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LILRARY Michigan State University This is to certify that the dissertation entitled THE 15°Sm(t,3He)15°Pm* AND 150Nd(3He,t)15°Pm"' REACTIONS AND APPLICATIONS FOR 2v and 0v DOUBLE BETA DECAY presented by Carol Jeanne Guess has been accepted towards fulfillment of the requirements for the Doctoral degree in Physics and Astronomy .‘f /' ’- W L I Major Professor’s Signature A 2 (57 7a /5/ 2 0/0 ’ I Date MSU is an Affinnative Action/Equal Opportunity Employer .-—.—----—-—--—-—-.-.-.—u-o-u-u-o-n-o-o-u—nfi.-.-o-c--u-o--o-.-.—-.—.—.-.—-.-u-I-v-n_u-o-o—n-o—o-n-n-o-I-o-n-v-I-I- PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 K'lProj/AocsPres/CIRCIDateDue.indd THE 15OSm(t,3He)150Pm* AND 150Nd(3He,t)15OPm* REACTIONS AND APPLICATIONS FOR 21/ AND 01/ DOUBLE BETA DECAY By Carol Jeanne Guess A DISSERTATION Submitted to Michigan State University in partial fulfillment Of the requireIIIeIIts for the degree Of DOCTOR OF PHILOSOPHY Physics and AstrOIIOIIIy 2010 ABSTRACT THE 1508m(t,3He)150Pm* AND 150Nd(3He,t)150Pm* REACTIONS AND APPLICATIONS FOR 21/ AND 01/ DOUBLE BETA DECAY By Carol Jeanne Guess In models Of 2143,13 and 01/63 decay, the transition is described as proceeding through “virtual” states of the intermediate nucleus. Knowledge of the location and popula- tion strength of these levels is crucial for constraining the nuclear matrix elements Of the transition. Charge—exchange (CE) experiments at intermediate energies can be used to extract. the GaIIIOW-Teller strength for both legs of this transition, as well as additional information on dipole and quadrupole excitations. The {3,3 decay of 150Nd to 150Sm was probed in two experiments: 150Nd(3He,t)15017’111"< at RCN P. Osaka, Japan, and 150SIn(t,3He)ISOPm’I‘ at NSCL/MSU, East Lansing, .\Iichigan, USA. Gamow-Teller strength distributions and dipole and quadrupole cross section distributions have been extracted using multipole decomposition techniques, includ- ing a strong GT state in 150F111 at 0.11 MeV. Applying the extracted Gamow-Teller strength from both experiments in this region, the single-state dominance hypothe- sis predicts a 21/83 decay half life of 10.0 d: 3.7 ><1018 years. This is a reasonable result, but the presence of other low-lying Gamow-Teller strength requires further investigation using QRPA or other theoretical techniques. The extracted strength distributions should constrain the nuclear matrix elements for both 21/53:} and 01/33 decay. In addition, an excess of Gamow-Teller strength in the 150Sm(t,3He) ex- periment is attributed to the population of the IV SGMR. Data are compared with deformed QRPA calculations from V. Rodin. Copyright by CAROL JEANNE GUESS 201i) To Victor and Lois Guess, who have always supported and (’71,(T()'tl'l'(l._(](?d my interest in science. iv ACKNOWLEDGMENTS NO thesis is completed in a vacuum, even in a field like nuclear physics where vacuum systems are a. part of life. There are many people who have contributed to the experiments described here and to the thesis itself. First, I’d like to thank my adviser, Dr. Remco G.T. Zegers. I knew upon my arrival at the cyclotron that any professor carrying a. mug depicting the Muppets Stat‘ler and Waldorf would be an interesting person to talk to, and I was not disappointed. He puts an amazing amount of time into his research students and is always ready to listen and give comments and suggestions. I appreciate his deep knowledge Of physics, patience with students as we learn (and forget, and learn again), and attention to details that others overlook. His sense of humor, informal manner, and perceptive questions help to make the charge-exchange group an invigorating environment in which to learn physics. All chapters of this thesis were drastically improved due to Remco’s insistence on reading things again and again until they were completely correct and well-organized. Drs. Alex Brown, Alexandra Gade, V‘Vayne Repko, and Stuart Tessmer agreed to serve on my guidance committee. I thank them for their assistance over the last few years and for helpful comments on this thesis. I have benefited from many friendships with the members of the CE group over the last 5 years. Though the group members have changed, we have a convivial atmosphere and I’ve benefited from interacting with all of you. Sam Austin‘s persis- tent questions during group meetings have been very valuable, as has Daniel Bazin's knowledge of the 8800 spectrometer. Arthur, Yoshi, George, and Masako have been postdocs in our group and were always willing to answer my questions. Meredith and Wes have done previous triton beam experiments and their experiences were in- valuable in the planning process for my (t,3He) experiment and helping me get on my feet with the PAW code. W ‘S has 'iven me many detailed ex )lanations of CE . S . l phenomena, and I appreciate his insights. Rhiannon has been very supportive and always has constructive criticism when I need it, and was very helpful while locating interview clothes. Her policy of eating anything that I cook or bake has been very entertaining. Amanda is the perfect source for a completely logical viewpoint when I need to bounce ideas Off Of someone, and I appreciate her sense of humor and generos- ity. Jenna has boundless entlmsiasm and attention to detail no matter what she is working on, even when grading papers and working out the minutiae of an electronics diagram. Shawna’s tendency to ask the first question is inspirational. Thanks for taking SO many midnight shifts. I hope we can work together again soon. The other graduate students at the cyclotron have been unfailingly friendly and helpful, and I appreciate the good conversations I have had with many of you. Many thanks go to my amazing Office-mates. Despite having five people in the office. we have always managed to get. along. I’ll miss the interesting conversations and good food that occur so frequently in the Office. The cyclotron staff and my collal'IOI'ators at NSCL and RCNP were very helpful during planning and execution of my two experiments. Thank you for the explana- tions, help with design and setup, great beam tunes, and for coming in to fix the beam line magnets at 4 am. H. Fujita, and T. Adachi were invaluable while setting up the 150Nd(3He,t) experiment, and I also appreciate insightful discussions with Y. Fujita, H. Ejiri, M. Fujiwara, D. Frekers, and J. Thies. J. Yurkon and N. Verhanovitz made the 150Sm target, and J. Honke designed the target frames. Daniel, Mauricio. and Remco ensured that the dispersion—matching was correct. H. Sakai and S. Noji allowed us to use their 13C target for calibrations. Vadim Rodin has answered many questions about his QRPA calculations. Any mistakes in my descriptions of his work are mine. The National Science Foundation and the Joint Institute for Nuclear Astrophysics provided the funds for my graduate career, bought my targets. sent me to Japan twice, vi and gave me money to present at conferences. The Capital Area District Library and the Curious Book Shop have helped to keep me sane. When I arrived at the cyclotron after receiving a music minor, I hoped to stay active in the music community. The Grand Canonical Ensemble has been a source of bad physics jokes. laughter, and friendship over the last five years and a wonderful break from nuclear physics. I miss singing with you already. Many thanks go out. to Amanda, Megan. Ania, and Dan for fun and eventually productive homework sessions. Somehow, we always managed to get work done, even when it was delayed by spherical jousting horses, quantum mechanics Pac—Man. pumpkin carving, and discussions about stuff completely unrelated to physics. Ania, I’ll miss our food trades. My friends in Dinner Club have shared so many meals of delicious food, good wines, pifiatas, plays, poetry, and friendship. Thanks to Kim, Erin, Neil, Ana, and Catherine and those who have come to our meetings from time to time. I look forward to future meetings in our separate states and countries. Johanna and JOlm have been extremely helpful during beam time. big exams, and stretches of concentrated writing. Thank you for the extra. dishes and cat care. Victor and Lois Guess were very understanding about my lack of visits home. even though I lived a 2-hour drive away from them. Mom and Dad, thank you. Neil, thank you for all of your encouragement, love, and support. vii TABLE OF CONTENTS List of Figures ................................ List of Tables ................................. Introduction . . 1.1 Motivation ................................. 1.2 Organization ............................... DoubleBetaDecay 2.1 21/ and 01/ Double Beta Decay ...................... 2.1.1 Introduction ............................ 2.1.2 Implications ............................ 2.2 Detection Challenges ........................... 2.2.1 Detection Methods ........................ 2.2.2 Detectors for 150er 43:3 decay .................. 2.3 Nuclear Matrix Elements ......................... 2.3.1 Half—Life Calculation ....................... 2.3.2 The Shell Model Approach .................... 2.3.3 QRPA ............................... 2.3.4 Constraining N MES with charge-em‘hange experiments Charge-Exchange Reactions . . . . . 3.1 Introduction to Charge-Exchange Re artions .............. 3.2 Reaction Theory ............................. 3.2.1 DWBA ............................... 3.2.2 One-body transition densities .................. 3.2.3 The nucleon-nucleon interaction ................. 3.2.4 FOLD ............................... 3.2.5 DWHI ............................... 3.2.6 The unit cross section and B(GT) ................ 3.3 Giant Resonances ............................. 150Nd(3He,t)15OPm* at RCNP 4.1 RCNP Experimental Setup and Procedure ............... 4.1.1 Beam preparation and tuning .................. 4.2 Calibrations ................................ 4.2.1 Sieve Slit Calibrations ...................... 4.2.2 Beam rate Calibration and Cross Section Calculation ..... 4.3 Analysis of Data ............................. viii xiv car—1t .—| Oflflmhrfish TABLE OF CONTENTS List of Figures ................................ List of Tables ................................. Introduction . 1.1 Motivation ................................. 1.2 Organization ............................... Double Beta Decay 2.1 21/ and 01/ Double Beta Dec av ...................... 2.1.1 2.1.2 Introduction ............................ Implications ............................ 2.2 Detection Challenges ........................... 2.2.1 2.2.2 Detection Methods ........................ Detectors for 150Nd 1H decay .................. 2.3 Nuclear Matrix Elements ......................... 2.3.1 2.3.2 2.3.3 2.3.4 Half-Life Calculation ....................... The Shell Model Approach .................... QRPA ............................... Constrainin T N MES with ('ll'dl‘U‘O-G‘Xt'll'dll re ex )eriments O Charge-Exchange Reactions 3.1 Introduction to Charge-Exchange Reactions .............. 3.2 Reaction Theory ............................. 3.2.1 DWBA ............................... 3.2.2 One-body transition densities .................. 3.2.3 The nucleon—nucleon interaction ................. 3.2.4 FOLD ............................... 3.2.5 DWHI ............................... 3.2.6 The unit cross section and B(GT) ................ 3.3 Giant Resonances ............................. 150Nd(3He,t)150Pm* at RCNP 4.1 RCNP Experimental Setup and Procedure ............... 4.1.1 Beam preparation and tuning .................. 4.2 Calibrations ................................ 4.2.1 Sieve Slit Calibrations ...................... 4.2.2 Beam rate Calibration and Cross Section Calculation ..... 4.3 Analysis of Data ............................. viii xiv car—A NNQJA-i—sih 10 11 11 15 16 18 21 21 26 26 28 29 32 33 34 creams—.41 IQIQIOIQN 1C1 .14__ 4.3.1 FOLD calculations for 150Nd(3He,t) .............. 4.3.2 The Optical Potential and the IAS ............... 4.3.3 Multipole Decomposition Analysis ................ 4.3.4 Resonance fits ........................... 4.3.5 Extrapolation to q=0 ....................... 4.3.6 Calculation of the Gamow-Teller strength ........... 4.3.7 Other Multipole Excitations ................... 4.4 Comparison with Theory ......................... 4.4.1 Cross sections and Giant Resonances .............. 4.4.2 QRPA calculations ........................ 5 150sm(t,3He)150Pm* at the NSCL . 5.1 Experimental Setup and Procedure ................... 5.1.1 Production of a Triton Beam .................. 5.1.2 The S800 Focal Plane ....................... 5.2 Calibrations ................................ 5.2.1 CRDC Mask Calibrations 5.2.2 Beam Rate Calibration ...................... 5.2.3 Calculation of the Excitation Energy of 150F111 ........ 5.2.4 Acceptance Corrections ...................... 5.2.5 Background and Hydrogen Subtraction ............. 5.2.6 Calculation of the Cross Section ................. 5.3 Data Analysis ............................... 5.3.1 FOLD calculations 5.3.2 Multipole Decomposition ..................... 5.3.3 Extrapolation to q=0 ....................... 5.3.4 Calculation of the Gamow-Teller strength ........... 5.3.5 Other Multipole Excitations ................... (I! at; Comparison with Theory ......................... 5.4.1 AL=0 Cross sections and the IVSGMR .............. 5.4.2 QRPA calculations ........................ 6 Application to 21/88 decay . 6.1 Low- lying states and the SSD hypothesis ................ 6.2 Calculating the 21/83 decay half life In the SSD ............ 7 Conclusions and Outlook Bibliography ix 58 58 61 65 77 79 81 88 88 90 95 95 95 100 102 102 102 104 106 109 112 115 115 118 123 123 129 139 139 140 . 145 145 146 . 149 . 152 2.1 2.2 2.4 2.6 2.7 3.1 3.2 3.3 3.4 3.6 LIST OF FIGURES Images in this dissertation are presented in color The two methods of double beta decay. 2—neutrino and 0—neutrino. Neutrino mixing and two possible arrangements of the hierarchy. . Simulation of the 13.3 decay summed electron spectrum in a direct counting experiment. A SuperNemo module Prototype of DCBA. The SNO detector. I o I r a Population of Intermediate states 111 1"’UPIII With charge—exchange re- actions. Schematic of charge—exchange on a subset of the chart of nuclides. Isospin in charge-exchange reactions. Pauli blocking in charge-exchange reactions. Operator strengths for the UT and 7' t-matrices. The Fermi and GaIII(_)w-Teller unit cross sections as a function of A. Giant Resonances 9 11 12 13 to Cf! to C1 2.1 2.2 2.3 2.4 2.6 2.7 3.1 3.2 3.3 3.4 3.5 3.6 LIST OF FIGURES Images in this dissertation are presented in color The two methods of double beta decay, 2—neutrino and (I-neutrino. Neutrino mixing and two possible arrangements of the hierarchy. . Simulation of the 133 decay summed electron spectrum in a direct counting experiment. A SuperNemo module Prototype of DCBA. The SNO detector. | I I O r I Po )ulation of Intermediate states 111 IOUPIII w1th char 'e-exchanU'c re— (-3 actions. Schematic of chargruexchange on a subset of the chart of nuclides. Isospin in charge-exchange reactions. Pauli blocking in charge-exchange reactions. Operator strengths for the or and T t-matrices. The Fermi and Gamow-Teller unit cross sections as a function of A. Giant Resonances CH 11 12 to C”! to C1 3.7 3.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.17 IVSGMR schematic for 1SUNd —+ 1'50Pm ................ IVSGMR schematic for 1508111 —> 150Pm ................ The “S beamline at RCNP ........................ Ion optical modes for high-resolution spectrometers .......... Dispersion—matched beam image at the target. of the Grand Raiden spectrometer. ............................... The 150Nd targets ............................ The Grand Raiden Spectrometer .................... PID for the Grand Raiden ........................ Angular acceptance of the Grand Raiden ................ Elastic scattering on 150Nd. ....................... Images Of the sieve-slit as measured in the focal plane of the Grand Raiden spectrometer ............................ Reconstructed sieve slit spectrum after the determination Of raytracing parameters. ................................ r r Cross sections for the 1')0.\'d(‘3He.t) experiment ............ 1' Cross sections from 0-2 MeV for the 1‘30,\'(l(3He.t) experiment. . . . . - - - . 150 3 . - Angular distributions from Nd( He.t) as calculated With FOLD. The angular distribution Of the optical potential compared to the IAS o o a 1 I r lV'IultIpole decompOSItion of the first two peaks in 130F111 ....... Multipole decomposition of the 5-6 MeV excitation energy bin. . . . . Multipole decomposition summary for each half-degree angular bin. cont. hilultipole decomposition summary for each half-degree angular bin ...................................... xi 40 41 44 46 47 48 49 50 51 60 63 64 66 4.17 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 5.1 5.2 5.3 .5.4 5.5 5.6 5.7 5.8 cont. Multipole decomposition summary for each half—degree angular bin ...................................... 68 cont. Multipole decomposition summary for each half—degree angular bin ...................................... 69 Resonance fit to the excitation energy spectrum ............. 72 Angular distributions of the giant resonances. ............. 78 Ratio of the cross section at 6:00 and 0 linear momentum transfer to that. of 0 linear momentum transfer, as calculated in DWBA ...... 80 GT strength from the MDA ....................... 81 Low—lying GIT strength from the MDA ................. 84 Dipole cross sections from the MDA ................... 85 Quadrupole cross sections from the MDA ................ 85 QRPA calculations for 1'50Nd(3He.t) .................. 91 Gamow-Teller strength in 150F111 via 150Nd(3He,t) .......... 93 Dipole cross sections and strength in 150F111 via 1'50Nd(3He,t) . . . . 94 Schematic of the Coupled Cyclotron Facility and the S800 Spectrograph 96 Dispersion—Iiiatched triton beam image at the target of the S800 spec- trometer ................................... 98 The 150Sm target .............................. 99 Layout Of the CRDCS in the focal plane of the S800 spectrometer 100 Ytac spectrum .............................. 105 PID at the S800 focal plane ....................... 107 Background subtraction for 150Sm(t,3He) ............... 110 Hydrogen subtraction for 1508111( t,3He) ................. 111 xii 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 6.1 Cross sections for the 150Sin(t,3He) experiment (300 keV bins) Cross sections for the 150Sm(t,3He) experiment (1 MeV bins) Angular distributions from 150Sm(t.,3He) as calculated with FOLD, at Q=0 ..................................... Angular distributions from 150Sm(t,3He) as calculated with FOLD. at Q=20. ................................... MDA for the 0-1 MeV excitation energy bin .............. MDA for the 20-21 MeV excitation energy bin ............. MDA for the 0-0.3 MeV excitation energy bin ............. MDA for the. 0.1-0.2 MeV excitation energy bin ............ Multipole decomposition summary for each angular bin ......... Multipole decomposition summary for each angular bin (0-6 MeV). Ratio of the cross section at 0:00 and 0 linear momentum transfer to that of 0 linear momentum transfer. as calculated in DVV BA ...... Extracted GT strength. 0-26 MeV .................... Extracted GT strength, 0-6 MeV .................... Extracted AL=1 cross section. 0-26 MeV ................ Extracted AL=2 cross section, ()-26 MeV ................ Raw QRPA calculations for 150Sm(t,3He) ............... Gamow-Teller strength in 150Pm via. 150Sm(t.3He) .......... Dipole cross sections in 150Pm via 150Sm(t.3He) ........... ,_ B(GT) strength in 100F111 at low energies ............... xiii 113 114 116 117 119 120 121 138 141 143 144 147 LIST OF TABLES 2.1 Half-life values for 8,3 decay ........................ 10 3.1 Charge—exchange excitations and their quantum numbers ....... 22 3.2 Giant resonances ............................. 39 4.1 Parameters of the Grand Raiden Spectrometer ............. 48 4.2 Optical potentials for the elastic scattering data and the angular dis- tribution of the IAS. ........................... 61 4.3 Parameters used in calculating the quasi-free curve. .......... 77 4.4 GaIIiow-Teller strengths from resonance fits. .............. 80 4.5 Gai’now-Tcller strength distribution from the MDA, 0-30 MeV . . . . 82 4.6 Gamow-Teller strength distribution from low-lying states. 0-2 MeV. . 83 4.7 Dipole cross sections from the MDA. ()-30 MeV ............ 86 4.8 Quadrupole cross sections from the MDA. 0-30 MeV ......... 87 4.9 Dipole cross sections from giant resonance fits .............. 88 4.10 Quadrupole cross sections from giant resonance fits ........... 88 4.11 Exhaustion of the full normal mode strength for AL=0 ........ 89 4.12 Exhaustion of normal mode strength for AL=1 ............ 90 4.13 Exhaustion of normal mode strength for AL=2 ............ 90 xiv 4.14 Values of I32, gpp, and gph as adopted in the QRPA calculations. . . 90 5.1 Parameters of the S800 Spectrometer .................. 101 5.2 Important S800 parameters in the analysis of the 150Sm(t,3He) exper- iment ..................................... 103 5.3 Parameters for the 150Sm(t,3He) missing mass calculation ...... 108 5.4 Gamow-Teller strength distribution from the MDA, 0-26 MeV . . . . 131 5.5 Gamow-Teller strength distribution from the MDA, 0-6 MeV ..... 133 5.6 AL=1 cross sections from the MDA, 0-26 MeV ............. 136 5.7 AL=2 cross sections from the MDA, 0-26 MeV ............. 137 6.1 Calculation of the 21/1311 decay matrix element. assuming single-state dominance ................................. 147 XV Chapter 1 Introduction 1. 1 Motivation Double beta decay is currently the focus of a great deal of interest from within the physics community. 121/1313 decay occurs when two neutrinos and two electrons are Simultaneously emitted from a nucleus. This process occurs only when other de- cay methods are forbidden, and the half-lives associated with it are extremely long 017 years). Much of the interest in 8,3 decay is centered around the (greater than 1 second possible mode, which is 01/1'31'3 decay. The emission of two'electrons without two neutrinos would violate the Standard Model, breaking the conservation of lepton number, and would prove that neutrinos are Majorana rather than Dirac in nature. A Majorana neutrino is its own antiparticle. Half life values for the 01/ mode. of decay are several orders of magnitude higher than the 21/ mode and 01/ events could easily be overshadowed by 21/ events, so successful detection of this 01/ mode would be a major experimental feat. If the measurement is exact enough. it should be possible to extract the Ma jorana neutrino mass from the half life. 1313 emitters tend to be heavy nuclei, which makes them hard to model. Theorists working on this problem must model the location and strength of an enormous number of states, and little to no data exists to constrain these models for several nuclei. The decay of 150Nd to 150Sm (through 150Pm) is one of these cases. A quantity called a nuclear matrix element contains the physics of two simultaneous beta. decays, from 150Nd to 150Pm and then to 150Sm. and this quantity must. be known with an error less than 20% to design the experiments that measure the decay half life and then to successfully extract the neutrino mass from a half life measurement [1]. Knowing the location of the levels in the ii‘itermediate nucleus, as well as how strongly they may be populated, can place constraints on the models used to describe 13,3 decay. Charge-exchange experiments are an excellent tool for this, since they allow us to measure the location and strength of Gamow-Teller, Fermi. dipole, and quadrupole transitions along the same paths taken by beta decay. A charge—exchange reaction is characterized by a change in isospin (AT) of 1. When performed at intermediate energies (energies between 100 and 500 MeV/ u), the reaction can be modeled as a single-step process and Gamow—Teller transitions are preferentially excited. This thesis describes two charge-exchange experiments designed to constrain the nuclear matrix elements for the 138 decay of 150Nd to 150Sm. Both populate excited states in 150Pm. The first experiment, 1'50.\3(l(3He,t), took place at RCN P (Osaka, Japan) with a primary 3He beam. The second experiment was 150S1ii(t.3He) and took place at the NSCL (East Lansing, Michigan. USA) with a secondary triton beam. Gamow-Teller strengtl‘is were extracted from both experiments, along with information on dipole and quadrupole strengths and the population of several giant resonances. The results of these. two experiments will be immediately useful for 13.3 decay theorists and for several experiments that are planned to directly search for 01/138 decay signals [2. 3, 4] from 150Nd. 1.2 Organization This work is divided into chapters by topic. Double beta decay is introduced in Chapter 2, followed by an introduction to charge-exchange reaction theory in Chapter 3. Chapters 4 and 5 discuss the two experiments and make up the bulk of this work. Chapter 6 briefly ties the two experiments together, and Chapter 7 summarizes the findings of both experiments and provides an outlook for similar experiments and future charge-em'haiige techniques. Chapter 2 Double Beta Decay 2.1 21/ and 01/ Double Beta Decay 2.1.1 Introduction Fermi introduced his theory of beta (11’) decay in 1934 [5. 6]. One year later. half lives for two-neutrino double-beta decay (21/1313) were first calculated by M. Goeppert- Mayer [7]. She correctly predicted half lives to be on the order of 1017 years or more. Four years later, M. Furry built upon this work by also considering zero-neutrino double beta decay (01/131'3) [8], which was possible only using Majorana symmetry concepts, a departure from Fermi’s Dirac model. While double electron capture [9] has also been considered, much of the subsequent experimental and theoretical focus has been on 21/131'3 and 0u1313 decays. Figure 2.1 shows a schematic of both types of decay. 2143,13 decay is modeled as two simultaneous .13 decays. making it a second- order weak interaction within the standard model. 21/1313 (,lecay: N(A, Z) _, N(A, Z + 2) + 21)“ + 217,. (2.1.2—V) (2.1) N(A, Z) —> N(A. Z — 2) + 212+ + 211,. (2.13;) (2.2) 4 v- .\.. ZVBB OvBfl Figure 2.1: The two methods of double beta decay. 2-neutrino and O-neutrino. is permitted in the standard model, while 01/1’313 decay N(A, Z) —1 N(A, Z + 2) + 2c- (2861/) (2.3) N(A, Z) —> N(A. z — 2) + 2e+ (2.331!) (2.4) would require physics beyond the standard model. {38 half lives are between 1017 to 1026 years, and 813 decay is only observed in Situations where single— 1'3 decay and other decay modes are forbidden. This can have two causes: extremely high angu- lar momentum transfer between mother and daughter (e. g. 48Ca). or parent nuclei where decay to the ,8 daughter has a. positive Q value and decay to the .33 daughter a negative Q value. All 813 mother and daughter nuclei have ground state J7T of 0+. and decay from ground state to ground state is more common than that to excited states, because the phase space is reduced in decay to excited states [10. 11]. Cur- rently, decay to excited states has only been measured in 100Mo and 153(le (see [12] and references within). Some exotic models predict, other causes and variants of 3.3 5 decay, such as the simultaneous emission of a Majoran X particle, which is a hypo- thetical Goldstone boson associated with the breaking of lepton number symmetry [13]. However, the experimental spectrum of summed electron energy would have a shape that is predicted to differ from both 01/88 and 21/19’13 decays [14] (Figure 2.3 shows this spectrum for 0 and 21/813 decay only). 21/138 and 01/313 decays are the most. frequently considered and studied modes for 13,3 decay. 2. 1 .2 Implications Signatures of neutrino oscillation were first. seen in atmospheric neutrinos during the Super-Kamiokande experiment [15] and were confirmed by the SNO experiment [16]. The scientific community then turned to questions of the absolute mass scale, how the flavors change, the nature of the mass hierarchy, and whether neutrinos are their own antiparticle. Neutrino flavor eigenstates and mass eigenstates are linked through the Pontecorvo-h’Iaki-Nakagawa-Sakata (PMNS) unitary mixing matrix: ”6 Uel U82 U€3 V1 ”M = U/11 U112 U113 V2 (2-5) ”7' UTI UT2 UT3 I’3 where e, 11, and T are flavor eigenstates and 1, 2, and 3 are mass eigenstates. The U no matrix elements contain the neutrino mixing angles and three charge-parity (CP) vi- olating phases (one Dirac phase and two Majorana phases, none of which have been determined yet). Neither the absolute scale nor the hierarchy of mass eigenstates is well known. Oscillation experiments were able to determine the. squared differences between squares of mass eigenstates, but not their order. Figure 2.2 shows the two options for the neutrino hierarchy. Successful detection of neutrinoless double. 13 de- cay would allow bounds to be placed on the absolute mass scale and hierarchy [17] if 6 normal hierarchy inverted hierarchy U3 1.2—— ssew Buiseeioui Ue electron neutrino flavor U . p E muon neutrino flavor UT D tau neutrino flavor Figure 2.2: Two possible configurations of the neutrino mixing and hierarchy. If combined with improved measurements of the neutrino mixing angles and the mass squared differences. a successful measurement of double beta decay can constrain the absolute mass scale and hierarchy the mass squared differences and mixing angles are also known [18]. An observation of 011.3,} decay would break the conservation of lepton number and also immediately confirm that neutrinos are their own antiparticle (Majorana) rather than being two distinct particles (Dirac). An unprecedented number of expm'iments are being devel- oped to measure this decay. 2.2 Detection Challenges 2.2.1 Detection Methods There are three techniques used to detect evidence of .53.} decay: geochemical. radio- Chemical. and direct detection. In geochemical experiments. samples of very old ore are carefully analyzed for the presence of 3.3 emitters and their daughters. Since this Inethod looks only at the presence of past decays. it measures a total rate for com- 7 ‘-‘ 'I‘ -‘ bined 2113/3 and (ll/dd decays [19]. In raciicwhemical studies. a 40—50 year old sealed sample containing a .23..) emitter is chemically purified and analyzed for evidence of i313 decay [20. 21]. Like geochemical analysis, this method is sensitive to a. total decay rate. The two methods have been used to set lower limits on the half lives of three isotopes. The. most common method of measuring dd decay half lives is that of direct. counting experiments. In this method. a large quantity of an isotope is placed in a low- background environment and decay electrons analyzed. Many experiments take place underground and are built from extremely low—background material. Direct counting experiments can distinguish between the two decay methods. 21AM decay gives off a total of four particles: two electrons/positrons and two neutrinos/antine1.1trin(_)s. Some of the decay energy is lost to the neutrinos, so the total decay energy of the electrons is therefore a continuous distribution. In (ll/.33 decay (without emission of a Major-an x). the. neutrino is reabsorbed. and the sum of the two decay electrons must equal the total Q value for the reaction. Poor experimental energy resolution can lead to the tail of the 214133 decay overpowering a. small (ll/.3.) decay signal. so accurate models of the detector response. and simulated 21A?) signal are important. Figure 2.3 shows a schematic for the a total 13.3 decay electron energy spectrum. A plethora of experimental techniques exist for direct detection experiments. The CANDLES [22] project searches for the decay of 48Ca using CF2(Eu) scintillators. CARVEL [23] is a competing experiment using 48Ca\VO4 crystals with an expected sensitivity of .04—.09 eV. CUORE (a larger version of CUORICINO) [24] uses bolome- ters to detect thermal energy from electrons emitted by the decay of 128‘130Te. CO- BRA, made of cadmiurn-zinc-telluride (CZT) detectors. contains five 13.}— emitters and four ,l3d+ emitters [14]. MAJORANA [25] is constructed of segmented Ge detec- tors enriched in 76Ge. and together with GERDA [26] (TGGe diodes) it will test the controversial claim for Gulf decay detectitni made by the HEIDELBERG-MOSCOW 8 > 0) .‘j> N O I /.e N 9 .3 0" I ..r‘“ O I it Ke/Q / \ 0.901.001.10 O 1 “5...”. /. dN/d(Ke/Q) .0 U1 1 “N. 1 0.2 0.4 0.6 0.8 1 .0 Ke/ Q Figure 2.3: Simulation of the [3,6 decay summed electron spectrum in a direct counting experiment, taken from reference [18]. K6 is the electron kinetic energy and Q is the Q value. The Oufifl events fall at Ke=Q=1, while the 211/313 events have a wider energy distribution. In the inset, the size of the 01166 decay spectrum is normalized to 10_6 of the 21435 decay amplitude. Detector energy resolutions of 57(- are folded into the simulation. (see text) .0 o o Isotope Q value (MeV) z2o5 G2V(1/y) ([30]) TI/2 (y) ([12, 31]) 4804 4.274 5.7x105 4.0x10—17 4.41:8? x1019 76Ge 2.039 3.6x104 1.3x10—19 1,510.1 x1021 8288 2.995 2.8x105 4.3x10—18 0.92i0.07 x1020 962r 3.347 6.7x105 1.8x10—17 2.3i0.2 x1019 100Mo 3.035 4.5x105 8.9x10—18 7.13:0.4 x1018 11606 2.004 7.4x104 7.4x10_18 28:20.2 x1019 124Sn 2.287 1.6x105 1.5x10’18 21.0i02 x1017 128Te 0.865 1.3x103 8.5x10“22 1.93504 x1024 130Te 2.530 2.8x105 4.8x10-18 6.81“]? x1020 136Xe 2.468 2.7x105 4.9x10—18 28.1 x1020 150Nd 3.368 1.6x106 1.2x10“16 8.2i0.9 x1018 Table 2.1: Recommended half—life values for {3‘23- emitters. Q values are from NNDC. experiment [27]. EXO [28] uses liquid xenon calorimeters to detect the 3:3 decay of 136Xe. MOON [29] is a tracking calorimeter device that looks for the decay of 100310. Several more experiments are either planned or have completed their run. using some combination of these techniques. Table 2.1 lists the most recent recommended values for some double-beta half lives. 2.2.2 Detectors for 150Nd fifl decay In the case of [3,5 decay from 150Nd, there are three high-sensitivity direct count- ing experiments planned: SuperNemo [32], DCBA [33], and SNO+ [4]. DCBA is a magnetic tracking chamber that will be able to trace three—dimensional electron paths. The detector is still in development, but it should be able to distinguish a neutrino mass as low as 0.1 to 0.5 eV. See Figure 2.5 for a picture of the prototype. SuperNemo is the successor to NEMOIII, which contained small slices of several dif- ferent [3‘6 emitters. SuperNEMO is a. calorimetry—based experiment and will look at either 8“Se or 150Nd 1n more detail. Sens1t1v1ty is expected to be around 70 meV. 10 Figure 2.4: One module of the SuperNemo detector. Picture credit: [2]. Figure 2.4 shows a single SuperNEMO module. SNO+ is a successor to the SNO neutrino oscillation experiment. where the heavy water neutrino detector has been drained and will be replaced with Nd—loaded liquid scintillator. This detector aims for a sensitivity of around 100 meV [34]. A schematic of SNO+ is shown in Figure 2.6. 2.3 Nuclear Matrix Elements 2.3.1 Half-Life Calculation r . o . 1QONd IS a popular ch01ce of nucleus because 1t has a short 21x13 decay half life and 1s expected to also have a short half life for (IL/(7’0, decay. In order to z-u-curately predict. 11 Figure 2.5: A prototype of the DCBA detector. Image taken from [3]. what. direct detection experiments might see. all parameters of the half life equation must be well known. The 21AM decay half life is [2 [YQV(0+‘—+0+)[—1==GQV(EDHZHAQ%% 1/2 (2.6) where G2” is a. phase space factor and is proportional to ZQQO. It can be calculated r . . . . . r ,0 exactly. and 1003 d has the Inghest value of this quantlty. 22(2" and Cr” values for {3,3 — nuclei are shown in Table 2.1. MQV can be represented by a double Gamow—Teller matrix element: a sum over the 1+ states in the intermediate nucleus. <(flfuarnitt><1t'naru0+:> 7‘ J J I E] + (213/2 — E0 4am: 97> J The double Gamow—Teller matrix element is the combination of Gamow-Teller matrix elements for each leg of the decay and comes from second-order 1.)erturbation theory. Chapter 3 will discuss the 07 (Gamow-Teller) operator in greater detail. In the 12 Figure 2.6: The SNO detector. SNO+ will feature the same acrylic vessel. but will be held down With a series of ropes to offset the density difference between liquid scin- tillator and water. Photo credit: Lawrence Berkeley National Lab (Roy Kaltschmidt. photographer) 13 denominator, E0 is the energy of the initial ground state, Q3}, is the Q value for 7313 decay, and E j is the energy of the intermediate state. Contributions from the various states may interfere either constructively or destructively, so theory must be used to calculate the relative phases. Because of this. experimental information about transitions in each leg can constrain but not. replace theory. Since the phase space factor G3 is well known (see Table. 2.1) and the 214733 decay half life has been measured experimentally [35. 36]. theorists can check their calculations of the summed nuclear matrix elements directly. Abad et al. [37] first hypothesized that the presence of a single low-lying state in the intermediate nucleus was sufficient to predict the 21433 decay half life. The idea of single-state dominance (SSD) has become a significant question in the field. It seems to apply to some nuclei but not. to others, and it is not known whether higher—lying states simply do not contribute to the total matrix element or whether their contributions cancel [38]. thn‘nickgj et al. [39] recently proposed that single—state dominance would not. be realized in the decay of 150Nd unless a low-lying 1+ state were measured in 15OPm. thinking higher-state dominance (HSD) to be more likely. The DU mode of decay is much more complicated than the 21/ mode. A neutrino reabsorbed in Oi/Hd decay can have a very large virtual excitation energy in the intermediate nucleus with an associated momentum transfer around 50-100 MeV/c [14], because the interaction occurs at a very short range. Therefore. the 91/33 decay process can go through any intermediate state rather than just 1+ states. The half life equation is [T[’)’2(()+ a 0+)1-1 = GUI/(13],, Z)|MOV 1.14 + 4191’+ ' 1,31%qu ,,>° (28) 9:4 where GOV is known [30]. Matrix elements for Gamow-Teller (Mg/T). Fermi (3125/). 01/ and tensor (MT ) transitions must be calculated. The final term is the effective 14 Majorana neutrino mass: 3 "2.133 = 2 U 24‘1" k (2.9) k=1 where U is the. unitary neutrino mixing matrix from section 2.1.2 and 111 is the neutrino mass eigenstate. Accurate half life calculations are important when planning direct decay experiments, but if a positive signal of neutrinoless 13,53 decay is found. half life must be known to an error of 15-20‘70 to allow for the extraction of the Ma jorana neutrino mass [1] with high enough precision to discern the correct neutrino mass scale and hierarchy [40]. This requires additional work on the nuclear matrix elements (NMEs. 2.3.2 The Shell Model Approach The large—scale shell model can be used to calculate the nuclear matrix elements of 73/3 emitters. 2143,73 decay in 48Ca can be calculated without any truncations to the pf model space [41]. Recently, Horoi et al. have extended this effort to the calcula- tion of 48Ca’s (II/i353 decay matrix elements [42]. though they assumed that negative parity states in the intermediate nucleus could safely be neglected. Unfortunately, the prohibitively large model spaces required for heavier nuclei restrict the reach of the shell model and do not allow for full calculations of these nuclei within complete model spaces. Caurier et 01. did 0M3}? decay calculations in limited model spaces for 7'6Ge and 828e when only considering the ground-state-to-ground-state transition [43]. The Interacting Shell Model has recently allowed for 01x33 decay calculations in masses 11p to 136 [44], but these calculations (as well as many QRPA calculaticms) rely on the closure approximation. Since the 011.37} decay calculation is so complex. attempts have been made to reduce the dependence of the calculated nuclear matrix elements on extensive knowledge of the intern'it-‘diate nucleus. The closure approxima- tion [45] collapses the sum over intermediate virtual states to a single matrix element. and approximates the difference in their excitation energies as an average energy. The rationalization for this approach is that the virtual neutrino's high momentum (100 MeV) drowns out. the smaller differences in nuclear excitation energy [46]. Errors from using the closure. approximation are estimated to be approximately 10% [47], but this is still a concern when the matrix elements overall need to be known to 15-20‘7t. More accurate calculatimis are certainly desirable. 150Nd is both heavy and deformed. and shell model calculations are not yet available even if the closure approximation is applied, although work on the projected shell model may produce results in the future [48]. Calculations in the Interacting Boson .\Iodel can provide another tool to calculate 7'33 decay matrix elements [49]. 2.3.3 QRPA The QRPA (quasiparticle random phase approximation) is based on the RPA (random phase approximation) method of calculz-ition. Quasiparticles are fern‘iions constructed from particles and holes via a canonical Bogoliubov transformation. The addition of quasiparticles to the RPA reproduces ground state pairing correlations more closely than with particles alone [50]. A full discussion on techniques for solving the QRPA equations will not be tn‘esented here. (See references [46. 51. 50. 52].) However. I will give a brief overview of recent developments in the field that are of i111p(_)1‘tance to nuclear matrix element calculations for 13,3 decay. The QRPA model was developed to accurately describe collective states. such as giant resonances. In the words of reference [46]. “. . . in the QRPA and RQRFA (relativistic QRPA) one can include essentially unlimited set of single-particle states ...but only a limited subset of configurations (iterations of the particlehole. respec- tively two—quasiparticle configurations), in contrast to the nuclear shell model where the opposite is true.” Two important. variants of QRPA are the anBI’A (proton-neutron QRI’A) and 16 cQRPA (continuum QRPA). The anRPA [53, 54] was developed to model .3 decay and Gamow~Teller excitations in nuclei, and is now one of the most popular techniques for calculating .13 decay nuclear matrix elements. Particle-particle and particle—hole residual interactions are required [55]. The CQRPA [56. 57. 58] allows for the consid- eration of particle-unbound states and the study of widths and decay properties of isovector giant. resonances (see section 33). 2V3.) decay calculations in the anRPA and cQRPA are very sensitive to the chosen values of the parameter gm). This parameter represents the strength of the particle—particle part of the proton-neutron t.wo-l:)ody interaction [54, 59]. It is deter- mined by the ratio of the particle-}_)art.icle and particle-hole interaction strengths [60]. and should be on the order of l. A successful reproduction of the 2V3} decay half life is often used to check the feasibility of the more difficult (ll/.33 decay calculation. There are two main ways to determine the value of gm): one can fit it to matrix elements derived from experimental data on the 21/{2’33 decay half-life [61], or one can use information from single .13 decay [62]. Most calculatitms use the first method. but increased use of charge-exchange experiments to constrain nuclear matrix elements may change this. The QRPA’s sensitivity to gpp is a cause for concern [42]. but using available data from single— and 21/343 decay should constrain the term enough that calculations for 014533 decay can be successfully performed. Large deformations in some {3.3 emitters (76Ge. 1SUNd) have posed a serious chal- lenge to theorists [63, 64]. Deformation differences between the mother and daughter nuclei are thought. to decrease the 6’43 decay nuclear matrix elements because of re- duced overlap in their wavefunctions [65, 66] in comparison to transitions from one spherical nucleus to another. Introducing deformations into the QRI’A calculations changes both the location and the shape of Gamow-Teller strengths in the inter- mediate nucleus. Information on these intermediate states is necessary for accurate calculations of the nuclear matrix elements, and this can be done with the use of 17 charge-exchange experiments. The group of Vadim Rodin (University of Tiibingen) has provided new QRPA cal- culations for the Gamow—Teller and dipole strengths in 15Ol’m from both 150Nd and 150Sm. These results will be presented and compared with experiment in Chapters 4. 5, and 6. 2.3.4 Constraining NMEs with charge-exchange experiments Intermediate—energy charge—exchange experin’ients (see Chapter 3) can be used to preferentially populate Gamow-Teller transitions in the intermediate nucleus between the [353 mother and daughter. Transitions in the 73+ direction may take place using the (up). (d,2He), (t,3He), or (7Li,7Be) reactions. and transitions in the ,3- direction may use the (p.11) or (3He.t) reactions. Since all 73:3 mothers and daughters have a ground state J7r of 0+. the Gamow- Teller transitions (AL=0,AS=1) go to 1+ states. 21/7‘373 decay should proceed largely through 1+ states. and knowledge of the exact location and the strengths with these states are populated is important for accurate nuclear matrix element calculations. Charge-exchange experiments will also populate other multipoles. such as dipole and quadrupole transitions, which are significant in calculations of (ll/.33 decay matrix elements [61]. F igure 2.7 shows the. population of intermediate states in 1SOPm via charge-exchange reactions on 1'50 Nd and 150 Sm. A collaborative effort is underway to systematically measure charge-exchange tran- sitions in 713,73 decay nuclei. Older (p.11) and (up) data sets are being augmented by new data, and this approach allows for Gamow-Teller contrilgmtions to be measured up to high excitation energy. 48Ca( p.11) and 48Ti(n.p) were recently re-measured by Yako et al. [67]. Unfortunately. (n.p) measurements suffer from poor (~l .\IeV) en- ergy resolution, which makes spectroscopy of low-lying states very challenging. Use of more complex probes (such as (t.3He) and ((1.2He)) has brought (n.p)-direction reso- 18 ‘1', —----—---- ‘50 Nd(0+) OVBB/ZVBB > ‘ 15°Sm(0+) (3H6,t) (t,3He) Figure 2.7: Population of intermediate states in 15OPm via the (t.3He) and (3He.t) charge-exchange reactions. This figure is a schematic, and levels shown do not corre- spond to the location of actual levels. Figure by R.G.T. Zegers. 19 lutions down to 110-300 keV. In the (p.11) direction, high-resolution beams of 3He are regularly produced at RCNP, and (3He,t) experiments can achieve a 20—40 keV resolu- tion. Recent. measurements include 96l\Io(d,2He) [68], 7’GSe(d.2He) [69]. 64Zn(d,2He) [70], 100Mo(3He,t) and 116Cd(3He.t) [71], 48Ti(d.21~le) [72]. and 48Ca(3He,t) [73]. Data on several more nuclei exist but have not yet been publisluad. The measure— ments of 150Nd(BHet) and 1'50Pm(t,3He) described in this document are the first such measurements to address the i353 decay of 150Nd. Chapter 3 Charge-Exchange Reactions 3.1 Introduction to Charge-Exchange Reactions Extensive programs in charge-exchange (CE) reactions have been (l(,'vel(.)[')e(_l in the last half century (see [74, 75] and references therein) to probe the spin-isospin response of nuclei. Charge-exchange reactions are characterized by an isospin transfer (AT) of 1, and can excite a number of different transitions. Table 3.1 provides a partial list. In hadronic charge-exchange, a proton (neutron) transitions into a neutron (proton). The process can be modeled by the exchange of 77 (and other) mesons between the projectile and the target. where the projectile may consist of a single nucleon or be a composite probe. Pion charge-exchange has also been used as a probe [76]. but will not be discussed in any detail here. Although charge-exchange is mediated by the strong interaction and .73 decay by the weak interaction. the same final and initial states are populated. The Fermi and Camow-Teller transitions correspoml to the two types of allowed (3 decay, and the other transitions correspond to various types of forbidden ,{3 decays. In a 73 decay experiment, states may be seen in an excitation energy region from 0 MeV up to the Q value of the reaction. but higher-lying states will not be accessible. Charge-exchange reactions allow for the excitation of higher- ()1 H AL AS ha; 0+—+fl 0 0 0 0+ Fermi 0 l 0 1+ Camow-Teller 1 0 1— dipole l. 1 (0.1.2)_ spin-dip(_)le 2 0 0.2 2+ quadrupole 2 1 0.2 (1,2,3)+ spin-quadrupole 3 0 1.3 3‘ octupole 3 1 1,3 (2.34)". spin-octupole 4 0 0,2,4 4+ hexadecapole Table 3.1: Charge—exchange excitations and their quantum numbers. All have ATzl. A 0+ ground state is assumed. The hw column refers to a transition between major oscillator shells (i.e. a Ahw=1 could represent a transition between the sd- and pf- shells). lying states and give a complementary description of the s]*)in-isospin response of a nucleus. Gamow-Teller (GT) strength is represented by B(GT). The GT transition is me- diated by the or operator. If the general forms of a CE particle-hole operator are OAT: Z a}, (r) )rij (3.1) for no—isovector spin—flip transitions and 04C”: :74 J. [m (f) ..ojmti, (3.2) .7 for isovector spin—flip transitions. setting A to 0 results in the Fermi and GT operatm's from l3—decay: Ztij and zafij [77]. (3.3) 3‘ j - /\ corresponds to AL+An. where. An is the change in major oscillator shell. 22 Equation 3.4 gives the relationship between B(GT) and the or operator in :3- decay. U31 and t, F are the initial and final nuclear states. and {14 is the axial-vector coupling constant of the weak interaction. B(GT) 1 |§j< ll *1] >2 (30 = —— (7-7. .5 i 2.1+1,‘F 11" J In 1963, Ikeda et al. [78] developed a non-energy-weighte(l sum rule for the total amount of CT strength that should be seen in CE transitions from a given nucleus. Ikeda’s model-independent sum rule is .flfw—mffl=wN—Z] mm Fermi strength has a similar sum rule: fiffl—flfiflzN—Z cm The GT sum rule provides a useful upper limit on the amount of strength an exper- imentalist is likely to see. although in most cases only 50-60% of the expected sum rule strength can be accounted for (for an example, see reference [79]). This is known as the quenching problem [80, 81, 82, 83]. and will be discussed in Chapters 4 and 5. Sum rules also exist for higher multipole excitations (see reference [77] for examples). A charge-exchange reaction can go in either of two directions: Angil. AT3=+1 corresponds to an (n,p)-type reaction, which goes diagonally down and to the right on a chart of nuclides. AT:=-1 corresponds to a (p.11)-type reaction. which goes diagonally up and to the left on a. chart of nuclides. Figure 3.1 shows both types superimposed upon a small section of the chart. of nuclides of relevance for this thesis. Figure 3.2 shows a more thorough picture of isospin in CE reactions. The target nucleus has Tz=(N-Z)/2. For a (p.11)—type transition. a T0 ground state in the target 23 p 1508m 1518m 1528m 4 I(n,p), (d,2He) or(t,3He) ATZ=+1 149Pm 150pm 151pm 148Nd 149Nd 150Nd > n Figure 3.1: Schematic of charge—exchange on a subset of the chart pf nuclides. Nu— clei of interest (100Nd,Pm,Sm) are shown. The transition from 1"308111 to 150Pm Q 0 a I f V represents an lsospm change of AT z=+1. and the trans1tion from 1'30.\d to 1'50Pm represents an isospin change. of AT32-1. has an analogue T=T0 state (the Isobaric Analogue State) in the residual. In general. a (p,n) transition can populate T0+1. T0. and TO-l states in the residual nucleus. In an (n,p)-type transition, the residual has a minimum isospin of T=T0+1. so only states with isospin of T0+l can be populated. Figure 3.3 shows the microscopic picture of CE reactions as excitations of proton- holes/neutron—particles (AT3=+1) and neutron—holes/proton—particles (Ang-l ). In medium—to-heavy stable nuclei with a significant. neutron / proton asymmetry. the neu- tron single-particle orbits are filled above the proton Fermi level. Pauli blocking constrains the single-particle orbits involved in a transition: excitations of lp-lh components in the same oscillator shell are hindered in the (up) direction. 24 M ’/’ T0+1 I I I’ ” To M I I T0+1 I’ I _"""‘ TO-l ‘ ATZ = +1 AT2 : ’1 (n,p) type (p,n) type Figure 3.2: Isospin symmetry in charge-exchange reactions. States of like isospin (analogue states) are shown in like colors. In the (p.11) direction. the IAS is populated from a T0 to T0 transition. but no such transition can occur in the (up) direction. (3He,t) (t,3H9) . 0‘ ,4 . ‘ ' ::'~O I" of-X» Protons Neutrons Protons Neutrons Figure 3.3: Pauli blocking is strong for the (t.3He) reaction and reduces transition strengths, but is not as significant in the (‘3He.t) direction. 25 3.2 Reaction Theory Cross section calculations in this thesis are 1,)erformed using DVV BA (Distorted “’ave Born Approximation) methods with the. code FOLD [84]. The incoming and outgoing waves are distorted by the nuclear mean field of the target. An effective potential (Ve f f) describes the interaction between nuclei in the target and the projectile. The cross section can be determined from the square of the amplitude of the outgoing spherical wave. A T-matrix represents the transition l‘)etween final and initial states. Input from single-particle wave functions, one—body transition densities (OBTDs), the nucleon-nucleon interaction. and optical potentials result in calculated angular distributions for each type of charge-exchange transition listed in Table 3.1. These angular distributions are then compared to data. Absolute Gamow-Teller and Fermi strengths are calculated with the help of a pliei’iomenological unit cross section. 3.2.1 DWBA The scattering potential (V) is separated into two pieces: the distorting potential from the nuclear mean field (L11) plus a residual interaction (U2) containing the physics of interest. The Schrodinger equation is then (E — T — U1 — U2)(," 2 0 (3.7) and the wavefunction may be written as a partial Lippmann-Schwinger equation z.» = o + 63(0’1 + ((2)0. (3.8) 26 where (,0 is the homogeneous solution to the Schrédinger equation and GS— is a Green's function equal to (E — T)_1. The resulting T-matrix is m < If, Ttot : _ ’1! (,— c", > . (3-9) When V is expanded into U1 + U2. the expression for the T-matrix can be simplified to 72/4: , _ _ —%;flm=rm+r%u«a]mu>+, on» where X is 0‘) after being distorted by the mean field of the nucleus: -= C?“ U 3 11 x e + 0 1X- ( - ) Expanding X into a, series yields the Born Series r 2 , 7f2z—ff (39) H ‘ . . ._ 7 . . where G1]— 18 equal to (E — T — U1) 1. Tl’l can be ignored. smce U1 does not connect the initial and final states. The T matrix for DWBA calculations is then , x' 2,11. _. TmHM=—7—. . an) 71 k (Notation and equation sequence largely taken from reference [85].) In many cases. the Born series is truncated at. the first term. and this approximation is known as a first order Distorted Wave Born Approximation (DWBA). The reaction cross section is proportional to the square of the T-matrix element governing the transition between (10 [1. 21‘] __ z __ T (AZ (gfih27 A ].f 1. initial and final states 2 . a. min 27 When U1 of the DVV BA is a central optical potential, TUl is 0 and TU2=Tf,-. . U 2 , Tfi = T 2 = ——- < XiUQ[X >. (3.15) DWBA calculations in this work were carried out using the FOLD code. FOLD [84] is a three—part program for charge—exchange reaction calculations originally developed by J. Cook and J .A. Carr in 1988. The three separate sections of this code are called W SAW, FOLD, and DWHI. W SAW uses numerical methods to solve for single-particle radial wave functions of relevance to the DWBA calculation. A VVood-Saxon potential is used to represent the volume section of the total potential, and Coulomb and spin—orlnt potentials are also taken into account. The input of WSAW consists of binding energies and shell model quantum numbers for single—particle orbits. Output wave function files are then read into the FOLD code along with other input paramr;>ters. 3.2.2 One-body transition densities Wave functions from WSAW are single-particle wave functions. The DVV BA nuclear structure input for each calculation involves a combination of 1p—1h transitions be.— tween single—particle orbits. The relative weight of each 1p-lh transition is given by its OBTD. OBTDs contain information on the overlap between the final and initial nuclear states [51] and must be calculated for both the projectile/ejectile and the target / residual systems. A nuclear structure code (often a shell model code like OXBASH [86] or NuShellX [87]) calculates the importance of each single-particle transition. calculates phase factors, incorporates all of the necessary angular momentum coefficients. and returns 28 an OBTD. The OBTD formula in an isospin framework is < fT ll [0].” c; (SMVHAT || 1T’ > \/(2,\ + 1)]2AT+ 1) where aJr and a are single-particle creation and annihilation operators, f and i repre- sents the final and initial quantum numbers, /\ is the rank of the operator. [(0.13 are final and initial isospin states. and AT is the change in isospin. 150Nd and 150Sm are too heavy to calculate the OBTDs in the shell model because the model space is too large. A normal modes formalism [89] is used instead. Normal modes are the most coherent superposition of l-particle l-hole states for a particular operator in a given particle-hole basis. They exhaust. full (non-energy- weighted) sum rule strengths and give a set of OBTDs for each type of transition associated with the operator 0907-. However, the downside of this method is that no information is provided on the strength distribution as a function of excitation energy. The following bases were used in calculations in this thesis: 1508m (150Pm)(150Nd) was assumed to have 32 (31)(30) protons (4 (3)(2) in the 2p 3/2 shell) and 88 (89)(90) neutrons (6 (7)(8) in the 111 9/2 shell). The neutron space included the 1b 11. / 2 level for the 150Sin—+150Pm calculation to allow for CT transitions —— without. this modi— fication, Pauli blocking would prevent. all GT transitions. To accommodate all of the transitions relevant for this work, the model space was allowed to include orbits up through 1i 11/2. 3.2.3 The nucleon-nucleon interaction The free nucleon-nucleon interaction V12 takes the form —> v12 = V0012) + #45012) L G" + News”. (3.17) 29 VLS where V0 is the central potential, is the spin- -orbit potential. VT is a tensor potential, and 1 and 2 refer to the two interacting Imcleons. L - S is the spin- orbit operator and 812 is the tensor operator. In their 1981 paper, Love and Franey [90, 91, 92] determine V12 with the use of a. large body of nucleon-m1cleon scattering data. They decompose Vf(r), VLS(1'). and VT(r) in terms of Yukawa potentials with Q; T ,chosen for their similarity to the one— pion exchange potential (OPEP). t he form The three potentials become 7V0 .C ,. _ C, L I (”—2.21% y(17,7 VLSm ZS VLSV(’ ) (3.18) :vT 0;?— 1) N; These sums run over Yukawa potentials with different ranges that. reflect the ranges of the 77, p. and 2-7r meson exchange. The result of Love and Franey’s work was a set of effective nucleon-nucleon t-matrix interaction strengths applicable to a. wide variety of nucleon-nucleus scattering techniques, such as (p,p") and (p,n). Vt'hile the full effective interaction has many terms, the ones important for charge exchange are Var = 2 V9 14%?) VC 113.96 «71> + Jessa—st - S I] (3.19) (The sum over i and j runs over all m1cleons in the projectile and target.) Love and Franey showed that the UT component is preferentially excited at energies above 100 IV’IeV/u and below 500 MeV/u. where the 7' contribution is at a minimum (see 30 I Strength of the or t-matrix interaction I Strength of the 1: t-matrix interaction 2.5 — _1 a a. E 5 2.0 — g 93 :1,- a: V «E 1.5 — J 1.0 - 0.5 *- 00 i g l I I ° 0 200 400 600 800 1000 E/A(MeV) Figure 3.4: Relative operator strengths for the or and T t-matrices, from reference [91]. The 07 t-matrix is significantly stronger than the T t-matrix at energies above 100 MeV/u, which allows charge-exchange experiments to preferentially populate 07 transitions over T ones. Values come from reference [91]. Figure 3.4). In addition to the dominance of the or term above 100 MeV, this energy regime also features decreased contributions from multi-step processes and decreased distortion effects from the central isoscalar potential. Near zero momentum transfer, the LST term is so small it is negligible (it. is also taken out of FOLD calculations). Contributions from the TT interaction are small, but must be taken into account. for non-zero momentum transfer as they create amplitudes that interference with amplitudes mediated by V07. A two-body interaction between nucleons is represented by “direct" and “ex- change” terms. The exchange term represents amplitudes due to processes where 31 a nucleon in the target is struck and ejected and the projectile nucleon is captured [77]. The exchange term contains non—local effects. which makes it much more difficult to calculate. A short—range (no-recoil) approximation is often used [90] to deal with the exchange terms, although it is known to underestimate the destructive exchange contributions for reactions involving complex probes like (t,3He) and (3He,t) [94. 95]. 3.2.4 FOLD The Love-Franey interaction. is an effective nucleon-nucleon interaction, but CE with complex probes involves a. nucleus-nuclei1s interaction. In order to calculate the cor- rect T-matrix, the effective nucleon-nucleon interaction must. be double-folded (i11- tegrated) over the. transition densities of the projectile/ejectile and target /residual systems to create a form factor: F('I') = (12(1‘) =<(tga‘y-[Veff('l')[(1101) >, (3.20) where (163,311,!) represent. the ejectile. residual, target, and projectile wavefunctions. respectively. The FOLD code carries out the double—folding procedure and produces this form factor. Each type of transition requires its own FOLD input file. For the 1SONd and 1508m experiments, the use of normal-mode OBTDs means that. only one form fac- tor is available for each type of J7r transition. (If QRPA or shell model transition J7r could be calcu- densities were available, form factors for many states of the same lated.) Depending on the excitation, several form factors might have to be calculated: these correspond to different units of angular momentum transfer between the target and the projectile. Contributions to the cross section from each form factor are then added. As an example, in a GT transition the relative change in total angular mo- mentum is AJ =1 (AL—=0. AS=1) for both the projectile/ejectile and target/ residual 32 systems. The relative angular momentum transfer can be calculated from that in the projectile and target JR: Jp-i-JT. (3.21) In this case, JP=1 and JTzl. so JR can be either 0 or 2 (1 is forbidden due to parity conservation). A form factor is calculated for each J R: Calculations were done for all of the multipoles listed in Table 3.1 using OBTDs from NORMOD (with the exception of octupole transitions, where only the 3- spin- flip octupole was calculated). 3.2.5 DWHI As mentioned in section 3.2, the incoming and outgoing particles are represented by plane and spherical waves distorted by the optical potential. DWBA calculations ac- count for this effect. Form factors are integrated with the distorted waves to calculate the T—matrix T =< ,XfIFtrllxr >. (3.22) which is then used in Equation 3.14 to calculate the cross section. A common way to determine optical potential parameters is to take elastic scat- tering data with the same experimental setup used for the experiment you wish to apply it to — using the same projectile. the same target, and the same beam energy. Optical potentials are fit to the cross sections from this elastic scattering data. Pro— grams such as ECIS (used here) [96] or SFRESCO [97] are used. Real and imaginary VVood-Saxon functions (volume. radius, and diffuseness parameters) are used as the base for the fit, and the complexity of the fit can increase if extra functions are in- cluded to account for surface or spin-orbit potentials. As D\V'HI is set up to handle only volume-type optical 1.)(‘)t.entials. these. extra functions were not used. In many cases, optical potentials are not available: they may be very difficult 33 to measure, or it may be almost impossible to get. beam time to do the measure- ment. Fit parameters are then extrapolated from known potentials. Some efforts have been made to establish global potentials from simultaneous analysis of many elastic scattering experiments. For the two ex1'1eriments discussed in this work. the 150Nd(3He,3He) optical potential was measured following the 150Nd(3He.t) exper- iment. A measurement of the 150Sm(t.t) reaction was not feasible, so the 150Nd optical potential from was scaled by 85% for the 15USm(t,3He) experiment (following reference [98]). This is a purely phenomenological solution and has been employed in other (t,3He) experiments. More details of the optical potential measurement. will be discussed in section 4.3.2. 3.2.6 The unit cross section and B(GT) Taddeucci ct al. found a qua1'1titative description [99] of the proportionality l’)etween the Fermi and Gamow-Teller cross sections and beta decay: do . '2 __ : 1 [NY , = A B F . ‘ '2‘ (152 (1:0 1 Ir] B(F) 0F f 7 (3 3) ‘11. _ 1m 1 [28(GT) — a B(GT) [99] (3 24) (IQ (1:0— J 7 0T — GT ‘ l where (3 F and [TGT are phenonrenological unit cross sections. N is a distortion factor (the ratio of distorted to plane waves, and here the transformation to q=0 has been included) _ GD”, ((1 =0) aPl‘l’tq = 0) N [99. 100]. (3.25) K is a kinematical factor that includes the momenta (Is, and If) and reduced energies (E3 and E f) for both the entrance and exit channels 1. 2 £34.52: (3.20 (1121-276.— 1.7, 34 and Jar (or Jr) is the volume integral of the corresponding effective interaction (see equation 3.19). In his derivation, one of Taddeucci’s assumptions was that the Eikonal approxmation was valid —- both the projectile and ejectile trajectories are well-represented by straight lines and the beam energies are much higher than the excitation energy. These conditions are satisfied for CE experiments at E> 100 MeV/ 11 and qz0. The Eikonal approximation comes into play because its use allows the different components of the T-matrix to be factorized into a nuclear structure and a nuclear reaction part [101, 99]. While the factorization is only exact in the plane wave approximation [101]. experiments have shown that it also works well for distorted waves in AL=0 transitions [99, 95]. The proportionality of Equation 3.24 can be checked using GT strengths obtained from beta decay experinwnts [102]. The unit cross section for both Fermi and Gamow- Teller transitions are simple functions of the mass number A [99]. Figure 3.5 shows this dependence for the (3He,t) reaction [95]. Where there are differences in the calculated and measured unit cross sections. interference between V§T and V; is thought to partially explain this difference. One example is the case of 58Ni. where removing contributions from the tensor interaction (based on theory) restored the proportionality of Equation 3.24 and brought the data point back to the phenomeno- logical curve [103]. 3.3 Giant Resonances In a macroscopic picture, giant resonances are defined as a. density oscillation of proton-neutron nuclear fluid. The two fluids form overlapping spheres in the nucleus. Density oscillations of these spheres can fall into one of two categories: isoscalar. if the two fluids move in phase. or isovector, if they move out of phase. Isovector resonances are further separated by whether particles with opposite spins move in or 3' (mb/sr) Figure 3.5: T0 the left. the Fermi unit cross section as a function of mass number for (3He,t) data taken at 420 MeV. At the right, the dependence of the Gamow-Teller unit cross section as a function of mass number for (3He,t) data taken at 420 MeV. The data point completely off of the fit line corresponds to the case of 58M, and the difference is due to the interference of V; with V9; [103]. Both figures are adapted O . Fermi GT 1.06 _ Gem—72m F . : C E C E _ 0.65 L Germ—109m .- L ‘ g ' “exp _ o i i l l l 4 1 10 from reference [95]. 3G out of phase. Those resonances with a spin dependence are known as ism-rector spin giant resonaiiices (represented by an extra S in the abbreviation). Figure 3.6 shows the oscillatory modes for several giant resonances. A microscopic. picture of giant resonances may be constructed by ctmsidering a coherent superposition of many particle-hole excitations. If a large number of particle- hole pairs are excited (collective excitatimis). single-particle characteristics are washed out[77l The amount of collective motion present may be observed by comparing the total multipole strength seen with that predicted by a sum rule. Examples include the Fermi and Gamow-Teller sum rules mentioned in section 3.1. A general expression for the non—energy-weigl1ted sum rule (NEW’SR) for transitions with multipolarity /\ Zlis 2J+1 A] /\J _ __ S : S_ + 27r 2 . . (w—Z<%*» mm. my) Spin transfer is ignored in this equation. Giant resonzmces are defined to fulfill over 50% of the relevant NEVVSR [77]. Table 3.2 provides the quantum numbers, approximate centroid. and excitation energy for monopole, dipole. and quadrupole giant resonances. The resonances are categorized based on angular momentum. spin, and isospin transfer. The single as- terisks of Figure 3.6 indicate that. the spin polarizations of the IVSGMR, IVSGDR. and the IVSGQR can be drawn in two different ways. For example, in the IVSGMR you can have protons with spin up and neutrons with spin down or protons with spin down and neutrons with spin up. Isovector giant resonances have three isospin conmonents in the ("SHe.t) direction. The total strength will be split between T0+1. T0. and Tn-l isospin levels. The relative population strengths of each level is dependent on the isospin of the target 1 (To+1><2Tu+1>‘ nucleus. The strength of the T0+1 level is weighted with a factor of 37 P p! (IS)GMR IVGMR IVSGMR* AL=1 ’ . ISGDR** IVGDR (IV)SDR* A “A «a» «I» , "v v v ISGQR IVGQR 'VSGQR* AT=O AT=1 AT=1 AS=O AS=0 AS=1 Figure 3.6: Giant resonance modes from Table 3.2. p and 11 represent protons and neutrons, and the small triangles indicate spin. Arrows show the proton and neutron directions of motion. Single asterisks denote resonances for which more than one accurate picture can be drawn: the spin polarization of protons and neutrons can be either spin up or spin down. The double asterisk by the ISGDR indicates that this is a second-order resonance. Neither this figure nor Table 3.2 is meant to be exhaustive; higher-order resonances exist but will not be discussed in this work. See [77] for more detailed informaticm. The figure was modified from [105]. 38 Resonance name AL AS AT E X (MeV) ISGMR 0 0 0 80A—1/3 IVGMR 0 0 1 59.2A“1/6 IVSGMR 0 1 1 * ISGDR 1 0 0 120A-1/3 IVGDR 1 0 1 31.2A—1/3+20.6A_1/6 IVSGDR 1 1 1 * ISGQR 2 O O 64.7A"1/3 (heavy nuclei) ) IVGQR 2 0 1 130A—1/3 IVSGQR 2 1 1 * Table 3.2: Isoscalar and isovector giant resonances, from [77]. IS stands for isoscalar. IV for isovector, and IVS for isovector-spin. Likewise, GMR stands for giant monopole resonance, GDR for giant dipole resonance, and GQR for giant quadrupole resonance. The excitation energy given is approximate and based on the hydrodynamic model. and may be changed by significant deformation. See Figure 3.6 for a drawing of the various modes. * The IVSGDR and IVSGQR have three spin components (the middle resonance will have an excitation energy similar to the no—spin-flip resonance of the same type), while the IVSGMR has one spin component. All resonances in the (p.11) direction also have three isospin components. but one is preferentially populated (see text for more details). 39 // '——""_ To VOA-113 T /. ,/// lIAS ° 150Nd 150Pm o I r V r I O Flgure 3.7: IVSGMR schematic for 1"().\d —+ 1”()Pm. Monopole exc1tatlons are n o l I I r V r represented by very thick lines. and isobaric analogue levels in 1" ).\d and 1”UPm are connected by thin dotted lines. the T0 level with a factor of 7.0171. and the TU-l level with a factor of $31}; For the 1'50Nd(3He.t)15017111 case (the target isospin is 15), these correspond to values of 0.002, 0.0625, and 0.9355, so we expect the TO-l resonance to dominate. In the (t.3He) direction. all of the strength goes into the T()+1 isospin compmient because it is the only one that can be populated. The IVSGMR has only one spin component. but the IVSGDR and IVSGQR resonances have. three spin components on top of their isospin coinimnents: from a 0+ ground state. 0“. 1_. and 2_ states can be populated through the IVSGDR, and 1+, 2+, and 3+ states through the IVSGQR. For example, let‘s examine the population of the IVSGMR in 150Pm as excited from 150Nd and 150Sm. In the (3He.t) direction. Bohr and Mottelson [100] predict that the (TO-1) IVSGMR can be found at an energy of 6=(T() + 1)l’1A—1 lower than the T0 monopole resonance: Ep’SGMR : VOA—U3 + 51445 (T) _ (5 (3.28) where V is around 155 and V is 55 107. 105 . This er uation )redicts a resonance 0 1 1 centroid of 37.5 MeV. Figure 3.7 shows the relevant schematic. 40 .._\__T0. 1 \ ‘s To .. T011 413 VOA X \\ 150Eu 15°Sm 150pm Figure 3.8: IVSGMR schematic for 1508111 —> 150F111. Monopole excitations are rep- resented by very thick lines, and isobaric analogue levels are connected by dotted lines. Differences in excitation energy between analogue levels are close to the Coulomb dis- placement. The excitation energy of the IVSGMR in 150Pm is represented by the. quantity X, which has isobaric analogues in both 1508111 and 150En. The situation in the (t,3He) direction is more complicated. Figure 3.8 shows this situation. There are no other quantities in the residual nucleus with which to calculate the excitation energy. but the analogue state in 150Eu can be calculated r l 0 100F111 based on isospin and that excitation energy can then be extrapolated back to . . . 5' . , _ symmetry. The excitatlon energy 111 1J0Eu 1s 6=(T())l 1A 1 above the T0 monopole . . . r . . resonance. and the excitation energy 111 1”Pm can be found by subtracting twice the energy of the Coulomb (.lisplacemeut from the 150Eu value. EfVSG-M R (15017111) = VOA—U3 + 51:45 (r)(1ul50rzu) + e — 21b (3.29) and .22 +1 0.76 AV = 0.70415—7—(1 — —,—, C A1/5 22/5 For values of VC=16.5 MeV and 624.77 .\-IeV, the Eg-VSG‘UR in 1501’111 is predicted ). (3.30) to be around 15 MeV. Centroids of the dipole and quadrupole resonances can be calculated in a similar fashion. 41 Chapter 4 150Nd(3He,t)150Pm* at RCNP 4.1 RCNP Experimental Setup and Procedure 4.1.1 Beam preparation and tuning The Research Center for Nuclear Physics (RCNP) in Osaka, Japan, has a. well- developed program of experiments with intermediate—energy 3He2+ beams. The AVF and the Ring cyclotrons are coupled to accelerate a beam of 3He nuclei to 420 MeV 010 and achieve beam intensities of up to 5x1 particles per second. The faint-l)eam method is used to check dispersion-matched tuning [108, 109] in the “S beam line [110, 111] (see Figure 4.1 for the W'S floorplan). Excitation energy resolutions of 20-40 keV can be achieved. The Grand Raiden Spectrometer [112] (see Figure 4.5) is used to analyze the mo- mentum of tritons from (3He,t) experiments taking place at RCNP. It contains three dipole magnets, two quadrupoles, one sextupole, and one multipole magnet. The multipole magnet can produce dipole, quadrupole. oct upole, sextupole. and decapole fields to correct for aberrations in the ion optics. One magnet. the DSR (dipole mag- net for spin rotation). is meant for polarized beam experiments and was not used in this work. The Grand Raiden focal plane contains two sets of .\lulti-\\'ire Drift 42 Chambers (MVVDCs), which were used to collect position and angle information. Each MWDC has two planes of anode wires in between its three cathode planes: the X layer has wires perpendicular to the “medium plane” of the spectrometer and the U plane has wires at. a 48.190 angle [112] with respect to that plane. “Potential“ wires are charged to create a uniform electrical potential [113], and “sense.” wires are grounded and detect ionization electrons. Cathode voltages were set to -5.6 kV and potential wire voltages were set to -0.3 kV. The MVVDCs were filled with a mixture of 71.4% argon, 28.6% isobutane, and a very small amount of isopropyl alcohol [113, 114]. Drift times from the. four sets of anode wires give position resolutions around 300nm in each plane. A set of two 10mm-thick plastic scintillators placed behind the drift chambers is used to measure energy loss and time-of—flight information for each hit. and the first scintillator triggers the data acquisition system and serves as the start of the time of flight measurement. The cyclotron RF provides the stop signal. A 1 mm aluminum plate placed between the scintillators improves the particle identification (PID) by increasing the. energy lost in the second scintillator (see Figure 4.6). For more information on the parameters of the Grand Raiden, see Table 4.1. Event rates were such that the data acquisition live time during the experiment was 96%. One of the primary considerations in planning a charge-exchange experiment is to optimize the measured energy resolution of the ejectile, which (in cmnbination with angular resolution) determines the excitation energy resolution in the residual. This is accomplished by carefully considering the type of probe and target thickness. but use of dispersion-matched rather than focused beam optics prior to the target can increase the resolution by up to a factor of 3 (for a 3He beam at RCNP _ the increase is closer to a factor of 5 for the tritium beam at the NSCL). Dispersion- matching techniques were enmloyed for both the 1501\'d(3He.t) and 1"13()S111(t.3He) experiments (see Chapter 5). The beam is momenttun-dispersed on the target to match the dispersion of the spectrometer. and the spectroi’neter then focuses the beam 43 05320 ~55 coho—um ocE mSE 32m :. Cozumm ‘1 rs 1 Q2220 , a \ . . é 55$. $.20 . _ . .. . 9.25220 .... a . ... “Xv \. eczema. 69m... . -.. ... Figure 4.1: The W S beamline at RCNP. Figure taken from reference [111]. 44 Magnetic fl :3; : . spectrometer Target 12> .......... Figure 4.2: Ion optical modes for high-resolution spectrometers. The beam trajecto- ries represent different incoming momenta. a) shows focus mode. which focuses the beam at the target and disperses the ejectiles throughout the focal plane. b) shows dispersion-matched mode with lateral dispersion-matching only. The momentum- dispersed beam hits the target. and creates a. large beam spot in the dispersive di- rection, but the ejectiles have different. angles coming into the focal plane. c) shows a dispersion—matched mode with both lateral and angular dispersion-matching. The ambiguity in the ejectile angle at the focal plane is cmisiderably reduced. The figure is taken from reference [109]. at a single point in the focal plane. In a lateral dispersion—matched tune. the beam coming into the target area is focused along the spectrometer’s non-dispersive axis and momentum-dispersed along the spectrometers dispersive axis. This produces a long, thin beam spot. The lateral dispersion-matching technique can result in angular ambiguities [109] in the dispersive direction miless angular dispersion matching is also applied. In this process, the beam line is tuned so that tracks incident at different angles to the target have the same angle in the focal plane [110]. Both types of dispersion-matching were used in the 150l\ld(3He.t) experiment. Figure 4.2 shows the beam optics for focus matching, lateral dispersion—matching. and simultaneous lateral and angular dispersion-matching. Figure 4.3 shows an image of the beam spot from the 150Nd(3He.t) experiment. The faint beam method is used to tune the dispersion—matching. In this method, an attenuated beam is sent directly into the spectrometer without hitting a target 45 Figure 4.3: Dispersion-matched beam image at the target of the Grand Raiden [112] spectrometer. The viewer dimensions are circled in red. Targets are around 2 cm by 2 cm or slightly smaller, and the beam spot is a bit less than 1 cm long. foil. The dispersion of the beam can then be fine-tuned by optimizing the image in the focal plane, and the absence of a target foil ensures that the measured resolution is intrinsic to the beam rather than energy straggling in a target. Four target foils were used to aid in the beam tuning in the 150Nd experiment: a ZnS viewer, 27Al to check energy resolution, and 197Au and 13C to optimize the angular calibrations. Once tuning was complete, strong Gamow-Teller transitions from a natMg target were used to calibrate the triton momentum. Nd foils oxidize quickly, so a special container was used to keep the foils in vacuum during the transfer from a glove box to the target chamber. Both foils were made from 150Nd enriched to 96% purity (see Figure 4.4). They were thin and self-supporting, with thicknesses of 1 mg/cm2 and 2 rug/(#1112. Excellent beam tuning and thin foils allowed us to achieve an energy resolution of 32 keV FWHM. The difference in energy loss between the 3He++ particles and tritons in the target is 8 keV (as calculated in LISE++ [115]), and does not limit the resolution. 46 Figure 4.4: The 150Nd targets used in the experiment, 1 mg/cm2 and 2 mg/cm2. To optimize the angular resolution. an optical technique called “over-focus mode" was used [108]. The ion optics of the Grand Raiden are such that the non-dispersive angular magnification between target and focal plane is 0.17 [114] in focused mode. A large non—dispersive angle at. the target (Om) corresponds to a small non-dispersive angle (Ofp) in the focal plane in focus mode. which. for a given angular resolution in the focal plane, makes it difficult. to reconstruct em with good precision. To improve this situation, the field of the Ql quadrupole magnet is changed to place the non-dispersive optical focus outside of the focal plane. producing a large y fp (non- dispersive position at. the focal plane) that is directly proportional to Om and has a small relative error. The proportionality changes as a fimction of Xfp' When centered around 00. the Grand Raiden can accept particles in the ranges of 02". To increase the angular range available for this experiment. the spectrometer was rotated to take data at Oh0.,.=2.5o and OhOT=4O. Figure 4.7 shows the Grand Raiden’s angular acceptances for the three data sets. Each rectangle represents an angular range of :l: 20 mrad horizontally and d: 40 mrad vertically. for a total solid angle of 3.2 msr. After the 150.\'d(3He.t) data was taken. one day was allocated for 47 foffflf D2 , “I1 / “a! we “111111 MP NM“ fl“! iii “' (“RN #1141111] I “’ “DSR / " Faraday Cup 'i.‘ g; /F:al Plane 01._1__1_11 2 3 m I)! Q1 ’43 Target Figure 4.5: Schematic of the Grand Raiden Spectrometer. Image from Ref. [113]. Parameter Value momentum resolution (Ap/ p) 2.7 x10"5 energy resolution (AE/E) 4.5 X10-5 position resolution maximum Bp maximum B (D1 and D2) maximum magnetic gradient (Q1) maximum magnetic gradient (Q2) momentum range focal plane tilt mean orbit radius total deflection angle angular range horizontal magnification (x—-x) vertical magnification (y—y) maximum momentum dispersion horizontal acceptance angle vertical acceptance angle solid angle weight 300 11.111 (both horizontal and vertical) 5.4 Tm 1.8 T 0.13 T/cm 0.033 T/cm 5% 450 3m 1620 -5 to 900 -0.417 5.98 15.45 m 21:20 mr 21:40 mr (in over-focus mode) 5.6 msr (3.2 in over-focus mode) 600 tons Table 4.1: Parameters of the Grand Raiden Spectrometer 48 A1 200 annels 1 000 800 600 e2 scintillator (ch 400 200 0 0 200 400 600 800 1000 1200 e1 scintillator (channels) Figure 4.6: Particle identification from the plastic scintillator signals at the Grand Raiden focal plane. Tritons are circled in red, and the charge state is circled in yellow. Particles clustered near channel 0 in both axes are due to cosmic rays and other noise in the scintillators. These events are. not associated with reconstructed tracks in the MWDCs. 49 2.0 9hor I -2.0 \03. 0.7 1.7 3.2 3.3 4.7 Figure 4.7: The angular acceptance of the Grand Raiden Spectrometer for the three angular settings used. The scattering angle. as measured from the beam axis, is shown with circles for 0.5-degree angular bins. Three rectangles represent the angular coverage accessible when the spectrometer is rotated around its pivot point. Red: angular acceptance for 0-degree measurements. Blue: angular acceptance for 2.5 degree measurements. Green: angular acceptance for 4—degree measurements. In areas where two angle settings overlap, the measurement with best statistics and most complete coverage of the angular range is used, as indicated by the numerical label. 50 A 0 (JO 1 u—A O N [III Trill! do/dQ (mb/sr) _\ O Tilt] -1 10 :— ’9 1A1114141111L -2” O.3.5...5.-.7...5...10.i.2:5 15 17.5 202325 Gem (deg) 10 Figure 4.8: Elastic scattering on 15ONd, including an ECIS fit from which the optical potential was extracted. The fit shown was used in all FOLD calculations. See section 4.3.2 for further details. a (3He,3He) elastic scattering measurement. Twenty-seven data points were taken at angles between 8 and 21 degrees in the center of mass. The angular distribution of the cross section was fit with the code ECIS [96] as shown in Figure 4.8, resulting in an optical potential that will be discussed in section 4.3.2. 4.2 Calibrations 4.2.1 Sieve Slit Calibrations Field maps that provide detailed information on distribution and strength of the magnetic field emanating from a magnet do not exist with sufficient precision for the elements of the Grand Raiden Spectrometer to use them for the purpose of particle track reconstruction. However. the ability to ion—optically ray-trace particles from the focal plane back to the target is important for any experiment. The solution is to perform a sieve—slit measurement. A block with a distinctive hole pattern is inserted into the beam line ~ 60 cm behind the target and data is taken. Polynomials to are IllE‘Il fOllIlCl tO I'BCOIISII'UCt 8})()7‘i~()llf(ll dependent on Xfpr y fp’ and Of p her" 1? 3071 tal and 9m . based on the hole pattern in the sieve slit. These polynomials go up to vertzcal - the 6th order. The sieve slit is then removed from the beam line and the polynomials are used to reconstruct target angles for the rest of the data. See Figures 4.9 and 4.10 for more details on this procedure. In this experiment, the triton energies were calibrated from known states in 24’25’26Mg. Strong states in the natMg data were matched to known values of the excitation energy to extract a relationship between the triton energy and the (lis- persive position and angle in the focal plane, and this relationship was then applied. to the 150Nd data. A second correction accounted for the difference in recoil energy as a function of scattering angle between the nang and 150Nd. Differences in energy losses for different targets were calculated using LISE++ [1‘15] and accounted for as 52 ®fp Figure 4.9: Images of the sieve-slit as measured in the focal plane of the Grand Raiden spectrometer. The axes are y versus 0 p for four situations: (top left) no cuts on x, (t0p right) x between -100mm and 100mm, (bottom left) x between 100mm and 300mm, and (bottom right) x between 300mm and 500mm. 53 3 g C o .5 ®§ O -2 -2 -1 O 1 2 ®vertical (deg) Figure 4.10: Reconstructed sieve slit spectrum after the determination of raytracing parameters (dispersive angle vs. non—dispersive angle at the target) well. Run-dependent shifts (due to small changes in beam parameters) in the focal plane angles and the beam energy over the course of the experiment were monitored and corrected for so that all of the data could be viewed at once with optimal angu- lar and energy resolution. The energy resolution was 33 keV FW’HM. The angular resolution of the laboratory scattering angle was 0.420, which is based on the angular widths of the 3He+ charge state as observed in the focal plane. 4.2.2 Beam rate Calibration and Cross Section Calculation Beam line polarimeters (BLPs) at two stations were used to cross-check the Faraday cups used to integrate the beam charge at different angular settings. In each BLP. two plastic scintillator detectors at 480 and 170 were placed around a retractable 14pm CH2 foil in the beam line. The incident beam undergoes elastic scattering on the protons in the CH2, and this yield can then be used to cross-calibrate Faraday cups 54 for the (3He,t) measurements. Three different Faraday cups were used to measure the beam rate: the 00 cup was inside the D1 dipole, the 20 cup is by the Q1 quadrupole, and 40 cup is inside the scattering chamber (see Figure 4.5). No rescaling had to be applied. Cross sections were calculated with the equation do Y _ = . . . (4.1) (19 NbA/telczdfl where Y is the total number of counts in an angular bin, Nb is the number of nuclei in the beam, N t is the number of nuclei in the target, (IQ is the opening angle, (‘1 corrects for the lifetime of the data acquisition system (DAQ) (which was around 96%), and 62 corrects for the target purity (also 96%). 4.3 Analysis of Data Data for the three angular settings of the Grand Raiden were merged. The laboratory angular range of 0-50 was sliced into ten half-degree bins, as shown in Figure 4.7. Figure 4.11 shows the excitation energy spectrum of 15OPm for every other angular bin. Shape changes occur as a function of angle due to the angular distrilmtions of the isobaric analogue state (IAS, at 14.35 MeV), Gamow-Teller resonance (GTR. centered at 15.25 MeV), and the spin—dipole resonance (SDR or IV SGDR. centered at 22.8 MeV). The non-spin-flip giant resonances (IVGMR and IVGDR) are very small and do not contribute significantly to the spectrum. The peak due to the IAS exceeds the y-axis scale in Figure 4.11 and peaks at 0.43 mb/sr/5keV (at 00). A strong, forward—peaked discrete state is visible at 0.11 .\IeV, and a weaker pair of dipole states is present near 1.6 MeV (both are easier to see in Figure 4.12). 9 CI! lsr O '_. N T (mb 0 0.08 :— G/deE «U005 } 0.04 i— . (alillbl‘bflllb- ] 0.02 3 .‘ _- ‘ \ l L 1 4L L L L L L L L A l #4 #4 1 l 0' 5 10 15 2015025 30 Ex( Pm) (MeV) Figure 4.11: Cross sections for the 15().\'4 are not included in the analysis. because the angular distributions peak beyond 50 in the center of mass and data was taken in the range of 0-50. The population of transitions with high AL values is reduced near q=0, and contributions from these transitions (with AL>4) are effectively absorbed into AL=3 and 4. 4.3.2 The Optical Potential and the IAS As mentioned at the end of Section 4.1.1. optical potential parameters were fit to elastic scattering data using the code ECIS [96]. The first optical potential tried (potential 1 of Table 4.2) was an extrapolation of parameters obtained from elastic scattering on other targets (such as 90Zr and 208Pb). This potential produced a cross section that had a magnitude 60% higher than the IAS seen in the data. This is consistent with what has been seen for other targets. but in this case the calculated angular distribution for the IAS did not. match the data. A second set of optical model parameters (potential 2 in Table 4.2) was deduced from a fit to the elastic 58 1 .4 f —— AL=O 1.2 l AL=2 arbitrary units l> 1- _1_1‘ 1 ----- AL=4 0.8 O ° 0.6 0.4 0.2 0 00.54 1.5 2 .5 3 3.5 4 4.55 Gem (deg.) Figure 4.13: Angular distributions from 150Nd(3He,t) as calculated with FOLD. Relative scaling of the distributions is arbitrary and chosen solely to better display the function shape. 59 .3 N ’5 . B . g 1 C1 10 — E 8 8 — 0 data 6 L— — potential2 4 1.. 1 2 _ O .1..i....i....1....l. ....1....1....1....1.... 00.511. 22.533.54 .55 9cm(deg.) Figure 4.14: The-angular distribution of the optical potential. compared to the IAS. Table 4.2 contains the parameters for potential 2. 60 Potential V r a W r-w ra 1 31.79 1.34 0.83 36.4 0.94 1.28 2 58.57 1.134 1.032 66.7 1.0925 0.94 Table 4.2: Optical potentials for the elastic scattering data and the angular distri- bution of the IAS. Potential 1 was the standard potential at the beginning of the iterative process and fit neither the elastic scattering data or the IAS. Potential 2 was chosen for use in DWHI calculations because it. was a reasonable fit to both. scattering data in Figure 4.8. \Vhen this potential was scaled and applied to the IAS. the location of maxima and minima were well reproduced (see Figure 4.14). This set of optical model parameters was used in the rest of the analysis, since the main purpose of the calculated angular distribution is to provide input for a multipole decomposition analysis. While the W (imaginary volume term) is unusually large in potential 2, lower values of W could not reproduce the elastic scattering curve. This may be because of 150Nd’s deformation, or it may be a sign that the functional form of the optical potential should include other components, such as surface or spin-orbit terms. Given the limited angular coverage of the elastic scattering measurement and the absence of polarization observables, it was not possible to perform a fit to a more complicated optical potential. 4.3.3 Multipole Decomposition Analysis The 150Nd(BHeI) data was analyzed with two separate methods. The first method is a multipole decomposition analysis (MDA)[116]. Since 150Pm‘s level density is high (it. is a heavy odd-odd nucleus), a peak-by~peak analysis is impossible. even at low energies. Instead, the spectrum is split up into 1 MeV bins. and the angular distribution of the measured cross section is fit with a linear combination of the calculated cross sections from DWBA. Angular (,listributions for each discernable peak in the region of 0-2 MeV were fit in a similar fashion. In the following equation 61 for a multipole decomposition, 0’- represents the calculated angular distribution from Figure 4.13 for AL=i and the capital letters represent the fit parameters. at0t=A>k01+B*02+C*03+D*04+E*05 (4.2) In the 150Nd analysis, five functions were used in the multipole decomposition. rep- resenting AL=0,1,2,3,and 4. Contributions from certain AL values were crmsistent with 0 for some angular bins. As shown in Figure 4.15 (top), transitions to the ground state peak around 1.50, which implies it is dominated by dipole (AL=1) contributions. A significant AL=3 contribution is also observed. Unless two states exist at energies too close to separate, this fit indicates that the ground state may have a J7r of 2_. Barrette et a]. [117] studied the decay of 150Pm to 1508m and give a tentative J7r of 1- or 1+ to this level, but this is based entirely on arguments that no states of high spin in 1508111 were observed to be directly fed by the 150 Pm ground state decay. H(_)wever. a large number of levels from that experiment were not placed in a level scheme. and their evidence for a J of 1 is incomplete. The MDA from the current experiment shows that J7T contributions of 27' or 1_+3_ (in the case of two inseparable states) are possible. but 1+ is ruled out. Since we don’t. have OBTDs for individual transitions, we cannot fully exclude the possibility that this is a 1— state with an angular distribution that has been modified due to the influcuice of the TT interaction. A very strong state at 0.11 MeV (see the bottom of Figure 4.15) peaks at 0 degrees and decreases in strength at higher angles, which is a sign of a sizable AL=0 (GT) component. There are AL=1 and 2 contributions to this angular distributirm. but it is dominated by AL=O contributions. The extracted AL=1) cross section is 0.565 350.085 mb/sr. The AL=2 contribution may be from the GT transition. and both the. AL=1 and 2 components may be indicative of a second state at the same excitation 62 0.08 . g E 0 data E 0.07:— —— AL=1 g ; —— AL=2 B 0.06. U . 0.05 0.04: 0.03: 0.025 0.01 00”0.5”‘1”‘1.5”2”‘2‘.‘5m3‘ 3.5 44.5 5 6) (deg) a 0.7, °"‘ 3 : C data g 0.6_ AL=0 g AL=1 g 0.5: AL=2 . sum 0.4 _ 0.3 0.2 : 0.1: 0 0511.5 2 2.5 3 3.5 4 4.5 5 @cm(deg) 0 Figure 4.15: Multipole decmnposition of the first two single peaks in 1SUPm: the ground state and the first excited state. The differential cross section for each peak is fitted to a linear combination of angular distributions associated with different units of angular momentum transfer. 63 C data do/dQ mb/sr Figure 4.16: l\1ultipole decomposition of the 5-6 MeV excitation energy bin. The AL=0,2 components are strongest. energy or the presence of small states nearl'1y. Other states below 2 MeV that could be identified were analyzed in the same fashion. They are all much weaker than the 0.11 MeV state. Some states, such as the three peaks near 1.3 MeV. can be modeled well as AL=0 states. and others, such as the two peaks near 1.63 MeV. as AL=1 states. After dividing the excitation energy spectrum into 1 MeV energy bins, angular distributions were created and MDA fits were performed for the entire energy range of the experiment (0—30 MeV). Figure 4.16 gives an example for the 5-6 .\IeV energy bin. Results from the MDA for each energy bin can be combined to show the excitation energy distribution of different types of transition strengths. Figure 4.17 shows the results for all angles and excitation energies. The Gammv-Teller resonzmce peaks around 15 MeV, along with some strength at 5 and 10 MeV, but the long tail at higher excitation energies may come from the spin-flip giant monopole resonance ( IVSGMR) 64 and/ or high-lying GT strength associated with coupling to 2p-2h configurations [77]. As noted in Chapter 3, the resonance centroid of the IVSGMR is expected to be around 37 MeV. In a similar ex1_)eriment on 120Sn, low-lying AL:0 strength was attributed to “core 1,)olarization spin—flip” (j = l i % ——> j = l i :12) and "back spin flip” (j = l — % —> j = 1+ %) [118, 119] modes. The strength at 5 and 10 MeV is likely due to the same types of contributions. These peaks are referred to as pygmy resonances [120, 58]. The spin-dipole resonance dominates between 1.5 and 20. No resonance structures are seen (or expected) for AL=2,3, and 4. Giant resonances associated with these AL values are expected to peak at higher excitation energies. Information from the AL=3,4 distributions may represent higher multipoles as well, as noted in Section 4.3.1. 4.3.4 Resonance fits A second method of analyzing the 150Nd data was that of resonance fits. The goal of this method is to completely reproduce the spectrum with a combination of (res- onance) base functions. Many studies of nuclear giant resonances have been done in this manner [77]. and the location of some resonances (such as the IAS. GTR.. and the IVSGDR) is quite well—known. The IAS and GTR both peak around 15 MeV. Vt’hile the IVSGDR has three components (0_, 1‘, and 2—), they are not distin- guishable from one another in the data because they overlap and have the same AL value and angular distribution. Instead, the summed strength peaks around 22 MeV. The IVGDR occurs around the same excitation energy, but it is much smaller in mag- nitude than the IVSGDR (around a factor of 1 / 30). To model the entire excitation energy spectrum, one should include the GTR., IAS, IVSGDR, the fragmented GT strength at 5 and 10 MeV, and a function for quasi-free (QF) processes. Quasi—free charge exchange can occur when the 3He projectile interacts with a single neutron in the target. in such a way that the rest of the nucleus can be considered 65 W O 1 .. 005° 0 data — AL=0 - as 25 r __ _3 — 4 d‘fx/deE (mb/sr 1 MeV) 5‘. 8 U1 d iTfij—VTerYTYWrW—TYfYfifrVYY YTT' 0 PA L41 1;; LALL ‘ '1 l A a L J LA Aimmaa 6201 25 30 Ex(150Pm) (MeV) O U'I O _.| U'l O 0'54 0 data AL: ‘ it: 0 — AL: - AL: sum W O 1 I N Ln bylaw-1O i r f dzo/deE (mb/sr 1 MeV) '5’ 15 10 5 0LLAIIII1.1L1111.1L114411L4141 0 5 10 15 20 25 30 Ex(150Pm) (MeV) Figure 4.17: Multipole decomposition summary for each 0.50 angular bin. The IAS was removed from the fit, .causing the visible discontinuities around 14 MeV in ex- citation energy, and the 160 impurity cases a second discontinuity around 16 MeV. The GT resonance dominates the spectrum between 005°. but rapidly diminishes and is replaced by the IVSGDR at 1.5—2O. Higher multipoles (or quasifree processes) take over at higher angular bins. 66 P; E H50 0 data 2 30 f " AL=0 ~ . - 251 525 F — AL;3 D - AL=4 £20 : sum LLI '0 Ci '3 b N .0 % ; 15"” C data 2 30 (r - AL=O «- . - a; e 25 f — AL;3 g 20 5 sum u.1 f '8: 1s 3 '2 E ND 10 :— 'o . 5 O A n x .1; A A .1. L 1 A 1‘4. .1. . Agl +L1 A 0 5 10 15 20 25 30 Ex(‘5°Pm) (MeV) Figure 4.17: Multipole decomposition summary for each 0.50 angular bin. The IAS was removed from the fit, causing the visible discontinuities around 14 MeV in ex- citation energy, and the 160 impurity cases a second discontinuity around 16 MeV. The GT resonance dominates the spectrum between 005°, but rapidly diminishes and is replaced by the IVSGDR at 1.5—2°. Higher nmltipoles (or quasifree processes) take over at higher angular bins. 67 % , 2'2'50 C data E 30[ _ AL=0 ~ » - as 52“ - 21123 £20; sum .. 1 a 15 f 2 E Nb10; '5 : 5: 0 i. 0 E t 2 30 E a 25 E \ -° 1 E20: Lu 1 .0 c} 15 ~ '3 ND 10 .0 5 O LirnlmmxmLmr..JhmnxLi...l..1. 0 5 10 15 20 25 30 Ex (15°Pm) (MeV) Figure 4.17: Multipole decomposition summary for each 0.50 angular bin. The IAS was removed from the fit,‘causing the visible discontinuities around 14 MeV in ex- citation energy, and the O impurity cases a second discontinuity around 16 MeV. The GT resonance dominates the spectrum between 005°, but rapidly diminishes and is replaced by the IVSGDR at. 1.5-2O. Higher multipoles (or quasifree processes) take over at higher angular bins. 68 - 0 33.5 C data Yfii‘r 30 r — AL=0 : - as 25 g” — AL;3 C — AL=4 20 ? sum 15 E 10 dzo/deE (mb/sr 1 MeV) o 5 10 15 20 25 30 Ex(‘5°Pm) (MeV) A O a : 3'5'4 C data 2 30 f — AL: '— 1 - as e 25 E — AL;3 g 20 f ' sum 1.1.1 Z t a 15 f 32 E No 10 f 13 1 ‘.'..".'|IIIIIIDJ 5 i 0 i * . 1‘... ..'.T‘Tf7 0 5 10 15 20 25 30 Excsopm) (MeV) Figure 4.17: Multipole decomposition summary for each 0.5 “'0 angular bin. The IAS was removed from the fit causing the visible discontinuities around 14 \IeV in ex- citation energy and the 160 impurity cases a second discontinuity around 16 \IeV. The GT resonance dominates the spectrum between 0- 0..) '0, but rapidly diminishes and IS replaced by the IVSGDR at 15 -.20 Higher multipoks (or quasifree processes) take over at higher angular bins. 69 - O AL=O AL=1 AL=2 AL=3 AL=4 U) 0 111 N u: T lllll. sum d 7 Y T ff Y—rj Y I V V dzo/deE (mb/5r 1 MeV) ES 23 S _ 0 414111141L41q111111111111L1 o 5 10 15 20 25 30 Ex (‘50Pm1 (MeV) % I 4.5-50 . data 2 30 : - AL=O ~ 1 - as c 25 F — AL=3 fig 1 - 15L: V 20 f sum 15 cg 15 P 33 F ”.8 10 p 5 3 g . O h 1 1 1 1 i 1 1 1 1 l 1 1 Lmj 1 1 1 l 1 1 1 1 1 1 1 1 O 5 1O 15 A20 25 3O 51(1 5°Pm) (MeV) Figure 4.17: cont. Multipole decomposition summary for each half-degree angular bin. The IAS was removed from the fit, causing the visible discontinuities around 14 MeV in excitation energy, and the 160 impurity cases a second discontinuity around 16 MeV. The GT resonance dominates the spectrum between 0-0.5°, but rapidly diminishes and is replaced by the IVSGDR at 1.5-2°. Higher multipoles (or quasifree processes) take over at higher angular bins. 70 a spectator and the neutron a free particle except for its binding energy. The neutron is transformed into a proton and “knocked out,” which requires the process to take place only above the proton separation energy. As the energy transferred to the residual gets higher. neutrons in deeper shells (with greater binding energy) can be removed. Superimposing the energy of the knocked-out protons results in the characteristic. QF shape (see Figure 4.18). In this work, the QF cmitribution was modeled with Erell‘s [76] semi-phenomenoh)gical function (12., 1 _ .1/T1 -— = N , (4.3) dEdQ 1 + [(13th — .1: — qu)/11]2 initially developed for pion charge-exchange. This function has since been applied to (3He.t) [118. 119] and other types of CE experiments. Three parameters are fit: N is an overall normalization factor different for each angular bin. W represents the Fermi motion of the nucleon within the nucleus, and T is a temperature parameter. E0 is the energy at which the QF curve crosses the x (excitation energy) axis. The exponential represents the effects of Pauli blocking. The remaining para111eters are described or derived from values in Table 4.3. The GTR and IVSGDR are represented by Gaussians, because using Lorentzians creates non-physical long tails. The IAS was modeled with a Lorentaian. In addition to the quasi-free curve. two small Gaussian functions (G5 and G10) were added to the fit near 5 MeV and 10 MeV to represent the pygmy GT resonances and other low-lying strength. With these six functions, an 18-parameter fit was perfm'med using MINUIT [121]. Figure, 4.18 shows the result for all ten angular bins. The fit resonances were integrated for each angular bin to produce an angular distribution. These were. then decomposed into AL components with the functions used in the MDA (see Figure 4.13). Comparisons between the two 111cthods could then be made (see Figure 4.19). As expected, the function labeled IAS was entirely A 0'16» "T O E, . _ GTR » 00.5 12 0.14? IAS . g 0.12;— - IVSGDR E 0.1’ ”6 c: 0.08“ E 006» q: ' 0.04 0.02» O 150 Ex( Pm) (MeV) 0.16_ j—’ 0 E : _ GTR _ 0.5-1 x L , m 0'14. | AS , $0.12} — IVSGDR i E, 0.1: ' QF 53 ; GS 008— g \ 0.06 - Nb '0 0.04 0.02 E x(‘50Pm) (MeV) Figure 4.18: Resonance fit to the excitation energy spectrum. The sum of the six fit functions reproduces the original shape of the data. 72 A 0.16? 1 . -6--- _) E LE — GTR 1-1.5 ( m 0.14? IAS , g 0.12: - IVSGDR ( E 0.1} ‘ QF ( kg : — 65 0.08~ g : \ 006— Nb '0 0.04- 0.02» 00 A 0.16 > : g 014"— m ' ; IAS E 0.12} — IVSGDR '0 l - OF E, 0.15 55 : - GS C} 0.08: 610 1 g 0.06 ;—» sum N " 3 U 004» .1 0.02- J o 5 10 15 20 25 30 Ex(150Pm) (MeV) Figure 4.18: cont. Resonance fit to the excitation (-11’1ergy spectrum. The sum of the six fit functions reproduces the original shape. of the data. A . O E : _ GTR 2-2.5 x 0.14»— m IAS % 0.12: — IVSGDR u.1 l _ G5 '0 _ C: 0.an _ G10 '2 0.06_L sum Nb ; '0 0.04— 0.02 O “#1.! 0 5 A 0.16_ 3 1 — GTR x l— m 0'141 IAS % 0.12} — IVSGDR E, 0.1 E 7 OF 5161 ; — GS C} 0.08:— _ (310 g 0.06} sum N : '5 0.045 0.02 - .1 0 5 10 15 20 25 30 150 Ex( Pm) (MeV) Figure 4.18: cont. Resonance fit to the excitation energy spectrum. The sum of the six fit functions reproduces the original shape of the data. 0.16 A O E, _ GTR 3-3.5 m 0.14 IAS & 0.12 - IVSGDR .Q g 0.1 _ QF LU _ U 008 65 C: ' 610 13 B 0.06 sum , N .0 C _ GTR 3.5-4o 0.14 5 I AS 0.12 f— — IVSGDR dzo/deE (mb/5r 5 keV) Ex(‘5°Pm) (MeV) Figure 4.18: cont. Resonance fit to the excitation energy spectrum. The sum of the six fit. functions reproduces the original shape of the data. Kl C1 -nu-1q A 0.16 o E, l _ GTR 44.5 1.2 0.14:5 IAS 3 0.12} — IVSGDR g 0.1 :— " QF 95' 1 — GS C: 008-“ — G10 '3 1 b 0.06: sum NU L A 0'16» 0 m 0.14g IAS 3 0.12? _. IVSGDR g 01— ' QF -”6' _ — GS C} 0.08: _ G10 % 0.06: sum N l- '° 0.04 E X(‘ SOPm) (MeV) Figure 4.18: cont. Resonance fit to the excitation energy spectrum. The sum of the six fit functions reproduces the original shape of the data. 76 Parameter Definition/ Method of Calculation Value name N normalization, fit fit. Q Q value for 150Nd(3He.t) 0.105 MeV nQ Q value for 11(3He.t)p at. 420 .\Ie.V 0.764 MeV Ep’f‘Oj energy of the incoming 3He 420 MeV Et free energy of the free triton, Epro j + nQ 420.764 MeV Etgs ground state energy of the triton, E117“) } - Q 419.895 MeV Sp ' proton separation energy for 150Nd 9.922 MeV Beoul Coulomb barrier for the proton 12 MeV Earn, excitation energy of the neutron hole state. 2 MeV from [105] qu quasi-free energy. Et.f,,.€(.—(Sp+En1+B(.0u)) 396.864 MeV W width of resonance, fit. fit (around 22) T slope, fit fit (around 120) Table 4.3: Parameters used in calculating the quasi-free curve. The values of T and W' are comparable to those found in other works [105. 76]. AL=0, while the. GTR was overwhelmingly so and the IVSGDR primarily AL=1. This confirms that the resonance shapes and their locations are a good match for the real resonances. The QF, G5. and G10 distributions also contain significant AL=0 and AL=1 strength. 4.3.5 Extrapolation to q=0 These two analysis methods. resonance fitting and multipole decomposition. result in absolute cross sections for each type of multipole excitation for half-degree angular bins and 1 MeV excitation energy bins (with peak-by-peak resolution below 2 MeV). When contributions from the IAS are removed and possible contributions from the IVSGMR at high excitation energies are ignored. the L=0 cross section and equation 3.24 can be used to extract the Gamow-Teller strength for a given transit ion. However. the cross section must first be extrapolated to zero momentum transfer ((1:0). Using 77 do/dQ(mb/sr) do/dQ(mb/sr) 60 4° 1 20 _ l 0 [0.5.1 1.552 2.5 A; 3.599475 5 @Cm(deg) @cm(deg) A j“ * - -* *- *- 1: r ' 5* " __ —- "gram-1 5 120 1 , 0 IVSGDR {140 g Q" — AL=0 .Q ‘ -Q [ Q " AL=1. E E 120 y “ AL=2: v V ' . .\ - AL=3[ C: C ‘ ' ' a — AL=4 '5 'U ‘00 sum \ O B b 1 '0 ‘O @Cm(deg) ’5 25 i _"'65_‘ E 25 610 0 data 3225 :' a data [ 322-5 . - 11:0 g 20 l — ALzo g 20 3’ \_ - AL=1 C175 1 — AL=1 (317.5 f— - AL=2 ‘0 '0 1 ' — AL-3 \ ‘5 t. _ AL=2 \ 15 . ~. - .8 o — AL=3 .8 - AL=4 12.5 r _ AL=4 12.5 10 E 7-5 7.5 .g 5 I 5 gm . \ 2.5 3 2.5 of 1.....-1. _ ’ 0 ' _ . _ ‘. ‘\ W— 00.511.522.533.544.55 00.511.522.533.544.55 @Cm(deg) ecm(deg) Figure 4.19: Angular distrilgmtions of the giant resonances. Components from con- tributing AL values are included. The GTR can be reconstructed with only AL=0 and 2. the IAS with only AL=0. and the IVSGDR with only AL=1 and 3. because the shapes are accurate representations of the actual resonances. The pygmy reso- nances and the QF are nmch more complex (though both have significant AL=0,2 contributions). 78 equation 4.4. do ((7,?le 2 00~ Q = 0) (In 0 d6 = I (10 o 1 X Eff) 2 0 Q) (4.4) ‘ q—10 mfg = 0 1(2) Dll'BA (arpcrinwnt the ratio of the DWBA cross-section to that at 00 is calculated and multiplied by the experimental cross section. The ratio is shown in Figure 4.20 and is well-descrilml by a. polynomial: ”Mn-,0 = 1.00057 —— 0.0236(2 +1).001s5(22 — 4.607 x 10—5Q3 +1.20? >< 10—5Q4. (4.5) where Q is the Q value. Application of this ratio is straightforward for the individual states and the multipole decomposition analysis. since the states are well defined in their excitation energy. Giant resonance fits are spread over a much larger energy range. A convolution of the correction ratio and resonance shape was used to extract {71% for each resonance. (1—10 4.3.6 Calculation of the Gamow-Teller strength After extrapolating the experimental cross sections to q=0. the unit cross section (67) and equation 3.24 are. used to calculate the B(GT). Table 4.4 shows results from the resonance fit method. The total extracted B(GT) is 56.62 :1: 6.16. including both statistical and systematic errors. The [V SGDR and the IAS are excluded from this calculation. because they cannot contain any GT strength. but the QF. G5. and G10 resonz-mce shapes are included because of the large amount of L20 strength present and because they are composites of more than one multipole. The MDA yields a total GT strength of 50.01 3: 1.69 (combined statistical and systematic errors). Associated statistical errors from the. fitting procedure are quite small, but systematic errors (including contributions from the optical potential and 79 6 2- 1 €31.75 — J 92, ' ,0 13:515 7 . B o. U125 ”‘ .. A 1 ..o‘ o 1W II q L "00.75 ~ é (30.5 — 2 80.25 — O .11111..1111111....1....1.... 0 5 10 15 20 25 30 Q(MeV) Figure 4.20: Ratio of the cross section. at 6:00 and 0 linear momentum transfer to that of 0 linear momentum transfer, as calculated in DWBA. See equation 4.5. Resonance B(GT) B(GT) stat. error B(GT) syst. error total error G5 2.506 0.442 0.380 0.585 G10 3.812 0.656 0.574 0.872 GTR 23.072 1.170 3.484 3.676 QF 22.706 1.326 4.055 4.349 total 56.62 2.11 5.390 5.790 Table 4.4: Gamow-Teller strengths from resonance fits. Quasi-free contributions must be included because of the large amount of high-lying L=0 strength. (Inclusion of the QF section also makes comparisons with the MDA method feasible.) The IVSGDR and the IAS, by definition, do not contain GT strength. Systematic errors are taken to be 15% of the GT strength, which is consistent with methods used in the MDA. 80 5 25 ‘ ° B(GT) 5 T l. 4 T i 3 _ i l 2 L l [ : if I911111“ 1 : ii 0 1.111114111.11111111111111111 0 5 1O 15 20 25 30 15x (‘50Pm1 (MeV) Figure 4.21: GT strength from the MDA the nucleon-nucleon interaction) are 15% as applied to each 1-.\IeV fit of the an- gular distributions (systematic errors from different energy bins are assumed to be independent of each other). Table 4.5 and Figure 4.21 give detailed values. Directly comparing the low-lying states. the peak-by—peak analysis sums to a total strength of 0.370 i 0.023. while the 0-2 MeV section of the MDA sums to 0.46 :1: 0.050. The two methods are barely consistent. which makes sense because the peak- by—peak analysis is ignoring everything in between what was chosen as a peak. while the full MDA takes those regions into account. Details are shown in Table 4.6 and Figure 4.22. 4.3.7 Other Multipole Excitations While GT and Fermi strengths are. directly 1..)1'oportional to the CE cross section at q=0, no such relationship is proven to exist for higher nudtipolcs. MDA dipole and 81 Ex.(MeV) B(GT) stat. error syst. error total error 0.5 1.5 2.5 3.5 99051939“? moammmmmm 6;: in .CI! .01 in h—‘D—‘h—‘h—‘P—‘h—‘P—‘r—‘P—J OOHCDCJ‘JAOJEOF-‘C in 511111 0.2698 0.1908 0.2728 (1.3829 0.4866 0.5067 0.5486 0.6764 0.9031 1.1693 1.4595 1.7731 2.2785 3.2420 4.5515 4.4297 4.1008 3.2501 2.5150 2.0486 1.7192 1.5975 1.4426 1.3301 1.3273 1.3620 1.3912 1.4868 _ 1.5896 1.7075 50.01 0.0037 0.0037 0.0039 0.0043 0.0048 0.0049 0.0051 0.0054 0.0059 0.0065 0.0072 0.0048 0.0088 0.0104 0.1433 0.0128 0.0129 0.0126 0.0122 0.0125 0.0128 0.0135 0.0142 0.0147 0.0154 0.0161 0.0169 0.0177 0.0177 0.0201 0.1563 (10405 (10286 ()()4(M) (10574 (10730 (10760 (10823 (11015 (11355 (11754 (12189 (12660 (13418 (14863 (16827 (16645 (16151 (14875 (13773 (13073 (12579 (12396 (12164 (11995 (11991 (12043 (12087 (12230 (12384 (12561 1.68 (10406 (10289 (10411 (1057' (10731 (10762 (10824 (11016 (11356 (11755 (12190 (12660 (13419 (14864 (16976 (16646 (16153 (14877 (13774 (13075 (12582 (12400 (12169 (12001 (11997 (12049 (12094 (12237 (12392 (12569 1.69 Table 4.5: Gamow-Teller strength distribution from the MDA. 0-30 .\IeV Ex. (MeV) B(GT) stat. error syst.. error total error 0.11 0.1334 0.0023 (10202 (10203 0.19 0.0226 0.0012 0.0034 0.0036 0.282 0.0128 0.0008 0.0019 (1.0021 0.40 (10155 0.0009 0.0023 0.0025 0.497 0.0133 0.0014 0.0020 0.0025 0.592 0.0086 0.0015 0.0013 0.0020 0.667 0.0085 0.0013 0.0013 0.0018 0.725 0.0069 0.0013 0.0010 0.0017 0.86 0.0066 0.0014 0.0010 0.0017 (1904 0.0108 (10015 (10016 0.0022 1.0 0.0064 0.0013 0.0010 (10016 1.14 0.0066 0.0011 0.0010 (1.0015 1.225 0.0064 0.0020 0.0010 0.0022 1.267 0.0152 0.0003 0.0023 0.0023 1.319 0.0131 0.0013 0.0020 0.0024 1.368 0.0131 0.0009 (1.0020 0.0022 1.397 0.0050 0.0001 0.0008 0.0008 1576* 0.0199 0.0021 0.0030 0.0037 1684* 0.0181 0.0020 0.0027 0.0034 1.831 0.0219 0.0018 0.0033 0.0037 1.949 0.0041 0.0008 0.0006 0.0010 sum 0.3699 (10065 (10220 0.0229 Table 4.6: Gamow-Teller strength distribution from low-lying states. ()—2 MeV. Exci- tation energies of these states come from the fit. and may be off by :t 10 keV. The states listed with asterisks at 1.576 and 1.684 MeV are primarily dipole states. 83 $0.16 k .. m 0.14 E " 0.12 o B(GT) 0.1 i 0.08 0.06 [— 0.04 g : i 0021 i 5;”. :5. ”in. H i. 0 o 10.125; 0:5 103753 i 61.1253 115 1.1753 2 Ex(15°Pm) (MeV) Figure 4.22: Low-lying GT strength from the MDA quadrupole cross sections (at. their peak angles). respectively. are given in Tables 4.7 and 4.8 and Figures 4.23 and 4.24. Cross sections from the resmiance fit method are given in Tables 4.9 and 4.10. Cross sections for the (resonance fit method are slightly higher than for the MDA method of analysis. This discrepancy may arise for several reasons: the shape assumed for the quasi-free curve and other resonances may not be correct. and the resonance fit method assigns significant AL=0 and 1 strength to large regions without. allowing for AL=3 and 4 contributions. Since the assmnptions used to perform the resonance fit analysis have inherent ambiguities that are difficult. to quantitatively test. the results of the MDA will be used for the remainder of the analysis. 84 t 10 j dipole dzddeE (mb/sr 1 MeV) co +— —-._ l I A A 1 J l A A L 1 L L l o 5 10 15 20 25 3o Ex(15°Pm) (MeV) Figure 4.23: Dipole cross sections from the MDA 5 4.5 4 . 3.5 ; 1 .3 + 245 2 +++ IrT 11771 g C r j—TT—T —.— + ——.— quadrupole dzo’deE (mb/sr 1 MeV) 1J5 g ++ 1 E .¢++++¢i+ 0.5 r... O 1 1 1 11L 1 1 111 1 1 1 1 14 1 111 1 1 111 1 a 1 O 5 1O 15 20 25 30 Ex (‘5°Pm) (MeV) Figure 4.24: Quadrupole cross sections from the MDA Ex. (MeV) Cross Section (mb/sr) stat. error syst. error total error 0.5 ....1 C11 CJ'I CI! 01 C51 010710"! Hp—a v-‘O. mm C11 p—d N 01 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 sum 0.5270 0.7445 0.6236 0.5599 0.6033 0.6122 0.7589 0.8910 1.0410 1.2193 1.4205 1.7795 2.2994 3.0044 4.7925 5.3794 6.3067 7.1589 7.9162 8.7827 9.1426 9.2700 8.8952 8.0133 7.0217 5.9347 4.9326 4.1357 3.3305 2.7587 119.8559 0.0174 0.0224 0.0153 0.0108 0.0095 0.0093 0.0120 0.0148 0.0188 0.0226 0.0262 0.0322 0.0409 0.0501 0.8003 0.0759 0.1084 0.1029 0.1175 0.1268 0.1233 0.1180 0.1046 0.0994 0.0864 0.0758 0.0685 0.0600 0.0509 0.0469 0.8850 (10791 (11117 (10935 (10840 (10905 (10918 (11138 (11337 (11562 (11829 (12131 (12669 (13449 (14507 (17189 (18069 (19460 1J0738 1.1874 L3174 1.3714 13905 1.3343 1.2020 110533 (18902 (17399 (16204 (14996 (14138 :11514 0.0809 0.1139 0.0948 0.0847 0.0910 0.0923 (11145 0.1345 0.1573 0.1843 0.2147 0.2689 0.3473 0.4535 1.0758 0.8105 (1.9522 1.0788 1.1932 1.3235 1.3769 1.3955 1.3384 1.2061 1.0568 (1.8934 0.7431 (16232 (15022 0.4165 4.2447 Table 4.7: Dipole cross sections from the MDA. 0-30 MeV. taken at the peak of the angular distribution (1—1.5O ). 86 Ex.(MeV) Cross Section (mb/sr) stat. error syst. error total error 0.5 1.5 to 0'! Cf! 010101 997“ ?>9nrh 99. CH CI! 29.5 811111 (13187 (13766 (14758 (16285 (18355 (19229 (19079 (18976 (1856 (18916 (19631 lfl0426 1.1673 1J3769 1.7089 2.0772 1.8104 2.1822 2.2443 2.5682 2.9635 113910 £18348 £19978 £12886 z4.2716 £12687 411340 4.1184 #11496 (1137 (10116 (10105 (10163 (10257 (10350 (10399 (10329 (10293 (10252 (10242 (10241 (10225 (10215 (10220 (12161 (10229 (10173 (10187 (10173 (10187 (10212 (10247 (10295 (10348 (10432 (10506 (10623 (10707 (10884 (11132 (13079 (1(1478 (10565 (10714 (10943 (11253 (11384 (11362 (11346 (11284 (11337 (11445 (11564 (11751 (12065 (12563 (13116 (12716 (13273 (13366 (13852 (14445 (15087 (15752 (15997 (16433 (164077 (16403 (16201 (16178 (16224 2(11h3 0.0492 (10575 (10732 (10977 (11301 (11441 (11401 (11378 (11308 (11359 (11465 0.1580 (11764 (12077 (13353 (13124 (12721 (13279 (13371 (13857 (14450 (15092 (15760 ().(3()()7’ (16447 (16427 (16433 (16241 (16241 (16326 2.1163 Table 4.8: Quadrupole cross sections. 0—30 MeV. taken at the peak of the angular distribution (2-2.5O). 87 Resonance %q2 (mb/sr) stat. error syst. error total error IVSGDR 117 15 17.55 23.1 G5 3.5 0.5 (1525 0.725 G10 1.0 0.5 0.15 0.52 QF 38 10.0 5.7 11.51 sum 159.5 18.04 18.46 25.81 Table 4.9: Dipole cross sections from giant resonance fits. Resonance 5% (mb/sr) stat. error syst. error total error G5 6.5 (175 0.975 1.23 G10 6.25 0.5 0.94 1.06 GTR 7.0 2.0 1.05 2.26 QF 68.0 8.0 10.2 12.96 sum 87.75 8.3 10.3 13.3 Table 4.10: Quadrupole cross sections from giant, resonance tits. 4.4 Comparison with Theory 4.4.1 Cross sections and Giant Resonances The extracted cross section of the IAS at. 00 and (1:0 is 9.18 :l:0.25 mb/sr (see Figure 4.14 and apply Equation 4.5). and its Fermi strength is equal to (N—Z)=30. The deduced Fermi unit cross section is 0.31 :l: .01 nib/sr. This matches the value from the phenomenological equation [95] for A2150 within 10%: 72 This match between the phenomeiiological and measured (3 F gives us confidence that the PheIIOIDGDOlOgical dGT = 4.19 is also ai')plicable. Based on reference [95]. an error f10‘7~f A " -ll 0 (1 0r O’GT 1s reasona ,) e. Table 3.2 gives a list of 1,)ertinent giant resonz’mces. ()ne way to measure the ex— 88 GT IVGMR IVSGMR sum 0.606 22.395 0.933 0.361 Table 4.11: Exhaustion of the full normal mode. strength for AL=0. under the as- sumption that each of the resonances is the only contribution to the measured L=0 strength. Data could fulfill 60% 0f the GT sum rule, 90% of the IVSGMR sum rule. and over 22 times the IVGMR sum rule. When combined, 36% of the possible sum rule strength from all resonances is seen; however. it is known that the IVGMR and IVSGMR peak at higher excitation energies and that the GTR thus makes up the bulk of the strength. ha ustion of normal mode strength (see section 3.2.2) is to compare the measured cross section (from the MDA) with the calculated cross section in each excitation energy bin. For example. measured AL=0 strength can be attributed to a comlfination of the GTR, IVGMR, and the IVSGMR (assuming the IAS is analyzed separately). al- though very little strength is expected from the IV GMR. Since all three resonances have the same angular distribution. they cannot be separated. The measured strength can be compared separately to the calculated strength for each resonance and then again to the combined resonances. (AL=0 data. is not adjusted to (120.) Results are shown in Tables 4.11, 4.12. and 4.13. \Ve certainly do not expect. to see very much strength from the IVSGMR at low excitation energies. but it is feasible to see some of it since it peaks around 37 MeV and it is very broad. The Fermi unit cross section calculated in D\\'BA is (12025. This difference from the experin'iental value of 0.31 occurs because of the large imaginary volume term in the optical potential. Since the actual unit cross section has been measured. values for the exhaustion of normal mode. strength (Tables 4.11. 4.12. and 4.13) have. been scaled by 0.2025/0.31 = 0.66 under the assumption that it applies for all transitions alike. 89 IVSGDRO— IVSGDRl- IVSGDRZ- IVGDR sum 12.048 0.761 1.148 13.495 0.427 Table 4.12: Exhaustion of normal mode strength for AL=1. IVGQR IVSGQR1+ IVSGQR2+ IVSGQR3+ sum 16.467 5.213 1.638 1.629 (1677 Table 4.13: Exhaustion of normal mode strength for AL=2. 4.4.2 QRPA calculations The group of Vadim Rodin at. the University of Tiibingen has used QRPA methods to calculate the GT and dipole strengths in 150Pm in both the (3He.t) and (t.3He) directions [122. 55. 123]. This research group is able to incorporate nuclear defor- mation into their model. and will use data from this thesis to test their calculations. Table 4.14 [122, 124] gives a list of relevai'it parameters. Raw calculations for three different values of K (projection of angular momentum onto a deformed axis of sym- metry) are shown in Figure 4.25. K is a good quantum number in deformed nuclei (J is not). GT strength is predicted around the region of the experimental GTR.. and dipole strength is anticipated mostly at higher excitation energies (coinciding with the expected location of the IVSGDR). Nucleus 7’32 912]) 9p}: 150Nd 0.183 1.11 1.16 1508111 0.114 1.11 1.16 Table 4.14: Values of the deformation parameter .132 for 15031.1 and 1508111 as adopted in the QRPA calculations. along with the fitted values of the p— p strength parameter gpp. The 9p}; value is found by fitting the position of GT resonance [124]. Quenching of the GT strength is taken into account when gpp is fitted. 90 _o O .1 l l l l I l | l l l l l l 9 q-#* ,_ ‘#_-_fi NA 5 [l > E 9 ; — QRPA1 v l — v l m 8 QRPA1 5 8 .r | 7 —- QRPA2 [ gt 7 E [ 6 r “d [ : t ' 1;; 6 ;; ‘ 5 l g c [ ' o 5 ‘1 4 L l .9- 4 l» ' l ' '0 7 [ 3 l 1 <5 3 3 ‘ 2 [ l 2 f f 1 [ l t . 1 I 1 t [ I J l 0 (L... "1.4.. , . 0 m1“- 1.____1--. 11...... 114 o s 10 15 20 25 3o 0 5 10 15 15020 25 30 150 Ex( Pm) (MeV) Ex( Pm) (MeV) 235 &— ——-——~— —~ ~. A 3 [— —-————~ — 2. NE ‘ “ QRPA‘ l ”g — QRPA1 "L‘, 3" t r [ _: - QRPA2 _: 2.5 l H . H ! 2‘25 1 g : — QRPAZ ' e ; w 2 * 171' 2 i a l 2 [ 2 15 — QRPA3 [ o . o ' 9.1.5 1' .9- l '0 i '0 ‘* I I I 1 '— '— 1 [ N 1 , E ' l 0.5 : 05 [ 0 #1.,“ . 0 Ir 0 5 10 1515020 25 3o 0 5 10 1515020 25 30 Ex( Pm) (MeV) Ex( Pm) (MeV) Figure 4.25: Raw QRPA calculations for 1501\'d(3He.t). K values are as follows: black represents K20. red represents K21. and blue represents K22. Top left: unquenched GT strength. top right: 0_ dipole cross sections and strength. bottom left: 1" dipole cross sections and strength. bottom right: 2_ dipole cross sections and strength. 91 To facilitate comparison with data. these calculations are smeared with Gaussians to represent. the effects of spreading not included in the calculation. For the GT strength. the smearing widths used were 0.035 MeV (FVVHM) (for the region between 0—2 MeV) or 4.7 MeV (FVV HM) Gaussians (for the. region above 2 MeV). The dipole cross sections were smeared with widths of 3.5 MeV (FVVHM). All smearing widths were tuned to the data. The calculations were then put into 1 MeV excitation energy bins. Contributions from all values of K are summed. and a GT quenching factor of 0.56 is added to the GT distribution. This GT quenching factor is the standard value of 0.75:2 [101] applied to all theoretical calculations of GT strength. Figure 4.26 shows a superposition of the calculation and 150Nd data. as well as the cumulative (running sum) Gamow—Teller strength. The experimental AL20 strength does not drop off as much as predicted by theory at. higher excitation energies. which may indicate the presence of high-lying GTR. 2p—2h strength and/ or the lower tail of the IVSGMR. The experimental GT strength in the bin between 0—1 MeV is 9 times lower than predicted. Figure 4.27 shows the same type of comparison for dipole states. In the absence of a known unit. cross section. experimental cross sections (between 1 and 1.50) and calculated dipole strengths are superimposed. The smeared calculations for dipole strength predict three distinct peaks rather than the one seen. suggesting that the placement. and strength of levels could be improved. This (glifference results in a slight mismatch in the shape of the cumulative strength distributions. B(GT) D1: CI 1 1 DU¢¢ [ 39.. El Dhaa1 J 1 1 1 1g 1 1 m 1 1 1D1D1D1UDDr-I O 5 10 15 20 25 30 E x(1 SOPm) (MeV) U1 0 cumulative B(GT) w A O O N O _| O L l 1 L 1 4 L A L 1 n L L k L L 0 S 10 15 20 25 30 E X(‘ 50Pm) (MeV) Figure 4.26: Gamow-Teller strength in 150F111 via 150.\'d(3I-le.t). Data is in black and QRPA is in red. The strength distribution and cun‘iulative strength are shown. Data and theory disagree at very low excitation energy. where the QRPA predicts much more strength. and at the region between 20-30 MeV. where data sees more strength than predicted. See text. 93 ’5 10 5- 10 o. B I A data ' g g l 1] total QRPA ‘30 l a c 8 f U -‘ 8 m -- ‘ 1:1 ‘0' ‘~ V 1-QRPA 1:) g s 6 : D“ 5 ‘9. a .‘ O-QRPA 00 4 3. o l 0000 O V' c: b 4 1 1:1 1') *‘ 4 3 ‘ N 2 + V O 42 v O 2 V O " . .9‘ 2 t 5138 'V' O ' 2 13 QEEQEAAA .v"' 1“ 00‘ $931-13!v.11!!!.11..t 1 ‘5‘. _ . . . . Q 0 o 5 10 15 20 25 30 E x(‘ SOPm) (MeV) $120 120 9, — 'o m 2100 .— 100 2 6 ‘2 l 3 1% E 80 80 g s E «2 -.:. LU 6O 60 5' 2 '0 [ =2 3 Cl 3 E D 40 40 V1») 3.3 U 'o 20 . 20 l o 0 5 10 15 20 25 30 150 Ex( Pm) (MeV) ‘ I I I I r I Fi ‘ure 4.27: Comparisons of dipole cross sections and strength in 1"”1’111 via 1‘) .\d(‘3He.t) shown for the excitation energy distribution and for a cumulative dis- tribution. Data is in black and QRPA calciilati(_)iis are in color. Data shows a cross section distribution that is smoother than predicted. 94 Chapter 5 15OSm(t,3He)150Pm* at the NSCL 5.1 Experimental Setup and Procedure 5.1.1 Production of a Triton Beam The first triton beams at the NSCL were produced from the fragmentation of a pri- mary alpha beam. Following the coupling of the K500 and K1200 cyclotrons [125]. this was no longer the optimal method. Decoupling and recoupliiig the cyclotrons to produce a primary alpha. beam is associated with high overhead time. and greater triton intensity can be achieved using 161180 beams because less ambient neutron radiation is produced. Hitt ct al. [126] performed a systematic study of triton produc- tion for beams of 161180 impinging upon a range of primary Be targets. and found that a 345 MeV/u 160 beam on 3526 ing/cm2 was the optimal method of triton production (for triton energies over 100 MeV) within the constraints imposed by the Bp of the available beam lines. All subsequent triton beams have been produced this way. Before the 150Sin(t.3He) experiment. small geometric iiiisaligiiiiients of analysis line magnets were discovered and subsequently corrected. The realignment increased triton transmission from the focal plane. of the A1900 to the object of the S800 from 95 c288. $55398 8mm 9.: seamen 953 .88 wIm 3:51.28“ 3 w M? . .. \> 55302 $93 cozuanoa 5522 m: E :\>ms_ mp O 0. Figure 5.1: Schematic of the Coupled Cyclotron Facility and the S800 Spectrograph 96 y- y— . . . r ‘ ‘ . around 50% to 85%. Beam purity at the beginning of the 10081110331111) experi- . re- . . . - . . 6 9 - . ment was ~8o/6. The contaminants were He and Li. A small. flat. l.)ackground was noticed in the excitation energy spectrum. and halfway through the experiment we determined that it came from the (6He ——-+ 3He + 311) breakup reaction on the secondary target (target directly before the S800. as compared to the triton produc- tion target). A 195 mg/cm2 wedge inserted into the A1900 removed this impurity. producing a background-free spectrum for the second half of the experiment. Slits in the A1900 were set at a momentum bite of i0.25% (total 0.5%). The analysis line of the S800 spectrometer [127] was operated in dispersion—matched mode (see Section 4.1.1) to achieve the best resolution. In dispersion-matched mode. the beam is dispersed over the target to match the dispersion of the spectrometer (11 cm/% dp/ p). With a momentum acceptance of 0.5% in the A1900. a ~55 cm tall beam spot is created at. the target. In the i'ion-dispersive direction. the beam spot is ~1 cm wide. Figure 5.2 shows an image of the dispersi(_)ii-iiiatched beam on the viewer. The Bp of the analysis line was set to 4.8 Tm. which is close to the maximum possible in dispersion—matched mode. The spectrometer Bp was set to 2.3293 Tin. Triton rates of 107 pps were achieved at the target. during the (‘XpCI‘llllt‘lll’ because of the high triton transmission. To allow for ease in target. changes. the Large Scat— tering Chamber was installed and two remote—controlled. retractable arms placed at the target position. One arm contained a. 1 min-thick plastic scintillator for beam rate measurements. and the other held the 18 ing/cm2 150Sm target. a 10 ing/cm‘2 12CH2 target. and a viewer (piece of aluminum covered in ZiiO. which finoresces when hit. by the beam). The thickness of the 12C was chosen so that the energy loss in the target would be close to that of the 150Sin target. Near the end of the experiment, an 18 ing/cm2 13CH2 target [128] was installed for further calibrations but also produced interesting physics results. Data from this 13C calibration target was published [129] but. will not be discussed here. 97 Figure 5.2: A (118p01‘51011-llle-1H‘llt‘tl triton beam image is shown incident on a ZiiO target (viewer) at the entrance to the S800 [127] spectroiiietei‘. Targets must have dimensions of about 2.5 cm by 7.5 cm to accommodate the large licaiii spot. 98 . ,r' . . Figure 5.3: The 1J0Sm target. crafted by J. Yurkon and .\. Verlianovnz. The tar- get was sandwiched between the two halves of the frame and is shown sideways. Dimensions are 2.5 cm by 7.5 cm. The 12’13C targets were a full 5 cm across. but the high cost of the 1508m material provided sufficient motivation to reduce the width to 2.5 cm (2.0 cm was visible within the boundaries of the frame). Creating the 1508111 target was challenging. Dr. John Yurkon and Dr. Nate Verhanovitz produced the target. ' ’ ”"8111 was used to test two methods of target production: rolling and evaporation. Samarium is a brittle metal. and an attempt. to roll the target failed after reaching ~35 iii g/ c1112. twice the desired thickness. The evaporation procedure causes a significant fraction of the original material to be lost. but this method was successful. Several sequential evaporations produced a target of half the desired thickness and twice the desired width. so this was then folded in half to give the correct. target dimensions. Figure 5.3 shows the 150Sm target near the end of the framing process. The position of the beam on the target was a concern. so the sides of the target frame were painted with 2110 to monitor the beam centering on the 1"308m metal. This proved to be a very helpful technique and was applied in a subsequent (t.‘3He) experiment. 99 (X2, Y2) (X1,Y1) Integrated Image Charge Pad Number Figure 5.4: Layout of the CRDCS in the focal plane of the S800 spectrometer (shown sideways). A sample event is shown in red. relative to the central ,0 of the S800. The inset shows an electron distribution from an event. as it is induced on the cathode pads. This figure was created by J. Yurkon and modified by G.W. Hitt. 5.1.2 The S800 Focal Plane The 8800 Spectrograph contains two quadrupole and two dipole magnets. which focus and bend the 3He residuals into the focal plane of the spectrometer. Position and angle measurements are taken on an event-by-event basis with two Cathode Readout Drift Chambers (CRDCs) [130] spaced at a distance of 107.3 cm (See Figure 5.4). The x (y) coordinate is usually referred to as the dispersive (non-dispersive) coordinate. An overview of specifications for the 8800 appear in Table 5.1. In each CRDC. the ejectiles encounter field-shaping electrodes encased in a con- tinuously-renewed 80% /20% mixture of CF 4 and C4H10. This gas mixture has a high drift velocity. low avalanche electron spread and ages slowly [130]. Incoming 3He ions 100 Parameter _ Value momentum acceptance (AP/P) 5% energy resolution (intrinsic AE/ E) 1/10.000 angular resolution 3 2 mrad position resolution (15 min (both vertical and h(_)ri2(_)ntal) horizontal magnification (x—x) 0.74 focal plane tilt 28.50 maximum Bp (analysis line) 4.8 Tin maximum Bp (spectrograph) 4.0 Tin maximum dipole field (spectrograph) 1.6 T dipole bend radius 2.8 m dipole bend angle 750 angular range 0 to 600 solid angle 20 msr weight N250 tons Table 5.1: Parameters of the S800 Spectrometer. Values are taken from references [131]. [132]. [93]. and [127]. ionize the gas. producing ion-electron pairs. Newly-created ions drift toward the Frisch grid and are not recorded. but the more. quickly-iiioving electrons drift toward the anode wire. which is placed between two sets of cathode pads. The anode current induces charge on the cathode pads. resulting in an electron distribution over about 8- 10 pads. A Gaussian fit. (or center—of—gravity calculation) to this distribution provides the dispersive (x) position signal. The non—dispersive. (v) signal is calculated from the electron drift time to the anode wire. Track angles are. determined from the position differences for one event between the two CRDCs. Plastic scintillators placed behind the CRDCs measure the energy loss of the particles. The signal of one scintillator is used as the trigger for the data acquisition system and serves as the start of the time-of-fli ht measui'ei'nent. The cyclotron RF signal )rovides the sto ). v D 101 5.2 Calibrations 5.2.1 CRDC Mask Calibrations The drift velocity of particles going through the CRDCs must be calibrated so that particle hit positions and angles in the non—dispersive direction can be calculated. To accomplish this. tungsten masks with a distinctive and well-known pattern of holes and lines are placed in front of each CRDC in turn. A target with a high CE event rate (such as CH2) is installed. The ejectiles are steered across the focal plane during the run. For most ejectiles. only sections of the CRDCs directly behind holes and lines will detect the incoming particles. reproducing the mask pattern. 3He and other very light. ejectiles tend to punch through the mask. However, the particles that punch through have a different energy loss in the scintillator and can be rejected in the analysis of the run. The locations of the holes in the CRDC data in terms of channel number can be matched with their actual location on the mask, giving precise (~ 1 mm) ejectile positions. Mask runs must be taken every few days during an experiment to account for slight changes in the drift velocity. but in the 150Sm(t,3He) experiment very little drift was observed. The drift can be monitored between mask runs by monitoring positions and angles in either charge states. if present, or the reconstruction of the recoil energy for a light target nucleus. Reactions on hydrogen impurities in the 150Sm target were used for this purpose. 5.2.2 Beam Rate Calibration The number of 160 ions accelerated through the cyclotrtms must be crmverted into the number of tritons at the 8800 target. A retractable plastic scintillator can count particles at the target position. but. it saturates at high beam rates (>10'5 tritons/s). A Faraday cup at the end of the cyclotrons can measure absolute beam rates. and non-intercepting probes (Nll’s) in the beam line monitor beam rate and are continu- 102 Parameter name Description afp bfp xfp yfp tof(c) ata(c) bta(c) yta(c) dta 6' lap/down. e2up/(Iotcn. E0 6 dispersive angle at the focal plane nondispersive angle at the focal plane dispersive position at. the focal plane nondispersive position at the focal plane time of flight of the ejectile between the start sig- nal (focal plane scintillator) and the stop signal (cyclotron RF) dispersive angle at the target nondispersive angle at the target nondispersive position at. the target momentum deviation from the central Bp track in the 8800. related to the total energy of the ejectile by EO.Uf=(dta*E0+EU)+1110 (me is the mass ofthe ejectile) energy in the top/bottom of the first scintillator energy in the top/ bottom of the second scintillator central energy for a particle following the 3800 Bp track scattering angle as defined from the beam axis. calculated from the nondisrwrsive and dispersive angles at the target , , . r ‘ ‘ . Table 5.2: Important 8800 parameters in the analysis of the 1')()Stii(t.‘5He) expen- ment. A c placed at the end of a parameter indicates it has been corrected for a dependency on another parameter (see section 5.2.4 for more details). 011st read during an experiment. Rate calibrations were performed every twenty-four hours during the experiment: an attenuated beam was measured at the Faraday cup. the NIPs. and the scintillator. and the relationships between the. probes at different attenuator settings allowed us to (alculate an absolute beam rate at the target for the entire experiment. The rate calibrations gave very consistent results and little variation was observed. 1 0 3 5.2.3 Calculation of the Excitation Energy of 150Pm Table 5.2 defines many of the raw 8800 parameters important in any experimental analysis. Using the parameters dta. ata. and bta (see Table 5.2 for further informa- tion), it is possible to calculate the total energy and scattering angles of the outgoing 3He nuclei on an event-by-event basis. The ion-optical code COSY Infinity [133] is paired with accurate magnetic field maps of the S800 to ray—trace focal parameters (from the CRDCs) to angles. the non- dispersive position. and energy at the secondary target. COSY gives a fifth-order in- version matrix which is used in the analysis software. The use of dispersion-matching optimizes the energy resolution achievable from the 8800. In this ex1*)eriment, energy resolution was ~ 300 keV F WHM. Around 160 keV of that was due to the difference in energy loss between 3He particles and tritons in the target. Using data from the 13C(t.3He) reaction. the absolute error in the excitation energy was estimated to be 50 keV. A CE experiment in dispersion-matched mode is very sensitive to the placement. tuning, and energy of the incoming beam. as well as changes in detector response. Small changes in any of these can 1.)ro(‘luce noticeable effects in the data that must be corrected for on a run-by—run basis. The scintillat(_)r energy and time of flight were corrected for correlations with the dispersive position and angle at the focal plane. Corrections were applied to the energy loss of 3He ions in the plastic scintillator. slight changes in the beam energy or the, Bp of the S800. the placement of the beam on the target. and the time of flight. In addition. further corrections must be done. to minimize the effects of an imperfect ray-tracing matrix and imperfect dispersion- matching. These correctitms included making small shifts to center the dispm‘sive and non-dispersive angles at the target. as well as correcting the excitation energy’s dependence on the time of flight and both the dispersive and non-dispersive angles 104 22500 COUNtS 20000 17500 15000 12500 10000 7500 5000 2500 p— p. l L l L 1 O t 1 1 r 1 l 1 l -0.03 -0.02 -0.01 O 0.01 0.02 0.03 ytac (m) Figure 5.5: A ytac spectrum is shown for all runs on the 1508111 target. The two side peaks represent events from tritons incident on the aluminum target frame, while the events in the center are from the 150Sm metal. While the fraction of beam particles impinging upon the frame was small. the frame was very thick and therefore the yield was high. Frame events were removed from the analysis. at. the target. Once all of these. corrections were done. many runs could be combined together. Figure 5.5 shows the reconstructed non-dispersive hit position at the target (ytac) spectrum for all runs on the 1508111 target. Events on the 1508111 are in the center. and events from the frame are shown on the sides. Frame events were removed with a cut. on the acceptance. and the acceptance correction procedure discussed in Section 5.2.4 was used to account for real events discart’led in this manner. The particle identification spectrum for 3He nuclei in the 8800 focal plane is shown in Figure 5.6. This plot shows the corrected energy loss in the second scintillator as a function of the corrected time of flight. The blob at the left (coordinates (0.500)) is 3He, and the area. at the lower right may be deuterons or from a 1*)ackground process (such as the triton beam scattered off of the first dipole magnet chamber in the 8800). The excitation energy of the residual nucleus is found using a missing mass calcula- tion. Parameters for this calculation are shown in Table 5.3. Using the incoming and outgoing energy and momenta of the projectile and ejectile. one can apply the con- servation of energy and momentum to calculate how much momentum / energy went into exciting the residual nucleus. This is called the “missing" momenturn/energy. and the excitation energy is calculated directly from these quantities and the mass of the residual nucleus (see. the bottom two lines of Table 5.3). K) o Er : \/E;nis — Putts-(4)2 — "’1' (5'1) 5.2.4 Acceptance Corrections The function governing the acceptance of the 8800 is complex: it depends on the momentum and scattering angle of the outgoing particle as well as the hit position on the target. Previous CE experiments were able to use only a portion of the lalmratory scattering angle avail-able (0 to ~350 [134]). The CE group has created a Monte— 106 w A U10 \1 CO OO O CO OO O e2_up (corrected) (channels) N O O 100 O -100 -50 0 50 100 150 200 250 300 tofc (channels) Figure 5.6: Final particle ID in the focal plane of the S800. 3He is in the upper left corner. 107 MM Parameters Definition 11m, 931.5 MeV 111,, mass number A mass excess 111p 3*um+Ap (mass of projectile (triton)) 111(2 3*um+Ae (mass of ejectile (31%)) mt 150*um+At mass of target mr 150*um+Ar mass of residual E0 central energy for a particle following the 3800 Bp track Ebmm Energy of the incoming beam in MeV Em Ebea ,,.l+r11p+1nt (total incoming energy) Eout (dta*E0+E0)+1ne (total outgoing energy) E-mz's Ein'Eout (“missing” energy) Pf(4) 2 - 7112,. (total final momentum) out ‘ Pf(1) Pf(4)*sin(utuc) (dispersive component. of final momen- t11111) Pf(2) Pf(4)*sin(btn.c) (non-dispersive component of final mo- ment um) Pf(3) Pf-(4)*cos(()) (compm1ent of final momentum along the beam axis) Pi(1) 0 (assumed) (dispersive component of initial momen- tum) Pi(2) 0 (assumed) (non-dispersive component of initial mo- 111entum) P.,:(3) 1439 MeV (beam axis component of initial momentum) PI-(i) 1439 MeV (total initial momentum) P771215“) Pl-n(1)—Pf(l) (dispersive component of missing momen- tum) Pm.,-S(2) Pin (2)-Pf(2) (non-dispersive component of missing mo- mentum) P‘II'2.i.S'(3) Pm(3)’Pf(3) (beam axis component of missing momen- tum) Pun-3(4) \/sz's]1)2 + Pun-SOP + Pun-5(3)2 (total missing mo- 111ent11111) mm E72,!“ — Pun-8(4)2 E1. nun—m,- Table 5.3: Parameters used 111 the 1J0Sm(t.3fle) 1111ss1ng mass calculation. applicable. to any target used in the experiment. This calculation is done on an event—by-event basis. so all momentum and energy variables (except for Ebmm) refer to individual particles. 108 Carlo model of the 8800‘s a.cce1.)tance to allow safe use of a greater angular range. The model creates a three-tlimensional acceptance matrix based 011 the scattering angle. dta. and yta. W hen applied to an experimental event. this matrix returns a correction factor that weights the event based 011 its probability of acceptance. The si111ulation 1213(3). was successfully tested for cases where the differential cross section is known ( 150Sm. and was then applied to Use of this acceptance weighting factor allowed us to extend our angular range to 50 in the laboratory frame. 5.2.5 Background and Hydrogen Subtraction As mentioned in Section 5.1. a small lmckground was evident in the data during the first half of the experiment. Vt’e eventually determined that the 6He impurity in the triton beam was breaking tip on the target. forming 3He and three neutrons. This 3He momentum distribution is very broad and overlaps with 3He particles produced in the (t.3He) reaction. A 195 mg/cm2 thick alumimun wedge was inserted into the intermediate image of the A1900 fragment separator halfway through the experiment in an attempt to purify the beam. Insertion of the wedge produced a nearly pure triton beam and removed the flat background. This backgror111d-free data set (""\\'\V” for “with wedge”) was then used to determine the shape of and remove background from the rest of the data ("NVW for "no wedge"). Excitation energy spectra were produced in 100 keV bins for the purpose of back- ground subtraction. Bins of 300 keV (which correspomls to the energy resolution) and 1 MeV are used in the physics analysis. Using the number of counts in the hydrogen peak (events from the small H impurity in the 1'50Sm target). the “WV data was scaled to the NW data and subtracted. The resulting background shape proved to be well—represented by a flat distribution for all angles. so the line fit was subtracted from the NW data to produce a background-free spectrum. See Fig. 5.7 for an example. W’l‘lile error bars are not shown in the figure to allow for 111axi1num 109 100 Ta) 75 counts O . . 100 d) 75 so 25 O 5 10 15150 20 25 Ex( Pm) (MeV) \ Figure 5.7: Background subtraction for the 1—20 angular bin. a) The NW excita- tion energy spectrum for 150pm in 100 keV bins. b) The WW excitation energy spectrum for 150Pm in 100 keV bins c) After scaling the WW spectrum to the NW spectrum using the ratio of counts in the hydrogen peak, a line is fit to the subtracted spectrum. The WW data has slightly better resolution than the NW data. so the H peak is slightly narrower. d) The NW excitation energy spectrum after background subtraction. 1 1 0 counts 20} 10} O 1 1111 1 1 11 11 1 111 -5 0 5 10 15150 20 25 Ex( Pm)(MeV) Figure 5.8: Hydrogen subtraction for the 1-20 angular bin: the WW excitation energy spectrum for 150Pm in 100 keV binning. The hydrogen peak is shown in blue, and the final spectrum shown in black. 111 clarity. they are calculated and carried through the whole analysis. . ‘ n o u r w ‘ There was very 11ttle. l'1ydrogen ctmtannnatlon 111 the 1005111 target. but. the 1H(t.3He) cross section is large and a second subtraction procedure was necessary. For a given scattering angle. the recoil energy of a neutron produced in this reaction is 150 times larger than that. of the 150P111 produced in the 150Sm(t.3He) reaction. As the Q- value difference. for the two reactions is only 2.67 MeV. some events from reactions 011 1H begin to bleed into the 1508111 data at angles greater than 20. The two sets of data are completely separate for scattering angles below 20. Since the CH2 cali- bration target contained significant amounts of H and the Q-value difference between the 1H(t.3He) and 12C(t.3He) reactions is large (12.59 MeV). the 1H shape could be cleanly modeled from the CH2 data. This situation was predicted when the experi- ment was planned. and the CH2 target thickness was chosen such that the differential energy loss between 3He and tritons was the same as that of the 150Sm target. A double sigmoid function was found to reproduce the H shape well. and it was scaled to and subtracted from the l'50S1n data. Figure 5.8 shows the WW 1-20 excitation energy spectrum before and after the hydrogen peak subtraction. 5.2.6 Calculation of the Cross Section Differential cross sections were calculated using .1_.. _ Y (152 _ Nb1vt1-queg' Y is the total munber of counts. Nb is the number of nuclei in the beam. .\'t is the number of nuclei in the target. (IQ is the opening angle. 61 corrects for the dead time in the data. acquisition system (96.7%~ live time). and 62 corrects for the purity of the 1508111 target (96%). F1 ‘ures 5.9 and 5.10 show the cross sections for all five angular bins in 300 keV .. D 112 TITTTTIY? dzo/deE (mb/sr 300 keV) O O O —\ —k _L _x .h. CD m —‘ N -§ 0) CD 11,11 Y T Y ._ 15 1_ :13 .o N 1 ._.-L O o 5 1o 15 15020 25 Ex( Pm) (MeV) Figure 5.9: Cross sections for 1508111(t.3He)150Pm from 0—26 MeV. in 300 keV bins. Data is grouped into 10 angular bins: 0-10 in black, 1-20 in red. 2-30 in green. 3-40 in dark blue. and 4—50 in light blue. and 1 MeV excitation energy binning. Individual states are not visible. Unlike experiments in the (3He.t) direction. experiments in the (t.3He) direction experience strong Pauli blocking (for N>>Z). so much less GT strength is expected and GT transitions are. not obviously present in the spectrum. Similarly. the IVSGDR strength is also reduced. The centroid of the IVSGMR is predicted to be around 15 MeV (see equation 3.29) and should contribute to the spectrum. particularly at small scattering angles. See Figure 3.8. 113 dzo/deE (mb/sr1 MeV) l | I 0 % L L 1 L l L L L L l L L L l L J_ P; l o 5 1o 15 150 20 25 Ex( Pm) (MeV) Figure 5.10: Cross sections for 150Sm(t.3He)150Pm from 0-26 MeV. in 1 MeV bins. Data is grouped into 10 angular bins: 0-10 in black. 1—20 in red, 2—30 in green. 3-40 in dark blue. and 4-50 in light blue. 114 5.3 Data Analysis 5.3. 1 FOLD calculations The FOLD code was introduced in Chapter 3. Calculated angular distributions for various multipole transitions in the 150Sm(t.3He) reaction are shown in Figures 5.11 and 5.12, where an arbitrary scale factor is applied to make each function easier to see. Except for the AL=0 curve. shapes of the angular distributions change significantly as the excitation energy increases. This was not the case for the 1501\Id(3He.t) reaction. Although this phenomenon is still under investigation outside the scope of this thesis. it is believed to be due to the effects of the Coulomb force in the reaction process. As the incoming projectile approaches the target. it is (:lecelerated by the repul- sive Coulomb force between the projectile and target. After the reaction. the ejectile is accelerated by the Coulomb force between the ejectile and target / residual. The “effective” linear momentum transfer ((16. f f) is defined as the difference in linear momentum between the projectile and ejectile at the interaction point. whereas the “asymptotic” linear momentum transfer ((10511) is the difference in the calculated linear momentum transfer between the projectile and ejectile far away from the in- teraction point. If we ignore the effect. of the Coulomb force. (16 f f equals (Iggy and should increase with the Q-value of the reaction in a similar fashion for experiments in both the (t..3He) and (3He.t) directions. Accounting for the Coulomb force. causes qeff <(lasy in the (3He.t) direction and qeff >(lasg in the (t.3He) direction. The larger qe f f for the (t.3He) reaction results in stronger contributions from amplitudes with AL>0. This leads to significant cl‘1anges in angular distributions. as shown in Figures 5.11 and 5.12. Collection of data for the 1'50S1n(t.t‘) optical potential was not possible for two reasons: the S800 cannot bend 345-MeV tritons. and a measurement with the rela- tively low beam intensity (comj‘mred to a pri111ary beam) would require a very long 115 r .. I} urn uni-.11. 1.4 P — AL=0 arbitrary units .5 4 4.5 5 Gcm(deg.) AIALALLLLLIALA 0‘ ‘05 1 1.5 ‘2“2.5H3m3 . . . . l' V . Figure 5.11: Angular distributmns from 1‘)Oblii(t,3He) as calculated with FOLD, at Q=0. Relative scaling of the distributions is arbitrary and chosen solely to better display the function shape. 116 :2 g 14 ~ AL 0 a: ----- AL=1 g 12 ~ AL=2 .0 a 11L 1+1! L l o 0.5 1“1‘.“‘2‘2.5‘*3*“3.5 4 4.5 5 e cm Figure 5.12: Angular distrilimtions from 150Sm(t.3He) as calculated with FOLD, at Q=20. Relative scaling of the distributions is arbitrary and chosen solely to bet- ter display the function shape, which have changed significantly for AL=1.2.and 4 compared to Figure 5.11. 117 bean‘itime to gain enough statistics. Therefore, the optical potential parameters do- duced from the 1"’ONd JHe.3He elastic scatterimr measurement were ada )ted for the. ('3 present ant-ilysis as well (see Sections 3.2.5 and 4.3.2). 5.3.2 Multipole Decomposition The level density of 150pm is expected to be quite high, because it is a heavy odd-odd . . 7 . . ‘ _ . 1,50 . .3 . nucleus. This was ev1dent even in the lugl1—resolut1(_)n .\e( He.t) data. where only some individual levels could be discerned at low excitation energy. Because of this high level density and the energy resolution of 300 keV F\\"H.\l. we cannot resolve any individual peaks in the 150Sm(t,3He)150F111 excitation energy spectrum. Instearl. the angular distributions of the data were created from each bin ((l-‘ZU MeV in excitation energy) of Figure 5.10 and the first 20 bins ((l—(j MeV in excitation energy) of Figure 5.9 and multipole contrilmtions were decomposed using the FOLD calculations. A linear combination of multipole shapes was fit to each angular distribution in each bin [116] with the equation 0t0t=A>l<01+B*(72+C*03+D*04. (5.3) The best fit. results were obtained using AL=().1.2. and 4 as shown in, Figures 5.11 and 5.12. AL=3 was left out because the limited statistics allowed for the use of only five angular bins in this experiment. Contributions from AL=3 are effectively absorbed into contributions from AL=2 and AL=4. Figure 5.13 shows the MDA fit for the ()-1 MeV excitation energy bin. AL=1 strength dominates. but there is also considerable strength associated with £31.20 and 2. In contrast, Figure 5.14 shows the 20-21 MeV excitation energy bin. AL=4 strength dominates here. AL=1 strength is completely absent. and only small amounts of AL=0 and 2 strength appear. Many of the fits in this higher excitation energy 118 _L —‘L do/dQ (mb/sr) 0.8 0.6 0.4 0.2: Ooiliiin A 3“ 4H15 Gem (deg) Figure 5.13: MDA for the 0—1 .\IeV excitation energy bin. region are similar. Because of the previously-mentioned limitations in the MDA, it is likely that this AL=4 strength represents a combination of strength with ALZ3. Low-lying states are of 1,)articular interest for 21143.13 decay. Since a strong GT state was seen around 0.11 MeV in the 150Nd(3He.t) experiment. this region was closely examined for evidence of population through 1'508111(t,3He) as well. Figure 5.15 shows the angular distribution for the region between 0-300 keV. The AL=0 cross section at 00 is 0.08 mb/sr :t 0.05 mb/sr. However. this error bar can be improved upon. If the region between 100 and 200 keV (which is expected to include most of the 1+ strength) is examined, the AL=4 component is consistent with zero. This is shown in Figure 5.16. The cross section associated with AL=O in the lth-keV bin (0.08 :t 0.03 mb/sr) at ()0 is consistent with that in the 300-keV bin. Due to the limited energy resolution, some AL=O strength should appear in the bins innnediately below and above the IOU-200 keV bin. but the extracted AL=0 cross sections at O0 are. 0.018 119 ’5 B 5 E, s 4 b 'O 3 2 1 O . 1 . . . 4 5 ecm (deg) Figure 5.14: MDA for the 20-31 MeV excitation energy bin. €6ng mb/sr (1.)elow) and. 0.012 iggig mb/sr (above). and both are consistent with zero. Given our uncertainty in the absolute energy calibration of 50 keV and the fact. that. no other 1+ states appear within that. error margin in the 150.\'d(3He.t) data. we conclude that the GT strength associated with the 100—200 keV bin in the 150Sm(t,3He) data is likely associated with the 110 keV state in 150.\’d(3He.t) data. However. the possibility that the two transitions do not represent the same state in 150pm cannot be completely excluded. Figures 5.17 and 5.18 show the assignment of multipole strength deduced from the MDA for the full spectrum and the lower-lying states. respectively. AL=0 strength is seen in several areas (0.85 MeV. 2.25 MeV, 5 MeV. and 5.5 MeV). The strength in the 100—200 keV region does not stand out with these large bin sizes. For the full spectrum. quite a bit of AL=0 and ‘2 (which may include contributions from AL=3) strength appears to peak between 10-12 MeV, and AL=1 strength dominates 1‘20 do/dQ (mb/sr) .0 O .0 on is 01 .O N 0.1 Figure 5.15: MDA for the 0-0.3 MeV excitation energy bin. The AL=0 angular distribution at. 0° has a cross section of 0.08 mb/sr :1: 0.05 mb/sr. The significant AL=1 cross section may correspond in part to po )ulation of the 150pm ground state. which is shown to be. a dipole transition in the 1" 0.\id(3He.t) data. dat .3 03 r F A N T I Y 1 1 sum do/odfl émbgr) I; | D l— l" 3 4 A A 5 Gem (deg) Figure 5.16: MDA for the 0.1-0.2 MeV excitation energy bin. The AL=0 angular distribution at 00 has a cross section of 0.08 mb/sr :1: 0.03 mb/sr. AL=0 strength is enhanced in this bin and can be extracted with a smaller error than in a fit to the 300 keV bin. in lower regions. AL=4 strength (including contributions from AL=3. 5. and higher multipoles) makes up an increasing amount of the spectrum as the excitation energy increases. 5.3.3 Extrapolation to q=0 Absolute AL=0 cross sections from the MDA must be extrapolated to q=0 (zero asymptotic linear momentum transfer) with equation 4.4 before they can be used to calculate GT strengths. A fourth-order polynomial describes this ratio. which is shown in Figure 5.19 and Equation 5.4. Y. I ‘atio = 1.00946+0.01039Q+0.00857Q2 — 4.082 ><10—4Q3 + 1.843 ><10—5Q4 (5.4) The effective linear momentum. transfer q(, f f is different from the asymptotic lin- ear momentum transfer (lag-(l for the (t,3He) and (JHe.t) directions. as discussed in Section 5.3.1. This difference is reflected in Figures 5.19 and 4.20. 5.3.4 Calculation of the Gamow-Teller strength Once the cross section has been extrapolated to (1:0. the B(GT) can be extracted with Equation 3.24 using the phenomenological unit cross section [100]. Table 5.4 and Figure 5.20 show the results for 1 MeV bins up through 26 MeV in excitation energy, and Table 5.5 and Figure 5.21 show results for 300 keV bins for 0—6 MeV in excitation energy. Statistical/fitting errors are dominant in both choices of binning. Systematic errors in the extracted cross sections are estimated to be 15“( and are due to uncertainties in the optical model potential and the plicnomenological unit cross section. A large amount. of AL=0 strength is seen over the region of 5-20 MeV. Between 0-6 MeV, bins centered at 0.15. 0.75. 1.0. and 2.25 MeV show evidence of CT strength. While, all of the AL=0 strength is assumed to be CT for the purposes of this 123 A t O data > [ —— 1+ O-Ideg g6 r —— 1- » —— 2+ '- 5 ; -— 4+ '9 4 j fl :5, ~ 0' LL! 3 r ' "o : . Ci 2 l . E 3" Nb] L '0 t O b . . .1....i.i. r .L.i O 5 10 15 20 25 150 EX( Pm)(MeV) A7 ~ C data > i —— 1+ 1-2deg 0" 6 j -—-—-— 1- E . —— 2+ ‘— 5 L -— 4+ 3 l sum 3 4 £5, LLI 3 ~ '0 i C} ; E 2 Nb 1 r '5 i Orr-..in...1.n- ..AL 0 5 10 15 20 25 EXCSOPm) (MeV) Figure 5.17: Multipole decomposition sunnnary for each angular bin. Higher mul- tipoles (or quasifree processes) take over at the highest excitation energies. as was discussed earlier in this section. Sizable cross sections associated with AL=0 and 2 are centered around 10-12 MeV. and a smaller amount of AL=1 cross sections are centered between 0—10 MeV. 124 7 6 d A ata > —— 1+ 2-3deg g 6 ,— —— 1- » —— 2+ ‘— 5 r — 4+ 3, sum 3 4 _ é : 1.1.1 3 1 'C i , c: 2 : n , '2 if Nb 1 -' '0 i 0 5.. . .1. . . .11 . AlAl in L1 0 5 10 15 20 25 150 EX( Pm)(MeV) A7 ~ 1 data > I ——- 1+ 3-4deg (U 6 j _ 1- 2 ~ --——- 2+ h t sum m . \ .Q 4 : é LLI 3 ” U .. Ci 2 f '3 Nbl f '0 L O J .14_L .A ‘i O 5 10 15 20 25 EXCSOPm) (MeV) Figure 5.17: cont. Multipole decomposition summary for each angular bin. Higher multipoles (or quasifree processes) take over at the highest excitation energies, as was discussed earlier in this section. Sizable cross sections associated with AL=0 and 2 are centered around 10—12 MeV, and a smaller amount of AL=1 cross sections are centered between 0-10 "MeV. p...‘ [\3 CI! » data [ ——. 1+ 4-5 deg 6 e 1- 2+ _ + 5 sum dzo/dQ dE (mb/sr1 MeV) O 5 '10 15 20 25 EXCSOPm) (MeV) Figure 5.17: cont. Multipole decomposition summary for each angular bin. Higher multipoles (or quasifree processes) take over at the highest excitation energies, as was discussed earlier in this section. Sizable cross sections associated with AL=O and 2 are centered around 10-12 MeV. and a smaller amount of AL=1 cross sections are centered between 0-10 MeV. dzo/dQ dE (mb/sr 1 MeV) 9.0 000—- dE (mb/5r 1 MeV) 9.0 km S) E 0.2 ND '0 0.1 '. O i - . . . . . o 1 2 3 4 5 6 Ex(‘5°Pm) (MeV) Figure 5.18: Multipole decomposition summary for each angular bin (0-6 MeV). Cross section peaks associated with AL=0 are visible in the 0-10 plot. dzo/dQ dE (mb/sr 1 MeV) 4 S 6 x(‘50Pm1 (MeV) l'l'lw dzo/dQ dE (mb/sr 1 MeV) o 1 2‘ 3""‘4"s"6 EXCSOPm) (MeV) Figure 5.18: cont. Multipole decomposition summary for each angular bin (0-6 .\Ie\'). Cross section peaks associated with AL=0 are visible in the 0-10 plot. 128 dzc/dQ dE (mb/5r 1 MeV) 5 ‘1 2‘ 3 “£1 55 EXCSOPm) (MeV) Figure 5.18: cont. Multipole decomposition summary for each angular bin (0-6 MeV). Cross section peaks associated with AL=0 are visible in the 0-10 plot. calculation, the IV SGMR (which peaks around 15 MeV) is expected to contribute. The IVSGMR is expected to be very broad (~10 MeV) and cannot be separated from CT transitions by its angular distribution. It. should be a smooth function of the excitation energy, so any isolated low-lying states likely stem from GT transitions. The magnitude of the extracted AL=0 cross section in terms of the IVSGMR, will be further discussed in section 5.4. 5.3.5 Other Multipole Excitations Since 150Nd ——> 150Sm (11/:33 decays can be described as going through virtual states in 150pm of any J” and excitation energy. it is helpful to extract the strength distri- butions for dipole and quadrupole transitions. Dipole contributions. which peak at 1-20, are clustered between 0—10 MeV, with the largest cross sections present between 1 and 4 MeV. Figure 5.22 and Table 5.6 contain dipole cross sections. Quadrupole cross sections (the cross sections in the 0-10 bin are displayed) form a diffuse area 129 A 8 0. 1 . O o 7 - .1. : ' g 5 5 o B i 0 E 5 “ o A L . 3’ 4 r o q 1 O "o 3 — o. 11 Z O 8 .0. 8 1 in!“" 0 . . pl 1 1 i 1 . . 1 r r 1 O 5 1O 15 20 25 Figure 5.19: Ratio of the cross section at 6:00 and 0 linear momentum transfer to that of 0 linear momentum transfer. as calculated in DWBA. At a Q of 25 MeV, the effective monwntum transfer q is 0.3. 130 Ex.(MeV) B(GT)* stat. error syst. error total error 0.5 0.1052 0.0278 0.0158 0.0320 1.5 0.0613 0.0282 0.0092 0.0297 2.5 0.0952 0.0334 0.0143 0.0363 3.5 0.0921 0.0334 0.0138 0.0361 4.5 0.1592 0.0404 0.0239 0.0469 5.5 0.2381 0.0541 0.0357 0.0648 6.5 0.2542 0.0702 0.0381 0.0799 7.5 0.4252 0.0864 0.0638 0.1074 8.5 0.4303 0.1014 0.0645 0.1202 9.5 0.5872 0.1182 0.0881 0.1474 10.5 0.7088 0.1293 0.1063 0.1674 11.5 0.706 0.1524 0.1059 0.1856 12.5 0.753 0.1683 0.1130 0.2027 13.5 1.0289 0.1943 0.1543 0.2481 14.5 1.0186 0.2142 0.1528 0.2631 15.5 0.7394 0.2246 0.1109 0.2505 16.5 1.1134 0.2563 0.1670 0.3059 17.5 1.1147 0.2845 0.1672 0.3300 18.5 0.645 0.4275 0.0968 0.4383 19.5 0.8295 0.3441 0.1244 0.3659 20.5 1.2223 0.4138 0.1833 0.4526 21.5 0.4127 0.6374 0.0619 0.6404 22.5 0.73 0.4694 0.1095 0.4820 23.5 0.2313 0.2377 0.0347 0.2402 24.5 0.9893 0.5002 0.1484 0.5217 ‘ 5.5 0.6728 0.5167 0.1009 0.5265 sum 15.3637 1.4376 0.5275 1.5313 Table 5.4: Gamow-Teller strength distribut-ions have been extracted for excitation energies of 0-26 MeV. See Figure 5.20. The seemingly large scatter and large error bars at higher excitation energies are due to the q=0 correction factor, which sharply increases in this region and magnifies the statistical error. *The entire AL=0 cross section is assumed to be GT strength here, but most of the strength is expected to actually represent IVSGM R strength. 1331 E . 351.75 —. B(GT) .- 7- 1.5 ~ .. 1 1.25 ~ ,1. . 01)" 1 P 00 # L .1 " 4. 0 0.75 5 " 0 " o 0 H1 .. - 0 4. 0.25 — I” .. -- 0315.55.51.51. r o 5 1o 15 20 25 E 150 x( Pm) (MeV) Figure 5.20: Extracted GT strength distributions are shown in the region of ()-26 MeV. Low-lying strength is seen between 0-1 MeV. and higher-lying strength is dispersed between 5 and 20 MeV. The entire AL=0 cross section is assumed to be GT strength here. but most may be contributions from the IVSGMR resonance. See Table 5.4 for the same information in tabular form. Some values are consistent with zero. 132 Ex.(MeV) B(GT) stat. error syst. error total error 0.15 0.0165 0.009 0.0025 0.0093 0.45 0.0153 0.0114 0.0023 0.0116 0.75 0.0505 0.0126 0.0076 0.0147 1.05 0.0414 0.0127 0.0062 0.0141 1.35 0.005 0.0109 0.0008 0.0109 1.65 0.0162 0.0118 0.0024 0.0120 1.95 0.0159 0.0118 0.0024 0.0120 2.25 0.0709 0.016 0.0106 0.0192 2.55 0.0102 0.0128 0.0015 0.0129 2.85 0.0057 0.0127 0.0009 0.0127 3.15 0.0238 0.0138 0.0036 0.0143 3.45 0.0323 0.0136 0.0048 0.0144 3.75 0.0318 0.0142 0.0048 0.0150 4.05 0.0284 0.0137 0.0043 0.0143 4.35 0.0208 0.0143 0.0031 0.0146 4.65 0.0443 0.0171 0.0066 0.0183 4.95 0.0801 0.0207 0.0120 0.0239 5.25 0.0457 0.0186 0.0069 0.0198 5.55 0.1152 0.025 0.0173 0.0304 5.85 0.054 0.0239 0.0081 0.0252 sunl 0.724 0.0689 0.0304 0.0753 Table 5.5: Gamow-Teller strength distributions are shown for the region of 0-6 MeV (300 keV bins). See Figure 5.21. Even at these low excitation energies. the tail of the IVSGMR resonance may cmrtribute to the extracted GT strength. 133 B(GT) P I; l 0.12 . B(GT) 0.1 E 0.08 E .. 0.06 l ll ‘[ Ni ..ii Mimi ‘ O 1 2 3 4 5 6 Ex(1 5°Pm) (MeV) Figure 5.21: Extracted GT strength is shown for the region between 0-6 MeV. See Table 5.5 for the same information in tabular form. Some values are consistent with zero, and the tail of the IVSGMR is expected to contribute to the strength. However. isolated strength below 3 MeV is unlikely to be due to the IVSGMR. 134 O ‘3 2 ~ :5 1.75 ~ .0 . E 1.5 ~ c . .9. 1.25 ~ .. 1' g . 11' 1- v . g 1 l 0 ”l ‘ , 7' v T s 0.75 . " 0 ¢ i ~- .. _”| 0.5 ~ 0 0 ll 0 4 . o 0.25 5 " J 0 D 1 0 5 1L .. 0 11 l 0 c " «i t? 0 0 1 o 5 1o 15 20 25 EX(150Pm) (MeV) Figure 5.22: Extracted AL=1 cross sections are shown from 0-26 MeV (at 1-20). Some values are consistent with zero. of strength between 5 and 17 MeV. as shown in Figure 5.23 and Table 5.7. The quadrupole distribution is similar to the. AL=0 distribution, and could be indicative of an IV SGQR. In fact, the strength distribution of the IVSGQR and IVSGMR are expected to peak at nearly the same excitation energies and have similar widths [135]. However, the possible contributions from AL=i} strength in the extracted AL=2 cross section make it hard to draw strong conclusions on the magnitude of the IVSGQR. 135 Ex.(MeV) Cross Section (mb/sr) stat. error syst. error total error 0.5 0.8164 1.1871 1.1149 0.972 0.9144 0.9132 0.9105 0.7922 0.6780 0.5583 0.5539 0.5935 0.1767 0.0833 0.0819 0.1615 0.0008 0.0071 0.1672 0.0059 0.0054 0.2734 0.3811 0.4834 0.0034 0.0030 11.8385 0.1253 0.1550 0.1714 0.1718 0.1878 0.2364 0.2906 0.3474 0.3951 0.4266 0.4798 0.5589 0.5961 0.6601 0.7266 0.7891 0.5950 0.6816 0.3824 0.8209 0.8322 1.0079 0.7648 0.6850 0.9344 0.2206 2.9224 0.1225 0.1781 0.1672 0.1458 0.1372 0.1370 0.1366 0.1188 0.1017 0.0837 0.0831 0.0890 0.0265 0.0125 0.0123 0.0242 0.0001 0.0011 0.0251 0.0009 0.0008 0.0410 0.0572 0.0725 0.0005 0.0005 0.4592 0.1752 0.2361 0.2395 0.2253 0.2326 0.2732 0.3211 0.3672 0.4080 0.4347 0.4869 0.5659 0.5967 0.6602 0.7267 0.7895 0.5950 0.6816 0.3832 0.8209 0.8322 1.0087 0.7669 0.6888 0.9344 0.2206 2.8582 Table 5.6: AL=1 cross sections are shown from 0-26 MeV (at 1-20). 136 Ex.(MeV) Cross Section (mb/3r) stat. error syst. error total error 0.5 0.3995 0.1533 0.0599 0.1646 1.5 0.3796 0.1662 0.0569 0.1757 2.5 0.4927 0.1848 0.0739 0.1990 3.5 0.4516 0.1886 0.0677 0.2004 4.5 0.4664 0.2140 0.0700 0.2251 5.5 0.7456 0.2611 0.1118 0.2840 6.5 1.1218 0.3146 0.1683 0.3568 7.5 1.3661 0.3699 0.2049 0.4229 8.5 1.5730 0.4035 0.2360 0.4674 9.5 1.5728 0.4403 0.2359 0.4995 10.5 1.4995 0.5018 0.2249 0.5499 11.5 1.5064 0.5874 0.2260 0.6294 12.5 1.7716 0.6251 0.2657 0.6792 13.5 1.5544 0.6808 0.2332 0.7196 14.5 1.3727 0.7700 0.2059 0.7971 15.5 1.2286 0.8376 0.1843 0.8576 16.5 1.2695 0.3209 0.1904 0.3731 17.5 0.9608 0.3267 0.1441 0.3571 18.5 0.8545 0.7063 0.1282 0.7178 19.5 0.6068 0.3972 0.0910 0.4075 20.5 0.5803 0.6267 0.0870 0.6327 21.5 0.2184 0.3726 0.0328 0.3740 22.5 0.0243 0.5938 0.0036 0.5938 23.5 0.0083 0.4641 0.0012 0.4641 24.5 0.0005 0.4770 0.0001 0.4770 25.5 0.0005 0.1972 0.0001 0.1972 sunl 22.0262 2.4094 0.7806 2.5327 Table 5.7: AL=2 cross sections are shown from 0-26 .\IeV (at 0-10). A portion of the cross section is likely due. to AL=3 contributions. which were not accounted for in the MDA. 137 0,. 2.25 . 31.75 1 * 1 ‘E’ 1.5 t 1 ""00 " 5 1. .. 1 61.25 ” 11"1 3 ~ ‘1 .. .1 ._ a 1 7 .. o O 0 so. 5 ~ 1 N 9 o r1 3 <1 .0 NPV mm TTWIVI P—O—fi 1-——o——1 1—-o—-1 O 7 h ..L 0 0 1 o 5 1o. 15 20 25 Ex(150Pm) (MeV) Figure 5.23: Extracted AL=2 cross sections are shown from 0—26 .\1eV (at 0—10). Some values are consistent with zero. A portion of the cross section is likely due to AL=3 contributions, which were not accounted for in the MDA. 138 5.4 Comparison with Theory 5.4.1 AL=0 Cross sections and the IVSGMR Very little GT strength is expected to be observed in the 150Sm(t.3He) reaction due to the effects of Pauli blocking. According to the QRPA calculaticms discussed in Section 5.4.2. the total GT strength should be 0.5. However. if we assume that all of the extracted AL=0 strength from the data can be attributed to GT transitions. the. total strength is 15.4 :1: 1.5. This is unrealistically high. especially when compared with similar (n.p)-type -experiments [82. 136. 137. 138. 139]. The largest extracted GT strength in the (n.p) direction over a comparable energy range (30 MeV) was 6 [139] for the case of 116Sn(n.p). and Pauli blocking is stronger for 150Sm than 116Sn. Most references claim sunnned B(GT) values of 1-3. Transitions due. to the IVSGMR have similar angular distributions to GT transi- tions and the two can’t be experimentally distinguished from each other. The centroid of the IVSGMR was roughly predicted to be 15 MeV in Section 3.3. with a width of around 10 MeV [135]. Since this roughly matches the observed AL=1) distribution. the data strongly suggest that the bulk of the observed AL=0 strength is due to the excitation of the IVSGMR. To test this idea. the total IVSGMR cross section was calculated in DWBA using OBTDs from NORMOD and compared with the DWBA differential cross section per unit of B(GT) for GT transitions. The extracted B(GT) from the data corresponds to about 407? of the normal mode strength of the IVS- GMR, assuming that all experimental AL=0 strength is attributed to the IVSGMR and that. there is a proportionality between the IVSGMR cross section and IVSGMR strength. Although the uncertainties in this simple calculation are large. it shows that the extracted AL=1) strength is indeed likely due to the excitation of the IVSGMR. 139 5.4.2 QRPA calculations 0 o a n n o r ‘ QRPA calculatlons for the GT and dipole strength distrlbutions 1n 1")0S1u(t.‘5He) have been provided by Vadim Rodin's group to complement the calculations for 1' v ‘ ‘ . o . . 100.\d(‘3He.t). 'Ihese calculatlons 111clude. the effect of deformation. Table 4.14 111 Chapter 4 lists important parameters used in the reaction calculations. Individual states for the B(GT) and dipole transitions (.IW=0_,1_.2_) are shown in Figure 5.24. The. calculations incorporate three different values of K. which is a. good quantum number in a deformed nucleus (.1 is not). and the three colors corre- spond to these three values of K. In the plot of B(GT). the GT strength has not yet been quenched. The calculations are smeared to represent spreading in the strength distribution. which is not accounted for in the QRPA. and put into 1 .\IeV bins. Below 4 .\IeV. the GT calculations were smeared with the experimental resolution and re-binned into 1-;\IeV-wide bins. Above 4 MeV. the calculations were. smeared with Gaussians (FWHM = 4.7 MeV) so that the width of the GTR excited via the ”UM-M31191) reaction roughly matches the data. The dipole smearing widths (FVVHM = 3.5 MeV) were chosen to match those. used for the 1’5()1\'d(3He.t) data. Figure 5.25 shows the distrilmtion and cunmlative B(GT) strength. Figure 5.25 compares all of the experimental AL=0 strength to the GT strength distribution from QRPA. Because of reasons discussed in Section 5.4.1. no conclusions can be drawn about the validity of the QRPA calculations in terms of the total GT strength found. since most of the AL=0 strength found is likely due to the IVSGMR. It is notable that the summed B(GT) strength predicted in the 0-1 MeV energy bin does match the data. but a single strong GT transition predicted in QRPA was not seen in the data. Figure 5.26 compares the distribution and cumulative dipole cross section for the data and QRPA. As no proportionality between cross section and dipole strength has 140 0.08 5 ~ ~~— »——--—~~-»—. 7. 7 ,n . . 1 p . [ NA ‘ ffi *5 ‘ “‘— ’—‘ 5 K9 E : 350.07 [ — QRPA1 [ .5 09 3 - QRPA1 0.06 —— QRPAZ '51 0-3 g c 0.7 [- 0.05 9 1 0.04 i 2 i : 8. 0.5 0.03 {— [ :5 0.4 E 1 . 0.02 E : ° 0'3 '5 1 0.2 . 1| 0.01 : I [ 01 1 I t O ;JdJJA ' l I ......LL... 0 1 _, A_._A_‘A_i o 5 1o 15 20 25 30 o 5 1o 15 20 25 30 150 150 Ex( Pm) (MeV) Ex( Pm) (MeV) 1.2 ; ~ 44 Q4 _ . -..-__.___-_ .. NA _ — QRPA1 NE 1 l E . 5 g — QRPAZ ... [ ‘61 0'3 '” - QRPA3 g’ 0.8 c l 2 g 0.25 I. ‘3' 0.6 1 3 02 ‘1 6 1 '5 . o. 9 0.4 ~ =5 ‘0 [ I ' N 0.2 [ 1 “Ill til [[7 [[ j 0 ._ ._.1 A...‘ .....2. ., .. 1 ._‘_._x-.1_1 _. A _- l o 5 1o 15 o 25 30 o 5 1o 15 20 257—310 Ex1‘50Pm) (MeV) Ex1‘50Pm) (MeV) Figure 5.24: Raw QRPA calculations for 150Sm(t,3He). Top left: GT strength. top right: 0— dipole cross sections and strength. bottom left: 1_ dipole cross sections and strength, bottom right: 2— dipole cross sections and strength. The three colors represent three different values for K: black represents K=0. red represents K=1. and blue represents K=2. No quenching has (yet) been applied to the calculations of GT strength. 141 been established. we can only compare the shapes of cross section distribution from data with that of the calculated strength distribution. VVl‘iile the dipole cross section is correctly predicted to exist entirely below 15 MeV in excitation energy. the experimmrtal and theoretical distributions are somewhat different. The total QRPA strength distribution consists of two overlapping bumps: a small bump near 3.5 MeV and a larger one around 8.5 MeV. Dipole distributions from the data show a large bump around 2 MeV and a smaller bump around 7 MeV. 142 . 0 04 ~ 0 0 0 F: 14 E 0 :0 data 0 ES‘HZF “ {D QRPA ['0 1 r 7. > 0 1 0'8 :0 QRPAX3 e41 : 3 b 7 6 0.6 7 00; O t O CO 0 O 02 3 0031+ 0'3899 _ a DD 000 oo o 5 1o 15 20 25 E,.(150 Pm) (MeV) 18 cumulative B(GT) 8 6 4 2 O O 5 1110‘ A A 71157 A A 727077 7275 5x050 Pm) (MeV) Figure 5.25: Extracted Gamow-Teller strength in 15OPm via 150Sm(t.3He) compared with QRPA calculations. Data is shown in black. QRPA calculations in red. and in the top plot the QRPA is scaled by a factor of 30. The excess strength seen in data is attributed to the population of the IVSGMR resonance. 143 A 3 - me 4~— 4 3 a , o. E E DD 0 data ‘ ‘5' E 2.5 [ a [:1 total QRPA -; 2.5% .g . D v 1-QRPA 3 U : n O-QRPA to Q) [ DD :41 v, 1.5 . D 1.5; 3 c: 9 1 3 L) J '9 05 l o - ;* .9- :8 U . I. I! O _-_-__ ._.d ._ - 1 o 5 1o 15 20 25 E x(‘50 Pm) (MeV) 3 25 ———4- w; 25 q, £225 ; — data 22.5 9, B 20 3 ‘ 20 U Q‘— : Q. ‘5 5175 I 175 m m\ . . m >"ED 15 15 g 171 “125 125 3 _u_, - LO 3 . H- E78; 10 3 1o :- 31: 75 E 75 7'7“ U\ 7 i ' 3 (._.: 5 5 J’ 2.5 2.5 0 L.-nfih-r .-A_i . . -_u - 1-.. . r . 1.2.-. O o 5 1o 15 20 25 E x(150 Pm) (MeV) Figure 5.26: Extracted dipole cross sections in 1501’11‘1 via 150Sm(t.3He) compared with QRPA calculations. 144 Chapter 6 Application to 21x56 decay 6.1 Low-lying states and the SSD hypothesis Charge-exchange experiments can provide constraints for theory calculations aimed at modeling H13 decay processes. Both experiments discussed in this thesis populated states in l'501’111. A 2123.3 decay transition. under the single-state dominance hypoth- esis (SSD) [37]. is governed by a virtual two—step transition connecting the initial and final ground states through the first 1+ state in the intermediate odd-odd nucleus. All of the 21/12’3 decay strength is assumed to travel through this state and other inter- mediate states can be ignored. If the SSD hypothesis is not used. contributions from the intermediate states in Equation 2.6 can add either constructively or destructively. and these. phases must be accounted for within the calculation. The low-lying state in 150F111 as populated from 150.\.'d is centered at 0.11 MeV in excitation energy and has a B(GT) of 0.1334 i 0.0213 associated with it. A combina- tion of high level density and poorer resolution hindered our ability to distinguish the population of this state in the 150Sm(t.3He) (‘lirectioIL However. a small amount of GT strength (0.0195 i 0.0071) has been associated with the 0—300 keV region and ap- pears to peak between 0.1-0.2 .\leV. It is reasonable to assume that this GT strength 145 is due to the population of the 0.11 MeV state in 150.\'d(3He.t) (given the 50—keV systematic error in the excitation energy in the 150Sm(t.3He) data). However. there is also some GT strength located at 0.19 MeV in the 150311 data. and the effect. of a possible overlap of this state with the 1'50 Sm data must be checked with theoretical techniques. The 211,153,113 decay half life will be calculated here under the SSD with the assumption that the strength between 100-200 keV in the 150Sm(t.3He) data matches the 0.11 MeV state in the 150Nd(3He.t) data. r o o a 10013111 from both directions. It is Figure 6.1 shows low-lying GT strength in difficult to draw many conclusions here because of 150Pm’s high level density. the 1- . . . . 150 1 3 oO-keV uncertainty 111 the energy calibration for the ‘ Sm(t. He) data. and the differences in excitation energy bins between the two experiments. In other. similar analyses [67. 138]. GT strengths have been superimposed over a much larger energy range, but. since much of the “GT” strength in the (LJHe) direction is likely from the IVSGMR. this would not be appropriate in this situation. 6.2 Calculating the Zufifl decay half life in the SSD Four quantities are needed to calculate the 21113.3 decay half-life: the phase space. factor G2”. a sum of several energies (the denominator of Equation 2.6). and the extracted B(GT)s for the 0.11 MeV state from the 150Sin(t.3He) and 150.\'d(3He.t) experiments. The phase space factor of 1.2 ><10716 was taken from [30]. The en- ergy denominator includes the Q-value for 1317 decay. the excitation energy of the intermediate state. and the energy difference between the ground states of the initial and intermediate nuclei. These values are 3.3677 MeV. 0.11 .\IeV. and 0.024 MeV respectively. The double GT nuclear matrix element is then 0.029 i 0.006 and the 211/313 decay half life is 10.0 :1: 3.7 X 1018 years (see Table 6.1). The currently recommended value of 8.2 i 0.9 ><1018 years from Barabash [12] 146 0.16 55 0.14 '— 0 150Sm(t,3He) 0.12 O 150Nd(3He,t) 1 ITTTI 0.08 3 0.06 :— 0.04 E— 0.02 0.5 0.75 1 1.25 1. 1.75 2 1:x (lSOPm) (MeV) 4 O 0.25 Figure 6.1: B( GT) strength in 150F111 at low energies from both experiments. Note that binning is different for the two experiments: the points from 1SONd are from a peak-by-peak analysis, while the. 1508111 data have been put into 300 keV bins. B(GT) from 150Sm B(GT) from 1'SUNd M2,, Tl/2 (Y) 0.0194i 0.0070 0.1344 3: 0.0203 002893200056 10.0 i3.7 X1018 Table 6.1: Calculation of the 2i/J’d decay matrix element assuming single-state dom- inance. The energy dt—‘nominator of Equation 2.6 is calculated to be 1.77 .\le\" and is discussed in the text. 147 is consistent. with our result. as is the recommended value from \\ DC of 7.9i0.7 X1018 years [140]. The error in the extracted half-life from the CE data, (assuming the SSD hypothesis) would be significantly reduced if the error in the 150Sin—+150I’m B(GT) could be decreased. This would require. a high-rate. high-resolution (n,p)—type CE experiment and could benefit by the addition of ‘7—1'ay (‘letectors for coincidence measurements so that. the excitation of the 110-keV state can be unambiguously observed. We can conclude from our measurements that the SSD hypothesis is a plausible explanation for the 21/131‘3 decay half life of 150 Nd. However. it. is important to note that the transition seen in the 100-200 keV bin in the 15()Siii(t.3He) data may not be the 110 keV state seen in the 150Nd(3He.t) data. because of the 50-keV systematic uncertainty in the 150Sm(t.3He) excitation energy and the existence of a small state seen at 190 keV in the 150Nd(3He.t) data. In this case the SSD hypothesis cannot. be applied. Furthermore. the possibility of many higher-lying states c(_)iitributing constructively or destructively to the half life cannot be ruled out. Future theoretical work, such as that. by V. Rodin presented in the previous chapters. will hopefully provide further insights. 148 Chapter 7 Conclusions and Outlook Cl’large-exchange reactions are an effective tool to probe the two lu'anches of a double beta decay transition. The 1'50N('1(3He,t)150Pm and 15()Slll(t.3H(‘)150Plll reactions , _ . 150 . . . 7. . , have been used to populate states 111 Pm. the intermediate nucleus 111 the decay of 150Nd to 1508111. Doing the experiment at intermediate energies (over 1()() MeV/u) allowed for the extraction of the GT strength distribution as well as cross sections for dipole and quadrupole transitions. Comparing the exact location of states between the two experiments is very difficult in this work because of the high level density in 150F111 and the differences in energy resolution. but the results are important for constraining theoretical models of both the. 21153;} and (ll/3.3 decays of 150 Nd. Tables and figures of the extracted GT strength distributions and dipole. and (pimlrupole cross section distributions in 150F111 have been presented for excitation energy ranges of 0—30 .\IeV from the 150Nd target and ()—26 MeV from the 1508111 target. A strong GT state with low excitation energy has been identified in the 1"r)(i).\'d(3He,t) experiment, and a. small amount of strength is seen in the same location in the 15USm(t.3He) experinu—‘nt. If the single-state dominance hypothesis is applied. the resulting Quid decay half life is 10.0 i 3.67 X“)18 years, which is consistent with the accepted value from direct decay measurements. This. in conjunction with non- 149 zero GT strength present at several other locations in 150Pm. suggests that the SSD hypothesis needs to be carefully examined in this case. Reference [39] states that “[unless] there is an unknown 1+ low-lying state of 150F111. the experimental measurement should confirm [that the higher—state dominance (HSD) hypothesis reg- ulates the] Quid decay of 150Nd." \Vhile this low-lying 1+ has been shown to exist in this work and the SSD seems to adequately explain the 211.33 decay half life. the contribution of these higher-lying GT states cannot be ruled out. Giant resonances have been identified in regions of higher excitation energy in the. 1'50Nd(3He.t) experiment. including the IAS. GTR. IVSGDR. and what may be the tail of the IVSGMR. In the 150Sm(t.3He) experiment. the vast majority of the AL=0 strength seen is very likely due. to the population of the IVSGMR rather than GT excitations. Because the two types of excitations have similar angular distributions. they are not experimentally separable under current experimental configurations. The two experiments discussed in this work provide a new basis on which to test calculations of theoretical nuclear matrix elements for both the {Zr/.33 and the - 150 . . . . . . - OI/JJ’ decay of Nd. Those NNlEs can then be used to design the next generation of 011de decay direct counting experiments. such as SNO+ [141]. DCBA [33]. and SuperNEMO [‘2]. and any positive signals from those will again use the NMEs to calculate the Majorana neutrino mass. Collab<_)rative efforts (see. Section 2.3.4) to systematically measure GT transitions in dd decay nuclei are underway and have borne fruit. 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