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Lifetime Measurements of Excited States in the Semi-Magic Nucleus

94

Ru

AYŞEGÜL ERTOPRAK

Licentiate Thesis in Physics

Stockholm, Sweden, 2017

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TRITA-FYS 2017:73 ISSN 0280-316X

ISRN KTH/FYS/–17:73–SE ISBN 978-91-7729-601-0

KTH, AlbaNova University Center SE-106 91, Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläg- ges till offentlig granskning för avläggande av Teknologie Licentiatexamen torsdag den 23 November 2017 klockan 14.00 i Sal FB55, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm.

© Ayşegül Ertoprak, November 2017 Tryck: Universitetsservice US AB

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iii

Abstract

Lifetimes of highly excited states in the semi-magic (N=50) nucleus94Ru were deduced from an analysis of the Doppler broadened transition line- shapes. Excited states in 94Ru were populated in the 58Ni(40Ca, 4p)94Ru fusion-evaporation reaction at the Grand Accélérateur National d’Ions Lourds (GANIL) accelerator complex situated in Caen, France. Doppler Shift At- tenuation Method (DSAM) lifetime analysis was performed on the Doppler broadened peaks in energy spectra from γ-rays emitted while the residual nu- clei were slowing down in a thick 6 mg/cm2metallic58Ni target. In total eight excited-state lifetimes in the angular momentum range I = (13 − 20)~ have been measured, five of which were determined for the first time. The deduced corresponding B(M1) and B(E2) reduced transition strengths are discussed within the framework of large-scale shell model calculations.

Keywords: nuclear structure, lifetime measurements, Doppler Shift At- tenuation Method, thick target

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iv

Sammanfattning

Livstider för exciterade tillstånd i den semimagiska (N = 50) atomkär- nan 94Ru har uppmätts från en analys av Doppler-breddade övergångslin- jer. Exciterade tillstånd i94Ru har populerats i58Ni(40Ca, 4p)94Ru fusion- evaporationsreaktioner vid partikelacceleratorn Grand Accélérateur National d’Ions Lourds (GANIL), Caen, Frankrike. Livstidsanalys utfördes med hjälp av Dopplerskiftsattenueringsmetoden (DSAM) på linjer i energispektra upp- mätta från gammastrålning som emitterades medan de högt exciterade atom- kärnorna bromsades ned i ett homogent metalliskt strålmål av 6 mg/cm2 tjocklek. Mätmetoden har verifierats med hjälp av data från en mätning utförd med en alternativ metod. Livstider för sammanlagt åtta exciterade tillstånd i spinn-intervallet I = (13 − 20)~ har uppmätts, varav fem bestämdes för första gången. Ur dessa har härletts B(M1) och B(E2) övergångssannolikheter vilka diskuteras inom ramen för storskaliga skalmodellsberäkningar.

Keywords: kärnstruktur, livstidsmätningar, Dopplershiftattenuering, tjockt strålmål

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v

Acknowledgements

First of all, I would like to thank my supervisor Prof. Bo Cederwall, for giving me the opportunity to work with KTH nuclear physics group and for his scientific sup- port. Many thanks to Prof. Roberto Liotta who has helped me not only to improve my knowledge on the nuclear theory but also proof-reading this thesis. I would like to thank Prof. em Arne Johnson, Prof. Ayşe Ataç Nyberg, Prof. Ramon Wyss, Doc. Chong Qi and Dr. Torbjörn Bäck. I am also grateful to Dr. Maria Doncel for her positive attitude, advice, and help with the proof-reading of the papers and this thesis. I would also like to thank my colleagues Özge Aktaş, Sara Changizi, Dr. Xin Guan, Daniel Karlsson, Dr. Malin Klinjeford, Dr. Hongna Liu, Xiaoyu Liu, Sanya Matta, Karl Sallmén, Pranav Subramaniam, Dr. Debora Trombetta and Wei Zhang.

Finally I would like to thank my family for their continual encouragement and support. None of this would have been possible without your unconditional love.

Ayşegül Ertoprak, Stockholm, 2017

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Contents

title-1

Contents vi

List of Figures viii

List of Tables x

List of Acronyms xi

1 Introduction to Nuclear Structure and Scientific Motivation 1

1.1 What is Nuclear Physics? . . . 1

1.2 Nuclear Shell Model . . . 2

1.3 Shell model parameters in the100Sn region . . . 4

2 Experimental Techniques 7 2.1 Heavy-Ion Fusion-Evaporation Reactions . . . 7

2.2 Experimental set-up . . . 8

2.2.1 EXOGAM . . . 9

2.2.2 Neutron Wall . . . 10

2.2.3 The DIAMANT charged particle detector . . . 11

2.2.4 Trigger Condition . . . 12

3 Data Analysis 13 3.1 Measurement of Nuclear Level Lifetimes . . . 13

3.2 The Doppler Shift Attenuation Method . . . 13

3.2.1 Stopping Powers . . . 15

3.2.2 LINESHAPE software package . . . 15

3.2.3 Cross-section dependence . . . 16

3.2.4 DSAM Lineshape Analysis . . . 18

4 Results 25 4.1 Results for the level lifetimes . . . 25

vi

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CONTENTS vii

4.2 Reduced Transition Probabilities and Comparison with Large Scale Shell Model Calculations . . . 26 5 Summary of Papers and the Author’s Contributions 29 5.1 Paper I . . . 29 5.2 Paper II . . . 30 5.3 Author’s Contributions . . . 30

Bibliography 31

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List of Figures

1.1 Shell model orbitals important for the description of nuclei in the region around100Sn, and single particle energies (SPE) relative to100Sn. The SPE values given in the right-hand part of the figure are in MeV. The differences between the values for protons and neutrons are mainly due to the electromagnetic repulsion between protons. Levels representing the SPE energies are plotted symmetrically with respect to the center of the N = 50 and Z = 50 shell gaps. . . 5 2.1 Schematic representation of the58Ni(40Ca,xnyp) fusion-evaporation re-

action. . . 8 2.2 Left: Photograph of the EXOGAM Ge-array (right side of the yellow

support frame) and Neutron Wall detector array (left side). Right:

Close-up photograph of the Exogam detectors taken with the Neutron Wall and the downstream hemisphere of the target chamber and beam- line retracted. An empty target frame is mounted at the center of the target chamber (Photos by B. Cederwall). . . 9 2.3 Left: Neutron Wall seen from downstream the target, here consisting of

50 liquid scintillator detectors. Right: DIAMANT detector array (Photos by A. Ertoprak). . . 10 2.4 A schematic drawing of the DIAMANT detector array (courtesy of

B. M. Nyakó) . . . 11 3.1 Basic principle behind the Doppler Shift Attenuation Method for lifetime

determinations of short-lived nuclear states. Due to the slowing down of the fusion products in the target, the velocity at which the γ-rays from excited states are emitted (magenta, blue, and red, respectively) decreases on average with the emission time. . . 14 3.2 Experimental fusion excitation function for the 40Ca +58Ni reaction,

taken from D. Bourgin et al. . . 16 3.3 The intrinsic asymmetry of the peak shape for completely stopped 756 keV

γ-ray transition. . . 20 viii

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List of Figures ix

3.4 Detailed view of the level scheme obtained from the present work. Levels with measured lifetime are highlighted in blue. . . 22 3.5 Experimental γ-ray energy spectra and fits to the Doppler shifted shapes

for the 1898 keV (13+1 →12+1) transition. The spectra were produced by setting a narrow gate on the stopped component of the 725 keV transition decaying from the 4717 keV state in the Eγ1−Eγ2 coincidence matrix. Here Eγ1 refers to energies of γ-rays detected at 90 or 135 with respect to the beam direction while Eγ2 refers to energies of any γ-rays detected in coincidence with these γ-rays. . . 23 4.1 Comparison between the experimental and theoretical transition strengths

for M1 and E2 transitions in94Ru. The top panel shows B(M1) values and the bottom panel gives the B(E2) values. The transition labels show the initial and final states for the transitions. Experimental values are marked by circles with error bars. . . 28

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List of Tables

3.1 Production cross section as a function of target depth, divided into ten bins, the ten thickness boundaries for defining the bins and the rela- tive yield in each bin. The cross section, beam energy in the center of mass frame, and the laboratory frame are given in the first, second and third column respectively. The target thickness boundaries are given in column 4. . . 17 3.2 Peak asymmetries fitted for selected γ-ray transitions in94Ru, see text.

The peak shape asymmetry parameter R = 15 is proportional to the asymmetry of the peak shape (see text). The spin-parity assignments and γ-ray energies are given in the first and second column, respec- tively. The Beta parameter values (Beta is the skewness parameter for the skew Gaussian component and varies slowly with γ-ray energy) are shown in the column 3. Previous lifetime values or limits for the initial state are given in the column 5. The smooth step function components were fixed to zero during the fitting procedure. Uncertainties are given within parentheses. The asymmetry parameter R is found to be con- stant around R = 15 for peaks corresponding to transitions known to occur when the nuclei are at rest in the target, while transitions from states which have previously been reported [11] to have short lifetimes of the order of picoseconds and no slow feeding systematically have R values of 20 and higher. . . 18 3.3 Relative γ-ray intensities for 94Ru measured in the present work. . . 21 4.1 Lifetimes of excited states in94Ru from the present work in comparison

with previously reported values and limits. . . 25

x

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List of Acronyms

BGO bismuth germanate

CsI cesium iodide

DSAM Doppler Shift Attenuation Method FWHM full width at half maximum

GANIL Grand Accélérateur National d’Ions Lourds

ho Harmonic oscillator

HPGe high-purity germanium

QCD quantum chromodynamics

QED quantum electrodynamics

RDDS Recoil Distance Doppler Shift SPE single particle energies

SRIM Stopping and Range of Ions in Matter

TOF time-of-flight

ZCO zero-crossover

WS Woods-Saxon

xi

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Chapter 1

Introduction to Nuclear Structure and Scientific Motivation

1.1 What is Nuclear Physics?

The main aim of nuclear physics research is to achieve a better understanding of the structure and dynamics of atomic nuclei. The atomic nucleus is a quantum mechanical system which contains strongly interacting protons and neutrons. Such systems constitute more than 99.9% of the mass of the “ordinary” matter of the Universe (which, incidentally, is only a relatively small fraction of its total mass energy; the missing mass and energy being called dark matter and dark energy, respectively).

In the 1960’s and 70’s a theory emerged that could explain the origin of the strong force acting within hadrons, i.e. between the constituent quarks. That is Quantum Chromo Dynamics (QCD). Nucleons are not the elementary particles that they were thought to be before QCD was invented. As is now well known, a nucleon consists of quarks and gluons. The quarks are the elementary particles in- teracting via the strong force in the Standard Model and they are massive fermions.

The gluons are massless bosons that transmit the force, binding quarks together.

There is a resemblance between QCD and the field theory that explains quantum electrodynamics, namely QED. Here the electrons are massive fermions that inter- act through the exchange of the massless bosons which are the photons. However there are several profound differences between QCD and QED. From the nuclear physics point of view an important difference is that the QED coupling constant α is indeed a constant with value α ≈ 1/137, while in QCD the coupling constant is energy dependent. Thus, at high energy, which corresponds to short distances, the quarks interact weakly, while at low energy, which corresponds to large distances, the interaction among quarks is strong. This explains why no free quarks have been observed. At the small distances inside a nucleon, i.e. . 1 fm, the quarks are nearly free, but across the nuclear volume, which is a “large” distance of several fm

1

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2

CHAPTER 1. INTRODUCTION TO NUCLEAR STRUCTURE AND SCIENTIFIC MOTIVATION in the context of QCD, the interaction among quarks is very strong. As a result, the energy needed to extract a quark from a nucleon is so large that, instead of the quark becoming free, it is the vacuum which is excited producing a quark-antiquark pair, i. e. a meson, like the pion. Hence, the manifestation of the strong force be- tween nucleons in the nucleus can be viewed as an exchange of mesons of different character.

From a nuclear physics viewpoint the most important drawback of QCD is that it is not known how to solve exactly the QCD equations of motion regulating the motion of quarks. The only first-principles approach available is to attempt to solve the equations numerically, using what is called lattice QCD. This requires complicated algorithms and very powerful computers. However, the most compli- cated system that could be solved in this fashion is just a free nucleon. It is for this reason that the study of low-energy nuclear properties is performed by means of effective forces which are easier to apply and may explain some nuclear features with reasonable accuracy. The first of such features is that the nuclear force is of a very short range. Therefore, nucleons inside a nucleus move following relatively long mean free paths. In other words, in first approximation one may considered the dynamics of nucleons inside a nucleus as that of free spin 1/2 particles obeying the Pauli principle (this is called Fermi gas). Since the nucleons are tightly bound inside the nucleus the most simple assumption is to consider the free nucleons mov- ing inside a schematic potential like a square well or, more realistically, inside a harmonic oscillator or Woods-Saxon potential. This is the basis of the nuclear shell model. Within this extremely simple picture it was possible to explain the lowest of the so-called “magic” numbers which is discussed in the following section.

1.2 Nuclear Shell Model

Properties of the atomic nucleus, such as an increase in nucleon separation energies for nuclei with specific number of nucleons, reveal that nuclei with proton or neutron numbers equal to 2, 8, 20, 28, 50, 82, 126 are more stable than predicted by the Bethe-Weizsäcker semi-empirical mass formula [1]. Already referred to above, these numbers are known as “magic” numbers and their existence is one of the strongest motivations for the development of the shell model concept.

The Hamiltonian of a nuclear system can be written as:

H = T + V =

A

X

k=1

Ti+X

i6=j

Vij (1.1)

where Tidescribes the kinetic energy of each nucleon and Vij describes the interac- tion between valence nucleons and where three-body interactions and higher-order interactions are neglected. Assuming that a central single-particle field U exists so that H = T + U + V − U with v = V − U a small (residual) interaction and

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1.2. NUCLEAR SHELL MODEL 3

H0= T + U a single-particle Hamiltonian, Eq.(1.1) becomes,

H = H0+ v (1.2)

The single-particle nuclear shell model assumes that neutrons and protons move under the influence of H0, obeying the Pauli exclusion principle [2]. The corre- sponding nuclear wave function, that is the wave function of the A free nucleons in the nucleus, is the antisymmetrized product of all the A independent-particle eigenstates. This ensures that the nuclear wave function obeys the Pauli principle.

It is a “Fermi” gas.

The central potential H0 is often taken to be an Harmonic oscillator (ho) po- tential. With appropriate choice of the parameters defining the ho potential one can described the lowest magic numbers, as already mentioned. It was found that to get all the known magic numbers one has to include a spin-orbit interaction.

A potential which is more realistic than the ho one is the Woods-Saxon potential (WS) [3] which has the form,

VW.S.(r) = −V0[1 + exp(r − R0

a )]−1 (1.3)

Here, r = |r| is the absolute value of the radius vector from the origin centered at the nucleon center of gravity and hence equation 1.3 describes a spherical potential.

But even with the WS potential only the lowest magic numbers can be explained.

Also in this case one has to add a spin-orbit potential of the form f(r)~l·~s, as was proposed by Goeppert-Mayer and Jensen in 1949 [4] and for which they received the Nobel Prize in Physics in 1963 (together with Wigner). Nearly simultaneously the shell model was also introduced by Haxel, Jensen and Suess [5].

The disadvantage of using the Woods-Saxon potential is that, in contrast to the ho potential, the single-particle wave functions can not be derived analytically.

However in both cases one can characterize the orbits by the quantum numbers n, l, j; i. e. the principal quantum number, the orbital angular momentum and the total angular momentum j = l + s, s being the nucleonic spin. The spin- orbit coupling term splits the levels through the term j = l ± 1/2, which presses down high-j and lifts up low-j spin-orbit partners [2]. This “single-particle” shell model now reproduces all the known magic numbers correctly. Besides predicting the correct magic numbers, the shell model also correctly predicts the spins and parities of many excited states in nuclei across the Segré chart. We now know that these magic numbers are not constants of Nature. They can disappear and appear as we move far away from the beta stability line in the nuclear chart, in until recently unexplored territory.

When many valence particles (i. e. particles outside the magic core) are con- tained in a nucleus, then the shell-model Hamiltonian (1.2) has to be solved. Most importantly one has to choose a proper residual interaction v. A large number of such effective interactions have been used during the last decades and it is one of the tasks of modern nuclear theory to better capture the essential physics of the

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4

CHAPTER 1. INTRODUCTION TO NUCLEAR STRUCTURE AND SCIENTIFIC MOTIVATION nucleon-nucleon interaction into effective residual forces in large scale shell model calculations.

The spherical shell model works well at predicting nuclear properties for a select group of nuclei which are near closed shells. Nuclei near closed shells generally do not have enough valence particles to facilitate collective motion at low energies, and their structure can be interpreted in terms of single particle excitations. As valence nucleons are added, the shell model wave functions become complicated and are more difficult to model. As indicated above, the simple shell model was developed with a spherically symmetric potential. The expected energy levels corresponding to the solutions of the Schrödinger equation for such a system would shift if the shape of the potential is changed. Such “nuclear deformation” is not considered in this work.

Exotic, proton-rich nuclei form a “laboratory” in which many aspects of nuclear structure can be tested. In general, as they contain similar numbers of protons and neutrons, the two kinds of nucleons are placed in similar or identical states, i.e.

“orbitals” within the shell model framework. Hence, their spatial wave functions to a large extent overlap, leading to simplifications and interactions which often cannot be observed in other parts of the nuclide chart. Especially interesting are the studies of nuclei in the vicinity of double closed shells, so called doubly-magic nuclei, in which the observable properties can be interpreted as a result of a motion of a few valence particles situated outside a more or less inert, “magic”, core. In a first approximation the valence particles can be regarded as moving independently, to second order they interact with each other via residual two-body interactions, and to third order possible excitations of the core are included. The interplay of such single-, two-, and many-body effects studied in the relatively speaking “simple objects” that constitute nuclei with just a few valence particles provides rich oppor- tunities to stringently test nuclear models. At the crossing points of the N=Z line with the major shell closures, particularly interesting regions of the nuclear chart appear. The arguably most interesting such region is that around the doubly- magic nucleus 100Sn, which is the heaviest self-conjugate, doubly-magic nucleus that is bound.

1.3 Shell model parameters in the

100

Sn region

In order to further our understanding of nuclear structure in the vicinity of100Sn, a wealth of experimental information on excited states of nuclei in this region should be collected. This will enable verification of the fundamental parameters of the model: the single particle energies (SPE) and residual interactions between valence nucleons. The shell model orbitals important for the description of nuclei in the region of 100Sn are presented in Fig. 1.1. A recent review of the experimental and theoretical achievements in the region has been published in Ref. [6].

γ-ray spectroscopy of nuclei close to100Sn can provide information on the prop- erties of the doubly-magic N=Z=50 gap, even though direct experimental studies

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1.3. SHELL MODEL PARAMETERS IN THE100Sn REGION 5

Figure 1.1: Shell model orbitals important for the description of nuclei in the region around100Sn, and single particle energies (SPE) relative to100Sn. The SPE values given in the right-hand part of the figure are in MeV. The differences between the values for protons and neutrons are mainly due to the electromagnetic repulsion between protons. Levels representing the SPE energies are plotted symmetrically with respect to the center of the N = 50 and Z = 50 shell gaps.

of excited states in100Sn are not yet possible. Experiments in this region of the nuclidic chart are technically extremely challenging since the nuclei of interest can only be produced with small cross sections. However, thanks to dramatic develop- ments in detection sensitivity in recent years, significant progress has been made in elucidating the structure of nuclei approaching 100Sn. In addition, due to the rapid increase in the performance of modern supercomputers, it is gradually be- coming possible to model excitations extending beyond the standard shell model valence space with a100Sn core. It is now also possible to investigate directly the N= Z = 50 shell gaps by calculating the properties of “core excited” states at high excitation energy and comparing with their experimental properties.

Recent results on excited states in96Pd and97Ag [7, 8] as well as on E1 transi- tions in the N = 50 isotones94Ru [9] and95Rh [10] illustrate how this is possible and indicate that the odd-parity orbitals 1p3/2and 0f5/2 couple to the excitations of the doubly-magic N = Z = 50 core, and are necessary for the description of core-

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6

CHAPTER 1. INTRODUCTION TO NUCLEAR STRUCTURE AND SCIENTIFIC MOTIVATION excited states. Measurements of excited state lifetimes, which is a powerful way to characterize nuclear wave functions, has previously been used, e.g. by Jungclaus et al. Ref. [11] to characterize excited states, including core excited states in the semi-magic nuclei95Rh and94Ru using the Recoil-Distance Doppler Shift method.

This method uses a device, a so called plunger, to measure the differential change of Doppler shifted γ-ray lines as a function of target-stopper distance.

This licentiate thesis is focused on the measurement of somewhat shorter life- times of medium-high-spin states in the semi-magic nucleus 94Ru using a special version of the Doppler Shift Attenuation technique and uses large-scale shell model calculations to compare between experiment and theory. “Semi-magic” here means that the neutron number N = 50 is “magic”, and hence the nucleus should exhibit a structure that is easier to describe theoretically and therefore is a good testing ground for theoretical models, in particular with respect to the core-excited states starting at angular momentum beyond I = 12~.

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Chapter 2

Experimental Techniques

In this chapter, some experimental techniques that have been used to investigate the excited states of the94Ru nucleus are described. In the experiment outlined in this thesis nuclei are populated in high-lying states via fusion-evaporation reactions.

γ-rays emitted during the de-excitation of the residual nuclei were detected using highly efficient germanium detectors while charged particles and neutrons were detected in order to help to identify the fusion products.

2.1 Heavy-Ion Fusion-Evaporation Reactions

Excited states in nuclei are commonly populated and studied using fusion-evaporation reactions, in which highly excited compound nuclei are created through collisions between lighter nuclei. The first step is the formation of a compound nucleus.

Within typically 10−18 s, light particles such as protons, neutrons and alpha par- ticles evaporate from the “hot” compound nuclei, followed by γ decays. The prob- ability of neutron emission from the compound nucleus is, due to the absence of Coulomb barrier, higher than the emission of protons and α particles. Massive particles, like neutrons, carry away large amounts of energy but only few units of angular momentum, hence the residual nucleus can stay in a high-angular momen- tum state. When the excitation energy is below the separation energy of neutrons and the Coulomb barrier for charged particles the residual nucleus may de-excite by emitting a cascade of γ transitions connecting different excited states towards the ground state. Typically, the beam currents for the fusion-evaporation exper- iments are of the order of a few particle nano amperes (pnAs), i. e. a few times 6.109 ions per second. The energy of the beam should be sufficiently high to ex- ceed the Coulomb barrier and produce the desired excitation energy of the final nuclei. A detailed description of the fusion-evaporation process can be found in Refs. [12, 13, 14].

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8 CHAPTER 2. EXPERIMENTAL TECHNIQUES

Figure 2.1: Schematic representation of the 58Ni(40Ca,xnyp) fusion-evaporation reaction.

2.2 Experimental set-up

The experiment was set up in the G2 experimental cave at the GANIL accelerator facility. The detectors consisted of the DIAMANT [15, 16] charged particle detector in conjunction with the Neutron Wall detector [17] for the detection of neutrons and the EXOGAM Ge γ-ray spectrometer array [18] and was employed in the experiment resulting in paper I and paper II. The main aim of the experiment was to study 96Cd and other extremely neutron deficient nuclei in the region of nuclear masses A = 90 − 100, close to100Sn. One particular goal was to test shell model predictions in terms of the spin-aligned coupling scheme [19]. In this work, we focus on the “side channel” corresponding to94Ru. Excited states in94Ru were populated via the 58Ni(40Ca, 4p)94Ru fusion-evaporation reaction. A schematic view of the production of the compound nucleus98Cd and possible reaction channels after particle evaporation is depicted in Fig. 2.1.

The40Ca ions were accelerated to an energy of 150 MeV by the CSS1 cyclotron, degraded to 128 MeV in a thin Ta foil, and used to bombard target foils consisting of 99.9% isotopically enriched 58Ni with areal density of 6 mg/cm2, sufficient to stop the fusion products of interest. The beam intensity varied between 5 - 10 pnA with an average of 7 pnA during 14 days of irradiation time. The neutron detectors were placed in front of the target while the other angles were used for the Ge detectors as shown in Fig. 2.2.

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2.2. EXPERIMENTAL SET-UP 9

Figure 2.2: Left: Photograph of the EXOGAM Ge-array (right side of the yellow support frame) and Neutron Wall detector array (left side). Right: Close-up photograph of the Exogam detectors taken with the Neutron Wall and the down- stream hemisphere of the target chamber and beamline retracted. An empty target frame is mounted at the center of the target chamber (Photos by B. Cederwall).

2.2.1 EXOGAM

The γ-rays produced in the reactions were detected by the EXOGAM spectrometer array [18], consisting of 11 Compton-suppressed segmented HPGe clover detectors placed in two rings; seven at 90 and four at 135 relative to the direction of the beam. The segmentation of the clover detectors is useful for reducing Doppler broadening when the emitting nucleus recoils with a large velocity since it allows a more precise determination of the interaction position of the γ-rays. Each seg- mented Ge detector is surrounded by an escape suppression shield which rejects the events when it detects γ-rays scattered out of the Ge crystals. The escape suppres- sion shields are typically made of a high-Z material with high density, in the case of EXOGAM Bismuth Germanate, Bi4Ge3O12 (BGO) and thallium doped Cesium Iodide (CsI(Tl)). This technique is used to minimize the Compton background and therefore to improve the γ-ray spectrum quality. Photos of the experimental set-up are shown in Fig. 2.2 and 2.3. The data were collected in files on hard drives, one every few hours, and sorted in (Eγ − Eγ) coincidence matrices according to the detector angle. The energy calibration of each Ge detector was performed using standard radioactive sources (60Co,152Eu). The γ-ray spectrum of each detector was checked off-line and carefully aligned file-by-file in order to eliminate the effects of gain shifting during the experiment.

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10 CHAPTER 2. EXPERIMENTAL TECHNIQUES

2.2.2 Neutron Wall

Neutron Wall [17] is an array of 50 closely packed organic liquid-scintillator detec- tors constructed for the detection of fast neutrons. The array covers a 1π solid angle in the forward direction. The distance between the target and the detectors is approximately 50 cm and the thickness of each detector is 15 cm. In the present experiment, it was taken great care to discriminate between neutrons and γ-rays in Neutron Wall in order to select the most exotic neutron-deficient species which were formed after emission of one or two neutrons from the compound nucleus. For the neutron-γ discrimination the zero-crossover (ZCO) method was used in combi- nation with the time-of-flight (TOF) of γ-rays and neutrons [17, 20, 21]. The pulse shape discrimination was completed by putting two-dimensional gates on the TOF vs zero-crossover time matrix sorted for each detector. The Neutron Wall detection efficiency is typically 25% for detection of a single neutron.

Figure 2.3: Left: Neutron Wall seen from downstream the target, here consisting of 50 liquid scintillator detectors. Right: DIAMANT detector array (Photos by A.

Ertoprak).

Neutron Wall has a time resolution of about 1 ns which can be used as a time reference. The hardware trigger condition for recording events for subsequent off- line analysis was one detected γ-ray in any of the Ge clover detectors and one detected neutron-like event in Neutron Wall. Note that for the present work the analysis of γ-rays from the nucleus 94Ru, which is populated by emission of four protons from the compound nucleus, did not involve detection of neutrons. This will be discussed in section 2.2.4.

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2.2. EXPERIMENTAL SET-UP 11

2.2.3 The DIAMANT charged particle detector

The charged particles, such as protons and α particles, are subjected to Coulomb force from electrons or nuclei and they are swiftly stopped and slowed down within the detector material. Emission of light charged particles was detected by the nearly 4π detector array DIAMANT [15, 16], consisting of 80 CsI(Tl) scintillators and used in “slave mode” with respect to EXOGAM and Neutron Wall. The output signals from the DIAMANT detector are energy, time and a Particle IDentification (PID) signal. The latter is produced to reflect the pulse shape of the signals similar to the zero-crossover time used for the Neutron Wall detector signals. It uses the fact that CsI has several scintillation light components in Ref [22] with different time constants resulting in different pulse shapes for different types of charged particles.

The proton- and α-particle efficiencies are around 55% and 48%, respectively. A picture of the DIAMANT detector can be seen in the right part of Fig. 2.3 and a drawing of the detector arrangement is shown in Fig. 2.4.

Figure 2.4: A schematic drawing of the DIAMANT detector array (courtesy of B. M. Nyakó)

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12 CHAPTER 2. EXPERIMENTAL TECHNIQUES

2.2.4 Trigger Condition

Data were collected event-by-event triggered by at least one γ-ray detected in EXOGAM in coincidence with a “neutron-like” event in Neutron Wall. The con- dition for the neutron-like events was determined by a hardware threshold on the zero-crossing time of the signals from the Neutron Wall shaping amplifiers and was sufficiently relaxed to allow also a sizeable fraction of γ-ray induced signals. As a result, the majority of the events collected were due to the most prolific, pure charged particle evaporation reactions, such as the 58Ni(40Ca, 4p)94Ru reaction channel studied in the present work. For the off-line charged particle selection, individual two-dimensional gates on the “particle identification” and “energy” pa- rameters of the DIAMANT detectors enabled the identification of γ-rays as belong- ing to specific charged particle evaporation channels. A 50 ns wide time gate was applied to the time-aligned Ge detector timing signals in order to select prompt γ-ray emission.

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Chapter 3

Data Analysis

This chapter covers the procedures used to analyse lifetimes of excited states in the94Ru nucleus using the Doppler Shift Attenuation Method (DSAM). After cal- ibration and gain matching, the data were sorted according to detector angle and analysed offline. Due to the use of a thick target, γ-rays from short-lived states were emitted from the recoiling nuclei while they were slowing down or stopped inside the target. As the target was thick enough to stop the recoils (and because the lifetimes of high-spin states are in the same order as the slowing down time of the recoiling ions in the target) Doppler-broadened line shapes were produced.

Lifetimes of excited states and their related transition probabilities have been de- termined from the analysis of such line shapes from data obtained in the E623 (96Cd) experiment at GANIL.

3.1 Measurement of Nuclear Level Lifetimes

The lifetimes of excited nuclear states are related to the reduced transition prob- abilities to lower-lying states, and rather sensitive to details of the structure of the states. Knowledge of level lifetimes and hence transition probabilities gives details about transition matrix elements and nuclear wave functions. Therefore, such measurements can represent crucial tests for theoretical models. Numerous techniques are available for the measurement of reduced transition strengths, each with a particular time range of applicability.

3.2 The Doppler Shift Attenuation Method

The Doppler Shift Attenuation Method is one of the most commonly used exper- imental techniques to measure lifetimes of excited-states in the range from 10 fs to around 10 ps [23, 24]. It is based on the determination of the Doppler broad- ened line shapes in energy spectra obtained from the emission of γ-rays while the residual nuclei are slowing down in the medium. The idea behind this technique

13

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14 CHAPTER 3. DATA ANALYSIS

Figure 3.1: Basic principle behind the Doppler Shift Attenuation Method for lifetime determinations of short-lived nuclear states. Due to the slowing down of the fusion products in the target, the velocity at which the γ-rays from excited states are emitted (magenta, blue, and red, respectively) decreases on average with the emission time.

is illustrated graphically in Fig 3.1. The velocity distribution of the recoils when undergoing a certain γ-ray transition can be deduced by analysing the correspond- ing Doppler broadened γ-ray line shapes. The shifted photon energies (according to the laboratory frame) can be written, by the standard Doppler shift formula as

Eγ0 = Eγ

p1 − β2

1 − β cos θ (3.1)

where Eγ0 and Eγ are the energies of the photon in the laboratory frame and center of mass frame, respectively. The velocity, v, of the recoiling nuclei is represented by β which is equal to v/c, where c is the speed of light and θ is the angle between the direction of the beam and the detector in the laboratory frame. The moving nucleus of interest is slowing down inside the target as given by the stopping power of the target medium. During this slowing down process, the velocity of the recoil reduces from its initial value to zero (when it’s completely stopped). The centroid of the total lineshape analysis measures the average velocity of the recoils when the certain γ-ray was emitted. The centroid shift for a state with lifetime τ (assuming fast feeding of the state) is given by the Doppler Shift Attenuation Factor F (τ);

F(τ) = vav v0 = 1

v0τ Z

0

v(t)exp(−t

τ)dt (3.2)

where v0is the initial recoil velocity and vavis the average velocity when the γ-ray is emitted. The slowing down of the recoil as a function of time (v(t)) can e.g. be found

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3.2. THE DOPPLER SHIFT ATTENUATION METHOD 15

in the table of stopping powers by Ziegler [25]. Hence, the lineshape, corresponding to the velocity distribution during emission of the γ-ray, gives information about the lifetime of the initial state and the feeding time of it. The lineshape of the transition of interest as a function of detector angle in combination with the known beam energy and stopping powers make it possible to obtain the lifetime of the corresponding nuclear state. Converting the lineshape into a lifetime value requires the simulation of number of recoils as a function of time (β(t)) taking also into account the feeding times which can be known or modeled to give the best fit with the experimental data. The LINESHAPE code was developed by Wells and Johnson [26] for this specific purpose.

3.2.1 Stopping Powers

The slowing down of the recoiling nuclei in the target is caused by two effects:

electronic and nuclear stopping. In the electronic stopping, the recoils interact with atomic electrons and lose their energy. In the heavy-ion fusion-evaporation reactions the amount of recoiling nuclei is much less than the electrons in the medium. The energy loss is relatively small per interaction. In the nuclear stopping the recoils lose energy in a small number of discrete steps through collisions with nuclei along the target. The stopping power can be divided into three different velocity regions: (i) the “low” recoil velocity (β < 0.005 ) region where the nuclear stopping power is dominant; (ii) the “high” recoil velocity (β > 0.02) region where the electronic stopping power is dominant and (iii) the “intermediate” velocity region (0.005 < β < 0.02 ) where the two stopping process compete with each other to be dominant. The SRIM software package [27] which is based on the stopping powers deduced by Ziegler has been used in the present analysis to calculate the nuclear stopping powers in each case.

3.2.2 LINESHAPE software package

A modified version of the LINESHAPE package, see Ref. [28], has been used to calculate the expected Doppler shape for a given γ-ray transition at a particular detector angle and perform a least-square fit in order to extract the level lifetime (τ).

It consists of three subprograms: DECHIST, HISTAVER and LINESHAPE. The DECHIST program [29, 30] uses Monte Carlo techniques to simulate the velocity distribution of the recoiling nuclei in the target and backing material by combining a set of stopping powers either (i) Ward’s effective charge [31] or (ii) Ziegler’s pro- ton stopping powers [25] or (iii) shell-corrected Northcliffe and Schilling stopping powers [32]. In the present case a thick homogeneous metallic58Ni target was used without a backing material. The second program HISTAVER uses the velocity his- tory from DECHIST and calculates the time-velocity distribution (D(v, t)). The lineshapes were stored in a “shape vs time” two dimensional matrix. The LINE- SHAPE program uses the “shape vs time” matrices as an input, converts them to the lineshapes, and performs the fits to the experimental data to determine the

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16 CHAPTER 3. DATA ANALYSIS

lifetimes. During the fitting process the lifetime value of the nuclear state (τ), the lifetime value of the sidefeeding (τSF) and some other parameters related with the spectrum have to be taken into account. It uses the least-squares minimiza- tion routine MINUIT [33] to find the best fit. Uncertainties are calculated using the MINOS routine by χ2 minimisation. The fitting procedure is maintained by adjusting the parameters until the best fit is obtained.

3.2.3 Cross-section dependence

In standard DSAM measurements, a thin target on a thick backing material en- sures that the production cross section of the residual nuclei, which usually depends strongly on the kinetic energy of the beam particles, is approximately constant across the target. Use of a thick homogeneous target results in a substantial change (decrease) in the energy of the beam while traversing the target with an associated change in the production cross section of the residues along the depth of the target.

The knowledge of the residue production rate as a function of target depth follows from the information on the cross section dependence on the beam energy (“excita- tion function”) together with the relevant stopping powers. The excitation function can be obtained from experimental data and/or from statistical model calculations using, e.g., the PACE4 code [34].

Figure 3.2: Experimental fusion excitation function for the 40Ca +58Ni reaction, taken from D. Bourgin et al.

Since in the present case, the kinetic energy of the 40Ca ions incident on the target, 128 MeV, was very close to the Coulomb barrier, the majority of the fusion products were induced by the beam particles in the first thin fraction (< 1 mg/cm2)

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3.2. THE DOPPLER SHIFT ATTENUATION METHOD 17

of the target. The cross section for the production of, e.g., the94Ru residues changed dramatically in this part of the target while the rest of the target thickness acted as a stopping medium, i.e. as a traditional “backing”. Due to the strong variation of the fusion cross section as a function of beam energy close to the Coulomb barrier, we rely on the detailed experimental fusion cross section data obtained by Bourgin et al. [35], see Fig.3.2.

The original version of the DECHIST program calculates the stopping histories of a given number of recoiling residues and simply divides the target thickness (provided by user) into ten bins of equal thickness. The residues are produced with a constant cross section uniformly throughout the target. This is valid when the change in the beam energy across the target is not important as in a traditional

“thin target + backing” DSAM measurement. The scenario is completely different for a thick-target measurement as in the present case. The LINESHAPE package has therefore been modified to consider the changing production cross section while the beam traverses the target. The evolution of the beam energy as a function of target depth was calculated using the SRIM code [27]. In the present version of DECHIST up to ten bins can be defined along the target thickness and relative cross sections for the residue of interest are given for each bin. The values used in the present work can be found in Table 3.1.

Table 3.1: Production cross section as a function of target depth, divided into ten bins, the ten thickness boundaries for defining the bins and the relative yield in each bin. The cross section, beam energy in the center of mass frame, and the laboratory frame are given in the first, second and third column respectively. The target thickness boundaries are given in column 4.

σ(mb) ECM (MeV) ELab (MeV) Thickness (mg/cm2)

115 75.5 127.6 0.16

80 74.1 125.2 0.26

65 73.2 123.7 0.33

45 72.6 122.7 0.41

30 71.9 121.5 0.50

19 71.1 120.1 0.58

8 70.4 118.9 0.65

0 69.7 117.8 0.65

0 69 116.6 0.65

0 68.2 115.2 0.65

It can be noted that for this reaction, the last three target bins have a vanishing cross section. This means that the corresponding reaction channel is closed starting from the beam energy reached in the eighth target bin. Since the production stops after the seventh bin, the last three bins have been given the same boundary as the seventh bin, i.e. they are merged to one.

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18 CHAPTER 3. DATA ANALYSIS

3.2.4 DSAM Lineshape Analysis

As the present experiment employed a very low beam energy, close to the Coulomb barrier of the fusion-evaporation reaction, and a thick homogeneous production tar- get it was rather unexpected that useful lifetime values for medium-high spin states could be extracted from the data. Nevertheless, a systematic trend of increasing lineshape asymmetry as a function of decreasing “effective” lifetime (taking into account feeding time) was found in the data as summarized in Table 3.2. Here the results of fits to representative peaks using the RADWARE graphical analysis package [36] have been made using the standard RADWARE function consisting of a sum of a symmetric Gaussian function with a skew Gaussian function with rel- ative heights 1−R and R, respectively. The additional smoothed step function that is sometimes applied in RADWARE to increase the background on the low-energy side of the peak was neglected in this case.

Table 3.2: Peak asymmetries fitted for selected γ-ray transitions in94Ru, see text.

The peak shape asymmetry parameter R = 15 is proportional to the asymmetry of the peak shape (see text). The spin-parity assignments and γ-ray energies are given in the first and second column, respectively. The Beta parameter values (Beta is the skewness parameter for the skew Gaussian component and varies slowly with γ-ray energy) are shown in the column 3. Previous lifetime values or limits for the initial state are given in the column 5. The smooth step function components were fixed to zero during the fitting procedure. Uncertainties are given within parentheses.

The asymmetry parameter R is found to be constant around R = 15 for peaks corresponding to transitions known to occur when the nuclei are at rest in the target, while transitions from states which have previously been reported [11] to have short lifetimes of the order of picoseconds and no slow feeding systematically have R values of 20 and higher.

Iiπ → Ifπ Eγ (keV) Beta (fixed) R τef f

13+1 →12+1 1898.2 4 22.3(15) 1.26(17) ps

2+1 →0+1 1431 4 15(7) >94 ns

201 →181 1113 4 30.2(12) <2.6 ps 131 →111 1080 4 12.5(8) 2.89(32) ps

71 →51 1033 4 16(8) >1.1 ns 15+1 →14+1 615.6 3.5 21.8(13) <0.4 ps 18+1 →17+1 486 3.5 19.9(10) 0.52(3) ps 12+1 →10+1 725 3 15(5) 34.3(16) ps 14+1 →13+1 543.4 3 21(11) 0.48(6)

This encouraged us to proceed with the lifetime analysis of several excited states in94Ru using the Doppler Shift Attenuation Method. However, the low residue ve- locities before slowing down and stopping in the target naturally lead to small

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3.2. THE DOPPLER SHIFT ATTENUATION METHOD 19

Doppler effects, even for very short-lived states with lifetimes of a few picosec- onds and less. A further complication was also the intrinsic asymmetric lineshapes present in the γ-ray energy spectra due to charge trapping induced by neutron damage in the germanium detectors. Such effects could potentially overshadow any Doppler-induced lineshape effects and needed to be accounted for carefully in the analysis.

The Doppler broadened line shapes in the γ-ray energy spectra were analyzed using a modified version of the LINESHAPE program [26, 28]. The intrinsic asym- metry of the peak shapes was determined from a systematic study of transitions for which the residual nuclei were completely stopped and fitted to a superposition of gaussian functions, see Fig. 3.3 for an example. The asymmetric peak shape for stopped peaks is given by

f(e) = N · 1 σ

2π(e−(e−c)2/2σ2+ a · e−(e−c−k·22ln2σ)2/2σ2) (3.3) where N is a normalization factor proportional to the total peak area (i.e. γ-ray intensity), c is the centroid of the symmetric Gaussian component, k = 1.29 and a= 0.07 are constants obtained from the fits, and σ is an energy dependent width determined by fitting the full-width-at-half-maximum, F W HM = 2

2ln2σ to sev- eral known stopped peaks using the function F W HM(e) =

A+ B · e + C · e2 in the standard RADWARE fashion [36].

The procedure in which a gate on a lower transition with respect to the tran- sition of interest is made, developed by Brandolini and Ribas for the analysis of DSAM measurements [37] was employed in the present work. Therefore, the side feeding coming from higher-lying excited states has been taken into account for the lifetime determination. In the present case the analysis is facilitated by the fact that several lifetimes in94Ru for the relevant spin range have been determined pre- viously using the Recoil Distance Doppler Shift technique [11] as discussed further below. Branching ratios for the γ-decays into and out of the states of interest were determined accurately in the present work. The situation is summarized in Fig. 3.4 and Table 3.3.

Due to the low energy employed for the reaction, many of the states of interest receive significant amounts of direct (and therefore unobserved) feeding. For exam- ple, for the highest-lying states considered here; the 19+ excited state at 9921 keV and the 20excited state at 11041 keV, the feeding transitions that have previously been observed from states that are situated at 1.9 - 3.8 MeV higher excitation en- ergy [38], were not observed in the present experiment, presumably due to the significantly lower excitation energy in the present reaction. It is therefore reason- able to assume that the 19+ and 20 states receive direct, very fast feeding. This conclusion is supported by the fact that the previously known lifetimes of states measured by Jungclaus et al. [11], which lie below the 19+and 20states, are well reproduced in the present analysis. However, for the yrast positive-parity states in the spin range I = (13 − 18)~ analyzed in this work, the Bateman equations, where the side feeding is included, have been applied. In addition, an iterative process

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20 CHAPTER 3. DATA ANALYSIS

Figure 3.3: The intrinsic asymmetry of the peak shape for completely stopped 756 keV γ-ray transition.

has been performed in such a way that once the lifetime of a certain state has been determined it has been used as an input value for the next lower level in the γ-ray cascade, and so on. Intermediate verifications could be obtained due to the known values for the lifetimes of the 18+ and 14+ excited states, measured by Jungclaus et al. [11]. As it can be seen in Table 4.1, Chapter 4, also the limits established for the 15+, 16+, and 17+excited states in Ref. [11] are in agreement with the values obtained in the present measurement. As an example, for the determination of the lifetime of the 18+excited state, the lifetime value determined for the 19+ excited state (3.08(69)) as well as the lifetime value of the 18excited state (5.04(34)) have been considered, giving a value of 0.55(7) ps in agreement with the 0.52(3) ps given in Ref [11]. This value has been used as input, together with the lifetime value determined for the 19+ excited state, for the determination of the lifetime of the 17+ excited state, giving a value of 0.86(13) ps.

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3.2. THE DOPPLER SHIFT ATTENUATION METHOD 21

Table 3.3: Relative γ-ray intensities for94Ru measured in the present work.

Positive Parity Negative Parity Iiπ → Ifπ Eγ (keV) Iγ Iiπ → Ifπ Eγ (keV) Iγ

14+1 →13+1 543.4 45.3(9) 181 →161 932 4.8(3) 15+1 →14+1 615.6 28.2(7) 201 →181 1113.5 6.1(2) 16+1 →15+1 638.5 17.7(4) 172 →152 1288 0.0030(7) 17+1 →16+1 630.1 21.7(5) 152 →14+1 1344 4.5(2) 18+1 →17+1 486.0 13.6(4) 152 →14+2 462 1.1(3) 19+1 →18+1 394.5 9.9(3) 153 →15+1 964 0.98(8) 15+1 →13+1 1159 4.0(1) 162 →15+2 1225 0.8(1) 16+1 →14+1 1254 1.4(1) 181 →18+1 402 <0.001 15+2 →13+1 1296 3.4(1) 163 →154 610.6 0.028(4) 17+1 →15+1 1269 1.5(1) 171 →161 257.3 11.7(6) 18+1 →16+1 1115 0.02 172 →161 792.4 4.1(2) 19+1 →17+1 880 2.3(1) 172 →163 325 0.012(1) 15+2 →14+1 753 9.2(3) 191 →171 1190.4 0.029(2) 16+1 →14+1 501 6.4(2) 201 →191 597.5 1.39(9)

10+1 →8+1 1347 100 162 →153 398.1 1.6(1) 161 →152 496 21.9(8) 162 →154 281.6 1.1(1) 172 →162 654 0.80(7) 181 →171 674 1.7(1) 182 →181 616.4 0.085(8) 191 →181 515.6 3.2(3)

The fitted lineshape of the 1898 keV transition de-exciting the 13+1 state is shown in Fig. 3.5. The resulting lifetime obtained in the present work, τ131 = 1.36(17) ps is in agreement with the value previously obtained by Jungclaus et al. τlit= 1.26(17) ps [11]. The fact that the lifetime values obtained by Jungclaus et al. were deduced using a different technique, i.e. the Recoil-Distance Doppler Shift (RDDS) method, further verifies the validity of the method employed in the present work.

For the DSAM analysis, the relative intensities of the transitions in the cascades are needed in order to properly model the side feeding. The transition intensities have therefore been measured in the present work. They are listed in Table 3.3.

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22 CHAPTER 3. DATA ANALYSIS

Figure 3.4: Detailed view of the level scheme obtained from the present work.

Levels with measured lifetime are highlighted in blue.

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3.2. THE DOPPLER SHIFT ATTENUATION METHOD 23

Figure 3.5: Experimental γ-ray energy spectra and fits to the Doppler shifted shapes for the 1898 keV (13+1 →12+1) transition. The spectra were produced by setting a narrow gate on the stopped component of the 725 keV transition decaying from the 4717 keV state in the Eγ1− Eγ2 coincidence matrix. Here Eγ1 refers to energies of γ-rays detected at 90 or 135 with respect to the beam direction while Eγ2 refers to energies of any γ-rays detected in coincidence with these γ-rays.

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Chapter 4

Results

4.1 Results for the level lifetimes

The lifetime values which have been deduced in the present work are summarized in Table 4.1. For comparison with the previous measurements, lifetime values and limits determined in Ref. [11] are also included in Table 4.1. The lifetimes for the 13+ excited state at 6614 keV, the 14+ excited state at 7157 keV, and the 18+ excited state at 9526 keV as well as upper limits on the lifetimes of the 15+excited state at 7773 keV, the 16+ excited state at 8411 keV, the 17+ excited state at 9041 keV, the 19+excited state at 9921 keV and the 20excited state at 11041 keV were known from the work of [11]. The lifetimes of the latter five states have been determined for the first time in the present work.

Table 4.1: Lifetimes of excited states in94Ru from the present work in comparison with previously reported values and limits.

Positive Parity

Ex(keV) Iiπ→ Ifπ Eγ (keV) τ(ps) τlit(ps) B(M1↓) (µN2) B(E2↓) (e2fm4) 6614 13+1 →12+1 1898.2 1.36(17) 1.26(17) 0.0061(7)

7157 14+1 →13+1 543.4 0.43(6) 0.48(6) 0.82(11)

7773 15+1 →14+1 615.6 0.32(4) <0.4 0.64(10) 178(22) 8411 16+1 →15+1 638.5 0.69(10) <1.0 0.21(3) 34(5) 9041 17+1 →16+1 630.1 0.86(13) <2.0 0.24(4) 31(5) 9526 18+1 →17+1 486.0 0.55(7) 0.52(3) 0.89(11)

9921 19+1 →18+1 394.5 3.08(69) <4.9 0.24(5) 99(22) Negative Parity

Ex(keV) Iiπ→ Ifπ Eγ (keV) τ(ps) τlit(ps) B(M1↓) (µN2) B(E2↓) (e2fm4) 11041 201 →181 1113.5 2.20(31) <2.6 0.028(4) 166(23)

25

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26 CHAPTER 4. RESULTS

4.2 Reduced Transition Probabilities and Comparison with Large Scale Shell Model Calculations

Once the lifetime, τ, of an excited state has been measured, the γ-ray transition probability, T(λ), can be obtained. The reduced transition probabilities for a state decaying from Ji to state Jf, can be deduced as follows:

B(Eλ) = 4.57 × 10−22λ[(2λ + 1)!!]2 8π(λ + 1) (197

Eγ)2λ+1T(Eλ) [e2f m] (4.1)

B(Mλ) = 4.15 × 10−20λ[(2λ + 1)!!]2 8π(λ + 1) (197

Eγ)2λ+1T(Mλ) [µN2

f m2λ−2] (4.2) where B(Eλ : Ji → Jf) called “reduced matrix element”, Eγ is energy in MeV and λ is the decay constant and inversely related with the τ value. The reduced transition probabilities, in this study B(M1) and B(E2), have been deduced from the measured lifetimes in this way. The equations that given above can also be written as:

B(E2) = 1

τ(1 + α)1.22 × 109E5γ [e2f m4] (4.3)

B(M1) = 1

τ(1 + α)1.76 × 1013Eγ3 N2] (4.4) where the lifetime τ is in sec, Eγ is in MeV, and α is the internal conversion coefficient. The internal conversion coefficients were calculated using BrIcc [39]. In the present work, we use the fact that the M1 transition offers a special opportunity to test the many-body wave function, in particular with respect to cross orbital excitations. The magnetic dipole operator only links single-particle orbitals with the same orbital angular momentum [2], i.e. spin-orbit partners or states within the same single-j subshell. This provides a powerful tool to test the theoretically predicted configurations of excited states in nuclei in the region of 100Sn. In the particular case of the semi-magic nucleus 94Ru, the M1 transition properties can be expected to be dominated by coupling within the g9/2 proton subshell and the possible excitation of nucleons from g9/2 to its g7/2 spin-orbit partner across the N = 50 shell gap. Here, the significantly larger gyromagnetic factor for states involving these orbitals compared with those of p1/2or p3/2 plays a key role.

We have performed shell model calculations for the M1 transitions in94Ru in the

“fpgd” model space with the same Hamiltonian as described in Ref. [9]. The lowest three eigenstates for each spin and parity have been calculated and the reduced transition strengths for all possible M1 (and E2) transitions were evaluated. For the calculations of B(M1) reduced transition strengths we used both the bare and the effective spin gyromagnetic factors with gs= 0.7·gs(free) while effective electric

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4.2. REDUCED TRANSITION PROBABILITIES AND COMPARISON WITH

LARGE SCALE SHELL MODEL CALCULATIONS 27

charges eπ= 1.5e and eν = 0.8e were used for protons and neutrons, respectively, to calculate the E2 transition probabilities.

The experimental transition probabilities, [B(M1), B(E2)] which are deduced from the present lifetime analysis are compared with large shell model calculations performed by Chong Qi, together with those reported by Jungclaus et al. [11] as

“SM2”, which were deduced from the work of Johnstone and Skouras [40], see Fig 4.1.

The results clearly point to the presence of core excitations above the yrast 12+ state. While this state is dominated by the coupling within the g9/2 subshell in the shell model calculations, the first 13+ state is predicted to be the lowest-lying neutron core-excited state. The comparison between experiment and theory for the B(E2) transition strengths points in a similar way to the importance of neutron core excitations for the medium- to high-spin states in94Ru. Some discrepancies that are observed between theory and experiment are attributed to mixing of the wave functions that is not reproduced by the calculations.

References

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