Dani MS’S’WV LFB‘RTARY Michigan State University This is to certify that the dissertation entitled COLLECTIVITY IN LIGHT NEUTRON-RICH NUCLEI NEAR N=20: INTERMEDIATE-ENERGY COULOMB EXCITATION PhD. OF 32“”Mg, 3536M, AND 37sr presented by Jennifer Anne Church has been accepted towards fulfillment of the requirements for the degree in Physics T. Whats Major Professor’s‘Signature My: /02 . Date MSU is an Affirmative Action/Equal Opportunity Institution 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 6/01 cJCIRCIDateDuepss-ots COLLECTIVITY IN LIGHT NEUTRON-RICH NUCLEI NEAR N220: INTERMEDIATE-ENERGY COULOMB EXCITATION OF 32’34Mg, 35,36A1 AND 378i. By Jennifer Anne Church A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 2003 ABSTRACT COLLECTIVITY IN LIGHT NEUTRON-RICH NUCLEI NEAR N.—.20: INTERMEDIATE-ENERGY COULOMB EXCITATION OF 32’34Mg, 35’36Al AND 378i. By Jennifer Anne Church Collectivity in the neutron-rich nuclei 32’34Mg, 35MA] and 37Si has been studied via intermediate-energy Coulomb excitation in an early experiment at the National Superconducting Cyclotron Laboratory’s Coupled Cyclotron Facility. Reduced tran- sition probabilities and quadrupole deformation parameters were extracted from the measurements. 32"3“Mg are members of a group of nuclei near N = 20 which exhibit collective characteristics unexpected for nuclei at or near a shell closure. The observations are successfully reproduced by shell model calculations that allow a Zhw intruder config- uration as the ground state, in which 2 neutrons are promoted from the Vld3/2 level to the V1f7/2 level. Models based on valence spaces limited to the Chi» configuration which describe the stable nuclei well are unsuccessful in calculating observables for these nuclei. Because of this inversion of the Ohw and 2m.) configurations in energy, the group has been named the “Island of Inversion,” and is predicted by the 2m.) shell model to extend to 10 S Z S 12 and 20 S N S 22. The possibility for static shape deformation has also been considered, and models utilizing a deformed potential are successful in calculating excited state energies and reduced quadrupole transition probabilities for this group. Recently, the neutron boundary has been predicted to extend to N = 24 by the 3d — pf Monte Carlo Shell Model, and several mean-field calculations predict shape coexistence for 32Mg. The energy of the first 2+ state in 32Mg was measured to be 885(18) keV, and found to be in agreement with the adOpted value of 885.5(7) keV. A second gamma- ray at 1436 keV was also observed with low statistics. The reduced electric quadrupole transition probability, B(E2; 0;, —> 2?), of 447(57) e2fm4 and a Iflzl value of 0.51(3) were extracted from the measurements without consideration of feeding from the 2321 keV state. A B (E2; 0; , —+ 21+) value of 328(48) ezfm“ and a resulting quadrupole deformation parameter I B2| 2 O.42(3) were obtained after consideration of a minimum correction for feeding of the 885 keV state by the 2321 keV state via the 1436 keV gamma-ray. Both values are in agreement with the adopted value, and in contrast with the measurement by Chisté et al. which yielded 622(90) ezfm4 for the excitation strength. Our measurement agrees with the calculations of the 2hw shell model. A first excited-state energy of 659(14) keV for 34 Mg was also measured via Coulomb excitation. The value extracted for the reduced quadrupole transition probability of 541(102) ezfm4 with a corresponding lfigl value of 054(5) indicates that 34Mg is more collective than 32Mg as expected for a nucleus away from the closed shell. However, the value is also considerably higher than that calculated by the Ohw shell model. One other observed value for the B(E2; 0;, —> 2;") has been reported by Iwasaki et al. of 631(126) e2fm4. Our measurement is slightly lower than this. Both are in agreement with the Zhw shell model calculations. The remainder of the constituents of the 34Mg cocktail beam were also observed. Gamma-rays were observed at 985(21) keV for 35Al, 1437(30) keV for 37Si, and 647(14) keV and 967(20) keV for 36Al. Because there are no adopted values for the spin and parity of the odd-nucleon nuclei 35"‘6A1 and 37Si, J " values were calculated by the OXBASH shell model. Maximum values were then obtained for E1, M1, and E2 reduced transition probabilities and compared to listed recommended upper limits. The B(7r/\) values for 36A] were previously unmeasured. E1, M1 and E2 cannot be excluded as possibilities for the transitions in all three nuclei from our analysis. for my mom and dad, and my brother dan iv ACKNOWLEDGMENTS I acknowledge the work of the Gamma group. It was the idea of Thomas Glas- macher and Boris Pritychenko to investigate the Island of Inversion in depth. The graduate students in no particular order helped perform and set up the experiment: Ben Perry, Chris Campbell, Dan-Cristian Dinca, Katie Yurkewicz (Miller), Heather Olliver, and Alexandre Obertelli. Katie Miller wrote the Physica portion of the cal- ibration routines, and Heather Olliver and I compared adaptations of Boris Prity- chenko and Heiko Scheit’s angular distribution codes as a double check. Heather also gave me the m-state plots shown in the figure. Ben Perry dis- and reassembled the APEX N aI(Tl) array and took the great majority of the calibration data during the final pre—experiment calibration (there were 4 months or so of pre-experiment cal— ibrations). (I wish the graduate students the best of luck. You will need it). The post doctoral researchers Zhiqiang Hu, Wilhelm Mueller and Joachim Enders worked on the electronics. Zhiqiang Hu wrote the SpecTcl code which was later adjusted. Alexandra Gade made comments pertaining to the thesis itself. Robert Janssens (Argonne Nat. Lab) had the idea for the bismuth target. Dave Sanderson and Andy Thulin worked with me setting up the vacuum system. This was one of the first experiments to utilize the Coupled Cyclotron Facility and would not have been possible without it. It took a lot of hard work by too many people to list to complete the upgrade. Their work is duly acknowledged. Alex Brown, Gregers Hansen, Bernard Pope, Wayne Repko, and Horace Smith participated as members of my thesis committee. Now I move on to people I would like to personally thank. Thanks to four all around nice guys: Len Morris for working with me on the detector support design, Dale Smith for all the help with the phone dialer, Jim Wagner for fixing the hydraulics and investigating the ticking in the ceiling, and Jon Bonofiglio for the Malt-O-Meal and the carbon stripper mechanism. A special thank you to the computer group at the NSCL: Ron, Jay, Barb, Milt and the helproom staff. Thanks Milt for being strangely normal. Thanks Katie Miller for organizing a great get- together and the help with the thesis template, and Pat Lofy and Michelle Ouellette for all the rides to the airport. Thank you Professor Pollack, Professor Kovacs and Debbie Simmons for all of your understanding and support starting with the divorce, and continuing until even now. Thank you to my friends. Thanks Waz, Karen, Jen, and Cherami for calming me down and helping me through the hard parts, and Chris, AJ, and Jon for being there after my divorce. To Chung’s Taekwondo, thanks for letting me take my frustrations out on you all. (I hope you find out someday exactly how Grandmother Chung got to be so fast.) When a person grows up in Westwood, California, (population 2000, elevation 5000 ft.), the entire faculty and staff of the schools she attends become her extended family (some of whom actually are her parents). So I thank my old teachers for their enduring support, Mr. Church, Senora Iglesia, Mr. Kindig, Mrs. Brydon, Mr. Bryant, Mr. Hasselwander, Mr. Clark and Mr. Growden. Thank you to some family friends, Arvid and Lynne. Thanks Frank and Bill (I think). Most of all I thank my family. My Mom and Dad, Patricia and Randall Church, my brother Daniel Church, my nephews Tyler and Nicolas, and my niece Alissa are irreplaceable. In this world of broken homes, and unsolid ground, my family has been my foundation, the solid ground under all of the mountains I climb. They remind me that sometimes my mountain is really just a rocky knoll somewhere out in the middle of Nevada, complete with dragonflies and aliens and stuff. Thanks Morn for always encouraging me to keep on, for being positive at all the right moments, you are the one who keeps me on track. Thanks Dad, for all the late night talks, for the philosophy and the perspective, you are the one who opens my eyes. Thanks Dan for being my biggest fan, you are the one who reminds me it is worth it. Thanks Tanya vi for trying to get me to be peppy (haha). Thanks Tara for always being supportive, no matter what. Thanks Ty and Nic and Ali for all the singing and laughing and goofing around (with Aunt Lucky). Thanks also to all the Uncles and Aunts and cousins (there are many) for just being there. Lastly, I give thanks to God (I) for getting me through this, and of course for all the nuclei—exotic, stable, and the rest. vii Contents 1 Motivation 1.1 The Atomic Nucleus ........................... 1.2 Electromagnetic Transitions and Collectivity .............. 1.2.1 Transition Strengths ....................... 1.2.2 Collectivity ............................ 1.3 Nuclear Models .............................. 1.3.1 Collective Models and Quadrupole Deformation ........ 1.3.2 Single-Particle Models ...................... 1.4 The Island of Inversion .......................... 1.4.1 Early Experiments ........................ 1.4.2 Early Theory ........................... 1.5 32Mg .................................... 1.5.1 Experimental Background .................... 1.5.2 Theory ............................... 1.6 34Mg .................................... 1.6.1 Experimental Background .................... 1.6.2 Theory ............................... 1.7 Experiment Objective .......................... 2 Method 2.1 Coulomb Excitation ............................ 2.1.1 Relativistic Coulomb Excitation at Intermediate Energies . . . 2.1.2 Angular Distributions ...................... 2.2 Experiment ................................ 2.2.1 The Rare-Isotope Beam ..................... 2.2.2 Coulomb Excitation Targets ................... 2.2.3 Particle and Gamma-ray Detection ............... 2.2.4 Coincidences and Constraint of the Minimum Impact Parameter ......................... 2.2.5 Data Flow ............................. 2.2.6 Consideration of Relativistic Effects ............... 3 Data Analysis 3.1 Characterization of the APEX NaI(Tl) Array ............. 3.1.1 Calibrations ............................ 3.1.2 In-beam Energy Calibration ................... viii mahwooHi-t 7 13 13 14 15 15 19 21 21 23 24 25 25 26 29 30 30 32 35 42 43 47 50 50 51 58 3.1.3 Energy Resolution ........................ 58 3.1.4 Efficiency ............................. 62 3.2 Gamma-ray and Particle Sorting ..................... 76 3.2.1 Projectile Nuclei ......................... 77 3.2.2 Time Cuts ............................. 78 3.3 Error Analysis ............................... 8O 4 Observations 82 4.1 Degraded 96Mo .............................. 82 4.2 The 48Ca Primary Beam ......................... 84 4.3 The Test Case 26Mg ........................... 86 4.4 32Mg and 197Au .............................. 90 4.4.1 Background and Compton Contribution ............ 91 4.4.2 19"An ................................ 92 4.4.3 32Mg ................................ 92 4.5 34Mg .................................... 96 4.5.1 Nuclear Contribution ....................... 98 4.6 35"36M and 37Si .............................. 98 4.6.1 37Si ................................. 101 4.6.2 35Al ................................ 101 4.6.3 36A1 ................................ 103 4.6.4 Upper Limits ........................... 105 5 Discussion 107 5.1 The Island-of-Inversion Nuclei 32’34Mg .................. 107 5.1.1 32Mg ................................ 107 5.1.2 34Mg ................................ 109 5.2 3536Al and 37Si .............................. 110 5.3 Summary ................................. 112 A Weisskopf Single-Particle Estimates 115 B Energy Resolutions of the APEX NaI(Tl) Detectors 118 Bibliography 122 ix List of Figures 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 3.1 3.2 3.3 3.4 3.5 Simplified chart of the nuclides. ..................... Energy levels after spin-orbit splitting. ................. Ohm and Zhw configurations for 20 neutrons. .............. N ilsson energy levels for light nuclei. .................. The Island of Inversion as predicted by the 25w shell model. ..... Diagram of the Coulomb excitation reaction ............... The Coupled Cyclotron Facility ...................... Particle identification at the focal plane of the A1900 fragment separator. Experiment apparatus. .......................... The APEX NaI(Tl) Array ......................... The zero degree detector and positioning frame. ............ Diagram of the basic electronic setup for the APEX NaI(Tl) array. . . Electronics layout for the zero degree detector .............. Diagram of the trigger timing ....................... Energy spectra for 32Mg demonstrating the Doppler reconstruction. . 88Y position versus energy spectra demonstrating the Doppler recon- struction. ................................. Position calibration setup for the APEX N aI(Tl) array. ........ Position calibration for a single N aI(Tl) bar. .............. Position spectra for the central region of a single NaI(Tl) detector. . . Example of the position-dependent energy calibration for one detector. Reconstructed energy spectra for the sources 88Y and 137Cs as mea- sured by one slice of one of the APEX NaI(Tl) array detectors. . . . . 16 26 31 33 36 37 41 43 44 45 46 47 52 53 54 56 57 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 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 5.1 5.2 Calibrated and uncalibrated position vs. energy spectra. ....... 58 Energy resolution versus energy for the APEX N aI(Tl) array ...... 60 The virtual efficiency rings for the APEX N aI(Tl) array. ....... 63 GEANT geometry. ............................ 68 The efficiency curve for the APEX N aI(Tl) array. ........... 71 Simulated thresholds in the efficiency curve. .............. 72 Angular distributions of Coulomb de-excitation 'y—rays of 26Mg, 32Mg, and 34Mg. ................................. 73 Photoabsorption cross sections versus 7-ray energy. .......... 76 Particle identification contours on the energy-loss versus time-of-flight spectrum taken by the zero degree detector. .............. 78 Prompt time cuts .............................. 79 Gamma-rays in coincidence with 96M0 and fit. ............. 83 48Ca energy spectrum. .......................... 84 Low energy 7-ray background ....................... 87 Gamma rays in coincidence with 26Mg, 32Mg, and 34Mg. ....... 89 54Mn and 137Cs energy spectra showing Compton edges ......... 90 Energy spectrum and fit for 197Au. ................... 93 32Mg energy spectrum and fit ....................... 94 Gamma-rays in coincidence with 34Mg. ................. 96 ECIS calculation for 34Mg ......................... 99 Energy spectra in coincidence with 35“Al and 37Si. .......... 100 3"’Si energy spectrum and fit ........................ 102 35Al energy spectrum and fit. ...................... 103 36A] energy spectrum and fit. ...................... 104 Adopted, measured and calculated B(E2; 0;, —> 21*) values for 32Mg. 108 Compilation of experimental results, theoretical calculations and adopted B(E2;0;, —+ 2?), ,82, and E(2l+) values for the magnesium isotopes fromN=10toN=24. ......................... 113 B.1 Energy resolutions of the individual APEX NaI(Tl) detectors. . . . . 119 IMAGES IN THIS DISSERTATION ARE PRESENTED IN COLOR. xii List of Tables 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 A.1 8.1 B2 Beam parameters .............................. Coulomb excitation of the secondary targets ............... Minimum impact parameters. ...................... Calibration sources used for the position, position-dependent energy and efficiency calibrations. ........................ Energy resolutions of the APEX N aI(Tl) array. ............ Photoabsorption fit coefficiencts ...................... Final efficiencies for the APEX N aI(Tl) array. ............. Uncertainties for the 26Mg measurements. ............... Coulomb excitation cross section parameters for 26’32'34Mg and 96Mo. Extracted and adopted B(E2 T) values for 263234Mg and 96Mo. . . . . Coulomb excitation cross section parameters for 3536A] and 37Si. . . . Single-Particle Estimates ......................... Energy resolutions of the individual APEX N aI(Tl) detectors, January 2002 ..................................... Energy resolutions of the individual APEX NaI(Tl) detectors January 2002 including uncertainty due to the Doppler reconstruction. xiii 32 34 42 55 61 75 77 80 97 98 106 116 120 121 Chapter 1 Motivation 1.1 The Atomic Nucleus The field of nuclear physics has grown rapidly since the discovery of the atomic nucleus in the early 19008 [1—5]. As shown in Figure 1.1, a simplified chart of the nuclides, we now know of thousands of nuclei, only 270 of which are stable to decay [6]. Each nucleus is comprised of a number of protons (Z) and neutrons (N). These nucleons are quantum-mechanical particles and thus the nucleus itself is a quantum-mechanical many-body system. A current goal of nuclear physics is the unified microsc0pic description of all nuclei, stable and unstable [8]. This is approached by comparing experimental obser- vations to predictions made by theoretical models of the nucleus. Because the nuclear Hamiltonian is not known exactly, there are many nuclear models with many forms for the nuclear Hamiltonian. The comparison of the nuclear models to experimental observables is key to obtaining a realistic description of the nucleus. Most nuclear models accurately predict the quantum-mechanical observables for stable nuclei. However, with the production of rare-isotope beams, discrepancies be- tween what is expected from observed trends in the stable nuclei and the measured properties of neutron-rich nuclei have become obvious. For example, mass measure— 120 - 3 § Proton Number 40 20 _ I — 0 Ii 20 40 60 80 [00 120 l 40 160 180 Neutron Number Figure 1.1: A simplified chart of the nuclides. Proton number vs. neutron number are plotted. Nuclei in black are stable. The dashed lines indicate the magic numbers as predicted by the standard shell model of Mayer and Jensen [7]. Although they ex- tend beyond stability in the figure, recent experimental and theoretical developments indicate that these numbers of neutrons and protons may not indicate a closed shell for nuclei away from stability. ments of 31’32Na in 1975 by Thibault et 01. revealed larger two-neutron binding en- ergies than seen in trends for all of the other nuclei at the time [9]. The strong E1 transition in 11Be measured by Millener et al. in 1983 is only reproduced by integrat- ing out to an unprecedented nuclear radius of 35 fm [10]. Measurements by Tanihata et al. in 1985 showed 11Li to have a much larger interaction radius than expected from the comparison to other light nuclei [11]. In the context of the present work, the collectivity in 32Mg, as evident in the large B(E2) value measured by Motobayashi et al. in 1995, was not expected for a nucleus with a magic number of neutrons neither from the shell model nor the trends in the stable nuclei [12]. Although much work has already been done, data are still needed for nuclei that are closer to the proton- and neutron-drip lines in order to fully develop the framework for these rare isotopes. The problem is being attacked by making nuclei far from stability accessible through upgrades to existing accelerators such as the Coupled Cyclotron Facility (CCF) at the National Superconducting Cyclotron Laboratory (NSCL) [13], and by the proposed construction of new facilities such as the Rare Isotope Accelerator (RIA). 1.2 Electromagnetic Transitions and Collectivity In order to reconcile theory and experiment for a complete description of the atomic nucleus, it is ideal to compare nuclear structure properties that can be measured in an essentially model—independent way, and at the same time, be robustly predicted by theory. Electromagnetic transitions between nuclear states are particularly well-suited for this. Reduced electromagnetic transition probabilities are readily calculated by most models and can be measured via intermediate-energy Coulomb excitation [14]. 1.2.1 Transition Strengths For an electromagnetic transition, the multipolarity /\ of the operator 0,\ connecting the nuclear state Ii) to the state | f) depends on the angular momentum J and parity 1r selection rules, where the subscript z' indicates the initial state, and the subscript f indicates the final state, E indicates an electric transition and M indicates a magnetic transition [15]: [Ji—Jfl 3A3 Ji+Jf (1.1) and (_1)A) EA 7rf7r, = (1.2) (—1))‘+1, MA. The transition rate W for the selected multipole is approximated as W(7r)\; J,- —+ Jf) = 87r(x\+1) (19h); (n+1) M [(2A + 1)!!]2 7‘) B(“i Ji —’ Jr) (1-3) where E7 is the energy of the gamma ray emitted in the decay, and he is 197.329 MeV/fm, with the reduced transition probability 1 B(7T/\;Ji ‘9 Jf) :- 2.1+]. | [2) the reduced transition probability is denoted B(7r)\ I) and for the transition [2) —> | f) the reduced transition probability is denoted B(1r/\ T), and 2Jf+1 2J5+1 B(m T) = am i). (1.5) For a nucleus with even numbers of protons and neutrons, the ground state generally has spin and parity 0+ because all of the nucleons are paired. The spin and parity of the first excited state are usually 2+ for these even-even nuclei, although there are some exceptions. From the selection rules, the transition between these states is an electric quadrupole (E2) transition. 1.2.2 Collectivity A nuclear transition from an initial state to a final state may involve one nucleon, or it may involve the coherent motion of many of the nucleons in the nucleus. When the transition includes motion of many of the nucleons, the states are called collective states, and the transition is called a collective transition. The transition probability for a collective electromagnetic transition is generally larger than that for an electromag- netic transition in which only one particle changes its state. [16]. Thus the comparison of the measured reduced transition probability to an estimate for the single-particle reduced transition probability gives an indication of the “collectivity” of the nucleus. The estimates for single-particle transition rates are outlined in Appendix A. Gener- ally, at shell closures, nuclei are less collective than at mid-shell because the nucleons have to overcome a large energy gap to form an excited state. Even-even closed-shell nuclei tend to exhibit high 21+ excitation energies and low reduced electric quadrupole transition probabilities to their first 2+ state as compared to those at mid-shell [17]. 1.3 Nuclear Models Nuclear structure properties such as the reduced transition probability B (2r/\) can be calculated by two general types of models. In collective models such as the rotational model, the Hamiltonian describing the nucleus is written in terms of the coordinates of the nucleus as a whole. In single-particle models such as the nuclear shell model, the nuclear Hamiltonian is written in terms of the coordinates of the individual nucleons. 1.3.1 Collective Models and Quadrupole Deformation The collective models describe collective excitations as the deformation or a vibration of the nucleus as a whole. Models that describe the statically deformed nuclei are called rotational models while vibrations are described by vibrational models. Within the framework of the rotational model for a ground state band and an axially symmetric deformation, the reduced electric quadrupole transition probability is related to the intrinsic quadrupole moment Q0 of the nucleus through [18] BE2-0+ 2+ —— 5 2 2 16 ( tg.s._)1)__1_6—7_[:6Q0' () The quadrupole deformation parameter 62 is defined to be related directly to the intrinsic quadrupole moment Q0 of the nucleus. The relationship is written to first order in fig [18]: 3 = Z 2 . 1.7 Q0 m R0182 ( ) Combining 1.6 and 1.7, the quadrupole deformation parameter is related to the B (E2) value by w | — 4” B(E2) 1 (18) 2 _ 3 Z6123, ' where R0 = 1.2/1V3 fm and 82 : 1.44 MeV/fm [18]. The distortion parameter 6 is also used to describe deformation and is related to 62 through 6 2 0945/5. (1.9) It can also be shown that the same relationship between the quadrupole deformation parameter and the B(E2) value is obtained through the vibrational model of the nucleus [18]. Therefore, the reduced transition probability alone does not indicate whether or not the nucleus is deformed, but rather gives the strength of the collective excitation. A distinction can be made between rotational and vibrational collectivity by mea- suring the intrinsic quadrupole moment of the ground state of the nucleus directly. The ratio of the energy of the first 4+ state to the energy of the first 2+ state in the nucleus, E(4f)/E(2f) also gives an indication of the source of the collectivity [19]. From a model assumption of a rigid rotor, the E(4'1*) / E(2f) ratio is 3.3, while from the assumption of a harmonic vibration the ratio is 2. By convention negative ,82 values indicate nuclei that are flattened oblately, and positive values of 62 are nuclei that are elongated in a prolate shape. The shapes are depicted in Figure 1.4, where the energy levels are plotted as a function of the distortion parameter 6 in the Nilsson model which will be discussed in Section 1.3.2. The second type of nuclear model is the single-particle model. This type of nuclear model includes the degrees of freedom of the nucleons individually. Since the B(7r)\) value can be calculated from these models, the calculated degree of collectivity can be compared to experimental values. Subsequent conclusions drawn from the comparison oftentimes elucidate the single-particle structure of the nucleus. 1.3.2 Single-Particle Models Single-particle models attempt to describe the nucleus in terms of the individual nu- cleons. The Hamiltonian arises from the interaction of one nucleon with an average field generated by the remainder of the nucleons in the nucleus. The nuclear wave- functions 4),, are expressed in terms of the individual nucleons, and combinations of these functions form the antisymmetrized many-body basis states which are written as a Slater determinant [15], ¢1(1‘1) ¢1(I‘2) ¢1(l‘3) ¢2(1'1) ¢2(P2) (152(1‘3) 1 ‘I’k(r1,r2,l‘3a"' #34) = W (1-10) ¢A(1‘1) ¢A(r2) (PAPA) where the ¢,(rJ-) are the single-particle wave functions for A nucleons and r,- are the positions of the individual nucleons. The single-particle Schr6dinger equation is h(l‘i)¢k(l‘i) = €k¢k(rt), (1.11) where h(r,-) is the single-particle Hamiltonian and ck are the single-particle energy eigenvalues. The many-body Hamiltonian can then be written H = ihm) + i 17(r,-,rj) (1.12) 121 #321 where 17(r,-, r,) is the residual 2-body interaction. The variety of single-particle models in existence today are based on the early Shell Model of Mayer and Jensen [20,21]. This model is often referred to as the Stan— dard Shell Model [6]. In 1948, Mayer noticed from the experimental evidence of the time that nuclei with proton or neutron numbers 20, 50, 82 and 126 are particularly stable [7]. The following year, Mayer [20] and Jensen [21] added the spin-orbit term nl nlj , 1f... 2 ._ /:.... 2pl/Z p :;c ‘ E ' 2pm 1f7/2 1,, ZS ‘ " : ' = f V 3” ld : Z _ ~ ‘ 231/2 . 1d5/2 1 ————— 1pm p @ ...... @ 1173/2 ls ----- ls,,2 Harmonic Oscillator + Spin-orbit Figure 1.2: Energy level schemes for the harmonic oscillator including an 12 term (left) and including the spin-orbit splitting (right). Major shell gaps up to N = 28 are displayed. to the harmonic oscillator Hamiltonian. The addition of the spin-orbit term to the harmonic oscillator results in the splitting of the harmonic oscillator levels as is de- picted in Figure 1.2. Adding the spin-orbit term results in major gaps in energy for the proton and neutron numbers 2, 8, 20, 28, 50, 82, and 126. These numbers of nucleons are called the “magic” numbers. Although the simple shell model fits the overall observed trends in stable nuclei for measurements such as the first excited- state energy accurately, it has been found to diverge from the results of measurements on neutron-rich and neutron-deficient nuclei, see for example, Reference [9]. The single-particle energy levels are filled according to the Pauli exclusion princi- ple, and the neutrons (V) and protons (7r) fill separate sets of levels. Here the levels are labelled by 72,1 and j, where n is the major harmonic oscillator quantum number, I is the angular momentum quantum number and j = l + s is the total angular mo- Ohm Zita) 2pm 2pm 1f7/2 .. lf7,2 0.00 .000 .. 12d3/2 .. 12d3/2 S S one" 161;: one" 161;: 00 1 co 1 .... [71/2 .... 171/2 1pm 1173/2 .. 151/2 .. 151/2 Figure 1.3: Oliw and 2m configurations for 20 neutrons. The 2hw configuration in- cludes the promotion of two nucleons over the major energy gap in the standard shell model ordering. In this example, two neutrons are promoted over the N = 20 gap to the pf shell and two holes (open circles) are left in the 1d3/2 level. mentum, where sis the spin of the nucleon. A level is full when it has 2j +1 protons or neutrons since j has 2 j + lm-substates (projection of the angular momentum along 2), and the nucleon in each m state can be 3 = 1/2 or s = —1/2 (spin up or spin down) [15]. Configurations of nucleons with respect to this standard shell model energy level ordering are referred to as nhw configurations. The It refers to the number of neutrons or protons that have been promoted from the 3d shell to the pf shell, leaving 11 holes in the original level. For example, a 2m.) configuration promotes 2 neutrons or protons and leaves 2 holes in the original level. These are also referred to as n-particle n—hole (npnh) configurations. In stable nuclei the Ohw configuration generally describes the ground state well. However, for nuclei away from stability, this is not always the case. The idea of the 2hw configuration as the ground state is known as an intruder configuration and is depicted in Figure 1.3 [6]. A variety of single-particle models have been developed. The majority of them are shell model or mean-field approaches, although there are also attempts to go beyond the mean field. The models are currently written in a variety of forms with spheri— cal or deformed potentials, pairing and correlation energies, and even incorporating relativistic formalism in which the nucleons interact via meson exchange. For the shell models, the single-particle states are generated and grouped into core and valence parts. The nuclear wave functions are generated in the valence shell. For a state of total angular momentum J there are 2J + 1 M -substates, and the number of determinants for a given value of M gives the shell model dimension. The dimensions get quite large, around 10 billion for an exotic nucleus near N = 20, and truncation or other methods such as the Monte Carlo treatment are necessary to reduce the space such that it can be managed by the current computing technology [6,22]. The Hamiltonian is generally a combination of a spherical potential and an effective interaction that accounts for the interactions excluded by the truncation. Complete descriptions can be found in References [6,22]. One common mechanism utilizing the mean field are Hartree-Fock (HF) calcu- lations. HF is a calculus of variations approach involving self-consistent calculations of the eigenstates such that the majority of the interactions are contained in the single-particle Hamiltonian [15]. The goal is to make the residual Hamiltonian small. There are a variety of interactions that may be included in this treatment. Another mean-field approach is the relativistic mean field framework (RMF). In the RMF, the nucleons are treated as Dirac spinors and interact with each other via meson exchange [23]. It is possible to make calculations from a basis and potential defined for a statically deformed nucleus. The deformed shell model developed by Sven Nilsson [24] was the first to do this and is still used as a first approach to modelling a deformed nucleus [25]. The spin-orbit term is included in the deformed potential written here 10 for a quadrupole field: 1 /1 h(r) z hm) — 35W§r2 351mm, as) + a1 - s + b12 (1.13) where hgo is the harmonic oscillator potential (spherical), al - 3 includes the spin- orbit splitting, bl2 gives proper ordering to the single-particle states, and 6 is related to the quadrupole deformation parameter by 6 m 0.955 to first order [18]. Due to the statically deformed shape of such a nucleus, the quantum numbers in this model differ from the spherical shell model. N is the major harmonic oscillator number, 122 the 2 component of that number, and K1r is the projection of the total angular momentum J onto the symmetry axis. The projections of j, l and 8 onto the symmetry axis are Q, A and 2 respectively. K = A :l: E and I} = 1 / 2. The energy eigenvalues depend on the degree of axial deformation and are called N ilsson levels. They are labelled by K "[N n, A] and are displayed in what is known as a Nilsson diagram as a function of the axial quadrupole deformation. An example of such a diagram for light nuclei taken from [25] is displayed in Figure 1.4. Toward the prolate and oblate limits, the energies of the spherical levels begin to change, and in some cases the level ordering is actually inverted. When comparing the experimental data to the models, the starting point is gen- erally the spherical potential, Ohw ground-state, standard shell model standpoint. In such an extension of the trends for the stable nuclei, the nucleus is expected to be rather stable at and near a closed shell (a magic number of nucleons). Compared to a nucleus away from a closed shell, the first excited state energy should be higher and the probability for an electric quadrupole transition should be lower for the closed shell nucleus. Compared to the closed shell nucleus, two-neutron separation energies (82") for a nucleus with one more than the magic number of nucleons should drop off sharply [6]. 11 0011/21 1000521 4301312] ‘ [3123/2] - [3101/21 . [300712] I [20230.] - [3211121 . [333% [2025/21 [321312] [211 112] [330112] 4 [211 312] ‘ [101 112] ‘ [2201/21 [101 912] , . . . . . 001/21 -o.0 .04 —02 0.0 0.2 0.4 0.0 Oblate Spherical Prolate Figure 1.4: The N ilsson diagram pertinent to neutron levels for light nuclei near N : 20. Positive values of the distortion parameter 6 m 0.94532 to first order in the quadrupole deformation parameter 62 indicate a prolate shape while negative indicate an oblate shape by convention. Representative shapes are depicted below the Nilsson diagram. The diagram is taken from [25]. 12 1.4 The Island of Inversion From the viewpoint of the standard single-particle shell model, nuclei at the magic number N = 20 should be spherical and are expected to exhibit non—collective or solely vibrational characteristics [6]. As a reference point, note that the doubly-magic nucleus 4OCa has E(2f) = 3904.38(3) keV and B(E2; 0;, —-) 2]) 2 99(17) e2fm4 [17]. As the rare isotopes began to be experimentally explored, it was thought that the neutron-rich N = 20 isotones would also exhibit such characteristics. However, in 1975, experimental evidence to the contrary for a group of nuclei centered at Z = 11 and N = 21 began to appear [9]. 1.4.1 Early Experiments The first indication of an anomaly came with the results from Thibault’s mass mea- surements showing the two-neutron separation energies 52,, of 31MN a to differ from those expected for a Ohw ground state, with 32Na having a larger value than 31N a [9]. This discovery created a stir in the experimental and theory communities and much effort was subsequently put into the investigation of these nuclei. The fl‘n decay of 33Na in 1984 [26] and Coulomb excitation in 1995 [12] showed an anomalously low first excited state energy for 32Mg and it seemed as though this N = 20 closed shell nucleus was indeed collective. The suspicions were confirmed when a high reduced transition probability B(E2; 0;, —> 21+) was discovered for 32Mg [12]. The extent of the region was investigated through several Coulomb excitation experiments at the NSCL. 31Na was found to have a low-lying first excited state energy of 350 keV [27]. 28Ne [25] and 34Al [28] also showed evidence for low-lying states. Nuclei that did not demonstrate the apparent collective anomaly were the N = 20 nucleus 34Si [29] with a first excited-state energy of 3305(55) MeV and a B(E2 T) value of 85(33) ezfm". 13 1 .4.2 Early Theory The experimental discoveries in the region of the “Island of Inversion” [30] led to a substantial development in the theory community as well. The general consensus among the early models is that the anomalous B (E 2) values and first excited state energies can be explained by a shape deformation caused by an intruder state with a 2122:) configuration in which one pair of 1d3/2 neutrons is promoted to the 1f7/2 level. This intruder configuration would be a highly excited state in nuclei near stability and yet is found to be lower in energy than the state with the standard Ohm configuration for nuclei in the Island of Inversion. The expected configurations are in this way inverted. In 1975, Campi first proposed that the high two—neutron separation energies 82,, in the sodium isotopes were due to nuclear deformation. He showed that spherical Hartree-Fock calculations cannot explain the discontinuity in the curve of two-neutron separation energies 82,, in the sodium isotopes while deformed Hartree—Fock theory was successful [31]. In 1981, Wildenthal and Chung showed that the standard shell model with neutron d5/2 — 31/2 — d3/2 (3d) levels only as the valence space, could not accurately predict the 31Na and ”Mg mass measurements [32]. The same year, Watt added the 1f7/2 shell to the neutron sd valence space, restricted protons to 1d5/2 and by allowing (0+2)hw mixed configurations was able to reproduce the measured binding energies [33]. Poves and Retamosa added the 2103/2 energy level to the neutron basis and also restricted protons to the 1d5/2 orbit. They were able to produce the experimental results for the excited-state energy, and claim that 30Ne, 31Na, ”Mg are deformed due to the large correlation energy of these configurations [34]. The boundaries of this Island of Inversion were first predicted by Warburton, Becker and Brown in 1990 [30]. With a full sdrfp shell model space, and a weak coupling model, they diagonalized the nhw configurations separately and found that for the nuclei shown in Figure 1.5 , with 10 5 Z s 12 and 20 g N S 22, the 2hw configuration is lower in energy than the Ohw configuration which does not allow 14 promotions over the neutron 1d3/2 — 1f7/2 gap. They described the mechanisms for this level inversion to be a small reduction in the single-particle energy gap, increased neutron-neutron pairing energy Em, and an increase in the proton-neutron interaction energy Em [30]. 1.5 ”Mg Recent experiments confirm the early indications of the collectivity for the nuclei in the Island of Inversion. Support for the suspected static deformation in the region has been reported by Reference [35] in a comparison of the intrinsic quadrupole mo- ments extracted from heavy ion scattering and an ECIS calculation with spectroscopic quadrupole moments measured through [3 — N MR by Reference [36] for the sodium isotOpes. For the N = 20 nucleus, 31Na, the intrinsic quadrupole moment deduced from the scattering of 28“MN a differs from the spectroscopic quadrupole moment, in- dicating triaxiality as a possibility for the nucleus. In another recent development, the E(21+) value of 30Ne has been measured at 791(26) keV [37]. This is lower than that for ”Mg and indicates that the collectivity is enhanced with increasing neutron excess along N = 20 [37]. The first excited state energy and B(E2; 0;, —+ 21*) values for 34Mg have also been recently measured at RIKEN [38,39]. Of all of the recently researched nuclei in the Island of Inversion, ”Mg has been the most extensively in- vestigated nucleus both experimentally and through nuclear modelling. Because the focus of our work is the magnesium isotopes, 34Mg and the well-studied ”Mg, their experimental and theoretical background will be outlined in detail. 1.5.1 Experimental Background Measurements of the lowest energy level in ”Mg show it to be collective in nature. The first 2+ level at 885 keV is well-known [12, 25, 38—41] and found to be unexpectedly low-lying for an N = 20 closed-shell nucleus. The corresponding B(E2; 0;, ——> 2:“) 15 Stable Nuclei / N=20 325 33s 345 353 36S 378 383 398 MP 32p 33p 34}: 35p 36p 37p 38p +2698 288i 293i 305i 313i 323] 333i 348i 3531 368i 373i +1816 Hog +2261 27A1 28A] 29A] 30Al 31A] 32A] ”Al 34A] 35A] 36A] “ +2214 +854 +786 *** *** 26Mg 27Mg 28Mg 29Mg ”Mg ”Mg . ”Mg +2104 +780 -966 L1090 -685 *** 25Na 26Na 27Na 28Na 29Na 3"Na 11 a a 34Na +776 - -502 -.‘1295~ -427 +480 24Ne 25Ne 26Ne 27Ne 28Ne 29Ne 1 +609 ' -698 ' -891 {-128 23F 24F 25F 26F 27F 29F +1286 Z 22 O 230 240 N 4 Figure 1.5: The Island of Inversion as predicted by Warburton, Becker and Brown [30]. The nuclei in the shaded boxes are predicted to exhibit a 2hw ground state configuration. Excitation energies calculated from the weak-coupling model for the 2hw ground state relative to the Ohm state are indicated by the numbers in the boxes in keV. Nuclei with asterisks were beyond the calculation. Stable nuclei are outlined in bold. value is also found to be uncharacteristically large [12, 25, 38,39,41]. A second 'y-ray at 1436 keV has also been seen by Refs. [25,38,41—43], and found to be in coincidence with the 885 keV line in the B-decay studies of Refs. [26,40,42], the '7 — '7 coincidences of Yoneda et al. [38], and the 7—ray spectroscopy following fragmentation performed by Belleguic et al. [43]. These findings place the second level at 2321 keV and show that it feeds the 21+ state. This feeding has been treated differently in each of the B(E2 T) value measurements since its spin and parity are not well-known. In 1995, Motobayashi et at. were the first to measure the B(E2;0;, —-) 2+) for ”Mg at RIKEN [12]. Two scattering experiments were performed, 2°8Pb(”Mg,32 Mg'y), and 12C(”Mg,” Mgcy). The analysis assumes the second excited state in ”Mg to be at 2858 keV with J1r = 4*, as given in a shell-model calculation [44]. A coupled-channels calculation predicted that the 4+ —> 2+ transition would be 5% of the 2+ —-) 0+ in- tensity and so the Coulomb excitation cross section for the 2°8Pb(”Mg,” Mg'y) was reduced by 5%. To extract the B(E2 T) value, an average of four 62 ([30, which is the Coulomb deformation parameter) values were used in an ECIS79 [45] calculation for which EN 2 BC, one from an optical potential based on 170 [46,47], one extracted from a 208Pb(“°Ar,40 Ar7) run [48] and two extracted from the 12C run with the same optical potentials as the 17O and 208Pb(“°Ar,40 Ar'y). From the average 62 value of 0.512(44), the final B(E2;0;, —> 2+) value is 454(78) e2fm4. In 1999, the B(E2; 0;, —) 2+) for the nucleus was measured at the NSCL by Prity- chenko et al. [25] via the intermediate-energy Coulomb excitation 197Au(” Mg,” Mg'y). A RELEX coupled-channels calculation [49] excluded the possibility of a 2-step exci- tation of the 2321 keV level, and so the authors assumed that only direct excitation to that level could occur [27]. If the B (E2 T) value is constrained to the upper limits listed by Endt [50], the J’r for the 2321 keV level should be 1‘ or 2*. The feeding cross sections for the 1436 keV 7-ray for each of these transitions, E1 and E2 respec- tively, give essentially the same value, resulting in a correction of 24%. Without the correction, the B(E2;0;, ——> 2+) reported is 440(55) ezfm“ in agreement with [12], 17 and with it 333(70) e2fm4. N ilsson calculations also reported by this reference report fig 2 0.31 and predict that the ”Mg is prolate. In 2001, a significantly higher value for the B(E2; 0;, —) 2+) for ”Mg was measured via inelastic scattering by Chisté et al. [41]. Cross sections from both 208Pb(”Mg,” ng), and 12C(”Mg,” Mgry) were measured. Feeding contributions from the 2321 keV level were estimated by scaling the feeding cross-sections from all pos- sible states of 24Mg by the one-proton and one-neutron separation energies. This is based on the assumption that the total feeding cross-section should be proportional to the highest energy that still decays by 7-emission. They then extracted fig and fiN using an ECISQ4 [51] calculation with a combination of the optical potential of Barrette [47] and the proximity potential [52] and found the charge and mass defor— mations to be equal. Their extraction of the B (E2; 0;, —~> 21*) for ”Mg is reported as 622(90) e2fm4 [41] in disagreement with the adopted value of 390(70) e2fm4. A similar inelastic nuclear scattering experiment by Mittig using a 28Si target in place of the carbon also measured the 2321 keV state and claim that it is an excellent candidate for a J’r = 3‘ lplh state [53]. Although ”Mg is obviously collective, the B(E2 T) alone cannot determine whether or not it is deformed. The ratio E(4f)/E(2f) is one method which gives a good in- dication about the deformation. Higher-lying states were investigated by "Tr-ray spec- troscopy following fragmentation by Belleguic et al. [43, 54, 55] in 2001. Using BaF2 and Ge detectors, they report 'y rays at 885, 1430, 2870, and 1950 keV. Their find- ings agree with levels predicted by shell model calculations [56] and indicate that the levels up to J = 4 are prolately deformed. Since the 2870 keV photon is a result of a decay directly to the ground state, it is a candidate for a second 2+ level and indicates possible shape coexistence in ”Mg. They assign J"’r :2 4+ to the 2315 keV level. Two more experimental papers in 2001 report on ”Mg. Reference [39], whose main focus is ”Mg, lists a B(E2; 03:, —+ 2f) value of 449(53) ezfm4 with no feeding correction thus reaffirming [12, 25]. Yoneda et al. [38] report the coincidence of the 18 885 keV and the 1438 keV ’y-rays. From the above review, it can be seen that there are several different measurements for the B(E2 T) for ”Mg which are not all in agreement. Three similar values for the B(E2 T) are obtained from the three Coulomb excitation measurements [12, 25,39] while the consideration of feeding of the 885 keV state by the 2321 keV state lowers the value [25]. Finally, the value obtained via inelastic scattering by [41] is clearly larger than the others. 1.5.2 Theory The comparisons of the measured B(E2; 0;, —~> 2;) values, 454(78) e2fm4 [12], 440(55) ezfm“ [25], 449(53) ezfm4 [39], which are all similar, 333(70) e2fm4 [25] taking into account feeding of the 885 keV state by the 2321 keV state, and the high value of 622(90) ezfm4 [41], for ”Mg, to shell model, Hartree Fock, mean field, and beyond mean field calculations are presented in this section. The fig values for the measure- ments are 0.51, 0.50, 0.43, 0.6, and 0.51 respectively, and are extracted according to Equation 1.8. Shell Model The shell model calculations explain the anomaly at 10 3 Z g 12, 20 S N g 22 by including nhw promotions over the 1f7/2 — 1d3/2 gap in the ground state. The pairing interaction between neutrons seems to aid the deformation by creating valence particles and holes [44]. Wildenthal’s Ohw model, including only neutron sd levels in the valence space, uses an empirical effective interaction and predicts the energy of the first 2+ state to be greater than 2 MeV. Of the shell model calculations, this prediction is the furthest from the experimental values [32]. Another Ohw, sd—space calculation predicts E(2l+) = 1.69 MeV, and a B(E2; 0;, —> 2:) value of 180 ezfm" with the deformation parameter fig equal to 0.32 [57]. With the same model allowing 2hw promotions the 19 E(2f) value is 0.93 MeV, B(E2 T) value is 490 e2fm4 and the fig value 0.5 [57]. They also find that the 2p2h configuration dominates over the 4p4h. Warburton et al. also allowed 2hw configurations and obtain a B(E2 T) value of 340 ezfm4 with the fi2 value 0.44 [30]. It was also thought that mixing the Ohw and 2hw configurations would provide a better description of the ground state. The values obtained by Poves and Retamosa et at. with the full sd-pf valence, mixing (O+2)hw configurations, led to an E(2f) value of 0.3 MeV, B(E2 T) of 205 ezfm“ and fig of 0.34 [34]. Allowing 0 through 4hw promotions for the first time, Fukunishi predicted the first excited state energy to be 1.17 MeV, the B(E2 T) value to be 449 e2fm4 and the deformation parameter fig to be 0.51 [44]. Other efforts have been made to reduce the valence space through Monte Carlo treatments [22, 56]. The number of p-h excitations is not restricted, intruder mixing is enhanced and angular momentum projection is used to restore the total angular momentum. The calculated results for ”Mg are E(2f) = 0.885 MeV, a B(E2 T) = 454 e2fm4 and a fig = 0.51 [22,56], which are very close to the measured values. This calculation indicates shape coexistence, showing four minima in the potential energy surface, as compared to the two found in many of the mean field treatments (see below). The ratio E(4f)/E(2f) is predicted to be 2.55 by this model, which does not give preference to a rotational or vibrational configuration [22,56] . In summary, the shell model calculations allowing the maximal possible number of np-nh promotions and with the largest valence space give predictions closest to the ex- perimental values. In these models, the 2p2h configuration dominates a configuration which promotes 2 pairs of neutrons over the gap [56]. Mean Field and Beyond It has been suggested experimentally [43] by the existence of a possible second 2+ low-lying state that ”Mg may be a candidate for shape coexistence. Indeed in many cases, Mean-Field models agree and find more than one local minimum for the total 20 energy as a function of the deformation parameter in their calculations. For the Hartree Fock models there are over 80 different Skyrme parametrizations that may be included in the self-consistent calculations [58]. A few representative models are presented here. ”Mg is found to exhibit coexistence of prolate deformed and a spherical shapes in HF plus 10 different Skyrme interactions all described in [58], SLy4, SkO, SkT6, SkI4, Zo, SkI3, SkP, SkM*, SkIl, or SkO’. The shape preference between the interactions depends on the surface coefficient where a large coefficient favors a spherical shape and the unpaired calculations give rise to a prolate minimum [58]. A density-dependent monopole plus HF also predicts mixed spherical and prolate shapes with for the prolate state, a fig value of 0.54, and a B(E2 T) value of 507 e2fm4 [59]. The nucleus is found to be spherical (,62 = 0) with HF- Bogoliubov (HF B) +SLy4 in the transformed harmonic oscillator (THO) basis [60], and the deformed HFB calculation [61]. Relativistic Mean Field models predict a spherical shape [23, 62, 63]. Beyond the mean field, the Gogny force is used followed by the Angular Momentum Projection and Configuration Mixing (AMPCM) treatment revealing a mixture of prolate and oblate minima [64]. The quadrupole deformation parameter for the prolate minimum is calculated to be 0.36 by these models. 1.6 34Mg 1.6.1 Experimental Background ”Mg has eight more neutrons than the stable 26Mg, and because of this has been diffi- cult to access experimentally until recently. The first attempts to measure the first 2+ energy and B(E2; 0;, —> 21*) value for ”Mg were made at the N SCL in 1999 by Pri- tychenko et al. [25] via intermediate-energy Coulomb excitation 197Au(”Mg,” Mg 7) in inverse kinematics. Because of the limited number of projectile nuclei, Pritychenko et al. were limited statistically to placing only an upper limit on the B(E2 T) value of 21 670 e2fm4. Recently however, upgrades to the rare-isotope beam facilities have made it possible to access the more neutron-rich members of the Island of Inversion at significantly improved intensities. In February 2001, Yoneda et al. reported on the double fragmentation reaction used to access ”Mg at RIKEN [38]. From the initial fragmentation of a 40Ar pri- 9Be 36Si was selected and then in a second reaction fragmented on mary beam on anohter 9Be target. The de-excitation 7-rays from ”Mg were detected in 66 rect- angular N aI(Tl) crystals for 'y-ray detection and silicon telescopes for the particle identification. Yoneda et al. report a first excited-state energy of 660(10) keV, the lowest of the region, and a second "y-ray at 1460(20) keV which was observed at rather low statistics and give a tentative J’r assignment to the second state at 2120 keV of 4*. The ratio E(4f)/E(2f) is then 3.2, indicating that the nucleus may be statically deformed. In December 2001, Iwasaki et al. reported the B(E2;0;, —> 2;) value for ”Mg [39]. Iwasaki et 01. performed Coulomb excitation on 44.9 MeV/ nucleon /nuc34Mg also using 66 rectangular NaI(Tl) detectors and silicon telescopes for the experimen- tal apparatus. Iwasaki et (11. performed two experiments, one with a thick lead target for the Coulomb excitation 2°8Pb(”Mg,” Mg '7) and the other with a carbon target l2C(”Mg,” Mg '7) to determine the nuclear excitation contribution to the cross sec- tion. With the ECIS calculation they determine the deformation parameter fig to be 058(6) after averaging over four values obtained in a similar fashion to Motobayashi above. To extract the B(E2 T) value, an average of 4 fig (fic, which is the Coulomb deformation parameter) values were used in an ECIS79 [45] calculation, one from an optical potential based on 17O [46,47], one extracted from a 208Pb(4°Ar,40 Ar'y) run [48] and two extracted from the 12C run with the same optical potentials as the 17O and 4oAr. In the end, they report a first 2+ energy at 656(7) keV and the large B(E2; 0;, —+ 2+) value of 631(126) ezfm“. 22 1.6.2 Theory Comparisons of the measured E(2f), B(E2;0;, —> 2;), and fig values 0.660(10) MeV [38]; 0.656(7) MeV, 631(126) e2fm4, 058(6) [39]; and < 670 e2fm4 [25] for ”Mg to shell model, Hartree Fock, mean field, and beyond mean field calculations are presented in this section. Shell Model In the Ohm sd shell model of Ref. [57], the E(2fr ) value is calculated to be 1.09 MeV, the B(E2 T) value to be 375 ezfm4 , and the fig value to be 0.45. The same model with 2hw promotions to the fp shell gives an E(2f) value of 0.660 MeV, a B(E2 T) value of 655 e2fm4 , and a fig value of 0.6. In the Quantum Monte Carlo Shell Model (QMCSM) calculation for ”Mg [56], the first excited state energy E(2'f) is calculated to be 0.620 MeV, the B (E2 T) value to be 570 ezfm4 , and the quadrupole deformation parameter fig to be 0.55. Both the QMCSM and the 2hr» shell model predict that this nucleus is very collective and possibly statically deformed. However, while the 2hw model of Ref. [30] predicts the limit of the island to be ”Mg at N = 22, the QMCSM extends the line to N = 24 [56]. Mean Field and Beyond The Hartree-Fock calculations for ”Mg generally predict it to be prolate deformed with a fig value of 0.3-0.4 (HF+Skyrme) [58], 0.22 (DHFB) [61], 0.46 with a B(E2 T) value of 404 ezfm4 (density dependent monopole) [59]. Stoitsov predicts an oblate shape with a fig value of -0.1 (HFB+Sly4, THO) [60]. RMF models predict either a prolate deformed nucleus with a fig value of 0.3 [63] or one that is close to being spherical with the deformation parameter equal to 0.162 [62] and 0.17 [23]. The AMPCM predicts it to have the same deformation parameter as ”Mg, a value of 0.36 [64]. 23 In general, both shell model and mean field models calculate values for the de- formation parameter fig which indicate ”Mg is deformed and prolate in shape. The shell models predict a larger degree of collectivity for ”Mg than ”Mg while several of the mean field models above calculate similar values for the reduced transition probability and the quadrupole deformation parameter between the two nuclei. 1 . 7 Experiment Objective The focus of our experiment was the measurement of the energy of the first 2+ level and the reduced transition probability B(E2;0;, —+ 23‘) for the N = 22 nucleus ”Mg. The nucleus is predicted to be a boundary of the Island of Inversion by the 2hw shell model [30], and is expected to exhibit the 4p2h intruder state as its ground state configuration, with a large B(E2 T) value and a low first excited state energy [56]. In order to accomplish the measurement, we utilized intermediate-energy Coulomb excitation to populate the 2; state of ”Mg, and by measuring the de-excitation ’y-rays were able to extract the energy and probability for the quadrupole transition. The method also allows the simultaneous measurement of the de-excitation 'y-rays from the other nuclei in the secondary cocktail beam, and so we also set out to measure 35”Al and 37Si. A secondary goal was the measurement of the E(2'f) and B(E2 T) values, again via Coulomb excitation, for the N = 20 nucleus ”Mg. 26Mg and 96Mo served as test cases for the Coulomb—excitation studies. 24 Chapter 2 Method 2.1 Coulomb Excitation The technique of intermediate-energy Coulomb excitation is used in experimental nu- clear structure physics with exotic beams [14]. Through scattering, the interaction of a projectile and target via their Coulomb fields produces excited states in both pro- jectile and target nuclei. For pure Coulomb excitation, the subsequent de-excitation of the nuclei is an electromagnetic transition whose probability can be measured. These probabilities along with the 7-ray energies give important information about the energy level structure of both of the nuclei (Section 1.2). The process is ideal in that it is almost completely model-independent, though it does depend on the model of intermediate~energy Coulomb excitation, and through inverse kinematics provides a method with which to measure these electromagnetic properties for neutron-rich and neutron-deficient nuclei which cannot be made into long-lived targets. Coulomb excitation has been widely used, with particular success for the investigation of light neutron-rich isotopes, for example [25, 29, 65]. The developments by Winther and Alder [66] in which the semi-classical straight line approximation is adapted to rela- tivistic energies are necessary for the rare-isotope beams studied in the present work, and the main points of the theory are outlined here. Discussions and applications can 25 If m " Naif??? ’fi'f‘w'ii Zero Degree Detector [' NaI(Tl) 1 L Figure 2.1: A depiction of the Coulomb excitation process for ”Mg on 209Bi. The angle of emission, 0, of the 7-ray and the impact parameter, b are shown. The NaI (Tl) 7~ray detectors and particle detector (zero degree detector) are also shown for this experiment. be found in Refs. [12, 14, 27, 67—71]. Coulomb excitation was originally developed by considering the excitation of a target (Zg) in the field of a projectile (Z1) [3, 72]. However, it may also be thought of in the inverse situation, simply by exchanging the target nucleus and the projectile nucleus as in the following outline. Figure 2.1 depicts the Coulomb excitation setup. 2.1.1 Relativistic Coulomb Excitation at Intermediate Ener- gies Since intermediate-energy radioactive beams have velocities (v/c z 0.3-0.4) which are above the classical limit, relativistic effects must be considered both in the Coulomb excitation mechanism and, as outlined in Section 2.2.6, the detection of the de- excitation 'y-rays. The relativistic quantities fi and '7 are defined as 1 ”ZMW’ and a z (‘2’). (2.1) 26 In the relativistic limit, the adiabaticity parameter is the ratio of the collision time and the time of internal motion of the nucleus, {(b)_ _ wfz £71), (2.2) where w 2 Efi/h and E f,- is the difference in excitation energies between the final and initial states. The impact parameter b is b : %cot (Q?) , (2.3) where ZzZ1€2 : — 2.4 mm)2 ’ ( ) and mo is the reduced mass of the projectile and target, mo 2 mlmg / (m1 + mg). At intermediate beam energies the straight-line trajectory assumed for a relativistic projectile is modified by the recoil of the target. As a result, the impact parameter b is modified to 7ra0 b . +27 (2.5) In the theory of intermediate-energy Coulomb excitation, the Coulomb excitation cross section is written [66] 01—4} : (22:22) kaA— 1) (82(7I'A, J, —7) Jf)/€2) «Au we 9,.(6 (bmin))1 (2-6) where g.. A) ”bgnin ( ) (2.10) hc 82b . b _ mm 2ln(fif:) for A — 1 which shows the proportionality 0,-_,f O( B(7r/\). (2.11) Thus, the B(7r/\) can be extracted directly from the measurement of the Coulomb excitation cross section through detection of the number of de—excitation 7-rays. Ex- perimentally, this cross section is determined by (2.12) where N3; is the number of de-excitation ’y-rays emitted, N, is the number of target 28 nuclei, and Np is the number of projectile nuclei. The constraint of b such that the majority of the 'y-rays detected result from Coulomb excitation requires practical considerations which will be discussed in Section 2.2.4. 2.1.2 Angular Distributions Coulomb excitation results in a non-isotropic distribution of de-excitation 7-rays. This is important for detection considerations which will be discussed in Section 3.1.4. The derivation of the 'y-ray angular distribution may be found in [70,72] with final results presented here. The 'y-ray angular distribution is given by we...) = Z |a.,.(§)|29.<€)(—)“(: f” ’3) (2.13) I: even, p L, L’ I I k x{/\f ; I }Fk(L,L’,If,If)\/2k+1Pk(cos(0m))6L6U. where 9,, are the Winther and Alder functions of Equation 2.7, Pk(cos(6)) the Legen- dre polynomials, the 6L 6),: are Kronecker Delta functions, and the 7 — '7 correlation functions Fk(L, L’, 1,, 1;) are given by Fk(L,L’,Il,12) = (—)’”"2‘1\/(2k+1)(212+1)(2L+1)(2L’+1) (2.14) x L L’ k L L’ k 1 —1 0 12 I2 11 ' The relation is generally expressed mom.) = 2 aka. (cos (9....)). (2.15) I: even These angular distributions are folded with the intrinsic efficiency of the 'y-ray de- tectors for the final photopeak efficiency used in determining the number of v-rays 29 emitted from the Coulomb—excited state. 2.2 Experiment The experiment was performed at the National Superconducting Cyclotron Labo- ratory at Michigan State University and designated NSCL experiment 01017. The recent coupling of the two superconducting cyclotrons, the K500 cyclotron to the K1200 cyclotron, has made extremely exotic nuclei such as 34Mg accessible at sta- tistically significant intensities [13]. The facility layout is displayed in Figure 2.2. The primary beams are accelerated in the K500 cyclotron, and then injected into the K1200 cyclotron. Upon injection into the K1200 superconducting cyclotron they are stripped of the majority of their electrons, and subsequently accelerated to final energies. The advantage of the coupled system over the K1200 in stand-alone mode is the production of primary beams with greater intensities at higher energies. This results in a higher number of secondary fragment products and thus better statistics for experimenters. The fragment separator was also upgraded from the A1200 to the A1900 by increasing the angular acceptance of the beam line. As one of the first ex- periments to utilize beams produced by this new Coupled Cyclotron Facility (CCF), Coulomb excitation of three different secondary beams and two degraded primary beams was performed. The experiment apparatus was set up in the N3 vault. An array of 24 NaI(T1) detectors (the APEX array [73,74]) was used for 7-ray detection, a fast / slow plastic phoswich detector and PIN Si detectors for particle identification, and two parallel plate avalanche counters (PPACs) for beam location. 2.2.1 The Rare-Isotope Beam Two primary beams were used over the course of the experiment. 48Ca ions were produced in the room temperature ECR source ARTEMIS by heating solid 48Ca with a resistive heating element mounted radially on the plasma chamber [75]. The 30 ECR K500 A1900 N1 N2 N3 N4 7‘! A -\'1 ‘5. _ C: '1: ‘Tfiz‘ \jj \ \ 1 K1200 Primary ImageZ Focal Extended SI 82 S3 Target Plane Focal Plane Figure 2.2: The Coupled Cyclotron Facility at the NSCL. The experiment was per~ formed in the N3 vault. ions were first accelerated to 10 MeV/ nucleon in the K500 and then injected into the K1200 where they were immediately stripped to a charge state of 19+ by a 0.2 mg/cm2 12C foil. After stripping, the primary beam was accelerated to a final energy of 110 MeV/ nucleon. The radio frequency (RF) of the cyclotrons was 21 MHz (3. period of 47.5 ns) and the intensity of the “Ca primary beam was on average 15 pnA. 96Mo was also utilized as a test case for the experimental setup. The ions were produced by sputtering. They were stripped to a charge state of 38+, and accelerated to 110 MeV/ nucleon. Three separate secondary beams and the two degraded primary beams were stud- ied. The three cocktail beams were produced by fragmentation of the 48Ca19+ primary beam on a primary target of 9Be. The 9Be primary target was also used to degrade the 48Ca and 96Mo primary beams. The nuclei of interest were selected from the frag- mentation products with the A1900 fragment separator [76]. The beam characteristics (grouped by secondary beam) for 26Mg and ”Al; 32Mg and 33Al; 34Mg, 35'“Al and 37Si; and the degraded 96Mo and 48Ca are listed in Table 2.1. At the dispersive focal plane (Image 2) of the A1900, a 116 mg/cm2 aluminum wedge reduced the number of light nuclei that would reach the focal plane and physical slits allowed particles to be 31 Table 2.1: Secondary beam parameters for NSCL experiment 01017. Bp5 is the rigidity of the beamline to the N3 vault at the exit of the A1900. The thicknesses of the primary fragmentation target 9Be are given in mg/cmZ. The intensity is measured in the N3 experimental vault as is N B, the number of projectile nuclei. The amount of energy prior to interaction with the secondary target is also listed. Bp5 9Be Intensity (N3) Total N B Energy Nucleus (Tm) (mg / cmz) (particles / 3) (particles) (MeV / nucleon) 26Mg 2.822 376 5500 2.51 x 108 78.6 32Mg 3.531 587 150 5.25X106 81.1 34Mg 3.637 795 8 2.14 X 106 76.4 35Al 3.637 795 23 6.22 x106 84.2 36Al 3.637 795 11 3.06X106 79.8 37 Si 3.637 795 21 5.39 x 106 87.3 48C3. 3.127 587 9750 1.30X 108 71.2 96M0 2.723 376 14100 1.44 X 108 67.1 filtered by their momentum. Particle identification was accomplished by measuring energy loss in a PIN Si detector located at the focal plane, and the time-of-flight with respect to the K1200 cyclotron’s radio frequency as shown in Figure 2.3. After the de- sired nuclei were selected, they were guided to the N3 vault and energy loss was again recorded in a second PIN Si detector, which was retracted for the main experiment. Spectra from the the N3 PIN and the A1900 PIN were compared to ensure integrity of the beam. During one transmission test, the beam transmission was approximately 55% from the A1900 focal plane to the PIN Si detector in N3 for 34Mg with a primary beam intensity of 7 pnA and and a production rate of 0.36 pps/pnA. On average, the 34Mg cocktail composition in N3 was 11% 34Mg, 33% 35A], 22% 36Al, and 34% 37Si. 2.2.2 Coulomb Excitation Targets For a Coulomb excitation experiment, the ideal target has a high Z and is monoiso— topic. Since both projectile and target nuclei are excited in the process, the energy of the target’s de-excitation 7-rays, as well as its Coulomb excitation cross sections, must be considered. Because the target peaks are broadened in the frame of the 32 l l l l I -.1.l.l . I l l I I I"? l l 1 l l 1‘. .1 7.1 .‘l. l l l 1- 1.. .11"; 1; 100 200 300 400 500 600 700 800 900 time-of-flight [a.u.] 400 .— 350 — ‘3 . ill 30° .— . t‘ "‘5' 'fi #3 373i 5- ..’ v. ' 36 _. ~ "*- Al in: / 25° 7. * t '- }‘ ‘ @A//35Al @200 3"" ' “v .- .7 .‘ A E) : "‘9 5* 4‘ _ _ ”i Q - 0150;" *Q’* .’§ 100 ; 'I ' 50 — .I-v Figure 2.3: The particle identification at the focal plane of the A1900 fragment sepa rator for the 34Mg cocktail beam. The energ loss was measured with a 470 mg/cmz- thick PIN Si detector and is plotted versus time-of—flight in arbitrary units. Two rf pulses are shown and the nuclei selected for the Coulomb excitation experiment are labelled for one of the pulses only. projectile, they have the potential to interfere with the analysis of the projectile de- excitation 7—ray, especially if the Coulomb excitation cross section for the target is relatively large. If the energy does not interfere however, the energy and reduced transition probability obtained for the transition in the target through the analysis of the laboratory—frame spectrum can sometimes be a good test case. A common choice for the Coulomb excitation target is 197Au [25,28] for its high Z. 33 Table 2.2: The Coulomb excitation targets, cross sections, and expected detected number of target de-excitation 7-rays (fo, where d indicates detected versus emitted) are listed for each projectile nucleus. Thickness N pro,- Cross section Expected Target (mg/cmz) NW9“ Projectile (particles) (mb) N: 2093i 980 2.82 X 1021 26Mg 2.51 x 108 025(3) 45 34Mg 2.14x 106 040(4) 0.359 35A1 6.22 x106 0.47(5) 1.23 36Al 3.06x106 046(5) 0.599 37Si 5.39x106 0.51(5) 1.17 480a 1.30x 108 062(6) 33.9 197Au 968 2.96x1021 32Mg 5.25x106 20(2) 90 The number of target nuclei is determined by Ntarget = d- —1—— - NA, (2.16) M where d is the surface mass density of the target in g/cmz, M is the molar mass in mol/ g of the target material, and N A is Avogadro’s number. In this experiment, a target of 197Au was used for the 32Mg and 96Mo beams and a target of 209Bi for the 26'34Mg and 48Ca beams. 197Au has an adopted B(E2; 3/2+ —-) 7/2+) value of 4488(408) efifm4 for the 547.5(3) keV transition from the ground state to the first 7 / 2* state [77]. The expected cross section extracted from these values for this transition in the Coulomb excitation reaction with 3"’Mg (parameters are listed in Table 2.3) is 20(2) mb, which corresponds to an expected 90 counts to be detected in the photopeak. In contrast, 209Bi has an adopted B(E2; 9/2" —> 7/2‘) value of 80(9) e2fm4 for the 896.28(6) transition [78], yielding an expected cross section of 0.40(4) mb for the 34Mg reaction. Due to the small cross section, the number of counts in the 209Bi de-excitation photopeak was also expected to be small at 0.36 counts total in the reaction with 3“Mg spread over a range of 715 keV to 1256 keV. The small contribution is negligible in the estimation of the background for the analysis as 34 described in Section 4.2. The 209Bi was chosen so that the target 547.5(3) keV 'y-ray would not interfere with any photopeaks in the projectile frame with the anticipated 660(10) keV 34Mg photon [38]. The interference by the 547 keV 7-ray from the 197Au target in the projectile-frame spectrum of 32Mg can be seen in Figure 2.10. The number of target nuclei in both the 209Bi and 197Au targets used here are listed in Table 2.2, Coulomb excitation cross sections and expected number of de-excitation 7-rays. 2.2.3 Particle and Gamma-ray Detection After completing the isotope identification, the N3 PIN was retracted and the nuclei were focused through 2 PPACs onto the secondary target. Gamma rays produced from the resulting reactions were detected with an array of 24 position-sensitive NaI(Tl) detectors first used as part of the trigger for the Atlas Positron EXperiment (APEX) collaboration at Argonne National Laboratory [73,74] (the APEX array). The scat- tered particles were stopped in a 4-inch-diameter fast / slow plastic phoswich detector centered at zero degrees (ZDD) which is described below. The setup is displayed in Figure 2.4. Gamma-ray Detection In order to accomodate the low intensities expected for the 34Mg secondary beam, 7- ray detection was accomplished with an array of N aI(Tl) detectors. While segmented germanium detectors such as the SeGA array at the N SCL [79] have better energy resolution, the NaI(Tl) crystals are significantly more efficient. For a 660 keV ry— ray, SeGA has a photopeak efficiency of 3.6% for v/c = 0.35 with 8 detectors at a scattering angle of 90° and 7 detectors at a scattering angle of 37° [80] while the photopeak efficiency for a 'y-ray of the same energy and v/c of 0.36 for the Na.I(T1) array used here is 22%. The NaI(T1) array was recently brought to the NSCL from Argonne National 35 NaI(Tl) Array (APEX) l Seg. Ge Plastic Phoswich Detector - ‘ ' tv . .~ 5| . II. 1 film : I I II. __ It“ W “1.2.! Secondary Target E: Figure 2.4: The Coulomb excitation detector setup as explained in Section 2.2.3. Although segmented germanium detectors (a portion of SeGA [79]) remained in the vault after a previous experiment, they were not used for this experiment. Laboratory and cleaned and reassembled by BC. Perry [74]. The array is shown in Figure 2.5 and consists of 24 long, trapezoidal N aI(Tl) crystals with photomultiplier tubes (PMTs) at each end to facilitate position sensitivity. The dimensions of each crystal are 55.0 x 6.0 x 5.5(7.0) cm3 (L x H x W). They are encased in an evacuated 0.4 mm thick stainless steel housing. At each end of the crystal, there are 1.1 cm thick quartz windows that are 4.4 cm in diameter. Harnamatsu H2611 PMTs were used in order to reduce gain drifiting due to the high magnetic fields present in the experiment at Argonne, although that feature was not necessary for this application. The PMTs are 5.08 cm in diameter and coupled with optical couplant to the quartz 36 . . , ,-/:,i:::::::::' N’Ppfi/ Na; T1) ,1 :12“ (j PV:Q],.,, Figure 2.5: The APEX NaI(Tl) array. The top panel shows the assembled array with- out shielding. The bottom panel indicates the length L and the photon-interaction position X for a single detector. windows. Light—tight PMT shields cover the junctions. Exponential light attenuation in each crystal is achieved by diffusing the surface of the crystal via grinding and coating the crystal with an optical reflector. The diffusion causes the angle of reflection to be independent of the angle of incidence and the selection of the coating allows adjustment of the attenuation coefficient while maintaining the energy resolution of the detector [73]. Because the light attenuation is exponential, the energy of the 7-ray and position of its interaction with the crystal can be reconstructed from the PMT signals. Above 100 keV, a 7 ray interacting with the N aI(Tl) produces a number of scintillation pho- tons, NS, proportional to the energy deposition [81]. Below 100 keV, the number of scintillation photons N5 produced is no longer linearly related to the energy deposi- tion. We assume that half the scintillation photons contribute to the signal in each of 37 the PMTs. The number of photoelectrons released from the photocathodes are N123“) oc fife—”(Mn)” (2.17) NPE,u (x %e—u(L/2—X) where N p3,.) and N193,” are the number of photoelectrons released from the down- stream PMT and the upstream PMT respectively, X is the distance of the interaction point measured from the center of the crystal, L is the length of the crystal, and p is the attenuation coefficient (see Figure 2.5). The pulse heights of each of the PMTs (Yu and Yd) are proportional to the number of scintillation photons N; by a factor dependent on the gain of the electronics, Yd CX NPE,d (2.18) Y“ O( NPE,u- The PMTs are gain matched by placing a 60C0 collimated source (also used for the position calibration see Section 3.1.1) at the center of the array and first adjusting the PMT biases so that the output signal amplitudes match on an oscilloscope. The dispersion is fine-tuned by adjusting shaper gains to the range desired, in this case 2 keV/ channel [74]. Gain matching is important in order to avoid loss of dynamic range in the amplifier. After gain matching, position and energy can be reconstructed from the pulse heights: 1 Ya : — —-—- 2.1 X 2H [In Yd] , ( 9) and E = (mi/d. (2.20) The position and energy calibrations, efficiency, and energy and position resolutions 38 are discussed further in Chapter 3. For a position resolution of 3.0 cm, the attenuation coefficient is p = 0.047cm‘1 [73] and the energy resolution (FWHM/centroid) for a section at the center of the array is 16% for the sum of all the detectors at 662 keV. When the array is assembled, the detectors are arranged in a cylinder supported by two stainless steel rings. The inner diameter of the array is 42.8 cm, the outer diameter is 56.7 cm and it is shielded by a steel case and lead plates. For '7 rays originating from the center, 75% of 47r is covered over the full length of the detectors. Some of the physical features detract from the energy resolution of the APEX array. Near the ends of the detectors, optical-light collection efficiency decreases due to the trapezoidal shape. The junctions between the NaI(T1) crystal, the quartz window and the PMT also cause a loss of light due to the size mismatch. Particle Detection After passing through the PPACs for relative beam position monitoring, and inter- acting with the secondary target, projectiles were detected by a fast / slow plastic phoswich detector (ZDD) [82]. The maximum scattering angle 0m was determined by the detector placement. The ZDD provides a means for projectile identification and counting as well as allowing the constraint of the impact parameter (Section 2.2.4). The ZDD must be carefully placed in the beam pipe to ensure centering on the beam axis, and correct distance from the target. The ZDD is 10 cm (4 inches) in diameter and consists of two thin (1.5 and 1 mm) pieces of BC400 fast plastic and a thick (100 mm) piece of BC444 slow plastic covered in mylar and black tape to prevent light leakage. The thickness of the fast plastic was chosen such that 19% energy was lost for 34Mg, and 24% for 32Mg. The estimate was simulated using LISE [83] and SRIM [84] with the material BC400 (polyvinyltoluene with an H to C atom ratio of 1.103). The plastic is optically coupled through a light guide to two THORN EMI PMTs [82]. Particles detected in the detector are identified by their energy loss in the fast plastic vs. time of flight, or the full energy deposited 39 in the slow plastic vs. time of flight. Both signals can be detected with one PMT by pulse shape discrimination. Over the course of the experiment, several target changes were made including scintillation targets for beam tuning and the actual Coulomb excitation targets. In order to conserve beam time, it was important to make these changes as quickly as possible. Because the only access to the secondary target was from the downstream end of the setup, beyond the ZDD placement, it was also important to position and remove the ZDD itself as quickly as possible with the smallest uncertainty. To acco- modate this, an Al frame to support the ZDD within the beam pipe was constructed such that the ZDD could be levelled on the x-z plane prior to placement in the beam pipe, where z is the beam axis, and x is the horizontal direction. In previous experi- ments, it was necessary to adjust the vertical and horizontal positions for each corner of the detector each time it was placed in the beam pipe. This was a tedious process and took a substantial amount of time. The frame is displayed in Figure 2.6. It consists of two 1.27 cm thick Al rings 23.5 cm in diameter and 20.32 cm apart. They are connected by a 23 x 11 x 2 cm3 level Al platform. Two identical carts with wheels were constructed to fit into grooves on the platform. The alignment cart holds wire crosshairs spaced 16.5 cm from a pin hole drilled in a block normal to the cart surface. It is displayed in the top panel of Figure 2.6. The ZDD cart holds the zero degree detector and is displayed in the bottom panel of Figure 2.6. The center of the ZDD face corresponds exactly to the center of the crosshairs and pin hole on the alignment cart. In order to align the ZDD to the beam axis, the frame was inserted into the beam pipe with the alignment cart in place. The crosshairs and pin hole were then lined up with a beam telescope further downstream that had previously been centered on the beam axis. This was accomplished by adjusting the vertical and horizontal position of the platform of the frame via one dial for the vertical adjustment and two screws for the horizontal adjustment. Since the platform on the frame and the beam 40 Figure 2.6: The zero degree detector and positioning frame. The frame is shown in the top panel with the alignment cart in place. In the bottom panel, the alignment cart has been replaced with the zero degree detector on the ZDD cart. pipe were levelled in advance of the experiment, only one set of horizontal and vertical adjustments was necessary. After the frame was aligned, the alignment cart was easily removed and the ZDD cart with the zero degree detector was put in its place on the frame, cables were connected, the beam pipe evacuated and the experiment continued along in an efficient fashion. 41 Table 2.3: Impact parameters for NSCL experiment 01017. The energy, v/c, and bmin are values at the middle of the secondary target. 03,,“ is the maximum scattering angle in the projectile frame. Energy 0,0,3; bmin Nucleus (MeV / nucleon) v/c (deg) (fm) 26Mg 66.8 0.36 2.68 20.4 32Mg 71.9 0.37 2.63 15.4 34Mg 67.3 0.36 2.77 15.5 35Al 74.4 0.38 2.78 14.7 36A] 70.1 0.37 2.79 15.2 373i 77.1 0.38 2.80 14.5 48Ca 51.0 0.32 2.93 24.2 96M0 55.9 0.34 3.69 20.3 2.2.4 Coincidences and Constraint of the Minimum Impact Parameter In order to exclude direct nuclear reactions from the recorded data, the impact pa- rameter must be constrained such that no events are accepted at parameters less than the sum of the target and projectile radii plus two to four fm [85]. This minimum impact parameter bmin (Equation 2.3) is related to the maximum laboratory angle by max _ 2Z1Z262 1 lab — 777111)? bmin (2.21) where the projectile is denoted by the subscript 1 and the target by 2, and lab indicates the laboratory frame. The radii can be estimated by R = 1.25/1” 3 fm. In order to practically accomplish this constraint, the zero degree detector is placed at a distance that limits 67m to a value dictated by Equation 2.21 through R L tan 03"” : —, (2.22) where R is the radius of the plastic phoswhich detector, and L is the distance from the plastic phoswich detector to the target. The master trigger is (particle AND 7) 42 Ngfgba fist/Slow ADC —> Readout ”mg ~— DAQ Clear Amp 1 —> Readout CFD Delay *— Sealer l TDC Register H Readout 70R i DAQ Clear Start -Masl€r.iive Readout -DAQ Clear Figure 2.7: The basic electronics layout for the APEX N aI(Tl) array. OR (downscaled particle) and binning of the 7 rays is performed by placing a software gate on the appropriate particle and using that as a condition for sorting 7 rays. 03,,“ and bun-n are listed in Table 2.3 for the nuclei studied in this work. The software gates are discussed in Section 3.2. 2.2.5 Data Flow Detector signals were processed by a series of nuclear electronics which were read out by the SpectroDAQ data acquisition system and sorted utilizing SpecTcl [86] both online and offline. The APEX array The electronics diagram for the APEX array is shown in Figure 2.7. The 7—ray in- teraction with the NaI(T1) crystal produces scintillation photons which are converted to photoelectrons by the PMTs at each end of the crystal. Signals from both pho— tomultiplier tubes for each detector in the APEX N aI(Tl) array were sent to CAEN N568B shaping amplifiers. The slow branch of the signal was digitized by Phillips Scientific model 7164H peak-sensing 16-channel analog to digital converters (ADCs) and then read out for the energy information. The fast pickoff signal was directed to LeCroy 3420 16—channel constant fraction discriminators (CFDs) for discrimination. 43 ZDD Attenuator—— Delay L—l Splitter/ — FERA --—-> Readout PMT Attenuator (fast) ZDD Splitter/ FERA All at *— — —‘> R 'd 1 PMT em or Dclay Attenuator (slow) ea ou Delayed Particle I TDC ‘-——> Readout Attenuator —— Discr. 0000 | f Sealer ——> Readout Downscaler Downscliled Particle Figure 2.8: The basic electronics layout for the zero degree detector. The particle, downscaled particle, and delayed particle are indicated. Particle The logic-or was sent to the trigger logic unit (P/ S 752) as the APEX OR. It was also delayed and split to a 4434 sealer module which was read out and a 16-channel 7186H P/ S time to digital converter (TDC). The TDC was read out and the signal from the TDC was sent to a LeCroy 4448 bit register and read out. Thresholds were placed on the CFDs such that, for a certain detector, if any one PMT signal was above the threshold, and the trigger was set, both PMTs’ energy and time signals were read out. The thresholds were different for every detector, but were set to exclude signals below the 133Ba line at 356 keV. The gain was set in the ADCs such that the shapers began to saturate at 2 MeV. The ZDD The zero degree detector electronics are shown in Figure 2.8. Signals from both of the PMTs on the zero degree detector were also read out. For the energy information, the signal was sent to a splitter. Each of these signals was then attenuated (Lecroy A101 attenuator), delayed by 500 ns, and sent through an MSU splitter/ attenuator to a Lecroy 4300B FERA (fast encoding readout ADC), one for the slow signal, and the other for the fast and subsequently read out. The timing information signal was 44 Aparticle = 200ns particle 7 OR Aparticle + 40ns particle A 7 ——> bit register —> sealer delayed particle A(delayed particle) = 40ns trigger p A r Aparticle + 40ns downscaled particle (delay and gate) —> bit register —> sealer delayed particle trigger d/s p Figure 2.9: The gate widths for the trigger logic. OR is indicated by A, particle by p and downscaled by d/s. The final Masterlive is an AND of the trigger d/s p and the data aquisition not busy. sent to a discriminator (P / S 711) and split. One signal went to the trigger logic unit as the Particle signal. Another signal went to a downscaler, an MSU quad gate and delay generator (QDGG) and then the trigger logic unit as the Downscaled Particle signal. Another went directly to the QDGG and was delayed. The delayed signal was sent to the logic unit as the Delayed Particle and also to a P / S 7186 TDC and the readout as the stop. The Particle, Downscaled Particle, and Delayed Particle signals were all read out as sealers. 45 60 50 4 0 J v = 40 ‘ 547 keV . t 650-1115keV 30 . l % 20. .54 o 1 2' 10 - E o ' 5 60 - ' 1 434-745 50‘. keV 885 keV v = 0.37e 40- l J 30 20- 10- 0 . . a . . . 0 1000 1500 2000 2500 Energy (keV) Figure 2.10: Energy spectra for 32Mg demonstrating the Doppler reconstruction. The top panel is the energy spectrum without Doppler reconstruction. The target de- excitation 7-rays are prominent while the 885 keV 7-rays from the projectile are smeared over 500 keV. The bottom panel shows measured energies that have been reconstructed event-by-event in the moving frame of the projectile. Gamma-rays orig- inating in the projectile are seen as a prominent peak, while the photopeak from the target nuclei is broadened. 'Ii‘iggers The trigger timing is shown in Figure 2.9. Phillips Scientific 752 and 755 4-fold logic units were used to set the live trigger. The Masterlive was the particle-7 coincidence (((7-OR AND particle) OR (downscaled particle)) AND delayed particle) AND NIM not busy. The NIM busy gate start was Masterlive OR data aquisition not busy. The stop was the data aquisition clear. The 7-OR was an OR between the APEX CFDs. 46 8) Position (cm) 45_ 1 1 1 1 1 1 1 1 0 100 200 300 400 Ener channels b) gy( ) ’s‘ 8 C .2 '5 a. 45 1 l l l l l l 0 100 200 300 400 Energy (channels) Figure 2.11: Position-versus—energy spectra for decay 7-rays emitted from 88Y (898 and 1836 keV). The spectrum is shown without Doppler reconstruction of the energy in Panel a), and in panel b) the 7-ray energies are reconstructed in the projectile frame. The position is measured in the laboratory frame in both spectra. The energy dispersion is 8 keV/ channel. 2.2.6 Consideration of Relativistic Effects Because intermediate-energy radioactive beams have velocities above the classical limit, experimenters utilizing them must consider relativistic effects. In particular, de— excitation 7-rays originating in the projectile nucleus are detected in the laboratory frame. In order to obtain the energy of the 7-rays originating from the projectile at their values in the rest frame of the projectile, the change in frequency from one inertial frame to the next is taken into consideration. 47 A full development of the relativistic kinematics pertinent to experiments at in- termediate energies can be found in [70]. The results utilized in this experiment will be summarized here. The collision of a projectile moving at B = v/c with a stationary target is described by the following relationships between angles and energies of the massless photons in the center of mass (cm) and laboratory (lab) frames: sin 61d, tan 0m, : 2.23 7(cos9za1- m ( ) tan 0 sin 06'" lab 7 (cos 0m + 6), and Ecm = 7Elab (1 — ,BCOS 0m) (2.24) Elab 2‘— ’)‘Ecm (1 + ,BCOS 0cm) . The observables for the experiment are the energies and positions of the 7-rays de- tected in the laboratory frame. For purposes of illustration, the relationships between energy, scattering angle and velocities in the laboratory and projectile rest-frames can be thought of in the following manner. For the target (tar), v = 0 with respect to the lab frame and Eltgg : Ex. (2.25) On the other hand, the projectile (pro) has some velocity v with respect to the laboratory frame and the quantity detected is 135;," = 7135;: (1 + 6605.95,?) (2.26) In order to determine the energy of the 7-rays originating from the projectile, the 48 observed Eff: must be boosted to the projectile frame which is moving at velocity v: Eff: : 7E3: (1 — Bcos 05:). (2.27) There are uncertainties in the measurement of 5, and the angle 010), which make small contributions to width of the energy peak. Section 3.1.3 discusses the energy resolution of the APEX array in detail, and Table 3.2 lists the contributions to the overall width of the energy peak by these uncertainties in B and 0,01,. Gamma-rays originating from the target can also be boosted to the projectile frame moving at v: 13...: = 7 {21(1— ficosetzz). (2.28) This effect broadens the energy in the laboratory frame for the projectile and in the projectile frame for the target. 49 Chapter 3 Data Analysis After the experiment, a detailed analysis of the raw data leads to the final Coulomb excitation cross section: N5 003 = W, (3.1) where NI,3 is the number of emitted de-excitation 7-rays, N; is the number of target nuclei, and Np is the number of incoming beam nuclei. In order to arrive at the final cross section, the energy and emission angle of the de-excitation 7 rays detected by the APEX N aI(Tl) array are sorted event-by-event in coincidence with the projectile nuclei. These bombarding nuclei are also identified and counted. The overall analysis involves characterization of the N aI(Tl) array, data sorting, fitting and error analysis. 3.1 Characterization of the APEX NaI(Tl) Array In order to examine previously unmeasured nuclei, the 7-ray detectors must be un- derstood. Calibrations for both energy and position are performed and the energy resolution and efliciency of the array are carefully examined. 50 3.1. 1 Calibrations The first step in the data analysis is the calibration of the APEX NaI(Tl) array. Position and position-dependent energy calibration data taken before the test 96Mo beam, and before and after the main experiment are processed through calibration routines. The results are then included in SpecTcl for the analysis of the data collected in the experiment. Calibration procedures are outlined below. Position Calibration Position-sensitivity of the 7-ray detectors becomes important for experiments at in- termediate energies. Beam velocities such as those utilized here (5 rx.4c) require that energies of the de-excitation 7-rays be D0ppler corrected, and therefore the angle of emission of the 7-ray must be known (see Equation 2.24). This emission angle can be determined from the position information given by the APEX N aI(Tl) array, whereby the point of 7-ray interaction with the crystal is reconstructed from the photomul- tiplier tube signal strengths (see Section 2.2.3). Position calibrations are performed immediately before and after the experiment. A position calibration is carried out by placing a 10.45 pCi 60Co source in a collimator and making measurements of the 1332 keV 7-ray along the length of the array [65,82]. The collimator is made up of two Heavymet cylinders which are 14 cm in diameter and 7.62 cm long, with a 4.6 mm gap between the two cylinders. Heavymet is an alloy made of tungsten copper and nickel. The collimator is equipped with rollers and a long bar to facilitate movement inside the beam pipe (see Figure 3.1). With some effort, the collimator is placed inside the beam pipe at the upstream end of the array, and data is taken at 47 positions in 1.27 cm increments through to the downstream end of the array. The detector multiplicity is restricted to 1, meaning that events are not accepted if more than one discriminator triggers. The length increments are measured with a measuring tape attached to the long bar on 51 [MT] NaI(Tl) T121117] (I ] 'Y 7 L Collimated 60Co Beam pipe EMT[ Na1(Tl) EMT] Figure 3.1: Longitudinal section depicting the position calibration for the APEX N aI(Tl) array. the collimator. At each position, data is taken simultaneously by all 24 detectors for approximately 5 minutes. This follows closely the method used in [82]. The position is reconstructed from the measured pulse heights of both upstream and downstream photomultiplier tubes (PMTs) [73, 74]. (See section 2.2.3). In this experiment we used the expression, Yu PR : 2000 + 1000 loglo (17—) , (3.2) d where PR is the reconstructed position, Yu is the signal from the upstream PMT and Yd is the signal from the downstream PMT. Centroids of the position spectra are determined for all 47 positions for each of the 24 detectors using the fitting program GF2 [87]. The reconstructed position is then calibrated to the actual physical position through the position calibration routine. The routine utilizes a Tcl [88,89] script to extract GF2 fit information, and Physica [90] to make the fits of the calibration curves. As an example, the position calibration is shown for Detector 16 in Figures 3.2 and 3.3. Figure 3.2 shows the relationship between the reconstructed position and the physical position. The light collection at the ends of the detectors is non-uniform, and so portions of the detectors at each end have been excluded from the calibration and do not enter into the data analysis. The amount of crystal excluded depended 52 40- 35a 30— Physical Position (inches) 25~ 20- I I I I I 1000 1500 2000 2500 3000 2000 + 1000 1og(Yu,,/\(,0w,,) Figure 3.2: The position calibration for Detector 16 of the APEX NaI(Tl) array. There is a linear dependence between the physical and reconstructed positions. on the detector itself, and was on average 7 cm at each end. The fit is linear for the remainder of the positions (shown). Figure 3.3 shows position spectra for several intervals at the center of the detector. The position resolution (FWH M / centroid) is approximately 3 cm. The position calibration makes it feasible to bin SpecTcl parameters by position interval. The position intervals are referred to as “slices” and are created by placing software gates on the calibrated reconstructed position spectra along the longitudinal axis of the detectors. Each detector can be virtually divided into any number of these slices. For any quantity, such as the 7-ray energy, the sum of all the individual slice spectra for one detector will.result in the total spectrum for that detector. The sum may also be performed for one slice over all the detectors, as is done to create the efficiency rings (see Section 3.1.4). By dividing the data in this way, it is possible to consider the position dependence of the raw energy, take into account any imperfect detector segments, and as mentioned, partition the array virtually into efficiency rings. 53 Counts 120 60‘ 120 60J 1201 60- 120 60~ l20 601 l20 60- AAl‘h‘- . 0 " “‘1 w w— “W 1000 I 1500 r 2000 2500 17cm 19cm 22 cm 24 cm 27 cm 30 cm 2000 + 10001116ng HQ...) Figure 3.3: Representative reconstructed position spectra for Detector 16. These are a sample from the central region of the detector. The positions in the right margin indicate the actual physical distance from the downstream end of the array. 54 Serial Energy A0 (pCi) Branching , , Source Number (keV) (Date) Ratio Calibration 203Hg kf301 279.2 2.00(8) (12/1/01) 0.81 Energy/ Efficiency 133Ba kf297 356.0 106(3) (12/ 1 / 01) 0.62 Efficiency 137Cs Z7097 661.7 103(4) (6 / 15 / 01) 0.85 Energy / Efficiency 5“ Mn kf302 834.8 105(3) (12 /1 / 01) 0.99 Energy / Efficiency 88Y kf304 898.0 103(3) (12 / 1 / 01) 0.90 Energy / Efficiency 88Y kf304 1836.1 103(3) (12 / 1 / 01) 0.96 Energy/ Efficiency 228Th E2881 2614.5 13.91(63) (10/ 1 / 88) 0.36 Energy/ Efficiency Table 3.1: Calibration sources used in the energy and efficiency calibrations. The branching ratios for the two Yttrium lines reflect the multiplicity 1 condition. A0 is the initial activity of the source at the date listed. Position-Dependent Energy Calibration To accomodate the division into slices, the energy calibration must be position depen- dent; every position for each detector must have its own energy calibration. For this experiment, one position interval is approximately 2.54 cm wide and 24 slices were calibrated per detector. It was assumed that there were no unresponsive portions in the middle of the crystals, and of the 24 slices, the only slices excluded were those excluded by the position calibration. The position-dependent energy calibration is carried out by placing a known ra- dioactive source at the secondary target position in the center of the array and mea- suring decay 7-rays for 2-6 hours. Data is taken separately for several different sources over a range of energies, again with multiplicity 1. The sources used in this experiment are listed in Table 3.1. The energy is reconstructed from the raw signals of the upstream (Yu) and down- stream (Yd) PMTs (see also Section 2.2.3) as E3, E3 : MY“ - Yd. (3.3) Centroids of the reconstructed energy spectra are fit with GF2 [87], and calibrated 55 2000 . 1836.052 keV 88Y 1500~j : f; : Q) It a : 31000-2 54 6 5 Mn 898.036 keV ,5, 3 ‘37Cs 834.848 keV .1 500 _: 209 661.657 keV : Hg 3 279.194ch 0dIIIIIIIIT‘IIIIT‘U‘I'ITTIIIIIUIIIUIIIIIlIIjI'rI—UIIIIIUI 0 200 400 600 800 1000 Channels Figure 3.4: The position-dependent energy calibration for one slice in Detector 7 of the APEX NaI(Tl) array. This particular slice is approximately 18 cm upstream of the center of the array. Gamma-ray sources are indicated. to the actual 7-ray energy with a position-dependent energy calibration routine. As with the position calibration, the routine makes use of Tel scripts and Physica to arrive at the final energy calibration coefficients. Examples of the energy calibration for Detector 7, Slice 1700 are shown in Figures 3.4 and 3.5. The slice number is simply a naming convention, and the location of slice 1700 is actually 18 cm upstream from the center of the array. Figure 3.4 shows the actual calibration over the range of 7-ray energies used. The fit is first-order. Recon- structed energy spectra for the 137Cs and 88Y sources are shown in Figure 3.5 with the corresponding decay schemes. The gain was set such that there were 2 keV / channel in the ADCs. The importance of the position-dependence of the calibration is shown in Figure 3.6. Over the entire array, there were 21,000 peaks fit in total to accomplish the calibration. After the calibration data is taken and processed through the calibration routines, the position and position-dependent energy calibration coefficients are written to XML files which are incorporated into the SpecTcl code. 56 140 BOY 120 2734 keV 100 80 Counts 60 40 20 0 200 400 600 800 1000 1200 1400 Channels 350 5 300 —; 250 -; 200 -: Counts 150 -: 100 -':' so-fi 0 100 200 300 400 500 600 Channels Figure 3.5: Reconstructed energy spectra for the sources 88Y and 137Cs as measured with one slice of the APEX N aI(Tl) detector number 17. The slice is 18 cm upstream of the center of the array. The corresponding level diagrams are shown. The dispersion is 2 keV per channel. 57 Uncalibrated Position (cm) Energy (channels) Figure 3.6: The position vs. energy plots for Detector 18. The 133Ba line at 356 keV is shown on the left in the uncalibrated spectrum and on the right in the calibrated spectrum. The dispersion is 16 keV/ channel. 3.1.2 In-beam Energy Calibration The initial energy calibration was performed as outlined above with the data taken from calibration sources. A second energy calibration was performed using the energy of the de-excitation 7-rays from the projectile nuclei 26Mg, 32Mg and the target 197Au, at 1808 keV, 885 keV and 547 keV respectively. These are all well-known energies [17,77]. The relationship was again linear for the spectra contracted by 20: y = 0.04832 + 0.0846 (3.4) where a: is the channel number, and y is the energy in keV. 3.1.3 Energy Resolution Resolving the energy of the detected 7 rays is another important part of the data analysis. The response of a scintillation detector such as N aI(Tl) is expected to be imperfect [81]. In other words, the energy emitted is detected not as a single energy, but rather as a range of energies with a Gaussian distribution. The resolution, R is defined as the full width at half the maximum height FWH M of this distribution 58 divided by the centroid H0 of the same distribution. FWHM R H0 III (3.5) The energy resolution R in a scintillation detector is discussed in [81], and summa- rized here. Charge collection statistics, electronic noise, variations in detector response over the volume of the detector, fluctuations in PMT gains may all contribute to the response. In general, beyond the intrinsic resolution of the crystal, charge collection statistics in the PMTs are the most significant additional factor in determining the resolution. Although these factors are beyond the scope of this work, it can be shown that the gain of the APEX detectors did not fluctuate over the course of the 34Mg run by comparing the centroids of the reconstructed energy peaks for all slices of detector both before and after the experiment. For detectors with a complicated shape, such as the trapezoidal crystals that make up the APEX NaI(Tl) array, the uniformity in the light collection is not as great as that of a crystal with a simple shape, and as a result, the energy resolution R broadens. For example, the resolution of a 662 keV line for a solid right cylinder of N aI coupled to a PMT can be about 6-7% [81]. For a crystal with a well, this resolution increases by 3% [81], and for a long cylinder the resolution is about 8% [65]. In the present work with the APEX N al(T1) array, the resolution at 662 keV is measured to be 18%. Energy resolutions of the APEX array were also measured in two previous reports [73, 74] which are discussed below. As is also discussed in [81], the intrinsic energy resolution is dependent on the 7-ray energy. For a scintillator with a simple shape, the energy resolution should be inversely proportional to the square root of the 7-ray energy, R = —— (3.6) ,/E,’ where C is a constant of proportionality, and E7 is the 7—ray energy. When crystals 59 0.4 * * Rdop A A AR(10 rings) 0.3- O O Raine 5) __ Empirical fit a: 0.2_ 0.1- f 1 1 1 1 0 500 1000 1500 2000 2500 3000 Energy (keV) Figure 3.7: Energy resolution R for APEX NaI(Tl) source measurements for Ring 5 (filled circles) and all 10 rings (triangles). The solid line through the measured points for Ring 5 is an empirical equation pertaining to the energy range shown. The stars show the resolutions after the uncertainty in the Doppler reconstruction has been included Rdop (Equation 3.9). of more complicated shapes are taken into account, the energy dependence changes and the resolution worsens as is the case with the APEX array. Energy resolutions for the measurements taken with the APEX NaI(Tl) array are shown in Figure 3.7. Resolutions R are shown for measurements including ring 5 only (center of the detectors), all 10 rings, and with the corrections due to uncertainties in the Doppler corrections (Rdop). The solid line is an empirical equation that is a close fit to the source resolutions for Ring 5 in the energy range 356-2614 keV: 1.368 R = 0 355. (3.7) (E, —— 268.76) - The necessary Doppler correction Ecm = 7E1“ (1 — ,BCOS 9105) , (3.8) also contributes to the energy resolution through the spread in velocity of the pro- 60 Table 3.2: Expected energy resolutions of the APEX NaI(Tl) array. The values are determined through interpolation of the source measurements via empirical fits. The corrected energy resolution Rdop is determined from the values interpolated from measurements including all 10 rings at 90° and corrected for uncertainties due to B = v/c R5 and the opening angle 0 R9 in the Doppler correction. The nuclei are grouped according to their cocktail beams and mid-target energies are listed. Unless otherwise noted, E, are the adopted values for the first excited state energy. E, Ebeam R R RB R9 R40? Nucleus (MeV) (MeV) (ring 5) (all rings) @ 90° @ 90° @ 90° 26Mg 1.808 66.8 10.1% 10.6% 1.6% 6.0% 12.4% 32Mg 0.885 71.9 14.0% 15.0% 1.7% 6.5% 16.4% 34Mg 0.658 a 67.3 16.5% 18.0% 1.7% 6.3% 19.1% 35Al 1.006 b 74.7 13.1% 14.0% 1.8% 6.6% 14.7% 36A] 0.967 C 70.1 13.4% 14.3% 1.7% 6.4% 15.7% 373i 1.437 b 77.1 11.1% 11.2% 1.8% 6.7% 13.2% 48Ca 3.832 51.0 7.5% 7.6% 1.4% 5.6% 9.7% 96M0 0.778 55.9 15% 16.1% 1.5% 5.8% 17.3% “ Ref. [38]. b Ref. [29]. C Present work. jectile nuclei, and the uncertainty in the opening angle of the array. The corrected resolution becomes 2 __ c0519“), 2 2 2 (Rd...) — (1_flcosolab—fl”r) A13 . 2 1.3.2253...) ..2.(.)2. <39) As will be discussed in Section 4, the widths of the Doppler corrected energy peaks play a significant role in unfolding the 34 Mg spectrum. It is important to note that the energy range of interest for this experiment, including all Doppler shifts, is 487-2264 keV. The interpolated expected resolutions are shown in Table 3.2. There were two previous reports on the APEX N aI(T1) array [73, 74]. In the first, Kaloskamis et al. report an energy resolution R of 13% for the 1022 keV reconstructed 61 photopeak for annihilation photons and a position resolution between 3.0 and 3.6 cm. The second is a report by Perry et al. who measured an energy resolution R of 13% at 1332 keV and a position resolution of 3 cm. In the present work, the energy resolution R for all 10 rings at 1332 keV is expected to be 12% while at 1022 keV, the expected energy resolution R for all 10 rings is 13.9%. 3.1 .4 Efficiency Another crucial step in measuring the Coulomb excitation cross section is the estima- tion of the number of 7 rays emitted from the Coulomb excitation reactions (Equation 3.1). In order to do this correctly, the overall photopeak efficiency of the 7-ray de- tectors must be understood. For this experiment, the number of 7 rays detected is assumed to depend on the intrinsic efficiency of the APEX N aI(Tl) array, the angular distribution of the emitted 7 rays, and the photoabsorption in the secondary target. The intrinsic efficiency was determined by comparing a GEAN T [91] simulation of the array to the values obtained from several radioactive source measurements. Pho- toabsorption cross sections for the target material were taken from [92]. For the final determination of the number of emitted Coulomb de-excitation 7-rays, the angular distribution was calculated and folded with the probablity for absorption and the intrinsic detector efficiency using MATHEMATICA [93]. Efficiency by Rings Gamma-rays produced by Coulomb excitation have a non-isotropic angular distribu- tion [14,66, 70—72, 82]. To account for this distribution, as well as crystal variations over the length of a single detector, the array was subdivided virtually into 10 rings transverse to the beam axis (Figure 3.8). First, each detector was divided into 10 transverse slices by placing software gates on the reconstructed position. Slice bound- aries were common over all detectors with dead portions toward the ends of the crystals excluded through the position calibration. To constitute a ring (72,-), each of 62 ” lllllll/i " / .. 1,;11115/5/1/ Figure 3.8: a) End view of the volumes defined in the GEANT simulation of the APEX Nal(Tl) array. b) Side View. The dashed line represents the z-axis, with the arrow indicating the positive direction. the slice spectra (8,) were then summed over all 24 detectors, 24 72.- : 2(1):), (3.10) 1:1 In the laboratory (lab) frame, each virtual ring, R,- is spanned by the solid angle (2?" where Qfiabzsin Bladdfiabdflab. The discussion will assume that all angles are lab-frame angles unless otherwise noted, and the superscript lab will be dropped. All 10 rings have their symmetry axis along z, unique boundaries in t9, and a common range in d) from 0 to 21r. The intrinsic efficiency for R,- can be written: n,- : (number of 7 rays detected in (2,) 0" — (number 0 f 7 rays emitted into 9,) ’ e (3.11) where the superscript denotes the element of solid angle over which 7-rays are de- tected, and the subscript indicates the element of solid angle over which 7-rays are 63 emitted. The sum of all 10 intrinsic ring efficiencies gives the total intrinsic efficiency Tot intr e of the array: 10 263: = 6:33.. (3.12) i=1 The number of 7 rays emitted into Q, depends on the angular distribution of the de-excitation 7-rays, as well as the photon absorption cross section for the target material. The angular distribution is defined in the center-of-mass frame of the excited nucleus. For target nuclei this is the same as the laboratory frame, so Wzab(0) : W(6cm) and dfllab = d526,". Due to beam velocities of 5 z 0.40 the relationship between Wzab(9) and W(6m,) for the projectile requires that 06m must be boosted according to, sin 01,11, tan 06m = , 3.13 7 (90801.11, - fl) ( ) where 6: (v/c) and 1 (3.14) Due to the change in variables, the differential solid angles must also be related by ”i“ (sin 0a,? + 72 (cos 0m + B)2) dflcm : 7 (1 + ficosdcm) dam. (3.15) The derivation can be found in [70]. For the number of 7-rays emitted into 9,- (Nf') we have an - I N31: (/ W, 6 -——‘1"—"-dQ,/ ——dQ,-) N30t 3.16 6.. ”(l 49. 6.10 l l where Wlab(6) is the angular distribution in the laboratory frame, I is the intensity of 7-rays half-way through the target, as described in Section 3.1.4, and 10 is their initial intensity. Thus the number of 7-rays detected in the element of solid angle 9,- 64 (N?) can be written: (430 W,ab(9 —°—d::"”dfl / 11612,) N30t (3.17) 0 where / W,a,,(9) 5111969 = 1. (3.18) 0 Summing over all 10 rings, the total number of 7—rays detected is: 1° )d9 1 NT“: 6 )/ Wa(0 C—m-‘dfl / —dn N3“. 3.19 (;(n n: 1 b d9 1—0 ) ( ) For an isotropic distribution, such as is emitted from a radioactive source, the number of 7-rays emitted into one element of solid angle 9,- is a fraction 52,- /41r of the number of 7 rays emitted into 47r. The intrinsic ring efficiency becomes: . N“ 4 2 (5’1) 653;, (3.21) and enters the calculation of the final corrected efficiency at Ecm 7(1— )Bcosdzab) EM, 2 (3.22) With this, the final expression for the total number of Coulomb de-excitation 7-rays detected by the Apex N aI(T1) array is 10 NJ“: (2G,— 7:62)] 114.9):‘352146 [a 110.16..) N3“, (323) £21 and denoting the quantity in the parenthesis by 6cm» the Coulomb excitation cross 65 section becomes (Nd ) C _— ——————, .24 Determining 63: (E,) To determine the intrinsic efficiency of the array as a function of 'y-ray energy, mea- surements were made with 5 calibrated radioactive sources and then compared to a GEANT simulation of the array. Placed at the target position, the sources 137Cs, 203Hg, 54Mn, 223Th, and 88Y (see Table 3.1) were measured for 2-6 hours each. The de- tector multiplicity was restricted to 1, meaning that an event was counted if and only if no more than one detector triggered. This is the same constraint that was placed on the detectors for the analysis of the experiment data. After position and position- dependent energy calibrations were included (Section 3.1.1) by SpecTcl, these effi- ciency runs were sorted. For each of the 10 rings, SpecTcl output energy spectra in the gf2 format. GF 2 was used to fit the spectra for each source and the efficiency was extracted from the peak areas as N (hr) (9 (A)’ (3'25) 6:. where N is the area under the photopeak, br is the branching ratio for that particular 7-ray, t is the duration of the measurement, and the A is the activity of the source at the time of the measurement. The activity is given by A = A0 exp (_ 1112 ~t) , (3.26) t1/2 or, A = A0 (1/2)‘/‘1/2, (3.27) where A0 is the initial activity of the source, t1 )2 is the half-life, and t is the time elapsed since A0 was determined. 66 The GEANT Simulation GEANT 3.21 is a powerful detector simulation tool available from CERN [91]. It simulates the interaction of particles with all user-defined volumes and reports the energy deposited in any volume specified as active. Most known 7-ray interactions are included. For this experiment, the N aJ(Tl) crystals were defined as the active volume [74], while the target, target frame, vacuum, beam pipe, air, quartz windows, and aluminum housing were included as inactive volumes (Figure 3.9). GEANT facilitates user-control of particle kinematics and also allows for simulation of data flow after the interaction has taken its course. In order to simulate the efficiency, 7 rays ranging from 0 to 5000 keV are emitted from the center of the array in a random fashion. One 7 ray is fully processed before the next is triggered. The positions of the interactions in the crystal, and the energy deposits at those interaction points are recorded until none of the initial energy is left. For each of the 24 detectors, every energy deposit is converted into the light output expected to be seen by the photomultiplier tubes at each end of the detector [73]. These light outputs are incremented until there are no more energy deposits resulting from that event. The final position and energy of the initial 7 ray are then reconstructed from these two quantities. Following the reconstruction, the histograms are filled. Thresholds are placed on the energy to simulate those on the CFDs and the shapers in the actual setup. The values for the thresholds are determined by direct comparison of the simulated raw energy with the actual raw energy for each photomultiplier tube. The virtual efficiency rings are also recreated at this stage in order to exclude the dead portions of the crystals. The simulated widths of the PMT signals Yu and Yd are extracted from the mea- sured width of the total reconstructed energy spectra (E). The reconstructed energy 67 Nal crystal and Al housing Air Target Al beam pipe b) Nal crystal and Al housing Target beam pipe Al target frame Figure 3.9: The geometry specified in the GEANT simulation of the APEX Na.I(Tl) array. 68 is a combination of Gaussian distributions, 2 E 2 m ~ (cap (1:2 .. 321.». (3.28) 2 on ad If the 7 ray is deposited at the center of the array, the light attenuation in either direction can be assumed to be the same. If we assume both PMTs to be identical, the signals Yd and Y“, Y. = gem—9m + X». (3.29) and Y. = gum—”(m — x». (3.30) where g is a simulated gain factor, p: 0.047 is the attenuation coefficient obtained from [73], L is the length of the detector, and X is the position of interaction of the 7—ray, and their widths an and ad are also equal. If this is true, 0,, = 0.; = a, and we can extract the width 0 from the total width of the reconstructed energy distribution 03. Combining the gaussians Yu and Yd leads to the relationship, and the FWHM = 2.3540 [94]. The reconstructed energy source measurements for Ring 5 (center of the array) were fit and the energy dependence of this full reconstructed energy resolution is given in Equation 3.8. The form is an empirical function that fits the resolutions of the energy source measurements from 356 keV to 2614 keV and was discussed in Section 3.1.3. For the final simulated resolution, each of the PMT signal widths were extracted individually using equation 3.31 and equation 3.8. The full detected energy was then reconstructed from these two PMT distributions. In order to simulate the efficiency, only photopeak events with multiplicity one, 69 above lower and below upper energy thresholds , and within one of the 10 specified rings were binned. As in the actual data collection, the energies are binned by ring, and then the rings are summed for the total spectrum. This spectrum was then divided by the emitted energy spectrum to obtain the efficiency curve. Even with these constraints, there is a discrepancy between the GEAN T prediction and the measured source points 662, 834, 898, 1836, 2614 keV. The discrepancy can be accounted for by considering the difference of backgrounds between the simulation and the measured values, and the exclusion of slices through the position calibration since the simulation was necessarily treated differently than the data in this regard. For example, the simulated efficiency (31.87%) for 137Cs at 662 keV differs from the measured efficiency (22.2%) by 43%. When the measured and simulated peaks are fit with CF 2, there is a 17% difference between the areas of the two. Another 22% is accounted for in the difference of excluded position slices between the simulation and the measurement. For the final intrinsic efficiency, the values for the efficiency simulated by GEANT were scaled to fit the source data. A X2 analysis was performed to find a scaling factor of 0.813(24). For the final analysis, the scaled simulated efficiency was used in the extraction of the Coulomb excitation cross section. It is compared to measured values in Figure 3.10. The discrepancy between the 356 keV measured point and the simulation is beyond the lower limit of the energy range for this experiment, 487-2264 keV. The disagreement is due to the difference between the simulated thresholds and actual thresholds. Figure 3.11 shows that varying the thresholds in the simulation does not affect the shape of the efficiency curve for the range of energy of interest to this work. Angular Distribution Coulomb excitation produces a non-isotropic distribution of de-excitation 7-rays. This is well-understood and has been applied and discussed in several cases [69—71,82]. The 70 Geant Simulation 0 0 0Measurement '00 Efficiency /" 0 ' 10'00 ' 20'00 ' 3000 4000 5000 Energy (keV) Figure 3.10: The scaled, simulated efficiency curve compared to the measured values. This is the intrinsic efficiency which is later folded with the angular distribution and photon absorption to obtain the final value for the efficiency. Thresholds in this simulation are matched to actual experiment threshold values. full derivation can be found in [66,72]. The application in this case follows [27,70] and only the final expressions will be stated here for continuity. Since the APEX NaI(Tl) detectors are arranged symmetrically around z, we integrate over (1) and from the minimum impact parameter bmin to 00, and the angular distribution is expressed as a function of 0m, we...) = Z [0...(5 )| syn—m )“(2 f k) (3.32) keven, u [I 0 L, L’ I I X{ [if I f}Fk(L,LI,Iff,If)\/2k+1Pk(COS(0m))6L6LI. 1 keV Actual — Thresholds 1 keV _4 Actual Thresholds Thresholds 500 keV 0 ¢ 0 Measurement 3 .3 500 keV c: .2 O a E .2 .1 .0 ' l ' I ' l ' l ' l 0 1000 2000 3000 4000 5000 Energy (keV) Figure 3.11: The shape of the efficiency curve does not vary with changes in the simulated thresholds for the energy range 487—2264 keV. where g,,({) are the Winther and Alder functions (Equation 2.7), G A), (c/v) are given in equations 2.9 and 2.9, Fk (Equation 2.15) expresses 7—7 correlations, the Pk (cos (9cm)) are the usual Legendre polynomials, and the 6,, 6L: are Kronecker Delta functions. This is usually expressed: W(9.,,.) = Z 9.13,. (cos (9....» (3.33) Iceven The angular distributions for 26Mg, 32Mg, and 34 Mg are shown in Figure 3.12 along with the corresponding bmi" and [3. The bottom panel shows the m-state distributions as calculated by [95]. The middle panel shows the distinction between the laboratory frame and the center-of-mass frames, and the top panel shows the difference between a boosted isotropic distribution and that of the true W(0) of the projectile, in the laboratory frame. The range of effective detector coverage in 0w: is indicated in both Figures. The a], calculation was taken from [27,70]. For all three of these examples, 72 34Mg [3 = 0.36 bmin 2 15 fm Gmax = 0.048 rad E] = 0.658 MeV Effective Detector 32Mg [3 = 0.37 bmin = 15 fm Omax = 0.046 rad E; = 0.885 MeV Effective Detector 26Mg (3 = 0.36 bmin = 20 fm Omax = 0.047 rad 1;; = 1.808 MeV Effective Detector Range Range Range A . A . A .l e e e .E- .E - .E '08 / a) 3’1. 5’1 . 5Q -06 / 3_ a _ a .02 0 0(rad) 1)) r" l [ ‘2 1: . c . , E .41 . .9 AF .g . 1 ‘- .3. ‘6 -3: a ? a 1 _ . C) -' 1 "' ’~ :3' ‘ 3 2 3 2 D. : Q" 8' 1 O 8.1 n..l Q- 01 . *3“. W4 0 ‘ , . -2 -l 0 l 2 -2 m m m Figure 3.12: The Coulomb excitation angular distributions are presented for 26Mg, 32Mg, and 34Mg at the listed velocity, 6, and minimum impact parameter, bmin. a) The importance of including the angular distribution as a correction, where the dashed line indicates W(0) and the solid line is the boosted isotropic distribution. The effective detector range after exclusion of the ends of the detectors is indicated. b) Parametric plots highlighting the difference between the center-of-mass indicated by the dashed line, and the boosted, indicated by the solid line, frames. c) The population of the m-states as determined by [95]. 73 it can be seen that the distribution is close to that expected for an I : 2, m = :1:2 state. Photon absorption in the target As lifetimes for excited states considered here are on the order of picoseconds [17], it is likely that the Coulomb de-excitation 7-rays will be emitted somewhere inside the target material. Interactions of photons with the target material through the photoelectric effect, Compton scattering, and pair production reduce the fraction of photons that leave the target, but do not decrease their energy. As was shown in [82], these interactions are a significant factor in the estimate of the final number of 7- rays detected in a Coulomb excitation experiment. Here, the probability for a single photon to survive the target thickness is folded with the angular distribution and the measured efficiency for each ring in the determination of the final number of 7 rays detected by the APEX array. For a beam of parallel photons, the intensity loss after passing through a material of thickness d is exponential: ;d_ lab, 2 1022/2, (3.34) where 10 is the initial intensity of the beam, and d1/2 is the half-thickness of the material [96]. The half-thickness is the thickness of material through which half the intensity of the photon beam is lost: A log2 NA P0 (E7), (335) 611/2 = where A is the mass number and p is the density of the absorbing material, N A is Avogadro’s number, E7 is the 7-ray energy and a is the photon absorption cross section (b/ atom): 0' : ¢plwto + ZUC + Tpaira (336) 74 which depends on the photoelectric effect ((bphow), Compton scattering (ac), and pair production (Twir) cross sections. As there are Z electrons per atom, the Compton scattering cross section is multiplied by Z [81,97]. In terms of the virtual rings and the solid angle Q,- for the APEX N aI(Tl) array, the probability that the photon exits the material in that solid angle becomes: PT : / labs dfli 0.- 10 —d : / 2mm... (3.37) 51" Photon absorption cross section values taken from [92] have been fitted for both the 197Au and 209Bi targets in Figure 3.13 with the second order exponential [82,92]: .(E,):A.B...(—gv)wan—57). (3.38) where A, B,C, D, and F are fit parameters and E.y is the energy of the 7 ray. The coefficients are shown in Table 3.3. Values taken from the fits were folded with the intrinsic efficiency and the angular distribution to determine the final number of detected 7 rays. The Final Efficiency Estimation The final efficiency at each 7-ray energy of interest is folded together using MATHE— MATICA. For each ring,-, W(0) (Equation 3.33) is calculated and integrated over 52,-. LTarget] A4 B l C ] D J F ] 197.40 7.7424 87.515 324.59 1958.4 75.364 20982' 8.4175 104.76 325.66 2111.0 76.724 Table 3.3: Photoabsorption cross section fit parameters. 75 10 , . .. . , 4 2098i 4 197AU 2 1O 1 7 - A L‘ E .4 O -< *4 -1 O \ 1 « .0 V -( J O b 1 1O 1 i _ .1 0 1O 2 I I IIIIII]3 I I III—ITI 2I I I'IIIIII3 I I IIIIII4 10 10 10 10 10 Energy (keV) Figure 3.13: Photon absorption cross sections 0,, for the gold and bismuth secondary targets. It is then folded with the photon absorption probability (Equation 3.37) and finally with the intrinsic efficiency for that ring and when combined with the number of 7 rays detected, give the number of emitted de-excitation 7-rays: 10 d9 - I NT"t = e“? / W 9 “n" (152,] ibidn. NS“. 3.39 .. (g...) m () d0. “‘10 < > Table 3.4 lists the efficiencies for the range of energies pertinent to this study. 3.2 Gamma-ray and Particle Sorting Once the calibrations, energy resolutions and efficiencies are understood, the actual experiment data can be processed to obtain the number of detected de-excitation 7-rays and the number of projectile nuclei. Using the histogrammer SpecTcl, several contours are placed on the data and used as software gates in order to distinguish between the particle-7 coincidences for different isotopes in the cocktail beam. Ener- gies, emission angles, and counts of de—excitation 7-rays are recorded in addition to the numbers of identified projectile nuclei. 76 Table 3.4: The efficiencies for the APEX NaI(Tl) array are listed. The 6(E1ab) values are raw efficiencies for Doppler corrected energies integrated over the Q,- for each Ringi. Corrections are made for W(0) and photon absorption in the target. The sums over all 10 rings are performed for the listed values. The values used in the analysis are listed in the right-most column. The EC", (E,) are taken from the sources listed in Table 3.2. E2 transitions are assumed, and the 7-ray energies for 34Mg, 35MAI and 37Si were measured in the present work. Isotope Em, (keV) EM, (keV) 233:, (151—7) 6235“,.) w ( @0536]; 8:172th I... 26Mg 1808 1344—2264 12.2% 11.6% 10.7% 32Mg 885 650-1115 20.47% 20.45% 17.9% 34Mg 659 487-822 25.5% 26.2% 21.8% 35A1 985 734-1272 18.9% 17.7% 14.7% 36A1 967 713-1216 19.2% 18.0% 15.1% 37Si 1437 1043-1822 14.3% 13.3% 11.1% 96Mo 778 589-964 22.1% 22.3% 21.4% 3.2.1 Projectile Nuclei The particle identification is achieved by placing contours on the energy loss (AE) versus time-of-flight spectrum built from the zero degree detector (Section 2.2.3) fast plastic signal and the cyclotron radio frequency (rf). Four sets of these particle software gates were required over the entire “Mg,”36 Al,37 Si run to account for small shifts in the time-of—flight. For the remainder of the runs 96Mo,26 Mg, and 480a, only one set was necessary. Each set consists of one contour per isotope per rf pulse. Figure 3.14 shows one set of the particle gates for the 34Mg beam. The radio frequency was 21 MHz corresponding to 47 .5 ns between pulses. The particle contours are used as a condition for binning the 7 rays. A de— excitation 7-ray is binned as originating from a particular nucleus only if it is detected in coincidence with a projectile nucleus that falls into that particular particle’s soft- ware gate. When the data is completely scanned, integrating the AE/time—of-flight spectrum also gives the number of incoming projectile nuclei per particle gate (Np) 77 446 ~— ' 1 7‘? § 696 J: a .33 340 3 a E 296 L5 246 _ 196 l I I l 286 536 586 636 686 Time of flight (rt) (channels) Figure 3.14: Contours on the AE vs. time-of—flight spectrum serve as software gates identifying the isotope species. The 34Mg cocktail is shown here. divided by the zero degree detector’s down—scale factor, as listed in Table 2.2. 3.2.2 Time Cuts Accidental coincidences with 7 rays originating from the zero degree detector’s plastic can also be excluded through software gating. They can be excluded from the 7-ray energy spectra by making cuts on the time spectra. This is illustrated in Figure 3.15 which depicts the following example and shows the time spectrum and effects of the time cuts on the energy spectra. In the case of 26Mg, the distance from the target to the array is 25 cm, from the target to the zero degree detector 122 cm, and from the zero degree detector to the center of the array is 124 cm. The speed of the projectiles after the target is 0.30. Assuming the 7 rays to be traveling at the speed of light, we can deduce from these values that 7 rays originating from the zero degree detector should be detected 15 us after those originating at the secondary target position. The delay is evident in the time spectra in which the TDC stop is binned. The 15 ns delay is resolvable in this spectrum and so cuts may be placed on the 7 rays originating at 78 200 t l . Target 300 ‘ Zero Degree 3 g _ S 200 Detector 8 100 8 '( 100 0 ;J_L_lfi 0 l l 0 1000 2000 3000 O 1000 2000 3000 Energy (keV) Energy (keV) :9 c: :1 o O 500 1000 1500 Time (channels) Nal(Tl) I t1 t2 Zero 26 Mg : ll : Degree d = 1-22 m Detector Target Figure 3.15: The time cuts for 26Mg are shown. The top panels show the effects of the time cut on the Doppler-corrected energy spectra. On the left is the spectrum resulting from the cut made on 7 rays originating at the target position. On the right are the 7 rays originating at the zero degree detector. The time spectrum is shown in the middle panel with 10 channels per nanosecond (1 ns/ 10 ch). The origination of the 7 rays is depicted in the drawing at the bottom. 79 [ 26Mg ] ] TOTAL ] E, Fit Systematic <2% <0.5% < 12.1% 0 C E N7 6intrinsic 6corr 2.8% 2.9% 1% 14.8% B(E2) 003 0m :t 0.1° 4.8% 7.3% :1:8.8% Table 3.5: Uncertainties in 7-ray energy, E,, Coulomb excitation cross section, 003, and the reduced quadrupole transition probability B(E2) value for 26Mg. the target and applied to the rest of the 7-ray spectra. 3.3 Error Analysis Uncertainties in the measurements and extracted B(E2) values are listed in Table 3.5. The largest contributions to the error were the statistical uncertainties associated with peak fitting. Systematic errors resulted from target placement (i5mm) and the efiiciency estimation. The statistical uncertainty in E, was primarily due to the large intrinsic resolution (Section 3.1.3) of the APEX N aI(T l) array. Contributions via Doppler broadening by the beam spread and uncertainty in the energy loss in the target are insignificant when compared to the energy resolution. The statistical error is then fully determined by the peak fit. Systematic errors arise from the energy calibration which contributes less than 2%, and the placement of the target with a contribution less than 0.5%. The uncertainty in the Coulomb excitation cross section results from counting the number of detected 7 rays N,, the intrinsic efficiency estimation, and the corrections to the efficiency by photoabsorption in the target and the angular distribution. For N,, the entire uncertainty comes from counting statistics as determined using the fitting program GF 2. Uncertainty in the intrinsic efficiency has a combination of contributions. This 80 includes the uncertainties in the half-life t1 )2, initial activity, A0, and the peak fits, which were folded together through error propagation [94]. The final uncertainty in the intrinsic efficiency is 2.9%. The corrections to the intrinsic efficiency due to photoabsorption in the target and the angular distribution, W(6), were on the order of 10%. Uncertainties on these 10% corrections were 10% for the absorption cross sections and negligible for W(0). An uncertainty of 10% on a 10% correction results in a 1% contribution to the overall uncertainty in the efficiency. Together, for the example of 26Mg in Table 3.5, the uncertainty in the Coulomb excitation cross section is 4.8% after the error propagation. Uncertainties in the num- ber of target nuclei, N t, and the number of projectile nuclei, N p, are small compared to those of the cross section and the efficiency, and may be omitted. For the B(E2) value, errors arise from the cross section, and the constant of proportionality (equation 2.10) which depends heavily on the maximum scattering angle. 6m was taken to vary by i0.1° and when combined with the uncertainties in the cross section, 003, the total uncertainty in the B(E2) value for 26Mg is 8.75%. The analysis of the uncertainties in the other nuclei follow the same treatment, but will not be shown. The values for the first excited—state energy, Coulomb excitation cross section and B(E2 1‘) were obtained with the method outlined here, and will be presented in the next chapter. 81 Chapter 4 Observations Observed Coulomb excitation cross sections, and excited state energies were measured and B (E2 T) values were extracted from the experimental data with the general analysis methods outlined in Chapter 3. In total, nine nuclei were analyzed: the secondary fragments 26Mg, 32 Mg, 34Mg, 3536Al, and 37Si, two degraded primary beams 96M0, and 48Ca, and the target nucleus 197Au. The well-known nuclei 96Mo, ”Mg, and 197Au were used as test cases. Beam production and parameters can be found in Table 2.1, and reaction specifics in Table 4.1 and Table 4.3. 4.1 Degraded 96Mo Prior to the main experiment, the B(E2; 0;, —+ 2]) and E(2f) values of a degraded 96Mo primary beam were measured via Coulomb excitation with a 0.184 g/cm2 197Au target as a first test. 96M0 is an even-even Z = 42, N = 54, stable nucleus with a relatively low first-excited state energy and high B(E2 T) value. The adopted values for E(2f) and the B(E2; 0;, —-> 2?) are 778.245(12) keV and 2711(50) e2fm4 respec- tively [17]. The nucleus was measured several times via Coulomb excitation [98—101] in which 96Mo participated in the reaction as a target nucleus, rather than the projectile as is the case here. 82 197 Au(96M0,96 Mo Y) 1000 t 800 L 600 ’ 400 L 200 r ] v = 0.34c Counts / 40 keV 800 ~ 778(16) keV / 600 ~ 400 L 200 l .xxxl....l.xxalx...l..r. 0 400 800 1200 1600 Energy (keV) Figure 4.1: Gamma-rays in coincidence with degraded 96Mo nuclei. Energies without Doppler reconstruction are displayed in the top panel while those following Doppler reconstruction in the projectile frame are displayed at the bottom. The mid-target velocity is listed and is assumed to be the velocity at the time of 7-ray emission. Gamma-rays originating from the 197Au target are expected at 547 keV in the upper panel but are not distinguishable from the low energy background. Coulomb de-excitation 7-ray energy spectra for each of the positions in the APEX array were obtained by the methods outlined in Chapter 3. The spectra for each of the slices not excluded by the position calibration were then summed over all 24 detectors to create the final total energy spectrum. Both the projectile-frame (v = 0.34c) and laboratory-frame (v = 0) spectra are shown in Figure 4.1. The photopeak at 778(16) keV was fit on top of a continuous quadratic background of the form N = 200 — 15.82: + 0.5496 where N is the number of counts in channel :12. The fit yields 3273(134) counts in the photopeak corresponding to a Coulomb excitation 83 500 48 48 2°93“ Ca, Ca 7) v = 0 400 - 300 v- 200 - 100— Counts / 20 keV 400 ' 300 “P 200 T- . 3831.72 keV 100 : 0 l 000 2000 3000 4000 Energy (keV) Figure 4.2: Gamma-rays in coincidence with 48Ca nuclei after the reaction 209Bi(“8Ca,48 Ca 7). The arrow marks the location of the expected 3831.72 keV peak. cross section of 182(11)mb, and a B(E2;0;, ——> 2;“) value of 2640(260) ezfm“. Both first excited state energy and reduced transition probability values agree with the adopted E(2f) value of 778.245(12) keV and the B(E2; 0+ —) 2?) value of 2711(50) 9.3. ezfm“ [17]. 4.2 The 48Ca Primary Beam All secondary beams were produced by fragmentation of the “Ca primary beam. In addition, the 48Ca itself was degraded and directed to the N3 vault for Coulomb excitation with the 0.980 g / cm2 209Bi target also used with the 26Mg, and 34Mg beams. 48Ca is a stable, even-even nucleus with 20 protons and 28 neutrons. 480a has a 84 high E(2f) of 3831.72 keV [17], and the known B(E2;0;,. —> 2]) value of 95(32) ezfm4 [17] yields a Coulomb excitation cross section of only 10.4 mb for this setup. This corresponds to 114 expected counts in the photopeak with an expected width of 410 keV (FWHM) for the total 1.3 x 108 incoming projectile 48Ca nuclei. The energy spectra of the 7-rays detected in coincidence with the 48Ca are shown in Figure 4.2. The spectrum is the sum of the energy spectra of all 24 detectors after particle and time and position gates are applied. For the projectile-frame spectrum, the energies are Doppler shifted into the projectile frame on an event-by-event basis. The goal of the analysis of the 48Ca energy spectrum is to obtain a background spectrum. In order to estimate the contributions to the overall spectrum by the 48Ca de-excitation 7-rays, the interaction of the known 3831.72 keV 7-ray corresponding to the 0 —) 2+ transition was simulated using GEAN T. The contribution to the energy spectrum by 7 rays originating in 48Ca over the energy range covering the lowest and highest energy Doppler shift in the whole experiment, 487-2264 keV, which is the range of interest for the experiment, was estimated by integrating the simulated spectrum in that energy range and comparing it to the integral of the photopeak. For 10,000,000 simulated counts overall, there were 4.0 x 105 counts in the photopeak and 6.7 x 105 counts in the energy range of 487-2264 keV. From the adopted B(E2) value of 95 e2fm“, 114 counts are expected for the photopeak in the experiment. Scaling the simulated spectrum to this expected number, 2.1 counts per 20 keV from the first 2+ de-excitation 7-rays are expected in the experimental data for Figure 4.2 in the energy range 487-2264 keV. The range is below the single-escape, double- escape and photopeaks and so the contributions can be mainly attributed to Compton scattering, the target de-excitation and accidental background radiation. The random background has not been subtracted from the spectrum, and we use this spectrum as part of the background that will be scaled to each experiment. There is a small cross section of 0.617 mb for the Coulomb excitation of 209Bi in this reaction with 48Ca as well (Section 2.2.2), and 7-rays originating there make 85 a similarly small contribution to the overall spectrum. For the 48Ca experiment we expect 33.9 detected de-excitation 7-rays originating from the 209Bi target spread from 700 to 1235 keV in the projectile—frame spectrum. Because both the 48Ca and the target 209Bi make only small contributions to the full 48Ca energy spectrum in the energy range of 487-2264 keV, the majority of the 7-rays in that range for the 48Ca spectrum are described as room and beam-related background counts. A fit energy spectrum of 209Bi("8Ca,48 Ca 7) over 600-2400 keV is therefore used in determining the shape of the background for the other nuclei in the experiment. The form of the fit is N = 51.73 — 1.03551: + 0.02079:2 where N is the number of counts in channel :12. For each of the other nuclei in the experiment, this background form needs to be scaled by the number of target nuclei times the number of incoming beam particles Nth. Below 600 keV, the energy spectra show a dependence on A and Z which cannot be extracted consistently from the data available here. The dependence is shown in Figure 4.3 which shows the low-energy 7-ray spectra of 197Au(96Mo,96 Mo 7), 209Bi(“8Ca,48 Ca 7), and 209Bi(26Mg,26 Mg 7) after normalization to the target and projectile nuclei, N, and N,,. In order to understand the shape of the low energy background for Z = 12, the 48Ca fit was scaled by the quantity Nth to the 26Mg spectrum. This scaled fit, N = 100 — 2:1: + 0.04:1:2 where N is the number of counts in channel :12, describes the shape of the 26Mg spectrum in the low-energy range, and was used as the background shape after scaling by the number of target and projectile nuclei, N, and N,,, for the Mg, Al, and Si isotopes. 4.3 The Test Case 26Mg Coulomb excitation of the stable nucleus 2'i’Mg on a 0.980 g / cm2 209Bi target provided a test of the experimental apparatus and data analysis method. Gamma-ray energies and B(E2;O;,_ —> 21+) values were checked against the adopted [17] E(2f) value 86 1057 96 - 4 fix/ Mo [3 — 0.34 10 - f ,7 48Ca B = 0.32 10,] 7 ”Mg 6:036 Counts / 20 keV / ( Nt pr 10'29 ) 0 ' 800 - 1600 ' 2400 V 3200 Energy (keV) Figure 4.3: Energy spectra for 197Au(”Mo,96 M07) at 55.9 MeV/ nucleon, ”9Bi(48Ca,48 Ca 7) at 51.0 MeV/ nucleon, and 209Bi(”Mg,” Mg 7) at 66.8 MeV / nucleon are shown on a log scale normalized by the number of target and pro- jectile nuclei, M, N,,. The plot shows the low-energy background for nuclei of different A and Z. Mid-target velocities are listed for each nucleus. of 1808.73(3) and B(E2;0;,. ~+ 2f") value of 305(13) e2fm4 for ”Mg. In addition, the energy resolution of the ”Mg peak was used to incorporate in-beam data and the uncertainty in the Doppler reconstuction into the characterization of the energy resolution of the APEX NaI(Tl) array as discussed in Section 3.1.3. Gamma rays in coincidence with the stable nucleus ”Mg are shown in Figure 4.4. The background was not subtracted, instead, the parametrization of the 48Ca background was used to fit the spectrum in addition to a Gaussian for the photopeak. The 48Ca background was scaled by the number of target and incoming projectile nuclei Nth from the 48Ca fit described in Section 4.2 by a factor of 2.1. This resulted in a quadratic background N = 100 — 29: + .042:2 for the ”Mg energy spectra, where N is the number of counts in channel 2:. In the ”Mg projectile-frame spectrum there are 3376(116) counts on top of the background in the 1808(38) keV photopeak for ”Mg corresponding to a Coulomb excitation cross section of 44.4(19) mb, a B(E2;0;_,. —> 21‘) value of 315(28) ezfm“ and the deformation parameter [62] value of 049(2). Both energy and reduced tran- sition matrix element values are in agreement with the adopted [17] E(2f) value of 87 1808.73(3) and B(E2; 0;, ——> 2;) value of 305(13) ezfm". Two peaks of unknown origin are visible in the spectrum corresponding to ener- gies in the laboratory frame. One peak is seen at 926 keV, corresponding to counts from the 9/2‘ —9 7/2‘ 896.28(6) keV transition in 209Bi plus additional background contribution. Additional 7-ray flux of unknown origin apparently at rest is observed at 528(6) keV. In addition to these are the ”Mg de-excitation photons which are spread over a range of 1340-2264 keV, and the ”Mg Compton continuum. In order to estimate the contribution of the 528(6) keV and 926(15) keV peaks to the background estimation, they were fit using GF 2 varying the width, position and height on a con- tinuous quadratic background. The cross sections for the 528(6) keV and 926(15) keV peaks are 4.8(4) mb and 5.0(10) mb respectively. While the origin of these peaks is unknown, we conclude that they originate from sources at rest in the laboratory and thus we expect them to contribute to the energy spectra in coincidence with 32'3‘4Mg as well. We estimate the contribution by scaling the ”Mg spectrum. The cross sections for the 528(6) keV and 926(15) keV peaks are 4.8(4) mb and 5.0(10) mb respectively. This corresponds to 19(2) counts in the 528(6) peak and 12(2) counts in the 926(15) keV peak for 32Mg. For 34Mg the estimate is 8(1) and 5(1) counts respectively. In the projectile frame, the Doppler shift spreads the 528 keV peak over 300 keV and the 926 peak over a range of 512 keV on average for the ”’32’34Mg spectra. For 32Mg therefore, there are 0.060(6) counts per keV over the range of the 528 keV peak and 0.024(4) counts per keV from the 926 keV peak. In the 34Mg projectile-frame spectrum, the estimates are 0.030(3) counts per keV and 0.010(2) counts per keV for the 528 keV and 926 keV peaks, respectively. The contributions by these 528(6) keV and 926(15) keV peaks in the 323“Mg spectra are minimal. 88 80 ] 2093“ 26Mg,26Mg Y) ] I97Au( 32Mg,32M g Y) ~ «528(6) keV I 547(11) keV 60 ] _ 926(5) keV , v - 0 | x0.067 ] 4 40 ’ ' v = 0 l I I I I 20 > f , 32 E I l ‘ 3 0’.m..............- lam E x 0.067 ] v = 0.37c » v = 0.36c E: V = 0.360 I ’ g 1808 38 k V. 885(18)keV U 60 ’5 ) e ] ” ] 659(14) keV I I I / I I I I 40 _ l I II I I . I 1- F i 20 ” | I I : w I . I I I") ~ . O ....l...al....l ........ 1....1 ........ l....1 JI LLI 0 800 1600 2400 0 800 1600 2400 0 800 1600 2400 Energy (keV) Figure 4.4: Gamma-ray energy spectra are shown for ”Mg, ”Mg, and 34Mg, left, middle and right respectively. Spectra are displayed in the top panel without Doppler reconstruction, and in the bottom panels after they have been Doppler reconstructed to the projectile frame. Velocities listed for each of the reactions are mid-target values. The ”Mg spectra are scaled by 0.067 for display purposes. In the top panel for ”Mg the two background peaks as described in the text are shown. In the top panel for ”Mg, the 197Au 7-rays are seen. In the top panel for 34Mg, the 659(14) keV 7-rays are Doppler boosted to 611(13) keV at 90°. The bottom panels show the reconstructed energies in the frame of the moving projectile. 89 834.85 keV 1400000 - / 54M“ Compton Edge > 600000 - .12 0 v a 0 A 1 I J: {53’ 661.66 keV 137 2200000 . / CS 1400000 -I Compton Edge 600000 - 0 1L . . . - . 3 0 400 800 1200 1600 Energy (keV) Figure 4.5: Measured energy spectra for the calibration sources 54Mn (top) and 137Cs (bottom). The spectra are used to determine the height and position of the Compton edge at these energies. The solid lines are GF2 fits. The energies listed are the adopted values [17]. 4.4 32Mg and 197Au In the analysis of the 197Au(” Mgf‘2 Mg7) reaction, de-excitation 7-rays resulting from the Coulomb excitation of both the projectile ”Mg and the target 197Au were binned for the sum of the 24 detectors and are shown in Figure 4.4. For each spectrum, the same-frame photopeak, Compton scattered ”Mg events, and the Doppler broadened opposing-frame photopeak were fit on top of a quadratic background using GF2. In the 7-ray spectrum for ”Mg we see a peak in the laboratory frame at 547 keV and another after the energies have been reconstructed to the projectile frame corresponding at 885 keV. The 547 keV photopeak is broadened after Doppler reconstruction to the projectile frame and is also visible. The 885 keV photopeak is visible in the laboratory frame as a broad peak as well. 90 4.4.1 Background and Compton Contribution The shape of the background was taken from the 48Ca fit described in Section 4.2 scaled by 0.042 the number of projectile nuclei and the number of target nuclei, N,, N,,. The final quadratic for the background was N = 2.15 - 043016 + .00051:2 for the ”Mg and 197Au energy spectra, where N is the number of counts in channel 1:. The contribution to the spectra by Compton scattered ”Mg events was estimated by fitting a Gaussian to the Compton edge for the calibration source 54Mn which has a photopeak at 834 keV. The 54Mn fit is displayed in Figure 4.5. The FWHM of this Gaussian is 2.10 times the FWHM of the photopeak, the height is 0.25 times the height of the photopeak. The height of the ”Mg Compton edge was fixed by these parameters in the final ”Mg and 197Au fits. Because of the energy resolution of the APEX array and the close proximity of the ”Mg and 197Au photopeaks, it is important to consider the width of the ”Mg photopeak in the laboratory frame and the 197Au photopeak in the projectile frame. The de-excitation 7-rays emitted by the ”Mg projectile are spread over 650— 1115 keV in the laboratory frame, while those emitted by the 197Au target have a range of 434-745 keV in the projectile frame. In addition to the Doppler spread, the energy resolution of the N aI(T1) array is taken into account when determining the width of the ”Mg peak in the laboratory frame spectrum and 197Au in the projectile frame spectrum. The range including the largest Doppler shifted value at its expected width due to the resolution of the array, and the smallest Doppler shifted value at its expected width is 310-890 keV for 197Au target in the projectile frame and 508-1285 keV for the projectile ”Mg in the target frame. These energy ranges are used to determine the width of the opposing-frame photopeak in the final fit. 91 4.4.2 197Au The peak observed at 547 keV is a superposition of contributions from the 197Au photopeak, the Doppler broadened ”Mg and the ”Mg Compton edge. All of these contributions are considered. From the adopted B(E2 T) value of 4488(408) e2fm4 [77], the expected cross section for the 3/2+ —> 7/2+ 547 keV transition to the first excited-state in 197Au is 20 mb and the number of expected 197Au de-excitation 7- rays is 90 for the reaction with ”Mg. The energy resolution of the 197Au peak in the target frame is expected to be 18.5% based on the analysis in Section 3.1.3. This corresponds to a FWHM of 101 keV and the standard deviation, 0, of 43 keV. The 547 keV peak is spread over 460-633 keV for a range of 21:20. This peak along with the Doppler smeared ”Mg peak was fit on top of the 48Ca scaled background. In order to estimate the full contribution due to the Compton scattering of ”Mg, the height of the Gaussian estimated from the 54Mn Compton was 25% of the photopeak. In ”Mg this corresponds 0.430(2) counts per keV. This height of 0.430(2) counts was multiplied by the range extending to $20 of 461-633 keV of the 197Au peak, resulting in a flat contribution by the Compton scattered events of 70(3) counts over the 172(7) keV. This is taken to be a maximum Compton contribution. There were 159(16) counts under the fit of the full Gaussian and subtracting the estimated Compton contribution resulted in 89(10) counts. This corresponds to a lower limit for the cross section for the Coulomb excitation of 197Au of 21(3) mb and a lower limit for the B(E2 T) of 4609(876) e2fm4. 4.4.3 3""Mg The previously-measured and predicted E(2;L ) and B(E2; 0;, —1 2?) values are dis- cussed in Section 1.5. The adopted E(2f) value is 885.5(7) keV with a B(E2 T) value of 390(70) ezfm4 [17] for the 0;, —> 2;" transition. Of the measurements of the B(E2 T), all of the B(E2; 03.3. —-> 21+) values agree with the adopted value when the 92 40 . 193°)“(32M8132M8 II) - 547(ll)keV v=0 30 f ‘ 885(18)keV > 1 Q) .14 a F I— \ 20 H g; . C'- '. :3 . O 10 ~ V I I A 0 I 'llll 0 400 800 1200 1600 2000 Energy (keV) —T Figure 4.6: 197An energy spectrum showing the fit determined with GF2. The Doppler- broadened photopeak of ”Mg is included with the 197Au photopeak. data are analyzed the same way, with the exception of [41] who measured the large value of 622(90) ezfm4 [41]. The energy spectrum for ”Mg after reconstruction to the projectile frame shows two peaks, one at 885 keV and a broad peak at 547 keV corresponding to the transition in the 197Au target nuclei. The ”Mg peak was fit in a similar manner to the 197Au peak. The background was again scaled by Nth from the 48Ca fit and the Doppler broadened 197Au peak and ”Mg Compton edge fit with the parameters determined above. The 885 keV photopeak is also fit with a Gaussian, and while the 1436 keV 7-ray [25,38,42,43] is noticable, there are only a few counts in the peak. The fit is shown in Figure 4.7. The 197Au peak spread over 21:20 was 351-842 keV. The Compton contribution by ”Mg was estimated by multiplying the height of the Compton edge expected from the 54Mn source measurement over that range. The height of the 54Mn Compton edge is 25% the height of the photopeak, corresponding 93 : 19kt“ 32Mg,32 Mg Y) 50 ,7 v=0.37c [ 547(ll)keV 40 _ 885(18)keV I/ > . :2 I N I *3 I I , 320: I . * ‘1' “M | I [L] 01“ . 0 400 800 1200 1600 2000 Energy (keV) Figure 4.7: ”Mg energy spectrum after Doppler reconstruction in the projectile frame showing the fit determined using GF2. to 0.430(2) counts at the height of the Compton edge for ”Mg. This height multiplied by the range of the 197Au peak leads to an expected contribution to the spectrum in the range 351-842 keV of 207 counts by the Compton edge of ”Mg and the Doppler broadened 197Au peak together. The result of the fit of the ”Mg photopeak with no correction for feeding from the 2321 keV state is 252(25) counts in the ”Mg photopeak at 885(18) keV, and an average of 101(33) counts due to the Doppler broadened 197Au peak. The Coulomb excitation cross section is 91(10) mb and the B(E2; 0;, —> 2?) value extracted from our data for ”Mg is 447 (57) e2fm4 which results in a value for [62] of 0.51(3). We treat the feeding of the 885 keV state by the 2321 keV state as in [25]. Although the number of counts in the 1436 keV peak is small compared to the photopeak of the ”Mg which has 252(25) counts, a maximum area under this peak of 41 counts is estimated by multiplying the height of the highest channel above the background, approximately 12 counts, times a width of 20 above and below 1436 keV, where 94 0 = 68 keV is the standard deviation obtained from the expected FWHM of 160 keV. Because the actual area of the peak does not extend over the full area of the rectangle defined by the width 143631220 keV times the height of the highest channel over the background, the area of this rectangle was then divided by 2 to obtain an estimate for the maximum number of counts under the 1436 keV peak. Because we do not know the spin and parity assignment for the 2321 keV state, we compare values for transition probabilities for E1, M1 and E2 transitions extracted from our estimated maximum number of counts in the 1436 keV peak to those listed as recommended upper limits by Reference [50]. The reduced transition probabilities extracted from our measured maximum number of counts corresponding to a cross secition of 23 mb for this setup in the 1436 keV peak are B(E1;0;,_ —> 1‘) = 0.004 e2fm2, B(M1;0;3. —9 1+) = 0.36 111%), B(E2;0;8, —) 2+) = 115 ezfm4 and B(M2;0;8. —) 1‘) = 12 pfifmz. The recommended upper limits are B(E1;09+_,, —> 1‘) = 0.06 e2fm2, B(M1;O;,, ——> 1+) = 8.95 #126, B(E2;0;,. —+ 2+) = 600 e2fm4 and B(M 2; 0;, —> 1“) = 830 pfifmz. The only transition for which our measured value exceeds the recommended upper limit is the M2 transition. The number of 7-rays actually emitted at 1436 keV can be estimated by making an efficiency correction to the estimated number of counts in the 1436 keV peak in our spectrum. The value for an efficiency-corrected number of counts emitted at 1436 keV is 372 counts. For each of these 372 counts, a 7-ray is emitted at 885 keV and detected with an efficiency of 18%. This corresponds to a maximum 67 counts detected in the 885 keV photopeak that are caused by feeding from the 2321 keV level. With this correction, the number of 885 keV de-excitation 7-rays becomes maximum 185(25), which yields a B(E2 T) minimum value of 328(48) e2fm4 and the minimum deformation parameter [62] of O.42(3) including the maximum correction for feeding from the 2321 keV state. 95 40 - 209131641111,“ng Y) : v=0.36c 30 L I g 659(14)keV a, . g- / .2 O o ' U Q 20 ~ :9 F g . O U 10 111.11111l..111111111. 0 0 400 800 1200 1600 2000 Energy (keV) Figure 4.8: 34Mg energy spectrum. The Compton edge estimate is shown for demon- stration purposes only. The actual fit did not include this Gaussian for the Compton edge. 4.5 34Mg The previously-measured E(2f) and B(E2; 0;, ——> 21+) values are discussed in Section 1.6. These measurements indicate that 34Mg is more collective than ”Mg [38,102]. The energy spectra for 34Mg are shown in Figure 4.4. One peak is visible in each of the spectra. In order to fit the photopeak after Doppler reconstruction to the projectile frame, a Gaussian was fit on top of the parametrized 48Ca background. The background was taken as the form of the 48Ca fit scaled by the number of target and projectile nuclei, Nth, a factor of 0.016 for 34Mg. The quadratic background is described by N = 0.86 — .01732: + .00032:2 for the 34Mg energy spectra, where N is the number of counts in channel 2:. Gamma-ray spectra gated on both time- and particle-gates in coincidence with 34Mg are shown in Figure 4.4. The contribution to the spectra by Compton scattered 34Mg events was estimated 96 Table 4.1: Coulomb excitation cross section parameters for the 0;, -—> 2f" transitions in ”*””Mg and 96Mo. E, is the 7-ray energy, N, is the number of nuclei in the target, N B is the number of incoming projectile nuclei, N, is the number of detected 7 rays, 0 is the Coulomb excitation cross section, and the B(E2 T) is the reduced electric quadrupole transition probability. E 06m 0 Nucleus (keff) deg. v/c N, NB N, (mb) ”Mg 1808(38) 2.68 0.36 2.82x1021 2.51x108 3376(116) 44.4(19) 96Mo 778(16) 3.69 0.34 5.63x10” 1.44x108 3273(134) 182(11) ”Mg 885(18) 2.63 0.37 2.96x1021 5.25x106 252(25) 90.6(95) ”Mg 659(14) 2.77 0.36 2.82x1021 2.14x106 166(28) 126(22) by fitting a Gaussian to the Compton edge for the calibration source 137Cs which has a photopeak at 662 keV. The 137Cs fit is displayed in Figure 4.5. The FWHM of this Gaussian is 2.13 times the F WHM of the photopeak, the height is 0.22 times the height of the photopeak. The height of the Compton edge was fixed by these parameters in order to estimate the Compton contribution to the final ”Mg fit. Contributions to the spectrum by 209Bi target excitations can be neglected. Less than one count total is expected from target excitations over the energy range of 715-1210 keV. The final fit of ”Mg was performed by constraining the width of the peak to 19.1% FWHM, according to the analysis in Section 3.1.3. For an energy near 650 keV as is seen in the ”Mg spectrum, the energy resolution of the APEX array is expected to be 0.19. The peak fit on top of the quadratic background given above is shown in Figure 4.8 with an additional display of a Gaussian representing the Compton edge. The Compton edge was not included in the final fit. The number of 7 rays in the ”Mg photopeak is 166(28), at an energy of 659(14) keV. The resulting cross section is 126(22) mb and the B(E2;0;3. —1 21*) value extracted from our data is 541(102) ezfm“ corresponding to the deformation parameter [)82] = 054(5). 97 Table 4.2: Extracted and adopted B(E2 T) values for the 03“,. —) 2? transitions in ”’32’”Mg and 96Mo. E, is the 7-ray energy, 0 is the Coulomb excitation cross section, and B(E2 T) is the reduced electric quadrupole transition probability. The ”Mg extracted value listed has not been corrected for feeding of the 885 keV state by the 2321 keV state. E Extra A ed 2 Nude“ (key) (1:15) “2335‘“ T) (10132276113513 T) ”Mg 1808(38) 44.4(19) 315(48) 305(13) ”Mo 778(16) 182(11) 2640(260) 2711(50) ”Mg 885(18) 90.6(95) 447(57) 390(70) ”Mg 659(14) 126(22) 541(102) none 4.5.1 Nuclear Contribution The possibility for contribution to the cross sections of ”I”I”Mg by direct nuclear reactions was evaluated through an ECIS88 coupled-channels calculation [103]. The calculation assumes isoscalar behavior for which the Coulomb deformation 50 is as- sumed to be the same as the nuclear deformation EN, 30 = )BN. The optical model parameter set is taken from that determined for the 170+2”le reaction [47]. Figure 4.9 shows the calculated differential cross sections for 3‘lMg, with Coulomb only con- tributions and with Coulomb plus nuclear contributions, versus scattering angle. The calculations show the nuclear contributions to be at maximum 7% for all 3 nuclei. Similarly small contributions have also been reported in References [12] and [82]. 4.6 3536A] and 37Si In addition to the even-even nuclei already discussed, three odd nuclei were con- stituents of the ”Mg cocktail beam. Gamma-rays resulting from Coulomb excitation reactions of 35”Al and 37Si with the 0.980 g/cm2 209Bi target were recorded and ana- lyzed following the general analysis procedure outlined above. Excited state energies and B(1rA) values were measured and extracted. Particle identification was performed by placing software gates on the plastic phoswich detector’s energy loss versus time— 98 of-flight spectrum. The energy spectra were constrained by these particle gates as well as prompt time cuts and then fit using GF 2. The form of the background was again taken from the 48Ca background scaled by the number of projectile nuclei and the number of target nuclei. The spectra are shown in Figure 4.10 Of these three nuclei, only 35 Al and 37Si have been observed via Coulomb excitation before. Excitation energies and B(E2 T) values were observed [29] to be 1006(19) keV and 142(52) ezfm4 for 35Al and 1437(27) keV and 101(45) ezfm“ for 37Si. The ground state spin and parity for 37'Si has a value of (7 / 2‘) [104] while for 35”Al there are no adopted values for the spin and parity. The ground state spin and parity for 35”Al and 3"'Si have been predicted by an OXBASH shell model calculations [105] to be 5/2+ for 35Al, 4‘ or 5‘ for 36A1 and 7/2‘ for 37Si. Excited state energies and B(1r/\) values for 36Al have not been measured previously. 1W I I I I I I I ‘ Coulomb + Nuclear 10(X)0 :7 ‘2 __ - / g “”0 :- Coulomb 3 . ca 100 E g . 10 r E ; 59° ‘ “i 0‘1 r Maximum 02/ r m 0.01 h l J 1 l l l 1 0 1 2 3 4 5 6 7 0cm(deg) Figure 4.9: ECIS calculation for ”Mg on ”9B1 at 67.3 MeV/nucleon. The differen- tial cross section is shown as a function of center-of-mass scattering angle for both Coulomb only and Coulomb plus nuclear contributions. 99 80 2°93“ 35A1, ”A1 7) 2°"Bi( ”Al, "in 7) 209Bi( 37Si, 3si 7) =0 =0 60r ’ v ' v 4o 20 > .3 3 O ....1...-1 ....... Am: -1 r g v = 0.38c v = 0.37c O U 60 . 985(21) keV l / 40 674(14) keV 142,760) keV l 967(20) keV / 20 . + 0‘ A ‘3‘00‘ A food ‘2‘40‘0‘ M d ‘ sod ‘1A60AO‘ 3460‘ d A 30611506 A A2460” Energy (keV) Figure 4.10: Energy spectra are shown for 35Al, 36Al, and 3"’Si, left, middle and right respectively. The bottom panels show the 'y-rays after they have been Doppler re- constructed in the projectile frame. Velocities listed for each of the reactions are mid-target values. 100 4.6.1 37Si One prominent 7-ray peak in coincidence with 37Si is observed at 1437(30) keV. The projectile-frame spectrum is shown in Figure 4.11 as it was fit with GF2. The background was scaled from the 48Ca background by a factor of 0.041 obtained by scaling by the number of target nuclei times the number of incoming projectile nuclei, Nth. The form of the quadratic background is N = 2.12 — 0.0421: + 0.0008332, where N is the number of counts in channel 2:. The width of the peak was again fixed to the value obtained from the expected energy resolution of the array discussed in Section 3.1.3. Contributions by the Compton continuum to the energy spectrum were also estimated in the same manner as the 32Mg analysis from the 54Mn source peak. The 37Si photopeak fit yielded a total of 144(22) 'y-rays for the 1437(30) keV transition. Cross sections were extracted for assumed E1 (7/2‘ ——> 9 / 2+), Ml (7/2‘ —) 9/2‘), E2 (7/2‘ —+ 9/2‘) and E2 (7/2‘ —> 11/2+) transitions resulting in 90(14) mb, 91(14) mb, 90(14) mb, and 85(13) mb respectively. The 'y-ray angular distributions do not affect cross sections significantly. From shell model calculations [105], the ground state is expected to have a spin and parity of 7/2‘. The corresponding B(7n\) values are B(E1;7/2‘ —-> 9/2+) = 0.078(12) ezfm2, B(M1;7/2‘ —+ 9/2‘) = 0.0079(11) p’f’v, B(E2; 7/2“ —> 9/2‘) : 380(63) ezfm4 and B(E2;7/2‘ —> 11/2‘) 2 360(62) ezfm“. 4.6.2 35Al One clearly distinguishable peak is observed at 985(21) keV in coincidence with 35Al and are shown in Figure 4.12 with the GF2 fit. The lower energy 'y-rays centered at approximately 650 keV are of unknown origin as they are visible in the laboratory- frame spectrum as well, and so were not fit. Gamma rays in coincidence with 35A1 are shown in Figure 4.12 with the GF2 fit. The background was scaled from the 48Ca background by a factor of 0.048 obtained by scaling by the number of target nuclei times the number of incoming projectile nuclei, Nth and is of the form N = 101 4O 7 209 37 37 ll Bi( Si, Si y) v=0.380 30 ‘ 1437(30) keV I 20 Counts / 20 keV 10 V T j Y K V V Compton 0 ....1....1....1....1.m.r-. 0 400 800 1200 1600 2000 Energy(keV) Figure 4.11: Energy spectrum after Doppler reconstruction in the projectile frame in coincidence with 37Si. The fit includes both the photopeak and the estimate for the position and height of the Compton edge. 2.48 - 0.0499: + 0.001332, where N is the number of counts in channel 3:. The width of the peak was again fixed to the value obtained from the expected energy resolution of the array discussed in Section 3.1.3. Contributions by the Compton continuum were also estimated in the same manner as the 32Mg analysis from the 54Mn source peak. The peak fit yielded a total of 193(25) 7 rays corresponding to the 985(21) keV transition. Cross sections were extracted for assumed El (5/2+ —-> 7/2‘) , M1 (5/2+ —+ 7/2+), E2 (5/2+ -—> 7/2+) and E2 (5/2+ —> 9/2+) transitions resulting in 80(11) mb, 81(12) mb, 80(12) mb and 75(10) mb, respectively. We assume the spin and parity of the ground state to be 5/2+ as calculated with the OXBASH shell model code [105]. The corresponding B(7r/\) values are B(El;5/2Jr —> 7/2‘) = 0.053(7) ezfmz, B(Ml; 5/2+ —> 7/2+) 2 0.0053(8) a}, B(E2;5/2Jr ——> 7/2+) : 330(51) ezfm“, and B(E2;.5/2Jr ——) 9/2+) = 310(45) e2fm4. 102 40 u ”I 2093“ 35Al,35Al y) i 985(21)keV - ' / 30 ' v=0.380 Counts / 20 keV 20’ w 31 v Compton 0....1.. luminu 0 . . l i . l . 400 800 1200 1600 2000 Energy (keV) Figure 4.12: Energy spectrum after Doppler reconstruction in the projectile frame for 35Al. The fit includes both the photopeak and the estimate for the position and height of the Compton edge. 4.6.3 36A1 Two 'y-ray peaks are observed in coincidence with 36A] at 674(14) keV and 967(20) keV. The 36A] spectrum and fit are shown in Figure 4.13. The background was scaled from the 48Ca background by a factor of 0.024 obtained by scaling by the number of target nuclei times the number of incoming projectile nuclei, Nth. The form of the quadratic background is N = 1.24 — 0.0252: + 0.0005232, where N is the number of counts in channel :13. The width of the peaks were again fixed to the value obtained from the expected energy resolution of the array discussed in Section 3.1.3. Contri- butions by the Compton continuum were estimated from the 137Cs source for the 674(14) keV line, and from the 54Mn peak for the 967(20) keV peak. The lower energy 674(14) keV peak fit yielded a total of 60(11) 7-rays. Cross sections were extracted for assumed E1, M1 and E2 transitions resulting in the values 103 30 m | 2""13i(3‘3c\1,36.4\1 y) I v=0.37c 25 f _ l 674(14)keV I?) 20:” l / .24 . r 967(20)keV $3. _ l .l / \. 15 ‘ ,| E (I 1 :3 o . L l U 10 i “‘1 l l . ls‘, . l “M 'vl J 5 j \ i ,, | l I’. l L Compton , OI‘L . o 7100 800112001600 .2000 Energy (keV) Figure 4.13: Energy spectrum after Doppler reconstruction in the projectile frame in coincidence with 36A]. The fit includes both the photopeak and the estimate for the position and height of the Compton edge. listed in Table 4.3. Shell model calculations for this nucleus reveal two possibilities for the spin and parity of the ground state [105]. It could either have a J1r value of 4— or a value of 5‘. The corresponding B(7rA) values for a 4‘ ground state are a B(El; 4‘ —> 5+) value of 0.02(4) e2fm2, B(M1;4‘ —> 5‘) = 0.0020(3) [1%, B(E2;4‘ —> 5‘) = 160(33) e2fm4 and B(E2;4‘ —> 6‘) :- 150(27) ezfm“. For a 5‘ ground state, the values extracted from the measured cross sections are B(E1;5‘ —> 6+) = 0.02(4) e2fm2, B(Ml; 5‘ —> 6‘) 2 0.0020(3) ufv, B(E2; 5‘ —+ 6‘) 2 160(31) e2fm4 and B(E2;5‘ —> 7‘) 2: 150(27) e2fm4. The higher energy 967(20) keV peak was found to have 89(12) counts. Cross sections were extracted for assumed E1, M1 and E2 transitions resulting in the values listed in Table 4.3. For the assumption of a 4‘ ground state the transition probabilities are B(El;4‘ —> 5+) = 0.047(7) e2fm2, B(Ml; 4‘ —> 5‘) 2 0.0050(7) pi“ B(E2; 4‘ —-+ 5‘) = 310(45) and B(E2;4‘ —> 6‘) = 280(41) e2fm4, while those for an assumed 5‘ 104 ground state are B(E1;5‘ —> 6*) = 0.047(7) e2fm2, B(Ml;5‘ —+ 6‘) 2 0.0050(7) pfv, B(E2; 5‘ —+ 6‘) = 300(43) e2fm4, and B(E2;5‘ —> 7‘) = 280(41) e2fm4. The reduced transition probabilities and deformation parameters extracted from our observations are discussed as they relate to theoretical calculations in Chapter 5. 4.6.4 Upper Limits Because the spin and parity of the higher-lying states are unknown, the B(7n\) val- ues extracted from our measured cross sections were compared to the recommended upper limits for the electromagnetic transitions listed in References [50,106]. The rec- ommended upper limits for this mass region are a B(E 1) value of 0.1 W.u., a B(Ml) value of 5 Wu. and a B(E2) value of 100 Wu The smallest B(E2) values in this mass region are on the order of 3 Wu Because the largest M1 transition strengths correspond to the the smallest E2 strengths, an observation of a strong transition can be assumed to be E2 in nature. Here we compare the recommended upper limits with our measurements analyzed under the assumption of M1, E1 and E2 transitions. The cross sections and extracted transition probabilities for each of the considered transitions are listed in Table 4.3. For 37Si the recommended upper limits are 0.07 e2fm2, 8.95pr and 732 ezfm4 for the corresponding B(7rA) values. From our measured values listed above, we cannot exclude any of the transitions. For 35A], the recommended upper limits are 0.07 ezfmz, 8.95 pg, and 680 ezfm4 for the corresponding B(7rA) values. Again, we cannot exclude E1, M1 or E2 transitions. For the 674(14) keV peak in 36Al the recommended upper limits are 0.07 e2fm2, 8.95 Mg, and 706 ezfm4 respectively for the corresponding B(7rA) values, none of which can be excluded. Finally, for the 967(20) keV peak in 36Al, the recommended upper limits are 0.07 e2fm2, 8.95 pi, and 706 ezfm“ for the corresponding B (7M) values. Comparing our measurements to these upper limits, we again cannot exclude E1, M1 or E2 transitions as possible origins of the observed 'y-rays. 105 Table 4.3: Coulomb excitation cross section parameters for transitions assumed for 35'36Al and 37Si. E1, M1 and E2 are considered as possible transitions. Assumed transitions are listed and explained in the text. E, is the 'y-ray energy, N, is the number of nuclei in the target, N,, is the number of incoming projectile nuclei, N7 is the number of detected 7 rays, 0 is the Coulomb excitation cross section, and the B(7rA T) is the reduced transition probability. The units for the B(1rA T) values are ezfm” for EA transitions and 1212,,fm2"‘2 for M A transitions. E Assumed a Nucleus (kezl) 'IYansition N” N" (mb) B(fl)‘ T) 35A1 985(21) E1; 3* —> g‘ 6.22x106 193(25) 80(11) 0.053(7) M1; 3+ —> g+ 81(12) 0.0053(8) E2; 3* —+ 9 80(12) 330(51) E2; 3* —+ g+ 75(10) 310(45) 36A] 674(14) E1; 4- —> 5+ 3.06x106 60(11) 37(7) 002(4) M1; 4- —> 5- 38(7) 0.0020(3) E2; 4- —+ 5- 36(7) 160(33) E2; 4- —> 6‘ 35(6) 150(27) E1; 5- —> 6+ 37(7) 002(4) M1; 5- —> 6‘ 38(7) 0.0020(3) E2; 5- —+ 6‘ 37(7) 160(31) E2; 5- —> 7— 35(6) 150(27) 967(20) E1; 4- —> 5+ 89(12) 73(10) 0.047(7) M1; 4- —> 5- 74(10) 0.0050(7) E2; 4- —> 5- 73(9) 310(45) E2; 4- —> 6‘ 68(9) 280(41) E1; 5- —5 6+ 73(10) 0.047(7) M1; 5- —> 6‘ 74(10) 0.0050(7) E2; 5- —+ 6‘ 72(9) 300(43) E2; 5- —> 7- 69(9) 280(41) 378i 1437(30) E1; g“ —> 3* 5.39x106 144(22) 90(14) 0.078(12) M1; '5" —+ 3‘ 91(14) 0.0079(11) E2; 5 —> 3" 90(14) 380(63) E2; ; —> %— 85(13) 360(62) 106 Chapter 5 Discussion Gamma-rays resulting from the transition from states populated via intermediate- energy Coulomb excitation and the ground state have been observed for the nuclei 2632““ Mg, ”’36 A1, 37Si, 96 Mo and 197Au. Energies of the observed 7-rays were measured and reduced electric quadrupole transition matrix elements were extracted from the measured Coulomb excitation cross sections. The observables reported in Chapter 4 are discussed here as compared to adopted values, previously measured values and theoretical calculations. 5.1 The Island-of-Inversion Nuclei 32’34‘Mg 5.1.1 32Mg Our measured E(2f) value of 885(18) keV and extracted B(E2;0;,. ——> 21+) value without a feeding correction from the 2321 keV state of 447(57) ezfm“ agree with the other measured values [12, 25,38, 39]. However, our value is not in direct agreement with the 622(90) e2fm4 B(E2; 0;, —> 21‘) value measured by Reference [41], although it still agrees to within 20. The adopted, measured and calculated B(E2; 0;, —-) 2f) values for 32Mg are displayed in Figure 5.1. Several experiments have reported the 1436 keV 7-ray to be in coincidence with the 885 keV '7 ray [26,38,40,42,43], and the 107 Adopted Measured Calculated 800— a I n + H - “400- T,, g +0“) @1200— m I 0 llIlI l I l l Tl 78%%.%%.@%%%“g8 $6310}-%’%% w- 8a%‘%°%-gg $0 %5$-°?é* $938,, U“; Figure 5.1: Reduced electric quadrupole transition probabilities, B(E2; 0;, —-> 2f“), for 32Mg. Values measured by Motobayashi et al. [12], Pritychenko et al. [25], Chisté et al. [41], Iwasaki et al. [39] and the present work are compared to the adopted value [17] and a Ohw shell model calculation [107], the 3d — pf Monte Carlo Shell Model (MCSM) [22,56], and an example of a Hartree-Fock calculation (HF) [59]. Asterisks indicate that the feeding of the 885 keV state by the 2321 keV state has been included in the analysis. feeding of the 2? state is treated differently among the reports. The B(E2 1‘) value of 328(48) e2fm4 we obtain after including a contribution to the 885 keV peak by feeding from the 2321 keV state, agrees with the feeding-corrected value of 333(70) ezfm4 obtained by Ref. [25]. As discussed in Section 1.5, this N = 20 nucleus is significantly more collective than expected for a closed-shell nucleus. Although the investigation of the quadrupole moments in the sodium isotopes provides evidence that the region is most likely stat- ically deformed [35], recent fragmentation studies by Ref [43] report an E(4f)/E(2f) ratio of 2.6 for 32Mg. This is not definitively close to either the rotational limit of 108 3.3 nor the vibrational of 2. In addition, their reported 2870 keV 7—ray is a decay directly to the ground state, and a good candidate for a second 2* level indicating a possibility for shape coexistence in the nucleus [43]. We cannot draw a conclusion about the possibility of a coexisting spherical excited state from our data. Because the feeding from the 2321 keV state is not well-understood, we will com- pare our uncorrected values to the theory. Our E(2{L), B(E2;0;,. —-> 2f) and ,82 values agree with shell model calculations which include 2hw intruder configurations in the ground state [30, 57] and as expected, deviate from Ofiw models [32]. The best agreement comes with the sd — pf Quantum Monte Carlo Shell Model [22,56], which predicts an E(2f) value of 885 keV, the B(E2;0;,. —) 2?) value to be 454 ezfm“ and a deformation parameter, ,32, of 0.51. The E(4f)/E(2f) ratio predicted by this model is 2.55, and is in agreement with the measurement of Ref [43] contributing to the argument that the nucleus exhibits shape coexistence. Hartree Fock models predict either a spherical shape [60,61] or shape coexistence with more than one minimum in the plot of energy versus deformation parameter [58,59]. The deformation parameter, 32 = 0.51(3) extracted from our measurement does not agree with the example of the spherical nucleus 40Ca which has an adopted 62 value of 0.123(11) [17]. An example of the shape coexistence is found in Reference [59] predicting a 52 value of 0.54 and the B(E2;0;,. —-> 2?) value of 507 ezfm“ for the prolate shape. Both of the calculations are in agreement with our measurement. Relativistic Mean Field calculations find the nucleus to be spherical [23, 62,63]. The Angular Momentum Configuration Mixing model predicts a combination of prolate and oblate minima with a 62 value of 0.36 for the prolate minimum [64]. Our measured value for mg] is larger than this at 0.51(3). 5.1.2 34Mg Three other experiments have been performed previously on 34Mg. The first [25], placed an upper limit on the B(E2'0+ —> 2f) value of 670 e2fm4. The second a 9.3. 109 measurement [38], found an E(2f) value of 660(10) keV via a double fragmentation experiment at RIKEN. Finally Ref. [39] used Coulomb excitation 208Pb(3"Mg,34 Mg7) to measure the E(2f) value of 656(7) keV, B(E2; 0;, —) 2?) = 631(126) ezfm“, and a 52 2 058(6). Our measured E(2f) value of 659(14) agrees. Our B(E2; 0;, -—> 2]) value of 541(102) e2fm4, and fig value of 054(5) are slightly lower but still agree. The second 7-ray at 1460 keV seen by [38] does not appear in our spectrum. All of the measurements agree with the 3d— pf Quantum Monte Carlo Shell Model (QMCSM) predictions of a first 2+ excited state energy of 620 keV, a B(E2; 0;, ——> 21‘) value of 570 e2fm4, and the ,62 value of 0.55 [22,56] as well as the 21102 shell model predictions of a 660 keV E(2f) value, a B(E2; 0;, —> 23‘) value of 655 e2fm4, and a ,32 of 0.6 [57]. The 011w values [32] do not agree with the measurements. From this evaluation it is apparent that the intruder-configuration ground state explanation is supported by the experimental data, and 34 Mg can decidedly be included in the Island of Inversion. Hartree Fock models predict a range of both prolate and oblate deformation pa- rameters. The prolate predictions calculate 62 values of 0.3-0.4 [58], 0.22 [61], and 0.46 [59] and the oblate calculation is a 62 value equal to -0.1 which is close to spheri- cal [60]. We measured a [B2] = 054(5). Our measurement also does not agree with the Relativistic Mean Field theory calculations for the deformation parameter of 0.3 [63], 0.162 [62], and 0.17 [23]. We find 34Mg exhibits more collectivity than 32Mg, contrary to the prediction of the Angular Momentum Projection and Configuration Mixing model which predicts a 32 value of 0.36 [64] for both 32Mg and 34Mg . 5.2 35MA] and 37Si Reduced transition probabilities were also extracted from measurements of the elec- tromagnetic transitions in 3 neutron-rich nuclei with odd numbers of nucleons, 378i and 35’36A1. 36Al was until now unmeasured via Coulomb excitation. The spin and 110 parity of the states for all 3 nuclei are also unmeasured and so were assumed from an OXBASH Shell Model calculation [105]. The B («A T) values extracted from assumed E1, M1 and E2 transitions were therefore compared to the recommended upper limits listed by Endt [50]. Our measured low-lying excited state energies for 37Si and 35Al, 1437(30) keV and 985(21) keV, are in agreement with those of Ibbotson et al. [108], 1437 (27) keV and 1006(19) keV respectively. However, the values for the B(E2 T) extracted from our data, 362(62) ezfm4 and 360(24) ezfm“, differ significantly from Ibbotson’s of 101(45) ezfm“ and 142(52) ezfm". The discrepancy may be due to the low statistics in the Ibbotson experiment, as described in the reference. As mentioned, the 674(14) keV and 967(20) keV 7—rays in the 36Al energy spectrum have not been measured before via Coulomb excitation. The adopted values for the nearest even-even nucleus 368i, with N = 22 and Z = 14, are a B(E2;03‘_,. —> 2f) value of 190(60) ezfm“, an E(2f) value of 1399(25) keV and a deformation parameter ,3; of 0.259(42) [17]. This nucleus has two protons more than 34Mg and is significantly less collective. As discussed in Reference [29], the quick onset for the difference in collectivity is also found for the N = 20 isotopes 34Si and 32Mg, and experimentally marks the boundary of the Island of Inversion. In comparison to these even-even nuclei, we can consider the strength of the E2 transitions extracted from our measurements of the odd nuclei to be lower limits of the collectivity in these odd nuclei. Our extracted B(E2;5/2+ —> 9/2+) value for 35A1 of 310(45) ezfm4 and B(E2; 7/2‘ —> 11/2‘) = 360(62) ezfm4 for 37Si are higher than those of the even-even nuclei, while the observed states for 36Al have B(E2; 4‘ —> 6‘) values that are comparable, at 150(27) e2fm4 and 280(41) ezfm4 for the two observed 7-rays. 111 5.3 Summary In summary, first 2 i excited states of the projectiles 26'32’34Mg, 96Mo, and 48Ca have been populated via intermediate energy Coulomb excitation in one of the early ex- periments at the Coupled Cyclotron Facility at the NSCL. Low-lying states of the odd-nucleon nuclei 35”A1 and 37Si and the target nucleus 197Au were also populated by the same method. The energy and number of de-excitation 7-rays were observed using the APEX NaI(T1) array and reduced transition probabilities were extracted from the observed quantitites, thereby providing a means for examination of the ex- cited states of these nuclei. ”Mg and 34Mg are predicted to be members of the Island of Inversion, and as such were expected to exhibit characteristics indicative of a 2m intruder ground state. The reduced quadrupole transition probabilities extracted from the observations of the 09., —> 21+ electric transitions in 32”Mg indicate that they are extremely collective nuclei. The values for the B (E2; 0;, —) 2]) and quadrupole deformation parameters, [32, agree with calculations made with shell models that include a 2hr» ground state configuration in the model space. Of the mean-field calculations our measurements agree with those that calculate a prolate deformed shape, or possibly shape coex— istence in ”Mg, although our data do not reveal the coexisting state. Beyond the mean-field, models tend to predict smaller fig values than our measurements. We find 34Mg to have a higher degree of collectivity than ”Mg and that our measurement agrees with that of Iwasaki et al. [39]. As compared to the theory, our measurement contrasts with models which predict a spherical shape for 34Mg as well as the Ohw shell model. The measurement is in agreement with models that include the 2110.: intruder as the ground state such as the 8d — pf Quantum Monte Carlo Shell Model. There is a slightly better agreement with the QMCSM calculations as compared to the 2hw shell model in Reference [30]. These results are summarized in Figure 5.2 in which our measurements are compared with adOpted and calculated 112 ) (ezfm4) 2? ; 0g? | 5.1 8912 ) (MeV) + l E(2 800 - 600 - 400 ~ 200 - sd—prCSM 0 Present Work --------- Ohm A RIKEN — — — HF I Adopted O Pritychenko 0.6 - 0.5 ~ 0.4 3 0.3 - 0.2 0.5 A l 20 22 24 26 28 30 32 34 36 l I l I I fl I Mass Number Figure 5.2: Reduced quadrupole transition probabilities, B(E2' 0+ 3 9.3. 113 —> 2f), quadru- pole deformation parameters, 62, and first excited state energies, E (2:), for Mg iso- topes with even numbers of neutrons. Values measured in the present work (circles), at RIKEN (triangles), and by Pritychenko et al. [25] (open circle) are compared to the adopted values (squares) as well as theoretical calculations (lines). The RIKEN measurement of ”Mg is reported in Reference [12], while those for 3“Mg are the work of References [38,39]. The 3d — pf Monte Carlo Shell Model (MCSM) [22, 56] values are connected by the solid line, the Ohw shell model calculation (Ohw) [107] by the dotted line, and an example of a Hartree-Fock calculation [59] by the dashed line. values for the reduced quadrupole transition probabilities, quadrupole deformation parameters and first 2+ excited state energies for the magnesium isotopes. The figure demonstrates the evolution of collectivity in the magnesium isotopes with increasing numbers of neutrons, and thereby confirms the inclusion of 34Mg in this anomalous group of collective nuclei near the magic number N = 20. Although much work has already been done in the region of the Island of Inversion, many new opportunities for exploring the region have been created with the increased intensities of the rare-isotope beams currently being developed world-wide. With these increases, considerable advancements can be made through the measurement of the higher-lying states, such as the 4f state in ”Mg, in order to confirm the suspected deformation of the region. Spectroscopic information on the N = 24 nucleus 36Mg will provide information confirming the neutron boundary predicted by many models. While the high-Z boundary has been explored, the low-Z boundary also remains to be investigated. This further experimental exploration of the Island of Inversion may be feasible, should the construction of new rare-isotope beam facilities be completed, and is necessary in order to continue the investigation into the structure of these extremely rare, neutron-rich, anomalously-collective nuclei. 114 Appendix A Weisskopf Single-Particle Estimates A standard for estimating transition strengths between two states are transition prob- ability estimates made from the extreme single-particle viewpoint. These are called Weisskopf single-particle estimates and involve moving a single nucleon from one state to the next without affecting the rest of the nucleus. The derivation for the single-particle estimates is found in Reference [109]. The results are outlined here. For the reduced electric transition probability, the Weisskopf single-particle esti- mate, or one Weisskopf unit (W.u.), is 1 BW(E/\) : —( 3 A+3 2 4W ) (1.2A1/3)2’\e2fm”, (A.1) where /\ is the multipolarity of the transition. The Weisskopf single-particle estimate for the reduced magnetic transition probability is written similarly, _10 —7l' 2 Bw(M/\) ( 3 ) (1.2A1/3)2"—2;1§fm2*—2, (A.2) A+3 115 Table A.1: Electric and magnetic Weisskopf single-particle estimates for transition probabilities up to the first four multipoles. Order in A Ww(EA) (s‘l) Ww(MA) (s‘l) 1 1.02x1014 AWE}; 3.15x1013 ES; 2 7.28x107 114/3E3 2.24x107 AWE?7 3 3.39x 10 AZE; 1.04x10 AWE; 4 1.07x10-5 A8/3Eg 3.27x10-6 A2Ej9, where my is the nuclear magneton, _ eh #N _' 2MPC, (A.3) for which he 2 197.329 MeV - fin, and MI, is the mass of the proton, 938.3 MeV/c2. Thus, 11%, can be written in terms of e2fm2 through the conversion my 2 0.105 efm. (A.4) Weisskopf estimates for the transition probabilities WW are for electric transitions _ 871(A+1) 1 1 ”+1 1 3 2 1/3 2, m, WW(E’\)‘“hC,\[(2A+1)!!JZE(E) 450—173 (I'M )E ’ (A5) and for magnetic transitions WWW” : “54211292 A§72r§A++1)1!)!]2f1 (firm (A6) 10 3 2 X __ __ 12A1/3 2A—2E2A-fl. 7r (A + 3) ( ) The values for the first three orders of A are listed in Table A1 Comparison of these single-particle estimates with observed values is a good indication of the collective nature of the transition. If the observed value is much greater than the single-particle estimate, the nucleons are acting together collectively to produce the transition. On 116 the other hand, in the case that the observed value is in agreement with the single— particle estimate, the excitation must be largely single-particle in nature. 117 Appendix B Energy Resolutions of the APEX NaI(Tl) Detectors Energy resolutions (F WHM / centroid) are presented here for each of the individual N aI(Tl) detectors of the APEX N aI(Tl) array. The first table contains measurements taken immediately following the reconditioning at the NSCL by BC. Perry. The second and third tables are values measured after N SCL Experiment 01017 in January of 2003. The detectors are numbered from 1 to 24. 118 Figure B.1: Energy resolutions measured immediately following reconditioning under laboratory conditions. The figure is the work of BC. Perry [74]. DETECTOR RESOLUTION AFTER RECOUPLING NUMBER 1173 keV 1332 keV 1 15.1% :1: 0.1% 13.3% 0.1% 2 N/A :1: N/A N/A N/A 3 15.9% :1: 0.1% 14.0% 0.1% 4 14.4% :t 0.1% 12.7% 0.1% 5 12.6% d: 0.1% 11.2% 0.1% 6 16.8% :t 0.2% 14.9% 0.2% 7 13.7% :1: 0.0% 12.1% 0.0% 8 13.5% d: 0.1% 12.0% 0.1% 9 14.8% i: 0.1% 13.2% 0.1% 10 14.8% :t 0.1% 13.0% 0.1% 11 15.6% i 0.1% 13.8% 0.1% 12 15.2% i: 0.1% 13.4% 0.1% 13 15.7% :1: 0.1% 13.9% 0.1% 14 14.0% :t 0.1% 12.4% 0.1% 15 19.1% i 0.3% 17.0% 0.3% 16 12.7% i 0.1% 11.0% 0.1% 17 13.8 % d: 0.1% 12.2% 0.1% 18 13.7% i 0.1% 12.1% 0.1% 19 13.2% i 0.1% 11.7% 0.1% 20 14.0% i 0.1% 12.4% 0.1% 21 14.9% 3: 0.1% 13.2% 0.1% 22 17.8 % i 0.2% 15.7 % 0.2% 23 15.0% i 0.1% 13.3 % 0.1% 24 16.6% i 0.2% 14.7 % 0.2% AVERAGE 14.6% d: 0.1% 12.9% 0.1% 119 Table B.1: Energy resolutions for the individual APEX N aI(T1) detectors measured January 31, 2002 in the NSCL Experiment 01017 setup. 137Cs 88y 88Y Detector 662 keV 898 keV 1836 keV 01 19.8(1)70 17.8(3)7o 12.0(3)70 02 23.8(2)70 19.7(4)70 13.8(4)70 03 22.8(1)70 16.3(3)7o 10.9(3)70 04 20.2(1)70 16.6(3)70 11.4(3)70 05 17.3(1)70 14.1(3)% 10.2(3)% 06 19.6(1)% 15.8(3)% 11.1(3)% 07 20.1(1)70 16.7 (3)70 11.0(3)70 08 19.9(l)70 16.0 (3)70 11.1(3)% 09 19.2(1)70 18.0 (4)70 12.4(3)70 10 20.0(1)70 16.8 (3)70 12.6(3)% 11 18.5(1)70 15.8 (3)70 l2.1(3)70 12 21.0(l)70 16.8 (3)70 12.0(3)70 13 20.3(1)70 18.2 (4)70 13.3(3)70 14 21.3(1)% l6. 4 (3)70 11.9(3)70 15 24.3(2)70 18.6 (4)%14.9(4)% 16 14.3(1)% 12. 7 (3)70 9.1(2)70 17 18.5(1)70 15. 6 (3)7011.7(3)70 18 l7.7(1)70 14. 6 (3)7010.1(3)% 19 17.4(1)70 15.3 (3)0 10.3(3)70 20 19.7(1)70 15. 8 (3)%11.9(3)% 21 16.5(1)70 14.9 (3)7010. 5(3 )70 22 22.7(2)70 18. 6 (4)% 13. 9(4)% 23 18.9(1)70 16.4 (3)7011.8(3)70 24 20.6(1)% 16.8 (3)70 12. 6(3)% 120 Table B.2: Energy resolutions with uncertainties due to Doppler reconstruction at 90° and v/c=0.36 taken into account. 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