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Michigan State 2615 University This is to certify that the dissertation entitled Angular distributions of gamma rays with intermediate-energy beams and spectroscopy of ”Mg presented by Heather Zwahlen has been accepted towards fulfillment of the requirements for the PhD. degree in Dept of Physics and Astronomy T ' flflmwi/v’ Major Professor’s Signature ll/S/zoos 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 2/05 p:/C|RC/DaleDue.indd-p.1 Angular distributions of gamma rays with intermediate-energy beams and spectroscopy of 32Mg By Heather Zwahlen 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 2005 ABSTRACT ANGULAR DISTRIBUTIONS OF GAMMA RAYS WITH INTERMEDIATE-ENERGY BEAMS AND SPECTROSCOPY OF 32Mg By Heather Zwahlen The feasibility of using 'y-ray angular distributions and 7—7 angular correlations with intermediate-energy beams to assign multipolarities of 'y-ray transitions is ex- plored theoretically and experimentally. The formalism for 7-ray angular distributions and 7-7 angular correlations with intermediate-energy beams is presented. At the N a- tional Superconducting Cyclotron Laboratory (NSCL) the 7-ray angular distribution of the first excited state in 40Ar was measured in an intermediate-energy Coulomb excitation experiment. The observed y-ray angular distribution agrees with the cal- culated distribution. The nucleus 32Mg was studied in a two-proton knockout experiment performed at the NSCL. Nine 7 rays were observed; five for the first time. Here, excited state energies and partial cross sections to those states are presented. A level scheme is con- structed based upon ’y-ry coincidence data and 'y-ray intensities. The results confirm that the ground state wavefunction of 34Si is dominated by the sd shell configuration. Previous experiments observed a 7 ray around 1438 keV. The interpretations of those experiments resulted in possible contradictions in spin and parity assignments for an excited state in 32Mg. In the present experiment two peaks were identified around 1438 keV. The existence of two distinct levels may resolve these contradictions. Calcu- lations of the two-proton knockout reaction, assuming two correlated protons knocked out of the (15/2 shell, predict three pure sd states that are not observed in the present experiment. These 3d states are each thought to fragment over a few MeV to create six of the observed excited states. my father iii ACKNOWLEDGMENTS I would like to thank many people for their guidance, help and support during my time as a graduate student at the NSCL at MSU. I’d like to thank my advisor Thomas Glasmacher for guiding me through graduate school and teaching me nuclear physics as well as the practical aspects of being an experimentalist. I would like to thank Michael Thoennessen, Alex Brown, Wayne Repko, and Joey Huston for serving on my guidance committee. I am grateful to the National Science Foundation, MSU and the NSCL for their financial support. The ’gamma group’ members, past and present, were essential to my success as a graduate student. I’d like to thank all of them, true group members and ’honorary’ ones: Ana Becerril, Chris Campbell, Jenny Church, Jon Cook, Andrew Davies, Dan- Cristian Dinca, Joachim Enders, Alexandra Gade, Zhiqiang Hu, Wilhelm Mueller, Ben Perry, Russ Terry, and Katie Yurkewicz. Being a member of the gamma group when I first arrived at the NSCL meant trips to Argonne, learning to use the labelmaker, and daily meetings. From the liquid nitrogen filling system with the old Labview ’brain’, to the old hardware with new brain, to the new hardware and new brain, Chris and then Jon, were both great to work with. We never did quite figure out how to count male and female VCR connectors. I’d like to thank the entire staff at the N SCL for their time and effort in everything. I’d especially like to thank Renan Fontus for designing the new SeGA detector stand and Kelly Davidson for programming and working with us on the filling systems. Thanks to the entire gamma group for setting up and taking down my complicated setup multiple times and for taking countless numbers of shifts during my experi- ments. Thanks to the operators, A1900 staff, Daniel Bazin and Alexandra Gade and the entire staff at the NSCL for making my thesis experiments possible. I’d like to thank Russ, Chris and Alexandra who were very helpful when I was working on my analysis. Thanks to Alexandra for reading my thesis. Discussions with Andrew Stuch- iv bery during his time at the NSCL were essential and enlightening. Thanks also to Gregers Hansen, B. Alex Brown and Jeff Tostevin for calculations and discussions pertaining to the work in this thesis. Without the friends I’ve made at MSU my graduate experience would have been much less exciting. During the first couple years when many hours were spent doing homework in the NSCL conference rooms with Joe, Josh, Nate, Andrew, and Russ I’ll never forget the Britney Spears outbursts by Josh and Andrew, the paper wars, and the yelling of “Hi Jacque” when my cell phone would ring. Also unforgettable are when Ryan and Pete gave Debbie Simmons flowers our first year, Ben’s parties, and mine, Maria, and Lucy’s Halloween party. Ryan, Pete and Russ could always be counted on to go out to lunch or to get Cold Stone cakes. Katie, you helped bring me up to speed in the gamma group and became a good friend. Thanks to Michelle Ouellette and Debbie Davies for being there for me when I wanted to chat. Debbie, we have a weakness for Sparty’s. Jon and Debbie, in the spring I know I was particularly crazy trying to finish everything and thank you for putting up with me in the office. My friends from Geneseo would remind me there is something else to life besides graduate school. Thank you Bridget, Kelly, and Sarah for being good friends. Brid- get, thank you for never telling me that you knew I wouldn’t be finished before the wedding! Thanks to Maria for being such a good roommate for 2 years, getting me to enter 5k races, and introducing me to all of your friends. I should have taken the opportunity to learn some Spanish! Thank you Fr. Mark for your encouragement and stability. St. John’s is a wonderful place and you bring God’s message and peace and big picture into real life. Without the love and upbringing of my parents I never could have gotten this far. Thank you mom and dad for everything. Thanks Jim for being such a great ’little’ brother. You almost got your masters degree before I finished my PhD! I’ll ride your roller coasters some day. I wouldn’t recommend pursuing a Ph.D. 700 miles away from the one you love, but the phones and plane tickets helped make it possible. Jacque, without you I couldn’t have done this. Thank you for everything, for your love, for all of you. vi Contents 1 Introduction 1.1 The atomic nucleus ............................ 1.2 The nuclear shell model ......................... 1.3 7-ray angular distributions and 7-7 angular correlations ........ 1.4 The nucleus ”Mg ............................. 1.4.1 Previously observed ”Mg 7 rays ................. 2 7-ray angular distributions and 7-7 angular correlations 2.1 Angular correlation formalism ...................... 2.1.1 Notation and terminology .................... 2.1.2 Angular correlation function .................. 2.2 Angular distribution function ...................... 2.3 Alignment ................................ 2.4 W(l9) for transitions of identical multipolarity ............. 2.5 Different reaction mechanisms ...................... 2.5.1 Fusion evaporation ........................ 2.5.2 Fragmentation reactions ..................... 2.5.3 Nucleon-knockout reactions ................... 2.5.4 Intermediate-energy Coulomb excitation ............ 3 Experimental setup 3.1 Coupled Cyclotron Facility ........................ 3.2 A1900 fragment separator ........................ 3.3 SEgmented Germanium Array ...................... 3.3.1 Configuration ........................... 3.3.2 Standard coordinate system ................... 3.3.3 Detector positions ......................... 3.3.4 Energy and efficiency calibrations ................ 3.3.5 Doppler reconstruction ...................... 3.3.6 GEANT simulations ....................... 3.4 S800 Spectrograph ............................ 3.5 'Ifigger ................................... 4 Intermediate-energy Coulomb excitation of 40Ar 4.1 Intermediate-energy Coulomb excitation ................ 4.2 Experiment setup ............................. vii 4.3 Analysis of the 7-ray spectrum ...................... 68 4.4 7-ray angular distribution ........................ 70 Study of ”Mg via two-proton knockout 73 5.1 Two-proton knockout ........................... 73 5.2 Experiment setup ............................. 74 5.2.1 Trigger ............................... 75 5.3 Analysis of the 32Mg spectrum ...................... 76 5.3.1 Particle identification ....................... 76 5.3.2 Doppler correction ........................ 78 5.3.3 Doublet parameter study ..................... 82 5.3.4 Fitting the ”Mg 7-ray energy spectrum ............ 86 5.4 Results and Discussion .......................... 91 5.4.1 Inclusive Cross Section ...................... 91 5.4.2 Proposed level scheme ...................... 97 5.4.3 7-ray angular distributions .................... 101 5.4.4 Partial cross sections ....................... 107 Summary 116 A Subroutine used by GEANT 118 GEANT input file 135 Bibliography ................................... 163 viii List of Figures 1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 Schematic diagram of energy levels in the spherical shell model . . . . ”Mg level schemes from 6 decay and 9Be(36S,”Mg 7)X ........ Level notation used for angular correlation formalism .......... Level notation used for angular distribution formalism ........ Plots of W(6) with 100% oblate or prolate alignment at different beam velocities .................................. Plots of W(01, (92, ) with no alignment at different beam velocities Three-dimensional 7-ray angular distributions with different percent- ages of oblate alignment at different beam velocities .......... Plots of W(0) for 100% and 50% oblate or prolate alignment for two different quadrupole transitions at 6 = 0.4 ............... Plots of W(6) with different percentages of oblate alignment at different beam velocities. Each panel contains a quadrupole, dipole and octupole transition ................................. Plots of W(6) with different percentages of prolate alignment at two different beam velocities. Each panel contains a quadrupole, dipole and octupole transition ............................ Plots of W(6) with different percentages of oblate alignment at two different beam velocities. Each panel contains only one transition - quadrupole, dipole or octupole ...................... Plots of W(6) with different percentages of prolate alignment at differ- ent beam velocities. Each panel contains only one transition - quadru- pole, dipole or octupole .......................... Plots of W(61, 62, ) for 20% oblate alignment ............. Plots of W(61, 62, (D) for 20% prolate alignment ............ Plot of W(6) for a 1n knockout reaction at 6 = 0.4 and inset containing the m substate distribution of the excited state ............ Plots of W091, 62, ) for the one-neutron knockout reaction of 9Be(”Ca, 51Ca 7)X at B = 0.4 ........................... The m substate distribution of the 7/2‘L excited state of 51Ca produced from the ln knockout reaction 9Be(”Ca, 51Ca 7)X at B = 0.4 Three-dimensional plot of the relative difference between two 7—ray transitions in 51Ca from the ln knockout reaction of 9Be(”Ca, 51Ca 7)X at 6 = 0.4 .............................. Plots of W(9) for a Coulomb excitation reaction at two different beam velocities. m substate distributions for the excited states are shown ix 4 9 12 17 20 21 23 24 29 30 32 33 34 34 36 37 37 37 40 2.18 2.19 2.20 2.21 2.22 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 Plots of W(0) for a Coulomb excitation reaction at different minimum impact parameters, bmin and plots of W(0) for different Coulomb ex- citation reactions at the same bmm ................... Plots of W(0) for quadrupole Coulomb excitation reactions with dif- ferent excitation energies ......................... Plots of W(0) for dipole Coulomb excitation reactions with different excitation energies ............................ Plots of W(0) for octupole Coulomb excitation reactions with different excitation energies ............................ Plots of W091, 62, ) for the intermediate-energy Coulomb excitation reaction 209Bi(56Ni,56Ni 7) ........................ Schematic of the Coupled Cyclotron Facility including the ion sources, two cyclotrons and A1900 fragment separator. ............. Photograph of SeGA at the target position of the S800 spectrograph . Sketch of standard coordinate system ................... Front-view of SeGA detector geometry displayed using GEANT Side-view of SeGA detector geometry displayed using GEANT . . . . Top-view of SeGA detector geometry displayed using GEANT . . . . Absolute efficiency of the 60° SeGA angle pair ............. Energy resolution versus half-life for a simulated 7-ray peak ..... Activity versus distance in the secondary target ............ Schematic of the S800 spectrograph ................... Number of scattered 40Ar nuclei versus scattering angle ........ 40Ar 7-ray spectrum of the 60° SeGA angle pair fit with GEANT re— sponse functions .............................. Angular distribution of the intermediate-energy Coulomb excited 40Ar 1460 keV 7 ray .............................. 34Si —» ”Mg 2p knockout schematic ................... ”Mg particle-identification spectra ................... 32Mg 7-ray spectrum containing sum of all detectors .......... Energy resolution versus 6 ........................ Energy resolution versus zo ........................ Energy resolution versus energy ..................... Three-dimensional plots of doublet parameter study results (1) . . . . Three-dimensional plots of doublet parameter study results (2) . . . . Three-dimensional plots of doublet parameter study results (3) . . . . ”Mg 7-ray spectrum of the 40° SeGA angle pair fit with GEANT response functions ............................. Plot of ratio of 1460 keV 7 rays in and out-of-beam versus SeGA de— tector angle ................................ Fragment yield versus the x-position distributions of ”Mg and 31Mg fragments in the S800 focal plane .................... ”Mg level scheme observed from the 2p knockout reaction 9Be(3"Si,”Mg 7)X ..................................... 41 42 43 43 45 48 50 51 53 53 54 57 59 60 63 69 71 72 74 77 78 79 80 81 83 84 85 87 89 94 98 5.14 ”Mg 7-ray spectrum in coincidence with the 885 keV 7 ray ...... 99 5.15 ”Mg 7-ray spectrum in coincidence with the 1964 keV 7 ray ..... 100 5.16 ”Mg 7-ray spectrum in coincidence with the 2405 keV 7 ray ..... 102 5.17 2405 keV 7-ray angular distribution ................... 105 5.18 885 keV 7-ray angular distribution ................... 112 IMAGES IN THIS DISSERTATION ARE PRESENTED IN COLOR. xi List of Tables 1.1 2.1 5.1 5.2 5.3 5.4 5.5 5.6 Previously observed ”Mg 7 rays ..................... Table of relative differences of W(0) between quadrupole and dipole transitions with oblate or prolate alignment .............. Summary of corrections to ”Mg 7—ray intensities ........... Numbers used for 9Be(3“Si,”Mg)X inclusive cross section calculation Results of fits of experimental ”Mg 7—ray angular distributions . . . . Ranges of the uncertainties on the experimental 7-ray angular distri- butions of the nine SeGA detector angle pairs ............. Observed ”Mg 7 rays ........................... Level energies and partial cross sections for ”Mg ........... xii 106 108 110 Chapter 1 Introduction 1.1 The atomic nucleus The atomic nucleus is a many-body system composed of nucleons bound together by the strong force. Protons (Z) and neutrons (N) are spin 1 / 2 fermions and are two different states of the nucleon in isospin representation. The mass number, A, is the sum of the number of protons and neutrons in a nucleus. Protons have charge e and neutrons have zero charge. The size of the nucleus increases as the number of protons and neutrons increase. The radius in femtometers (fm) can be approximated by 1.2 x A1/3. The study of nuclear physics aims to understand the force between nucleons, study the structure of atomic nuclei, and understand the interactions between nucleons and other subatomic particles. Studying the properties of atomic nuclei is the focus of nuclear structure research. These properties include energies, spins and parities of excited states, nuclear shapes, and electromagnetic moments and transition rates. Trends in some of these properties become apparent when examined over the entire nuclear chart. Nuclei with a ’magic’ number - 2, 8, 20, 28, 50, 82 or 126 - of protons and/ or neutrons exhibit distinct properties from those of neighboring nuclei. This will be discussed in the next section. Interactions between nucleons are primarily governed by the strong force but also are influenced by the weak and electromagnetic interactions. The strong force is poorly understood and different theoretical models are used to approximate the behavior of atomic nuclei. These models aim to reproduce the properties of known nuclei that have been measured experimentally and accurately predict the properties of nuclei that have not yet been studied. Experimental nuclear structure information is used to test and improve these theoretical models. The current nuclear structure models can be divided into two major categories: collective and single-particle. Collective models focus on how the nucleons act together using collective degrees of freedom. One such model is the vibrational model which describes nuclear excitations as vibrations of the nucleons as a whole. Another collective model is the rotational model which describes nuclear excitations as rotations of statically-deformed nuclei. Single-particle models focus on the individual nucleons using the individual nu- cleon’s degrees of freedom. The most well-known single-particle model is the nuclear shell model. 1.2 The nuclear shell model The nuclear shell model is a single-particle model where each nucleon is assumed to move in a potential created from the other A — 1 nucleons. If the potential is es- timated using a three-dimensional harmonic oscillator, the resulting single-particle energy spectrum has large energy gaps between groupings of energy levels (shells). The first few shell gaps appear at nucleon numbers 2, 8, 20, 40 and 70. Experimen- tally the magic numbers, or the number of nucleons needed to fill up the last shell before a large energy gap, occur at 2, 8, 20, 28, 50, 82 and 126. The nucleons have an intrinsic spin, 3, and orbital angular momentum, l. Coupling l and s, to a total spin j, and adding it to the three-dimensional harmonic oscillator potential results in a potential that accurately reproduces the experimentally observed magic numbers: 2, 8, 20, 28, 50, 82 and 126. This ’spin—orbit’ term was first suggested in 1949 [1,2]. Fig. 1.1 shows a schematic diagram of energy levels calculated in the spherical shell model. In general, nuclei are spherical at closed shells (where the number of protons and/ or neutrons are equal to a magic number). This can be understood because if a shell is filled with 23' + 1 protons or neutrons, then the total projection of the spin along the quantization axis is zero. At closed shells nuclei typically also have large first excited state energies, large proton and neutron separation energies and small reduced transition probabilities, B(E2T), to the first excited state. Examples of nuclei with large first excited state energies in the N=20 region are, 3.328 MeV for 34Si, 3.291 MeV for 36S and 2.167 MeV for 38Ar. For a given Z, as N increases the two- neutron separation energy (32,.) decreases smoothly until a shell closure is reached. At the crossing of a shell closures there is a sharp drop in 82,, because the new neu- tron shell has less binding energy. Mid-shell nuclei (where the numbers of protons and neutrons are in between the magic numbers) tend to have smaller first excited state energies and larger B(E2T) values. Larger B(E2T) values are indicative of a col- lective behavior - rotational or vibrational excitations. In between closed shells there are many single-particle states available and therefore it is often more energetically favorable for a nucleus to assume a deformed shape. The spherical shell model potential can be modified to reproduce experimental observables for deformed nuclei. The deformed shell model, or Nilsson model [3], uses the potential of the spherical shell model plus a deformation term. The deformation term typically is due to a quadrupole field. Since in the Nilsson model nuclei are no longer assumed to be spherical, j is no longer a good quantum number, but the projection of 3' along the quantization axis is a good quantum number. Therefore each single-particle level splits into (2 j + 1) / 2 states. The resulting calculated single-particle energies as a function of the deformation of the nucleus, for non-zero deformation parameters, can change significantly. For example, the energy of the fig level can drop below the d3/2 level essentially eliminating the N=20 shell gap. This occurs in f‘ "111/2 12 ’2d3/2 4 3s ,_,:1—-3S1/2 2 4 2d 4:-,,”12:7/2 8 19 (l’ - 5/2 6 \ ‘ \ \ \ \ ~ \ ‘199/2 10 2p I’1f5/2 6 ‘~-_ I 3 ‘;af‘~ZP1/2 2 1f <: ‘293/2 4 ‘1f7/2 8 ,w1d3/2 4 2s ---,’ 2 1d ,” “-281/2 2 """"" 1d5/2 6 _--4 1 1p :::- 91/2 2 “-193/2 4 0 ts ------- 181/2 2 N nl nlj 2j+1 Figure 1.1: Schematic diagram of energy levels calculated in the spherical shell model, shown up to the 1h11/2 level. On the left is the single-particle spectrum produced by an isotropic, three-dimensional harmonic oscillator potential. On the right is the single-particle spectrum with the spin-orbit interaction included. N is the number of harmonic oscillator quanta in a major shell. The radial quantum number, n, increases each time a particular I j combination occurs in order of ascending energy. The angular momentum quantum number, I, equals 0, 1, 2, 3, 4, 5, 6... corresponding to s, p, d, f, g, h, i, The projection of the total spin, j, is equal to 1+ 3. The numbers in the right-hand column are the maximum number of nucleons each level can contain. The magic numbers are indicated in each majolr shell gap. ”Mg as discussed in Section 1.4. The Nilsson model in the context of the vanishing N=20 shell gap is further discussed in [4]. In discussions of protons and neutrons occupying levels in the shell model, 7t and V will be used in the text to indicate protons and neutrons, respectively. The p shell refers to the 1p3/2 and 1p1/2 group of single-particle levels. The sd and [19 shells refer to the 1d5/2, 231/2, and 1d3/2 and the 1f7/2, 2p3/2, 2p1/2, and 1 f5/2 groups of single-particle levels, respectively. As used here, Ohw configurations refer to nucleons occupying the sd shell with no nucleons in the fp shell. N hw configurations refer to N nucleons in the fp shell. 1.3 7-ray angular distributions and 7-7 angular cor- relations Below the nucleon separation energies, nuclei in excited states can decay by 7-decay, fl- decay, a-particle emission, conversion electron emission, and fission. Electromagnetic decay, or the emission of 7 rays, is the dominant mode of decay for low-lying excited states in nuclei. Measuring the energies and intensities of 7 rays emitted as nuclei deexcite from excited states to their ground states can give information on the excited state structure and shape of nuclei. The excited states have spin I and parity 7r, denoted by I 7'. The deexcitation 7 ray from an initial state 2' to a final state f can have multipolarities l with, II,- — I f] S l S II,- + I f] (where l 2 1 because a photon has spin 1), based on angular momentum coupling. The parities of the initial and final states are constrained by the parity selection rule, 7mg = (—1)1 for electric (El) transitions and may = (—1)’+1 for magnetic (Ml) transitions. Determining the multipolarity of 7-ray transitions can constrain the possible If and I; values. If the ground state is known to be a 0+ state and a 7-ray decay to the ground state is observed, measuring the multipolarity of the 7 ray would give the spin of the initial state, 1,. Measuring 7-ray angular distributions and 7-7 angular correlations from excited nuclei are important tools in nuclear structure studies because of the ability to as— sign multipolarities and multipole mixing ratios of 7-ray transitions. In-beam tech- niques are well established for beams with energies near the Coulomb barrier (see for example [5]), but have not been thoroughly investigated for intermediate-energy (v/c z 0.3 — 0.8) exotic beams. An advantage of measuring 7-7 angular correlations from excited nuclei over measuring 7-ray angular distributions is that there can be an anisotropy in the angular correlation distribution even if there is no initial orientation of the nuclei. For various excitation methods, the m substate distributions can be calculated or estimated. From the m substate distribution the 7-ray angular distributions and 7-7 angular correlations can be calculated. Fusion evaporation, fragmentation, intermediate- energy Coulomb excitation and nucleon knockout will be discussed in Chapter 2. The feasibility of using 7-ray angular distributions with fast exotic beams is shown in Chapter 4 for the intermediate-energy Coulomb excitation of 40Ar. Shown in Chap- ter 5 are 7-ray angular distributions measured in the two-proton knockout reaction 9Be(3“‘Si,”Mg 7)X. A discussion regarding the feasibility of using 7-ray angular dis- tributions to determine multipolarities of 7 rays in future nucleon-knockout and frag- mentation reactions is in Section 5.4.3. 1.4 The nucleus ”Mg The first observation of ”Mg was in 1977 [6]. A low first 2+ excitation energy of 885 keV was determined for ”Mg in the fi decay of ”Na in 1979 [7] (see also Table 1.1 and references therein). Mass measurements of ”Mg [8,9] revealed a larger mass (and 32..) than predicted with N=20 assumed to be a good shell closure. A large B(E2T) value [4,10] was measured for ”Mg. A simple sd shell model picture [11] of the nucleus ”Mg could not explain the small first 2+ energy, large B(E2T) and relatively large 82,; value. These anomalies have been studied in shell model [ll—22] and mean-field [23—31] calculations. In ”Mg, the binding energy of 2hw configurations, where two neutrons are excited across the N=20 shell gap to the fp shell, is lower than ’normal’ 0hw configurations, where there are no neutrons excited across the N =20 shell gap. This ’inversion’ - where the 2m configurations are lower in energy than the Oh.) configurations - is thought to be due to the reduction of the N=20 shell gap between the ud3/2 and uf7/2 orbitals due to a large excess of neutrons and proton—neutron interactions between the 7rd5/2, ud3/2 and V f7/2 orbitals [12, 15, 22]. Assuming an axially symmetric shape, the intrinsic quadrupole moment and quadrupole deformation parameter can be calculated from the B(E2T) value. In shell model calculations [15, 19, 20] the deformation is in this way predicted to be prolate. The Nilsson model calculations in [4], in agreement with the experiment [4], predict prolate deformation for ”Mg. This inversion effect is most pronounced for nuclei with Z=10—12 and N =19—22. This so—called Island of Inversion [15] was first discovered with the measurement of the masses of 31Na and ”Na [32]. The experimental masses (and S2,, values) were greater than expected if N=20 was a good shell closure. This measurement prompted the study of the entire region, including the nucleus ”Mg. The east (most neutron- rich) boundary of the Island of Inversion is not clear [20]. The 21w, 1hr.) and Ofiw configurations are all in close competition for low excitation energies [19,33]. The ground states of ”Ne, 31'”Na and ”Mg are all dominated by 2hr.) configurations [12, 14,15,17—20,22]. At N=19 and N=22, the energy differences between normal and intruder configurations are nearly degenerate [19]. Two recent experiments showed ”Mg to be in the Island of Inversion [34] and 30Mg to be outside the Island of Inversion [35]. According to the N ilsson model, the nucleus must be deformed in order for the f7/2 orbital to be in close competition at low excitation energies with the d3/2 orbital. How- ever, deformation is not a necessary condition to reproduce the properties seen in ”Mg Table 1.1: Previously observed ”Mg 7 rays. If there are no uncertainties listed, they were not quoted in the respective papers. Ref. Year Reaction Energy (keV) of observed 7 rays assigned to ”Mg decay [10] 1995 203Pb(”Mg,”Mg 7) 890 [4] 1999 197Au(”Mg,”Mg 7) 885(9), 1438(12) [37] 2001 9Be(”Si,”Mg 7)X 885, 1430 [38—43] 20002002 gBe(”S,”Mg 7)X 885(10), 1430(10), 1950(20), 2870(40) [44] 2003 9Be(3“Si,”Mg 7)X 880(20), 1430(20) [45] 2002 ”Si(”Mg,”Mg 7)X 860(50), 1460(50) [7] 1979 e decay of 32Na. 885.7(2.0), 1233(4), 1440(4), 1970(5) [46] 1984 a decay of ”Na 885.5(O.7), 1782.4(0.9), 1973.0(1.2), 2151.5(0.7), 3934.8(18)a [47] 1993 H decay of 32N8. 885, 1232, 1436, 1783, 1973, 2152, 2551, 3935 [34] 2001 fi-n decay of ”Na 885.3(1), 1437.0(3), 1972.9(5), 2152.4(1), 2551(1) “Also observed were 7 rays at the following energies: 239.5(1.2), 694.4(l.2), 1436.1(1.0), 1232.2(1.2), 2550.7(1.0). The last three have been seen and assigned to ”Mg in later beta decay experiments [7, 34, 47] listed in the table. according to a recent calculation. In [31], Hartree—Fock—Bogoliubov(HFB) [36] plus quasiparticle random phase approximation calculations reproduce the large B(E2T) value and small first 2+ energy in ”Mg assuming no deformation effects but only neutron-pairing correlations between the 11 f7 /2, 1423/2 and ”PI/2 levels. The 2hw con- figuration can be generated by deformation effects or neutron-pairing effects. In [31] neutron-pairing effects are shown to be essential. An older HFB calculation [26] and relativistic mean-field calculation [28] both predict ”Mg to be spherical as well. 1.4.1 Previously observed ”Mg 7 rays In ”Mg, the 2+ —+ 0;, 7-ray transition at 885 keV was first observed in the 6 decay of ”Na in 1979 [7]. Since then eight other 7 rays have been observed in ”Mg as summarized in Table 1.1. The level schemes as observed in 6 decay [47] and in fragmentation [40] reactions are shown in Fig. 1.2. One 7 ray of interest is the 7 ray at approximately 1430 keV. log it 4-6 - 4820 4.7 3037 5.1 2858 2870 i 60 5.4 2551 5.4 2321 2315 :15 5.8 21 17 V Q L Y 885 i 885 V 0 keV V 0 keV 32Mg 32Mg Figure 1.2: Level schemes from 6 decay [47] (left panel) and the fragmentation reaction 9Be(”S,”Mg 7)X as reported in [40] (right panel). The 885 keV 7 ray has been observed in coincidence with a 7 ray at approxi- mately 1430 keV in Refs. [37,47] and [38] (also reported in [39—43]). The 1430 keV 7 ray has been observed in intermediate-energy Coulomb excitation [4,10], fragmenta- tion [37, 38], nucleon-knockout [44], beta-decay [7, 34,47] and inelastic-scattering [45] measurements. The assignment of the 1430 keV 7 ray to a 4+ —> 2+ transition [38] (also stated in [39—43]), in agreement with [20] and a calculation by F. Nowaki in [39—41,43], is not consistent with the intermediate-energy Coulomb excitation ex— periments in which the 1430 keV 7 ray was seen. In the intermediate energy Coulomb excitation experiment in [10] (and also reported in [48]), a 7 ray around 1430 keV is clearly visible in the published 7-ray spectrum, but is not discussed in the text. In intermediate-energy Coulomb excitation a direct E4 excitation from the 031,. to a 4+ state is unlikely and in [4] the observed 7 ray yields were inconsistent with multi- step excitations which are suppressed at intermediate beam energies. In 5 decay, the 1436 keV 7 ray deexcites a level at 2321 keV. The 2321 keV level has a measured log ft value of 5.4 [47]. (The ft-value is proportional to the fl-decay half-life.) The log ft value of 5.4 means the transition is allowed. The selection rules for an allowed 5 decay are that AJ = 0 or 1, AT = 0 or 1 and that there is no parity change. AT is the change in isospin from the initial to the final state. Therefore, assuming ”Na has a negative parity ground state, the 2321 keV level must have negative parity. This contradicts the tentative assignment of 4‘L and also contradicts two of the tentative spin assignments from the intermediate-energy Coulomb excitation experiment [4]. One goal of this work is to determine the multipolarity of the 1430 keV 7 ray using 7-ray angular distributions. As will be seen in Chapter 5, the 1430 keV 7 ray is more complicated than originally expected. However, this may resolve the spin and parity contradictions of the exited state around 2321 keV. A comprehensive description of ”Mg, as the prototypical nucleus in the Island of Inversion, will be presented. 10 Chapter 2 7-ray angular distributions and 7-7 angular correlations 2.1 Angular correlation formalism 2.1.1 Notation and terminology In this workl, the notation of [51] is used. Fig. 2.1 shows the initial 1,, middle 1m and final I f spins of a two step 7-ray cascade. The lowest multipolarities of the 7—ray transitions are l and l’ for the spin 1,- —+ Im transition and lmf and (in, for the spin 1m -) I f transition. I,-,- is the spin of the initial state before the nucleus is excited and la is the orbital angular momentum of the excitation. The general formalism for spin cascades with more than two 7-ray transitions is addressed elsewhere [5, 52, 53] for low energy beams. Some of the following equations can be found in the literature (see for example [5, 51—54]), but for completeness they are presented here as well. In this work the use of the terminology 7-7 angular correlations is used when referring to cases when there is no alignment in the initial excited state as well as 1Reprinted excerpts and figures with permission from the following two articles: H. Olliver, T. Glasmacher, A.E. Stuchbery, Physical Review C, 68:044312, 2003. Copyright 2003 by the American Physical Society. [49] H. Olliver, T. Glasmacher, A.E. Stuchbery, Physical Review C, 69:024301, 2004. Copyright 2004 by the American Physical Society. [50] (The maiden name of the author of this thesis, Heather Zwahlen, is Olliver.) 11 n / \ I, 1': 1+1 1m ’f 1e excitation Iii Figure 2.1: Notation used in angular correlation formalism. I.-, Im and I f are the initial, middle and final spins of the 7-ray cascade respectively. The lowest possible multipolarities of the 7-ray transition are l and l’ = l + 1 for the spin I,- —>, Im transition and lmf and lfinf = [my +1 for the spin Im —+ I f transition. I,-,- is the spin of the initial state before the nucleus is excited and la is the orbital angular momentum of the excitation. when there is some alignment in the initial excited state. The latter is also referred to as directional correlations from oriented nuclei (DCO). In this work the acronym DCO is not used because the term DCO can be associated with DCO ratios in which ratios of different correlation curves are taken as the physical observable instead of the correlation itself as is examined here. For the illustrative purposes of this chapter, only pure transitions of the lowest allowed multipolarity are considered. The notation I,- —> [m —+ I I will be used to denote 7—ray cascades from states with spins I,- to Im to I f. 2.1.2 Angular correlation function The 7-7 angular correlation function wc'm'(l9§'m', 03m, (1)) specifies the relative prob- ability to observe, in the center-of-mass, a 7-ray transition from spin I,- to Im at spherical angles 91 and 051 in coincidence with a 7-ray transition from spin [m to I f at spherical angles 02 and 432. (I) is defined as 051 — (152. 01 and 02 are measured with respect to the beam axis. The beam axis is the z—axis in the coordinate system used. The center-of-mass frame is the moving frame in which the 7 rays are emitted. This is nearly identical to the projectile frame in the fragmentation, nucleon-knockout and 12 intermediate-energy Coulomb excitation reactions discussed here. For intermediate- energy Coulomb excitation this is a good approximation for light nuclei, but for heavy nuclei the projectile and center-of-mass frames should be distinguished. Thus, in the center-of-mass the angular correlation function has the form 21,- 21m 41T(—1)’\ 1 AA A wc.m.(0(l:.m.,0§.m.,q))_ 2’: Z 2 AA2 1 A _0 "l-o’\2_0 \/2/\ +1 A A1 A A2 XAAgBAl Z q=-)l 0 q “q WWW,4.)Y§;"(0§'m-.42). (2.1) The A12” coefficients (defined in Eq. (2.6)) depend on 1,, Im, l, l’ and 6. The A A2 coef- ficients (defined in Eq. (2.9)) depend on 1,", I f, lmf, 1;”, and 6mf. The multipole mixing ratios for the 7-ray transitions, 6 and 6",], are defined in Eq. (2.7). The B A1 coefficients (defined in Eq. (2.11)) depend on 1,, I,-,- and l... The Y"(9°'m , 451) and YA2 ”(63m ,¢2) terms in Eq. (2.1) are spherical harmonics. To transform wc'm'(l9'f'm',0§'m',t1>) from the moving frame in which the 7 rays are emitted into the laboratory frame in which they are observed, two solid angle corrections are applied and the spherical harmonics are written as a function of the laboratory frame angles, 01 and 02. The angle (1) is not changed by the Lorentz transformation. Thus, in the laboratory frame wc'm'(Bf'"", 6‘2"”, (1)) becomes 21 21 I m 47r(— 1)’\ 1 AA A “1091,92,” =: Z: 2 AA2 1 1 _0 A1_0A2_0,/27\+1‘ A A1 /\ A2 q cm > )=:ZZ\/_2_(T:2_]_A:A AAA A: 0 A1=0A2=0 Aeven A AAzBAl 2: A1 A A2 x (47r)2Bo wa‘f-m, (151) q=-A 0 q “q 1—fl2 l—fl2 XYA2—qwg’m.’ ”2) [fl cos(01) — 1]2 [fl cos(62) — 1]2 (2'4) and is normalized to 271‘ 21r / f / Whoa 5) x sin(l91) sin(62) d¢1 d¢2 d61 d62— — 1. (2.5) Thus, the probability of observing two 7—ray transitions in coincidence over all space is one. In Eq. (2.4), dividing by (470280 fulfills the normalization condition in Eq. (2.5). If the population parameter P(m) (introduced in Eq. 2.11 and defined for different reaction mechanisms in Section 2.5) is normalized then B0 is unity. If P(m) is not normalized, as in intermediate-energy Coulomb excitation discussed in Section 2.5.4, then dividing by Bo is necessary to normalize the angular correlation function. In Eqs. (2.1), (2.2) and (2.4) the A values are even because 7 ray decay is an electromagnetic 14 process in which parity is conserved. A1 and A2 can have even or odd values but in order for the angular correlation function to be non-zero the sum of A1, A2 and A must be even. If only the directional correlation of two 7 rays is observed, as is discussed here, then A1 and A2 must be even as well as A. Note that with the normalization used here, the A = A1 = A2 = 0 term in the correlation function is 1/(47r)2, not unity, when 5 = 0. In Eqs. (2.1), (2.2) and (2.4), A2221 are generalized angular distribution coefficients with the form AA 1 AA AA21=1+62[FA21(lalalmali) +2618,"2A1 (r, 1', 1m, 1,) + 62Ff2"1(1',z',1m,1,)] . (2.6) These angular distribution coefficients consider only two multipolarities, l and l’ = l + 1, contributing to the 7-ray transition from spin 1,- -+ Im corresponding to the lowest two multipolarities that are allowed by angular momentum coupling. For pure transitions where l is the only multipolarity that contributes to the 7-ray transition A2231 = F£2A1(l,l,1m, 1,). The multipole mixing ratio, 6 is defined as 5 _ < Im||7r(l+1)||1,- > _ <1m||7r’(l)||11> ’ (2.7) where 7t and 7r’ specify the type of radiation, electric or magnetic. 6",, is defined likewise and is used in Eq. (2.9). The generalized F-coefficients in Eq. (2.6) are defined as F:2A1(l,l’,1m,1i)= (_1)l'+/\2+/\1+1(2Im +1)1/2 x(2I,-+1)‘/2(21+1)1/2(2l'+1)1/2(2/\ + 01/2 I... z I, z 1' A x(2A2+1)1/2(2A1+1)1/2 1m 1' I,- . (2-8) 1 —1 0 A2 A A1 15 In Eqs. (2.1), (2.2) and (2.4), A A, are angular distribution coefficients with the same form as Eq. (2.6), but with ordinary F-coefficients in place of the generalized F- coefficients and 1",, I f, 1m}, 1;", and 6mf in place of 1,, Im, l, l’ and 6 respectively. Thus, the A A2 coefficients have the form 1 AA2 : T1?! [FA2(lmf1lmf)If)Im) +26mfFA2(lmf1 :rnfslfalm) +612anAQUfrnft fnftIfalm)] (2’9) with the ordinary F-coefficients defined as FAUmI’ infilvam) = (—1)If+lm+l(2>\ +1)”2 ><(21mr + 1)‘/’(211.., +1)1/2(21m + 1)”2 1 —1 0 I... I... I, (2.10) This is the limiting case of the generalized F-coefficients when only one 7 ray is observed with respect to a fixed direction. The BA coefficients in Eqs. (2.1), (2.2) and (2.4), are orientation parameters which have the form I 1' BA = (2). +1)1/2(2I,-+1)1/2 Z (—1)’i+"‘ m=—I,- I,- I,- A x P(m). (2.11) —m m 0 As can be seen in Eq. (2.11), the BA coefficients are directly related to the population parameters P( m ), which specify the initial m substate distribution of a nucleus in an excited state with spin 1,- and are related to the amount of alignment in the excited state. If a nucleus is in an excited state with 100% alignment, the angular momentum of the nucleus is fully oriented either perpendicular to or parallel and anti-parallel to 16 /\ "' l, l' = [+1 ’f excitation 1e Iii Figure 2.2: Notation used for angular distribution formalism. I,- and I f are the initial (excited) and final spins of the 7—ray transition, respectively. The lowest possible multipolarities of the 7-ray transition are l and l’ = l + 1. 1,.- is the spin of the initial state before the nucleus is excited and la is the orbital angular momentum of the excitation. the beam axis. A percentage of alignment quoted as less than 100% is measured or calculated with respect to the B2 value for the fully aligned cases. 2.2 Angular distribution function The 7-ray angular distribution function is a specific case of the 7-7 angular corre- lation function in which only one 7 ray is observed. It is the same as if the 7-ray transition from spins Im —> I f in Fig. 2.1 is not observed. Therefore, Fig. 2.2 shows the comparable notation if just one 7 ray is observed from spins I,- —* I f. Throughout this work the notation W(l91,62,<1>) containing 61, 02 and will indicate a 7—7 angular correlation function and W(0) will indicate a 7-ray angular distribution. The 7-ray angular distribution function wc'm°(0°°"") specifies the relative proba- bility to observe, in the center of mass, a 7-ray transition from spin I,- to I f at an angle 0“" with respect to the beam axis and has the form 21' mam-(om) = Z AABAPA(cos(HC'm')). (2.12) A=0 A even The A A coefficients depend on I, l’, I,- and I I. They are a special case of the generalized 17 angular distribution coefficients (defined in Eq. 2.6) where A1 and A2 are equal to zero. The A A coefficients are defined in Eq. 2.9, but for the 1m —» I f transition as needed for the 7-7 angular correlation function. Therefore the A A coefficients as needed for the 7- ray angular distribution function can be calculated by replacing, A2, Jmf, lmf, [inf and Im with A, 6, l, l’ and 1,, respectively. The BA coefficients (defined in Eq. 2.11) depend on I“, I,- and la. The PA(cos(r9¢'m')) terms in Eq. 2.12 are Legendre polynomials. The 7-7 angular correlation function in Eq. 2.4 contains spherical harmonics. Spherical harmonics can be written in terms of associated Legendre polynomials. The initial excited state has axial symmetry about the quantization axis (in 05), so q (in Eq. 2.4) is equal to zero. This reduces the spherical harmonics to the Legendre polynomials seen in Eq. 2.12. To Lorentz boost wc'm'(0c""') into the laboratory frame, a solid angle correction is applied and the Legendre polynomials are written as a function of the laboratory frame angle 6 with respect to the beam axis. Thus, 2” CJn. 1'_-z32 211(9) = g AABAPA(cos(6 )) (fl cos(0) _ 1),, (2.13) A even where cos(r9°"“) can be transformed to cos(6) in the laboratory frame using the rela- tion found in Eq. 2.3. For 6 = 0, 20(9) is equal to wc'm'wc'm'). The angular distribution functions in Eqs. 2.12 and 2.13 are not necessarily nor- malized. Dividing by 47erBo normalizes 10(6) and wc'm'(6‘3'm') to unity, with respect to integrating over all space, because the even Legendre polynomials of order two and greater integrate to zero over the interval 0 g 0 S 7r. As with the normalized 7-7 angular correlation function (Eq. 2.4), if the presented formulas are used and P(m) is normalized, then A0 and Bo are unity. The normalized angular distribution function 18 W(l9) has the form 21’ 28(0) 1 AIBIP1(cos(0C""')) W = : — (6) 477/1080 47f 1 + AZ—E A080 A glen 1 — 52 X moose) — 02’ (2'14) and is normalized to 27r/ W(0)sin(0)d0 = 1. (2.15) 0 Normalizing wc'm'wc'm') is similarly performed. The sums in Eqs. 2.12, 2.13 and 2.14 extend to twice the multipolarity of the 7-ray transition, l’ = l + 1, with the odd A terms equal to zero. As introduced in Eq. 2.11, the BA coeflicients contain the population parameters P(m) which are related to the amount of alignment in an excited state. Both the amount of alignment in an excited state and the velocity of the center-of-mass frame with respect to the laboratory frame can affect the resulting 7-7 angular correlation or 7-ray angular distribution. Fig. 2.3 illustrates the effect of the Lorentz boost on W(6) for beam velocities of B = 0.4 and B = 0.6 with 100% oblate or prolate align- ment, defined in Section 2.3. It will be seen that a Lorentz boost for 6 = 0.4 or 0.6 has a smaller effect on W(0) than a reduction in the alignment of an excited state. The amount of alignment in an excited state and the Lorentz boost are independent and have different effects on W(0) and III/(61, 02, Q). A reduction in the amount of align- ment in an excited state reduces the anisotropy of W(0) (and changes the anisotropy of W091, 02, Q)) and a Lorentz boost forward focuses W(0) and W(01,62, Q). 2.3 Alignment For an unoriented initial state (one with no alignment in the initial excited state) a 7- ray angular distribution shows no anisotropy, but a 7-7 angular correlation can show 19 100% oblate alignment 0 A 0.2 - r "" ' ' 0.5 3 7.99 B = 0.0 x‘ '.' 1' r3 = 0.4 (3 = 0.6 A 0.15 i ' ' i 0-3 i '. g 3 ‘. 2 -> 0 .’ i I I g 0.1 " “ ,fkr \, ‘3 0.2 " '| f l 1’ I. i 5‘ l .' ’ I ‘r 0.05 [1 ’3 4.2 ' ' .’ f i- 0.1”! g ,"\ 'x_ ,' .‘ '. .‘ .1 .‘ ‘ ‘ - \' \ ‘1; \ ..... 1.- .....’...' ' .li --.’..‘.T.‘~- 30 90 150 30 90 150 9(d99) 6(deg) 100% prolate alignment is _ \ _ l3 = 0.0 0.25 X B _ 0.4 0.4 ,\ s _ 0.6 0.15 ‘ \ a 3 _, 2 , . \ 0.2 A 0.3 ‘\ v "' ~ ’ I ’ \ " \ ‘ \ \ I. \ l I .1 I .\ \ g 0 1 \ ’ - I 0 5 [\x ‘x 0 2 i“ ‘ o 05 2 -) 7“ \ _, II \.\ 0.1 I. \\ ‘\ j \\ \.\ . .l ' I. \ ‘ — —. _____ ] 0.1 I \x \ ' I 3 _) 0 \\ 0.05 ,” \\ "o 2 ________ 30“”90“”1‘50 “80‘“90 ”1‘50 ‘80”“9‘0‘”150 9(degl 6(deg) 9(d69) Figure 2.3: Plots of W(6) at 5 = 0.0, 0.4 and 0.6 for quadrupole (solid curves), dipole (dashed curves) and octupole (dash-dotted curves) transitions with 100% oblate or prolate alignment. 20 0% alignment Projectile frame ([3 = 0.0) 01: 60 deg 91 = 120 deg 91 = 150 deg 0.007 0.007 3 0.0065 _ 0.0065 g, 0.006 -. .. .. 0.006 CD 9': 0-035 ~.91=90deg 91:120deg 91:150deg 00,2 3 0°25 - _ 0.008 0.015 " ~ , 0.004 0.005 A A A A A A A A A A 30 ‘90 ‘1‘50 ‘30 ‘00 ‘1‘50 30 90 150 30 90 150 30 ‘90 ‘1‘50 02 (deg) 92 (deg) 92 (deg) 92 (deg) 02 (deg) 0% alignment Laboratory frame ([3 = 0.4) Figure 2.4: Plots of W(61, 92, ) for 4 —> 2 —* 0 (solid curves) and 3 —+ 2 —* O (dash- dotted curves) correlations at fl = 0 (top row) and [3 = 0.4 (bottom row) with no alignment. In each plot (I) and 01 are fixed; <1) E (1)1 — ¢2 = 0° and the value of 61 is as labeled on each plot. The angular correlation function, W091, 62, ), is normalized as in Eq. 2.5. a large amount of anisotropy. In Fig. 2.4, W(61, 02, ) is shown for spin transitions 4 ——+ 2 —> 0 and 3 —> 2 —» 0 with no alignment and beam velocities of fl = 0 and fl = 0.4. The 4 —+ 2 and 2 —> 0 transitions are taken to be pure quadrupole transitions. The 3 —> 2 transition is taken to be a pure dipole transition and the 2 ——> 0 transition is taken to be a pure quadrupole transition. For all the 7-7 angular correlation plots in this chapter is equal to 0°. This was chosen because the dependence of on W091, 62, ) is such that the maximum anisotropy of W(01, 02, ) occurs for (I) = 0°, 180°, when fl = O. The 4 —> 2 —> O and 3 —> 2 —» 0 correlations can be easily distinguished when 6 = 0. For B = 0.4 the relative difference between the 4 —+ 2 —> 0 and 3 —+ 2 —+ 0 correlation functions, seen in Fig. 2.4, is as large as 20% when 01 = 02. The alignment produced in a nucleus in an excited state can be one of two types. If the angular momentum of the nucleus is aligned perpendicular to the beam axis, then the m = 0 substates are preferentially populated. This alignment, referred to as oblate, is observed in fusion-evaporation (heavy-ion, :m) reactions. If the angular momentum of the nucleus is aligned parallel and anti-parallel to the beam axis, then 21 the m = :l: I,- substates are preferentially populated. This is referred to as prolate alignment. A prolate-aligned nucleus produces a different 'y-ray angular distribution or '7-7 angular correlation for a given y-ray transition than an oblate—aligned nucleus. Three-dimensional plots of W(6) are shown in Fig. 2.5. The top row shows a pure quadrupole transition, the middle row a pure dipole transition and the bottom row a pure octupole transition. The first column shows the transitions at rest with 100% oblate alignment. The second column shows the transitions at B = 0.4 with 100% oblate alignment. The third column shows the transitions at 3 = 0.4 with 20% oblate alignment. As can be seen in the third column of Fig. 2.5, a reduction in the percentage of alignment combined with a Lorentz boost for fl = 0.4 washes out much of the anisotropy in the 'y-ray angular distribution. A similar reduction in anisotropy occurs in the 7—ray angular distribution for a prolate aligned nucleus. However, distinguishing quadrupole, dipole and octupole transitions is still experimentally possible and will be discussed quantitatively in section 2.5. In Chapters 4 and 5 experimental 7-ray angular distributions will be shown and discussed. The percentage of alignment for an initial excited state, 1,, given the population parameters P(m), is defined as follows for prolate and oblate alignment. For prolate alignment, 1. ' 3712 — 1,-(Ii + 1)P(n) = 1 . 2.16 AP ”:24. 1421‘ _ 1) x 00 ( ) For oblate alignment, ’i 2 ._ . . A2” = ":24. 3n Ifz((Ift-:11))P(n) x 100 for I,- integer, (2.17) Ii - 3n2 — I-(I-+1)P(n) , half—mt ____ 3 1 _ _ A0 E LU" + l) _ 3/4 x 100 for I, half integer. (2.18) n=— ,- 22 Figure 2.5: Three-dimensional 'y-ray angular distributions for three *y-ray transitions from an initial state with oblate alignment. The top row of figures shows pure quadru- pole transitions (2—>0), the middle row shows pure dipole transitions (3——>2) and the bottom row shows pure octupole transitions (3—.0)_ The first column shows all three transitions, quadrupole, dipole and octupole, at [i = 0.0 with 100% oblate alignment. The second column shows the same transitions Lorentz boosted for (3 = 0.4. The third column shows the same transitions Lorentz boosted for fl = 0.4 and with the alignment reduced to 20% oblate alignment. In all the plots, the beam axis is the z-axis (up is positive 2) with the beam direction towards positive 2. 23 100% oblate alignment 50% oblate alignment 0.25 » B = 0_4 0.25 . B = 0.4 2 -+ O 2 —9 0 0.2 . 0.2 . 8 V 0.15 . 0.15 . 3 0.1 . 0.1 , 4 -—> 2 0.05 . 0.05 . 100% prolate alignment 50% prolate alignment 0.15. 0:04 0.15.240 B=0-4 a? 4 —+ 2 ‘3’ 0.1 . 2 2.0, 2 0 1 . 0.05 » 0.05 » 30 L l A 90 U 41550 U30 A l A 1910441 i 150 J 6 (deg) 0 (deg) Figure 2.6: Plots of W(l9) for 100% and 50% oblate or prolate alignment for two different quadrupole transitions at fl = 0.4. The top row shows that 4—>2 and 2—+0 transitions are significantly different at 100% oblate alignment. However, at 50% alignment they are similar. The bottom row shows that 4—+2 and 2—>O transitions are identical with 100% prolate alignment and at 50% alignment they are similar. The dipole transitions, 3——»2 and 1-+0, and the octupole transitions, 6—>3 and 3—>O, behave similarly to the two quadrupole transitions shown. 2.4 W09) for transitions of identical multipolarity Examples of 4—>2 and 2—>0 quadrupole transitions at 100% and 50% oblate or prolate alignment are shown in Fig. 2.6. For 100% prolate alignment different spin I,- —+ I f transitions with the same multipolarity are nearly identical at B = 0.4. Angular distributions for quadrupole 2—>0 and 4—>2 transitions are equivalent, for dipole 3—>2 and 1—>0 are equivalent and for octupole 6—»3 and 3-—>O are equivalent. However, for 100% oblate alignment those same three pairs of transitions are significantly different. The l—>0, 2—+0 and 3—>0 transitions vanish at the origin and have more features than 24 the 3—->2, 4—+2 and 6—>3 transitions, respectively. As the amount of oblate alignment decreases, the same-multipolarity different-spin I,- —» I f transitions become more alike. As the amount of prolate alignment decreases, same-multipolarity different-spin I,- ——> If transitions do not all approach an isotropic Lorentz-boosted distribution the same way, but the differences between those same multipolarity transitions are small enough not to impede distinguishing quadrupole, dipole and octupole transitions. 2.5 Different reaction mechanisms In an in-beam experiment, partial alignments will be observed. Four reaction mecha- nisms, namely fusion evaporation, fragmentation, nucleon-knockout and intermediate— energy Coulomb excitation, will be discussed in this chapter. The latter three have become standard in-beam spectroscopy tools with 7—ray detection, using intermediate- energy (fl = 0.3—0.8) exotic beams, but have not been thoroughly explored. Fragmen- tation reactions were pioneered in 1979 [55], but did not utilize '7-ray detection in the early implementations. Nuclear break-up reactions (including fragmentation and nucleon-knockout reactions) were first used in conjunction with the detection of 7- rays as a spectroscopic tool in three pioneering experiments [37, 56, 57]. The first intermediate-energy Coulomb excitation experiments were performed in 1995 [10,56]. 2.5.1 Fusion evaporation Non-relativistic fusion evaporation reactions will first be discussed. Yamazaki [58] approximated the population parameter P( m ), in Eq. 2.11, for non-relativistic (heavy- ion, am) reactions using a Gaussian distribution centered at m = 0, based upon work done by Diamond et al. [59]. Normalized, P( m) then has the form e—m2/2a2 P(m) = 1. (2.19) _2 2 2:611/20 n=— i 25 The width of the Gaussian distribution, or, is related to the amount of oblate align- ment in the excited state. For an excited state with less than 100% alignment, 0 > 0 such that 32/85“; x 100 is equal to the percentage of alignment, where Bgm is the 82 coefficient corresponding to the fully aligned case. The percent of alignment, in terms of P(m), is defined in Eqs. 2.16 and 2.17. We have reproduced the tabulated angular distribution coefficients of Yamazaki [58] and der Mateosian and Sunyar [60]. A reduction in the amount of alignment produced in a nucleus in an excited state sig- nificantly decreases the anisotropy of the 'y-ray angular distribution as seen previously for oblate alignment in Fig. 2.5. 2.5.2 Fragmentation reactions With the intermediate-energy beams available today, fragmentation reactions have become a spectrosc0pic tool for creating and studying exotic nuclei. In fragmentation reactions the amount and type of alignment produced is not well studied. Investiga- tions of alignment and polarization produced in intermediate-energy fragmentation re- actions [61—64] are not consistent. The alignment produced in [62-64] ranges from less than 1% to 35%. In the experiment where less than 1% alignment was produced [62], the fragment was five protons and ten neutrons removed from the incoming projectile, so little alignment was expected. Each of the 15 nucleons carried off some amount of linear and angular momentum in a random direction, thus leaving the fragment with a significantly lower angular momentum and destroying the alignment. Reactions using intermediate-energy beams were 9Be(22Ne,18N)X [62], 9Be(180,14B 7)X [64], 12C(13C,”B)X [62] and 9Be(‘“‘Ti,""‘""Sc 7)X [63]. The first two reactions were per- formed at 60 MeV/ nucleon and were 3p1n removal reactions. In those two reactions, 14.4% prolate alignment and less than 5% prolate or oblate alignment, respectively, were observed in the center of the longitudinal momentum distribution. In the wings of the momentum distributions, oblate alignment was observed in both experiments. In 12C(13C,”B)X [62], 4.7% prolate alignment was observed in the center of the momen- 26 tum distribution. In 9Be(46Ti,“3'"Sc 7)X [63], performed at 500 MeV/nucleon, 35% prolate alignment was observed in the center of the momentum distribution and 15% oblate alignment was observed in the wing of the momentum distribution. Though the magnitudes of the alignment observed differ significantly, the type of alignment produced as a function of the longitudinal momentum distribution is consistent. In [62—64] the alignments produced from the reactions were prolate in the center of the momentum distributions and oblate in the wings (or tail) of the momentum distribu— tions. This can be understood in terms of a simple kinematical model [63, 64] where the projectile fragment acts as a spectator while the nucleons in the overlapping vol- ume with the target are removed. The outgoing momentum of the fragment is then directly related to the angular momentum of the fragment, thus giving prolate or oblate alignment values. Reactions where more nucleons are removed have broader momentum distributions [65]. Thus the m substate distributions are flatter, resulting in less alignment. This supports the result in [62] where 15 nucleons were removed and less than 1% alignment was observed. In contrast to the above experiments, in [61] an alignment of 30%-70% was observed, though the type of alignment was not reported. The reaction was 9Be(“‘BCa,“6Ar 7)X at 60.3 MeV/ nucleon. In addition to “Ar, many of the fragments from the reaction were analyzed. In order to reproduce the approx- imate ratios of the 'y-ray intensities of the quadrupole and dipole transitions in Fig. 2 of [61], we had to assume oblate alignment. The ratios observed in [61] seem to be incompatible with any percentage of prolate alignment. Extending the formalism of Yamazaki [58] and Diamond at al. [59] for oblate population distributions, two Gaussian distributions centered at m = :l: 1.- are used for approximating prolate alignment in fragmentation reactions [54]. Normalized, P(m) then has the form e-(I.-Iml>2/2o2 P(m) (2.20) : Ii . Z e-(I.—In|>2/2a2 n=—h 27 Table 2.1: Relative difference of W(6) between a quadrupole, 2—+0, and dipole, 3—.2, transition (with respect to the dipole transition) for 20% oblate or prolate alignment at 5 = 0.4 and 0.6 for a few selected angles, 6, with respect to the beam axis. The angles were selected to sample the range of angles from 0° to 180°. Relative difference (%) .3 0 Oblate Prolate 0.4 0° 22.5 9.4 25° 10.7 13.7 60° 10.7 16.8 150° 18.9 12.4 0.6 0° 22.5 9.4 25° 4.7 10.0 60° 10.5 16.4 150° 20.4 11.3 In fragmentation reactions the target is viewed as a means of removing nucleons from the projectile. Thus, we consider the center-of-mass frame to be the projectile frame. In order for 'y-ray angular distributions to be a useful experimental tool for measuring y-ray multipolarities and thus spins of excited states, there must be enough alignment in the excited state to distinguish between different multipolarity transitions. As the amount of alignment decreases, the difference between the angular distribution curves for different multipolarity 7—ray transitions decreases as both curves approach Lorentz-boosted, isotropic distributions. Typical intermediate-energy beam velocities (NSCL, RIKEN, GANIL, GSI) are between 0.3c and 0.8c. At the proposed RIA facility [66], beams will have velocities of approximately 0.6c. Figs. 2.7 and 2.8 show plots of W(6) calculated with 50%, 20% and 10% oblate and prolate alignment, respectively, for quadrupole, dipole and octupole transitions. The plots are shown for incoming beam velocities of 0.4c and 0.6c. Table 2.1 lists the relative difference between the quadrupole and dipole transitions with 20% alignment, as seen in Figs. 2.7 and 2.8, for selected angles. Table 2.1 illustrates that for a fragmentation reaction where only 20% prolate or oblate alignment is produced the difference between a quadrupole and dipole transition is large enough to distinguish between the two and to make the measurement of 'y-ray angular distributions experimentally feasible. To 28 0.25 » 0.05 0.05 » 0.25 0.05 . Figure 2.7: Plots of W(6) for 50%, 20% and 10% oblate alignment with an incom- ing beam velocity of 0.4c (left column) and 0.6c (right column). Quadrupole transi- tions (solid curves), dipole transitions (dashed curves) and octupole transitions (dash- dotted curves) are shown. The order of the transitions, as labeled on the upper left \. 50% oblate alignment -§*0 B=04 20% oblate alignment [3: 0.4 10% oblate alignment B= 0.4 30 60 90 120150180 9(d69) 0.3» 0.3» 0.3» 0.2» 0.1» 50% oblate alignment B= 0.6 20% oblate alignment B= 0.6 A '30 ‘60. ‘90 ‘120 ‘150 ‘180 9(deg) plot, is the same for all of the plots in the figure. 29 0.25 > 0.05 50% prolate alignment B = 0.4 0.25 - 0.05 » 0.255 0.2 » 0.1 » 0.05 20% prolate alignment B: 0.4 ‘3’ 0.15: ' 10% prolate alignment B: 0.4 120 150 180 30 60 90 9(deg) 0.3 0.3 » 0.3 ; 50% prolate alignment B: 0.6 20% prolate alignment B= 0.6 10% prolate alignment B: 0.6 30 60 90 120150180 0(deg) Figure 2.8: Plots of W(6) for 50%, 20% and 10% prolate alignment with an incom- ing beam velocity of 0.4c (left column) and 0.6c (right column). Quadrupole transi- tions (solid curves), dipole transitions (dashed curves) and octupole transitions (dash- dotted curves) are shown. The order of the transitions, as labeled on the upper left plot, is the same for all of the plots in the figure. 30 better illustrate the large effect of the percentage of alignment on W(6), W(6) is shown for quadrupole, dipole and octupole transitions with 100%, 50%, 20%, 10% and 0% oblate and prolate alignment in Figs. 2.9 and 2.10, respectively, at ,8 = 0.4 and 0.6. The curves with 0% alignment are isotropic, Lorentz-boosted distributions. The curves with the most features correspond to 100% alignment. A reduction in the percentage of alignment has a much larger effect on W(0) than a Lorentz boost for 0 = 0.4 or 0.6. For P(m) approximated by a Gaussian distribution(s), transitions with a given multipolarity but different spins I, —» I f are similar for approximately 50% oblate or prolate alignment. Thus stretched quadrupole, dipole and octupole transitions can be distinguished from one another. In this chapter, the same amount of alignment in all excited states has been assumed in comparing different multipolarity transitions. Fragmentation reactions may produce different percentages of alignment for different initial spins, 1,, in the same nucleus. However, such differences are likely to be small and would only lead to ambiguities if one state was prolate-aligned and another state was oblate-aligned -— a situation that is unprecedented. To distinguish quadrupole, dipole and octupole transitions pr0per care must be given to detector placement. At the angles around 40° and 100°, W(l9) is the same for different multipolarity transitions at B = 0.4. Avoiding the range of angles around 40° and 100°, the relative difference between a quadrupole (2—>0) and dipole (3—.2) transition is on the order of 10—20% for fragmentation reactions with 20% prolate or oblate alignment at 5 = 0.4. A minimum of approximately 600 counts with negligible background, corresponding to an uncertainty of i4%, in the ’y-ray peak at a given angle is needed to distinguish between a quadrupole and dipole transition that are 10% different at that angle. See Sect. 3.3.1 for the discussion of the detector setup for the experiments performed. The previous discussions of fragmentation reactions and alignment also apply to 7-7 angular correlations. Figs. 2.11 and 2.12 show 'y-y angular correlations for 31 0 - 100% oblate alignment B = 0.4 0.3» § 0.2L 3 Ef§1\ 0.1 7 ’ ‘x‘§ «3333335 0 - 100% oblate alignment B = 0.4 0.35 g 0.2 3 0.1» 0 100% oblate alignment .' ', B: 0.4 0,3. ; 'g "I" '. , i \ ‘~ § 0.21113 ‘3 g ’7“..\\-\' l ,' Rx} .‘ 0.1 .' I. \:‘§.‘- , . i ‘. 03‘5“ ,r' "‘. i ‘ .’ - ““.*~'*'=‘;;.;.-:::- ,' ‘. .I \ ‘/ -\"-:=--'i ‘30”‘60‘ '90 120 '150 .180 9(deg) 0 - 100% oblate alignment 04_ 0:05 0.3E§ _ :\ 0.2 p ’ I \ \ \ \ 0.1 . \\ 0 - 100% oblate alignment 04’ 0:06 0.3I 0.2. 0.1 ._l ‘3 0 100% oblate allgnment 0.4» fl ' B=0.6 ~11 '~ I if.) ', 0.3 » 7-‘33‘! .' "(K 0.2.;l ”I. i I, \3\..~. .' I ‘33»). 0.1 r,’ I f ‘33‘;\ ' . l \x". I' " \ f |‘ .I. .\ XWG-l-ffi-E‘; A ‘ ‘ 310’ ‘60 A 90 '1'20 '150 ‘180 6(deg) Figure 2.9: Plots of W(6) for dipole ((3—+2), dashed curves, top row), quadrupole ((2—>0), solid curves, middle row) and octupole ((3—->0), dash-dotted curves, bottom row) transitions, each with 100%, 50%, 20%, 10% and 0% oblate alignment at 0 = 0.4 (left column) and 0.6 (right column). The curves with the most features are those corresponding to 100% alignment. The flatter curves are those corresponding to 0% alignment (an isotropic Lorentz—boosted distribution.) 32 0.25 I 0.2 I § 0.15» 0.1 0.05 0.25 I 0.2 Q 0.15. 0.1 . 0.05 0.25 0.2 I § 0.15 0.1 I 0.05 ~\ 0 - 100% prolate alignment \ B = 0.4 vvvvvvvvvvvvvvvvv 0 - 100% prolate alignment B = 0.4 L J L A A 0 - 100% prolate alignment B = 0.4 I \ ‘-~ ‘ . 60‘ ‘60 00 120 I50 180 9(d99) 0.4 0.4 I 0.3 I 0.2 I 0.1 I 0.4 I 0.3 . 0.2 I 0.1 . 0 - 100% prolate alignment 0 - 100% prolate alignment B=0.6 L 1m 1 1 A 1L 0 - 100% prolate alignment B=0.6 9(d99) Figure 2.10: Plots of W(6) for dipole ((3—>2), dashed curves, top row), quadrupole ((2—>0), solid curves, middle row) and octupole ((3—I0), dash-dotted curves, bottom row) transitions, each with 100%, 50%, 20%, 10% and 0% prolate alignment at 5 = 0.4 (left column) and 0.6 (right column). The curves with the most features are those corresponding to 100% alignment. The flatter curves are those corresponding to 0% alignment (an isotropic Lorentz-boosted distribution.) 33 A ‘AK91,92,¢> VVGH.92.¢0 01:25deg 91=6Odeg 61:90deg 61=120deg 91=150deg 0.0075 _, 2 _, _ 0.0065 ‘_ , ' 0.0055 344,510 0.03 0.02 0.01 20% oblate alignment Projectile frame (B = 0.0) 9090150 30 9‘0 ‘ 150 30 90150 3090150 3090150 92(d69) 02(d99) 20% oblate alignment Laboratory frame (B = 0.4) Figure 2.11: Plots of W(01,62,) for 4 —+ 2 —) 0 (solid curves) and 3 —> 2 —+ 0 (dash-dotted curves) correlations at 6 = 0 (top row) and fl = 0.4 (bottom row) with 20% oblate alignment. In each plot (I) and (91 are fixed; (D E 051 — 452 = 0° and the value of 61 is as labeled on each plot. The angular correlation function, W(01, 02, (1)), is normalized as in Eq. 2.5. 20% prolate alignment Projectile frame (B = 0.0) 92(d90) 92(d99) 92(d99) 0.0075 _ 00065 0.0055 0.01 0.006 0.002 0.008 0.007 0.006 . . 0.005 01 = 25 deg 01=60deg 01 = 120 deg 01= ’ o 150 deg 0.008 0.007 0.006 0.005 0.025 0.015 0.005 01 =L-A1‘20‘de‘g .“ \ Q . \ o 0.01 0.006 0.002 3090150 3090150 “30590150 3090150 3090150 92(d99) 92 (deg) 20% prolate alignment Laboratory frame (B = 0.4) Figure 2.12: Plots of W(01,62,) for 4 -—> 2 -+ 0 (solid curves) and 3 —-> 2 —-> 0 (dash-dotted curves) correlations at H = 0 (top row) and fl = 0.4 (bottom row) with 20% prolate alignment. In each plot and 01 are fixed; (I) E 051 — (152 = 0° and the value of 01 is as labeled on each plot. The angular correlation function, W(01, 02, (1)), is normalized as in Eq. 2.5. 92(d99) 34 92(d99) 92(d99) 4 —> 2 —+ 0 and 3 —+ 2 —+ 0 transitions at 0 = 0 and B = 0.4 with 20% oblate or 20% prolate alignment, respectively. The larger the reaction induced alignment in an excited state, the greater the anisotropy in the '7-7 angular correlation. For nuclei with 20% oblate or prolate alignment at B = 0.4, as seen in Figs. 2.11 and 2.12, the relative difference between the 4 —> 2 —» 0 and 3 —> 2 —I 0 correlation functions is as large as approximately 30%. A minimum of approximately 50 counts with negligible background, corresponding to an uncertainty of i14%, in the 7-ray peak at a given set of angles (61, 02 and 2 0°) would be needed to distinguish between a 4 —> 2 -—> 0 and 3 —> 2 —> 0 correlation where the correlation function, W(01, 62, (1)) differs by 30%. To give a specific example, measuring 4 —+ 2 —I 0 and 3 —-> 2 —> 0 correlations from nuclei with 20% oblate alignment could be accomplished by having detectors at the following angles: 17°, 29°, 88°, 102°, 112°, 139° and 155°. For (61,02,) = (17°,29°,0°) the relative difference between the 4 -—> 2 —> 0 and 3 ——> 2 —> 0 correlation functions is 28%. Likewise, for (102°,88°,0°) the relative difference is 20%, for (139°,112°,0°) it is 27%, for (155°,139°,0°) it is 30% and for (155°,155°,0°) it is 29%. 2.5.3 Nucleon—knockout reactions The category of nucleon-knockout reactions includes the removal of one to a few nucleons. For one-nucleon knockout reactions, it is possible to calculate P( m). The multipolarity, le, of the transition from I,-,- to I,- is deduced from the momentum dist- ribution of the knocked-out particle as well as knowledge of 1a- (In nucleon-knockout reactions 1,,- is usually the spin of an excited state in another nucleus, not the ground state spin of the nucleus of interest as may be implied in Figs. 2.1 and 2.2.) P(m) can be calculated using an extension of the eikonal model [67] to obtain m sub— state dependent cross sections. The model utilizes the black-disk limit: the projectile wavefunction is unchanged throughout space except for a cylinder of a given radius where it is set to zero. This is discussed in more detail in [68]. In Fig. 2.13, W(9) 35 939 (28Mg’ 27Mg’ y) X = 0.4 0.25 I 0.2 0.15I W(0) 0.1 I 0.05 I ‘30‘ ‘ 60 ‘ ‘ 90‘ ‘1i20‘ 15051130 9 (deg) Figure 2.13: W(6) for the one-neutron knockout reaction of 9Be(28Mg,27Mg 7)X at 6 = 0.4-for the 5 / 2+ to 1/2+ quadrupole transition and 5 / 2+ to 3 / 2+ dipole transition. The inset shows the m substate distribution for the 5 / 2+ excited state which corresponds to 27% prolate alignment. is shown with the corresponding calculated m substate distribution [69] for the one- neutron removal reaction of 9Be(28Mg,27Mg 7)X at 5 = 0.4. The anisotropy of the 'y-ray angular distribution can be significantly increased by selecting momenta from the center region of the momentum distribution where the greatest contribution is from the m = :l: I,- substates. In Fig. 2.14, W(01, 62, (1)) is shown for the one-neutron knockout reaction of 9Be(52Ca, 51Ca 7)X at 6 = 0.4 for the 7/2+ —> 5/2+ —> 3/2+ and 7/2+ —> 3/2+ ——> 3/2” 7-ray transitions in 51Ca. The m substate distribution [69] for the 7/2+ excited state in 51Ca is shown in Fig. 2.15. The relative difference between the 7/2+ —> 5/2+ —> 3/2+ and 7/2+ —+ 3/2+ —> 3/2+ correlation functions is shown in a three-dimensional plot in Fig. 2.16 and is as large as 40%. The relative difference between quadrupole and dipole transitions for nucleon- knockout reactions is similar to fragmentation reactions when the entire momentum distribution is used. Thus the number of counts needed at a given angle is approxi- mately the same as stated at the end of the previous section. If a cut is made on the 36 9Be(52Ca,51 Ca 7) X Laboratory frame (B = 0.4) 01 =25 deg 01 =60 deg 01 =90 deg 91 = 120 deg 01 = 150 deg 0.01 0 .: 0'02 \ 0-006 9; 0.01 . ‘- . _. 3 ~~~~~ 0.002 30 90 30 90 150 30 90 150 92 (deg) 92 (deg) 92 (deg) 92 (deg) 92 (deg) Figure 2.14: Plots of W091, 02, (1)) for the one-neutron knockout reaction of gBe(520a, 51Ca. 7)X at 0 = 0.4 for the 7/2+ —> 5/2+ —+ 3/2+ (dash-dotted curves) and 7/2+ —> 3/2+ —> 3/2+ (solid curves) 'y-ray transitions in 51Ca. In each plot and 01 are fixed; E d), —¢2 = 0° and the value of 01 is as labeled on each plot. The angular correlation function, W(01, 02, ), is normalized as in Eq. 2.5. .0 N 989(52Ca,51 Ca 7) X o 9 ’-‘ —L 0'1 Population probability 8 U1 -7/2 -5/2 -3/2 -1/2 72 3/2 5/2 7/2 m Figure 2.15: The m substate distribution of the 7/2+ excited state of 51Ca produced from the one-neutron knockout reaction 9Be(52Ca, 510a 'y)X at 0 = 0.4. The distri- bution corresponds to 25% prolate alignment. (p4:- 380° 0 Relative difference (%) 0 100 7 (deg) 150 Figure 2.16: Three-dimensional plot of the relative difference between the 7/2+ —-> 5/2+ —» 3/2+ and 7/2+ —> 3/2+ —> 3/2+ 'y-ray transitions (with respect to the 7/2+ —I 5/2+ —> 3/2+ transition) in 51Ca from the one-neutron knockout reaction of 9Be(520a, 510a 7)X at 5 = 0.4. 37 center of the momentum distribution, the relative difference between quadrupole and dipole transitions can be increased significantly. In this work, the beam axis is used as the quantization axis for calculations of ’y-ray angular distributions and 7—7 angular correlations. It has been shown [70] for nucleon-knockout reactions that if the recoil direction of the knocked—out nucleon is used as the quantization axis instead of the beam axis, a large amount of anisotropy is produced in the angular distribution. Unlike in fragmentation reactions, in nucleon- knockout reactions the excited state P(m) can be estimated from the momentum distribution of the knocked-out nucleon. Looking at the 'y-ray angular distribution with the recoil direction as the quantization axis gives a much larger anisotropy than using the beam axis as the quantization axis and is worth pursuing further. 2.5.4 Intermediate-energy Coulomb excitation In Coulomb excitation P( m ) can be calculated from a known Hamiltonian for given electromagnetic transitions. P(m) can be calculated using the theory of Alder and Winther [71] for intermediate-energy Coulomb excitation. Its form is 16 P(m) = Z 0.01, 5, 100201, 3, le)g(u,€) #=—le x(—1)‘"“"(—1)’\(2)\ +1) Ii Ii A la le A If I; A x (2.21) —mm0 u—pO lele I,~ where 7r specifies either electric or magnetic transitions and 1,, is the multipolarity of the Coulomb excitation. The 09,01,618) and g(p, 5) functions are defined in [71]. The adiabacity parameter, 5, specifies a straight-line trajectory with a correction that takes into account that the distance of closest approach is increased due to Coulomb deflection. 5 depends on the atomic number and mass of the target and projectile, the impact parameter, the velocity of the incoming projectile, and the 38 excitation energy of the nucleus. In the plots in Figs. 2.17 through 2.21 the impact parameter of the projectile is integrated from the minimum impact parameter, bmm, to infinity. This is experimentally realized by selecting events in which the angle of the scattered projectile is less than 0m. 0m“. is analytically related to the minimum impact parameter, bmin, for a specific reaction at a specific energy. In intermediate- energy Coulomb excitation reactions the target is viewed as a means of exciting the projectile. Thus, we consider the center-of—mass frame to be the projectile frame. This is a good approximation for light nuclei, but for heavy nuclei the center-of—mass and projectile frames should be distinguished. With typical minimum impact parameters in Coulomb excitation (touching spheres plus a few femtometers), and excitation energies of up to a few MeV, a Coulomb-excited nucleus exhibits prolate alignment. For illustrative purposes, it is worth noting that for very large minimum impact parameters (i.e. bmin z 100 fm), the alignment becomes oblate. For typical impact parameters in Coulomb excitation, as the velocity of the incoming beam increases, the amount of prolate alignment increases. The 7-ray angular distribution curves for 209Bi(5°Ni,5°Ni 7) at 85 MeV/nucleon (fl = 0.4) and 233 MeV/nucleon (fl = 0.6) for quadrupole, dipole and octupole transitions with their corresponding m substate distributions are shown in Fig. 2.17. The 7—ray angular distribution plots in Figs. 2.17 through 2.21 are calculated with E2 and E3 excitations. A dipole transition is 1,,- = 0 —» I,- = 3 —> I f = 2 (0—I3—I2), quadrupole is 0—>2—>0 and octupole is 0——>3—>0. For a given multipolarity, transitions between different spins, I,- —> I f, are very similar at 5 = 0.4 for intermediate-energy Coulomb excitation reactions. In intermediate-energy Coulomb excitation, the effect on W(0) of varying the min- imum impact parameter, bmm, is negligible for typical minimum impact parameters (touching spheres plus a few femtometers). As can be seen in the top row of Fig. 2.18, W(l9) is shown for 209Bi(5°Ni,5°Ni '7) at 85 MeV/nucleon (fl = 0.4) for three differ- ent minimum impact parameters: touching spheres plus 2 fm, touching spheres plus 39 0.25. 0. 2. GD v 0.15. 3 0.1. 0.05 55Ni + 2098i bmin = 13.7 fm 0=04 \‘\0—’3"2 0.4 [i=3 \ 0.2 30 ' 60 ‘ 90‘ 150 180 0(deg) A 0' f 56Ni+209Bi 0. [:2 bmin = 13.7 "TI 02 ‘ 0max=1.1 deg 0 0 3 B=0-6 -2-1012 ' ‘ ‘paaaz m \ IK 0.4 [i=3 3; 0.2 0 3 o 2 '. -3-2-10123 m 0.1 30 60’ 90 120 150 180 Figure 2.17: W(9) for the Coulomb excitation reaction of 209Bi(5°Ni,5°Ni7) at 85 MeV/ nucleon (fl = 0.4) (t0p plot) and 233 MeV/ nucleon (fl = 0.6) (bottom plot) with a minimum impact parameter, bmm, of touching spheres plus 2 fin and an excitation energy of 1 MeV. GM is the maximum scattering angle of the projectile in the laboratory frame corresponding to bmm. The insets show m substate distributions for the excited states. 40 \ 0 25 \ 56Ni + 2098i \ 55Ni + 2098i 5$Ni + 2098i ' . bmin = 13.7fm ‘. bmin = 21.7fm bmin = 111.7 fm 02 \\ emax = 3.0 deg \\ emax = 1.9 deg “ \ emax = 0.4 deg 8 015 \\ ."\‘ 3:0'4 \\ .“-\ B=0.4 3:0'4 ; . \ . 0.1 0.05 ------- -.\ 56Nj ... 2098i x 443 ... 2093i ‘\ 32Mg + 2093i 0-25 1‘ bmin = 13.7fm I\ 5min = 13.4fm \\ bmin = 12.9 fm 02 \‘ emax = 3.0 deg \\ emax = 2.3 deg \\ emax = 2.4 deg A \ 5:0.4 \ B=0.4 \ 0:0.4 8 \ ,’-\‘ \ ’.\. \ ,’.\‘ g 0 15 \. X a ‘. 0.05 " . ------- .' , ....... ,- 7», ...... . o . n . D - A_A14.1AAI..A..-L_A_AA ..... .. ..... 1.1—.AA AA . AAAAAAAAA ”cm—LA 30 90 150 30 90 150 30 90 150 Oldeg) 9(d69) 9(d69) Figure 2.18: W(6) for Coulomb excitation reactions at 85 MeV/nucleon (fl = 0.4). The top row shows W(l9) for three different minimum impact parameters, bmin. The bottom row shows W(0) for three different reactions. The minimum impact parameter is touching spheres plus 2 fm for all three reactions. 6m is the maximum scattering angle of the projectile in the laboratory frame corresponding to bmin- As in previ- ous plots, the quadrupole transitions are depicted by solid curves, dipole transitions are dashed curves and octupole transition are dash-dotted curves. The order of the transitions as labeled on the upper left plot is the same for all of the plots in the figure. 10 fm and touching spheres plus 100 fm. The difference between W(0) for the first two cases is minimal. In an experiment, as long as the minimum impact parameter corresponds to only Coulomb excitation reactions, the effect of the size of the mini- mum impact parameter used is negligible on the percentage of alignment produced. For a minimum impact parameter of touching spheres plus 100 fm, W(0) is signif- icantly different for each of the three transitions, however, the Coulomb excitation cross section at such a large minimum impact parameter is essentially zero. In general, as the minimum impact parameter increases, the amount of prolate alignment slowly decreases until the alignment becomes oblate. The second row of Fig 2.18 shows three 41 56Ni + 2098i Ex =3 1 MeV bmin = 13.7fm 83D l3=°-4 Ex=2MeV .2 02 Ex=10MeV 00 0-05 Ex = 1 MeV ‘30‘ ‘60‘ ‘90‘ 120’ ‘150‘ ‘180 9(d69) Figure 2.19: W(0) for quadrupole Coulomb excitation reactions with 1, 2, 5 and 10 MeV excitation energies at 85 MeV/nucleon (fl = 0.4). The minimum impact parameter is equal to touching spheres plus 2 fm. GM is the maximum scattering angle of the projectile in the laboratory frame corresponding to bmm. plots of W(9) for three different Coulomb excitation reactions. The minimum impact parameters are all touching spheres plus 2 fm. As can be seen, there is no detectable difference between 209Bi(32Mg,32Mg 'y), 209Bi("“S,4“S 'y) and 209Bi(56Ni,56Ni 7). For different light projectiles (from 14N to 115In), with a heavy (197Au or 209Bi) target, the effect on P( m ) is negligible. In Coulomb excitation not only do the velocity of the beam, impact parameter and nuclei involved affect P(m), but the excitation energy also affects P(m), which in turn influences W(6). As just seen, the minimum impact parameter and nuclei involved have a small effect (within typical impact parameters for Coulomb excitation) on W(0), but the excitation energy of the nucleus can have a fairly large effect on W(0). In Fig. 2.19, W(0) is shown for a quadrupole transi- tion with 1, 2, 5 and 10 MeV excitation energy with the corresponding m substate distributions. W(6) is plotted using ,8 = 0.4 and a minimum impact parameter equal to touching spheres plus 2 fm. Figs. 2.20 and 2.21 are likewise, except they show dipole and octupole transitions respectively. As the excitation energy increases, the 42 Ex = 1 MeV 56"“ + 2093 E :§\ bmin a 13.7fm 8-3 0.25. \ em. = 3.0 deg 15 p=o.4 J L l A n l 1 L 1 1 1 l 1 1 J l 4 30 60 90 1 20 1 50 180 9(d99) Figure 2.20: W(0) for dipole Coulomb excitation reactions with l, 2, 5 and 10 MeV excitation energies at 85 MeV/ nucleon (fl = 0.4). The minimum impact parameter is equal to touching spheres plus 2 fm. 0m is the maximum scattering angle of the projectile in the laboratory frame corresponding to bmm. 56M + 20931 Ex . 1 MeV bmin = 13.7fm 0.4 0.25» 6mm = 3-0 “99 002 B = 0'4 Ex = 2 MeV Ex = 10 MeV 0.4 0.2. .2- l 0 § - ~ - Ex - 5 MeV 3 0-‘5’ f1: X 0.2 , ’5' ‘. ' \ \‘r 0 01 ' 1' Ex! , I. ’2: \ :\ o l l l } 0.05» ,y 30‘. 11.5"?" 1’ 1. MP". . . 1 . . TIT-3- - 30 60 90 120 9(d99) Figure 2.21: W(6) for octupole Coulomb excitation reactions with 1, 2, 5 and 10 MeV excitation energies at 85 MeV/nucleon (,8 = 0.4). The minimum impact parameter is equal to touching spheres plus 2 fm. OM is the maximum scattering angle of the projectile in the laboratory frame corresponding to bmin. 43 amount of prolate alignment decreases until the alignment becomes oblate. As can be seen in Fig. 2.19, at energies between 5 and 10 MeV there exists an excitation energy where there is 0% alignment. This is significant experimentally because for certain excitation energies where there is no alignment an isotropic angular distribution will be observed, thus yielding no information about excited state spins. Different mul- tipolarity transitions can be distinguished clearly as long as excitation energies are on the order of a few MeV. Therefore, in intermediate-energy Coulomb excitation, using ’y-ray angular distributions to determine multipolarities of '7-ray transitions is feasible. For intermediate-energy Coulomb excitation performed at 0 = 0.4 with an ex- citation energy of 1 MeV and a minimum impact parameter of touching spheres plus a few femtometers, the relative difference between a quadrupole (0—+2—+0) and dipole (0—>3—>2) transition ranges from 88% to 20% if the range of angles around 40° and 100° is avoided. Thus for intermediate-energy Coulomb excitation a minimum of 150 counts with negligible background in the 7-ray peak at a given angle would be needed to distinguish quadrupole and dipole transitions with a relative difference of 20%. If detectors are placed at very forward (less than 16°) or backward (greater than 142°) angles, the relative difference between a quadrupole and dipole transition is greater than 60%. See Chapter 4 for the angular distribution of 1460 keV 'y rays from intermediate-energy Coulomb excited 40Ar. The above discussion of intermediate-energy Coulomb excitation is valid for 747 angular correlations as well. Fig. 2.22 shows 7-7 angular correlations for 209Bi(5°N i,5°N i '7) at 3 = 0.4 in the center-of-mass and in the laboratory frames for I,-,- = 0 —> I,- = 4-—>Im=2—>If=0(0—+4-—>2——»0)and0——>3—>2—)000rrelations.The minimum impact parameter is equal to touching spheres plus 2 fin and the excitation energy is 1 MeV. For intermediate—energy Coulomb excitation performed at fl = 0.4 with an excita- tion energy of 1 MeV and a minimum impact parameter of touching spheres plus a few 44 209Bi(56Ni,56Ni 7) Projectile frame (B = 0.0) 001“ 01:25deg 91=60deg 01:90deg 91=120deg 01:150deg 0-014 0.01 .,".-.“\ I".--‘. 0.01 ’5‘ 0-006 :"3 2249 """ ,. ----- .. “““ 0-006 cg“: 0.002 4.2.2-2-.I‘x . - AW - - - --2 . - -- -. . ‘ 0.002 .3- 01=25deg 01=60deg 01: 91:120deg 01=150deg 8 :‘h‘. 90deg 001 g 0.025 ,. .. ,. . ,3 -\‘\ .‘ "I '\.\. ir "\_‘ f,- \.‘I\‘ 0.006 0.005 ’ i " 0.002 3c'9‘o”1‘50‘3b‘9b‘150 ab‘eb‘i‘so 311211150 31) 90 1550 92 (deg) 92 (deg) 92 (deg) 92 (deg) 92 (deg) 209Bi(56Ni,56Ni 7) Laboratory frame ([3 = 0.4) Figure 2.22: Plots of W(61,02,) for the intermediate-energy Coulomb excitation reaction 209Bi(5°Ni,5°Ni 7) with the population parameter calculated using 0 = 0.4. The top row shows W(01, 62, (P) in the projectile frame and the bottom row shows W091, 62, 4)) in the laboratory frame. The excitation energy of the nucleus is 1 MeV and the minimum impact parameter is touching spheres plus 2 fm (13.7 fm) which corresponds to a maximum scattering angle of the projectile of 30°. The initial spin, I,,, of 5°Ni is zero. 0 —» 4 —+ 2 —> 0 correlations are solid curves and 0 ——> 3 —1 2 —+ 0 correlations are dash-dotted curves. In each plot and 01 are fixed; <1) 5 (1)1 — (152 = 0° and the value of 61 is as labeled on each plot. The angular correlation function, W(01, 02, (P), is normalized as in Eq. 2.5. 45 femtometers, as seen in Fig. 2.22, the relative difference between the 0 —> 4 —+ 2 ——» 0 and 0 —» 3 —> 2 —> 0 correlation functions is as large as 99%. Thus, with proper detector placement, distinguishing 0 —> 4 —+ 2 -—+ 0 and 0 —> 3 ——> 2 —+ 0 correlations from one another is possible. 46 Chapter 3 Experimental setup 3.1 Coupled Cyclotron Facility To study radioactive nuclei they must be produced, transported to the experimen- tal area, and studied before they decay. The Coupled Cyclotron Facility (CCF) at the National Superconducting Cyclotron Laboratory (NSCL) utilizes projectile frag- mentation to produce radioactive nuclei. The CCF consists of two superconduct- ing cyclotrons capable of accelerating stable atomic nuclei ranging from hydrogen to uranium with energies up to 200 MeV/ nucleon. At the CCF, a primary beam of a stable isotope is produced in either a room-temperature (RT) or superconducting (SC) electron-cyclotron resonance (ECR) source and then accelerated in the first su- perconducting cyclotron (K500). Once the K500 has fully accelerated the isotopes, they are injected into the second superconducting cyclotron (K1200) where they are stripped of additional electrons and accelerated further. The beam of stable isotopes is then bombarded onto a production target with a low atomic number (such as 9Be). Many different radioactive isotopes are produced and the desired isotopes are mag- netically analyzed and separated using the A1900 fragment separator. Fig. 3.1 shows a. schematic of the ion sources, the two cyclotrons and the A1900 fragment separator at the CCF. 47 RT-ECR K500 20 ft ion sources ' " - . /- ° 'o ”W sc-eca ‘. 10m \\ K1200 _ -' é l @‘i:"\ §_ “9” WW J production . 1:, a - stripping ta 0! $ . ‘. foil r9 " \ wedge Figure 3.1: Schematic of the CCF including the ion sources, two cyclotrons and A1900 fragment separator. 3.2 A1900 fragment separator The A1900 fragment separator [72] physically selects the isotope(s) of interest using a series of quadrupole and dipole magnets. The nuclei are selected by their mass-to— charge ratio. The A1900 contains a wedge-shaped degrader, typically made of alu- minum or acrylic that can be placed in the path of the beam to purify the beam, and slits to block unwanted isotopes. Slits are located at focal points between each dipole magnet. The resulting ’secondary’ beam can contain one or many different isotopes travelling at 0.3c to 0.50. The momentum spread of the secondary beam can be controlled with momentum slits and can range up to :l:2.5%. The fragments are identified and their rates are measured using a series of detectors at the focal plane of the A1900. For specific information on the the secondary beam energy, production target, wedge and slits used for the 40Ar intermediate-energy Coulomb excitation ex- periment and the 9Be(3"‘Si,32Mg 7)X two-proton knockout experiment see Sections 4.2 and 5.2, respectively. 48 3.3 SEgmented Germanium Array 3.3.1 Configuration The SEgmented Germanium Array (SeGA) is an array of eighteen thirty-two—fold segmented high-purity germanium detectors [73]. The high degree of segmentation is necessary to event-by-event Doppler correct the 'y rays emitted by the excited nuclei travelling at fl = 0.3 — 0.5. To correct for the Doppler boosting of the emitted '7 rays, the first interaction point of the '7 ray must be known. If the SeGA detectors were not segmented the uncertainty in that position would be the size of the entire germanium crystal, dominating the resolution of the Doppler reconstructed ’y-ray peak. See the more detailed discussion of the contributions to the resolution of a Doppler reconstructed 7-ray peak in Section 3.3.5. The detectors of SeGA were arranged in a unique configuration in the experiments presented here. The detectors were positioned such that the array was optimized for detecting 'y-ray angular distributions and 7-7 angular correlations. The eighteen detectors were at the following nine (approximate) angles 0 with respect to the beam axis: 24°, 29°, 40°, 60°, 78°, 90°, 126°, 139° and 147°. There were two detectors per angle. The distance from the center of each germanium crystal to the target was 24.5 cm. For all but the two detectors at 24°, the detectors were oriented with the long side of the cylindrical crystal perpendicular to the target. (The detectors at 24° were nearly perpendicular to the target, but physical constraints prevented them from being completely perpendicular.) Fig. 3.2 shows a top-view photograph of the SeGA detectors arranged in the configuration described. See also Figs. 3.4, 3.5, and 3.6 in the discussion of the detector positions in Section 3.3.3 for a front-view, side-view and top-view of the detectors generated by GEANT. For the experiments presented here, SeGA was positioned at the pivot point (target position) of the S800 spectrograph. The secondary beam impinged upon the secondary target at the center of SeGA and the fragments and scattered beam continued into the S800 spectrograph while SeGA 49 Beam direction towards S800 Figure 3.2: Photograph of SeGA at the target position of the S800 spectrograph. The black line drawn on the beampipe in the center of the array indicates the approximate position of the secondary target inside the beampipe. detected the 'y rays emitted from excited nuclei produced in the reactions. 3.3.2 Standard coordinate system The standard coordinate system used in the experimental setup, the data analysis program SpecTcl, and the detector simulation program GEANT [74] is shown in Fig. 3.3. The angles, 0 and d) in degrees are defined below. cos 0 = —z—_ 3.1 ( ) (96)2+(y)2+(2)2 ( ) arctan(y/x) if y > 0 and a: > 0 arctan(y/:r) + 180 if y > 0 and z < 0 arctan(y/z) if y < 0 and a: > 0 arctan(y/x) + 180 if y < 0 and x < 0 50 downstream beam axis CD > \ Y , \ «s‘° \ \ Q \ 90$? ‘1’ \ 9° x Figure 3.3: Sketch of the standard coordinate system used in the experimental setup, SpecTcl, and GEANT. 3.3.3 Detector positions In order to Doppler correct the '7-ray energy spectra as will be discussed in Sec- tion 3.3.5, the (x,y,z) positions of the center-of-mass of the segments of each SeGA detector with respect to the target position must be known. As discussed in Sec- tion 3.3.1, the configuration of the SeGA detectors was new for the experiments presented here. The arrangement was also more complicated than previous configu- rations. The procedure used to generate the (x,y,z) position of the center-of-mass of each segment of each SeGA detector will be outlined below. First, the detectors were measured with respect to the frame that held them in the arrangement described in Section 3.3.1. This was done using two theodolites. Each theodolite measured the horizontal and vertical angle of the point focused on. This gave the position of that point with respect to the coordinate system set up by the theodolites. Three to four points on the end cap of the crystal were measured as well as three places on the frame. This was done for each SeGA detector. Using 51 MicroStation [75], the horizontal and vertical angles from the theodolites, for each point, were transformed into (x,y,z) coordinates with the origin at the center of the array (at the target position). The (x,y,z) positions of the three to four points on the end cap of each crystal then had to be transformed into center-of-mass coordinates for each segment of each SeGA detector. This was done using a Mathematica [76] script that, for each detector, folded the (x,y,z) positions of the points on the end cap with the known [77] crystal segment positions with respect to the detector can. The Mathematica script was modified from a previous script [78] so that it could function with three (instead of four) end cap position points as input. (When using the theodolites, because of the complicated configuration, four end cap points were not always visible, so only three points could be measured.) The (x,y,z) positions of the center-of-mass of each segment of each SeGA detec- tor in the current arrangement were then used in SpecTcl and GEANT. (GEANT is discussed in Section 3.3.6.) In previous detector configurations GEANT only used the (x,y,z) center-of-mass coordinate of each segment for Doppler reconstruction. The SeGA detector configuration was put in separately by hand. For the current arrange- ment, manually inputting to GEANT the detector configuration would have been prohibitively tedious and prone to errors. Therefore a Fortran subroutine was written that took the (x,y,z) positions of the center-of-mass of each segment of each SeGA detector as input and gave as output the detector positions in space in the format required by GEANT. The subroutine is shown in Appendix A. This subroutine can be used for any detector geometry, not just the current configuration. It requires the (x,y,z) positions of the center-of-mass of each segment of each SeGA detector. For each detector configuration this information must be known for SpecTcl, therefore to run GEANT simulations now with a new configuration there is no additional work to be done. It has been tested for the previous, simpler, configuration and the re— sults match those where the geometry was put in manually. Figs. 3.4, 3.5, and 3.6 52 vvvvvvv (I. 33561-90 "ii “‘\\ mm} null) ‘ a...) . t. - [W 'iwiiiiiiiii‘iiiiii l\ [\ mm. \ --. mm m: -llll- if" llll- "W“ i". [it] '4']! 1111- Lil/11125;" 3”»‘3‘ GEANT. 3133's; .. ; 5““ 4%., ”1.1K. . . E '[ ”IN“? 7"] \Vii'im .- Figure 3.5: Side—view of the SeGA detector geometry. The figure was generated in GEANT. The beam direction is indicated by the arrow. show the front-view, side-view and top-view, respectively, of the SeGA detectors in GEANT in the configuration as described in Section 3.3.1. The figures were generated in GEANT by displaying the geometry of the SeGA detectors after the subroutine described above was called. The SeGA detector frame had to be aligned such that the target would be at the target position. SeGA was positioned such that the target position of the frame was at 53 Figure 3.6: Top-view of the SeGA detector geometry. The figure was generated in GEANT. The beam direction is indicated by the arrow. 54 the pivot point of the 8800 spectrograph. The target position is where the secondary beam was focused. To align the frame theodolites were used as well as reference points on the frame and wall (to mark the position of the pivot point). Once the frame was aligned, the target was inserted into the beampipe with a measured stick such that the target was at the target position. 3.3.4 Energy and efficiency calibrations In order to accurately determine the energies of the ”y rays observed in the de- excitation of a nucleus, the SeGA detectors must be energy-calibrated with standard y-ray sources. In order to determine the number of 7 rays emitted in the de—excitation of a nucleus the efficiency of each SeGA detector in the experimental setup must be determined. The energy and efficiency calibrations were performed using two standard calibra- tion sources, 152Eu (NSCL ID: E2880) and 5600. They were chosen because they have many strong 7 rays between 245 keV and 3272 keV. (The 122 keV '7 ray in 152Eu was below the low-energy thresholds of the SeGA detectors.) The sources were placed at the target position of SeGA for two hours each. The correct placement of the sources is essential for the measurement of the efficiency of the detectors because the effi- ciency is dependent on distance from the source and physical materials between the source and the detectors (i.e. the beampipe). The energy and efficiency calibrations were done using PAW [79]. For the energy calibration, the 'y-ray peaks were fit with a Gaussian peak plus a quadratic back- ground for each detector. The centroids were then fit to the calibrated energies using a quadratic polynomial to get the calibration parameters. Using these calibration pa- rameters, calibrated energy spectra were created for each detector and checked with the source data. For the efficiency calibration the area of the y—ray peaks must be accurately deter- mined. The 7-ray peaks have slight low-energy tails that become larger with increasing 55 energy. In order to measure accurate peak areas it was necessary to parameterize the peak shapes. This was not necessary for the energy calibration because the low-energy tails did not affect the accurate determination of the peak centroids. The total fit func- tion for the efficiency calibration is a Gaussian distribution plus a skewed Gaussian plus a quadratic background. The parameters include a Gaussian width, skew width and skew height. The skew width was fixed to be 1.25 times the Gaussian width. First the Gaussian width was parameterized, letting the skew height and Gaussian width vary. Then using the parameterized Gaussian width, the skew height was parameter- ized. Once the skew height was determined, the skew shape could then be fixed for each 7-ray peak in each detector. Keeping the skew parameters fixed, another fit was performed letting the Gaussian width vary in order to refine the width parameteri- zation. Once the peak shape was completely parameterized as a function of energy for each detector, final fits were performed fixing the Gaussian width, skew width and skew height, only letting the peak height, peak centroid and three quadratic background parameters vary. Once all the 7-ray peaks from both the 152Eu and 56Co sources were fit in each detector, separate efficiency curves for both sources were fit with the following func- tion, P1 EfiEy — (E7 _ P2 + e—O.269*E7)P3 (3.3) where E, is the energy of the 7 ray and P1, P2 and P3 are the fit parameters. This function is only able to accurately fit the experimental source efficiency above ap- proximately 150 keV. The 152Eu source was a calibrated source, but the 56Co source was not a calibrated source. To combine the two efficiency curves, the 56Co source efficiency curve was scaled to the 152Eu source efficiency curve by minimizing the square of the difference between the two curves. The combined efficiency source data was then fit to get the final efficiency calibration parameters. For the analysis pre- sented here, the efficiency calibration was done for each SeGA angle pair. Fig. 3.7 56 0.8 ' 0.6 . 0.4 - Absolute efficiency (%) 0.2 . A A4 A A A A A A A A A A A I A A L A A A A A A L AAAAAAAA 500 1000 1500 2000 2500 3000 Energy (keV) Figure 3.7: Plot of the absolute efficiency in percent versus 7-ray energy in keV for the 60° SeGA angle pair. The values of the parameters in Eq. 3.3 for the 60° angle pair are P1 = 39.9, P2 = 36.0 and P3 = 0.69 shows an efficiency curve and fit parameters for the 60° SeGA angle pair. The effi- ciency curve shown is representative of all angle pairs in SeGA in the configuration described in Section 3.3.1. The total photo-peak efficiency for the entire SeGA at 1 MeV is (3.01 :t 0.04)%. 3.3.5 Doppler reconstruction For the experiment prasented here, the secondary beam energies were approximately 0.4c. When an excited nucleus emits a 7 ray traveling at relativistic speeds, the energy of the 7 ray as seen by the SeGA detectors is Doppler boosted. The 7-ray energy spectra must be Doppler corrected to reconstruct the energy of the emitted 7 ray. The following equation is used, E;“b(1 — B cos(6¢ab)) m where Elf" is the energy of the 7 ray in the projectile frame, Elf" is the energy of the pro_ E7 — (3.4) 7 ray in the laboratory frame, 3 is the velocity of the projectile frame with respect to the laboratory frame relative to the speed of light at the point of 7-ray emission, 57 and 0,0,, is the laboratory angle of the emitted 7 ray with respect to the beam axis. In order to perform the Doppler correction, Eff”, fl and 0101, must be known. Un- certainties in the 0 and 610;, parameters produce a resolution in the Doppler corrected 7-ray peaks that is larger than is seen for 7—ray source peaks. The square of the energy . AE . . resolution, ———7- 1S given by, E, ’ AE.y 2 _ flsin(0¢ab) 2 (E) _ (1 — flCOSWzabl) (A6”)? —3 + cos(6,ab) 2 2 ( A E;ntr)2 + ((1— A32)(1— 30046101,») (A8) + (E,)2 ‘ (3.5) The uncertainty of the beam velocity, A0, is primarily due to the beam slowing down in the secondary target and the inability to determine the exact velocity at the point of 7-ray emission. The uncertainty in the angle of the emitted 7 ray, A61“, is dominated by the finite size of the segments of the SeGA detectors. With the detectors in the configuration discussed in Section 3.3.1, if the point of 7 ray emission was known exactly and was within approximately 5 mm of the center of the target, then A010,, would be approximately 2.34°. However, the position of the 7-ray emission is not known exactly. The z position is most dependent on the half-life of the excited state. If the half-life is less than approximately 50 ps, the additional contribution from the uncertainty in the 2 position of the 7-ray emission to A0105 is small. Fig. 3.8 shows a plot of energy resolution versus half-life, to illustrate the contributions from A3 and ABM, to the energy resolution. The half-life affects A0 and A010,, because it affects where the de—excitation occurs. If the de-excitation occurs as soon as the nucleus is excited that will give a different energy loss than if the de-excitation occurs 20 ps after the nucleus is excited. Fig. 3.9 illustrates the affect of the half-life on the z position of 7-ray emission for a given target thickness. The intrinsic resolution, AE;""/E.,, contributes the least to the resolution of the Doppler corrected 7-ray peaks. The intrinsic resolution for an individual SeGA de— tector is approximately 0.30% (FWHM) at 1408 keV. 58 2.85 . . l 2.75 fl . l 2.65 r 2.6 [ ll l 2.55 E 2.5 l Energy resolution (FWHM) (%) 0 ”20‘ ‘40 ‘60 A 80 A 100 t1/2(DS) Figure 3.8: Energy resolution of a GEANT simulated 1436 keV 7 ray versus half-life t1/2 in picoseconds (ps). Each point was generated by a GEANT simulation by set- ting the half-life and velocity associated with that half-life as used for the Doppler reconstruction. All other GEANT input parameters stayed the same between simu- lations. The resolution is the energy resolution of the 1436 keV peak in the summed spectrum of all the detectors in SeGA. From 0 ps to approximately 15 ps the energy resolution drops sharply. This is because the energy loss in the target (376 mg/cm2 9Be) is greatest at 0 ps and thus the contribution from A0 is very large. Once the decays happen primarily outside the target (for half-lives greater than approximately 20 ps) the contribution from A0 is much smaller and stays mostly constant because there is negligible energy loss after the target. Once the half-life is long enough that A0101, increases, around approximately 50 ps, the energy resolution starts to increase. 59 1.2 - 11,2 = 0.0001 ps 1.21 mg = 1.0 ps .1: 1.0 .1: 1.0» (D (I) 9, 0.8» 9; 0.8; I; 0-6’ g 0.6- 2 0.4. 2 0.4 0.2. 0.2 I “0 1‘ 2 ‘0 ‘1‘ “s ”0 1 ‘2‘ 3 Q1 5 z(mm) z(mm) 12' 11,2 = 5.0 ps 1-2’ 11,2 = 12.0 ps :7" 1.0. {P 1-0: (I) (I) 3 0.8. 3 0.8. 3%: 0.6» g 0.6. 2 0.4. 2 0.4 0.2. 0.21 o 1 i ‘2‘ a 21 5 OF 1 é a ‘71”? z(mm) z(mm) Figure 3.9: Plots of 7-ray activity versus distance for a target thickness of 2 mm. The calculations assume a constant probability for excitation over the thickness of the target, a linear decrease in velocity over thickness of the target and no absorption in the target. 60 The Doppler correction is performed event-by-event. As stated above, to perform the Doppler correction, Eff”, B and 0101, must be known. For each event the segment of the SeGA detector that was hit first by the de—excitation 7 ray is selected and the position coordinates of that segment are used in conjunction with a 3:0 yo and 20 to get the angle 0,0,, in Eq. 3.4 using 2 - 20 \/($ _ $0)2 + (y _ yo)2 + (z - Zo)2 cos(610b) = (3-6) where (x, y, z) is the coordinate of the first hit segment and (:00, 310,20) is the coordi- nate of the emission of the 7 ray. Typically, so, ya and 20 are each less than 1 cm. Since (1120, ya, 20) cannot currently be determined event-by-event, the ($0,310, 20) used is the average ($0,310,250) determined from the Doppler correction as described below. Currently the segment that is considered to be hit first by the 7 ray is taken to be the segment in which the greatest amount of energy is deposited. For each nucleus discussed here the energy, E5”, of the largest peak in the 7-ray energy spectrum was known. Therefore, the data was scanned with a mid-target fl and x0 = yo = zo = 0. In the case of the 32Mg 7-ray spectrum, the 885 keV 7-ray peak from the 2+ —» 0;; transition was fit with a Gaussian peak in each detector. For 40Ar, the 1460.8 keV 2+ -* 03“,. 7-ray transition was used. Using Mathematica, the difference between the known peak energy and the Doppler corrected centroids were minimized letting fi, :00, yo and 20 vary. The data was re—scanned with the new 3, $0, yo and 20 values and the procedure was repeated until the difference between the known peak energy and the Doppler corrected centroids were within the uncertainty in the energy calibration. For 32Mg, this deviation was 21:4 keV for 16 of the 18 detectors. The other two detectors were i6 keV. For 40Ar, this deviation was :t8 keV for 12 of the 17 detectors. The deviations for other five detectors were —12 keV for three detectors and +17 keV for two detectors. (One of the detectors could not be used as discussed in Section 4.3.) The 4oAr Doppler corrected 7-ray spectrum with these 61 deviations was sufficient since there was only one 7-ray peak in the 40Ar spectrum. 3.3.6 GEANT simulations The Doppler correction, described in Section 3.3.5, corrects the 7-ray energy spectrum observed in the laboratory frame to the projectile‘frame such that the photopeak en- ergy centroids are the emitted 7-ray energies. But, the correction from the laboratory frame to the projectile frame does not result in a spectrum that would be observed from a source at rest. Compton scattering, pair production and scattering and inter- actions with the surrounding equipment all produce features that are spread out in_ the Doppler corrected spectrum. Also, the efficiency of the SeGA detectors are energy dependent as seen in Section 3.3.4. In order to accurately fit the Doppler-corrected energy spectra, especially if there are multiple 7-ray peaks with similar intensity close (less than approximately 500 keV apart) in energy, a simulation of the responses of the detectors is necessary. GEANT [74] performs a Monte Carlo simulation which event-by-event emits and then subsequently ’detects’ a given number of 7 rays. Input includes, but is not limited to, the following parameters: 7-ray energy, incoming beam velocity, target thickness, target material, outgoing beam velocity, velocity spread in the incoming beam, and SeGA detector geometry. A sample xml input file used by GEANT is shown in Appendix B. For the GEANT simulations used in the fitting of the 32Mg and 40Ar spectra, ten billion events were emitted at each 7—ray energy, with the input parameters taken from experimental observables To verify that GEANT accurately reproduced the SeGA detector responses, the simulated peak shape for the 885 keV and 1460 keV peaks were compared to the data for the 32Mg and 40Ar datasets, respectively. 62 Focal Plane Figure 3.10: Schematic of the S800 spectrograph. 3.4 3800 Spectrograph The S800 Spectrograph [80—83] is a high—resolution spectrometer containing six de- tectors at its focal plane capable of giving the energy loss, energy, x and y position, scattering angle and time—of—flight of the detected nuclei. Those detectors include two cathode readout drift chambers (CRDCs) three plastic scintillators and an ion chamber [81]. See Fig. 3.10 for a schematic of the spectrograph. At the object posi- tion is a plastic scintillator used for timing. Due to physical constraints in the setup described in Section 3.3.1, the S800 was operated in focused mode rather than dis- persion matched mode. In focused mode the secondary beam is focused at the target position, in the center of SeGA, and dispersed in the S800 focal plane. In dispersion matched mode the secondary beam is dispersed at the target position and focused in the 8800 focal plane. 63 3.5 Trigger The choice of what data to record during an experiment is crucial to it’s feasibility and success. If uninteresting or not useful information is being recorded in place of physically interesting and necessary information then the experiment will be inefficient and the goal of the experiment may not be obtained. This includes what information of the available information is read for each event and what events are read. Selecting what events to read is possible with a trigger; events are only selected if certain conditions are true. In the setup described in Section 3.3.1 with SeGA at the target position of the 8800 spectrograph, each of the following could be a trigger during an in-beam experiment: downscaled 3800 particle singles, particle—7 coincidences or downscaled 7 singles. Each will be described below. For all the trigger settings, the data will only be recorded if the data acquisition is not busy. The data acquisition is busy for a finite amount of time while an event is being processed. The time the data acquisition is busy varies according to the time it takes to read out the event for a particular trigger. The rate of what is being recorded (particles or 7 rays for example) also affects whether the data acquisition is busy when the next event to be recorded occurs. If the particle rate in the 8800 spectrograph is very low (the length of time between events is longer than the recording time), the data acquisition is unlikely to be busy by the time the next event comes. The 8800 particle singles trigger means that data is recorded whenever a particle is detected in the plastic scintillator in the 8800 focal plane. In this trigger setting, the 7 ray(s) that were detected in SeGA in coincidence with that particle are also recorded. The readout of a 8800 particle singles event takes approximately 800 ps. The particle-7 coincidence trigger means that data is recorded whenever a particle and a 7 ray are detected in coincidence. The readout of a particle-7 coincidence event takes approximately 800 as. The 7 singles trigger means that data is recorded whenever a 7 ray is detected in SeGA. The readout of a 7 singles event takes approximately 64 200 as. The three triggers can be used in combinations; the or of any two, or even all three of the triggers can be used as the final trigger for the data acquisition. The particle-singles and 7-singles trigger each have a downscale value associated with them which can be set from one to 999. A downscale value of a: means instead of taking every particle singles (or 7 singles) event, take only every 2:“ event. This is useful when combinations of triggers are used and the rate of one trigger is dominating the others or when only one trigger is used but the rate (i.e. of particles) is too large for the data acquisition. Usually if just one trigger is used and the rate is too large for the data acquisition, the beam rate is actually lowered instead of downscaling the trigger. The fraction of time the data acquisition is unable to take data, the deadtime, must be quantified in order to correct the number of observed 7 rays and particles to the number that would have been seen if every final event trigger could have been taken. This is necessary in making cross section measurements or any measurement where absolute numbers are desired. The downscale value for a trigger increases the livetime through a reduction in rate, but also introduces a correlation in the data such that the number of events occurring in close succession are reduced. With the or of multiple downscaled triggers comprising the final trigger, the understanding of the concept of livetime becomes complicated. The probability distribution of the duration between two downscaled random (Poisson distributed) events is given by a gamma distribution. For a downscale value of one, the distribution reduces to an exponential distribution with maximum prob- ability at time zero. As the downscale value is increased the peak of the probability distribution moves farther from time zero. This is shown in Chapter 3, Section VII of [84]. Therefore, the act of downscaling one or more triggers comprising the final trig- ger creates different livetimes for different triggers. The livetime of a downscaled trigger can actually be greater than the busy time of the data acquisition because the downscaler effectively separates the minimum time between two events. This re- 65 duces the number of events that can fall within the readout time. This applies to the 9Be(3“Si,32Mg 7)X experiment where the or of three triggers were used. Two of the triggers had downscale values greater than one. See further discussion applicable to that experiment in Section 5.2.1. 66 Chapter 4 Intermediate-energy Coulomb excitation of 40Ar 4.1 Intermediate-energy Coulomb excitation Intermediate-energy Coulomb excitation has become an established tool for measuring properties of exotic nuclei [85]. In intermediate-energy Coulomb excitation the target and projectile are excited in the Coulomb field of the other. The 7 rays emitted from the excited target and projectile nuclei can be distinguished because the target 7 rays are emitted at rest, while the projectile 7 rays are emitted at the speed of the projec- tile. The formalism for 7-ray angular distributions of intermediate-energy Coulomb excitation reactions is presented in Section 2.5.4. Using 7-ray angular distributions to identify and confirm transition multipolarities is common for experiments where the beam energy is lower than the Coulomb barrier. However, this technique has not been explored with intermediate-energy Coulomb excitation reactions. Here the 7- ray angular distribution of the decay of the first excited state of intermediate-energy Coulomb excited 40Ar is presented. 67 4.2 Experiment setup For the intermediate-energy Coulomb excitation reaction measurement of 40Ar, the primary beam was 40Ar and a 9Be production target with a thickness of 1151 mg/cm2 was used to degrade the primary beam from 140 MeV/ nucleon to the desired beam energy. SeGA was positioned at the target position of the S800 spectrograph as dis- cussed in Section 3.3.1. At the center of SeGA, the 83.6 MeV/nucleon 40Ar beam impinged upon a 350.8 mg/cm2 197Au target. The trigger was set to: downscaled particle singles or particle—7 coincidences or downscaled 7 singles. 4.3 Analysis of the 7-ray spectrum The particle identification of the 4"Ar fragments in the S800 focal plane was unam- biguous because the secondary beam was a pure 40Ar beam and the magnetic field of the S800 spectrograph was set to the 4"Ar fragments. Other isotopes that may have been produced in the gold target by nucleon knockout and fragmentation reactions could make it to the S800 focal plane but the cross sections of those reactions on a gold target are very small compared to the scattering of 40Ar. To ensure that the 40Ar 7 rays came from Coulomb excitations and not nuclear excitations, the 40Ar fragments with scattering angles corresponding to a minimum impact parameter of touching spheres plus 2 fm were selected. Therefore, the maxi- mum scattering angle was Omm, = 2.97° = 51.8 mrad. Shown in Fig. 4.1 is the 40Ar scattering angle distribution. A gate from 0 — 51.8 mrad in the scattering angle spec- trum in Fig. 4.1 was applied to the 40Ar 7-ray spectra. The scattering angle gate only reduced the number of 40Ar particles by 1.3%, but reduced the number of 1461 keV 7 rays by 20—35%. As discussed in Section 3.3.5 the Doppler correction was performed using the known 1460.8 keV 2+ —+ 0:5 7 ray. One of the 126° detectors could not be used in this analysis because its gain was shifting within runs. Therefore the 126° spectrum 68 12000 — 10000 8000 6000 Counts/(mrad) 4000 2000 75 90 Scattering angle (mrad) Figure 4.1: Number of scattered 40Ar nuclei versus scattering angle (mrad). The spec- trum includes both 40Ar nuclei in coincidence and not in coincidence with a 7 ray. 69 only contained one SeGA detector. The 40Ar 7-ray spectrum for each SeGA angle pair was fit from 800-2000 keV with one GEANT simulated response function and an exponential background. Fig. 4.2 shows the 60° SeGA angle pair 7—ray spectrum. After each SeGA angle pair was fit as described, two corrections had to be made to the fit results to obtain the experimental 7-ray angular distribution. The corrections were the GEANT scale factor and the correction for the effect where the detectors became paralyzed after certain events. Both corrections are described in Section 5.3.4. 4.4 7-ray angular distribution In Fig. 4.3 is the 7-ray angular distribution of the intermediate-energy Coulomb excited 40Ar 1460.8 keV 2+ —> 0;], 7 ray. The experimental points are fitted with a 1,,- = 0 —> I,- = 2 —+ I f = 0 (0 —» 2 —» 0) 7-ray angular distribution (black curve) calculated as outlined in Section 2.5.4. The only parameter that was allowed to vary in the fit was an overall scaling factor. This was due, in part, because the two additional corrections described in Section 5.3.4, coincidence trigger livetime and data loss, were not included because they are not dependent on the SeGA detector angle. The value of the scale factor is 0.0163 2t 0.0015. Fitting the experimental 7-ray angular distribution with a 0 —> 3 —> 2 spin transition or a 0 —’ 3 ——» 0 spin transition resulted in much larger xfm." values of 29.3 and 20.2, respectively compared to the 0 ——> 2 ——> 0 spin transition fit where xfm-n = 6.7. The shape of the 0 —-> 3 —> 2 and 0 ——+ 3 —-+ 0 calculated 7-ray angular distributions are different than the observed 7- ray angular distribution. This can be clearly seen in Fig. 4.3. Therefore, 7-ray angular distributions can be used as a method for determining the multipolarity of observed 7 rays for intermediate-energy Coulomb excitation reactions. 70 Pair of SeGA detectors at 60 degrees Counts/(10 keV) \x. WII 0 \ Al.n/\/\M A/\/lfA/\ /\Jl m AV. ”/5va AV NIL/[AAA A.“ -KVYV.. ”lib/V1 .[11 .1 800 10100 1200 1400 16001800 2000 Energy (keV) Figure 4.2: 40Ar 7—ray spectrum of the pair of SeGA detectors at 60° fit with a GEAN T response function. The spectrum was fit from 800—2000 keV. The spectrum was fit with one GEANT response function and an exponential background. The fit (blue), response function (black) and exponential background (black) are shown on the spectrum. The residual (red) between the fit and the data points is shown below the spectrum. 0.002 L‘ '1' 0.0015 * \ W(6) 0.001 1 x - ~ _ 0.0005 i .’ 25 50 75 100 125 150 175 9c.m.(d99) Figure 4.3: 7-ray angular distribution of the intermediate-energy Coulomb excited 4OAr 1460.8 keV 2+ —> 0;, 7 ray. The points are the experimental results and the solid (black) curve is the O ——> 2 —) 0 calculated 7-ray angular distribution, calculated as outlined in Section 2.5.4. The only parameter that is allowed to vary in the fit of the curve to the data is an overall scaling factor as described in the text. The dash-dotted (red) and dashed (blue) curves are 0 —+ 3 —-> 0 and 0 —> 3 —+ 2 7—ray angular distributions, respectively. They were scaled with the same scale factor as the 0 —+ 2 —> 0 curve. 72 Chapter 5 Study of 32Mg via two-proton knockout 5. 1 Two-proton knockout One-nucleon knockout reactions have been established as direct reactions and have been used to study the single-particle degree of freedom in exotic nuclei [68]. A recent two-proton knockout experiment [44] gives evidence that two-proton knockout reac- tions of neutron-rich nuclei also proceed as direct reactions. Calculating two-nucleon knockout cross sections is more complicated than calculating one-nucleon knockout cross sections and is addressed in [86]. In one-nucleon knockout the reaction and struc- ture contributions can be separated, thus simplifying the calculation of the partial cross section to a given state. This is not possible in two-nucleon knockout because the reaction and structure terms are coupled. 32Mg was studied using the two-proton knockout reaction 9Be(34Si,32Mg 7)X. A schematic of the two-proton knockout from 34Si to 32Mg is shown in Fig. 5.1. The sequential two-step process where one proton is knocked out of 34Si to create 33Al and another proton is evaporated to form 32Mg is excluded. Removing one proton from 343i leaves 33Al in a state with enough energy to emit a neutron, but not enough energy to emit a proton. Therefore, the two-proton 73 1t — — — —+20.9M6V +15MeV— — — — 1t — — — +5.809 MeV +5.51 MeV — — _ 32M9 33A' 19.2 MeV 11.5 MeV -1p 18.74 MeV 348i Figure 5.1: 34Si —> ”Mg two-proton knockout schematic. The reaction, 9Be(3“Si,”Mg 7)X proceeds as a direct reaction. knockout from 34Si to ”Mg proceeds as a direct reaction. 5.2 Experiment setup The two-proton knockout reaction 9Be(3“Si,”Mg 7)X was used to study ”Mg. The 140 MeV/ nucleon 40Ar primary beam impinged upon a 705 mg/cm2 9Be production target. In the A1900 fragment separator, the secondary beam of 34Si was physically selected, with the aid of a 750 mg/cm2 acrylic wedge. SeGA was positioned at the target position of the S800 spectrograph as discussed in Section 3.3.1. At the center of SeGA, the 81.05 MeV/nucleon 34Si beam hit a 376 mg/cm2 9Be target. The 8800 74 spectrograph was operated in focused mode and the 32Mg fragments were centered in the focal plane. 5.2.1 Trigger General information about the trigger for the the configuration with SeGA at the tar- get position of the 8800 spectrograph is discussed in Section 3.5. For the 9Be(34Si,”Mg 7)X experiment, the trigger was set to: downscaled particle singles or particle-7 coin- cidences or downscaled 7 singles. The particle-7 coincidence trigger was used because the goal of the experiment was to measure the 7 rays emitted from the excited states in 32Mg. Every 7 ray that was in coincidence with a particle made a trigger and was read out if the data acquisition was not busy. The downscaled particle singles trigger was used to obtain a subset of the complete particle data without bias toward particles in coincidence with 7 rays and without contributing significantly to the data acquisition deadtime. The particle downscale value was set to 10 throughout the experiment. The particle singles data is also used for particle identification, and analyzing momentum distributions and scattering angle distributions of scattered particles. The 7 singles trigger was used in order to record 7 rays not necessarily in coincidence with a beam particle. A measure of the 1460 keV background 7 ray was needed for a correction to the measured 7-ray intensities as described in Section 5.3.4. The 7 singles trigger was first downscaled to 100 for 17 runs, then 25 for 54 runs, and then 10 for the last 108 runs. The reason the downscale value was changed throughout the experiment was due to a misunderstanding of the deadtime calculation during the experiment. The goal was to set the downscale value such that there would be enough 1460 keV 7 rays per run to make the correction described in section 5.3.4, but to not significantly affect the deadtime. As discussed in Section 3.5 the deadtime must be calculated in- dependently for each trigger. During the experiment it was not calculated this way so the downscale value for the 7 singles trigger was set lower (twice) because it appeared that it did not affect the deadtime significantly. However, the livetime of the other 75 triggers were affected by the subsequent high 7 singles rate. To calculate the livetime of any one of the triggers comprising the final trigger (i.e. particle-7 coincidence trig- ger) only runs with the same downscale values can be used. This ultimately applies to the calculation of the partial cross sections in Section 5.4.4. To measure the inclusive cross section only particle data is needed. Therefore, in the runs used to measure the inclusive cross section of 9Be(34Si,”Mg)X the trigger was particle singles only. The downscale value was set to one in order to accept all the particle data. See Section 5.4.1 for the description of the calculation of the inclusive cross section. 5.3 Analysis of the ”Mg spectrum 5.3. 1 Particle identification , In the S800 spectrograph focal plane there were other isotopes besides the 32Mg fragments of interest. They were primarily generated from reactions from the 34Si secondary beam and the 9Be target. To identify the ”Mg fragments, first the Mag- nesium isotopes were identified in a time-of-flight versus energy-loss spectrum. The time-of-flight was measured between the object scintillator at the entrance to the S800 analysis line and a plastic scintillator in the S800 focal plane. Fig. 3.10 in Sec- tion 3.4 shows a schematic of the S800 spectrograph. The energy loss was measured in the ion chamber located in the S800 focal plane. The magnesium fragments were clearly separated by isotope in a time—of-flight versus angle in the focal plane spec- trum which had the first particle identification gate applied. The angle in the focal plane was measured using the two CRDCs in the S800 focal plane. Fig. 5.2 shows the two particle-identification spectra. 76 700 - " r " ’ r 600 F - <— 343] E . g . s 5°° " ~ .5 : Magnesium V _ Kisoto as a _ p ; 400 — E, . m . C ”J I 300 _ 700 750 850 950 Time-of-flight (channels)900 , 0.075 _ 0.025 :— Angle in the focal plane (rad) O O 0.05 3 0.075 E l i l . l n l n l 200 250 300 350 400 450 500 Time-of-flight (channels) Figure 5.2: 3"’Mg particle—identification spectra. The spectrum in the lower panel is gated on the gate drawn on the spectrum in the upper panel. 77 Ifllllf I 103 fillfllll g G) x o l" I; \ U) E 102 l.“ :3 F O l" 0 C b | l 10 ' rrrrTr] I 1000 2000 3000 4000 5000 Energy (keV) Figure 5.3: ”Mg 7-ray spectrum containing the sum of all 18 detectors from SeGA. Below approximately 250 keV, there is a loss of counts due to the low-energy threshold of the SeGA detectors. 5.3.2 Doppler correction As discussed in Section 3.3.5 the Doppler correction was performed using the known 885 keV 2+ —> 03.... 7 ray. Fig. 5.3 shows the Doppler corrected ”Mg 7—ray energy spectrum. Three of the 7-rays have been previously observed, namely the 885 keV, 1964 keV and 1438 keV 7 rays. (See Section 1.4.1 for a discussion of the previously observed 7 rays in 3"’Mg.) The accepted half-life for the known (885.3 d: 0.1) keV 2+ —+ 03.3. 7-ray transition in ”Mg is (11.4 :1: 2.0) ps [87]. Since the energy spectra 78 6 I 1964 keV * [ * 1438 keV , o 885 keV Energy resolution (FWHM) (%) i—lv—l "-—- «k """""""'T—;f""""""" HH He PM W .1 Q l—f-i't—H 0.35 0.36 0.37 0.38 0.39 1 I Figure 5.4: Energl resolution versus 6 for three prominent 7 rays in the ”Mg 7- ray spectrum. The energy resolution of the 1438 keV peak is the energy resolution assuming it is one peak. The horizontal dashed line illustrates the GEANT energy resolution of the 885 keV 7-ray peak (with a half-life of 11.4 ps) and also the GEANT energy resolution of the 1438 keV 7-ray peak (with a half-life of 0 ps). The vertical dashed lines indicate the position of the target. The larger 6 values correspond to the upstream beam direction. were Doppler corrected using the 885 keV peak, those 7 rays which do not have a half-life of approximately 11.4 ps will have a larger energy resolution and be offset in energy. The energy centroid is only affected by a few keV and will be addressed in Section 5.3.4. The 7-ray peak resolutions as a function of 0 can give information about the half-life of the observed 7-ray transitions. As discussed in Section 3.3.5, the Doppler correction has four parameters which must be optimized: 6, are, ya and 20. Fig. 5.4 shows energy resolution versus 6 and Fig. 5.5 shows energy resolution versus zo for three prominent 7 rays in the ”Mg spectrum. As can be seen in Fig. 5.5, the zo position does not affect the energy resolution within :t0.2 cm. Horn Fig. 5.4, it is obvious that the ,6 value that gives the smallest energy resolution for the 885 keV 7-ray peak is lower than the 6 value that gives the smallest energy resolution for the 79 4 l l l l l l l g 3.75} 3 E 1964K v 9:, 3.5- " e c at 1438 keV {-3) 325; o 885 keV 6 E (I) . 9 3 r I a z a) 2.75: C “J l 2.5 : -0.7 -0.6 -0.5 04 20 (cm) Figure 5.5: Energy resolution versus 20 for three prominent 7 rays in the ”Mg 7- ray spectrum. The energy resolution of the 1438 keV peak is the energy resolution assuming it is one peak. 1438 keV 7-ray peak. The 0 value which produces the minimum energy resolution for the 1438 keV peak (assuming it is one peak) is 6 = 0.3770 :1: 0.0047. That value of 0 is consistent with a 7-ray transition with a half-life of (0 — 3.5) ps. A half-life of 0 ps, versus 3.5 ps, gives the greatest energy resolution (because of the largest contribution to A0). Using GEANT, the energy resolution of the entire SeGA of a single 1438 keV 7 ray with a 0.0 ps half-life is 2.76 :l: 0.02% (FWHM). As can be seen in Fig. 5.4, the minimum experimental energy resolution is 3.41 :l: 0.10% (FWHM) for the sum of the entire SeGA. This gives strong evidence that the peak around 1438 keV is a doublet. To confirm that the 1438 keV peak was not one peak, it was fit with a 0.0 ps half-life simulated response. The response was too narrow for the data. The minimum experimental energy resolution of the entire SeGA of the 885 keV 7-ray peak is (2.913 :t 0.030)% (FWHM) and the GEANT energy resolution is (2.782 :1: 0.004)% 80 O 885 keV, data 0 1438 keV, data 0 1964 keV, data {a * 885 keV, GEANT L 3.6 ' 1t 1438 keV,GEANT 3 , i * 1964 keV, GEANT I 3.4 E 3.2 . C .g 3.0 - Q .2 0.0ps 0 2.8 - “-4 PS 4/ g i A/ : 4.4 ps 4/ 0.0 ps § 2.6; 54/135” (“Ape E 2.4- A/ZOPS 1000 A 1200 1400 i 1600 A 1800 A 2000 Energy (keV) Figure 5.6: Energy resolution versus energy for three prominent 7 rays in the ”Mg 7— ray spectrum. The experimental data is shown by diamonds. The GEANT simulated resolutions are shown for a few different half lives for the 1438 keV and 1964 keV peaks as labeled on the figure. (FWHM). The small difference between the GEAN T simulated energy resolution and the experimental energy resolution for the 885 keV peak cannot account for the large difference observed for the 1438 keV peak between simulation and experiment. This is illustrated in Fig. 5.6. Shown are the experimental and simulated, by GEANT, energy resolutions of the entire SeGA for the 885 keV, 1438 keV and 1964 keV 7 rays. The GEANT simulations are for a few different half lives for the 1438 keV and 1964 keV peaks. Therefore, it is clear that the peak at 1438 keV is a doublet. The energy resolution of the sum of the two peaks around 1438 keV corresponds to an effective half-life of (0 — 3.5) ps (as discussed above). The half-lives of the individual peaks are not known. The unknown doublet parameters are then related 35, (Ia/Ib)Tf/2 + ”/2 (0 _ 3'5) pg 2 (I./I.) +1 ’ (5.1) where a and 0 denote the two peaks comprising the doublet, Tf/z and le/z are the 81 half lives in ps and la and 1;, are the peak intensities. Another unknown parameter is the energy spacing dE between the peaks. 5.3.3 Doublet parameter study To gain insight on the individual peaks of the 1438 keV doublet a parameter study was performed. The experimental 1438 keV peak in the summed spectrum of all nine SeGA detector angle pairs was fit with two GEANT simulated peaks, a and 0, each with a given half life (Tla/2 and le/2)1 intensity (Ia and 1;, = 1 — Ia) and energy spacing between the two peaks (dE). The parameters looped over all combinations allowed by Eq. 5.1 of {/2 = 0, 1,2,3, 4, 5, 6, 7,8 ps, le/z = 0, 1,2,3, 4, 5, 6, 7,8 ps, Ia = 0.03, 0.06, 0.09, ..., 0.87, 0.9 and dE = 0,5, 10, ..., 55, 60 keV. An initial set of fits was performed with coarser half-life increments up to 30 ps which showed the mini- mum X2 occurred when {/2 and Tf/2 were both less than 8 ps. The intensities were chosen such that each peak could range to approximately 30 times the other in mag- nitude. A maximum energy spacing, dB, of 60 keV was chosen because a peak shape characteristic of two peaks would have been apparent in the experimental summed energy spectrum if the energy spacing was greater than approximately 60 keV; this was not observed. For each combination of parameters allowed by Eq. 5.1, the fit was performed and the X2 value was recorded. For a given parameter, the 10 uncertainty was determined by fixing the other three parameters and looking at what values of the parameter gave X2 values up to XEm'n + 1. The resulting uncertainties in the dB and 1., parameters were on the order of the step size. The uncertainties in Tf/2 and le/2 are on the order of the step size, but the generated GEANT response functions are not sensitive to half-life increments smaller than 1 ps. Therefore a final set of fits was performed where Tf/2 and Tf/2 looped over the same values listed above, but Ia and dE varied with Ia = 0.75, 0.76, ..., 0.80, 0.81 and dE = 25, 26, ...,34, 35 keV. The values of the parameters which give a global minimum, with their 10 error bars, are 1‘72 = 33‘) ps. T152 = 1:“? ps, Ia = 0.77t8;33, and dE = 29:? keV. Peak a is the 82 dE 30 keV 150.8 1392 139.2 139.2 _ . ._ 1276 127.6 127-6 X2 115, __ ' 2116-0 116.0 1 116.0 104.4 2. 104.4 212 1°44 .20 92.8 . 92-8 b '1.0 0.608 812 a ' 812 T s 0.4 - . 1/2(P) 0.0 0.2 l x2 T1/2(pS) x2 a T12=3PS /_ 319 13.078 ._ 290 290 319 261 261 - , 232 203 'Zi?---~ 2” 203 - x2 203 x2 203 145 1.- iii: 174 145 :- 174 87 145 87 145 116 4.0 116 2.0 87 50.0 87 T32 (ps) 0.0 3 ' 2 T172 (ps) 100 3°") 112 dE (keV) X dE (keV) dE=30 keV la = 0.78 _ g 82.94 82.94 ‘...-f:;jj:ZZIZ3-:;j 8236 81.78 - ' 8‘78 x2 80 62 . 81.2 ' ...... 80.62 79.46 " 80 04 78-3 79.46 3'00 40 78.88 a 1.0 - T1/2 (p3) 0_0 0.0 2'0 278.3 5 T1/2(DS) Figure 5.7: Three-dimensional plots illustrating the goodness-of—fit x2 values ver- sus different parameters, for the experimental 7—ray spectrum fit with GEANT simulated response functions in which the four parameters looped over Tf/z = 0, 1,2,3,4, 5,6, 7,8 ps, le/z = 0, 1, 2, 3,4, 5,6, 7,8 ps, Ia = 0.03,0.06,0.09, ...,0.87,0.9 and dE = 0,5, 10, ..., 55,60 keV. Each plot shows the region around the global mini— mum as a function of two of the four parameters. The two fixed parameters are given in the upper left of each plot. 83 a T1a/2=3ps T1/2=3l3s Tf/z =1 ps 464 T1” =1 ps__....---- 188.5 464 406 174 323 . 343 159.5 2 290 145 x 232 ;_ 290 130.5 1173 -- 232 116 0.8 174 101.5 0. 4 116 25 30 35 87 la '02 15 20 x2 dE (keV) x2 dE(keV) T1a/2=3ps b 71/2 "1 ps-..---~“ 188.5 174 ‘74 ‘ 159.5 x2145 - 145 116 130.5 87 116 08 101.5 0. 87 0E X2 (keV) Figure 5.8: Three-dimensional plots illustrating the goodness-of—fit X2 values ver- sus different parameters, for the experimental 7-ray spectrum with GEANT sim— ulated response functions in which the four parameters looped over Tf/z = 0, 1,2,3,4, 5,6, 7,8 ps, le/Z = 0, 1, 2,3, 4, 5,6, 7,8 ps, In = 003,006,009, ...,0.87,0.9 and dE = 0,5, 10, ..., 55,60 keV. For each plot the x and y axes are dE and Ia, re- spectively. Each successive plot is over a smaller energy range than the previous plot (going left to right, top to bottom). 84 a Wm=3m 81.5 81 80.5 80.5 2 X m5 w 785 79.5 79 78.5 T1/2(FJS) 0 076 2 la 11 b 87 Ta =3 s “3:1” 12 P 86 |a= 85 _ 84 87 84 83 “H I” 83 ......... 83 82 x2 83 81 81 80 79 80 79 V 79 33 2 a31 29 43 x T1b/2 (105)2 027 29 31 2 T172008) dE (keV) dE(keV) x a fic=3m b T1/2=1ps 81 """" 80.5 80.5 ----- 825 .,_ 80 81.5 x2 79.5 79.5 X2 80-5 78.5 7 79.5 9 78.5 5 30 31 78.5 3 2 28 29 2 Tb 1 3 dE (keV) X 172(ps) Ta(ps) Figure 5.9: Three-dimensional plots illustrating the goodness—of—fit x2 values ver— sus different parameters, for the experimental 7-ray spectrum with GEANT sim— ulated response functions in which the four parameters looped over Tf/z = 0,1,2,3,4,5,6,7,8 ps, Tf/z = 0, 1,2,3, 4, 5,6, 7,8 ps, 1., = 075,076, ...,0.80,0.81 and dE = 25,26, ...,34,35. Each plot shows the region around the global minimum as a function of two of the four parameters. The two fixed parameters are given in the upper left of each plot. 85 higher energy peak and peak 0 is the lower energy peak. Figs. 5.7, 5.8 and 5.9 show three-dimensional plots illustrating the goodness-of—fit x2 values versus different pa- rameters, for the experimental 7-ray spectrum fit with GEANT simulated response functions. In all of the figures the z axis and color indicate X2. The figures show selected slices or sections of the parameter space that illustrate the location of the global minimum. 5.3.4 Fitting the ”Mg 7-ray energy spectrum For each pair of SeGA detectors at a given angle 0, with respect to the beam axis, the ”Mg 7-ray spectrum was fit from 650 to 3800 keV with simulated response functions from GEANT plus an exponential background. To generate simulated response func- tions for the two peaks around 1438 keV, the half-lives and energy separation results from the parameter study were used. (See Section 5.3.3 for the results of the parame- ter study.) When fitting the spectra all peak magnitudes were allowed to vary freely. The intensity ratio between the two peaks around 1438 keV could not be fixed for the SeGA angle pair spectra because the parameter study used the summed spectrum of all the SeGA detectors. If the two 7 rays around 1438 keV have angular distributions, then fixing the intensity ratio for each SeGA angle pair would be incorrect. When fit- ting the spectra, offsets (in energy) for each GEANT response function were allowed to vary to account for unknown energies and half-lives of most of the 7-ray peaks. The calculation of the energy centroids of the 7-ray peaks is discussed at the end of this section. Fig. 5.10 shows the 40° SeGA angle pair 7—ray spectrum fitted with nine simulated response functions plus an exponential background. The other eight SeGA angle pairs were fit likewise. The summed spectrum, of all nine SeGA angle pairs, was fit likewise except the intensity ratio between the two peaks around 1438 keV was fixed from the parameter study results. The fit to the summed spectrum was used to obtain the energies of the 7-ray peaks and the partial cross sections of the two states which decay by the two 7 rays around 1438 keV. The fits to the SeGA angle pair 86 Counts/(10 keV) Pair of SeGA detectors at 40 degrees 120 100 — 80— 600 , 500 Z Counts/(10 keV) 00 O O -~/\ MMA. /\/\ n 700 800 900 1000 1100 1200 Energy (keV) Alufmmmmh , j | u I "" - l1. llll W I], T. . 1 l _ l l l l 1 l l' l 1500 2000 2500 3000 3500 Energy (keV) Figure 5.10: ”Mg 7-ray spectrum of the pair of SeGA detectors at 40° fit with GEANT response functions. The spectrum was fit from 650-3800 keV. The spectrum was fit with nine GEANT response functions and an exponential background. The inset shows the fit from 650-1200 keV. In the figure, the fit (blue), individual response functions (black) and exponential backgound (black) are shown on the spectrum. The residual (red) between the fit and the data points is shown below the spectrum. 87 spectra were used to obtain 7-ray angular distributions and partial cross sections for the other observed 7 rays and excited states, respectively. After the ”Mg 7-ray energy spectra were fit with GEAN T response functions, four corrections had to be applied to obtain the 7-ray intensities and partial cross sections shown in Tables 5.5 and 5.6. Two of the corrections must be applied to each detector angle pair. One is a GEANT scale factor. The detector efficiency as simulated by GEANT does not exactly match the experimental source efficiency for each detector angle pair. The GEANT scale factor corrects for this discrepancy. The scale factor is dependent on the angle of the detector to the beam axis (6) and the detected 7-ray energy. The scale factors for the different angle pairs for the 885 keV 7 ray ranged from 0.89 to 1.05. The other correction that must be applied to each detector angle pair is a correction for an effect where the detectors became ’paralyzed’. This happened after certain events and caused the particular SeGA detector affected to be unable to detect another 7 ray for a finite amount of time. It is thought to be caused by scattered particles hitting the SeGA detector crystal and saturating the preamplifier. Cursory inspection of the preamplifier signal for one of the SeGA detectors at 24° indicates the preamplifier is saturated for approximately 1 ms. The detectors at forward angles were affected much more than detectors at backward angles. This effect was quantified by measuring the peak area of the background 7 ray at 1460 keV in and out-of-beam for each detector. The area of the 1460 keV 7-ray peak per unit time should stay constant in and out-of-beam. Measuring the discrepancy gives a measure of the amount the detectors are paralyzed. Fig. 5.11 shows the ratio of 1460 keV 7 rays in and out—of- beam as a function of detector angle. The points are normalized such that the 147° detectors have a value of one. This introduces a systematic error of 5.6% that is not shown on the error bars in the figure. For the two peaks around 1438 keV the fit to the summed spectrum was used to obtain the partial cross sections for those peaks (as stated above). The GEAN T scale factor used for these two peaks was not the GEANT scale factors for the individual 88 —L l——-O—1 1—-O—l l——O—1 1-—-¢—4 ’1? >8 ] 8.9 095 [ [ gas .éség 0"’ >~>— [ 53 531185 » 33 i 55 0.3» L 20 40 60 80 100 120 120 9(deg) Figure 5.11: Plot of the number of in-beam 1460 keV 7 rays divided by the number of out-of-beam 1460 keV 7 rays as a function of detector angle. angle pairs, but a scale factor calculated by comparing the measured efficiency of the sum of the entire SeGA to the efficiency of the entire SeGA as simulated in GEANT. For the problem of the detectors becoming paralyzed, an ’effective’ correction for the entire array was calculated using the five largest 7 rays (excluding the two peaks around 1438 keV) in the 7-ray spectrum. For those five 7 rays, using the fit from the summed spectrum (of all nine SeGA angle pairs) and the fits of the SeGA angle pair spectra an effective detector ’paralyzation’ value could be calculated. The other two corrections to the to the 7—ray intensities, which are detector angle independent, are a livetime correction and a correction due to data loss. The particle- 7 coincidence trigger was not 100% efficient. For the runs used to obtain the 7-ray intensities the coincidence trigger livetime was 0.7755 :1: 0.0006. During the ”Mg experiment some coincidence trigger data was lost due to a readout programming error. The amount of data lost was independent of detector angle. The amount of data lost for the runs used to get the 7-ray intensities was (11.5 :1: 0.1)%. In Table 5.1 is a summary of the four corrections to the 7-ray intensities and their magnitudes The 7-ray peak energies were calculated from the results of the fit to the summed spectrum of all nine SeGA angle pairs. The energy centroids for the 7-ray peaks were 89 Table 5.1: Summary of corrections to ”Mg 7-ray intensities. See text for description of corrections. Correction Value GEANT scale factor 0.89 - 1.05 (detector angle and 7-ray energy dependent) correction for paralyzed detectors 0.82 - 1.0 (detector angle dependent) coincidence trigger livetime 0.7755 :1: 0.0006 data loss 0.115 :1: 0.001 calculated for each 7 ray by taking the centroid of the GEANT simulated 7-ray peak and subtracting the fitted offset for that peak. The fitted offsets for all the 7 rays were 3 6.1 keV. From the parameter study the spacing between the two doublet peaks, dE, was determined (see Section 5.3.3 for the results of the parameter study) and this was implemented in the fitting of the doublet. Therefore in the calculation of the energy centroid of the 1414 keV 7 ray, dE was also subtracted. The total uncertainty on the energy centroids of the 7 rays reported in Table 5.5 include the uncertainty in the centroid of the GEANT simulated 7—ray peak (3 0.14 keV for all 7 rays), uncertainty in the fitted offset (3 5.6 keV for all 7 rays), and an uncertainty due to not knowing the half-lives of the 7 rays (explained below). For the 1414 keV 7 ray there is an additional uncertainty of 2 keV due to the 29 keV spacing between the two peaks comprising the doublet. All of the uncertainties are added in quadrature. As mentioned in the beginning of Section 5.3.2, since the Doppler correction of the ”Mg 7-ray energy spectrum was performed using the 885 keV 7 ray, those 7-ray transitions with different half-lives than the 885 keV 7 ray will be Doppler corrected to energies a few keV above or below the emitted 7 ray energy. The uncertainties on the centroids reported in Table 5.5 have a systematic uncertainty folded in to account for the fact that the half-lives of the 7 rays in the ”Mg energy spectrum are not known. The magnitude of the uncertainty was estimated using GEANT simulations in the following way. In GEANT, simulations of 7 rays were performed using mismatched fl and half-life values and the deviation in the energy centroids of the simulated 7 rays versus the inputted 7 ray energies were recorded. For example, a simulated 90 2000 keV 7 ray with a half-life of 40 ps was Doppler reconstructed with a B value corresponding to a 0 ps half-life. The centroid of the resulting simulated peak was recorded. Likewise a simulated 2000 keV 7 ray with a half-life of 0 ps was Doppler reconstructed with a ,8 value corresponding to a 40 ps half-life. This was done for multiple half-life and fl combinations for two different energy 7 rays. The maximum deviation of the simulated energy centroid from the energy of the 7 ray that was inputted into GEANT scaled linearly with energy. Therefore for the 1964 keV 7 ray the magnitude of the uncertainty of the energy centroid due to not knowing the half- life is :l:8.69 keV. (The resolution of the 1964 keV 7-ray peak in the 7-ray spectrum of all SeGA angle pairs is approximately 2.5% which corresponds to a FWHM of 50 keV.) Scaling with energy, the uncertainty of the 2912 keV 7 ray is 3:12.89 keV. The uncertainties are scaled likewise for the other 7 rays. 5.4 Results and Discussion 5.4.1 Inclusive Cross Section The inclusive cross section, and, for the two-proton knockout of 3“Si to ”Mg was calculated by taking the ratio of the number of livetime and efficiency corrected ”Mg fragments detected in the 8800 focal plane to the number of incident ”Si nuclei, normalized to the number of livetime and efficiency corrected incident ”Si beam particles and multiplied by the target number density, Nf<32Mg)/(Nf(348i total) * Lia’s... * 302.8(32Mg) * E05..- (3481)) 0"“ Z N(34Si)/(N(34Si total) 4 L738“, mg * Efiadw(34Si) * Efiobj(34Si)) * 0“” (5.2) where At * 1027 afar — m. (5.3) 91 In the calculation of the target number density, am. in barns, At is the atomic mass of the 9Be target in g/mol, t is the thickness of the target in g/cm2 and N A is Avo- gadro’s number (mol’l). In the numerator of Eq. 5.2, the number of 32Mg fragments, N f (32Mg), was determined by the number of counts in the particle identification gate. (Section 5.3.1 contains a discussion of the 32Mg particle identification.) The total number of 348i nuclei, N f (348i total), was determined using the object scin- tillator at the entrance to the S800 analysis beam line. The livetime of the trigger, the efficiency of the CRDCs and the efficiency of the object scintillator—LT;800 mg, Efi'irm and Efiibj respectively—were included. In the denominator, the number of 34Si fragments, N(34Si), was determined by a set of data where the S800 spectrograph magnetic fields were set to the rigidity of the incident 348i beam. The total number of 34Si nuclei was determined using the object scintillator. The numerator and de- nominator were calculated using different sets of data because the spectrograph was set to different magnetic field settings. (The setting where the 32Mg fragments are centered in the focal plane denoted by the superscript f on the terms in Eq. 5.2 and the setting where the 34Si fragments are centered in the focal plane, respectively.) Therefore, the trigger livetime and two efficiency terms in the numerator do not can- cel out with those in the denominator in Eq. 5.2. The magnitudes of these terms are not negligible. In Table 5.2 are the particle numbers, livetimes, and efficiencies used in the calculation of the 9Be(34Si,32Mg)X inclusive cross section in Eq. 5.2. For the 32Mg fragment setting, the rate on the object scintillator was 33557 particles/ second and the rate in the 8800 focal plane was 32 particles/ second. For the 34Si setting, the rate on the object scintillator was 291 particles / second and the rate in the 8800 focal plane was 154 particles/ second. The inclusive cross section calculated from Eq. 5.2 must be corrected for loss of acceptance in the S800 spectrograph focal plane. There are two effects. Firstly, at the edges of the focal plane the momentum distribution gets slightly distorted. Secondly, the entire 32Mg momentum distribution did not fit into the acceptance of the focal 92 Table 5.2: Particle numbers, livetimes, and efficiencies used for the 9Be(3“Si,321\/Ig)X inclusive cross section calculation in Eq. 5.2. Variable Value Nf(32Mg) 2333 i 48 Nf(34Si total) 252967878 i 15905 LTSf800 mg 0.985 :1: 0.003 Efigdcs(32Mg) 0975 i- 0.022 Egg, .(34Si) 0.956 :t 0.019 N(§4Si) 92039 i 303 N(34Si total) 203008 :1: 451 [1‘3ng 0.875 :1: 0.004 mama‘s» 1.0 i: 0.0(1 Efiob,(343i) 1.000 i 0.005 “The CRDCs were not used in the identification of the 34Si fragments therefore the CRDC efficiency correction is identically one. plane. These corrections were estimated based upon a previous measurement of the same reaction [44,88]. The total acceptance correction was 11.5%. Therefore, the inclusive cross section for the 9Be(3“Si,32Mg)X reaction is (0.86 i 0.08) mb. This is in agreement with the inclusive cross section of (0.76 :t 0.10) mb determined in a previous measurement [44]. In addition to the 32Mg fragments, 31Mg fragments were also detected in the S800 focal plane as seen in Fig. 5.2. The observed inclusive cross section of the reaction 9Be(34Si,3ll\/Ig)X is (0.403 :1: 0.026) mb. This measurement represents a lower limit on the true inclusive cross section for two experimental reasons. Firstly, in the S800 focal plane 31Mg was severely cut off in acceptance because the 32Mg fragments were cen- tered in the focal plane. Secondly, the momentum distribution of the 31Mg fragments is expected to be wider than the distribution of the 32Mg fragments since the reaction process leading to 31Mg has a non-direct component. The location of the center of the 31Mg momentum distribution cannot be determined in the data. But, because the momentum distribution of the 31Mg fragments is expected to be wider than the momentum distribution of the 32Mg fragments, it can be concluded that more than half of the 31Mg momentum distribution does not fit in the acceptance of the 8800 93 2000: 1500: 1000 ; Counts/(128 channels) 500 ; 1600 E 1200 800 f Counts/(128 channels) — l l l 0 i000 2000 3000 4000 CRDC1 x-position (channels) Figure 5.12: Fragment yield versus the x—position distributions of 32Mg and 31Mg fragments in the S800 focal plane. Both spectra are gated on the and of the Magnesium isotopes gate drawn on the top panel of Fig. 5.2 and the corresponding isotope from the bottom panel of Fig. 5.2. focal plane. Therefore, the 0.403 mb lower limit of the inclusive cross section for the production of 31Mg is significantly less than what the true inclusive cross section for the reaction 9Be(34Si,31Mg)X would be if 31Mg was not being cut in the S800 focal plane. Fig. 5.12 shows the x—position distributions of the 32Mg and 31Mg fragments in the S800 focal plane. The reduction of counts in the highest and lowest channels in the x-position distributions in Fig. 5.12 is due to a loss of acceptance at the edges of the CRDC. 94 Discussion Using the theory of two-nucleon stripping reactions as presented in [86], with shell- model wavefunctions calculated by B. A. Brown [89] and nuclear radii from B. A. Brown’s (Skyrme SKX) Hartree—Fock calculations [90], J. Tostevin [91] calculates an inclusive cross section for the 9Be(3"Si,32Mg)X reaction of 2.76 mb. The shell-model calculation predicts 0+, 2‘L and 4+ sd states at 2.41 MeV, 4.09 MeV, and 5.31 MeV with partial cross sections of 0.42 mb, 0.98 mb, and 1.36 mb, respectively. The sum of the partial cross sections to the 0+, 2+ and 4+ states is 2.76 mb. This method of calculating the inclusive cross section by the stripping of two correlated nucleons has matched experimental inclusive cross sections reasonably well. Examples include two-proton and two-neutron knockout from 12C at three different energies [92,93], two-proton knockout from 28Mg to 26Ne [44,86], two-proton knockout from 46Ar to 44S [94] and two-proton knockout from 44S to 42Si [94]. The 0+, 2+, and 4‘L sd states at 2.41 MeV, 4.09 MeV and 5.31 MeV are not observed in the experiment. The excited sd states are each thought to fragment over a few MeV to create the observed excited states. The ground state of 34Si is dominated by the Ohw configuration [95]. The deformed excited states in 32Mg are not expected to be directly populated because there is very little overlap between the spherical and deformed state wavefunctions. In the calculation of the two-nucleon knockout reaction cross sections, diffraction should also be included in addition to the stripping components. This is just be- ginning to be explored for two-nucleon knockout [91,96]. In the calculation of the two-nucleon knockout inclusive cross sections, diffraction contributions seem to be approximately the same size as the stripping contributions. In one-nucleon knockout, in comparing the calculated inclusive cross section, including stripping and diffrac- tion parts, to the experimental inclusive cross section for a given reaction, a reduction factor of approximately 0.5-0.7 [68,97] is observed for normally to loosely-bound nu- clei. These observed reduction factors give a measure of the correlations that are 95 absent in, or only approximately described by, the nuclear shell-model. These correla- tions arise from the repulsive core and tensor part of the nucleon-nucleon interaction (short-range) and also couplings to low-lying excitations and higher-lying giant reso- nances (long-range) [98,99]. In [96] it was suggested that the reduction factor needed for two-nucleon knockout reactions are approximately equal to the reduction factor for one-nucleon knockout squared. A reduction factor of approximately 0.5 for two- nucleon knockout is then reasonable. Including diffraction components is currently under investigation [91] and will not be pursued further here. For the two-proton knockout of ”Si to ”Mg the calculated inclusive cross section of 2.76 mb is much larger than the measured inclusive cross section of (086320.08) mb. A possible explanation for this difference is that some of the predicted strength to the 2+ and 4+ sd states is fragmented over a few MeV such that a significant fraction of the 2* and 4+ strength could be above the ”Mg neutron separation energy (8,.) of 5.809 MeV. For excitations above 8,, a neutron would be emitted, leading to 31Mg. The observed inclusive cross section of (0.403 d: 0.026) mb for the 9Be(”Si,3lMg)X reaction has contributions from two processes. One is the two-step process where ”Mg is produced then one neutron is emitted and the the other process is where 31Mg is made directly in fragmentation from ”Si. It is not possible to know how much of the observed inclusive cross section for the production of 31Mg is due to which process. The observation of 31Mg with an inclusive cross section of at least 0.403 mb is a plausible explanation why the observed inclusive cross section of 9Be(”Si,”Mg)X is less than the expected cross section. The strength from the 2+ and 4+ sd states exciting levels above the neutron separation energy in ”Mg leads to the emission of a neutron and observation of 31Mg. If half of each of the partial cross sections that lead to the 4+ and 2+ sd states go into 31Mg, then the expected inclusive cross section of 9Be(”Si,”Mg)X would be (2.76 — 1/2(1.36) — 1/2(0.98) = 1.59) mb. The observed inclusive cross section is (0.86 :l: 0.08) mb. More than half of the unobserved cross section can thus be explained. However, a better understanding of the structure of 96 32Mg is needed to accurately predict the observed inclusive cross section and observed excited states. 5.4.2 Proposed level scheme Fig. 5.13 shows the ”Mg level scheme observed from the two-proton knockout reaction 9Be(”Si,”Mg 7)X. The placement of the 7 rays and energy levels was based upon 7-7 coincidence analysis and the observed 7-ray intensities. 7-7 coincidence data can give valuable information on the excited state level scheme of a nucleus. ”Mg 7-7 coincidence data were obtained by requiring two prompt 7 rays in two different SeGA detectors and a 3r‘zMg particle in coincidence. When analyzing the 7-7 coincidence spectra, two criteria were used to assert that two 7 rays were in coincidence. Given two 7 rays A and B, the first criterion compares the calculated with the observed number of 7 rays of A in the 7-7 coincidence spectrum gated on B. The second criterion is the reverse; it compares the calculated with the observed number of 7 rays of B in the 7-7 coincidence spectrum gated on A. If each of the two comparisons agree within 10, then it is asserted that two 7 rays are in coincidence. The fits to the SeGA angle pair spectra were used to obtain the 7-ray intensities, L,, (as outlined in Sections 5.3.4 and 5.4.4) and are presented in Table 5.5. The last column in the table summarizes what 7 rays were seen in coincidence with the listed 7 ray. Placement of each 7 ray is discussed below. Fig. 5.14 shows the ”Mg 7-ray energy spectrum in coincidence with the 885 keV 2]” —> 0:3. 7-ray transition. While background was subtracted, the spectrum contains a large density of 7 rays (compared, for example, to a spectrum gated on a higher- lying transition as shown in Fig.5.15) since all of the observed excited states deexcite through the 2+ state at 885 keV. In the 885 keV 7—7 coincidence spectrum seven 7 rays can be seen; the 1800 keV, 1964 keV, 2405 keV, 2599 keV, 2912 keV, and 3258 keV 7 rays, and the doublet around 1438 keV. The 885 keV 7 ray was the only 7 ray observed in the 7—7 coincidence spectra of the 1800 keV, 1964 keV, 2599 keV 97 (”A o m-D I 8 3 :3 3g 393 g 8 SE c. ca 0. 0. 0. . 0°. . o- o" o o o o oo oo o — C H +l +1 +I +1 +4 +l +| +l 1" 52.9 8 v- N 00 we N00 0 92 :13 :0 v- co NN veto co 0 mm 0. o. '1 0. "1°. "'0. 0. o' O. a) o o o o oo oo o . + + N o I I or) | v- . H | $3 0) I N I ' o I 1- «H I N o F o I H 00 ... . g — 1- — — — —> l g} > , ) £2 I :7 +I l 2 In I :2 :3; a, l +I 2 I g ‘- y a I l I to +l I :2 3 I +l , g '— i I N I > [N I +I v I 5 | ’— —'-— —' 8 co 9 l\ g mm coo: to > 8 2 .- 2 v £8 8% 8 °’ 0 Vt v to on N01 N01 2’; a? + + + V N o 2 U) I .8 o o o 1- O) 1- B 3 " N Figure 5.13: ”Mg level scheme observed from the two-proton knockout reaction 9Be(”Si,”Mg 7)X. The 7-ray energies and uncertainties in keV are labeled to the right of each 7 ray. The excited state energies in keV are labeled to the left of each level. The partial cross sections in mb (discussed in Section 5.4.4) for each level are labeled to the right of each level. The sd states in keV, calculated in the shell-model (SM), are shown to the left of the level scheme. 98 30 _— 1414 keV 885 keV ; 1443 keV Hay gated 25 l J 9 20 L 1800 keV g : I 2599 keV O )- 9‘, 15 _ a _ 1964 keV E J 2912 keV 3 8 10 ~ 2405 kevV I ; 3258 keV 5‘ l 0 ..I l i l 1 L .[I l 500 1000 1500 2000 Energy (keV) 1 . . 2500 3000 3500 Figure 5.14: ”Mg 7—ray energy spectrum in coincidence with the 885 keV 7 ray. The background-subtracted spectrum contains a large density of 7 rays; those placed in the level scheme are labeled. 99 1964 keV 885 keV y—ray gated )- )- y. 15” 20 l Counts/(20 keV) 0 ”J I l l n I _ . l l 1 500 1000 1500 2000 2500 3000 3500 Energy (keV) J_ L 1 Figure 5.15: Background-subtracted ”Mg 7-ray energy spectrum in coincidence with the 1964 keV 7 ray. The 885 keV 7 ray, in coincidence with the 1964 keV 7 ray, is labeled. and 3258 keV 7 rays. This can be seen for the 1964 keV 7 ray in Fig. 5.15 which shows the 1964 keV 7 ray gated, background subtracted, 7-7 coincidence spectrum. The 1800 keV and 3258 keV 7 rays are dashed on the level scheme because one of the two above discussed comparisons between the 1800 keV and 885 keV 7 rays, and the 3258 keV and 885 keV 7 rays, respectively, did not agree within 10. The 1414 keV and 1443 keV 7 rays comprise the doublet discussed in Sections 5.3.2 and 5.3.3. The 1443 keV 7 ray is in coincidence with the 885 keV and 2405 keV 7 rays as shown in the level scheme. The placement of the 1414 keV 7 ray is uncertain. The more intense 1443 keV 7 ray can be solely responsible for the observed 7-7 coincidence agreements with the 885 keV 7 ray. However, it can not be excluded that the 1414 keV 7 ray is in coincidence with the 885 keV 7 ray. If the 1414 keV 7 ray is the 7 ray seen in the intermediate-energy Coulomb excitation experiments (as 100 proposed in Section 5.4.4), then there is no additional (besides the present experiment) experimental evidence for the 1414 keV 7 ray to be in coincidence with the 885 keV 7 ray. Therefore, the 1414 keV transition is dashed in the level scheme. The 2405 keV 7 ray is in coincidence with the 1443 keV and 885 keV 7 rays and deexcites a state at 4733 keV. Fig. 5.16 shows the 2405 keV 7 ray gated 7-7 coincidence spectrum containing the 885 keV and 1443 keV 7 rays. The 2405 keV 7 ray was determined to be in coincidence with the 1443 keV 7 ray and not the 1414 keV 7 ray for two reasons. Firstly, the intensity of the 2405 keV 7 ray is greater than the intensity of the 1414 keV 7 ray and less than the intensity of the 1443 keV 7 ray. Secondly, the energy of the 7 ray in the 2405 keV 7-7 coincidence spectrum agrees with 1443 keV. The 2912 keV 7 ray is in coincidence with the 885 keV 7 ray and deexcites a state at 3797 keV. There is some weak evidence that the 2912 keV 7 ray may be in coincidence with the 1414 keV 7 ray and thus feed the 2299 keV level. Only one of the 7-7 coincidence comparisons (described above) between the 2912 keV and 1414 keV 7 rays agreed within 10. Also, the 2912 keV 7 ray is almost twice as intense as the 1414 keV 7 ray. Therefore, it is assumed that the 2912 keV 7 ray directly feeds the 885 keV state. 5.4.3 7-ray angular distributions As discussed in Chapter 2, 7-ray angular distributions can determine the multipolar- ities of 7 rays. The ”Mg 7-ray energy spectra were fit for each SeGA detector angle pair as described in Section 5.3.4. Many of the 7-ray angular distribution spectra for the different 7 rays had large error bars due to low statistics and angle-dependent corrections to the fitted intensities described in Section 5.3.4. The experimental 7-ray angular distributions were fit with calculated W(0) curves with fixed initial and final spins I,- and I f, respectively, optimizing for the alignment, the multipole mixing ratio 6 and the overall amplitude a. The multipole mixing ratio was in general not known. 101 885 keV _ f 2405 keV 12 — ‘ “Hay gated 10 _ 1443 keV Counts/(20 keV) [l]. . HHWJHMMUILWLL background gate above 10 * 2405 keV y—ray applied Counts/(20 keV) 500 1 000 1 500 2000 2500 3000 35 30 Energy (keV) Figure 5.16: The ”Mg 7—ray energy spectrum in coincidence with the 2405 keV 7 ray is shown in the upper panel. The 885 keV and 1443 keV 7 rays, in coincidence with the 2405 keV 7 ray, are labeled. The background is shown in the lower panel. 102 Also, for each 7—ray transition the m substate distribution of the initial state, I .-, was not known. This alignment of a state can be oblate or prolate. (Section 2.3 defines oblate and prolate alignment.) As discussed in Section 5.4.1, .1. Tostevin calculated the energies and partial cross sections of the sd states in ”Mg from the 9Be(”Si,”Mg 7)X reaction. The 2+ and 4+ excited states are predicted to have 15% and 24% prolate alignment, respectively. The alignments calculated for one-nucleon and two-nucleon knockout reactions are prolate and of similar magnitude. Section 2.5.3 briefly discusses one-nucleon knockout reactions and the calculated alignments. It is highly unlikely that different excited states could have oblate alignment. Therefore, in fitting the experimental 7-ray angular distributions the type of alignment was assumed to be prolate, but the amount of alignment was allowed to vary. The error bars for the intensities at each angle for the different 7 rays were too large to confine the amount of prolate alignment. For different I,- and If spin com- binations, 6 and a values could be found such that the X2 was similar for all of the spin combinations tried. Table 5.3 gives a summary of the angular distribution fits performed and the results. The 7-ray angular distribution of the 2405 keV 7 ray is shown in Fig. 5.17 and illustrates the above mentioned issues. Table 5.3 shows the 7- ray angular distributions are inconclusive. The different fits for each 7-ray transition have similar x2 values and are therefore indistinguishable from each other. For each 7-ray transition fit with a 4 —> 2 and/ or a 6 —> 4 spin transition, except the 885 keV 7 ray, the 7-ray angular distribution data was also fit with 4 —+ 2 and 6 ——> 4 spin transitions where 6 was fixed to zero and only a and a were allowed to vary. The results of those fits gave the same a values which are in the table and gave a values which were consistent with the results in the table for the respective 7-ray energy and I,- and I f spins. The X2 values were similar to the x2 values of the other spin transition fits. (I.e. The x2 from the 3 ——» 2 spin transition fits.) Therefore, for all the 4 —-> 2 and 6 —> 4 transitions, fixing 6 = 0 did not significantly change the results of the other fit parameters or significantly change the x2. 103 Table 5.3: Summary of results of fits of experimental 7-ray angular distributions. Listed are the 7—ray energies E,, initial I,- and final I f spins, intensities a, measures of the degree of prolate alignment 0, the percentages of prolate alignment that 0 corresponds to, the multipole mixing ratios 6 and the X2 values of the fits. In each fit a, a and 6 were allowed to vary unless otherwise noted. I,- and I I were fixed to the values listed. As can be seen by the large error bars for a and 6, and the similar X2 values for each set of fits for a given E7, the present experiment is not able to distinguish different multipolarity transitions. E, (keV) I,- If a a % alignment 6 X 885 2 0 0.540 i 0.017 4.70537 2.2733 0“ 1.10 1800 3 2 0.017 i 0.003 0-02‘:°T’0195 1007:?00 0.22:3}; 3.05 1800 2 2 0.017 i 0.003 1.0233,“,9 36.323; 3.67 i 00 3.03 1800 4 2 0.017 i 0.003 1.63f‘f’07 37.7fm 4.34:2? 2.99 1964 3 2 0.081 i 0.004 0.45fgjjgg 92013-298 0.23 i 0.05 2.59 1964 2 2 0.081 3: 0.004 0.67f33g3gg 62.4%: 4.401122 2.59 1964 4 2 0.080 i 0.004 2.62:3;73 17.7f}§;3 2.34f3° 2.40 2405 3 2 0.054 :t 0.004 0057:3339, 10012,,07 0.29 i 0.06 1.55 2405 4 3 0.054 :1: 0.004 0.133%9 1007:2378 0.27 i 0.05 1.53 2405 2 2 0.054 i 0.004 0.923537 42.23%; 1.70:{;$§ 1.54 2405 4 2 0.054 3: 0.004 1.20333 536:3}: —5.63f§'2"7]77 1.35 2405 6 4 0.054 i 0.004 1.63323 5462:3333 —30.69:‘§3~92 1.52 2599 3 2 0.054 i 0.004 3.61385, 64:3? 0.341139 15.96 2599 2 2 0.055 i 0.004 1.04:3?61 35.3733 6.25330 15.49 2599 4 2 0.055 t 0.004 1.78335 3323;311:113 —0.64t3;fl 14.64 2912 2 2 0.076 d: 0.004 2.68_‘1’f‘54 6.74,,- 0.0 i 00 4.08 2912 3 2 0.076 :1: 0.004 4.6173852 4.032,“ 2.99 i 00 4.07 2912 4 2 0.076 t 0.004 6103‘?22 3.63}? 3.89 :1: oo 4.07 3258 3 2 0.040 3: 0.003 001333;],g 100.01%?2 1945130 11.92 3258 2 2 0.040 3: 0.003 0.95:0;353 40.53;: 1.941333: 14.16 3258 4 2 0.040 i 0.003 1.151322 55832;? 411341;, 13.44 “6 was set to zero and not allowed to vary because the 885 keV 7 ray is known to be a 2+ —o 0;,_ (pure E2) transition. 104 4.5 ’ X2 3.5: 251 L5’ - - - . . 0.05 ‘ 0.055 ‘ 0.06 ‘ o 2 4 6 8 10 a 0' 45+ :15 x2 . 215’ L5» (12 (125 (13 (135 (14 5 3+2 black 2+2 red 4+3 green 4+2 blue 6+4pmk 25 50 75 100 125 4150 175 9cm. (deg) Figure 5.17: Plot of goodness-of-fit x2 values versus different parameters for the 2405 keV 'y-ray angular distribution 3 —> 2 spin transition fit shown together with a plot of the angular distribution versus angle shown in the bottom row. In the 2405 keV '7-ray angular distribution versus angle plot, the points are the data and the lines are the different spin transition fits as described in the text and Table 5.3. 105 Table 5.4: Ranges of the uncertainties AW(0)/W(0) on the experimental 7—ray an- gular distribution points, of the nine SeGA detector angle pairs. (The two 7 rays comprising the doublet are not listed.) The detectors at forward angles have the smaller uncertainties because they have more statistics due to relativistic kinematics. E. Amen/we) 885 940% 1800 34-66% 1964 14-20% 2405 19—29% 2599 14—50% 2912 14—21% 3258 18—30% If the m substate distribution could be calculated explicitly for each observed excited state and if the error bars on the observed 7-ray angular distributions were smaller, then with only the multipole mixing ratio and an overall scale factor allowed to vary for each 7 ray, the fits to the observed 7-ray angular distributions would be able to identify the multipolarities of the 7-ray transitions. For future fragmentation and nucleon-knockout experiments with fast beams, the number of unknowns in the fits to the experimental 7—ray angular distributions must be reduced and the error bars on the intensities must be smaller than the ones in the present experiment. The ranges, over all nine SeGA detector angle pairs, of the uncertainties on the experimental angular distribution points for the observed 7 rays (excluding the two 7 rays comprising the doublet) are shown in Table 5.4. For all but the 885 keV 7 ray, 1,, I f, a, 0' and 6 were unknown. Assuming a fit is always done by fixing I.- and If and letting a vary, something about a or 6 needs to be known (or assumed) to fit the 7-ray angular distribution and get information about the multipolarity of the 7-ray transition. Even if the error bars were infinitesimally small, a and 6 can not both be free to vary over all possible values for either oblate and prolate alignment. For example, a 3 —> 2 prolate distribution and a 3 —> 2 oblate distribution are essentially identical from approximately 20° to 160° with certain values of a and 6. If the type of alignment is constrained, as was done for the fits to the 32Mg 7-ray angular 106 distributions, something else still must be known or assumed about either the amount of alignment, 0, or 6 (or I,- and I f). For example, a 3 —» 2 prolate distribution and a 4 —> 2 prolate distribution are clearly distinguishable if 6 = 0, but a value of 6 can be found such that the distributions are very similar. If the alignment of either is less than 100%, then the distributions can become nearly identical. This is illustrated in the fit results in Table 5.3 and can be seen in Fig. 5.17. In Chapter 2, only pure transitions of the lowest allowed multipolarity were discussed. In Section 2.5.2 an uncertainty of 4% was quoted as needed to measure 7—ray angular distributions of fragmentation or nucleon-knockout reactions if there is only 20% prolate or oblate alignment and fl = 0.4. That would allow one to tell the difference between a quadrupole and dipole whose relative difference is 10-20%. However, this is not true if the transition is not a pure transition of the lowest possible multipolarity. The actual level of uncertainty needed to make a measurement is then dependent on what is known. Therefore, the type of alignment and some knowledge or plausible assumptions about a, 6, I.- or I I must exist to successfully use 7-ray angular distributions with fragmentation and nucleon-knockout reactions at intermediate energies to determine multipolarities of 7—ray transitions. 5.4.4 Partial cross sections In Table 5.5 the observed 32Mg 7-ray energies and intensities are presented. Table 5.6 lists the excited state energies and partial cross sections. The calculation of the ener- gies of the 7—rays is outlined in Section 5.3.4. To calculate the 7—ray intensities and the partial cross sections the results of the fits of the SeGA angle pair spectra were used and the procedure is as follows. The fitting of the 7-ray spectra for each SeGA angle pair is described in Section 5.3.4. The 7-ray angular distributions were then fit as described in Section 5.4.3 and the results are shown in Table 5.3. For each 7 ray, 107 Table 5.5: ”Mg 7 rays observed from the 9Be(3“Si,”Mg 7)X reaction. I.- and I f are the initial and final excited state spins for the levels connected by the respective 7-ray transition with energy, E7. The 17 values are the intensities of the 7-ray transitions relative to an observed ”Mg particle. The column labeled coincidences contains the 7 rays that were observed to be in coincidence with the respective E7. An energy in brackets denotes that in its 7-7 coincidence spectrum the 7 ray with energy E, was not observed. E, (keV) I,- I f I7 (%) coincidences“ 885 :l: 4 2+ 0+ 96.4 :l: 3.2 [1414], 1443, 1800, 1964, 2405, 2599, 2912,3258 1414 :l: 7 7.4 a: 1.2 [885] 1443 :t 6 26.1 :1: 2.6 885, 2405 1800 :l: 10 3.0 :l: 0.6 885 1964 :l: 9 14.5 :1: 1.0 885 2405 :l: 11 9.6 :l: 0.8 885, 1443 2599 :l: 12 9.6 :l: 0.8 885 2912 :l: 13 13.6 a: 1.0 885, [1414] 3258 d: 16 7.1 :l: 0.6 885 “For the 1414 and 1443 keV 7 rays it was not possible to create separate 7-7 coincidence spectra. Therefore, the 7 rays listed as coincidences for these 7 rays are those deduced from calculations as described in the text. 0 o d a the 1ntensrty I: , lS , (5-4) 18d=(ix107x 1 x 1 )x 1 7 47? LTcoinc trig (1 — d) N (”Mg) X 6138800 trig/LTSBOO trig where 107 is the number of 7 rays emitted in the GEANT simulation for each 7 ray, LTCOinc mg is the coincidence trigger livetime, d is the data loss, N (”Mg) is the number of ”Mg fragments, LTssoo mg is the 8800 particle singles trigger livetime and dSSgoo trig is the S800 particle singles trigger downscale value. (The coincidence trigger livetime and data loss are described in Section 5.3.4. The different livetimes for the different triggers are discussed in Sections 3.5 and 5.2.1.) In Eq. 5.4, for a given E7, a is the intensity from the fit of the 7-ray angular distribution where the GEANT scale factor and correction for paralyzed detectors (both described in Section 5.3.4) were taken into account. Those corrections had to be made to the intensities obtained by fitting the SeGA angle pair spectra prior to fitting the 7- 108 ray angular distributions because they are detector angle dependent. The efficiency of the CRDCs and efficiency of the object scintillator are not included in Eq. 5.4 because they affect the a and N (”Mg) terms equally. There were 179 good runs (approximately 88 hours) in the 9Be(3“Si,”Mg 7)X dataset. The entire dataset was used when fitting the 7-ray spectrum for each SeGA detector angle pair. However, different trigger downscale values were used for some runs. This was addressed as follows. A subset of 108 runs (approximately 54 hours) with one trigger setting (and set of downscale values) was used to accurately obtain the intensity for just the 885 keV 7 ray. The intensity of the 885 keV 7 ray from the subset of runs, [385, was then taken as an absolute reference for the other intensities. The coincidence trigger livetime and data loss values listed in Table 5.1 are the values from this subset of runs and are used in the calculation of 1835. For the subset of runs the number of 3"’Mg fragments, corrected for the S800 particle singles trigger livetime and downscale factor, was N (”Mg) XdSSgoo trig / LTssoo mg = 347108 :t 2266. The livetime of the S800 particle singles trigger, LTssoo trig, was approximately 68%. Therefore, for each 7 ray the absolute intensity 1, given in Table 5.5 is I, = 13" x % (5.5) Calculating partial cross sections from the 7-ray intensities was possible once a level scheme was constructed. The level scheme was constructed using the observed 7-ray intensities and 7-7 coincidence analysis as discussed in Section 5.4.2. For an excited state that was not fed by higher-lying 7 rays, the partial cross section was calculated from the intensity of the 7 ray which decays from that state by multiplying the intensity by the inclusive cross section. For an excited state fed by higher lying 7 rays, the intensities of the feeding 7 rays were subtracted from the intensity of the 7 ray from the excited state whose partial cross section is being calculated. The value is then multiplied by the inclusive cross section to get the partial cross section. For the 109 Table 5.6: Level energies E and partial cross sections in ”Mg corresponding to the level scheme shown. E (keV) Partial cross section (mb) 0.0 —0.019 :1: 0.030“ 885 :l: 4 0.080 :l: 0.043“ 2299 :l: 8 0.063 :t 0.012 2328 :t 8 0.142 :t 0.027 2685 d: 11 0.026 i 0.005 2849 :l: 10 0.124 :1: 0.014 3484 :l: 13 0.083 i 0.011 3797 :l: 14 0.117 :t 0.014 4143 :t 16 0.061 :t 0.008 4733 :l: 13 0.083 :t 0.011 Sum = 0.760 :l: 0.066 “Includes feeding from 7—ray strength from 3400-4924 keV. 9Be(3“Si,”Mg 7)X reaction, the partial cross sections were calculated according to the level scheme shown in Fig. 5.13; the intensities of all the 7 rays that are shown to feed the 885 keV level, for example, were subtracted from the 885 keV 7-ray intensity when calculating the partial cross section for the 885 keV level. The partial cross sections are presented in Table 5.6. Some excess 7—ray strength above 3400 keV that could feed the 885 keV 7 ray was quantified. 7 rays above 4924 keV cannot feed the 885 keV level because 8,, = 5809 keV. A 3400-4924 keV gated, 7-7 coincidence spectrum showed the 885 keV 7 ray. In the ”Mg 7-ray summed spectrum of all SeGA detectors it is not clear how many peaks are between 3400 and 4924 keV. Regardless of whether one or two peaks were assumed from 3400-4924 keV, using GEAN T and the 7—7 coincidence spectrum a cross section for the 7 ray(s) strength between 3400 and 4924 keV was estimated to be (0.05 :l: 0.01) mb. Therefore, assuming feeding from above 3400 keV, the partial cross section of the 885 keV level is (0.080 i 0.043) mb and the partial cross section of the ground state is (—0.019 :1: 0.030) mb. If the 1414 keV 7 ray is not in coincidence with the 885 keV 7 ray, the partial cross section of the 885 keV level, including the feeding from above 3400 keV, would be (0.143 :l: 0.044) mb. 110 Discussion The 885 keV first 2+ level is well established from previous experiments and its struc- ture has been interpreted as being a 2hw excitation. (See [46] and other references in Section 1.4.1.) In the reaction 9Be(3“Si,”Mg 7)X, two protons are removed from the 7rd5/2 orbital in the ground state of 34Si which is dominated by the 0hr.) configu- ration [95]. Therefore, due to a very small overlap between these two configurations, little direct feeding is expected in this reaction to the 2+ 885 keV level in ”Mg. Within 20, the observed partial cross section of the 885 keV level of (0.080 :t 0.043) mb is consistent with no direct feeding to the 885 keV level. The partial cross section of the ground state of (—0.019 :l: 0.030) mb is also consistent with no direct feeding to the ground state. This is expected because, like the 2+ state, the ground state is dominated by a 25..) configuration. If the 1414 keV 7 ray was not in coincidence with the 885 keV 7 ray, the partial cross section of (0.143 2120.044) mb for the 885 keV level would imply that the 885 keV level is not a 2hr.) configuration, but has some small Ohw admixture. The 885 keV 7-ray angular distribution is shown in Fig. 5.18. The 885 keV 7 ray is the lowest excited state and is fed by at least eight observed 7 rays and appears to have little or no direct feeding. Therefore, one would expect its 7-ray angular distribution to be isotropic. The fit of the 7—ray angular distribution, as can be seen in Table 5.3, is consistent with an isotropic distribution. The (1443 :t 6) keV 7 ray agrees in energy with the (1437.0 :l: 0.3) keV [34] 7 ray observed in 5 decay. Within 20, the (1414 :t 7) keV 7 ray agrees in energy with the (1438 :l: 12) keV 7 ray seen in intermediate-energy Coulomb excitation [4]. If the 6 decay and intermediate-energy Coulomb excitation experiments observed two different 7 rays near 1430 keV as suggested here, then there is no contradiction in possible J7r values for their initial states. (The possible J1r values are discussed in Section 1.4.1 and below.) Two recent experiments give contradictory information on whether the 1414 keV 111 100' x2 60’ 20‘ 0.51 7.5 ‘ 0.753 A a 12.5 0.55 A 17.5 0.57 ‘ W(6) 2.4 1 2.0 ’ 1.6 ' 1.2' 0.05 » 0.0475» 0.045 5 0.04251 0.04 5 0.0375 25 50 75 100 125750 175 9cm. (deg) Figure 5.18: Plots of goodness-of-fit x2 values versus different parameters for the 885 keV 7-ray angular distribution 2 ——> 0 spin transition fit shown together with a plot of the angular distribution versus angle shown in the bottom row. The points are the data and the line is the 2 —* 0 spin transition fit as described in the text and Table 5.3. 112 7 ray and 885 keV 7 ray are in coincidence. The two published measured ”Mg B(E2T ) values of 454(78) ezfm4 [10] and 333(70) ezfm“ [4] are different because the latter takes into account feeding into the 885 keV level from an observed 7 ray around 1430 keV. Without the feeding correction, the B(E2T) value in [4] is equal to 440(55) ezfm“ [4] which agrees well with [10] which has no feeding correction. A recent livetime measure- ment [100] measures a half-life with corresponds to a B(E21) value of 327(87) e2fm4. This value best agrees with the feeding corrected B(E2T) value. However, another recent Coulomb excitation measurement at low energy [101] reports a preliminary B(E2T) value of 434(52) ezfm“. Assuming that the 1414 keV 7 ray is the 7 ray observed in the intermediate- energr Coulomb excitation experiment and that the 1414 keV and 885 keV 7 rays are in coincidence, from [4] the possible J1r values of the 2299 keV state are 1‘, 1+ or 2+. If the 1414 keV 7 ray decays directly to the ground state from a level at 1414 keV and it was observed in intermediate-energy Coulomb excitation, possible J1r values are 1‘, 1+ or 22; because A = 1 and A = 2 excitations are the most probable in Coulomb excitation. The (1964 d: 9) keV 7 ray agrees in energy and level scheme placement with the (1972.9:l:0.5) keV 7 ray observed in 6 decay [34]. Because the 1443 keV and 1964 keV 7 rays agree in energy with the 7 rays seen in 6 decay, it is concluded that the excited state levels, (2327.9 :1: 7.5) keV and (2848.9 :1: 9.7) keV are the same levels as the (2322.3 :1: 0.3) keV and (2858.2 :1: 0.5) keV levels, respectively, seen in the fi-decay measurement [34]. The 2328 keV and 2849 keV levels have the largest partial cross sections, (0.142 d: 0.027) mb and (0.124 :1: 0.014) mb respectively, and thus are the most strongly fed levels observed in the two-proton knockout reaction. The fl-decay experiment of [47] reported a log ft value of 5.4 for the fl-decay transition to the 2321 keV level and a log ft value of 5.1 for the fl-decay transition to the 2858 keV level. Generally, log ft values corresponding to allowed fl-decay transitions range from 113 2.9-10 and log ft values corresponding to first forbidden transitions range from 5- 19 [102]. In the mass region around ”Mg, log ft values of 5.1 and 5.4 seem to corre- spond to allowed fi-decay transitions. From a compilation of log ft values for nuclei with Z < 80, a lower limit of log ft = 5.9 for first forbidden transitions can be sug- gested [103]. In [103] eight exceptions are reported where the log ft values are in the range of 5.5 to 5.7; these are from decays of 96Y, 142Cs, 143Cs, 150Ho, 158Er, 1‘5sz and 163Yb. The ground state of ”Na is thought to be negative parity from shell-model calculations [12] and the 6 decay in [47] finds a negative-parity ground state for ”Na more plausible than a positive-parity ground state. Shell-model calculations in [15] present configurations for the ground state of ”Na to be positive parity or negative parity, but say that a negative parity ground state is consistent with fl-decay re- sults [46]. Assuming the 3"’Na ground state is negative parity and the log ft values correspond to allowed transitions, then the 2849 keV level and the 2328 keV level must each have J1r S 5‘. This contradicts the simple picture of the two-proton knockout reaction 9Be(3“Si,”Mg 7)X which cannot explain the strong population of negative parity levels; direct removal of two 415/2 protons does not yield negative parity excited states. Some possibilities as to what might be occurring are presented. The first is that the two negative-parity levels are not directly fed in the two-proton knockout reaction, but are fed by higher-lying 7 rays. In the ”Mg 7-ray energr spectrum containing the sum of all the detectors, when fit with nine simulated response functions plus an exponential background, there is a small excess of counts around 2300 and 2700 keV corresponding to a partial cross section of (0.055 :1: 0.008) mb. This excess of counts is too small to account for the two very strong 7 rays at 1443 keV and 1964 keV. The second possibility is that the ground state of ”Na is not negative parity as expected, but is positive parity. The possibility of a positive-parity ground state of ”Na is supported by shell model calculations [15,19], but is very dependent on the magnitude of the N=20 shell gap. If the ground state of ”Na is positive parity then 114 the 2328 keV and 2849 keV levels would be positive parity and strongly populating positive parity states are what would be expected. The third possibility is if in the fi-decay experiment some 7-ray strength was missed due to feeding from unobserved higher-lying 7 rays in ”Mg, the 6 branch intensities would change and small changes in the intensities have large impacts on the log ft values. If a small amount of 7- ray strength in ”Mg feeding the 2328 keV and 2849 keV levels was not observed, the log ft values would increase and correspond to first forbidden transitions. If the ground state of ”Na is assumed to be negative parity, then first forbidden transitions would make the 2328 keV and 2849 keV levels have positive parity. A fourth possibility is that the log ft values of 5.4 and 5.1 for the fi-decay transitions to the 2321 keV and 2858 keV levels, respectively, correspond to first forbidden transitions. First forbidden transitions may be enhanced in the neutron-rich region around 3"’Mg. Two examples of first forbidden transitions in this mass region, with relatively small log ft values are a 6 decay from the 4‘ ground state of 3“‘Al to a 2+ state in 34Si with a log ft > 5.5 [104] and a B decay from the negative parity ground state of 38P to a 2’“ state in 38S with a log ft = 5.9 :l: 0.2 [105]. Lastly, a more remote possibility for the strong population of the 2328 keV and 2849 keV levels, assuming they are negative parity, is that many higher-energy, positive-parity states are weakly populated resulting in a shower of weak, unobserved 7 rays feeding the 2328 keV and 2849 keV levels. These unobserved 7 rays must be very weak and contribute only to the exponential background because they are not seen in the observed 7—ray spectrum. 115 Chapter 6 Summary The feasibility of using 7-ray angular distributions and 7-7 angular correlations with intermediate-energy beams was explored theoretically. The angular distribution of 1460 keV 2+ —> 0;, 7 rays emitted from intermediate-energy Coulomb excited 40Ar was measured. The nucleus, ”Mg was studied in a two—proton knockout reaction from 348i. The formalism for 7-ray angular distributions and 7-7 angular correlations with intermediate—energy beams was presented. Measuring angular distributions and angu- lar correlations of 7 rays from different reaction mechanisms, including intermediate- energy Coulomb excitation, was discussed. Measuring 7-ray angular distributions and 7—7 angular correlations is feasible with intermediate-energy beams. This was shown for the intermediate-energy Coulomb excitation of 40Ar. The measured angular dist- ribution for the 1460 keV 2+ —* 0;, 7 ray agrees with the calculated 0 —> 2 —-> 0 distribution. ”Mg was studied in the two-proton knockout reaction, 9Be(3“Si,”l\/Ig 7)X. Nine 7 rays were observed; five were observed for the first time. For seven of the nine 7 rays, 7-ray angular distributions were measured. No spin and parity assignments could be made from the distributions due to not knowing the magnitude of the prolate alignment and the relatively large error bars on the experimental points. 116 A level scheme for ”Mg was constructed using the observed 7-7 coincidence data and 7-ray intensities. In the two-proton knockout reaction, 9Be(3“Si,”Mg 7)X, no direct feeding to the ground state of ”Mg was observed and no direct feeding to the first excited 2+ state at 885 keV was observed within 20. This supports that the ground state of 34Si is predominantly Ohw because the ground and first excited states of ”Mg are known to be deformed, 21w configurations. The sd states are each thought to fragment over a few MeV to create the observed excited states. Six excited states were deduced from the observed 7 rays which are thought to be from the fragmented sd states. The other two excited states at 2849 keV and 2328 keV appear to have negative parity and different possibilities to their population have been presented. Two of the observed 7 rays were at (1414.0:l:6.6) keV and (1443.0:t6.4) keV. These two 7 rays may resolve the possible spin and parity conflicts arisen from previous measurements of what was thought to be one 7 ray around 1438 keV. The observed inclusive cross section for the 9Be(3“Si,”Mg)X reaction is (0.86 :l: 0.08) mb. Calculations of the two-proton knockout reaction, assuming two corre- lated protons knocked out of the d5/2 shell, predict three pure sd states that are not observed in the present experiment. The predicted inclusive cross section for the 9Be(34Si,”Mg)X reaction is 2.76 mb. Some of the calculated 3d strength appears to be lost due to being above the neutron separation energy in agreement with the ob- servation of 31Mg. Thus the predicted cross section more closely reflects the observed inclusive cross section, though a better understanding of the structure of ”Mg is needed to accurately predict the inclusive cross section and observed excited states. 117 Appendix A Subroutine used by GEANT Below is the Fortran subroutine described in Sect. 3.3.3. It takes the x,y,z center-of- mass of each segment of each SeGA detector from the rml file used by SpecTcl and generates the x,y,z positions and rotation angles for the geometry of SeGA as needed for GEANT. GEANT requires three theta and three phi angles for each detector as well as a position. The variables used below are thx, phix, thy, phiy, thz, phiz, xx, yy, and 22 for each detector. SUBROUTINE calcrotm(thx,phix,thy,phiy,thz,phiz,xx,yy,zz) * User common blocks INCLUDE ’uinc/uoutput.ins’ INTEGER detnum(18) REAL x(18,576), y(18,576), z(18,576) REAL Offset/0.8/ REAL aox/O./, aoy/O./, aoz/O./ REAL box/0./, boy/1./, boz/0./ REAL cox/1./, coy/O./, coz/O./ 118 REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL ax(18), ay(18), az(18) bx(18), by(18), bz(18) cx(18), cy(18), cz(18) rot(18,3,3) thx(18), phix(18), thy(18), phiy(18), thz(18) phiz(18) PI/3.14159265359/ det(18,3,3), orig(3,3), detinv(18,3,3) originv(3,3) dettrans<18,3,3), rottrans(18,3,3) origtrans(3,3), origtransinv(3,3) denoml, denom2, denomlo, denom2o determ(18), determo, norma(18), normb(18) normao, normbo, normco, normc(18) xx(18), yy(18), zz(18) v1(18,3), v2(18,3), v3(18,3) v1v2(18), v1v3(18), v2v3(18) sumx(18), sumy(18), sumz<18) avgx(18), avgy(18), avgz(18) sumx12(18), sumy12(18), sumz12(18) avgx12(18), avgy12(18), avgzl2(18) sumx14(18), sumy14(18), sumzl4(18) avgx14(18), avgy14(18), avgzl4(18) last3x(18), 1ast3y(18), 1ast32(18) avglast3x(18), avglast3y(18), avglast3z(18) distance(18) 119 aoz=-0ffset D0 I=1,18 * Convert the DetPos data to the format used in this subroutine. * DetPos are the detector positions as read in from the *.xml * file by a GEANT subroutine that is called previous to this one. DD J=1,32 x(I,J)=DetPos(I,J,1) yCI,J)=DetPos(I,J,2) z(I,J)=DetPos(I,J,3) END DD * Begin calculations DO J=1,32 sumx(I)=sumx(I)+x(I,J) sumy(I)=sumy(I)+y(I,J) sumz(I)=sumz(I)+z(I,J) END D0 avgx(I)=sumx(I)/32 avgy(I)=sumy(I)/32 avgz(I)=sumz(I)/32 distance(I)=(avgx(I)**2+avgy(I)**2+avgz(I)**2)**.5 120 sumx12(I)=x(I,1)+x(I,2)+x(I,5)+x(I,6)+x(I,9)+x(I,10)+x(I,13)+ x(I,14)+x(I,17)+x(I,18)+x(I,21)+x(I,22)+x(I,25)+x(I,26)+ x(I,29)+x(I,30) avgx12(I)=sumx12(I)/16 sumy12(I)¥y(I,1)+y(I,2)+y(I,5)+y(I,6)+y(I,9)+y(I,10)+y(I,13)+ y(I,14)+y(I,17)+y(I,18)+y(I,21)+y(I,22)+y(I,25)+y(I,26)+ y(I,29)+y(I,30) avgy12(I)=sumy12(I)/16 sumzl2(I)=z(I,1)+z(I,2)+z(I,5)+z(I,6)+z(I,9)+z(I,10)+z(I,13)+ z(I,14)+z(I,17)+z(I,18)+z(I.21)+z(I,22)+z(I,25)+z(I,26)+ z(I,29)+z(I,30) avg212(I)=sumzl2(I)/16 sumx14(I)=x(I,1)+x(I,4)+x(I,5)+x(I,8)+x(I,9)+x(I,12)+x(I,13)+ x(I,16)+x(I,17)+x(I,20)+x(I,21)+x(I,24)+x(I,25)+x(I.28)+ x(I,29)+x(I,32) avgx14(I)=sumx14(I)/16 sumy14(I)=y(I,1)+y(I,4)+y(I,5)+y(I,8)+y(I,9)+y(I,12)+y(I,13)+ yCI,16)+y(I,17)+y(I,20)+y(I,21)+y(I,24)+y(I,25)+y(I,28)+ y(I,29)+y(I,32) avgy14(I)=sumy14(I)/16 sumzi4(I)=z(I,1)+z(I,4)+z(I,5)+z(I,8)+z(I,9)+z(I,12)+z(I,13)+ z(I,16)+z(I,17)+z(I.20)+z(I.21)+z(I,24)+z(I,25)+z(I,28)+ z(I,29)+z(I,32) 121 avgz14(I)=sum214(I)/16 D0 J=21,32 1ast3x(I)=1ast3x(I)+x(I,J) 1ast3y(I)=last3y(I)+y(I,J) 1ast32(I)=1ast32(I)+z(I,J) END DD avglast3x(I)=1ast3x(I)/12 avglast3y(I)=1ast3y(I)/12 avglast3z(I)=1ast32(I)/12 ax(I)=-(avglast3x(I)-avgx(I)) ay(I)=-(avg1ast3y(I)-avgy(I)) az(I)=-(avg1ast3z(I)-avgz(I)) bx(I)=avgx12(I)-avgx(I) by(I)=avgy12(I)-avgy(I) bz(I)=avgzl2(I)-avgz(I) cx(I)=avgx14(I)-avgx(I) cy(I)=avgy14(I)-avgy(I) cz(I)=avgzl4(I)-avgz(I) norma(I)=SQRT(ax(I)**2+ay(I)**2+az(I)**2) normb(I)=SQRT(bx(I)**2+by(I)**2+bz(I)**2) normc(I)=SQRT(cx(I)**2+cy(I)**2+cz(I)**2) normao=SQRT(aox**2+aoy**2+aoz**2) normbo=SQRT(box**2+boy**2+boz**2) 122 DOOM normco=SQRT(cox**2+coy**2+coz**2) deth,1,1)=ax(I)/norma(I) det(I,1,2)=ay(I)/norma(I) det(I,1,3)=az(I)/norma(I) det(I,2,1)=bx(I)/normb(I) det(I,2,2)=by(I)/normb(I) det(I,2,3)=bz(I)/normb(I) det(I,3,1)=cx(I)/normc(I) det(I,3,2)=cy(I)/normc(I) det(I,3,3)=cz(I)/normc(I) Printing to screen the dot products of the det vectors They should all be zero if orthonormal PRINT *, det(I,1,1)*det(I,2,l)+det(I,1,2)*det(I,2,2)+ det(I,1,3)*det(I,2,3), det(I,1,1)*det(I,3,1)+ det(I,1,2)*det(I,3,2)+det(I,1,3)*det(I,3,3), det(I,3,1)*det(I,2,1)+det(I,3,2)*det(I,2,2)+ det(I,3,3)*det(I,2,3) orig(1,1)=aox/normao orig(1,2)=aoy/normao orig(1,3)=aoz/normao orig(2,1)=box/normbo orig(2,2)=boy/normbo orig(2,3)=boz/normbo orig(3,1)=cox/normco 123 .bwro orig(3,2)=coy/normco orig(3,3)=coz/normco END D0 D0 I=1,18 D0 J=l,3 DO K=l,3 dettrans(I,J,K)=det(I,K,J) END D0 END D0 END D0 D0 I=l,3 D0 J=1,3 origtrans(I,J)=orig(J,I) END D0 END D0 D0 I=1,18 denom1=det(I,1,3)*det(I,2,2)*det(I,3,1)- det(I,1,2)*det(I,2,3)*det(I,3,1)- det(I,1,3)*det(I,2,l)*det(I,3,2)+ det(I,1,1)*det(I,2,3)*det(I,3,2)+ det(I,1,2)*det(I,2,1)*det(I,3,3)- det(I,1,1)*det(I,2,2)*det(I,3,3) denom2=-det(I,1,3)*det(I,2,2)*det(I,3,1)+ det(I,1,2)*det(I,2,3)*det(I,3,1)+ 124 det(I,1,3)*det(I,2,1)*det(I,3,2)- det(I,1,1)*det(I,2,3)*det(I,3,2)- det(I,1,2)*det(I,2,1)*det(I,3,3)+ UlrbCOIO det(I,1,l)*det(I,2,2)*det(I,3,3) detinv(I,1,1)=(det(I,2,3)*det(I,3,2)-det(I,2,2)*det(I,3,3))/ denom1 detinv(I,1,2)=(det(I,1,3)*det(I,3,2)-det(I,1,2)*det(I,3,3))/ denom2 detinv(I,1,3)=(det(I,1,3)*det(I,2,2)-det(I,1,2)*det(I,2,3))/ denom1 detinvCI,2,1)=(det(I,2,3)*det(I,3,1)-det(I,2,1)*det(I,3,3))/ denom2 detinv(I,2,2)=(det(I,1,3)*det(I,3,1)-det(I,1,1)*det(I,3,3))/ denom1 detinv(I,2,3)=(det(I,1,3)*det(I,2,1)-det(I,1,1)*det(I,2,3))/ denom2 detinv(I,3,1)=(det(I,2,2)*det(I,3,1)-det(I,2,1)*det(I,3,2))/ denom1 detinvCI,3,2)=(det(I,1,2)*det(I,3,1)-det(I,1,1)*det(I,3,2))/ denom2 detinv(I,3,3)=(det(I,1,2)*det(I,2,1)-det(I,1,1)*det(I,2,2))/ denom1 The determinant should be 1 if the normalized vector lengths are conserved. determ(I)=denom2 125 END D0 denomlo=orig(1,3)*orig(2,2)*orig(3,1)- 1 orig(1,2)*orig(2,3)*orig(3,1)- 2 orig(1,3)*orig(2,1)*orig(3,2)+ 3 orig(1,1)*orig(2,3)*orig(3,2)+ 4 orig(1,2)*orig(2,1)*orig(3,3)- 5 orig(1,1)*orig(2,2)*orig(3,3) denom2o=-orig(1,3)*orig(2,2)*orig(3,1)+ 1 orig(1,2)*orig(2,3)*orig(3,1)+ orig(1,3)*orig(2,1)*orig(3,2)- orig(1,1)*orig(2,3)*orig(3,2)- #0310 orig(1,2)*orig(2,1)*orig(3,3)+ 5 orig(1,1)*orig(2,2)*orig(3,3) originv(1,1)=(orig(2,3)*orig(3,2)-orig(2,2)*orig(3,3))/denom10 originv(1,2)=(orig(1,3)*orig(3,2)-orig(1,2)*orig(3,3))/denom2o originv(1,3)=(orig(1,3)*orig(2,2)-orig(1,2)*orig(2,3))/denomlo originv(2,1)=(orig(2,3)*orig(3,1)-orig(2,1)*orig(3,3))/denom2o originv(2,2)=(orig(1,3)*orig(3,1)-orig(1,1)*orig(3,3))/denom10 originv(2,3)=(orig(1,3)*orig(2,1)-orig(1,l)*orig(2,3))/denom2o originv(3,1)=(orig(2,2)*orig(3,1)-orig(2,1)*orig(3,2))/denomlo originv(3,2)=(orig(1,2)*orig(3,1)-orig(1,1)*orig(3,2))/denom2o originv(3,3)=(orig(1,2)*orig(2,1)-orig(1,1)*orig(2,2))/denom10 denom10=origtrans<1,3)*origtrans(2,2)*origtrans(3,1)- 1 origtrans(1,2)*origtrans(2,3)*origtrans(3,1)- 2 origtrans(1,3)*origtrans(2,1)*origtrans(3,2)+ 3 origtransCl,1)*origtrans(2,3)*origtrans(3,2)+ 126 4 origtrans(1,2)*origtrans(2,1)*origtrans(3,3)- 5 origtrans(1,1)*origtrans(2,2)*origtrans(3,3) denom2o=~origtrans(1,3)*origtrans(2,2)*origtrans(3,1)+ 1 origtrans(1,2)*origtrans(2,3)*origtrans(3,1)+ M origtrans(1,3)*origtrans(2,1)*origtrans(3,2)- 3 origtrans(1,1)*origtrans(2,3)*origtrans(3,2)- 4 origtrans(1,2)*origtrans(2,1)*origtrans(3,3)+ 5 origtrans(1,1)*origtrans(2,2)*origtrans(3,3) origtransinv(1,1)=(origtrans(2,3)*origtrans(3,2)-origtrans(2,2)* 1 origtrans(3,3))/denomlo origtransinv<1,2)=(origtrans(1,3)*origtrans(3,2)-origtrans(1,2)* 1 origtrans(3,3))/denom2o origtransinv(1,3)=(origtrans(1,3)*origtrans(2,2)-origtrans(1,2)* 1 origtrans(2,3))/denom1o origtransinv(2,1)=(origtrans(2,3)*origtrans(3,1)-origtrans(2,1)* 1 origtrans(3,3))/denom2o origtransinv(2,2)=(origtrans(1,3)*origtrans(3,1)-origtrans(1,1)* 1 origtrans(3,3))/denom10 origtransinv(2,3)=(origtrans(1,3)*origtrans(2,1)-origtrans(1,1)* 1 origtrans(2,3))/denom2o origtransinv(3,1)=(origtrans(2,2)*origtrans(3,1)-origtrans(2,1)* 1 origtrans(3,2))/denom1o origtransinv(3,2)=(origtrans(1,2)*origtrans(3,1)-origtrans(1,1)* 1 origtrans(3,2))/denom2o origtransinv(3,3)=(origtrans(1,2)*origtrans(2,1)-origtrans(1,1)* 1 origtrans(2,2))/denomlo The determinant should be 1 if the normalized vector lengths 127 are conserved. determo=denom2o D0 I=1,18 D0 J=l,3 D0 K=1,3 rot(I,J,K)=0.0 D0 L=1,3 rot(I,J,K)=rot(I,J,K)+dettrans(I,J,L)* 1 origtransinv(L,K) END DO END DO END D0 END D0 Fixing rounding problems DO I=1,18 D0 J=1,3 D0 K=l,3 IF ((rot(I,J,K).GT.1.0).0R.(rotCI,J,K).LT.-1)) THEN IF (rot(I,J,K).GT.l.0) THEN rot(I,J,K)=1.0 END IF IF (rotCI,J,K).LT.-1) THEN 128 rot(I,J,K)=-1.0 END IF END IF END D0 END D0 END D0 The GEANT angles, (thx, thy, thz, phix, phiy, phiz), are calculated from rottrans(I,J,K) D0 I=1,18 D0 J=l,3 D0 K=l,3 rottrans(I,J,K)=rot(I,K,J) END D0 END D0 END D0 D0 I=1,18 thx(I)=ACOS(rottrans(I,1,3))*180.0/PI thy(I)=ACOS(rottrans(I,2,3))*180.0/PI thz(I)=ACOS(rottrans(I,3,3))*180.0/PI IF (rottrans(I,1,1).GT.0.0) THEN phix(I)=ATAN(rottrans(I,1,2)/rottrans(I,1,1))*180.0/PI ELSE IF (rottrans(I,1,1).LT.0.0) THEN 129 phix(I)=ATAN(rottrans(I,1,2)/rottrans(I,1,1))*180.0/PI+ 180.0 ELSE IF (rottrans(I,1,1).EQ.0.0) THEN phix(I)=90.0 END IF END IF END IF IF (rottrans(I,2,1).GT.0.0) THEN phiyfI)=ATAN(rottrans(I,2,2)/rottrans(I,2,1))*180.0/PI ELSE IF (rottrans(I,2,1).LT.0.0) THEN phiy(I)=ATAN(rottrans(I,2,2)/rottrans(I,2,1))*180.0/PI+ 180.0 ELSE IF (rottrans(I,2,1).EQ.0.0) THEN phiy(1)=90.0 END IF END IF END IF IF (rottrans(I,3,1).GT.0.0) THEN phiz(I)=ATAN(rottrans(I,3,2)/rottrans(I,3,1))*180.0/PI ELSE IF (rottrans(I,3,1).LT.0.0) THEN phiz(I)=ATAN(rottrans(I,3,2)/rottrans(I,3,1))*180.0/PI+ 180.0 130 ELSE IF (rottrans(I,3,1).EQ.0.0) THEN phiz(I)=90.0 END IF END IF END IF The below IF statements are so the final angles are all positive (GEANT requires positive angles) IF (thx(I).LT.O.) THEN thx(I)=thx(I)+360.0 END IF IF (thy(I).LT.O.) THEN thy(I)=thy(I)+360.0 END IF IF (thz(I).LT.O.) THEN thz(I)=thz(I)+360.0 END IF IF (phix(I).LT.0.) THEN phix(I)=phix(I)+360.0 END IF IF (phiy(I).LT.0.) THEN phiy(I)=phiy(I)+360.0 END IF IF (phiz(I).LT.O.) THEN phiz(I)=phiz(I)+360.0 131 END IF END DO Calculate if the rotation matrix GEANT creates from the above calculated angles is orthonormal - all the dot products should be zero 00 I=1,18 v1(I,1)=SIN(thx(I)*PI/180)*COS(phix(I)*PI/180) v1(I,2)=SIN(thx(I)*PI/180)*SIN(phix(I)*PI/180) v1(I,3)=CDS(thx(I)*PI/180) v2(I,1)=SIN(thy(I)*PI/180)*COS(phiy(I)*PI/180) v2(I,2)=SIN(thy(I)*PI/180)*SIN(phiy(I)*PI/180) v2(I,3)=COS(thy(I)*PI/180) v3(I,1)=SIN(thz(I)*PI/180)*COS(phiz(I)*PI/180) v3(I,2)=SIN(thz(I)*PI/180)*SIprhiz(I)*PI/180) v3(I,3)=COS(thz(I)*PI/180) v1v2(I)=v1(I,1)*v2(I,1)+v1(I,2)*v2(I,2)+v1(I,3)*v2(I,3) v1v3(I)=v1(I,1)*v3(I,1)+v1(I,2)*v3(I,2)+v1(I,3)*v3(I,3) v2v3(I)=v2(I,1)*v3(I,1)+v2(I,2)*v3(I,2)+v2(I,3)*v3(I,3) PRINT *, ’v1v2, v1v3, v2v3’, I, v1v2(I), v1v3 and tags) are taken from experimental observables.
segasim0509061902.xml 2005 09 06 19 02 135 Heather Zwahlen 1.0
Random number seeds (2 values). RNDM Two random number seeds should be defined. 2342307421 4572456119 Version of GEANT code VERSIDN This is the version of the SeGA GEANT code that will be used for this simulation. 2 Filename for SeGA segment position SEGPOS 136 The valid angle files should have a .xml or .ang extenstion. segang041105.xm1 Run mode for GEANT MODE i -- interactive mode b -- batch mode b Number of events to emit TRIG This key only works for batch mode. 1000OOOO Show events tracks in display window. SHTRACKS 137 If =1, show event tracks. This is useful only in interactive mode. O Detector setup option (2 values). SETOPT v1=0 v2= : Very simple cylinder of Ge (For testing only) v1=1 v2=0: 1 SeGA det, source is at DISTANCE from front. v1=1 v2=1: 1 SeGA det, source is at DISTANCE from side. ROTATE spins detector around cylindrical axis v1=1 v2=2: 2 SeGA dets at DISTANCE from source in middle v1=1 v2=3: 1 ring of 7 SeGA dets. (DISTANCE adjusts radius) v1=1 v2=4: 2 rings of 7 SeGA dets. (DISTANCE adjusts radius) v1=1 v2=5: 1 ring of 4 SeGA dets. (DISTANCE adjusts radius) v1=1 v2=6: 1 ring of 6 SeGA dets at DISTANCE from source, and steel plate at the end of detectors (ANL setup) v1=1 v2=7: 18 detector SeGA array (fixed configuration) v1=1 v2=8: 18 detector SeGA array in configuration specified by the SEGPOS filename 1 8 138 Detector availability array (18 values) USE These flags are only valid for SETOPT:(1,7) and (1,8). For SETOPT (1,7). values 1-8 are positions: F8000, F8045, F8090, . . ., F8315 values 9-18 are positions: FAOOO, FA036, FA072, . . ., FA324 For SETOPT (1,8), (numbering 1-6, 9-14, 17-22): 1, 17 ~24deg; 9, 14 ~29deg; 2, 22 ~40deg; 6, 18 ~60deg; 10, 13 ~78deg; 3, 21 ~90deg; 11, 12 ~126deg; 19, 5 ~139deg; 20, 4 ~147deg; The flag definitions are as follows: 0: Remove detector 1: Detector present but not operational 2: Detector present and operational 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8eam pipe size (2 e1ements) PIPSIZ Element 1: Inner radius (in cm) Element 2: Outer radius (in cm) 139 Some sample dimensions: deltaSeGA: inner 4.9149, outer 5.08, A1 classicSeGA: inner 7.366, outer 7.62, A1 LH2SeGA: inner 7.41, outer 7.62, Fe (steel really) 4.9149 5.08 Beam pipe materia1 PIPMAT Material of the beam pipe. Options: 0: No beam pipe 3: Aluminum 5: Iron 3 Distance adjustment (in cm). DISTANCE The key is valid only for certain setups (see SETOPT for details). The distance is always measured in centimeters from the edge of the ge crystal (whether it is from the front or side). 140 0.0 Rotation around detector symmetry axis (in degrees). ROTATE The key is valid only for certain setups (see SETOPT for details) 0.0 Position of target on beam axis (in cm). TARPOS TARPOS = 0.0 is the center of the array. Gamma-rays will be emitted based on this target position. -0.554936 Target material type. 141 TTYPE The medium id number for the target. Sample id’s are: 0 -- no target 1 -- liquid hydrogen 16 -- gold 17 -- lithium 18 -- berylium 19 -- carbon 20 -- bismuth 21 -- polypropelene 22 -- nickel Notes: (a) The no target option also removes any beam lines (b) The Liquid Hydrogen option adds the appropriate beam line mechanics for the target. 18 Target thickness (in mg/cm“2). TTHICK Thickness of the experimental target. 376. 142 Intrinsic germanium energy-resolution parameters (4 values) GERES The intrinsic resolution is definded as: (1-R/100)*exp(-0.5*((x-x0)/siga)**2) + (R/100)*exp((x-x0)/sigma)*erfc((x-x0)/(sqrt(2)*sigma) + 1/sqrt(2)) where sigma = CERES(1)*x0 + GERES(2) R = GERES(3)*x0 + GERES(4) 0.000205 1.31 0.0162 5.19 Intrinsic germanium-segment energy-resolution parameters (2 values) SEGRES The intrinsic resolution is definded as: FWHM = v1*energy + v2, where energy is the detected energy in keV. 0.0007123 2.5 143 Intrinsic silicon energy-resolution parameters (2 values) SIRES The intrinsic resolution is definded as: FWHM = v1*energy + v2, where energy is the detected energy in keV. 0.0013534 0.3973 Intrinsic plastic energy-resolution parameters (2 values) PLARES The intrinsic resolution is definded as: FWHM = v1*energy + v2, where energy is the detected energy in keV. 0.3111111 11.1111 Intrinsic NaI energy-resolution parameters (2 values) 144 NAIRES The intrinsic resolution is definded as: FWHM = v1*energy + v2, where energy is the detected energy in keV. 0.3111111 11.1111 Segment smear standard deviation SEGUNC A value of 1 means 68% of the events take the selected segment. A value of 0.5 means 95% of the events take the selected segment. 0.0 Options for event creation. KINOPT 0 -- Emit PARTICLE with ENERGY in four pi. (Kinematics disabled). 1 -- Emit PARTICLE with ENERGY from 0 to THETA 145 (isotropic in phi). (Kinematics disabled). Emit PARTICLE with ENERGY at THETA and PHI (Kinematics disabled). Emit gamma ray in 4pi with randomw energy between 0 and 10000 (record only photpeak events) (Kinematics disabled). Emit gamma ray from predefined list of energies in 4pi. (record only photpeak events) (Kinematics disabled). Emit gamma rays with ENERGY in four pi. (Kinematics ENABLED). Emit gamma rays in 4pi with random energy between 0 and 10000 (Kinematics ENABLED). Emit gamma rays with ENERGY in a distribution defined by a probability distribution in the file WGFUNC (Kinematics ENABLED) 5 Angle of emission of particle (in degrees) THETA This key is only valid for certain event creation options (see KINEOPT). 90 146 Angle of emission of particle (in degrees) PHI This key is only valid for certain event creation options (see KINEOPT). 90 Options for beta particle emission BETOPT This key is only valid for particle types 2 or 3. 0 -- Emit realistic beat spectrum with Qbeta = ENERGY 1 -- Emit monoenergetic beta of ENERGY. 0 Number of particles emitted in coicidence. NUMPART 147 The maximum number of particles is 10. 1 Particle type array (10 values) PARTICLE The array of particle types to be emitted in coincidence. This array is ignored in some event creation options (see KINEOPT). Some particle types: 1 -- gamma ray. 2 -- positron. 3 -- electron. 1 0 O 0 0 0 0 0 O 0 Emitted energy array (10 values) (in keV). ENERGY Array of energies of emitted particles. This array is ignored in some event creation 148 options (see KINEOPT). 1965. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 File that contains the W(theta) function WGFUNC The file name with a W(theta) function for emitting gamma rays Format of the file should be: qunc1 Theta1(deg) qunc2 Theta2(deg) (repeat for a maxmium of 1800 lines) obalign1002t00.dat Initial velocity of the beam (in v/c). VBEAM This key is only valid when kinematics are enabled 149 in KINEOPT. .39204 Velocity of the beam after the target (in v/c). VEND This key is only valid when kinematics are enabled in KINEOPT. .36374 Velocity spread of the beam incomming beam (in v/c). VUNC Standard deviation of the incomming beam velocity (gaussian distribution). .00302 150 Velocity to use in the Doppler reconstruction (in v/c). VUSE This key is only valid when kinematics are enabled in KINEOPT. 0.3787 Focus option of the beam. FOCUSOPT This key is only valid when kinematics are enabled in KINEOPT. 1 -- Focussed mode BSPOT(1): Spot size sigma (circular distribution). BSPOT(2): Not used in this mode. BSPOT(3): Vertical position of the center of the beam spot BSPOT(4): Horizontal position of the center of the beam spot 2 -- Dispersion-matched mode BSPOT(1): Vertical distribution (isotrOpic with sharp cutoff). BSPOT(2): Horizontal distribution (sigma of gaussian). 151 BSPOT(3): Vertical position of the center of the beam spot BSPOT(4): Horizontal position of the center of the beam spot 1 8eam spot size on the target (4 values) (in cm). BSPOT This key is only valid when kinematics are enabled in KINEOPT. Meaning of values depends on the beam optics (see FOCUSOPT). 0.388 0.0 0.465202 0.381167 Maximum scattering angle of the beam (in degrees). SCATMAX This key is only valid when kinematics are enabled in KINEOPT. This key can be positive or negative. 152 If it is positive then the particle scattering is assumed to be gaussian with its peak at PSCAT and a width WSCAT. If it is negative then the particle scattering is assumed to be lorentzian with the following type of distribution: ang1e**(PSCAT)*exp((WSCAT - angle)/WSCAT) -5.16 The peak of the beam scattering angle (in degrees). PSCAT This key is only valid when kinematics are enabled in KINEOPT. If SCATMAX is positive the following description applies: While the beam scattering angle is naturally peaked at zero, the scattering angle at which gamma-ray emission is peaked is typically not at zero. This situation is approximated by a gaussian peaked at PSCAT with a width of WSCAT. If SCATMAX is negative the following description applies: PSCAT is the exponent in the Lorentz distribution 153 described in SCATMAX and WSCAT is the exponential decay of the Lorentzian. 1.88975 Width of the beam scattering distribution (in degrees). wSCAT This key is only valid when kinematics are enabled in KINEOPT. See PSCAT for more details. 0.523103 Half life of the emitting energy level (in picoseconds). HALFLIFE This key is only valid when kinematics are enabled in KINEOPT. The actual location of gamma-ray emission depends on the beam velocity and the level half life. 0.0 154 Save event increment. OUTPUT The hbook file will be saved after every OUTPUT events. In this way, if GEANT crashes for some reason after near the end of a simulation, you would still have most of your data. 100000 Central contact energy threshold (in keV) CCTHRESH Central contact energy threshold (in keV) below which the event is not recorded. This threshold applies the same to all SeGA detectors. O Gamma-ray angle determination 155 option. GAMANGOPT This key is only valid when kinematics are enabled in KINEOPT. 0 -- Select segment with greatest energy deposited. 1 -- Use the WFM selection procedure. O Uncertainty in the germanium positions (in cm). GEUNC GangUnc is assumed to be the radial standard deviation of the germanium positions. This deviation is applied to each detector at the beginning of the simulation. 0.3 Beam tracking option. TRKOPT This key is only valid when kinematics are enabled 156 in KINEOPT. O -- Take the center of the target for angle reconstruction. 1 -- Take the actual target position with an uncertainty of TRACKUNC. O Uncertainty in the beam position (in cm). TRKUNC This key is only valid when kinematics are enabled in KINEOPT. This is the one sigma uncertainty. 0.0 The target position and beam centroid assumed in the reconstruction (3 values) (in cm). UTARPOS This key is only valid when kinematics are enabled in KINEOPT. 157 UTARPOS(1): Position along the beam line (2) UTARPOS(2): Vertical position (x) UTARPOS(S): Horizontal position (y) -O.554936 0.465202 0.381167 Maximum energy in the gamma-ray histograms (in keV). ENMAX 8000.0 Number of channels for the energy histograms. ENCHAN Note: For KINEOPT 4 the number of channels is forced to int(ENMAX). 8000 158 GEANT defined parameter (do not change). RUNG You should only change this value if you fully understand GEANT. 1 1 GEANT defined parameter (do not change). CUTS You should only change this value if you fully understand GEANT. 0.000010 0.000010 0.000010 0.000010 0.000010 0.000010 0.000010 1.94 1.94 0.000010 1.e10 0. 0. O. 0. 0. GEANT defined parameter (do not change). MUNU 159 You should only change this value if you fully understand GEANT. 0 GEANT defined parameter (do not change). HADR You should only change this value if you fully understand GEANT. 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