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The World Ends Tomorrow and
YOU MAY DIE!Well, no, probably not. . . but whatever you do, just keep reading!
Cover page illustration by:
Karin Rönmark
The theory of everythingKonstfack University College of Arts, Crafts and Design spring exibition 2009.
http://www.karinronmark.se/
List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I P.-A. Söderström, et al. Spectroscopy of Neutron-Rich 168,170Dy:
Yrast Band Evolution Close to the NpNn Valence Maximum. PhysicalReview C, 81:034310, 2010
II G. M. Tveten, P.-A. Söderström, et al. The neutron rich isotopes167,168,169Ho studied in multi-nucleon transfer reactions. In manuscript.
III P.-A. Söderström, et al. Interaction Position Resolution Simulationsand In-beam Measurements of the AGATA HPGe detectors. NuclearInstruments and Methods in Physics Research, A638:96, 2011.
IV P.-A. Söderström, J. Nyberg, and R. Wolters. Digital pulse-shape
discrimination of fast neutrons and γ rays. Nuclear Instruments andMethods in Physics Research, A594:79, 2008.
V E. Ronchi, P.-A. Söderström, J. Nyberg, E. Andersson Sundén,S. Conroy, G. Ericsson, C. Hellesen, M. Gatu Johnson, M. Weiszflog.An artificial neural network based neutron-gamma discriminationand pile-up rejection framework for the BC-501 liquid scintillationdetector. Nuclear Instruments and Methods in Physics Research,A610:534, 2009.
Reprints were made with permission from the publishers.
Major publications not included in this thesis.
1. K. Straub, et al. Decay of drip-line nuclei near 100Sn. Submitted to theGSI Scientific Report 2010.
2. A. Pipidis, et al. The Genesis of NEDA (NEutron Detector Array):
Characterizing its Prototypes. Submitted to the LNL Annual Report 2010.
3. F. C. L. Crespi, et al. Measurement of 15 MeV γ rays with the AGATAcluster detectors. Submitted to the LNL Annual Report 2010.
4. M. Senyigit, et al. AGATA Demonstrator Test with a 252Cf Source:Neutron-Gamma Discrimination. Submitted to the LNL Annual Report2010.
5. D. D. DiJulio, et al. Electromagnetic properties of vibrational bands in170Er. Eur. Phys. J., A47:25, 2011.
6. S. Hirayama, et al. Production of protons, deuterons, and tritons fromcarbon bombarded by 175 MeV quasi mono-energetic neutrons Prog.Nucl. Sci. Tech., 1:69, 2011.
7. B. Cederwall, et al. New spin-aligned pairing phase in atomic nucleiinferred from the structure of 92Pd. Nature, 469:68, 2011.
8. T. S. Brock, et al. Observation of a new high-spin isomer in 94Pd. Phys.Rev. C, 82:061309, 2010.
9. A. Blazhev, et al. High-energy excited states in 98Cd. J. Phys. Conf. Ser.,205:012035, 2010.
10. R. Wadsworth, et al. The northwest frontier: Spectroscopy of N ∼ Z nucleibelow mass 100. Acta Phys. Polon., B40:611, 2009.
11. P.-A. Söderström, et al. AGATA: Gamma-ray tracking in seg-mented HPGe detectors. In Proceedings of the 17th InternationalWorkshop on Vertex detectors, PoS (VERTEX 2008), page 040. Sissa, 2009.
12. U. Tippawan, et al. Studies of neutron-induced light-ion production withthe MEDLEY facility. In O. Bersillon, et al. (editors), Proceedings of theInternational Conference on Nuclear Data for Science and Technology2007, page 1347. EDP Sciences, 2008.
13. M. Hayashi, et al. Measurement of light-ion production at the new Uppsala
neutron beam facility. In O. Bersillon, et al. (editors), Proceedings of theInternational Conference on Nuclear Data for Science and Technology2007, page 1091. EDP Sciences, 2008.
14. M. Hayashi, et al. Neutron-induced proton production from carbon at
175 MeV. In T. Hazama and T. Fukahori (editors), Proceedings of the2007 Symposium on Nuclear Data November 29-30, 2007, Ricotti, Tokai,Japan, volume JAEA-Conf 2008-008, page 62. Japan Atomic EnergyAgency, Tokai-mura, Japan, 2008.
15. P.-A. Söderström. Detection of fast neutrons and digital pulse shapediscrimination between neutrons and γ rays. In A. Covello, et al.
(editors), Proceedings of the International School of Physics ’EnricoFermi’, volume 169 Nuclear Structure far from Stability: new Physics and
new Technology, page 551. SIF, Bologna and IOS Press, Amsterdam, 2008.
16. M. Hayashi, et al. Effect of nuclear interaction loss of protons in the
response of CsI(Tl) scintillator. Engineering Sciences Reports, KyushuUniversity, 29:374, 2008.
Contents
1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Nuclear structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 The nuclear landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Spin and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Nuclear astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 S-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 R-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 P-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Rp-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.5 νp-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Radioactive ion beam facilities . . . . . . . . . . . . . . . . . . . . . . . . . 10
Part I: Physics2 Theory of deformed nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Nuclear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Nilsson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Particle plus triaxial rotor model . . . . . . . . . . . . . . . . . . . 17
2.1.3 Cranking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Variable moment of inertia model . . . . . . . . . . . . . . . . . . 20
2.2 Nuclear deformations in the r process . . . . . . . . . . . . . . . . . . . 21
2.3 Heavy-ion induced nuclear reactions . . . . . . . . . . . . . . . . . . . . 24
3 PRISMA and CLARA experiment . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 LNL accelerator complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 PRISMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 MCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Quadrupole and dipole magnets . . . . . . . . . . . . . . . . . . . . 32
3.2.3 MWPPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.4 Ionization chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.5 Mass determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 CLARA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Dysprosium isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Holmium isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Evolution of collectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Variable moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Deformations in odd-A nuclei . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Rigidity and backbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Part II: Technology5 The AGATA HPGe spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 HPGe crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3 Pulse-shape analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Tracking of γ rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.6 Position resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.6.1 Reaction selection and simulations . . . . . . . . . . . . . . . . . . 635.6.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.7 Neutrons in AGATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Neutron detector NEDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1 SPIRAL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 The Neutron Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.3 The neutron detector array NEDA . . . . . . . . . . . . . . . . . . . . . . 72
6.3.1 The BC-501A and BC-537 liquid scintillators . . . . . . . . . 736.3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3.3 Detection of scintillation light . . . . . . . . . . . . . . . . . . . . . 74
6.3.4 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4 Digital pulse shape analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4.1 Charge comparison and zero cross-over . . . . . . . . . . . . . . 766.4.2 Artificial neural networks . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4.3 Time resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Part III: Discussion7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.1 AGATA at LNL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 AGATA at GSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3 AGATA at SPIRAL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939 Kollektiv kärnstruktur hos neutronrika sällsynta jordartsmetaller och
nya instrument för gammaspektroskopi . . . . . . . . . . . . . . . . . . . . . . 9510 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Contribution to the papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
List of Acronyms
ACTAR Active Targets
ADC analog-to-digital converter
AGATA Advanced Gamma Tracking Array
AGAVA AGATA VME Adapter
ALICE A Large Ion Collider Experiment
ANN artificial neural network
APD avalanche photodiode
ATC AGATA triple-cluster
BCS Bardeen, Cooper and Schrieffer
BGO bismuth germanate
B2FH Burbidge, Burbidge, Fowler and Hoyle
CARMEN Cells Arrangement Relative to the Measurement of Neutrons
CERN Organisation Européenne pour la Recherche Nucléaire
CIME Cyclotron pour Ions de Moyenne Energie
CLARA Clover Detector Array
CNO carbon, nitrogen and oxygen
CPU central processing unit
DANTE Detector Array for multi-Nucleon Transfer Ejectiles
DAQ data aquisition system
DESCANT Deuterated Scintillator Array for Neutron Tagging
DESIR Désintégration, Excitation et Stockage des Ions Radioactifs
DSP digital signal processor
ESS European Spallation Source
FAIR Facility for Antiproton and Ion Research
FAZIA Four pi A and Z Identification Array
FET field-effect transistor
FOM figure-of-merit
FPGA field programmable gate array
FRS fragment separator
FWHM full width at half maximum
GANIL Grand Accelerateur National d’Ions Lourds
GASPARD Gamma Spectroscopy and Particle Detection
GDR giant dipole resonance
GRETA Gamma Ray Energy Tracking Array
GSI Gesellschaft für Schwerionenforschung mbH
GTS global trigger and synchronization
GUI graphical user interface
HELIOS Helical Orbit Spectrometer
HPGe high-purity germanium
IBFM interacting boson-fermion model
IBM interacting boson model
IC ionization chamber
ILL Institut Laue-Langevin
IReS Institut de Recherches Subatomiques de Strasbourg
ISAC Isotope Separator and Accelerator
ISOL isotope separation on-line
LHC Large Hadron Collider
LINAC linear accelerator
LNL Laboratori Nazionali di Legnaro
LYCCA Lund-York-Cologne Calorimeter
MCP micro-channel-plate
MS/s megasamples per second
MSU Michigan State University
MWPPAC multi-wire parallel-plate avalanche counter
NARVAL Nouvelle Acquitision temps-Reel Version. 1.6 Avec Linux
NEDA Neutron Detector Array
NFS Neutrons for Science
NIM Nuclear Instrumentation Module
PARIS Photon Array for studies with Radioactive Ion and Stable beams
PC personal computer
PMT photomultiplier tube
PSA pulse-shape analysis
PSD pulse-shape discrimination
QCD quantum chromodynamics
RHIC Relativistic Heavy Ion Collider
RIB radioactive ion beam
RIKEN Rikagaku Kenkyusho
RISING Rare Isotope Spectroscopic Investigation at GSI
S3 Super Separator Spectrometer
SPIRAL Système de Production d’Ions Radioactifs en Ligne
TOF time-of-flight
TRIM Transport of Ions in Matter
TRS total Routhian surface
VME Versa Module Europe
VMI variable moment of inertia
WMAP Wilkinson Microwave Anisotropy Probe
ZCO zero cross-over
List of Figures
1.1 Illustration of the particles of the standard model and their
tree-level interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Selection of nuclear structure topics in the nuclear chart andin the E − J plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Periodic table from an astronomer’s point of view and a geol-ogist’s point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Collective excitations of an atomic nucleus . . . . . . . . . . . . . . . 14
2.2 Nilsson diagram for protons, 50 ≤ Z ≤ 82. . . . . . . . . . . . . . . . 17
2.3 Solar r-process abundances, contours of constant neutron sep-aration energy and constant β -decay rates . . . . . . . . . . . . . . . . 22
3.1 The region of interest, for this work, in the Segré chart . . . . . . 28
3.2 The PRISMA and CLARA set-up . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Sketch of the PRISMA spectrometer . . . . . . . . . . . . . . . . . . . . 31
3.4 Positions of the reaction fragments as measured by the MCP . 31
3.5 The positions and time-of-flight in the MWPPAC . . . . . . . . . . 34
3.6 Partial energy loss and range of the fragments with respect tothe total energy loss of the fragments in the IC . . . . . . . . . . . . 36
3.7 Relative mass of the fragments with respect to their positionin the MWPPAC and the bending radius with respect to theenergy used to obtain the absolute mass . . . . . . . . . . . . . . . . . 36
3.8 Mass spectrum from PRISMA of target-like fragments gatedon krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.9 The CLARA HPGe detector array . . . . . . . . . . . . . . . . . . . . . 39
3.10 Doppler-corrected spectra from beam-like and target-like
fragments as recorded by CLARA . . . . . . . . . . . . . . . . . . . . . 40
3.11 Time-of-flight with respect to γ-ray energy for a selection on168Dy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.12 Spectrum of γ-ray energies from target like fragments gated
on 168Dy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.13 Spectrum of γ-ray energies from targetlike fragments gated on170Dy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.14 Ground state rotational bands for dysprosium isotopes with
N = 94−104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.15 Ground state rotational bands for 167Ho and 169Ho and the
level scheme for 168Ho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1 Quadrupole deformation parameters β2 for even-even nuclei
according to the Möller and Nix calculations and experimen-
tally measured β2 deformations . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Experimentally measured β2 values and Harris parameters for
a selection of even-even nuclei . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Projections of the particle plus triaxial rotor parameter spacefor 100 000 random points . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Total Routhian Surface calculations of Dy isotopes with160 ≤ A ≤ 170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Total Routhian Surface calculations of Ho isotopes with
161 ≤ A ≤ 171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1 The AGATA HPGe crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Signals from the core, the segment with the primary hit, andfrom the mirror charges for a γ-ray interaction in a six fold
segmented HPGe detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Simulated interaction points of 30 γ rays of energyEγ = 1.33 MeV in the (θ ,φ sinθ ) plane of an ideal
germanium shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Effects of the position resolution when determining the angleused to correct for the Doppler shifts of the γ rays . . . . . . . . . . 62
5.5 The difference between the calculated interaction position res-
olutions and the mgt smearing parameter . . . . . . . . . . . . . . . . 66
5.6 The AGATA detector in position for the first commissioningexperiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.7 Gamma-ray spectrum measured using the 30Si+12 C reactionwith the AGATA triple cluster detector . . . . . . . . . . . . . . . . . . 68
5.8 Interaction position resolution as a function of γ-ray energy . . 69
5.9 The AGATA and HELENA detectors in position for the neu-
tron experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1 Pulse shapes from a BC-501 liquid scintillator from a γ rayand a neutron interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2 Possible geometry of the NEDA detector array . . . . . . . . . . . . 75
6.3 Weighting function for digital and analogue charge compari-
son PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.4 Difference between the integrated rise time of a γ-ray and a
neutron pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.5 FOM and R for the ZCO and charge comparison based meth-ods as a function of energy, bit resolution and sampling fre-
quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.6 Fraction of incorrectly identified γ-ray and neutron events as a
function of the deposited energy for the artificial neural network 816.7 Sampling of a Gaussian function with a time between sam-
pling points equal to the σ of the Gaussian function . . . . . . . . 82
6.8 The measured times, T , as a function of T1 and the time dis-tributions due to the finite sampling frequency . . . . . . . . . . . . 83
6.9 Time distributions folded with a typical Gaussian time resolu-tion of a liquid scintillator detector plus PMT . . . . . . . . . . . . . 84
7.1 Grazing calculations of production cross-sections for dyspro-
sium isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.2 Calculated production cross-sections of the Dy isotopic chain
in the FRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1. Background
“Thence come the maidens mighty in wisdom,Three from the dwelling down ’neath the tree;
Urðr is one named, Verðandi the next,On the wood they scored, and Skuld the third.
Laws they made there, and life allottedTo the sons of men, and set their fates.”
– Prophecy of the Völva
Since the dawn of time women and men have thought about the world, why
it looks the way it does and what it is actually made of. In order to understand
nature, observations have been made and based on these (and a fair amount
of imagination to fill in the blanks) conclusions have been drawn about the
universe. One example of the early beliefs about the world is from the Nordic
countries, where the three norns Urðr, Verðandi and Skuld weaved the fabric
of reality at the roots of the world tree Yggdrasill. Nothing could be found
outside of this weave, since it contained the entire cosmos [1].
These days, the number of observations is much larger and the scientificmethod has been developed. Thus, our understanding of the universe is muchmore accurate today than during the days of the Vikings. There are still, how-ever, many blanks to fill. In order to fill some of these blanks, the presentfront-line of theoretical physics considers another weave. Using a mathemat-ical formulation of the most fundamental constituents of matter as tiny multi-dimensional Planck scale threads making up space-time itself, the reality canbe modelled through these threads different vibration patterns. This theory isknown as the supersymmetric string theory, or simply superstring theory [2].However, even if this theory aims at describing the universe from first princi-ples it is very much under development and still far from being experimentallytestable. So at the moment, the most fundamental description of the world isinstead given by an effective theory called the standard model [3–5].
The standard model describes the universe in terms of the electromagnetic,
the weak and the strong nuclear interactions that govern the dynamics between
the fundamental particles: quarks1 and leptons that make up the matter and
bosons to mediate the interactions between these. Furthermore, the standard
model contains an additional boson, the Higgs boson [8, 9], that generates
1The name quark was given by Murray Gell-Mann and is a actually nonsense word from the
book Finnegan’s Wake, by James Joyce, where the sentence “Three quarks for Muster Mark” is
sung by a chorus of sea birds [6, 7].
1
Figure 1.1: Illustration of the particles of the standard model and their tree-level inter-
actions. The leptons in the standard model are the electron (e), the muon (μ), the tauon
(τ) and their corresponding neutrinos (νe,νμ ,ντ ). The quarks of the standard model
are the up (u), down (d), charm (c), strange (s), top (t) and bottom (b) quarks. The
quark-gluon structures of the proton (uud) and neutron (udd) are also shown. Figure
from [10].
mass to some of the particles. The Higgs boson is the only particle of the
standard model yet to be discovered. See figure 1.1 for an illustration of the
particles of the standard model and their respective interactions.All matter that we encounter in everyday live is made up of atoms. The
atoms in turn are composed of an atomic nucleus, with a specific number
of protons [11, 12] and neutrons [13] (together commonly referenced as
nucleons), surrounded by a cloud of electrons. Nuclear matter is governed
by the interaction between the fundamental standard model particles quarks
and gluons. This part of the standard model is known as the quantum
chromodynamics (QCD) [14] and could in principle be used to calculate
any feature of the nuclear matter, although this is a very complicated task.
As of today, lattice QCD has successfully been used to calculate the mass
of the proton2 [16]. Even if this is a great achievement it is still a long way
to go before QCD can be used to understand the complex dynamics of a
many-body nuclear system.
To reduce the complexity of these calculations one can construct effectivenucleon-nucleon interactions from QCD or experiments. Using these it is pos-sible to calculate the properties of any nuclear system from the first principles,ab initio, of this effective theory. For practical reasons one is, however, limited
2There is, however, some controversy about this statement. Since the calculations are carried
out for heavier quarks masses the results must be scaled down to the physical masses through
chiral extrapolation [15].
2
to few-body systems. When the system becomes too large, also the ab initiocalculations become unfeasible. The largest system that so far has been stud-
ied ab initio using an importance-truncated no-core shell model is 40Ca [17].To study heavier nuclear systems, but also to simplify the study of lighter sys-tems, different models based on phenomenological observations of the nuclearmatter are introduced. Some of these models will be presented in further detailin chapter 2.
1.1 Nuclear structure
As mentioned earlier, each atomic nucleus is made of a specific number ofprotons and neutrons. The nucleus is usually denoted AXx, where Xx is the
element label and A is the number of nucleons in the nucleus. For lighternuclei the number of protons and neutrons are approximately the same, whileheavier nuclei consist of more neutrons than protons. The chart of nuclides,or the Segré chart, is a plot of the number of protons versus the number ofneutrons, see figure 1.2. As seen in the Segré chart, the number of stable nucleiare very few. Arranged in a bent line called the line of β stability, there areonly about 250 of them.
Many more nuclei can, however, be constructed either in laboratories onearth or in violent astrophysical events like supernovae explosions. About3000 elements have up to now been created and observed in laboratories, buttheorists predict that more than 6000 bound nuclei can exist between the neu-tron and proton drip-lines, which are defined as the limits of nuclear existence.The physics of these very exotic nuclides is to a large extent unknown andmany surprises probably await in this terra incognita.
1.1.1 The nuclear landscape
One of the most notable features in figure 1.2, the line of β stability, roughly
follows a pattern of lines with certain values of the number of protons and
neutrons. These numbers, called the magic numbers, represent the nuclei in
nature that are most tightly bound and form closed shells where the main
structure properties comes from the behaviour of the nucleons outside these
shells. The shell model of spherical nuclei is one of the most fundamental of
nuclear physics and is well described in many standard references [18–24].
Some calculations also suggest the existence of new magic numbers larger
than has been observed so far. These new magic numbers could cause an “Is-
land of Stability” of superheavy elements where completely new long-lived
chemical elements would appear. For further details regarding the discovery
of super-heavy elements see reference [25] and references therein.To test the nuclear shell model it is of much interest to explore the proton-
rich side of the line of β stability. The proton-rich region around 100Sn is,
3
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Figure 1.2: Selection of nuclear structure topics in the nuclear chart, with proton num-
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4
for example, an ideal testing ground for another important aspect of nuclear
structure physics, the proton-proton, neutron-neutron and proton-neutron pair-
ing [26]. The nucleus 100Sn is the heaviest self-conjugate doubly-magic nu-cleus that is expected to be bound, first discovered by two independent experi-ments [27, 28] and recently studied using the fragment separator (FRS) at GSI[29, 30].
On the neutron rich side of the Segré chart, the only area where the drip-line has been reached is the area containing the light neutron-rich nuclei. Thefirst time this area was explored the results showed that some neutron-richnuclei, in particular 11Li, had an abnormally large size. In fact, 11Li has thesame size as 208Pb despite the much fewer number of nucleons. This was
interpreted as 11Li being a so called Borromean3 halo nucleus consisting of a9Li core surrounded by a halo of two neutrons, a picture that was confirmed
by further measurements on 11Li [31, 32]. Future efficient and precise neutrondetectors, see chapter. 6, can provide an opportunity to further understandthe structure, radii, masses and reaction probabilities of neutron-rich exoticnuclei. Such understanding is crucial for the knowledge of how the chemicalelements we are made of are created through a process called the r-process,further discussed in section 1.2.
1.1.2 Spin and energy
The study of the structure of nuclei often involves measuring the characteristicenergies and angular distributions of particles (for example electrons, neutronsor α particles) or γ rays emitted from these nuclei. In order for particles or γrays to be emitted from a nucleus it must be provided with some excess en-
ergy. For closed shell nuclei this excess energy can be understood in terms of
rearrangement of the nucleons within the shell structure. For collective sys-
tems like a rigid, deformed nucleus excitation energy could for example go
into nuclear rotation, increasing the angular momentum, or spin, of the nu-
cleus. When all excess energy goes into angular momentum and no energy
into other excitation modes, for example shape vibration, the nucleus is said
to be in an yrast4 state. The yrast line is illustrated in figure 1.2, together withsome phenomena that can occur when providing the nucleus with excitationenergy.
Even if the collective properties give a good description of the nucleus at
low spin and excitation energy, the properties of high spin states will show that
the nucleus cannot simply be described by only collective motion, but that the
3This name comes from the Borromean family crest which is made of three rings entwined such
that if one is removed the entire system falls apart. In the same way the Borromean halo nuclei
consist of a nucleus and a halo of two neutrons, while neither system of the specific nucleus
and one neutron halo nor the system of two neutrons are bound.4The name yrast originates from a Swedish play with words. Literally it translates to “most
dizzy”, which you of course become if you spin as fast as you can.
5
properties of the individual nucleons are also very important. One example of
this is when the nucleon-nucleon pairs break and the angular momentum is
reduced with increasing excitation energy, an effect called backbending that
was first discovered in 1971 [33]. The influence of single particles on the
collective behaviour of nuclei also gives rise to other effects. One of these
being when the single particles of the broken pairs have a large spin along the
axis of nuclear symmetry, giving rise to long lived excited states called high-Kisomers [34]. These high-K isomeric states close to the yrast line results from
the nuclear system being well ordered with clear rules for how it can decay. At
higher excitation energies the nuclear system is fully chaotic. This region of
order-to-chaos transition is another example of where the nuclear many-body
system is yet to be fully understood [35, 36].But not all high energy excitations show this chaotic behaviour. At high ex-
citation energies with small angular momentum other kinds of collective be-
haviour can be observed where the nucleus can be interpreted as separated into
proton and neutron fluids and, for example, oscillate against each other [37] or
have rotational oscillations with opposite phase around a common axis [38].
This kind of collective behaviour is usually referred to as giant resonances,
and will be briefly discussed in section 5.7.This is just a small selection of the different phenomena shown in figure 1.2,
which in turn is just a small selection of the different phenomena that occur in
the atomic nuclei. It should, however, be clear that nuclear many-body systems
are very complex and that there is a long way before its phenomenology can
be explained from the first principles of QCD.
1.2 Nuclear astrophysics
To answer the question where matter, as we know it, originates from we should
go back to the beginning of the universe. The current model of the universe
says that it originated from a singularity that expanded into its current size.
This model is referred to as the Big Bang, based on an idea of Lemaître [39]
that was first confirmed by Hubble [40] and have been confirmed many more
times after that, most recent by the Wilkinson Microwave Anisotropy Probe
(WMAP) measurement of the cosmic microwave background [41] to test the
cosmic inflation theory.
The very first moment is currently beyond our physical understanding andwould require a theory that combines quantum field theory with gravity, forexample the previously mentioned superstring theory [2]. However, the uni-verse expanded and about 10−12 s after the Big Bang the four forces took theircurrent form. The universe then consisted of hot quark-gluon plasma. This isthe first point in the time line of the universe where nuclear physics, althoughat very high energies, plays an important role in our understanding of nature.The study of the quark-gluon plasma is a task that has been undertaken at the
6
H He
C N O Ne
Mg Si S
Fe
H
Li Be B C N O F
Na Mg Al Si P S Cl
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br
Rb Sr Y Zr Nb Mo Ru Rh Pd Ag Cd In Sn Sb Te I
Cs Ba 1La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi
2Ac
Figure 1.3: Periodic table from an astronomer’s point of view (top) and a geologist’s
point of view (bottom). The size of the box is proportional to the element abundance.
From reference [45].
Relativistic Heavy Ion Collider (RHIC) [42] and that will be further studied
by ALICE at the Large Hadron Collider (LHC) [43].As the universe expanded and cooled further, neutrons and protons formed
and together fused into the light elements. A first attempt to describe this
early nucleosynthesis was made by Alpher, Bethe and Gamow [44]. After the
Big Bang nucleosynthesis, the universe consisted of about 75 % hydrogen,
25 % helium and some traces of heavier elements. This is approximately true
also today from an astronomical point of view, apart from a small amount of
heavier elements, see figure 1.3.From a geological point of view, however, the picture is very different. If
we look around us we see many elements other than hydrogen and helium.
The distribution of elements in the earth’s crust is illustrated in figure 1.3.
These elements must have been created somewhere else than in the Big Bang,
for example in the fusion processes of a star. In the beginning of a stars life it
burns with the same process as in the Big Bang nucleosynthesis, that is proton-
proton fusion into deuterium and further into helium [46] or, if these elements
exist, a thermonuclear cycle involving carbon, nitrogen and oxygen (CNO)
[46, 47]. When the core of the star has run out of hydrogen it will instead
7
start to burn helium, creating 12C5. The carbon can then be further used as a
starting point to synthesise heavier elements through reactions such as 12C +4He, 16O+4He, 12C+12C, 16O+16O and similar. For details, see for examplereference [48]. There is, however, still a problem with this model. The nucleithat are most tightly bound are the nuclei around iron, thus the star cannotgain energy by fusing heavier nuclei. A solution to this problem came withthe breakthrough paper by Burbidge, Burbidge, Fowler and Hoyle (B2FH)[49]. In that paper, three different processes to synthesise heavier nuclei wereproposed, called the s-process, the r-process and the p-process.
1.2.1 S-process
The first process that was proposed in the B2FH paper was the slow neutron-
capture process, or s-process. This is a process that occurs at locations with
a low density of neutrons and an intermediate temperature, for example in
a type of red giant stars called asymptotic giant branch stars. It starts with a
seed nucleus from the iron area, the endpoint of the charged particle reactions.
These stable nuclei capture neutrons, turning them into radioactive isotopes of
the same element. The half life of this radioactive isotope is then determining
the probability for the nucleus to capture another neutron, or decay. When
the neutron capture probability becomes so small that the average time for
neutron capture is larger than the half life, it will β decay to the next heavier
element, turning one neutron into a proton and an electron. In this way, the
line of β stability is followed closely up to 209Bi, the heaviest stable element.
Since this process involves stable and close-to-stable nuclei most of them are
accessible for detailed measurements of neutron capture cross sections and βdecay half-lives.
1.2.2 R-process
As the s-process can only create nuclei up to 209Bi it cannot explain the exis-
tence of the heavier naturally occurring elements like 232Th, 235U and 238U.Furthermore, the s-process does not reach stable neutron-rich intermediate-mass nuclei further away from the line of β stability. The process suggested
in the B2FH paper for the creation of these elements was instead the rapidneutron-capture process, or r-process. While the s-process occurs at areas inthe universe with a low density of neutrons so that the average time for neu-tron capture in general is smaller than the half life of the radioactive isotope,there are other areas of so high neutron densities that the half life of the ra-dioactive isotope is negligible. These very high neutron density areas can befound in core-collapse supernovas like the famous supernova SN1987A in theLarge Magellanic Cloud [50, 51]. In these areas the neutron capture rapidly
5As 8Be is unbound, three helium nuclei are required in this process.
8
creates very neutron rich nuclei that, when the process stops, either β decays
or fissions back to β stability. Since this process occurs close to the neutrondrip-line in the terra incognita, very little is known of the exotic nuclei par-ticipating in this process. As discussed in section 1.1 many surprises prob-ably awaits there, some of which will be discovered at the radioactive ionbeam (RIB) facilities discussed in section 1.3.
1.2.3 P-process
The third process proposed in the B2FH paper was the proton-capture process,or p-process, to explain the heavy proton-rich nuclei that cannot be producedby the s- or r-process, for example 190Pt or 168Yb. It occurs in similar areasas the r-process, core-collapse supernovas, but instead of neutron-capture theγ rays in this high temperature environment removes neutrons from the nucleiand thus increases the proton ratio of the resulting nucleus. It is not actuallyso much of a proton-capture process, but more of a neutron-removal process.
1.2.4 Rp-process
The rapid proton-capture process, or rp-process [52], is not one of the originalB2FH processes, but the process in which the lighter proton-rich nuclei arecreated. Although the end point of the rp-process is not precisely known it isestimated to be located in the area around nuclei with 100 nucleons and at leastless than tellurium [53]. To establish this endpoint is one of many motivationsfor the study of atomic nuclei around 100Sn [29]. The rp-process takes placeright after the thermonuclear explosion of a binary system consisting of aneutron star that is accreting hydrogen and helium from another star. Thisexplosion causes a very hot and proton rich environment where consecutiveproton captures can occur.
1.2.5 νp-process
A quite recently suggested astrophysical process to complement the four clas-sical processes is the νp-process [54, 55] that have emerged from advances in
core-collapse supernova simulations. When a core-collapse supernova occurs
the γ radiation and proton-rich matter is accompanied also by a large neutrino,
ν , and antineutrino, ν , emission [51]. This emission can cause the total dis-integration of heavy nuclei into protons and neutrons. When these recombineinto nuclei in the hot environment after the explosion, similar as the fusionprocesses in stars, elements up to iron are recreated. However, antineutrinocapture in these nuclei will result in protons transforming into neutrons, al-lowing the process to flow past iron creating intermediate-mass proton-richnuclei.
9
1.3 Radioactive ion beam facilities
To study the physics described in previous sections specific tools and methodsare needed. The very first use of an ion beam for studies of nuclear reactionswas done by Rutherford when he used a collimated 226Ra source of α parti-cles together with a nitrogen gas [56]. This work was followed by Cockcroftand Walton who designed a machine that, instead of using radioactive decayas a source for particles, used electric fields to accelerate ions to study nuclearreactions [57]. Around the same time the neutron was discovered [13] and in1942 the first artificial nuclear reactor was built under the stands of a footballstadium at the University of Chicago6 [60]. After this, the fission of uranium
made it possible for a large number of neutron-rich radioactive isotopes to be
produced and studied. For example, by bombarding a uranium target with a
particle beam it was possible to produce radioactive noble-gas isotopes and
study them in an electromagnetic isotope separator [61], a method that is now
called isotope separation on-line (ISOL). The ISOL method was further re-
fined in Louvain-la-Neuve where the isotope separator was connected to the
existing accelerator complex, creating an accelerated beam of radioactive ions
[62].Another method, developed at Oak Ridge National Laboratory, to separate
fission fragments of different types from each other is to use magnetic fields
that give a mass separation of the fragments when they are emitted in-flight
from a thin target [63]. This in-flight fragmentation technique has now been
developed so that instead of using the fission energy to give the fragments the
necessary kinetic energy, a heavy-ion beam is accelerated to high energies,
typically to about 50% of the speed of light. To produce the RIB, one lets the
primary heavy ion beam hit a light target like beryllium after which it will
either fragment or fission into a cocktail of reaction products. The reaction
products are then separated by a system of dipole magnets, quadrupole mag-
nets and energy degraders. This technique is implemented at, for example, the
FRS at GSI [64].Many modern facilities for RIB production have recently been built or will
be built in the future. For example the RIB factory at RIKEN [65], the new
RIB facility at Michigan State University (MSU) [66], the upgrade to ISAC-
III [67] at TRIUMF, the upgrade to HIE-ISOLDE at CERN, the upgrade to
SPIRAL2 [68] and the future EURISOL at GANIL, the upgrade to the Super-
FRS at FAIR and the new RIB facility SPES [69] at Laboratori Nazionali di
Legnaro (LNL). These new facilities will require powerful experimental set-
ups, two of which that are discussed in chapter 5 and chapter 6 of this thesis.
6This reactor was also the starting point of the era when nuclear physics entered world politics
through the infamous bombings of Hiroshima and Nagasaki [58, 59].
10
Part I:
Physics
2. Theory of deformed nuclei
“Non-Euclidean calculus and quantum physicsare enough to stretch any brain; and when onemixes them with folklore, and tries to trace astrange background of multi-dimensional real-ity behind the ghoulish hints of the Gothic talesand the wild-whispers of the chimney corner,one can hardly expect to be wholly free frommental tension.”– H. P. Lovecraft, The Dreams in the Witch
House
As mentioned in section 1.1, one of the most successful descriptions of
the structure of nuclei is the nuclear shell model, further discussed in sec-
tion 2.1.1. However, as also mentioned in section 1.1, it quickly becomes
more difficult to make accurate predictions using the shell model when mov-
ing away from the closed shells. Instead, it is the interplay between the macro-
scopic shape degrees of freedom and the microscopic nature of the underlying
single-particle structure of the shell-model orbitals that offers an explanation
for the nuclear behaviour. As this work focuses on the collective structure
of nuclei, a couple of macroscopic descriptions, and their interplay with the
nuclear shell model, will be introduced in the following sections.
2.1 Nuclear deformation
The surface of an atomic nucleus can be described in terms of spherical har-
monics, Yμλ , of order μλ , by the equation
R(θ ,φ) = Rα
(1+
∞
∑μ=1
μ
∑λ=−μ
αμλYμλ (θ ,φ)
), (2.1)
where (R,θ ,φ) are the parameters of a standard spherical coordinate system
and Rα is related to the radius of a sphere with the same volume as the nucleusto be described [24]. If all coefficients αμλ = 0 the nuclear surface becomes
spherical. The only collective excitation of a spherical nucleus is the vibra-
tions around this spherical shape. As each vibration quanta carries the same
energy, this will result in an excitation spectrum with equal distance between
13
0
2
024
0
2
4 2
02
4
6
8
10
Figure 2.1: Collective excitations of an atomic nucleus. From left to right is the spher-
ical vibrator, the γ-soft vibrator and the rigid rotor.
the energy levels where all magnetic substates to an energy level will be en-ergy degenerate. If the nucleus is not spherical but stretched in one direction,α20 �= 0, the vibrations can occur both along the symmetry axis, so called βvibrations, and perpendicular to it, so called γ vibrations. See below for a dis-
cussion about β and γ . Furthermore, a deformed nucleus can also rotate witha rotational energy, E ∝ J(J +1), where J is the spin of the rigid rotating nu-
cleus. The schematic excitation energy spectra for a spherical vibrator, a γ-softvibrator and a rigid rotor are shown in figure 2.1.
The coefficients αμλ are, however, not in general a convenient way to de-
scribe nuclear shapes. Instead one usually defines the shape in terms of Euler
angles. Assuming axial symmetry for all deformations of higher order than
μ = 2, the most important Euler angles, βμ and γ = γ2, of the nucleus can bedefined as
a20 = β2 cosγ, (2.2)
a22 = β2 sinγ = a2−2, (2.3)
a40 = β4, (2.4)
a60 = β6. (2.5)
Unfortunately, different conventions exist and are frequently used regardingthe notation of the deformation parameters. Two of these notations will be
used in this work, βμ and εμ . For moderate deformations (−0.2 � β2 � 0.4and −0.05 � β4 � 0.15) βμ and εμ are approximately related as
ε2 ≈0.944β2 −0.122β 22 +0.154β2β4 −0.199β 2
4 , (2.6)
ε4 ≈−0.852β4 +0.141β 24 +0.122β2β4 +0.295β 2
2 . (2.7)
14
For further discussions on the relation between βμ and εμ , see reference [70].
Different conventions also exist for the definition of βμ and γ , but in this workthe Lund convention will be used, where β2 > 0 and γ = 0◦ corresponds to anaxially symmetric prolate nucleus.
2.1.1 Nilsson model
To describe the deformed shell model, or the Nilsson model, we first needto return to the spherical shell model. As is evident from a large number ofexperimental observables, for example the binding energies, some nuclei withcertain number of protons and neutrons, the so called magic numbers, aremore strongly bound than other nuclei. This is an effect of the mean fieldnuclear potential. A model of the nuclear potential can be made very complexto reproduce all subtle effects of the mean field, but also simplified potentialsare known to give good agreement with experimental data. The most commonof these simplified potentials are the square well potential, the Woods-Saxonpotential and the harmonic oscillator potential.
For a quantum mechanical harmonic oscillator potential with frequency ω ,
the Hamiltonian of a particle with mass m moving in this potential can bewritten
H =p2
2m+
mω2x2
2. (2.8)
The energy eigenstates, with the principal quantum number n, of the particleis
E = hω(
n+3
2
), (2.9)
with degeneracy (n + 1)(n + 2). It is easy to see that already this crude ap-proach reproduce the first three magic numbers (N,Z = 2,8,20). For a more
realistic nuclear potential that becomes flat in the center and at large distances
an orbital angular momentum term, �2, can be added to the potential. The
angular momentum of the orbital is usually denoted using the spectroscopic
notation from atomic physics, where � = 0,1,2,3,4,5,6, . . . correspond to thes, p, d, f, g, h, i, . . . symbols. The �2 term has two effects, to lower the en-
ergy of the high lying states with high angular momentum and to break the(n+1)(n+2) degeneracy so that states with different � is no longer energy de-
generate. Finally, by introducing a coupling between the intrinsic spin and the
orbital angular momentum, as introduced by Goeppert-Meyer [18], � · s, where
states with parallel � and s are favoured, also the orbital angular momentumdegeneracy breaks and the magic numbers are completely reproduced. Thefull modified oscillator potential is now
H =p2
2m+
mω2x2
2−C� · s−D(�2 −〈�2〉). (2.10)
15
Returning to the deformed shell model, instead of the spherical shell model
potential the modified oscillator potential can be written [71]
H =p2
2m+
m(ω2x x2 +ω2
y y2 +ω2z z2)
2−C� · s−D(�2 −〈�2〉), (2.11)
where the oscillator frequencies are
ωx = ω0(ε2,γ)(
1− 2
3ε2 cos
(γ +
2π3
)), (2.12)
ωy = ω0(ε2,γ)(
1− 2
3ε2 cos
(γ − 2π
3
)), (2.13)
ωz = ω0(ε2,γ)(
1− 2
3ε2 cosγ
). (2.14)
Here ε2 determines the strength of the quadrupole deformation and γ the devi-
ation from axial symmetry. Higher order terms, like the hexadecapole defor-
mation ε4, can also be included in the model. The relation between ε2,4 andthe deformation parameters β2,4 are discussed in section 2.1. The deforma-
tion of the nuclear potential will further break the degeneracy of the spherical
shell model. In the prolate deformed potential, equatorial orbitals (with angu-
lar momentum vector parallel to the z axis) will require a higher energy due tothe steeper potential relative to the polar orbitals, due to the softer potential.In figure 2.2, the energy levels for protons, 50 ≤ Z ≤ 82 is shown as a func-tion of the deformation parameter, ε2, in a so called Nilsson diagram. Whenthere is no deformation, ε2 = 0, the orbitals become energy degenerate and the
spherical shell model gaps are reproduced. The different orbitals are labelled
according to the Nilsson labels, Kπ [nnzΛ], where K is the projection of the to-
tal angular momentum on the z axis, π is the parity of the orbital, n is the totalnumber of nodes in the wave function, nz is the number of these nodes in the zdirection and Λ = K±1/2 is the component of the orbital angular momentumalong z. For example, the 7/2−[523] orbital with spin and parity Kπ = 7/2−,where the spin momentum is anti-aligned with the angular momentum, orig-inate from the n = 5 harmonic oscillator shell. It has nz = 2 nodes in the zdirection and an orbital angular momentum of Λ = 3 along z, giving a total
angular momentum of � = 5. The two nodes in the z direction, nz = 2, togetherwith the projection of the total angular momentum on the z axis K = 7/2 gives
a total spin of 11/2 for the orbital. Thus, this correspond to the negative par-
ity h11/2 orbital with an orbital plane at an angle of sin−1 7/211/2
≈ 39.5◦ with
respect to the z axis.
16
Es.
p. (
h)
50
82
1g9/2
1g7/2
2d5/2
1h11/2
2d3/2
3s1/2
3/2[301]
3/2[541]
5/2[303]
5/2[532]1/2[301]
1/2[301]
1/2[550]
1/2[440]3/2[431]
3/2[431]
5/2[422]5/2[422]
7/2[413]
7/2[413]
9/2[4
04]
1/2[431]
1/2[431]
3/2[422]
3/2[422]
5/2[413]
5/2[413]
7/2[40
4]
7/2[4
04]
7/2[633]
1/2[420]
1/2[420]
3/2[411]3/2[411]
3/2[651]
5/2[4
02]
5/2[642]1/2[550]
1/2[550]
1/2[301]1/2[541]
3/2[541]
3/2[541]
3/2[301]
5/2[532]
5/2[532]
5/2[303]
7/2[523]7/2[523]
9/2[514]
9/2[514]
11/2
[505
]
11/2
[505
]
1/2[411]
1/2[411]
1/2[660]
3/2[402]
3/2[651]
3/2[411]
1/2[400]
1/2[660]
1/2[411]
1/2[541]
1/2[301]
3/2[532] 5/2[523]7/2
[514
]
9/2[5
05]
1/2
[660
]
1/2
[400
]
1/2[651]
3/2[651]
3/2
[402
]
3/2
[642
]
5/2[642]
5/2
[402
]
7/2[633]
7/2[404]
11/2
[615
]
13/2
[606
]
1/2[530]
3/2[521]
7/2
[503
]
1/2[770]
3/2
[761
]
1/2[640]
Figure 2.2: Nilsson diagram for protons, 50 ≤ Z ≤ 82 (ε4 = ε22 /6). Reprinted from
[72].
2.1.2 Particle plus triaxial rotor model
The basic deformed rotor or vibrator model, described in section 2.1, onlyapply to collective even-even nuclei where all nucleons are paired. To treatodd-A nuclei the model needs to be modified. One way to do this is to create amodel where a single odd nucleon is treated explicitly outside a core consist-
ing of all other paired nucleons. The Hamiltonian is for such a model can be
17
written as
H = Hcore +Hsp +Hint, (2.15)
where Hcore is the Hamiltonian of the even-even core, Hsp is the Hamiltonian
of the single particle and Hint is the interaction between the core and thesingle particle. The requirements of the core is that it is collective and thatpolarization effects are negligible or included in Hint. If this is fulfilled, thecore can in principle be chosen as any physical collective system; for examplea vibrator, an axially symmetric rotor [23, 73], a triaxial rotor [23, 73–75] orbe described in terms of the interacting boson model (IBM) which gives theinteracting boson-fermion model (IBFM) [76]. Note that, in general, one isrestricted to a fixed core shape used as input and that the model cannot predictthe deformation parameters of this input.
The particle plus triaxial rotor model describes an odd particle outside aneven-even rigid triaxial rotor. This means that one can use the deformed shellmodel Hamiltonian, similar to the Nilsson treatment, as a core. The first step isthe creation of, and diagonalization of, the deformed shell model Hamiltonianwith the desired parameterization of the mean field, most commonly a Woods-Saxon potential or modified oscillator potential [70, 77, 78]. The particle-holeinteraction is included in Hsp and Hint together. Once the deformed shell
model orbitals has been constructed, the particle-particle pairing interaction is
included using the Bardeen, Cooper and Schrieffer (BCS) method [19, 23, 79]
by transforming the system to quasiparticles and creating the corresponding
creation and destruction operators. To obtain a BCS vacuum giving the correct
particle number of the one quasiparticle intrinsic state of the odd nucleus,
the average of the two neighbouring even-even nuclei is used. Finally, we
will have a Hamiltonian that acts on the system as a whole, the intrinsic one
quasiparticle state alone as well as a Coriolis term that couple the intrinsic and
rotational motions.
2.1.2.1 Coriolis attenuation parameterA well known problem in the particle plus triaxial rotor calculations is that the
mixing between intrinsic states due to the Coriolis term is too large compared
to experimental data, especially for high-� orbitals. To take care of this an
ad hoc attenuation parameter, 0 < ζ < 1, has been introduced that multipliesthe Coriolis matrix elements. The parameter ζ is known to go down as far as
ζ = 0.5 in some nuclei. A number of possible explanations for the Coriolisattenuation have been proposed. Some of these are that it is a finite particlenumber effect [80], a Pauli principle effect manifested as the boson-fermionexchange term in the IBFM [81], that a better treatment of pairing without theBCS method is needed [82] or that it origin from neglected octupole couplings[82]. However, there is no clear consensus of the interpretation.
18
2.1.3 Cranking model
A complementary model to the particle plus triaxial rotor is the crankingmodel [83, 84]. This model treats all the nucleons equivalently as particlesmoving in a rotating mean field and makes no distinction between core andvalence particles, thus allowing for multi-quasiparticle states. This is impor-tant to describe high spin effects such as, for example, backbending. Further-more, the cranking model calculates the deformation parameters instead ofusing them as an input to the model. The model does, however, not do sowell for low-spin and non-yrast states or for estimation of the electromagneticproperties of the nucleus [82].
In the cranking model, the total energy, Etot,n, of a n-quasiparticle config-uration is given by the contribution from the macroscopic properties of thenucleus, Emacr, as well as the microscopic properties of the nucleus, Emicr,n, as
Etot,n(Z,N; x) = Emacr(Z,N; x)+Emicr,n(Z,N; x). (2.16)
The macroscopic part varies smoothly with particle number and is usuallytaken from the liquid drop model. In this simple approach to the binding en-ergy the semi-empirical mass formula developed by von Weizsäcker [85],
EB = avA−asA2/3 −aCZ2
A1/3−ai
(N −Z)2
A+δ (A), (2.17)
is used as a starting point. The terms in eq. (2.17) are related to the volume ofthe nucleus (av), the surface energy (as), the Coulomb repulsion of the protons(aC), the proton-neutron symmetry (ai) and the nuclear pairing energy (δ (A)).Already this expression reproduces the bulk behaviour of the relatively heavy
nuclei very well1, but for detailed variations of masses and deformations, mi-
croscopic effects needs to be included. The proton-neutron symmetry term can
be modified according to reference [86] and the pairing energy can be calcu-
lated using the Lipkin-Nogami approach of the BCS method [87]. The main
difference between the Lipkin-Nogami approach and the regular BCS approx-
imation is that the role of particle-number violating terms is minimized. Usu-
ally the Strutinsky shell correction [88, 89] method is applied to obtain Emicr,n,where the microscopic total energy is calculated, for example using the Nils-son method, and added to the liquid drop energy Emacr.
When the system is defined, it is rotated by a frequency, hω . The effects on
the single-particle orbitals in the rotating potential are calculated. In the rotat-
ing frame the inertial forces will influence these orbitals. Due to this treatment
some things are worth pointing out. The eigenstates of the Hamiltonian will
1Actually this semi-empirical formula gives very good prediction. For example one can, if one
adds a gravitational term to the eq. (2.17), within reasonable limits predict the critical mass of a
neutron star. It is very impressive that the fitted values still are quite valid after an extrapolation
from 1 < A < 250 to A ≈ 5 ·1055, over 53 orders of magnitude. See Box 7a in reference [20].
19
not be the energies in the lab system, but energies in the rotating frame referred
to as Routhians. Neither will the total angular momentum or the angular mo-
mentum projection on the symmetry axis be good quantum numbers any more.
The only good quantum numbers remaining in the cranking Hamiltonian are
the parity and signature (π,α). The total energy in the lab frame is calculatedas the sum of the single particle contributions and the spin is the projectionof the total angular momentum on the rotation axis. After each calculation,the results are renormalized to the liquid drop model. These calculations areperformed on a grid in the deformation space, (β2,γ,β4), creating an energysurface, or a total Routhian surface (TRS). The equilibrium deformation isobtained by minimizing the Routhian in the used deformation space. As thecalculations are made at a specified rotational frequency, not at a specifiedspin, and the energies calculated are the Routhians, not the lab frame energy.The spin and lab frame energy can vary across the TRS deformation spacegrid, so one should be careful when transforming the TRS into a physicalenergy and spin surface.
2.1.4 Variable moment of inertia model
A model that is closely related to the cranking model is the variable momentof inertia (VMI) model. The VMI model does not have any predictive powersabout the structure of the nucleus under study, but it can serve as a useful toolto extract physics from the energy spectrum of the nucleus, a reference forhigh spin states, and highlight deformation systematics [90].
In the VMI model, the energy of a deformed nucleus is written as
E(x, I) =I(I +1)2J (x)
+V (x) =I2
2J (x)+V (x), (2.18)
where x is the intrinsic configuration of the nucleus, I =√
I(I +1) is the spin,J is the moment if inertia and V is the intrinsic energy. When the system is
rotated with frequency ω , the Routhian can be written as
R(x,ω) = V (x)− 1
2J (x)ω2. (2.19)
If we are interested in describing the yrast states of the nucleus the first deriva-tive of the Routhian with respect to x must be zero, and if we are interested
in the equilibrium state the first derivative of the intrinsic state energy with
respect to x must also be zero. This makes it possible to write the spin of thesystem as
I = ω(J0 +ω2J1), (2.20)
20
and the energy of the system as
E =1
2ω2
(J0 +
3
2ω2J1
). (2.21)
For a full derivation, see reference [90]. The ground-state rotational band canthus be represented by the two constants [J0,J1] called the Harris parameters
[91, 92]. These parameters are defined as
J0 = J (x)− xJ ′(x), (2.22)
and
J1 =J ′(x)2
2V ′′(x), (2.23)
where
x =1
2
J ′(x)V ′′(x)
ω2. (2.24)
These expressions give a very good representation of the first members of theground state band in deformed even-even nuclei [93, 94]. For odd nuclei, how-ever, one needs to add a non-collective spin component, i, to equation (2.20)and a bandhead energy, E0, to equation (2.21) [90].
Besides creating a reference for studies of the structure of deformed nuclei,the parameters [J0,J1] have some interesting physical meanings. For example,J0 is directly related to the moment of inertia of the system, and thus also
the deformation of the nucleus. For rotational nuclei, J0 can per definition notbecome negative so a negative J0 either imply that the nucleus is becoming
vibrational [94] or that higher order expansions of I and E in terms of ω areneeded. The parameter J1 is related to the rigidity of the system. The small-
est values of J1 occur when the deformed prolate minimum in the potential-energy surface is most pronounced [90].
2.2 Nuclear deformations in the r process
One example of where the theory of deformed nuclei outlined in this chaptercan be applied is the astrophysical r-process. As described in section 1.2.2,the r-process occur in high neutron density areas in the universe, where theaverage time for neutron capture is smaller than the half life of the radioactiveisotope, and a (n,γ)–(γ ,n) equilibrium between neutron capture and photo-disintegration has established itself. After this so called steady phase of ther-process, the free neutrons disappear and the nuclei β decay back to stabil-ity, a process called freeze out. The most dominant features of the abundance
distribution of elements created during the steady phase and freeze out are
21
Figure 2.3: Calculated (line) and measured (crosses) solar r-process abundances (top).
Contours of constant neutron separation energy in MeV (solid lines) and constant
β -decay rates in s−1 (dashed lines). The inset is a schematic of two such contours
with the arrows depicting the flow of nuclei into the region containing the separation-
energy kink (bottom). Reprinted with permission from [95] (top) and [96] (bottom).
Copyright by the American Physical Society.
the large peaks at A ≈ 80, A ≈ 130 and A ≈ 195 that are due to the r-process
flow through closed shells. The second most pronounced feature is the peak at
A ≈ 160 in the rare-earth region. See figure 2.3 for a selection of the r-process
abundance distribution.
22
While the closed shell peaks are well understood to be formed during the
steady phase [49], it has been argued that the A ≈ 160 peak is due to thedeformation in the nuclei created after the steady phase freeze out [96]. Anexplanation that has been proposed is that nuclei deform when deformationincreases stability. As the deformation maximum is reached the nucleus can-not deform more so the next heavier nucleus will be less stable, an effect thatcan mimic closed shells.
However, the process proposed to explain the abundance peak is slightly
different from the process behind the closed shell peaks. As long as the sys-
tem is in (n,γ)–(γ ,n) equilibrium the r-process path will follow a contour ofconstant separation energy,
Sn(Z,Nmax) = −kT ln
(ρNAYn
2
(2π h2
mnkT
)3/2)
. (2.25)
In this equation kT is the temperature, ρ the density, NA Avogadro’s number
and Yn the free neutron abundance per nucleon. If the temperature and den-sity does not change dramatically, the r-process stays in (n,γ)–(γ ,n) equilib-
rium even after steady phase is over. Thus, the path will continue to lie along
contours of constant neutron separation energy as it moves towards stability.
See figure 2.3 for an example of some r-process paths during freeze out. It is
during this time that the peak in the rare-earth nuclei is formed. Besides the
large kinks at the closed neutron shells, a kink in the separation energies at
N ≈ 104 is clearly seen, corresponding to the deformation maximum in thecalculations.
This kink causes the peak to form in two ways. One way is similar to theclosed shell isotopes, that the kink produces a concentration of populated iso-topes close together. The other way is due to the effect that the free neutronabundance per nucleon falls much more rapidly than the temperature and den-sity as the path moves towards stability by β decay, and that the contours of
constant β decay rate does not coincide with the contour of constant separa-tion energy in this region, see figure 2.3. Below the kink the nuclei along thecontour are farther from stability and decay faster than average. So a β decayfollowed by a neutron capture will cause their value of A to increase. The nu-
clei above the kink, however, decay slower than average, allowing them time
to photodisintegrate and thus decrease their value of A.As seen in figure 2.3, this reproduces the peak quite well. However, one can
note that there is a small overestimation of the nuclei with higher mass num-bers and a small underestimation of the nuclei with lower mass numbers. This
could be related to the assumption of N = 104 as the deformation maximumall the way down to Z = 50. A better understanding of the evolution of nuclear
deformations in this region might improve the agreement between calculated
and observed abundances.
23
2.3 Heavy-ion induced nuclear reactions
Nuclear reactions can be classified in many different ways, for example ac-cording to the time during which the reaction occur, the energies involved inthe reaction or the so called impact parameter that determine how close thenuclei are during the reaction. For example, in Coulomb excitation reactions,which are not actually nuclear reactions as such, the Coulomb field from theinteracting nuclei is used to excite both the beam and the target. Coulombexcitation experiments are usually carried out at low beam energies, but canas well be carried out at relativistic beam velocities. Since the nuclei in thisreaction do not come into direct contact, the impact parameter will be large.See, for example, the recent experiment carried out at LNL to search for two-phonon excitations in 170Er [97].
The opposite reaction to Coulomb excitation, in terms of the impact param-eter, is the fusion reactions where the two nuclei fuses together and de-excitethrough γ-ray emission or by particle evaporation. These types of reactions
usually involve head-on collisions between the nuclei at relatively low en-
ergies, but above the Coulomb barrier, and with long enough reaction times
so the compound nucleus has time to form. Since most angular momentum
goes into the spin of the compound nucleus, the fusion-evaporation reactions
are well suited to study high-spin states. Together with a good neutron de-
tector array for evaporated particles it is also a valuable tool for experiments
aiming at studies of proton-rich nuclei. See for example reference [26] where
a fusion-evaporation reaction was used together with the Neutron Wall, de-
scribed in section 6.2. Fusion-evaporation reactions were also used in the first
commissioning run with the AGATA array, see chapter 5.At the other extreme, regarding the energies involved, are the fragmentation
reactions. In this case heavy nuclei at relativistic velocities are fragmented by
letting them hit a light nucleus. The nuclear fragments can then be studied
by themselves, see for example refs. [98, 99] for some recent results, or via
secondary reactions like Coulomb excitation.In between these reactions are the deep inelastic collisions, or multi-nucleon
transfer reactions. These are fast nuclear reactions at intermediate energies,
where the surfaces of the nuclei come into contact at a grazing distance, al-
lowing for a fast redistribution of neutrons and protons among the nuclei.
Thus, nucleons are transferred between the beam and the target, but the frag-
ments produced in the reaction keeps resemblance to the original beam and
target nuclei. The primary fragments de-excite through evaporation of neu-
trons, protons and α particles, and emission of γ rays. Heavier fragments can
also fission. These collisions can be used to populate neutron rich parts of the
Segré chart not reachable by fusion-evaporation reactions. Angular momen-
tum is transferred from the relative orbital motion to intrinsic spin through
three different relative motions, sliding, rolling and sticking. The sliding pro-
cess is the simplest of these as the nuclei slide with respect to each other and,
24
thus, does not transfer any angular momentum. The rolling mode is when the
beam nucleus is rolling on the target nucleus and, thus, by a strong frictional
force deposits some of its angular momentum to the target nucleus, causing
it to rotate in the opposite direction. The sticking mode is when the nuclei
stick together and begins to rotate around a common centre of gravity, while
each fragment has the same rotational velocity around their own centre. The
maximum of the angular distribution of the binary cross-section when the dis-
tance between the two nuclei is equal to the sum of their radii is known as the
grazing angle of the reaction. For more details on experimental and theoretical
aspects of deep inelastic collisions see for example refs. [100–102].
25
3. PRISMA and CLARA experiment
“You do research now? Want a cappuccino anda pack of cigarettes to go with it?”– Buffy, the Vampire Slayer
As discussed in chapter 2, one important approach to the nuclear many-body problem is the macroscopic approach, based on the collective propertiesof nuclei. The regions in the Segré chart where quadrupole collectivity is mostprominent are around the doubly mid-shell nuclei, with many valence parti-cles and far away from closed shells. The distance from the closed shells issometimes quantified in terms of the product of valence nucleons NpNn [103],which is equal to the number of neutron-proton interactions outside the shell.Neglecting any potential sub-shell closures, the nucleus with A < 208 that hasthe largest number of valence particles is 170
66 Dy104. Accordingly, it should be
one of the most collective of all nuclei in its ground state [104]. However,
sub-shell closures, such as those at Z = 64 [105, 106], and at Z = 76 [107]
as well as microscopic effects complicates the simple NpNn relationship andit is not clear where the maximum of collectivity is located. The amount ofcollectivity has been shown to have a smooth dependence on both the energyof the first excited state, E(2+), and the reduced transition probability fromthe first state to the ground state, B(E2:2+ → 0+), as well as the energy ratio
of the first excited 4+ and 2+ states, E(4+)/E(2+) [105, 107–109].Furthermore, it has been suggested that 170Dy could be the single best case
in the entire Segré chart for the empirical realization of the SU(3) dynamical
symmetry of the IBM [110]. Although many deformed nuclei show some of
the predicted signatures of the SU(3) IBM symmetry, to date, no nucleus has
been demonstrated to show them all.Previous experimental work in this region has been limited due to a lack of
suitable conventional nuclear reactions able to populate such exotic neutron-
rich nuclei. The only experiment that has reported production of 170Dy, before
this work, was using fragmentation of a 1 GeV per nucleon 208Pb beam atGSI [111]. The use of deep inelastic transfer reactions have been successfulin populating 168Dy as described in this chapter and reference [112, I], 169Ho
[113, 114], 172Er [115], 174Er [116] and 178Yb [117]. The location of 170Dy inthe region of interest in the Segré chart is shown in figure 3.1.
In this chapter, an experiment aiming to study the structure of 170Dy, andthe neighbouring nuclei 168Dy, 167Ho, 168Ho and 169Ho will be described.
The experiment was carried out at the PRISMA and CLARA set-up, shown
27
166Yb 167Yb 168Yb 169Yb 170Yb 171Yb 172Yb 173Yb 174Yb 175Yb 176Yb 177Yb 178Yb 179Yb 180Yb 181Yb 182Yb
165Tm
164Er
163Ho
162Dy
161Tb
160Gd
159Eu
158Sm
166Tm
165Er
164Ho
163Dy
162Tb
161Gd
160Eu
159Sm
167Tm
166Er
165Ho
164Dy
163Tb
162Gd
161Eu
161Sm
168Tm
167Er
166Ho
165Dy
164Tb
163Gd
162Eu
162Sm
169Tm
168Er
167Ho
166Dy
165Tb
164Gd
163Eu
163Sm
170Tm
169Er
168Ho
167Dy
166Tb
165Gd
164Eu
164Sm
171Tm
170Er
169Ho
168Dy
167Tb
166Gd
165Eu
165Sm
172Tm
171Er
170Ho
169Dy
168Tb
167Gd
166Eu
166Sm
173Tm
172Er
171Ho
170Dy
169Tb
168Gd
167Eu
167Sm
174Tm
173Er
172Ho
171Dy
170Tb
169Gd
168Eu
168Sm
175Tm
174Er
173Ho
172Dy
171Tb
170Gd
169Eu
169Sm
176Tm
175Er
174Ho
173Dy
172Tb
171Gd
170Eu
170Sm
177Tm
176Er
175Ho
174Dy
173Tb
172Gd
171Eu
171Sm
178Tm
177Er
176Ho
175Dy
174Tb
173Gd
172Eu
172Sm
179Tm
178Er
177Ho
176Dy
175Tb
174Gd
173Eu
160Sm
180Tm
179Er
178Ho
177Dy
176Tb
175Gd
174Eu
173Sm
181Tm
180Er
179Ho
178Dy
177Tb
176Gd
175Eu
174SmZ64
6668
70
N 98 100 102 104 106 108 110 112Figure 3.1: The region of interest, for this work, in the Segré chart. Light grey boxes
represent short lived nuclei and dark grey boxes represent stable nuclei. White boxes
with black text represent nuclei where no previous information of excited stated were
known before this work, and with grey text nuclei that have not been observed at all.
Figure 3.2: The PRISMA and CLARA set-up. Photo from reference [118].
in figure 3.2, at the XTU Tandem-ALPI-PIAVE accelerator complex at LNL
using multi-nucleon transfer reactions between 82Se and 170Er. The beamwas 82Se at an energy of 460 MeV and an intensity of ∼25 enA (∼2 pnA)
for effectively ∼ 3.5 days. This beam was incident on a 500 μg/cm2 thickself-supporting 170Er target. Beam-like fragments were identified using the
PRISMA magnetic spectrometer, placed at the grazing angle of 52◦. The γ-
28
ray energies from both the beam-like and target-like fragments were measured
using the CLARA array, in this experiment consisting of 23 Compton sup-
pressed clover detectors.
3.1 LNL accelerator complex
The main accelerator complex for nuclear physics at LNL consists of three
machines that accelerate the ions delivered by an electron cyclotron reso-
nance ion source [119]. The Tandem XTU accelerator, that was used in this
experiment, has been in operation since 1982 and nominally runs at a ter-
minal voltage of 18 MV. The charge stripping is usually obtained using a 5μg/cm2 carbon foil. Together with the XTU accelerator there is also a su-
perconducting resonant cavity post-accelerator, ALPI [120], for heavy ions.
ALPI can provide ion beams up to uranium with energies between 5–20 MeV
per nucleon. Instead of the Tandem XTU, there is also the option to use PI-
AVE [121], which is an injector for ALPI based on superconducting radio
frequency quadrupoles and quarter wave resonance structures. For this exper-
iment, only the XTU tandem and the ALPI post-accelerator was used.
3.2 PRISMA
As no experimental information about the structure of 170Dy was known be-fore this experiment it was not possible to identify the reactions leading to170Dy from its emission of γ rays. Instead, 170Dy had to be identified from thereaction products of the 170Er(82Se,82 Kr)170Dy reaction. The identification
of the reaction products was also an important tool for the other nuclei stud-
ied in this experiment, such as 168Dy where the 4+ → 2+ and the 2+ → 0+
transitions had been previously observed [122], as a means of reducing thebackground from other, more strongly populated, reaction channels.
To identify the beam-like reaction fragment the large-acceptance magnetic
spectrometer PRISMA [123–125] was used. PRISMA is designed to iden-
tify nuclei populated in heavy-ion binary reactions at kinetic energies of 5–
10 MeV/A. It covers a solid angle of 80 msr, has a momentum acceptance
of Δp/p = ±10 %, a dispersion of ∼ 4 cm per percent in momentum and
a counting rate capability of 50–100 kHz. The total flight distance from the
start detector to the focal plane is ≈ 6.5 m. It can be positioned at anglesbetween −20◦ to 130◦ relative to beam line. The spectrometer consists of
two large magnets, one 50 cm length and 32 cm diameter quadrupole mag-
net and one dipole magnet with 1.2 m radius of curvature in a 60◦ bending
angle for the central trajectory. Together with the magnets a set of start and
focal plane detectors was also used. The start detector is a micro-channel-
plate (MCP) located right after the target chamber. The focal plane detectors
29
consist of a multi-wire parallel-plate avalanche counter (MWPPAC) for (x,y)position measurements of the fragments exiting the dipole and an ionization
chamber (IC) grid for total kinetic energy, E, and partial kinetic energy, ΔE,measurements of the fragments. The atomic number (Z) resolution in this ex-
periment was Z/ΔZ ≈ 65 and the mass resolution (A) was A/ΔA ≈ 200 forelastic scattering of 82Se. The set-up of PRISMA is illustrated in figure 3.3.
3.2.1 MCP
The first detector in PRISMA, 25 cm after the reaction chamber, is a rectan-gular MCP with a size of 80×100 mm2 mounted in Chevron (or ’V’ shaped)
configuration. It is located in a stainless steel box, tilted 45◦ with respect tothe optical axis, with two transparent windows. At the exit window there isa 20 μg/cm2 carbon foil with a voltage of −2300 V. The MCP detects the(x,y) positions, with an accuracy of 1 mm, of the secondary electrons pro-
duced when the beam-like fragments passes the carbon foil before entering
the quadrupole magnet. The MCP also registers the time when the fragment
enters PRISMA with an accuracy of 400 ps, which is later used together with
the MWPPAC to extract the time-of-flight (TOF) of the fragment through the
spectrometer. The efficiency of the MCP is � 100% in typical experiments.
For more details, see reference [126]. It can be calibrated in x and y using athin cross of wires and two nails located downstream in PRISMA, see fig-ure 3.4.
3.2.1.1 Calibration of the MCPSince the calibration of the MCP is the most critical part of the data sortingprocedure and was done in a non-standard way it will be described here insome more detail, following the matrix method in reference [127]. As seenin figure 3.4, a reference cross with four flags is placed on the carbon foil.The reference cross is slightly deformed due to the magnetic field that accel-erate the secondary electrons from the carbon foil to the MCP. Furthermore,the mask is rotated by 2–3◦, as seen on the projection of two nails in thequadrupole magnet. Due to this, a linear calibration containing both the x andy parameter is used as
x′ = ax +bxx+ cxy, (3.1)
y′ = ay +byx+ cyy, (3.2)
where (x′,y′) is the calibrated position in mm and (x,y) is the uncalibrated
position. A coefficient matrix, A, containing the positions of the reference
30
Dipole
QuadrupoleMCP
MWPPAC
IC
250
mm
540
mm
3279 mm 720 mm
Figure 3.3: Sketch of the PRISMA spectrometer.
x (ch)1000 2000 3000
y (c
h)
1000
2000
3000
Figure 3.4: Positions of the reaction fragments accepted by the PRISMA spectrometer
as measured by the MCP, before calibration. The reference cross for calibration and
the projection of two nails in the quadrupole magnet is clearly visible.
31
points can thus be defined as
A =
⎛⎜⎜⎜⎜⎜⎜⎝
1 x1 y1
1 x2 y2
1 x3 y3
1 x4 y4
1 x5 y5
⎞⎟⎟⎟⎟⎟⎟⎠
. (3.3)
The covariance matrix, V , of the fitting parameter vector, θ T = (a,b,c), canbe calculated from the matrix expression
V (θ) =[ATA
]−1=
⎛⎜⎝ Vaa Vab Vac
Vba Vbb Vbc
Vca Vcb Vcc
⎞⎟⎠ , (3.4)
this gives the parameters
θx = V (θx)ATx′ =
⎛⎜⎝ ax
bx
cx
⎞⎟⎠ . (3.5)
Similarly for y,
θy = V (θy)ATy′ =
⎛⎜⎝ ay
by
cy
⎞⎟⎠ . (3.6)
3.2.2 Quadrupole and dipole magnets
The PRISMA quadrupole magnet is located 54 cm from the target. It has an
aperture diameter of 32 cm and an effective length of 50 cm, giving length-
over-diameter ratio of ≈ 1.5 which is on the limit for the fringing field to be
small enough with respect to the inner field. The purpose of the quadrupole
magnet is to focus the fragments along the y axis to increase the acceptance
of PRISMA. At the same time the fragments are defocused along the x axis,due to the properties of the quadrupole field. The maximum field strength ofthe quadrupole is BQ = 0.848 T. In this experiment, the field strength wasBQ = 0.645 T.
The main component of PRISMA is the large dipole magnet that sepa-
rates fragment trajectories with respect to their magnetic rigidity. It is located
160 cm from the target, 60 cm from the quadrupole, and has a curvature ra-
dius of 1.2 m. It has a pole gap of 20 cm height and 1 m width with a maxi-
32
mum magnetic rigidity of BDρ = 1.2 Tm. The field strength of the dipole was
BD = 0.707 T in this experiment.
3.2.3 MWPPAC
At the focal plane of PRISMA, the first detector is a 1 m wide MWPPAC
located 327.9 cm from the dipole. The MWPPAC is divided into three elec-
trodes with a total active focal plane area of 100 × 13 cm2, mounted in a
stainless steel vacuum vessel filled with isobuthane, C6H10, in this experi-ment at a pressure of 6.5 mbar. On each side of the vacuum vessel along theoptical axis is a 1.5 μm thick Mylar window, located 3 mm from the anode
wire planes, supported by 100 μm thick stainless steel wires with a distance of3.5 mm between them. Both the central cathode and the x-position wire plane
are divided into ten independent sections each, while the y-position wire planecovers the entire width of the MWPPAC. Each x-position anode has 100 gold-
plated tungsten wires of thickness 20 μm, giving a spacing of 1 mm. Eachcathode has 330 gold-plated tungsten wires, giving a spacing of 0.3 mm. They-position anode has 136 gold-plated tungsten wires giving a spacing of 1 mm.
From the electrodes, the xleft and xright signal in the focal plane is read out, aswell as the stop signal for the TOF measurement and the amplitude of thecathode signal. The yup and ydown signals are also read out and used mainlyfor beam positioning. From the difference in xleft and xright the x position of
the fragment at the focal plane, xFP, can thus be extracted. The x positions inthe MWPPAC and the TOF between the MWPPAC and the MCP are shownin figure 3.5.
3.2.4 Ionization chambers
The last detector in the focal plane of PRISMA is the IC array located 72 cmafter the MWPPAC. The IC array consists of four segments (z-direction) of100(x)×20(y)×250(z) mm3 in ten sections (x-direction), see figure 3.3. This
segmentation makes it possible to use ΔE −E techniques to resolve the dif-ferent atomic numbers of the fragments. Furthermore, there are two more out-ermost sections used as veto detectors for fragments that leave the chamberwithout depositing all their energy in the active volume. Each section is readout by a common cathode and a common Frisch grid but every segment has itsindividual anode. The Frisch grid is made of 1200 gold-plated tungsten wiresof thickness 100 μm and the entrance window is made of a 1.5 μm thick My-
lar foil supported by 1000 stainless steel wires of thickness 100 μm with adistance of 1 mm between them. To provide a high drift velocity and, thus, agood energy resolution the IC array is filled with methane gas, CH4, in this
experiment at a pressure of 62 mbar.
33
(ch)rightx0 1000 2000 3000 4000
(ch
)le
ftx
0
1000
2000
3000
4000
(a)
(mm)fpx0 200 400 600 800 1000
Cou
nts
0
20000
40000
60000
(b)
TOF (ns)100 200 300 400 500
Cou
nts
10
210
310
410
510
(c)
(mm)fpx0 200 400 600 800 1000
TO
F (
ns)
100
200
300
400
500(d)
Figure 3.5: The right versus left position in the MWPPAC (a) and the xFP position at
the focal plane (b). The time-of-flight between the MCP and the MWPPAC (c) and
the time-of-flight versus the xfp position at the focal plane (d).
The energy loss of the fragment in the IC detectors is governed by theBethe-Bloch equation,
dE(z)dz
∝MZ2
E(z), (3.7)
where z is the distance along the beam trajectory, M is the mass of the frag-ment, Z the atomic number of the fragment and E the energy at a given mo-ment. This means that the energy loss of different fragment species in thedifferent segments of the IC array will vary. The Z can be obtained by com-paring the energy deposited in the first two layers of the IC, ΔE, to the total
energy, E. This is illustrated in figure 3.6. In a similar way, the range, r of
34
the fragment in the IC will also depend on the energy loss as
r(E) =∫ E
0
(−dE
dz
)−1
dE, (3.8)
where E is the total energy of the fragment when entering the IC. The rangeand energy is also compared in figure 3.6. In the analysis in this work, r–E is
the quantity that was used as it made it possible to resolve Z down to lowerincident energies.
3.2.5 Mass determination
One of the most important features of the PRISMA spectrometer is its abil-ity to separate fragments of different mass by reconstructing their trajectorythrough the magnets, following the procedure of references [128, 129]. The(x,y) position where a fragment with charge q enters the quadrupole is ob-
tained from the MCP. The forces, F , acting on the fragment when it is passingthrough the quadrupole are
Fx = qvbx, (3.9)
Fy = −qvby, (3.10)
where v is the, so far unknown, velocity of the fragments and b is the magnetic
field gradient. The equations of motion will thus be
d2xdz2
= k2x, (3.11)
d2ydz2
= k2y, (3.12)
where
k2 =qbMv
, (3.13)
for a fragment of unknown mass, M. Denoting dxdz = x′ and dy
dz = y′, the solu-
tions to equations (3.11)–(3.12) are
x(z) = Asinh(kz)+Bcosh(kz), (3.14)
y(z) = C sin(kz)+Dcos(kz), (3.15)
x′(z) = Ak cosh(kz)+Bk sinh(kz), (3.16)
y′(z) = Ck cos(kz)−Dk sin(kz), (3.17)
where the boundary conditions causes the coefficients A, B, C and D to depend
on the position and velocity vector of the fragment entering the quadrupole
magnet, as measured by the MCP. Thus, using equations (3.14)–(3.17), the
35
E (a.u.)1000 1500 2000 2500
E (
a.u.
)
2500
3000
3500
4000
4500
5000
5500(a)
E (a.u.)500 1000 1500 2000 2500
r (a
.u)
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800(b)
Z=34
Z=35
Z=36
Figure 3.6: Partial energy loss (a) and range of the fragments (b) with respect to the
total energy loss of the fragments in the IC. The selection used for selenium (Z = 34),
bromine (Z = 35) and krypton (Z = 36) isotopes are also shown in panel (b).
(mm)fpx0 200 400 600 800 1000
M/q
(a.
u.)
280
300
320
340
360
380
400
420
440
460(a)
E (a.u.)1000 1500 2000 2500
Rv
(a.u
)
1200
1400
1600
1800
2000
2200
2400
2600
2800
(b)
+20+21
+22
+24+23
+27
+26
+25+24+23
Figure 3.7: Relative mass, M/q, of the fragments with respect to their position in
the MWPPAC (a) and the bending radius, Rv, with respect to the energy (b) used to
obtain the absolute mass, M, of the fragments. The different charge states for selenium
(Z = 34) is shown in panel (b).
36
position and velocity vector of the fragment entering the dipole magnet can
be calculated. The radius, R of the trajectory of the fragment in the dipole canin principle be calculated from the Lorentz force,
Mv2
R= qvBD, (3.18)
but as the mass, M, and the atomic charge, q, are not yet known the value
of R needs to be guessed. By default, the first guess is R = 1200 mm. Usingthis guess and equations (3.14)–(3.18), the position where the fragment hitsthe MWPPAC can be calculated. The velocity, v, is obtained from the TOF
information and the total length of the guessed track. The calculated position
in the MWPPAC is compared to the measured position in the MWPPAC and
the guessing procedure is iterated until the difference is less than 1 mm and,
thus, the correct values of R, v and M/q have been obtained. The M/q ratio
for the selenium (Z = 34) isotopes is shown in figure 3.7.To obtain the absolute mass of the fragment its charge state also needs to be
known. This can be achieved from the relation
E =1
2mv2, (3.19)
which together with eq. (3.18) gives
E ∝ qRv. (3.20)
A plot of E together with Rv that separates different q is shown in figure 3.7.
The absolute identification of each charge state is obtained by comparing
the intensities in figure 3.7 with the transmission calculations from the
reaction code available at LNL.
By selecting each charge state, q, and multiplying this with M/q, a massspectrum of the detected fragments can be obtained. Such a spectrum is shownin figure 3.8 for krypton (Z = 36) isotopes.
However, the fragments of interest in this experiment were not the beam-
like fragments, but the target-like fragments. Unfortunately, the target-like
fragments were too heavy to be resolved by the PRISMA spectrometer. The
A and Z of the target-like fragments can be calculated directly assuming a
binary reaction without any particle evaporation. The velocity vectors of the
target-like fragments, which are important for the Doppler correction, can be
calculated using standard two-body kinematics, again assuming a binary reac-
tion. The mass limits on the target-like fragments are also shown in figure 3.8.
37
Kr mass81 82 83 84 85 86 87 88 89
Co
un
ts
0
200
400
600
800
1000
1200
1400
1600
170A
169A
168A
167A
166A
165A
164A
Figure 3.8: Mass spectrum from PRISMA of target-like fragments gated on krypton
(Z = 36). The masses (A) of the corresponding dysprosium isotopes are also shown.
Reprinted from reference [130].
3.3 CLARA
To measure the γ rays emitted by the fragments, the CLARA [131] high-purity
germanium (HPGe) array was used. CLARA is an arrangement consisting of
the clover detectors from the EUROBALL III and EUROBALL IV arrays1
[133]. Each Clover detector is composed of four HPGe crystals, mounted in
a single cryostat, with a diameter of 50 mm and surrounded by a bismuth
germanate (BGO) shield to suppress Compton scattering out from the detector
and increase the peak-to-total ratio. In its full configuration, CLARA consist
of 25 Clover detectors closely packed in a hemisphere around the target posi-
tion of PRISMA, at angles between 104–256◦ with respect to the entrance ofthe spectrometer. A photo of the CLARA array is shown in figure 3.9.
In the full configuration, the total photopeak efficiency of CLARA is 3.3 %for single 1 MeV photons and the peak-to-total ratio is 48 %. In this exper-iment 23 Clover detectors were mounted. Doppler correction was performedevent-by-event using the velocity vectors measured by PRISMA. This gave anenergy resolution of 4.4 keV (0.7 %) at 655 keV for the beam-like fragmentsand 5.8 keV (1.1 %) at 542 keV for the target-like fragments reconstructed ac-
1The EUROBALL Cluster detectors are now assembled into the RISING array at GSI [132] and
the EUROBALL Phase I detectors are used in the JuroGam array at the University of Jyväskylä
physics laboratory. At the time of writing, the University of Jyväskylä also has the Clover
detector as CLARA has been decommissioned and replaced with the AGATA Demonstrator
array, see chapter 5
38
Figure 3.9: The CLARA HPGe detector array. Photo from reference [118].
cording to the procedure in section 3.2.5. Doppler-corrected spectra for beam-
like and target-like fragments are shown in figure 3.10. The γ-ray spectraobtained from CLARA were then analysed using both in singles mode and
using the γγ-coincidence technique.
39
(keV)E0 100 200 300 400 500 600 700 800 900 1000
Cou
nts
210
310
410
510
Beam like
Se
Br
Kr
(keV)E0 100 200 300 400 500 600 700 800 900 1000
Cou
nts
210
310
410
510
Target like
Er
Ho
Dy
Figure 3.10: Doppler-corrected spectra from beam-like and target-like fragments as
recorded by CLARA. The spectra shows all isotopes for each Z without any mass
gate.
3.4 Dysprosium isotopes
As mentioned in section 3.2.5, the analysis of the target-like fragments was
carried out using the assumption of binary reactions, that no particles were
evaporated neither in the beam-like fragments nor in the target-like fragments.
This is not true, however, since the experiment was carried out on nuclei far
in the neutron-rich region and, thus, neutron-evaporation channels all the way
up to four neutrons are strongly populated. At the CLARA and PRISMA set-
up it was not possible to identify the evaporated neutrons or in any other way
identify evaporated particles event-by-event. One way to solve this is to com-
pare neighbouring selections on A and Z and, by looking at which peaks thatappear and disappear depending on the cuts, conclude which γ-ray peaks that
belong to which isotope.
40
(keV)E130 140 150 160 170 180 190 200 210 220 230
TO
F (
ns)
220
230
240
250
260
270
280
Figure 3.11: Time-of-flight with respect to γ-ray energy for a selection on 168Dy as-
suming a binary reaction. The two-neutron evaporation peak is seen at 243 ns and the
four-neutron evaporation peak is seen at 250 ns. The continuous band at 181 keV is
corresponding to random background events from the target, 170Er.
For example, by making a selection on 170Dy (82Kr) one would expect to seealso 168Dy from two-neutron evaporation and 166Dy from four-neutron evap-
oration. Making a selection on 168Dy (84Kr) would make the 170Dy γ raysdisappear, but would show 166Dy from two-neutron evaporation and 164Dy
from four-neutron evaporation. Another way is to use the TOF, corresponding
to the total kinetic energy, of the fragments through PRISMA. For the four iso-
topes 164Dy, 166Dy, 168Dy and 170Dy the two-neutron separation energies are,
S2n = 13.93 MeV, S2n = 12.76 MeV, S2n = 12.12 MeV and S2n = 11.24 MeV,respectively [134–137]. The neutron evaporation cannot take place unless thisamount of energy is transferred from the kinetic energy of the fragments whichmeans that the TOF is increased ≈ 2.7% for two-neutron evaporation channels
and ≈ 5.5% for four-neutron evaporation channels. As the TOF of the kryp-ton isotopes in this experiment is ≈ 237 ns, this would correspond to a mini-mum TOF of 243 ns for the two-neutron evaporation channels and a minimumTOF of 250 ns for the four-neutron evaporation channels. The γ-ray energy isshown against the TOF in figure 3.11, for a gate on 168Dy (84Kr) where three
distinct peaks is clearly visible, corresponding to binary reactions (173 keV),
two-neutron evaporation reactions (177 keV) and four-neutron evaporation
reactions (169 keV). A continuous band at 181 keV is also seen, correspond-
ing to random background events from the 4+ → 2+ transition in the target,170Er. The γ-ray spectrum for 168Dy, with a TOF gate to suppress neutron
evaporation events, is shown in figure 3.12 where a rotational band is clearly
seen. This rotational band was verified by the γγ coincidence technique, see
reference [112, I] for details.The same was also done for 170Dy. However, in this case the high back-
ground from the very strong γ-rays from the target made it not possible to
41
Energy (keV)0 50 100 150 200 250 300 350 400 450 500
Cou
nts/
1 ke
V
0
10
20
30
40
50
60
70
Figure 3.12: Spectrum of γ-ray energies from target like fragments gated on the beam-
like fragments 84Kr plus a short time of flight. The transitions identified as the rota-
tional band in 168 Dy are marked with solid lines. Reprinted from reference [112,
I].
unambiguously identify the 4+ → 2+ transition in 170Dy and as no excited
states were known, the traditional γγ coincidence technique could not be em-ployed. Instead, a gate on the 777 keV γ-ray in the binary partner, 82Kr [138],was used to reduce the background. This made it possible to do a tentativeidentification of the 4+ → 2+ transition at 163 keV. The results are shown infigure 3.13. The yrast bands of the dysprosium isotopes with N = 94−104 are
shown in figure 3.14.
3.5 Holmium isotopes
The analysis of the 167Ho and 169Ho isotopes was carried out in the sameway as for the dysprosium isotopes. Even if the cross sections were higherfor production of the Ho nuclei, the many excited states with similar energiesand the Z resolution of PRISMA made it more difficult to produce clean γ-ray
spectra. The details of the analysis are presented in reference [114, II] and the
level schemes from the analysis are shown in figure 3.15.A number of γ rays was also observed in 168Ho. The level scheme from
reference [143] has been revised and is shown in figure 3.15. An interesting
result from the analysis in this experiment is the observation of the previously
known 143 keV γ-ray that in reference [143] was assigned to the (1)− → 3+
transition with a half life longer than 4 μs. This lifetime is not compatible with
the observation of the transition in this work. Furthermore, the previously ob-
42
Cou
nts/
1 ke
V
02468
10121416 163 Er170
Dy168
Energy (keV)0 50 100 150 200 250 300 350 400 450 500
Cou
nts/
3 ke
V
01234
Figure 3.13: Spectrum of γ-ray energies from target-like fragments gated on the beam-
like fragments 82Kr plus time-of-flight (top). Coincidence γ-ray spectra gated on the
beam-like fragments 82Kr, time-of-flight plus the γ-ray energy 777 keV in the beam-
like fragments (bottom). An estimation of the background using adjacent gates is also
shown (filled histogram). The background gates are about 20 times the width of the
gates on the γ-ray peaks and normalized relative to the size of the peak gates. The
tentative γ ray associated with the yrast 4+ → 2+ transition in 170Dy is marked with a
solid line. Reprinted from reference [112, I].
0 02 87
4 284
6 581
8 967
0 02 81
4 266
6 549
8 921
10 1375
0 02 734 242
6 501
8 843
10 1261
0 02 774 254
6 527
8 892
10 1341
0 0(2 ) (75)(4 ) (248)
(6 ) (516)
(8 ) (873)
(10 ) (1315)10 1429
0 (0)(2 ) (72)(4 ) (235)
87197
297
386
81185
283
372
454
73169
259
342
418
77177
273
365
449
75173
268
357
442462
(72)(163)
DyDyDyDyDy94 96 98 100 102160 162 164 166 168 Dy104
170
Figure 3.14: Ground state rotational bands for dysprosium isotopes with N = 94−104
from [134, 135, 139–141] and the current work for 6+–10+ in 168Dy and the 4+ →2+ transition in 170Dy. The 2+ → 0+ transition in 170Dy is from the calculations in
reference [142]
43
3
1
7/2
11/2
15/2
19/2
9/2
13/2
17/2
21/2
9/27/2
11/2
15/213/2
17/2
217
301
384
260
343
97120
140161
182
202
230
192167
143122
98
133143187
352483
193
167Ho 168Ho 169Ho
Figure 3.15: Ground state rotational bands for 167Ho and 169Ho and the level scheme
for 168Ho from this work.
served 488 keV γ-ray is observed together with a close lying γ ray of 483 keV.When gating on the γ ray at 488 keV, the 143 keV γ-ray is not seen, while it isseen when gating on the 483 keV γ ray. This is an indication that the ordering
of levels is likely different than previously suggested.
44
4. Evolution of collectivity
One of the standard references for nuclear masses, and deformations, is
the calculations made by Möller and Nix using the finite range liquid drop
model [144]. This reference has for example been used in the calculations
in Refs. [95, 96] and, thus, to obtain the results shown in figure 2.3. The β2
deformation parameters from reference [144] are shown in figure 4.1 for
even-even nuclei in the region 50 ≤ Z ≤ 82 < and 82 ≤ N ≤ 126. As seen inthis figure, the deformations behave smoothly with a maximum in the regionaround N ≈ 104. In figure 4.1, the evaluated experimental deformations from
reference [145] are also shown, but the calculations from reference [144]
does not appear to follow the same pattern as the experimental data. Besides
the larger absolute variation in deformations for different Z values, thedeformation maximum for each Z does not appear to be as stable around
N ≈ 104 as in the Möller and Nix calculations. Rather, the deformationsseem to peak at lower values of N for lower values of Z. This has beendiscussed in reference [90] as an effect originating from the contributions tothe moment of inertia when the Fermi energy is around the middle of thei13/2 neutron shell, similar to the orbitals shown in figure 2.2. As described in
section 2.1.1, the nuclear deformations causes some orbitals to move down
relative to the lower lying orbitals in the spherical shell model so that, in this
case, the i13/2 neutron shell starts to be filled at lower values of N.
4.1 Variable moment of inertia
As the experimental data on nuclear deformation is quite sparse other waysof understanding the evolution of nuclear deformations have to be investi-gated. One other way is to use the excitation energy spectrum together withthe VMI model described in section. 2.1.4. The VMI model is based on theHarris parameters [J0,J1] and not the deformations directly. However, J0 can
be related to the deformation of the nucleus and J1 can be related to the amountof freezing of the internal structure, that is the rigidity. In this way it is pos-sible to obtain a much richer set of data than when using only experimentaldeformations directly. The VMI fits have been made according to the proce-dure in reference [90], with the inclusion of the data on 168Dy [112, I] and170Dy [112, 142]. The experimental data from reference [145] is shown to-
gether with the Harris parameters J0 in figure 4.2. Note that the parameters
45
N85 90 95 100105110115120125
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N85 90 95 100105110115120125
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Sn (50)
Te (52)
Xe (54)
Ba (56)
Ce (58)
Nd (60)
Sm (62)
Gd (64)
Dy (66)
Er (68)
Yb (70)
Hf (72)
Os (74)
W (76)
Pt (78)
Figure 4.1: Quadrupole deformation parameters β2 for even-even nuclei according to
the Möller and Nix calculations [144] (left) and experimentally measured β2 defor-
mations [145] (right).
N90 95 100 105 110 115 120
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N90 95 100 105 110 115 120
)-1
MeV
2 ( 0J
0
5
10
15
20
25
30
35
40
45
50
Sm (62)
Gd (64)
Dy (66)
Er (68)
Yb (70)
Hf (72)
Os (74)
W (76)
Pt (78)
Figure 4.2: Experimentally measured β2 values (left) and J0 Harris parameters (right)
for a selection of even-even nuclei.
might differ slightly from reference [90] due to different choices of cut-off andthat the J0 for 170Dy is based only on two data points, the tentative 4+ → 2+
transition from reference [112, I] and the calculated 2+ → 0+ transition fromreference [142]. These new data points actually suggest that the apparent shiftin the location of the deformation maximum to lower N for lower Z could
due to the lack of experimental data in combination with the sub-shell closure
at N = 98, and that the N = 104 deformation maximum could be reasonably
stable from, at least, Dy to Hf, 66 ≤ Z ≤ 72.
46
4.2 Deformations in odd-A nuclei
In figure 4.1, only the deformations of the even-even nuclei are shown forthe Möller and Nix calculations, since the only experimental data available inreference [145] is for even-even nuclei. The reason for this is that the defor-mations are usually obtained from measuring the quadrupole moment throughthe B(E2) values in Coulomb excitation experiments. Coulomb excitation in-vestigations of the odd-A isotopes are more challenging since these nucleihave a high density of low excitation-energy states that are, in most of thecases, connected by low-collectivity transitions. However, recent experimentsat REX-ISOLDE have been carried out to study both odd-A [146] and odd-
odd [147] nuclei. If the experimental data can be expanded to include also
the odd-A nuclei, this would be an important increase in the number of data
points available for systematic investigations on the evolution of deformations
and collectivity.In reference [114, II] this has been discussed using a minimization proce-
dure of the experimental data with the particle plus triaxial rotor model. To
evaluate the stability of this fitting procedure a function defining the average
difference between the calculated results and the experimental results can be
defined as
e =N
∑i=0
|Eexp −EPTR|NEexp
. (4.1)
Note that the function in equation (4.1) is not the same as the functionused in reference [114, II]. This function has been evaluated at 100 000 ran-dom points in the four-dimensional (ε2,γ,ε4,ζ ) space. The different two-dimensional projections of this four-dimensional space are shown in figure 4.3for 169Ho. As is seen in the figure, there is no correlation between γ and ε4 orγ and ε2. The only strong correlation that is seen is between ε2 and ζ , while
weaker correlations exist between ε2 and ε4, γ and ζ , and ε4 and ζ . The strongcorrelation between ε2 and ζ makes it impossible to extract a definite resultof the deformation, however, the weak correlations make it possible to put re-strictions on the fit. By restricting γ ≈ 0 and |ε4|� 0.05 it is possible to obtainconverged values of ε2 and ζ .
To restrict the values even further one could use the 1/2−[541] orbital by
setting requirements on the energy of the bandhead of the Jπ = 12
−band.
As this orbital has a very strong dependency on ε2 it will limit the fit con-
siderably. This, however, requires experimental knowledge of the Jπ = 12
−
bandhead which was not obtained in the experiment in reference [114, II].Thus, the results are too rough for a systematic comparison along the isotopicchain. One conclusion is, however, that the deformations are smaller than inthe neighbouring Z nuclei, which is not unexpected since the presence of aspectator nucleon can modify the deformation parameters, as discussed in ref-erence [90] for odd-A Os isotopes in terms of the VMI model.
47
2
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-30
-20
-10
0
10
20
30
2
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
4
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
2
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-30 -20 -10 0 10 20 30
4
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-30 -20 -10 0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
4
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Figure 4.3: Projections of the particle plus triaxial rotor parameter space for 100 000
random points and a requirement that the average error between the experimental and
theoretical energy levels are less than e < 1%.
48
4.3 Rigidity and backbending
Another important aspect of the collective structure of the nuclei in this regionis the rigidity of the nucleus, closely related to the J1 Harris parameter of the
VMI model thoroughly discussed in reference [90]. For a very rigid nucleus
the moment of inertia is expected to remain unchanged with increasing rota-
tional frequency and can also manifest itself through a delayed backbending.The experimental observables, energy and angular momentum, can be
translated into the TRS observables through the well known canonicalrelation between rotational frequency (hω) and angular momentum,
hω =∂H
∂ I≈ Ei −Ef
Ii − If
, (4.2)
and the quantum mechanical relation between moment of inertia (J (1)) and
angular momentum,
J (1) =I
hω≈ Ii + If
2hω, (4.3)
where Ei (Ef) and Ii (If) are the energy and spin of the initial (final) state,respectively, discussed in more detail in section 2.1.3 and section 2.1.4. Usingthese relations, the excitation energy spectrum can be compared to the TRScalculations, shown in figure 4.4 for Dy isotopes with 160 ≤ A ≤ 170 (94 ≤N ≤ 104) and in figure 4.5 for Ho isotopes with 161 ≤ A ≤ 171 (94 ≤ N ≤104).
As seen in these figures, the backbending region has been reached exper-
imentally, for 160Dy, 161Ho and 163Ho. The backbendings in the TRS calcu-lations are, however, clearly occurring at lower rotational frequency than ob-served experimentally. For 162Dy–166Dy and 165Ho, the predicted backbend-ing region has been reached experimentally, but no backbending has yet beenobserved experimentally, why it is difficult to draw conclusions about howclose the TRS calculations are to the experimental backbending for these nu-clei. For the nuclei studied in this thesis, 168Dy, 170Dy, 167Ho and 169Ho, the
predicted backbending region was not yet reached experimentally.
49
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Dy160
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Dy162
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Dy164
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Dy166
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Dy168
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Dy170
Figure 4.4: Total Routhian Surface calculations of Dy isotopes with 160 ≤ A ≤ 170
with zero quasiparticles (solid line) compared to available experimental data (circles).
50
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Ho161
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Ho163
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Ho165
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Ho167
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Ho169
(MeV)0.1 0.2 0.3 0.4 0.5
)-1
MeV
2 (
(1)
J
0
10
20
30
40
50
60
70
80
90
100Ho171
Figure 4.5: Total Routhian Surface calculations of Ho isotopes with 161 ≤ A ≤ 171
with one quasiparticle in the (π,α) = (−,− 12 ) configuration (solid line) and the
(π,α) = (−,+ 12 ) configuration (dashed line) compared to available experimental data
with (π,α) = (−,− 12 ) (filled circles) and (π,α) = (−,+ 1
2 ) (open circles).
51
Part II:
Technology
5. The AGATA HPGe spectrometer
“It’s the apparent change in the frequency ofa wave caused by relative motion between thesource of the wave and the observer.”– Sheldon Cooper
As seen in chapter 3, we are currently approaching the limit of what is
achievable with stable ion beams and the current detector systems. To reach
further out in the terra incognita of the Segré chart new exotic nuclei will be
produced at the second generation RIB facilities. These nuclei will be pro-
duced with very low cross sections and in a high γ-ray background environ-ment, which makes the weak γ-ray transitions extremely difficult to detect
with existing spectrometers. To distinguish these rare events, new instruments
with higher efficiency and resolving power are required. The sophisticated
high-resolution HPGe arrays, crowned by EUROBALL [133] and GAMMA-
SPHERE [148], have given access to very weak signals from high-spin states,
unravelling many new nuclear structure phenomena. To increase the detec-
tion efficiency in experiments at the first generation RIB facilities and for
reactions with low γ-ray multiplicity, HPGe detector arrays like MINIBALL
[149] and EXOGAM [150], which are using segmented HPGe detectors to
obtain a higher detector granularity, have been built. To increase the detection
power even further, the next generation HPGe detector arrays are based on
the γ-ray tracking technique, which currently is being developed within the
AGATA [151–154] and GRETA [155] projects.One source of background in previous HPGe detector arrays is the true
γ rays that are detected, but only deposit parts of their energy in the detec-
tor when they are Compton scattered out from the crystal. To suppress this
background the detectors can be surrounded with active shields, usually high-
efficiency BGO detectors. If a γ ray is Compton scattered the BGO shield willdetect it and the event is discarded. This is, however, not a very economicmethod since it means that the array will cover a smaller solid angle due tothe dead space of the BGO shields. Furthermore, the Compton-scattered γ-rays contain information and a method that recover Compton-scattered γ-rays
instead of rejecting them would dramatically increase the efficiency.To recover this information the AGATA detector will instead use a method
based on γ-ray tracking. By removing the BGO shields and instead deter-mine each interaction position in the crystals it is possible to recreate the γ-
ray track and by adding the energy deposition in each interaction point the
55
original γ-ray energy can be obtained. Furthermore, a precise knowledge of
the position of the interaction points will increase the precision of the cor-
rections for the Doppler effects during in-beam γ-ray spectroscopy and thusalso increase the energy resolution. The complete AGATA detector will con-sist of 180 asymmetric hexagonal, tapered and encapsulated HPGe crystals.The HPGe crystals are available in three slightly different shapes that togetherform an AGATA triple-cluster (ATC) detector [153]. In its full configurationAGATA will form a 9 cm thick HPGe shell with an 82% solid angle coverage.
5.1 HPGe crystals
The AGATA array will consist of tapered closed-end coaxial n-type HPGe
crystals, with an impurity concentration of 0.4 ·1010–1.8 ·1010 cm−3, in threeslightly different asymmetric shapes. The crystals typically weigh 2 kg eachand has a diameter of 80 mm at the rear and a length of 90 mm. The taperingangle is 8◦, creating a hexagonal shape at the front. The bore hole in the crys-tals has a diameter of 10 mm and extends to 13 mm from the front surface.Each crystal is electrically segmented into 36 segments which are read outby 36 outer and one inner (core) contact. There are six angular segments andsix segments in the z direction with thicknesses 8, 13, 15, 18, 18 and 18 mmstarting at the front face of the detector. The crystals are encapsulated into alu-minium canisters with a 0.8 mm wall thickness. See references [151, 153, 154]and figure 5.1 for a more detailed description of the AGATA HPGe crystalsand the ATC.
5.2 Electronics
Due to the extreme conditions AGATA will run in, all electronics for the sys-tems have been developed specially for this array. Closest to the crystal are thelow-noise silicon field-effect transistor (FET) preamplifiers [156, 157] that op-erates both with a cold part in the detector cryostat and a warm part outsidethe cryostat. The preamplifiers prepare the signal for the digitizers that contin-uously samples each segment, and the core signal, with a 100 MHz samplingfrequency and 14 bit resolution in the analog-to-digital converter (ADC). Thesignals from all segments are sent to the pre-processing electronics where theenergy, time and the digitized leading edge of the traces of all the 36 seg-ments are extracted for the pulse-shape analysis (PSA) (see section 5.3). Thecore signal is also pre-processed and used as an input into the global triggerand synchronization (GTS) system which provides the system clock and thetrigger for the array. The core signal from the GTS system and the interac-tion points obtained from the PSA are then tracked and merged, together with
56
Figure 5.1: The AGATA HPGe crystals. Reprinted from reference [151].
Figure 5.2: Signals from the core, the segment with the primary hit (Seg 4), and from
the mirror charges for a γ-ray interaction in a six fold segmented HPGe detector.
Figure from [152].
57
data from ancillary detectors that is prepared using a special AGATA VME
Adapter (AGAVA) interface, in the event builder.
5.3 Pulse-shape analysis
To determine the interaction position in a crystal segment, pulse-shape in-
formation from the segment which was hit, the mirror charges in the neigh-
bouring segments, and the core signal is used, as illustrated in figure 5.2. The
azimuthal position in the crystal segment is obtained by comparing the ampli-
tudes of the mirror charges. The radial position is obtained from the shape of
the signal of the hit segment.This is done by comparing the sampled pulse-shapes to a database, with
more than 30 000 basis sites per crystal, of pulse shapes for different inter-
action positions. Each grid point in the PSA basis contains a reference pulse
shape with several time steps within the digitizer resolution of 10 ns. For a
2×2×2 mm3 grid with a time step of 1 ns the size of each PSA basis file isroughly 1 GB per crystal. One of the main problems of the PSA algorithmsis to obtain the database of pulse shapes. When delivered, the crystals arescanned using radioactive sources [158]. This process gives a very precisedatabase of pulse shapes but it is unfortunately a very slow process. To com-plement the scanned pulse shapes, codes for calculating them are being devel-oped. These calculations are much faster, but the main challenge is to obtainrealistic pulse shapes for the complex electric field within the geometry of theAGATA crystal and large uncertainties in the impurity concentration. Further-more, high electric fields and low temperatures will cause the charge carrierdrift velocities to become anisotropic with respect to the crystallographic lat-tice orientation [159], which further complicates the situation.
The performance of the γ-ray tracking will depend strongly on the qual-
ity of the PSA. Accuracy better than 5 mm has to be achieved using algo-
rithms fast enough for real-time application. Interaction point coordinates are
obtained by comparing the detected pulse shapes to the signal basis. The sim-
plest method is the grid search [160] algorithm that works well for experi-
ments when there is only one γ ray interaction a time in the same segment. In
the grid search algorithm the sum of the squared difference between measured
and calculated signals is compared and the interaction point with the best value
of this quantity is selected. More sophisticated algorithms have also been de-
veloped. Since these algorithms have different advantages and disadvantages a
dispatcher code that can distribute the events to the optimal algorithm depend-
ing on the event properties is planned within the collaboration. The algorithms
evaluated so far are the extensive grid search [161], the particle swarm opti-
misation [161], the matrix method [162], genetic algorithms [163], recursive
subtraction [164] and neural networks [161].
58
Figure 5.3: Simulated interaction points of 30 γ rays of energy Eγ = 1.33 MeV in the
(θ ,φ sinθ ) plane of an ideal germanium shell with an inner radius of 15 cm and an
outer radius of 24 cm. Circles are correctly and squares incorrectly identified clusters.
Figure from [152].
5.4 Tracking of γ rays
The output from the PSA will contain the energy, time and three-dimensionalposition of each identified interaction point, located in a, so called, world mapfor each event. A typical such world map, where the interaction positions areshown in terms of the (θ ,φ sinθ ) coordinates relative to the center of the ar-ray is shown in figure 5.3. The purpose of the tracking algorithms is to recon-struct the trajectories of the γ rays and disentangle them from their world map.There are two ways to accomplish this, the so called backtracking or forwardtracking methods. The backtracking algorithm was developed in Stockholm[165, 166]. This algorithm uses the information that the final, photoelectric,interaction point most probable has an energy deposition between 100 to 250keV. Starting from this assumed final interaction point, other interaction pointsare searched for within a distance based on the interaction length in germa-nium for γ rays of that energy. This procedure is repeated until the track isterminated by the source location.
In the forward tracking algorithm the interactions caused by a certain γ rayare assumed to be clustered together within a certain angular spread in the
(θ ,φ sinθ) plane, which depends on the total number of hits registered in the
59
various detectors, see figure 5.3. These clusters are searched for and identified.
Initially the source location is labelled to be the zeroth interaction point. The
first and second interaction points are chosen randomly and the γ-ray energyafter scattering is determined from the measured energy depositions in thecluster. This energy is compared to the energy calculated by the Comptonscattering formula
E ′γ =
Eγ
1+ Eγmec2 (1+ cosθγ)
, (5.1)
where E ′γ is the energy after Compton scattering, mec2 the electron rest mass
and θγ the angle between the incoming and scattered γ ray. The angle θγ isdetermined from the positions of the three interaction points. A figure-of-merit(FOM) based on the agreement between the two energies is calculated. Allpossible permutations of interaction points are evaluated and the one with thehighest FOM is chosen. This procedure is repeated with the identified firstinteraction point as a starting point until all interaction points in a cluster havebeen assigned to the track. An interaction is allowed to be a member of severalclusters but only the cluster with the best FOM is accepted. Pair productionevents are identified if two tracked γ rays each with an energy of 511 keV are
found within a cluster.Other types of clustering and tracking have also been considered within the
AGATA array. One example it to use fuzzy logic to identify well separated
groups in a multidimensional space [167]. The optimal position of the clus-
ter centres and the degree of membership of each point is identified using a
fuzzy logic algorithm. After the identification, a defuzzification is carried out
in which every point is switched from being a member of a certain degree to
every cluster to just belonging to one cluster. This is, however, still a work in
progress.
5.5 Data acquisition
The data aquisition system (DAQ) in AGATA has two functions. The first of
these is to process the data flow from the AGATA front end electronics and the
ancillary detectors to the local data storage. The local storage is used during
the data analysis after which the data is archived at a LHC Grid [168] Tier-1
computing centre in Bologna. The second task is to control and monitor the
whole system and the data flow during experimental campaigns.
To handle the large data flow in AGATA a well structured DAQ is needed.The DAQ developed for AGATA is called NARVAL [169]. It is written inAda1, but the actors can load C++ shared libraries. NARVAL is based on
1The Ada language is named after Ada Lovelace, one of the first computer programmers, and
was developed for the United States Department of Defence. It was designed for critical sys-
60
an abstract class, called actor, which comes in three types: producers, inter-
mediary and consumers. A producer is an actor that collects data from the
hardware, such as the VME crates. The intermediary actors performs tasks
like PSA, building the event and tracking. The consumers store the data to
disk and builds histograms. Finally, there is also the chef d’orchestre actorthat handles the state machine and the gestion d’erreurs actor that handles
and recovers errors. These actors can be written separately and included in the
DAQ so that it is always adapted to the needs of the particular experiment in
question. This also means that NARVAL is not a one program DAQ, but more
a society of interacting actors that can run on many different computers in
parallel. The DAQ chain is controlled by the user through the Cracow graph-
ical user interface (GUI) [170], originally developed for RISING. The GUI
is not a part of NARVAL, but is an independent program that communicates
with NARVAL through web services and can thus be run from any computer
connected to the network.
5.6 Position resolution
As mentioned in section 5.3, in order to have a high detector peak efficiency a
position resolution of � 5 mm is required [152]. Previously, two methods havebeen used to determine the interaction position resolution of segmented HPGedetectors. The first method utilizes imaging techniques with a 60Co source[171]. The second method makes use of the Doppler effect by measuring theenergy resolution of peaks in Doppler corrected γ-ray spectra recorded in in-
beam experiments in which the γ rays are emitted in-flight by the movingnuclei [172–174]. In the experiments in references [172–174], the positionresolution has been obtained by comparing the experimental data with MonteCarlo simulations of the set-up. In this work, and reference [175, III], a MonteCarlo model independent method to obtain the position resolution from anin-beam experiment is presented.
The well known Doppler effect is the apparent change in frequency an ob-server experience when a source is moving with respect to the observer [176].The energy of a Doppler shifted γ ray emitted by a nucleus moving at rela-
tivistic velocities is given by the expression
Eγ = Eγ0(1−β 2)1/2
(1−β cosθ). (5.2)
where Eγ is the energy of the emitted electromagnetic radiation, β is thesource velocity as a fraction of the speed of light and θ is the angle between
tems like avionics, thermonuclear weapon systems and satellites. This is the reason why Ada
programs are usually very stable. This is also the reason why the first program one learns to
write in Ada is often called “Goodbye, World!”.
61
r
pW
d
1pE
2pE
3pE
4pE
E
z
Figure 5.4: Effects of the position resolution when determining the angle used to
correct for the Doppler shifts of the γ rays. Reprinted from [175, III] with permission
from Elsevier.
the direction of the source and the direction of the electromagnetic radiation,see illustration in figure 5.4. Here Eγ0 is the energy of the γ ray in the rest
frame of the nucleus. The energy resolution of the detected γ rays will thusbe given by the expression
W 2Eγ0
=(
∂Eγ0
∂Eγ
)2
W 2Eγ +
(∂Eγ0
∂β
)2
W 2β +
(∂Eγ0
∂θ
)2
W 2θ , (5.3)
where WE is the full width at half maximum (FWHM) of the detected energy,Wθ is the angular resolution of the detector, Wβ the spread in velocity of the
recoil ions and WEγ the intrinsic resolution in the detector and electronics. The
62
partial derivatives in equation (5.3) are given by
∂Eγ0
∂Eγ=
1−β cosθ(1−β 2)1/2
, (5.4)
∂Eγ0
∂β= Eγ
β − cosθ(1−β 2)3/2
, (5.5)
∂Eγ0
∂θ= Eγ
β sinθ(1−β 2)1/2
. (5.6)
It can be shown [175, 177] that the position resolution, Wp, is given as,
W 2p =
1
b2
(W 2
Eγ0,c−W 2
Eγ0,f
)( 1
r2c
− 1
r2f
)−1
, (5.7)
where
b2 =(
∂Eγ0
∂θ
)2
. (5.8)
The interaction position resolution can, thus, be extracted from equation (5.7)by performing two measurements of WEγ0
at a close (c) and far (f) distance
[177].
5.6.1 Reaction selection and simulations
A first preliminary survey of reactions possible to use to measure the position
resolution experimentally was done using the TALYS code [178], that calcu-
late cross sections for reactions with γ rays, neutrons, and light ions up to 4He
in the energy range 0.1–200 MeV and a mass range of 12 ≤ A ≤ 339. Thecontribution of the target thickness to the total FWHM was also studied us-ing the TRIM code [179, 180]. For more details, see reference [181]. A fewcases were selected for further simulations using the evapOR code [182, 183],which is a fusion-evaporation code that calculates the fusion cross-section fortwo arbitrary nuclei, followed by the evaporation of particles from neutronsto 6Li. Furthermore, evapOR generates a Monte Carlo dataset from these
cross sections that can be used in further simulations. The requirements on
the reactions are discussed in more detail in reference [175, III], but can be
summarized in the following points:• minimize the spread of the velocity vector of the residual nuclei (as small
Wβ as possible),
• enhance the effects of the Doppler broadening due to the γ-ray detectionby maximizing β , Eγ0 and sinθ ,
• select transitions with upper and lower limits on the effective life-time, and
• choose a reaction which gives strong and clean peaks in the γ-ray spectra.
63
5.6.1.1 The 2H(81Br,n)82Kr reactionThe first selected reaction was 2H(81Br,n)82Kr with a 1044 keV 4+ → 2+
transition in 82Kr. The main advantages of this reaction are the very lighttarget and that only one neutron is evaporated. Due to this, the velocity vectoris expected to have a very narrow distribution in both magnitude and direction.
There are, however, several long-lived states in 82Kr, for example the yrast(8+) state has a half life of t1/2 = 96 ps. Calculations using evapOR show
that only < 0.1% of the compound nucleus cross-section feeds states withtotal angular momentum J ≥ 17/2.
5.6.1.2 The 9Be(136Xe,3n)142Ce reactionThe second selected reaction, with 142Ce as the final nucleus, has the advan-
tage that the chosen 2+→0+ ground state transition is very strongly populated.The heavy projectile and the relatively light target also gives narrow β and θdistributions of the velocity vector despite the 3n evaporation. A disadvan-
tage is that the γ-ray energy is rather low, only 641 keV. The yrast 2+ and 4+
states in 142Ce have lifetimes of 8 ps and 11 ps [184], respectively, which areshort enough to fulfil the lifetime criterion. All other states with known life-times are short lived [184] and do not influence the interaction position res-olution determination. However, only states up to spin 6+ are known in thisnucleus [184], while in the proposed reaction states of considerably higherspin are populated. Possible long lived (� 15 ps) and strongly populated butso far unobserved high-spin states may therefore influence the measurement.
5.6.1.3 The 12C(82Se,3n)91Zr reactionThe third selected reaction has 91Zr as the final nucleus. The 21/2+ state in
this nucleus is isomeric with a half life of 4.35 μs [185]. The selected tran-sitions must therefore occur between states above this isomer. For the sim-ulations we chose the (23/2−)→21/2+ transition, which has an energy of2141 keV [186–188]. A plunger measurement of lifetimes above the isomerwas performed using the same inverse kinematic reaction as selected in thiswork [189]. A state at 5741 keV, which feeds the (23/2−) state at 5308 keV,has a lifetime of 45 ps. The distance travelled by the 91Zr residues during this
time is about 1 mm. The selected 2141 keV will, therefore, be slightly influ-
enced by the lifetime of the 5741 keV. All other observed states above the
21/2+ isomer had lifetimes and feeding times which were much smaller than45 ps [189]. Above the 21/2+ isomer there are several other high-energy tran-
sitions, which are not fed by the long lived 5741 keV level and which also may
be used in the measurement. The relative intensity of the (23/2−)→21/2+
transition in the proposed reaction is not known. We estimate it to be of the
order of 10% of the total reaction channel population.
64
5.6.1.4 The 12C(82Se,4n)90Zr and 12C(30Si,np)40K reactionsTwo reactions, for which experimental results exist on interaction position res-
olution measurements, have also been included in the simulations. The firstone is the reaction 12C(82Se,4n)90Zr, which was used in a test of a GRETAHPGe crystal [173]. The second reaction is 12C(30Si,np)40K, which was used
in the first in-beam commissioning experiment of an asymmetric AGATA
triple cluster detector [190, 191], see section 5.6.2.
5.6.1.5 The reference reactionA reference reaction with β = 10.0%, Wβ = 0 and Eγ0 = 50 keV− 5 MeVwas also used in the simulations in order to study systematic errors.
5.6.1.6 ResultsThe evapOR data sets were used as input to the AGATA GEANT4 simulation
program [154, 192, 193]. This program simulates γ-ray interactions in theAGATA HPGe detectors and produces an output file containing the energy andposition of the interaction points within the detectors. The interaction pointsproduced by the GEANT4 program were used as input to the mgt trackingcode [194, 195], which is based on the forward tracking algorithm [196]. Theresults from these simulations, see figure 5.5, show that the proposed methodworks well. For a more detailed discussion of the results, see reference [175,III].
5.6.2 Experiment
The first in-beam commissioning test of the AGATA detectors and infrastruc-ture was performed in March 2009. The proposed reaction to use for the exper-iment was the 12C(82Se,3n)91Zr reaction. Unfortunately, it was not possible
to produce a 82Se beam with high enough energy and intensity at the time ofthe experiment. Therefore the reaction 12C(30Si,np)40K was chosen instead.
A 30Si beam with energy of 64 MeV was produced by the Tandem Accelera-tor at LNL, see section 3.1. The γ radiation was detected by the first AGATA
ATC [153], see figure 5.6. Data were taken using two distances between the
front face of ATC and the target, dc = 55 mm and df = 55 mm [175, III].The complete front-end electronics and NARVAL DAQ [169] of the
AGATA Demonstrator was used in the experiment where dedicated NARVAL
actors were performing the on-line PSA and γ-ray tracking in real time.
The system included the autofill system, the low-voltage power supply,
preamplifiers, the digitizers, the pre-processing electronics, the GTS system,
the computer farm for PSA, event building and on-line γ-ray tracking, andthe disk storage array. The high-voltage was provided by a standard CAENSY527 system. As this was the first time in which the full system wasrunning, the processed and tracked spectra were only shown on-line whileparts of the digitized pulse shapes were stored to disk for later replay.
65
s (mm)0 2 4 6 8 10
- s
(m
m)
pW
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
spW
Reference
s (mm)0 2 4 6 8 10
- s
(m
m)
pW
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Kr82
s (mm)0 2 4 6 8 10
- s
(m
m)
pW
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Ce142
s (mm)0 2 4 6 8 10
- s
(m
m)
pW
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Zr91
s (mm)0 2 4 6 8 10
- s
(m
m)
pW
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Zr90
s (mm)0 2 4 6 8 10
- s
(m
m)
pW
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2K40
Figure 5.5: The difference between the calculated interaction position resolutions,
Wp, and the mgt smearing parameter, s, (that simulates the position resolution) shown
as a function of s for the simulated reactions listed in reference [175, III]. The sys-
tematic error, estimated by a fit of a constant W sp , is shown as a solid line. Reprinted
from [175, III] with permission from Elsevier.
The event replay was carried out off-line using the NARVAL emulator run-
ning on a dual quad-core Intel E5520 2.27 GHz CPU computer with 24 GB ofRAM and a 10 Gbit Ethernet connection to the data storage disk. The pulse-shape analysis running in the NARVAL emulator used the adaptive grid-searchalgorithm [160] with a signal basis grid having a resolution of 2×2×2 mm3
and which was calculated by the ADL code [197–199]. The off-line γ-raytracking was performed by the mgt program. A tracked γ-ray spectrum mea-sured at the close distance is shown in figure 5.7. The spectrum is Dopplercorrected by using the position of the first interaction point and an averagevalue of β = 0.048. As seen in figure 5.7, several γ rays not originating from40K are also populated. Especially the one-proton evaporation channel (41K)and the one-neutron evaporation channel (40Ca) were found to provide useful
data points at low and high energies.The results obtained for the interaction position resolution Wp are listed in
table 5.1 and shown in figure 5.8 for the six selected γ rays. As seen, the
interaction position resolution varies roughly linearly as a function of γ-rayenergy from 8.5 mm at 250 keV to 4 mm at 1.5 MeV, and has an approximately
constant value of about 4 mm in the γ-ray energy range from 1.5 MeV to
66
Figure 5.6: The AGATA detector in position for the first commissioning experiment.
Reprinted from reference [191].
4 MeV. See reference [175, III] for more detailed discussion on the positionresolution results.
5.7 Neutrons in AGATA
The experiment in section 5.6.2 measured the position resolution of theAGATA ATC up to about 4 MeV. Although, in some nuclear structureexperiments the energies of the γ rays emitted are much higher than this.
For example, the γ rays emitted by a giant dipole resonance (GDR) in therare-earth region will have an energy around 10–15 MeV [200, 201]. Anothercrucial point for GDR experiments is the possibility to discriminate betweenneutron and γ rays in AGATA [200]. This is, furthermore, important whendoing experiments on neutron-rich nuclei at RIB facilities like FAIR andSPIRAL2.
Neutrons can be detected in AGATA either via elastic scattering,
Ge(n, n)Ge, inelastic scattering, Ge(n, n′γ)Ge, or nuclear reactions [202].The expected probability of neutrons with energies 1–10 MeV to be detected
by AGATA is about 50% [203].
67
(keV)0E0 1000 2000 3000 4000 5000
Cou
nts
10
210
310
410
510
610 246 770 1352 18232333 3905
242 251
K41
760 790
K40
1339 1362
K40
1800 1850
K40
2305 2355
K40
3860 3960
Ca40
Figure 5.7: Gamma-ray spectrum measured using the 30Si +12 C reaction with the
AGATA triple cluster detector placed at the close distance (dc ≈ 55 mm). The spec-
trum was created by performing γ-ray tracking using mgt and by applying a Doppler
correction based on the position of the first interaction point and an average value of
β = 0.048. Expanded views of the six peaks selected for the analysis are also shown
in the figure. Reprinted from [175, III] with permission from Elsevier.
A method has recently been developed by Monte Carlo simulations to dis-criminate between neutrons and γ rays in AGATA [204]. In that work, thediscrimination was done using the forward tracking algorithm with respectto three parameters: the energy deposited in the first and second interactionpoints, the incoming direction of the γ ray, and the FOM from the tracking
algorithm. Differences in pulse shapes between neutron and γ-ray interactionshas also been studied in non-segmented HPGe detectors with negative results[202]. It is, however, not known if this is also true for segmented HPGe detec-tors with much smaller volume per segment than a non-segmented detector.
An experiment to check and further develop these methods has been carried
out using AGATA, the HELENA array and a 252Cf source, see figure 5.9.The HELENA BaF2 detectors were placed closed to the 252Cf source with a
thick lead shield in-between the source and AGATA to reduce the number γ
68
(keV)0E1000 2000 3000 4000 5000
(m
m)
pW
0
2
4
6
8
10
12
Figure 5.8: Interaction position resolution as a function of γ-ray energy. The error
bars due to statistical errors only. The estimated systematic deviations for the data
set are shown as the filled histogram. Reprinted from [175, III] with permission from
Elsevier.
Figure 5.9: The AGATA and HELENA detectors in position for the neutron experi-
ment.
69
Table 5.1: Experimental values of the interaction position resolution Wp obtained for
the six selected γ rays. The columns in the table are the residual nucleus, the energy
of the γ ray, the FWHM of the peak at close and far distance, the parameter b and
the interaction position resolution. The experimentally determined average interaction
distances, needed for the calculation of Wp according to equation 5.7, have the values
rc = 100 mm and rf = 270 mm.
Residual Eγ0 WEγ0,c WEγ0,f b Wp
nucleus (keV) (keV) (keV) (keV) (mm)41K 246 2.326±0.023 2.148±0.013 11.6 8.6±0.640K 770 5.159±0.024 4.732±0.015 36.2 6.10±0.2140K 1352 7.14±0.04 6.734±0.027 63.5 3.93±0.2340K 1823 10.31±0.04 9.851±0.025 85.7 3.76±0.1840K 2333 11.92±0.05 11.365±0.033 110 3.49±0.18
40Ca 3905 19.32±0.14 18.4±0.1 184 3.53±0.32
background in AGATA. The TOF between HELENA and AGATA was usedto determine whether a neutron or a γ ray was detected in AGATA. The datafrom this experiment is still under analysis [205].
70
6. Neutron detector NEDA
6.1 SPIRAL2
One of the new RIB facilities that will be constructed in Europe, as dis-
cussed in chapter 1.3, is SPIRAL2 at GANIL. The SPIRAL2 facility will
be an ISOL facility based on a high power superconducting driver linear
accelerator (LINAC) that will deliver a 40 MeV deuteron beam with a beam
current of 5 mA and 14.5 MeV per nucleon heavy-ion beams with a beam
current of up to 1 mA. The deuteron beam will be bombarding a UCx target
that produces the isotopes for the secondary RIB. The intense stable beams
can either be used directly in experiments or for creation of proton-rich RIBs
via fusion-evaporation reactions.Two of the experiments that will be located before the uranium target is
the Super Separator Spectrometer (S3) and the Neutrons for Science (NFS)
project. The S3 is a magnetic spectrometer that will use the high intensity sta-ble ion beams to study, for example, super-heavy nuclei and nuclei beyond thedriplines. The NFS facility will be a compliment to other high intensity neu-tron sources, like at Institut Laue-Langevin (ILL) and the European SpallationSource (ESS), to be used for both fundamental and applied neutron research.It will use the protons and deuterons from SPIRAL2 to generate the neutrons.These neutrons will then be used at different experimental setups, for exam-ple CARMEN [206] or the MEDLEY setup previously located in Uppsala[207–209].
After the UCx target and the CIME post-accelerator cyclotron a number ofexperimental setups will be built for nuclear physics studies using the RIB.Among the planned setups are an experimental area called DESIR, to studyexotic nuclei through laser spectroscopy, decay spectroscopy and mass spec-trometry. Furthermore, ACTAR, an active-target detection system, FAZIA, a4π detector array for isospin studies, GASPARD, an array for reaction studies,PARIS, an array to study giant resonances, shape changes and hyperdeforma-tion, and HELIOS a superconducting solenoidal spectrometer are projected.
Finally, the existing HPGe array at GANIL, EXOGAM [150], will be up-
graded to EXOGAM2. To further increase the detection efficiency it will be
possible to run EXOGAM2 together with the HPGe detector array AGATA,
described in chapter 5. A new neutron detector array, NEDA [210], is also
projected to replace the existing Neutron Wall [211].
71
6.2 The Neutron Wall
The Neutron Wall is the current neutron detector array at GANIL for usein nuclear structure experiments [211]. The Neutron Wall was originally de-signed for experiments together with EUROBALL [133] at LNL and IReSStrasbourg. Since 2005 it is located at GANIL where it is used together withEXOGAM [150], where it in recent experiments [26] has been used togetherwith the charged particle detector array DIAMANT [212, 213]. It consists of15 hexagonal detectors of two different shapes and one pentagonal detector.The detectors are assembled into a closely packed array covering about 1π ofthe solid angle. They are filled with the liquid scintillator BC-501A to a totalvolume of 150 litre. The 16 detectors are in turn divided into 50 segments intotal. The hexagonal detectors are subdivided into three individual segments.Each of the segments contain 3.23 litre of scintillation liquid and is read out bya 130 mm Philips XP4512PA photomultiplier tube (PMT). The pentagonal de-tector is subdivided into five individual segments of 1.07 litre each and is readout by a 75 mm Philips XP4312B PMT. The total neutron efficiency of theNeutron Wall is about 25% in symmetrical fusion-evaporation reactions. Thepulse-shape discrimination (PSD) in the Neutron Wall is done by NIM elec-tronic units of the type NDE202 [214] based on the zero cross-over (ZCO)discrimination technique.
6.3 The neutron detector array NEDA
As mentioned in chapter 6.1, one of the SPIRAL2 instruments being devel-oped is NEDA, that will replace the Neutron Wall. NEDA will be used fornuclear structure studies both on the neutron-rich side and proton-rich side ofthe line of β stability. In a typical experiment on proton-rich nuclei using a
fusion-evaporation reaction, the reaction channel of interest is when two or
three neutrons are evaporated. These channels are often very weak compared
to the one neutron evaporation channel. This means that high detection effi-
ciency is required to separate the reaction channels with two or more neutrons
evaporated from one neutron reaction channels. This requires a good way to
distinguish multiple-neutron events from one-neutron events that have been
misidentified as multiple-neutron events due to scattering between detectors,
or neutron cross-talk. Furthermore, an excellent discrimination of neutrons
and γ rays is required, especially when using a RIB. It has been shown thateven a small amount of γ rays misinterpreted as neutrons dramatically reduce
the quality of the cross-talk rejection [215, 216]. The suggested specifications
of NEDA, compared to the Neutron Wall, is listed in table. 6.1.
72
Table 6.1: Proposed specifications of NEDA compared to the Neutron Wall. Efficien-cies are estimated for symmetric fusion-evaporation reactions.
Parameter NEDA Neutron Wall
Type of detector Liquid scintillator Liquid scintillator
Type of liquid BC501A or BC537 BC501A
Number detectors 150–350 50
Solid angle coverage ∼ 2π 1πTarget-detector distance ∼100 cm 50 cm
Detector thickness 20 cm 15 cm
Scintillation light detector PM, SiPM, APD 5" PM
Electronics Fast sampling ADC Analogue NIM units
PSA algorithm Digital Analogue
1n efficiency 30–50 % 20–25 %
2n efficiency 3–15 % 1–3 %
6.3.1 The BC-501A and BC-537 liquid scintillators
As mentioned in chapter 6.2 and table 6.1, the Neutron Wall uses a liquidscintillator to detect neutrons. The basic processes of scintillation in organicmaterials are thoroughly described in refs. [217, 218]. A very popular scintil-lator for neutron detection is BC-501A1, based on xylene or dimethylbenzene,
C6H4(CH3)2, which is the liquid that is currently used in the Neutron Wall. InDESCANT [219, 220], a neutron detector array project at ISAC similar toNEDA, it is proposed to use a deuterated liquid, BC-537, instead.
The liquid BC-501A has a light output that is about 78% of anthracene, amaximum emission wavelength of 425 nm and a hydrogen to carbon ratio of1.287. It has three decay components with 3.16 ns, 32.3 ns and 270 ns decaytimes according to reference [221]. An experiment to verify these decay timeshas been carried out [222] but is still under analysis. Since the relative amountof the slower components compared to the fast component are different fordifferent particle species, see figure 6.1, BC-501A has very good PSD prop-erties. However, practical problems with xylene, like that it is flammable witha flash point (the temperature where it can form an ignitable mixture in air) of24◦C and can cause neurological damage at high exposures, makes searchesfor an alternative detection material interesting for future arrays.
The liquid BC-537 is made of purified deuterated benzene, C6H6, and has alight output that is about 61% of anthracene, a maximum emission wavelengthof 425 nm, a deuterium to carbon ratio of 0.99 and a deuterium to hydrogen
1In older detector arrays a similar liquid called BC-501 was used. These are manufactured by
Saint-Gobain Ceramics & Plastics. There is also an equivalent liquid from Nuclear Enterprise
known as NE-213
73
Time [ns]0 50 100 150 200 250 300 350 400 450
Am
plit
ud
e [a
.u.]
-310
-210
-110
1 -ray
neutron
Figure 6.1: Pulse shapes from a BC501 liquid scintillator from a γ-ray and a neutron
interaction. The decay times are 3.16 ns, 32.3 ns and 270 ns. Reprinted from [223,
IV] with permission from Elsevier.
ratio of 114. Its flash point is −11◦C. BC-537 also have PSD properties andit may give some additional energy resolution and cross-talk rejection prop-erties, which could make it an option to use BC-537 instead of BC-501A inNEDA, despite the lower light output.
6.3.2 Geometry
It is very important to carefully study different geometries both to maximizeefficiency and to minimize cross talk. The cross talk is due to the effect thatneutrons deposit energy mainly by elastic scattering with the protons in theliquid. The energy of the recoil protons can have values from zero up to theincoming neutron energy, depending on the scattering angle. Because of thisthere exists a probability that the neutron will scatter into a neighbouring de-tector and interact again, causing severe errors in the counting of the num-ber detected neutrons. Several methods attempting to correct for this exist[215, 216]. For example a method based on the TOF difference between thedifferent segments has shown to give good results [216] when the segmentsare sufficiently far away while neighbouring segments still are problematic.This is important to take into account when designing the geometry of a newarray. A suggested geometry is shown in figure 6.2.
6.3.3 Detection of scintillation light
One way to improve the efficiency of the neutron detector array is to improvethe efficiency of the readout of the scintillation light. The standard way to readout a liquid scintillator is to couple the liquid to a PMT. The PMT typically
74
Figure 6.2: Possible geometry of the NEDA detector array as seen at an angle from
the front (left) and from the back (right).
consist of a photocathode of a bialkali metal alloy, like Sb-Rb-Cs or Sb-K-Cs [224], a series of dynodes to multiply the photoelectrons and an anode toread out the signal. The bialkali photocathode has a quantum efficiency ofmaximum ∼ 25% and sensitivity well matched to the most common scintil-lator materials. Another very promising technology is the ultra-pure bialkaliphotocathode PMTs that have been made available quite recently by Hama-matsu. By using a fine tuned deposition process one could achieve ultra-purephotocathode materials and reach a quantum efficiency of up to 43% [225].
One alternative to a PMT for readout is to use either a regular photodiode
or an avalanche photodiode (APD). Today regular photodiodes can be manu-
factured up to sizes of about 20× 20 mm2 while APDs are limited to about10×10 mm2. The quantum efficiency of the photodiodes can be very high, up
to 85% at the peak of maximum sensitivity, but the spectral response of this
kind of diodes usually shows a very sharp drop at lower wavelengths. There
are, however, also some examples of large-area short-wavelength APDs, like
the Hamamatsu S8664 which is planned to be used for the readout for the
PANDA electromagnetic calorimeter [226].Silicon PMTs are another kind of position sensitive readout that has gained
popularity in some fields lately [227]. These consist of an array of around 1000independent APDs per mm2. The quantum efficiency and gain of this kindof readout is quite similar to regular PMTs. The main advantages are goodtime resolution, no dependency on external magnetic fields and an extremelycompact design.
6.3.4 Electronics
With the fast development of digital electronics there is an opportunity touse much more sophisticated PSA algorithms than with analogue electron-ics, see chapter 6.4. The term digital electronics means, in this context, thatthe detector signal is digitized with a fast sampling ADC and then processed
using a programmable device like a digital signal processor (DSP), field pro-
75
grammable gate array (FPGA) or even a personal computer (PC). However,
digital PSA also has limitations regarding, for example, computing time and
signal reconstruction. The influence of the bit resolution and sampling fre-
quency of the ADC on the PSD has been investigated in refs. [218, 223].
6.4 Digital pulse shape analysis
In both analogue and digital PSA the difference in pulse shape between an in-
teracting γ ray and an interacting neutron, see figure 6.1, is used to distinguishbetween these types of interacting particles.
Several sophisticated methods for digital PSD have been developed by var-ious research groups lately. For example, using variable gates in the chargecomparison method has been investigated in reference [228]. Another of thesemethods uses a previously measured standard pulse shape from the detector.By defining a correlation function of a sample pulse shape and the standardpulse shape one can discriminate between neutrons and γ rays [229]. Anothermethod that uses standard pulse shapes has also been developed and is beingimplemented in a FPGA [230, 231]. In this method the measured pulse shapeis fitted with “true” neutron and γ-ray pulse-shapes and the χ2 values of these
fits are compared. It has been suggested that one can use fuzzy logic to obtain
the true pulse shapes used in these methods [232]. A fourth method that has
been developed for digital electronics is the pulse gradient method [233, 234].
In this method two sampling points from the tail of the pulse are selected
and the slope between these two points is calculated. All these methods have
yielded good results regarding the discrimination of neutrons and γ rays.In reference [223, IV] two methods have been developed to perform digital
PSD and to study the effects of the ADC bit resolution and sampling fre-
quency on the PSD quality. These methods were developed to be numerically
simple, such that the limitations from computing time should be not too large.
They were also selected due to their similarity to well studied analogue meth-
ods. The two methods were also compared to a standard Neutron Wall NIM
electronic unit of the type NDE202 [214] based on the ZCO discrimination
technique.
6.4.1 Charge comparison and zero cross-over
In the charge comparison method two integration gates are set on the fast and
slow decay components of the pulse. By comparing these integrals with each
other one will get a separation between neutrons and γ rays. Using digitalelectronics this method can be generalized to concern evaluating the integral
S =∫ T
0p(t)w(t)dt, (6.1)
76
Time [ns]0 50 100 150 200 250 300 350 400 450
Am
plit
ud
e [A
.U.]
-210
-110
1
Figure 6.3: Weighting function w(t) for digital (solid) and analogue (dashed) charge
comparison PSD shown together with an average neutron pulse (dotted). Reprinted
from [223, IV] with permission from Elsevier.
where T is the time to evaluate the pulse p(t), w(t) is a weighting function
and S is a quantity that differs between neutrons and γ rays. To enhance thePSD maximally, the best choice of w(t) has been shown [235] to be
w(t) =n(t)− γ(t)n(t)+ γ(t)
, (6.2)
where n(t) and γ(t) are the average neutron and γ-ray pulse shapes, respec-
tively. See figure 6.3 for an example of how w(t) looks like for the experimentin reference [223, IV].
The other analogue method that has been adapted digitally is the ZCOmethod. The ZCO method is described in refs. [236, 237]. Integrating thepulse and taking the rise time of the integrated pulse has been shown to beequivalent to shaping and taking the zero-crossing time [238]. In figure 6.4the principle of the integrated rise-time method is shown. It was found thatfor the setup in reference [223, IV] the best result was obtained by using the10–72% rise time.
To quantify the results, two different quantities were constructed. The firstis the standard FOM (see reference [239]),
FOM =|Xγ −Xn|Wγ +Wn
, (6.3)
77
Time (ns)0 50 100 150 200
Inte
gra
l (a.
u.)
0
2
4
6
8
Figure 6.4: Difference between the integrated rise time of a γ-ray (dotted) and a neu-
tron (solid) pulse. The points at 10 % and 72 % of the pulse height are indicated by
dashed lines. Reprinted from [223, IV] with permission from Elsevier.
where Xγ (Xn) is the centre and Wγ (Wn) is the width of the S distribution for γrays (neutrons). To complement the FOM a parameter, R, was defined as
R =Nb
Nn −Nb, (6.4)
based on the estimated number of γ-ray background counts, Nb, relative to thenumber of neutron counts, Nn, in the neutron peak. See reference [223, IV]
for details on how to calculate R and refs. [218, 223] for a discussion of theadvantages and drawbacks of the two quantities.
The results from the digital PSD are shown in figure 6.5. As seen, the
digital PSD gives at least as good separation as the analogue PSD in the en-
tire energy range. The ZCO based PSD method is shown to work better than
the charge comparison based. The FOM saturates around 9 bits for the charge
comparison based method and at about 10 bits for the ZCO based method2.
It should be noted that these values are for the dynamic range of this exper-
iment with γ-ray energies between Ee = 15–700 keV and recoil proton ener-gies between Ep = 250–2700 keV. Increasing the bit resolution by one unit
would double the dynamic range which implies that a bit resolution of 12 bits
would be adequate for most experiments since this allows for PSD up to re-
coil proton energies of Ep = 12 MeV. For the low energy pulses the FOMshows no strong dependence of the sampling frequency above 100 megasam-
2However, it is important to remember that the effective number of bits an ADC can use is
somewhat smaller than the number of bits in the full range of the ADC.
78
E (keV)20 30 210 2102
FO
M
0
0.2
0.4
0.6
0.8
1
1.2
1.4
a)
bit6 8 10 12 14
FO
M
0
0.2
0.4
0.6
0.8
1
1.2
1.4
c)
MS/s100 200 300
FO
M
0
0.2
0.4
0.6
0.8
1
1.2
1.4
e)
bit6 8 10 12 14
R (
%)
-110
1
d)
MS/s100 200 300
R (
%)
-110
1
f)
E (keV)20 30 210 2102
R (
%)
-110
1
b)
Figure 6.5: FOM (a, c, e) and R (b, d, f) for the ZCO (circles) and charge compari-
son (squares) based methods as a function of energy (a, b), bit resolution (c, d) and
sampling frequency (e, f). The solid line (a, b) is from analogue reference data using
NDE202 electronics. Filled and empty symbols (c, d, e, f) correspond to an electron
(proton) energy gate of Ee = 500–700 keV (Ep = 2200–2700 keV) and Ee = 50–
57 keV (Ep = 500–540 keV) respectively. Reprinted from [223, IV] with permission
from Elsevier.
79
ples per second (MS/s). For high energy pulses the FOM increases slowly
above 100 MS/s. The R values saturate already around 75 MS/s. The appar-ently odd frequency behaviour for low-energy pulses is due to asymmetries inthe distributions of the PSD parameters.
6.4.2 Artificial neural networks
To make full use of the information from the digitized pulse shape one canapply a method based on an artificial neural network (ANN) [240], as used inreference [241, V]. An ANN is a computational simulation of the biologicalneural network that our brains are made of [242–244]. The ANN consist of anumber of neurons arranged in layers. The first layer is the input layer, thenthere are a number of hidden layers and finally an output layer. Each neuronin a layer has its output connected to the input of the neurons in the next layerwith a certain weight. By adjusting these weights the network can be trained togenerate a desired output pattern for a certain input pattern. Furthermore, eachneuron has an internal transfer function that can be used to tune the networkfor the desired type of application. The most common transfer functions arelinear (no modification), a threshold function or a logistic sigmoid function,
P(t) =1
1+ e−t . (6.5)
The data sets from reference [223, IV] has been analysed using an ANN in
reference [241, V] where the simplest type of configuration, the feed forward
configuration, was used together with a logistic sigmoid function as transfer
function for the neurons. This gave a significant increase in separation quality,
especially in the region of small deposited energy, as shown in figure 6.6,
which typically contains the majority of the events where the parameter P is
defined as
P =√
ε2n + ε2
γ , (6.6)
where ε2n and ε2
γ are the fractions of misidentified neutrons and γ rays, respec-tively.
6.4.3 Time resolution
In a fully digital system the TOF measurement should also be determined bythe digitized pulse in the neutron detector relative to a time reference. In ref-erence [223, IV] a test of the influence of the finite sampling frequency on theachievable time resolution was made. Two digitizer channels, each recordedwith a sampling frequency of 100 MS/s, were compared and a timing parame-ter Δt21 was defined as the difference between the extracted leading edge timesof each of the two channels. The distribution of Δt21 was used as an estimate
80
Figure 6.6: Fraction of incorrectly identified γ-ray and neutron events as a function
of the deposited energy for the charge comparison method (open circles), zero-cross
over based method (filled circles) and the artificial neural network (crosses). Reprinted
from [241, V] with permission from Elsevier.
of the time resolution due to the finite sampling frequency. A FWHM of 1.7
ns was extracted and by comparing this to the achievable intrinsic time reso-
lution of a liquid scintillator detector plus PMT, typically FWHM = 1.5 ns or
larger, it was concluded that the contribution of the finite sampling frequency
to the total FWHM of the time resolution should be almost negligible already
at 200 MS/s.
To verify this conclusion one can construct an analytical expression for themeasured time distribution due to the finite sampling frequency; see figure 6.7for a definition of the parameters. Using linear interpolation between the sam-pled points for the timing measurements, the time distribution can be shown[218] to be
T = T1 −ΔTf (T1)− p0
f (T1)− f (T1 −ΔT ), with T0 < T1 < T0 +ΔT , (6.7)
where f (t) is the pulse shape from the detector.
In figure 6.8 two examples of time distributions, using a Gaussian pulse
f (T1) = exp
(− T 2
1
2σ2T1
), (6.8)
with p0 = 0.05 and p0 = 0.5 for 100 MS/s, are shown. Assuming that the
time between the sampling points and when the pulse passes the threshold is
random and uniform one can use equation (6.7) to generate the time response
due to the finite sampling frequency. As seen in figure 6.8, the finite sam-
81
Time (a.u.)-2 -1 0 1 2
Am
plit
ud
e
00.10.20.30.40.50.60.70.80.9
1
1T
T-1T0TT
0p
Figure 6.7: Sampling (circles) of a Gaussian function (dashed line) with a time be-
tween sampling points, ΔT , equal to the σ of the Gaussian function. The threshold,
p0, is crossed at a time T0. The first sampling point above the threshold occur at time
T1 and the measured time after sampling is T . Reprinted from reference [218].
pling frequency will result in a non Gaussian time distribution. Folding thesetime response distributions with a typical intrinsic time resolution of a liq-uid scintillator detector plus PMT of FWHM=1.6 ns, gives the distributions
in figure 6.9 for the different sampling frequencies used in reference [223,
IV]. As can be seen in the figure, the conclusion in reference [223, IV], that
200 MS/s sampling frequency is enough for a negligible contribution to the
time resolution, is verified for a threshold of p0 = 0.5.
82
1T/1T0 0.5 1
1T
T/
-0.4
-0.2
0
100 MS/s =0.050
p
Time (ns)0 1 2 3
Co
un
ts
0
50
100
310
1T/1T0 0.5 1
1T
T/
-0.1
0
100 MS/s =0.50
p
Time (ns)0 1 2 3
Co
un
ts
0
50
100
310
Figure 6.8: In the left panels the measured times, T , as a function of T1 are shown
for 100 MS/s and thresholds p0 = 0.05 (upper panel) and p0 = 0.5 (lower panel).
The times are normalized to σ of the Gaussian pulse. In the right panels the time
distributions due to the finite sampling frequency are shown for the parameters used
in the corresponding left panels. Reprinted from reference [218].
83
Time (ns)-4 -2 0 2 4 6 8 10 12 14
Co
un
ts
0
200
400
600
800
1000310 =0.05
0p
Time (ns)-4 -2 0 2 4 6 8 10 12 14
Co
un
ts
0
200
400
600
800
1000310 =0.5
0p
Figure 6.9: Time distributions for p0 = 0.05 and p0 = 0.5 folded with a typical Gaus-
sian time resolution of a liquid scintillator detector plus PMT with a time resolu-
tion of FWHM=1.6 ns. The widths of the distributions are decreasing with increasing
sampling frequency. The distributions are for the frequencies 300 MS/s, 200 MS/s,
150 MS/s, 100 MS/s, 75 MS/s, 60 MS/s and 50 MS/s. The widest distributions corre-
spond to a sampling frequency of 50 MS/s and the narrowest distributions to a sam-
pling frequency of 300 MS/s. Reprinted from reference [218].
84
Part III:
Discussion
7. Outlook
“ In the eyes of those who anxiously seek per-fection, a work is never truly completed – aword that for them has no sense – but aban-doned; And this abandonment, of the book tothe fire or the public, whether due to wearinessor to a need to deliver it for publication is sortof accident,”– Paul Valéry, Au Sujet du Cimetiere Marin
7.1 AGATA at LNL
The AGATA Demonstrator, described in chapter 5, is at the moment of writing
running at LNL. This is a good opportunity to study the neutron-rich rare-earthregion furhter. According to the current time schedule this campaign will con-tinue until the end of 2011, with the AGATA Demonstrator then consisting of5 ATC detectors. Taking advantage of this new detector technology at LNL
and the possibility to use a heavier beam, like 136Xe, both the detection powerand production cross-section of the experiment described in chapter 3 couldbe improved considerably. Thus, it would be possible to extend the systematicstudies of collectivity further into the neutron-rich region. Simulations showthat at γ-ray energies above a few 100 keV both the Doppler correction capa-
bility and photo-peak efficiency of AGATA is much better than of CLARA. At
a γ-ray energy of 800 keV GEANT4 simulations give a FWHM of 2.7 keV of
the full energy γ-ray peak and a tracked peak efficiency of 7.3% for AGATAat 15 cm, to be compared to 5.2 keV and 4.0%, respectively, for CLARA.
The production cross-section of dysprosium isotopes for three different
types of ion beams calculated using the grazing code [101, 102] are shownin figure 7.1. Besides the higher production cross-section, the advantageof using a 136Xe beam is the existence of isomers in the binary partners of168Dy (134Ba) and 170Dy (136Ba) with half-lives of 2.63 μs [245] and 91 ns
[246], respectively. By gating on the strongly populated delayed γ rays belowthe isomers it is possible to identify decays in the binary partners. Using theDANTE [247, 248] detector array and detection of known delayed γ rays in
AGATA (isomer tagging) the target-like fragments can be Doppler corrected
using their average velocity and by their angle of emission. Since AGATA has
no collimators the efficiency to detect delayed γ rays emitted by fragments
87
A155 160 165 170 175 180 185
Cro
ss s
ectio
n (m
b)
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 7.1: Grazing calculations of production cross-sections for dysprosium isotopes
using an 170Er target and a 48Ca beam with an energy of 230 MeV (dotted), a 82Se
beam with an energy of 460 MeV (dashed) and a 136Xe beam with an energy of
1000 MeV (solid).
not located at the target position but at various positions in the target chamber,
whose radius is about 10 cm, will be larger than with BGO shielded HPGe
arrays.
Based on the tentative identification of 170Dy in reference [112, I], the esti-mated number of counts in the 4+ → 2+ and 10+ → 8+ peaks in 170Dy, gatedby various conditions, are shown in table 7.1. Efficiencies of 3% for detectionof the projectile-like fragments in PRISMA and 30% for detection of target-like fragments in DANTE are assumed. The double and triple coincidence ef-ficiencies used for the estimates in table 7.1, were obtained as the single γ-rayefficiency squared and to the third power, respectively. The relative intensitiesof the 10+ → 8+ and 4+ → 2+ transitions in 170Dy are assumed to be 0.25,which is the same value as obtained for 168Dy in [112, I]. In the calculationsof the number of counts shown in table 7.1 six effective days of beam timeand a target thicknes of 0.5 mg/cm2. As seen, the statistics for 170Dy would beincreased significantly, and it will also be possible to study nuclei further outin the neutron-rich rare-earth region. Cross sections relative to 170Dy calcu-lated using grazing are: 1/4 for 172Dy, 1/20 for 174Dy, 1/20 for 166Gd and
1/40 for 168Gd.
88
Table 7.1: Number of counts in the γ ray #1 peak for different conditions on the
γ-ray and ancillary detectors. The 170Dy 4+ → 2+ transition has been estimated to
have Eγ ≈ 160 keV. The 170Dy 10+ → 8+ transition has been estimated to have Eγ ≈450 keV. The 136Ba condition correspond to a sum of five gates below the 10+ isomer
using isomer tagging. See text for details.
Condition γ ray #1 γ ray #2 γ ray #3 Counts
γ-PRISMA-TOF 170Dy 4+ → 2+ - - 8300
γγ-PRISMA 170Dy 4+ → 2+ 170Dy 10+ → 8+ - 400
γγγ-DANTE 170Dy 4+ → 2+ 136Ba 136Ba 4100
γγγ-DANTE 170Dy 4+ → 2+ 170Dy 10+ → 8+ 136Ba 500
It is also of interest to study the collective structure and the evolution of nu-clear deformations for nuclei above the Z = 82 shell closure. In particular, the
possibility that many superheavy nuclei may have ground state deformations
of 0.50 < β2 < 0.60, which is in the superdeformed regime, has been debatedin Refs. [249–251]. Even if high-spin spectroscopy of superheavy elements iscurrently out of experimental reach, an important step toward the understand-ing of these nuclei is the experimental investigations of heavy actinides. Thedata in this region is very sparse but also very important if one wants to doextrapolations into the superheavy element region. Furthermore, the actinideregion is also very interesting in itself as many theoretical predictions havebeen made [252] but not studied experimentally. The fission of the heavy el-ements also has an important role in the astrophysical r-process, described insection 1.2.2 and section 2.2. Previous experiments using thick targets [253]shows that it is possible to use multi-nucleon transfer reactions for studiesof actinides and references [112, 114] show that it is possible to obtain dataon target-like fragments by gating on the beam-like fragments identified in amagnetic spectrometer.
7.2 AGATA at GSI
Following the physics campaign at LNL, AGATA will move to GSI to beused with relativistic ion beams in the PRESPEC campaign. This campaign isplanned for 2012 and 2013 where AGATA will, at the end of the campaign,consist of up to 12 ATC detectors. This campaign can provide a valuable com-plement to the measurements at LNL by the possibility to establish B(E2) val-
ues to determine the electric quadrupole moments and the degree of triaxial
deformation, and thus the evolution of quadrupole collectivity, for a range of
neutron-rich rare-earth nuclei [254]. By using relativistic Coulomb excitation
it would be possible to determine the B(E2 : 0+ → 2+1 ), the B(E2 : 2+
1 → 2+2 )
and B(E2 : 0+ → 2+2 ) for even-even nuclei between 164Dy and 196Os. In such
89
A140 150 160 170 180 190
Cro
ss s
ectio
n (m
b)
-1510
-1410
-1310
-1210
-1110
-1010
-910
-810
-710
-610
-510
-410
-310
-210
-1101
10
Figure 7.2: Calculated production cross-sections of the Dy isotopic chain in the FRS
for the primary beams 176Yb (solid), 186W (dashed), 197Au (dotted) and 198Pt (dash-
dotted).
an experiment the secondary beams from the FRS would impinge on a 197Autarget at beam energies of ∼ 100 MeV per nucleon using the technique de-
scribed in references [132, 255, 256]. After the target, the nuclei could be
tagged using the LYCCA [257, 258] spectrometer.
In figure 7.2 the production cross-sections, calculated by the LISE++ code[259, 260], for a couple of primary beams with energies of 800 MeV per nu-cleon and projectile fragmentation on a 4 g/cm2 beryllium target are shown.
At a recoil energy of ∼ 100 MeV per nucleon and at forward angles, the γ-rayenergy resolution should be about 3% [132]. A problem with this kind of mea-surement, however, is the large background from bremsstrahlung radiation forγ-ray energies Eγ � 300 keV, why detailed simulations are required.
7.3 AGATA at SPIRAL2
After the construction of SPIRAL2 is completed, AGATA will be moved to
GANIL in 2014 and be used together with EXOGAM and NEDA. In this
phase it is anticipated that AGATA will consist of up to 20 ATC detectors. The
AGATA, EXOGAM and NEDA set-up will be designed to study, for example,
90
the shell structure of nuclei near the proton drip-line using fusion-evaporation
reactions [26] and the halo properties of light neutron-rich nuclei [261].The main requirements of the NEDA array are described in section 6.3.
The current focus in the development of NEDA is to decide on a geometrythat gives a high efficiency for the physics goals. The neutron-γ discrimina-tion for digital versions of the standard PSD algorithms are described in ref-erence [223]. However, a detailed comparison of the advanced digital PSDalgorithms described in section 6.4 would be of interest. Furthermore, a criti-cal parameter in the design of Neutron Detector Array (NEDA) is the rejectionquality for neutron cross-talk. A careful study of different methods for cross-talk rejection, for example by applying an artificial neural network would bevery valuable for the design of the digital electronics.
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8. Concluding remarks
I was once attending a series of lectures on nuclear structure given by profes-
sor Rick Casten, while he was visiting the University of Surrey. During one
of these lectures he told us that when he wrote the first edition of his famous
book [21] the publishing company was delaying the publication, which made
him afraid that the field would die out before the book was published. When
he wrote the second edition of his book, the publishing company was again
delaying the publication. But this time it meant that he would have more time
to add everything new that was going on. What had happened between the first
and second edition of this book? It was the development radioactive ion beam
facilities.
In recent years the technical developments of the radioactive ion beam facil-ities means that we will soon have access to radioactive ion beams of high in-tensity. In Europe the planned facilities are: NUSTAR/FAIR in Germany, SPI-RAL2 in France, HIE-ISOLDE in Switzerland and SPES in Italy. Experimentsperformed at these advanced facilities will require new advanced equipmentto study the weak signals of exotic nuclei. Two of these instruments underdevelopment in Europe are the high-purity germanium spectrometer AGATAand the neutron detector array NEDA.
AGATA is a spectrometer based on an entirely new technology called γ-
ray tracking. Instead of, as in previous spectrometers, accepting only the γrays which deposit all their energy into a crystal and to reject any γ rays scat-tered from the crystal, AGATA will have no anti-Compton shields that rejectγ-rays that scatter between crystals. By using digital electronics and pulse-shape analysis the position and energy of each γ-ray interaction will be deter-
mined. These positions and energies will allow the tracks of the γ rays to bereconstructed. From these tracks, the energies and angles of the individual γrays can be disentangled, thus dramatically increasing the energy resolution
and efficiency of the spectrometer. For this method to work satisfactorily, one
must be able to measure the interaction position of the γ rays with an accuracy
of at least five millimetres. One part of this thesis has focused on measuring
the interaction position resolution of the array using a new method based on
how well the detector can correct for the Doppler shift of the γ rays at differentdistances between the detector and the source.
NEDA is a neutron detector project under SPIRAL2 and will be used to-gether with other detector systems at this facility, for example γ-ray spec-trometers like AGATA and EXOGAM2, which is an upgrade of the existing
93
γ-ray spectrometer at GANIL. One of the physics focuses of this set-up is the
proton-rich nuclei around the double magic nucleus 100Sn. These nuclei willbe studied using fusion reactions between two nuclei, where the compoundnucleus subsequently evaporates a certain number of protons, neutrons and αparticles. NEDA will detect the evaporated neutrons. One problem with neu-
tron detection is that it is difficult differ between neutrons and γ rays in the de-
tector. New methods for discriminating between neutrons and γ rays utilizingdigital electronics has been one part of this thesis. By moving from analogueto digital electronics, it is possible to use more advanced and efficient algo-rithms for the analysis of the pulse shape from the detector. Digital versionsof the more common analogue methods have in this work yielded as good, orbetter, results as obtained with analogue electronics. A sophisticated digitalmethod, artificial neural networks, has also been applied to the experimentaldata. It is shown that the neural network can distinguish between neutrons andγ rays even more efficiently.
Furthermore, this thesis deals with the evolution of collective structure ofneutron-rich nuclei in the rare-earth metals. These are among the most col-lective nuclei that can be found in nature. In particular, the nucleus 170Dy is
expected to be the nucleus where the collective structure is maximized. This
has important implications for the astrophysical r-process, since it has been
suggested that this maximum plays an important role in the abundances of the
rare-earth elements that are created in supernova explosions. By performing
an experiment at the laboratory LNL in Italy using the magnetic spectrome-
ter PRISMA and the γ-ray spectrometer CLARA we have been able to studythe structure of the five nuclei 168,170Dy and 167,168,169Ho. These results have
been compared to calculations by the variable moment of inertia model, the
cranked shell model and the particle plus triaxial rotor model. Furthermore the
possibility to use the new radioactive ion-beam facilities to reach further into
the neutron-rich area has been discussed.
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9. Kollektiv kärnstruktur hos neutron-rika sällsynta jordartsmetaller och nyainstrument för gammaspektroskopi
Syftet med grundforskning är att förstå världen, hur den fungerar och varförden fungerar som den gör. Detta gäller oavsett vilket forskningsfält det rörsig om: sociologi, psykologi, biologi, kemi eller fysik. Det som skiljer des-sa olika discipliner åt är vilka frihetsgrader man betraktar och vilka effektivainteraktioner man väljer att arbeta med. En effektiv interaktion är ett utmärktverktyg för att behandla komplexa problem men som samtidigt döljer den un-derliggande dynamik som finns. En sociolog ser, till exempel, den mänskligainteraktionen som grundläggande utan att ta hänsyn till detaljer i de kemiskaprocesser i hjärnan som styr den mänskliga interaktionen i grunden. Inom fy-siken arbetar vi istället med att förstå några av de mest grundläggande interak-tionerna i universum, väl medvetna om att de aldrig i praktiken kan användasför att beskriva effekterna av ett kraschat förhållande eller anledningen till enrevolution i Egypten. Däremot hoppas vi att vi så noga som möjligt kunnabeskriva var vi kommer ifrån och hur det kommer sig att universum ser utsom det gör. Och historien har visat oss att denna kunskap ofta omdanar detmänskliga samhället även på makroskopiska nivåer.
När universum skapades i Big Bang skapades även de lättaste grundäm-nena, det vill säga väte och helium. Efter ett tag började vätet och heliumetklumpa ihop sig och bilda stjärnor, som fortfarande förbränner dessa ämnentill tyngre grundämnen ända upp till järn (26 protoner och 30 neutroner). Närstjärnan sedan dör blåser den ut dessa grundämnen i universum, och det ärdessa ämnen som vi ser omkring oss. Men om vi tittar runtomkring oss ser viju att det finns många tyngre ämnen än järn: koppar, silver, guld, bly och uranbara för att nämna några. Varifrån kommer då de tunga elementen från järntill uran? Tyngre grundämnen än järn bildas när vissa stjärnor på ett våldsamtsätt slutar sitt liv genom supernovaexplosioner. Supernovaexplosioner inne-bär väldigt extrema miljöer med mycket neutroner som driver kärnfysikaliskaprocesser som bildar de tyngre ämnena. Så, vi består alla av stjärndamm frånett par generationer stjärnor som dött en våldsam död. En av de processer, r-processen, som skapar dessa tunga kärnor går nästan helt genom okända ochväldigt neutronrika kärnor, ett område som brukar kallas terra incognita. Föratt förstå detaljerna i denna process är det viktigt att förstå hur dessa neutron-
rika atomkärnors struktur ser ut.
95
Inom kärnfysiken säger man att det finns två olika typer av strukturer hos
atomkärnorna, dels partikelstruktur och dels kollektiv struktur. De allra lät-
taste grundämnena beskrivs till stor del endast av sin partikelstruktur, men
när man studerar tyngre grundämnen kommer både kollektiv struktur och par-
tikelstruktur att samverka. Eftersom det är kvantmekanikens lagar som styr
atomkärnorna ordnar protoner och neutroner (nukleoner) sig i något som bru-
kar kallas en skalstruktur, där de fyllda skalen agerar som ett kollektiv och där
enskilda nukleoner har mycket liten inverkan på kärnans struktur. Detta gör att
enskilda nukleoner utanför dessa slutna skal blir de som bestämmer atomkär-
nans egenskaper. För atomkärnor som ligger långt ifrån dessa slutna skal ser
situationen helt annorlunda ut. Där är det de kollektiva frihetsgraderna som
till stor del bestämmer atomkärnans struktur medan de enskilda partiklarnas
inverkan endast kan ses genom subtila effekter i den i övrigt jämna utveck-
ling av kärnstruktur mellan olika isotoper som brukar karaktärisera kollektiva
kärnor. Atomkärnan är alltså ett tydligt exempel på ett kvantmekaniskt system
där både de kollektiva frihetsgraderna och partikelfrihetsgraderna interagerar
och måste tas hänsyn till för full förståelse av kärnmaterien.Den här avhandlingen behandlar strukturen hos atomkärnorna i de sällsynta
jordartsmetallerna, vilka är bland de mest kollektiva kärnor som man kan hitta
i naturen. I synnerhet atomkärnan 170Dy är en kärna som man kan anta ligger
precis på, eller i alla fall ganska nära, den plats där den kollektiva struktu-
ren är maximerad, precis i mitten mellan två fyllda protonskal och två fyllda
neutronskal. Detta är något som bland annat har betydelse för vår förståel-
se av den tidigare nämnda r-processen eftersom det har föreslagits att detta
maximum spelar en viktig roll i hur mycket av de olika isotoperna av säll-
synta jordartsmetaller som bildas i denna. Genom att utföra experiment vid
laboratoriet LNL i Italien med två instrument som heter PRISMA, en mag-
netisk spektrometer för tunga joner, och CLARA, en germaniumspektrometer
för detektion av gammastrålning, har vi kunnat studera strukturen hos de fem
atomkärnorna 168,170Dy och 167,168,169Ho.Problemet inför framtiden är att mer neutronrika kärnor ä så är väldigt svå-
ra att komma åt med dagens anläggningar. Det finns olika begränsningar förde kärnreaktioner vi har i vår arsenal för att skapa, och studera, dessa exo-tiska, kortlivade kärnor, något som är fundamentalt för att förstå hur tyngregrundämnen bildats och var vi kommer ifrån. Med dessa reaktioner börjarvi närma oss gränsen för hur neutronrika atomkär vi experimentellt kan nå.även om det finns mycket kvar att upptäcka och finjustera i vår förståelse aväven kända områden, är det av stor vikt att nå ut i det okända för att öka vårförståelse av kärnmaterien och hur grundämnena bildats. En av de stora be-gränsningarna är att vi bara har ungefär 250 långlivade kärnor att använda ivåra experiment. På senare år har den tekniska utvecklingen gått framåt medhög fart, vilket resulterat i att vi snart kommer att ha tillgång till radioaktiva
96
jonstrålar1 av hög intensitet vid nya experimentanläggningar i Europa: FAIR i
Tyskland, SPIRAL2 i Frankrike, HIE-ISOLDE i Schweiz och SPES i Italien.
Med andra ord kommer arsenalen på ungefär 250 långlivade kärnor uppgrade-
ras till tusentals olika kärnor med en livstid från allt från år ner till bråkdelar
av en sekund. Helt plötsligt har vi en unik möjlighet att tränga ut i helt okän-
da områden. Dessa nya avancerade anläggningar kräver dock ny avancerad
utrustning för att vi ska få ut så mycket som möjligt av dem. Två av de instru-
ment som är under utveckling i Europa är germaniumspektrometern AGATA
och neutrondetektorn NEDA.AGATA är en spektrometer som baseras på en helt ny teknologi som hand-
lar om att spåra gammastrålning. Istället för att, som i tidigare germanium-spektrometrar, mäta den strålning som lämnar all sin energi i en kristall ochförkasta all strålning som sprids till närliggande detektorer kommer AGATAatt även acceptera den strålning som sprids mellan kristallerna. För att kunnamäta strålningen på ett korrekt sätt använder vi digital elektronik och puls-formsanalys för att, genom tidsstrukturen på spänningspulsen från detektorn,ta reda på exakta positioner för gammainteraktionerna. Genom att veta dessapositioner och vilken energi gammastrålningen deponerat på de olika ställenakan man återskapa spåret, eller vilken väg gammastrålningen spridits i detek-torn. Från dessa spår kan man då sortera ut vilka energier gammastrålningenhar haft samt vilken vinkel de skickats ut i och därmed dramatiskt öka spekt-rometerns energiupplösning och effektivitet. För att denna metod ska funge-ra tillfredsställande måste man dock kunna mäta interaktionspositionerna förgammastrålningen med en noggrannhet på minst fem millimeter. En del avden här avhandlingen handlar om att undersöka just hur noggrant vi kan mätainteraktionspositionen, det vill säga vilken positionsupplösning detektorn har,genom att använda en ny metod som grundar sig i hur väl detektorn kan kor-rigera för Dopplereffektens inverkan på gammastrålningen vid olika avståndmellan detektor och stålkälla.
NEDA är ett delprojekt under SPIRAL2 och kommer att användas tillsam-mans med andra detektorsystem vid denna anläggning. I första hand är dessagermaniumspektrometrar AGATA och EXOGAM2, som är en uppgraderingav den befintliga germaniumspektrometern på GANIL. Ett av de fysikområ-dena som man kommer att fokusera på vid denna uppställning är protonrikakärnor runt den dubbelmagiska kärnan 100Sn. Detta område kommer att stu-deras med hjälp av fusionsreaktioner mellan två kärnor där den sammansattakärnan evaporerar ett visst antal protoner, neutroner och alfapartiklar. NEDAkommer här att detektera de evaporerade neutronerna. Ett problem med dettaär dock att det är svårt att i detektorn se skillnad på vad som är neutronstrål-ning och vad som är gammastrålning, varför nya metoder för att avgöra dettahar varit en del av denna avhandling. Genom att övergå från analog till digitalelektronik kan man använda mer avancerade och effektiva algoritmer än tidi-
1Till skillnad från den totalt felaktiga och missvisande termen ”radioaktiv strålning” som media
använder för att beskriva joniserande strålning, som i verkligheten inte är det minsta radioaktiv.
97
gare för att analysera spänningspulserna från detektorn. Digitala versioner av
de vanligaste analoga metoderna har i detta arbete visat sig ge minst lika bra,
eller bättre, resultat som den tillgängliga analoga elektroniken. En sofistikerad
digital metod, artificiella neurala nätverk, har också applicerats på testdata och
det har visats att denna kommer att göra det möjligt att ännu effektivare skilja
på neutroner och gammastrålning i neutrondetektorer av den typ man planerar
att använda i NEDA-projektet.
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10. Acknowledgements
First of all, I would like to thank my supervisors, Johan Nyberg and Ayse Ataç.
You have managed to both steer me in the right direction when needed and at
the same time you have given me a lot of scientific freedom and supported me
when I wanted to follow my own path and my own ideas.The third person I would like to thank is professor Paddy Regan, who has
been a valuable collaborator in many aspects of this work.I would also like to thank Smålands nation and Anna Maria Lundins fund
for financing my trip to and participation in the 180Yb experiment at iThemba
LABS, Cape Town.For parts of this thesis I have also benefited by the hard work of the un-
dergraduate students that have been working in our group. Big thanks to both
Ali Al-Adili and Paula Salvador Castiñeira. I would also like to thank all the
other collaborators I have worked with to obtain the results presented in this
thesis. This, of course, includes everyone in the 170Dy, AGATA and NEDAcollaborations but I would like to especially mention the following people:Gry Tveten, a very valuable colleague for the data analysis; Jose Javier Va-liente Dobòn for valuable support in all parts of this thesis and the coordina-tor for the NEDA project; Ryan Kempley for discussions about AGATA dataand the movie night at LNL; Grzes Jaworski for a good time at many differ-
ent places in Europe and James Ollier for all discussions about rigid nuclear
structure and sharing the apartment at iThemba LABS.In order for the thesis to be as well written as possible I have also had help
from the proofreaders of my manuscript. These people are Henrik Jäderström,
Kristofer Jakobsson and Mikael Höök. Great comments from all of you!
Furthermore, that I even got started in the field of experimental nuclearphysics is much thanks to the MEDLEY crew: Chai Udomrat, RiccardoBevilacqua, Vasily Simutkin, Stephan Pomp, Masateru Hayashi, Jan“Bumpen” Blomgren, Alexander Prokofiev and Yukinobu Watanabe.
I would also like to thank all former and present doctoral candidates, master
students, project workers and senior physicists that have worked this depart-
ment. I will not mention you all by name, in case I forget someone, but you
are all included. A few colleagues have been extra close to me, however, and
these I would like to mention: Elias Coniavitis, Oscar Stål, Martin Flechl, So-
phie Grape, Erik Thome, Olle Engdegård, Magnus Johnson, Peder Eliasson,
Arnaud Ferrari, Camille Bélanger-Champagne, Claus Buszello, Henrik Jäder-
99
ström, Henrik Petrén, Karin Schönning, Mattias Lantz, Bengt Söderbergh and
Kristofer Jakobsson.There are also a couple of people who have been a very valuable resource
in keeping track of both the univerisy administrative procedures and computerissues, these are Inger Ericsson, Annica Elm, Ib Kôersner and Teresa Kupsc.
My time as a doctoral candidate has not, however, only been about research.
I have also been lucky enough to have the opportunity to work together with
many other people on issues regarding doctoral candidates in every level from
the faculty to the European Union. I have had the great opportunity to work
together with the following great people in the UUS party: Caroline Erland-
son, Johan Gärdebo, Jonas Boström, Klas-Herman Lundgren, Karin Nord-
lund, Michel Rowinski, Michael Jonsson, Mathias Johansson, Niclas Karls-
son, Kristina Ekholm and many more that I have probably forgot to mention.
The PhD Students’ Council of the Faculty of Science and Technology hasalso done a great job for our local doctoral candidates, especially Marta Kisiel,Adrian Bahne, Kristofer Jakobsson, Riccardo Bevilacqua and Seidon Alsaody.During the year I spent in the doctoral council at the Uppsala Student UnionI have also very happy to have gotten to know Marta Axner, Carl Nettelbladand Per Löwdin.
Through the Swedish National Union of Students I am also happy to have
had the opportunity to collaborate with Odd Runevall, Moa Ekbom, Martin
Dackling, Kristina Danilov, Melissa Norström and Lars Abrahamsson. And
also the collaboration with the undergraduate representatives Klas-Herman
Lundgren, Robin Moberg, Thomas Larsson, Beatrice Högå and Elisabeth
Gehrke. Finally I would like to acknowledge the wonderful work done by
the people in the European Council of Doctoral Candidates and Junior
Researchers, then especially the work by Izabela Stanisławiszyn, Ing-Marie
Ahl, Marisa Alonso Núñez and Sverre Lundemo.To Klas-Herman Lundgren and Hanna Victoria Mörck. Your friendship dur-
ing this time of trial has been valuable beyond words. Thank you for feeding
me at late nights, playing video games with me and putting up with my some-
times edgy and emotionally instable state of mind during the period of my
thesis writing.There is also other friends that have been very valuable to me. Karin Neil
Persson, Jenny Arnberg and Henrik Jäderström have been a great support. Carl
Lowisin, Mikael Höök and Caroline Marjoniemi have also been very good
friends, both in China and in Sweden. Fredrik Gunnarson and his never-ending
enthusiasm for computerized sound. And my great dance teacher Emma Rå-
dahl. I would also like to thank my flatmates Tova Dahné and Naomi Reniutz
Ursoiu.During these years, I have also had the opportunity to travel the world and
meet many people that have influenced my life and inspired me. Unfortu-
nately, listing every one of you would just end up as an exercise in futility.
But I will go for it anyway: Yao Hailin for showing me the beauty of China;
100
Alma Joensen for introducing me to Faroese-Danish-Icelandic cuisine; Car-
mens Dobocan for the dinner and the broccoli at the old castle in Krakow;
Diana Zubko for exploring the Institut Français with me; Liliya Ivanova for
just being a nice person; Elif Ince for the wonderful boat ride through Istanbul;
Madeleine Laurencin for dancing waltz with me in Vienna while the demon-
strations were raging outside; Khotso Mokoenya for keeping me company
when driving through Lesotho; Amineh Kakabaveh, my favourite mountain
partisan and MP and Noriaki Oba for the Japanese-Swedish food crossover
experiments involving surströmming and natto fermented soy beans.Sist men inte minst vill jag också tacka Leena Söderström-Blom och Tage
Blom.
101
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Contribution to the papers
Paper I
This paper concerns the identification of the yrast band of 168Dy up to spin
10+ and the tentative identification of the 4+ → 2+ transition in 170Dy. Theexperiment was carried out with the PRISMA and CLARA set-up at LNLwhere I participated in the data taking. For this paper I was responsible for theanalysis of the experimental data and, together with F. R. Xu and J. Y. Zhu, theinterpretation of the data in terms of TRS calculations. I was also responsiblefor writing the paper.
Paper II
The second paper is from the same PRISMA and CLARA experiment as Pa-per I. For this paper, both G. M. Tveten and I were responsible for the dataanalysis, carried out in parallel to the data analysis in Paper I in order to cross-check the results. The paper concerns the identification of the yrast bands of167Ho and 169Ho, as well as excited states in 168Ho. The interpretations of theresults was made in collaboration between G. M. Tveten and myself, wheremy main responsibility was the TRS calculations and for writing parts of thepaper.
Paper III
This paper presents simulations of, and results from, the first commissioningexperiment of the AGATA detector array in LNL. The experimental set-upand data taking was carried out by many people from the AGATA collabora-tion. I have been responsible for the simulations of the experiment, togetherwith A. Al-Adili that was a masters student in our group at the time, the dataanalysis, the interpretation of the results and for writing the paper.
121
Paper IV
The fourth paper is about discrimination between neutrons and γ rays in liquidscintillator detectors. The experiment was prepared by R. Wolters when hewas an Erasmus student in our group. I have had the main responsibility fordata taking, data analysis and interpretation of the results presented in thispaper and writing the paper.
Paper V
The final paper is an extension of Paper IV, where the data has been reanalysedby E. Ronchi and myself using an artificial neural network. E. Ronchi was atthe time a doctoral candidate in applied nuclear physics working with plasmadiagnostics for the nuclear fusion group. The analysis was carried out in acollaboration between E. Ronchi and myself, while E. Ronchi was responsiblefor the model set-up, the interpretation of the results and the writing of thepaper.
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