a pure transition with δ 1 = 0, and that γ 2 could potentially be a mixed transition described by δ 2. The experimental data was fit with two different variations of Eqn For the experimental fit, the A 2 and A 4 parameters were allowed to vary. Whereas for the theoretical fit, the A 2 and A 4 parameters were fixed to theoretical values for the particular cascade listed taking into account δ Table 6.1 The predicted amount of the πd 3/2 component contained in the groundstate wavefunction for odd-a K isotopes from shell model calculations [47]. For 47 K and 49 K there is a near degeneracy of the πd 3/2 and πs 1/2 orbitals, therefore those ground states contain a large contribution of the πs 12 orbital. Numbers taken from Ref. [47] Table 6.2 Calculated proton occupancies for the πd 3/2, πs 1/2, πf 7/2, and πp 3/2 orbitals for predicted 2 states in 46 Ca [51]. In this calculation 6 protons were distributed amongst these orbitals. The shown occupation numbers show how many of the 6 protons occupied that particular orbital in that particular level xii

13 List of Figures Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 The experimentally observed ionization energies for atoms. The increased energy required to remove an electron from the noble gas elements indicates that they are particularly stable. This is due to their closed-shell configurations of electrons. This figure has been reprinted with permission from [2] Two-nucleon separation energies across a range of different nucleon configurations. The lowest Z member of each sequence is labelled. (Top) Two-proton separation energies for isotones with constant N. (Bottom) Two-neutron separation energies for sequences of different isotopes. This data shows the difference between the measured values and theoretical predictions. The sudden changes in the smooth variation of the behaviour of the data indicate the presence of the magic numbers. This figure was reprinted with the permission of Ref. [2] Plots of three different nuclear potentials that can be used to derive the energies of the single-particle orbitals for the nuclear shell model. This includes a square-well (V sq ), a harmonic-oscillator (V ho ), and a Woods-Saxon (V W S ) potential. Figure reprinted from Ref. [4] (Left) Nuclear energy levels that were calculated with the Woods- Saxon potential shown in Eqn This form of the potential does not correctly reproduce the experimentally observed magic numbers above 20. (Right) Nuclear energy levels calculated with the inclusion of the spin-orbit interaction. The spin-orbit interaction causes additional splitting of the levels, which exactly reproduces the correct magic numbers. This figure is reprinted with permission from [2].. 6 The nuclear forces as described by chiral effective field theory. Here, the nucleons (solid lines) interact via the exchange of pions (dashedlines). The different contributions at successive orders for each force are diagrammed. The many-body forces are highlighted in gold along with the year of their derivation. This figure is reprinted with permission from Ref. [12] xiii

14 Figure 1.6 (Top) The energy of the excited state for even-even calcium isotopes. The sudden jumps in excitation energy from one neutron Figure 1.7 Figure 1.8 Figure 1.9 number to the next is an indication of a neutron-shell closure. (Bottom) Experimental B(E2: ) values. Figure reprinted with permission from Ref. [5] Different theoretical reproductions of the experimentally observed energy for the state in even-even calcium isotopes. (a) Energies obtained from phenomenological models. (b) Energies obtained using NN-only theory. (c) Energies obtained when the contributions from 3N forces are included with NN. (d) Energies from NN and 3N forces where the single-particle energies for 41 Ca are calculated consistently. Figure reprinted from Ref. [14] A schematic representation of the attractive interaction between the πf 7/2 and νf 5/2 orbital. As protons are removed from π f 7/2 there is less attraction between the two orbitals as is shown in (a)-(d). The πf 7/2 orbital is completely empty for the calcium isotopes, creating two exotic shell closures at N=32 and N=34. Figure reprinted with permission from Ref. [18] Previously reported decay schemes for the β decay of the 2 ground state of 46 K to low-lying states in 46 Ca [24]. There are notable discrepancies present between the two decay schemes. Particularly, there is large variance between the reported energies of the gamma rays and the intensities of the beta feeding. (right) The decay scheme as reported by Parsa and Gordon [25]. (Left) The decay scheme as reported by Yagi et al. [26] Figure 2.1 Figure 2.2 Figure 2.3 An artist s rendition of the ISAC experimental hall located at TRI- UMF in Vancouver, BC. Reprinted with permission The GRIFFIN spectrometer located at TRIUMF-ISAC. When fully populated, the array contains 16 HPGe clover detectors that can be closely packed together for optimal solid angle coverage. GRIFFIN can also be coupled to a suite of ancillary detectors Each GRIFFIN clover detector contains 4 HPGe crystals that are 60 mm in diameter and 90 mm in length. The tapered edges of the crystals allow them to be closely packed together within the array. 27 xiv

15 Figure 2.4 Figure 2.5 Figure 2.6 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 The GRIFFIN spectrometer has a rhombicuboctahedral geometry containing 18 square faces and 8 triangular faces. 16 of the 18 square faces can be occupied by HPGe clover detectors, and the other two square faces are left open to allow for the entry of the beam and for the exit of the tape system. The triangular faces can be used to mount ancillary detection systems The PACES array for the detection of conversion electrons includes 5 lithium-drifted silicon detectors During this experiment, the downstream half of SCEPTAR was installed behind the MYLAR tape. The 10 thin plastic scintillators are used to tag beta particles Linear fit of channel number versus gamma-ray energy data listed in Table Absolute efficiency versus gamma-ray energy using standard calibration sources of 60 Co, 152 Eu, 133 Ba, and 56 Co as well as simulated high-energy data points for the 15 HPGe GRIFFIN detectors present during the time of the experiment. The experimental data points are shown as red circles and the simulated data points are represented by purple squares % confidence interval for the addback-clover absolute efficiency data as determined by IGOR-Pro The maximum percent difference between the calculated 95% confidence interval and the efficiency for energies between 121 kev and 7.5 MeV. This percent difference is equivalent to the percent error on the efficiency. Select data points (filled-red circles) were fit with a second-order polynomial whose parameters are presented in Table The use of leading edge triggering will create a time-walk effect. The signal for a given event does not trigger the data-acquisition system until it passes a set threshold. Therefore, smaller amplitude signals such as B appear to arrive later in time (Top) A plot of the time difference between gamma ray events detected in coincidence with the 1332 kev gamma ray. There is a clear time variance for low-energy events due to the walk effect created by the LED. (Bottom) Walk -corrected time spectrum. The time resolution for these events was reduced from 400 ns to 250 ns. 43 xv

16 Figure 3.7 A plot of the time difference between pairs of gamma rays observed in this data set. The time-random events between 400 and 650 ns have been subtracted from the prompt events between 0 and 250 ns. These good coincidences are presented in Fig Figure 3.8 Time-random background subtracted gamma-gamma matrix constructed with a 250 ns time window. The x and the y axis show the energies of the two gamma rays that are in coincidence, and the z axis shows how many times those gamma rays were observed in coincidence during the experiment Figure 3.9 The gate taken on the 1346 kev gamma ray that show the quantities of the gamma rays observed in coincidence (within 250 ns) with it. The upper spectrum shows the coincident gamma rays with energies between 0 and MeV, and the lower spectrum shows the coincident gamma rays with energies between 4 and 6 MeV. The 4029 kev gamma ray is labelled in both spectra to show the scale of counts of the lower spectrum as compared to the upper spectrum.. 47 Figure 3.10 Gamma-singles spectrum for the beta decay of 46 K into low-lying states of 46 Ca. When possible the observed peak areas for gamma rays present in this spectrum where used to determine gamma-ray intensities. Select peaks are labelled in energies of kev Figure 3.11 A simple level scheme in which a gamma ray (γ 1 ) populates an excited state that is then depopulated by γ 2. A gate taken from below γ 1 on γ 2 would show how many times γ 1 was observed in coincidence with γ Figure 3.12 The beta continuum as observed for the decay of the 46 K 2 ground state as observed with the PACES array. The Q β value for this decay is (16) MeV. The absence of beta intensity beyond 6 MeV indicates that there is no population of the 46 Ca ground state. 60 Figure 4.1 The 46 K beta decay curve as was published by P. Kunz et al. [27]. The residuals indicate that there are no significant dead time distortions at higher count rates. The fit of this decay curve resulted in a reported T 1/2 = (79) s half-life for the beta decay of 46 K 46 Ca. This figure is reprinted with permission of Ref. [27] Figure 4.2 The E γ versus time matrix used to establish the beta-decay T 1/2 of the 46 K 2 ground state xvi

17 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 5.1 Figure 5.2 The decay of the 1346 kev gamma ray is plotted in black with the fit of Eqn. 4.2 shown in yellow. The events shown in red are Comptonscatter background, with the fit of Eqn. 4.1 shown in black. From this fit, T 1/2 = (66) s The associated residuals of the initial fit of the 1346 kev gamma ray decay from 300 to 1200 s given the fit parameters of Eqn. 4.2 listed in Table 4.3. The behaviour of the residual is consistent throughout the boundaries of the fit A visual plot of the chop analysis data for fits with varying start time between 300 and 600 s. There is a clear systematic decrease in the resultant T 1/2 from fits with that are performed with later start times. The fit T 1/2 = s is drawn with the black solid line, with the T 1/ s represented by the purple dashed line. To create agreement between the fit T 1/2 and the resultant fits of the chop analysis, an additional systematic error of s must be taken into account. The total error of 0.10 s is drawn as the blue dashed line The resultant T 1/2 fit for the strongest gamma ray depopulating each excited state observed in 46 Ca (when statistically possible). The data plotted here is listed in Table 4.6. Comparisons of these values to the value of T 1/2 =96.41(10) s, the plotted solid black line, show no indications that there is any contamination or isomeric states present. The dashed blue line represents is the 1σ error of 0.43 s from the weighted average of T 1/2 =96.50(7) s considering all of the fit values A depiction of how the decay J i J f is actually the result of decays between m-states. Each transition between pairs of m-states is emitted with an anisotropic angular distribution, but if all the initial m-states are evenly populated, then only an isotropic distribution of radiation will be observed. Reprinted with permission of Ref. [44].. 75 (Left) An angular correlation measurement of a cascade of successive gamma rays γ 1 and γ 2. The z-axis is defined as the direction of the emission of γ 1, and θ 2 is the direction of detection of γ 2 with respect to the z-axis. (Right) A simple level scheme showing the different m-states that are involved in the decay of the cascade. This figure has been reprinted with permission from Ref. [44] xvii

18 Figure 5.3 A plot of the angular correlation expected to be observed for a gamma ray cascade Figure 5.4 The two-dimensional array of angular index versus gamma-ray energy created when a gate was taken on the 1346 kev gamma ray. The array shows the number of times a gamma ray was detected in coincidence (within a 250 ns time window) with the 1346 kev gamma ray at each of the 51 angular indexes Figure 5.5 A simple partial level scheme showing the direct succession of the and 1346-keV gamma rays in 46 Ca. Note that the level scheme is not to scale Figure 5.6 A zoomed-in version of Fig. 5.4 to show the number of events of the 3706 kev gamma ray in coincidence with the 1346 kev gamma ray at each of the 51 angular indexes Figure 5.7 Plot of the fit peak areas of the 3706 kev gamma ray in coincidence with the 1346 kev gamma ray at each angular index Figure 5.8 The uncorrelated angular distribution between the 3706 and 1346 kev gamma rays created from the event-mixed background. These events are used to normalize the events shown in Fig 5.7, to give the angular correlation presented in Fig Figure 5.9 The measured angular correlation between the successive 3706 and 1346 kev gamma rays for the 51 GRIFFIN angular indexes. The data has been corrected for experimental effects with the use of the event mixing technique, and has been fit with Eqn The error bars on the experimental data points are statistical. This correlation is discussed further in sec Figure 5.10 Theoretically predicted A 2 and A 4 coefficients for a gamma-ray cascade given different values of δ for each gamma ray. Reprinted with permission of Ref. [44] Figure 5.11 The angular correlation for the kev gamma-ray cascade. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table xviii

19 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 The angular correlation for the kev gamma-ray cascade that depopulates the 2575 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 1229 was allowed to vary, and the best fit was achieved for δ 1229 = The angular correlation for the kev gamma-ray cascade that depopulates the 3021 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = for the 2 2 gamma ray. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 1675 was allowed to vary, and the best fit was achieved for δ 1675 = -0.15(8) The angular correlation for the kev gamma-ray cascade that depopulates the 3611 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2264 was allowed to vary, and the best fit was achieved for δ xix

20 Figure 5.18 Figure 5.19 Figure 5.20 Figure 5.21 Figure 5.22 Figure 5.23 The angular correlation for the kev gamma-ray cascade that depopulates the 3638 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = 0.14 for the 2 2 gamma ray. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2292 was allowed to vary, and the best fit was achieved for δ 2292 = 0.14(11) The angular correlation for the kev gamma-ray cascade that depopulates the 3857 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = 0.04 for the 4 2 gamma ray. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2510 was allowed to vary, and the best fit was achieved for δ 2510 = 0.03(15) The angular correlation for the kev gamma-ray cascade that depopulates the 4257 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0.37 for the 1 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0.11 for the 1 2 gamma ray. (c) The fit shown in red is for a cascade with δ = 0.38 for the 3 2 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2911 was allowed to vary. Best fits were achieved for δ 2911 = 0.11(10) for the cascade, and for δ 2911 = 0.38(20) cascade xx

21 Figure 5.24 Figure 5.25 Figure 5.26 Figure 5.27 The angular correlations for cascades of successive gamma rays depopulating the 4386 kev state. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = -0.1 for the 2 3 gamma ray. (b) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = 0.7 for the 3 3 gamma ray. (c) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = 0.02 for the 2 2 gamma ray. (d) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = 0.6 for the 3 2 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev (left) and kev (right) gamma-ray cascades. The values of δ were allowed to vary for the 776 and 3086 kev gamma rays. (Left) Best fits were obtained for values of δ 776 = -0.1(3) for the cascade and δ for the cascade. (Right) Best fits were obtained for values of δ 3040 = 0.02(5) for the cascade and δ for the cascade.. 99 The angular correlation for the kev gamma-ray cascade that depopulates the 4405 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of the kev gammaray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 1830 was allowed to vary, and the best fit was achieved for δ xxi

22 Figure 5.28 Figure 5.29 Figure 5.30 Figure 5.31 Figure 5.32 The angular correlation for the kev gamma-ray cascade that depopulates the 4428 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ -0.3 for the 2 1 gamma ray. (b) The fit shown in red is for a cascade with δ -0.1 for the 2 2 gamma ray. (c) The fit shown in red is for a cascade with δ = -0.4(3) for the 3 2 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 624 was allowed to vary. Best fits were obtained for values of δ for the cascade, δ for the cascade, and δ 624 = -0.4(3) for the cascade The angular correlation for the kev gamma-ray cascade that depopulates the 4432 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ 0.25 for the 2 4 gamma ray. (b) The fit shown in red is for a cascade with δ = 0 for the 3 4 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade that depopulates the 4432 kev level. The value of δ 1858 was allowed to vary, and best fits were obtained for values of δ for the cascade and δ 1858 = 0 for the cascade The angular correlation for the kev gamma-ray cascade that depopulates the 4487 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0.15(10) for the 2 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0.30(15) for the 3 2 gamma ray xxii

23 Figure 5.33 Figure 5.34 Figure 5.35 Figure 5.36 Figure 5.37 A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to kev gamma-ray cascade that depopulates the 4487 kev state. The value of δ 3142 was allowed to vary, and best fits were obtained for values of δ 3142 = 0.15(10) for the cascade and δ 3142 = 0.30(15) for the cascade The angular correlation for the kev gamma-ray cascade that depopulate the 5052 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 2 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0.4(6) for the 4 2 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade that depopulates the 5052 kev level. The value of δ 3706 was allowed to vary, and best fits were obtained for values of δ 3706 = 0 for the cascade and δ 3706 = 0.4(6) for the cascade Angular correlation for gamma-ray cascades that depopulate the 5375 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 3 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0 for the 3 4 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev (left) and kev (right) gamma-ray cascades. The values of δ were allowed to vary for the 4029 and 2800 kev gamma rays. (Left) Best fits were obtained for values of δ for the cascade and δ 4029 = 0.01(10) for the cascade. (Right) Best fits were obtained for values of δ 2800 = 0.2 for the cascade and δ 2800 = 0 for the cascade xxiii

24 Figure 5.38 Figure 5.39 Figure 5.40 Figure 5.41 Figure 5.42 Figure 5.43 Angular correlation for the gamma-ray cascade that depopulates the 5414 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 2 3 gamma ray. (b) The fit shown in red is for a cascade with δ 1 for the 3 3 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 1804 was allowed to vary, and best fits were obtained for values of δ 1804 = 0 for the cascade and δ for the cascade The angular correlation for the kev gamma-ray cascade that depopulates the 5535 kev state. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = 0.05(5) for the 3 2 gamma ray. The resultant fit parameters are listed in Table A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 4189 was allowed to vary, and best fits were obtained for values of δ 4189 = 0.05(5) for the cascade Angular correlation for the gamma-ray cascade that depopulates the 5712 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 2 2 gamma ray. (b) The fit shown in red is for a cascade with δ 0.5 for the 3 2 gamma ray A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 4366 was allowed to vary, and best fits were obtained for values of δ 4366 = 0 for the cascade and δ for the cascade. 118 xxiv

25 Figure 5.44 Figure 5.45 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 4765 was allowed to vary, and best fits were obtained for values of δ 4765 = 0 for the cascade, δ for the cascade, δ 4765 = -0.1(3) for the cascade The angular correlation for the kev gamma-ray cascade that depopulates the 6111 kev state. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 1 2 gamma ray. (b) The fit shown in red is for a cascade with δ 0.75 for the 2 2 gamma ray. (c) The fit shown in red is for a cascade with δ = -0.1(3) for the 3 2 gamma ray Systematics of select low-spin state for the even-even Ca isotopes are presented [24]. These isotopes are each populated from a potassium-parent nucleus of the same mass A. Beta feeding intensities (marked in red) to the ground, first-excited, and most intensely populated higher-lying states are shown. Note that the majority of the beta feeding populates the ground states of Ca, whereas very little feeding is observed to the ground state of 48 Ca, and no feeding is observed to the 46 Ca ground state Experimental energies for the first observed 1/2 + and 3/2 + states in odd-a K isotopes. Note the inversion of spin for 47 K and 49 K. This is due to the evolution of the πsd orbitals with varying energy. This figure is reprinted with permission from Ref. [47] Calculated proton occupancies for the πd 3/2 and πs 1/2 orbitals for K isotopes from shell model calculations using two different effective interactions. For K there are 3 protons in the πd 3/2 orbital and 2 protons in the πs 1/2 orbital. There is a sudden shift in occupation numbers for 47 K when there are 4 protons in the πd 3/2 orbital and 1 proton in the πs 1/2. This figure is reprinted with permission from Ref. [47] The first two excited states are presented for each even-a K isotopes. The first excited state observed for 46 K is at much higher energies than in any of the neighbouring isotopes xxv

26 Figure 6.5 Figure 6.6 This calculation utilized a unique valence space that included the s 1/2 and d 3/2 orbitals from the closed core in an attempt to observe any possible excitations across the Z = 20 shell gap [51] Shell model calculations have been performed for 46 Ca that include NN and 3N forces. Shown here are the excited state calculated with the use of different valence spaces. The calculation presented on the left only used orbitals between f 7/2 and g 9/2 [15], whereas the current calculation uses the valence space shown in Fig. 6.5, which allows for excitations from the s 1/2 and d 3/2 orbitals [51]. These calculations are shown in comparison to current experimental data from this work.129 xxvi

27 Chapter 1 Introduction As humans, we want to understand the world around us. We want to be able to explain not only what we can see with our own eyes, but also phenomena that are either too small or too large for us to observe directly. This desire has sprouted numerous fields of study ranging from the scrutiny of single particles to investigations of entire galaxies. Typically, our initial explanations for anything are based on observations of cause and effect. We assume that every effect must have an associated initial cause, and we expect the same cause to always produce the same effect. Such explanations are referred to as models. In science, models are developed to theoretically explain what is observed experimentally. Effective scientific models should reproduce what has already been measured, as well as accurately predict what has yet to be observed. Models are key to linking theoretical ideas with experimental work, as they can be used to guide and focus experimental research programs for further theoretical understanding of complex ideas. It is the constant feedback between experimental measurements and theoretical models that continues to drive science forward. 1.1 The Shell Model The Atomic Shell Model The atomic shell model was established in 1913 when Niels Bohr suggested that electrons occupied quantized orbitals [1]. He proposed this model to explain how electrons could stably orbit a positively charged nucleus without spiralling and colliding into it. Additionally, the experimentally observed discrete emission spectra of different atoms were explained by the ability of an electron to move from one orbital to another by either emitting or absorbing a quantized amount of radiation. All that was needed to derive the model was to consider the electrostatic interactions that took place between an individual electron with the positively charged nucleus and the negative charges of all of the other electrons present in the atom. The electron orbitals can 1

28 Figure 1.1: The experimentally observed ionization energies for atoms. The increased energy required to remove an electron from the noble gas elements indicates that they are particularly stable. This is due to their closed-shell configurations of electrons. This figure has been reprinted with permission from [2]. then each be defined by a set of quantum numbers. Each orbital has a specified electron occupancy and is filled by electrons of increasing energy under the observation of the Pauli principle, such that no two electrons are defined by the same quantum numbers. The electron orbitals cluster together to form shells separated by energy gaps. Atoms that possess closed-shell electron configurations, referred to as the noble gases, are particularly stable as it requires a lot of energy to either add or remove an electron from the system. This is supported by the experimentally observed large ionization energies for these atoms, as is shown in Fig Typically in atomic structure, atoms are modelled as an inert core made up of filled shells of electrons coupled to some number of active valence electrons. It is well understood that the properties of an atom are dependent on its particular configuration of valence electrons The Nuclear Shell Model The nucleus has proven to be much more difficult to model than the atom. This is primarily due to the self-propagating nature of the nuclear force, which is not as well understood as the Coulomb force. However, there were many experimental indications that shell structure was also present in nuclei. Studies of two-nucleon separation energies reveal how bound nucleons are in a particular nucleus. These separation energies are analogous to the ionization energy 2

29 required to remove an electron from an atom. There was an observed deviation of the measured values from theoretical predictions for nuclei with either N or Z = 2, 8, 28, 50, 82, as is presented in Fig This result revealed that those particular nucleon numbers were more stable than the configurations of the neighbouring nuclei. The smooth variations in the data suggest the gradual filling of a nuclear shell, and any sudden deviation is indicative of a closed-shell system. The experimentally observed special configurations of nucleons became known as magic numbers. It then became desirable to reproduce these numbers theoretically with a nuclear shell model. Ideally, a nuclear shell model would be able to reproduce the experimentally observed magic numbers, and be able to predict other experimental observables that have yet to be measured. However, there was some initial doubt in the idea. It was reasonable to believe that electrons were free to move in defined spatial orbits without colliding with other electrons, but it was difficult for some to accept that the same could be true for nucleons due to their relatively large size compared to the nucleus itself. But, if the assumption is made that nucleons occupy quantized orbitals contained within a potential well, then the Pauli principle can be used to justify the existence of definite spatial orbits. For instance, if two nucleons were to collide within the well, there would be a transfer of energy between them. If such a collision occurred at the bottom of the potential well, with all of the above orbitals already occupied by nucleons, then the only option would be for one of the colliding nucleons to be excited all the way up to the valence level. Clearly, this would take a tremendous amount of energy and would be unlikely to occur. Therefore, it is reasonable to believe that nucleons do not collide with each other and can move freely in spatial orbits [3]. To calculate the energies of the nucleon orbitals it is necessary to choose an appropriate potential and then to solve the Schrödinger equation. It is assumed that the nucleons themselves create an average potential that govern the motion of each single nucleon. Today this model is often referred to as the single-particle shell model solution as it considers each nucleon individually in a fixed potential. This description cannot describe all the different aspects of nuclear structure, but it does give some insight as to the essential ideas of nuclear physics. In a simple approximation of the nuclear shell model, the nuclear potential can be described either as a square well, a harmonic oscillator, or as a Woods-Saxon potential. Each of the three potentials are shown in Fig Both the square well and harmonic oscillator potentials require an infinite amount of energy to remove a nucleon out of the well, which is unreasonable. The Woods-Saxon potential is the most realistic choice of the three as it does not have a sharp edge, but rather smoothly approaches zero beyond the expected average nuclear radius (R). This potential (V (r)) can be written as V (r) = V exp[(r R)/a], (1.1) 3

30 Figure 1.2: Two-nucleon separation energies across a range of different nucleon configurations. The lowest Z member of each sequence is labelled. (Top) Two-proton separation energies for isotones with constant N. (Bottom) Two-neutron separation energies for sequences of different isotopes. This data shows the difference between the measured values and theoretical predictions. The sudden changes in the smooth variation of the behaviour of the data indicate the presence of the magic numbers. This figure was reprinted with the permission of Ref. [2]. 4

31 Figure 1.3: Plots of three different nuclear potentials that can be used to derive the energies of the single-particle orbitals for the nuclear shell model. This includes a square-well (V sq ), a harmonic-oscillator (V ho ), and a Woods-Saxon (V W S ) potential. Figure reprinted from Ref. [4]. where V 0 is the depth of the potential well ( 50 MeV to give the proper separation energies), R is the nuclear radius (R = 1.25A 1/3 fm), and a is the assumed skin thickness of fm. Given this potential V( r ) = V(r), the energies of the nuclear orbitals can be calculated with the time-independent Schrödinger equation, [ 2 2m 2 + V (r)]ψ nlml ( r) = EΨ nlml ( r). (1.2) It is possible to separate Eqn. 1.2 into radial and angular coordinates, such that the solutions for the wavefunctions Ψ nlml are written as Ψ nlml ( r) = Ψ nlml (rθφ) = 1 r R nlψ nlml (θφ). (1.3) Here, n is the radial quantum number, l corresponds to the orbital angular momentum, and m l is the eigenvalue of the z component of l z. For a given value of l, m l takes on the integer values from -l l. By convention, different values of l are labelled l = 0, 1, 2, 3, 4, 5 s, p, d, f, g, h (1.4) The resultant energies of the nuclear orbitals are presented in Fig The degeneracy of each level with orbital angular momentum l is 2(2l + 1), where the (2l + 1) degeneracy originates from the set of m l states that exist for each l state, and the additional factor of 2 comes from two possible values of the spin quantum number m s ± 1/2. Note that these derived orbitals are filled with increasing energy only the magic numbers 2, 8, and 20 are correctly reproduced. There is a further modification to the nuclear potential necessary in order to correctly calculate the higher magic numbers. 5

32 Figure 1.4: (Left) Nuclear energy levels that were calculated with the Woods-Saxon potential shown in Eqn This form of the potential does not correctly reproduce the experimentally observed magic numbers above 20. (Right) Nuclear energy levels calculated with the inclusion of the spin-orbit interaction. The spin-orbit interaction causes additional splitting of the levels, which exactly reproduces the correct magic numbers. This figure is reprinted with permission from [2]. In the atomic shell model, the correct atomic energy levels are only theoretically reproduced when the interaction between an electron s spin and its orbital angular momentum are considered. In atoms, the effect of this spin-orbit interaction is very small, but in nuclei, the effect would be much larger due to the strength of the strong force. The nuclear potential can be modified to take this into account as V (r) V (r) f(r) l s, (1.5) where f(r) is a function of the position of the nucleon and l and s are the orbital angular momentum and nucleon spin, respectively. The spin-orbit interaction contributes the most on the edge region of the nucleus where the nucleon density is sporadic compared to the 6

33 homogeneity of density in the interior of the nucleus. Therefore, f(r) is written as f(r) 1 r so that that a minimal correction is applied for small values of r [1]. dv (r), (1.6) dr States derived from with the inclusion of the spin-orbit interaction are labelled with their total angular momentum j, where j = l + s. For nucleons s = 1/2, so the possible values of j are either l + 1/2 and l 1/2. So, each l state is split into two sub-states corresponding to the two allowed values of j (note that when l = 0 only j = 1/2 is allowed, as j cannot be negative). To determine the magnitude of the splitting, the expectation values of j 2, l 2, and s 2 are <j 2 > = j(j + 1) 2 <l 2 > = l(l + 1) 2 <s 2 > = s(s + 1) 2, (1.7) where j 2 can also be j 2 = l 2 + s l s. (1.8) where, l s can be rewritten as Then the expectation value for < l s> is l s = 1 2 (j2 l 2 s 2 ). (1.9) < l s> = 1 [(j(j + 1) l(l + 1) s(s + 1))]. (1.10) 2 Now, the energy difference between two j states is found to be V 2l + 1. (1.11) by substituting s = 1/2 into Eqn The potential is slightly different for the parallel and anti-parallel orientations of l and s. From Eqn. 1.5 the state j = l+1/2 will be lower in energy than the state j=l - 1/2. The resultant energy levels with their respective occupancies are shown in Fig The additional splitting of the l states, as a result of the inclusion of the spin-orbit interaction, causes a rearrangement of the energy orbitals. Specifically, higher-lying j orbital are brought down in energy to create new shell gaps. The nucleon configurations that correspond to these new shell closures correspond to the experimentally observed magic numbers. 7

34 1.1.3 The Evolution of Shell Structure in Nuclei The nuclear shell model, as described in Sec , is often referred to as the spherical shell model due to the spherical potential chosen in its derivation, or as the independentparticle shell model as it ignores any possible nucleon-nucleon interactions that may occur if more than one valence nucleon is present. This model has been incredibly successful in reproducing the experimentally observed nuclear structure for isotopes that exist at stability, especially near closed-shell configurations. These nuclei are typically spherical and have a well-balanced number of protons and neutrons. However, as recent experimental advances have allowed for the exploration of nuclei that have large excesses of neutrons, it has been discovered that the standard magic numbers do not necessarily persist throughout all nuclei [5]. This was first observed in investigations of the N = 20 shell gap for nuclei far from stability [6, 7, 8, 9]. There is also evidence for the disappearance of the standard shellclosure at N = 28 in 42 Si [10], and new shell closures have emerged, as was found for N = 16 in the exotic oxygen isotopes [11]. In the derivation of the independent-particle shell model, nucleon-nucleon interactions were simply generalized for all nuclei by the use of a Woods-Saxon potential. But, in reality the interactions between nucleons vary drastically from one nucleus to the next. This is primarily due to the radial distribution, angular momentum, and spin orientation of the different nucleon orbitals involved in the interaction. In general, it can be shown that the residual interaction between two coupled nucleons is greatest when there is a large spatial overlap between the involved single-particle orbitals. This means that there is a strong attractive pull between orbitals that have similar n and l values. An example of the attractive residual interaction between neutrons in the νf 5/2 orbital and protons in the πf 7/2 orbital is discussed in Sec The force felt by an individual nucleon is dependent on how many nucleons are in that particular nucleus, and which orbitals those nucleons occupy. Therefore, the nuclear potential of an exotic nucleus can be very different than that of a stable nucleus. If the nuclear potential varies from nucleus to nucleus, then as a natural extension, the energies of the single-particle orbitals will also fluctuate, and the locations of the shell-closures will change. The evolution of shell structure across the nuclear landscape is currently of great interest to both experimental and theoretical nuclear physicists alike. In particular, to understand how robust the standard magic numbers are in exotic nuclei, and to be able to identify and predict potential new shell closures Modern Nuclear Models To address the evolution of shell structure across the nuclear landscape, more realistic models are needed that take into account the variance of nucleon-nucleon interactions from one nucleus to the next. Ideally, a successful model would be able to reproduce the location 8

35 Figure 1.5: The nuclear forces as described by chiral effective field theory. Here, the nucleons (solid lines) interact via the exchange of pions (dashed-lines). The different contributions at successive orders for each force are diagrammed. The many-body forces are highlighted in gold along with the year of their derivation. This figure is reprinted with permission from Ref. [12]. and magnitude of the shell closures for nuclei at stability and for exotic neutron-rich systems, as well as adequately predict new shell closures beyond current experimental limits. There have been recent breakthroughs in the understanding of nuclear forces from chiral Effective Field Theory (EFT), which describes nuclear forces on the resolution scale of the pion mass built on the symmetries of Quantum ChromoDynamics (QCD) [12]. In this description, nucleons can interact either through direct short-range contact or via pion 9

36 exchange. A pion is a particle that is composed of two quarks. This type of theory is useful because only the necessary energy scale and degrees of freedom for a particular system need to be considered. The nuclear force includes contributions from one or multiple pion exchanges that can occur at short or long distances. These different interactions are organized as being either two-nucleon interactions (NN), three-nucleon interactions (3N), or four-nucleon interactions (4N) etc., as is shown in Fig Then the nuclear Hamiltonian (H(Λ)), for a particular energy scale Λ, is written as H(Λ) = T (Λ) + V NN (Λ) + V 3N (Λ) + V 4N (Λ) +... (1.12) to show the contribution of each component. Early theoretical calculations with this method only included contributions from NN interactions [13]. However, it has recently been shown that the inclusion of 3N forces in many-body calculations is necessary to be able to adequately describe neutron-rich nuclei [5, 12]. These forces arise as a natural extension of chiral EFT. The inclusion of these forces creates additional interactions between nucleons and can increase the spin-orbit splitting. Understanding these forces is key towards a theoretical description of shell evolution. 1.2 The Calcium Isotopes The calcium isotopes are an excellent testing ground to investigate the evolution of shell structure. They have a closed shell of 20 protons and a large range of neutron numbers that extend from neutron-deficient (N = 14) to very neutron-rich examples (N = 38). Any drastic change in the observed nuclear structure from one isotope to the next would be the result of the change in neutron number. From a theoretical perspective, these nuclei are simple enough that their structure can be accurately calculated and then compared to experimental observations [14]. In order to adequately describe the evolution of shell structure in the calcium isotopes, it is necessary for the structure of each to be well-studied experimentally and then for the structure to be correctly reproduced theoretically. This would confirm that the underlying physics is understood. To continue to investigate this region, more detailed experimental data is necessary to aid the advancement of the shell model calculations [15]. In the even-even calcium isotopes, the spin of the state is generated by breaking a pair of neutrons within an orbital, and then re-coupling them to a total angular momentum of 2 +. If there is a neutron-shell closure present, more energy is required to break the pair of neutrons, thus driving up the energies of the resultant state [3]. The measured energies for the even-even calcium isotopes are displayed in Fig There is a clear enhancement of energy for the doubly-magic 40 Ca and 48 Ca, which each contain standard neutron shell closures of N = 20 and N = 28, respectively. To confirm the doubly-magic 10

37 Figure 1.6: (Top) The energy of the excited state for even-even calcium isotopes. The sudden jumps in excitation energy from one neutron number to the next is an indication of a neutron-shell closure. (Bottom) Experimental B(E2: ) values. Figure reprinted with permission from Ref. [5]. nature of these nuclei, theoretical calculations can attempt to reproduce the energies of these excited states. Initially, calculations in the calcium region were performed based on well-established NN forces. These were successful in reproducing the doubly-magic nature of 40 Ca, but could not predict the standard N = 28 shell closure in 48 Ca [16, 17]. This result prompted the first application of three-nucleon (3N) forces in the region. Only with the use of these forces, were theorists successful in reproducing the Fig energy in 48 Ca [14] as is shown in Recent measurements indicate potential exotic shell closures in 52 Ca and 54 Ca at N = 32 and N = 34, respectively. Investigations of 52 Ca have revealed the onset of a large energy gap at neutron number N = 32. This is evidenced by the sudden enhancement of the energy relative to that for 50 Ca as is shown in Fig. 1.6 [19]. Observed drops in two-neutron separation energies based on precision mass measurements in 53 Ca and 54 Ca relative to 52 Ca further solidify the existence of the exotic shell closure at N = 32 [20]. The observed energy of the state in 54 Ca is very similar to that observed for 52 Ca. Shell model calculations can only reproduce this energy if N = 34 is also a magic number was also observed in 54 Ca [18]. The exotic shell closures at N = 32 and N = 34 are a direct consequence of the nucleonnucleon interaction between protons in the πf 7/2 orbital and neutrons in the νf 5/2 orbital 11

38 Figure 1.7: Different theoretical reproductions of the experimentally observed energy for the state in even-even calcium isotopes. (a) Energies obtained from phenomenological models. (b) Energies obtained using NN-only theory. (c) Energies obtained when the contributions from 3N forces are included with NN. (d) Energies from NN and 3N forces where the single-particle energies for 41 Ca are calculated consistently. Figure reprinted from Ref. [14]. Figure 1.8: A schematic representation of the attractive interaction between the πf 7/2 and νf 5/2 orbital. As protons are removed from π f 7/2 there is less attraction between the two orbitals as is shown in (a)-(d). The πf 7/2 orbital is completely empty for the calcium isotopes, creating two exotic shell closures at N =32 and N =34. Figure reprinted with permission from Ref. [18]. 12

39 [18]. When both orbitals are fully occupied they are strongly attracted to each other, bringing the πf 7/2 orbital up in energy and the νf 5/2 orbital down in energy. But when protons are removed from the πf 7/2 orbital, there is a weakening of the attractive interaction between the two orbitals such that the πf 7/2 -νf 5/2 energy gap will increase. This effect is illustrated in Fig. 1.8(a-d) for even-even nuclei with ranging proton number Z = and a constant neutron number of N = 34. When the πf 7/2 orbital doesn t contain any protons, as is the case for the calcium isotopes, then the νf 5/2 orbital would be pushed above the νp 1/2 orbital. This results in sizeable energy gaps at N = 32 and N = Previous Investigations of 46 Ca Although 46 Ca is stable, its structure has not been studied in as great detail compared to its calcium neighbours. This is due to its very low natural abundance of 0.004%. Many excited states in 46 Ca have been identified by various reaction mechanisms, most notably from (p, p ) [21], (t, p) [22], and (p, t) [23]. However, many spins have only been tentatively assigned or have not been measured at all [24]. The low-lying structure of 46 Ca has been investigated by three previous beta-decay measurements. There are two early measurements from the late 1960 s by Parsa and Gordon (1966) [25] and Yagi. et al. (1968) [26], and a more recent measurement by Kunz et al. in 2014 [27]. There are large discrepancies present between the two early measurements of the 46 K beta decay. The Parsa and Gordon measurement produced 46 K via the 48 Ca(d, α) 46 K reaction. They measured gamma-ray singles with a lithium-drifted germanium Ge(Li) detector, gamma-gamma coincidences with sodium iodide detectors, and the beta energy distribution with a plastic scintillator. The Yagi et al. measurement produced 46 K via the 48 Ca(γ, pn) 46 K reaction, and only measured gamma-singles with a Ge(Li) detector. The decay schemes reported by the two measurements are displayed in Fig 1.9. The more recent measurement of Kunz et al. [27] utilized a radioactive beam of 46 K created at TRIUMF, and observed 33 gamma lines believed to belong to the 46 Ca daughter with a single germanium detector. However, as the measurement was not equipped to measure gamma-gamma coincidences, these gamma rays could not be placed in the level scheme. The intensities of these observed gamma rays are listed in Table 1.1 along with the results of the two previous measurements. Additional measurements are required on 46 Ca to investigate its nuclear structure and to aid in the advancement of shell model calculations in the calcium region. 13

40 14 Figure 1.9: Previously reported decay schemes for the β decay of the 2 ground state of 46 K to low-lying states in 46 Ca [24]. There are notable discrepancies present between the two decay schemes. Particularly, there is large variance between the reported energies of the gamma rays and the intensities of the beta feeding. (right) The decay scheme as reported by Parsa and Gordon [25]. (Left) The decay scheme as reported by Yagi et al. [26].

41 Table 1.1: of 46 K. Previous measurements of gamma-ray intensities observed following the decay Kunz et al. [27] Parsa & Gordon [25] Yagi et al. [26] E γ (kev) Int.(%) E γ (kev) Int.(%) E γ (kev) Int.(%) (4) (23) (6) (18) (6) (8) (1) (5) (5) (1) (11) (9) (1.3) (6) (6) (29) (30) (32) (66) (6) (29) (42) (1.3) (5) (6) (5) (1) (5) (4) (4) (9) 15

42 Chapter 2 Spectroscopy with The GRIFFIN Spectrometer 2.1 Decay Spectroscopy In order to study the structure of a nucleus it is necessary to populate its excited states. This can be done using different experimental techniques, including nuclear reactions, Coulomb excitation, fusion evaporation, and radioactive decay. Each method will populate different excited states in the resultant daughter nucleus. In transfer and knock-out reactions, nucleons are either placed in or removed from specific orbitals, which makes this a good method to populate single-particle states. In Coulomb excitation the nucleus is excited from its ground state due to the Coulomb interaction, which populates low-energy low-spin states. In fusion evaporation, a compound nucleus is formed, which then evaporates particles to create many diverse excited daughter nuclei in high-energy high-spin states. Finally, in radioactive decay one nucleus will transform into another more stable nucleus, either by emitting an alpha or beta particle, selectively populating states given various selection rules. When a nucleus is in an excited state it will de-excite by emitting gamma rays or conversion electrons. Gamma rays can easily be observed with high-purity germanium (HPGe) detectors, due to their good energy resolution ( 2 kev at 1.3 MeV) and fast timing capabilities ( 100 ps). When multiple HPGe detectors are used it is possible to make coincidence measurements for the observation of weak gamma rays, and to make angular correlation measurements. Many facilities, including TRIUMF (Canada s National Laboratory for Particle and Nuclear Physics) [28], now house large HPGe arrays dedicated for high-statistics beta-decay measurements. Such measurements yield critical information on the low-lying structure of the populated daughter nuclei, including gamma-ray intensities and branching ratios, the beta feeding of excited states, and spin and parity assignments of excited states. In this 16

43 work, the excited states of 46 Ca were populated in the beta decay of the 46 K 2 ground state, and the resultant gamma rays were observed with the Gamma-Ray Infrastructure for Fundamental Investigations of Nuclei (GRIFFIN) [29] experiment at TRIUMF Beta Decay Beta decay is a process that is driven by the weak force, in which a quark will transform from one flavour to another with the exchange of a W boson [2]. As nucleons are made up of three quarks, this transformation results in the conversion of a neutron into a proton, or vice-versa. Nuclei undergo beta decay, when it is energetically favourable, to achieve a better balance of neutron number (N) and proton number (Z), while maintaining a constant mass number (A). There are three different types of beta decay processes that occur; β, β +, and Electron Capture (EC), shown below in Eqn. 2.1, Eqn. 2.2, and Eqn. 2.3, respectively. A NX Z A N 1 Y Z+1 + β + v e (2.1) A NX Z A N+1 Y Z 1 + β + + v e (2.2) A NX Z + e A N+1 Y Z 1 + v e (2.3) As with any nuclear process, each different type of beta decay has an associated Q-value, which represents the energy available for the process to occur. The energy originally held in the parent nucleus is distributed amongst the products of the beta decay. After a beta decay occurs, the daughter nucleus may be left in an excited state that will then de-excite by emitting gamma rays. The equations to determine the Q value for each beta decay process are shown in Eqns below, where M P and M D are the atomic masses of the parent and daughter nuclei, respectively, and m e is the electron rest mass. Each of these beta-decay processes will occur spontaneously when Q >0. Q β = (M P M D )c 2 (2.4) Q β + = (M P M D )c 2 2m e c 2 (2.5) Q EC = (M P M D )c 2 (2.6) The transition rate for a beta decay between an initial state (i) in the parent nucleus and a final state (f) in the daughter nucleus can be described by Fermi s Golden Rule: Here M fi is the nuclear matrix element given by λ = 2π M fi 2 dn de. (2.7) M fi = Ψ d HΨ p d 3 r, (2.8) 17

44 and dn de is the density of final states. If a given nucleus has A nucleons at coordinates ( r 1, r 2,..., r A ), then it can be assumed that the beta decay of a specific neutron into a proton, followed by the emission of an electron and a antineutrio, can be described as a point interaction g represented by the Hamiltonian H = gδ( r n r p )δ( r n r e )δ( r n r νe )(Ô(n p)), (2.9) where the operator (Ô(n p)) transforms a neutron into a proton [30]. The wavefunctions of the parent (Ψ p ) and daughter (Ψ d ) nuclei can be written as Ψ p ( r p1, r p2,..., r p,z ; r n,1,..., r n,n ), (2.10) and Ψ d ( r p1, r p2,..., r p,z, r p,z+1 ; r n,2,..., r n,n ), (2.11) to show that a neutron at coordinate r n,1 has been transformed into a proton at coordinate r p,z+1. The matrix element M fi can then be re-written as M fi = Ψ d( r p1, r p2,..., r p,z, r p,z+1 ; r n,2,..., r n,n )Ψ e ( r e )Ψ ν e ( r νe ) H Ψ p ( r p1, r p2,..., r p,z ; r n,1,..., r n,n )d r e d r νe d r i, (2.12) where d r i shows integration over all possible nucleon coordinates. This can be reduced to M fi = g ϕ D( r) p Ψ e ( r)ψ ν e ( r)ô(n p)ϕ p( r) n d r, (2.13) if the coordinates of all of the other nucleons involved in the decay (A-1) are unchanged such that Ψ P and Ψ D can be written as and Z N Ψ P ( r i ) = ϕ P ( r p,i ) ϕ P ( r n,j ), (2.14) i=1 j=1 Z+1 N Ψ D ( r i ) = ϕ D ( r p,i ) ϕ D ( r n,j ). (2.15) i=1 j=2 The electron and antineutrino wavefunctions be considered as a plane wave that can be written as a Taylor expansion with k = p/h, Ψ e ( r)ψ νe ( r) = e i( k e + k νe ) r = 1 + i( k e + k νe ) r + i 2 ( k e + k νe ) r (2.16) Since their wavelengths are much larger than the nuclear radius R, such that R/λ e R/λ νe << 1, only the first terms of the expansion need to be considered. The resultant and 18

45 M fi is M fi = g[ ϕ D( r) p Ô(n p)ϕ p ( r) n d r + i( k e + k νe ) ϕ D( r) p rô(n p)ϕ p( r) n d r ], (2.17) where the integral in the first term represents the overlap between the single-particle wavefunctions of the parent and daughter nuclei. This term only contributes to M fi when the parities (π) of ϕ P and ϕ D are identical. In the second term of Eqn will only contribute to the M fi when ϕ P and ϕ D have opposite parity due to the factor of r that is present. This result shows that there is a parity selection rule present for beta decays. The existence of angular momentum selection rules can be observed if the Ψ e ( r) and Ψ νe ( r) plane waves are expanded in spherical harmonics: e i( k e + k νe ) r = L,M(4π)i L j L (kr)yl M ( k e k νe ) Y M L (ˆr), (2.18) where k k e + k νe and ˆr (θ r, ϕ r ) show the angular variables [30]. Now, M fi can be written as M fi = g ϕ ( r) p Ô(n p)y M L (ˆr)ϕ P ( r) n j L (kr)d r, (2.19) where the wavefunctions ϕ P and ϕ D can be separated into radial and angular components, ϕ P ( r) = R P (r)y M P L P (ˆr) ϕ D ( r) = R D (r)y M D L D (ˆr). (2.20) This allows for the matrix element to also be split into radial and angular parts: Y M D L D (ˆr)Y M L (ˆr)Y M P L P (ˆr)dΩ R D (r)j L (kr)r p (r)r 2 dr. (2.21) The angular momentum selection rules come from the angular part of Eqn. 2.21, as it is necessary that L P = L D + L. This puts constraints on possible values of L for the outgoing electron and antineutrino. Parity is dependent on angular momentum as π = (-1) L. conserved in this process it is necessary that In order for parity to be π D π l π P = 1, (2.22) where π D, π l, and π P are the parities of the daughter, lepton, and parent wavefunctions, respectively. When π l is even then there was no change in parity between the parent and daughter wavefunctions. If π l is odd, then there was a change in parity. 19

46 Table 2.1: Selection rules and logf t values for beta decay transitions of varying forbiddenness, where J shows the allowed change in angular momentum between the decaying state of the parent and the populated state of the daughter, and π shows whether or not a change in parity is allowed [2]. Forbiddenness J π logf t Superallowed No Allowed 0,1 No st Forbidden 0,1,2 Yes nd Forbidden 1,2,3 No rd Forbidden 2,3,4 Yes >15 The radial component of the matrix element is also dependent on L, as j L (kr) (kr)l (2L + 1)!!. (2.23) From eq is is clear that the value of j L drastically decreases with increasing values of L. This means that transitions that require a large change in L will have smaller contributions to the total matrix element, and are less likely to occur. The transitions that are most likely to occur are known as allowed transitions and those that are less likely to occur are referred to as forbidden transitions. Allowed transitions have a maximum spin change of one unit of angular momentum and no change in parity, whereas forbidden transitions can have larger changes in angular momentum as well as changes in parity. The decay probability of a state is dependent on its half life (t 1/2 ) and gives information on the forbiddenness" of a transition. The comparative half life (ft 1/2 ), written as 2π 3 7 ft 1/2 = ln(2) g 2 M f i 2 m 5 ec 4, (2.24) where g is the strength parameter, is used to compare beta-decay probabilities. The t 1/2 of different beta decays span a very large range from fractions of a second to billions of years, so it is more practical to report and compare the ft value as logft. The angular momentum ( J) and parity ( π) selection rules along with the accepted ranges of logf t values are presented in Table 2.1 for each type of beta transition Gamma Decay Most nuclear reactions and decays leave the resultant daughter nucleus in an energetically excited stated, such that the nucleons are not in their ground-state configuration. When a nucleus is in an excited state it will de-excite by emitting a gamma ray, or a cascade of gamma rays until the nucleus is in its ground state configuration. This type of transition 20

47 represents a re-ordering of the nucleons in the nucleus from one configuration to another. To determine the energy (E γ ) of the gamma ray emitted, it is necessary to consider the initial (E i ) and final (E f ) energies of the participating nuclear states as well as the recoil momentum (p r ) of the final nucleus and its kinetic energy (T r ) [2]. energy and momentum, the following relationships must be observed: For conservation of E i = E f + E γ + T R (2.25) 0 = p r + p γ. (2.26) In this two-body process the nucleus will recoil with a momentum that is equal and opposite that to the gamma ray, such that E = E i E f = E γ + E2 γ 2Mc 2, (2.27) where M is the mass of the nucleus, and T r = E2 γ. Generally, E ranges between 50 kev 2Mc 2 to 10 MeV, whereas the rest energies Mc 2 are much larger ranging upwards of 10 4 MeV. This means that the correction to the energy of the emitted gamma ray from the recoil of the nucleus is negligible, so it can be assumed that E γ = E. (2.28) The participating initial and final states each have an associated angular momentum and parity (J π ) that must be conserved when the gamma ray is emitted. As there are multiple different ways that the angular momenta of these states can couple, there are many different possible values for the angular momentum of the emitted gamma ray (referred to as its multipolarity). If the initial state has an angular momentum I i and the final state has an an angular momentum I f, then the possible values of the angular momentum L for the gamma ray are (I i I f ) L (I i + I f ). (2.29) It is important to note that a gamma ray cannot be emitted with multipolarity L = 0. There cannot be a monopole transition, as the monopole moment is just the electric charge itself, and does not vary with time. The initial and the final states involved in the gamma decay are each a distribution of matter and charge. During the gamma-decay process, the rearrangement of nucleons can result in a shift in the distribution of charge or a shift in the distribution of current, which would give rise to an electric or a magnetic field, respectively. The parity (π) of the emitted radiation field is dependent on L as is seen in Eqns and Electric and magnetic 21

48 multipoles of the same order always have opposite parity. π(ml) = ( 1) L+1 (2.30) π(el) = ( 1) L (2.31) The electric or magnetic type of the emitted gamma ray can be determined by the relative parity change of the initial and final states. If there is no change in parity, then the radiation field must have even parity. If the parity does change, then the radiation must have odd parity. From Eqn. 2.30, electric radiation has even parity if L is even, and, from Eqn. 2.31, magnetic transitions have even parity when L is odd. The different types of possible gammaray transitions are listed in Table 2.2. For any gamma decay, many different gamma-ray types may be emitted as per the selection rules discussed previously. Table 2.2: Selection rules and multipolarities for different types of gamma-ray transitions. Multipolarity Name L π E1 Electric Dipole 1 Yes M1 Magnetic Dipole 1 No E2 Electric Quadrupole 2 No M2 Magnetic Quadrupole 2 Yes E3 Electric Octupole 3 Yes M3 Magnetic Octupole 3 No E4 Electric Hexadecapole 4 No M4 Magnetic Hexadecapole 4 Yes The probability that a gamma ray will be emitted is given by the transition rate λ(δl) (δ = E or M): λ(δl) = where m fi (δl) is the matrix element ( ) 2(L + 1) ω 2L+1 ɛ 0 [(2L + 1)!!] 2 [m fi(δl)] 2, (2.32) c m fi (δl) = ψ f m(δl)ψ i dv (2.33) with m(δl) as the multipole operator [2]. The matrix element can be evaluated if it is assumed that the gamma-ray transition is the result of a single proton moving from one nuclear orbital to another. The estimated matrix elements are m fi (EL) W eiss = 1.22L 4π ( ) 3 2 A 2L/3 e 2 (fm) 2L, (2.34) L

49 Table 2.3: Weisskopf estimates for the transition rates (δl) for a given gamma ray with energy E γ of either electric (E) or magnetic character [2]. L λ(el) (s 1 ) λ(ml) (s 1 ) x A 2/3 Eγ x Eγ x 10 7 A 4/3 Eγ x 10 7 A 4/3 Eγ x 10 1 A 2 Eγ x 10 1 A 4/3 Eγ x 10 5 A 8/3 Eγ x 10 6 A 2 Eγ x A 10/3 Eγ x A 8/3 Eγ 11 and m fi (ML) W eiss = 10 ( ) 3 2 π (1.2)2L 2 A (2L 2)/3 µ 2 L + 3 N(fm) 2L 2, (2.35) for electric and magnetic transitions, respectively. These estimates were initially derived by Victor Weisskopf, with the assumption that the nucleus had a uniform density with radius R 0 = 1.2A 1/3 fm, so the resultant transition strengths are given in Weisskopf units (W.u.). Transition rates λ(δl) determined with the Weisskopf estimates for select electric and magnetic transitions of varying multipolarity L are presented in Table 2.3 [2]. From the Weisskopf estimates it is clear that lower multipolarity transitions are more probable than higher multipolarity transitions. Also, for a given multipolarity L there is a preference for the transmitted radiation to have electric character over magnetic character. In many cases the emitted gamma ray will not be purely electric or magnetic, and it will contain a mixture of both electric and magnetic character. Due to the nature of the parity selection rules for gamma decay, the mixture will typically be between multipolarities M L and E(L + 1). The mixing ratio δ is written as δ = ψ f E(L + 1) ψ i, (2.36) ψ f ML ψ i to compare the relative matrix elements of the electric (E) and magnetic (M) components for a particular gamma-ray transition. 2.2 Rare-Isotope Beam Production at TRIUMF TRIUMF [28] is a rare-isotope beam facility located in Vancouver, British Columbia that is home to the world s largest cyclotron. This cyclotron is capable of producing proton beams up to energies of 500 MeV with a current of 100 µa. This beam can be delivered to several different experimental halls within the facility, such as the Isotope Separator and Accelerator Facility (ISAC), where it can be used directly or to create secondary beams. 23

50 24 Figure 2.1: An artist s rendition of the ISAC experimental hall located at TRIUMF in Vancouver, BC. Reprinted with permission.

51 The ISAC facility, schematically drawn in Fig. 2.1, utilizes the high-energy proton beam to produce high-purity radioactive ion beams with the isotope separator on-line (ISOL) technique. These beams are then transported directly to different experimental facilities on site, such as the GRIFFIN facility. To create the ion beam, the proton beam is accelerated onto a production target where several secondary reactions can occur, including spallation, fission, and fragmentation. These reactions create all nuclei with masses less than or equal to that of the specific target material used. In some cases heavier elements can be produced from the capture of neutrons or other fragments. TRIUMF uses several different target materials that range in proton number from silicon (Z = 14) to uranium (Z = 92), as different isotopic yields are obtained from different materials. Once the isotopes are created they are selectively ionized and transported through the facility via electric or magnetic fields. Different ion sources currently available include surface ion sources, forced electron beam arc-discharge (plasma) ion sources, and resonant laser ionization sources. In this experiment, the surface ion source was used to ionize the created radioactive species. This source has the simplest design of those that are in use at ISAC; a rhenium foil is inserted into the transfer tube after the production target and heated up to 2200 C. When an atom leaves the target chamber it will interact with the hot foil, potentially leaving an electron behind to become a positively-charged ion. This source is most efficient for elements that have an ionization energy below 6 ev, which are mainly the alkali elements. To filter out the ion of interest, the ionized cocktail beam is passed through an electric field caused by a potential difference ( V ) to accelerate the ions. The velocity of each ion is dependent on its mass (m ion ) and charge (Q ion ) as is shown in Eqn v ion = 2Q ion V m ion (2.37) The accelerated ions then enter the u-shaped mass separator where they will pass through a perpendicular magnetic field (B) causing them to trace a circular arch. The radius (r) of their curved path is r = 2m ion V Q ion B 2. (2.38) The potential difference ( V ) is fixed and the charge of the desired ion (Q ion ) is known, so all that is necessary to select an isotope of desired mass is to adjust the magnetic field (B). This allows only the selected isotopes to have the correct curved path to exit the mass separator and be delivered to the chosen experimental set-up. The resolution of the mass separator is 2000 m/ m. There is always the possibility that there will be isobaric contaminants in the beam; none were observed in this experiment. 25

52 Figure 2.2: The GRIFFIN spectrometer located at TRIUMF-ISAC. When fully populated, the array contains 16 HPGe clover detectors that can be closely packed together for optimal solid angle coverage. GRIFFIN can also be coupled to a suite of ancillary detectors. 2.3 GRIFFIN and its Ancillary Detectors The GRIFFIN spectrometer [29] is a state-of-the-art HPGe detector array that can be selectively coupled to a suite of ancillary detectors (pictured in Fig. 2.2). GRIFFIN s highly efficient detectors and flexible set-up make it ideal for decay spectroscopy investigations. When radioactive beam is delivered to GRIFFIN it is implanted on aluminized-mylar tape at the center of the array in a vacuum-sealed chamber, and the resultant radioactivity is observed by the surrounding detection systems. GRIFFIN was installed in TRIUMF s ISAC-I hall in the summer of 2014 with commissioning beams delivered to GRIFFIN in the fall of that year. The data set discussed in this work was collected in December of 2014 as part of a small campaign investigating the calcium isotopes. This beam time also included the measurement of the beta decay of 47 K [31]. 26

53 2.3.1 GRIFFIN HPGe Clover Detectors When fully populated, GRIFFIN contains 16 clover HPGe gamma-ray detectors. Each clover houses four large-volume germanium crystals, that are 90 mm in length and 60 mm in diameter with the outer edges tapered at 22.5 over the first 30 mm of their length, as is shown in Fig Each HPGe crystal is 40% efficient and has an energy resolution of 1.9 kev for the detection of a 1.3 MeV gamma ray. Figure 2.3: Each GRIFFIN clover detector contains 4 HPGe crystals that are 60 mm in diameter and 90 mm in length. The tapered edges of the crystals allow them to be closely packed together within the array. The faces of each clover detector can be closely-packed together within the array in a rhombicuboctahedral geometry that contains 18 square and 8 triangular faces (Fig. 2.4). In its optimal closely-packed configuration, each clover detector is 110 mm from the center of the array, which allows for good solid-angle coverage. In GRIFFIN, 16 of the square faces are occupied by HPGe clover detectors, one of the faces is open to allow for the radioactive ion beam to enter the array, and one of the faces is used as the exit point for the in-vacuum tape system. The triangular faces are used to mount various ancillary detection systems. Figure 2.4: The GRIFFIN spectrometer has a rhombicuboctahedral geometry containing 18 square faces and 8 triangular faces. 16 of the 18 square faces can be occupied by HPGe clover detectors, and the other two square faces are left open to allow for the entry of the beam and for the exit of the tape system. The triangular faces can be used to mount ancillary detection systems. 27

54 When a gamma ray interacts with one of the HPGe crystals it may undergo Comptonscatter, which causes only a fraction of its energy to be deposited in the initial crystal of the interaction and the remaining portion of its energy may be deposited in another crystal within the array, or may avoid detection all together. It is possible to add back these of multiple interactions into a single event in the off-line analysis of the data. It is necessary to set constraints on the position and time for the detection of the two scattered events. This technique, referred to as addback", greatly increases the efficiency of the detection system. In this analysis, the two gamma-ray events were required to be detected in two crystals within the same clover detector within a 250 ns time window in order for an addback event to be reconstructed. The increased detection of gamma rays at specific energies, referred to as addback factors (F AB ), are listed in Table 2.4. These are experimental values fit from source data taken of 152 Eu, 133 Ba, 60 Co, and 56 Co calculated with F AB = A AB A CS. (2.39) Here, A AB is the peak area of a given gamma ray including the reconstructed addback events, and A CS is the peak area of that same gamma ray from the crystal-singles data when the addback events are not included. As high-energy gamma rays are more likely to scatter, they have the largest addback factors The Ancillary Detectors The GRIFFIN array can be coupled to a suite of different ancillary detectors to compliment the gamma-ray data of a specific experiment. The detection systems that were used in this experiment will be briefly discussed. When a nucleus de-excites it may emit a conversion electron instead of a gamma ray. These electrons have an energy that is equivalent to the energy of the de-excitation, minus the binding energy of the electron itself. The Pentagonal Array for Conversion Electron Spectroscopy (PACES) [32] is an array of 5 lithium-drifted silicon detectors used for the detection of conversion electrons. Each detector has a 200 cm 2 surface area and is cooled to liquid-nitrogen temperatures. It is possible to achieve an energy resolution of <2 kev for electrons with energies between 15 and 2000 kev [33]. PACES (Fig. 2.5) was installed in the upstream position of the chamber during this experiment. 28

55 Table 2.4: The addback factors (F AB ) as calculated from Eqn for experimental source data taken of 152 Eu, 133 Ba, 60 Co, and 56 Co. The errors on these values are purely statistical. Source Gamma-Ray Energy (kev) F AB 152 Eu (2) 133 Ba (3) 133 Ba (2) 152 Eu (1) 133 Ba (1) 133 Ba (2) 152 Eu (5) 152 Eu (2) 56 Co (2) 152 Eu (5) 152 Eu (2) 56 Co (6) 152 Eu (2) 60 Co (2) 152 Eu (9) 56 Co (2) 60 Co (2) 152 Eu (2) 56 Co (6) 56 Co (17) 56 Co (10) 56 Co (7) 56 Co (11) 29

56 Figure 2.5: The PACES array for the detection of conversion electrons includes 5 lithiumdrifted silicon detectors. The Scintillating Electron-Positron Tagging Array (SCEPTAR) [33] occupied the downstream position within the vacuum chamber. In this configuration, SCEPTAR consisted of 10 thin plastic scintillators used to tag beta particles, pictured in Fig There is also an upstream SCEPTAR detection system that contains an additional 10 scintillators, which was not used during this experiment due to the presences of PACES. SCEPTAR can be used to dramatically improve the peak-to-background ratio by requiring a coincidence between a beta particle and a gamma ray within a specified time window. Figure 2.6: During this experiment, the downstream half of SCEPTAR was installed behind the MYLAR tape. The 10 thin plastic scintillators are used to tag beta particles. 30

57 Chapter 3 Beta Decay Scheme of 46 K 3.1 Details of the Measurement A high-statistics data set of the beta decay of the 46 K 2 ground state into low-lying levels of 46 Ca was measured with the GRIFFIN spectrometer, located at TRIUMF-ISAC, in December of A 9 µa 500 MeV proton beam was impinged upon a uranium carbide target, the resultant radioactive species were surface ionized and a high-resolution mass separator was used to select singly-charged A = 46 ions. A beam of 4x10 5 pps of 46 K was delivered and implanted at the center of the GRIFFIN array onto the Mylar tape. At the time of the experiment, the array operated with 15 HPGe detectors used for the detection of gamma rays. The array was also coupled to PACES in the upstream position for the detection of conversion electrons, and SCEPTAR in the downstream position for beta particle identification. As the beam was implanted onto the Mylar tape, it was possible to conduct the experiment in a cycling mode. The data was collected with the following cycle condition: tape move, background measurement, beam-on implantation, and beam-off decay. Different cycle lengths were used throughout the experiment as is shown in Table 3.1. When the tape was cycled outside of the array is was stored behind lead shielding to avoid any possible contamination to the measurement. The data was collected un-filtered, with no trigger conditions applied, continuously over the course of forty hours. 31

58 Table 3.1: The different cycle conditions used during the measurement of the 46 K beta decay with GRIFFIN. 32 Run Number Number of Sub-runs Tape Move Time (s) Background Time (s) Beam-On Time (s) Beam-Off Time (s)

59 3.2 Gamma-Ray Energy Calibration During the experiment each HPGe crystal measured an electrical pulse amplitude that was proportional to the energy deposited by a gamma ray. The electrical pulse was converted to a digital signal with an analog-to-digital converter (ADC), and each pulse was placed into channel numbers based on its height. The gain for each crystal was adjusted such that the signal sent to the ADC corresponded to roughly 0.3 kev per channel. The relationship between channel number (x) and gamma-ray energy (E γ ) can be described by: E γ = a + bx (3.1) The fit parameters a and b were determined with the use of standard calibration sources that emit gamma rays with energies that have been measured to high precision [34, 35, 36, 37]. At the time of the experiment, data was taken with the GRIFFIN array for the standard calibration sources 56 Co, 60 Co, 133 Ba, and 152 Eu. These sources emit intense gamma rays transitions with energies that span a range of 50 kev to MeV. A special high-energy calibration was constructed for energies greater than MeV, discussed in Sec Each individual HPGe crystal was gain matched so that each would bin the same energy (E γ ) into the same channel number (x). This was performed so that the channel numbers roughly corresponded to 1 kev per channel. This allowed the energy spectrum from each crystal to be summed together into a single gamma-ray spectrum. A linear fit to determine the energy calibration was performed on the channel (x) vs. energy (E γ ) data from the calibration sources (listed in Table 3.2). presented in Figure 3.1 with the fit parameters listed in Table 3.3. The resultant fit is The uncertainty on the E γ associated with the linear fit can be described with the standard error propagation formula σ E = σ 2 a + (bx) 2 [( σ b b )2 + ( σ x x )2 ]. (3.2) The uncertainties σ a and σ b correspond to the uncertainties in the fit parameters a and b, and the uncertainty σ x corresponds to the error of the channel number x. 33

60 Energy (kev) Channel Number Figure 3.1: Linear fit of channel number versus gamma-ray energy data listed in Table 3.2.

61 To determine the systematic uncertainty in the energy calibration, high-precision energies of transitions from the room background decay of 40 K and from contamination of 207 Bi present in the GRIFFIN chamber were used. The difference between the measured and the accepted energy values [38, 39] for these transitions is presented in Table 3.4. An additional systematic uncertainty of 0.25 kev is necessary to make the measured energies agree with the accepted high-precision values. Table 3.2: Channel numbers that correspond to the energies taken from [34, 35, 36, 37] of the gamma-ray transitions from 56 Co, 60 Co, 133 Ba, and 152 Eu chosen for the purpose of the energy calibration. Channel Number Energy (kev) Table 3.3: Fit parameters obtained from a linear fit of the data presented in Table 3.2 with their associated uncertainties. Parameter Value Uncertainty a b e-05 35

62 Table 3.4: The difference between the measured (E M ) and accepted (E A ) [38, 39] gammaray energies. An additional systematic uncertainty of 0.25 kev is required to make all the measured values consistent with the accepted values of these transitions. Source E A E M E A E M 207 Bi (2) (7) 0.32(7) 207 Bi (3) (8) 0.012(8) 40 K (6) (9) 0.32(9) Calibration for Energies Above MeV Several gamma rays were observed with energies above 3.2 MeV, therefore a high-energy calibration had to be created in addition to the low-energy calibration previously described. Many of these high-energy transitions decay directly to the 46 Ca ground-state, such that their energies correspond to the energy of the excited state that they depopulate. Given the measured energies of all of the gamma-ray transitions observed below MeV, it was possible to determine the energies of each of the observed excited states. The excited state energies (E level ) that equate to high-energy gamma-ray energies are presented in Table 3.5 with the corresponding observed channel number (x) of each gamma ray. These data points were fit with the polynomial and the resultant fit parameters are presented in Table 3.6. E level = a + bx + cx 2, (3.3) Table 3.5: Channel numbers for high-energy gamma-ray transitions that have energies that correspond to excited state energies. These excited state energies are from an initial χ 2 - minimization fit of the gamma-ray energies observed below MeV. Channel Number Excited State Energy (kev) (1) (128) (15) (9) (6) (10) (6) (10) (19) (10) (6) (12) (14) (14) (8) (22) (24) (18) (24) (39) 36

63 Table 3.6: Fit parameters obtained from a second-order polynomial fit of the data presented in Table 3.5 with their associated uncertainties. Parameter Value Uncertainty a b e-04 c e e-08 For a piece-wise energy calibration, it is important that the overall energy calibration is continuous from 50 kev to kev. The low- and high-energy calibrations intersect at MeV. It was necessary to adopt a systematic uncertainty of 0.82 kev for the highenergy calibration to create agreement between the two calibrations (Table 3.7). Table 3.7: Energy of the MeV transition determined from the low-energy (A) and high-energy (B) calibrations respectively. To ensure agreement between the two calibrations, a systematic uncertainty of 0.82 kev was adopted for values obtained from the high-energy calibration. Channel Number Energy A (kev) Energy B (kev) Difference (kev) Gamma-Ray Efficiency Calibration In order to determine the intensities and branching ratios of observed gamma-ray transitions it is necessary to understand the detection efficiency of the 15 GRIFFIN HPGe detectors that were present in the array at the time of the experiment. The efficiency is energy dependent as high-energy gamma rays are more likely to scatter, which results in missing events that need to be corrected for. To determine the efficiency of the GRIFFIN array data was taken immediately following the experiment using standard calibration sources 56 Co, 60 Co, 133 Ba, 152 Eu. These sources have gamma-ray transitions with intensities that have been measured to high precision and are tabulated in refs. [34, 35, 36, 37]. The absolute efficiency of the array at the energies of the 60 Co gamma-ray transitions were determined given the activity of the source at the time the data was collected. The relative efficiencies of transitions from the 152 Eu, 133 Ba, and 56 Co sources were determined by dividing the measured areas of the photopeaks by the intensities of the corresponding gamma-ray transitions. These were normalized to the absolute efficiencies of the transitions from 60 Co with sequential polynomial fits to minimize the reduced-χ 2 /ν and to give a continuous fit. This is discussed in more detail in Appendix A. 37

64 3.3.1 High-Energy Efficiency The highest-energy gamma ray from the measured sources is the 3273 kev transition from the decay of 56 Co. In the analysis of this data, gamma rays were observed with energies greater than 3273 kev, so it was necessary to use simulated high-energy efficiency data points. These simulated data points were obtained from the GRIFFIN efficiency simulator, based on GEANT4 simulations, which is included in the GRIFFIN toolkit [40]. The simulated efficiencies were calculated with the inclusion of 16 HPGe detectors positioned 11.0 cm from the source with 20 mm Delrin absorbers. As only 15 HPGe detectors were in use during the experiment, the obtained values from the efficiency calculator were multiplied by ; these are listed in Table 3.8. These data points were normalized to the sixth-order polynomial fit of the source efficiencies. The resultant set of data points was then fit with an 8th-order polynomial, the normalization factors along with the reduced χ 2 /ν values for the fit are listed in Table A.4. Table 3.8: Simulated high-energy absolute efficiency (ɛ sim ) data points calculated with the GRIFFIN efficiency calculator for the addback-clover data mode for 15 clover detectors. Energy ɛ sim Final Absolute Efficiency Fit Parameters The final fit of the absolute efficiency (ɛ abs ) data points for the addback data was obtained with a ninth-order polynomial ɛ Abs = a + bx + cx 2 + dx 3 + ex 4 + fx 5 + gx 6 + hx 7 + ix 8 + jx 9. (3.4) The fit is presented in Figure 3.2, and the final fit parameters are listed in Table

65 39 Figure 3.2: Absolute efficiency versus gamma-ray energy using standard calibration sources of 60 Co, 152 Eu, 133 Ba, and 56 Co as well as simulated high-energy data points for the 15 HPGe GRIFFIN detectors present during the time of the experiment. The experimental data points are shown as red circles and the simulated data points are represented by purple squares.

66 Figure 3.3: 95% confidence interval for the addback-clover absolute efficiency data as determined by IGOR-Pro. [H] Table 3.9: Parameters for the gamma-ray absolute efficiency fit for Equation 3.4 for the addback-clover data. Parameter value a b c d e f g h i j e e e e e e e e e-34 With the use of the fitting program Igor-Pro [41] it was possible to calculate a 95% confidence interval over the full range of the fit, such that one would expect to find 95% of all possible efficiency values within this interval given the errors on the fit parameters. The calculated confidence interval is shown in Figure 3.3. The maximum percent difference between the confidence interval and the efficiency for energies between 121 kev and 7.5 MeV is plotted in Figure 3.4. Select values were fit with the second-order polynomial %Err = a + bx + cx 2, (3.5) 40

67 Figure 3.4: The maximum percent difference between the calculated 95% confidence interval and the efficiency for energies between 121 kev and 7.5 MeV. This percent difference is equivalent to the percent error on the efficiency. Select data points (filled-red circles) were fit with a second-order polynomial whose parameters are presented in Table Table 3.10: Fit parameters obtained for the second-order polynomial fit used to describe the error on the efficiency for a given energy. [H] Parameter a b c Value e e e-10 to determine the % error on ɛ abs for a given E γ. The resultant fit parameters are listed in Table Gamma-Ray Time Gates To construct the gamma-gamma coincidence events the timing data of the HPGe crystals as output by the GRIF-16 modules was used LED "Walk" Correction The GRIF-16 modules each contain a Leading-Edge Discriminator (LED) that emits a logic pulse once the leading edge of the input signal meets a pre-set threshold value. However, it is possible to observe a timing variance dependent on the size of the input signal referred to as a "walk". This effect causes low-energy events (with small pulse amplitudes) to be 41

68 Figure 3.5: The use of leading edge triggering will create a time-walk effect. The signal for a given event does not trigger the data-acquisition system until it passes a set threshold. Therefore, smaller amplitude signals such as B appear to arrive later in time. detected late as is seen in Figure 3.5. It is necessary to correct for this "walk-effect" in order to obtain precise timing resolution. To correct for the "walk" effect in the GRIFFIN HPGe data, gamma-gamma coincidences from the decay of 60 Co were investigated. Figure 3.6 shows the gamma-ray events detected in coincidence with the 1332 kev transition. There is a clear time variance for the lowenergy events, as the time differences are much larger than for high-energy events. These low-energy events were fit with the function T = A + B E C, (3.6) where T is the time difference between the low-energy event and the 1332 kev gamma ray, and E is the energy of the low-energy event. The resultant fit parameters are presented in Table This function was then used to correct the timestamps of the detected gammaray events. The "walk" corrected spectrum is presented in Figure 3.6(bottom). The time resolution for these events was reduced from 400 ns to 250 ns. 42

69 Energy of Coincident Gamma Ray (kev) Time Difference (10 s of ns) Figure 3.6: (Top) A plot of the time difference between gamma ray events detected in coincidence with the 1332 kev gamma ray. There is a clear time variance for low-energy events due to the walk effect created by the LED. (Bottom) Walk -corrected time spectrum. The time resolution for these events was reduced from 400 ns to 250 ns. Table 3.11: Resultant fit parameters for Eqn These parameters were used to correct the timestamps of low-energy events to overcome the walk effect. Parameter Value A B C

70 Prompt Events Time-Random BG Events Number of Events Time Difference (10 s ns) Figure 3.7: A plot of the time difference between pairs of gamma rays observed in this data set. The time-random events between 400 and 650 ns have been subtracted from the prompt events between 0 and 250 ns. These good coincidences are presented in Fig Gamma-Gamma Coincidence Timing A plot of the time difference between pairs of gamma rays observed in this data set is presented in Fig The events between 0 and 250 ns are prompt events, which represent two gamma rays in coincidence, whereas events beyond 250 ns are time random events where the two gamma rays are not from the decay of the same nucleus. These time random events were also present within in the prompt event time window, and needed to be removed to ensure that only good coincidences are analysed. To account for this, time-random background events between 400 and 650 ns were subtracted from the prompt events. A spectrum of the resultant good time-random background subtracted gamma-gamma coincidences are presented in Fig The 46 Ca Level Scheme The 46 Ca level scheme was constructed by analysing the gamma-gamma coincidence data. This was done by gating on gamma rays to confirm their placement in the level scheme. The term gating means to observe the quantities of all of the other gamma rays that are in coincidence with the gamma ray gated on. A gate taken on the 1346 kev gamma ray is presented in Fig. 3.9 shows all of the gamma rays that are observed in coincidence with it. In the initial analysis of the gamma-gamma coincidence data, gates were taken on gamma rays that had been previously placed in the 46 Ca level scheme [24] to both confirm the placement of those gamma rays and to look for possible new gamma rays and excited 44

71 Energy γ 2 (kev) Energy γ 1 (kev) Figure 3.8: Time-random background subtracted gamma-gamma matrix constructed with a 250 ns time window. The x and the y axis show the energies of the two gamma rays that are in coincidence, and the z axis shows how many times those gamma rays were observed in coincidence during the experiment. 45

72 states. Then to expand the level scheme, further gates were taken on newly observed gamma rays to determine their placement until all possible coincidences had been investigated. From the analysis of this data the 46 Ca level scheme now contains 194 gamma rays (150 newly observed) and 42 excited states (24 newly observed). Only 14 of the previously reported 44 observed gamma rays believed to belong to the 46 Ca level scheme had been previously placed. Currently, 39 of these transitions were observed in this work, and all have been placed in the level scheme (given the exception of the 3676 kev transition, which is a single-escape peak of another observed transition in the data set). 3.6 Determination of Gamma-Ray Intensities The intensity of a gamma ray can be determined either from the gamma-singles or the gamma-gamma coincidence data. Ideally, all of the intensities would be measured from the gamma-singles events, but often it was not possible to observe weak gamma rays over the Compton background. In these cases the intensities were extracted from the gamma-gamma coincidence events. By convention, the intensities of all of the gamma rays are reported relative to the most intense gamma ray observed in the data set, which, in this case is the 1346 kev gamma ray. The intensities of all of the gamma rays belonging to the 46 Ca level scheme are presented in Table

73 Figure 3.9: The gate taken on the 1346 kev gamma ray that show the quantities of the gamma rays observed in coincidence (within 250 ns) with it. The upper spectrum shows the coincident gamma rays with energies between 0 and MeV, and the lower spectrum shows the coincident gamma rays with energies between 4 and 6 MeV. The 4029 kev gamma ray is labelled in both spectra to show the scale of counts of the lower spectrum as compared to the upper spectrum. 47

74 Table 3.12: There are 194 gamma rays and 42 excited states that were observed in the 46 Ca level scheme from the beta decay of 46 K 2 ground state. These are presented per level, where the intensity (I γ ) of each gamma ray has been determined relative to that of the 1346 kev transition. Absolute intensities can be obtained by multiplying the listed value by (6). The gamma-ray branching ratio (BR γ ) of a given gamma ray is the ratio of the intensity of that gamma ray relative to the intensity of the strongest gamma ray that depopulates the same level. Spin-parity assignments (J π ) are from Ref. [24] unless otherwise stated. a spin (J) assignments are the result of the angular correlation analysis presented in Sec b parity (π) assignments suggested from determined logft values as discussed in Sec c spin and parity (J π ) assignments suggested from determined logft values. E inital (kev) J π E γ (kev) E final (kev) I γ BR γ (14) (26) (22) (26) (9) (16) (26) (12) (14) (26) (6) (26) (6) 76.7(25) (15) (26) (8) 0.416(16) (26) (7) 6.03(18) (26) (14) (84) (98) 0.945(160) (15) (26) (15) (84) (18) 8.02(33) (17) (26) (18) (26) (33) 88.2(47) (16) (3 ) (26) (5) 26.8(18) (26) (24) (15) (1,2,3,4) + c (27) (49) 5.27(39) (30) (13) 8.1(9) (26) (7) (15) (2,3) + ab (26) (3) 5.69(18) (26) (4) 76.5(24) (26) (7) 5.35(28) (83) (7) (83) (4) 4.28(19) (15) (21) (37) (74) (26) (7) 0.304(18) (26) (23) 0.950(56) (26) (6) 2.37(15) Continued on the next page 48

75 Table 3.12 continued from previous page E inital (kev) J π E γ (kev) E final (kev) I γ BR γ (26) (5) 28.2(15) (83) (20) (14) (1,2,3) a (26) (9) 13.6(8) (26) (7) 88.1(59) (26) (9) 18.7(10) (20) (30) 54(29) (83) (4) (16) 3 + ab (26) (15) 26.5(12) (26) (70) 7.01(51) (26) (23) 3.55(18) (22) (29) 66.0(36) (26) (3) (83) (7) 19.0(6) (16) (2,3) + ab (83) (8) 33.9(19) (83) (19) (21) (2 + ) (27) (19) (16) 2 ab (26) (18) 0.250(9) (26) (11) 0.475(34) (26) (3) 0.465(17) (26) (12) 5.61(38) (26) (8) 1.26(4) (26) (44) 0.295(16) (26) (6) 3.89(20) (26) (14) 8.43(49) (83) (10) (17) (2,3) + ab (83) (5) (18) (26) (96) 27.2(41) (27) (66) 40.3(33) (30) (13) (40) (45) 16.9(20) (16) (3 ) (26) (8) 2.94(19) (26) (4) 1.38(9) (26) (16) 7.04(37) (26) (5) 2.95(12) (26) (6) 2.05(14) (26) (24) 1.24(6) Continued on the next page 49

76 Table 3.12 continued from previous page E inital (kev) J π E γ (kev) E final (kev) I γ BR γ (26) (5) 2.67(12) (26) (50) 1.85(12) (26) (4) 11.6(1) (83) (2) (15) (2,3) ab (26) (33) 0.360(27) (26) (31) 2.92(24) (26) (19) 1.96(16) (29) (13) 0.797(93) (27) (10) 0.797(93) (26) (35) 3.44(26) (26) (11) 13.9(9) (26) (28) 2.36(20) (26) (12) 1.57(10) (26) (7) (26) (28) 37.4(24) (83) (27) 42.5(25) (84) (7) 0.988(63) (17) 3 ab (26) (14) 0.613(51) (26) (10) 5.34(36) (26) (8) 5.56(33) (26) (22) 0.860(74) (26) (10) 6.34(39) (27) (18) 0.945(65) (28) (12) 8.60(50) (83) (15) (99) (12) 0.772(50) (16) (2,3) + ab (26) (33) 0.832(65) (26) (9) 25.1(18) (27) (9) 2.05(18) (30) (12) 1.89(20) (28) (12) 2.35(21) (26) (21) 4.13(37) (30) (22) 0.396(38) (26) (10) 33.9(21) (26) (17) 4.79(34) (83) (29) 100 Continued on the next page 50

77 Table 3.12 continued from previous page E inital (kev) J π E γ (kev) E final (kev) I γ BR γ (167) (1,2,3,4) + c (26) (91) 0.551(77) (26) (51) 64.2(51) (27) (68) 58.3(45) (27) (9) 7.78(64) (27) (16) 0.952(99) (26) (6) (32) (42) 3.15(36) (84) (20) 31.0(21) (18) (1,2,3,4) + c (27) (11) 20.2(17) (27) (40) 5.57(56) (27) (7) 14.9(11) (29) (13) 32.0(22) (26) (46) 5.53(62) (83) (38) (17) (4 + ) (27) (31) 13.0(10) (26) (17) 63.8(56) (27) (9) 32.5(29) (26) (9) 37.4(30) (27) (11) 57.2(41) (84) (19) (84) (5) 27.6(21) (28) (1,2,3) ab (28) (35) 0.869(90) (75) (74) 1.09(18) (83) (21) (83) (16) 78.1(56) (16) (4 + ) (26) (14) 4.90(45) (27) (14) 0.641(49) (29) (13) 3.25(39) (26) (28) 11.9(9) (26) (11) 4.84(37) (27) (47) 1.47(15) (26) (43) 26.7(17) (26) (33) 19.4(13) (84) (15) 7.23(54) (83) (10) 6.05(40) (83) (17) 100 Continued on the next page 51

78 Table 3.12 continued from previous page E inital (kev) J π E γ (kev) E final (kev) I γ BR γ (16) (1,2,3,4) c (26) (14) 6.44(57) (32) (26) 0.84(10) (26) (40) 23.2(18) (27) (54) 2.89(23) (27) (54) 2.36(22) (30) (54) 1.91(21) (27) (27) 13.3(11) (26) (6) 4.47(30) (83) (9) 5.38(41) (83) (11) (18) (27) (24) 16.7(11) (65) (11) 36.5(33) (27) (13) 41.3(38) (26) (9) 28.4(25) (26) (21) 5.19(56) (26) (14) 64.0(47) (83) (16) 60.7(49) (83) (22) (18) (1,2,3,4) + c (27) (29) 12.4(11) (30) (60) 19.7(21) (32) (27) 8.64(9) (28) (29) 8.43(10) (26) (16) (60) (1,2,3,4) + c (83) (19) (84) (12) 68.8(57) (19) (1,2,3,4) + c (27) (13) (28) (45) 30.9(33) (29) (30) 23.7(24) (29) (51) 46.6(44) (226) (1,2,3,4) + c (29) (41) 25.0(27) (28) (45) 5.73(75) (83) (37) 11.2(9) (83) (7) 16.6(15) (83) (35) (84) (1,2,3,4) + c (83) (7) (21) (1,2,3,4) c (28) (22) 3.14(33) Continued on the next page 52

79 Table 3.12 continued from previous page E inital (kev) J π E γ (kev) E final (kev) I γ BR γ (32) (45) 4.71(62) (30) (56) 9.55(90) (83) (9) 16.6(15) (83) (35) 6.59(59) (83) (16) 46.9(36) (83) (43) (25) (1,2,3,4) + c (33) (22) 12.5(17) (84) (7) (30) (1,2,3,4) + c (40) (14) 11.9(14) (83) (6) (84) (43) 41.1(45) (24) (1,2,3,4) + c (43) (142) 1.81(35) (30) (31) 7.59(83) (36) (126) 2.30(32) (83) (23) (60) (1,2,3,4) + c (83) (16) 45.0(57) (83) (27) (26) (1,2,3,4) + c (31) (14) 23.7(30) (32) (19) 31.8(39) (83) (36) (29) (2 + ) (84) (59) (84) (73) 60(11) (37) (1,2,3,4) + c (36) (13) 32.7(50) (92) (22) Intensities from the Gamma-Singles Data It was possible to fit the intensities of some of the observed gamma rays directly in the gamma-singles data. Often, these were intense transitions that were clearly visible above the Compton background. The gamma-singles spectrum is presented in Fig 3.10, roughly 900 million gamma-singles events were detected in this experiment. The intensity (I γ ) of a gamma ray was obtained by fitting the area (A p ) of the corresponding peak in the gammasingles data, and correcting for the efficiency (ɛ γ ) of the array at that particular energy with eqn I γ = A p ɛ γ (3.7) 53

80 1346 Counts Counts Counts DEP SEP 6111 SEP Energy (kev) Figure 3.10: Gamma-singles spectrum for the beta decay of 46 K into low-lying states of 46 Ca. When possible the observed peak areas for gamma rays present in this spectrum where used to determine gamma-ray intensities. Select peaks are labelled in energies of kev. 54

81 3.6.2 Intensities from the Gamma-Gamma Coincidence Data Often, a gamma ray was not clearly visible in the gamma-singles spectrum. This may be because the gamma ray was too weak to be observed above the Compton background or possibly because the peak area of the desired gamma ray was obscured by that of another gamma ray with a similar energy. In this situation it was necessary to determine the intensity of that gamma ray from the gamma-gamma coincidence data. The gamma-gamma matrix is presented in Fig. 3.8, roughly 430 million gamma-gamma coincidence events were detected in this experiment. These intensities were determined by gating from below the gamma ray of interest, as was developed by Kulp et al. [42] in their investigation of the decay of 154 Eu. A simplified level scheme presented in Fig shows a gamma ray (γ 1 ) populating an excited state that is depopulated by a second gamma ray (γ 2 ). A gamma spectrum created by taking a time-correlated gate on γ 2 would show a peak of γ 1. In the gating from below method the gamma-ray intensity (I γ1 ) of γ 1 is determined by eqn. 3.8 where N γ1 is the area of the peak corresponding to the number ofγ 1 coincidences observed in the gate taken on γ 2, ɛ γ1 and ɛ γ2 are the efficiencies of the GRIFFIN Spectrometer at the energies of γ 1 and γ 2 respectively, BR γ2 is the gamma-ray branching ratio of γ 2, and N is a scaling factor specific to the data set. Figure 3.11: A simple level scheme in which a gamma ray (γ 1 ) populates an excited state that is then depopulated by γ 2. A gate taken from below γ 1 on γ 2 would show how many times γ 1 was observed in coincidence with γ 2. I γ1 = N γ1 ɛ γ1 ɛ γ2 BR γ2 N (3.8) It is possible to calculate the scaling factor N if I γ1 and BR γ2 have already been measured from the gamma-singles data for a pair of coincident gamma rays by re-arranging Eqn. 3.8 into Eqn N = N γ1 ɛ γ1 ɛ γ2 BR γ2 I γ1 (3.9) 55

82 Table 3.13: Determination of N from the intensity (I γ ) of the 1229 kev fit from the gammasingles data, given the number of counts in the 1229 kev peak (N 1229 ) observed in a gate taken on the 1346 kev gamma ray. The efficiency of GRIFFIN at 1229 (ɛ 1229 ) and 1346 kev (ɛ 1346 ) are also included. I 1229 N 1229 ɛ 1229 ɛ 1346 BR 1346 N 6.29(12) (1358) 1.07(1)x (1)x (590561) In this analysis, the intensity of the 1229 kev gamma ray was fit directly from the gamma-singles data and BR γ for the 1346 kev gamma ray is 1, so N can be determined (Table 3.13). To ensure that N was energy independent, different pairs of coincident gamma rays with varying energies were evaluated. For each pair, the intensity of the feeding gamma ray was fit in the gamma-singles data. This allowed for the intensity determined from the gating from below method to be compared. Select pairs of gamma rays are presented in Table In all cases the difference between the intensities measured from the gammasingles and the gamma-gamma coincident data is approximately zero, as the error on the difference is larger than each individual difference. 56

83 Table 3.14: For each of the following pairs of coincident gamma rays, the gamma-gamma intensity was determined with the gating from below method, where a gate was taken on the draining gamma ray so that the intensity of the feeding gamma ray could be determined. In each case the gamma-gamma intensity is equivalent to the measured intensity of the feeding gamma ray from the gamma-singles data. 57 Draining Gamma Ray (kev) Feeding Gamma Ray (kev) Gamma-Gamma Int. Gamma-Singles Int. Difference (11) 0.173(3) 0.007(11) (18) 0.422(8) 0.007(20) (5) 1.42(3) 0.02(6) (11) 3.11(6) 0.05(12) (18) 4.14(13) 0.13(23) (9) 2.27(4) 0.026(10) (15) 0.298(6) 0.003(17)

84 Table 3.15: The total gamma ray intensity observed feeding the 46 Ca ground state from 15 gamma rays. Gamma-Ray Energy (kev) Relative Intensity (6) (98) (18) (4) (4) (4) (19) (5) (7) (12) (16) (12) (35) (43) (22) Total Intensity (7) Absolute Intensity Normalization The intensities of all of the gamma rays placed in the 46 Ca level scheme in Table 3.12 are reported relative to the intensity of 1346 kev gamma ray. The absolute intensity of each gamma ray per 100 decays of the 46 K nucleus was also determined. Table 3.15 shows the total gamma-ray intensity feeding the 46 Ca ground state from 15 observed gamma rays. This total observed feeding to the ground state was normalized to 100 decays of the 46 K. An absolute intensity normalization factor (A γ ) was determined for the 1346 kev gamma ray from eqn This normalization factor can also be applied to all of the gamma rays in the 46 Ca level scheme. From this work A γ is (6). Previous works have reported this value to be 1.00 [26] and 0.90 [25]. A γ = I 1346 Iγgs (3.10) 3.7 Beta Feeding To determine the beta feeding from the 46 K 2 ground state to excited states in 46 Ca the gamma-ray intensity balance of each excited state was investigated. Any unobserved intensity populating an excited state was then attributed to beta feeding from the parent nucleus.the intensity populating and depopulating each excited state is presented in Table If the error on the missing intensity was found to be larger than the value itself, it 58

85 was assumed that there was no missing intensity feeding that particular excited state. The total missing intensity populating all of the excited states is equivalent to the total beta feeding intensity. The beta feeding percentages and logf t values are presented in Table 3.17 for each excited state. This use of this method requires that there is no beta feeding to the 46 Ca ground state. Previously, Parsa and Gordon [25] measured the beta continuum for this decay and did not observe any intensity above 6.5 MeV. Due to the Q β value of this decay of (16) MeV [24], this would mean that no states are fed with energies lower than the 1346 kev state. It was also possible to observe the beta continuum in the analysis of this data with the PACES detection system as is shown in Fig There is minimal beta intensity observed for energies greater than 6 MeV, so there is no beta feeding observed to be populating the 46 Ca ground state Parity Assignments It is possible to make parity (π) assignments for the excited states in 46 Ca observed to be populated from the beta decay of 46 K 2 ground state based on the previously discussed beta decay selection rules (Table. 2.1). The 46 Ca excited states are populated from the 2 ground state of the 46 K parent, which has negative π. This means that for any observed allowed beta transition (logft = 4.5-6), as there is no change in π between the involved states of the parent and daughter nucleus, any state populated in 46 Ca would have negative π. For any observed first-forbidden beta transitions (logft = 6-10), there is a change in π between the decaying state of the parent nucleus and the populated state in the daughter nucleus, so any state populated in 46 Ca would have positive π. These assignments are listed in Table Comparison to Previous Measurements There are three previous measurements of the beta decay of the 46 K 2 ground state, as was previously discussed in sec The large discrepancies present between the reported decay schemes of Parsa and Gordon [25] and Yagi et. al [26] can now be rectified by the results of this work. In the gamma-singles measurement from P. Kunz et al. [27] an additional 33 gamma-ray transitions were observed that had not been previously placed into the 46 Ca level scheme by either Parsa and Gordon [25] or Yagi et. al [26]. As a result of this analysis, 30 of these gamma rays were placed in the level scheme. The kev gamma ray was not observed in this data set and was therefore not placed in the day scheme. Additionally, it was determined that the kev gamma ray observed by P. Kunz et. al is a single-escape peak corresponding to the 4189 kev gamma ray. A comparison of the intensities observed as a result of this analysis are compared to those that were reported from the three previous beta-decay measurements in Table

86 60 Figure 3.12: The beta continuum as observed for the decay of the 46 K 2 ground state as observed with the PACES array. The Q β value for this decay is (16) MeV. The absence of beta intensity beyond 6 MeV indicates that there is no population of the 46 Ca ground state.

87 Table 3.16: The total gamma-ray intensity that populates and depopulates each excited state in 46 Ca is presented with any unobserved feeding. Any unobserved intensity populating an excited state is attributed to beta feeding from the 46 K 2 ground state. Excited State Energy (kev) Intensity In Intensity Out Missing Intensity (15) (25) (10) 0.437(8) 0.402(8) (8) 6.29(12) 1.01(15) (18) 5.50(13) 0.47(22) (2) 6.72(16) 0.299(159) (10) 0.704(17) 0.211(19) (5) 0.147(6) (81) (19) 0.581(29) 0.438(29) (35) 0.175(8) (9) (2) 5.54(13) 4.34(13) (12) 6.98(21) 4.75(24) (13) 0.355(8) (15) (11) 3.15(6) 2.92(6) (8) 0.758(22) 0.503(23) (3) (19) (19) (10) 40.3(10) 40.1(10) (7) 0.122(5) 0.952(5) (18) (17) (26) (28) 5.71(28) (8) 3.22(8) (15) 4.48(15) (3) 1.15(3) (9) 0.354(9) (4) 0.145(4) (4) 0.133(4) (28) 0.807(28) (19) 0.687(19) (23) 0.438(23) (9) 0.154(9) (18) (18) (25) (25) (16) (16) (42) (42) (25) (25) (5) 0.155(5) (8) (8) (12) (12) (25) (25) (31) (31) (44) (44) (97) (97) (37) (37) Total Missing Intensity 105.7(27)) 61

88 Table 3.17: The beta feeding observed in this data set compared to the two previous measurements. In total, 42 excited states were observed and 39 of them are populated by the beta decay of the 46 K 2 ground state. The J π values shown here are those that have been previously reported from Ref. [24]. E level J π % logft %[25] %[26] (25) 6.88(5) (13) 8.41(17) (14) 7.95(7) (21) 8.11(21) (15) 8.04(24) (2) 8.18(4) (3 ) 0.41(3) 7.69(4) (8) 7.86(2) (16) 6.47(2) (30) 6.40(3) (9) 6.62(2) 4487 (4 + ) 0.476(2) 7.35(2) (2 + ) 0.021(2) 8.37(4) 5052 (4 + ) 37.9(13) 5.08(2) (5) 7.57(3) (3 ) 5.39(14) 5.96(2) (11) 5.91(2) 5535 (4 + ) 4.24(11) 5.67(2) (4) 6.09(2) (1) 6.50(2) (5) 6.94(2) 6032 (4 + ) 0.125(3) 6.70(2) 6111 (2 + ) 0.76(3) 5.90(3) 6245 (4 + ) 0.65(2) 5.75(3) (2) 5.88(4) (9) 6.23(4) (2) 6.56(3) 6540 (0 + ) 0.057(3) 6.43(4) (2) 6.54(4) (4) 6.09(4) (7) 6.94(5) (6) 5.74(4) (8) 6.66(4) (12) 6.49(4) (3) 6.03(4) (2) 7.43(6) (5) 6.62(3) 7034 (2 + ) (9) 6.76(6) (37) 6.69(5) 62

89 Table 3.18: A comparison of select gamma rays observed in this data set to the three previous measurements. The intensities are reported relative to the 1346 kev transition. This Work Kunz et al.[27] Parsa & Gordon [25] Yagi et al. [26] E γ (kev) Int.(%) E γ (kev) Int.(%) E γ (kev) Int.(%) E γ (kev) Int.(%) (5) (11) (19) (13) (4) (23) (9) (6) (14) (18) (5) (6) (7) (8) (13) (1) (7) (5) (6) (5) (5) (12) (1) (11) (11) (10) (9) (14) (1.3) (16) (6) (10) (6) (5) (29) (5) (30) (7) (32) (6) (66) (4) (6) (3) (29) (3) (42) (12) (1.3) (4) (5) (16) (6) (4) (5) (8) (1) (3) (5) (4) (4) (3) (4) (8) (6) (9) 63

90 Chapter 4 46 K Half-Life Measurement 4.1 Previous Measurements Previous to this study, there were three measurements of the 46 K beta decay, as was previously discussed in Section 1.3. Each measurement reported a half-life (T 1/2 ) for the decay of the 46 K 2 ground state, as are listed in Table 4.1. Currently, the Nuclear Data Sheets for A = 46 [24] reports T 1/2 = 105 s for the decay of 46 K, as an average of the two earliest measurements from Parsa and Gordon [25] and from Yagi et al. [26]. This value does not take into account the recent precision measurement of (79) s from P. Kunz et al. [27]. This recent measurement is clearly in better agreement with the value reported by Yagi et al.. The beta decay curve of 46 K from P. Kunz et al. is presented in Fig Table 4.1: Previous measurements of the half-life (T 1/2 ) of the beta decay of 46 K 46 Ca. Currently the Nuclear Data Sheets for A = 46 reports T 1/2 = 105 s [24], as an average of the two earliest measurements from Parsa and Gordon [25] and from Yagi et al. [26]. Measurement T 1/2 (s) Parsa and Gordon (1966) [25] 115(4) Yagi et al. (1968) [26] 95(5) P. Kunz et al. (2014) [27] (79) 4.2 Measurement from this Analysis In a beta decay measurement it is possible to indirectly measure the T 1/2 of the decay of the parent nucleus by directly measuring the decay of an excited state populated in the daughter nucleus via the changing intensity of the depopulating gamma ray. This is due to the fact that T 1/2 of the excited state in the daughter nucleus is very short and insignificant relative to the T 1/2 of the beta decay itself. 64

91 Figure 4.1: The 46 K beta decay curve as was published by P. Kunz et al. [27]. The residuals indicate that there are no significant dead time distortions at higher count rates. The fit of this decay curve resulted in a reported T 1/2 = (79) s half-life for the beta decay of 46 K 46 Ca. This figure is reprinted with permission of Ref. [27]. 65

92 Decay Time (s) Energy (kev) Figure 4.2: The E γ versus time matrix used to establish the beta-decay T 1/2 of the 46 K 2 ground state. In this measurement, the beta decay T 1/2 of the 2 46 K ground state was determined by measuring the decay of the 1346 kev excited state in 46 Ca. The 1346 kev excited state is directly populated in the beta decay with an intensity of 32.3(25)%, see Table 3.16, and is solely depopulated by the kev gamma ray. To determine the T 1/2, the change in the intensity of the 1346 kev gamma ray was fit over a given decay period. The data from Run-2298 was chosen as its decay time of 1200 s was the longest of any run collected during the experiment. This run contains 22 decay periods, which were summed together into a matrix of gamma-ray energy versus decay time (in 1 s bins) as is plotted in Fig A gate was taken between the energies of kev and the decay spectrum of the 1346 kev gamma ray was extracted. To account for the Compton-scatter events that were also present in the gate, a second gate was taken between energies The time behaviour of the Compton-scatter events was fit with A γ (t) = Ae Bt + C, (4.1) where A γ (t) is the number of scatter events at any given point of time (t). The resultant fit parameters for A, B, and C are listed in Table 4.2. The ground state of 46 Ca is stable and there was no evidence of any isobaric contamination in the 46 K beam, therefore no additional background decays were taken into account while fitting the decay of the 1346 kev gamma ray. 66

93 Table 4.2: Resultant fit parameters for Eqn. 4.1 with χ 2 /ν=1.14. Parameter Value A 2180(74) B (9) C 15.9(2) Table 4.3: The final parameters fit with Eqn. 4.2, with χ 2 =0.97, for the decay of the 1346 kev gamma ray from 300 to 1200 s. Parameter Value A (330) events λ (5) /s The decay of the 1346 kev gamma ray was then fit with A γ (t) = A 0 e λt + BG (4.2) to extract the decay rate λ. Here, A 0 is equivalent to the initial number of events of the 1346 kev gamma ray, and BG is the correction to the fit with Eqn. 4.1 where A, B, and C are the fit parameters for the Compton-scatter background listed in Table 4.2. The fit of the decay of the 1346 kev gamma ray was done between 300 and 1200 s to extract the decay rate λ. The final fit is plotted in Fig 4.3 and the fit parameters are listed in Table 4.3. The associated residuals are plotted in Fig. 4.4, which show a consistent behaviour between 300 and 1200 s. The T 1/2 of the decay of 46 K was then determined to be s with a statistical error from the fit of s. This was calculated from T 1/2 = ln2 λ, (4.3) given that λ = (5) /s. This is in agreement with the previous precision measurement of (79) s from P. Kunz et al. [27]. To investigate any potential systematic effects on this measurement, additional fits of the decay of the 1346 kev gamma ray were made with the data binned into 2 and 4 s bins and a chop analysis was performed. This probes any rate-dependent effects that may be present by changing the fit region used. This type of analysis has been used previously for the measurements of T 1/2 values for neutron-rich cadmium isotopes with GRIFFIN [43]. The start-time of the fit was allowed to vary between 300 and 600 s with a fixed end-time of 1200 s; the resultant T 1/2 from each fit is presented in Table 4.4. Additionally, the analysis was performed with the end-time of the fit allowed to vary between 600 and 1200 s with a fixed start time of 300 s; the resultant T 1/2 from each of these fits are presented in Table

94 Decay of the 1345 kev Gamma Ray Events Compton-Scatter Events Decay Time (s) Figure 4.3: The decay of the 1346 kev gamma ray is plotted in black with the fit of Eqn. 4.2 shown in yellow. The events shown in red are Compton-scatter background, with the fit of Eqn. 4.1 shown in black. From this fit, T 1/2 = (66) s. Residual Decay Time (s) Figure 4.4: The associated residuals of the initial fit of the 1346 kev gamma ray decay from 300 to 1200 s given the fit parameters of Eqn. 4.2 listed in Table 4.3. The behaviour of the residual is consistent throughout the boundaries of the fit. 68

95 Each analysis was performed with time binned into 1, 2, and 4 s bins to see if there was a statistically significant change in the resultant fit of the T 1/2. There is a clear systematic effect on the resultant T 1/2 dependent on the start time of the fit as it systematically decreases with increasing start time (Fig. 4.5). To account for this effect an additional systematic error of s was added to create agreement between the value of T 1/2 =96.418(66) s from the s fit of the decay of the 1346 kev gamma ray and the T 1/2 values that result from the chop analysis. Therefore, the reported T 1/2 of the beta decay of the 2 ground state of 46 K from this measurement is 96.41(10) s. The weighted average of the reported 46 K half-lives from the three previous measurements with this newly reported value is (79) s. Table 4.4: Chop analysis performed for the fit of the decay of the 1346 kev gamma ray with varying start times and a constant stop time of 1200 s to look for possible systematic effects associated with the start time of the fit. This analysis was performed with time binned into 1, 2, and 4 s bins. It is clear that the resultant T 1/2 is dependent on the start time of the fit, as it systematically decreases with increasing start time. The data presented here is plotted in Fig. 4.5 for visual inspection of the systematic behaviour of the data. 1 s/bin 2 s/bin 4 s/bin Starting Bin (s) T 1/2 (s) T 1/2 (s) T 1/2 (s) (7) 96.41(7) 96.42(7) (8) 96.30(8) 96.31(8) (10) 96.36(10) 96.36(10) (12) 96.32(12) 96.32(12) (15) 96.17(15) 96.16(15) (17) 96.13(18) 96.13(18) (23) 96.16(23) 96.15(23) Measurements of Other Gamma-Rays This work has greatly expanded the 46 Ca level scheme with the addition of many new gamma rays and excited states. It is possible to fit the decay of these excited states, as was described previously in Sec. 4.2, to confirm that they are from the decay of the T 1/2 =96.41(10) s 2 ground state of 46 K. This analysis would also expose any possible long-lived isomeric states that may be present in the level scheme. It was statistically possible to fit the decay of the strongest gamma ray depopulating 30 of the 42 excited states observed in the level scheme. The resultant T 1/2 of each fit are presented in Table 4.6 and are plotted in Fig 4.6. The weighted average of these fits is T 1/2 =96.50(7) s, this value includes the additional systematic error of s for each gamma ray fit. There are no indications that any of these excited states do not belong to the 46 Ca level scheme or that there are any isomeric states present. 69

96 Table 4.5: Chop analysis performed for the fit of the decay of the 1346 kev gamma ray with a constant start time of 300 s and a varying stop time to look for possible systematic effects associated with the end time of the fit. This analysis was performed with time binned into 1, 2, and 4 s bins. This shows that there is no systematic behaviour of the resultant T 1/2 associated with the end time chosen for the fit. 1 s/bin 2 s/bin 4 s/bin Ending Bin (s) T 1/2 (s) T 1/2 (s) T 1/2 (s) (7) 96.41(7) 96.42(7) (7) 96.41(7) 96.41(7) (7) 96.41(7) 96.42(7) (7) 96.41(7) 96.41(7) (7) 96.42(7) 96.42(7) (7) 96.42(7) 96.40(7) (7) 96.43(7) 96.41(7) (8) 96.42(7) 96.41(7) (8) 96.42(7) 96.42(7) (9) 96.47(8) 96.42(8) (9) 96.46(9) 96.44(8) (10) 96.51(9) 96.47(9) (12) 96.56(10) 96.50(9) 70

97 71 Figure 4.5: A visual plot of the chop analysis data for fits with varying start time between 300 and 600 s. There is a clear systematic decrease in the resultant T 1/2 from fits with that are performed with later start times. The fit T 1/2 = s is drawn with the black solid line, with the T 1/ s represented by the purple dashed line. To create agreement between the fit T 1/2 and the resultant fits of the chop analysis, an additional systematic error of s must be taken into account. The total error of 0.10 s is drawn as the blue dashed line.

98 Table 4.6: The resultant T 1/2 fit for the strongest gamma ray depopulating each excited state observed in 46 Ca (when statistically possible to fit). The listed numbers (#) correspond each fit to the plot shown in Fig # E Level (kev) E γ (kev) T 1/2 (s) # E Level (kev) E γ (kev) T 1/2 (s) (10) (28) (16) (53) (37) (33) (51) (64) (27) (21) (25) (14) (92) (13) (15) (11) (14) (48) (41) (55) (47) (36) (70) (46) (85) (84) (24) (30) (16) 72

99 73 Figure 4.6: The resultant T 1/2 fit for the strongest gamma ray depopulating each excited state observed in 46 Ca (when statistically possible). The data plotted here is listed in Table 4.6. Comparisons of these values to the value of T 1/2 =96.41(10) s, the plotted solid black line, show no indications that there is any contamination or isomeric states present. The dashed blue line represents is the 1σ error of 0.43 s from the weighted average of T 1/2 =96.50(7) s considering all of the fit values.

100 Chapter 5 Angular Correlations of Successive Gamma Rays 5.1 Theoretical Description There is a directional correlation between two gamma rays that are emitted in succession, which is unique to the multipolarities of the transitions involved in the cascade. Experimental measurements of these correlations can give insight as to the spins of the participating nuclear states. Each nuclear state with angular momentum J contains a set of m-states, m J = J, J + 1, J + 2,...J 2, J 1, J, (5.1) that have degenerate energies [2]. The de-excitation of one nuclear excited state to another is actually a transition between different pairs of m-states as is shown for a dipole transition in Fig For a dipole transition, there are three possible m-state transitions that can occur. Each of these is emitted with a characteristic anisotropic angular distribution. The observed angular distribution of radiation emitted from the transition of m i m f is W (θ) = p(m i )W mi m f (θ), (5.2) m i Table 5.1: The specific angular distribution W mi m f (θ) for emitted radiation between pairs of m-states in Fig m i m f W mi m f (θ) (1+cos2 θ) 0 0 sin 2 θ (1+cos2 θ) 74

101 Figure 5.1: A depiction of how the decay J i J f is actually the result of decays between m-states. Each transition between pairs of m-states is emitted with an anisotropic angular distribution, but if all the initial m-states are evenly populated, then only an isotropic distribution of radiation will be observed. Reprinted with permission of Ref. [44]. where p(m i ) is the population of the initial state. For the example shown in Fig. 5.1, the specific angular distribution W mi m f (θ) for each transition between pairs of m-states are listed in Table 5.1. If each initial state (m i ) is equally populated, such that p(m 1 ) = p(m 0 ) = p(m 1 ) = 1 3, then the observed angular distribution W (θ) 1 3 [1 2 (1 + cos2 θ)] (sin2 θ) [1 2 (1 + cos2 θ)], (5.3) is constant at all angles θ; no anisotropic angular distribution will be observed as the gamma rays are emitted isotropically. To observe an anisotropic distribution of radiation, it is necessary to create an uneven population in the initial m-states. This can be done by detecting the previous radiation that preciously populates those initial m-states [2]. To continue with the example of the dipole radiation in Fig. 5.1, suppose that J i was populated from another excited state with spin J 0 = 0, so that there is a cascade of gamma rays γ 1 and γ 2 (drawn in Fig. 5.2). If γ 1 is observed in a detector at a particular location, then a z-axis can be defined along the emission axis of γ 1, such that γ 2 would then be observed at an angle θ 2 with respect to that axis (shown in Fig. 5.2). By definition, the angle of γ 1 (θ 1 ), with respect to the z-axis, is 0. When θ 1 =0, then W m0 m 0 (θ 1 ) = sin 2 θ 1 = 0, which means that m j =0 cannot be populated. Now an uneven population of m-states corresponding to J i has been created. The observed anisotropic angular distribution is W (θ 2 ) 1 2 [1 2 (1 + cos2 θ 2 )] + 0(sin 2 θ 2 ) [1 2 (1 + cos2 θ 2 )] 1 + cos 2 θ 2, (5.4) 75

102 Figure 5.2: (Left) An angular correlation measurement of a cascade of successive gamma rays γ 1 and γ 2. The z-axis is defined as the direction of the emission of γ 1, and θ 2 is the direction of detection of γ 2 with respect to the z-axis. (Right) A simple level scheme showing the different m-states that are involved in the decay of the cascade. This figure has been reprinted with permission from Ref. [44]. which is not constant at all angles (plotted in Fig. 5.3). as In general, the angular correlation between two successive gamma rays can be written W (θ) = 1 + A 2 P 2 (cosθ) + A 4 P 4 (cosθ), (5.5) where P 2 and P 4 are Legendre polynomials, and the coefficients A 2 and A 4 are dependent on the spins of the nuclear states involved in the cascade, as well as the angular momenta and the respective mixing ratios δ of each gamma ray [45]. Each type of cascade of gamma rays (0 1 0, 0 2 0, etc.) has a unique angular correlation with characteristic A 2 and A 4 coefficients. Examples for select cascades are presented in Table 5.2. A comparison of theoretically predicted A 2 and A 4 coefficients to fits of experimental data can be used to make spin assignments for the nuclear states involved in that particular cascade. 5.2 GRIFFIN Angular Properties and Analysis Methods When two successive gamma rays are detected in coincidence, the angle between them is simply the angle between the two detectors in which those gamma rays were detected. The GRIFFIN spectrometer contains 64 HPGe crystals for the detection of gamma rays, such that there are 4096 (64 x 64) possible detector pairs. Given the geometry of GRIFFIN, these detector pairs can be grouped into 52 unique angles ranging between 0 and

103 W(θ) cos(θ) Figure 5.3: A plot of the angular correlation expected to be observed for a gamma ray cascade. Table 5.2: Theoretical A 2 and A 4 coefficients used to describe the angular correlation of each listed cascade as per Eqn Cascade A 2 A

104 These angles are listed in Table 5.3 along with the corresponding number of detector pairs (weights) of each. Note that angular index 0 is a detector paired with itself, such that the two gamma rays would both interact with the same detector. In this case the HPGe crystal would not be able to distinguish the two energies and would only detect their sum. For this reason, angular index 0 was not used for the angular correlation analysis, so only 51 angles were considered. Table 5.3: The 52 angular indexes (labelled 0 to 51) that exist between the 64 GRIFFIN HPGe crystals. The angle in degrees and the number of detector pairs (weights) at each angle are listed. Angular Index Angle (degrees) Weight Angular Index Angle (degrees) Weight In the analysis of this data, 51 gamma-gamma coincidence matrices were created that corresponded to each of the 51 unique angles present between detector pairs given the GRIFFIN geometry (ignoring angular index 0). These matrices were created with the requirement that the two gamma rays be detected in coincidence within 250 ns of each other. Time-random background was subtracted from each matrix to eliminate counts from 78

105 3 2 Figure 5.4: The two-dimensional array of angular index versus gamma-ray energy created when a gate was taken on the 1346 kev gamma ray. The array shows the number of times a gamma ray was detected in coincidence (within a 250 ns time window) with the 1346 kev gamma ray at each of the 51 angular indexes. false coincidences. Once the matrices were created, it was then possible to investigate angular correlations for pairs of successive gamma rays. The angular correlations were created by taking a gate on one of the two gamma rays in each of the 51 angular matrices, then the peak area of the second coincidence gamma ray was fit in each gate. It was more advantageous to take the initial gate on the weaker of the two gamma rays, and then fit the peak area of the more intense gamma ray. The resultant gates from each angular index were then summed together into a two-dimensional array. As an example, the array created when a gate was taken on the 1346 kev gamma ray is shown in Fig The kev Gamma-Ray Cascade To continue the discussion of the analysis technique it is useful to highlight a particular cascade of successive gamma rays. The direct coincidence between the 1346 and 3706 kev gamma rays, the most intense cascade present in the 46 Ca level scheme, serve as an excellent example. The 3706 kev gamma ray directly populates the 1346 kev 2 + level from the

106 (4 ) Figure 5.5: A simple partial level scheme showing the direct succession of the and 1346-keV gamma rays in 46 Ca. Note that the level scheme is not to scale. kev (4 + ) state as is shown in Fig A zoomed-in version of Fig. 5.4 displaying the coincidence events of the 3706 kev gamma ray is shown in Fig 5.6. To create the angular correlation between the and kev gamma rays, the peak area of the 3706 kev gamma ray is fit in the projection of each angular index. The the fit peak area per angular index is plotted in Fig To extract the final angular correlation it is necessary to correct for the differences in detection efficiency between each HPGe crystal, the different weights of detector pairs present at each angular index, and for other possible experimental conditions that could effect the measured angular distribution. In the analysis of this data the angular correlations were corrected for these effects with the use of an event mixing technique [46], discussed in Sec Event Mixing Technique An angular-correlation measurement is simply the measurement of the number of coincidence events between a pair of successive gamma rays detected at different angles relative to the source of the radiation. However, it is necessary to take into account any potential systematic effects that may affect the shape of the observed angular distribution. Any systematic effects observed for correlated events would also be present in an angular correlation constructed for uncorrelated events. An uncorrelated event contains gamma rays that are from the decays of two separate nuclei, rather than from a successive decay in a 80

107 Figure 5.6: A zoomed-in version of Fig. 5.4 to show the number of events of the 3706 kev gamma ray in coincidence with the 1346 kev gamma ray at each of the 51 angular indexes. single nucleus. There is no angular correlation present between two uncorrelated gamma rays, and the resultant observed distribution of such events is isotropic. The event mixing technique (EMT) was originally established to determine the magnitude of uncorrelated background events present in an experimentally measured angular correlation [46]. The basis of the technique is to artificially generate an uncorrelated background by creating a data set of uncorrelated events, where each event is specifically taken from a different decay of the nucleus. Any measured angular distribution between a pair of these uncorrelated events will contain all of the same systematic experimental effects present for correlated gamma rays. The event-mixed background can be used to correct for such effects. The experimentally observed angular distribution between two gamma rays with energies E a and E b can be given as w(θ, E a, E b ) = θ= θ i θ j ij ɛ i (E a )ɛ j (E b ) X ij (θ)w (θ)dθ, (5.6) where w(θ, E a, E b ) is the measured experimental angular correlation, ɛ i (E a ) and ɛ j (E b ) are the efficiency corrections for specific detector pairs ij at energies E a and E b, respectively, X ij (θ) is the weight of detector pairs ij at each angle θ, and W (θ) is the theoretical angular correlation. Then, the observed angular distribution between two uncorrelated gamma rays 81

108 Figure 5.7: Plot of the fit peak areas of the 3706 kev gamma ray in coincidence with the 1346 kev gamma ray at each angular index. with the same energies E a and E b from generated the event-mixed background can be given as y(θ, E a, E b ) = θ= θ i θ j ij ɛ i (E a )ɛ j (E b ) X ij (θ)y (θ)dθ, (5.7) where y(θ, E a, E b ) is the observed experimental angular distribution and Y (θ) is the theoretical isotropic distribution. Note that the theoretical isotropic distribution Y (θ) in Eqn. 5.7 is modified by the same experimental factors (ɛ i (E a ), ɛ j (E b ), and X ij (θ)) as the theoretical angular distribution W (θ) in Eqn As both distributions were measured by the same detector system, it is possible to correct for these experimental effects in w(θ, E a, E b ) by dividing by y(θ, E a, E b ). The corrected experimental angular correlation w corr (θ, E a, E b ) is written as w corr (θ, E a, E b ) = w(θ, E a, E b ) y(θ, E a, E b ) = Xij (θ)w (θ)dθ Xij (θ)y (θ)dθ. (5.8) As a continuation to the discussion of the analysis of the successive cascade of the 3706 and 1346 kev gamma rays from Sec , the uncorrelated angular distribution created from the event-mixed background between the two gamma rays in shown in Fig In order to obtain the corrected angular correlation between the 1346 and 3706 kev gamma rays it is necessary to divide the distribution presented in Fig. 5.7 by the event-mixed background in Fig The resultant final angular correlation, fit with Eqn. 5.5, is drawn in Fig A further discussion of this cascade and the results of the fit of the angular correlation is presented in Sec

109 Figure 5.8: The uncorrelated angular distribution between the 3706 and 1346 kev gamma rays created from the event-mixed background. These events are used to normalize the events shown in Fig 5.7, to give the angular correlation presented in Fig χ 2 /ν Minimization Analysis To make spin assignments for the observed nuclear states in the level scheme of 46 Ca a χ 2 /ν minimization analysis was performed. As was mentioned in Sec. 5.1, there is a theoretically predicted angular correlation W theo (θ) with a unique combination of A 2 and A 4 coefficients (Eqn. 5.5) for each type of successive gamma-ray cascades. If the spins of the nuclear states involved in the gamma-ray cascade are not known, then the experimental data can be fit with different theoretical angular correlations to determine which has the best fit to the data. The fit function used was W (θ) = A 0 + A 2 P 2 (cosθ) + A 4 P 4 (cosθ), (5.9) where A 2 and A 4 were fixed to the theoretically predicted values for specific pairs of J π, and A 0 was allowed to vary. A goodness-of-fit test was performed to extract the reduced χ 2 /ν between the theoretical W exp (θ i ) and experimental W theo (θ i ) values at each angle θ i. The reduced χ 2 /ν is then calculated from χ 2 /ν = 1 (W exp (θ i ) W theo (θ i )) 2 ν σw exp (θ i ) 2, (5.10) i 83

110 Figure 5.9: The measured angular correlation between the successive 3706 and 1346 kev gamma rays for the 51 GRIFFIN angular indexes. The data has been corrected for experimental effects with the use of the event mixing technique, and has been fit with Eqn The error bars on the experimental data points are statistical. This correlation is discussed further in sec where σw exp (θ i ) is the variance in W exp (θ i ). As was discussed in Sec , it is possible for gamma rays to contain a mixture of electric and magnetic character. For a particular gamma ray the ratio of electric to magnetic character is defined as the mixing ratio δ (Eqn. 2.36). The value of δ for each gamma ray involved in an angular correlation affects the theoretically predicted A 2 and A 4 coefficients used to describe that decay. An example of the drastically different theoretical angular correlations predicted for a cascade with different values of δ used to describe the gamma rays is shown in Fig In performing the χ 2 /ν minimization analysis to make spin assignments, it was necessary to take into account different values of δ for the gamma rays involved in each potential cascade type. To simplify this analysis, cascades were chosen where the multipolarity and δ of at least one of the gamma rays was previously reported. Plots of χ 2 /ν versus δ were created for each cascade type per pair of successive gamma rays analysed to determine which W theo (θ i ) was the best fit to the experimental data. 84

111 Values δ 1 Figure 5.10: Theoretically predicted A 2 and A 4 coefficients for a gammaray cascade given different values of δ for each gamma ray. Reprinted with permission of Ref. [44]. 5.3 Spin Assignments for Excited States in 46 Ca In the analysis of this data, 42 excited states were observed in the level scheme of 46 Ca. Previous spin assignments have been made for 19 of these states from transfer reactions [24], most notably from (p, p ) [21], (t, p) [22], and (p, t) [23] reactions. To make spin assignments for these states, angular correlations were created and normalized with eventmixed background for pairs of successive gamma rays as was discussed in Sec and Sec Then a χ 2 /ν minimization analysis (Sec ) was performed to deduce the best fit of the experimental data to theoretical angular correlations. This analysis was limited by statistics, so it was not possible to create angular correlations for all pairs of cascading gamma rays observed in the data set. It was possible to investigate cascades depopulating 18 of the 41 excited states (not including the 1346 kev state). The final results are presented in Table 5.4. A discussion of the results of this analysis per investigated excited state follows. 85

112 Table 5.4: A summary of the angular correlation analysis from this work. The spin assignments of 18 excited states were investigated by looking at angular correlations from a cascade of two coincidence gamma rays γ 1, and γ 2. There are 12 reported spin assignments (J P ) from previous measurements [24]. The spin assignments (J) reported from this work are shown in comparison. It was assumed that γ 1 was a pure transition with δ 1 = 0, and that γ 2 could potentially be a mixed transition described by δ 2. The experimental data was fit with two different variations of Eqn For the experimental fit, the A 2 and A 4 parameters were allowed to vary. Whereas for the theoretical fit, the A 2 and A 4 parameters were fixed to theoretical values for the particular cascade listed taking into account δ Experimental Fit Theoretical Fit E level J P J Cascade γ 1 (kev) γ 2 (kev) δ 2 A 2 A 4 χ 2 /ν A 2 A 4 χ 2 /ν (41) 1.03(4) (3) 0.015(5) (8) 0.349(7) 0.021(9) (5) (6) (11) 0.146(13) 0.034(17) (15) 0.122(10) (13) (9) (13) (10) (20) , (3) 0.022(13) 0.010(16) (5) 0.232(6) (75) x (25) (4) ,2, (41) (53) (3) Table 5.4 continued on next page

113 87 Table 5.4 continued from previous page Experimental Fit Theoretical Fit E level J P J Cascade γ 1 (kev) γ 2 (kev) δ 2 A 2 A 4 χ 2 /ν A 2 A 4 χ 2 /ν (22) (29) (4) 2, (10) 0.134(11) 0.014(15) (15) (4) (2) 0.002(3) (6) (3) (10) (59) 0.011(8) (2) (27) , (18) (24) (4) (82) (10) , (17) (20) ,2, (21) 0.058(28) (3)

114 5.3.1 The 2422 kev State In previous measurements, the 2422 kev state was assigned J π = 0 + [24]. In this work it was possible to create an angular correlation for the kev gamma-ray cascade to investigate this assignment. The angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit of theoretical gamma-ray cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. This analysis confirms the spin assignment of J = 0 for the 2422 kev state. Figure 5.11: The angular correlation for the kev gamma-ray cascade. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table The 2575 kev State In previous measurements, the 2575 kev state was assigned J π = 4 + [24]. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate this assignment. The angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit of theoretical angular correlation of a cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. A χ 2 /ν minimization analysis was performed that varied δ 1229 of the 1229 kev gamma ray between -1 and 1, with the assumption that the 1346 kev gamma ray is a pure E2 transition. The resultant χ 2 /ν versus δ 1229 plot is drawn in Fig The best fit of χ 2 /ν = 2.13 for the experimental data was obtained at δ 1229 = 0. This analysis confirms the spin assignment of J = 4 for the 2575 kev state. 88

115 Figure 5.12: The angular correlation for the kev gamma-ray cascade that depopulates the 2575 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table 5.4. Figure 5.13: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 1229 was allowed to vary, and the best fit was achieved for δ 1229 = 0. 89

116 5.3.3 The 3021 kev State In previous measurements, the 3021 kev state was assigned J π = 2 + [24]. An angular correlation for the kev gamma-ray cascade was created to investigate the spin J = 3 assignment. The resultant angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit corresponding to a theoretical cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. It was assumed that the 1346 kev gamma ray is a pure E2 transition and a χ 2 /ν minimization analysis was performed that varied the δ 1675 between -1 and 1. The resultant χ 2 /ν versus δ plot is drawn in Fig The best fit was obtained when δ 1675 = -0.15(8) and χ 2 /ν = This analysis confirms the spin assignment of J = 2 for the 3021 kev state The 3611 kev State In previous measurements, the 3611 kev state was assigned J π = 3 [24]. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate the spin J = 3 assignment. The angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit of theoretical angular correlation of a cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. A χ 2 /ν minimization analysis was performed that varied δ 2264 between -1 and 1, with the assumption that the 1346 kev gamma ray is a pure E2 transition. The resultant χ 2 /ν versus δ plot is drawn in Fig The best fit of χ 2 /ν = 1.53 was obtained when δ This analysis confirms the spin assignment of J = 3 for the 3611 kev state The 3638 kev State In previous measurements, the 3638 kev state was assigned J π = 2 + [24]. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate the spin J = 2 assignment. The angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit of theoretical angular correlation of a cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. A χ 2 /ν minimization analysis was performed varying δ 2292 between -1 and 1, with the assumption that the 1346 kev gamma ray is a pure E2 transition. The resultant χ 2 /ν versus δ plot is drawn in Fig The best fit of χ 2 /ν = 1.43 was obtained when δ 2292 = 0.14(11). This analysis confirms the spin assignment of J = 2 for the 3638 kev state The 3857 kev State In previous measurements, the 3857 kev state was assigned J π = 4 + [24]. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate this assignment. The angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit 90

117 Figure 5.14: The angular correlation for the kev gamma-ray cascade that depopulates the 3021 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = for the 2 2 gamma ray. The resultant fit parameters are listed in Table 5.4. Figure 5.15: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 1675 was allowed to vary, and the best fit was achieved for δ 1675 = -0.15(8). 91

118 Figure 5.16: The angular correlation for the kev gamma-ray cascade that depopulates the 3611 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table 5.4. Figure 5.17: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2264 was allowed to vary, and the best fit was achieved for δ

119 Figure 5.18: The angular correlation for the kev gamma-ray cascade that depopulates the 3638 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = 0.14 for the 2 2 gamma ray. The resultant fit parameters are listed in Table 5.4. Figure 5.19: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2292 was allowed to vary, and the best fit was achieved for δ 2292 = 0.14(11). 93

120 Figure 5.20: The angular correlation for the kev gamma-ray cascade that depopulates the 3857 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = 0.04 for the 4 2 gamma ray. The resultant fit parameters are listed in Table 5.4. Figure 5.21: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2510 was allowed to vary, and the best fit was achieved for δ 2510 = 0.03(15). 94

121 of theoretical angular correlation of a cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. A χ 2 /ν minimization analysis was performed varying δ 2510 between -1 and 1, with the assumption that the 1346 kev gamma ray is a pure E2 transition. The resultant χ 2 /ν versus δ plot is drawn in Fig The best fit of χ 2 /ν = was obtained when δ 2510 = 0.03(15). This analysis confirms the spin assignment of J = 4 for the 3857 kev state The 4257 kev State There is no previous spin assignment for the 4257 kev state. An angular correlation was created for the kev gamma-ray cascade to investigate possible spin assignments, as is shown in Fig A χ 2 /ν minimization analysis was performed that varied possible δ 2911 values given potential spins of 1, 2, 3, and 4 for the 4257 kev state (Fig. 5.23). The minimum χ 2 /ν observed for each potential spin assignment were roughly all the same, such that it is not possible to make a firm spin assignment based on the analysis of this data set. However, it is possible to rule out the 4 spin assignment, as that would require δ 2911 = 0.1. This would indicate only 10% E2 character, which is unrealistic as a transition should contain mainly E2 character. The values of δ 2911 for each potential spin assignment are listed in Table The 4386 kev State The 4386 kev state has not been observed in any previous measurements, therefore there is no previous spin assignment. It is possible to rule out a spin 4 assignment for this state, as it is observed to de-excite to the 46 Ca ground state via gamma-ray emission. To investigate potential spin assignments for this excited state, angular correlations were created for two different cascades of gamma rays, specifically the kev and kev gamma-ray cascades (presented in Fig. 5.24). A χ 2 /ν minimization analysis was performed by varying the values of δ for the 776 and 3040 kev gamma rays given potential spin assignments of 1, 2, and 3 for the 4386 kev state. The results are presented in Fig It is possible to rule out the spin 1 assignment due to the poor resultant fit of the kev cascade. However, it is not possible to rule out either of the spin J = 2 or spin J = 4 assignments, as there is a similar minimum χ 2 /ν achieved for fits of both spin types for both cascades of gamma rays. Therefore, a spin assignment of 2 or 4 is suggested as a result of this analysis. The corresponding A 2 and A 4 coefficients and δ values are presented in Table

122 a cascade b cascade c cascade Figure 5.22: The angular correlation for the kev gamma-ray cascade that depopulates the 4257 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0.37 for the 1 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0.11 for the 1 2 gamma ray. (c) The fit shown in red is for a cascade with δ = 0.38 for the 3 2 gamma ray. 96

123 Figure 5.23: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 2911 was allowed to vary. Best fits were achieved for δ 2911 = 0.11(10) for the cascade, and for δ 2911 = 0.38(20) cascade. 97

124 a cascade b cascade 98 c cascade d cascade Figure 5.24: The angular correlations for cascades of successive gamma rays depopulating the 4386 kev state. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = -0.1 for the 2 3 gamma ray. (b) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = 0.7 for the 3 3 gamma ray. (c) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = 0.02 for the 2 2 gamma ray. (d) The angular correlation for the kev gamma-ray cascade, with the theoretical fit shown for a cascade where δ = 0.6 for the 3 2 gamma ray.

125 Figure 5.25: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev (left) and kev (right) gamma-ray cascades. The values of δ were allowed to vary for the 776 and 3086 kev gamma rays. (Left) Best fits were obtained for values of δ 776 = -0.1(3) for the cascade and δ for the cascade. (Right) Best fits were obtained for values of δ 3040 = 0.02(5) for the cascade and δ for the cascade The 4405 kev State In previous measurements, the 4405 kev state was assigned J π = 3 [24]. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate this assignment. The angular correlation is shown in Fig with the fit of Eqn. 5.9 and the fit of theoretical angular correlation of a cascade. The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. It was assumed that the 1229 kev gamma ray is a pure E2 transition and a χ 2 /ν minimization analysis was performed varying the δ 1830 between -1 and 1. The resultant χ 2 /ν versus δ 1830 plot is drawn in Fig The best fit was obtained when δ and χ 2 /ν = This analysis confirms the spin assignment of J = 3 for the 4405 kev state The 4428 kev State The 4428 kev state has not been observed in any previous measurements, therefore no previous spin assignments have been made. This state decays directly to both the ground state and the 2422 kev state, so it is possible to rule out a spin J = 4 assignment. It was possible to create an angular correlation for the kev gamma-ray cascade 99

126 Figure 5.26: The angular correlation for the kev gamma-ray cascade that depopulates the 4405 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade. The resultant fit parameters are listed in Table 5.4. Figure 5.27: A plot of LOG(χ 2 /ν) versus δ for fits of the kev gamma-ray cascade angular correlation with theoretical A 2 and A 4 coefficients. The value of δ 1830 was allowed to vary, and the best fit was achieved for δ

127 a cascade b cascade c cascade Figure 5.28: The angular correlation for the kev gamma-ray cascade that depopulates the 4428 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ -0.3 for the 2 1 gamma ray. (b) The fit shown in red is for a cascade with δ -0.1 for the 2 2 gamma ray. (c) The fit shown in red is for a cascade with δ = -0.4(3) for the 3 2 gamma ray. 101

128 Figure 5.29: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 624 was allowed to vary. Best fits were obtained for values of δ for the cascade, δ for the cascade, and δ 624 = -0.4(3) for the cascade. 102

129 to investigate potential spin assignments for this state as is shown in Fig A χ 2 /ν minimization analysis was performed varying δ 624 with the assumption that the 4428 kev gamma ray is a pure E2 transition. The results of this analysis are presented in Fig There was no difference between χ 2 /ν minima observed for potential spin assignments of 1, 2, and 3. Therefore, from the analysis of this data it is not possible to make a firm spin assignment for the 4428 kev state. The A 2 and A 4 coefficients and δ 624 values for each possible spin assignment are listed in Table The 4432 kev State In previous measurements, the 4432 kev state was assigned J π = 2 + [24]. It was possible to create an angular correlation for the kev gamma ray cascade to investigate this assignment, as is presented in Fig A χ 2 /ν minimization analysis was performed varying the value of δ 1858 with the assumption the 1229 kev gamma ray is a pure E2 transition. The results of this analysis are presented in Fig It is clear that the minimum possible χ 2 /ν is achieved for a spin J = 3 assignment of the 4432 kev state (χ 2 /ν = 0.89), rather than for a spin J = 2 assignment (χ 2 /ν = 4.99). Therefore, the 4432 kev state is given a spin J = 3 assignment as the result of this work. The A 2 and A 4 coefficients and δ 1858 values for each possible spin assignment are listed in Table 5.4. The previous spin J = 2 assignment for the 4432 kev state came as the result of the 44 Ca(t, p) transfer reaction, which populated a state at 4429(10) kev. In this work, two excited states are observed at nearly degenerate energies of 4428 and 4432 kev, with this being the first observation of the state at 4428 kev. The firm assignment of spin J = 3 given as a result of this analysis for the 4432 kev state suggests the possibility that the previous spin J = 2 assignment may in fact be for the excited state observed at 4428 kev. This work was not able to make a firm spin assignment for the 4428 kev state from the χ 2 /ν minimization analysis, but a spin J = 2 assignment is very likely The 4487 kev State In previous measurements, the 4487 kev state was assigned J π = (4 + ) [24]. In this work, the 4487 kev state was observed to decay to the ground state via gamma-ray emission, which rules out a spin J = 4 assignment. It was possible to create angular correlations for the kev gamma-ray cascade to investigate other potential spin assignments, these are presented in Fig A χ 2 /ν minimization analysis was performed varying the value of δ 3142 with the assumption that the 1346 kev gamma ray is a pure E2 transition. The results of this analysis are presented in Fig The minimum χ 2 /ν value observed for a spin J = 1 assignment is almost twice as large ( 2) than is observed for potential spin J = 2 and spin J = 3 assignments, such that a spin J = 1 assignment can be ruled out. The results of this analysis do not indicate whether or not a spin J = 2 or spin J = 103

130 a cascade b cascade Figure 5.30: The angular correlation for the kev gamma-ray cascade that depopulates the 4432 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ 0.25 for the 2 4 gamma ray. (b) The fit shown in red is for a cascade with δ = 0 for the 3 4 gamma ray. 104

131 Figure 5.31: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade that depopulates the 4432 kev level. The value of δ 1858 was allowed to vary, and best fits were obtained for values of δ for the cascade and δ 1858 = 0 for the cascade. 105

132 a cascade b cascade Figure 5.32: The angular correlation for the kev gamma-ray cascade that depopulates the 4487 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0.15(10) for the 2 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0.30(15) for the 3 2 gamma ray. 106

133 Figure 5.33: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to kev gamma-ray cascade that depopulates the 4487 kev state. The value of δ 3142 was allowed to vary, and best fits were obtained for values of δ 3142 = 0.15(10) for the cascade and δ 3142 = 0.30(15) for the cascade. 107

134 3 assignment would be more probable. Therefore, the 4487 kev state is likely either spin J = 2 or spin J = 3. The A 2 and A 4 coefficients and δ 3142 values for each possible spin assignment are listed in Table The 5052 kev State In previous measurements, the 5052 kev state was assigned J π = (4 + ) [24]. It was possible to create an angular correlation for the kev gamma ray cascade to investigate this assignment, as is presented in Fig This particular cascade has been previously discussed in Sec A χ 2 /ν minimization analysis was performed varying the value of δ 3706 with the assumption that the 1346 kev gamma ray is a pure E2 transition. The results of this analysis are presented in Fig It is clear that the minimum possible χ 2 /ν is achieved for a spin J = 2 assignment of the 5052 kev state (χ 2 /ν = 2.6), rather than for a spin J = 4 assignment (χ 2 /ν = 6.0). Therefore, the 5052 kev state is given a spin J = 2 assignment as the result of this work. The A 2 and A 4 coefficients and δ 3706 values for each possible spin assignment are listed in Table 5.4. The previous tentative spin J = (4) assignment for the 5052 kev state in the evaluated nuclear data sheets came as the result of the 48 Ca(p, t) knock-out reaction [23], which populated a state at 5052(5) kev. However, a state at 5047 kev was observed to be populated in a previous measurement of the beta decay of the 46 K 2 ground state [25], and a tentative spin assignment of J = 2 or 3 was suggested based on measured logft values. As there were large discrepancies reported between the two previous beta decay measurements [25, 26], it is likely that this result was ignored in the evaluation of the data available for the 46 Ca level scheme The 5375 kev State In previous measurements, the 5375 kev state was assigned J π = (3 ) [24]. It was possible to create angular correlations for the and kev gamma-ray cascades to investigate this assignment. The angular correlations are shown in Fig The A 2 and A 4 coefficients for the experimental and theoretical fits are listed in Table 5.4. A χ 2 /ν minimization analysis was done that varied the values of δ for the 4029 and 2800 kev transitions while assuming the 1229 and 1346 kev transitions to be pure E2. These results are presented in Fig There is no difference between the minimum χ 2 /ν values resulting from spin assignments 2 and 3 for the kev gamma-ray cascade. However, there is a clear minimization of the resultant χ 2 /ν value for the spin J = 3 assignment in the fit of the kev gamma-ray cascade. Therefore, a spin J = 3 assignment is given to the 5375 kev state as a result of this work, which is in agreement with previous measurements. 108

135 a cascade b cascade Figure 5.34: The angular correlation for the kev gamma-ray cascade that depopulate the 5052 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 2 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0.4(6) for the 4 2 gamma ray. 109

136 Figure 5.35: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade that depopulates the 5052 kev level. The value of δ 3706 was allowed to vary, and best fits were obtained for values of δ 3706 = 0 for the cascade and δ 3706 = 0.4(6) for the cascade. 110

137 a cascade b cascade Figure 5.36: Angular correlation for gamma-ray cascades that depopulate the 5375 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 3 2 gamma ray. (b) The fit shown in red is for a cascade with δ = 0 for the 3 4 gamma ray. 111

138 Figure 5.37: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev (left) and kev (right) gamma-ray cascades. The values of δ were allowed to vary for the 4029 and 2800 kev gamma rays. (Left) Best fits were obtained for values of δ for the cascade and δ 4029 = 0.01(10) for the cascade. (Right) Best fits were obtained for values of δ 2800 = 0.2 for the cascade and δ 2800 = 0 for the cascade. 112

139 a cascade b cascade Figure 5.38: Angular correlation for the gamma-ray cascade that depopulates the 5414 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 2 3 gamma ray. (b) The fit shown in red is for a cascade with δ 1 for the 3 3 gamma ray. 113

140 Figure 5.39: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 1804 was allowed to vary, and best fits were obtained for values of δ 1804 = 0 for the cascade and δ for the cascade. 114

141 The 5414 kev State There are no previous measurements that give a spin assignment for the 5414 kev state. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate possible spin assignments for this state, shown in Fig It was possible to rule out a spin J = 4 assignment as this state was observed to decay directly to the 46 Ca ground state via gamma-ray emission. A χ 2 /ν minimization analysis was done that varied the values of δ 1804 with the assumption that the 2264 kev transition is pure E1. The results are plotted in Fig In this scenario it was possible to rule out a spin J = 1 assignment, as there was no possible value of δ 1804 for the 1804 kev gamma ray which gave appropriate theoretical A 2 and A 4 coefficients that were comparable to those fit from the raw experimental data. A similar minimum χ 2 /ν value is found for spin J = 2 and spin J = 3 assignments. Therefore it is not possible to make a firm spin assignment for the 5414 kev state as a result of this analysis. The A 2 and A 4 coefficients and δ 1804 values for each possible spin assignment are listed in Table The 5535 kev State In previous measurements, the 5535 kev state was assigned J π = (4 + ) [24]. In this work, the this state was observed to decay directly to the ground state via gamma-ray emission. This rules out a potential spin J = 4 assignment. It was possible to create an angular correlations for the kev gamma-ray cascade to investigate other potential spin assignments, as is presented in Fig A χ 2 /ν minimization analysis was performed varying the value of δ 4189 with the assumption the 1346 kev gamma ray is a pure E2 transition. The results of this analysis are presented in Fig There is a clear minimization of δ 4189 observed for this cascade for a spin J = 3 assignment. The A 2 and A 4 coefficients and δ value for this spin assignment are listed in Table The 5712 kev State There are no previous measurements that give a spin assignment for the 5711 kev state. It was possible to create an angular correlation for the kev gamma-ray cascade to investigate possible spin assignments for this state, shown in Fig A χ 2 /ν minimization analysis was done that varied the value of δ 4366 with the assumption the 1346 kev transition is a pure E2. The results are plotted in Fig In this scenario it was possible to rule out a spin J = 1 assignment, as there was no possible value of δ 4366 for the 4366 kev gamma ray which gave appropriate theoretical A 2 and A 4 coefficients that were comparable to those fit from the raw experimental data. A similar minimum χ 2 /ν value is found for spin J = 2 and J = 3 assignments. Therefore it is not possible to make a firm spin assignment for the 5712 kev state as a result of this analysis. The A 2 and A 4 coefficients and δ 4366 values for each possible spin assignment are listed in Table

142 Figure 5.40: The angular correlation for the kev gamma-ray cascade that depopulates the 5535 kev state. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients that correspond to a cascade with δ = 0.05(5) for the 3 2 gamma ray. The resultant fit parameters are listed in Table 5.4. Figure 5.41: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 4189 was allowed to vary, and best fits were obtained for values of δ 4189 = 0.05(5) for the cascade. 116

143 a cascade b cascade Figure 5.42: Angular correlation for the gamma-ray cascade that depopulates the 5712 kev level. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 2 2 gamma ray. (b) The fit shown in red is for a cascade with δ 0.5 for the 3 2 gamma ray. 117

144 Figure 5.43: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 4366 was allowed to vary, and best fits were obtained for values of δ 4366 = 0 for the cascade and δ for the cascade. 118

145 Figure 5.44: A plot of LOG(χ 2 /ν) versus δ for fits of angular correlations corresponding to the kev gamma-ray cascade. The value of δ 4765 was allowed to vary, and best fits were obtained for values of δ 4765 = 0 for the cascade, δ for the cascade, δ 4765 = -0.1(3) for the cascade The 6111 kev State In previous measurements. the 6111 kev state was assigned J π = (2 + ) [24]. In this work, the this state was observed to decay directly to the ground state via gamma-ray emission. This rules out a potential spin 4 assignment. It was possible to create an angular correlations for the kev gamma-ray cascade to investigate other potential spin assignments, as is presented in Fig A χ 2 /ν minimization analysis was performed varying the value of δ 4765 with the assumption the 1346 kev gamma ray is a pure E2 transition. The results of this analysis are presented in Fig There is no difference observed for the resultant χ 2 /ν minima observed for potential spin assignments of J = 1, 2, and 3. Therefore, it is not possible to make a firm spin assignment for the 6111 kev state as a result of this analysis. The A 2 and A 4 coefficients and δ 4765 value for this spin assignment are listed in Table

146 a cascade b cascade c cascade Figure 5.45: The angular correlation for the kev gamma-ray cascade that depopulates the 6111 kev state. The experimental data points are shown with two different fits of Eqn The fit shown in blue allowed all the fit parameters to vary, and the fit shown in red used fixed theoretical values for the A 2 and A 4 coefficients. The resultant fit parameters for each plot are listed in Table 5.4. (a) The fit shown in red is for a cascade with δ = 0 for the 1 2 gamma ray. (b) The fit shown in red is for a cascade with δ 0.75 for the 2 2 gamma ray. (c) The fit shown in red is for a cascade with δ = -0.1(3) for the 3 2 gamma ray. 120

147 Chapter 6 The Structure of 46 K 6.1 Systematics of the Even-Even Ca Isotopes There are three even-even calcium isotopes between the doubly-magic 40 Ca and 48 Ca nuclei: 42 Ca, 44 Ca, and 46 Ca. Each of these nuclei are populated in the beta decay of the ground state of the potassium isotope with the same mass number. Select low-lying levels of these calcium isotopes are shown in Fig. 6.1 with the observed beta-feeding intensities from the corresponding potassium parent shown in red. Large beta-feeding intensities indicate a large overlap between the wavefunctions of the decaying state in the parent nucleus and the populated state in the daughter nucleus. Therefore, these intensities can be used to deduce structural information on the two nuclei involved in the decay. The beta decays of 40 K, 42 K, and 44 K primarily populate the ground states of their daughter calcium isotopes, whereas in the decays of 46 K and 48 K there is no population of the 46 Ca ground state and very little population of the 48 Ca ground state, respectively. Alternatively, in both instances the most intense beta feeding is primarily to excited states with J π = 2, and secondly to the first excited state. These systematics indicate that there is a large overlap between the wavefunctions of the ground states of 40 Ca, 42 Ca, and 44 Ca with the wavefunctions of the ground states in the corresponding parent nuclei, but that there is very little overlap between the ground state wavefunctions of 46 Ca and 48 Ca with that of their potassium parents. The ground states of the even-even Ca isotopes each contain a full proton πd 3/2 orbital paired with coupled neutrons in the νf 7/2 orbital. For K, the ground state configurations are simply a proton hole in the πd 3/2 orbital coupled to an odd neutron in the νf 7/2 orbital. The ground state to ground state beta decay of K to Ca is simply the transformation of the odd neutron located in the νf 7/2 orbital into a πd 3/2 proton. However, from this simple description, there is nothing unique about the structures of 46 K and 46 Ca, compared to that of the other isotopes, that would hinder the ground state to ground state beta decay transition. 121

148 4-40 K β β β β 42 K 44 K 46 K 48 K β (2 ) 4947 (2 ) 4409 (38% β) (2 ) (2 ) (2 ) (23% β) (21% β) (16% β) (28% β) (6% β) (18% β) (2% β) (31% β) (89% β) (82% β) (34% β) (0% β) (<1% β) Ca 42 Ca 44 Ca 46 Ca 48 Ca 0 0 Figure 6.1: Systematics of select low-spin state for the even-even Ca isotopes are presented [24]. These isotopes are each populated from a potassium-parent nucleus of the same mass A. Beta feeding intensities (marked in red) to the ground, first-excited, and most intensely populated higher-lying states are shown. Note that the majority of the beta feeding populates the ground states of Ca, whereas very little feeding is observed to the ground state of 48 Ca, and no feeding is observed to the 46 Ca ground state. In the ground state configuration of 48 K, the odd neutron is located in the νp 3/2 orbital and is coupled to a proton hole in either the πd 3/2 or the πs 1/2 orbital. This is due to the near degeneracy of the πd 3/2 and πs 1/2 orbitals in 48 K [47], which will be discussed further in Sec The distribution of the proton hole across the πd 3/2 and πs 1/2 orbitals could be 122

149 Figure 6.2: Experimental energies for the first observed 1/2 + and 3/2 + states in odd-a K isotopes. Note the inversion of spin for 47 K and 49 K. This is due to the evolution of the πsd orbitals with varying energy. This figure is reprinted with permission from Ref. [47]. what hinders the ground state to ground state beta decay of 48 K to 48 Ca. Potentially the πd 3/2 orbital is so strongly bound that it acts as a closed shell configuration. It might not be possible for the πd 3/2 orbital to accept another proton to form the ground state configuration of 48 Ca. Then it might only be possible to beta decay to an excited configuration of 48 Ca where the newly-formed proton would occupy the πf 7/2 orbital above the Z = 20 shell gap. Note that the coupling between a proton in the πf 7/2 orbital (which has -π) to a proton hole in the sd-shell (which has +π) would result in an excited state of negative parity, which are the types of states that are primarily populated in 48 Ca from the beta decay of 48 K. In order to explain the hindrance of the ground state to ground state 46 K to 46 Ca decay, a closer look at their nucleon configurations is necessary. The systematics of the low-lying states in the even-even calcium isotopes presented in Fig. 6.1 does not show any obvious difference in the structure of 46 Ca relative to the other calcium isotopes. Also, the closedproton shell present makes any misinterpretation of the 46 Ca ground-state configuration unlikely. Therefore, it is necessary to investigate possible misinterpretations of the structure of 46 K. Currently, there are contradictory reports as to whether or not there is near degeneracy of the πd 3/2 and πs 1/2 orbitals in 46 K as is suggested for 48 K [48, 47]. The clear similarities in the observed beta-feeding intensities for the decays of the 46 K and 48 K ground states, suggest that the behaviour of the πd 3/2 and πs 1/2 orbitals needs to be further investigated in the structure of 46 K. 123

150 6.2 Evolution of the sd-shell in the Potassium Isotopes Over the past decade, nuclei with Z < 20 and 20 N 28 have been extensively investigated to study the evolution of the πsd orbitals with varying neutron number [5, 49]. In the odd-a potassium isotopes, the spins of the ground and first excited states are due to the spin of the nuclear orbital that contains the unpaired proton, as all neutrons are paired and do not contribute to the spin. If the unpaired proton is in the πd 3/2 orbital than the resultant state will have spin 3 + 2, and if the proton is in the πs1/2 orbital then the spin of the resultant state would be The experimentally observed energies for the first and states in the odd-a potassium isotopes are shown in Fig For K the ground states have spin 3 + 2, and the first excited states have spin Note that there is a gradual decrease in the first excitation energy for these isotopes with increasing neutron number, until there is suddenly an inversion of these spin states for 47 K where the ground state is observed to have spin and the first excited state has spin The same inversion exists for 49 K, where there is near degeneracy between the two states, but in 51 K the ground state is once again observed to have a spin of The observation of the re-ordering of these spin states in the potassium isotopes is believed to be due to the interaction between neutrons in the νf 7/2 orbital with the unpaired proton in the πd 3/2 orbital [5]. As the νf 7/2 orbital is filled with more neutrons, the interaction becomes stronger, making the πd 3/2 orbital more bound such that it is pushed down in energy towards the πs 1/2 orbital. When the νf 7/2 orbital is completely filled, the πd 3/2 and πs 1/2 orbitals are nearly degenerate. This occurs at 47 K, where the νf 7/2 orbital is full and the inversion of spins has been measured [50]. The theoretically predicted contribution of the πd 3/2 orbital to the ground state configurations of these potassium isotopes are listed in Table 6.1. Note that only 13% of the ground state configuration of 47 K is believed to originate from an unpaired proton in the πd 3/2 orbital. The rest of the ground-state configuration is from a proton hole in the nearly degenerate πs 1/2 orbital. Table 6.1: The predicted amount of the πd 3/2 component contained in the ground-state wavefunction for odd-a K isotopes from shell model calculations [47]. For 47 K and 49 K there is a near degeneracy of the πd 3/2 and πs 1/2 orbitals, therefore those ground states contain a large contribution of the πs 12 orbital. Numbers taken from Ref. [47]. Isotope πd 3/2 % 39 K 100% 41 K 95% 43 K 92% 45 K 90% 47 K 13% 49 K 15% 51 K 93% 124

151 Figure 6.3: Calculated proton occupancies for the πd 3/2 and πs 1/2 orbitals for K isotopes from shell model calculations using two different effective interactions. For K there are 3 protons in the πd 3/2 orbital and 2 protons in the πs 1/2 orbital. There is a sudden shift in occupation numbers for 47 K when there are 4 protons in the πd 3/2 orbital and 1 proton in the πs 1/2. This figure is reprinted with permission from Ref. [47]. This effect is easily observed experimentally in the odd-a potassium isotopes due to their unpaired proton. The relative energies of the πd 3/2 and πs 1/2 orbitals cannot be as readily observed in the even-a potassium isotopes, as the spins of the nuclear states are due to the coupling of an unpaired proton with an unpaired neutron. The level schemes for the low-lying states of the even-a K isotopes are presented in Fig In most instances, the first excited state in these isotopes is observed to be quite low in energy, with the exception of 46 K. In 46 K the first excited state is observed at 587 kev, which is much higher in energy than in any of the neighbouring isotopes. This indicates that it takes a relatively large amount of energy to excite 46 K out of its ground state configuration. Perhaps it is possible the ground state configuration of 46 K contains more πs 1/2 character than previously believed. If it is strongly mixed with the πd 3/2 orbital, as is the case for 48 K, then it may be possible to explain the hindrance of the ground state to ground state 46 K to 46 Ca beta decay. The occupations of the πd 3/2 and πs 1/2 orbitals for the ground state configurations of K isotopes have been previously calculated (see Fig. 6.3 [47]), and do not show any such mixing for 46 K. Strong mixing between the πd 3/2 and πs 1/2 orbitals is only suggested for an even-a potassium isotope in the case of 48 K. 125

152 (1,2,3) K 42 K 44 K 46 K 48 K Figure 6.4: The first two excited states are presented for each even-a K isotopes. The first excited state observed for 46 K is at much higher energies than in any of the neighbouring isotopes. It is clear that from the beta-decay intensities observed populating 46 Ca and 48 Ca that there are likely strong similarities between the ground state configurations of 46 K and 48 K. There is minimal beta feeding to the 46,48 Ca ground states, and in each case a majority of the feeding is observed to states with J π = 2. The nucleon configuration of the strongly fed 5052 kev 2 state in 46 Ca might give further insight as to the structure of the 46 K ground state, as the intense beta feeding to that states indicates a large overlap of the two wavefunctions participating in the decay. If the ground state configuration of 46 K did contain an admixture of πd 3/2 and πs 1/2 character, then the populated 2 excited state in 46 Ca would likely have very similar character. 6.3 Shell Model Calculation of 46 Ca A shell model calculation was performed to investigate the 46 Ca level scheme [51]. This calculation utilized NN and 3N forces from chiral EFT to create an effective valence space 126

153 Figure 6.5: This calculation utilized a unique valence space that included the s 1/2 and d 3/2 orbitals from the closed core in an attempt to observe any possible excitations across the Z = 20 shell gap [51]. Hamiltonian that calculated shell model single-particle energies. This calculation utilized a unique valence space that included the s 1/2 and d 3/2 orbitals from the closed 40 Ca core in an attempt to observe any possible excitations across the Z = 20 shell gap (Fig. 6.5). Previously, a similar calculation was performed that did not include possible excitations from the s 1/2 and d 3/2 orbitals, so it could not predict any negative parity states in 46 Ca [15]. A comparison of the current [51] and previous calculations [15] to experimental data of select excited states in 46 Ca is presented in Fig The comparison of the calculated energies to those that have been observed experimentally are slightly underestimated in energy, but in general are in good agreement. Of interest to the current discussion are the calculated nucleon configurations that give rise to excited states with J π = 2 in 46 Ca. The energies of the calculated 2 states are listed in Table 6.2 with their respective occupation numbers. In all cases the 2 spin is due to coupling between the πd 3/2 and πp 3/2 orbitals. It is unlikely that such a configuration would be strongly populated from the beta decay of the 46 K ground state, as it is not energetically favourable to create a proton in the πp 3/2 orbital. From this calculation there is no evidence of any proton excitations from the πs 1/2 orbitals in the level scheme of 46 Ca. It may be necessary for a calculation to include the entire sd-shell for such excitations to be observed. Such a calculation would be very computationally difficult due to the large size of the valence space necessary. 6.4 Suggestions for Future Work The results of this work have brought into question the nature of the 46 K ground state due to the the lack of beta feeding observed to the 46 Ca ground state and the intense beta 127