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I ' This is to certify that the dissertation entitled HIGH SPIN STATES IN 158Yb AND 128Ce presented by José Luiz de Santana Carvalho has been accepted towards fulfillment of the requirements for Ph.D. degree in Chemistry W2 Hm! J Major iarofessor DateMarch 19 , 1982 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES ” RETURNING'MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. Cr/I/d HIGH SPIN STATES IN 168Yb AND 128Ce By Jose’ Luiz de Santana Carvalho A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry Program in Nuclear Chemistry 1982 ABSTRACT HIGH SPIN STATES IN 168Yb AND 128Ce By Jose” Luiz de Santana Carvalho The high spin states of 168Yb and 128Ce have been studied using in-beam gamma ray spectrosc0py. The A ~ 128 region is regarded to be transitional between prolate-deformed and spherical shape. In such deformed regions, the observed anomalous behavior at large angular momentum values in the yrast sequence (backbending) results from the crossing of the ground state rotational band with an aligned quasi- particle band. For 158Yb, i13,2 neutrons are involved in this phenomenon, and for 128Ce, h11,2 protons. Several two-quasiparticle bands were observed in 168Yb through the 166Er (a,2n) reaction. Of particular interest is the 5' isomer and associated band structure identified to 11', which was determined to be two-quasineutron with some proton character mixed in. The v-vibrational band was extended up to spin 11+, and displayed a per- turbed structure that probably results from mixing with nearby bands. CAUTION This document has not been fiven final patent clearance and the dissemination of its information is only for official use. No release to the public shall be made without the approval of the Law Department of Union Carbide Corporation. Nuclear Di- vision. Jose Luiz de Santana Carvalho The levels in 128Ce were populated via the 112Cd(2°Ne,4n) reaction, and the excitation v-rays were observed through a multi-Ge detector array. The yrast sequence was observed up to spin 24+, and a strong backbend is observed around fiw = 0.31 MeV. A pair of two- quasiparticle bands were determined at 1890 keV and 1989 keV, for the lowest observed level. For the former, a h11,2 two-quasiproton band structure has been tentatively assigned. Both isotOpes were used to test the applicability of the Cranking Shell Model in the different deformed regions. The behavior of the yrast band was generally well predicted, while for the side- bands difficulties were encountered in trying to describe the amount of relative alignment as a function of angular momentum. DEDICATION To my wife, Suely ("pretinha"), to my daughter, Gabriella, and to my parents, Alexandrina, Santana, Daisy, and Castor. ii ACKNOWLEDGMENTS It is with great pleasure that I thank Fred Bernthal for his patience and guidance as my advisor, and his interest in me as a person. I would also like to thank George Leroi for being my "substitute" adviser in the last few months and actually reading this dissertation! Several people were also important factors in the preparation of this work, and I wish to thank them: Phil Walker, Noah Johnson, Jorma Hattula, Lee Riedinger, and I. Y. Lee. They devoted much time and effort to many discussions about the theory and the data, as well as to the experiments performed. Words are not enough to express my thankfulness and gratitude towards them. I would like to thank Reg Ronningen, Wayne Bentley and Steve Faber for their cooperation in the experiments at Michigan State. At Oak Ridge, Heinz Ower, Matthew Fewell, Bill Atkins, and Bill Milner are warmly acknowledged for their help in software development and experimental setup. I am grateful for the financial assistance during my graduate studies by the National Science Foundation (National Superconducting Cyclotron Laboratory) and the Chemistry Department. The Coordenacao iii iv de Aperfeicoamento de Pessoal de Nivel Superior (Brazil) is also acknowledged for its financial assistance in the latter part of my studies. I would like to express my gratitude to the Oak Ridge National Laboratory through Paul Stelson and Ed Gross for the use of the ORIC facilities and the financial support for the typing of this dissertation. I also would like to thank Ginny Hill, Chris Wallace, Lynda Hawkins, Mary Sue Long, and Betty McHargue for patiently typing this thesis with a deadline in mind and sight. A special thanks goes to Ginny for her kindness and genuine concern about my family and me. Finally, I take much pleasure in thanking my wife for being so patient and loving, and for giving me another reason to push forward, our daughter Gabriella. TABLE OF CONTENTS 3393 LIST OF TABLES vii LIST OF FIGURES viii CHAPTER I INTRODUCTION....... ........... . ........ ............... 1 CHAPTER II THEORETICAL REFLECTIONS............................... 5 2.1 Introduction..................................... 5 2.2 The Deformed Shell Model......................... 7 2.3 The Cranking Shell Model.........................12 2.3.1 The Cranking Shell Model Hamiltonian......12 2.3.2 Pairing Correlations and Quasiparticles...12 2.3.3 The Quasiparticle Hamiltonian and Its Eigenvalue Equation...................l3 2.3.4 Important Quantities and Properties.......17 2.3.5 The Experimental Routhians and Related quantitTESeeeeeeeaeoeeeeoee000000.21 2.3.6 Interaction and Crossing of Bands.........25 CHAPTER III YTTERBIUM-168 EXPERIMENTAL DETERMINATIONS.............28 3.1 Gamma-Gamma Coincidences.........................28 3.2 Gamma-Gamma-Time Spectra.........................44 3.3 Angular-Distribution Measurements................54 5 CHAPTER IV CHAPTER V CHAPTER VI CHAPTER VIII APPENDIX REFERENCES YTTERBIUM-168 EXPERIMENTAL RESULTS....................56 4.1 The Ground State Rotational Band.................56 4.2 The y-Vibrational Band...........................61 4.3 The K"=3' Band...................................67 4.4 The K"=5’ Band...................................74 4.5 The Cranking Shell Model Calculations............80 CERIUM-128 EXPERIMENTAL DETERMINATIONS................93 5.1 Gamma-Gamma Coincidences.........................93 5.2 Angu1ar DistributionSoeeooeeeeeoea00000000000000106 CERIUM-128 EXPERIMENTAL RESULTS......................117 6.1 The Yrast Band..................................117 6.2 The Sideband 1..................................139 6.3 The Sideband 2..................................141 6.4 The Cranking Shell Model and the Observed Bands in IZBCEOOOOOOOOOOOO.0.00.00.00.0143 CONCLUDING REMARKSOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0.0.0.153 Some Important Gated Coincidence Spectra 0f TranSitionS in 168Yb.OOOOOOOOOOOOOOOOO00.00.00.300155 C.OOOOOOOOOOOOOO0.0...0..0..0...0.0.0.000000000000000175 vi 3.1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 LIST OF TABLES Page Energies (EY), Intensities (IY), Angular Distribution . Coefficients and Spin Assignments for Electromagnetic TranSitionS in 168Yb ..OOOOOOOOOOOOOOOOOOOOO0.00.00.00.000033 Angular distribution coefficients for the ground State rotational band OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOIO0.0.0.57 List of experimental and theoretical B(E2) ratios for I. + I transitions between the y-band and the ground siate fictationa] band eeeaeoeeeeoeeeaeaeeaeoeoeeoeeeaooaose.65 Mixing ratios and (g - ) values for K" = 3+ band in 168Yb using the planEr Detector intensity results ..........69 Calculated JbgK-QRI values for the Klr = 5' isomeric band of 158 OCOOOOOOOOOOOOOOOOO..OOOOOOOOOOOOOOOCO0.0.0.0.78 Experimental Routhian e' values for the 5' isomer band in 168Yb at different rotational frequencies (see Figure 4.7) compared to summed Routhians .from 16 Yb 0000......0.0.0....OOOOOOOOOIOOOOOOOOOOO..0.0.0.085 Experimental i values for the 5‘ isomer band in 168Yb at different rotational frequencies (see Figure 4.8) compared to summed values for 167Yb ........................89 Comparison between ex erimental and theoretical . calculations for Rout ian e' (in MeV) as a function ogarotational frequency hm for the 5' isomer in Yb0.00.0...0..0........OOOOOOOOOOOOOOO0.00.00.00.00...0.090 Comparison between experimental and theoretical calculations for alignment gain i as a function of rotational frequency no for the 5' isomer in 168Yb..........91 Final reaction products observed in the bombardment Of IIZCd With a 103.2-Mev zoNe beam OOOOOOOOOOOOOOOOOOOOO0.101 Corrected experimental Azls and A4's for observed tranSitionS in lzace....0...OOOOOOOOOOOOOCOOOOOOOO000......114 vii 2.1 2.2 2.3 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 LIST OF FIGURES Page Nilsson level diagram for 168Yb using eZ=O.255 and 62:0.11...OOOOOOOOOCOOO...00.00.000.00...OOOOOOOOOOI...00.0.11 Theoretical Routhian e' versuS‘hw diagram for 168Yb.........18 BaCkbending p10t for 160Yb..C.....OOOOCOOOOOOCOOCOOCOOO0.0.026 Experimental setup for the 166Er(a,2n)153Yb reaction........3O Projected total events spectrum for the large Ge(Li) detector in the 168Yb coincidence experiment................32 Leve] SCheme for 168Yb...0.O.....OCCOOOOOOOOCOOCCCOOOOO00.0.45 TAC spectrum for the 168Yb coincidence experiments..........46 "Early" TAC time slices for the 168Yb timing experiments....47 "Delayed" TAC time time slices for the 168Yb timing eXPerimentSOC0.00.00.00.00.COO...OCCOOOOOOOOOCOOCOOOOOOO0.0.48 Radioactive_"growth" curve for the 111-keV transition 1" the K =5 isomer band in 168Yb00..OOOOOOOOOOOOOOOOOOO0.0.50 Radioactive :degay“ curve for the 229-keV transition OUt of the K =5 isomer bandhead...OOOOOOOOOOOOOOO0.0.0.000050 Time spectra for some important A£=1 transitions for the K =5 isomer in 168Yb00000......OOOOOOOOOOOOOOOOOOOOOOO.51 Time_spectra for some important transitions out of the K =5 isomer bandhead in 168Yb. The decay slope in the "positive" time region is equal to t“2 of the isomer.......52 P10t Of Ix versus‘fiw for 168Yb.OOOOOOOIOOOOOOOOOOOOOOO0.0.0.58 Plot of AI/Aw versus‘hzw2 for the g-band in 168Yb...........60 Plot of AE/ZI versus 12 for the y-band in 168Yb.............66 viii 4.4 4.5 4.6 4.7 4.8 4.9 5.1 5.2 5.3 5.4 5.5 5.6 5.7 K“ = 3* levels in the A ~ 170 region. The thick horizontal bars represent the experimental I = 4 levels; below each is given its energy in keV, and above is given the cross section at 125° (in ub/sr) from 12 MeV deuteron inelastic scattering. Calculated two-quasiparticle energies are shown by the thin horizontal bars, for the {7/2+ [404] 1/2+[411] } (p1), {5/2+[402] , 1/2+[4113 } (p2). {5/2'E5123n, 1/2- 5121"} (n1) and {7/2'[514]n, 1/2'[521]n} (n2) singlet configurations....72 Plot of 2 U/hz VS.‘h2w2 for the 5' isomer and the ground state rotational bands for 168Yb; the 4 isomer for 17°Yb and the 6' isomer for 172Yb are also plotted..........76 Plot of E' (level energies in the rotating frame) versus ‘fiw for some of the quasiparticle bands related to the observed quasineutron bands in 168Yb........................83 Experimental Routhian e' versus hm for observed quaSineutron bands in 168Yb...OOOOOOOOOOOOOOOOOO0.0.0.00000084 Plot of Ix versu5‘fim for the observed quasineutron bands in 168Yb000.00.00.00.00.O..0...OOOOOOOOOOOOOOOOOOOOO.087 Plot of i versus‘fiw for the observed quasineutron bands in 168Yb00OGOO...O...O0.00..OOOOOOOOOOOOOOOOOOOOOOO.0088 Calculated excitation functions for the bombardment of 112Cd With ZONE ”Sing ALICEO0.00.00000000000000000000000.0.094 Positioning of the Ge and NaI detectors for the bombardment of 112Cd with 20Ne..............................96 Block diagram of the experimental setup for the bombardment Of 112Cd With ZONeOOOOOOOOOOOOOOOOOO0.0...00.00.98 Block diagram for the data transfer from the experimental area to the Perkin-Elmer 3220 computer.........99 Total (7 ra 3 energy spectrum for the bombardment of 112de1th Ne.00.00.0000...0......0.0.0.0....000000000000102 The 2++0+ gate spectrum for 128Ce obtained in the 7-7 COinCidence experiment.............O..........C............105 "Singles" spectrum of 0° for the bombardment of 112Cd With ZONECCOOOOC..OOOOCOCOCOCOCOC00.00.0000....000000000000108 ix 5.8 5.9 5.10 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 Page Angular distribution curves for several ground state rotational band transitions in 128Ce........................111 Theoretical angular distribution using an average of several theoretical E2 transitions.......................113 Angular distribution curve for some At = 1,2 transitions in CEOOOOOCOOOO0....CO...OI.OOOOOOCIOOOOOOOOOOOOOOO0.0.0.115 Leve] SCheme for128C€eeeeaeeeeeooeeeeeeeeeoeeeeeeeeee000000118 Schematic diagram for one possible deca path for the continuum into the yrast line for the 1 2Cd(2°Ne,4n)128Ce react-ion..0.0.0.0.0.....00...OOOOOOCOOOOOO0.0.0.0000...0.00.120 Total coincidence spectrum for the bombardment of 112Cd with 2°Ne uSTng a 4-particle total energy gate..............121 The 20++18+ and 18++16+ transitions observed in some gated Spectra inlzece...C.....C.................0..........122 "Sum spectrum" for the 128Ce coincidence experiment.........124 Expanded portion of the "Sum spectrum" showing the new yrast transitions (marked with an asterisk).................125 Intensity for the yrast transitions versus I" from the "sum Spectrum..............I...OOOOOOOOOOOOOCOOOOCOO0.0.0126 The 20++18+ and the 18++16+ gated spectra from the 128Ce coincidence experiment................................127 Angular distribution for the 18316+ yrast transition.......128 Spectrum gated on the 12++10+ transition in 128Ce...........13O Different energy slices of a muliplet in the total coincidence spectrum for the 128Ce coincidence experiment...131 Plot Of ijz versus flzwz for128-13“C60000000000000000000132 Plot of E} versus I(I+1) for the g.s.b. of 128'134c9........134 Plot Of Ix versusm for 128-13uC60eeoooeeoeoooe000.000.000.135 Schematic drawing for coupling and decoupling of a pair 0f h11/2 nucleons in time-reversed orbits............137 6.16 6.17 6.18 6.19 6.20 6.21 6.22 A1-A12 Page Nilsson level diagram for 128Ce using ez=0.260 and ek=0000 daformation.C...CCOOOOOOOO..COOOOOCCCCOOOOO0....00.138 Plot of E versus I(I+1) for sideband 1 in 128Ce...........142 I Experimental Routhian e' versu5‘hw for 128Ce...............145 Experimental i versus fiw for 128Ce. Also, 13°Ce side— bands are p1otted.00.0.0.0...0.00.0.0.....CCOCCOOOOCCOC0.0.146 Theoretical Routhian e' versus fiw diagram for protons in CECCOOOOOOIOOOOOOOOO...O.....OOOOOOOOCOOOOOOOOOOO0.0.148 Theoretical Routhian e' versu5‘ho diagram for neutrons in CEOOOOOOOOOOOOOO....0....0.000000000000000000000000.. 49 Schematic diagram for two possible excitations for SidEband 1 in 128CEOO.O.0....000......OOOOOOOOOOOOOOOOOO..0151 Sidefeeding transitions between the v- and the ground State bands.00......00......OOOOOOOOOOOOOOOOOOOOOO0.0.0....156 A13-A19 Intraband transitions for the K"=3+ and the K"=5' bands....168 xi CHAPTER I INTRODUCTION The main goal of experimental nuclear spectroscopy is to construct the intrinsic, vibrational and rotational spectra of nuclei and relate them to the dynamics of the nucleus according to current ideas and theories. Accordingly, the purpose of this study was to extend any of the previously observed bands, and identify and characterize any addi- tional multiparticle structures of two nuclei, 168Yb and 1230e. The deformed rare-earth 168Yb isotope is a slightly neutron- deficient nucleus whose ground state rotational band has been exten- sively studied, theoretically and experimentally. Yet not nuch is known about the side-bands of this nucleus as well as of the other neighboring isotopes. With that in mind, a series of studies on several Yb isotopes was started at Michigan State University. These experiments were performed at the M.S.U. sector-focused isochronous cyclotron running in the N=2 harmonic mode for alpha-particle beams in the 25-50 MeV energy range. The investigative technique, in-beam v-ray Spectroscopy, was used because of the detailed information it yields on the intra- and interband transitions, and therefore on the association of the observed bands with certain Nilsson (deformed orbitals) levels and related nuclear behavior. Most of the computer codes used in this work were developed by previous students. Especially useful codes were the three-parameter event recording and sorting programs for off-line analysis. The high resolution detectors used were Ge(Li) and Ge (intrinsic) and were of reasonable timing quality and efficiency. The Yb experiments were performed during the last year of cyclotron operation before the construction of a new superconducting cyclotron (K=500) was initiated. When this work was initiated, not much was known about the yrast band and nothing about the sidebands of the highly neutron-deficient 128Ce. Therefore, a series of studies was started at Oak Ridge National Laboratory to investigate the lighter Ce isotopes through detailed in-beam y-ray spectroscopy. The experiments performed are representative of a new trend in experimental nuclear spectrosc0py where the reaction chamber is surrounded by a battery of Ge(Li) and Ge detectors. Since (Heavy Ion, xn) reactions in this region involve high multiplicity events, such multi-Ge detector arrangement is useful in increasing the data collection efficiency. This is especially true when the data acquisition rate of the system as a whole has an upper limit set by the software of the computer. By using large total y-ray energy filters it is possible to greatly decrease the intensity of the unwanted channels in the data, as well as the contribution to the background from Coulomb excitations. This is necessary because in this deformed region, unfortunately, the compound nucleus reactions produce many reaction channels due to unfavorable balance between pro- tons and neutrons. In the Ce experiments 7 Ge(Li) detectors and 2 Na(I) detectors (acting as a total y-ray energy filter) were used, and coincidence and angular distribution results could be obtained in one comprehensive experiment. Exhaustive and long analysis is still necessary to sort out the data. For these experiments, the Oak Ridge Isochronous Cyclotron was used at one of its lowest energy settings for 20Ne beams. The data taking was done with three different computers that operated the basic electronics, the CAMAC crate and the event-recording programs. Especially useful were the Perkin-Elmer 3220 computer and the assorted peripherals that enable close monitoring of the experiment. Some of the computer codes used were developed for these experiments, and since then software improvements have been made to make the codes more flexible and powerful. Much information was obtained for both nuclei studied here, and many new states were sorted out from the data and placed into single- particle or collective bands. This work has also revealed additional experiments that would be interesting in order to obtain more infor- mation about other side bands for both nuclei. Some of the theoreti- cal calculations performed involved the Cranking Shell Model. This model has extended the concepts involved in the Nilsson deformed potential to include the rotation of the nucleus, and how this rota- tion affects the nuclear orbitals in order to explain the behavior of the nucleus at high excitation energy. 4 Since 1979 more intense theoretical work has been done with this model, and its success has been quite remarkable, especially in the rare-earth deformed region, in explaining many features of single- particle and rotational spectra. But very few experimental calcula- tions had been made in the A ~ 130 region. In this dissertation, however, the Cranking Shell Model is used to explain some features of the observed bands in 128Ce. The general organization of this dissertation will consist of: 1) a discussion of the theory as it relates to deformed nuclei; 2) experimental determinations for each nucleus; and 3) a discussion of the results and how the data fit to the band systematics and struc- ture of the region. CHAPTER II THEORETICAL REFLECTIONS 2.1 Introduction A better understanding of nuclear structure requires appropriate mathematical treatments of a simplified mechanical system. Therefore, our concept of the nucleus evolves as the properties of different models emphasize many of the characteristics of the nucleus [In69]. In the mid-1930's N. Bohr compared the nucleus to a liquid drop [8036], giving rise to the idea that collective types of motion would correspond to fundamental modes of excitation. An apparently contra- dictory approach introduced by M. Mayer [Ma49] and others [Ha49] was the nuclear shell model (for several previous attempts concerning the shell model approach, see [Be36], [Be37], and [L137]) that brought the idea that nuclear orbitals could be described in terms of the motions of individual nucleons in an average force field (this potential represents the interaction force between a nucleon and all others collectively within the nuclear volume). These two models approached different characteristics of nuclear structure and dynamics, namely, nucleonic collectivity and individuality, but J. Rainwater [R350] was able to combine these two 5 ideas by suggesting that a nonspherical nuclear shape would arise from the single-particle motion in anisotropic orbitals by including the nuclear deformation as in the liquid-drop model. The mathematical treatment of this unified model was given by A. Bohr and B. Mottelson [8052], [8053], who were then able to explain simple rotational spectra and many collective properties of the nucleus. With this unified model as a starting point, S. Nilsson [Ni55] set out to describe the behavior of the intrinsic (nucleonic) states under increased deformation of the nuclear potential, and the single-nucleon orbitals obtained, along with some refinements [Gu67], are widely used today in classifying intrinsic spectra. At about the same time, D. Inglis [In54] suggested that the nuclear moment of inertia could be taken as the summation of the iner— tial effect of each particle in a rotating deformed potential, there- fore giving the appearance that the potential is externally "cranked". A few years later, Thouless and Valatin [Th62] further enhanced this idea by using time-dependent Hartree-Fock equations to obtain an approximation of the self-consistent behavior of a many-body system. Therefore, particle motion could be thought of in terms of a rotating frame of reference (timeedependent), as opposed to a stationary frame of reference (time-independent). Although this self-consistent cranking shell model, also called Hartree-Fock-Bogoljubov Cranking Shell Model (HFBC model), includes the pairing effect (the effect of coupling particles with resulting zero angular momentum), these calcu- lations are very difficult and time-consuming [Che]. In the mid-1970's, with a wealth of new information about the high-spin states in a wide mass region, it became evident that rota- tional spectra could be understood in terms of configurations of quasiparticles (a mixture of particles and holes; a more detailed description can be found in Chapter II, Section 3.2) in a rotating potential [8e77], giving origin to a simplified Cranking Shell Model [8e79b]. These simplifications are mainly concerned with realistic values for nuclear shape and nucleon pairing. The Cranking Shell Model has been successfully applied in the rare-earth region [Ri80] and is just beginning to be tested in the region far from p—stability [Nol]. 2.2 The Deformed Shell Model The Hamiltonian for an ensemble of nucleons moving in a deformed potential can be written as H = Hintr(qsp) + Hrot,a(Pw) 9 (2'2°1) which corresponds to intrinsic and collective motions. It also indicates that these two types of motions are separable [8075]. The intrinsic motion can be described by the body-fixed coordinates q and conjugate momenta p, while the rotational Hamiltonian depends on the conjugate angular momenta Pm (w is the angular variable) and may depend on the intrinsic state quantum number a. For the case of a particle strongly coupled to a rotating core, Equation (2.2.1) can be rearranged to(variables are omitted): H = Hcol + Hcoup + Hint (2'2'2) where the first and last terms refer to collective and individual par- ticle (intrinsic) motions, respectively, while the second term (coupling) contains the interaction between the particle and rota- tional motions. The collective Hamiltonian can be described as 2 _'h a2 .2 and the coupling Hamiltonian as Hcoup = =3;- (1+j- + I-j+)- (2.2.4) The moment of inertia of the nucleus is defined by CI while I is the total angular momentum operator. The intrinsic Hamiltonian, i.e., the Nilsson Hamiltonian, yields = °A" 62" A2 00 Hint Ho + 2Kfiw[£ s u(2 (2 >)] , (2 2 5) where H0 is the deformed harmonic oscillator [NiSS] such that, at zero deformation, it corresponds to a spherical harmonic oscillator. The potential parameters K and u are obtained by empirically adjusting Hint to the available data on the intrinsic spectra of the deformed nuclei [8075]. The 12 term splits the degeneracy in each major oscillator shell (N) by favoring large n values; therefore, in the N=6 shell, the li-subshell is energetically favored over the Zg-subshell. The t-s term further splits the subshell into j=gis degenerate orbits; hence, the 1i-subshell is separated into 1i11,2 and 1i13,2 orbits. With increased deformation, the orbits (like 1i13,2) display a degeneracy of (2j+1)/2 orbitals. For a nucleon in an orbital, the projection of its angular momentum on the symmetry axis of the nucleus can have the values 9=j,j-1,...,1/2. For prolate deformation, the low n-values are energetically favored. The (22> term is a constant for each (spherical) oscillator shell so that the average energy dif- ference between the oscillator shells is not affected by the £2 term. The eigenstate of the Hamiltonian is a product wave function of the type VIIKM = 49am ¢aIKM(‘”)' (2.2.6) on where, for each intrinsic state a, a sequence of rotational levels arise. The rotational wave function can be further normalized in the body-fixed coordinates (intrinsic coordinates) specified by the three‘ angular momentum quantum numbers I (total angular momentum), M (projection of I on a space-fixed axis), and K (total angular momentum projection on the symmetry axis): ¢IKM(‘”) g ($1,1’29M) . (2.2.7) where @MK are the rotation matrices [8075]. The intrinsic Hamiltonian must be invariant with respect to a 180° rotation about the x axis; 10 99x $110199 ="l’IKM (2'2°8) The final product wave function is [8069]; = H21+1 1/2 0.)) I+K w— I— (.0 WIKM( 161,2) [‘pK ”’9'; H + ('1) ‘PK (“@1115 l]. (2.2.,9) where K implies negative values of K (time-reversed orbits). By using the intrinsic Hamiltonian, H. on the product wave nt’ function, a set of eigenstates having energies which are dependent on the nuclear deformation can be determined. Each eigenstate (Nilsson level) is labelled with the Q[anA] quantum numbers where n is the intrinsic angular momentum projection on the symmetry axis. Along the symmetry axis, the label 112 represents the number of nodal planes, and A is the projection of the intrinsic orbital angular momentum. A Nilsson level diagram is shown in Figure 2.1 for neutron and proton single-particle deformed orbitals at 22 = 0.255 and e“ = 0.110 deformation(calculated according to the Strutinsky method [N169] and the systematics of the mass region). The collective Hamiltonian, Hcol’ yields the rotational band energies, Erot=%[1(1+1)-K2] . (2.2.10) and the c0upl1ng Ham1lton1an, Hcoup’ y1elds - 2 _ . ._ Emu" - g5 HITKI)(IJ"'<1+1)]U2[(J¢QI)(3”“91HHU2 5K1.K2215o1.92:1f (2.2.11) where f is a pairing reduction factor. 11 mcobamc .fifi.ou~a we. www.cuua anew: n>om_ so» Eucmmwv pm>mp commpwz H.~ assume mcogoa at. mm «.m. We. 92 5.9 33S WES. :3. S 0.33. mag ll)“ 33 S 2m. 3. ma. S. Ea S :5 S 85 Q— $an WES =33 Ea S :3. N: was S a... Na we. 3 :2... N: 0K lllllllllll U da 3 t'\. m m (Mu/3) AEms; apnmd 81605 6,: u: to 12 2.3 The Cranking Shell Model 2.3.1 The Cranking Shell Model Hamiltonian In the cranking model [In54] the average nuclear potential is considered to rotate at a constant frequency, w, about the x axis (rotational axis). It was suggested again [8077], [8e77] in 1977 that it was possible to calculate excitation eigenvalues (excited states in the rotational spectrum) through the appropriate cranking Hamiltonian by including a frequency dependence: H' = H - mix , (2.3-1) where H is the deformed shell model Hamiltonian and 5x yields the total angular momentum projection on the rotational axis. H' is also called the Routhian Hamiltonian, since it refers classically [R005] to the rotating frame of reference. In the analysis, the transformation from the body-fixed to rotating frame of reference allows us to calculate excitation energies (relative to the ground state) involved with valence par- ticles and holes. This leads to the discussion of the concept of quasiparticles. 2.3.2 Pairing Correlations and Quasiparticles The description of pairing correlations in a superconductor by Bardeen, CoOper and Schreiffer [8a57a], [8a57b] can be used for other many-body correlations. As pointed out by Migdal [Mi59] and Belyaev 13 [8e59], the nuclear pairing effect can be treated in the context of the 805 domain. In this model, a pair of nucleons in time-reversed deformed orbits interacts with each other through the pairing force. In con- sidering the valence nucleons, the resulting ground state of the nucleus is lowered in energy due to such pairing force. Also, the excited states nay be characterized as a mixture of unoccupied orbits (holes) and occupied orbits (particles). One may refer collectively to the holes and particles as quasiparticles, and the related excita- tion as the quasiparticle excitation [8069]. Then, the ground state of an even-even nucleus nay be thought of as a zero-quasiparticle state or the quasiparticle vacuum, and the low-lying states of odd-A nuclei as single quasiparticle states. 2.3.3 The Quasiparticle Hamiltonian and Its Eigenvalue Equation' The importance of pairing correlations leads to a Hartree-Eock- Bogoljubov (HFB) approach to the Routhian Hamiltonian H' (the HFB cranking approach), where quasi-particle equations in the rotating system are obtained. Therefore, the quasiparticle HFBC Routhian Hamiltonian can be written as: “aa=h3m.+MN+P)-m moi) where the single-particle Routhian Hamiltonian [Ne76] is h's.p. = hSopo (E) " (03x 0 (20303) 14 The single-particle Hamiltonian, hs.p.(€)a is the modified har- monic oscillator [An76] with a dependence on the deformation parameter (£2,6u), and 3x operation yields the single-particle angular momentum projection on the x axis. The P1 and P terms are the annhilation and creation operators for the pair field, and A is the gap parameter relating paired and unpaired ground states. The A term is a Lagrange multiplier (also called the chemical potential or Fermi surface), and N is the particle number. A more convenient representation of h'q.p. can be obtained by expanding the terms of Equation (2.3.2) into I: AT A A1. at A A . = o- 600 - +1 A +c C h q.p. 2:“‘31 A) 11' “3x . Jciaci'a (M) Z (ciacia ——) ’ 10 (111' 101,1 (1 10. 10 (2.3.4) where the particle motion in the cranked (deformed) potential is described in the first term while the second term describes the scat- tering of particles into holes and vice versa (quasiparticles) under the pairing field [Fr81a]. The term e1 stands for the single-particle energy, while the ET and 2 represent the quasiparticle creation and annhilation operators. The label i stands for a set of particle quan- tum numbers (npj) and'TE indicates time-reversed ia. The label a is a symmetry quantum number that depends on the particle number (it is also called the signature of the quasiparticle state). 15 A state wave function of the Bardeen, Cooper and Schrieffer (BCS) type can be used to describe the lowest-energy quasiparticle state as well as other states. The wave function BCS> does not describe a state with a fixed number of nucleons but rather it is an admixture of states with different numbers of nucleons. This is achieved by assuming that the pair state (pa,§5) is occupied with probability Via, and is empty with probability Uia. To find this energy minimum, the mean particle number has to be equal to N, namely, <8CS | N I BCS> = N where N = :61 6. . .ia 10 1G Therefore, by adjusting Vua and Una, the expectation value of l (BCSI hq p | BCS> is minimized. In introducing the quasiparticle operators (Bogoljubov transformation), 10 101 6:, =Z(U‘,‘ 21+ v‘f E, ) , (2.3.5) 1. 101 101 h; p becomes diagonalized, and Equation (2.3.4) is simplified to hq.p. = (1/2):E:E;ab:abua+(ll2) 2E:(ei-A-ijia 1.0), (2.3.6) no 10 : representing a system of independent quasiparticles. Note that the Bogoljubov transformation is unitary. The set of operators bud has twice the dimension of the single-particle space. Therefore, a new set of quasiparticle operators, av“, is restricted to the appropriate dimension. These new operators obey the commutation relations. The quasiparticle vacuum can now be defined by 16 3,, BCS >vac = 0 (2.3.7) Any other state is an excitation with reSpect to the quasiparticle vacuum and'can be described by the operators 3:“. Then the quasipar- ticle Routhian Hamiltonian, Equation (2.3.6), can be further transformed to h' =ZE' 5+5 +Z(e°-A-w3 ) -(1/2)ZE' . (2.3.8) q.p. va va va * 1 Xia,ia va W1 W1 V0. The last term is a constant that describes a reference state like the quasiparticle vacuum (the ground state). The quasiparticle spectrum is built by states of the type 5" |Bcs> , 5* 3". “309 , V0 V3 V0 VG Va C C also called one-, two-, ... quasiparticle states. Accordingly, the energy of a quasiparticle configuration will be the sum of the l 1 q.p. corresponds to an yrast state and the yrast line is the sequence of individual quasiparticle energies. The lowest eigenstate of h these yrast states with increasing w. By using Equation (2.3.6) accordingly, some of the parameters described above can be explicitly evaluated [0971], [Pr75]: 2 ”:01 = (1/2) (1 a 3.1.121) (2.3.9) via EQOpo 1 An yrast state of a nucleus is the level of least energy at a given angular momentum; Grover [Gr67] suggested the term because there is no graceful English language equivalent in expressing the adjective for rotation in its superlative form. The original Swedish adjec- tive is "yr", meaning "dizzy", and its superlative ”yrast" can be literally translated as “dizziest”. 17 where Ea p. = [(es.p.-A)2 + A2)]1/2 , (2.3.10) and A can be calculated as a first approximation by [Ni69] 12 A =AT72 MeV (2.3.11) Also, the systematic mass difference between even and odd neighboring isotopes (associated with the pairing field) can be used to calculate A [8075]. The mean number of particles, N, can be calculated by N =2” _ E5.13.4) , (2.3.12) ii' q.p. The above calculations are time-consuming and long, since most of them have to be repeated for different m, hence, computer codes are used as described in the next section. 2.3.4 Important anntities and Properties In order to better understand the important quantities and prop- erties associated with the Cranking Shell (CS) model, a quasiparticle Routhian diagram is drawn in Figure 2.2 for 168Yb. The quasiparticle Routhians were calculated by solving the quasiparticle equations discussed in the previous section. A computer code, referred to here as the "CSM Code" [C578] was used to obtain numerical solutions for the quasiparticle Routhians, e', as a function of hm. Also included in the figure are the Nilsson quantum numbers [N155] at hm = 0 MeV as additional labels. The a deformation parameters were calculated 2’e4 18 O N o —“ - ._._.__-_.—s_—: ‘ — - / NN~~ ’ I -q 50(512]—+.-—~\ /’ \\ snag—o “*‘K— ___ __ __ __ flF_/___ ... 5052 §§‘\ %\:‘\~~. I, WEI)? \~§\-\~:“\\ \~$‘~ 0‘ ‘ "\‘\§_"\ h"-'—-—--— \ \- \\ -\.\ \ \ \ —“‘ \_\$ 8 ‘\\_ ’ ~ ———— o \ ,‘K‘ \ \ d A \ -B/ \N \ e’/f1 0.1. 0 i \ \ \ / / /’ / ’ ~‘ I I ~_’ I / / / O / ' 0 B / A / /‘/ c1 - / ' ' / / ./ - / I, s a -A , é- / v ’a" o ’ é \ I is the angular momentum "aligned" along the x-axis. In keeping w fixed in the calculations, the constraint = Ix (2.3.18) is fulfilled [Be79a]. By using a reference state (such as the quasi- particle vaccum), the (relative) aligned angular momentum, i, will be 1(0)) = IX(W) - Ix,rEf(w) 0 (203019) The gain in alignment for an excited state composed of several quasiparticles can now be simply stated as to be equal to the sum of the individual valence's angular momentum. 21 Some important properties are: i) Additivity: By using a reference state like the quasiparticle vacuum, all quasiparticle quantities are additive, e.g., excitation energy, angu- lar momentum and signature. ii) Conjugate orbits: If a quasiparticle orbit is occupied, its conjugate partner is unoccupied. This is pointed out pictorially in Figure 2.2 as the orbits -A and A, respectively. iii) The reference configuration: In order to use relative quantities, a reference configuration is needed. The reference configuration, also called the g-configuration if the ground state is used, corresponds (in Figure 2.2) to having all levels below zero occupied at low w, and therefore the vacuum and g- configurations are identical [8e79a]. iv) The parity of an excited configuration is the product of the parities of the quasiparticle configurations involved in the excitation. Also, the g-configuration of an even-even nucleus has fl = +, a pr0perty of the parity operator [Fr81b]. 2.3.5 The Experimental Routhians and Related Quantities One of the keys of the CS model is the additivity of energy and other quantities for the excited states in the quasiparticle spectrum; namely, a certain quasiparticle excitation (and the rotational band 22 based on it) is obtained by adding up the contributions of the dif- ferent quasiparticles involved. For example, a two-quasiparticle state is formed by adding two single-quasiparticle states. In order to calculate the experimental Routhians, the rotational frequency, w, is determined by the canonical relation [Fr81b]: _ dE(I) (L) " , 203020 which is approximated by (0x = w is used for simplicity), 6(1) = E(I+11'E(I'11 . (2.3.21) Ix(I+1)-Ix(I-1) The IX represents the total angular momentum projection on the x «2 axis. Therefore, the expectation value of Ix will be: «2 <1x> = 1(1+1) - K2 (2.2.22) and . 2 (11> = Ix . (2.3.23) Since physical nuclear rotors are three-dimensional [8077], the magnitude of the total angular momentum can be approximated, through a Taylor series expansion, to the following [I(I-I-1)]1’2 = (1+1/2) for large 1. (2.3.24) Substituting Equations (2.3.24) and (2.3.22) into (2.3.23), we have: Ix =‘[(I+1/2)2 - KZJIIZ . (2.3.25) 23 Since only AI = 2 transitions are considered due to symmetry pro- perties (signatures), a more precise definition of 0(1) is prescribed as an average value of the two frequencies related to the levels I+1 and I-1. Graphically, 0(I) is an interpolation between the discrete values of w(I+1) and m(I-1). As pointed out previously, one of the keys of the CS model is its additivity property; thus, an excitation configuration is related to a g-configuration (or quasiparticle vacuum). The angular momentum of an excited band is referred to the g-configuration as i(w) = Ix(w) - ng(w) . (2.3.26) where ng is the core's angular momentum (also called R). Then one must find the reference ng(w). Because the g-configuration changes since there is a gradual destruction of pairing among all members with increasing rotation, a variable moment of inertia (VMI) approach [Ma69] is used as a reference instead: 1x90.) = '16 (2.3.27) or ng(‘”) = (30 + 0162).. . (2.3.28) After rearranging and taking the derivative in relation to the frequency, 3.1.339)...— 50 + 3 5162 . (2.3.29) Equations (2.3.28) and (2.3.27) are equal if the nucleus behaves like a rigid rotor but usually it does not. Therefore, one must estimate 24 '3 (Equation 2.3.27) in the best way possible using an appropriate reference configuration which can be the g-configuration or the 5- configuration (the s-band is an aligned quasiparticle band that crosses the ground state band and becomes the continuation of the yrast band). Experimentally, a reasonable estimate of the moment of inertia for a g- or s-band is obtained by AIX -A—w— : '30 + 3 31(02 , (293030) from two consecutive AI = 2 transitions . For the g-band, Ix/w can also be used because it will give the same 90 and 31 as in Equation (2.3.30). Because w is an interpolation, as pointed out in Equation (2.3.21), one way to extract m2 more accurately is by using: w2 = Lfl [w(I) + w(I-2)]2 . (2.3.31) Finally, the eigenvalues for the Routhian Hamiltonian, also called the Routhians, are defined as e'(w) = E'(w) - Eé(w) , (2.3.32) where, experimentally, E'(w) = L2 [E(I+1) + E(I-1)] - me (2.3.33) and 5&0») = '/ng(w)dw = '1/2 (30012) '1/4 (3100“) + (112/830)‘ (2°3'34) The information obtained from using the above equations is discussed in the next section. 25 2.3.6 Interaction and Crossing gf Bands Discontinuity in the smooth trend of rotational spectra with increasing angular momentum has been termed "backbending" [Jo72] due to the fact that, in the graphical representation of Zj/h2 (moment of inertia) vs.‘h2m2, a bending backwards of the plotted curve is seen. In Figure 2.3 the backbending effect is observed to occur between spins 12+ and 14+ for 15°Yb [Ri80]. In the rare-earth region this backbending phenomenon has been explained microscopically with a quasiparticle-plus-rotor model. That is, backbending occurs as a result of a crossing between a ground state rotational band and a quasiparticle band. Since the quasi- particle band has a larger aligned angular momentum, the yrast line displays a change in the effective moment of inertia. In this model, high-j quasiparticles are needed for effective band crossing at low frequencies, such as i13,2 and hg,2 quasineutrons and h“,2 and h9,2 quasiprotons. The occurrence of the backbending effect is not common to all nuclides in the deformed regions. For example, consider the case of the N=98 isotones where there is a small upbend, as opposed to backbend, indicating that there is a crossing with large mixing between the ground state and the quasiparticle bands. Therefore, it is more appropriate phenomenologically to interpret backbending or no-backbending by the amount of interaction between two bands. For strong "backbenders", there is a small interaction, while for "non-backbenders", large interaction. 26 1501 130‘ llCP '> 0 90‘ Z N s; \ h 70 ‘ N 50- ,/ 30‘ 10‘ (3 . . . 0 005 010 015 2. hie} (MeV ) Figure 2.3 Backbending plot for 160Yb. 27 In the cranking quasiparticle-plus-rotor approach, the valence nucleons form quasiparticle orbits that can interact to a greater or lesser extent. The magnitude of these interactions is measured by the Ae' of closest proximity between the quasiparticle orbits, as noted out in Figure 2.2 by the vertical connecting lines. For large Ae', an upbend behavior will be observed. For those small interactions that cause an unoccupied quasiparticle orbit to dip below e'=0 (see Figure 2.2), a sharp change in the rotational sequence of the band should occur (backbending). In a simplistic picture, a nucleon associated with a quasiparticle orbit slowly gains more alignment as the nucleus rotates faster until its excitation energy no longer allows it to stay in the same quasiparticle space; suddenly, it changes to a different, previously unoccupied orbit which has a certain amount of angular momentum alignment. In other words, there exist even at relatively low m highly aligned two-quasiparticle bands, but they are not yrast. Only at “c (crossing frequency) does this two-quasiparticle band become yrast; i.e., the gain in energy due to the Coriolis alignment overcomes the loss in energy due to exciting the two-quasiparticle band. With the alignment gain in angular momentum, there is a corresponding change in moment of inertia, and, depending on the magnitude of Ae', there will be backbending or not. CHAPTER III YTTERBIUM-168 EXPERIMENTAL DETERMINATIONS 3.1 Gamma-Gamma Coincidences In order to obtain 168Yb nuclei, oxide targets of enriched (98.5%) 166Er were bombarded with 27-MeV alpha-particles producing the reaction 166Er(a,2n)153Yb. These oxide targets were made by weighing small amounts (1.0-1.5 mg) of the enriched erbium, mixing with distilled water, depositing them on a thin layer of Formvar and letting the targets dry overnight in a desiccator. The resulting targets were about 3 mg/cm2 and were able to withstand up to two days of bombardment without breaking apart. The Michigan State University (MSU) sector-focused isochronous cyclotron was used to produce the alpha-particle beams in the K=50 energy mode, and the RF signal was used for timing purposes. As a preliminary guideline, the optimum bombarding energy for different (a,xn) reaction channels was calculated theoretically by using the computer code CS8N [Sik]. Also, the systematics of the reactions to produce neighboring Yb isotopes was used. A qualitative measurement of the excitation functions was performed and the bombarding energy of 27 MeV was chosen to minimize contaminant y rays. One prominent 28 29 interfering reaction channel was (o,n) and several y rays coming from 169Yb appeared in the singles spectrum. The beam current averaged about 3 nanoamperes. The detectors used were a large-volume (90 cc) closed-end detec- tor with 16 percent efficiency and a high-purity planar detector with a resolution of 650 eV FWHM at 122 keV. They were placed at 90° to the beam direction and as close as possible (8 cm) to the reaction chamber, as shown in Figure 3.1. The planar detector was especially useful in resolving low-energy y rays between 80 and 300 keV. Thin foils of copper (0.1 mm thick) and lead (0.5 mm thick) were used in front of the detectors to reduce x rays coming mostly from the erbium target. The deexcitation of 168Yb nuclei causes a cascade of y rays and those which are in coincidence were recorded on a magnetic tape as yl-vz-time using the computer program IIEVENT [Au72]. Some 18 million events were recorded and, in the off-line analysis, were sorted out by restricting two parameters using digital gates and pro- jecting the third parameter either as energy or as time spectra. This sorting was done by the computer code KKRECDVERY [M075], where up to sixty 4096-channel spectra could be obtained in a single sorting. Total events spectra were obtained by placing a gate on the prompt peak of the TAC spectrum and an open gate on the energy spectrum of one of the solid-state detectors. The projected energy spectrum con- tains essentially all y rays observed in the experiment, as shown in 30 Beam Target GE Detector | // L GE Detector To Beom Dump Figure 3.1 Experimental setup for the 166Er(a,2n)158Yb reaction. 31 Figure 3.2. From such a spectrum, it was possible, through repeated sortings, to make some 300 digital gates for both detectors. Some important gates are shown in Appendix A. Compton background subtrac- tion was done by choosing low and high energy regions near the gated peak where there would be no peaks, and which therefore would be representative of the background in that section of the spectrum. In order to make energy and intensity calibrations, a standard spectrum for each detector was taken during bombardment by measuring the y rays emitted by 152Eu and 133Ba. Through internal comparison with the standard sources, the 168Yb cascade y rays were calibrated. To analyze these spectra efficiently, a computer code called SAMPO [R069] was used by first fitting the most intense standard y-ray peaks with Gaussian curves and varying exponential tails. The resulting standard "shape" parameters were used to fit Gaussians to the v-ray peaks on all gated spectra. This code was especially useful in resolving up to six neighboring peaks (approximate channels for the centroids are manually put in, and the code maximizes the fitting procedure through an algorithm that involves the "shape" parameters and the centroids). A list of all Y rays above 80 keV identified in this experiment is in Table 3.1. The energies and the relative inten- sities of the transitions, as well as the angular distribution coef- ficients and the transition assignments, are listed also and will be discussed in later sections. 32 7449- 6392' 5335' 4270- 322b 2164' 1 107- 50 1443' 1284' 1125' couw1s ‘ 3 Lmbn; ‘ M. j T 1470 1670 1070 1270 250 she £50 35° 3%” I anon. Figure 3.2 Projected total events spectrum for the large Ge(Li) detector in the 168Yb coincidence experiment. 33 m+a + m+m Nm.Hz Ammvmmo.o- Aevmm.o Reve.m o.m- mm-e . 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HAHNNHV-m . m-mu mm.Hz fiflavemo.o- Aomvmmo.o- Am.fivo.mm “.mNN hm-m + ANNNNVH e.eo.e- AN.HVN.© » . m+e + Amewfiv-he mvu e.o¢.m- eeemEem_mm< o<\.< o<\~< A.oee ..aev A>exv pcmwuwmwmou rfi arm cowumnwgumwo Lapzmc< A.e.peeev H.m a_eee 37 Nu Aowvmwo.o- Aevmm.o Ao.va.NH e.Hmm Nu RomHveNo.o Amvwfi.o vam.mfi «.mmm NA hoofivfimo.o- Aevmm.o mAmVN.HH m.H~m » r +m + +5 Nu Aevmm.o- onem.o Aevo.mfi 3.6Hm m+m . m+e Nu efimveeo.o- efifivom.o A~.Hve.m o.HHm m-~ a m-m NA eAemvoH.o- caefivem.o Ae.mvw.m m.mcm m m . . . . +N + +3 Nu afifimvwfl o- eAmHvem o AN my“ New “.maw eeeaEeumm< o<\.< o<\N< A.ee_ ._aev A>axv “empoweeeou rfi e»m cowpmawsum_o mezmc< A.6.eeoev H.m a_eae mAmmmHv-Am.av + m-mu NA.H2 ANHVaN.o- RHHVAH.C- Amve.e c.3oa N+m + m+m sem.awm m a +8 + +m NA Amvom.o- Amvma.o Am.mvm.eoe I +.emm r > +~ + +6 .ea.mmm “omefiv + m-m . em.w+m 8 > a. +m + Acmefiv 6H.wem + + “H.va.mm 2 +6 + +m em.eem m-m + m-cH , ue.m+m secesem+mm< o<\+< o<\~< A.oe+ ._eev A>exv Aee+e++eeeo »H e»m cowumn_gum_o gmpzm=< A.6.eeeuv A.m e_eae m m 39 +~H + +eA Nu AoHvHN.o- “ovoe.o Aa.Hvo.~m “.mmm r m+m + Amemfiv-fio.mvu Hm Aemvmfio.o- Amvmm.o- AN.HVm.oa m.mmm m m +oH + +NH Am.ofimv Nu Aawvmeo.o- AHva.o Amvm.m m.+me Nu.Hz Amovemo.o- Amvmfl.o- AH.va.NN N.Hme r r +m + +oH NA fimvofl.o- Amvmm.o Am.mvm.esm ~.mma r r +m + +HH Nu AoVeH.o- Amvmm.o Ao.Hve.o~ m.oee r r +m + +oH Nu Aevemo.o- Afivmm.c Amvm.HH o.oHe gucmE:m_mm< o<\s< c<\~< A.u:+ .Fmgv A>mxv eee+e_e+eou »H 6+“ cowumn_gumwo Lopsmc< A.u.peoev H.m a_eee 4O 6 . +6 + +6 66 62 66666.6- 66666.6- 66.666.66 6.666 6 6 . +6 + +6 66 62 66666.6- 66666.6- 66.666.66 6.666 6 6 +6 + +6 6666.6 6.666 6 6 +66 + +66 U6.666 66+6 + 6666666 666666.6- 6666.66 6.666 6 6 . +6 + +6 66 A: 6666666.6 6666666.6- 66.666.66 6.666 6 6 . +6 + +6 66 A: 666666.6 66.666.66 6.666 66 6 + 6666666 . + 66 666 > 6+6 + 6666666 66 66666.6 66666666.6 66.666.66 6.666 66665666666 666+< 66666 6.666 .6666 66666 #:06066600u 66 arm :owpmnwtumwo Lapsoc< 6.6.66666 6.6 66666 41 +6 + +6 6.6666 6 6 66.66 666666.6 666666.6- 66.666.666 +6 + +6 6.6666 G P a 0 +6 + +6 66 66 66666666 6- 6.6666 66.6: 66666.6- 6666666.6- 66.666.66 66.6666 6 6+6 + 6666666 66.6666 6 6 . +6 + +6 66 Hz 66666666.6- 66.666.66 6.666 6 6 +6 + +6 6666.66 6.666 6 6+66 + 6666666 06.666 6 6 . +6 + +6 66 66 66666.6- 66666.6- 6666.66 6.666 66666666666 66666 66666 6.666 .6666 66666 pcwwowwmmou 66 666 6066m6666669 666366< 6.6.66666 6.6 66666 42 6 . +6 + 6+6 66 6: 666666.6- 66.666.66 6.6666 6 . +6 + 6+6 66.666.66 6 6666 m > a o o o o 0 +66 + +66 66 62 66666 6- 66666 6- 66 666 66 6.66 6666 6 + 6 . . +6 +6 66 6: 666666 6- U6.6666 6 +6 + 666666 66.666.66 6.6666 G 0 0 6+6 + 666666-66 66 666 66 U6.66: 6 6 +6 + +6 66 666666.6- 666666.6 6666.66 6.6666 6 6+6 + 6666666 66.666.66 6.6666 6 6 +6 + +6 66.666.66 U6.6666 66666666666 66666 66666 6.666 .6666 66666 66660666mou 66 666 :o6umn6gum6o 666=mc< 6.6.66666 6.6 66666 43 .66666666666 :6 cm>_m 66 666666 666 .6605666666 666666 6 666 6666 6 «:06663 mpw>op 660:6 6cm; .6666 6666656660o x6666: mvwxo Lo 66 66 .p<66 .6666 .6666666 :o6666666666 6666666 «66666666: 6666666 666 6u_=s\umpa:oo6 .6666 66666566606 x6666e muwxo La 66666 .6<66 .666 an 6666626 .6660066 666666-660 6666: 66660666 666666 popamppss\uopn:oou .Eaguomam ump6m :6 uw>gwmnoo .o u 6<\6< :66: umsgoveoq 663 60666660660 606666666666 6666666 066 .606666 6<\6< o: 6°66 .>mx m.o m6 60666666660 zmgmcw on» :6 6666666602: 6566 6 6+6 + 666666-66 66 66666.6- 66.666.66 6.6666 6 . +6 + 6+6 66 Hz 6666666.6- 66.666.66 6.6666 6 . +6 + 6+6 66 66 666666.6- 66.666.66 06.666 66.6: 666666.6- 66.666.66 6.6666 6666566666< 6<\6< 6<\6< 6.666 .Fmgv A>mxv 6606066600u 66 666 cowumn6gumwo 66~=mc< 6.6.66666 6.6 66666 44 From these coincidence gates it was possible to construct most of the level scheme that is drawn in Figure 3.3. The K"=2+ band is strongly populated, and another band based on the K"=3+ bandhead was observed. One of the most interesting features of the level scheme is the K"=5‘ isomer band. Since its intraband transitions are weak and low in energy, it is difficult to ascertain their existence from the coincidence gates alone. These transitions were better observed by making use of Y1'Y2't spectra which will be discussed in the next section. 3.2 Gamma-Gamma-Time Spectra One feature of the pulsed-beam experiment is its timing content and it can be useful in determining y rays that are not prompt in the time limits imposed by the electronics and the detectors used. The time between each beam burst (0.5 ns wide) was 63.8 t 0.2 ns, and an experimental timing resolution of 10 ns FWHM was obtained from the TAC spectrum (Figure 3.4). Pictorially, one can describe the two off- prompt regions of a TAC spectrum as either "early" or "delayed", that is, corresponding to the y rays "growing" in or decaying out of the isomer, respectively. Therefore, by taking time-slices from the TAC spectrum and an open gate on one of the energy spectra, it is possible to see which y rays are "growing" in or decaying out of an isomer as a function of time. Figure 3.5 shows the 111-, 112-, 128-, 146-, 164-, 182-, 198-, 224-, 240-, 274-, 310-, and 346-keV peaks growing in, and Figure 3.6 shows the 157-, 180-, 229-, 325-, 349, and 401-keV peaks 45 _u_‘_fi__ 2920.0 I «m ‘ ' 2730-. on I 299.: ”I n. 2112 an ' r 2335-: nu 2239.: 3’” no I“ .4" 2111...,__I.‘ In 5‘ "ll we; : In I :90 5: ‘ WOO.) till. m "S . . \ m, I7 mun b.4434 a: W nu ..u nu) m.o\ mu \ ., 411M ‘ m Ill 0* a. Lo Figure 3.3 Level scheme for 168Yb. 46 IZO l comrs mo” $ A4—. OAALA I J ‘T 1 101 201 301 401 501 601 701 W Figure 3.4 TAC spectrum for the 168Yb coincidence experiments. 47 l28ns l92ns COUNTS / CHANNEL W 255ns .2 > 4 no “AWL. ENERGY (keV) Figure 3.5 “Early" TAC time slices for the 168Yb timing experiments. 48 "DE LAYE D" 255 us 226 15-9 » 1 .2. I . l92 n: l28 M COUNTS / CHANNEL 2".“ 37 can 64ns' z i I” 40' ENERGY (keV) Figure 3.6 "Delayed" TAC time time slices for the 168Yb timing experiments. ' 49 decaying out of the 5' isomer. By taking the area under a y-ray peak and plotting it against time, a typical radioactive growth or decay curve is obtained (Figures 3.7 and 3.8) and its half-life (the stan- dard deviation is obtained from weighted least-squares fit) can be determined. For the "early" region, all y rays with the same half- life indicate that they are members of the same isomer band, and for the "delayed" region it would indicate that they are coming from the same isomer bandhead. It is also possible to obtain a time spectrum for each y ray by placing a gate on its energy, in coincidence with an open gate for the other detector, and the projected time spectrum gives a prompt peak as well as two off-prompt regions where a decay curve indicates the existence of an isomer. In Figure 3.9, the time spectra for some of the AI = 1 transitions of the K“=5' isomer are shown. They clearly show a long decay and the STOpe is equal to the t1,2 of the isomer. Similarly, in Figure 3.10, some of the transitions coming out of the K"=5‘ isomer are plotted, and the "decay" slope in the "positive" time is equal to the t1,2 of the isomer. The "negative" time region in Figure 3.10 suggests another isomer. A TAC spectrum was obtained by gating on two relatively prompt transitions like the 2; + 0; versus the 4+ + 2+ 9 9 effect is due to the long time taken for charge collection in the where a similar behavior was displayed indicating that the large Ge detector. A list of the half-lives obtained by these two methods is in Table 3.2. By using the above information, as well as 50 1001 80‘ 1/ 60. / T ‘0. l 111 Kev slope: 8813 ns COUNTS /I so 160 1E0 266 250 360 TIME (ns) Figure 3.7 Radioactive_"growth" curve for the 111-keV transition in the K =5 isomer band in 168Yb. 100‘ 80‘ 50‘ \ 229 KeV a sl0pe: 8513 ns 4m \\\\\\\\\ 26 \\\\\\\\\fl\\\\\ 1° 1 T T v v 6 0 50 100 150 200 250 300 TIBflEE (11$) COUNTS Figure 3.8 Radioactive_"decay" curve for the 229-keV transition out of the K =5 isomer bandhead. 51 10 182 novcynE 1o2 10‘ 1 ‘ , , ‘ ‘ 164 102 1o1 3.,llhllll 11.. .1 146 2 __, 1o . u: 2 1 z 10 1 4 111M 1: L) \ 128 (12 102 . 1 z: 3 1 O c) 1111M I 1 * ll 11 10 ' , ' 1 10 I“. '4 Ill” 6 3'4 150 152 235 :39 3'33 437 TIME (11sec) Figure 3.9 Time_spectra for some important A£=1 transitions for the K =5 isomer in 168Yb. COUNTS / CHANNEL 10 10 10’ 10 10 10 10 10 10 10° 52 IJJHMI b 1|.lllilh. ‘1 '| l‘liil 349 kOV GETE Tin: hl 229» manATE . 4|1l4|d1 1 IOO ltd/GATE 1 , 1 f 1443 n 4 V I5? NV GATE | 1 l 'l“ 3‘11 Ill 303 31's?” 62 Ea 6'4 6 34.130432455419411: TIME'(nsec) Figure 3.10 Time_spectra for some important transitions out of the K =5 isomer bandhead in 168Yb. "positive" time region is equal to t The decay slope in the 1,2 of the isomer. 53 TABLE 3.2 Half-lives for the K"=5' isomer obtained by two different methods described in the text. 11K + ITK (EEV) t1(%§§) tlffiég) 5'5 + 5'5 111.5 88 i 3 75 i 2 7'5 + 5'5 128.5 a ’ 70 i 2 8'5 + 7'5 145.9 a 57 i 3 9'5 + 8'5 164.1 - 65 :_4 10-5 + 9'5 181.5 a 68 i 4 7'5 + 5'5 240.1 75 i. 7 - 8'5 + 5'5 274.3 a - 5'5 + 5+ 157.0 75 _+_ 5 68 i 6 5-5 + (1770)b 228.7 85 + 3 79 + 4 a 19F or oxide matrix contaminant lines are present. b Neither spin nor parity have been assigned; energy level is given in parenthesis. NOTE: Standard deviations on the half-life are derived from fitting errors from SAMPO. 54 the coincidence results, the K"=5' isomer band was constructed up to spin 11' and its half-life was calculated to be 74 i 5 ns. More detailed discussion about the K"=5' isomer band is left to Chapter IV. 3.3 Angular-Distribution Measurements In (HI,xn) reactions, the resulting compound nuclei will have their spins highly aligned along the plane perpendicular to the beam axis [Di66]. The initial spin alignment of such compound nuclei will remain essentially the same, even after the evaporation process is over, since the exiting neutrons carry little angular momentum. Therefore, the y rays depopulating the discrete states of the "cold" residual nuclei will be anisotropic, and the angular distribution of the radiation field will depend on the initial and final states as well as on the multipolarity of the Y rays emitted. By experimentally determining the distribution curve shape, it is possible to confirm the placement of the y rays in the level scheme previously done using the coincidence experiments. To measure these intensities, the coaxial Ge(Li) detector (main detector) was placed on the movable arm of the MSU goniometer at 25 cm away from the target. Singles spectra were taken at 90°, 105°, 115°, 125°, 135°, and 145° with respect to the beam direction and were normalized against the x rays emitted isotropically by the erbium target nuclei. During these measurements, an additional fixed detec- tor was used as a backup normalization method. A pulser was also used 55 to monitor dead-time corrections, but these and the solid-angle corrections were less than two percent, according to the tables of solid-angle corrections of Camp gt_al: [Ca69]. The v-ray yields at each energy were fitted by the normalized angular distribution function W(e) = 1 + A2P2(cos e) + A,P,(cos e) , (3.1) where A1 is the angular distribution coefficient and Pi(cos e) is the Legendre polynomials. A more in-depth discussion of the angular distribution equation is left for Chapter V, Section 2. From the fitted curves, the angular distribution coefficients were extracted. In a general way, for the 168Yb decay scheme obtained, positive A2 indicated A2=2 multipolarity while negative A2 indicated A£=1,2. Whenever large negative A2 were obtained, mixed multipolarity (M1,E2) was assumed according to the tables of der Mateosian 33 31. [de74] and Yamazaki [Ya67]. In Table 3.1, the angu- lar distribution coefficients measured are listed. The large error bars in some of the coefficients are due to poor peak fitting done by SAMPO because these transitions were weak. CHAPTER IV YTTERBIUM-168 EXPERIMENTAL RESULTS 4.1 The Ground State Rotational Band In the deformed rare-earth region (a,xn) reactions can be used to populate the ground state rotational band (g.s.b.) up to a spin of about 16+, and they can be favorably used to study side bands up to 14+ [Tj68]. Accordingly, in the 168Er(a,2n) experiments performed here, it was possible to construct the g.s.b. of 168Yb up to 14+. Also, the angular distribution coefficients measured for the g.s.b. are in good agreement with previously reported results elsewhere [0072], as shown in Table 4.1. The N=98 isotones are "non-backbenders" (large interaction between the g- and s-bands), as exemplified by 168Yb in a plot of Ix versu5'h6 (Figure 4.1). Therefore, the moment of inertia to be used in the experimental calculations of the different pr0perties involved in the Cranking Shell Model calculations for the side bands can be extracted from the g-band as pointed out in Chapter II, Section 3.5. Usually, the s-band is used as a reference configuration because it fulfills the two criteria representative of the quasiparticle vacuum: i) it should be yrast or close yrast; and ii) it should vary smoothly and 56 TABLE 4.1 Angular distribution coefficients for the ground state rotational band Transition Energy Angular Distribution Coefficients (keV) Az/AO A4/A0 AZ/AO A4/AO (this work) (reference) 2+ 0* 87.8 0.011(40) -0.036(6) 0.012(5) -o.10(4) 4+ 2+ 199.0 0.27(1) -0.053(9) o.25(4) -o.o2(3) 6+ 4+ 298.7 0.34(13) -O.18(21) O.28(4) —o.03(3) 8* 6* 384-7 043(2)b -o.20(3) 0.30(4) -o.05(3) 10+ 8+ 455.2 0.35(2) -o.10(3) 0.32(4) -o.o4(3) 12+ 10+ 510.5a 14+ 12+ 552.7 0.40(5) -o.21(10) 0.31(4) -o.12(3) a Contains contributions from the annihilation radiation. b Contains contributions from multiplet. 58 m6 .656: .68 an. 6:266, x_ 66 66:. 16 6.53... :62 3: 6H6- .. .6 . .m— flo— .ON (9) x1 59 slowly with the rotational frequency. For the former, the reference state is assured to exist as an individual state and it is usually known very well for being in the yrast region (therefore, easily accessible by different reactions). For the latter, a sudden change of the reference at certain rotational frequency is readily observed as an irregularity (like the backbending) at the same frequency for all excitations. For 168Yb, a "non-backbender", the s-band is not useful as a reference state because the mixing between the g- and s-bands is large and stretches over several spins in the s-band. Therefore, it becomes difficult to use the extrapolation technique described in Chapter II, Section 3 (Equation 2.3.30) because the s-band is not "pure" up to the measured spins. Then, the extrapolated ‘50 and '31 values for the s-band will reflect this mixing and will be transferred into the func- tions of e' and i of the excited configurations (these are relative functions obtained by using a reference state) whose absolute E' and Ix functions are regular. To avoid this difficulty, the g-band is used as a reference because the lower spin states of this band are not so strongly affected by the mixing as the s-band is, and so better']o and '31 values can be obtained. In Figure 4.2 a plot of measured AIx/Aw versus‘fizw2 for the g-band of 168Yb is shown, and the extracted values for '30 and '31 are 33.6'hMeV'1 and 146 h3MeV’3, respectively. It can also be seen in this figure that above‘fi262 = 0.03 MeV the mixing starts to strongly affect the g-band so that the upper spin states deviate from the extrapolated '30 and ‘31 values. (aha/MeV) Ak/Am 80' 60' 50' 40" 301 10' 60 1 = 33.51.14 ev" / ’ ’1.=17.5€Mcv" 0.01 0.02 0.03 007 0.65 056 1‘1sz (MeV') Figure 4.2 flotsof AI/Am versus 0262 for the g-band 61 4.2 The y-Vibrational Band A band that was strongly populated in this experiment is believed to be the y-vibrational band. Previously, the spins up to 6+ [Ch70], and possibly 7+ [Je70] were identified. In both experiments the data were obtained by using (a,5n), (p,2n) and (d,3n) reactions, and the multipolarity of the interband transitions as well as the band parity were determined. In this work it was possible to firmly establish the spin 7+ state and to extend the band up to spin 11*. The intraband transitions observed were mostly AI = 2 crossover and a few AI = 1 cascade tran- sitions could be seen. Virtually no intraband activity was observed for the lower members of the band. This latter behavior can be explained by the fact that interband Sidefeeding into the ground-state rotational band (g.s.b.) was strongly favored at high spins, greatly decreasing the population of the lower members of the band. These lower levels could only be observed because of the Sidefeeding to the lower spins of the y-band by other bands of higher energy. One interesting characteristic that is seen in this band relates to the Sidefeeding pattern where a preference of the transitions between like-spins, e.g. 4; + 4; (M1,E2 transition) as opposed to 4: + 2; (pure £2 transition), was observed. These data, together with the angular distribution results, can be used for a comparison with the Sidefeeding pattern predicted by the simple rotational model. The idea is to obtain a ratio of the photon transition probability 62 example, the 4: + 4; over the 4: + 2;, and compare to the theoretical values. Deviations that occur can be due to some perturbation like band mixing with the ground state rotational band. In this model the transition probability for emission of a photon of energy EY is pr0portional to (for E2 transitions): 5 T(E20,11 + If) a EY B(E2,Ii + If) (4.2.1) where 2 f: 1 is the reduced transition probability for E2 transitions. Similar equations can be obtained for M1 transitions. Experimentally, the tran51t10n probability ratio between (Ii + If)mixed and [Ii + (I'z)f]pure can be expressed as a trans1t10n inten51ty ratio, namely, I T(E2) + T(Ml) -JL£3 = m m (4.2.3) I T(E2) YsP P where p and m subscripts are used to simplify the equations. For a mixed M1,E2 transition, one can obtain a ratio of T(E2) over T(Ml) that is equal to the mixing ratio 6 of the transition, i.e., 5 = [um/1mm“2 . (4.2.4) By substituting equation (4.2.4) into (4.2.3), the following is obtained: 63 I I”, (T(E2)m + mam/52) Iv.p T(E2)P (4.2.4) where the mixing ratios can be obtained from the angular distribution results and the tables in reference [de74]. Equation (4.2.5) is further simplified by substituting T(E2) using equation (4.2.1): 1 m 13(132)m 125,“ Y’ = ( E’ )(1 +.l ) (4.2.5) 11.0 B(Ez)p 51.9 52 Rearranging the equation, the following is obtained: Iv,m)( 82 ‘) 4.2. (I 1 + 62 ( Q) 5 13(52)m (Em) B(Ez)p - E 1.9 Ysm Therefore, the experimental ratio between B(E2)mixed and B(E2)pure is obtained from the intensities of the mixed M1,E2 and the pure E2 of the Ii + If transitions between y-band and the g.s.b.. The theoreti- cal ratios are obtained from the matrix elements (see equation 4.2.2), namely, 2 B(E2) M(E2) 2'" - m (4.2.7) B E M E2 ( )p ( )ID The matrix elements M(E2) are obtained from Clebsch-Gordan coefficients. 64 In Table 4.2 a list of the experimental and theoretical, mixed/pure B(E2) values using the simple rotational model are shown. The experimental values are in near agreement, considering the error bars. The larger deviation corresponding to the spin 6+ could imply that some effect is perturbing the band behavior as discussed below. Also, another interesting feature is the deviation from the rota- tional I(I+1) dependence. This is readily observed in a plot of (E /21 versus I2 as shown in Figure 4.3. It can be seen that 1' EH) the even members are depressed in energy in relation to the odd members, and are said to be "favored". One possible explanation for the two features above is that they result from the non-adiabatic coupling of the intrinsic and rotational motions of the nucleus. This is, from the mixing of the y-vibrational band with the g.s.b. as a first-order effect. The simple rotational model [8053] assumes that the nucleus is an axially-symmetric rotor, and that the rotational and intrinsic motions do not perturb each other. But for bands with good K quantum number, it is possible to have strong coupling through non-adiabatic interaction between the rotational and intrinsic motions. This non-adiatic coupling mixes the wave function for the rotational and vibrational bands and leads to deviations on the B(E2) values. The deviations are reflected on both intensity and ener- values for the interband transitions between the y-vibrational and ground state rotational bands [Ri69]. The exact form of this coupling might be understood in terms of the Coriolis forces. That is, the 65 TABLE 4.2 List of experimental and theoretical B(E2) ratios for I + I transitions between the y-band and the ground state rotati6nal Band In Experimental Theoretical 1. B(E2)mixed/B(E2)pure B(E2)mixed/B(E2)pure 4* 3.57 i. 1.21 ‘ 2.94 5* 7.59 i 3.10 3.74 Note: Values for other spins could not be obtained because of unreliable angular distribution values coming from resolved multiplets. 66 14' 3+ / f 13- 4. > 6 2 V :74. F3 / \ UJ . G 12 '. 6 9+ 11, . 11‘ 8* 10’ 10 2'0 4'0 5'0 8'0 160 120 110 1’ (5’) Figure 4.3 Plot of 55/21 versus 12 for the y-band in 16815. 67 Coriolis forces have a tendency to align the core's angular momentum along the rotational axis, and this implies a tendency of the nucleus to break away from axial symmetry. This will cause a coupling between the rotational motion to the collective vibrations around the axial symmetry, hence, a coupling between the v-vibrational and ground state rotational bands [8075]. Also, this perturbation can be further observed experimentally as a compression of the level spacing in the v-band (Figure 4.3) because the band mixing pr0pagates through the even spin members of the vibrational band since both the ground state and the B-vibrational bands have even spins only. 4.3 The K“ = 3+ Band The K‘ = 3+ band was previously known up to 5+ and possibly to 6+ [Ch72], and it was assumed to be an even-parity rotational band built on the 3*, 1451.8-keV level. Also, in reference [Ch70], the ratio of the reduced transition probability from the 4+ level at 1551.4 keV to the g.s.b., namely, the B(E2,I K + 2+O)/B(E2,I“K + 4+0), was calcu- lated to be equal to 0.31(6). Together with the intraband transitions, it was suggested this level be assigned to the K"I = 3+ band. Although the existence of intraband transitions was mentioned in [Ch72], no intensity values were reported. It was possible to confirm these levels in the present work, and firmly place the 6+ level at 1819.5 keV, the 7+ level at 1985.4 keV, and possibly the 8+ level at 2174.8 keV. Information on the 68 quasiparticle structure of the band may be obtained from an analysis of the mixing ratios, 6, for cascade and crossover transitions within the band, using the Nilsson [Kh73] and rotational models. The mixing ratios can be obtained from the branching ratios of the 166Er(a,2n) coincidence and angular distribution experiments. For the former, 52/(1+52) = [21K2(21-1)/(I+1)(I-1+K)(I-1-K)]. [EI+-1/E (4.3.1) 5 I+I-2] where A = IY(I+I-2)/IY(I+I-1). The sign of 6 is obtained from the angular distribution experiments [de74]. According to the rotational model the g-factor in a rotational band, gK (gyromagnetic ratio for a K" band) is constant in relation to the core g-factor, gR. That 15. the difference gK-gR is constant for all the rotational states in an unperturbed (not mixed) band. The gyromagnetic ratio difference can be calculated according to _ = 2 1/2 where the energies are given in MeV, the quadrupole moment 00 is approximately equal to 7.4 barns [L070], and gR is equal to 0.32 [Kr68]. Since the 5 value is very sensitive to the transition inten- sities, the peak areas obtained from the planar detector were used in the intensity calculations because of the detector's resolution at low v-ray energy. In Table 4.3 the calculated 6's are listed as well as the (gK-gR) values. The relative constancy of the (gK-gR) values affirm the placement of the levels in a single rotational band. The average value for (gK-gR) is taken to be 0.57(6), and, therefore, the Mixing 69 TABLE 4.3 ratios and (g -g ) values for K“ = 3... band in 168Yb using the planar detector intensity results I: - I; Enfigay IY 5 (9K ' 9R) 5+ + 4+ 0.1230 51.5(4.1) 0.38(4) 0.45(5) 5+ . 3+ 0.2227 32.8(4.3) 5+ + 5+ 0.1450 48.8(3.4) 0.30(4) 0.55(5) 5+ + 4+ 0.2580 48.9(6.4) 7+ . 5+ 0.1550 41.8(4.2) 0.29(5) 0.58(6) 7+ + 5+ 0.3110 57.1(7.4) 70 value for gK is +0.85(9). In general, for a singlet two-quasiparticle state, the gK values are: 9K 0 for neutrons and 9K 1 for protons. Hence, the calculated gK value above indicates that this band has predominantly a two-quasiproton structure with about 15% neutron admixture. The possible configuration for the 3+ band is {7/2+[404]p, 1/2+[411]p} as it will be discussed below. This result contrasts with the corresponding band in 172Yb with a 9K = 0.2 [Wa80] where the dominating (80%) {5/2'[512]n, 1/2'[521]n}, two-quasineutron structure is also evident from particle transfer [Bu67] experiments. The contrasting features of the band structure for these two isotopes is further highlighted by the angular distribu- tion coefficients having different signs indicating different quasi- particle structure domination. For this deformed mass region it is possible to obtain several two-quasiparticle configurations that could give origin to the observed K" = 3... bands. In calculations performed by Walker, Carvalho and Bernthal [Wal] it is shown that the lowest K" two-quasiparticle configuration are all of the singlet type (intrinsic spin projections cancel) for the 9862 66 266» A66\6= :66 emu“ 66 6066666 66666 666 66>66 66 6>o66 6:6 .>66 :6 666666 666 66>6m 66 6666 36666 wm~6>6— 6 u _ ~66=6EP666x6 66» 666666666 6666 p6666666o6 6666» 666 .666666 oNH z < 666.6. mp6>6p +m u ex 6.6 666666 . mo~_ ow; .c6E_66nxm man. a man. . m~ 6.6.— . 666. ._ . 666. 466 . .8.— mnm. 8m 3 I / . c N 6: \ - // E ... 3 6 .6 LII-1111111. 1111111 / o \ / \\ ..ON.N a —Q 11111 —Q 5n x) “J 1 Na [\1111 , NC //\x :- OQN \I . II; \ / W III N“ /\\ / .Q towN a . x/ , um \ / .86 . .m "L: 5:23.60 / N: . x 2 .8.6 . //III\\\1111\\\111 .c~.m lev .69 No— .69 No. co. mm NQ oo— mm A11 z 461.1: 66.6 .1 36:66.6 . r .6886 . 73 energy trends for the different configurations and in part be explained phenomenologically as a result of the proton-neutron admix- ture where the participation of the neutron in the band structure generally lowers the band energies. It is known that inelastic deuteron scattering will preferen- tially excite collective states, and in [Bu67] the 4+ state of the 3* band was observed in several Yb isotopes. This would indicate that a certain amount of collectivity is present in the 4+3 level. There- fore, some interaction between the collective and two-quasiparticle degrees of freedom should be present in order to account for both expe- rimental results described above. The possible collective motion that could be associated with the low-lying K" = 3+ bands would be the hexadecapole vibration. That is, in the A ~ 170 region, there are several low-lying two-quasiparticle configuration with the same K1r that would favor the K = 3 hexadecapole vibration. In this deformed region other vibrational modes usually are low- lying in energy but are incompatible with a K = 3 band assignment as in the case of the B-vibration (zero unit of angular momentum along the symmetry axis) with a K = O and the v-vibration (two units of angular momentum along the symmery axis) with a K = 2. The octupole vibration (zero to three units of angular momentum along the symmetry axis) is an odd-parity excitation, hence incompatible with the K“ = 3+ band. Additionally, the K = 3 hexadecapole vibration would be expected 74 in this mass region since the pair of quasiparticles that contribute to the low-lying K" = 3+ bands have asymptotic quantum members so that Anz = 1 (favors the hexadecapole vibration). In summary, the K" = 3+ band has a predominantly (85%) { 7/2+ [404]p, 1/2+ [411]p }, two-quasiproton structure in contrast with the corresponding K“ = 3+ band in 172Yb which has a predominantly (80%) { 5/2' [512]", 1/2' [521]n }, two-quasineutron configuration. In addition, the band is strongly excited by inelastic deuteron scat- tering implying some interaction between intrinsic motion and vibra- tional hexadecapole oscillations. 4.4 The K" = 5' band The level at 1999.3 keV has been tentatively assigned as a KTr = 5' bandhead [Ch70] and as having a half-life of 81.7 t 4.5 ns [Ch73]. The level at 2110.8 keV was established as a 6' rotational state of the isomer [Ch72]. The long half-life indicates a large hindrance for the transitions depopulating this level and, therefore, a large K value would be expected for the band. This isomer has been assigned a Nilsson configuration { 7/2' [523]p, 1/2+ [411]p } by [Ch72] based on similar trends in 15”Er and B+-decay of the { 7/2+ [404]p, 5/2‘ [523]n }, the ground-state of 168Lu. In the present 166Er(a,2n) experiments, it was possible to extend the band up to 10' and possibly 11' spin. Crossover and cascade 75 intra-band transitions were observed more clearly in the delayed yy- coincidence analysis. Angular distribution results were also used to confirm these placements. Some Sidefeeding from another band into the isomer band was observed. The deexcitation of this band occurs through six transitions including those to the K" = 3+ band but none to the g.s.b. (K=O) or to the y-vibrational band (K=2), indicating a large K value for the isomer band. All the decay branches have been seen previously [Ch70] in the radioactive decay of 168Lu. The rotational structure of the band built on the 5' level is reasonably regular as shown in Figure 4.5 where 2 4,52 versusfizw2 is plotted. Also plotted are the long-lived isomers, based on 4‘ (360 ns) for 170Yb [Wa79] and 6' (3.6 us) for 172Yb [Wa80]. Their suggested structures are { 7/2+ [633]", 1/2’ [521]n } and { 7/2+ [633]", 5/2’ [512]n } reSpectively. It can be seen that the 5‘ isomer displays one of the general features of the other two iso- mers, namely, a much higher 2 9/fi262 value than the g.s.b. (only the g.s.b. of the 168Yb has been plotted in order to simplify the picture but the g.s.b. for 170:172Yb behave nearly the same as that of 168Yb at low spins). This indicates similarities in the band structure for the isomers. It has been demonstrated for the isomers in the neigh- boring Yb isotopes, EWa79] and [Wa80], that this feature results from the Coriolis effects on an i13/2 neutron. The Coriolis forces tend to decouple a nucleon pair in a high-j orbital and align their angular momentum with the rotation of the core [St75]. Therefore, a high 76 .66uaopa om_6 666 a>~6_ 666 66566? um 63» 6:6 6666. 666 665666 -6 6;» 6666 . 666 66:66 .666666666 66666 6::O6m 6:6 666 665666 um 66% 66$ 636:..m> ~c§w.~ 66 66—6 m.¢ 6666.6 . (is; 6.3 6..» .lmbo. . Own. one cmc. en OmG. Ono. 0... duo TOO. . .IIIIIIII .IIIIIIlI-I:\II. . \HWIIIIIIIIIIIIIIII\\\\IIJUHITTlllll:\\\\llllllvlllr :Illlllqlllllllll. rO—. ION. (know); A: 3 I00- '0'— 77 effective moment of inertia (109 MeV"1 for 168Yb) results from such alignment contributions. This behavior indicates the presence of an 113/2 neutron in the band structure of the 5' isomer. iixing of the isomer band with other suitable, nearby bands can be reflected in a compression of the isomer band levels as seen in Figure 4.5 for the 4’ isomer in 170Yb. This is not the case for the 5‘ isomer in 168Yb since it has a fairly constant effective moment of inertia as compared to the 4' isomer in 17oYb. Observation of tran- sitions between the isomer band and other bands might indicate the presence of such bands available for mixing, but no such bands were observed. Using the branching ratios, it is possible to calculate the ng-gR | value, as pointed out previously, with the sign determined by the angular distribution coefficients for the cascades in the band. In Table 4.4 the| gK-gRl values are listed using both coincidence and TAC results. Using the most consistent set of results, a value of | 9K’9Rl was calculated to be equal to 0.27(5). With the angular distribution results and gR = 0.32 [Kr68], the value of gK obtained is equal to +0.05(5). This implies that this band has predominantly a two-quasineutron structure with a small proton admixture. Using the Nilsson level diagram (Figure 2.1) in Chapter II, the lowest possible configuration consistent with the K" = 5' band head is { 5/2+ [642]“, 5/2’ [523]n }. There are two possible two-quasiproton band structures that could be used for band mixing, namely, the 78 TABLE 4.4 Calculated I 9K'9RI values for the K" = 5' isomeric ‘ band of 168Yb. I | 9K'9R I Coincidence TAC(1) TAC(2) 7 0.29(3) 0.26(3) 0.26(3) 8 0.28(4) 0.27(3) 0.28(3) 9 0.34(6) 0.31(5) 0.34(5) Note: Two sets of TAC results(different time slices) were used in order to obtain a more statistically significant average. 79 { 9/2' [514]p, 1/2+ [411]p } and the { 7/2' [523]p, 3/2+ [411]p }. The former band configuration is energetically more favorable than the latter since it involves valence nucleons and Nilsson orbitals near the Fermi surface. For the latter configuration the Nilsson orbitals are well below the Fermi surface and, therefore, it would require much energy in order to have the correct two-quasiproton excitation. This conclusion is in contrast with the suggestion of Charvet et al. [cn72], namely, { 7/2- [523]p, 1/2+ [411]p }. This suggested con- figuration gives a band of only K=4, and not the necessary K=5 assignment for the isomer. The half-lives reported previously for this isomer are 531?: ns [Dr77] and 81.7 i 4.5 ns [Ch73], and in the 166Er(6,2n) delayed coin- cidence experiments the half-life obtained was 74 i 5 ns as calculated in Chapter III, Section 3. This relatively long half-life can be explained in terms of K-forbiddenness (transitions with AK > A multipolarity are hindered) which requires comparable K values for the bands to which the isomer decays. This is the case for K" = 5' isomer where all the decay branches go to bands with K < 3. Besides the known K" = 3+ band, the other decay branches of the isomer seem to populate some members of what appears to be K" = 0' and 2' octupole bands according to previously neasured reduced transition probabili- ties [Ch70] for the band members. 80 4.5 The Cranking Shell Model Calculations In the deformed rare-earth even-even mass region the yrast spectrum is predominantly of collective nature at low angular momentum. At around I = 12-14‘fi and h6 = 0.25 - 0.27 MeV, the yrast structure changes, and an excited quasiparticle band becomes yrast. This is the so-called backbending effect. Additionally, these are other quasipar- ticle excitations near the yrast line (on which rotational bands are built on) that are of much interest because they represent the breaking up of nucleon pairs which can affect the nuclear motion and shape. Studies on these quasiparticle bands leads of a better understanding of the nuclear dynamics at high excitation energy and angular momentum. The Cranking Shell Model has been quite successful in the theoretical analysis of both the yrast and near-yrast rota- tional spectra by using the idea that these quasiparticle excitations (and the corresponding rotational bands) can be decribed by the motion of a particle in a cranked deformed Nilsson potential. Such calcula- tions are used in this section to describe the 5' isomer band. In order to simplify the description of a quasiparticle band it is useful to label the rotational configurations by the conserved quantum numbers, namely, the parity n and the signature 6 (the parity of a multi-quasiparticle state is the product of the parity of all par- ticipant quasiparticles, and a is the sum of all the participant 6's). Therefore, the ground state rotational band has a (n,a)=(+,0) quasiparticle configuration. Consider the case of the three recently observed [R081] one-quasineutron bands in the neighboring odd-A 167Yb. 81 The 167Yb ground state rotational band has the (-,1/2) quasineutron configuration, and it corresponds to the Nilsson 5/2' [523] single- particle configuration at‘fi6=0 MeV. The other two bands, (+,1/2) and (+,-1/2), correspond to the decoupled Nilsson 5/2+ [642] single par- ticle bands at'fi6=0 MeV. A two-quasineutron band configuration can be obtained by coupling these two bands as [(—,1/2), (+,:1/2)], that is, [(—,1/2)x(+,1/2)] = (-,1) and [(-,1/2)x(+,-1/2)] = (-,0). These two-quasineutron configurations are the two lowest possible con- figurations for the 5' isomer in 168Yb as discussed later in this section. In terms of the usual Routhian e' versus‘fiw diagram as in Figure 2.2, a two-quasiparticle band is formed, for example, by placing one quasiparticle in the orbital A and another in the orbital C. The band configuration would be (-,1). Since A is an i13,2 orbital, Coriolis forces should strongly align it with the increased rotation of the core and, consequently, lowering it in energy. This is immediately observed by the sharpness of slope along the orbitals in the e' versus hm diagram because the amount of alignment, i, is equal to -de'/d6. Using Figure 2.2 two of the possible lowest quasiparticle con- figurations for the 5' isomer band in 168Yb are (-,0) and (-,1) as mentioned before. There are two reasons for the choice: 1) correct band parity (negative); and 2) both quasiparticle configurations, (-,0) and (-,1), contain an i13,2 quasiparticle that is necessary in order to explain some of the band features like high alignment gain as pointed out previously in this Chapter. 82 In order to survey the general behavior of the observed quasi- neutron rotational bands in 167:163Yb, it is useful to plot E' (level energies in the rotating frame) versus hm as in Figure 4.6. It is readily seen in such a plot that the bands (-,0) and (-,1) are on the average 1.8 MeV above the (+,0), the g.s.b. for 168Yb. As discussed in Chapter II, E' can be further transformed into a relative quantity by using the appropriate reference state, and a relative excitation (e') is obtained more readily. A comparison between the experimental and the theoretical results obtained by the Cranking Shell Model can be performed more easily. In Chapter IV, Section 1, the choice of the reference state, the g-band, was discussed and the values obtained for go and ‘31 are 33.6“hMeV'1 and 146‘h3MeV'3, respectively. By plotting the experimental Routhian e' versus hw, as in Figure 4.7, the quasiparticle excitation energy of the rotational bands at different frequencies is easily obtained. By comparing with the theoretical Routhian e' versus hm for neutrons (Figure 2.2), it can be seen that all three one-quasiparticle bands are in close agreement with the pre- dicted e' values. Using the additivity rules, the sum of the experi- mental Routhian e' of the (-,1/2) with the (+, 1/2) bands are approxi- mately equal to the experimental Routhian e' of either the (-,0) or the (-,1) bands. In Table 4.5, the above results are listed according to rotational frequencies. It should be pointed out that the theoretical e' values (from the CSM) lie generally lower than the experimental e' values since the former are calculated with a pair potential, A, that is about 20% lower than the full value of A obtained from the even-odd mass difference that is used in the 83 E’ (MeV) -2, (+n/2) -3 , 1 f 6 oo Q1 oz 03 04 flw (MeV) Figure 4.6 Plot of E' (level energies in the rotatin frame) versus ‘fiw for some of the quasiparticle bands re ated to the observed quasineutron bands in 168Yb. .n>66_ =_ 66:62 :666=6:_m6:c 66>66mao 666 3c m:m66> .6 :6wnuaom P66665666axu 6.6 66=m_6 omd A >6Zv 36+ mmo PS 6.6.6 Amhi 5...: .../fl .3... .((.//I././ all. ... (AalN) ,9 85 TABLE 4.5 Experimental Routhian e' values for the 5' isomer band in 168Yb at different rotational frequencies (see Figure 4.7) compared to summed Routhians from 157Yb. e' (MeV) at‘fiw (MeV) Quasiparticle 0.08 0.10 0.12 0.14 0.16 Configuration [(-,1/2),(+,1/2)] 1.72 1.62 1.52 1.41 1.32 (-,1) 1.88 1.85 1.81 1.76 1.70 [(-,1/2),(+,-1/2)] 1.74 1.65 1.57 1.48 1.40 (-.0) 1.88 1.85 1.81 1.78 _- 86 experimental calculations. Also, useful information can be obtained by plotting Ix (angular momentum along the rotational axis) versuS‘fiw (Figure 4.8) since the energy of rotation is wao It is assumed that the excited rotational quasiparticle bands and the g-band do not differ significantly in collective moment of inertia but in angular momentum at a certain rotational frequency. This difference is called the aligned angular momentum i of the excited band relative to the g-band (Chapter II, Section 3.4). In Figure 4.9, i versus‘fiw is plotted for some of the observed quasiparticle bands in 167:153Yb. It is also seen that for both (-,1) and (-,0) bands, the aligned angular momentum i of the two- quasineutron band is much smaller than the value for either [(-,1/2), (+,1/2)] or [(-,1/2), (+,-1/2)] combination at different frequencies (Table 4.6). The above results indicate then that the theoretical calculations for the Routhian e' are close to the experimental Routhians (Table 4.7) but there is a certain discrepancy for the alignment gain (Table 4.8). It has been pointed out [Rie] that CSM calculations with high-K bands (K > 4) have not been quite effective in predicting the observed band behavior in the rare-earth deformed region, and that more theore- tical development is needed in this direction. Additionally, the 5' isomer band is not a pure quasineutron band, having a small proton admixing. This admixing could be affecting the alignment gain considerably. Unfortunately, proton-neutron admixing has not yet been considered in the literature impeding more reasonable conclusions 87 15- E _‘x 10‘ 5. (+,O) G . . . _fi (10 0.1 02 0-3 01. he: (MeV) Figure 4.8 Plot of I versuS'hw for the observed quasineutron bands in x168Yb. '88 .n>wwH cw mvcmn 60663656636 v6>66mno 65 606 3:. m:m66> .F mo uoE 6.6 66:36 9an 3 w ONMU m mo one mod 0 o e .F AN\_.;\.\ 8 7\r:.-v u s x \a\ AN\T.+v lllllllblllxlllllllllllliIIIlllllHHHHH\\\\\\ rm \ ollll\\l\\\\\ Ami L Illllllllil. r... (H) 89 TABLE 4.6 Experimental 1 values for the 5' isomer band in 153Yb at dif- ferent rotational frequencies (see Figure 4.8) compared to summed values for 167Yb. i (h) at fib (MeV) Quasiparticle 0.08 0.10 0.12 0.14 0.16 Configuration [(-.1/2),(+,1/2)] 4.52 4.91 5.17 5.39 5.58 (',1) 1077 2005 2.35 2063 2090 [(-,1/2),(+,-1/2)] 4.08 4.25 4.38 4.48 4.54 (-,0) 1.77 2.05 2.35 2.63 2.90 90 TABLE 4.7 Comparison between experimental and theoretical calculations for Routhian e' (in MeV) as a function of rotational frequency hm for the 5 isomer in 168Yb. e' (MeV) Quasiparticle at hm (MeV) Configuration 0.074 0.111 0.149 ('sl)theoretical 1'89 1°78 1-51 ('a1)experimental 1'90 1°83 1.74 (-.0)theoretica1 1.89 1.79 1.53 (':O)experimental 1'90 1°83 1°75 Note: The theoretical (-,1) and (-,0) results for 168Yb are obtained from the coupling of corresponding one-quasineutron in 167Yb. 91 TABLE 4.8 Comparison between experimental and theoretical calculations for alignment gain i as a function of rotational frequency hm for the 5' isomer in 168Yb. 1'03) Quasiparticle at hm (MeV) configuration 0.074 0.111 0.149 (‘:1)theoretical 4.23 5.05 5.58 ('s1)experimental 1.66 2.21 2.69 (-,0)theoretical 4.19 4.90 5.26 (-,0)experimental 1.66 2.21 2.68 Obs. The theoretical (-,1) and (-,0) results for 168Yb are obtained from the coupling of corresponding one-quasineutron bands in 167Yb. 92 about how it influences alignment gain. There is hardly any distinction between the (-,0) and (-,1) bands because of the very small signature splitting between (+,1/2) and (+,-1/2) at the measured low frequencies. The (-,1) configuration would seem to be favored because of the lower Routhian of the (+,1/2) component but not enough high spins were observed to make a more definite conclusion. In summary, the (-,0) and (-,1) two quasineutron configurations are reasonable choices for the 5' isomer band configuration. The experimental Routhians e' are in close agreement with the calculated theoretical values at low frequencies but the measured i differ somewhat. This deviation is probably due to proton admixing (CSM theoretical calculations for proton admixing have not been considered yet in the literature). CHAPTER V CERIUM-128 EXPERIMENTAL DETERMINATIONS 5.1 Gamma-Gamma Coincidences In order to populate high-spin levels of 128Ce, the Oak Ridge Isochronous Cyclotron (ORIC) was used to provide beams of 20Ne (6+ charge state) at about 103 MeV bombarding energy. The reaction used was 112Cd(2°Ne,4n)128Ce, and enriched (98.5%), self-supporting foil targets of 1.0-1.3 mg/cm2 thickness were employed. A preliminary bombarding energy was chosen by making calculations with the statistical model ORNL computer codes ALICE [P177] and PACE [Ga80]. The excitation functions obtained from ALICE for the different reaction channels are shown in Figure 5.1. Similar reactions performed in this nuclear region by the ORNL Nuclear Structure Group were also used as a general guideline. The final bombarding energy was chosen by measuring the excitation functions with a series of nickel degrader foils (added to a final thickness of 25 micrometers corresponding to about 7.5 MeV energy loss). To avoid beam spreading and decrease of intensity, the absorber foils were removed, and the beam energy was changed to 103.2 MeV, corresponding to the maximum yield in the excita- tion function. The beam current averaged 4.0 electrical nanoamperes 93 94 a4n 4n 4.5» 5n 2 /’F’ “\\\\ [O 1- a3n 03" 4n 4n a' «Zn (mborn) / 3 32:1 p4n / l 3n IO - 5,, 93" a4n p2n k? - 4 92" 3" ‘ EM) END ICK) 1K) l2!) ELABWeV) Figure 5.1 Calculated excitation functions for the bombardment of 112Cd with 20Ne using ALICE. 95 (namps) and it was fairly constant throughout the experiment. The difference between the ALICE and the actual experimental values can be expected [Gav] in this region since the optimization of the many input parameters is difficult because both the mass tables used and the Coulomb barrier calculations have been extrapolated for such low masses. The PACE calculations gave even poorer results than ALICE when compared to the bombarding energy of the reactions per- formed previously in the region. To make use of the high y-ray multiplicity [Ri81] of the (Heavy Ion, xn) reactions and to greatly enhance the accumulation rate of coincidence data, seven Ge detectors were placed in the reaction plane (Figure 5.2). Also, two large NaI detectors (25 cm x 25 cm) and (22 cm x 30 cm) were utilized, with one placed above the target and one below to act as a "total (y ray) energy spectrometer," as explained later in this section. Absorber sheets of lead, tin and copper (23 combinations of sheets, varying in thickness from 102 to 406 micrometers) were placed in front of and around all the detectors to decrease the intensity of x rays, produced by the target material and by the interaction of the evaporated neutrons with the lead beam stopper and absorber foils, and also to decrease the Compton scat- tering of y rays between the detectors. Since the count rate in such an experiment is usually high, con- siderable care was given in devising a system of analog and digital electronic modules which could handle such counting rates gracefully. 96 l, .5. 4/ TOP view of Ge detectors 9003 L ‘2 '03. (NaI's are ..L to paper) @ 3 Distances to target : 8.0-0.5 cm 49, 4 46° Relative sumo-20% 00 ’ Na 1 Side view of N01 detectors earn . (Go's are.L to paper) No: Distances to target '. 8.5- I0.0cm Figure 5.2 Positioning of the Ge and NaI detectors for the bombardment of 112Cd with 20Ne. 97 A block diagram of the system is shown in Figure 5.3. In general, a coincident event required at least three Ge detectors and one of the two NaI detectors to fire at the same time. The coincident y rays were required to undergo pulse shaping and timing (about 150 nanose- cond windows), multiplicity check (at least four y rays in a coin- cident event) and detector "firing" identification (which detectors were "fired"), before they could be fed as an event into a Computer Automated Measurement and Control (CAMAC) crate which passed them along to a Perkin-Elmer 3220 Computer to be recorded on magnetic tapes. One-dimensional projections of energy and time Spectra were being accumulated continuously in memory during the experiment and displayed on video monitors, so that problems like changes in amplifier gain and timing could be checked at any time. More specifically, as far as the data transfer is concerned, the PACE system (which digitized the energy signal of the Ge detectors) sent the data in a digital form to an event handler where event-by- event match (through machine language) of the PACE system to the CAMAC/Perkin-Elmer system could be done. In the Perkin-Elmer 3220 Computer, an Acquisition Task called HAC [Mi81a] continuously read the data, by block (8192 bytes) transfer and wrote them onto magnetic tape for later off-line analysis. Also, a histogramming task called DISMO [Mi81b] was used to display the data on a video terminal during the data acquisition (see block diagram in Figure 5.4). An important factor was the dead time of the system, which came mainly from the software of the acquisition task HAC because it had to 98 . _ .mzZ 53 EN: we 223353 of .8» 953 _Scostoaxm 93 we 533% v.88 m.m 8:3... Lllfimw ...ch . flaw] 99 .cmuzasou cmmm gospmuc_xcma w u on coca 222.2030 9: act 33.5.5 3% m5 .8.— 53 En xooE e..m 95m: 320 8.25 5...... mos“. . II II .30 oon... T28 curm— E 328.235 .I I I I I I I L i I l I l l .J ...—...d}—.— 3.35 .2: hv.m.c\s.lu....... ...mu 2 aoamfi . “”5233... .2358 8mm. beatima 100 delete the extra information recorded by the PACE system that was not needed by the Perkin-Elmer system. The information coming from the PACE system consisted of a string of 24-bit words with each energy parameter corresponding to a word. Each word was composed of the data in bits 1-13, the stretcher (consequently, the detector) ID in bits 14-16, a "No True" flag in bit 17, and zeros in bits 18-24. The "No True" flag indicated if the data in the word were indeed part of the event or not. HAC then identified the flag and placed zeros in the data bits whenever the "No True" flag was set, therefore reducing the number of useful bits to 16 which corresponded to the stretcher (Ge detector) 10 and the data. This restructuring of the data set the upper limit on the dead time of the system as a whole. The coin- cidence event count rate was approximately 2 kHz, while the average count rate for the NaI detectors was around 80 kHz and for the Ge detectors was about 8 kHz. The total y ray energy spectrometer is used in order to select the appropriate 4n-reaction channel since the light Ce isotopes are in a mass region that is highly defficient in neutrons, and therefore, the emission of protons and a-particles for the compound nucleus (for the reaction used in this experiment, 132Ce is such compound nucleus) becomes large enough to compete effectively with the 4n-reaction channel. The use of the spectrometer in this experiment can be described as follows. A series of rapidly emitted Y rays causes a pileup in the collection of charges in the NaI detector. The end result is a pulse out of the detector that is proportional to the sum 101 of the energy of all the y rays (total y ray energy) in the series. The reaction channels had broadened Gaussian distribution that peaked at different total y ray energies, and in this experiment, these NaI detectors were used to discriminate against different reaction channels. The energy window was set between 2 and 12 MeV. A total (y ray) energy spectrum is shown in Figure 5.5, where the range of the 5-, 4- and 3-particle reaction channels is shown approximately. The broad peak around 3 MeV comes mainly from the 4n reaction channel. In Table 5.1 are shown all the final reaction products observed. TABLE 5.1 Final reaction products observed in the bombardment of 112Cd with a 103.2-MeV 2°Ne beam Final Final Final 5-particle product 4-particle product 3-particle product 5n 127Ceb 4n 123Cea 3n 129Ce° a4n lzuaac a3n 1258ab aZn 125Bab p4n 127Laa p3n 128Lad p2n 129Lac 2P3" 127Bac 2p2n 128Bab 2pn --- ab Strong channel Medium channel C Weak channel 9 NProbably observed . Note er combinations of particles are not preferred as a reaction channel. 102 .m2o~ cue: vu~.. eo pcmevgmnEon one com Ezgpomam amcmcm Aha. r. Peach m.m mcamwu Jyéfi$fi ommm oucm cmmm swam omma omou 0mm cm “MI! I)- b p E b bl |h A O 19 :ex .. k??? . was . ...au... 3. .....ko . tin. ...r... . . we“ .88 ....Pflmfifi. “54... ....u m..." ....... \.. I «fit... . . kfififlffifltf. 58m 4 fl 158 33:34 83:8an 23:23 . figmn r .093 TBNNVHD/SLNOOD 103 At the end of the coincidence eXperiment, 79 magnetic tapes of data were obtained with an average of three million coincidence pair events per tape. In the off-line analysis, one sorting program [0w81a] was used to "crunch" and perform a preliminary ordering of the recorded data according to the "firing" sequence of the detectors. Also, only prompt events contained in the 4-particle reaction channel slice of the total energy spectrum were sorted out. The "crunching" and preliminary ordering were done by extracting from the event word structure (eighteen 16-bit words corresponding to the energy and time information of the seven Ge detectors and one NaI detector, plus begin- and end-of-event tags) the energy and the ID of the detectors "fired" and by rewriting this information on a new magnetic tape as a 32-bit word with its own ID. These new tapes are called “pre-scanned" tapes, and they can be used in a variety of ways that fall outside the scope of this work. The coincidence data were translated into a two-dimensional energy matrix by a scanning program [0w81b] which was used on the "pre-scanned" tapes where each 32-bit word had its energy information extracted and corrected for gain shift. The 2—0 matrix is obtained by taking the [71,72] double-coincidence event and placing it in an E, vs. Ej energy grid as a b12 event. By requiring at least three Ge detectors to "fire" together, the number of double-coincidence events .in the matrix is tripled because there are three different combinations of two detectors. Therefore three bij matrix points are obtained out of one triple-coincidence event. In the classical two-detector 104 experimental setup only one bij matrix point is obtained out of one double-coincidence event. To improve the statistics, the two-dimensional energy matrix was further folded onto itself around the 45-degree axis. By projecting the two-dimensional spectrum on one axis, the total projection spectrum was obtained. This spectrum contains all the counts (or coincident events) that were recorded during and sorted after the experiment. On this total projection spectrum, all energy gates corresponding to the y rays of interest were determined. By gating on a y ray along one of the axes of the energy matrix and projecting on the other axis, all y rays in coincidence with it were obtained in the projected spectrum. For example, an energy gate placed on the y ray correSponding to the 2+ + 0+ transition should show all the y rays that are in coincidence with it (Figure 5.6). Of primary importance is the correction of coincidence counts arising from the background lying underneath each gating transition in the total projection spectrum. Each y-ray peak in the total projec- tion spectrum lies on top of a background made up mostly of Compton scattered quanta coming from high-energy continuum y rays. Therefore, it is necessary to subtract this continuum contribution from the gated spectra. The technique used involved the construction of a Master Background spectrum which was subtracted from each gated spectrum. This Master Background was obtained by placing gates in different regions of the total projection spectrum, where there are no peaks, 105 4800i 4-00 4236* 3668 . 2’-o’ GATE 0 BKMD- ##4 25324 I74 I 202 2.2 l964' I229 ”2340 Omxnskflxnnd H39€L (328' IOO 3630 560 Enema/flew? Figure 5.6 The 2++0+ gate spectrum for 128Ce obtained in the y-y coincidence experiment. 106 and summing them together. A gated spectrum was obtained by multiplying the Master Background spectrum by a factor that is pr0por- tional to the background area underneath the gated peak and subtracting it from the gross gated spectrum. In the analysis of the gated spectra, the computer code SAMPO was used for energy calibration in a similar manner, as described pre- viously in Chapter III, Section 1, and the interactive computer code SPASM [Mi81c] was used for determining the peak areas. The intensity calculations were done by using the sum of angle-weighted efficiencies of all Ge detectors. The angle correction described in Section 2 of this chapter was used as the weighting factor for the efficiencies. The bulk of the analysis was done with SPASM because its interactive nature made "instant" and quick analysis much easier to perform than the slower SAMPO. The maximum difference in peak areas between SAMPO and SPASM was about five percent. This value is comparable to variations in peak areas caused by small shifts in determining the background level underneath a peak. 5.2 Angular Distributions The difference in angular distributions of A2 = 1,2 y-ray tran- sitions can be used, together with the general trends of sideband structure and feeding in this nuclear region, to place the transitions in the decay scheme. Therefore, angular distribution experiments can be very useful in determining the decay scheme and complement the 107 coincidence results. The classical experimental setup was described in Chapter III, Section 3, but a slightly different arrangement was used here. This consisted of using the multi-Ge array already in place and taking "singles" spectra in all detectors simultaneously, as opposed to taking "singles" spectra one angle at a time. A major problem encountered was the limited disk memory space of the PACE system. In order to circumvent this, three similar computers located in different areas of the laboratory were used. The Perkin-Elmer computer could not be used because it cannot build histograms at the high rates encountered in this experiment. In the off-line analysis many problems that were not perceived at the time of the experiment were found to be a hindrance in trying to understand the data. One of the problems was that the "singles" spectra were very complex due to the many reaction channels present in the experiment (Figure 5.7). Also, Doppler shifting and Doppler broadening of the photopeaks were problematic because in some cases they caused overlap of peaks. Extracting individual peak areas from multiplets is a quite dif- ficult task when there is Doppler broadening of the peaks. This is especially true when there are small peaks (e.g., the uppermost yrast and side-feeding transitions) which lie close to large peaks. These small peaks, which in many instances are the most interesting ones, very often lay underneath nultiplet envelopes, giving rise to very 108 I4 x IO” '3‘» 230' l2x ldt Singles at 0° HxlO“ 981:!03‘ 85xl03‘ . 2‘ 7lxl03‘ Cam's/crane! 57xl03‘ 43xl03‘ soxK? fl , , . 9| l9! ' 29: 391 49: Energy (keV) Figure 5.7 "Singlgs" spectrum of 0° for the bombardment of 112Cd with Ne. 109 large errors in the extracted areas. Most of the large peaks observed in the "singles" spectra came from the lower yrast transitions of the 4-particle reaction channel. Since adequate quantitative results could not be obtained from the "singles" spectra, other analytical paths can be employed. One possible initial way of treating the data consists of using the coin- cidence data and doing a "one-detector" sorting, i.e., choosing an individual detector and summing up, in an energy spectrum, all events considered coincident with the other detectors. For example, detector 1 might be taken as the individual detector and all events registered in detector 1 which were in coincidence with detector 2 were summed up to the coincident events with detector 3, and so on. The resultant spectrum would be the corresponding "singles" spectrum at the angle of detector 1 and will be referred to as "groupie". This is pictorially shown below. Individual Detector: 1 Ge Detectors Coincident events of: 1 ++ 2 +1++3 +1.+4 +1++5 +1.+5 +1++7 "singles" spectrum at angle 1. 110 Seven "groupies" were obtained in this manner, corresponding to the angles of the seven detectors. After efficiency corrections, the intensities for the individual y rays were plotted versus angle to obtain an angular distribution. The angular distribution plots shown in Figure 5.8 correspond to some of the most intense yrast transi- tions, where it can be seen that the distributions do not have the classical shape for A2 = 2, E2 transitions. The multiplicity of the reaction imposes a complexity on the angular distribution that is dif- ficult to surmount, but this behavior can be possibly explained in the following manner. The 4-fold multiplicity required that three y rays (corresponding to the Ge detectors) were in approximately the same spatial plane, and therefore an angular correlation among them is implied in a coincident event. Consequently, a triple angular corre- lation would have to be determined for every coincident event, which is a quite difficult and laborious software problem. Further corrections are made for the system efficiency, using well-known transitions such as the yrast E2 transitions shown in Figure 5.8. This is done by obtaining an averaged distribution curve over several theoretical E2 transitions and normalizing the experimen- tally known E2 distributions to it. That is, in the first step the theoretical angular distribution equation for partial alignment is calculated by using the following equation, w(e) = 1 + aZA'Z"ax P2(cos e) + aw?)ax Pu(cos e) , (5.2.1) Figure 5.8 111 43* + lllllllllLl } I { Jllllllllll l.5 ‘33 LO 9. II ’2? S E‘ g 0.5 93 C |—-0 GD .2: 4.. 2 IO 0) . O: 836‘ llllllnllLL 0.5 Angular distribution curves for several ground state not»? {I I lllllllllli 90° l30° |70° 90° I30° ITO° Angle rotational band transitions in 128Ce. 112 where Amax is the angular distribution coefficient for the total spin alignment. The attenuation coefficients ai (specifies the amount of alignment of the nuclear states) are averaged over several nuclear states at a constant partial spin alignment, and the angular distribu- tion coefficients for partial alignment are calculated by A1 = aiAr-inax o (5.202) The resultant theoretical angular distribution equation becomes W(e) = 1 + A2P2 (cos 9) + AHPH (cos 9) . (5.2.3) Figure 5.9 shows the theoretical distribution obtained in such a manner. In the next step, an average of the intensities of several yrast transitions was calculated at each angle, and an average experi- mental angular distribution was obtained. The correction factors were then obtained by dividing, at each angle, the average theoretical over the average experimental value. These correction factors were used on all experimental distributions, and corrected experimental Az's and Ak's were obtained. The corrected experimental Az's and Ak's deter- mined in this manner are shown in Table 5.2. A check on the validity of the method can be carried out by examining the angular distributions of known dipole transitions derived from odd-A nuclei produced in this experiment and by comparing them with the quadrupole transition distributions. They should have different angular distributions (Figure 5.10) and therefore different coefficients. 113 6 Relative Intensity Q U" I l J. IGCP EX? 1 l l l KXT’ IZCP Angle j l4£f Figure 5.9 Theoretical angular distribution using an average of several theoretical E2 transitions. 114 TABLE 5.2 Corrected experimental A215 and A4us for observed transitions in 128Ce. Angular Distributions EY Coefficient _ (keV) AZ/AO A4/A0 A551gnment 400 o.24(4) -o.11(5) 4; + 2; 551 0.31(4) -o.11(5) 6; . 4; 662 0.34(4) -o.84(56) 8; + 6; 711 o.52(6) -0.14(8) 10; . a; 577 o.43(10) -0.28(14) 12; + 10; 561 0.46(16) -o.11(4) 14; + 12; 202 -0.48(17) -o.14(15) a 222 -0.26(10) -o.20(10) 127Ce 272 -0.61(12) -0.18(9) 1258a 355 0.18(10) -D.08(13) a 425 -o.25(17) -0.18(17) (2245.6 keV) + (1820.5 keV) 491 o.54(10) -o.30(14) (2736.3 keV) + (2245.6 keV) 751 0.20(8) -o.41(11) (3766.9 keV) + (3678.9 keV) 857 0.58(12) -0.09(16) a No level assignment 115 l. W- 222 keV i 222 keV (Before Correction) { (After ((1 } LO .» } '0 I Q n - I m H (13‘ 0.5" { V ¢JIILIIILL llllllLiLJ Z‘ 272 keV 272 kev g (Before Correction) (After Correctlon) 9 E I . 1 l { ..— 9 1 CD 0: I 14144141§1 1111111111L 90° 130° (70° 90° 130° 170° Angle Figure 5.10 Angular distributgon curve for some A2 = 1,2 transitions in Ce. 116 The dipole transitions used come from 127Ce (222 keV) and 1258a (272 keV), and, for the latter, experimental angular distribution coefficients have been determined previously [Gi78]. As shown in Table 5.3, the coefficients obtained in the present experiment are in close agreement with those shown by Gizon. TABLE 5.3 Angular distribution coefficientszgor the M1 transition 13/2' + 11/2' in Ba. Angular Distribution E Coefficient Reference/ (kEV) AZ/Ao A4/A0 this work 272 -0.742(10) 0.095(6) [Gi78] 272 -0.61(12) -0.20(10) this work CHAPTER VI CERIUM-128 EXPERIMENTAL RESULTS 6.1 The Yrast Band The yrast states of 128Ce have been observed previously [Wa75] up to 16+, and the first backbend was found to exist in the vicinity of the 10+ and 12+ states. In this study, it was possible to extend the yrast band up to spin 22+ and possibly to 24*. In addition, two side- bands were also observed (see next sections). A level scheme for 128Ce is drawn in Figure 6.1. In order to observe such high spin states, it is first necessary to bring enough angular momentum into the compound system to assure population of such states. By using a 103-MeV 20Ne beam, around 45'fi of angular momentum is made available to the compound nucleus. Since 132Ce, the compound nucleus, is neutron-deficient and far from the line of beta stability, charged particle emission can compete effec- tively as a reaction channel. This competition leads to the par- titioning of the total cross section among the different reaction channels and consequently decreases the population of the desired reaction product [M076]. Using a heavier projectile such as “OAr to produce the same first reaction product would not necessarily mean that even higher spin yrast states could be populated [M076], as 117 118 egyL_1___af 1 I I : "no I I I MAL—1+ m7 ms;_JL__1f ”.2 Neu_____ .mn1_J___nf 191.9 468.2 1 0.9 721.3 an en: 33219—1 .... um ML... “.9 1319...: as uqe_1 «12 7321 "a ( 9+ Figure 6.1 Level scheme for 128Ce. 18.7 119 noted previously for 128Ce [Ha81]. The (“0Ar,4n) reaction will popu- late a continuum region different from that produced in the (2°Ne,4n) reaction (about SD'h of maximum input angular momentum for 140-MeV “oAr beam [Wa67]), but its yrast cascade entrance channels turned out to be the same as for the 20Ne reaction. This is because some of the populated excited levels in the continuum tend to decay through a series of collective bands lying above, but roughly parallel to, the yrast line via fast I+I-2 E2 transitions, in contrast to the decay through M1 transitions that will enter the yrast line at higher spin states [M076]. Thus, the competition between the E2 and M1 tran- sitions along the decay paths in the (“0Ar,4n) reaction resulted in a similar entrance point in the yrast cascade for both 20Ne and 1”Ar projectiles. In Figure 6.2, a schematic decay path from the continuum t0 the yrast line is drawn for the 112Cd (2°Ne,4n) 128Ce reaction. By making use of the known yrast transitions [Wa75] and the total coincidence spectrum (Figure 6.3), appr0priate v-v coincidence spectra were selected for a determination of the uppermost yrast transitions. The "12+-gate” Spectrum (the spectrum gated on the 12+ + 10+ transi- tion), as well as the 14+-gate and the 16+-gate Spectra, clearly showed the y-ray peak corresponding to the 18+ + 16+ transition (Figure 6.4). In addition, each of these gates showed a more poorly defined peak, corresponding to the 20+ + 18+ transition. When the intensity of the yrast transition versus energy was plotted, the intensity of the y-ray peak corresponding to the 18+ + 16+ transition followed the general downward trend expected in this coincidence spectrum. EXCITATION E NERGY 120 CONTINUUM I41 £52 £52 h41 é. V§ \> 4 A?" (15151 I (s;>in) Figure 6.2 Schematic diagram for one possible decay path for the continuum into the yrast line for the 112Cd(2°Ne,4n)128Ce reaction. 7lxl04‘ 63xl04‘ 55xl04‘ 4nuo‘ seuo‘. Counts/Channel 3lxl0“. 23xmf. l5 xld‘ < BOxIO3 IOO Figure 6.3 121 2°- 0 224 I!“ . ‘ 25‘ T0101 Comm 420 o o m 42- . n -12 I 3:3 "“0 12316 4 «0 ”go ' ‘04 634 “4 al. ‘ .... O. I I ...-0|: 0 O IO-OO 0 e 1"” 500 700 Energy (keV) Total coincggence spectrum for the bombardment of 1 Cd with Ne using a 4-particle total energy gate. 142.1? 6‘. e’ 122 16’» I4. GATE OCT-0 0. IOT-OIC. I 20"“. 286‘ 194 W :02 IO 0 CT-OQ .22.“) r. 6‘ I4’-o IZ'GATE I‘°-o I4O to“ to‘ I W») ...-.. 870 77l ‘ 668 « 565‘ 462 1 Counts/Chanel so; 500 359‘ J 256‘)“ 1531 ..-. 4. 14’» 12’ l WWW 550 0". 9° I2'—- IO’ GATE notoe‘ 600 700 750 600 630 900 950 Energy (keV) Figure 6.4 The 20++18+ and 18++16+ transitions observed in some gated spectra in 128Ce. 123 Another widely used technique is the summing of all the gated spectra of the yrast transitions producing a "sum spectrum" that will enhance all peaks, especially those that appear only weakly in the gated spectra (Figures 6.5 and 6.6). From the "sum spectrum", the intensity of the yrast transitions can be obtained and plotted as a function of spin. In Figure 6.7 such a plot shows that the new tran- sitions fit the general pattern of the yrast cascade intensities as indicated by the smooth downward trend. Some of the newly observed yrast transitions can be further con- firmed by studying the gated spectra of these y rays. The 18+-gate clearly shows the yrast transitions, while in the 20+-gate these tran- sitions are barely seen (Figure 6.8). For the 22+- and 24+-gates, the yrast cascade cannot be clearly distinguished from the background. These peaks are on top of a large background and are only two to three percent larger in counts per channel than the background, making the error on the background subtraction comparable to the number of counts in the peaks. The angular distribution for the 18+ + 16+ transition, as seen in Figure 6.9, displays a general upward trend indicating the A2 = 2 multipolarity of the transition. The large error bars reflect the rather poor statistics found in the corresponding y-ray photopeak. Since there were many reaction products, the existence of doublets, triplets, and even quadruplets was common in the total coincidence spectrum. This created an enormous difficulty in the analysis, since neny gates of interest were contaminated with inter- fering reaction products. Very few of the yrast gates were "clean" of 124 .ucwswgmaxw mucmku:_ou momNH mg» cow =Eacuumam Sam: m.m mgamwd Se: 52m 00m 00m 09 .................... . Onm < Sn» ooJo. . ammo 60.40! 60.00“. cote. .806 691.! 6' J‘ v no. x0. + O. xN. n out: .nO. um. . x ...—5:6on Eam. no. t QOJN . no. xON (auuoug/swnoo 125 .Axmwcwumm :6 saw: umxgmev mcor pmcmcu ummca 36: m u mcwzosm =Eacpuwawue=m= as» we cowucom uwucmaxm w.m mczmwu «>8: Sham CON. CON. On: 00: Ono. COO. 00m 00m 0mm 08 2..-; 8. . .1 1.1:... I; . ____. .mhm .00: lb 61 SI 1 ooh lauuouo/swnoo ...-... o N ka ,OONN mNmN ...l... 60% 126 Intensity 24’»22* 2 1 l I I I l l 1 1 1 1 l l0 2‘ 4’ 6' 8’ IO’ l2’ l4' (6’ (8'20‘22‘2 ' If Figure 6.7 Intensity for the yrast transitions versus 1“ from the "Sum spectrum". 127 .ucmewcmaxm mucwuwucwoo momNH mg» 5666 ocpumam 66666 +6H++wH 6;“ 6:6 +wfi.+o~ use 6.6 623621 gssxpagm 00m 00h COD mm vv 1 eNTIe! WFQG¢O_ 98 9016' kg 0N5 .. x .2 (.6. .613 NNN '5? mac .88.. 0.00 N0? 9n L .2 1.! 8c whqo .9169 80 80 WOUQ/SJU‘IOQ 128 . (8‘46. 2(3' (15' Relative Intensity (6,9. =l03°l '5 1 l l I l l l 1 90° 110° 130° 150° 170° Angle Figure 6.9 Angular distribution for the 18++16+ yrast transition. 129 contaminants, as can be seen in Figure 6.10. To better ascertain and evaluate such interferences, any peak of interest in the total coin- cidence spectrum was sliced in several ways in order to enhance the coincidence information from each of the components in the complex multiplets. The multiplet of y rays lying within the gate set on the 12* + 10+ transition (see Figure 6.10) was cut into 570-572, 573-575, and 576-580 keV slices and the corresponding three coincidence spectra are shown in Figure 6.11. It can be seen that the 230-, 422-, 694-, and 764-keV transitions are contaminant lines that are enhanced in the 576-580 keV slice. These are the ground state rotational band (g.s.b.) transitions in 12“Ba [C074], and appeared strongly in the 12+ + 10+ (577 keV) gate in 1280e because the 6+ + 4+ transition of 12“Ba is of the same energy. The 128Ce isotope displays a strong backbend between spins 10+ and 12+ as seen in a classical 2 EI/fi2 versu5‘f121u2 plot (Figure 6.12). The occurrence of the backbending would indicate that the interaction between the g- and s-bands is small. In a phenomenological analysis, this interaction becomes smaller with increasing number as indicated by the sharpness of the backbend from 128Ce to 13“Ce. The 128'13L'Ce isotopes are situated near the neutron closed shell N=82. This implies that these isotopes could be in a transitional region between prolate-deformed to spherical shape. In reference [De74] it was stated that the ratio B(EZ, 4+ + 2+)/B(E2, 2+ + 0*) for the g.s.b. indicates that 13“Ce is in a transitional region while 1309132Ce behave like (permanent) deformed nuclei. The rotational 130 I900 . 26.00 I669‘ I2°- IO GATE '4321 (Also '“Ba and '37Lo) ‘e’ze “95‘ ('2‘ Do) 958 ‘ 250 Comts/Chcnnel ". 1 “. 72" 2:2“) euc' 3600””) is‘ole‘ 22 '0‘... 4844 «411271..» |4ae w (mac) 1 7.4 ”m. 247‘ W h ‘ M I” sot-riot Iol. , . , . I l ICC 300 500 700 900 Energy (keV) Figure 6.10 Spectrum gated on the 12++10+ transition in 1 8Ce. 131 .pcmewcmaxm mocmuwucwou mumu. mg» so. Echumam mo:wu.u=.ou .mpou on» c. no.6..ae m .o mmowpm .mcmcm ucmcm.+.o ...o «gnaw. 521.a:!m 00m 00.. . .0 com ._ ._._ ... .._.. -- .-.. :_ ... .... .-.. ... 8:3 ultcfica 8.5 )3. N501 05$ .0» CO. -<_.o mm .mm L0 .w: .0! .vt . now Gnu ..151.11I... In9. mbnl who ol« 0. . ON. .OMN 6% one 08 6k@ .00... 0mm _ 6.1.0... . 66.. e e 0! J0. AUDVN$ e0 00600 "N0 .38.. can .919! 396% 61‘! eQOUO 096. 8.6 >8. 08195 ...... .88.. .013 m.— .OnN r men "WM/3WD” 132 110‘ 100‘ ‘300e 90‘ 13299 ‘\\:> 80 4‘ ' ‘_—””. 128Ce _r" 131.“ > 70‘ a: E N 50‘ 4: - t3 . N 50° / ) _ 40- /////’//::///' ,/) '\\\ 201 / ' 132Ce . 10 . . . . . . . 0 0.05 0.10 0.15 0.20 0.25 030 035 ’11sz (Mlth Figure 6.12 Plot of 2 9/h2 versus 6202 for 128"13‘*Ce. 133 behavior of the 1Z8'13‘*Ce isot0pes are readily observed by the level sequences as in Figure 6.13 where EI (level energy) versus I(I+1) is plotted. It can be seen that the yrast band displays rotational beha- vior both before and after the backbending region. The different deformation among the Ce isotopes is further deduced from the increase in energy of the first excited 2+ states as neutrons are added. (The 2+ + O+ transition energy increases from 207 keV for 128Ce to 409 keV for 131*Ce.) In addition, with increasing neutron number the measured lifetimes for the first 2+ levels of the 130'13‘*Ce isotopes [De74] decrease indicating that the heavier Ce isotopes are less deformed. The arguments above can be extrapolated to 128Ce leading to the conclusion that it is more deformed than 130Ce. Recent lifetime measurements for 128Ce performed at Oak Ridge [Jo81] seem to support this conclusion. Another illustrative way to demonstrate the behavior of the yrast sequence is in an Ix versus hm plot as shown in Figure 6.14. In this figure, the g-bands of the 128'13‘*Ce isotopes show a sequence where, at the same Ix, these bands have different rotational frequencies. This increase of rotational frequency indi- cates that at the same spin state (at low excitation energy), the more prolate-deformed 128Ce will rotate slower (higher moment of inertia) than the more spherical 13“Ce. Hence there is an indirect relation- ship between the difference in rotational frequency for the same spin among the g-bands and the deformation pattern among the isotopes. 134 ‘3OCe . 128Ce EI (MeV) Q? o r - . - . . O 100 200 300 400 500 600 I(I+1) (ha) Figure 6.13 Plot 2°f1§1 versus I(I+1) for the g. s. b. 8" Ce. (11) 28- 24* 20' 16‘ 121 81 [,1 135 0 130C? 9 I32Ce -( / / A. dz 02 d6 tux) (hdeVW Figure 6.14 Plot of 1x versus in» for 128’13“Ce. 136 The gain in alignment between the g- and s-bands can also be obtained semi-quantitatively from the Ix versus hm graph by the dif- ference in Ix near the crossing frequency. For 128,130,132s13hc9, the AIx values are (on the average) about 8.0, 9.0, 10.0 and 9.3‘h, respectively, at around‘fiw = 0.3 MeV. The large alignment gain indi- cates that the s-band is a high-j quasiparticle band. This is schema- tically seen in Figure 6.15. The h“,2 nucleons are coupled to zero angular momentum and the g.s.b. results from the collective motion of the nuclear core. With inreasing rotational frequency, the Coriolis forces will strongly decouple a high-j nucleon pair (in the case of an even-even nucleus) from time-reversed orbits, and align the pair along the rotational axis. The maximum alignment contribution from this decoupled h nucleon pair is 11/2 + 9/2 = 10 6. From the Nilsson 11/2 level diagram for protons (Figure 6.16) in 128Ce at £2 = 0.26 and e“ = 0.00 deformation, it is seen that the lowest unoccupied h”,2 orbital is near the Fermi surface, and easily accessible through low excitation energy. Hence, a large alignment gain can be obtained by decoupling the proton pair in the time-reversed h orbits. 11/2 One interesting feature of the Ix versus‘fiw plot (Figure 6.14) is the fact that the s-bands for 128’13°:132Ce are amazingly similar in Ix values up to flu ~ 0.5 MeV where the second backbend for 130Ce commences. Could this similarity indirectly imply that the defor- mation for the s-bands is the same for all these three Ce isotopes? This question can not be easily answered without a detailed calcula- tion about the nuclear shape at such excitation energy which is beyond 137 .mpwnso ummem>mcums.u c. mcom.u:: N...; .6 even 6 .o mcwpqaoumu new mampaaoo to. mcwzmcu ovumsmsom m..m ms:m.. mu. taco unseen eN—J meW._ .:111 00— ..III 6'71 11. m -. .31 061;.1. .0 —111|I mucosaom .mo.> 138 .ea..aEea.oe oo.ou.u ace oem.ouuu mcpma womu. so. Eocmm.v .m>m. commpwz o..m mgzmwu . 9.830: 9.36.... . om z§~e .m ...... S .2... S 6m. m3. S. :8. Q. .mm mm . :35. .3 RN... S. So... Na . 111111111111 .2... S . m m . ..K 8% 3 ...m no.3 SN \ EM. ~13. 111111111 Qo . .383. 9A . mm ”EMS . .. S . .m. .26 S em 5... S .. N6 . So... Q._ R ... Nxm . 5...... .2... 3 mm ADJSUB apguod alfiuis (729/3) 139 the sc0pe of this work. This suggests the possibility that the s-bands for these three Ce isotopes have similar band structure because of the resemblance displayed in the Ix versus hm plot. The adding of neutrons did not seem to significantly alter the structure of the s-bands, indirectly indicating that these bands nay possibly have a proton configuration. The s-band structure is well understood in the CSM, and further discussion is left to a later section. 6.2 The Sideband 1 The analysis of the sidebands was quite difficult due to the com— plexity of the total coincidence spectrum in the energy region of interest for intraband transitions (between 300-800 keV). Several factors contributed to this spectrum complexity, namely: 1) contamina- tion due to other reaction channels; 2) similarity in the energy of band transitions among neighboring final reaction products; and 3) restriction on the size of the two-dimensional matrix (2K by 2K) that could be built in disk storage (resolution was reduced). By using the analytical techniques described in the previous section, it was possible to build sequences of coincident transitions. The quality of each gated spectrum depended on: 1)if the transition of interest was part of a multiplet; and 2) how far up a specific gated transition is in the band sequence. Even though some 240 million coincident-pair events were recorded, the statistics in the gated spectra for the Sidefeeding transitions were not very high. Error bars could, sometimes, be as large as the peak areas of interest 140 in such a gated spectrum. One of the sequences built is seen in Figure 6.1 as the band at which the lowest observed level is 1890 keV. The angular distribution results indicate that the intraband transitions are of stretched E2 character, suggesting a AI = 2 sequence for the band. For the 425- keV Sidefeeding transition, which corresponds to the (2246 keV) + 8; decay, the angular distribution coefficients are consistent with the following transition possibilities: a) (M1,E2) mixed; b) pure or stretched E1; and c) pure or stretched M1. Because the 425-keV tran- sition decays to the 8; state, possibility a) indicates that the parent level would be 7*, 8+,or 9+ while (b) is consistent with 7' 0r 9'. The third possibility for the 2246-keV level suggests 7+ or 9* Spin assignment. Since the parity of the sideband cannot be deter- mined from an angular distribution experiment, the above results can be combined to a tentative overall Spin assignment of (7,8,9). A similar sideband was observed in 13oCe [NoP] (with a sequence of intraband transitions of 358-, 448-, 559-, 707-, 836-, and 896-keV). Its Sidefeeding pattern can also be used on a general guide in spin assignment for the Sideband 1 in 128Ce. The lowest observed level of the similar sideband in 13°Ce decays to the 6; level, while the next level up (which corresponds to the 2246-keV level in 128Ce) decays both to 8; and to 6;, placing an upper limit of spin 8 to the parent level (assuming that At > 3 multipolarities are not observed for tran- sitions in a low-lying Sideband in this mass region). Then, by comparison, the 2246-keV level can be assumed to also have an upper 141 limit of spin 8, reducing the tentative assignment from (7,8,9) to (7,8). Then, the lowest observed level of the sideband 1 in 128Ce is tentatively assigned spins (5,6). Using these spin assignments, a plot of E1 versus I(I+1) is drawn in Figure 6.17, and it can be seen that the sideband behaves like a rotational band with either assignment. A discussion of the band structure will follow in the Cranking Shell Model (Chapter VI, Section 4) discussions. 6.3 The Sideband g For the second sequence of coincident transitions, four levels were established (possibly five) and only one Sidefeeding transition was determined (Figure 6.1). The lowest level observed in this side- band has been placed at 1988.8 keV. The level at 2418.7 keV feeds into the 8; member of the ground band via a 598.2-keV transition which is coincident with a close-lying y ray at 597.2 keV. The latter tran- sition is a member of this sideband since it is observed in all gated spectra corresponding to the intra-band transitions. Reasonably good angular distribution results could not be obtained for this side- feeding transition because of the irregular behavior at different angles; therefore, no tentative spin assignment was given. This irre- gular behavior can be understood if it is assumed that the intraband transitions are E2 and that the observed Sidefeeding transition can be (M1,E2), pure E1 or pure M1, as argued for sideband 1. Then, the intensity of such a doublet will oscillate, depending on the intensity of the individual y rays at different angles. 142 EI (MeV) 2- /' 151 ('61 0 60 120 60 22.0 300 360 I (M) (’6’) Figure 6.17 Plot of EI versus I(I+1) for sideband 1 in 128Ce. 143 A similar Sideband has also been observed in 130 Ce [NoP] with a sequence of transitions 427-, 610-, 767-, 895-, and 977—keV as com- pared to the 430-, 598-, 751-keV and possibly 858-keV transitions in 128Ce. An interesting point in the 130Ce data is that transitions between the sidebands were observed while in 128Ce, no such tran- sitions were evident. The latter fact may Simply be due to the low statistics in the gated Spectra. These interband transitions in 130Ce could indicate that these two sidebands are decoupled bands due to Coriolis forces. Therefore, high-j quasiparticles would be necessary in order to form the band structure, and, consequently, large align- ment gain is obtained with increasing rotational frequency. 6.4 Ihg_Cranking Shell Model and the Observed Bands in 12903 One of the main advantages of the Cranking Shell Model is the sim- plification into compact diagrams of some of the characteristics of the nuclear behavior. The usual Routhian e' versuS‘fiw diagram indicates, for example, which quasiparticles are involved in a certain band struc- ture (and, consequently, the band parity), how much alignment gain and relative energy are involved in a quasiparticle excitation, the rota- tional frequency where two bands cross each other as well as and the interaction strength between the crossing bands. Therefore, such diagrams can be used to study the gradual changes in the valence- nucleon pairing. The rotating frame calculations can give an insight of the beha- vior of the observed bands in 128Ce. The reference band is taken to be the S-band Since it changes gradually with increasing rotational 144 frequency, and it is not affected as much by a backbending as the g-band. The "10 and ‘11 obtained are 16.5 ‘hMeV'1 and 24 ‘h3MeV'3, respectively. These parameters represent a compromise among different values that yield constant alignment for both the g- and s-bands. The experimental Routhians (e') can be obtained by using the extracted ‘30 and 91 values. In Figure 6.18, a plot of the experi- mental Routhian e' versuS‘fio is Shown. For the yrast band, the crossing frequency for the g- and s-bands iS‘fimC ~ 0.315 MeV, and the alignment gain is about 8.9 h (i = -de'/dm). The alignment gain is also conveniently observed in an i versus hm (Figure 6.19). This large alignment indicates the high-j quasiparticles are needed for the S-band structure. In the Nilsson level diagram for 128Ce (Figure 6.16) it is seen that, for both protons and neutrons, h11/2 orbitals are nearest to the Fermi surface and are logical choices for the s-band structure (high-j orbitals). One way to determine if the s-band is either quasiproton or quasineutron is by the observed blocking of the odd-proton in the neighboring l:ZLa [Na75]. The flat behavior of the (11/2') one-quasiproton band around the crossing frequency in the i versus hm plot (Figure 6.19) indicates that such proton (and its orbital) is involved in the s-band structure of 128Ce. Simplistically, the odd-proton blocks the excitation of a core-proton into its quasiparticle orbital (Pauli principle); therefore, a crossing quasiparticle band is not formed and no back- bending is observed in 127La at around‘fiwc ~ 0.315 MeV. In 128Ce, the quasiproton pair is excited into h11/2 quasiparticle orbitals, and 145 2 - ( 5) - 1 (5) \. \.§-\. ; . -\\. fig 0 \\.\\\\\ .. \.\\ -11 \ -3 E 0:3 015 0.7 Ti 00 (MeV) Figure 6.18 Experimental Routhian e' versus “T1111 for 128Ce. 1(6) 146 . I27 Q 10 ‘ «.1 band in 130C, ./. l- a 151 band in 13°C ' . . - S-bond 8- _( 6- .//,/”‘ O I ——-—’-/ "/z' i . ' ”_— (cl .. // ./ / U9 2‘ / ./ 0« -/ .._____,._. G—bond 2 . . fi. 0 0.2 0.4 0.6 ‘fiw(MeV) Figure 6.19 Experimental i versus ho for 128Ce. Also, 130Ce sidebands are plotted. 147 backbending occurs with the crossing of the g- and s-bands. Then, both quasiprotons contribute to the s-band structure. The theoretical Routhian diagram (e' versus hm) for protons in 128Ce is obtained from the Cranking Shell Model calculations, and is presented in Figure 6.20. The crossing frequency between the g- and s-bands is calculated to be around‘fiwC = 0.317 MeV(obtained as the rotational frequency of closest approach between curves A and -A as well as B and -B), and the alignment gain is about 5.0 + 3.4 = 8.4 h (obtained by adding the maximum slope of the curves A and B) which agrees well with the experimental data in the rotational frame (filoc = 0.315 MeV and 8.9‘h for the alignment gain). Additionally, the theoretical Routhian diagram for neutrons in 128Ce (Figure 6.21) yields fimc ~ 0.66 MeV for the first backbending, inplying, therefore, that the S-band is indeed a h11/2 two-quasiproton band. The sideband 1 is more difficult to understand in terms of the Cranking Shell Model. In the i versus hm plot (Figure 6.19), the side- band is shown to continuously gain alignment with rotation of the core, meaning that there is no blocking of h 2 quasiprotons and, 11/ therefore, this Sideband could have quasineutrons in its structure (see explanation of blocking particles in this section). There is also a small kink aroundiflm ~ 0.32 MeV that could be somehow related to the first backbend (hmC ~ 0.315 MeV). There are two possible explanations for the observed behavior of sideband 1. One is that sideband 1 starts as a (h1 ) two- 1/2d5/2 quasiproton band at low rotational frequency and crosses with a 148 .33. c. 336.... to. 56.66:. a: 23.8., .0 c6233. 18:950.: cm... 0.53... . . m8- . 2.9. 1 fig- jc (Aalnl) 1 a . m8 149 .mUou. c. mcosuzm: co. Eocmmwv 3: mzmcm> .o cmwzuaom .mu.pmcomze .N.m mcamwm 962. 3c m8 03 mod ,3 (ABW) 8.0 08 R~nwn .\ no... 3 1...... -.....o. 150 (h3 d ) four-quasiproton band (at a rotational frequency higher 11/2 5/2 than the backbending frequency seen in the yrast sequence) with a possible maximum alignment of 11'fl(adding the maximum slopes of curves A,B,C,and D). This is schematically seen in Figure 6.22(a) in an ideal i versus hm plot. The flat behavior of the sideband around we shows the blocking of a h 2 quasiproton, indicating the presence 11/ of this quasiproton in the sideband structure. This behavior is Simi- lar to 127La where the one-quasiproton g.s.b. crosses with a three- quasiproton band [NoP] at'hw ~ 0.50 MeV (Figure 6.19) and has a Ai of about S‘fi. One of the problems for this possibility is that Sideband 1 starts at low rotational frequency with an alignment (for either Spin assignment) that is smaller than what is observed for the h11/2 one-quasiproton band in 127La, suggesting that sideband 1 may not start (at low ho) as a twoquasiproton band. A more plausible explanation is that the sideband starts (at low‘hw) as a (h11/2d5/2) two-quasineutron band and crosses with a (h 2 ' e . 11/2dS/2)n(h11/2)p two-quas1proton-two-qua51proton band (Figure 6.22(b)). Therefore, this is a 2n + 2n2p excitation. Two "fingerprints" of the crossing can be seen in the i versus hm plot (Figure 6.19): 1) the crossing frequency - the two-quasineutron excitation is not blocked by the two-quasiprotons in the s-band, therefore, sideband 1 Should strongly backbend around hm ~ 0.32 MeV (actually, it only "upbends" slightly); 2) large alignment gain due to the presence of h quasiprotons in the four-quasiparticle exci- 11/2 tation - higher spins were not seen for the sidebands, but from 1 (f1) 1% ' 12 ‘ 12 ‘ 151 660 (MeV) 9 (1011/2 dis/2) 1“ (1211/2) : l M 117;.) — .../3‘13““1 (2332 050 ° ‘h w (M e V) Figure 6.22 Schematic diagram for two possible excitations for sideband 1 in 128Ce. 152 Figure 6.19 it is seen that the Sideband is still gaining alignment up to measured spins with yet no indication of reaching maximum alignment. The "upbend" and "backbend" behavior is a reflection of the interac- tion strength between the crossing bands, where the former indicates large interaction, as discussed in Chapter II. This difference in interaction ("upbend" for the sideband versus "backbend" for the yrast band) could possibly indicate a deformation change for the excitation related to the sideband. In summary, the experimental resuls for the crossing frequency of the first backbend (fiwc ~ 0.315 MeV) as well as the s-band structure (h two-quasiproton band) are reproduced very well by the Cranking 11/2 Shell Model calculations. For sideband 1, it is more difficult to obtain an accurate picture of its behavior with this model. One possi- ble reason for the observed behavior is that the band starts as a two- quasineutron band and crosses with a two-quasineutron-two-quasiproton band at'hwc ~ 0.32 MeV. The model can account for the alignment gain and crossing frequency, but the interaction strength is not so well predicted. For sideband 2, additional information about the spins of the levels is necessary in order to use the Cranking Shell Model. CHAPTER VII CONCLUDING REMARKS The nuclei studied in this work, 128Ce and 168Yb, represent two different regions of deformation. The former is in a prolate- spherical transitional region, while the latter is in a well-deformed (rare-earth) region. Therefore, their single-particle and collective spectra should differ somewhat because of the different valence orbi- tals involved in the Spectra. In this study it was possible to extend to higher spin states the known sidebands in 168Yb as well as to deter- mine the band members of the previously known 5' isomer. For 128Ce, the yrast band was extended well beyond the first backbend and two sidebands were also determined. A useful nuclear model should account for the observed spectra, such as those obtained for 168Yb or 128Ce. One such model that has been successfully applied in the well-deformed rare-earth region, especially for the light Yb isotopes like 160Yb, is the Cranking Shell Model. The main ideas in the CSM are that the excited states in the rota- tional spectra result from the arrangement of a few quasiparticle orbitals, and that these excited states are composed of quasi- particle(s) relative to a reference state. The nuclei studied in this work were good examples to test the wide applicability of the Inodel because of the different deformed regions involved and the 153 154 backbending/non-backbending behavior of the two isotopes. This work represents one of the first attempts to apply the model in the mass region around A ~ 128. For 168Yb, CSM was reasonably successful in predicting band structure, alignment gain, and excitation energy of the observed rota- tional bands. Also, the non-backbending behavior was accurately described. For 128Ce, the backbending behavior was predicted rea- sonably well, and the band structure and alignment gain of the S-band were accurately described. One of the sidebands could be only par- tially described, probably due to band structure changes at higher rotational frequency. It is clear that more theoretical work iS needed in relation to proton-neutron mixture and high-K bands in order to explain finer spectroscopic details of the rotational spectra. APPENDIX APPENDIX A Some Important Gated Coincidence Spectra of Transitions in 168Yb This section shows some of the important background subtracted gated Spectra that were obtained from a coincident experiment with two planar Ge and large volume Ge(Li) detectors. The peaks with a * could not be identified and those with a T are contaminant y rays originating from 169Yb, 19F, 27Al, or the oxide matrix. Since these spectra are compressed, statistical fluctuations in the background may add up to give a peak but these are easily iden- tified in the normal 4K spectra. The spectra numbered A1 through A12 are the "Sidefeeding tran- sitions between the v- and the ground state bands". The spectra num- bered A13 through A19 are "Intraband transitions for the K=3+ and K=5' bands". 155 $86833. .- '9 I 22511 Figure A1 ’0' 6‘4 1 1 l 1 1 1. 111'11111» 2450 l i . , 'l 111. . .' 17‘ star 1‘. 1! 11L 1 ‘ 1‘ ‘ 111111 11- 11 111.11 M 1= 11 2630 omen. 156 4‘02. 648.4 keV 1150 157 42.2. I 716.8 keV 2’» o’ 32.4’ I?“ was 21“ 19' 15- I2- | I, “ Ilifl T"! HIJ I if 1‘1! 1: 1 .7: [1111‘ 1‘1 I , 1'. H11. 24 2250 $0 2650 2050 ”50 Figure A2 158 x30' 4"? " 119‘ 4 1,, 822.4 keV 95‘ 4 MM 3‘ ’50 115. n? I ‘3- . 54' . O O 451 9’6 3' 2% , 1300 15“ 1700 1900 21” m 1 71 I‘- .. 12‘ ‘ IO- ‘ 0‘ d 9 1 .1 I 1 1H 1|? 1 - 1H! II‘ 1‘ ‘1“ “"1“!“ “I“ ”“ I “1?“ " 1| “I I” 2250 2450 2650 2850 350 m Figure A3 Figure A4 159 2‘. 4“ 2. '1 231‘ " "9 859.6kev 165' 4 132' 2“ d 1 ” C ‘6‘ 1 1“. c‘ 4. I” 1500 in I” 2’“ m 1 V 10 ‘ 12‘ ‘ 10' J o ‘ fl ‘ 11 1‘ 1‘ 1 1111111 N“ 3““*li I] '1gw‘wwli I 1 2359 2450 35. 2953 am ”5° Figure A5 160 t-OGO 16' 12- 1300 1m ”1 10‘ 15‘ 12" 1_. 1 :11 ‘1 .1 111-14111111111111 2430 2650 m 11 1ll‘iil1“1 fl—o 2. .1 8840/8841 keV 1 950 use 1024‘ 1 IIIIIIII ;11l 2t” 161 99. 220‘ 11 "J 895.6 keV ' w 1 w 1 4.. . ” ‘ n 1 to . 350 588 no 9511 use am ..1 1 121 i IOJ ‘ . . 1 1 1 ‘ 1‘ ‘ 1 1.11 » 111,11, 1 1 ~11:11w111mIIIII 11111 111111 11 1111 1111 11 I“ 15“ 1m 1m 21“ omen. 1 1.1 4 12- ‘ w ‘ o 6 4 Figure A6 Figure A7 q 2' 19‘ 15' 12‘ 1111“ 1 162 530 no om 304.8 304.9 304.1 I \ 15“ 583.4 1m w, ,1,.1..,,1 um 978.8 keV 4 use 21“ Figure A8 163 m. ‘Lz. . 1’ 3%- 1015.5 keV 2” 2401 Q 0 144- % 1 f 1 330 $0 730 93. 1150 . am 20' 2* 034° 20' d 1.11.11...111....11..L..... 1 _ mo 15” 1m 1908 210 amen. 1 1 ”I 4 241 20‘ 539.9 M! 16A “‘ c.2 33' 4 3,, 1032.0/103129kev 29' 4 2“ 230‘ 150' 1 I“ 1 91 1 350 50 no ”0 115. . um I” 3.04. 1 160‘ .1 I“ .1 1” «1 W 72' 8 02.8‘ 49' 24' me 1500 17. I” 21” m 1 '11 1“ 1 12" 10‘ .1 '1 ‘1 1 1 1 I 111 l ‘ 1 111 1 ' 1111111 111111111 0111. ‘1‘ . 1‘ 1.1. 1 1111 1 1111 11 111 111 1W1 . 11 1111! ‘III “ ‘~ llll r11 1 111 1111* «'1 ~I1I1111~1 2250 245° 2650 2830 v Figure A9 Figure A10 165 do" as- eko‘ . 3' “2 1082.8kev 25 20 15 :0- 350 as 750m 950 use 1 1 H 1H} ‘11 1’1 7 H 1 _ 1 no. me 1m 2m 1; I 12- .. .1 Figure A11 166 42.2’ 1158.1 keV 1 1 ‘ 1 ‘ ‘1 ;‘1 1| ‘I I IJI‘IIII 1| 1|1 |1|11 'J‘I‘ll‘l 13a 13“ 1700 I” 21” am 1.1 10‘ 2250 2650 2650 2850 350 L-_I V 167 1232.5keV 4%.? 29' “0’40 16' 12' 15” 1700 I” 21“ um ' T “I . t2- !0 < O- 6' 4.1 1 f I ‘ I l 1‘11 2250 2450 2650 2850 350 anon. Figure A12 491 42- 16" 12" Figure A13 28” 168 3.4.8 ISOQJ 99.6 keV GIID $3 169 49' 311.0 304.8 ‘2 123.0kev 4‘»? ”I 1 324.! 29 no.0 ' 21- ' ,Iull&..1.1u11u 1.. no 9” UN 11’ ~ 1 .AHJHiJ “1111111 1 300 um 16' 12' I 11 I 11."I. . I I I I1 1 1;I f .11 ‘ 111.1 11. ~11"...1 .~.?11 .... .. .111... 1250 1450 1650 1850 2030 am 1 up 201 2" 4 I266! 20- ‘. |564J Figure A14 1&1 15' 121 Figure A15 170 130‘ “,2. 4’-2° 1450/1459 kev ”‘5 mo 1‘- 97... ‘ 12 III 1111 11. 1300 15“ I?“ 1,1 III II III 1II 1I1I1 1.1.11 1‘ 'Il' IIIII11II1II 2259 233. “Owen 171 m7 + 1 I” 1641/1660 keV 19‘ 125' IIIIIIIIIIII III1'III.1II1 11I1Il|III LI..‘II11 l1I1. 11” mm 10129 I 2+ 2.1 “I 1205.2 1015: 1 12 / 'IIII I111 IIl I‘ II.» II I .I 1 I'IIliIII I; Illll III'I III III|CL I‘IIII II 2050 2250 2450 2650 2850 am Figure A16 Figure A17 Meat. 172 ‘1! ‘. ‘ 'mll H‘ w |"'1 2430 m H“ 1r 2650 2227 keV [36%| fl; 'JL 1 ”I“ 1 I‘I 2850 Figure A18 173 .2 I so 4.4. 2680 keV ‘ 3' 4 2434' we 24‘ no.0 2151‘ . w 254.7 L245 I an an no no no . am 20 no . 15' 4 12- * ‘ I W ‘ Wen I‘lllllill Ill 1950 1250 am 12' can | 1:94.: Figure A19 174 133' 4‘ m... 309.9/3ll0kev 304.8 200.7. 5ND ' . 1.! ml! :‘Is‘... I'l Hm“ .I .‘ 9. 307.6 l I III! I ‘l "I .m i m I“! N hm ll 1m I‘ll m 22“ 2‘5. 2530 m m REFERENCES 175 REFERENCES A [An76] Andersson, G., Larsson, S.E., Leander, G., Moller, P., Nilsson, 5.6., Ragnarsson, I., Aberg, 5., Bengtsson, R., Dudek, J., Nerlo-Pomoska, 8., Pomorski, K., and Szymanski, Z., Nucl. Phys. A268 (1976) 205. [Au72] Au, R., IIEVENT, National Superconducting Cyclotron Laboratory, Michigan State University, 1 72, unpublished. B [Ba57a] Bardeen J., Cooper, L.N., and Schreiffer, J.R., Phys. Rev. 19§_(1967) 162. [Ba57b] Bardeen, J., Cooper, L.N., and Schreiffer, J.R., Phys. Rev. lgfi (1957) 1175. [8e36] Bethe, H.A., and Bacher, R.F., Rev. Mod. Phys. §_(1936) 82. [8e37] Bethe, H.A., Rev. Mod. Phys. g_(1937) 69. [8e59] Begygev, S.T., Mat. Fys. Medd. Dan. Vid. Selsk. 81, N0. 11 [8e77] Bengtsson, R., and Frauendorf, 5., Proc. Int. 5 m . gg_High- Spin States and Nuclear Structure, Dresden,'E. rmany, September 1977: [Be79a] Bengtsson, R., and Frauendorf, 3., Nucl. Phys. 5815 (1979) 27. [8e79b] Bengtsson, R., and Frauendorf, S., Nucl. Phys. Egg; (1979) 139. [8036] 80hr, N., Nature lgl (1936) 344. [8052] Bohr, A., Dan. Mat. Fys. Medd. 26, No. 14 (1952). [8053] 803g§)A., and Mottelson, B.R., Dan. Mat. Fys. Medd. 21, No. 16 [8069] Bohr, A., and Mottelson, B.R., Nuclear Structure, Vol. I (Benjamin, New York, 1969). [8075] [8077] [Bu67] [C669] [Ch70] [Ch72] [Ch73] [Che] [C074] [CS78] [de74] [De74] [Di66] [Dr77] 176 80hr, A., and Mottelson, B.R., Nuclear Structure, Vol. II (Bemjamin, New York, 1975). 80hr, A., and Mottelson, B.R., Proc. Int. Conf. on Nucl. Struc., Tokyo, 1977. “ Burke, D.G. Elberk, 8., Mat. Fys. Medd. Dan. Vid. Selsk. 36, No. 6 (1967). "' C Camp, D.C. and Van Lehn, A.L., Nucl. Instr. Meth. 72 (1969) Charvet, A. Duffait, R., Emsallem, A., and Chery, R., Nucl. Phys. A156 (1970) 276. Charvet, A., Chery, R., Phuoc, D.H. Duffait R., Emsallem, A., and Marguier, G., Nucl. Phys. A197 (1972) 49 . Charvet, A., Chery R. Duffait, R., Morgue, M., and Sau, J., Nucl. Phys. A213 (1973) 117. Chen, Y.S., private communication. Conrad, J., Repnow, R., Grosse E., Homeyer, H., Jalschke, E., and Wurm, J.P., Nucl. Phys. A234 (1974) 157. Frauendorf, S., and Bengtsson, R., "CS Code", Copenhagen, 1978. D der Mateosian E. and Sunyar, A. Atomic Data and Nuclear w. Data Tables 13 (1974) 392; ibid. 408. Dehnhardt, N., Mills, S.J., Miller-Veggian, M., Neumann, U., Pelte, D., Poggi, G., Povh, 8., and Taras, P., Nucl. Phys. A255 (1974) 1. Diamond, R.M., Matthias E. Newton, J.O., and Stephens, F.S., Phys. Rev. Lett. 16 (1966) 1205. Dracoulis 6.0., Fer uson S.M. Newton, 0.0., and Slocombe, M.G., Nuci. Phys. A2 9 (1977) 251. [Elb] [Fr81a] [Fr81b] [Ga80] [GaVJ [Gi77] [Gi78] [6074] [Gr67] [Gu67] [Ha49] [Ha81] [1054] [In69] 177 Elbek, 8., unpublished. F Frauendorf, 5., Nuclear Physics Workshop, International Center for Theoretical Physics, Trieste, October 1981. Frauendorf, 5., "The CSM Bible Lectures", University of Tennessee, April 1981. G Gavron, A., Projection Angular Momentum Coupled Evaporation Monte Carlo Code (PACE), Oak Ridge National Laboratory, 1980, unpublished. Gavron, A., private communication. Gizon, J., and Gizon, A., Z. Phys. A281 (1977) 99. Gizon, J., and Gizon, A., Z. Phys. A285 (1978) 259. Goodman, A.L., Nucl. Phys. A230 (1974) 466. Grover, J.R., Phys. Rev. 151 (1967) 832. Gustafsson C. Lamm, I.L., Nilsson, 8., and Nilsson, S.G., Ark. Fys. 39 (1967) 613. H Haxe] 0., Jensen, J.H.D., and Suess, H.E., Phys. Rev. 15 (1949) 1766. Hattula, J., to be published. I Inglis, D.R., Phys. Rev. 2§_(1954) 1059. Inglis, D.R., Physics Today, 22 (1969) 29. [Je70] [Jo72] [Jo81] [Kh73] [Kr68] [Li37] [L070] [Ma49] [Ma69] [M159] [Mi816] [Mi81b] [Mi81c] [M075] [M076] 178 J Jett, J.H., and Lind, D.A., Nucl. Phys. A155 (1970) 182. Johnson, A., Ryde, H., and Hjorth, S.A., Nucl. Phys. A179 (1972) 753. “" Johnson, N.R., unpublished. K Khoo T.L. Naddington, J.C., and Johns, M.w., Can. J. Phys. .5; (i973) 2307. Krumlinde, J., Nucl. Phys. A121 (1968) 306. L Livingston, M.S., and Bethe, H.A., Rev. Mod. Phys. 9_(1937) Lobner, K.E.G., Vetter, M., and Honig, V., Nucl. Data Tables 51 (1970) 495. M Mayer, M.G., Phys. Rev. Z§_(1949) 209. Mariscotti, M.A.J., Schraff-Goldhaber, G., and Buck, 8., Phys. Rev. ll§_(1969) 1864. Migdal, A.B., Nucl. Phys. 13 (1959). Milner, w.T., Holifield Acquisition Task (HAC), Oak Ridge National Laboratory, 1981, unpublished. Milner, w.T. Disk Monitoring (DISMO), Oak Ridge National Laboratory, 1981, unpublished. Milner, w.T., Spectrum Analysis Program (SPASM), Oak Ridge National Laboratory, 1981, unpublished. Morgan, C.B., KKRECDVERY, National Superconducting Cyclotron Laboratory, Michigan State University, 1975, unpublished. Morinaga, H., and Yamazaki, T., In-Beam Gamma-Ray Spectroscopy, (North Holland, 1976). [Ne70] [Ne76] [N155] [Ni69] [Nol] [NoP] [0971] [Ow81a] [0w81b] [P177] [Pr75] [Ra50] [Ri80] [Ri81] [Rie] 179 Neergard, K., and Vogel, P., Nucl. Phys. A145 (1970) 33. Neergard, K., Pashkevich, V.V., and Frauendorf, S., Nucl. Phys. A262 (1976) 61. Nilsson, S.G., Dan. Mat. Fys. Medd. 29, N0. 16 (1955). Nilsson, S.G., Tsang, C.F., Sobiczewski, A., Szymanski, S., Wycech, C., Gustafsson, 6., Lamm, I.L., Moller, P., and Nilsson, 8., Nucl. Phys. A231 (1969) 1. Nolan, P.J., Todd, D.M., Smith, P.J., Love, D.J.G., Twin, P.J., Andersen, 0., Garrett, J.D., Hagemann, 6.8., and Herskind, 8., accepted for publication in Phys. Rev. Nolan, P.J., private communication. 0 Ogle, N., Wahlborn, S. Pilpenbring, R., and Frederiksson, J., Rev. Mod. Phys. 43 (1971) 424. Ower, H., Oak Ridge National Laboratory, 1981, unpublished. Ower, H., Oak Ridge National Laboratory, 1981, unpublished. p Plasil, R. ORNL/TM-6054, Oak Ridge National Laboratory, November 1977. Preston, M.A., and Bhaduri, R.K. Structure 9j_thg_Nucleus, (Addison-Wesley, Massachusetts, 1975). R Rainwater, J., Phys. Rev. 19 (1950) 432. Riedinger, L.L., Proc. Int. Conf. on Band Structure and Nuclear Dynamics, New Orleans, Louisiana, 1980, andTreferences therein. Riedinger, L.L., Phys. Scr. 24.(1981) 324, and references therein. Riedinger, L.L., private communication. [R005] [R069] [R081] [St75] [Sik] [Th62] [T168] [Wa67] [Wa75] [Wa79] [Wa80] [Wal] [Ya67] 180 Routh, E.J., A Treatise on the D namics of a_S stem of Rigid Bodies, Vol. 11, 6th Ea._(Macmilian, LonHEn, ; reprinted 5y Dover, New York, 1955. Ragtti, J.T., and Prussin, S.G., Nucl. Instr. Meth. zg_(1969) Roy, N., Jonsson, S., Ryde, H., Walus, W., Gaardhoje, J.J., Garrett, J.D., Hagemann, 6.8., and Herskind, 8., Internal. Report, University of Lund, Sweden, November, 1981. Stephens, F.S., Rev. Mod. Phys. 41_(1975) 43. Sikkeland, T., and Lebeck, 0., CSBN, University of California, unpublished. T Thouless, D.J., and Valatin, J.6., Nucl. Phys. §l_(1962) 211. Tjom, P.O., and Elbek, 8., Nucl. Phys. Algz_(1968) 385. W Ward, 0. Stephens, F.S., and Newton, J.D., Phys.Rev.Lett. 12.(1967) 1247. Ward, 0., Berschat, H., Butler, P.A., Colombani, P., Diamond, R.M., and Stephens, F.S., Phys.Lett. §§B_(1975) 139, and references therein. Walker, P.M., Faber, S.R., Bentle , W.H., Ronningen, R.M. Firestone, R.B., and Bernthal, F. ., Phys. Lett. §§§_(1979) 9. Walker, P.M., Faber, S.R., Bentley W.H. Ronningen, R.M., and Firestone, R.B., Nucl. Phys. A943 (1980) 45. Walker, P.M., Carvalho, J.L.S., and Bernthal, F.M., to be published. Y Yamazaki, T., Nuclear Data A3 (1967) 1.