A Thesis. Presented to. The Faculty of Graduate Studies. The University of Guelph ANDREW ALLAN PHILLIPS. In partial fulfilment of requirements

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1 STRUCTURE OF 186,188 Os STUDIED WITH ( 3 He,d) REACTIONS A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph by ANDREW ALLAN PHILLIPS In partial fulfilment of requirements for the degree of Doctor of Philosophy August, 2009 Andrew Allan Phillips, 2009

2 ABSTRACT STRUCTURE OF 186,188 Os STUDIED WITH ( 3 He,d) REACTIONS Andrew Allan Phillips University of Guelph, 2009 Advisors: Professor P.E. Garrett Professor C.E. Svensson The vibrational nuclear structure in Os isotopes has been the subject of debate for decades. In particular, the nature of the level was contested to be double quadrupole phonon (γγ) or a single hexadecapole phonon. The γγ view is based on collective B(E2) values from Coulomb excitation and lifetime studies. The single hexadecapole phonon interpretation is supported by a population of the state in single-proton (t,α) transfer reaction work and an enhanced E4 matrix element from inelastic scattering of α particles and protons. The proponents of each point of view offer criticisms to the other, but no new experimental work had been performed recently to aid in a resolution. We set out to add more data to this debate by performing a single-proton ( 3 He,d) transfer reaction experiment. The experiment was performed at the Maier-Leibnitz Laboratory in Garching, Germany. The 30 MeV 3 He beams provided by the tandem Van de Graaff accelerator bombarded 185,187 Re targets while the Q3D spectrograph analyzed the momenta of light ions and focused them onto a focal plane detector for identification and energy measurements. The spectrograph was rotated to angles between 5 and 50 for the transfer reaction work and cross sections were deduced for excited states in 186,188 Os. Results show a significant [402]π [402]π squared amplitude for the level consistent with Quasiparticle Phonon Model calculations predicting a dominant hexadecapole component and a large γγ component.

3 Acknowledgements Firstly, many thanks go to my advisors, Paul Garrett and Carl Svensson. They have always been willing to take the time to answer my questions and teach me more than just physics. I m very grateful to have had the opportunity of being their student, to learn new skills, and attend many experiments and conferences. I m especially thankful for their guidance, planning, and understanding during the process of finding a suitable research project. Secondly, I d like to thank my grandadvisor Dennis Burke for all of the perspective, knowledge, and feedback that he has given me over this work; it has been interesting to learn how the Os vibrational structure debate has evolved over the decades. Thanks must also go to Nicola Lo Iudice and Alexander Sushkov for performing new Quasiparticle Phonon Model (QPM) calculations in support of this work. Danke schön to my German colleagues from the Maier Leibnitz Laboratory (MLL) who worked tirelessly at keeping the experiment running. They were very helpful in teaching me what information from the Q3D spectrograph is needed to deduce cross sections: Thomas Faestermann, Ralf Hertenberger, Reiner Krücken, Rudi Lutter, and Hans-Friedrich Wirth. I d also like to thank Linus Bettermann and Norbert Braun (Universität zu Köln) for their help during the experiment. i

4 Life in grad school would have been dry and dull without the friendly and funny grad students and post-docs I had the pleasure to work with: Corina Andreoiu, Dipa Bandyopadhyay, Jack Bangay, Laura Bianco, Greg Demand, Paul Finlay, Katie Green, Geoff Grinyer, Bronwyn Hyland, Kyle Leach, Evan Rand, Michael Schumaker, Chandana Sumithrarachchi, Smarajit Triambak, Jose Javier Valiente Dobón, and lastly James W(r)ong whose name I may have spelt wrong. I would like to thank Erin O Sullivan and Katie Green for being the guinea pigs in testing my RADBUDDY program. It was also great to meet many undergraduate students through my position as a teaching assistant. I ve learned a lot from them through that experience. I would like to acknowledge the support by many members of the faculty and staff in the Department of Physics at the University of Guelph. In particular, my advisory committee members: Jimmy Law (Guelph) and Jim Waddington (who is my other grandadvisor from McMaster University). It s now time to give thanks to some people who I have never met or contacted. Thank you to Canadian taxpayers, who have supported me through NSERC and OGS awards. To Jorge Cham, the animator of PhD comics: many thanks for your hilarious portrayal of grad school life. To Alfred Aho, Peter Weinberger, Brian Kernighan and anyone else who has developed the AWK programming language: thank you for making my work much easier! Lastly, I d like to thank my family for their never-ending support through it all. To my sister and brother-in-law, Elizabeth and Wayne, thanks for being there for me. A special thanks to my parents, Cheryl and Roger: I couldn t have done it without you, I can t say thanks enough. Finally, to my Labrador retriever Emma: I look forward to tossing around a ball more often now that the thesis is done. ii

5 Contents Acknowledgements i 1 Introduction Background concepts Deformation Rotation Vibration Quasiparticles Motivation Timeline of events Criticisms of each view Overview Single-nucleon transfer reactions in deformed nuclei Single particle states in spherical nuclei Single particle states in deformed nuclei Elastic scattering Optical model parameter sets Transfer reaction cross sections iii

6 2.4.1 Transfer theory Performing transfer cross section calculations Refinements to single-nucleon transfer reaction theory Coriolis mixing Performing Coriolis mixing calculations Calculation of pairing factors Summary Materials and methods Online: experiment equipment Target properties Quadrupole-three-Dipole magnetic spectrograph (Q3D) Solid angle controls Focal plane detector Beam normalization and dead-time corrections Offline: Data analysis Peak fitting Components of the peak fitting function Fixing parameter trends Choosing the best polynomial degree Cross section measurements Energy calibration Effective excitation energy transformations Calculation of uncertainty in effective energy Corrections due to energy loss in targets Calibration results iv

7 3.5.5 Run-by-run energy calibration Summary Optical model parameters, DWBA calculations, and results He elastic scattering He optical model parameters ( 3 He,d) transfer reactions Deuteron optical model parameters Comparing ( 3 He,d) DWBA calculations and systematic uncertainties Coriolis mixing calculations ,188 Os cross section angular distributions Summary Nuclear structure of 186,188 Os Ground state bands γ bands K π = K π = bands bands K π = 3 bandheads Other results l = 0 strength Candidates for K π = 1 + band members Summary Conclusions Future research directions v

8 6.1.1 Optical model parameters Other nuclei in the vibrational structure debate Bibliography 173 A Excitation energy plots 180 vi

9 List of Tables 1.1 B(E2) values deduced from Coulomb excitation studies Calculating the momentum mismatch for 191,193 Ir(t,α) 190,192 Os reactions Calculating the momentum mismatch for 185,187 Re( 3 He,d) 186,188 Os reactions Terms in Hamiltonians describing transfer reactions Calculated pairing factors for spectroscopic strength and Coriolis mixing calculations for the 185 Re( 3 He,d) 186 Os reaction Calculated pairing factors for spectroscopic strength and Coriolis mixing calculations for the 187 Re( 3 He,d) 188 Os reaction Target properties Reaction Q values Summary of cross sections and excitation energies in 186 Os Summary of cross sections and excitation energies in 188 Os Uncertainty budget for a sample effective excitation energy calculation Particle kinetic energy loss corrections vii

10 4.1 3 He optical model parameter sets Deuteron optical model parameter sets Naming convention for groups of optical model parameter sets Comparison of spectroscopic strengths in 187 Re from different optical model parameter sets Comparison of spectroscopic strengths in 186 Os from different optical model parameter sets Comparison of spectroscopic strengths in 188 Os from different optical model parameter sets Spectroscopic strengths reported for 186 W( 3 He,d) 187 Re Unperturbed bandhead energies used in Coriolis mixing calculations for the 185,187 Re( 3 He,d) 186,188 Os reactions Unperturbed bandhead energies used in Coriolis mixing calculations for the 185,187 Re( 3 He,d) 186,188 Os reactions Spectroscopic strengths in 186 Os Spectroscopic strengths in 188 Os Rotational parameters for bands in 186,188 Os Properties of states in the ground state bands of 186,188 Os Ground state band B(E2) values. Data from ENSDF database Expected properties of a K π = [402]π [402]π band Properties of the ground state bands in 186,188 Os Properties of states in the γ bands of 186,188 Os γ band B(E2) values. Data from ENSDF database Expected properties of a K π = [402]π [400]π band Properties of the γ bands in 186,188 Os viii

11 5.10 Properties of states in the K π = bands of 186,188 Os Properties of the K π = in 186,188 Os Properties of states in the K π = bands of 186,188 Os Expected properties of a K π = [402]π [402]π band Properties of K π = 1 + band member candidates in 186 Os Reaction Q values ix

12 List of Figures 1.1 Deformed quadrupole shapes in nuclei Quadrupole harmonic vibrator A schematic of different types of phonon states with rotational bands Population of levels in the 191 Ir(t,α) 190 Os reaction Inelastic α scattering populating the states in 188,190,192 Os Ratios of intrinsic E2 transition matrix elements versus ratio of level energies A schematic of single-particle levels Nilsson model single particle states for protons in deformed nuclei with 50 Z Pairing factors in Re as a function of excitation energy Pairing factors in 186 Os as a function of effective 185 Re excitation energy MLL schematic floor plan Photograph of Q3D target ladder Diagram of the Q3D Q3D solid angle Cross-section diagram of the Q3D focal plane detector x

13 3.6 Distribution of events with cathode strip hits Anode wire signal peak height distributions Comparing the peak height distribution in the upper and lower anode wires E E plot (no gates) from anode and photomultiplier signals E E plot (with gates) from anode and photomultiplier signals E E 1 plot (no gates) from two different anode signals E E 1 plot (with gates) from two different anode signals E position plot (no gates) from the first anode signal E position plot (no gates) from the second anode signal E position plot (no gates) from the scintillator signal RADBUDDY screenshot Pt( 3 He,d) 195 Au calibration spectra Pt( 3 He,d) 196 Au calibration spectra Effective 186 Os energies for Au peaks Effective 188 Os energies for Au peaks Os calibration runs with a selection of level energies from the calibration polynomials Os calibration runs with a selection of level energies from the calibration polynomials Population of levels in the 185 Re( 3 He,d) 186 Os reaction Population of levels in the 187 Re( 3 He,d) 188 Os reaction Expected impurity from 13 C( 3 He,d) 14 N reactions He elastic scattering on 185 Re He elastic scattering on 187 Re xi

14 4.3 Deuteron elastic scattering cross sections[tji64] with DWBA calculations using different optical model parameters sets Deuteron elastic scattering cross sections[tji64] with DWBA calculations using different optical model parameters sets Deuteron elastic scattering cross sections[ynt59] with DWBA calculations using different optical model parameters sets Deuteron elastic scattering cross sections[ynt59] with DWBA calculations using different optical model parameters sets Intrinsic cross sections for the ground state in 186 Os calculated by DWUCK A comparison of Distorted Wave Born Approximation (DWBA) calculations using different optical model parameter sets in l=2 187 Re ground state angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=5 187 Re 390 kev angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=0 187 Re 512 kev angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=2 187 Re 591 kev angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=2 187 Re 648 kev angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=2 187 Re 775 kev angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=2 186,188 Os ground state angular distributions xii

15 4.15 A comparison of DWBA calculations using different optical model parameter sets in l=2 186,188 Os level angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=2 186,188 Os level angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=0 186,188 Os level angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=2 186,188 Os level angular distributions A comparison of DWBA calculations using different optical model parameter sets in l=5 186,188 Os 3 1 level angular distributions W( 3 He,d) 187 Re spectroscopic strengths Re( 3 He,d) 186 Os spectroscopic strengths Re( 3 He,d) 188 Os spectroscopic strengths Comparison of DWUCK4 and EVE calculations Comparison of DWUCK4 and EVE calculations Comparison of DWUCK4 and EVE calculations Comparison of DWUCK4 and EVE calculations Comparison of DWUCK4 and EVE calculations Comparison of DWUCK4 and EVE calculations Comparison of DWUCK4 and EVE calculations Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os xiii

16 4.35 Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 186 Os Cross sections for a selection of levels in 188 Os Cross sections for a selection of levels in 188 Os Cross sections for a selection of levels in 188 Os Cross sections for a selection of levels in 188 Os Cross sections for a selection of levels in 188 Os Cross sections for a selection of levels in 188 Os Cross sections for a selection of levels in 188 Os Cross sections for the kev level in 188 Os Single-proton states in rare-earth isotopes Moments of inertia in 186 Os Moments of inertia in 188 Os Comparison of B(E2) values from experiment with QPM calculations Other even-even nuclei of interest to a hexadecapole versus twoquadrupole phonon debate A.1 Energy versus angle plots for levels in 186 Os A.2 Energy versus angle plots for levels in 186 Os A.3 Energy versus angle plots for levels in 186 Os A.4 Energy versus angle plots for levels in 186 Os xiv

17 A.5 Energy versus angle plots for levels in 186 Os A.6 Energy versus angle plots for levels in 186 Os A.7 Energy versus angle plots for levels in 186 Os A.8 Energy versus angle plots for levels in 186 Os A.9 Energy versus angle plots for levels in 186 Os A.10 Energy versus angle plots for levels in 186 Os A.11 Energy versus angle plots for levels in 186 Os A.12 Energy versus angle plots for levels in 188 Os A.13 Energy versus angle plots for levels in 188 Os A.14 Energy versus angle plots for levels in 188 Os A.15 Energy versus angle plots for levels in 188 Os A.16 Energy versus angle plots for levels in 188 Os A.17 Energy versus angle plots for levels in 188 Os A.18 Energy versus angle plots for levels in 188 Os A.19 Energy versus angle plots for levels in 188 Os xv

18 List of Acronyms ASIC Application Specific Integrated Circuit BCS Bardeen Cooper Schrieffer CM Center of Mass DWBA Distorted Wave Born Approximation DWUCK4 DWUCK4 computer program ENSDF Evaluated Nuclear Structure Data File EVE EVE computer program gf3 germanium fit version 3 GNU GNUs Not Unix (a recursive acronym) GSL GNU Scientific Library IBA Interacting Boson Approximation LMU Ludwig Maximillians Universität MARaBOU A MBS and ROOT Based Online/Offline Utility MBS Multi Branch System xvi

19 MLL Maier Leibnitz Laboratory NNDC National Nuclear Data Center NSR Nuclear Science References ORNL Oak Ridge National Laboratory PWBA Plane Wave Born Approximation Q3D Quadrupole-three-Dipole magnetic spectrograph QPM Quasiparticle Phonon Model SRIM Stopping Range of Ions in Matter TUM Technische Universität of Munich WS Woods Saxon xvii

20 Chapter 1 Introduction The study of nuclear structure is in part a search for a better understanding of the individual and collective motions of nucleons. These motions, or wavefunctions, can be deduced in part through nuclear reactions involving the transfer of nucleons between an ion beam and a target. This thesis will focus on the measurement of wavefunction amplitudes in Os isotopes to investigate the vibrational nature of a particular excited state. Before the motivation of the experimental work is discussed, some background terminology and concepts will be explained. The next chapter gives an outline of transfer reactions in nuclei, followed by a detailed account of the apparatus and data analysis methods used. Results on the size of wavefunction components are presented and compared with calculations from theory. New research directions are proposed in the last chapter. 1

21 1.1 Background concepts Deformation If the nucleus is considered as a liquid drop, the shape of the nucleus can be given by a multipole expansion of the nuclear radial length: R(θ, φ) = R 0 (1 + λ=2 λ µ= λ α λµ Y λµ (θ, φ) ), (1.1) where the (2λ+1) α λµ coefficients control the size of the λ-pole term[hey99] and R 0 is chosen to enforce the conservation of nuclear volume[nil95]. The dipole term (λ=1) is ignored since it represents a shift in the centre of mass for small amplitudes α 1,µ. For quadrupole deformations, the coefficients are expressed as: α 22 = α 2 2 = 1 2 β sin γ (1.2) α 20 = β cos γ (1.3) α 21 = α 2 1 = 0, (1.4) where β is a measure of the magnitude of quadrupole deformation, and γ the axial symmetry[hey99]. The five coefficients have been reduced to two parameters because there is a freedom to choose the orientation of the nuclear shape using Euler angles. The change in the nuclear shape along three axes are given by: 5 δr k = R 0 (γ 4π β cos 2π3 ) k, (1.5) where the x, y, z axes are represented by k = 1, 2, 3. The parameter space of β > 0 and 0 γ 60 can describe all quadrupole shapes[hey99]. Prolate shapes, 2

22 like an American football, have γ = 0 while oblate shapes, like a Frisbee, have γ = 60. Triaxial nuclei, shaped similar to a kiwi fruit, have an intermediate value of γ. Figure 1.1 displays examples of these different quadrupole shapes. Quadrupole deformed nuclei will have a non-zero electric quadrupole moment in the intrinsic frame. For a nucleus with charge Z and deformation β, the moment is given to first order in β by[gre96]: Q 0 = 3ZeR2 0 β. (1.6) 5π Experiments do not measure the intrinsic moment, but the laboratory one. A deformed nucleus rotates with total angular momentum J in the laboratory frame, with a quantum number K denoting the projection of the total angular momentum onto the nuclear symmetry axis. The laboratory moment is determined from the intrinsic moment according to[gre96]: ( 3K Q JK = 2 J(J + 1) Q ) 5 (J + 1)(2J + 3) 7 π β. (1.7) It is interesting to note that the laboratory moment can be zero even if the intrinsic moment Q 0 is not; it is always zero for states with J = 0 and J = 1 2. In particular, the laboratory electric quadrupole moments of the states in 186,188 Os are 1.63(4) eb and 1.46(4) eb[bag03, Sin02]. Using the relations above, these laboratory moments indicate deformations with β values of 0.197(5) and 0.177(5). Later discussion in Chapter 2 on the single-particle states in deformed nuclei will introduce other conventions of expressing the size of a quadrupole deformation. Deformed nuclei exhibit excitations in the form of collective rotations and vibrations, which are introduced in the next two sections. 3

23 β = 0.0 γ = 0 Spherical β = 0.5 γ = 0 Prolate β = 0.5 γ = 60 Oblate β = 0.5 γ = 30 Triaxial Figure 1.1: Deformed quadrupole shapes in nuclei Rotation Collective excitations form when many nucleons participate in a motion, for example, if the nucleus is considered as a liquid drop it may have vibrations on the surface, and it may rotate if the shape of the nucleus is deformed. The rotational energy of an axially symmetric nucleus in a state rotating about an axis 4

24 perpendicular to the nuclear symmetry axis with total angular momentum J and the moment of inertia J is given by the well-known formula: E rot = h2 2J [J(J + 1) K(K + 1)], (1.8) where K is the projection of the total angular momentum onto the symmetry axis of the deformation. The discrete values of J give a series of levels called a rotational band. Rotational motion of the nucleus can be built on top of other types of excitations. It is typically assumed that the rotational motion and other forms of excitation can be separated so that the wave function may be given by: Ψ JM = 2J + 1 [ ] 16π 2 D JMK X K + ( 1) J K D JM K X K, (1.9) where the nucleus is in a state of total angular momentum J with projections M in the laboratory frame and K on the nuclear symmetry axis[cas00]. The rotational wave function is given by a D matrix and the other internal motions by X K. Note that the wavefunction is symmetrized with the degenerate K and K terms, and K = 0 rotational bands will only contain even members due to the ( 1) J K phase factor. In the analysis of experimental data, the energies of rotational band members are fit to an equation that may contain higher order terms in J, for example: E = E 0 + AJ(J + 1) + B[J(J + 1)] 2, (1.10) where the parameter A is called the rotational parameter and is related to the moment of inertia. In the case of 186,188 Os, the A parameters for the rotational 5

25 band built on the ground state are 20.4 kev and kev, while the B parameters have best fit values of kev and kev[bag03, Sin02] Vibration Vibrations are quantized into discrete excitations called phonons. These phonons are characterized by how much angular momentum they possess; the standard treatment of nuclear vibrations is to employ a multipole expansion of variations in the nuclear shape (Equation 1.1) in terms of how much angular momentum is carried (λ). The case of a spherical nucleus with quadrupole phonons each carrying 2 h of angular momentum is shown in Figure 1.2. Selection rules for transitions among phonon states can be formed from creation and annihilation operators[gre96]. To first order in the operators, transitions can only create or destroy a single phonon (the change in the number of phonons is one). In other words, states with phonons of different angular momenta are not likely to have strong transitions between them, for example, a single quadrupole phonon state and a single hexadecapole state. Strong transitions between phonon states of the same type are expected due to their collective nature. B(Eλ) values are related to transition matrix elements for an electric transition with λ units of angular momentum. Vibrations in deformed nuclei are more complicated since there is less symmetry to the nuclear shape. In the quadrupole case, the angular momentum of the phonon may have a projection of 0 or 2 along the symmetry axis. The projection of angular momentum on the symmetry axis is denoted by the quantum number K[Cas00]. A K=0 vibration, denoted by β, represents a vibration in the magnitude of the deformed nuclear shape, or in the β parameter of Equation 1.1, 6

26 # of phonons I π 3 6 +, 4 +, 3 +, 2 +, B(E2;I i I f ) = 3B(E2; gs) I f 4 +, 2 +, 0 + B(E2;I i ) = 2B(E2; gs) 0 B(E2; gs) 0 + Figure 1.2: Quadrupole harmonic vibrator. Adapted from Ref [Cas00]. The size of transition matrix elements is simply proportional to the number of phonons in a decaying state. while a K=2 γ vibration causes oscillations in the axial symmetry parameter γ. To digress, there is no K = 1 quadrupole vibration in a nucleus with a quadrupole deformation. This comes about by enforcing rotational invariance after applying a rotation operator to the wavefunction (Equation 1.9), where the end result is to have a ( 1) K factor multiplying the wavefunction[eis70, Gre96]. Invariance demands the K quantum number must be even, and positive due to reflection degeneracy. The harmonic oscillator picture of phonon states is useful to gain familiarity with some basic concepts, but in reality the many degeneracies of the multiphonon levels are broken by anharmonic effects[cas00]. Furthermore, in deformed nuclei, rotational bands will be built on each of the phonon states. A schematic of a more realistic picture is shown in Figure 1.3. In addition to the rotational bands built on vibrational states, rotational bands based on quasiparticle excitations also exist. 7

27 K= Hexadecapole band K= K=0 γγ bands K=2 γ band K=0 Ground state band K=0 ββ band K=0 β band K=2 βγ band K=2 3 K= K=3 Octupole bands K= Figure 1.3: A schematic of different types of phonon states with rotational bands. For display purposes, only four members are shown in each rotational band. The vertical position of bands is not meant to give an accurate energy. 8

28 1.1.4 Quasiparticles The shell model uses particles and holes to provide a framework to predict singleparticle excitations. However, nucleons exhibit a pairing interaction requiring some energy to break a pair. One consequence of the pairing interaction is that the occupation of levels becomes smeared around the Fermi surface so that a level is no longer either completely full or empty. The non-zero fullness (V 2 ) and emptiness (U 2 ) of orbits require that excitations be described with quasiparticles instead of particles and holes. A quasiparticle is a fermion which is part particle and part hole. The energy of a quasiparticle is given by: E qp = (ε λ) 2 + 2, (1.11) where ε is the energy of the original single-particle level, λ the Fermi energy, and the pairing gap parameter. The fullness and emptiness factors for an orbital are defined by Bardeen Cooper Schrieffer (BCS) theory to be: ( V 2 = ( U 2 = ) (ε λ) (ε λ)2 + 2 (1.12) ) (ε λ), (1.13) (ε λ)2 + 2 where the total probability of a level being empty or full is unity (U 2 + V 2 = 1). Phonons can be thought to be composed of a coherent linear combination of two quasiparticle excitations[elb69]. To put this in perspective for the work of this thesis, single nucleon stripping transfer reactions on odd-a targets are sensitive to populating states with two quasiparticle components, and provide a method of deducing the size of components in particular states. The concepts of phonons 9

29 and quasiparticles can be used to perform calculations of the expected transfer strengths via the Quasiparticle Phonon Model (QPM). The QPM takes the nuclear Hamiltonian composed of the mean field that a nucleon moves in from all other nucleons along with two-body interactions for particles and holes. The Hamiltonian is then recast in terms of phonon creation and annihilation operators along with terms describing phonon couplings[li08]. Diagonalization occurs in a space involving multi-phonon states with a variety of multipolarities. The end results of calculations allow for the prediction of level energies, wavefunction components, electromagnetic transition matrix elements, and transfer reaction strengths[li08]. The details of QPM formalism and calculations are beyond the scope of this thesis, but the interested reader may find Refs. [LI08, Sol92, Nes86] useful for further investigation. 1.2 Motivation There has been a long standing debate on the vibrational structure of the third 4 + state in the Os isotopes (denoted by ). There are two main interpretations: 1. Two quadrupole phonons (γγ), each with two units of angular momentum, coupling to a total spin of 4 +. This view is supported by B(E2) values and lifetime measurements. 2. One hexadecapole phonon with 4 units of angular momentum. This view is supported by population of the level in (t,α) reactions along with inelastic α and proton scattering measurements deducing a significant B(E4) value. At first glance, the two views are not compatible with the experimental evidence if the structure of the states are assumed to be purely either one type or the 10

30 other: 1. A two phonon γ-vibration consists of 4 quasiparticles, which would not be populated in a single-nucleon transfer reaction on odd-a targets. The odd- A target in a one quasiparticle state capturing the transferred nucleon will populate states with 2 quasiparticle components. 2. A B(E2) transition from a single hexadecapole phonon state to a single quadrupole phonon state violates the selection rule of a single phonon being created or destroyed. The proponents of each interpretation have criticized the other interpretation. These criticisms are discussed later in this chapter. It would appear that progress in the debate and finding a solution has paused as no new theoretical or experimental work has been performed in several years. This stagnation has come to an end with the collection and analysis of the new data presented in this thesis. The purpose of this work was to add more data to the debate with the hope of moving towards a solution by performing 185,187 Re( 3 He,d) 186,188 Os single proton stripping transfer reactions. These reactions allow tests of the earlier (t,α) transfer work, while avoiding the criticisms raised by the two-phonon proponents. Before the criticisms are discussed, a brief timeline of work and developments in this debate is presented for a historical perspective. 1.3 Timeline of events Early work proposes a two phonon γ-vibration based on the ratio of level energies[sha73, Sha76, Yat74b, Yat74a, Cas78, Cas84]. 11

31 The levels are strongly populated in 191,193 Ir(t,α) 190,192 Os reactions with 15 MeV tritons; a large 2 quasiparticle component infers a single hexadecapole phonon interpretation[bag77]. A spectrum is shown in Figure 1.4. α inelastic scattering results in 188,190,192 Os show excitations to the states yielding large B(E4) values consistent with a hexadecapole structure[bur78]. Later proton inelastic scattering in 192 Os confirms these observations to deduce an E4 matrix element of ±1080 e fm 4 [Bak89] or B(E4; gs) = 1.68 W.u. with the conversion formulae from Ref. [Mar82]. The uncertainty was estimated to be 25%. Cross-section angular distributions with calculations are displayed in Figure 1.5. Nesterenko et. al. perform QPM calculations over a range of nuclei with 158 A 188[Nes86]. In particular, predictions show approximately a 90% hexadecapole component in the states in 186,188 Os. Coulomb excitation studies of 192 Os[Osh93] and 186,188190,192 Os[Wu96b] reveal large collective B(E2) values with B(E2; γ ) 2B(E2;2 + γ 0 + gs) (see Table 1.1). The structure of the states are reinterpreted as double-γ vibrations. Note that Wu et. al. make a statement about (t,α) reaction mechanism uncertainties in their second paper[wu96b]. Burke defends the hexadecapole interpretation and performs a consistent set of DWBA calculations across (t,α) work on Os, Ir, and Pt targets[bur97]. The observed transfer strengths agree well with the calculations considering the standard 30% systematic uncertainty from optical model parameters. Burke encourages lifetime measurements to be performed for confirmation of the collective B(E2) values. 12

32 A study of 185,187 Re( 3 He,d) 186,188 Os is led by Paul Garrett in 1997, however the data are not published due to high levels of impurities in the Re targets making extraction of cross sections from particle spectra difficult. Preliminary data suggest that the states are strongly populated. Dennis Burke summarizes the conflict between the two interpretations with all available data as both transfer and Coulomb excitation experiments have a preferred interpretation. He goes into some depth in proposing methods of generating B(E2; γ ) strength without requiring a double-γ vibration[bur00]. Lifetime measurements confirm the earlier B(E2) values in 188,190 Os and continue to claim a dominant two phonon γ vibration component in the levels[wu01]. This brings the debate to a new level in which a comment by Burke[Bur02] restates the alternative sources for the large B(E2; γ ) values and raises the importance of considering all data rather than dismissing the transfer and B(E4) results. He proceeds to argue that a dominant hexadecapole vibration component can describe all of the transfer, B(E4), and B(E2) data and that further theoretical work should be performed to estimate the size of a two-phonon γ component in the Os states. Wu et. al. reply and reiterate that there may be a large systematic error in (t,α) work, on the other hand they also support further theoretical work in Os isotopes[wu02]. A plot of B(E2) value ratios versus energy level ratios is shown in Figure 1.6. Allmond et. al. perform calculations with the triaxial rotor model and are able to accurately reproduce many B(E2) values in 186,188,190,192 Os[All08]. They also discover from the trace of the E2 matrix elements that K = 4 E2 13

33 strength is missing, suggesting components outside their model in the K = 4 bands, such as those belonging to a hexadecapole phonon, may be present. Current Work: 185,187 Re( 3 He,d) 186,188 Os reactions definitely show a strong population of the levels and reveal strengths in qualitative agreement with the QPM calculations of Nesterenko[Nes86]. Motivated by the current work, Lo Iudice and Sushkov[LI08] perform new QPM calculations focusing specifically on Os isotopes. The results reveal a dominant hexadecapole phonon component (about 60%) and a large nondominant two-phonon γ-vibration component (about 30%). The current work supports the general observations of the earlier (t,α) transfer work, as well as avoiding criticism of two-phonon proponents by not using a reaction channel involving α particles. Perhaps even more importantly, the goal of moving the debate forward has also been met in the theoretical arena. The new QPM theoretical work by Lo Iudice and Sushkov[LI08] represents a major step forward in the debate as it predicts both large components of hexadecapole and γγ vibrations in the states with the hexadecapole being dominant. The new work builds on the older calculations by Nesterenko et. al. by including multipolar particle-particle interactions along with a more restricted set of parameters specific to Os isotopes[li08]. The new calculations also naturally encourage an interpretation of all available data: transfer populations, B(E4) values, and B(E2) values. 14

34 Figure 1.4: Population of the level in the single-proton pickup 191 Ir(t,α) 190 Os reaction. Note the large I π, K π = 4 +, 4 + peak. Source: Bagnell et. al. Ref. [Bag77] B(E2) values (W.u.) Intrinsic E2 matrix elements (eb) A 2 + γ 0 + gs γ K = 2 E2 K = 0 K = 4 E2 K = 2 Ratio (11) 0.67(29) 1.60(70) (17) 0.55(19) 1.37(48) ± (37) 0.53(6) 1.34(20) Table 1.1: B(E2) values deduced from Coulomb excitation studies. Intrinsic E2 matrix elements were found using Mikhailov plots. The expected value of the ratio of intrinsic E2 matrix elements for a harmonic vibration is 2. Data taken from Wu et. al. Ref. [Wu96a, Wu96b]. 15

35 Figure 1.5: Inelastic α scattering populating the states in 188,190,192 Os. The solid curve represents calculations for a pure E4 transition, while the lowest curves are for a two-step E2 transition. The other dashed and dotted curves are for constructive and destructive mixtures for these two processes. Clearly the onestep E4 transition dominates. Source: Burke et. al. Ref. [Bur78] 16

36 K = 4 E2 K = 2 / K = 2 E2 K = Os Coulex: Wu et. al. [Wu96a] Lifetimes: Wu et. al. [Wu01] Harmonic vibration 188 Os190 Os 192 Os E x (4 + 3 )/E x(2 + 2 ) Figure 1.6: Ratios of intrinsic E2 transition matrix elements versus ratio of level energies[wu96a, Wu01]. A perfect quadrupole harmonic oscillator would yield matrix elements with a ratio of 2 and levels with an energy ratio of 2. The arrows for 192 Os indicate a range from a mixing calculation. Note that the error bars are large, but the data suggest that the character of the states are consistent with a two-phonon interpretation. The lifetime data points are slightly shifted to the right for display purposes. 17

37 1.4 Criticisms of each view Firstly, the criticisms of the (t,α) work by the two-phonon proponents will be explained, followed by the opposing criticisms of the two-phonon interpretation by the hexadecapole workers. Wu et. al. claim that the (t,α) work is subject to large systematic uncertainties due to the reaction mechanism via uncertainties in the optical model parameters for the α particle[bag79]. A mismatch between the incoming and outgoing momenta exists in (t,α) and ( 3 He,α) reactions due to the large binding energy of the α particle[woo71]. A mismatch is undesirable since the DWBA treatment usually applied to transfer reactions assumes that the transfer process is a small perturbation to the much larger elastic scattering cross section, in which the incoming and outgoing momenta are perfectly matched. When a large mismatch occurs in a reaction involving an α particle, the partial wave space for the transfer is larger than the few terms needed for elastic scattering with the higher momentum waves being dominant[sto67]. This makes the calculations somewhat insensitive to the shape of the inner potentials determined from optical model parameters. Although the criticism of a momentum mismatch not being ideal is valid, there are reasons to have confidence in the reliability of (t,α) work. Burke makes a case in Ref. [Bur00] for the (t,α) reaction data by pointing out that the strengths seen from (t,α) work using different energies[bur97] agree well with (d, 3 He) work[bla82] so the criticism on reaction mechanism uncertainties is not as important as initially thought. Given this rebuttal, and the fact that the strong B(E4;0 + gs ) value is consistent with the (t,α) hexadecapole interpretation, it appears that the twophonon workers may be too hasty in dismissing the relevance of (t,α) work in 18

38 favour of their own. The work described in this thesis employs single-proton stripping transfer reactions 185,187 Re( 3 He,d) 185,187 Os to avoid the possible criticism of a reaction involving α particles. To quantify the improvement in using the ( 3 He,d) reactions, the mismatch in momentum can be expressed as an equivalent angular momentum: l R(k f k i ), (1.14) where R is the nuclear radius, and k i and k f are the wave numbers of the projectile and ejectile. Table 1.2 and Table 1.3 summarize the momentum mismatch of (t,α) and ( 3 He,d) reactions. The mismatch of the earlier (t,α) reaction work is about 2.5 times that of the ( 3 He,d) reactions used in this thesis. Particle triton α Kinetic energy (MeV) Momentum (MeV/c) Wave number (fm 1 ) l R(k f k i ) 6.1 Table 1.2: Calculating the momentum mismatch for 191,193 Ir(t,α) 190,192 Os reactions. A typical kinetic energy for α particles at a 0 scattering angle was used along with R = 6.9 fm. Particle 3 He deuteron Kinetic energy (MeV) Momentum (MeV/c) Wave number (fm 1 ) l R(k f k i ) 2.5 Table 1.3: Calculating the momentum mismatch for 185,187 Re( 3 He,d) 186,188 Os reactions. A typical kinetic energy for deuterons at a 0 scattering angle was used along with R = 6.9 fm. 19

39 The main criticism of the two-phonon interpretation comes about from the use of ratios of B(E2) values as the only data in favour of the claim. Dennis Burke presents arguments that B(E2; γ ) strength does not necessarily have to come from a two-phonon γ-vibration, or in other words the ratio: R = B(E2; I, Kπ = 4, 4 + I, K π = 2, 2 + γ ) B(E2; I, K π = 2, 2 + γ I, K π = 0, 0 + gs), (1.15) necessarily must be equal to 2 to make a pure two-phonon claim, but alone it is not sufficient to make the claim when conflicting data exists.[bur02]. Burke points out that several calculations[vi81, vi82, Lac92, Kuy94] with the sdg Interacting Boson Approximation (IBA) predict a low-lying K π = 4 + band constructed from g bosons resembling a hexadecapole phonon vibration. These calculations also predict large B(E2) values connecting the K π = 4 + to the γ bandhead. Thus, B(E2) values and their ratios R = 2 are not a signal which is unique to a two-phonon γ-vibration. Given this work, it is not clear what would constitute sufficient evidence for a two-phonon γ-vibration. In addition to examining the criticisms, it is also useful to look at which data has not been questioned. In particular, Burke begins his comments[bur02] on the lifetime work by praising the confirmation of the earlier B(E2) values, which build on his earlier acceptance of the Coulomb excitation B(E2) values[bur00]. In addition, no criticism of the B(E4; gs) values deduced from α and proton scattering is known by the author at the time of this writing. It appears that the criticisms and debate focus mostly on the interpretation of data and not on the accuracy of the data from different experiments. 20

40 1.5 Overview This thesis describes the work and data analysis of ( 3 He,d) single-proton stripping transfer reactions on 185,187 Re to populate states in 186,188 Os in order to study the nature of vibrational nuclear structure. In particular, the strength of quasiparticle components may be deduced by comparing measured cross sections to calculated cross sections using the DWBA. Chapter 2 will give the relevant background information on the physics of deformed nuclei and single-nucleon transfer reactions, while the experiment and data analysis methods are described in Chapter 3. Chapter 4 will discuss the analysis of elastic scattering data and the choice of optical model parameters for transfer data analysis. Chapter 5 provides a summary of the data collected in context of the present knowledge of Os isotopes. A final summary of conclusions and future directions is contained in Chapter 6. 21

41 Chapter 2 Single-nucleon transfer reactions in deformed nuclei The aim of this chapter is to provide the necessary concepts to understand the experimental work and data analysis discussed in later chapters. A short introduction to the single particle states in spherical nuclei will be given as a base on which a discussion of single particle states in deformed nuclei will be built. The physics of single nucleon transfer reactions will be developed to obtain the concepts used in the analysis of data from the 185,187 Re( 3 He,d) 186,188 Os experiment. 2.1 Single particle states in spherical nuclei Many standard introductory nuclear physics textbooks give an outline of the history of how the shell model of nuclear physics was developed based on the magic numbers of nucleons where increased binding energies were observed. The usual 22

42 approach[hey99] is to start with a 3D isotropic harmonic oscillator potential: V(r) = V mω2 r 2, (2.1) where mω 2 is effectively a spring constant for radial motion. This potential gives the well known result of equally spaced levels with energies denoted by an oscillator quantum number N: E = hω(n + 3/2) V 0. (2.2) The principal quantum number can be expressed in terms of other quantum numbers from Cartesian, cylindrical, or spherical coordinate systems[gre96]: N = n x + n y + n z = n z + 2n ρ + Λ = 2n + l, (2.3) where n x, n y, n z are the oscillator quanta on the x, y, z axes in Cartesian coordinates, n z and n ρ are the oscillator quanta in cylindrical coordinates, and n is the number of radial quanta in spherical coordinates. The angular momentum and projection on the nuclear symmetry axis are denoted by l and Λ. The next step towards describing the states in spherical nuclei is to use a more realistic Woods Saxon (WS) potential: V WS (r) = V 0 f (r), (2.4) ( 1 f (r) = 1 + exp[(r r 0 A )/a]) 1/3, (2.5) where V 0 is the potential depth, r 0 is a radial parameter with a typical value of 23

43 1.2 fm, A is the number of nucleons, and a is a surface diffuseness parameter. One effect of this potential is to break the l degeneracy compared to a harmonic oscillator potential. Unfortunately, a WS potential by itself does not reproduce the magic numbers of the shell gaps. The key ingredient to reproducing the magic numbers and properties of excited states of doubly-magic nuclei is the introduction of a spin-orbit interaction. A typical form of the potential derivative is used to focus the spin-orbit effects at the surface of the nucleus. The l s factor will split the degeneracy of orbits according to the total angular momentum j = l + s with the higher(lower) j orbit being moved down(up) in energy. Figure 2.1 shows a schematic of how the spin-orbit interaction is needed to produce the magic numbers. 2.2 Single particle states in deformed nuclei The deformed shell model was first developed by S.G. Nilsson in 1955[Nil55]. The nuclear potential was expressed as an axially symmetric anisotropic harmonic oscillator. Spin-orbit and l 2 terms are added to the Hamiltonian following the spherical shell model to account for the spherical magic numbers and flattening out the potential near the centre of the nucleus as in a WS potential. The discussion of the deformed shell model here will follow the approach taken by Refs. [Hey99, Gre96]. The axially symmetric harmonic oscillator Hamiltonian is given by: H = h2 2m m[ω2 (x2 + y 2 ) + ω 2 zz 2 ], (2.6) where is the Laplacian operator, and the oscillator frequencies ω and ω z are 24

44 70 3s 2d 82 3s 1/2 2d 3/2 1h 11/2 2d 5/2 N = 4 1g 50 1g 7/2 40 2p 1g 9/2 2p 1/2 2p 3/2 N = 3 1 f 28 1 f 5/2 1 f 7/ N = 2 2s 1d 2s 1/2 1d 3/2 1d 5/ N = 1 1p 1p 1/2 1p 3/ N = 0 1s 1s 1/2 Figure 2.1: A schematic of single-particle levels showing the reproduction of the magic numbers (2, 8, 20, 28, 50, and 82) by the addition of l 2 (middle) and spinorbit (right) effects to a harmonic oscillator potential (left). 25

45 chosen to match the deformation of the nucleus. The deformation of a prolate shape is characterized by a parameter δ 2 such that: ω = ω δ 2 (2.7) ω z = ω δ 2 (2.8) ω 0 ω 0 (1 + 2 ) 9 δ2 2, (2.9) where volume conservation (ω x ω y ω z = ω 3 0 ) has been enforced to second order in δ 2. The equivalent spherical nucleus has an oscillator frequency given by ω 0, where hω 0 = 41A 1/3 MeV[Gre96]. Substituting these expressions for oscillator frequencies into the Hamiltonian, the following expression is found: H = h2 2m + 1 [ 2 mω2 0 r 2 2 ] 3 δ 2(2z 2 x 2 y 2 ), (2.10) and the last term may be simplified using the definition of the quadrupole spherical harmonics to yield: H = h2 2m mω2 0 r2 mω2 0 16π 3 5 δ 2r 2 Y 20 (θ, φ), (2.11) where the first two terms represent an isotropic harmonic oscillator and all deformation effects appear in the last term. The coefficient in the deformation term is directly related to the β deformation parameter discussed in Chapter 1[Hey99]: β = π 5 δ δ 2. (2.12) 26

46 Another definition for deformation is given in terms of an elongation parameter: ε = ω ω z ω 0, (2.13) where the oscillator frequencies are expressed as[nil95]: ( ω = ω ) 3 ε (2.14) ( ω z = ω ) 3 ε (2.15) ( ω 0 ω ) 9 ε2. (2.16) The energies of this anharmonic oscillator are given by[nil95]: ( E = hω z n z + 1 ) ( + hω 2 (n + 1) = hω 0 N (n 2n z ) ε ), (2.17) 3 where n = n x + n y is the total number of oscillator quanta in the x and y directions, which can also be expressed as n = 2n ρ + Λ. n ρ is the number of oscillator quanta along the cylindrical coordinate radius and Λ the projection of the orbital angular momentum, l, on the nuclear symmetry axis. As in the case for spherical nuclei, an oscillator potential is not the only term in the Hamiltonian. The harmonic potential has an l 2 term added to flatten out the bottom which lowers the energies of levels with large l[gre96]. A spin-orbit l s term is also added. Therefore, the Hamiltonian for deformed nuclei, also called the Nilsson model, is given by: H = h2 2m + 1 ( 2 mω2 0 r2 mω0 2 βr2 Y 20 (θ, φ) hω 0 κ 2 l s + µl 2), (2.18) 27

47 where κ and µ are dimensionless parameters. The notation µ = κµ is also used in some literature[nil95]. The physics of deformed nuclei can be examined in two extremes: small deformation and large deformation. In the limit of small deformation, the quadrupole term in the Hamiltonian is treated as a perturbation and states are labelled with the quantum numbers for spherical nuclei: the oscillator number N, single-particle angular momentum j with projection Ω on the z axis, and orbital angular momentum l[cas00]. The perturbation shifts the energies relative to those of a spherical nucleus according to the matrix element[cas00]: E(NljΩ) = 2 3 hω 0 ( N + 3 ) [ 3Ω 2 j(j + 1) ] [ 3 δ 4 j(j + 1) ] 2, (2.19) 2 (2j 1)j(j + 1)(2j + 3) which reveals that the degeneracy among Ω states is broken with those of low projections lying lowest in energy. The splitting is also proportional to the deformation of the nucleus. In the limit of large deformation, the l 2 and l s terms are treated as a small perturbation using states with quantum numbers of the axially symmetric harmonic oscillator denoted by: Ω π [Nn z Λ], (2.20) where Ω is the projection of the total angular momentum onto the deformation symmetry axis, N the oscillator number, n z the number of oscillator quanta along the symmetry axis, and Λ is the projection of the orbital angular momentum on the symmetry axis. Note that the projection of the spin on the symmetry axis, denoted by Σ, can be found easily since Ω = Λ + Σ. The single-particle angular momentum and orbital angular momentum (j, l) are no longer good quantum numbers, however, the Nilsson wavefunctions (χ) may be expanded in the basis 28

48 of spherical shell model single-particle wavefunctions (φ)[elb69]: χ (Ω π [Nn z Λ]) = C jl φ(n, j, l, Ω), (2.21) jl where C jl are the coefficients describing which spherical wavefunctions are dominant. The Nilsson wavefunctions are normalized such that: C 2 jl = 1. (2.22) jl Figure 2.2 shows a selection of Nilsson single-particle states as a function of deformation. For example, the [402]π Nilsson wavefunction will have a large 2d 5/2 spherical wavefunction component in the expansion, particularly at low values of deformation. 2.3 Elastic scattering The elastic scattering of light-ions from heavy targets is modelled using potentials that are analogous to light scattering from a cloudy crystal ball[hod71]. Most of the light shining on a crystal ball (projectiles interacting with a target) will scatter at different angles (elastic scattering), while a small fraction will be absorbed by the material (other reaction channels). The absorption of light in optics is connected with an imaginary component to the index of refraction, so the optical model of nuclear scattering contains imaginary potentials to describe the removal of beam particle flux from the elastic scattering reaction channel into other channels, such as single-nucleon transfer reactions. The optical model potential may 29

49 Figure 2.2: Nilsson model single particle states for protons in deformed nuclei with 50 Z 82. The calculations use an additional hexadecapole deformation ε 4 related to the quadrupole deformation ε 2 via the formula ε 4 = ε 2 2 /6. The solid lines are positive parity states, while the dashed lines indicate negative parity states. Source: Firestone et. al. [Fir96]. 30

50 contain volume, surface, and spin-orbit terms with both real and imaginary components. The shape and size of these components are described by a set of optical model parameters tailored to the projectile and target nuclei at the projectile energy. The potentials described by the parameter sets may be used to perform computer calculations of elastic scattering cross sections. The procedure of obtaining cross sections via phase shifts from a potential is described in general to understand how calculations are performed. A wavefunction solution of a incoming plane wave with an outgoing spherical wave is assumed at large distances[hec00]: ψ(r, θ, φ) = ( ) 1 (2π) 3/2 e i k r + f (θ, φ) eikr, (2.23) r where k is the wave number of the particle, and f (θ, φ) is a scattering amplitude which determines the observed cross section given by: dσ dω = f (θ, φ) 2. (2.24) The usual assumptions of low-energy scattering from a localized potential are made suggesting that the oscillations of the incident wavefunction represented by a plane wave only differs from the outgoing spherical wave by a phase shift. The wavefunctions are broken down using the Rayleigh Faxen Holtzmark partial wave expansion[hec00]: ψ(r, θ, φ) = R l (kr)p l (cos θ), (2.25) l=0 31

51 where R l (kr) = w l (kr)/(kr) is a partial radial wavefunction and P l (cos θ) a Legendre polynomial. The behaviour of the radial wave function for large r is: ( w l (kr) a l sin kr lπ ) 2 + δ l(k), (2.26) where a l is the partial wave amplitude with phase shift δ l (k). The phase shifts are determined by matching the asymptotic wave function to the solution in the scattering region[gri95]. This expansion of the wave function in partial waves is used to express the scattering amplitude in terms of the phase shifts[hec00]: f (θ, φ) = 1 k e iδl(k) (2l + 1) sin δ l (k)p l (cos θ), (2.27) l=0 which can then be used to calculate the elastic scattering cross section from Equation Optical model parameter sets As stated above the potential describing elastic scattering may have a variety of terms. For elastic scattering of light-ions from heavy nuclei, the Coulomb potential along with a WS form is used[lu71]. The notation from the DWUCK4 transfer reaction computer program handbook[kun78] is used as it will apply to later calculations for transfer cross sections. The WS potential is described by: f (x i ) = 1/ [1 + exp(x i )] (2.28) x i = (r r 0i A 3 1 )/ai (2.29) g(x i ) = d f (x) dx, (2.30) x=xi 32

52 where r is the distance from the nucleus, r 0i A 1/3 is the nuclear radius, A the mass number of the nucleus, and a i the diffuseness of the potential. f (x i ) is a volume term, while g(x i ) is called a surface term since the derivative has a peak value near the nuclear radius. A volume WS potential with real and imaginary depths V R,vol and V I,vol is described by: V vol (r) = V R,vol f (x R,vol ) + iv I,vol f (x I,vol ), (2.31) while a surface WS potential with real and imaginary depths V R,sur and V I,sur is given by: V sur (r) = V R,sur g(x R,sur ) + iv I,sur g(x I,sur ). (2.32) One notation convention is to express the depths of the imaginary components as V I,vol = W 0 and V I,sur = 4W D [Lu71]. A spin-orbit potential is sometimes used, and may be described as: V so (r) = ( V R,so d f (x R,so ) i V I,so r dr r ) d f (x I,so ) l s, (2.33) dr or the potential can use the same shape as a volume term, but differing in magnitude given by parameters λ R and λ I : V so (r) = ( V R,vol r λ R d f (x R,vol ) i V I,vol 45.2 dr r λ I 45.2 ) d f (x I,vol ) l s. (2.34) dr The Coulomb potential between the projectile(1) and target(2) usually is taken to 33

53 be a uniformly charged sphere with radius R c [Gle83, Hod71]: V C (r) = [ 1 Z 1 Z 2 e 2 ( ) ] r 2 3, 4πε 0 2R c R c r < R c 1 Z 1 Z 2 e 2, 4πɛ 0 r r > R c. (2.35) This list of terms is not meant to be complete, but a description of the potentials used in this work. 2.4 Transfer reaction cross sections This section will outline the theory of single-nucleon transfer reactions in spherical and deformed nuclei with the goal of obtaining expressions for angular distributions of the cross-section of the final states which are populated. The approach taken here will draw heavily on Ref. [Elb69] which provides an excellent discussion of the theory of single-nucleon transfer reactions in deformed nuclei. The typical reaction presented is (d,p) single-neutron stripping-transfer reaction, but the theory is also equally valid for the ( 3 He,d) single-proton stripping-transfer reactions of this work. The notation here has been modified for use with ( 3 He,d) reactions. As an overview, the mechanism for a stripping reaction can be broken down into 3 stages[elb69]: 1. The population of reaction channels for 3 He projectiles interacting with a target is dominated by elastic scattering. The elastic scattering is described by a potential calculated with an optical model parameter set specific to the projectile, target, and beam energy. 34

54 2. The transferred proton is captured into an available single particle orbit in the target nucleus. The interaction is assumed to be point-like modelled by a Dirac δ function requiring the transferred proton and outgoing deuteron to be at the same location. The wavefunction of the target-like nucleus is said to be composed of a core wavefunction similar to the original target wavefunction and the single-proton wavefunction. An optical model parameter set is used to describe the transfer of a bound proton in the field of the target. 3. The outgoing deuteron wavefunction is dominated by elastic scattering in the potential of the target-like nucleus. Another optical model parameter set is used specific to the deuteron, and the target-like particle at the appropriate energy. The cross section for a single-nucleon transfer reaction can be represented as the probability of transfer to a specific single-particle state (spherical shell model) in the target multiplied by the single-particle angular distribution for the transfer process: dσ dω = Nσ l(θ)s jl, (2.36) where Nσ l is a normalized single-particle differential cross-section, and S jl called a spectroscopic factor is the chance of finding a proton in the state with total and orbital angular momenta j, l. The spectroscopic strength is a measure of the occupation of single particle states and yields important information on nuclear structure. The development of Equation 2.36 will be traced in Subsection In Chapter 1, the concept of quasiparticles was introduced demonstrating that 35

55 single-particle states are not completely full or empty. The smearing of the Fermi surface requires the modification of the transfer cross section Equation The spectroscopic factor becomes a measure of the fraction of the orbit capacity that is full or empty in the case of pickup or stripping reactions. It is expressed as S jl = (2j + 1)P 2 where P 2 is a pairing factor: U 2 for stripping and V 2 for pickup reactions. For even-even spherical targets, the cross sections are given by: dσ dω = (2j + 1)P2 Nσ l (θ), (2.37) where j and l are the total and orbital angular momentum of the transferred particle. In deformed nuclei, the total strength of the transfer for a given value of angular momentum is split across the single-particle Nilsson orbits which, from Equation 2.21, contain an appropriate (j, l) component in the linear combinations of the states in the spherical shell model basis. The total strength of a given angular momentum transfer over all Nilsson states with that component must equal the strength seen in the spherical case. Formally, this property is written as: 2 C 2 jl = 2j + 1, (2.38) Ω[Nn z Λ] where the sum is taken over all Nilsson orbits that possess the component j. The factor of two arises from the reflection degeneracy between Ω and Ω on the nuclear symmetry axis. The transfer cross section equations must be modified to give the cross section of a Nilsson orbit: dσ dω = 2C2 jl P2 Nσ l (θ). (2.39) 36

56 In the case of odd-a targets, angular momentum algebra allows many (j, l) values to contribute to the population of a final state. These contributions must be summed to obtain a cross section angular distribution. The general formula for both even-even and odd-a targets, with spin and projection (I i, K i ), is given by: dσ dω = g2 P 2 C 2 jl I i K i jω I f K f 2 Nσ l (θ), (2.40) j,l where the transferred particle carries (j, Ω) angular momentum to populate the final state (I f, K f ). The Clebsch-Gordan coefficients handle the angular momentum algebra and the sum is over all allowed values of transferred angular momenta. Lastly, g 2 = 2 if the target or final state is in a paired off K = 0 configuration, otherwise g 2 = 1. In addition, there is an important exception if the particle transferred is in the same orbit as the odd-particle in the target. If this is the case, then the opposite pairing factor is used since the orbital will be either completely full or empty and needs to be renormalized to be consistent with the quasiparticle character of the even-even ground state[bur67]. Equation 2.40 is further expanded once Coriolis mixing is discussed in Subsection Transfer theory The initial state consists of a 3 He projectile bombarding a target (T) at rest in the laboratory frame, while the final state consists of a scattered deuteron moving in the potential of a recoiling nucleus (target plus proton denoted by F). The cross section for single nucleon transfer reactions can be expressed in terms of a 37

57 scattering amplitude, S f i : dσ dω = µ iµ f (2π h 2 ) 2 ( k f k i ) S f i 2, (2.41) where the initial(final) state has reduced mass µ i (µ f ) and wave number k i (k f ). The sum represents an average over initial spin states and a sum over final spin states. The initial and final states have both internal structure among the particles and relative motion. The Hamiltonians for the initial and final states are given by Table 2.1. Hamiltonian Internal Kinetic Elastic Other Simplified structure energy scattering Initial H = H 3He + H T +T 3He +V optical 3He +V 3He,T H 0 i + V 3He,T Final H = H d + H F +T d +V optical d +V d,f = H 0 f + V p,d Table 2.1: Terms in Hamiltonians describing transfer reactions. V 3He,T describes transitions between the 3 He and target such as proton stripping or inelastic scattering, while V d,f is the interaction between the outgoing deuteron and the final nucleus. Due to the assumption that the final nucleus can be thought of as the original target plus a proton, most of the V d,f interaction is actually handled by the optical potential for elastic scattering. It can be replaced by the interaction of the deuteron with the proton that was transferred. This V p,d term is used in perturbation theory since it is responsible for the transfer of the proton[elb69]. The V p,d term is used to describe the transfer process between an initial wavefunction Ψ i (k i ) and a final wavefunction Ψ f (k f ), satisfying the following equations: HΨ i (k i ) = EΨ i (k i ) (2.42) HΨ f (k f ) = EΨ f (k f ), (2.43) 38

58 where H = H i = H f is the full Hamiltonian with the deuteron final nucleus interaction included. The usual approach to solving this problem employs the assumption that elastic scattering is the dominant reaction channel, and all other channels are small. This approximation is used to find solutions to a Hamiltonian without the V p,d interaction and instead using it in perturbation theory. A modified final wavefunction is constructed, satisfying the Hamiltonian H 0 f Φ f (k f ) = EΦ f (k f ), and the scattering amplitude for cross-section calculations is re-expressed as: S f i = Φ f (k f ) Vp,d Ψi (k i ). (2.44) Unfortunately, the initial wavefunction is not known either, so a modified wavefunction satisfying the Hamiltonian Hi 0Φ i(k i ) = EΦ i (k i ) is substituted in Equation The next step to calculating the scattering amplitude involves separating the internal structure of the particles and their external motion. In other words, the internal structure wavefunctions (χ) and the projectile and ejectile motion wavefunctions (ψ i (k i ) and ψ f (k f )) can be separated as: Φ i (k i ) = χ 3He χ T ψ i (k i ) (2.45) Φ f (k f ) = χ d χ F ψ f (k f ). (2.46) Introductory quantum mechanics textbooks would suggest using the Plane Wave Born Approximation (PWBA) to model the motion wavefunctions of the 3 He projectile and scattered deuteron as simple plane waves. The situation is not that simple. To obtain accurate cross sections, the distorting influence of the nuclear and Coulomb potentials on a plane wave must be considered[elb69]. Therefore, 39

59 ψ i (k i ) and ψ f (k f ) are represented by distorted waves in what is called the Distorted Wave Born Approximation (DWBA). The scattering amplitude is calculated by integration over all nuclear coordinates(τ): S f i = Φ f (k f ) Vp,d Φi (k i ) (2.47) = ψ f (k f)χ F χ d V p,dχ 3He χ T ψ i (k i )dτ d dτ F dτ 3He dτ T. (2.48) The wavefunction for the final nucleus, χ F, is expanded in terms of a product of wavefunctions: one for the shell model state of the transferred proton (φ Nlj ) and the other for the remaining nucleons (χ F ): χ F = α Nlj φ Nlj χ F. (2.49) lj The zero range approximation is used to assume that the transfer occurs when the 3 He, transferred proton, and deuteron are all in the same location: V p,d χ 3He = D 0 δ(r p r d )χ d, (2.50) where D 0 is a constant related to the normalization constant N in Equation 2.36, and δ is a Dirac δ function. Combining these expansions and approximations, the scattering amplitude becomes: [ ] [ ] S f i = D 0 α Nlj χ F χ Tdτ nucleus ψ f (k f)φ Nlj ψ i (k i )dτ trans f er. (2.51) lj This equation immediately displays the two integrals as important factors in each term in the sum. The first integral along with the α Nlj factor represents the overlap 40

60 of the target and final core wavefunctions and is equivalent to the spectroscopic factor in Equation The second integral depends on the nuclear structure of the transferred proton (Nlj) and gives the angular distribution of the cross section in Equation The D 0 factor is related to the normalization of the transfer cross section since it depends on the forms of the wavefunctions used. The DWUCK4 computer program manual recommends a value of D0 2 =4.42 for ( 3 He,d) reactions[kun78] Performing transfer cross section calculations Transfer cross sections using the DWBA may be calculated with the DWUCK4 computer program[kun78]. The program takes the masses, charges, and spins for the particles involved in the reaction along with optical model parameter sets for the projectile, ejectile, and transferred particle. The reaction Q value and projectile energy are also specified along with the shell model state quantum numbers to which transfer cross sections are desired. The output of the DWUCK4 program contains the elastic cross section angular distribution and all specified transfer cross section angular distributions. In addition, optional lists of scattering amplitudes, radial form factors, and log scale plots of cross sections can be output. The program completes the calculations in approximately a second on a laptop with a processor frequency in excess of 2 GHz and 2 gigabytes of memory. Chapter 4 discusses the choice of optical model parameter sets for ( 3 He,d) reaction DWUCK4 calculations and shows angular distributions of the transfer cross sections obtained. 41

61 2.4.3 Refinements to single-nucleon transfer reaction theory The DWUCK4 program allows two refinements to the cross section calculations to be made. Firstly, a finite-range correction beyond the zero-range approximation may be made by multiplying the form factor in the scattering amplitude by the function[kun78]: W FR (r) = exp [ A(r)] (2.52) A(r) = 2 h 2 m b m x m a R 2 [E b V b (r b ) + E x V x (r x ) E a + V a (r a )], (2.53) where for the reaction A(a,b)B with transferred particle x the energies(e) and potentials (V) are given with respect to the target nucleus (A). The finite range correction parameter is R which according to the DWUCK4 manual should have a typical value of fm for ( 3 He,d) reactions[kun78]. The second refinement is that the optical model potential is known to have non-local effects[gle83]. The non-local portions of the potential can be replaced by a local energy-dependent potential. The DWUCK4 program makes a correction to the scattering amplitude by multiplying the form factor for each particle by: W NL (r i ) = exp [ β 2 i 8 2m i h 2 V i(r i ) ], (2.54) with the non-local parameter β i. The DWUCK4 recommended parameters for 3 He, deuterons, and protons are 0.25, 0.54, and 0.85 respectively[kun78]. 42

62 2.4.4 Coriolis mixing One of the assumptions made is that the impact of nuclear rotation on single particle motion is negligible. To examine the validity of this claim, the coupling of the single particle motion to the rotation will be calculated through a Coriolislike interaction. The Hamiltonian describing Coriolis mixing in odd-a nuclei is given by[elb69, Cas00]: H Coriolis = h2 2J (J +j + J j + ), (2.55) where J is the nuclear moment of inertia, while J ± = J 1 ± ij 2 and j ± = j 1 ± ij 2 are the nuclear and single-particle angular momentum ladder operators. These operators give a mixing of states with K = 1, which can be thought of as adding some axial asymmetry to the nucleus thereby reducing the validity of K being a good quantum number[cas00]. Mixing of states with K = 2 does exist, but is much weaker since it must be considered as a sequential two-step process[cas00]. For an example, consider two states mixing with final wavefunctions given by Ref [Elb69]: ψ (1,2) I = a (1,2) I,K ψ I,K + a (1,2) I,K+1 ψ I,K+1, (2.56) where 1 and 2 represent the two final states, and a I,K with a I,K+1 are the amplitudes of the unperturbed states ψ I,K and ψ I,K+1. The amplitudes are chosen such that the final wavefunctions are normalized. The off-diagonal matrix elements of 43

63 the Coriolis Hamiltonian with pairing effects are[elb69, Cas00]: M = IK H Coriolis IK + 1 = A K (I K)(I + K + 1) (2.57) ( ) A K = h2 C K 2J jl CK+1 jl (j K)(j + K + 1) (U K U K+1 + V K V K+1 ), (2.58) j where the C jl factors are the coefficients of the spherical shell model states in the expansion of a Nilsson wavefunction, and U and V the pairing factors for the appropriate unperturbed states. The Hamiltonian is diagonalized to find the energy eigenvalues and amplitude eigenvectors. The energies of the final mixed states (1,2) are[elb69]: E (1,2) = E avg ± 1 2 E 2 + 4M 2, (2.59) where E avg = (E K+1 + E K )/2 is the average energy of the unperturbed states, and E = E K+1 E K is the energy difference. Note that the energy difference of the perturbed states is E 2 + 4M 2 with a minimum value of 2M. The amplitudes are determined through the following relationships[elb69]: a (1) I,K a (1) I,K+1 = a (2) I,K+1 a (2) I,K = 1 R + R (2.60) R = E 2M, (2.61) where a large value of R leads to small mixing, while a small value of R leads to a large amount of mixing with the new states being composed roughly of equal parts of the unperturbed states. To obtain an order of magnitude estimate for the size of Coriolis mixing, one 44

64 can assume that j is a reasonably good quantum number (which is good in unique parity states like 1h 11/2 ), so that the magnitude of the matrix element is given by[cas00]: M h2 Jj. (2.62) 2J For Re isotopes with a rotational parameter of about kev, Equation 2.62 gives a range of values up to a few hundred kev. Rotational bandheads based on Nilsson orbitals with K and K + 1 can be separated by several hundred kev, so the value of R is on the order of unity. The mixing of K and K + 1 states will change the observed cross sections in transfer reactions according to the size and phases of the amplitudes involved. The angular distribution is given by the following formula: [ dσ dω = g2 a i (C jl ) i P i I 0 K 0 j K I f K f j,l i ] 2 [ N dσ ] (θ, l, j) dω DW, (2.63) where the second sum is over the components in a Coriolis mixed state with amplitudes given by a. Note that in the case of strong mixing (a 1/ 2) in the two state problem, one state will have amplitudes with the same sign, while the other will have opposite signs. From the nature of Equation 2.63, and if all other factors are approximately the same, the state with opposite signs will have a dramatically reduced cross section while the other will receive roughly double the cross section after mixing[elb69]. 45

65 2.4.5 Performing Coriolis mixing calculations A computer program, EVE[War83], was used to perform Coriolis mixing calculations in Chapter 4. The program requires the Nilsson configurations with excitation energies along with pairing factors. The C jl coefficients are found from Nilsson calculations from specified κ and µ parameters. The EVE code calculates the Hamiltonian matrix elements, and diagonalizes the matrix to find the energy eigenvalues and the eigenvectors of a nucleus where Coriolis mixing occurs. Once the mixing calculations have been performed, the EVE program calculates the cross section according to Equation 2.63 for a convenient comparison to experimental angular distributions. 2.5 Calculation of pairing factors Pairing factors, U 2 or V 2, of the states in a target nucleus are required to extract spectroscopic strengths from the comparison of experimental transfer cross sections to DWBA calculations. The calculation of Coriolis effects also requires knowledge of pairing factors in both the target and final nuclei. This section describes in detail the methods and calculations used to deduce estimates of pairing factors for Re and Os isotopes. The pairing gap parameter,, may be calculated according to the following 3-point formula[dob01]: (N) = ( 1)N 2 [B(N + 1) + B(N 1) 2B(N)], (2.64) where B(N) denotes the negative binding energy of the isotope with N neutrons. 46

66 The formula yields pairing gaps of roughly 0.74 MeV and 1.0 MeV for Re and Os respectively. The main problem in calculating pairing factors is that the single-particle energies and Fermi level are not observable, however, approximations can be made based on the energies of excited states. For an odd-a nucleus in the ground state, by definition the Fermi level is located at the odd-particle orbital, and since it is half full, both of the squared pairing factors are 0.5. For odd-a excited states, one must consider quasi-particle energies[bur66]. When the odd particle is excited from a single-particle level with energy ε 0 to one of energy ε, the excitation energy of the nucleus is given by: E x = (ε λ) (ε 0 λ) (2.65) Note that the second term is the energy of the ground state (E 0 ), and is nearly equal to the pairing gap due to the proximity of the odd particle ground state orbital to the Fermi level. By making this approximation, and solving for the excited single-particle energy relative to the Fermi level, formulae for the pairing factors in terms of the nuclear excitation energy are given by: U 2 = V 2 = ( ) 2 (2.66) E x + ( ) 2. (2.67) E x + These formulae allow the calculation of pairing factors for 185,187 Re just by using the excitation energies of the levels and the pairing gaps found above. Figure 2.3 shows the dependence of the pairing factors on excitation energy. 47

67 Pairing factor U 2 V Excitation energy (MeV) Figure 2.3: Pairing factors in Re as a function of excitation energy. Calculating pairing factors for even-even nuclei is more challenging. The approach and approximations taken here assume that the single particle energies in Os are exactly the same as those in the Re target used in the transfer reaction. It is also assumed that the Os Fermi level (λ Os ) is now halfway in between the last filled orbital, which also happens to be the Fermi level in the odd-a Re nucleus (λ Re ) and the first available empty orbital(ε ). This Fermi level relationship is summarized as: λ Os = λ Re + (ε λ Re )/2, (2.68) 48

68 which may be substituted into Equation 1.13 to obtain: UOs 2 = 1 (ε λ 1 + Re (ε λ Re )/2) (2.69) 2 (ε λ Re (ε λ Re )/2) 2 + Os 2 = 1 X 1 +. (2.70) 2 X 2 + Os 2 The single particle energy in Os was assumed to be the same as in the Re, therefore it can be eliminated by an expression in terms of excitation energies in Re: X = ε λ Re (ε λ Re )/2 (2.71) = (E x,re + Re ) 2 2 Re 1 (E,Re + Re ) 2 2 Re, (2.72) where the Os and Re pairing gaps are denoted by respectively. Equation 2.69 calculates the emptiness of a configuration in Os using the corresponding excitation energy in Re (E x,re ) and the Re excitation energy of the lowest quasiparticle excitation. For example, the energies of the lowest excitations (E,Re ) in 185,187 Re are about 0.37 MeV and 0.21 MeV, while the [402]π bandheads are located at MeV and MeV. Plugging these values into Equation 2.70 yields emptiness factor values of 0.87 and Figure 2.4 displays the trend of the emptiness in Os isotopes using Equation Table 2.2 and Table 2.3 summarize the emptiness factors that were calculated for use in deducing spectroscopic strengths in the ( 3 He,d) reactions and Coriolis mixing calculations. 49

69 Pairing factor Figure 2.4: energy U 2 V Effective excitation energy in Re (MeV) Pairing factors in 186 Os as a function of effective 185 Re excitation Configuration E Re (MeV) U 2 Re U 2 Os [402]π [400]π [402]π [411]π Table 2.2: Calculated pairing factors for spectroscopic strength and Coriolis mixing calculations for the 185 Re( 3 He,d) 186 Os reaction. 2.6 Summary This chapter has introduced single-particle models for spherical and deformed nuclei and the DWBA theory of single-nucleon transfer reactions. Corrections to the theory and Coriolis coupling were discussed, and the method for calculating pairing factors was presented. The next chapter describes the experimental work and techniques used in performing 185,187 Re( 3 He,d) 186,188 Os reactions. 50

70 Configuration E Re (MeV) U 2 Re U 2 Os [402]π [400]π [402]π [411]π Table 2.3: Calculated pairing factors for spectroscopic strength and Coriolis mixing calculations for the 187 Re( 3 He,d) 188 Os reaction. 51

71 Chapter 3 Materials and methods 3.1 Online: experiment equipment The ( 3 He,d) single-proton stripping-transfer reaction experiment was performed at the MLL in Germany, which is located at the Garching campus of the Ludwig Maximillians Universität (LMU) and the Technische Universität of Munich (TUM). The laboratory contained a tandem Van de Graaff accelerator which provided 30 MeV 3 He beams with currents up to 1.2 µa directed towards the target chamber of the Quadrupole-three-Dipole magnetic spectrograph (Q3D) via beam steering elements. Figure 3.1 shows the layout of the laboratory Target properties Enriched Re target materials were purchased from the isotope sales division of the Oak Ridge National Laboratory (ORNL), while some Pt target materials were already present at the laboratory. The targets were prepared by an on-site target maker by vacuum evaporation of the metals onto carbon backings. The properties 52

72 Figure 3.1: MLL schematic floor plan. Courtesy of the MLL website. of the targets with backings are listed in Table 3.1. The targets were mounted on ladders that allowed a quick turn-around time for target changes during the experiment. A photograph of one of the target ladders is shown in Figure Quadrupole-three-Dipole magnetic spectrograph (Q3D) The Q3D shown in Figure 3.3, consists of a series of magnetic elements that focus charged reaction products onto the focal plane where the energy and particle type are determined by analyzing signals from detectors. The spectrograph is rotated relative to the beam axis by a motor control, and the angle determined by looking at division markers along an iron track on which the spectrograph moves. The 53

73 Ladder# Position# Target Purity (%) Backing Thickness (µg/cm2 ) Target Backing Re C Re 87.5 nat C Re nat C 75* Re nat C Re C 82* Re nat C Pt nat C Pt nat C 39 9 Table 3.1: Target properties. The target thickness values marked with an asterisk were deduced from elastic scattering and have an uncertainty of approximately 2.5%. The unmarked values are quoted from the target maker and have an uncertainty much larger, likely on the order of 20 30%. magnetic fields were automatically changed by computer whenever the target, spectrograph angle, or momentum bite settings were changed Solid angle controls The acceptance solid angle of the spectrograph was set using micrometres to adjust slits that create a mask over an aperture. A diagram of the aperture and slits is shown in Figure 3.4. The aperture is a diamond-like shape with height 2B, width 2A, at a distance D away from the target chamber. The values of these distances are A = 31.5 mm, B = mm, and D = mm. The offsets to motor control slit positions were mm in the vertical (y) direction and 1.02 mm in the horizontal (x) direction. The y micrometre setting was held at a constant 20 mm throughout the experiment, while the x slit was usually set at 20 mm but reduced to 10 mm, 5 mm, and 2 mm for lower angles of elastic scattering measurements to avoid reaching the high count rates which could damage the focal plane detector. The solid angle for x micrometre readings of 2 mm, 5 mm, 10 mm, and 20 mm 54

74 Figure 3.2: Photograph of thin metal targets with carbon backings mounted on a ladder for the Q3D target chamber. are 0.62 msr, 2.53 msr, 5.72 msr, and msr respectively. The solid angle from the Q3D slits may be calculated as follows: 1. Calculate the true x and y slit values using the motor control readings and micrometre offsets. 2. The main contribution to the solid angle assumes a square represented by F = 4xy. 3. If the slits are wide enough, the aperture corners will be masked from accepting ions. The size of the masked area is F = 2 x y where x = x A + Ay/B and y = y B + Bx/A. The condition for corner masking is if x/a + y/b > The solid angle is calculated to be (F F)/D 2 steradians. 55

75 Dipole 2 Multipole Dipole 1 Dipole 3 Focal plane Quadrupole Target 3 He beam Faraday cup E E position sensitive cathode-strip detector (particle ID + energy) Figure 3.3: Diagram of the Q3D (not to scale). The dashed lines indicate the focusing of charged particles from the target chamber onto the focal plane by magnets Focal plane detector The detector at the Q3D focal plane (Figure 3.5) is able to determine the energy and type of particles from a reaction. The energy is found by the particle position along the focal plane detector. The particle identification comes from measuring the amount of energy lost in the two proportional counters compared to the total energy found by stopping the particle in a 7 mm thick plastic scintillator, or by a comparing the energy losses in both proportional counters[wir08]. The first proportional counter has one wire signal denoted by E 1, while the second proportional counter has two wires with a summed signal denoted by E. The second proportional counter is coupled to a cathode-strip foil to measure the position of the ion, while a photomultiplier tube measures light from the scintillator. The detector proportional counter chambers are filled with 500 mbar isobutane gas, 56

76 X slit y axis B X slit Y slit A x axis Y slit Figure 3.4: Q3D solid angle denoted by grey region. The thick lines are the slits controlled by micrometres. The diamond represents the Q3D aperture with a width of 2A and a height of 2B, where A = 31.5 mm and B = mm with the aperture being a distance D = mm from the target. which when particles enter inside, electron avalanches occur. As an avalanche moves towards an anode wire in the second chamber, a positive charge signal is registered on the 3 mm wide strips of the cathode foil. The charge signals from the strips are sent for digital processing to Application Specific Integrated Circuits (ASICs) which determine if a valid event has occurred by looking for large signals from 3 to 7 neighbouring strips. Figure 3.6 show the multiplicity distribution of the strip events. The online data analysis computer program deduces the position of an ion along the focal plane from a centre of gravity fit of the strip charge histogram. Some key specifications of the focal plane detector are that the ion position can be determined to a precision better than 0.1 mm with a khz count rate and a 150 µs dead time per event[wir00]. The strip charge distributions are stored for later 57

77 Figure 3.5: Cross-section diagram of the Q3D focal plane detector. The ions from the Q3D enter the target chamber from the left and hit the chambers at an angle of Source: Ref.[Wir08] playback to allow a more accurate Gaussian fit to be performed by the MARaBOU data analysis software package[lut99] once 2D histogram particle gates have been applied (gates are discussed in Section 3.2). The two anode wire signals in the second proportional counter provide a way to adjust the height of the focal plane detector. The balance of events between the wires is very sensitive to the detector height. Sample distributions are shown for the wires with and without particle gates in Figure 3.7 and Figure Beam normalization and dead-time corrections To obtain accurate absolute cross section measurements, it is critical to know the total number of beam particles which bombard a target during a run, and if the true data has been altered by dead-time effects before it is recorded. A 58

78 Frequency Cathode strip multiplicity Figure 3.6: Distribution of events with cathode strip hits. The hits with zero multiplicity is an artifact of how the data acquisition system records deadtime. Faraday cup at 0 relative to the beam axis with a current integrator was used as a beam monitor to provide a normalization for the measurement of absolute cross sections. A scalar tracked the integrated current throughout each run allowing the calculation of the number of beam particles. This conversion only requires the beam ion charge and the integrator scaling factor. Dead-time corrections must be made for both the detector and the data acquisition system. They are calculated using scalars that keep track of dead versus live events. The average dead-time corrections were: 0.17% for the detector and 5% for the data acquisition system with standard deviations of 0.15% and 3% respectively. There were some runs with unusually large data acquisition dead-time corrections which were identified by looking for corrections at least 3 standard deviations away from the averages. The reason(s) for these large corrections were 59

79 not determined. These large dead-time corrections were dismissed and the averages of the dead-time corrections for all other normal runs were applied. The average corrections (above) were calculated after the abnormal values were removed. A dead-time correction uncertainty equal to the standard deviation was included in the subsequent cross section uncertainty calculations. 3.2 Offline: Data analysis Once the data has been initially sorted offline, it is necessary to select the reaction channel of interest by setting gates on 2D histograms. This gating was performed with the HistPresent and MARaBOU data acquisition software[lut99] and resorting the acquired data sets. The E E and E E 1 histograms are created with the ADC signals from the two proportional counters ( E and E 1 ) along with the plastic scintillator signal (E). The ADCs have a range of 10V[Wir08]. The 2D histograms are shown in Figure 3.9 through Figure 3.12 with and without E E and E E 1 gates applied. To show how the anode and scintillator signals are used to form the 2D histograms, the signals as a function of position are shown in Figure 3.13 through Figure The figures also include both X and Y projections to clarify the relationships between them. Gates were also applied on the other structures in the E E and E E 1 2D histograms, but the focal plane spectra did not show any peaks. Figure 3.17 and Figure 3.18 show typical focal plane counts once the deuteron gates have been applied. Each peak in the focal plane particle spectrum represents the population of an excited state in the recoiling nucleus of the reaction. 60

80 Upper wire (no gate) Upper wire (with gate) Frequency Pulse height (arb. units) Lower wire (no gate) Lower wire (with gate) Frequency Pulse height (arb. units) Figure 3.7: Anode wire signal peak height distributions. Note that applying gates does not change the distribution significantly. 61

81 Lower wire (no gate) Upper wire (no gate) Frequency Pulse height (arb. units) Lower wire (with gate) Upper wire (with gate) Frequency Pulse height (arb. units) Figure 3.8: Comparing the peak height distribution in the upper and lower anode wires. The detector is able to be moved up and down, to which the balance of events on each wire is very sensitive. 62

82 Energy Loss Rest Energy Figure 3.9: E E plot (no gates) from anode and photomultiplier signals. The X and Y projections are shown to demonstrate the relationship to other plots. 63

83 Energy Loss Rest Energy Figure 3.10: E E plot (with gates) from anode and photomultiplier signals. The X and Y projections are shown to demonstrate the relationship to other plots. 64

84 Energy Loss Energy Loss 1 Figure 3.11: E E 1 plot (no gates) from two different anode signals. The X and Y projections are shown to demonstrate the relationship to other plots. 65

85 Energy Loss Energy Loss 1 Figure 3.12: E E 1 plot (with gates) from two different anode signals. The X and Y projections are shown to demonstrate the relationship to other plots. 66

86 Energy Loss Position Figure 3.13: E position plot (no gates) from the first anode signal. The X and Y projections are shown to demonstrate the relationship to other plots. 67

87 Energy Loss Position Figure 3.14: E position plot (no gates) from the second anode signal. The X and Y projections are shown to demonstrate the relationship to other plots. 68

88 Rest Energy Position 1 Figure 3.15: E position plot (no gates) from the scintillator signal. The X and Y projections are shown to demonstrate the relationship to other plots. 69

89 3.3 Peak fitting Once the particle gates have been applied to create a spectrum showing counts which represent the population of a particular nuclear state in the reaction, the task turns to fitting the peaks. This section will describe the methods used to fit the peaks, while the remaining sections of this chapter explain the procedures used to deduce cross sections and excitation energies from peak areas and centroids. The peak fit function contains parameters describing the peak and background shapes. To obtain the best centroids and areas with uncertainties, it is important to fix these peak shape parameters to reduce the dimension of the space where chi-square minimization occurs. The techniques and software used to fix the peak shape parameters will now be described Components of the peak fitting function The program germanium fit version 3 (gf3) in the RADWARE software package version rw01[rad01] was used to display spectra and fit the observed peaks. This package was originally intended for use in analyzing γ-ray data, however, it fortunately contains a peak fitting subroutine that is relevant for this ( 3 He,d) work. The gf3 peak fit function has four components: 1. A symmetric Gaussian component describing the peak position (p), width (σ), and height (h). 2. A skewed Gaussian component describes the energy straggling of deuterons in the target foil and focal plane detector. The decay length scale of the tail is 70

90 denoted by β and the size of this component relative to the total peak height is denoted by R as a percent. 3. A quadratic polynomial to model the shape of a background. A user may set some coefficients to zero if a linear or constant background is desired. 4. A smoothed step function describing a background for γ rays. This component was not relevant for the work described in this thesis. The gf3 program allows a user to fix or free any peak shape parameter during a fit, and allows some shape parameters to be varied as a function of channel number in the spectrum. The R and β parameters can be fixed using a linear polynomial in channel number, while the width may be fixed by the following equation: σ 2 = F 2 + G 2 X + H 2 X 2, (3.1) where F, G, H are user-defined coefficients and X is the channel number divided by Fixing parameter trends In order to obtain the best values for peak centroids and areas, the values of R, β, and σ must be fixed. Data at a 50 scattering angle were used because of the ideal background conditions. Firstly, all of the relatively large peaks were fit with the shape parameters being free. The procedure for fixing shape parameters was aided by a computer program written by the author called RADBUDDY. The purpose of the RADBUDDY program was to automate the following stages of 71

91 analyzing peak fits: 1. Read in peak fit results from gf3 output files. 2. Perform polynomial weighted linear regression[bev03] on the peak shape parameters as a function of channel number. 3. Determine the best polynomial degree from the F-test (Subsection 3.3.3). 4. Allow the user to remove any outlying points and perform a new fit. 5. Output the shape parameter polynomial coefficients in a format that gf3 can read to quickly refit the original peaks with fixed peak shape parameters. A sample screenshot of the RADBUDDY program working on a Linux operating system is shown in Figure RADBUDDY uses the Qt4 cross-platform graphical user interface toolkit along with the GNU Scientific Library (GSL) for matrix inversion in the weighted linear regression calculations. The value of the R shape parameter was found to be best fit by a constant value of approximately 35.7%. Once the R parameter was fixed and the peaks refit, β was best described by a constant of 7.0 channels. These R and β best-fit values were then used for all runs of the experiment. The trends of the squared peak width were fit on a run-by-run basis. The best fit polynomial often varied between a constant and parabola depending on the run. One occasional problem did arise from the difference between the RAD- BUDDY squared width polynomial and the gf3 squared width formula (Equation 3.1). The RADBUDDY polynomial allows negative coefficients but the gf3 formula does not. When RADBUDDY recommended a set of coefficients with 72

92 Figure 3.16: A screenshot of the RADBUDDY program. 73